6.1.1 Efficient open surface reconstruction from lexicographic optimal chains and critical bases

Participant: David Cohen-Steiner.

In collaboration with André Lieutier (Dassault Systèmes) and Julien Vuillamy (Dassault Systèmes and Titane)

6.1.2 Lower bound on the Voronoi diagram of lines in ${ℝ}^d$

Participant: Marc Glisse.

6.1.3 ISDE : Independence Structure Density Estimation

Participant: Louis Pujol.

6.1.4 Hybrid transforms of constructible functions.

Participant: Vadim Lebovici.

In 47 we introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity results, while Euler calculus conveys topological information and allows for compatibility with operations on constructible functions. We conduct a systematic study of such transforms and introduce two new ones: the Euler-Fourier and Euler-Laplace transforms. We show that the first has a left inverse and that the second provides a satisfactory generalization of Govc and Hepworth's persistent magnitude to constructible sheaves, in particular to multi-parameter persistent modules. Finally, we prove index-theoretic formulae expressing a wide class of hybrid transforms as generalized Euler integral transforms. This yields expectation formulae for transforms of constructible functions associated to (sub)level-sets persistence of random Gaussian filtrations.

6.1.5 Computing persistent Stiefel-Whitney classes of line bundles.

Participant: Raphael Tinarrage.

In this work 24 we propose a definition of persistent Stiefel-Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its persistent Stiefel-Whitney classes. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot $(1,2)$.

6.1.6 Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure.

Participant: Alex Delalande.

In collaboration with Quentin Mérigot (Laboratoire de Mathématiques d'Orsay, Univ. Paris-Saclay).

This work 37 studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on ${ℝ}^d$, which we denote $T_{\mu }$. Assuming that the source density $\rho$ is bounded from above and below on a compact convex set, we prove that the map $\mu \to T_{\mu }$ is bi-Hölder continuous on large families of probability measures, such as the set of probability measures whose moment of order $p>d$ is bounded by some constant. These stability estimates show that the linearized optimal transport metric $W_{2,\rho }(\mu ,\nu )={\parallel T_{\mu }-T_{\nu }\parallel }_{L_2(\rho ,{ℝ}^d)}$ is bi-Hölder equivalent to the 2-Wasserstein distance on such sets, justifiying its use in applications.

6.1.7 Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport.

Participant: Alex Delalande.

We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport 38. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called Sinkhorn potentials) w.r.t. the regularization parameter, for which we ensure a better than Lipschitz dependence. Such facts may be a first step towards a mathematical justification of annealing or ε-scaling heuristics for the numerical resolution of regularized semi-discrete optimal transport. Our results also entail a non-asymptotic and tight expansion of the difference between the entropic and the unregularized costs.

6.1.8 Zeta functions and the topology of super level-sets of stochasic processes

Participant: Daniel Perez.

In this preprint 49, we study bar-codes of random functions on the real line using the lens of zeta functions of continuous processes, that encompasses the $Per{s}_p$ functional already in use in topological data analysis. We give a closed formula for the zeta function of Brownian motion. For $\alpha$-stable Lévy processes, the zeta function has a meromorphic extension with a unique simple pole at $\alpha$. This implies a sharp asymptotic expansion for the expected number of small bars of the corresponding random bar-codes. We use this theoretical information to propose a new statistical test for the parameter $\alpha$. A local version of the zeta function, counting bars beginning near a given value $x$, allows to analyse zeta functions of semi-martingales, and to give closed formulae in certain cases, like the Ornstein-Uhlenbeck process.

6.1.9 A uniformization theorem for closed convex polyhedra in Euclidean 3-space

Participant: Georg Grützner.

This work 42 deals with various aspects of Möbius geometry, including a discrete version. A new discrete uniformization theorem for 3-dimensional convex Euclidean polytopes is proved. Here, two polytopes are discretely conformal if certain cusped hyperbolic surfaces canonically associated to them are isometric. We then show that every convex polytope is discretely conformal to a polytope inscribed in a sphere, which is unique up to Möbius transformations between spheres.

6.2.1 Topologically penalized regression on manifolds

Participants: Olympio Hacquard, Gilles Blanchard.

In collaboration with C. Levrard (LPSM, Sorbonne Université), K. Balasubramanian (UC. Davis), W. Polonik (UC. Davis).

In this preprint 43 we study a regression problem on a compact manifold $ℳ$. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is "topologically smooth".

6.2.2 Minimax adaptive estimation in manifold inference.

Participant: Vincent Divol.

In this work 18, we focus on the problem of manifold estimation: given a set of observations sampled close to some unknown submanifold $M$ , one wants to recover information about the geometry of $M$. Minimax estimators which have been proposed so far all depend crucially on the a priori knowledge of parameters quantifying the underlying distribution generating the sample (such as bounds on its density), whereas those quantities will be unknown in practice. Our contribution to the matter is twofold. First, we introduce a one-parameter family of manifold estimators $\left(M_t\right)$, $t\ge 0$ based on a localized version of convex hulls, and show that for some choice of $t$, the corresponding estimator is minimax on the class of models of $C^2$ manifolds introduced in [Genovese et al., Manifold estimation and singular deconvolution under Hausdorff loss]. Second, we propose a completely data-driven selection procedure for the parameter $t$, leading to a minimax adaptive manifold estimator on this class of models. This selection procedure actually allows us to recover the Hausdorff distance between the set of observations and $M$ , and can therefore be used as a scale parameter in other settings, such as tangent space estimation.

6.2.3 Estimation and Quantization of Expected Persistence Diagrams.

Participants: Vincent Divol, Théo Lacombe.

Persistence diagrams (PDs) are the most common descriptors used to encode the topology of structured data appearing in challenging learning tasks; think e.g. of graphs, time series or point clouds sampled close to a manifold. Given random objects and the corresponding distribution of PDs, one may want to build a statistical summary-such as a mean-of these random PDs, which is however not a trivial task as the natural geometry of the space of PDs is not linear. In this article 27, we study two such summaries, the Expected Persistence Diagram (EPD), and its quantization. The EPD is a measure supported on ${ℝ}^2$, which may be approximated by its empirical counterpart. We prove that this estimator is optimal from a minimax standpoint on a large class of models with a parametric rate of convergence. The empirical EPD is simple and efficient to compute, but possibly has a very large support, hindering its use in practice. To overcome this issue, we propose an algorithm to compute a quantization of the empirical EPD, a measure with small support which is shown to approximate with near-optimal rates a quantization of the theoretical EPD.

6.2.4 A short proof on the rate of convergence of the empirical measure for the Wasserstein distance.

Participant: Vincent Divol.

We provide a short proof that the Wasserstein distance between the empirical measure of a n-sample and the estimated measure is of order $n^{\frac{-1}{d}}$ , if the measure has a lower and upper bounded density on the d-dimensional flat torus 39.

6.2.5 Reconstructing measures on manifolds: an optimal transport approach.

Participant: Vincent Divol.

Assume that we observe i.i.d. points lying close to some unknown d-dimensional $C^k$ submanifold $M$ in a possibly high-dimensional space. We study the problem of reconstructing the probability distribution generating the sample. After remarking that this problem is degenerate for a large class of standard losses ($L_p$ , Hellinger, total variation, etc.), we focus on the Wasserstein loss, for which we build an estimator, based on kernel density estimation, whose rate of convergence depends on $d$ and the regularity $s\le k-1$ of the underlying density, but not on the ambient dimension 40. In particular, we show that the estimator is minimax and matches previous rates in the literature in the case where the manifold $M$ is a $d$-dimensional cube. The related problem of the estimation of the volume measure of $M$ for the Wasserstein loss is also considered, for which a minimax estimator is exhibited.

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

Participant: Etienne Lasalle.

In 46, 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.

6.3.1 Optimal quantization of the mean measure and applications to statistical learning

Participants: Frédéric Chazal, Martin Royer.

In collaboration with C. Levrard (LPSM, Sorbonne Université.

This work 16 addresses the case where data come as point sets, or more generally as discrete measures. Our motivation is twofold: first we intend to approximate with a compactly supported measure the mean of the measure generating process, that coincides with the intensity measure in the point process framework, or with the expected persistence diagram in the framework of persistence-based topological data analysis. To this aim we provide two algorithms that we prove almost minimax optimal. Second we build from the estimator of the mean measure a vectorization map, that sends every measure into a finite-dimensional Euclidean space, and investigate its properties through a clustering-oriented lens. In a nutshell, we show that in a mixture of measure generating processes, our technique yields a representation in ${ℝ}^k$, for $k\in {\mathbb{N}}^{*}$ that guarantees a good clustering of the data points with high probability. Interestingly, our results apply in the framework of persistence-based shape classification via the ATOL procedure described in 31.

6.3.2 Optimizing persistent homology based functions.

Participants: Mathieu Carrière, Frédéric Chazal, Marc Glisse.

In collaboration with Yuichi Ike (Fujitsu) and Hariprasad Kannan.

Solving optimization tasks based on functions and losses with a topological flavor is a very active, growing field of research in data science and Topological Data Analysis, with applications in non-convex optimization, statistics and machine learning. However, the approaches proposed in the literature are usually anchored to a specific application and/or topological construction, and do not come with theoretical guarantees. To address this issue, we study the differentiability of a general map associated with the most common topological construction, that is, the persistence map 26. Building on real analytic geometry arguments, we propose a general framework that allows us to define and compute gradients for persistence-based functions in a very simple way. We also provide a simple, explicit and sufficient condition for convergence of stochastic subgradient methods for such functions. This result encompasses all the constructions and applications of topological optimization in the literature. Finally, we provide associated code, that is easy to handle and to mix with other non-topological methods and constraints, as well as some experiments showcasing the versatility of our approach.

6.3.3 Barcode Embeddings for Metric Graphs.

Participant: Steve Oudot.

In collaboration with Elchanan Solomon (Department of Mathematics, Duke University).

Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In 23 we study a rich homology-based invariant first defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in the barcode space. We prove that this invariant is locally injective on the space of metric graphs and globally injective on a GH-dense subset. Moreover, we show that is globally injective on a full measure subset of metric graphs, in the appropriate sense.

6.3.4 A Framework for Differential Calculus on Persistence Barcodes.

Participant: Steve Oudot.

In collaboration with Jacob Leygonie and Ulrike Tillmann (Mathematical Institute of Oxford).

In 20 we define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.

6.3.5 A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis.

Participants: Mathieu Carrière, Théo Lacombe, Steve Oudot.

In collaboration with Jacob Leygonie (Mathematical Institute of Oxford).

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 strata-of the ambient Euclidean space 20. 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.

6.3.6 Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs.

Participants: Mathieu Carrière, Frédéric Chazal, Marc Glisse, Théo Lacombe.

In collaboration with Yuhei Umeda (Fujitsu) and Yuichi Ike (Fujitsu).

Although neural networks are capable of reaching astonishing performances on a wide variety of contexts, properly training networks on complicated tasks requires expertise and can be expensive from a computational perspective. In industrial applications, data coming from an open-world setting might widely differ from the benchmark datasets on which a network was trained. Being able to monitor the presence of such variations without retraining the network is of crucial importance. In this article 28, we develop a method to monitor trained neural networks based on the topological properties of their activation graphs. To each new observation, we assign a Topological Uncertainty, a score that aims to assess the reliability of the predictions by investigating the whole network instead of its final layer only, as typically done by practitioners. Our approach entirely works at a post-training level and does not require any assumption on the network architecture, optimization scheme, nor the use of data augmentation or auxiliary datasets; and can be faithfully applied on a large range of network architectures and data types. We showcase experimentally the potential of Topological Uncertainty in the context of trained network selection, Out-Of-Distribution detection, and shift-detection, both on synthetic and real datasets of images and graphs.

6.3.7 ATOL: Measure Vectorization for Automatic Topologically-Oriented Learning

Participants: Frédéric Chazal, Martin Royer.

In collaboration with with Yuhei Umeda (Fujitsu), Yuichi Ike (Fujitsu) and Clément Levrard (Univ. Paris Diderot).

Robust topological information commonly comes in the form of a set of persistence diagrams, finite measures that are in nature uneasy to affix to generic machine learning frameworks. We introduce 31 a fast, learnt, unsupervised vectorization method for measures in Euclidean spaces and use it for reflecting underlying changes in topological behaviour in machine learning contexts. The algorithm is simple and efficiently discriminates important space regions where meaningful differences to the mean measure arise. It is proven to be able to separate clusters of persistence diagrams. We showcase the strength and robustness of our approach on a number of applications, from emulous and modern graph collections where the method reaches state-of-the-art performance to a geometric synthetic dynamical orbits problem. The proposed methodology comes with a single high level tuning parameter: the total measure encoding budget. We provide a completely open access software.

6.4.1 Fast rates for prediction with limited expert advice

Participant: Gilles Blanchard.

In collaboration with El Mehdi Saad (LMO, University Paris-Saclay)

Motivated by the question of frugal machine learning, we investigate in this paper 32 the problem of minimizing the excess generalization error with respect to the best expert prediction in a finite family in the stochastic setting, under limited access to information: we assume that the learner only has access to a limited number of expert advices per training round, as well as for prediction. Under suitable assumptions on the loss (strong convexity and Lipschitz) We design novel algorithms achieving fast rates in this setting, and show that it is necessary to query at least two experts per training round to attain such fast rates.

6.4.2 Error rate control for classification rules in multiclass mixture models

Participant: Gilles Blanchard.

Collaboration with T.Mary-Huard (AgroParisTech), V. Perduca (MAP5, Univ. Paris), M.-L. Martin-Magniette (AgroParisTech)

In the context of finite mixture models, in this paper 22 we consider the problem of classifying as many observations as possible in the classes of interest while controlling the classification error rate in these same classes using an appropriate adaptation of the FDR criterion. It is shown that finding an optimal classification rule boils down to searching an optimal region in the observation space where to apply the classical Maximum A Posteriori (MAP) rule, and we propose a heuristic to compute that optimal region and the classification rule in practice. It is shown on both simulated and real datasets that the FDR-like optimal rule may be significantly less conservative than the naive thresholded MAP rule.

6.4.3 High-Dimensional Multi-Task Averaging and Application to Kernel Mean Embedding

Participant: Gilles Blanchard.

Collaboration with: H. Marienwald (U. Potsdam), J.-B. Fermanian (LMO, U. Paris-Saclay)

In the multi-task averaging problem, the goal is the joint estimation of the means of multiple distributions using separate, independent data sets. This is also known as the "many-means estimation problem" which has a long history in statistics. In this paper 30 we are particularly interested in the setting where the ambient space is of large dimension and propose a method exploiting similarities between tasks, without any related information being known in advance. The principle is that each empirical mean is shrunk towards the local average of its neighbors, which are found by multiple testing. We prove theoretically that this approach provides a reduction in mean squared error. This improvement can be significant when the dimension of the input space is large, demonstrating a "blessing of dimensionality" phenomenon. An application of this approach is the estimation of multiple kernel mean embeddings.

6.4.4 On agnostic post hoc approaches to false positive control

Participant: Gilles Blanchard.

Collaboration with: P. Neuvial (CNRS, U. Toulouse), E. Roquain (LPSM, U Sorbonne université)

This article 33 is a chapter in the Handbook of Multiple Comparisons. It gives a partial survey on post hoc approaches to false positive control for multiple test procedures.

6.4.5 Nonasymptotic one-and two-sample tests in high dimension with unknown covariance structure

Participant: Gilles Blanchard.

Collaboration with: J.-B. Fermanian (LMO, U. Paris-Saclay)

Article to appear in the Festschrift in the honor of V. Spokoiny

In this paper 34 we are interested in the test of so-called relevant differences for closeness of two mean vectors in a two-sample testing setting, i.e. the null hypothesis allows for a certain distance (tolerance) between the mean vectors, making it a composite null. We give matching upper and lower bounds giving in particular the precise dependence of the minimax test separation distance as a function of the effective dimensionality of each sample, and of the tolerance parameter allowed in the null hypothesis. We construct a test achieving optimal performance also when the covariance matrix of both samples (which determines their effective dimensionality) is unknown.

6.4.6 Identifying homogeneous subgroups of patients and important features: a topological machine learning approach.

Participants: Frédéric Chazal, Mathieu Carrière.

In collaboration with Ewan Carr and Raquel Iniesta (King's College London) and Bertrand Michel (Ecole Centrale de Nantes).

This work 15 exploits recent developments in topological data analysis to present a pipeline for clustering based on Mapper, an algorithm that reduces complex data into a one-dimensional graph. We present a pipeline to identify and summarise clusters based on statistically significant topological features from a point cloud using Mapper. Key strengths of this pipeline include the integration of prior knowledge to inform the clustering process and the selection of optimal clusters; the use of the bootstrap to restrict the search to robust topological features; the use of machine learning to inspect clusters; and the ability to incorporate mixed data types.

6.4.7 Topology identifies emerging adaptive mutations in SARS-CoV-2.

Participant: Mathieu Carrière.

In collaboration with Michael Bleher (Heidelberg University), Lukas Hahn (Heidelberg University), Juan Patiño-Galindo (Columbia University), Ulrich Bauer (TUM), Raúl Rabadán (Columbia University) and Andreas Ott (Heidelberg University).

The COVID-19 pandemic has lead to a worldwide effort to characterize its evolution through the mapping of mutations in the genome of the coronavirus SARS-CoV-2. As the virus spreads and evolves it acquires new mutations that could have important public health consequences, including higher transmissibility, morbidity, mortality, and immune evasion, among others. Ideally, we would like to quickly identify new mutations that could confer adaptive advantages to the evolving virus by leveraging the large number of SARS-CoV-2 genomes. One way of identifying adaptive mutations is by looking at convergent mutations, mutations in the same genomic position that occur independently. The large number of currently available genomes, more than a million at this moment, however precludes the efficient use of phylogeny-based techniques. In this work 35, we establish a fast and scalable Topological Data Analysis approach for the early warning and surveillance of emerging adaptive mutations of the coronavirus SARS-CoV-2 in the ongoing COVID-19 pandemic. Our method relies on a novel topological tool for the analysis of viral genome datasets based on persistent homology. It systematically identifies convergent events in viral evolution merely by their topological footprint and thus overcomes limitations of current phylogenetic inference techniques. This allows for an unbiased and rapid analysis of large viral datasets. We introduce a new topological measure for convergent evolution and apply it to the complete GISAID dataset as of February 2021, comprising 303,651 high-quality SARS-CoV-2 isolates taken from patients all over the world since the beginning of the pandemic. A complete list of mutations showing topological signals of convergence is compiled. We find that topologically salient mutations on the receptor-binding domain appear in several variants of concern and are linked with an increase in infectivity and immune escape. Moreover, for many adaptive mutations the topological signal precedes an increase in prevalence. We demonstrate the capability of our method to effectively identify emerging adaptive mutations at an early stage. By localizing topological signals in the dataset, we are able to extract geo-temporal information about the early occurrence of emerging adaptive mutations. The identification of these mutations can help to develop an alert system to monitor mutations of concern and guide experimentalists to focus the study of specific circulating variants.

6.4.8 An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists.

Participant: Frédéric Chazal.

In collaboration with Bertrand Michel (Ecole Centrale de Nantes).

Topological Data Analysis (TDA)is a recent and fast growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. This paper 17 is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of TDA for non experts.

8.1.1 Visits of international scientists

8.2.1 ANR

8.2.2 Collaboration with other national research institutes

Metafora

Participants: Gilles Blanchard, Bastien Dussap, Marc Glisse, Louis Pujol.

- Type : Paris Region PhD² - PhD 2021.

- Title : Analyse de données cytométriques.

The Île-de-France region funds two PhD theses in collaboration with Metafora biosystems, a company specialized in the analysis of cells through their metabolism. The first is supervised by Pascal Massart (Inria team Celeste) and Marc Glisse, and its goal is to improve clustering for this particular type of data. The second one is supervised by Gilles Blanchard and Marc Glisse and aims to compare samples instead of analyzing just one sample.

9.1.1 Teaching

• Master: Frédéric Chazal, Analyse Topologique des Données, 30h eq-TD, Université Paris-Sud, France.
• Master: Marc Glisse and Clément Maria, Computational Geometry Learning, 36h eq-TD, M2, MPRI, France.
• Master: Frédéric Cazals and Mathieu Carrière, Foundations of Geometric Methods in Data Analysis, 24h eq-TD, M2, École Centrale Paris, France.
• Master: Frédéric Chazal and Julien Tierny, Topological Data Analysis, 38h eq-TD, M2, Mathématiques, Vision, Apprentissage (MVA), ENS Paris-Saclay, France.
• Master: Steve Oudot, Topological data analysis, 45h eq-TD, M1, École polytechnique, France.
• Master: Steve Oudot, Data Analysis: geometry and topology in arbitrary dimensions, 24h eq-TD, M2, graduate program in Artificial Intelligence & Advanced Visual Computing, École polytechnique, 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.
• Undergrad-Master: Steve Oudot, Algorithms for data analysis in C++, 22.5h eq-TD, L3/M1, École polytechnique, France.
• Undergrad: Marc Glisse, Mécanismes de la programmation orientée-objet, 40h eq-TD, L3, École Polytechnique, France.

9.1.2 Supervision

• PhD in progress: Vadim Lebovici, Laplace transform for constructible functions. Started September 2020. Steve Oudot and François Petit.
• PhD in progress: Christophe Vuong, Random hypergraphs. Started November 2020. Laurent Decreusefond and Marc Glisse.
• PhD in progress: Louis Pujol, Partitionnement de données cytométriques, started November 1st, 2019, Pascal Massart and Marc Glisse.
• PhD in progress: Bastien Dussap, Comparaison de données cytométriques, started October 1st, 2021, Gilles Blanchard and Marc Glisse.
• PhD in progress: Olympio Hacquard, Apprentissage statistique par méthodes topologiques et géométriques, 2020, Gilles Blanchard and Clément Levrard.
• PhD in progress: El Mehdi Saad, Efficient online methods for variable and model selection, started September 2019. Gilles Blanchard and Sylvain Arlot.
• 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: David Loiseaux, Multivariate topological data analysis for statistical machine learning. Started November 2021. Mathieu Carrière and Frédéric Cazals.
• PhD in progress: Wojciech Rieser, TDA for curve data. Started October 2020. 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: Etienne Lasalle, TDA and statistics on graphs. Started on October 2019. Frédéric Chazal and Pascal Massart.
• PhD in progress: Alex Delande, optimal transport. Started on December 2019. Frédéric Chazal and Quentin Mérigot.
• 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.

9.1.3 Juries

• Gilles Blanchard was a member for the following Ph.D. defenses: Batiste Le Bars (ENS Paris-Saclay), Fabrice Grela (Univ. Rennes), Timothée Mathieu (U. Paris-Saclay), Amandine Dubois (ENSAI), Tien-Dat Nguyen (U. Paris-Saclay).

International journals

• 12 articleS.Shreya Arya, J.-D.Jean-Daniel Boissonnat, K.Kunal Dutta and M.Martin Lotz. Dimensionality reduction for k-distance applied to persistent homology.Journal of Applied and Computational Topology5October 2021, 671-691
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• 13 articleG.Gilles Blanchard, A. A.Aniket Anand Deshmukh, U.Urun Dogan, G.Gyemin Lee and C.Clayton Scott. Domain Generalization by Marginal Transfer Learning.Journal of Machine Learning Research2222021, 1-55
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• 14 articleJ.-D.Jean-Daniel Boissonnat, R.Ramsay Dyer, A.Arijit Ghosh, A.Andre Lieutier and M.Mathijs Wintraecken. Local Conditions for Triangulating Submanifolds of Euclidean Space.Discrete and Computational Geometry662September 2021, 666-686
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• 15 articleE.Ewan Carr, M.Mathieu Carriere, B.Bertrand Michel, F.Frédéric Chazal and R.Raquel Iniesta. Identifying homogeneous subgroups of patients and important features: a topological machine learning approach.BMC Bioinformatics2021
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• 16 articleF.Frédéric Chazal, C.Clément Levrard and M.Martin Royer. Optimal quantization of the mean measure and applications to statistical learning.Electronic Journal of Statistics 151April 2021, 2060-2104
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• 17 articleF.Frédéric Chazal and B.Bertrand Michel. An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists.Frontiers in Artificial IntelligenceSeptember 2021
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• 18 articleV.Vincent Divol. Minimax adaptive estimation in manifold inference.Electronic Journal of Statistics December 2021
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• 19 articleR.Rémi Gribonval, G.Gilles Blanchard, N.Nicolas Keriven and Y.Yann Traonmilin. Compressive Statistical Learning with Random Feature Moments.Mathematical Statistics and Learning32August 2021, 113–164
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• 20 articleJ.Jacob Leygonie, S. Y.Steve Y. Oudot and U.Ulrike Tillmann. A Framework for Differential Calculus on Persistence Barcodes.Foundations of Computational Mathematics2021
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• 21 articleH.Hengrui Luo, A.Alice Patania, J.Jisu Kim and M.Mikael Vejdemo-Johansson. Generalized penalty for circular coordinate representation.Foundations of Data Science342021, 729-767
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• 22 articleT.Tristan Mary-Huard, V.Vittorio Perduca, M. L.Marie Laure Martin-Magniette and G.Gilles Blanchard. Error rate control for classification rules in multiclass mixture models.The International Journal of Biostatistics2021
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• 23 articleS.Steve Oudot and E.Elchanan Solomon. Barcode Embeddings for Metric Graphs.Algebraic and Geometric Topology2021
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• 24 articleR.Raphaël Tinarrage. Computing persistent Stiefel-Whitney classes of line bundles.Journal of Applied and Computational TopologyOctober 2021
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International peer-reviewed conferences

• 25 inproceedingsJ.-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.SoCG 2021 - 37th Symposium on Computational Geometry37th International Symposium on Computational Geometry (SoCG 2021)Buffalo, United StatesJune 2021
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• 26 inproceedingsM.Mathieu Carriere, F.Frédéric Chazal, M.Marc Glisse, Y.Yuichi Ike and H.Hariprasad Kannan. Optimizing persistent homology based functions.38th International Conference on Machine Learning (ICML) 2021.PMLR 139Proceedings of the 38th International Conference on Machine Learning, ICML 2021.Virtual conference, FranceJuly 2021, 1294-1303
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• 27 inproceedingsV.Vincent Divol and T.Théo Lacombe. Estimation and Quantization of Expected Persistence Diagrams.ICML - 38th International Conference on Machine LearningVirtual Conference, United StatesJuly 2021
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• 28 inproceedingsT.Théo Lacombe, Y.Yuichi Ike, M.Mathieu Carriere, F.Frédéric Chazal, M.Marc Glisse and Y.Yuhei Umeda. Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs.IJCAI 2021 - International Joint Conference on Artificial IntelligenceMontréal, CanadaAugust 2021
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• 29 inproceedingsC.Clément Maria and O.Owen Rouillé. Computation of Large Asymptotics of 3-Manifold Quantum Invariants.ALENEX 21 - SIAM Symposium on Algorithm Engineering and ExperimentsAlexandria / Virtual, United StatesJanuary 2021
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• 30 inproceedingsH.Hannah Marienwald, J.-B.Jean-Baptiste Fermanian and G.Gilles Blanchard. High-Dimensional Multi-Task Averaging and Application to Kernel Mean Embedding.Proceedings of Machine Learning ResearchAISTATS 2021 - International Conference on Artificial Intelligence and Statistics130Proceedings of Machine Learning ResearchVirtual, United States2021, 1963-1971
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• 31 inproceedingsM.Martin Royer, F.Frédéric Chazal, C.Clément Levrard, Y.Yuhei Umeda and Y.Yuichi Ike. ATOL: Measure Vectorization for Automatic Topologically-Oriented Learning.The 24th International Conference on Artificial Intelligence and Statistics (AISTATS 2021)The 24th International Conference on Artificial Intelligence and StatisticsVirtual conference, FranceApril 2021
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• 32 inproceedingsE. M.El Mehdi Saad and G.Gilles Blanchard. Fast rates for prediction with limited expert advice.NeurIPSAdvances in Neural Information Processing Systems 34Online conference, United States2021
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Scientific book chapters

• 33 inbookG.Gilles Blanchard, P.Pierre Neuvial and E.Etienne Roquain. On agnostic post hoc approaches to false positive control.Handbook of Multiple ComparisonsHandbooks of Modern Statistical MethodsChapman & Hall/CRCNovember 2021
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Reports & preprints

• 34 miscG.Gilles Blanchard and J.-B.Jean-Baptiste Fermanian. Nonasymptotic one-and two-sample tests in high dimension with unknown covariance structure.October 2021
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• 35 miscM.Michael Bleher, L.Lukas Hahn, J.Juanángel Patiño-Galindo, M.Mathieu Carriere, U.Ulrich Bauer, R.Raúl Rabadán and A.Andreas Ott. Topology identifies emerging adaptive mutations in SARS-CoV-2.October 2021
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• 36 miscD.David Cohen-Steiner, A.André Lieutier and J.Julien Vuillamy. Efficient open surface reconstruction from lexicographic optimal chains and critical bases.December 2021
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• 37 miscA.Alex Delalande and Q.Quentin Merigot. Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure.March 2021
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• 38 miscA.Alex Delalande. Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport.November 2021
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• 39 miscV.Vincent Divol. A short proof on the rate of convergence of the empirical measure for the Wasserstein distance.January 2021
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• 40 miscV.Vincent Divol. Reconstructing measures on manifolds: an optimal transport approach.February 2021
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• 41 miscM.Marc Glisse. Lower bound on the Voronoi diagram of lines in ℝᵈ.December 2021
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• 42 miscG. A.Georg Alexander Gruetzner. A uniformization theorem for closed convex polyhedra in Euclidean 3-space.December 2021
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• 43 miscO.Olympio Hacquard, K.Krishnakumar Balasubramanian, G.Gilles Blanchard, W.Wolfgang Polonik and C.Clément Levrard. Topologically penalized regression on manifolds.October 2021
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• 44 reportO.Olympio Hacquard, E.Etienne Lasalle and V.Vadim Lebovici. Challenge mathématiques et entreprises : Eurecam.Université Paris-SaclaySeptember 2021
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• 45 miscK.Kristóf Huszár. On the pathwidth of hyperbolic 3-manifolds.October 2021
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• 46 miscE.Etienne Lasalle. Heat diffusion distance processes: a statistically founded method to analyze graph data sets.October 2021
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• 47 miscV.Vadim Lebovici. Hybrid transforms of constructible functions.December 2021
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• 48 miscJ.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.September 2021
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• 49 miscD.Daniel Perez. $$-functions and the topology of superlevel sets of stochastic processes.October 2021
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• 50 miscD.Daniel Perez. On Sovacool's et al. study on the differences in carbon emissions reduction between countries pursuing renewable electricity versus nuclear power.March 2021
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• 51 miscM.Marie Perrot-Dockès, G.Gilles Blanchard, P.Pierre Neuvial and E.Etienne Roquain. Post hoc false discovery proportion inference under a Hidden Markov Model.May 2021
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• 52 miscL.Louis Pujol. ISDE : Independence Structure Density Estimation.November 2021
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• 53 miscE. M.El Mehdi Saad, G.Gilles Blanchard and S.Sylvain Arlot. Online Orthogonal Matching Pursuit.February 2021
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