2023Activity reportProject-TeamDATASHAPE

7.1.1 GUDHI

• Name:
  Geometric Understanding in Higher Dimensions
• Keywords:
  Computational geometry, Topology, Clustering
• Scientific Description:

  The Gudhi library is an open source library for Computational Topology and Topological Data Analysis (TDA). It offers state-of-the-art algorithms to construct various types of simplicial complexes, data structures to represent them, and algorithms to compute geometric approximations of shapes and persistent homology.

  The GUDHI library offers the following interoperable modules:

  . Complexes: + Cubical + Simplicial: Rips, Witness, Alpha and Čech complexes + Cover: Nerve and Graph induced complexes . Data structures and basic operations: + Simplex tree, Skeleton blockers and Toplex map + Construction, update, filtration and simplification . Topological descriptors computation . Manifold reconstruction . Topological descriptors tools: + Bottleneck and Wasserstein distance + Statistical tools + Persistence diagram and barcode

• Functional Description:
  The GUDHI open source library will provide the central data structures and algorithms that underly applications in geometry understanding in higher dimensions. It is intended to both help the development of new algorithmic solutions inside and outside the project, and to facilitate the transfer of results in applied fields.
• News of the Year:

  Below is a list of changes made since GUDHI 3.7.0 (december 2022):

  Perslay a TensorFlow layer for persistence diagrams representations.

  Cover Complex New classes to compute Mapper, Graph Induced complex and Nerves with a scikit-learn like interface.

  Persistent cohomology New linear-time compute_persistence_of_function_on_line, also available though CubicalPersistence in Python.

  Cubical complex Add possibility to build a lower-star filtration from vertices instead of top-dimensional cubes. Much faster implementation for the 2d case with input from top-dimensional cells.

  Hera version of Wasserstein distance now provides matching in its interface.

  Subsampling New choose_n_farthest_points_metric as a faster alternative of choose_n_farthest_points.

  SimplexTree SimplexTree can now be used with python pickle. A helper for_each_simplex that applies a given function object on each simplex A new option link_nodes_by_label to speed up cofaces and stars access, when set to true. A new option stable_simplex_handles to keep Simplex handles valid even after insertions or removals, when set to true.

  Čech complex A function assign_MEB_filtration that assigns to each simplex a filtration value equal to the squared radius of its minimal enclosing ball (MEB), given a simplicial complex and an embedding of its vertices. Applied on a Delaunay triangulation, it computes the Delaunay-Čech filtration.

  Edge collapse A Python function reduce_graph to simplify a clique filtration (represented as a sparse weighted graph), while preserving its persistent homology.

• URL:
  https://­gudhi.­inria.­fr/
• Publication:
  hal-01108461
• Contact:
  Marc Glisse
• Participants:
  Clément Maria, François Godi, David Salinas, Jean-Daniel Boissonnat, Marc Glisse, Mariette Yvinec, Pawel Dlotko, Siargey Kachanovich, Vincent Rouvreau, Mathieu Carrière, Clément Jamin, Siddharth Pritam, Frederic Chazal, Steve Oudot, Wojciech Reise, Hind Montassif, Hannah Schreiber, Martin Royer, David Loiseaux
• Partners:
  Université Côte d'Azur (UCA), Fujitsu

8.1.1 Fast persistent homology computation for functions on ℝ

Participant: Marc Glisse.

0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the connected components and we get the persistence diagram in time $O\left(n\alpha \right(n\left)\right)$. The running time is thus usually dominated by sorting the edges in $\Theta (nlog(n\left)\right)$. A little-known fact is that, in the particularly simple case of studying the sublevel sets of a piecewise-linear function on $ℝ$ or ${𝕊}^1$, persistence can actually be computed in linear time. This note 49 presents a simple algorithm that achieves this complexity and an extension to image persistence. An implementation is available in Gudhi.

8.1.2 Hausdorff and Gromov-Hausdorff Stable Subsets of the Medial Axis

Participant: Mathijs Wintraecken.

In collaboration with André Lieutier.

In 27 we introduce a pruning of the medial axis called the $(\lambda ,\alpha )$-medial axis (${\mathrm{ax}}_{\lambda }^{\alpha }$). We prove that the $(\lambda ,\alpha )$-medial axis of a set $K$ is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if $K$ and $K^{\text{'}}$ are close in the Hausdorff ($d_H$) sense then the $(\lambda ,\alpha )$-medial axes of $K$ and $K^{\text{'}}$ are close as metric spaces, that is the Gromov-Hausdorff distance ($d_{GH}$) between the two is $\frac{1}{4}$-Hölder in the sense that $d_{GH}({\mathrm{ax}}_{\lambda }^{\alpha }\left(K\right),{\mathrm{ax}}_{\lambda }^{\alpha }\left(K^{\text{'}}\right))\lesssim d_H{(K,K^{\text{'}})}^{1/4}$. The Hausdorff distance between the two medial axes is also bounded, by $d_H({\mathrm{ax}}_{\lambda }^{\alpha }\left(K\right),{\mathrm{ax}}_{\lambda }^{\alpha }\left(K^{\text{'}}\right))\lesssim d_H{(K,K^{\text{'}})}^{1/2}$. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis.

8.1.3 Tracing Isomanifolds in ${ℝ}^d$ in Time Polynomial in $d$ using Coxeter–Freudenthal–Kuhn Triangulations

Participant: Jean-Daniel Boissonnat, Siargey Kachanovich, Mathijs Wintraecken.

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ${ℝ}^d$ defined as the zero set of a multivariate multivalued smooth function $f:{ℝ}^d\to {ℝ}^{d-n}$, where $n$ is the intrinsic dimension of the manifold and we assume that 0 is a regular value. A natural way to approximate a smooth isomanifold $ℳ=f^{-1}\left(0\right)$ is to consider its Piecewise-Linear (PL) approximation $\widehat{ℳ}$ based on a triangulation $𝒯$ of the ambient space ${ℝ}^d$, whose longest edge has length $D$. In 12, we describe a simple algorithm to trace isomanifolds from a given starting point on each connected component. The algorithm works for arbitrary dimensions $n$ and $d$, and $D$. Our main result is that, when $f$ (or $ℳ$) has bounded complexity, the complexity of the algorithm is polynomial in $d$ and $\delta =1/D$ (and unavoidably exponential in $n$). Since it is known that for $\delta =\Omega \left(d^{2.5}\right)$, $\widehat{ℳ}$ is $O\left(D^2\right)$-close and isotopic to $ℳ$, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in $d$. Combining this algorithm with dimensionality reduction techniques, the dependency on $d$ in the size of $\widehat{ℳ}$ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art.

8.1.4 The reach of subsets of manifolds

Participant: Jean-Daniel Boissonnat, Mathijs Wintraecken.

Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift 176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended the reach $\mathrm{rch}\left(𝒮\right)$ from subsets $𝒮$ of Euclidean space to the reach ${\mathrm{rch}}_{ℳ}\left(𝒮\right)$ of subsets $𝒮$ of Riemannian manifolds $ℳ$, where is smooth (we'll assume at least $C^3$). Bangert showed that sets of positive reach in Euclidean space and Riemannian manifolds are very similar. In 13 we introduce a slight variant of Kleinjohann's and Bangert's extension and quantify the similarity between sets of positive reach in Euclidean space and Riemannian manifolds in a new way: Given and $p\in ℳ$, we bound the local feature size (a local version of the reach) of its lifting to the tangent space via the inverse exponential map (${\exp }_p^{-1}(𝒮$) at $q$, assuming that ${\mathrm{rch}}_{ℳ}\left(𝒮\right)$ and the geodesic distance $d_{ℳ}(p,q)$ are bounded. These bounds are motivated by the importance of the reach and local feature size to manifold learning, topological inference, and triangulating manifolds and the fact that intrinsic approaches circumvent the curse of dimensionality.

8.1.5 Local Criteria for Triangulating General Manifolds

Participant: Jean-Daniel Boissonnat, Mathijs Wintraecken.

In collaboration with Arijit Ghosh and Ramsay Dyer.

In 11, we present criteria for establishing a triangulation of a manifold. Given a manifold $M$, a simplicial complex $A$, and a map $H$ from the underlying space of $A$ to $M$, our criteria are presented in local coordinate charts for $M$, and ensure that $H$ is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on $M$. No Delaunay property of $A$ is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.

8.1.6 Local characterizations for decomposability of 2-parameter persistence modules

Participant: Vadim Lebovici.

In collaboration with Steve Oudot (Inria) and M. B. Botnan (Vrije Universiteit)

In 14, we investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so.

8.1.7 Discrete Morse Theory for Computing Zigzag Persistence

Participant: Clément Maria, Hannah Schreiber.

We introduce 23 a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes $K_i$, we introduce a zigzag Morse filtration whose complexes $A_i$ are Morse reductions of the original complexes $K_i$, and we prove that they both have same persistent homology. The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. We deduce an algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and show experimentally that it performs better in practice.

8.2.1 On the persistent homology of almost surely $C^0$ stochastic processes

Participant: Daniel Perez.

This paper 24 investigates the properties of the persistence diagrams stemming from almost surely continuous random processes on $[0,t]$. We focus our study on two variables which together characterize the barcode : the number of points of the persistence diagram inside a rectangle $]-\infty ,x]\times [x+\epsilon ,\infty [$, $N^{x,x+\epsilon }$ and the number of bars of length $\ge \epsilon$, $N^{\epsilon }$. For processes with the strong Markov property, we show both of these variables admit a moment generating function and in particular moments of every order. Switching our attention to semimartingales, we show the asymptotic behaviour of $N^{\epsilon }$ and $N^{x,x+\epsilon }$ as $\epsilon \to 0$ and of $N^{\epsilon }$ as $\epsilon \to \infty$. Finally, we study the repercussions of the classical stability theorem of barcodes and illustrate our results with some examples, most notably Brownian motion and empirical functions converging to the Brownian bridge.

8.2.2 Topological signatures of periodic-like signals

Participant: Wojciech Riese, Frédéric Chazal.

In collaboration with Bertrand Michel (Ecole Centrale Nantes)

8.2.3 Heat diffusion distance processes: a statistically founded method to analyze graph data sets

Participant: Etienne Lasalle.

In 21, we propose two multiscale comparisons of graphs using heat diffusion, allowing to compare graphs without node correspondence or even with different sizes. These multiscale comparisons lead to the definition of Lipschitz-continuous empirical processes indexed by a real parameter. The statistical properties of empirical means of such processes are studied in the general case. Under mild assumptions, we prove a functional Central Limit Theorem, as well as a Gaussian approximation with a rate depending only on the sample size. Once applied to our processes, these results allow to analyze data sets of pairs of graphs. We design consistent confidence bands around empirical means and consistent two-sample tests, using bootstrap methods. Their performances are evaluated by simulations on synthetic data sets.

8.2.4 Support and distribution inference from noisy data

Participant: Jérémie Capitao-Miniconi.

In collaboration E. Gassiat (LMO, Univ. Paris-Saclay) and L Lehéricy (Univ. Côte d'Azur)

8.2.5 A gradient sampling algorithm for stratified maps with applications to topological data analysis

Participant: Mathieu Carrière.

In collaboration J. Leygonie (Arda), T. Lacombe (Univ. Gustave Eiffel) and S. Oudot (Geomerix, Inria Saclay)

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

8.3.1 Choosing the parameter of the Fermat distance: navigating geometry and noise.

Participant: Frédéric Chazal, Laure Ferraris.

In collaboration with P. Groisman, M. Jonckheere, F. Pascal and F. Sapienza

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter $\alpha$ that greatly impacts the performance of subsequent tasks. Ideally, the value of $\alpha$ should be large enough to navigate the geometric intricacies inherent to the problem. At the same, it should remain restrained enough to sidestep any deleterious ramifications stemming from noise during the process of distance estimation. In 46, we study both theoretically and through simulations how to select this parameter.

8.3.2 MAGDiff: Covariate Data Set Shift Detection via Activation Graphs of Deep Neural Networks

Participant: Felix Hensel, Charles Arnal, Mathieu Carrière, Frédéric Chazal.

In collaboration with T. Lacombe (Univ. G. Eiffel), H. Kurihara (Fujitsu), Y. Ike (Kyushu Univ.)

Despite their successful application to a variety of tasks, neural networks remain limited, like other machine learning methods, by their sensitivity to shifts in the data: their performance can be severely impacted by differences in distribution between the data on which they were trained and that on which they are deployed. In 54, we propose a new family of representations, called MAGDiff, that we extract from any given neural network classifier and that allows for efficient covariate data shift detection without the need to train a new model dedicated to this task. These representations are computed by comparing the activation graphs of the neural network for samples belonging to the training distribution and to the target distribution, and yield powerful data- and task-adapted statistics for the two-sample tests commonly used for data set shift detection. We demonstrate this empirically by measuring the statistical powers of two-sample Kolmogorov-Smirnov (KS) tests on several different data sets and shift types, and showing that our novel representations induce significant improvements over a state-of-the-art baseline relying on the network output.

8.3.3 Label Shift Quantification with Robustness Guarantees via Distribution Feature Matching

Participant: Bastien Dussap, Gilles Blanchard.

In collaboration with Badr-Edine Chérief-Abdellatif (CNRS, LPSM, U. Paris-Sorbonne)

Quantification learning deals with the task of estimating the target label distribution under label shift. In 26, we established a unifying framework, distribution feature matching (DFM), that recovers as particular instances various estimators introduced in previous literature. We derived a general performance bound for DFM procedures, improving in several key aspects upon previous bounds derived in particular cases. We then extended this analysis to study robustness of DFM procedures in the misspecified setting under departure from the exact label shift hypothesis, in particular in the case of contamination of the target by an unknown distribution. These theoretical findings were confirmed by a detailed numerical study on simulated and real-world datasets. We also introduced an efficient, scalable and robust version of kernel-based DFM using the Random Fourier Feature principle.

This paper received the "Best student paper" award at the ECML/PKDD conference 2023.

8.3.4 Post hoc false discovery proportion inference under a Hidden Markov Model

Participant: Gilles Blanchard.

In collaboration with Marie Perrot-Dockès (MAP5, U. Paris), Étienne Roquain (LPSM, U. Paris-Sorbonne), Pierre Neuvial (CNRS, U. Toulouse)

We addressed in 25 the multiple testing problem under the assumption that the true/false hypotheses are driven by a Hidden Markov Model (HMM), which is recognized as a fundamental setting to model multiple testing under dependence since the seminal work of Sun and Cai (2009). While previous work has concentrated on deriving specific procedures with a controlled False Discovery Rate (FDR) under this model, following a recent trend in selective inference, we considered the problem of establishing confidence bounds on the false discovery proportion (FDP), for a user-selected set of hypotheses that can depend on the observed data in an arbitrary way. We developed a methodology to construct such confidence bounds first when the HMM model is known, then when its parameters are unknown and estimated, including the data distribution under the null and the alternative, using a nonparametric approach. In the latter case, we proposed a bootstrap-based methodology to take into account the effect of parameter estimation error. We showed that taking advantage of the assumed HMM structure allows for a substantial improvement of confidence bound sharpness over existing agnostic (structure-free) methods, as witnessed both via numerical experiments and real data examples.

8.3.5 Stable vectorization of multiparameter persistent homology using signed barcodes as measures

Participant: Mathieu Carrière, David Loiseaux.

In collaboration with Luis Scoccola (Oxford University), Steve Oudot (Geomerix, Inria Saclay) and Magnus Botnan (Vrije Universiteit)

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

8.3.6 A framework for fast and stable representations of multiparameter persistent homology decompositions

Participant: Mathieu Carrière, David Loiseaux.

In collaboration with Andrew Blumberg (Columbia University, NYC)

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is persistent homology, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on decompositions of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

8.4.1 An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number

Participant: Clément Maria.

In collaboration with Colleen Delaney (Berkeley), Eric Samperton (Purdue).

Quantum topology provides various frameworks for defining and computing invariants of manifolds. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work 47 we consider a restricted class of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. These categories are particularly simple, being entirely specified by three pieces of data: a finite abelian group, a bicharacter of that group, and a sign $\pm 1$. Despite being one of the simplest sources of state sum invariants, the computational complexities of Tambara-Yamagami invariants are yet to be fully understood. We make substantial progress on this problem. Our main result is the existence of a general fixed parameter tractable algorithm for all such topological invariants, where the parameter is the first Betti number of the 3-manifold with $\mathbb{Z}/2\mathbb{Z}$ coefficients. We also explain that these invariants are sometimes #P-hard to compute (and we expect that this is almost always the case). Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is easily computable in polynomial time.

8.4.2 Hard Diagrams of the Unknot

Participant: Clément Maria.

In collaboration with Benjamin Burton (University of Queensland), Hsien-Chih Chang (Dartmouth College), Maarten Löffler (TU/e - Eindhoven University of Technology), Arnaud de Mesmay (LIGM - Laboratoire d'Informatique Gaspard-Monge), Saul Schleimer (University of Warwick), Eric Sedgwick (DePaul University), Jonathan Spreer (The University of Sydney).

We present three “hard” diagrams of the unknot 15. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in S2. Both examples are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small “standard” example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in S2.

8.5.1 Variational Shape Reconstruction via Quadric Error Metrics

Participant: David Cohen-Steiner.

In collaboration with Tong Zhao, Pierre Alliez (Inria team Titane), Laurent Busé (Inria team Aromath), Tamy Boubekeur and Jean-Marc Thiery (Adobe Research)

Inspired by the strengths of quadric error metrics initially designed for mesh decimation, we propose a concise mesh reconstruction approach for 3D point clouds 32. Our approach proceeds by clustering the input points enriched with quadric error metrics, where the generator of each cluster is the optimal 3D point for the sum of its quadric error metrics. This approach favors the placement of generators on sharp features, and tends to equidistribute the error among clusters. We reconstruct the output surface mesh from the adjacency between clusters and a constrained binary solver. We combine our clustering process with an adaptive refinement driven by the error. Compared to prior art, our method avoids dense reconstruction prior to simplification and produces immediately an optimized mesh.

8.5.2 Two Lower Bounds for Random Point Sets via Negative Association

Participant: Marc Glisse.

In collaboration with Denys Bulavka (Charles University, Prague), Olivier Devillers (Inria team Gamble), Philippe Duchon (Laboratoire Bordelais de Recherche en Informatique) and Xavier Goaoc (Inria team Gamble)

We present 43 two lower bounds that hold with high probability for random point sets. We first give a new, and elementary, proof that the classical models of random point sets (uniform in a smooth convex body, uniform in a polygon, Gaussian) have a superconstant number of extreme points with high probability. We next prove that any algorithm that determines the orientation of all triples in a planar set of n points (that is, the order type of the point set) from their Cartesian coordinates must read with high probability $4nlogn-O(nloglogn)$ coordinate bits. This matches previously known upper bounds. Both bounds rely on a method due to Dubhashi and Ranjan (Random Structures and Algorithms, 1998) for obtaining concentration results via a negative association property.

8.5.3 Harmonic analysis on the boundary of hyperbolic groups

Participant: Georg Gruetzner.

In this paper 51 we show that a Möbius-structure $ℳ$ of dimension $Q$ has a minimal Ahlfors-David constant. This shows that a Möbius space is uniformly $Q$-Ahlfors-David regular. In summary, many classical theorems of harmonic analysis on ${ℝ}^n$ admit a Möbius-invariant formulation in the context of Möbius-geometry. We use this observation to show that the Knapp-Stein operator

is a continuous operator on the weighted $L^2$-space $L^2\left({\left(\frac{d^{\text{'}}}{d}\right)}^{\alpha }d{\mu }_d\right)$, with a norm independent of $d$ and $d^{\text{'}}$.

From here we construct a Sobolev space ${\mathscr{H}}_d^{-\alpha }$ on $s$-densities for a given $s$ as a function of $\alpha$. We would like to say that the construction is topologically independent of the metric $d$. In this paper we prove that the norms on a large class of functions are comparable.

The work is inspired by a paper by Astengo, Cowling, and Di Blasio, who construct uniformly bounded representations for simple Lie groups of rank 1. We formulate the problem in a much more general framework of groups acting on Möbius structures. In particular, all hyperbolic groups.

8.5.4 Constant regret for sequence prediction with limited advice

Participant: Gilles Blanchard.

In collaboration with El Mehdi Saad (CentraleSupelec, U. Paris-Saclay)

We investigated in 30 the problem of cumulative regret minimization for individual sequence prediction with respect to the best expert in a finite family of size $K$ under limited access to information. We assume that in each round, the learner can predict using a convex combination of at most $p$ experts for prediction, then they can observe a posteriori the losses of at most $m$ experts. We assume that the loss function is range-bounded and exp-concave. In the standard multi-armed bandits setting, when the learner is allowed to play only one expert per round and observe only its feedback, known optimal regret bounds are of the order $𝒪\left(\sqrt{KT}\right)$. We show that allowing the learner to play one additional expert per round and observe one additional feedback improves substantially the guarantees on regret. We provide a strategy combining only $p=2$ experts per round for prediction and observing $m\ge 2$ experts' losses. Its randomized regret (wrt. internal randomization of the learners' strategy) is of order $𝒪\left((K/m)log\left(K{\delta }^{-1}\right)\right)$ with probability $1-\delta$, i.e., is independent of the horizon $T$ (“constant” or “fast rate” regret) if ($p\ge 2$ and $m\ge 3$). We prove that this rate is optimal up to a logarithmic factor in $K$. In the case $p=m=2$, we provide an upper bound of order $𝒪(K^{2}log\left(K{\delta }^{-1}\right))$, with probability $1-\delta$. Our strategies do not require any prior knowledge of the horizon $T$ nor of the confidence parameter $\delta$. Finally, we show that if the learner is constrained to observe only one expert feedback per round, the worst-case regret is the “slow rate” $\Omega \left(\sqrt{KT}\right)$, suggesting that synchronous observation of at least two experts per round is necessary to have a constant regret.

8.5.5 Flagfolds

Participant: Blanche Buet.

In collaboration with Xavier Pennec (INRIA Team Epione, Sophia Antipolis)

In 42, by interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all $d$–dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

10.1.1 Visits of international scientists

10.1.2 Visits to international teams

10.2.1 ANR

10.2.2 Collaboration with other national research institutes

Metafora

Participant: Gilles Blanchard, Bastien Dussap, Marc Glisse.

- Type : Paris Region PhD 2021.

- Title : Comparaison de données cytométriques.

The Île-de-France region funds a PhD thesis in collaboration with Metafora biosystems, a company specialized in the analysis of cells through their metabolism. Bastien Dussap is supervised by Gilles Blanchard and Marc Glisse and aims to compare blood samples using statistics.

11.1.1 Scientific events: organisation

11.1.2 Journal

11.1.3 Invited talks

• Blanche Buet gave talks in several team seminars: Math department of Mulhouse, Analyse/EDP seminar of UCLouvain (Belgium), séminaire du pôle analyse du CMAP (Palaiseau) and conferences: Paris-London Analysis Seminar (UCL London, UK), Approximation Theory Workshop in FoCM 2023 (Paris, France), Off-the-grid workshop (IHP, France).
• Blanche Buet gave a 3h–short course on varifolds in Aussois (France).

11.1.4 Leadership within the scientific community

• Frédéric Chazal is the Scientific Director of the DATAIA Institute at Université Paris-Saclay.
• Frédéric Chazal is a member of the board of directors of the DIM project AI4IDF of the Région Ile-de-France.
• Clément Maria is co-head (with Théo Lacombe) of the GT GeoAlgo within the GdR IM.
• Clément Maria represents INRIA in the Steering Committee of the QuantAzur Federative Institute on quantum technologies.

11.1.5 Research administration

• Pierre Pansu was deputy director of the FMJH until August.
• Marc Glisse is president of the CDT at Inria Saclay.
• Blanche Buet is member of the CCUPS (Commission Consultative de l'Université Paris-Saclay), Laboratory council and "comité parité" of LMO.
• Blanche Buet is member of the FMJH postdoc selection committee.

11.2.1 Teaching

• Master: Frédéric Chazal, Analyse Topologique des Données, 30h eq-TD, Université Paris-Sud, France.
• Master: Clément Maria, Computational Geometry Learning, 18h eq-TD, M2, MPRI, France.
• Master: Frédéric Cazals and Mathieu Carrière, Foundations of Geometric Methods in Data Analysis, 24h eq-TD, M2, École CentraleSupélec, France.
• Master: Frédéric Cazals and Jean-Daniel Boissonnat and Mathieu Carrière, Geometric and Topological Methods in Machine Learning, 24h eq-TD, M2, Université Côte d'Azur, France.
• Master: Frédéric Chazal and Julien Tierny, Topological Data Analysis, 38h eq-TD, M2, Mathématiques, Vision, Apprentissage (MVA), ENS Paris-Saclay, France.
• Master: Gilles Blanchard, Mathematics for Artificial Intelligence 1, 70h eq-TD, IMO, Université Paris-Saclay, France.
• Master: Blanche Buet, TD-Distributions et analyse de Fourier, 60h eq-TD, M1, Université Paris-Saclay, France.
• Master: Marc Glisse, Conception et analyse d'algorithmes, 40h eq-TD, M1, École Polytechnique, France.
• Master: Blanche Buet, Short Course Géométrie et approximation at the Rentrée des Masters 2023, Université Paris-Saclay.
• Undergrad: Marc Glisse, Mécanismes de la programmation orientée-objet, 40h eq-TD, L3, École Polytechnique, France.
• Master: Nina Otter, Probabilités, 25h eq-TD, M1 Mathématiques Fondamentales, Laboratoire de Mathématiques d'Orsay, France.

11.2.2 Supervision

• PhD: Vadim Lebovici, Laplace transform for constructible functions. Defended in September 2023. Steve Oudot and François Petit.
• PhD: Christophe Vuong, Random hypergraphs. Defended in December 2023. Laurent Decreusefond and Marc Glisse.
• PhD in progress: Bastien Dussap, Comparaison de données cytométriques, started October 1st, 2021, Gilles Blanchard and Marc Glisse.
• PhD: Olympio Hacquard, Apprentissage statistique par méthodes topologiques et géométriques, defended in September 2023, Gilles Blanchard and Clément Levrard.
• PhD in progress: Hannah Marienwald, Transfer learning in high dimension. Started September 2019. Gilles Blanchard and Klaus-Robert Müller.
• PhD in progress: Jean-Baptiste Fermanian, Estimation de Kernel Mean Embedding et tests multiples en grande dimension. Started September 2021. Gilles Blanchard and Magalie Fromont-Renoir.
• PhD in progress: Antoine Commaret, Persistent Geometry. Started September 2021. David Cohen-Steiner and Indira Chatterji.
• PhD in progress: Lucas Brifault, Théorie de la mesure géométrique appliquée pour la modélisation de formes complexes. Started May 2022. David Cohen-Steiner and Mathieu Desbrun.
• PhD in progress: David Loiseaux, Multivariate topological data analysis for statistical machine learning. Started November 2021. Mathieu Carrière and Frédéric Cazals.
• PhD: Wojciech Reise, TDA for curve data. Defended in December 2023. Frédéric Chazal and Bertrand Michel.
• PhD in progress: Alexandre Guérin, Movement analysis from inertial sensors. Started on October 2021. Frédéric Chazal and Bertrand Michel.
• PhD in progress: Jérémie Capitao-Miniconi, deconvolution for singular measures with geometric support. Started on October 2020. Frédéric Chazal and Elisabeth Gassiat.
• PhD in progress: Charly Boricaud, Geometric inference for Data analysis: a Geometric Measure Theory perspective. Started on October 2021. Blanche Buet, Gian Paolo Leonardi et Simon Masnou.
• PhD in progress: Hugo Henneuse. Statistical Foundations of Topological Data Analysis for multidimensional random fields. Started on October 2022. Frédéric Chazal and Pascal Massart.
• PhD in progress: Laure Ferraris. Measure-dependent metric learning and applications in Topological Data Analysis. Started on October 2022. Frédéric Chazal.
• PhD: Georg Grützner. Espaces de Möbius et géométrie à grande échelle. Defended in May 2023. Pierre Pansu.
• PhD: Henrique Ennes. Computational Complexity Foundations of Quantum Topology. Started on October 2023. Clément Maria and Nicolas Nisse (INRIA).

11.2.3 Juries

• PhD defense jury: Nina Otter, PhD defense of Pepijn Roos Hoefgeest (December 2023, Free University Amsterdam, supervisor: Magnus Botnan)
• PhD defense jury: Blanche Buet, PhD defense of Elise Bonhomme (December 2023, LMO, supervisor: François Babadjian)

11.3.1 Interventions

• Clément Maria, c@fé-in INRIA-UCA, December 2023

International journals

• 11 articleJ.-D.Jean-Daniel Boissonnat, R.Ramsay Dyer, A.Arijit Ghosh and M.Mathijs Wintraecken. Local Criteria for Triangulating General Manifolds.Discrete and Computational Geometry692023, 156-191HALDOIback to text
• 12 articleJ.-D.Jean-Daniel Boissonnat, S.Siargey Kachanovich and M.Mathijs Wintraecken. Tracing Isomanifolds in R^d in Time Polynomial in d using Coxeter–Freudenthal–Kuhn Triangulations.SIAM Journal on Computing52April 2023, 452 - 486HALDOIback to text
• 13 articleJ.-D.Jean-Daniel Boissonnat and M.Mathijs Wintraecken. The reach of subsets of manifolds.Journal of Applied and Computational Topology2023HALDOIback to text
• 14 articleM. B.Magnus Bakke Botnan, V.Vadim Lebovici and S.Steve Oudot. Local characterizations for decomposability of 2-parameter persistence modules.Algebras and Representation Theory2023HALDOIback to text
• 15 articleB.Benjamin Burton, H.-C.Hsien-Chih Chang, M.Maarten Löffler, A.Arnaud de Mesmay, C.Clément Maria, S.Saul Schleimer, E.Eric Sedgwick and J.Jonathan Spreer. Hard Diagrams of the Unknot.Experimental MathematicsFebruary 2023, 1-19HALDOIback to text
• 16 articleJ.Jérémie Capitao-Miniconi and É.Élisabeth Gassiat. Deconvolution of spherical data corrupted with unknown noise..Electronic Journal of Statistics 171January 2023HALDOI
• 17 articleG.Guillaume Carlier, A.Alex Delalande and Q.Quentin Merigot. Quantitative Stability of Barycenters in the Wasserstein Space.Probability Theory and Related FieldsOctober 2023HAL
• 18 articleO.Otfried Cheong, O.Olivier Devillers, J.-W.Ji-Won Park and M.Marc Glisse. Covering families of triangles.Periodica Mathematica Hungarica872023, 86--109HALDOI
• 19 articleD.David Cohen-Steiner, A.André Lieutier and J.Julien Vuillamy. Delaunay and Regular Triangulations as Lexicographic Optimal Chains.Discrete and Computational Geometry70May 2023, 1–50HALDOI
• 20 articleA.Alex Delalande and Q.Quentin Merigot. Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure.Duke Mathematical Journal2023HAL
• 21 articleE.Etienne Lasalle. Heat diffusion distance processes: a statistically founded method to analyze graph data sets.Journal of Applied and Computational TopologyMay 2023HALDOIback to text
• 22 articleJ.Jacob Leygonie, M.Mathieu Carrière, T.Théo Lacombe and S.Steve Oudot. A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis.Mathematical Programming2022023, 199–239HALDOI
• 23 articleC.Clément Maria and H.Hannah Schreiber. Discrete Morse Theory for Computing Zigzag Persistence.Discrete and Computational GeometryNovember 2023, 538-552HALDOIback to text
• 24 articleD.Daniel Perez. On the persistent homology of almost surely $C^0$ stochastic processes.Journal of Applied and Computational TopologyJuly 2023HALDOIback to text
• 25 articleM.Marie Perrot-Dockès, G.Gilles Blanchard, P.Pierre Neuvial and E.Etienne Roquain. Post hoc false discovery proportion inference under a Hidden Markov Model.TestSeptember 2023HALDOIback to text

International peer-reviewed conferences

• 26 inproceedingsB.Bastien Dussap, G.Gilles Blanchard and B.-E.Badr-Eddine Chérief-Abdellatif. Label Shift Quantification with Robustness Guarantees via Distribution Feature Matching.Machine Learning and Knowledge Discovery in Databases: Research TrackECML-PKDD 202314173Lecture Notes in Computer ScienceTurin (IT), ItalySpringer Nature SwitzerlandJune 2023, 69-85HALDOIback to textback to text
• 27 inproceedingsA.André Lieutier and M.Mathijs Wintraecken. Hausdorff and Gromov-Hausdorff Stable Subsets of the Medial Axis.STOC 2023 - 55th Annual ACM Symposium on Theory of ComputingSTOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of ComputingOrlando (Florida), United StatesJune 2023, 1768–1776HALDOIback to text
• 28 inproceedingsD.David Loiseaux, M.Mathieu Carrière and A. J.Andrew J. Blumberg. A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions.NeurIPS 2023 - 36th Conference on Neural Information Processing SystemsAdvances in Neural Information Processing Systems 36New Orleans (LA), United StatesJune 2023HAL
• 29 inproceedingsD.David Loiseaux, L.Luis Scoccola, M.Mathieu Carrière, M. B.Magnus Bakke Botnan and S.Steve Oudot. Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures.NeurIPS 2023 - 36th Conference on Neural Information Processing SystemsAdvances in Neural Information Processing Systems 36New Orleans (LA), United StatesJune 2023HAL
• 30 inproceedingsE. M.El Mehdi Saad and G.Gilles Blanchard. Constant regret for sequence prediction with limited advice.Proceedings of The 34th International Conference on Algorithmic Learning Theory, PMLRAlgorithmic Learning Theory (ALT 2023)201Singapore, SingaporeFebruary 2023, 1343-1386HALback to text
• 31 inproceedingsE. M.El Mehdi Saad, G.Gilles Blanchard and N.Nicolas Verzelen. Covariance Adaptive Best Arm Identification.Advances in Neural Information Processing SystemsNeurIPS 2023 - Neural Information Processing Systems36New Orleans, United States2023HAL
• 32 inproceedingsT.Tong Zhao, L.Laurent Busé, D.David Cohen-Steiner, T.Tamy Boubekeur, J.-M.Jean-Marc Thiery and P.Pierre Alliez. Variational Shape Reconstruction via Quadric Error Metrics.SIGGRAPH 2023 - The 50th International Conference & Exhibition On Computer Graphics & Interactive TechniquesLos Angeles, United StatesAugust 2023HALDOIback to text

Scientific book chapters

• 33 inbookG.Gilles Blanchard and J.-B.Jean-Baptiste Fermanian. Nonasymptotic one-and two-sample tests in high dimension with unknown covariance structure.PROMS. 425Foundations of Modern Statistics : Festschrift in Honor of Vladimir Spokoiny, Berlin, Germany, November 6–8, 2019, Moscow, Russia, November 30, 2019Springer Proceedings in Mathematics & StatisticsSpringer International PublishingJuly 2023, 121-162HALDOI

Doctoral dissertations and habilitation theses

• 34 thesisG.Georg Grützner. Möbius spaces and large-scale geometry.Université Paris-SaclayMay 2023HAL
• 35 thesisO.Olympio Hacquard. From topological features to machine learning models : a journey through persistence diagrams.Université Paris-SaclaySeptember 2023HAL
• 36 thesisV.Vadim Lebovici. Two complementary approaches in multi-parameter persistence : interval-decompositions and constructible functions.Université Paris-SaclaySeptember 2023HAL

Reports & preprints

• 37 miscD.Dominique Attali, H. D.Hana Dal Poz Kouřimská, C.Christopher Fillmore, I.Ishika Ghosh, A.André Lieutier, E.Elizabeth Stephenson and M.Mathijs Wintraecken. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds.2022HALDOI
• 38 miscG.Gilles Blanchard, A.Alexandra Carpentier and O.Oleksandr Zadorozhnyi. Moment inequalities for sums of weakly dependent random fields.July 2023HAL
• 39 miscJ.-D.Jean-Daniel Boissonnat, F.Florestan Brunck, H. D.Hana Dal Poz Kouřimská, A.Arijit Ghosh and M.Mathijs Wintraecken. Simplicial subdivision of simplices of arbitrary dimension in a space of constant non-zero curvature with bounded quality.November 2023HAL
• 40 miscJ.-D.Jean-Daniel Boissonnat and K.Kunal Dutta. Dimensionality Reduction for Persistent Homology with Gaussian Kernels.January 2023HAL
• 41 miscJ.-D.Jean-Daniel Boissonnat, S.Siargey Kachanovich and M.Mathijs Wintraecken. Triangulating submanifolds: An elementary and quantified version of Whitney's method.2023HALDOI
• 42 miscB.Blanche Buet and X.Xavier Pennec. Flagfolds.May 2023HALback to text
• 43 miscD.Denys Bulavka, O.Olivier Devillers, P.Philippe Duchon, M.Marc Glisse and X.Xavier Goaoc. Two Lower Bounds for Random Point Sets via Negative Association.2023HALback to text
• 44 miscJ.Jeremie Capitao-Miniconi, É.Élisabeth Gassiat and L.Luc Lehéricy. Support and distribution inference from noisy data.April 2023HALback to text
• 45 miscE. W.Erin Wolf Chambers, C.Christopher Fillmore, E.Elizabeth Stephenson and M.Mathijs Wintraecken. Burning or collapsing the medial axis is unstable.November 2023HAL
• 46 miscF.Frédéric Chazal, L.Laure Ferraris, P.Pablo Groisman, M.Matthieu Jonckheere, F.Frédéric Pascal and F.Facundo Sapienza. Choosing the parameter of the Fermat distance: navigating geometry and noise.2023HALDOIback to text
• 47 miscC.Colleen Delaney, C.Clément Maria and E.Eric Samperton. An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number.November 2023HALback to text
• 48 miscU.Ulysse Gazin, G.Gilles Blanchard and E.Etienne Roquain. Transductive conformal inference with adaptive scores.October 2023HAL
• 49 miscM.Marc Glisse. Fast persistent homology computation for functions on ℝ.January 2023HALback to text
• 50 miscG. A.Georg Alexander Gruetzner. Asymptotic-Möbius maps.February 2023HAL
• 51 miscG. A.Georg Alexander Gruetzner. Harmonic analysis on the boundary of hyperbolic groups.February 2023HALback to text
• 52 miscO.Olympio Hacquard, G.Gilles Blanchard and C.Clément Levrard. Statistical learning on measures: an application to persistence diagrams.March 2023HAL
• 53 miscO.Olympio Hacquard and V.Vadim Lebovici. Euler Characteristic Tools For Topological Data Analysis.March 2023HAL
• 54 miscF.Felix Hensel, C.Charles Arnal, M.Mathieu Carrière, T.Théo Lacombe, H.Hiroaki Kurihara, Y.Yuichi Ike and F.Frédéric Chazal. MAGDiff: Covariate Data Set Shift Detection via Activation Graphs of Deep Neural Networks.May 2023HALback to text
• 55 miscH. D.Hana Dal Poz Kouřimská, A.André Lieutier and M.Mathijs Wintraecken. The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms.November 2023HAL
• 56 miscA.André Lieutier and M.Mathijs Wintraecken. Hausdorff and Gromov-Hausdorff stable subsets of the medial axis.April 2023HALDOI
• 57 miscD.David Loiseaux, M.Mathieu Carriere and A.Andrew Blumberg. Fast, Stable and Efficient Approximation of Multi-parameter Persistence Modules with MMA.June 2023HAL
• 58 miscW.Wojciech Reise, B.Bertrand Michel and F.Frédéric Chazal. Topological signatures of periodic-like signals.June 2023HALback to text

Other scientific publications

• 59 miscM.Mathijs Wintraecken. Translation of "Simplizialzerlegungen von Beschrankter Flachheit'' by Hans Freudenthal, Annals of Mathematics, Second Series, Volume 43, Number 3, July 1942, Pages 580-583.2023HALDOI

Scientific popularization

• 60 miscD.Davide Faranda, T.Théo Lacombe, N.Nina Otter and K.Kristian Strommen. Climate Science at the Interface Between Topological Data Analysis and Dynamical Systems Theory.February 2024, 267-271HALDOI