2024Activity reportProject-TeamGEOMERIX

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 122, 55, 56, 132. 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 111, 100, 83. 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 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 123, 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 130, 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 131 or a length cross-ratio 107 per edge) later formed the basis of circle patterns 59, 105 as well as conformal representations 137; the discrete Laplace-Beltrami cotan formula 126 also determines the edge lengths of a triangle mesh (and thus its discrete metric) up to a global scaling 149. More recently, generalized notions of metrics were proposed; for instance, 97 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 96. 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 119. 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 60.
• 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 84. 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 65. 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 142 and Taimiņa's crochet model). We are hoping that this intrinsic metric characterization can be investigated through recent discrete hyperbolic parametrization tools 92, 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 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 95 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 140, 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 79 or FEEC 53), 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 94 to provide a powerful Isogeometric Analysis (IGA). Recent developments 112, 93 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 76, 112 (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.

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 135: 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 109. 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 61, 75 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 73) or flux limiters in high resolution schemes 144 (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 it is an attractive alternative to non-linear turbulence modeling, extending the computationally-expensive Reynolds-Averaged Navier-Stokes (RANS 51), Large-Eddy Simulation (LES 103), or Detached-Eddy Simulation (DES 136) 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 115 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 128 through such a geometric treatment. Now that a variational formulation of nonequilibrium thermodynamics extending Hamilton's principle to include irreversible processes has been proposed 90, we are particularly interested in advancing further the arsenal of computational methods for physical simulation.

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 theorem 139. 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 125. 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 78, 82. 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 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 64, observer-based 117, and probabilistic 88 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 141, 52, 148 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 74 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 85, recent advances 57, 110 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 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 80, 81, 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.

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 69, 147, 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 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 132, 124.

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 77, 91, who proved that, in the special case where the data are sampled from some subanalytic compact sets in Euclidean space ${ℝ}^n$, 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 134, 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 101, 114, 121. 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 66, 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 58 and from sheaf theory 104. 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.

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 87, differential and integral invariants 127, heat or wave kernel signatures 54, 138, 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 120. 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 108, as well as towards optimizing its parameters using classical stochastic gradient descent 67. 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 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 70, which integrates a TDA-based feedback loop into density based algorithms to improve their stability and robustness; topological regularizers 72, 99, which add topology-based regularization terms to the loss in supervised statistical learning; topological layers 68, 89, 106, which are meant to be incorporated into neural networks. Meanwhile, geometry and topology have been used to analyze the behavior of neural networks 133, 63. 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 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 146, 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 50, 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 118 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.

6.2.1 MFS-chol

• Name:
  Lightning-fast Method of Fundamental Solutions
• Keywords:
  3D, Boundary element method
• Functional Description:
  The method of fundamental solutions (MFS) and its associated boundary element method (BEM) are commonly used due to the reduced dimensionality they offer: for three-dimensional linear problems, they only require variables on the domain boundary to solve and evaluate the solution throughout space, making them a valuable tool in a wide variety of applications. However, MFS and BEM have poor computational scalability and huge memory requirements for large-scale problems, limiting their applicability and efficiency in practice. By leveraging connections with Gaussian Processes and exploiting the sparse structure of the inverses of boundary integral matrices, we introduce a variational preconditioner that can be computed via a sparse inverse-Cholesky factorization in a massively parallel manner. We show that applying our preconditioner to the Preconditioned Conjugate Gradient algorithm greatly improves the efficiency of MFS or BEM solves, up to four orders of magnitude in our series of tests.
• Release Contributions:
  N/A
• Publication:
  hal-04589038v1
• Contact:
  Mathieu Desbrun
• Participants:
  Jiong Chen, Mathieu Desbrun

7.1.1 SING: Stability-Incorporated Neighborhood Graph

Participants: Pooran Memari, Steve Oudot.

In collaboration with Diana Marin, Stefan Ohrhallinger and Michael Wimmer (TU Wien) and Amal Dev Parakkat (Telecom, IP-Paris).

7.1.2 Stochastic Computation of Barycentric Coordinates

Participants: Mathieu Desbrun.

In collaboration with Fernando de Goes (Pixar).

In this work 22, we present a practical and general approach for computing barycentric coordinates through stochastic sampling. Our key insight is a reformulation of the kernel integral defining barycentric coordinates into a weighted least-squares minimization that enables Monte Carlo integration without sacrificing linear precision. Our method can thus compute barycentric coordinates directly at the points of interest, both inside and outside the cage, using just proximity queries to the cage such as closest points and ray intersections. As a result, we can evaluate barycentric coordinates for a large variety of cage representations (from quadrangulated surface meshes to parametric curves) seamlessly, bypassing any volumetric discretization or custom solves. To address the archetypal noise induced by sample-based estimates, we also introduce a denoising scheme tailored to barycentric coordinates. We demonstrate the efficiency and flexibility of our formulation by implementing a stochastic generation of harmonic coordinates, mean-value coordinates, and positive mean-value coordinates.

7.1.3 PoNQ: a Neural QEM-based Mesh Representation

Participants: Mathieu Desbrun, Nissim Maruani, Maks Ovsjanikov.

In collaboration with Pierre Alliez (Inria Sophia).

Although polygon meshes have been a standard rep resentation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning based applications. In this work 39, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neu ral shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the bound ary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target sur face alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.

7.1.4 Biharmonic Coordinates and their Derivatives for Triangular 3D Cages

Participants: Jiong Chen.

In collaboration with Jean-Marc Thiery and Élie Michel (Adobe).

This work 32 extends biharmonic coordinates, which were previously derived for 2D shape deformation, into three dimensions. The key contribution is deriving closed-form mathematical expressions for biharmonic coordinates and their derivatives specifically for 3D triangular cages. At the heart of this work is the development of closed-form expressions that calculate how the Euclidean distance integrates over a triangle, along with the derivatives of this integration. The significance of this advancement is twofold: it completes a gap in the theory of generalized barycentric coordinates, and it enables practical applications in 3D shape manipulation. These applications include creating various types of biharmonic deformations, solving shape deformation problems through variational methods, and providing efficient closed-form solutions for the recently developed Somigliana coordinates. This work ultimately bridges a theoretical gap while offering practical tools for 3D shape manipulation and deformation.

7.1.5 A Survey on Cage-based Deformation of 3D Models

Participants: Jiong Chen.

In collaboration with Daniel Ströter, Johannes Sebastian Mueller-Roemer, Sebastian Besler, Andre Stork and Dieter W. Fellner (Technical University of Darmstadt), Jean-Marc Thiery and Tamy Boubekeur (Adobe), Kai Hormann and Qingjun Chang (University of Italian Switzerland).

In this work 30, we review the advancement of 3D cage-based deformation. Cage-based deformation enables users to quickly manipulate 3D geometry by deforming the cage. Due to their utility, cage-based deformation techniques increasingly appear in many geometry modeling applications. For this reason, the computer graphics community has invested a great deal of effort in the past decade and beyond into improving automatic cage generation and cage-based deformation. Recent advances have significantly extended the practical capabilities of cage-based deformation methods. As a result, there is a large body of research on cage-based deformation. In this report, we provide a comprehensive overview of the current state of the art in cage-based deformation of 3D geometry. We discuss current methods in terms of deformation quality, practicality, and precomputation demands. In addition, we highlight potential future research directions that overcome current issues and extend the set of practical applications. In conjunction with this survey, we publish an application to unify the most relevant deformation methods. Our report is intended for computer graphics researchers, developers of interactive geometry modeling applications, and 3D modeling and character animation artists.

7.1.6 BallMerge: High-quality Fast Surface Reconstruction via Voronoi Balls

Participants: Pooran Memari.

In collaboration with Amal Dev Parakkat (Telecom, IP-Paris), Stefan Ohrhallinger and Michael Wimmer (TU Wien) and Elmar Eisemann (TU Delft).

7.1.7 DynBioSketch: A tool for sketching dynamic visual summaries in biology, and its application to infection phenomena

Participants: Pooran Memari.

In collaboration with Pauline Olivier, Tara Butler, Pascal Guehl, Renaud Chabrier and Marie-Paule Cani (LIX, Ecole Polytechnique) and Jean-Luc Coll (Institute for Advanced Biosciences, Grenoble Alpes University).

7.1.8 Ricci flow-based brain surface covariance descriptors for diagnosing Alzheimer’s disease

Participants: Pooran Memari.

In collaboration with Fatemeh Ahmadi, Mohamad-Ebrahim Shiri, Behroz Bidabad, Maral Sedaghat (Math department of Amirkabir University of Technology).

7.1.9

Participants: Pooran Memari.

In collaboration with Charline Grenier and Basile Sauvage (Strasbourg University).

7.1.10 Versatile Curve Design by Level Set with Quadratic Convergence

Participants: Jiong Chen.

In collaboration with Xiaohu Zhang, Shuang Wu, Hujun Bao and Jin Huang (Zhejiang University), and Yao Jin (Zhejiang Sci-Tech University).

In this work 33, we present an efficient and versatile approach to curve design based on an implicit representation known as the level set. While previous works have explored the use of the level set to generate curves with minimal length, they typically have limitations in accommodating additional conditions for rich and robust control. To address these challenges, we formulate curve editing with constraints like smoothness, interpolation, tangent control, etc., via a level set based variational problem by constraining the values or derivatives of the level set function. However, the widely used gradient flow strategy converges very slowly for this complicated variational problem compared to the classical geodesic one. Thus, we propose to solve it via Newton's method enhanced by local Hessian correction and a trust-region strategy. As a result, our method not only enables versatile control, but also excels in terms of performance due to nearly quadratic convergence and almost linear complexity in each iteration via narrow band acceleration. In practice, these advantages effectively benefit various applications, such as interactive curve manipulation, boundary smoothing for surface segmentation and path planning with obstacles as demonstrated.

7.2.1 Hybrid LBM-FVM Solver for Two-phase Flow Simulation

Participants: Mathieu Desbrun.

In collaboration with Wei Li and Xiaopei Liu (Shanghaitech University)

In this work 25, we introduce a hybrid LBM-FVM solver for two-phase fluid flow simulations in which interface dynamics is modeled by a conservative phase-field equation. Integrating fluid equations over time is achieved through a velocity-based lattice Boltzmann solver which is improved by a central-moment multiple-relaxation-time collision model to reach higher accuracy. For interface evolution, we propose a finite-volume-based numerical treatment for the integration of the phase-field equation: we show that the second-order isotropic centered stencils for diffusive and separation fluxes combined with the WENO-5 stencils for advective fluxes achieve similar and sometimes even higher accuracy than the state-of-the-art double-distribution function LBM methods as well as the DUGKS-based method, while requiring less computations and a smaller amount of memory. Benchmark tests (such as the 2D diagonal translation of a circular interface), along with quantitative evaluations on more complex tests (such as the rising bubble and Rayleigh-Taylor instability simulations) allowing comparisons with prior numerical methods and/or experimental data, are presented to validate the advantage of our hybrid solver. Moreover, 3D simulations (including a dam break simulation) are also compared to the time-lapse photography of physical experiments in order to allow for more qualitative evaluations.

7.2.2 Kinetic Simulation of Turbulent Multifluid Flows

Participants: Mathieu Desbrun.

In collaboration with Wei Li (Tencent)

Despite its visual appeal, the simulation of separated multiphase flows (i.e., streams of fluids separated by interfaces) faces numerous challenges in accurately reproducing complex behaviors such as guggling, wetting, or bubbling. These difficulties are especially pronounced for high Reynolds numbers and large density variations between fluids, most likely explaining why they have received comparatively little attention in Computer Graphics compared to single- or two-phase flows. In this work 24, we present a full LBM solver for multifluid simulation. We derive a conservative phase field model with which the spatial presence of each fluid or phase is encoded to allow for the simulation of miscible, immiscible and even partially-miscible fluids, while the temporal evolution of the phases is performed using a D3Q7 lattice-Boltzmann discretization. The velocity field, handled through the recent high-order moment-encoded LBM (HOME-LBM) framework to minimize its memory footprint, is simulated via a velocity-based distribution stored on a D3Q27 or D3Q19 discretization to offer accuracy and stability to large density ratios even in turbulent scenarios, while coupling with the phases through pressure, viscosity, and interfacial forces is achieved by leveraging the diffuse encoding of interfaces. The resulting solver addresses a number of limitations of kinetic methods in both computational fluid dynamics and computer graphics: it offers a fast, accurate, and low-memory fluid solver enabling efficient turbulent multiphase simulations free of the typical oscillatory pressure behavior near boundaries. We present several numerical benchmarks, examples and comparisons of multiphase flows to demonstrate our solver's visual complexity, accuracy, and realism.

7.2.3 Lightning-fast Method of Fundamental Solutions

Participants: Jiong Chen, Mathieu Desbrun.

In collaboration with Florian Schaefer (Georgia Tech)

7.2.4 TwisterForge: Controllable and Efficient Animation of Virtual Tornadoes

Participants: Jiong Chen.

In collaboration with James Gain (University of Cape Town), Jean-Marc Chomaz and Marie-Paule Cani (LIX, Ecole Polytechnique)

In this work 44, we introduce a layered approach for creating and animating realistic virtual tornadoes in computer graphics. The method centers on two types of cureves: a 3D curve to intitialize the tornado's core as a vortex filament and 2D profile curves to control the surrounding funnel shape. The core evolves dynamically subject to the Biot-Savart law, bending and twisting driven by its initial curvature, while the funnel profile represents the Stokes stream function and dictates the radial and axial air motion around the core. These two components, together, capture the rotation, sliding, and uplift of the tornado's air volume. Our method achieves visually plausible animations of tornadoes, capable of interacting with uneven terrain, destroying infrastructure, and transporting debris, offering a controllable and realistic solution for visual effects and interactive applications.

7.2.5 Volcanic Skies: coupling explosive eruptions with atmospheric simulation to create consistent skyscapes

Participants: Jiong Chen.

In collaboration with Cilliers Pretorius and James Gain (University of Cape Town), Maud Lastic, Damien Rohmer and Marie-Paule Cani (LIX, Ecole Polytechnique), and Guillaume Cordonnier (Inria).

Explosive volcanic eruptions rank among the most terrifying natural phenomena, and are thus frequently depicted in films, games, and other media, usually with a bespoke once-off solution. In this work 29, we introduce the first general-purpose model for bi-directional interaction between the atmosphere and a volcano plume. In line with recent interactive volcano models, we approximate the plume dynamics with Lagrangian disks and spheres and the atmosphere with sparse layers of 2D Eulerian grids, enabling us to focus on the transfer of physical quantities such as temperature, ash, moisture, and wind velocity between these sub-models. We subsequently generate volumetric animations by noise-based procedural upsampling keyed to aspects of advection, convection, moisture, and ash content to generate a fully-realized volcanic skyscape. Our model captures most of the visually salient features emerging from volcano-sky interaction, such as windswept plumes, enmeshed cap, bell and skirt clouds, shockwave effects, ash rain, and sheathes of lightning visible in the dark.

7.3.1 ADAPT: Multimodal Learning for Detecting Physiological Changes under Missing Modalities

Participants: Julie Mordacq, Steve Oudot.

In collaboration with Vicky Kalogeiton (Vista, LIX), Leo Milecki and Maria Vakalopoulou (Centrale Supelec).

7.3.2 Multimodal Learning for Detecting Stress under Missing Modalities

Participants: Julie Mordacq, Steve Oudot.

In collaboration with Vicky Kalogeiton (Vista, LIX), Leo Milecki and Maria Vakalopoulou (Centrale Supelec).

7.4.1 On the bottleneck stability of rank decompositions of multi-parameter persistence modules

Participants: Steve Oudot.

In collaboration with Magnus botnan (Vrije Universiteit Amsterdam), Steffen Oppermann (NTNU) and Luis Scoccola (University of Oxford).

7.4.2 Signed Barcodes for Multi-parameter Persistence via Rank Decompositions and Rank-Exact Resolutions

Participants: Steve Oudot.

In collaboration with Magnus botnan (Vrije Universiteit Amsterdam) and Steffen Oppermann (NTNU).

7.4.3 Intrinsic Interleaving Distance for Merge Trees

Participants: Steve Oudot.

In collaboration with Ellen Gasparovic (Union College), Elizabeth Munch (Michigan State University), Katharine Turner (Australian National University), Bei Wang (University of Utah) and Yusu Wang (University of California).

7.4.4 On the stability of multigraded Betti numbers and Hilbert functions

Participants: Steve Oudot.

In collaboration with Luis Scoccola (University of Oxford).

7.4.5 Efficient computation of topological integral transforms

Participants: Steve Oudot.

In collaboration with Vadim Lebovici (University of Oxford) and Hugo Passe (École Normale Supérieure de Lyon).

7.4.6 Differentiability and Optimization of Multiparameter Persistent Homology

Participants: Steve Oudot.

In collaboration with Luis Scoccola (University of Oxford), Siddharth Setlur (ETH Zürich), David Loiseaux and Mathieu Carrière (Datashape, Inria Sophia-Antipolis).

7.4.7 Fine-tuning 3D foundation models for geometric object retrieval

Participants: Maks Ovsjanikov.

In collaboration with Jarne van den Herrewegen, Tom Tourwé, and Francis Wyffels (from Oqton AI; and AI and Robotics Lab, IDLab-AIRO, Ghent University-imec, Belgium).

7.4.8 DeBaRA: Denoising-Based 3D Room Arrangement Generation

Participants: Léopold Maillard, Maks Ovsjanikov.

In collaboration with Nicolas Sereyjol-Garros, and Tom Durand (Dassault Systèmes).

7.4.9 Smoothed Graph Contrastive Learning via Seamless Proximity Integration

Participants: Maysam Behmanesh, Maks Ovsjanikov.

This work 34 introduces a smoothed graph contrastive learning (SGCL) technique for unsupervised representation learning on graphs, while leveraging proximity integration. Standard graph contrastive learning (GCL) aligns node representations by classifying node pairs as positives or negatives, typically treating all negatives equally in the contrastive loss. The Smoothed Graph Contrastive Learning model (SGCL) improves this by leveraging the geometric structure of augmented graphs to include proximity information for positive and negative pairs, regularizing the learning process. SGCL uses three smoothing techniques to adjust penalties in the contrastive loss and employs a batch-generating strategy to efficiently train on large-scale graphs. Extensive experiments show SGCL outperforms recent baselines in unsupervised settings across various benchmarks.

7.4.10 To Supervise or Not to Supervise: Understanding and Addressing the Key Challenges of Point Cloud Transfer Learning

Participants: Souhail Hadgi, Maks Ovsjanikov.

In collaboration with Lei Li (Technical University of Munich).

7.4.11 Self-Supervised Dual Contouring

Participants: Ramana Sundararaman, Roman Klokov, Maks Ovsjanikov.

This work 42 introduces a self-supervised dual contouring framework. Learning-based isosurface extraction methods offer robust alternatives to axiomatic techniques but often rely on supervised training with axiomatically computed ground truths, inheriting their biases. To address this, Self-Supervised Dual Contouring (SDC) introduces a self-supervised training scheme for the Neural Dual Contouring framework. SDC employs novel self-supervised loss functions to optimize mesh vertices by enforcing consistency with distances to the generated mesh. SDC surpasses data-driven methods in capturing intricate details and handling input irregularities. Additionally, the self-supervised objective regularizes Deep Implicit Networks (DINs), improving the quality of implicit functions and detail preservation across input modalities. SDC also enhances single-view reconstruction by enabling joint training of the predicted SDF and output mesh.

7.4.12 Back to 3D: Few-Shot 3D Keypoint Detection with Back-Projected 2D Features

Participants: Maks Ovsjanikov.

In collaboration with Thomas Wimmer and Peter Wonka (from Technical University of Munich and KAUST).

7.4.13 Unsupervised Representation Learning for Diverse Deformable Shape Collections

Participants: Souhaib Attaiki, Maks Ovsjanikov.

In collaboration with Sara Hahner and Jochen Garcke (Fraunhofer SCAI, Sankt Augustin, Germany, and Institute for Numerical Simulation, University of Bonn, Germany).

7.4.14 RIVQ-VAE: Discrete Rotation-Invariant 3D Representation Learning

Participants: Maks Ovsjanikov.

In collaboration with Mariem Mezghanni and Malika Boulkenafed (Dassault Systèmes).

7.4.15 Deformation Recovery: Localized Learning for Detail-Preserving Deformations

Participants: Ramana Sundararaman, Maks Ovsjanikov.

In collaboration with Nicolas Donati (Ansys, France), Simone Melzi (University of Milano-Bicocca, Italy), Etienne Corman (Université de Lorraine, CNRS, Inria).

7.4.16 Memory-Scalable and Simplified Functional Map Learning

Participants: Robin Magnet, Maks Ovsjanikov. This work 49 propose a memory-scalable and efficient functional map learning pipeline. Deep functional maps have become a leading framework for non-rigid shape matching, with recent advancements promoting consistency between functional and pointwise maps to improve accuracy. However, these methods rely on large dense matrices from soft pointwise maps, limiting efficiency and scalability. To address this, we propose a new method that avoids storing pointwise maps by leveraging the functional map structure. Additionally, we introduce a differentiable map refinement layer adapted from an axiomatic refinement algorithm, which can be used during training to enforce consistency between refined and initial maps. This approach is simpler, more efficient, and numerically stable, achieving near state-of-the-art results in challenging scenarios.

8.1.1 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).

8.1.2 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.

8.1.3 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.

9.1.1 Visits of international scientists

9.1.2 Visits to international teams

9.2.1 Horizon Europe

Contrat de recherche Inria - SHOM

Participants: Steve Oudot.

• Title:
  Traitement de nuage de points bathyémtriques (SMF et Lidar) par l'approche apprentissage automatique
• Partner Institution(s):
  • Service Hydrographique et Océanographique de la Marine (SHOM), Brest, France
• Date/Duration:
  2024-2028
• Additionnal info/keywords:
  Ce projet a pour objectif de mieux appréhender, à l’aide de l’apprentissage automatique, la donnée bathymétrique sous forme de nuages de points pour améliorer la description des fonds marins et des zones côtières. Ce sujet est en lien avec le traitement des erreurs ponctuelles de la donnée bathymétrique et également avec l’utilisation de cette donnée pour la génération de modèles numériques de terrain.

AEx PreMediT

Participants: Steve Oudot.

• 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.

10.1.1 Scientific events: organisation

10.1.2 Scientific events: selection

10.1.3 Journal

10.1.4 Invited talks

• Pooran Memari Invited talk at the Geometry & Computing conference, CIRM Luminy, October 2024.
• Mathieu Desbrun , invited talk at the annual GDR IFM meeting in 2024.
• Steve Oudot Invited speaker at the 9th European Congress of Mathematics (9ECM), Seville, July 2024.
• Steve Oudot Plenary speaker at the 21st International Conference on Representations of Algebras (ICRA), Shanghai, August 2024.
• Steve Oudot Invited speaker at the workshop Representation theory - combinatorial aspects and applications to TDA, NTNU, Trondheim, December 2024.
• Steve Oudot Seminar speaker in the Mathematical Institute of the University of Oxford, January 2024.

10.1.5 Research administration

• Maks Ovsjanikov Fellow of ELLIS, senior member of the European society for top AI researchers, since 2023.
• Pooran Memari Member of the Board of the French Chapter of Eurographics (EGFR), since October 2024.
• Pooran Memari Co-responsible for the Interaction, Graphics & Design (IGD) master’s program at IP-Paris, since September 2024.
• Steve Oudot member of the Conseil Académique of IP Paris, representing Inria

10.2.1 Teaching

• Master: Steve Oudot, Computational Geometry and Topology, 18h eq-TD, M2, MPRI;
• Master: Maks Ovsjanikov, Geometry Processing and Geometric Deep Learning, M2, MVA;
• Master: Steve Oudot, Topological data analysis, 45h eq-TD, M1, École polytechnique, France;
• Master: Mathieu Desbrun , Digital Representation and Analysis of Shapes, M2, École polytechnique, France;
• Master: Mathieu Desbrun , Computer Animation, M2, École polytechnique, France;
• Master: Pooran Memari, Digital Representation and Analysis of Shapes, M2, École polytechnique, France;
• Master: Pooran Memari, Computer Science refresher course at Artificial Intelligence and Advanced Visual Computing Master Program, 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.

10.2.2 Supervision

• 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: Souhaib Attaiki, LIX. defended: March 2024?. Maks Ovsjanikov.
• PhD in progress: Nasim Bagheri Shouraki, Application of neurocognition to study the effectiveness of geometric tactile 2D patterns in navigation maps and instructions for Visually Impaired Individuals, IP Paris. Start date: October 2024. Pooran Memari and Panos Mavros (Telecom Paris).
• PhD in progress: Theo Braune, École Polytechnique, Palaiseau. Mathieu Desbrun.
• PhD: Diego Gomez, École Polytechnique, Palaiseau. Defended: September 2024?. Maks Ovsjanikov.
• PhD in progress: Souhail Hadgi, École Polytechnique, Palaiseau. Maks Ovsjanikov.
• PhD: Robin Magnet, LIX. Defended: May 2024?. Maks Ovsjanikov.
• PhD in progress: Leopold Maillard, Dassault Systèmes. Maks Ovsjanikov.
• PhD in progress: Tim Scheller, École Polytechnique, Palaiseau. Maks Ovsjanikov.

10.2.3 Juries

• Mathieu Desbrun reviewer and jury member for Colin Weil-Duflos, Université Savoie Mont-Blanc.
• Pooran Memari Admission Jury for Masters & PhD-Track IGD (Interaction, Graphics & Design), IP-Paris, since 2020.
• Pooran Memari PhD Jury for Clément Chomicki, Université Gustave Eiffel, LIGM UMR8049 (28/11/2024).
• Pooran Memari PhD Jury for Bastien Doignies, Université Lyon 1 (25/11/2024).
• Pooran Memari Recruitment Committee for Assistant Professors at LORIA, Faculty of Sciences and Technologies, University of Lorraine (5/2024).
• Pooran Memari is a member of the Jury d’admission Masters & PhD Track IGD (Interaction, Graphics & Design), IP-Paris (Since 2020).

International journals

• 17 articleF.Fatemeh Ahmadi, M.-E.Mohamad-Ebrahim Shiri, B.Behroz Bidabad, M.Maral Sedaghat and P.Pooran Memari. Ricci flow-based brain surface covariance descriptors for diagnosing Alzheimer’s disease.Biomedical Signal Processing and Control93July 2024, 106212HALDOIback to text
• 18 articleM. B.Magnus Bakke Botnan, S.Steffen Oppermann, S.Steve Oudot and L.Luis Scoccola. On the bottleneck stability of rank decompositions of multi-parameter persistence modules.Advances in Mathematics451August 2024, 109780HALDOIback to text
• 19 articleM. B.Magnus Bakke Botnan, S.Steffen Oppermann and S.Steve Oudot. Signed Barcodes for Multi-parameter Persistence via Rank Decompositions and Rank-Exact Resolutions.Foundations of Computational MathematicsSeptember 2024HALDOIback to text
• 20 articleJ.Jiong Chen, F. T.Florian T. Schäfer and M.Mathieu Desbrun. Lightning-fast Method of Fundamental Solutions.ACM Transactions on Graphics434July 2024, 77In press. HALDOIback to textback to text
• 21 articleE.Ellen Gasparovic, E.Elizabeth Munch, S.Steve Oudot, K.Katharine Turner, B.Bei Wang and Y.Yusu Wang. Intrinsic Interleaving Distance for Merge Trees.La MatematicaDecember 2024HALDOIback to text
• 22 articleF.Fernando de Goes and M.Mathieu Desbrun. Stochastic Computation of Barycentric Coordinates.ACM Transactions on Graphics434July 2024, 13HALDOIback to text
• 23 articleJ.Jarne van den Herrewegen, T.Tom Tourwé, M.Maks Ovsjanikov and F.Francis Wyffels. Fine-tuning 3D foundation models for geometric object retrieval.Computers and Graphics122August 2024, 103993HALDOIback to text
• 24 articleW.Wei Li, K.Kui Wu and M.Mathieu Desbrun. Kinetic Simulation of Turbulent Multifluid Flows.ACM Transactions on Graphics434July 2024, 1 - 17HALDOIback to text
• 25 articleY.Yihui Ma, X.Xiaoyu Xiao, W.Wei Li, M.Mathieu Desbrun and X.Xiaopei Liu. Hybrid LBM-FVM solver for two-phase flow simulation.Journal of Computational Physics506June 2024, 112920HALDOIback to text
• 26 articleP.Pauline Olivier, T.Tara Butler, P.Pascal Guehl, J.-L.Jean-Luc Coll, R.Renaud Chabrier, P.Pooran Memari and M.-P.Marie-Paule Cani. DynBioSketch: A tool for sketching dynamic visual summaries in biology, and its application to infection phenomena.Computers and Graphics122August 2024, 103956HALDOIback to text
• 27 articleS.Steve Oudot and L.Luis Scoccola. On the Stability of Multigraded Betti Numbers and Hilbert Functions.SIAM Journal on Applied Algebra and Geometry81January 2024, 54-88HALDOIback to text
• 28 articleA. D.Amal Dev Parakkat, S.Stefan Ohrhallinger, E.Elmar Eisemann and P.Pooran Memari. BallMerge: High-quality Fast Surface Reconstruction via Voronoi Balls.Computer Graphics ForumApril 2024HALback to text
• 29 articleC.Cilliers Pretorius, J.James Gain, M.Maud Lastic, G.Guillaume Cordonnier, J.Jiong Chen, D.Damien Rohmer and M.-P.Marie-Paule Cani. Volcanic Skies: coupling explosive eruptions with atmospheric simulation to create consistent skyscapes.Computer Graphics Forum432April 2024HALDOIback to text
• 30 articleD.Daniel Ströter, J.-M.Jean-Marc Thiery, K.Kai Hormann, J.Jiong Chen, Q.Qingjun Chang, S.Sebastian Besler, J. S.Johannes Sebastian Mueller-Roemer, T.T. Boubekeur, A.Andre Stork and D. W.Dieter W. Fellner. A Survey on Cage-based Deformation of 3D Models.Computer Graphics Forum432April 2024, e15060HALDOIback to text
• 31 articleR.Ramana Sundararaman, N.Nicolas Donati, S.Simone Melzi, E.Etienne Corman and M.Maks Ovsjanikov. Deformation Recovery: Localized Learning for Detail-Preserving Deformations.ACM Transactions on Graphics436November 2024, 1-16HALDOIback to text
• 32 articleJ.-M.Jean-Marc Thiery, É.Élie Michel and J.Jiong Chen. Biharmonic Coordinates and their Derivatives for Triangular 3D Cages.ACM Transactions on Graphics43138July 2024, 17HALDOIback to text
• 33 articleX.Xiaohu Zhang, S.Shuang Wu, J.Jiong Chen, Y.Yao Jin, H.Hujun Bao and J.Jin Huang. Versatile Curve Design by Level Set with Quadratic Convergence.IEEE Transactions on Visualization and Computer Graphics2024. In press. HALDOIback to text

International peer-reviewed conferences

• 34 inproceedingsM.Maysam Behmanesh and M.Maks Ovsjanikov. Smoothed Graph Contrastive Learning via Seamless Proximity Integration.LoG 2024 - Learning on Graphs ConferenceVirtual, FranceNovember 2024HALback to text
• 35 inproceedingsC.Charline Grenier, P.Pooran Memari and B.Basile Sauvage. Ink-Edit: Interactive color-constrained textures editing.SA '24: SIGGRAPH Asia 2024 Technical CommunicationsSiggraph Asia, technical communications, Tokyo, Japan, décembre 2024Tokyo, JapanNovember 2024HALDOIback to text
• 36 inproceedingsS.Souhail Hadgi, L.Lei Li and M.Maks Ovsjanikov. To Supervise or Not to Supervise: Understanding and Addressing the Key Challenges of Point Cloud Transfer Learning.Lecture Notes in Computer ScienceECCV 2024 - European Conference on Computer VisionLNCS-15136Computer Vision – ECCV 2024 18th European ConferenceMiCo Milano, ItalySpringer Nature SwitzerlandOctober 2025, 146-163HALDOIback to text
• 37 inproceedingsV.Vadim Lebovici, S.Steve Oudot and H.Hugo Passe. Efficient Computation of Topological Integral Transforms.22nd International Symposium on Experimental Algorithms (SEA 2024)Vienna, FranceSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024HALDOIback to text
• 38 inproceedingsD.Diana Marin, A. D.Amal Dev Parakkat, S.Stefan Ohrhallinger, M.Michael Wimmer, S.Steve Oudot and P.Pooran Memari. SING: Stability-Incorporated Neighborhood Graph.SA 2024 - The 17th ACM SIGGRAPH Conference and Exhibition on Computer Graphics and Interactive Techniques in AsiaTokyo, JapanACMDecember 2024, 1-10HALDOIback to text
• 39 inproceedingsN.Nissim Maruani, M.Maks Ovsjanikov, P.Pierre Alliez and M.Mathieu Desbrun. PoNQ: a Neural QEM-based Mesh Representation.CVPR 2024 - IEEE/CVF Conference on Computer Vision and Pattern RecognitionSeattle, United StatesMarch 2024HALback to text
• 40 inproceedingsJ.Julie Mordacq, L.Leo Milecki, M.Maria Vakalopoulou, S.Steve Oudot and V.Vicky Kalogeiton. ADAPT: Multimodal Learning for Detecting Physiological Changes under Missing Modalities.Proceedings of Machine Learning ResearchMIDL 2024 - Medical Imaging with Deep LearningParis, FranceJuly 2024HALback to text
• 41 inproceedingsL.Luis Scoccola, S.Siddharth Setlur, D.David Loiseaux, M.Mathieu Carrière and S.Steve Oudot. Differentiability and Optimization of Multiparameter Persistent Homology.ICML 2024 - The Forty-first International Conference on Machine LearningWien, AustriaJune 2024HALback to text
• 42 inproceedingsR.Ramana Sundararaman and R.Roman Klokov. Self-Supervised Dual Contouring.CVPR 2024 - Computer Vision and Pattern RecognitionSeattle, United StatesJune 2024HALback to text
• 43 inproceedingsT.Thomas Wimmer, P.Peter Wonka and M.Maks Ovsjanikov. Back to 3D: Few-Shot 3D Keypoint Detection with Back-Projected 2D Features.2024 IEEE/CVF Conference on Computer Vision and Pattern RecognitionSeattle, WA, United StatesJune 2024HALback to text

Conferences without proceedings

• 44 inproceedingsJ.Jiong Chen, J.James Gain, J.-M.Jean-Marc Chomaz and M.-P.Marie-Paule Cani. TwisterForge: controllable and efficient animation of virtual tornadoes.MIG 2024 - The 17th Annual ACM SIGGRAPH Conference on Motion, Interaction and GamesArlington (USA), United States2024, 1 - 11HALDOIback to text
• 45 inproceedingsS.Sara Hahner, S.Souhaib Attaiki, J.Jochen Garcke and M.Maks Ovsjanikov. Unsupervised Representation Learning for Diverse Deformable Shape Collections.3DV 2024 - International Conference on 3D VisionDavos, SwitzerlandIEEEMarch 2024, 1594-1604HALDOIback to text
• 46 inproceedingsL.Léopold Maillard, N.Nicolas Sereyjol-Garros, T.Tom Durand and M.Maks Ovsjanikov. DeBaRA: Denoising-Based 3D Room Arrangement Generation.The Thirty-Eighth Annual Conference on Neural Information Processing SystemsVancouver (BC), CanadaNovember 2024, 1 - 16HALDOIback to text
• 47 inproceedingsM.Mariem Mezghanni, M.Malika Boulkenafed and M.Maks Ovsjanikov. RIVQ-VAE: Discrete Rotation-Invariant 3D Representation Learning.3DV 2024 - International Conference on 3D VisionDavos, SwitzerlandIEEEMarch 2024, 1382-1391HALDOIback to text
• 48 inproceedingsJ.Julie Mordacq, L.Leo Milecki, M.Maria Vakalopoulou, S.Steve Oudot and V.Vicky Kalogeiton. Multimodal Learning for Detecting Stress under Missing Modalities.WiCV 2024 - Women in Computer Vision workshop In conjunction with CVPRSeattle, United StatesJune 2024HALback to text

Reports & preprints

• 49 miscR.Robin Magnet and M.Maks Ovsjanikov. Memory-Scalable and Simplified Functional Map Learning.April 2024HALback to text

Cited publications

• 50 articleH.Hervé Abdi and D.Dominique Valentin. Multiple correspondence analysis.Encyclopedia of measurement and statistics242007, 651--657back to text
• 51 articleG.Giancarlo Alfonsi. Reynolds-averaged Navier--Stokes equations for turbulence modeling.Applied Mechanics Reviews6242009back to text
• 52 articleM. R.Michael R. Allshouse and J.-L.Jean-Luc Thiffeault. Detecting coherent structures using braids.Physica D: Nonlinear Phenomena2412012, 95--105back to text
• 53 articleD. N.Douglas N. Arnold, R. S.Richard S. Falk and R.Ragnar Winther. Finite element exterior calculus, homological techniques, and applications.Acta Numerica152006, 1–155back to text
• 54 inproceedingsM.Mathieu Aubry, U.Ulrich Schlickewei and D.Daniel Cremers. The wave kernel signature: A quantum mechanical approach to shape analysis.2011 IEEE international conference on computer vision workshops (ICCV workshops)IEEE2011, 1626--1633back to text
• 55 inproceedingsO.Omri Azencot, M.Mirela Ben-Chen, F.Fédéric Chazal and M.Maks Ovsjanikov. An operator approach to tangent vector field processing.Computer Graphics Forum325Wiley Online Library2013, 73--82back to text
• 56 articleO.Omri Azencot, M.Maks Ovsjanikov, F.Frédéric Chazal and M.Mirela Ben-Chen. Discrete derivatives of vector fields on surfaces--an operator approach.ACM Trans. Graph.3432015, 1--13back to text
• 57 articleB.Bogdan Batko, T.Tomasz Kaczynski, M.Marian Mrozek and T.Thomas Wanner. Linking combinatorial and classical dynamics: Conley index and Morse decompositions.Foundations of Computational Mathematics2052020, 967--1012back to text
• 58 articleU.Ulrich Bauer, M. B.Magnus B Botnan, S.Steffen Oppermann and J.Johan Steen. Cotorsion torsion triples and the representation theory of filtered hierarchical clustering.Advances in Mathematics3692020, 107171back to text
• 59 articleA. I.Alexander I. Bobenko and B. A.Boris A. Springborn. Variational Principles for Circle Patterns and Koebe's Theorem.Trans. Amer. Math. Soc.3562003, 659--689back to text
• 60 articleJ.-D.Jean-Daniel Boissonnat, R.Ramsay Dyer and A.Arijit Ghosh. Delaunay triangulation of manifolds.Foundations of Computational Mathematics1822018, 399--431back to text
• 61 articleF.F. Brogi, O.O. Malaspinas, B.B. Chopard and C.C. Bonadonna. Hermite regularization of the lattice Boltzmann method for open source computational aeroacoustics.Journal of the Acoustical Society of America14242017, 2332--2345back to text
• 62 articleM. M.Michael M. Bronstein, J.Joan Bruna, Y.Yann LeCun, A.Arthur Szlam and P.Pierre Vandergheynst. Geometric Deep Learning: Going beyond Euclidean data.IEEE Signal Processing Magazine3442017, 18-42DOIback to text
• 63 inproceedingsR.Rickard Brüel Gabrielsson and G.Gunnar Carlsson. Exposition and Interpretation of the Topology of Neural Networks.2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA)2019, 1069--1076back to text
• 64 articleM.Marko Budiši ć and I.Igor Mezić. Geometry of the ergodic quotient reveals coherent structures in flows.Physica D: Nonlinear Phenomena241152012, 1255--1269back to text
• 65 articleA.Alberto Cacciola, A.Alessandro Muscoloni, V.Vaibhav Narula, A.Alessandro Calamuneri, S.Salvatore Nigro, E. A.Emeran A Mayer, J. S.Jennifer S Labus, G.Giuseppe Anastasi, A.Aldo Quattrone, A.Angelo Quartarone and others. Coalescent embedding in the hyperbolic space unsupervisedly discloses the hidden geometry of the brain.arXiv preprint arXiv:1705.041922017back to text
• 66 articleG.Gunnar Carlsson and A.Afra Zomorodian. The Theory of Multidimensional Persistence.Discrete and Computational Geometry4212009, 71--93back to text
• 67 inproceedingsM.Mathieu Carriere, F.Frédéric Chazal, M.Marc Glisse, Y.Yuichi Ike and H.Hariprasad Kannan. Optimizing persistent homology based functions.Proc. International Conference on Machine Learning2021back to text
• 68 inproceedingsM.Mathieu Carrière, F.Frédéric Chazal, Y.Yuichi Ike, T.Théo Lacombe, M.Martin Royer and Y.Yuhei Umeda. PersLay: a neural network layer for persistence diagrams and new graph topological signatures.International Conference on Artificial Intelligence and Statistics (PMLR)2020, 2786--2796back to text
• 69 articleA. X.Angel X Chang, T.Thomas Funkhouser, L.Leonidas Guibas, P.Pat Hanrahan, Q.Qixing Huang, Z.Zimo Li, S.Silvio Savarese, M.Manolis Savva, S.Shuran Song, H.Hao Su and others. Shapenet: An information-rich 3d model repository.arXiv preprint arXiv:1512.030122015back to text
• 70 articleF. d.Fré déric Chazal, L. J.Leonidas J. Guibas, S.Steve Oudot and P.Primoz Skraba. Persistence-Based Clustering in Riemannian Manifolds.\bf Journal of the ACM6062013, 1--38URL: http://doi.acm.org/10.1145/2535927DOIback to textback to text
• 71 articleY.-C.Yen-Chi Chen, C. R.Christopher R. Genovese and L.Larry Wasserman. Statistical inference using the Morse-Smale complex.Electronic Journal of Statistics1112017, 1390--1433back to text
• 72 inproceedingsC.Chao Chen, X.Xiuyan Ni, Q.Qinxun Bai and Y.Yusu Wang. A topological regularizer for classifiers via persistent homology.The 22nd International Conference on Artificial Intelligence and Statistics (PMLR)2019, 2573--2582back to text
• 73 articleA.Albert Chern. A Reflectionless Discrete Perfectly Matched Layer.Journal of Computational Physics3812019, 91--109back to text
• 74 bookC. C.Charles C Conley. Isolated invariant sets and the Morse index.38American Mathematical Soc.1978back to text
• 75 articleC.Christophe Coreixas, G.Gauthier Wissocq, G.Guillaume Puigt, J.-F.Jean-François Boussuge and P.Pierre Sagaut. Recursive regularization step for high-order lattice Boltzmann methods.Phys. Rev. E9632017, 033306back to text
• 76 articleK.Keenan Crane, M.Mathieu Desbrun and P.Peter Schröder. Trivial Connections on Discrete Surfaces.Computer Graphics Forum (\emph{Symposium on Geometry Processing})2952010, 1525--1533back to text
• 77 articleJ.Justin Curry, S.Sayan Mukherjee and K.Katharine Turner. How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms.arXiv preprint arXiv:1805.097822018back to text
• 78 articleA.Anastasia Deckard, R. C.Ron C Anafi, J. B.John B Hogenesch, S. B.Steven B Haase and J.John Harer. Design and analysis of large-scale biological rhythm studies: a comparison of algorithms for detecting periodic signals in biological data.Bioinformatics29242013, 3174--3180back to text
• 79 incollectionM.Mathieu Desbrun, E.Eva Kanso and Y.Yiying Tong. Discrete Differential Forms for Computational Modeling.Discrete Differential GeometrySpringer2007back to text
• 80 article T. K.Tamal K Dey, M.Mateusz Juda, T.Tomasz Kapela, J.Jacek Kubica and M.Michal Lipiński. back to text
• 81 inproceedingsT. K.Tamal K. Dey, M.Marian Mrozek and R.Ryan Slechta. Persistence of the Conley Index in Combinatorial Dynamical Systems.36th International Symposium on Computational Geometry (SoCG 2020)164Schloss Dagstuhl--Leibniz-Zentrum für Informatik2020, 37:1--37:17back to text
• 82 inproceedingsM.Meryll Dindin, Y.Yuhei Umeda and F.Frédéric Chazal. Topological data analysis for arrhythmia detection through modular neural networks.Canadian Conference on Artificial IntelligenceSpringer2020, 177--188back to text
• 83 inproceedingsN.Nicolas Donati, A.Abhishek Sharma and M.Maks Ovsjanikov. Deep geometric functional maps: Robust feature learning for shape correspondence.Proceedings of the IEEE/CVF conference on Computer Vision and Pattern Recognition2020, 8592--8601back to text
• 84 inproceedingsP.Pierre Ecormier-Nocca, P.Pooran Memari, J.James Gain and M.-P.Marie-Paule Cani. Accurate synthesis of multi-class disk distributions.Computer Graphics Forum38(2)Wiley Online Library2019, 157--168back to text
• 85 articleR.Robin Forman. Combinatorial vector fields and dynamical systems.Mathematische Zeitschrift22841998, 629--681back to text
• 86 bookT.Theodore Frankel. The Geometry of Physics: An Introduction.Cambridge University Press2011back to text
• 87 inproceedingsA.Andrea Frome, D.Daniel Huber, R.Ravi Kolluri, T.Thomas Bülow and J.Jitendra Malik. Recognizing Objects in Range Data Using Regional Point Descriptors.European Conference on Computer VisionSpringer Berlin Heidelberg2004, 224--237back to text
• 88 inproceedingsG.Gary Froyland and K.Kathrin Padberg-Gehle. Almost-Invariant and Finite-Time Coherent Sets: Directionality, Duration, and Diffusion.Ergodic Theory, Open Dynamics, and Coherent StructuresSpringer New York2014, 171--216back to text
• 89 inproceedingsR. B.Rickard Brüel Gabrielsson, B. J.Bradley J Nelson, A.Anjan Dwaraknath and P.Primoz Skraba. A topology layer for machine learning.International Conference on Artificial Intelligence and StatisticsPMLR2020, 1553--1563back to text
• 90 articleF.François Gay-Balmaz and H.Hiroaki Yoshimura. From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective.Entropy2112019back to text
• 91 articleR.Robert Ghrist, R.Rachel Levanger and H.Huy Mai. Persistent homology and Euler integral transforms.Journal of Applied and Computational Topology212018, 55--60back to text
• 92 articleM.Mark Gillespie, B.Boris Springborn and K.Keenan Crane. Discrete Conformal Equivalence of Polyhedral Surfaces.ACM Trans. Graph.4042021back to text
• 93 articleF.Fernando de Goes, A.Andrew Butts and M.Mathieu Desbrun. Discrete Differential Operators on Polygonal Meshes.ACM Trans. Graph.3942020, Art. 110back to text
• 94 articleF.Fernando de Goes, M.Mathieu Desbrun, M.Mark Meyer and T.Tony DeRose. Subdivision Exterior Calculus for Geometry Processing.ACM Trans. Graph.3542016, Art. 133back to text
• 95 articleF.Fernando de Goes, S.Siome Goldenstein, M.Mathieu Desbrun and L.Luiz Velho. Exoskeleton: Curve network abstraction for 3D shapes.Comput. Graph.351February 2011, 112--121back to text
• 96 articleF.Fernando de Goes, B.Beibei Liu, M.Max Budninskyi, Y.Yiying Tong and M.Mathieu Desbrun. Discrete 2-Tensor Fields on Triangulations.Comp. Graph. Forum \emph{(Symposium on Geometry Processing)}335August 2014, 13--24back to text
• 97 articleF. d.Fernando de Goes, P.Pooran Memari, P.Patrick Mullen and M.Mathieu Desbrun. Weighted triangulations for geometry processing.ACM Transactions on Graphics (TOG)3332014, 1--13back to text
• 98 bookE.Ernst Hairer, C.Christian Lubich and G.Gerhard Wanner. Geometric numerical integration.31Springer Series in Computational MathematicsStructure-preserving algorithms for ordinary differential equationsSpringer-Verlag, Berlin2006back to text
• 99 inproceedingsX.Xiaoling Hu, F.Fuxin Li, D.Dimitris Samaras and C.Chao Chen. Topology-Preserving Deep Image Segmentation.Advances in Neural Information Processing Systems (NeurIPS)322019back to text
• 100 inproceedingsR.Ruqi Huang, M.-J.Marie-Julie Rakotosaona, P.Panos Achlioptas, L. J.Leonidas J Guibas and M.Maks Ovsjanikov. OperatorNet: Recovering 3d shapes from difference operators.Proceedings of the IEEE/CVF International Conference on Computer Vision2019, 8588--8597back to text
• 101 inproceedingsQ.Qitong Jiang, S.Sebastian Kurtek and T.Tom Needham. The Weighted Euler Curve Transform for Shape and Image Analysis.Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops2020, 844--845back to text
• 102 articleX.Xiaoye Jiang, L.-H.Lek-Heng Lim, Y.Yuan Yao and Y.Yinyu Ye. Statistical ranking and combinatorial Hodge theory.Mathematical Programming1272011, 203--244back to text
• 103 bookV.Volker John. Large eddy simulation of turbulent incompressible flows: analytical and numerical results for a class of LES models.34Springer Science & Business Media2003back to text
• 104 articleM.Masaki Kashiwara and P.Pierre Schapira. Persistent homology and microlocal sheaf theory.Journal of Applied and Computational Topology21-22018, 83--113back to text
• 105 articleL.L. Kharevych, B.B. Springborn and P.P. Schröder. Discrete conformal mappings via circle patterns.ACM Trans. Graph.2522006, 412--438back to text
• 106 inproceedingsK.Kwangho Kim, J.Jisu Kim, M.Manzil Zaheer, J.Joon Kim, F.Frédéric Chazal and L.Larry Wasserman. PLLay: Efficient Topological Layer based on Persistence Landscapes.Advances in Neural Information Processing Systems (NeurIPS)2020back to text
• 107 articleF.FENG LUO. COMBINATORIAL YAMABE FLOW ON SURFACES.Communications in Contemporary Mathematics652004, 765--780back to text
• 108 articleJ.Jacob Leygonie, S.Steve Oudot and U.Ulrike Tillmann. A framework for differential calculus on persistence barcodes.Foundations of Computational MathematicsTo appear2021back to text
• 109 articleW.Wei Li, Y.Yixin Chen, M.Mathieu Desbrun, C.Changxi Zheng and X.Xiaopei Liu. Fast and Scalable Turbulent Flow Simulation with Two-Way Coupling.ACM Trans. Graph.3942020, Art. 47back to text
• 110 article M.Michal Lipiński. back to text
• 111 inproceedingsO.Or Litany, T.Tal Remez, E.Emanuele Rodola, A.Alex Bronstein and M.Michael Bronstein. Deep functional maps: Structured prediction for dense shape correspondence.Proceedings of the IEEE International Conference on Computer Vision2017, 5659--5667back to text
• 112 articleB.Beibei Liu, Y.Yiying Tong, F.Fernando de Goes and M.Mathieu Desbrun. Discrete Connection and Covariant Derivative for Vector Field Analysis and Design.ACM Trans. Graph.3532016, Art. 23back to textback to text
• 113 miscE. R.Ephy R. Love, B.Benjamin Filippenko, V.Vasileios Maroulas and G.Gunnar Carlsson. Topological Deep Learning.2021back to text
• 114 inproceedingsC.Clément Maria, S.Steve Oudot and E.Elchanan Solomon. Intrinsic topological transforms via the distance kernel embedding.Proc. International Symposium on Computational Geometry2020back to text
• 115 bookJ. E.Jerrold E. Marsden and T. S.Tudor S. Ratiu. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems.Springer2010back to textback to text
• 116 articleJ. E.J. E. Marsden and M.M. West. Discrete mechanics and variational integrators.Acta Numerica102001, 357--514back to text
• 117 articleI.Igor Mezić. Analysis of Fluid Flows via Spectral Properties of the Koopman Operator.Annual Review of Fluid Mechanics4512013, 357--378back to text
• 118 articleT.Tomas Mikolov, K.Kai Chen, G.Greg Corrado and J.Jeffrey Dean. Efficient estimation of word representations in vector space.arXiv preprint arXiv:1301.37812013back to text
• 119 articleP.Patrick Mullen, P.Pooran Memari, F.Fernando de Goes and M.Mathieu Desbrun. HOT: Hodge-optimized triangulations.ACM Trans. Graph. (SIGGRAPH)3042011, 103:1--103:12back to text
• 120 bookS. Y.Steve Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis.AMS Mathematical Surveys and Monographs209American Mathematical Society2015back to textback to text
• 121 articleS.Steve Oudot and E.Elchanan Solomon. Barcode embeddings for metric graphs.Algebraic and Geometric Topology2021back to text
• 122 articleM.Maks Ovsjanikov, M.Mirela Ben-Chen, J.Justin Solomon, A.Adrian Butscher and L.Leonidas Guibas. Functional maps: a flexible representation of maps between shapes.ACM Trans. Graph.314July 2012, 30:1--30:11back to text
• 123 incollectionM.Maks Ovsjanikov, E.Etienne Corman, M.Michael Bronstein, E.Emanuele Rodolà, M.Mirela Ben-Chen, L.Leonidas Guibas, F.Frederic Chazal and A.Alex Bronstein. Computing and processing correspondences with functional maps.SIGGRAPH 2017 Courses2017back to text
• 124 articleM.Maks Ovsjanikov, W.Wilmot Li, L.Leonidas Guibas and N. J.Niloy J. Mitra. Exploration of Continuous Variability in Collections of 3D Shapes.ACM Trans. Graph.3042011back to text
• 125 articleJ. A.Jose A Perea and J.John Harer. Sliding windows and persistence: An application of topological methods to signal analysis.Foundations of Computational Mathematics1532015, 799--838back to text
• 126 articleU.Ulrich Pinkall and K.Konrad Polthier. Computing Discrete Minimal Surfaces and Their Conjugates.Experimental Mathematics21993, 15--36back to text
• 127 articleH.Helmut Pottmann, J.Johannes Wallner, Q.-X.Qi-Xing Huang and Y.-L.Yong-Liang Yang. Integral invariants for robust geometry processing.Computer Aided Geometric Design2612009, 37--60back to text
• 128 articleV.Vakhtang Putkaradze and F.Fraçois Gay-Balmaz. Exact geometric theory for flexible, fluid-conducting tubes.Comptes Rendus Mécanique34222014, 79--84back to text
• 129 articleM.-J.Marie-Julie Rakotosaona, P.Paul Guerrero, N.Noam Aigerman, N. J.Niloy J. Mitra and M.Maks Ovsjanikov. Learning Delaunay Surface Elements for Mesh Reconstruction.CVPR2021back to text
• 130 articleT.Tullio Regge. General relativity without coordinates.Nuovo Cim.1931961, 558--571back to text
• 131 articleI.Igor Rivin. Euclidean structures on simplicial surfaces and hyperbolic volume.Annals of Math.13931994back to text
• 132 articleR. M.Raif M. Rustamov, M.Maks Ovsjanikov, O.Omri Azencot, M.Mirela Ben-Chen, F.Frédéric Chazal and L.Leonidas Guibas. Map-Based Exploration of Intrinsic Shape Differences and Variability.ACM Trans. Graph.324July 2013back to textback to text
• 133 unpublishedB.Ben Sattelberg, R.Renzo Cavalieri, M.Michael Kirby, C.Chris Peterson and J.J. Beveridge. Locally Linear Attributes of ReLU Neural Networks.11 2020, arxiv.org/abs/2012.01940back to text
• 134 inproceedingsP.Pierre Schapira. Tomography of constructible functions.International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting CodesSpringer1995, 427--435back to text
• 135 articleX.Xiaowen Shan, X.-F.Xue-Feng Yuan and H.Hudong Chen. Kinetic theory representation of hydrodynamics: a way beyond the Navier--Stokes equation.Journal of Fluid Mechanics5502006, 413--441back to text
• 136 articleP. R.Philippe R Spalart. Detached-eddy simulation.Annual review of fluid mechanics412009, 181--202back to text
• 137 articleB.Boris Springborn, P.Peter Schröder and U.Ulrich Pinkall. Conformal equivalence of triangle meshes.ACM Trans. Graph. (SIGGRAPH)2732008, 77:1--77:11back to text
• 138 inproceedingsJ.Jian Sun, M.Maks Ovsjanikov and L.Leonidas Guibas. A concise and provably informative multi-scale signature based on heat diffusion.Computer graphics forum28(5)Wiley Online Library2009, 1383--1392back to text
• 139 incollectionF.Floris Takens. Detecting strange attractors in turbulence.Dynamical systems and turbulence, Warwick 1980Springer1981, 366--381back to text
• 140 articleJ.-M.Jean-Marc Thiery, P.Pooran Memari and T.Tamy Boubekeur. Mean value coordinates for quad cages in 3D.ACM Trans. Graph.3762018, 1--14back to text
• 141 articleJ.-L.Jean-Luc Thiffeault. Braids of entangled particle trajectories.Chaos202010, 017516back to text
• 142 bookW. P.William P Thurston. Three-dimensional geometry and topology.Princeton university press1997back to text
• 143 bookJ.Julien Tierny. Topological data analysis for scientific visualization.Springer2017back to text
• 144 articleS.Svetlana Tokareva, M. J.Mikhail Jurievich Shashkov and A. N.Alexei N. Skurikhin. Machine learning approach for the solution of the Riemann problem in fluid dynamics.Preprint2019back to text
• 145 articleM.Melanie Weber, J.Jürgen Jost and E.Emil Saucan. Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets.Axioms542016back to text
• 146 articleC.Chengyuan Wu and C. A.Carol Anne Hargreaves. Topological machine learning for mixed numeric and categorical data.arXiv preprint arXiv:2003.045842020back to text
• 147 inproceedingsZ.Zhirong Wu, S.Shuran Song, A.Aditya Khosla, F.Fisher Yu, L.Linguang Zhang, X.Xiaoou Tang and J.Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes.Proceedings of the IEEE conference on Computer Vision and Pattern Recognition2015, 1912--1920back to text
• 148 articleM.Melissa Yeung, D.David Cohen-Steiner and M.Mathieu Desbrun. Material Coherence from Trajectories via Burau Eigenanalysis of Braids.Chaos: An Interdisciplinary Journal of Nonlinear Science303Selected as \emph{Featured Article} online2020, 033122back to text
• 149 articleW.Wei Zeng, R.Ren Guo, F.Feng Luo and X.Xianfeng Gu. Discrete heat kernel determines discrete Riemannian metric.Graphical Models7442012, 121-129back to text