RESEARCH CENTRE

Inria Saclay Centre
at Institut Polytechnique de
Paris

IN PARTNERSHIP WITH:

CNRS, Institut Polytechnique de
Paris

2023
ACTIVITY
REPORT

Project-Team
GEOMERIX

Geometry-driven Numerics

IN COLLABORATION WITH: Laboratoire d’informatique de
l’école polytechnique (LIX)

DOMAIN

Perception, Cognition and
Interaction

THEME

Interaction and visualization

Contents

Project-Team GEOMERIX

1 Team members, visitors, external collaborators

2 Overall objectives

3 Research program

3.1 Geometry for Euclidean shape processing . . . . . . . . . . . . . . . . . . . . . . .
3.2 Geometry for simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Geometry for dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Geometry for data science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Application domains

5 Highlights of the year

5.1 Awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 New results

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6.1 Geometry for Euclidean shape processing . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Patternshop: Editing Point Patterns by Image Manipulation . . . . . . . .
15
6.1.2 Robust Pointset Denoising of Piecewise-Smooth Surfaces through Line Processes 16
16
6.1.3 Feature-Sized Sampling for Vector Line Art . . . . . . . . . . . . . . . . . .
6.1.4 Bio-Sketch: A new medium for interactive storytelling, illustrated by the
phenomenon of infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Somigliana Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Geometry for simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Building a Virtual Weakly-Compressible Wind Tunnel Testing Facility . . .
6.2.2 Fluid-Solid Coupling in Kinetic Two-Phase Flow Simulation . . . . . . . . .
6.2.3 High-Order Moment-Encoded Kinetic Simulation of Turbulent Flows . . . .
6.3 Geometry for data science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stable Vectorization of Multiparameter Persistent Homology using Signed
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Barcodes as Measures

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6.3.1

6.1.5

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6.3.4

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6.3.2 Local characterizations for decomposability of 2-parameter persistence modules 19
6.3.3 A Gradient Sampling Algorithm for Stratified Maps with Applications to
Topological Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shape Non-rigid Kinematics (SNK): A Zero-Shot Method for Non-Rigid
Shape Matching via Unsupervised Functional Map Regularized Reconstruction 20
6.3.5 Zero-Shot 3D Shape Correspondence . . . . . . . . . . . . . . . . . . . . . .
20
6.3.6 VoroMesh: Learning Watertight Surface Meshes with Voronoi Diagrams . .
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SATR: Zero-Shot Semantic Segmentation of 3D Shapes
6.3.7
. . . . . . . . . . .
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6.3.8
Spatially and Spectrally Consistent Deep Functional Maps . . . . . . . . . .
22
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6.3.9 TIDE: Time Derivative Diffusion for Deep Learning on Graphs . . . . . . .
6.3.10 ReVISOR: ResUNets with visibility and intensity for structured outlier removal 22
6.3.11 Assessing craniofacial growth and form without landmarks: A new automatic
approach based on spectral methods . . . . . . . . . . . . . . . . . . . . . .
6.3.12 Affection: Learning Affective Explanations for Real-World Visual Data . . .
6.3.13 Generalizable Local Feature Pre-training for Deformable Shape Analysis . .
6.3.14 Understanding and Improving Features Learned in Deep Functional Maps .
6.3.15 Scalable and Efficient Functional Map Computations on Dense Meshes . . .

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7 Bilateral contracts and grants with industry

7.1 Bilateral contracts with industry . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Contract with Sanofi Inc.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Contract with DASSAULT SYSTEMES . . . . . . . . . . . . . . . . . . . .
7.1.3 MEDITWIN with DASSAULT SYSTEMES . . . . . . . . . . . . . . . . . .

8 Partnerships and cooperations

8.1

8.2 European initiatives

International research visitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Visits of international scientists . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.2 Visits to international teams
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Horizon Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2 H2020 projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Dissemination

9.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1
Scientific events: selection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Journal
9.1.3
9.1.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Invited talks
9.1.5 Leadership within the scientific community . . . . . . . . . . . . . . . . . .
9.1.6 Research administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Juries
9.2.3

9.2 Teaching - Supervision - Juries

10 Scientific production

10.1 Major publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Publications of the year
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10.3 Cited publications

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Project GEOMERIX

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Project-Team GEOMERIX

Creation of the Project-Team: 2022 September 01

Keywords

Computer sciences and digital sciences

A3.4.1. – Supervised learning

A3.4.2. – Unsupervised learning

A3.4.4. – Optimization and learning

A3.4.6. – Neural networks

A5.5. – Computer graphics

A5.5.1. – Geometrical modeling

A5.5.4. – Animation

A6.1.4. – Multiscale modeling

A6.1.5. – Multiphysics modeling

A6.2.5. – Numerical Linear Algebra

A6.2.6. – Optimization

A6.2.8. – Computational geometry and meshes

A6.5.1. – Solid mechanics

A6.5.2. – Fluid mechanics

A8.3. – Geometry, Topology

A8.7. – Graph theory

A8.12. – Optimal transport

A9.2. – Machine learning

Other research topics and application domains

B9.2.2. – Cinema, Television

B9.2.3. – Video games

B9.5.1. – Computer science

B9.5.2. – Mathematics

B9.5.3. – Physics

B9.5.5. – Mechanics

B9.5.6. – Data science

2

Inria Annual Report 2023

1 Team members, visitors, external collaborators

Research Scientists

• Steve Oudot [Team leader, INRIA, Senior Researcher, HDR]

• Jiong Chen [INRIA, Starting Research Position, from Nov 2023]

• Mathieu Desbrun [INRIA, Advanced Research Position]

• Pooran Memari [CNRS, Researcher]

Faculty Member

• Maksims Ovsjanikovs [Ecole Polytechnique, Professor]

Post-Doctoral Fellows

• Maysam Behmanesh [Ecole Polytechnique]

• Anouar Abdeldjaoued Ferfache [Inria, Post-Doctoral Fellow, from Apr 2023]

• Roman Klokov [Ecole Polytechnique]

• Wei Li [Inria, Post-Doctoral Fellow, until Jan 2023]

• Vincent Mallet [Ecole Polytechnique]

PhD Students

• Souhaib Attaiki [Ecole Polytechnique, from Oct 2023]

• Elisa Ballantoni [Telecom Paris, from Oct 2023]

• Francesco Bellantoni [Telecom Paris, from Oct 2023]

• Theo Braune [Ecole Polytechnique]

• Lucas Brifault [Dassault Systèmes]

• André Freitas [Telecom Paris, from Oct 2023]

• Souhail Hadgi [Ecole Polytechnique]

• Vadim Lebovici [ENS Paris, until Aug 2023]

• Jingyi Li [Ecole Polytechnique, from Nov 2023]

• Robin Magnet [Ecole Polytechnique]

• Nissim Maruani [3IA Côte d’Azur]

• Julie Mordacq [Ministère des Armées]

• Tim Scheller [Ecole Polytechnique, from Oct 2023]

• Ramana Sundararaman [Ecole Polytechnique]

• Jiayi Wei [Ecole Polytechnique, until Sep 2023]

• Jiayi Wei [Inria, from Oct 2023]

Project GEOMERIX

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Interns and Apprentices

• Ethan André [ENS Paris, Intern, from May 2023 until Oct 2023]

• Diego Gomez [Ecole Polytechnique, from May 2023 until Jul 2023]

• Jingyi Li [Inria, Intern, from Mar 2023 until Sep 2023]

• Siddharth Setlur [Inria, Intern, until Jan 2023]

• Thomas Wimmer [FX Conseil, Intern, from Apr 2023 until Sep 2023]

Administrative Assistant

• Michael Barbosa [Inria]

Visiting Scientist

• Shreyas Samaga [Purdue University, from May 2023 until May 2023]

2 Overall objectives

Historical context. Geometry has been a unifying formalism for science: predictive models of
the world around us have often been derived using geometric notions which formalize observable
symmetries and experimental invariants. Tools such as differential geometry and tensor calculus
quickly became invaluable in describing the complexity of natural phenomena and mechanical
systems through concise equations, condensing local and global properties into simple relations
between measurable quantities. Today, geometry (be it Euclidean or not) is at the core of many
current physical theories: general relativity, electromagnetism (E&M), gauge theory, quantum
mechanics, as well as solid and fluid mechanics, all have strong underlying structures that are
best described and elucidated through geometric notions like differential forms, curvatures, vector
bundles, connections, and covariant derivative. Geometry also creeps up in unexpected fields such
as number theory and functional analysis, offering new insights and even breakthroughs, e.g., the
use of algebraic geometry to address Fermat’s last theorem.

Geometry in Digital Sciences. In sharp contrast, the role of geometry was mostly ignored
at the inception of computer science. Yet, it has now become clear that digital sciences are imbued
with an overwhelming amount of fundamentally geometric and topological concepts. Some are
rather obvious, when dealing with the modeling of Euclidean shapes in computer graphics or the
analysis of images in computer vision for instance; some are more subtle, such as the “manifold
hypothesis” underlying a number of supervised or unsupervised learning techniques; and some are
only nascent, such as the fields of Information Geometry (basically, the geometry used to understand
probability distributions), Geometric Statistics (new statistical methodology for non-Euclidean
entities), and Topological Data Analysis (where algebraic topology is used as a tool to enhance
data analysis pipelines). In fact, even the discretization of physical theories needed to offer fast
numerical simulation has brought geometry back to the forefront after it was understood that
the loss of numerical fidelity in standard numerical methods is due to a fundamental failure to
preserve geometric or topological structures of the underlying continuous models: partial differential
equations (PDEs) modeling our physical world are typically encoding invariants and structures that
are independent of the choice of coordinates used to express the equations and the tensors involved
in them; but invariance to the choice of basis is often lost during discretization, as numerical
approximations will in general not capture, let alone preserve, the key geometric structures that
exist in the continuous case. Seeing these numerical issues through the lens of geometry is thus not
just of academic interest: failure to maintain geometric invariants has serious consequences for the
accuracy and stability of solutions.

Rationale. Given the unusual reach of geometry and its rich literature, there is an opportunity
to assemble a team of experts in geometry and its vernacular, to help broadly impact digital science

4

Inria Annual Report 2023

and technology. We thus propose the creation of a new project-team whose core scientific
mission is to use geometry as a bedrock for the development of numerical tools and
algorithms: we wish to exploit the properties of infinite-dimensional and finite-dimensional spaces
that are related with distance, shape, size, and relative position, and bringing them to bear on
computational discretizations and algorithms for analysis, processing, and simulation. Adhering
to geometric structures and invariants as a guiding principle for computations is a rich source of
both theoretical and practical challenges, allowing to combine concepts and results from different
areas of geometry broadly construed to produce new computational tools with solid mathematical
foundations. While our team will be very focused in terms of the mathematical foundations and
tools upon which it builds, it will also be very broad in terms of applications given the pervasiveness
of geometry in sciences and technology. This makes for an unusual, yet powerful scientific setup that
will facilitate interdisciplinary projects through the common use of geometric foundations and their
specialized terminology. It will also allow us to contribute sporadically to pure and computational
mathematics when appropriate in order to push our scientific mission forward.

Positioning. We see GeomeriX as first and foremost Inria Saclay’s graphics team, but with
wider objectives afforded by the broad relevance of geometry. It is worth noting that graphics has
evolved to the point where it often intersects with applied mathematics, machine learning, vision,
and computational science in some of its efforts, and GeomeriX intends to continue this trend.

Objectives. Our project-team’s overall scientific objective is to contribute, through a geometric
perspective, both foundational and practical methods for geometric data processing. In particular,
we seek the development of predictive computational tools by drawing from the many facets of
geometry and topology: whether it be discrete geometry, basic differential geometry or exterior
calculus, symplectic geometry, persistent homology or sheaf theory, optimal transport, Riemannian
or conformal geometry, these topics of geometry inform and guide both our discretizations and
algorithmic designs towards computing. Note that we do not plan to merely adapt and exploit
geometric concepts and understanding for numerical purposes, as our focus on digital data may
even result in contributions to these mathematical fields, extending the current body of knowledge.
While we intentionally leave the range of our mathematical foundations open so as not to restrict
our potential team-wide explorations, we concentrate our research on four concrete themes,
which we believe can be most significantly impacted by a geometric approach to
developing new numerical tools:

1⃝ Euclidean shape processing: from computer graphics to geometry processing and vision,
the analysis and manipulation of low-dimensional shapes (2D and 3D) is an important
endeavor with applications covering a wide range of areas from entertainment and classical
computer-aided design, to reverse engineering and biomedical engineering. Our project-team
intends to lead efforts in this competitive field, with key contributions in shape matching,
geometric analysis, and discrete calculus on meshes.

2⃝ Simulation: traditional finite-element treatments of various physical models have had
tremendous success. Recently, a number of geometric integrators have upended the field,
either through structure-preserving integration which offers improved statistical predictability
by respecting the geometric properties of the exact flow of the differential equations, or
through novel discretizations of the state space. We intend to continue introducing novel
integration methods for increasingly complex multiphysics systems, as well as exploiting the
use of learning methods to accelerate simulation.

3⃝ Dynamical systems: we intend to leverage the geometric nature of dynamical systems to
investigate and promote high-dimensional data analysis for dynamics: the study of dynamical
systems from a limited number of observations of the state of a given system (for example,
time series or a sparse set of trajectories) offers a unique opportunity to develop scalable
computational tools to detect or characterize unusual features and coherent structures.
Meanwhile, the study of dynamical systems from a combinatorial point of view opens up the
possibility of characterizing their invariant sets and assessing their stability.

4⃝ Data science: finally, we are intent on exploring the underlying role of geometry in machine
learning and statistical analysis. This role has been put forward in the recent years, with
the emergence of approaches such as geometric deep learning or topological data analysis,

Project GEOMERIX

5

whose aim is to leverage the underlying geometry or topology of the data to enhance the
performance, robustness, or explainability of the methods used for their analysis. We
will pursue investigations toward this goal, concentrating our efforts on topics related to
explainable feature design, geometric feature learning, geometry-driven learning, and geometry
for categorical and mixed data types.

Evidently, our research efforts may at times lie across multiple of these themes given our multi-
disciplinary objectives, and it is our hope that we will all eventually participate in the four themes.

3 Research program

Below we introduce the details of our four research themes, in four separate subsections. In each
subsection, we first present the scientific focus and research objectives of the corresponding theme,
then we detail the research topics we intend to address and how we plan to leverage topology
and geometry for each one of them. For each theme, we list the most likely contributors, and
organize the various subtopics within each theme from short to long-term goals, based on our
current expectations and focus.

3.1 Geometry for Euclidean shape processing

Euclidean space is the default setting of classical geometry in two or three dimensions. Shapes in 3D
space are of particular interest as they represent the typical objects we interact with. Geometry
processing is an area of research focusing on these low-dimensional shapes in Euclidean space,
with the goal to design algorithms, data structures, as well as analysis tools for their digital
acquisition, reconstruction, analysis, manipulation, synthesis, classification, transmission, and
animation. Digital shapes are typically discretized through either point clouds, triangle meshes, or
polygonal meshes for surfaces, and through tetrahedron or polytopal meshes for volumes. Analyzing
and manipulating these digital representations already involve fundamental difficulties in terms of
efficiency, scalability, and robustness to arbitrary sampling, for which computational geometry and
computer graphics have generated a number of key algorithms. Simple surface meshes in 3D also
offer a simple context in which to define discrete notions of basic topological properties (quantities
preserved through arbitrary stretching, such as Euler characteristic, genus, Betti numbers, etc) and
relevant geometric properties (normal, curvatures, covariant derivatives, parallel transport, etc).
Yet the digital counterpart of the low-dimensional case of Euclidean geometry is far from being
settled or complete: it remains obviously relevant in a number of scientific fields on which we plan
to focus. A few research directions of particular interest are described below.

Operator-based methods for shape analysis We plan to develop novel approaches for
representing and manipulating geometric concepts as linear functional operators. Specifically we will
focus on tools for shape matching, design and analysis of differential quantities such as vector fields
or cross fields, shape deformation and shape comparison, where functional approaches have recently
been shown to provide a natural and discretization-agnostic representation [110, 43, 44, 120]. This
“functional” point of view is classical in many scientific areas, including dynamical systems (where
the pullback with respect to a map is closely related to the Koopman or composition operator,
allowing the study ergodicity or mixing property of non-linear maps through the spectral properties
of a linear operator), differential geometry (where vector fields are often defined by their action
on real-valued functions) and representation theory among others. However, it has only recently
been adopted in geometry processing with tremendous and constantly growing potential in both
axiomatic or even learning-based approaches [98, 88, 71]. We will continue developing efficient
and robust algorithms by considering shapes as functional spaces and by representing various
geometric operations as linear operators acting on appropriate real-valued functions. In addition
to the efficiency and robustness of methods obtained by considering this linear operator point of
view of geometry processing and dynamical systems, another very significant advantage of these
techniques is that they allow to express many different geometric operations in a common language.
This means, for example, that it makes it easy to define the pushforward of a vector field with

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Inria Annual Report 2023

respect to a map by simply considering a composition of appropriate discrete operators. Despite
the significant recent success of tools within this area, especially related to the functional map
framework [111], there does not exist a unified coherent theoretical framework in which different
geometric concepts can be represented and manipulated via their functional equivalents. Our
main long-term goal therefore would be to establish a novel field within geometry processing by
creating both a computational framework and a coherent theoretical formalism in which all of the
different basic geometric operations can be expressed, and thus in which different concepts can
“communicate” with one another. We believe that such a formalism and associated computational
tools, already quite well developed, will not only greatly extend the scope of applicability of many
existing geometry processing pipelines, but will also help expand this language to novel concepts,
and ultimately help pave the way towards representation-agnostic geometric data manipulation.

Discrete metrics and applications. While three-dimensional shapes are often encoded via their
Euclidean embedding, numerous research efforts have focused on studying and discretizing their
intrinsic metric. Regge calculus [118], an early approach to numerical relativity without coordinates,
proposed the use of edge lengths to encode a piecewise-Euclidean metric per simplex, from which the
Riemann curvature tensor can be easily computed to derive local areas or curvatures. This early work
led to a series of alternative metric representations: tip angles, for instance, are known to encode
the intrinsic geometry of a triangle mesh up to a scaling, while local measurements (an angle [119]
or a length cross-ratio [101] per edge) later formed the basis of circle patterns [47, 93] as well as
conformal representations [125]; the discrete Laplace-Beltrami cotan formula [114] also determines
the edge lengths of a triangle mesh (and thus its discrete metric) up to a global scaling [137]. More
recently, generalized notions of metrics were proposed; for instance, [85] presented a characterization
of an augmented discrete metric resulting from the orthogonal primal-dual structure of weighted
triangulations. Common to many of these various metric characterizations is the existence of convex
energies which allow to efficiently compute these metrics from various boundary conditions. We
intend to investigate the discrete treatment of metric for low-dimensional manifolds as a counterpart
to the discretization of antisymmetric tensors (differential forms), which is far less studied — and
a discrete theory unifying symmetric and anti-symmetric tensors remains elusive despite recent
advances [84]. Moreover, the metric of a surface is known in the continuous realm to induce Hodge
stars and a canonical torsion-free Levi-Civita connection (or parallel transport), but this picture
is far less clear for discrete manifolds, even if the construction of arbitrary-order discrete Hodge
stars and metric connections are well understood by now. A few research directions on generalized
metrics seem particularly interesting due to their likelihood of resulting in novel algorithmic and
computational frameworks:

• Metric-dependent meshing: Given a set of metric-based operators, optimized mesh structures
can be designed to offer optimal accuracy akin to Hodge-star mesh optimization for the
augmented weighted metric proposed in [107]. Another interesting research question is the
existence and construction of intrinsic Delaunay triangulation, the most common discrete
shape representation, with respect to a particular metric [48].

• Metric-aware sampling: Metric-dependent descriptors such as the pair correlation function
are particularly efficient in characterizing statistical properties of point distributions for
texture synthesis [72]. Extending this framework to arbitrary non-flat domains through
Multi-Dimensional Scaling (MDS) seem particularly promising.

• Shape characterization: Highly convoluted embeddings like the cortical surface of the brain
and its functional connectivity graph are naturally hyperbolic in nature [53]. However,
investigating a link between cortical folding and the volumetric fiber bundle structure from
a pure geometric viewpoint through a hyperbolic metric characterization has surprisingly
not be done in brain analysis, despite striking visual similarities between brain folding and
geometric realizations of the hyperbolic plane (see [130] and Taimin, a’s crochet model). We are
hoping that this intrinsic metric characterization can be investigated through recent discrete
hyperbolic parametrization tools [80], which may also lead to other shape classification
techniques in more general contexts.

• Piecewise-linear maps: We also wish to study the classification of the deformation of a triangle
mesh through its induced metric change in the embedding space. Developing an approach to

Project GEOMERIX

7

decompose such a diffeomorphic piecewise-linear map into canonical geometric transformations
through either linear algebra or convex minimization could offer new discrete equivalences
for conformal, equiareal, and curvature-preserving maps between triangulations, with direct
applications to mesh parameterization and more general processing of discrete meshes.
• Geodesic abstractions: curve-network representations [83] based on a few geodesics to describe
a shape provide a compact encoding of surfaces. While it is increasingly useful for artistic
depictions, we also want to study its relevance as a compact compression scheme from which
the shape and its metric can be derived with controllable precision.

• Metric-dependent cage: Finally, we also want to understand how to define optimized metric-
dependent cages for intuitive & expressive deformation and animation of complex shapes [128],
and how these cages can be understood as polygonal or polyhedral cells to locally simplify a
simplicial complex.

Discrete differential and tensor calculus. When working on low-dimensional spaces, the use
of meshes (as opposed to just point clouds) pays dividends as it allows for the development of
discrete versions of Exterior Calculus (see DEC [67] or FEEC [41]), where k-dimensional integrals
can be directly evaluated in k-cells, and differentiation can formally achieved through the boundary
operator: the concept of chains and cochains from algebraic topology forms the basis of a discrete
analog of Cartan’s exterior calculus of differential forms, providing crucial numerical tools such
as a discrete de Rham cohomology and a discrete Helmholtz-Hodge decomposition that precisely
mimick their continuous counterparts. Moreover, finite elements of arbitrary order can be associated
with these discrete forms through subdivision [82] to provide a powerful Isogeometric Analysis
(IGA). Recent developments [99, 81] have offered also a discrete approach to tangent vector
fields. While DEC encodes vector fields as 1-forms, processing tangent vectors and, more generally,
directional fields sampled at vertices of discrete surfaces requires the development of discrete (metric)
connections [64, 99] (which can be seen as discrete equivalent to the Christoffel symbols) to handle
the non-linearity of non-flat domains. From these connections can be derived the usual continuous
notions of covariant derivatives or Killing operator, and these discrete operators demonstrate the
same intimate link between geometry and topology as exemplified by the hairy ball theorem (Hopf
index theorem). While these operators apply equally well on discrete three-manifolds, much remains
to do: properly defining the notion of curvature matrix-valued 2-form or torsion vector-valued 2-form
in 3D and checking that these definitions provide consistent Bianchi identities (i.e., there exists
an exterior covariant derivative satisfying fundamental geometric and topological properties) is an
exciting research direction. Not only will it allow to deal with the line singularities in hexahedral
meshing robustly, but it will also provide a Bochner Laplacian (also called the vector Laplacian) in
3D devoid of the type of spurious modes that discrete Laplacians over flat domains can introduce if
one does not enforce a proper discrete deRham complex. Such a tensor calculus for three-manifolds
may allow us to explore possible applications in the context of general relativity in the longer
term. Finally, the design of simplicial or cell meshes that guarantee accurate computations while
approximating a given domain well remains an important endeavor for practical applications.

3.2 Geometry for simulation

Mathematical models of the evolution in time of mechanical systems generally involve systems
of differential equations. Simulating a physical system consists in figuring out how to move the
system forward in time from a set of initial conditions, allowing the computation of an actual
trajectory through classical methods such as fourth-order Runge-Kutta or Newmark schemes.
However, a geometric — instead of a traditional numerical-analytic — approach to the problem of
time integration is particularly pertinent [86]: the very essence of a mechanical system is indeed
characterized by its symmetries and invariants (e.g., momenta), thus preserving these geometric
notions into the discrete computational setting is of paramount importance if one wants discrete
time integration to properly capture the underlying continuous motion. Considering mechanics from
a variational point of view goes back to Euler, Lagrange and Hamilton [74], and Poincaré famously
stated that geometry and physics are “indissociable”. The variational principle most important
for continuous mechanics is due to Hamilton, and is often called Hamilton’s principle or the

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least action principle: it states that a dynamical system always finds an optimal course from one
position to another. One consequence is that we can recast the traditional way of thinking about
an object accelerating in response to applied forces, into a geometric viewpoint: the path followed
by the object between two space-time positions has optimal geometric properties, analogous to
the notion of geodesics on curved surfaces. This point of view is equivalent to Newton’s laws in
the context of classical mechanics, but is broad enough to encompass physical models ranging to
E&M and quantum mechanics [104]. While the idea of discretizing variational formulations of
mechanics is standard for elliptic problems using Galerkin Finite Element methods for instance, only
recently did it get used to derive variational time-stepping algorithms for mechanical systems [103].
These variational integrators have been shown to be remarkably versatile, powerful, and general for
simulations of physical phenomena when compared to traditional numerical time stepping methods:
the symplectic character of variational integrators guarantees good statistical predictability through
accurate preservation of the geometric properties of the exact flow of the differential equations. We
endeavor to continue contributing to this particular application of geometry and extend it further,
as we foresee a number of interesting scientific developments and industrial applications.

State-space discretization of statistical physics. Kinetic equations are used to describe a
variety of phenomena in various scientific fields, ranging from rarefied gas dynamics and plasma
physics to biology and socio-economics, and appear naturally when one considers a statistical
description of a large particle system evolving in time. In incompressible fluid simulation, kinetic
solvers based on the lattice Boltzmann method (LBM) have generated growing interest due to
their use of the Boltzmann transport equation and to its unusual state-space discretization based
on a computationally-efficient lattice [123]: compared to macroscopic solvers directly integrating
Navier-Stokes equations, LBM totally bypasses the difficult issue of discretizing advection to high
order, and absence of global pressure solves makes for extremely efficient parallel implementations,
which are now surpassing alternative discretizations [96]. However, the numerical treatment of
the collision operator of the Boltmann equation has not reached maturity; most surprising is the
complete absence of geometric approaches to deal with Boltzmann equations. One should be able
to formulate a variational approach to LBM based on Hamilton’s principle to derive a systematic
integrator with guaranteed accuracy and structure-preserving properties. Moreover, while dealing
with isothermal and incompressible flows is a good starting point, the kinetic standpoint of fluid
dynamics is not theoretically restricted to this case: far more complex physical systems, from
compressible flow (with shocks), to thermal conductivity, to even acoustics for example, can
be handled; but far less is known on how to handle these more involved cases computationally,
because no systematic numerical approach to handle Boltzmann equations is known. Success in
our geometric approach to LBM should offer a much better handle to deal with these difficult
cases: between new Hermite regularization tools [49, 63] and the recent introduction of variational
integrators for non-equilibrium thermodynamical systems mentioned above should provide the
necessary theoretical foundations to establish a geometric solver for this generalized case.

Learning-aided simulation. Computational physics is experiencing a tectonic shift as data-
driven approaches are quickly becoming mainstream. While we do not adhere to the idea being
floated that numerical integration could be simply “learned” to improve current solvers, the fact
is that many machine learning tools may have profound influence in practical applications using
simulation. Long standing problems such as the design of perfectly matched layers (PML, an
artificial absorbing layer for transport equations used to reduce the domain of simulation without
suffering from reflected waves [61]) or flux limiters in high resolution schemes [132] (to avoid
the spurious oscillations (wiggles) that would otherwise occur due to shocks or sharp changes)
could be found through training, and applied at very low numerical cost. We are curious to see
if geometry can help design better architectures or approaches for this type of learning-aided
simulation, by helping with better loss functions (with soft constraints) or better architectures (to
enforce hard constraints) that account for the importance of structure preservation. Learning the
highly non-linear and chaotic dynamics of fluids is also an interesting direction: we believe that one
can infer predictive high-frequency details of a turbulent flow from a low-resolution simulation as

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it is an attractive alternative to non-linear turbulence modeling, extending the computationally-
expensive Reynolds-Averaged Navier-Stokes (RANS [39]), Large-Eddy Simulation (LES [91]), or
Detached-Eddy Simulation (DES [124]) models used in CFD. Many other learning efforts in the
domain of simulation are being explored, in particular towards the goal of allowing real-time design
of shapes that satisfy some physical properties, such as lowest drag for improved aerodynamics or
highest stiffness for a light cantilever.

Geometric integration of physical systems and multiphysics. Although the use of geometric
integrators for differential equations in computational physics has recently brought off many
numerical improvements, the large body of knowledge in differential geometric mechanics remains
vastly under-utilized in discrete mechanics. Many mechanical systems require geometric objects
such as diffeomorphisms, vector fields, or (principal) connections for which no structure-preserving
discretization exists. Hydrodynamics, for instance, has well established and rich differential
geometric foundations, but rare are the numerical methods that take advantage of this rich
body of knowledge as yet. Yet, satisfying a form of “particle relabeling” symmetry [104] on a
discrete level could directly enforce Kelvin’s circulation theorem, a momentum preservation as
important as angular momentum preservation for rigid bodies. Relativity is another example, albeit
much more involved, where structure-preserving numerics would strongly impact the scientific
community: having discretizations automatically enforcing Bianchi’s identities would not only
simplify the numerical procedures involved in gravitational theory (as spectral accuracy would no
longer be required to avoid spurious modes), but could in fact result in conservation of energy and
angular momentum. Moreover, multiphysics (coupled mechanical systems involving more than
one simultaneously occurring physical field) can be consistently described through constrained
variational principles: a simple, yet already interesting example is the case of the equations of
motion for the garden hose, where rod dynamics coupled with fluid motion was only fully modeled
(along with its nonlinear solutions of traveling-wave type) a few years back [116] through such
a geometric treatment. Now that a variational formulation of nonequilibrium thermodynamics
extending Hamilton’s principle to include irreversible processes has been proposed [78], we are
particularly interested in advancing further the arsenal of computational methods for physical
simulation.

3.3 Geometry for dynamical systems

Dynamical systems – whether physical, biological, chemical, or social – are ubiquitous in nature, and
their study deals with the concept of change, rate of change, rate of rate of change, etc. Dynamical
systems are often better elucidated and modeled through topology and geometry. Whether we
consider a continuous-time dynamical system (flow) or discrete-time dynamical system (map),
the geometric theory of dynamical systems studies phase portraits: on the state-space manifold
(a geometric model for the set of all possible states of the system), the global behavior of the
dynamical system is determined by a cellular structure of basins enclosed by separatrices, each
basin being dominated by a different specific behavior or fate. A system’s trajectories on the
state-space manifold determine velocity vectors by differentiation; conversely, velocity vectors
determine trajectories by integration. Bifurcations can also be understood as geometric models for
the controlled change of one system into another, while the rate of divergence of trajectories in phase
space measures a system’s stability. Given this overwhelming relevance of geometry in dynamical
systems, we intend to dedicate some of our activities to develop geometry-based computational
tools to study time series and dynamical systems: while classic dynamical systems theory has
established solid foundations to study structures in steady and time-periodic flows and maps, new
tools are needed to analyze the complexity of time series or aperiodic large-scale flows from sampled
trajectories, and to automatically generate a simplified skeleton of the overall dynamics of a system
from input data. We discuss a few directions we are interested in further impacting next.

Time series. Geometric methods play an important part in the study of time series. Of particular
interest are time-delay embeddings, which are generically able to capture the underlying state
space and dynamics from which the time series data have been acquired, by the Takens embedding

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theorem [127]. Such embeddings transform discrete time series into point clouds in Euclidean space,
so that the underlying geometry of the point cloud reflects the geometry of the phase space the
data originate from. By doing so, questions related to the seasonality or anomalous behavior of
the time series are naturally reformulated into questions about the geometry or topology of their
embeddings [113]. Beside this approach, other more direct methods apply geometric or topological
tools in the original physical or frequency domain, which, despite its simplicity, has proven to be
relevant in various contexts [66, 70]. A common thread to all these developments is their restriction
to numerical time series, including (but not restricted to) data for which geometry plays an obvious
role—e.g. inertial or gyroscopic sensor data. With potential medical applications in mind, one
of our main long-term goals will be to adapt and extend these approaches to handle categorical
data, in connection to the item Geometry for categorical and mixed data types in the Geometry
for data science theme. We also plan to find principled methods to tuning the various parameters
involved in the techniques, e.g. the window size in time-delay embeddings: we will seek to optimize
or learn these parameters automatically, in connection to the item Geometry-driven learning in the
Geometry for data science theme. We will also seek to make these parameters adaptive, e.g. using
time-varying window sizes in time-delay embeddings of irregular time series, in order to obtain
more accurate data representations and improved learning performance.

Coherent structures. Another interesting area in need of new numerical methods concerns
coherent structures, i.e., persisting features of a flow over long periods that tend to favor or inhibit
material transport between distinct flow regions. While there is no universally agreed-upon definition
for coherent structures (there exist ergodicity-based [52], observer-based [105], and probabilistic [76]
approaches to their definition), most variants and associated computational methods assume a fine
knowledge of the Eulerian velocity field in space and time to deduce a good approximation of the
flow. However, flows are often known only as a set of sparse particle trajectories in time (an example
is the trajectory of buoys in the ocean). Such a sparse sampling of the dynamical system does
not lend itself well to a geometric analysis of transport, so topological methods have recently been
proposed to extract structures from a sparse set of trajectories by measuring their entanglement [129,
40, 136] based on the theory of braid groups, a classical area of topology. Coherent regions can then
be defined as containing particles that possibly mix with other particles within the region itself but
do not mix with particles outside the region; the set of trajectories arising from the particles within
a coherent region forms a coherent bundle. Even if the use of braid groups offers sound foundations
and numerical tools for the definition of coherent structures in 2D, there has been only limited
efforts in developing practical and scalable computational tools for the efficient analysis of flow
structures in 3D, offering a clear opportunity for us to try new geometric insights.

Invariant sets. Much of the theory of dynamical systems revolves around the existence and
structure of invariant sets, which by definition are subsets of the state space that are invariant
under the action of the dynamics. Invariant sets come in many different forms (stationary solutions,
periodic orbits, connecting orbits, chaotic invariant sets, etc), and their structure can be very
complicated and can undergo drastic changes under perturbations of the system, thus making their
study difficult. This is all the more true in practical applications, where the systems are only known
through space and/or time discretizations. Conley index theory [62] overcomes these issues by
restricting the focus to invariant sets that admit an isolating neighborhood, and by introducing a
topological invariant—the Conley index—that characterizes whether such isolated invariant sets
are attracting, repelling, or saddle-like. It is defined as the homotopy type of a pair of compact
subsets of the neighborhood, and it is proven to be independent of the choice of neighborhood—thus
characterizing the invariant set itself. We are interested in the study of invariant sets in the discrete
space and continuous time setting, where the space is typically described by a simplicial complex
and the dynamics by a combinatorial vector (or multivector) field. Building upon Forman’s seminal
work in combinatorial dynamical systems [73], recent advances [45, 97] have shown that isolated
invariant sets and their Conley indices can be properly defined even in this setting, and that they
can be related to the dynamics of some upper semicontinuous acyclic multivalued map defined
on the geometric realization of the simplicial complex; in simpler terms, not only can Conley

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index theory be adapted to the combinatorial setting, but it also connects to its classical analog in
the underlying space. Two important questions for applications arise from this line of work: (1)
how to compute the invariant sets and their Conley indices (including choosing relevant isolating
neighboroods) efficiently? (2) how do they behave under perturbations of the input vector field or
simplicial complex? These questions have just started to be addressed [68, 69], mostly through
the lens of single-parameter topological persistence theory, developed in the context of topological
data analysis. We intend to push this direction further, notably using multi-parameter persistence
theory to cope with some of the key difficulties such as the choice of isolating neighborhoods.

3.4 Geometry for data science

The last decade has seen the advent of machine learning (ML), and in particular deep learning
(DL), in a large variety of fields, including some directly connected to geometry. For instance,
DL-based approaches have become increasingly popular in geometry processing [117] due to their
ability to outperform state-of-the-art, domain-specific methods by leveraging the ever-increasing
amounts of available labeled data. On the downside, DL approaches suffer from a general lack of
explainability. Moreover, their performances can be disappointing on small data due to their large
numbers of parameters; this is especially true with end-to-end learning pipelines, which tend to
require humongous amounts of training data to learn the right data representation. Finally, DL is
by essence tied to Euclidean data representations, and as such it requires intermediate transforms in
order to be applicable to non-Euclidean data types such as graphs or probability measures. Because
of these limitations, we are seeing a rise of geometric and topological methods for data science
in general, and for ML and DL in particular, whose aim is to help address the aforementioned
challenges as well as others. For instance, geometric deep learning [50] tries to generalize deep
neural models to non-Euclidean domains. This includes for instance using information geometry to
apply deep neural models in probability spaces. Topological data analysis (TDA) [109] is another
popular approach to enhance ML and DL methods. It contributes to data science in at least
three different ways: first, by providing data mining tools that can help users uncover hidden
structures in data; second, by providing generic descriptors for geometric data that can be turned
into features for ML and DL with provable stability properties; third, by integrating itself deeply
into existing ML methods or DL architectures to enhance their performances or to analyze their
behavior [58, 100]. Other contributions of geometry to data science at large include: the use of
Forman’s Ricci curvature and its corresponding Ricci flow in networks, to understand the networks’
properties and growth [133]; the application of the Hodge-Hemholtz decomposition to statistical
ranking problems with sparse response data, with theoretical connections to both PageRank and
LASSO [90]; the use of Reeb graphs or Morse-Smale complexes in statistical inference [60] as well
as in data visualization [131]. These important developments reinforce our argument that geometry
and topology have their role to play in the elaboration of the next-generation data analysis tools.
We plan to focus on a few research directions related to these developments, which are of particular
interest in our view.

Deep learning for large-scale 3D geometric data analysis. We first propose to develop
efficient algorithms and mathematical tools for analyzing large geometric data collections using Deep
Learning techniques. This includes 3D shapes represented as triangle or quad meshes, volumetric
data, point clouds possibly embedded in high-dimensions, and graphs representing geometric (e.g.
proximity) data. Our project is motivated by the fact that large annotated collections of geometric
models have recently become available [57, 135], and that machine learning algorithms applied to
such collections have shown promising initial results, both for data analysis as well as synthesis. We
believe that these results can be significantly extended by building on recent advances in geometry
processing, optimization and learning. Our ultimate goal is to design novel deep learning techniques
capable both of handling geometric data directly and of combining and integrating different data
sources into a unified analysis pipeline. A key challenge in this project is the fact that geometric
data can come in a myriad different representations, such as point clouds and meshes among others,
with variable sampling and discretization. Furthermore, geometric shapes can undergo both rigid
and non-rigid deformations. Unfortunately, most existing deep learning approaches focus only

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on a particular type of representations and deformation classes (e.g., considering purely extrinsic
or purely intrinsic methods). Instead we propose to place special focus on designing learning
techniques capable of handing diverse multimodal data sources undergoing arbitrary deformations,
in a coherent theoretical and practical framework. Moreover we propose to develop novel powerful
interactive tools for analysis and annotation, to help harness user input, and also provide better
mechanisms for exploration of variability in the data [120, 112].

Explainable geometric and topological features for data. Another of our goals is to design
geometric and topological features that can capture richer content from the data, while maintaining
the robustness and stability properties that the existing features enjoy. If we can make our features
rich enough so that they characterize the input data (or their underlying geometric structures,
assuming such structures exist) completely, then we will be able to leverage them in the context of
explainable AI, to compute pre-images with guarantees on the corresponding interpretations. In
cases where our features cannot completely describe the data, we will study the geometry of the
fibers of the feature extraction step, in order to quantify the discrepancy that may appear between
different interpretations of the same feature. We envision two complementary approaches for this:

• The first approach relies on feature aggregation. In the context of TDA for instance, one
may consider using multiple filtrations (or filter functions on a fixed simplicial complex),
computing their corresponding topological descriptors, then aggregating these descriptors
together to form a feature vector.

• The second approach relies on more elaborate geometric and topological tools to design
the features. The idea is to encode the joint effect of multiple geometric and topological
constructions on the data, in a more integrated way than just by aggregating the corresponding
features. By encoding more complex effects, we hope to extract a richer content using smaller
constructions.

Research on the first approach in TDA started with [65, 79], who proved that, in the special case
where the data are sampled from some subanalytic compact sets in Euclidean space Rn, the compact
sets themselves are fully described by the aggregated features obtained by orthogonal projections
onto lines. This follows from a more fundamental result on the invertibility of the Radon transforms
of constructible functions [122], to which the above aggregated features belong. This initial result
has sparked a thriving new direction of research, exploring larger and larger classes of compact
sets [89, 102, 108]. Many important questions arise from this line of work, some of which have been
partially addressed, including: what kind of stability or robustness properties do these aggregated
features enjoy? Can the size of the collection of filter functions used be reduced, to become finite
and (more importantly) independent of the compact set under consideration? Can the aggregated
features be computed efficiently? Can non-Euclidean compact sets, such as manifolds or length
spaces, be considered as well, with similar guarantees?

The second approach is related to the development of multi-parameter persistence [54], which is
undeniably the most widely open and long-standing research topic in TDA today. The core challenge
is to define computationally tractable algebraic invariants that can capture as much of the joint
structure of multiple topological constructions as possible. The notorious difficulty of this question
comes from the fact that the algebraic objects underlying multi-parameter topological constructions
are significantly more complicated than the ones underlying single-parameter constructions. The
question also connects to notoriously hard problems in other areas of pure mathematics, such
as the classification of isomorphism classes of indecomposable poset representations in quiver
representation theory for instance. It can benefit from these connections, as mathematical tools that
have been developed for those problems can be imported into the TDA literature—several promising
such imports have been made in the recent past, including from representation theory [46] and
from sheaf theory [92]. In turn, mathematical and algorithmic advances made in multi-parameter
persistence may benefit these other areas of mathematics as well. This is clearly a high-risk and
long-term research topic, but if successful, it may eventually have an enormous impact on TDA
and related areas.

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Geometric feature learning. Geometry and topology have played a key role in the design of
feature extraction pipelines for certain types of data. The numerous existing geometric features
for geometry processing (shape contexts [75], differential and integral invariants [115], heat or
wave kernel signatures [42, 126], etc.) are a sign of the importance of this topic for the computer
graphics community. Meanwhile, the TDA community has developed generic feature extraction
pipelines, based on combinatorial constructions and their algebraic invariants, which have proven to
be useful in a variety of application domains [109]. All these approaches are, however, handcrafted,
with hyperparameters being tuned via manual, grid, or random search. Our goal is to make
these approaches transition from a paradigm of feature engineering to that of feature learning, in
order to set up end-to-end learning pipelines for improved performances and adaptability. Two
complementary directions are considered:

• designing piecewise-smooth variants of the existing pipelines, with a fine control over the
underlying stratification. This will make it possible to apply variational optimization methods,
typically stochastic (sub-)gradient descent, and to optimize the gradient sampling steps for
improved convergence rates.

• designing novel pipelines based on a combination of geometric/topological tools and deep

learning, in order to get the best out of both worlds.

Research in the first direction is still in its infancy. Promising theoretical advances were made
recently, towards understanding the piecewise differentiability of the basic topological persistence
operator in full generality [95], as well as towards optimizing its parameters using classical stochastic
gradient descent [55]. Can the knowledge gained in these studies about the underlying stratification
of the operator be leveraged to optimize the gradient sampling step and thus improve the convergence
rates? Can these results be extended to more advanced pipelines, such as the one for Mapper or for
zigzags and multi-parameter persistence?

The idea behind the second direction is to integrate topological or geometric layers into neural
network architectures such as auto-encoders or GANs for feature extraction — the challenge being
to determine how to do it in the appropriate way, so that we can make the most of this combination.
This question connects to the research topic Geometry-driven learning described further down in
this section.

Geometry-driven learning. Most of the contributions of geometry and topology to machine
learning until recently have been to the design of pre-processing steps (e.g. feature extraction) to
enhance the performances of the learning pipeline. There is now a thriving effort of the community
toward integrating geometric and/or topological computations deeper into the core of the pipeline.
This includes for instance: ToMATo [58], which integrates a TDA-based feedback loop into density
based algorithms to improve their stability and robustness; topological regularizers [59, 87], which
add topology-based regularization terms to the loss in supervised statistical learning; topological
layers [56, 77, 94], which are meant to be incorporated into neural networks. Meanwhile, geometry
and topology have been used to analyze the behavior of neural networks [121, 51]. This exciting line
of work is just emerging, and our intent is to push this direction further, in particular to address
the following important questions:

• How can we generalize the use of topological layers in neural networks? This question
is connected to the differentiability of the TDA pipeline, addressed in the research topic
Geometric feature learning. Inded, generalizing the current (nascent) framework for differential
calculus and optimization with the TDA pipeline will be key to designing both generic and
effective topological layers. Another more practical aspect of the question is to evaluate the
contribution of topological layers as initial or intermediate layers, depending on the neural
network architecture that they are combined with and on the data they are applied to.

• The same question arises for topological regularizers, with similar theoretical and practical

challenges.

• The development of richer families of geometric and topological descriptors, undertaken in the
item Richer geometric and topological features for data, will eventually lead to the question of

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generalizing the current differentiable framework to these new descriptors, in order to make
them as widely applicable as the current descriptors, and also to the practical question of
determining how to best combine them with existing loss functions, regularizers, or neural
network architectures.

• The aforementioned contributions and research directions concern mostly supervised learning.
Can we contribute as well to unsupervised learning problems, including clustering (as ToMATo
does already for density-based clustering), dimensionality reduction, or unsupervised feature
learning? This question connects also to the research topic Geometric feature learning
described previously. One direction we may explore is the design of geometric or topological
layers to be inserted in unsupervised neural network architectures such as auto-encoders or
GANs.

• Finally, as TDA is concerned primarily with topology, an obvious (yet still wide open) question
to ask is whether it can contribute to the current effort towards generating neural network
architectures automatically.

Geometry for categorical and mixed data types. Categorical data types are notoriously
hard to deal with in the context of ML and AI. Indeed, most of the existing ML toolbox has been
designed specifically to work with numerical variables, usually sitting in some vector or metric space.
By contrast, spaces of categorical data do not naturally come equipped with a linear structure
nor a metric. More importantly, these spaces are discrete by nature, so choices of metrics or
(dis-)similarity measures can be scarce, with limited effects on the learning efficiency. To make
things worse, categorical variables are often mixed with numerical variables, and choosing a proper
weighting for them is a challenge in its own right. Meanwhile, categorical variables play an important
part in many applications: for instance, in precision medicine, where the monitoring of patients
relies on collected longitudinal data that include not only numerical variables such as temperature
or blood pressure, but also categorical variables such as illness antecedents or symptoms lists. Thus,
handling categorical and mixed data types represents an important challenge today. Unfortunately,
with very few exceptions [134], it has been mostly overlooked so far in the development of topological
methods for ML and AI, so our goal will be to help fix this situation. The standard approach for
handling categorical variables is to define a proper vector representation, then to apply—either
off-the-shelf or with minor adaptations—an analysis method designed for numerical variables to
the new data representation. A prototypical instance of this approach is Multiple Correspondance
Analysis for dimensionality reduction [38], which applies classical PCA to the one-hot encoding
matrix of the input data. A variant of the approach replaces the vector representation by a suitable
metric or (dis-)similarity measure on the initial categorical variables or on some transformed version
of those. For instance, in clustering, one can define a metric on the input data, e.g. Jaccard or
Hamming distance, then apply a hierarchical bottom-up clustering algorithm such as single-linkage
to the resulting distance matrix. This variant seems quite appropriate for geometric or topological
methods, since the latter typically work with metric or (dis-)similarity spaces. The challenge is to
determine with which metrics or (dis-)similarity measures, and on which data types, geometric or
topological methods will be provably better.

A more refined version of the approach learns the new data representation instead of engineering
it, which is particularly relevant when end-to-end learning pipelines are sought for. The methods are
usually taylored to a specific data type, for instance word2vec [106] computes word embeddings for
text data using a two-layer neural network. Our developments in the research topic Geometry-driven
learning will make it possible to combine TDA layers with such networks, and thus to benefit from
the most recent advances on representation learning for these data types. The challenge will be to
understand when and how to make the most of this combination.

4 Application domains

Our work aims at a wide range of applications covering 3D shape analysis and processing, simulation,
and data science in general. While we typically focus on contributions that are of a fundamental,

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mathematical and algorithmic nature, we seek collaborations with academics and industrial from
applied fields, who can use our tools on practical and concrete problems. Here are a few examples
of collaborations:

• In the context of 3D geometry processing, we collaborate with Dassault Systèmes for a) the
PhD of Lucas Brifault on the design of novel geometric representations for shapes through
measure theory and b) the PhD of Mariem Mezghanni on the design of physical simulation
layers for 3D modeling.

• In the context of personalized medicine, we collaborate with statisticians and medical doctors
to incorporate our geometric and topological features into learning pipelines to design better
dynamic treatment regimens (AEx PreMediT).

• In a collaboration with the French Ministry of Defense, we seek to develop tools to analyze
multimodal time series data in order to predict the appearance of G-LOCs among fighter jet
pilots in training or in operation (PhD of Julie Mordacq).

Beside these few illustrative examples, GeomeriX also maintains regular collaborations with
Sanofi, EDF, Danone R&D, Immersion Tools, as well as with several key players in the world-wide
tech industry, including Ansys, Adobe Research, Disney/Pixar, NVidia.

5 Highlights of the year

5.1 Awards

• The paper “Functional Maps: A Flexible Representation of Maps Between Shapes” co-authored
by Maks Ovsjanikov and colleagues, has received the SIGGRAPH 2023 Test-of-Time award.

• Maks Ovsjanikov has received an ERC Consolidator Grant in 2023.

• The paper “Implicit fairing of irregular meshes using diffusion and curvature flow” co-authored
by Mathieu Desbrun and colleagues has been included in the ACM list of “Seminal Graphics
Papers: Pushing the Boundaries”.

5.2 Distinctions

• S. Oudot was a CAS fellow at the Norwegian Academy of Science and Letters for the academic

year 2022-2023.

• Maks Ovsjanikov became a fellow of ELLIS, the European Laboratory for Learning and

Intelligent Systems, bringing together top AI researchers in Europe.

• Mathieu Desbrun received an INRIA Chair on Geometry and AI.

6 New results

We list our new results for each of the four themes that our team is articulated around.

6.1 Geometry for Euclidean shape processing

6.1.1 Patternshop: Editing Point Patterns by Image Manipulation

Participants: Pooran Memari.

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In collaboration with Xingchang Huang, Hans-Peter Seidel and Gurprit Singh (MPI Saarbrucken)
and Tobias Ritschel (UCL).

Point patterns are characterized by their density and correlation. While spatial variation of
density is well-understood, analysis and synthesis of spatially-varying correlation is an open challenge.
No tools are available to intuitively edit such point patterns, primarily due to the lack of a compact
representation for spatially varying correlation. In this work [13], we propose a low-dimensional
perceptual embedding for point correlations. This embedding can map point patterns to common
three-channel raster images, enabling manipulation with off-the-shelf image editing software. To
synthesize back point patterns, we propose a novel edge-aware objective that carefully handles
sharp variations in density and correlation. The resulting framework allows intuitive and backward-
compatible manipulation of point patterns, such as recoloring, relighting to even texture synthesis
that have not been available to 2D point pattern design before. Effectiveness of our approach is tested
in several user experiments. Our proposed framework was patented under the reference M35655EP.
The corresponding code is however publicly available at https://github.com/xchhuang/patternshop.

6.1.2 Robust Pointset Denoising of Piecewise-Smooth Surfaces through Line Processes

Participants:

Jiayi Wei, Jiong Chen, Pooran Memari, Mathieu Desbrun.

In collaboration with Damien Rohmer (LIX).

Denoising is a common, yet critical operation in geometry processing aiming at recovering
high-fidelity models of piecewise smooth objects from noise-corrupted pointsets. Despite a sizable
literature on the topic, there is a dearth of approaches capable of processing very noisy and
outlier-ridden input pointsets for which no normal estimates and no assumptions on the underlying
geometric features or noise type are provided. In this paper [21], we propose a new robust-statistics
approach to denoising pointsets based on line processes to offer robustness to noise and outliers
while preserving sharp features possibly present in the data. While the use of robust statistics
in denoising is hardly new, most approaches rely on prescribed filtering using data-independent
blending expressions based on the spatial and normal closeness of samples. Instead, our approach
deduces a geometric denoising strategy through robust and regularized tangent plane fitting of the
initial pointset, obtained numerically via alternating minimizations for efficiency and reliability. Key
to our variational approach is the use of line processes to identify inliers vs. outliers, as well as the
presence of sharp features. We demonstrate that our method can denoise sampled piecewise-smooth
surfaces for levels of noise and outliers at which previous works fall short.

6.1.3 Feature-Sized Sampling for Vector Line Art

Participants: Pooran Memari.

In collaboration with Stefan Ohrhallinger (TU Wien) and Amal D. Parakkat (Telecom Patis).

In this work [27], by introducing a first-of-its-kind quantifiable sampling algorithm based on
feature size, we present a fresh perspective on the practical aspects of planar curve sampling.
Following the footsteps of ϵ-sampling, which was originally proposed in the context of curve
reconstruction to offer provable topological guarantees (Crust algorithm) under quantifiable bounds,
we propose an arbitrarily precise ϵ-sampling algorithm for sampling smooth planar curves (with a
prior bound on the minimum feature size of the curve). This paper not only introduces the first such

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algorithm which provides user-control and quantifiable precision but also highlights the importance
of such a sampling process under two key contexts: 1) To conduct a first study comparing theoretical
sampling conditions with practical sampling requirements for reconstruction guarantees that can
further be used for analysing the upper bounds of ϵ for various reconstruction algorithms with or
without proofs, 2) As a feature-aware sampling of vector line art that can be used for applications
such as coloring and meshing.

6.1.4 Bio-Sketch: A new medium for interactive storytelling, illustrated by the

phenomenon of infection

Participants: Pooran Memari.

In collaboration with Pauline Olivier, Renaud Chabrier, Marie-Paule Cani (LIX), Jean-Luc Coll
(INSERM, CNRS, Institute for Advanced Biosciences, Grenoble).

In the field of biology, digital illustrations play a crucial role in conveying complex phenomena,
allowing for idealized shapes and motion, in contrast to data visualization. In the absence of suitable
media, scientists often rely on oversimplified 2D figures or have to call in professional artists to
create better illustrations, which can be limiting. In this work [28] we introduce Bio-Sketch, a novel
progressive sketching system designed to ease the creation of animated illustrations, as exemplified
here in the context of the infection phenomenon. Our solution relies on a new progressive sketching
paradigm that seamlessly combines 3D modeling and pattern-based shape distribution to create
background volume and temporal animation control. The elements created can be assembled into a
complex scenario, enabling narrative design experiments for educational applications in biology.
Our results and first feedback from experts in illustration and biology demonstrate the potential
of Bio-Sketch to assist communication on the infection phenomenon, helping to bridge the gap
between expert and non-expert audiences.

6.1.5 Somigliana Coordinates

Participants:

Jiong Chen, Mathieu Desbrun.

In collaboration with Fernando de Goes (Pixar).

In this work [24], we present a novel cage deformer based on elasticity-derived matrix-valued
coordinates. In order to bypass the typical shearing artifacts and lack of volume control of existing
cage deformers, we promote a more elastic behavior of the cage deformation by deriving our
coordinates from the Somigliana identity, a boundary integral formulation based on the fundamental
solution of linear elasticity. Given an initial cage and its deformed pose, the deformation of the
cage interior is deduced from these Somigliana coordinates via a corotational scheme, resulting in a
matrix-weighted combination of both vertex positions and face normals of the cage. Our deformer
thus generalizes Green coordinates, while producing physically-plausible spatial deformations that
are invariant under similarity transformations and with interactive bulging control. We demonstrate
the efficiency and versatility of our method through a series of examples in 2D and 3D.

6.2 Geometry for simulation

6.2.1 Building a Virtual Weakly-Compressible Wind Tunnel Testing Facility

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Participants: Wei Li, Mathieu Desbrun.

In collaboration with Chaoyang Lyu, Kai Bai, Yiheng Wu, and Xiaopei Liu (all from Shanghaitech

University, China), and Changxi Zheng (Columbia University, USA).

Virtual wind tunnel testing is a key ingredient in the engineering design process for the automotive
and aeronautical industries as well as for urban planning: through visualization and analysis of
the simulation data, it helps optimize lift and drag coefficients, increase peak speed, detect high
pressure zones, and reduce wind noise at low cost prior to manufacturing. In this paper [18], we
develop an efficient and accurate virtual wind tunnel system based on recent contributions from
both computer graphics and computational fluid dynamics in high-performance kinetic solvers.
Running on one or multiple GPUs, our massively-parallel lattice Boltzmann model meets industry
standards for accuracy and consistency while exceeding current mainstream industrial solutions
in terms of efficiency Ð especially for unsteady turbulent flow simulation at very high Reynolds
number (on the order of 107) – due to key contributions in improved collision modeling and
boundary treatment, automatic construction of multiresolution grids for complex models, as well as
performance optimization. We demonstrate the efficacy and reliability of our virtual wind tunnel
testing facility through comparisons of our results to multiple benchmark tests, showing an increase
in both accuracy and efficiency compared to state-of-the-art industrial solutions. We also illustrate
the fine turbulence structures that our system can capture, indicating the relevance of our solver
for both VFX and industrial product design.

6.2.2 Fluid-Solid Coupling in Kinetic Two-Phase Flow Simulation

Participants: Wei Li, Mathieu Desbrun.

Real-life flows exhibit complex and visually appealing behaviors such as bubbling, splashing,
glugging and wetting that simulation techniques in graphics have attempted to capture for years.
While early approaches were not capable of reproducing multiphase flow phenomena due to their
excessive numerical viscosity and low accuracy, kinetic solvers based on the lattice Boltzmann
method have recently demonstrated the ability to simulate water-air interaction at high Reynolds
numbers in a massively-parallel fashion. However, robust and accurate handling of fluid-solid
coupling has remained elusive: be it for CG or CFD solvers, as soon as the motion of immersed
objects is too fast or too sudden, pressures near boundaries and interfacial forces exhibit spurious
oscillations leading to blowups. Built upon a phase-field and velocity-distribution based lattice-
Boltzmann solver for multiphase flows, this paper [16] spells out a series of numerical improvements
in momentum exchange, interfacial forces, and two-way coupling to drastically reduce these typical
artifacts, thus significantly expanding the types of fluid-solid coupling that we can efficiently
simulate. We highlight the numerical benefits of our solver through various challenging simulation
results, including comparisons to previous work and real footage.

6.2.3 High-Order Moment-Encoded Kinetic Simulation of Turbulent Flows

Participants: Wei Li, Mathieu Desbrun.

In collaboration with Tongtong Wang, Zherong Pang, Xifeng Gao, and Kui Wu (Tencent Lightspeed
Studios, China).

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Kinetic solvers for incompressible fluid simulation were designed to run efficiently on massively
parallel architectures such as GPUs. While these lattice Boltzmann solvers have recently proven
much faster and more accurate than the macroscopic Navier-Stokes-based solvers traditionally used
in graphics, it systematically comes at the price of a very large memory requirement: a mesoscopic
discretization of statistical mechanics requires over an order of magnitude more variables per grid
node than most fluid solvers in graphics. In order to open up kinetic simulation to gaming and
simulation software packages on commodity hardware, we propose a High-Order Moment-Encoded
Lattice-Boltzmann-Method solver which we coined HOME-LBM, requiring only the storage of
a few moments per grid node [17], with little to no loss of accuracy in the typical simulation
scenarios encountered in graphics. We show that our lightweight and lightspeed fluid solver requires
three times less memory and runs ten times faster than state-of-the-art kinetic solvers, for a
nearly-identical visual output.

6.3 Geometry for data science

6.3.1 Stable Vectorization of Multiparameter Persistent Homology using Signed

Barcodes as Measures

Participants:

Steve Oudot.

In collaboration with David Loiseaux (Inria, Datashape team), Luis Scoccola (Northeastern Univer-
sity), Magnus Botnan (Vrije Universiteit Amsterdam), and Mathieu Carrière (Inria, Datashape
team).

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted
graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most
applications of PH focus on the one-parameter case – where the descriptors summarize the changes
in topology of data as it is filtered by a single quantity of interest – and there is now a wide array of
methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable
vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH
(MPH) of data that is filtered by several quantities of interest encodes much richer information
than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so
far limited the available options for the stable vectorization of MPH. In this work [25] we aim to
bring together the best of both worlds by showing how the interpretation of signed barcodes – a
recent family of MPH descriptors – as signed measures leads to natural extensions of vectorization
strategies from one parameter to multiple parameters. The resulting feature vectors are easy to
define and to compute, and provably stable. While, as a proof of concept, we focus on simple
choices of signed barcodes and vectorizations, we already see notable performance improvements
when comparing our feature vectors to state-of-the-art topology-based methods on various types of
data.

6.3.2 Local characterizations for decomposability of 2-parameter persistence modules

Participants: Vadim Lebovici, Steve Oudot.

In collaboration with Magnus Botnan (Vrije Universiteit Amsterdam).

In this work [12] we investigate the existence of sufficient local conditions under which poset
representations decompose as direct sums of indecomposables from a given class. In our work, the
indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter

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persistence in topological data analysis. Our indecomposables of interest belong to the so-called
interval modules, which by definition are indicator representations of intervals in the poset. While
the whole class of interval modules does not admit such a local characterization, we show that
the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest
subclass to do so.

6.3.3 A Gradient Sampling Algorithm for Stratified Maps with Applications to

Topological Data Analysis

Participants:

Steve Oudot.

In collaboration with Jacob Leygonie (University of Oxford), Mathieu Carrière (Inria, Datashape
team), and Théo Lacombe (Université de Marne la Vallée).

In this work [15] we introduce a novel gradient descent algorithm refining the well-known
Gradient Sampling algorithm on the class of stratifiably smooth objective functions, which are
defined as locally Lipschitz functions that are smooth on some regular pieces—called the strata—of
the ambient Euclidean space. On this class of functions, our algorithm achieves a sub-linear
convergence rate. We then apply our method to objective functions based on the (extended)
persistent homology map computed over lower-star filters, which is a central tool of Topological
Data Analysis. For this, we propose an efficient exploration of the corresponding stratification
by using the Cayley graph of the permutation group. Finally, we provide benchmarks and novel
topological optimization problems that demonstrate the utility and applicability of our framework.

6.3.4 Shape Non-rigid Kinematics (SNK): A Zero-Shot Method for Non-Rigid Shape
Matching via Unsupervised Functional Map Regularized Reconstruction

Participants:

Souhaib Attaiki, Maks Ovsjanikov.

In this work [32], we present Shape Non-rigid Kinematics (SNK), a novel zero-shot method for
non-rigid shape matching that eliminates the need for extensive training or ground truth data. SNK
operates on a single pair of shapes, and employs a reconstruction-based strategy using an encoder-
decoder architecture, which deforms the source shape to closely match the target shape. During
the process, an unsupervised functional map is predicted and converted into a point-to-point map,
serving as a supervisory mechanism for the reconstruction. To aid in training, we have designed
a new decoder architecture that generates smooth, realistic deformations. SNK demonstrates
competitive results on traditional benchmarks, simplifying the shape matching process without
compromising accuracy.

6.3.5 Zero-Shot 3D Shape Correspondence

Participants: Ahmed Abdelreheem, Abdelrahman Eldesokey, Maks Ovsjanikov,

Peter Wonka.

In collaboration with Ahmed Abdelreheem, Abdelrahman Eldesokey, and Peter Wonka, from King
Abdullah University of Science and Technology (KAUST), Saudi Arabia.

In this work [29] we propose a novel zero-shot approach to computing correspondences between
3D shapes. Existing approaches mainly focus on isometric and near-isometric shape pairs (e.g.,

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human vs. human), but less attention has been given to strongly non-isometric and inter-class
shape matching (e.g., human vs. cow). To this end, we introduce a fully automatic method that
exploits the exceptional reasoning capabilities of recent foundation models in language and vision to
tackle difficult shape correspondence problems. Our approach comprises multiple stages. First, we
classify the 3D shapes in a zero-shot manner by feeding rendered shape views to a language-vision
model (e.g., BLIP2) to generate a list of class proposals per shape. These proposals are unified
into a single class per shape by employing the reasoning capabilities of ChatGPT. Second, we
attempt to segment the two shapes in a zero-shot manner, but in contrast to the co-segmentation
problem, we do not require a mutual set of semantic regions. Instead, we propose to exploit the
in-context learning capabilities of ChatGPT to generate two different sets of semantic regions for
each shape and a semantic mapping between them. This enables our approach to match strongly
non-isometric shapes with significant differences in geometric structure. Finally, we employ the
generated semantic mapping to produce coarse correspondences that can further be refined by
the functional maps framework to produce dense point-to-point maps. Our approach, despite its
simplicity, produces highly plausible results in a zero-shot manner, especially between strongly
non-isometric shapes.

6.3.6 VoroMesh: Learning Watertight Surface Meshes with Voronoi Diagrams

Participants: Nissim Maruani, Roman Klokov, Maks Ovsjanikov, Mathieu Des-

brun.

In collaboration with Pierre Alliez (Inria).

In this work [26], we present VoroMesh, a novel and differentiable Voronoi-based representation
of watertight 3D shape surfaces. From a set of 3D points (called generators) and their associated
occupancy, we define our boundary representation through the Voronoi diagram of the generators as
the subset of Voronoi faces whose two associated (equidistant) generators are of opposite occupancy:
the resulting polygon mesh forms a watertight approximation of the target shape’s boundary. To
learn the position of the generators, we propose a novel loss function, dubbed VoroLoss, that
minimizes the distance from groundtruth surface samples to the closest faces of the Voronoi diagram
which does not require an explicit construction of the entire Voronoi diagram. A direct optimization
of the Voroloss to obtain generators on the Thingi32 dataset demonstrates the geometric efficiency
of our representation compared to axiomatic meshing algorithms and recent learning-based mesh
representations. We further use VoroMesh in a learning-based mesh prediction task from input
SDF grids on the ABC dataset, and show comparable performance to state-of-the-art methods
while guaranteeing closed output surfaces free of self-intersections.

6.3.7 SATR: Zero-Shot Semantic Segmentation of 3D Shapes

Participants: Maks Ovsjanikov.

In collaboration with Ahmed Abdelreheem, Ivan Skorokhodov, and Peter Wonka, from from King
Abdullah University of Science and Technology (KAUST), Saudi Arabia.

In this work [30], we explore the task of zero-shot semantic segmentation of 3D shapes by
using large-scale off-the-shelf 2D image recognition models. Surprisingly, we find that modern
zero-shot 2D object detectors are better suited for this task than contemporary text/image similarity
predictors or even zero-shot 2D segmentation networks. Our key finding is that it is possible to
extract accurate 3D segmentation maps from multi-view bounding box predictions by using the

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topological properties of the underlying surface. For this, we develop the Segmentation Assignment
with Topological Reweighting (SATR) algorithm and evaluate it on ShapeNetPart and our proposed
FAUST benchmarks. SATR achieves state-of-the-art performance and outperforms a baseline
algorithm by 1.3% and 4% average mIoU on the FAUST coarse and fine-grained benchmarks,
respectively, and by 5.2% average mIoU on the ShapeNetPart benchmark.

6.3.8 Spatially and Spectrally Consistent Deep Functional Maps

Participants: Maks Ovsjanikov.

Joint work with Mingze Sun (Tsinghua Shenzhen International Graduate School, China) , Shiwei
Mao (Tsinghua Shenzhen International Graduate School, China) , Puhua Jiang (Tsinghua Shenzhen
International Graduate School, China, Peng Cheng Laboratory, China), Ruqi Huang (Tsinghua
Shenzhen International Graduate School, China).

In this paper [34], we investigate the utility of cycle consistency in Deep Functional Maps. We
first justify that under certain conditions, the learned maps, when represented in the spectral
domain, are already cycle consistent. Furthermore, we identify the discrepancy that spectrally
consistent maps are not necessarily spatially, or point-wise, consistent. In light of this, we present
a novel design of unsupervised Deep Functional Maps, which effectively enforces the harmony
of learned maps under the spectral and the point-wise representation. By taking advantage of
cycle consistency, our framework produces state-of-the-art results in mapping shapes even under
significant distortions. Beyond that, by independently estimating maps in both spectral and
spatial domains, our method naturally alleviates over-fitting in network training, yielding superior
generalization performance and accuracy within an array of challenging tests for both near-isometric
and non-isometric datasets.

6.3.9 TIDE: Time Derivative Diffusion for Deep Learning on Graphs

Participants: Maysam Behmanesh, Maks Ovsjanikov.

In collaboration with Maximilian Krahn (Aalto University, Finland).

In this paper [33], we present a novel method based on time derivative graph diffusion (TIDE)
to overcome the structural limitations of the message-passing framework in graph neural networks.
Our approach allows for optimizing the spatial extent of diffusion across various tasks and network
channels, thus enabling medium and long-distance communication efficiently. Furthermore, we show
that our architecture design also enables local message-passing and thus inherits from the capabilities
of local message-passing approaches. We show that on both widely used graph benchmarks and
synthetic mesh and graph datasets, the proposed framework outperforms state-of-the-art methods
by a significant margin.

6.3.10 ReVISOR: ResUNets with visibility and intensity for structured outlier removal

Participants: Maxime Kirgo, Maks Ovsjanikov.

In collaboration with Guillaume Terrasse and Guillaume Thibault (EDF R&D).

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In this paper [14], we make several contributions to address the problem of reflection-induced
outlier detection. First, to overcome the scarcity of annotated data, we introduce a new dataset
tailored for this task. Second, to capture non-local dependencies, we study and demonstrate, for
the first time, the utility of deep learning based semantic segmentation architectures for reflection-
induced outlier detection. By doing so, we bring together the fields of shape denoising/repair and
semantic segmentation. Third, we demonstrate that additional non-local cues in the form of laser
intensity and a computed visibility signal help boost the performance considerably. We denote our
pipeline as ResUNets with Visibility and Intensity for Structured Outlier Removal, or ReVISOR,
and demonstrate its superior performance against existing baselines on real-world data.

6.3.11 Assessing craniofacial growth and form without landmarks: A new automatic

approach based on spectral methods

Participants: Robin Magnet, Maks Ovsjanikov.

In collaboration with Kevin Bloch (Institut Necker Enfants-Malades), Maxime Taverne (Institut
Necker Enfants-Malades) , Simone Melzi (University of Milano-Bicocca), Maya Geoffroy (Institut
Necker Enfants-Malades), Roman Khonsari (Institut Necker Enfants-Malades)

In this paper [19] we present a novel method for the morphometric analysis of series of 3D
shapes, and demonstrate its relevance for the detection and quantification of two craniofacial
anomalies: trigonocephaly and metopic ridges, using CT-scans of young children. Our approach
is fully automatic, and does not rely on manual landmark placement and annotations. Our
approach furthermore allows to differentiate shape classes, enabling successful differential diagnosis
between trigonocephaly and metopic ridges, two related conditions characterized by triangular
foreheads. These results were obtained using recent developments in automatic non-rigid 3D
shape correspondence methods and specifically spectral approaches based on the functional map
framework. Our method can capture local changes in geometric structure, in contrast to methods
based, for instance, on global shape descriptors. As such, our approach allows to perform automatic
shape classification and provides visual feedback on shape regions associated with different classes
of deformations. The flexibility and generality of our approach paves the way for the application of
spectral methods in quantitative medicine.

6.3.12 Affection: Learning Affective Explanations for Real-World Visual Data

Participants: Maks Ovsjanikov.

In collaboration with Panos Achlioptas (Snap Inc.), Leonidas Guibas (Stanford University), Sergey

Tulyakov (Snap Inc.)

In this work [31], we explore the space of emotional reactions induced by real-world images. For
this, we first introduce a large-scale dataset that contains both categorical emotional reactions and
free-form textual explanations for 85,007 publicly available images, analyzed by 6,283 annotators
who were asked to indicate and explain how and why they felt when observing a particular image,
with a total of 526,749 responses. Although emotional reactions are subjective and sensitive to
context (personal mood, social status, past experiences)-we show that there is significant common
ground to capture emotional responses with a large support in the subject population. In light of
this observation, we ask the following questions: i) Can we develop neural networks that provide
plausible affective responses to real-world visual data explained with language? ii) Can we steer such

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methods towards producing explanations with varying degrees of pragmatic language, justifying
different emotional reactions by grounding them in the visual stimulus? Finally, iii) How to evaluate
the performance of such methods for this novel task? In this work, we take the first steps in
addressing all of these questions, paving the way for more human-centric and emotionally-aware
image analysis systems.

6.3.13 Generalizable Local Feature Pre-training for Deformable Shape Analysis

Participants:

Souhaib Attaiki, Lei Li, Maks Ovsjanikov.

In this paper [22], we analyze the link between feature locality and transferability in tasks
involving deformable 3D objects, while also comparing different backbones and losses for local
feature pre-training. We observe that with proper training, learned features can be useful in such
tasks, but, crucially, only with an appropriate choice of the receptive field size. We then propose
a differentiable method for optimizing the receptive field within 3D transfer learning. Jointly,
this leads to the first learnable features that can successfully generalize to unseen classes of 3D
shapes such as humans and animals. Our extensive experiments show that this approach leads to
state-of-the-art results on several downstream tasks such as segmentation, shape correspondence,
and classification.

6.3.14 Understanding and Improving Features Learned in Deep Functional Maps

Participants:

Souhaib Attaiki, Maks Ovsjanikov.

In this paper [23], we show that under some mild conditions, the features learned within deep
functional map approaches can be used as point-wise descriptors and thus are directly comparable
across different shapes, even without the necessity of solving for a functional map at test time.
Furthermore, informed by our analysis, we propose effective modifications to the standard deep
functional map pipeline, which promote structural properties of learned features, significantly
improving the matching results. Finally, we demonstrate that previously unsuccessful attempts at
using extrinsic architectures for deep functional map feature extraction can be remedied via simple
architectural changes, which encourage the theoretical properties suggested by our analysis. We
thus bridge the gap between intrinsic and extrinsic surface-based learning, suggesting the necessary
and sufficient conditions for successful shape matching.

6.3.15 Scalable and Efficient Functional Map Computations on Dense Meshes

Participants: Robin Magnet, Maks Ovsjanikov.

In this paper [20], we propose a new scalable version of the functional map pipeline that allows
to efficiently compute correspondences between potentially very dense meshes. Unlike existing
approaches that process dense meshes by relying on ad-hoc mesh simplification, we establish an
integrated end-to-end pipeline with theoretical approximation analysis. In particular, our method
overcomes the computational burden of both computing the basis, as well the functional and
pointwise correspondence computation by approximating the functional spaces and the functional
map itself. Errors in the approximations are controlled by theoretical upper bounds assessing
the range of applicability of our pipeline. With this construction in hand, we propose a scalable
practical algorithm and demonstrate results on dense meshes, which approximate those obtained
by standard functional map algorithms at the fraction of the computation time. Moreover, our

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approach outperforms the standard acceleration procedures by a large margin, leading to accurate
results even in challenging cases.

7 Bilateral contracts and grants with industry

7.1 Bilateral contracts with industry

7.1.1 Contract with Sanofi Inc.

Participants: Maks Ovsjanikov.

Title: Machine learning approaches for cryo-EM

Partner Institution(s): Sanofi Inc.

Date/Duration: 2023-2024

Additionnal info/keywords: Cryogenic electron microscopy (cryo-EM) allows the structure
determination of antibody fragments bound to pharmaceutically relevant targets to accelerate
drug discovery. The process of cryo-EM data analysis is time consuming and requires user
input. To accelerate the rate of structure solution by cryo-EM, this project investigates
machine learning approachesto fit and model the atomic coordinates of antibody fragments
into the cryo-EM density.

The project funds one post-doctoral researcher for 2 years, jointly between Sanofi Inc., and
Ecole Polytechnique (the employer of Maks Ovsjanikov).

7.1.2 Contract with DASSAULT SYSTEMES

Participants: Maks Ovsjanikov.

Title: Generative Models for the Guided Synthesis of Complex and Functional 3D Scenes

Partner Institution(s): DASSAULT SYSTEMES

Date/Duration: 2023-2026

Additionnal info/keywords: This thesis focuses on machine learning applied to 3D computer
vision, specifically addressing challenges related to the automatic synthesis of 3D environments.

The project funds one PhD student for 3 years.

7.1.3 MEDITWIN with DASSAULT SYSTEMES

Participants: Maks Ovsjanikov, Mathieu Desbrun.

Title: MEDITWIN: Virtual human twins for medical applications

Partner Institution(s): DASSAULT SYSTEMES

Date/Duration: 2023-2028

Additionnal info/keywords: In the context of IPCEI on Health called MEDITWIN, Geomerix
has started working on geometric measure theory and reduced models (Desbrun) and non-rigid
registration (Ovsjanikov), with one student and two postdocs to be hired soon.

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8 Partnerships and cooperations

8.1

International research visitors

8.1.1 Visits of international scientists

Other international visits to the team

Shreyas Samaga

Status PhD

Institution of origin: Purdue University

Country: USA

Dates: May-July 2023

Context of the visit: collaboration with Steve Oudot

Mobility program/type of mobility: research stay

8.1.2 Visits to international teams

Research stays abroad

Steve Oudot

Visited institution: Center for Advanced Study

Country: Norway

Dates: January-February 2023

Context of the visit: participation in the program Representation Theory: Combinatorial Aspects

and Applications

Mobility program/type of mobility: research stay as a fellow

Pooran Memari

Visited institution: Technische Universität Berlin

Country: Germany

Dates: February-June 2023

Context of the visit: Sabbatical leave, Visiting TU Berlin CG group,

Mobility program/type of mobility: research stay for collaboration initiation

8.2 European initiatives

8.2.1 Horizon Europe

Participants: Maks Ovsjanikov.

Title: Exploring Relations in Structured Data with Functional Maps

Partner Institution(s): • European Research Commission (ERC) Starting Grant

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Date/Duration: 2018-2023

Additionnal info/keywords: We propose to lay the theoretical foundations and design efficient
computational methods for analyzing, quantifying and exploring relations and variability
in structured data sets, such as collections of geometric shapes, point clouds, and large
networks or graphs, among others. In particular, we propose to depart from the standard
representations of objects as collections of primitives, such as points or triangles, and instead
to treat them as functional spaces that can be easily manipulated and analyzed. Since
real-valued functions can be defined on a wide variety of data representations and as they
enjoy a rich algebraic structure, such an approach can provide a completely novel unified
framework for representing and processing different types of data. Key to our study is the
exploration of relations and variability between objects, which can be expressed as operators
acting on functions and thus treated and analyzed as objects in their own right using the vast
number of tools from functional analysis in theory and numerical linear algebra in practice.

Participants: Maks Ovsjanikov.

Title: VEGA: Universal Geometric Transfer Learning

Partner Institution(s): • European Research Commission (ERC) Consolidator Grant

Date/Duration: 2024-2028

Additionnal info/keywords: In this project, we propose to develop a theoretical and practical
framework for transfer learning with geometric 3D data. Most existing learning-based
approaches, aimed at analyzing 3D data, are based on training neural networks from scratch
for each data modality and application. Our main goal will be to develop universally-applicable
methods by combining powerful pre-trainable modules with effective multi-scale analysis and
fine-tuning, given minimal task-specific data. The overall key to our study will be analyzing
rigorous ways, both theoretically and in practice, in which solutions can be transferred and
adapted across problems, semantic categories and geometric data types.

8.2.2 H2020 projects

Participants: Pooran Memari.

Title: Creating Lively Interactive Populated Environments

Partner Institution(s): • University of Cyprus, Universitat Politecnica de Catalunya, Uni-
versity College London, Trinity College Dublin, Max Planck Institute for Intelligent
Systems, KTH Royal Institute of Technology.

Date/Duration: 2020-2024

Additionnal info/keywords: This project designs new techniques to create and control interact-
ive virtual worlds and characters, benefiting from opportunities open by the wide availability
of emergent technologies in the domains of human digitization and artificial intelligence.

8.3 National initiatives

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AEx PreMediT

Participants:

Steve Oudot.

Inria Annual Report 2023

Title: Precision Medicine using Topology

Partner Institution(s): • CRESS, Hôtel-Dieu, France

Date/Duration: 2022-2025

Additionnal info/keywords: While recent advances in machine learning are opening promising
prospects for precision medicine, the sometimes small size, sparsity, or partly categorical
nature of the data involved pose some crucial challenges. The goal of PreMediT is to address
these challenges by integrating information about the geometric and topological structure of
the data into the machine learning pipelines.

ANR AI Chair AIGRETTE

Participants: Maks Osjanikov.

Title: Analyzing Large Scale Geometric Data Collections

Partner Institution(s): • ANR

Date/Duration: 2020-2024

Additionnal info/keywords: Motivated by the deluge of 3D data using geometric representations
(point clouds, triangle, quad meshes, graphs...) that are ill-suited for modern applications, we
are developing efficient algorithms and mathematical tools for analyzing diverse geometric
data collections.

9 Dissemination

9.1 Promoting scientific activities

9.1.1 Scientific events: organisation

General chair, scientific chair

• Mathieu Desbrun organized the Workshop Machine Learning for Geometry at the Institut

Poincaré (see https://ml4geo.sciencesconf.org/).

Member of the organizing committees

• Steve Oudot co-organized the Mini-Symposium Recent Developments in Multi-Parameter
Persistence at the SIAM Conference on Applied Algebraic Geometry (https://meetings.s
iam.org/sess/dsp_programsess.cfm?SESSIONCODE=77710).

9.1.2 Scientific events: selection

Chair of conference program committees

• Pooran Memari served as Program Co-Chair of Eurographics Symposium on Geometry

Processing (SGP) 2023.

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Member of the conference program committees

• Pooran Memari was a member of the Program Committee of Eurographics 2023 Conference.

• Mathieu Desbrun was a member of the Program Committee of Technical Papers for ACM

SIGGRAPH Asia 2023.

9.1.3 Journal

Member of the editorial boards

• Steve Oudot is a member of the Editorial Board of the Journal of Computational Geometry.

• Maks Ovsjanikov is a member of the Editorial Board of the IEEE Transactions on Visualization

and Computer Graphics journal.

• Pooran Memari is a member of the Editorial Board of the Journal of Computer Graphics

Forum (CGF).

• Pooran Memari is also a member of the Editorial Board of Graphical Models Journal, Elsevier.

• Mathieu Desbrun ss a member of the Editorial Board of the Journal of Geometric Mechanics.

9.1.4

Invited talks

• Maks Ovsjanikov gave a keynote talk at the Symposium on 3D Object Retrieval 2023

(3DOR’23) held in Lille, France.

• Maks Ovsjanikov gave a keynote talk at Pacific Graphics 2023, held in Daejeon, South Korea.

• Maks Ovsjanikov gave a keynote talk at the Machine Learning for Geometry Workshop, held

in Paris, France.

• Steve Oudot gave an invited talk in the special session The ubiquity of quivers and their

representations at the 29th Nordic Congress of Mathematicians, Aalborg, Denmark.

• Pooran Memari gave an invited talk at the LIB colloquium (Laboratoire d’Informatique de

Bourgogne) in Dijon, France, Nov. 2023.

• Mathieu Desbrun gave an invited talk at Ecole Normale Supérieure, France, Nov. 2023.

• Mathieu Desbrun gave an invited talk for the “Journée scientifique du groupe SMAI-SIGMA”

at Jussieu, France, Dec. 2023.

9.1.5 Leadership within the scientific community

• Pooran Memari is the local coordinator for the GT-MG (Modélisation Géométrique).

9.1.6 Research administration

• Steve Oudot is vice-president of the Commission Scientifique at Inria Saclay.

• Steve Oudot is a member of the Comité de Département in the CS Department of École

Polytechnique (DIX).

• Pooran Memari is a member of the comité de Web du LIX, École Polytechnique.

• Pooran Memari is a deputy member of the conseil de laboratoire du LIX, École Polytechnique.

• Mathieu Desbrun is a member of the Comité du Labo in the CS lab of École Polytechnique

(LIX)

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Inria Annual Report 2023

9.2 Teaching - Supervision - Juries

9.2.1 Teaching

• Master: Steve Oudot, Computational Geometry and Topology, 18h eq-TD, M2, MPRI;

• Master: Steve Oudot, Topological data analysis, 45h eq-TD, M1, École polytechnique, France;

• Master: Mathieu Desbrun, Digital Representation and Analysis of Shapes, M2, École poly-

technique, France;

• Master: Pooran Memari, Artificial Intelligence and Advanced Visual Computing, and Digital

Representation and Analysis of Shapes, M2, École polytechnique, France;

• Master: Maks Ovsjanikov, Artificial Intelligence and Advanced Visual Computing, École

polytechnique, France;

• Undergrad-Master: Steve Oudot, Algorithms for data analysis in C++, 22.5h eq-TD, L3/M1,

École Polytechnique, France.

• Master-PhD: Pooran Memari is a member of the Jury d’admission Masters & PhD Track

IGD (Interaction, Graphics & Design), IP-Paris (2020-2023).

9.2.2 Supervision

• PhD: Vadim Lebovici, Deux Approches Complémentaires de la Persistance Multiparamétrique:
Décompositions en Intervalles et Fonctions Constructibles, Université Paris-Saclay. Defended
in September 2023. Steve Oudot and François Petit (CRESS).

• PhD: Jiayi Wei, Robust Statistics for Geometry Processing – Detecting and Handling Discon-
tinuities and Dissimilarities in 3D Pointset Denoising and Mesh Parameterization. Defended
in December 2023. Pooran Memari and Damien Rohmer.

• PhD in progress: Julie Mordacq, Analyse Topologique des Données et Apprentissage Machine
pour analyser et prédire des transitions de phase en n-dimensions, Institut Polytechnique de
Paris. Started Sept. 2022. Steve Oudot and Vicky Kalogeiton (Vista, LIX).

• PhD in progress: Jingyi Li, Invariants algébriques effectifs pour la persistance multi-paramètre,

Institut Polytechnique de Paris. Started Nov. 2023. Steve Oudot.

• PhD: Mariem Mezghanni, Structural and Functional Learning for Industrial Design Automat-
ization. Defended in February 2023. Institut Polytechnique de Paris. Maks Ovsjanikov with
Malika Boulkenafed (Dassault Systèmes).

• PhD: Nicolas Donati, Robust representations for supervised and unsupervised 3D shape
matching. Defended in January 2023. Institut Polytechnique de Paris. Maks Ovsjanikov with
Etienne Corman (CNRS).

• PhD in progress: Souhaib Attaiki, 3D shape analysis with methods based on Deep Learning,

Institut Polytechnique de Paris. Started Nov. 2020. Maks Ovsjanikov

• PhD in progress: Robin Magnet, Robust Spectral Methods for Shape Analysis and Deformation
Assessment, Institut Polytechnique de Paris. Started February 2021. Maks Ovsjanikov

• PhD in progress: Souhail Hadgi, Transfer learning for 3D data, Institut Polytechnique de

Paris. Started January 2023. Maks Ovsjanikov

• PhD in progress: Ramana S Sundararaman, Analysis of large scale 3D shape collection with
learning based approaches, Institut Polytechnique de Paris. Started October 2021. Maks
Ovsjanikov

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• PhD in progress: Tim Scheller, Capturing 4D Plant Growth, Institut Polytechnique de Paris.
Started October 2021. Maks Ovsjanikov with Marie-Paule Cani (Ecole Polytechnique).

• PhD in progress: Nissim Maruani, Machine Learning for Geometric Modeling; Started October

2022. Mathieu Desbrun amd Pierre Alliez (INRIA Sophia-Antipolis).

• PhD in progress: Lucas Brifault, Geometric Measure Theory for Geometric Modeling; Started

April 2022. Mathieu Desbrun and David Cohen-Steiner (INRIA Sophia-Antipolis).

• PhD in progress: Theo Braune, Discrete Bundled-valued Exterior Calculus; Started October

2022. Mathieu Desbrun.

9.2.3 Juries

• Pooran Memari was an examiner for the Ph.D. defense of Guillaume Coiffier, LORIA,

Université de Lorraine, Dec. 2023.

• Mathieu Desbrun was an examiner for the PhD defense of Loï Paulin, LIRIS, Lyon, Apr.

2023.

10 Scientific production

10.1 Major publications

[1] K. Bai, C. Wang, M. Desbrun and X. Liu. ‘Predicting high-resolution turbulence details
in space and time’. In: ACM Transactions on Graphics 40.6 (Dec. 2021), p. 200. doi:
10.1145/3478513.3480492. url: https://hal.inria.fr/hal-03551723.

[2] J. Chen, F. Schäfer, J. Huang and M. Desbrun. ‘Multiscale Cholesky Preconditioning for Ill-
conditioned Problems’. In: ACM Transactions on Graphics 40.4 (19th July 2021), Art. 91. doi:
10.1145/3450626.3459851. url: https://hal.archives-ouvertes.fr/hal-03277277.

[3] N. Donati, A. Sharma and M. Ovsjanikov. ‘Deep Geometric Functional Maps: Robust Feature
Learning for Shape Correspondence’. In: CVPR. Seattle (virtual), United States, 14th June
2020. url: https://hal-polytechnique.archives-ouvertes.fr/hal-03001057.

[4] P. Ecormier-Nocca, P. Memari, J. Gain and M.-P. Cani. ‘Accurate Synthesis of Multi-Class
Disk Distributions’. In: Computer Graphics Forum 38.2 (2019). url: https://hal.inria
.fr/hal-02064699.

[5] P. Guerrero, Y. Kleiman, M. Ovsjanikov and N. J. M. Mitra. ‘PCPNET Learning Local
Shape Properties from Raw Point Clouds’. In: Computer Graphics Forum 37 (2018). url:
https://hal.science/hal-01741744.

[6] T. Lacombe, M. Cuturi and S. Oudot. ‘Large Scale computation of Means and Clusters for
Persistence Diagrams using Optimal Transport’. In: NIPS. Montreal, Canada, 2018. url:
https://hal.inria.fr/hal-01966674.

[7] J. Leygonie, S. Y. Oudot and U. Tillmann. ‘A Framework for Differential Calculus on
Persistence Barcodes’. In: Foundations of Computational Mathematics (2021). doi: 10.1007
/s10208-021-09522-y. url: https://hal.archives-ouvertes.fr/hal-02304300.
[8] W. Li, Y. Ma, X. Liu and M. Desbrun. ‘Efficient kinetic simulation of two-phase flows’. In:
ACM Transactions on Graphics 41.4 (2022), p. 114. doi: 10.1145/3528223.3530132. url:
https://hal.archives-ouvertes.fr/hal-03828392.

[9] P. Mullen, P. Memari, F. de Goes and M. Desbrun. ‘Hodge-optimized triangulations’. In:
ACM Transactions on Graphics 30.4 (Aug. 2011), pp. 103, 1–12. url: https://hal.teleco
m-paris.fr/hal-02286600.

[10] S. Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis. Mathem-
atical Surveys and Monographs 209. American Mathematical Society, 2015, p. 218. url:
https://hal.inria.fr/hal-01247501.

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[11] J.-M. Roufosse, A. Sharma and M. Ovsjanikov. ‘Unsupervised Deep Learning for Structured
Shape Matching’. In: ICCV. SEOUL, South Korea, 29th Oct. 2019. url: https://hal.sci
ence/hal-02305166.

10.2 Publications of the year

International journals

[12] M. B. Botnan, V. Lebovici and S. Oudot. ‘Local characterizations for decomposability of
2-parameter persistence modules’. In: Algebras and Representation Theory (2023). doi:
10.1007/s10468-022-10189-4. url: https://hal.science/hal-02995822.

[13] X. Huang, T. Ritschel, H.-P. Seidel, P. Memari and G. Singh. ‘Patternshop: Editing Point
Patterns by Image Manipulation’. In: ACM Transactions on Graphics 42.4 (26th July 2023),
pp. 1–14/53. doi: 10.1145/3592418. url: https://hal.science/hal-04235133.
[14] M. Kirgo, G. Terrasse, G. Thibault and M. Ovsjanikov. ‘ReVISOR: ResUNets with visibility
and intensity for structured outlier removal’. In: ISPRS Journal of Photogrammetry and
Remote Sensing 202 (2023), pp. 184–204. doi: 10.1016/j.isprsjprs.2023.05.027. url:
https://hal.science/hal-04352356.

[15] J. Leygonie, M. Carrière, T. Lacombe and S. Oudot. ‘A Gradient Sampling Algorithm
for Stratified Maps with Applications to Topological Data Analysis’. In: Mathematical
Programming 202 (2023), pp. 199–239. doi: 10.1007/s10107-023-01931-x. url: https:
//hal.science/hal-03330940.

[16] W. Li and M. Desbrun. ‘Fluid-Solid Coupling in Kinetic Two-Phase Flow Simulation’. In:
ACM Transactions on Graphics 42.4 (26th July 2023), pp. 1–14. doi: 10.1145/3592138.
url: https://inria.hal.science/hal-04174289.

[17] W. Li, T. Wang, Z. Pan, X. Gao, K. Wu and M. Desbrun. ‘High-Order Moment-Encoded
Kinetic Simulation of Turbulent Flows’. In: ACM Transactions on Graphics 42.6 (1st Dec.
2023). doi: 10.1145/3618341. url: https://inria.hal.science/hal-04223237.
[18] C. Lyu, K. Bai, Y. Wu, M. Desbrun, C. Zheng and X. Liu. ‘Building a Virtual Weakly-
Compressible Wind Tunnel Testing Facility’. In: ACM Transactions on Graphics 42.4 (Aug.
2023), pp. 1–20. doi: 10.1145/3592394. url: https://inria.hal.science/hal-0417429
5.

[19] R. Magnet, K. Bloch, M. Taverne, S. Melzi, M. Geoffroy, R. H. Khonsari and M. Ovsjanikov.
‘Assessing craniofacial growth and form without landmarks: A new automatic approach
based on spectral methods’. In: Journal of Morphology 284.8 (2023), e21609. doi: 10.1002
/jmor.21609. url: https://hal.science/hal-04352348.

[20] R. Magnet and M. Ovsjanikov. ‘Scalable and Efficient Functional Map Computations on
Dense Meshes’. In: Computer Graphics Forum 42.2 (23rd May 2023), pp. 89–101. doi:
10.1111/cgf.14746. url: https://hal.science/hal-04352328.

[21] J. Wei, J. Chen, D. Rohmer, P. Memari and M. Desbrun. ‘Robust Pointset Denoising of
Piecewise-Smooth Surfaces through Line Processes’. In: Computer Graphics Forum 42.2
(2023), pp. 175–189. doi: 10.1111/cgf.14752. url: https://polytechnique.hal.scienc
e/hal-04027983.

International peer-reviewed conferences

[22] S. Attaiki, L. Li and M. Ovsjanikov. ‘Generalizable Local Feature Pre-training for Deformable
Shape Analysis’. In: CVPR 2023 - The IEEE/CVF Conference on Computer Vision and
Pattern Recognition. Vancouver, Canada, 18th June 2023. url: https://hal.science/hal
-04352340.

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[23] S. Attaiki and M. Ovsjanikov. ‘Understanding and Improving Features Learned in Deep
Functional Maps’. In: CVPR 2023 - The IEEE/CVF Conference on Computer Vision and
Pattern Recognition. Vancouver, Canada: IEEE, 18th June 2023. doi: 10.1109/CVPR52729
.2023.00133. url: https://hal.science/hal-04352333.

[24] J. Chen, F. de Goes and M. Desbrun. ‘Somigliana Coordinates: an elasticity-derived approach
for cage deformation’. In: SIGGRAPH 2023 - Special Interest Group on Computer Graphics
and Interactive Techniques Conference. Los Angeles, CA, United States: ACM, 23rd July
2023, pp. 1–8. doi: 10.1145/3588432.3591519. url: https://inria.hal.science/hal-
04174303.

[25] D. Loiseaux, L. Scoccola, M. Carrière, M. B. Botnan and S. Oudot. ‘Stable Vectorization
of Multiparameter Persistent Homology using Signed Barcodes as Measures’. In: NeurIPS
2023 - 36th Conference on Neural Information Processing Systems. Advances in Neural
Information Processing Systems 36. New Orleans (LA), United States, 6th June 2023. url:
https://hal.science/hal-04133009.

[26] N. Maruani, R. Klokov, M. Ovsjanikov, P. Alliez and M. Desbrun. ‘VoroMesh: Learning
Watertight Surface Meshes with Voronoi Diagrams’. In: ICCV 2023 - International Conference
on Computer Vision. Paris, France, 28th Aug. 2023. url: https://inria.hal.science/ha
l-04199011.

[27] S. Ohrhallinger, A. D. Parakkat and P. Memari. ‘Feature-Sized Sampling for Vector Line
Art’. In: Pacific Graphics 2023 - The 31th Pacific Conference on Computer Graphics and
Applications. Daejeon, South Korea, 2023. url: https://hal.science/hal-04224967.

[28] P. Olivier, R. Chabrier, P. Memari, J.-L. Coll and M.-P. Cani. ‘Bio-Sketch: A new medium
for interactive storytelling illustrated by the phenomenon of infection’. In: Eurographics
Workshop on Visual Computing for Biology and Medicine. VCBM 2023 - 13th Eurographics
Workshop on Visual Computing for Biology and Medicine. Norrköping, Sweden, 19th Sept.
2023, p. 11. url: https://hal.science/hal-04216965.

Conferences without proceedings

[29] A. Abdelreheem, A. Eldesokey, M. Ovsjanikov and P. Wonka. ‘Zero-Shot 3D Shape Cor-
respondence’. In: SIGGRAPH ASIA 2023 - The 16th ACM SIGGRAPH Conference and
Exhibition on Computer Graphics and Interactive Techniques. Sydney, Australia, 10th Dec.
2023. doi: 10.1145/3610548.3618228. url: https://hal.science/hal-04352378.
[30] A. Abdelreheem, I. Skorokhodov, M. Ovsjanikov and P. Wonka. ‘SATR: Zero-Shot Semantic
Segmentation of 3D Shapes’. In: ICCV 2023 - International Conference on Computer Vision.
Paris, France, 2nd Oct. 2023. url: https://hal.science/hal-04352372.

[31] P. Achlioptas, M. Ovsjanikov, L. Guibas and S. Tulyakov. ‘Affection: Learning Affective
Explanations for Real-World Visual Data’. In: CVPR 2023 - IEEE/CVF Conference on
Computer Vision and Pattern Recognition. Vancouver, Canada: IEEE, 18th June 2023. doi:
10.1109/CVPR52729.2023.00642. url: https://hal.science/hal-04352344.

[32] S. Attaiki and M. Ovsjanikov. ‘Shape Non-rigid Kinematics (SNK): A Zero-Shot Method for
Non-Rigid Shape Matching via Unsupervised Functional Map Regularized Reconstruction’.
In: NeurIPS 2023 - 37th Conference on Neural Information Processing Systems. New Orleans
(Louisiana), United States, 10th Dec. 2023. url: https://hal.science/hal-04352383.

[33] M. Behmanesh, M. Krahn and M. Ovsjanikov. ‘TIDE: Time Derivative Diffusion for Deep
Learning on Graphs’. In: ICML 2023 - The 40th annual International Conference on Machine
Learning. Honolulu, United States, 23rd July 2023. url: https://hal.science/hal-0435
2364.

[34] M. Sun, S. Mao, P. Jiang, M. Ovsjanikov and R. Huang. ‘Spatially and Spectrally Consistent
Deep Functional Maps’. In: ICCV 2023 - International Conference on Computer Vision.
Paris, France, 2nd Oct. 2023. url: https://hal.science/hal-04352370.

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Doctoral dissertations and habilitation theses

[35] V. Lebovici. ‘Two complementary approaches in multi-parameter persistence : interval-
decompositions and constructible functions’. Université Paris-Saclay, 12th Sept. 2023. url:
https://theses.hal.science/tel-04235196.

Reports & preprints

[36] R. Marin, S. Attaiki, S. Melzi, E. Rodolà and M. Ovsjanikov. Smoothness and effective
regularizations in learned embeddings for shape matching. 14th Jan. 2023. doi: 10.48550/ar
Xiv.2112.07289. url: https://hal.science/hal-03938011.

[37] Y.-Q. Wang, L. Li, J.-B. Jiang, S. Arthaud, P.-H. Luppi and Z.-L. Huang. Glutamatergic
neurons in the lateral periaqueductal gray mediate wakefulness and REM sleep through
different pathways. 6th June 2022. doi: 10.21203/rs.3.rs-1650527/v1. url: https://ha
l.science/hal-04311796.

10.3 Cited publications

[38] H. Abdi and D. Valentin. ‘Multiple correspondence analysis’. In: Encyclopedia of measurement

and statistics 2.4 (2007), pp. 651–657.

[39] G. Alfonsi. ‘Reynolds-averaged Navier–Stokes equations for turbulence modeling’. In: Applied

Mechanics Reviews 62.4 (2009).

[40] M. R. Allshouse and J.-L. Thiffeault. ‘Detecting coherent structures using braids’. In: Physica

D: Nonlinear Phenomena 241 (2012), pp. 95–105.

[41] D. N. Arnold, R. S. Falk and R. Winther. ‘Finite element exterior calculus, homological

techniques, and applications’. In: Acta Numerica 15 (2006), pp. 1–155.

[42] M. Aubry, U. Schlickewei and D. Cremers. ‘The wave kernel signature: A quantum mechanical
approach to shape analysis’. In: 2011 IEEE international conference on computer vision
workshops (ICCV workshops). IEEE. 2011, pp. 1626–1633.

[43] O. Azencot, M. Ben-Chen, F. Chazal and M. Ovsjanikov. ‘An operator approach to tangent
vector field processing’. In: Computer Graphics Forum. Vol. 32. 5. Wiley Online Library.
2013, pp. 73–82.

[44] O. Azencot, M. Ovsjanikov, F. Chazal and M. Ben-Chen. ‘Discrete derivatives of vector

fields on surfaces–an operator approach’. In: ACM Trans. Graph. 34.3 (2015), pp. 1–13.

[45] B. Batko, T. Kaczynski, M. Mrozek and T. Wanner. ‘Linking combinatorial and classical
dynamics: Conley index and Morse decompositions’. In: Foundations of Computational
Mathematics 20.5 (2020), pp. 967–1012.

[46] U. Bauer, M. B. Botnan, S. Oppermann and J. Steen. ‘Cotorsion torsion triples and the
representation theory of filtered hierarchical clustering’. In: Advances in Mathematics 369
(2020), p. 107171.

[47] A. I. Bobenko and B. A. Springborn. ‘Variational Principles for Circle Patterns and Koebe’s

Theorem’. In: Trans. Amer. Math. Soc. 356 (2003), pp. 659–689.

[48] J.-D. Boissonnat, R. Dyer and A. Ghosh. ‘Delaunay triangulation of manifolds’. In: Founda-

tions of Computational Mathematics 18.2 (2018), pp. 399–431.

[49] F. Brogi, O. Malaspinas, B. Chopard and C. Bonadonna. ‘Hermite regularization of the
lattice Boltzmann method for open source computational aeroacoustics’. In: Journal of the
Acoustical Society of America 142.4 (2017), pp. 2332–2345.

[50] M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam and P. Vandergheynst. ‘Geometric Deep
Learning: Going beyond Euclidean data’. In: IEEE Signal Processing Magazine 34.4 (2017),
pp. 18–42. doi: 10.1109/MSP.2017.2693418.

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[51] R. Brüel Gabrielsson and G. Carlsson. ‘Exposition and Interpretation of the Topology of
Neural Networks’. In: 2019 18th IEEE International Conference On Machine Learning And
Applications (ICMLA). 2019, pp. 1069–1076.

[52] M. Budiši ć and I. Mezić. ‘Geometry of the ergodic quotient reveals coherent structures in

flows’. In: Physica D: Nonlinear Phenomena 241.15 (2012), pp. 1255–1269.

[53] A. Cacciola, A. Muscoloni, V. Narula, A. Calamuneri, S. Nigro, E. A. Mayer, J. S. Labus,
G. Anastasi, A. Quattrone, A. Quartarone et al. ‘Coalescent embedding in the hyper-
bolic space unsupervisedly discloses the hidden geometry of the brain’. In: arXiv preprint
arXiv:1705.04192 (2017).

[54] G. Carlsson and A. Zomorodian. ‘The Theory of Multidimensional Persistence’. In: Discrete

and Computational Geometry 42.1 (2009), pp. 71–93.

[55] M. Carriere, F. Chazal, M. Glisse, Y. Ike and H. Kannan. ‘Optimizing persistent homology

based functions’. In: Proc. International Conference on Machine Learning. 2021.

[56] M. Carrière, F. Chazal, Y. Ike, T. Lacombe, M. Royer and Y. Umeda. ‘PersLay: a neural net-
work layer for persistence diagrams and new graph topological signatures’. In: International
Conference on Artificial Intelligence and Statistics (PMLR). 2020, pp. 2786–2796.

[57] A. X. Chang, T. Funkhouser, L. Guibas, P. Hanrahan, Q. Huang, Z. Li, S. Savarese, M.
Savva, S. Song, H. Su et al. ‘Shapenet: An information-rich 3d model repository’. In: arXiv
preprint arXiv:1512.03012 (2015).

[58] F. d. Chazal, L. J. Guibas, S. Oudot and P. Skraba. ‘Persistence-Based Clustering in
Riemannian Manifolds’. In: Journal of the ACM 60.6 (2013), pp. 1–38. doi: 10.1145/25
35927. url: http://doi.acm.org/10.1145/2535927.

[59] C. Chen, X. Ni, Q. Bai and Y. Wang. ‘A topological regularizer for classifiers via persistent
homology’. In: The 22nd International Conference on Artificial Intelligence and Statistics
(PMLR). 2019, pp. 2573–2582.

[60] Y.-C. Chen, C. R. Genovese and L. Wasserman. ‘Statistical inference using the Morse-Smale

complex’. In: Electronic Journal of Statistics 11.1 (2017), pp. 1390–1433.

[61] A. Chern. ‘A Reflectionless Discrete Perfectly Matched Layer’. In: Journal of Computational

Physics 381 (2019), pp. 91–109.

[62] C. C. Conley. Isolated invariant sets and the Morse index. 38. American Mathematical Soc.,

1978.

[63] C. Coreixas, G. Wissocq, G. Puigt, J.-F. Boussuge and P. Sagaut. ‘Recursive regularization

step for high-order lattice Boltzmann methods’. In: Phys. Rev. E 96 (3 2017), p. 033306.

[64] K. Crane, M. Desbrun and P. Schröder. ‘Trivial Connections on Discrete Surfaces’. In:
Computer Graphics Forum ( Symposium on Geometry Processing) 29.5 (2010), pp. 1525–
1533.

[65] J. Curry, S. Mukherjee and K. Turner. ‘How Many Directions Determine a Shape and other
Sufficiency Results for Two Topological Transforms’. In: arXiv preprint arXiv:1805.09782
(2018).

[66] A. Deckard, R. C. Anafi, J. B. Hogenesch, S. B. Haase and J. Harer. ‘Design and analysis
of large-scale biological rhythm studies: a comparison of algorithms for detecting periodic
signals in biological data’. In: Bioinformatics 29.24 (2013), pp. 3174–3180.

[67] M. Desbrun, E. Kanso and Y. Tong. ‘Discrete Differential Forms for Computational Modeling’.

In: Discrete Differential Geometry. Springer, 2007.

[68] T. K. Dey, M. Juda, T. Kapela, J. Kubica and M. Lipiński. In: ().

[69] T. K. Dey, M. Mrozek and R. Slechta. ‘Persistence of the Conley Index in Combinatorial
Dynamical Systems’. In: 36th International Symposium on Computational Geometry (SoCG
2020). Ed. by S. Cabello and D. Z. Chen. Vol. 164. Schloss Dagstuhl–Leibniz-Zentrum für
Informatik, 2020, 37:1–37:17.

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[70] M. Dindin, Y. Umeda and F. Chazal. ‘Topological data analysis for arrhythmia detection
through modular neural networks’. In: Canadian Conference on Artificial Intelligence.
Springer. 2020, pp. 177–188.

[71] N. Donati, A. Sharma and M. Ovsjanikov. ‘Deep geometric functional maps: Robust feature
learning for shape correspondence’. In: Proceedings of the IEEE/CVF conference on Computer
Vision and Pattern Recognition. 2020, pp. 8592–8601.

[72] P. Ecormier-Nocca, P. Memari, J. Gain and M.-P. Cani. ‘Accurate synthesis of multi-class
disk distributions’. In: Computer Graphics Forum. Vol. 38(2). Wiley Online Library. 2019,
pp. 157–168.

[73] R. Forman. ‘Combinatorial vector fields and dynamical systems’. In: Mathematische Zeits-

chrift 228.4 (1998), pp. 629–681.

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<scratchpad>
- Main topic: Research Centre at Inria Saclay Centre at Institut Polytechnique of Paris
- Key points:
  - Project-Team GEOMERIX focuses on geometry-driven numerics
  - Four research themes: Euclidean shape processing, simulation, dynamical systems, and data science
  - The team aims to develop predictive computational tools using geometric notions
  - Geometric structures and invariants are used to preserve motion and simulate physical systems
  - The team also explores the application of geometry in machine learning and data science
- Structure: The document is organized into four sections: Activity, Contents, Research Program, and Application Domains
- Potential artifacts to ignore: HTML tags, links, and web-related elements
</scratchpad>

<final_summary>
The document presents the Research Centre at Inria Saclay Centre at Institut Polytechnique of Paris, specifically the Project-Team GEOMERIX, which focuses on geometry-driven numerics. The team explores four research themes: Euclidean shape processing, simulation, dynamical systems, and data science. They aim to develop predictive computational tools using geometric notions, preserving motion and simulating physical systems. The team also investigates the application of geometry in machine learning and data science, exploring its potential to enhance performance, robustness, and explainability of methods. By leveraging geometric structures and invariants, GEOMERIX seeks to contribute to the development of new numerical tools and algorithms, with a focus on interdisciplinary projects and applications in various fields.
The document presents the Research Centre at Inria Saclay Centre at Institut Polytechnique of Paris, specifically the Project-Team GEOMERIX, which focuses on geometry-driven numerics. The team explores four research themes: Euclidean shape processing, simulation, dynamical systems, and data science. They aim to develop predictive computational tools using geometric notions, preserving motion and simulating physical systems. The team also investigates the application of geometry in machine learning and data science, exploring its potential to enhance performance, robustness, and explainability of methods. By leveraging geometric structures and invariants, GEOMERIX seeks to contribute to the development of new numerical tools and algorithms, with a focus on interdisciplinary projects and applications in various fields.