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 3.9.0:

  - Persistence matrix > Matrix API is in a beta version and may change in incompatible ways in the near future. . Matrix structure for filtered complexes with multiple functionnalities related to persistence homology, such as representative cycles computation or vineyards.

  - Rips complex . Rips complex persistence scikit-learn like interface

  - Čech complex . A new utility to compute the Delaunay-Čech filtration on a Delaunay triangulation.

• 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

7.1.2 Multipers

• Name:
  Multiparameter Persistence for Machine Learning
• Keywords:
  Topology, Machine learning
• Functional Description:
  multipers is a Python library for Topological Data Analysis, focused on Multiparameter Persistence computation and visualizations for Machine Learning. It features several efficient computational and visualization tools, with integrated, easy to use, auto-differentiable Machine Learning pipelines, that can be seamlessly interfaced with scikit-learn and PyTorch. This library is meant to be usable for non-experts in Topological or Geometrical Machine Learning. Performance-critical functions are implemented in C++ or in Cython, are parallelizable with TBB, and have Python bindings and interface. It can handle a very diverse range of datasets that can be framed into a (finite) multi-filtered simplicial or cell complex, including, e.g., point clouds, graphs, time series, images, etc.
• URL:
  https://­davidlapous.­github.­io/­multipers/
• Publication:
  hal-04801544
• Contact:
  David Loiseaux
• Participants:
  David Loiseaux, Hannah Schreiber

8.1.1 Generalized Morse theory for tubular neighborhoods

Participant: Antoine Commaret.

We define a notion of Morse function and establish Morse theory-like theorems over offsets of any compact set in a Euclidean space at regular values of their distance function. Using non-smooth analysis and tools from geometric measure theory, we prove that the homotopy type of the sublevels sets of these Morse functions changes at a critical value by gluing exactly one cell around each critical point.

8.1.2 Persistent Intrinsic Volumes

Participant: David Cohen-Steiner, Antoine Commaret.

We develop a new method to estimate the area, and more generally the intrinsic volumes, of a compact subset $X$ of ${ℝ}^d$ from a set $Y$ that is close in the Hausdorff distance. This estimator enjoys a linear rate of convergence as a function of the Hausdorff distance under mild regularity conditions on X. Our approach combines tools from both geometric measure theory and persistent homology, extending the noise filtering properties of persistent homology from the realm of topology to geometry. Along the way, we obtain a stability result for intrinsic volumes.

8.1.3 Critical points of the distance function to a generic submanifold

Participant: Charles Arnal, David Cohen-Steiner.

In collaboration with Vincent Divol (CEREMADE)

8.1.4 On Edge Collapse of Random Simplicial Complexes

Participant: Jean-Daniel Boissonnat.

In collaboration with Kunal Dutta (Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland), Soumik Dutta (Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland), and Siddharth Pritam (Chennai Mathematical Institute, India).

We consider the edge collapse (introduced in 66) process on the Erdős-Rényi random clique complex $X(n,c/\sqrt{n})$ on $n$ vertices with edge probability $c/\sqrt{n}$ such that $c>\sqrt{{\eta }_2}$, where

For a given $c>\sqrt{{\eta }_2}$, we show that after $t$ iterations of maximal edge collapsing phases, the remaining subcomplex, or $t$-core, has at most

and at least

edges asymptotically almost surely (a.a.s.), where ${\left\{{\gamma }_t\right\}}_{|t\ge 0}$ is recursively determined by ${\gamma }_{t+1}=e^{-c^2{(1-{\gamma }_t)}^2}$ and ${\gamma }_0=0$. We also determine the upper and lower bound on the final core with explicit formulas. If $c<\sqrt{{\eta }_2}$, then we show that the final core contains $o\left(n\sqrt{n}\right)$ edges. On the other hand, if, instead of $c$ being a constant with respect to $n$, $c>\sqrt{2logn}$, then the edge collapse process is no more effective in reducing the size of the complex. Our proof is based on the notion of local weak convergence 65 together with two new components. Firstly, we identify the critical combinatorial structures that control the outcome of the edge collapse process. By controlling the expected number of these structures during the edge collapse process, we establish a.a.s. bounds on the size of the core. We also give a new concentration inequality for typically Lipschitz functions on random graphs which improves on the bound of 71 and is, therefore, of independent interest. The proof of our lower bound is via the recursive technique of 68 to simulate cycles in infinite trees. These are the first theoretical results proved for edge collapses on random (or non-random) simplicial complexes.

8.1.5 On Edge Collapse of Random Simplicial Complexes

Participant: Jean-Daniel Boissonnat.

In collaboration with Kunal Dutta (Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland.

Our proof utilizes the embedding of Chen and Phillips 67, based on the Random Fourier Functions of Rahimi and Recht 70, together with two novel ingredients. The first one is a new decomposition of the squared radii of Čech simplices computed using the GKPD, in terms of the pairwise GKPDs between the vertices, which we state and prove. The second is a new concentration inequality for sums of cosine functions of Gaussian random vectors, which we call Gaussian cosine chaoses. We believe these are of independent interest and will find other applications in future.

8.1.6 Sparsification of the generalized persistence diagrams for scalability through gradient descent

Participant: Mathieu Carrière.

In collaboration with Seunghyun Kim, Woojin Kim (KAIST, South Korea)

In this work, we introduce a novel method for optimizing the domain of the GPD through gradient descent optimization. To achieve this, we introduce a loss function tailored to optimize the selection of intervals, balancing computational efficiency and discriminative accuracy. The design of the loss function is based on the known erosion stability property of the GPD. We showcase the efficiency of our sparsification method for dataset classification in supervised machine learning. Experimental results demonstrate that our sparsification method significantly reduces the time required for computing the GPDs associated to several datasets, while maintaining classification accuracies comparable to those achieved using full GPDs. Our method thus opens the way for the use of GPD-based methods to applications at an unprecedented scale.

8.1.7 multipers: Multiparameter Persistence for Machine Learning

Participant: David Loiseaux, Hannah Schreiber.

multipers is a Python library for Topological Data Analysis, focused on Multiparameter Persistence computation and visualizations for Machine Learning. It features several efficient computational and visualization tools, with integrated, easy to use, auto-differentiable Machine Learning pipelines, that can be seamlessly interfaced with scikit-learn and PyTorch. This library is meant to be usable for non-experts in Topological or Geometrical Machine Learning. Performance-critical functions are implemented in C++ or in Cython, are parallelizable with TBB, and have Python bindings and interface. It can handle a very diverse range of datasets that can be framed into a (finite) multi-filtered simplicial or cell complex, including, e.g., point clouds, graphs, time series, images, etc.

8.1.8 Brillouin Zones of Integer Lattices and Their Perturbations

Participant: Mathijs Wintraecken.

In collaboration with Herbert Edelsbrunner (Institute of Science and Technology Austria) , Alexey Garber (Department of Mathematics, The University of Texas at Brownsville) , Mohadese Ghafari (Northeastern University [Boston]) , Teresa Heiss (Institute of Science and Technology Austria) , Morteza Saghafian (Institute of Science and Technology Austria)

For a locally finite set, $A\subseteq {ℝ}^d$, the $k$th Brillouin zone of $a\in A$ is the region of points $x\in {ℝ}^d$ for which $\parallel x-a\parallel$ is the $k$th smallest among the Euclidean distances between $x$ and the points in $A$. If $A$ is a lattice, the $k$th Brillouin zones of the points in $A$ are translates of each other, and together they tile space. Depending on the value of $k$, they express medium- or long-range order in the set. In 12, we study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in ${ℝ}^2$, and the convergence of the maximum volume of a chamber to zero for the integer lattice.

8.1.9 Tight Bounds for the Learning of Homotopy à la Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.

Participant: Mathijs Wintraecken.

In collaboration with Dominique Attali (Université Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab), Hana Dal Poz Kouřimská (Institute of Science and Technology Austria), Christopher Fillmore (Institute of Science and Technology Austria), Ishika Ghosh (Institute of Science and Technology Austria, Michigan State University [East Lansing]), André Lieutier (No Affiliation), Elizabeth Stephenson (Institute of Science and Technology Austria)

In 22 we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of $C^2$ manifolds with positive reach embedded in ${ℝ}^d$. We extend their results in the following ways:

• As the ambient space we consider both ${ℝ}^d$ and Riemannian manifolds with lower bounded sectional curvature.
• In both types of ambient spaces, we study sets of positive reach — a significantly more general setting than $C^2$ manifolds — as well as general manifolds of positive reach.
• The sample $P$ of a set (or a manifold) $𝒮$ of positive reach may be noisy. We work with two one-sided Hausdorff distances — $\epsilon$ and $\delta$ — between $P$ and $𝒮$. We provide tight bounds in terms of $\epsilon$ and $\delta$, that guarantee that there exists a parameter $r$ such that the union of balls of radius $r$ centred at the sample $P$ deformation-retracts to $𝒮$. We exhibit their tightness by an explicit construction.

We carefully distinguish the roles of $\delta$ and $\epsilon$. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.

8.1.10 The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition)

Participant: Mathijs Wintraecken.

In collaboration with Dominique Attali (Université Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab), Hana Dal Poz Kouřimská (Institute of Science and Technology Austria), Christopher Fillmore (Institute of Science and Technology Austria), Ishika Ghosh (Institute of Science and Technology Austria, Michigan State University [East Lansing]), André Lieutier (No Affiliation), Elizabeth Stephenson (Institute of Science and Technology Austria)

In our companion paper `Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds' 22 we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample $P$ of an input shape $𝒮$ (either manifold or general set with positive reach) such that one can infer the homotopy of $𝒮$ from the union of balls with some radius centred at $P$, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video 63 is to visualize and thus elucidate said construction in the Euclidean setting.

Note that a media contribution consists of a small description 21 (which is part of the proceedings of the conference) and the video itself 63.

8.1.11 The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms

Participant: Mathijs Wintraecken.

In collaboration with Hana Dal Poz Kouřimská (Institute of Science and Technology Austria), André Lieutier (No Affiliation)

8.1.12 The distance function to a finite set is a topological Morse function

Participant: Charles Arnal.

In 37, we show that the distance function to any finite set $X\subset {ℝ}^n$ is a topological Morse function, regardless of whether $X$ is in general position. We also precisely characterize its topological critical points and their indices, and relate them to the differential critical points of the function.

8.1.13 Distance-from-flat persistent homology transform

Participant: Nina Otter, Renata Turkeš.

In collaboration with Adam Onus (Queen Mary University of London).

In 60 we introduce a generalisation of the persistent homology transform (PHT) in which we consider arbitrary parameter spaces and sublevel sets with respect to any function. In particular, we study transforms, defined on the Grassmannian $𝔸𝔾(m,n)$ of affine subspaces of ${ℝ}^n$, which allow to scan a shape by probing it with all possible affine $m$-dimensional subspaces $P\subset {ℝ}^n$, for fixed dimension $m$, and by then computing persistent homology of sublevel set filtrations of the function $\mathrm{dist}(·,P)$ encoding the distance from the flat $P$. We call such transforms “distance-from-flat” PHTs. We show that these transforms are injective and continuous and that they provide computational advantages over the classical PHT. In particular, we show that it is enough to compute homology only in degrees up to $m-1$ to obtain injectivity; for $m=1$ this provides a very powerful and computationally advantageous tool for examining shapes, which in a previous work by Nina Otter and Renata Turkeš has proven to significantly outperform state-of-the-art neural networks for shape classification tasks.

8.1.14 Time-optimal persistent homology representatives for univariate time series

Participant: António Leitão, Nina Otter.

In 58 we introduce time-optimal PH representatives for time-varying data, that allow one to extract representatives that are close in time in an appropriate sense. We illustrate our methods on quasi-periodic synthetic time series, as well as time series arising from climate models, and we show that our methods provide optimal PH representatives that are better suited for these types of problems than existing optimality notions, such as length-optimal PH representatives.

8.2.1 Wasserstein convergence of Čech persistence diagrams for samplings of submanifolds

Participant: Charles Arnal, David Cohen-Steiner.

In collaboration with Vincent Divol (CEREMADE, Université Paris-Dauphine PSL)

8.2.2 Statistical estimation of sparsity and efficiency for molecular codes

Participant: Mathieu Carrière.

In collaboration with Jun Hou Fung, Andrew Blumberg (Columbia University, USA)

We show that in fact for information-theoretic reasons almost any sufficiently large collection of genes is able to disambiguate cell types, and that this property is not robust to noise. To quantify the discriminative properties of a molecular codebook in a more refined way, we develop new statistics – partition cardinality and partition entropy – borrowing ideas from coding theory. We prove these are robust to data perturbations, and then apply these in the C. elegans example and in cancer. In the worm, we show that the homeodomain transcription factor family is distinguished by coding for cell types sparsely and efficiently compared to a control of randomly selected family of genes. Furthermore, the resolution of cell type identities defined using molecular features increases as the worm embryo develops. In cancer, we perform a pan-cancer study where we use our statistics to quantify interpatient tumor heterogeneity and we identify the chromosome containing the HLA family as sparsely and efficiently coding for melanoma.

8.2.3 Subsampling, aligning, and averaging to find circular coordinates in recurrent time series

Participant: Mathieu Carrière.

In collaboration with Jun Hou Fung, Andrew Blumberg (Columbia University, USA) and Michael Mandell (Indiana University, USA)

We introduce a new algorithm for finding robust circular coordinates on data that is expected to exhibit recurrence, such as that which appears in neuronal recordings of C. elegans. Techniques exist to create circular coordinates on a simplicial complex from a dimension 1 cohomology class, and these can be applied to the Rips complex of a dataset when it has a prominent class in its dimension 1 cohomology. However, it is known this approach is extremely sensitive to uneven sampling density.

Our algorithm comes with a new method to correct for uneven sampling density, adapting our prior work on averaging coordinates in manifold learning. We use rejection sampling to correct for inhomogeneous sampling and then apply Procrustes matching to align and average the subsamples. In addition to providing a more robust coordinate than other approaches, this subsampling and averaging approach has better efficiency.

We validate our technique on both synthetic data sets and neuronal activity recordings. Our results reveal a topological model of neuronal trajectories for C. elegans that is constructed from loops in which different regions of the brain state space can be mapped to specific and interpretable macroscopic behaviors in the worm.

8.2.4 Resampling and averaging coordinates on data

Participant: Mathieu Carrière.

In collaboration with Jun Hou Fung, Andrew Blumberg (Columbia University, USA) and Michael Mandell (Indiana University, USA)

8.2.5 Topological Analysis for Detecting Anomalies (TADA) in Time Series

Participant: Frédéric Chazal.

In collaboration with Clément Levrard (Univ.Rennes) and Martin Royer (IRT System X).

8.2.6 Topological signatures of periodic-like signals

Participant: Wojciech Riese, Frédéric Chazal.

In collaboration with Bertrand Michel (Ecole Centrale Nantes)

8.2.7 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)

8.2.8 Deconvolution of repeated measurements corrupted by unknown noise

Participant: Jérémie Capitao-Miniconi.

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

Recent advances have demonstrated the possibility of solving the deconvolution problem without prior knowledge of the noise distribution. In 45, we study the repeated measurements model, where information is derived from multiple measurements of $X$ perturbed independently by additive errors. Our contributions include establishing identifiability without any assumption on the noise except for coordinate independence. We propose an estimator of the density of the signal for which we provide rates of convergence, and prove that it reaches the minimax rate in the case where the support of the signal is compact. Additionally, we propose a model selection procedure for adaptive estimation. Numerical simulations demonstrate the effectiveness of our approach even with limited sample sizes.

8.2.9 A Geometric Approach for Multivariate Jumps Detection

Participant: Hugo Henneuse.

53 addresses the inference of jumps (i.e. sets of discontinuities) within multivariate signals from noisy observations in the non-parametric regression setting. Departing from standard analytical approaches, we propose a new framework, based on geometric control over the set of discontinuities. This allows to consider larger classes of signals, of any dimension, with potentially wild discontinuities (exhibiting, for example, self-intersections and corners). We study a simple estimation procedure relying on histogram differences and show its consistency and near-optimality for the Hausdorff distance over these new classes. Furthermore, exploiting the assumptions on the geometry of jumps, we design procedures to infer consistently the homology groups of the jumps locations and the persistence diagrams from the induced offset filtration.

8.2.10 Persistence Diagram Estimation : Beyond Plug-in Approaches

Participant: Hugo Henneuse.

Persistent homology is a tool from Topological Data Analysis (TDA) used to summarize the topology underlying data. It can be conveniently represented through persistence diagrams. Observing a noisy signal, common strategies to infer its persistence diagram involve plug-in estimators, and convergence properties are then derived from sup-norm stability. This dependence on the sup-norm convergence of the preliminary estimator is restrictive, as it essentially imposes to consider regular classes of signals. Departing from these approaches, in 54 we design an estimator based on image persistence. In the context of the Gaussian white noise model, and for large classes of piecewise-constant signals, we prove that the proposed estimator is consistent and achieves parametric rates.

8.2.11 Persistence Diagram Estimation of Multivariate Piecewise Hölder-continuous Signals

Participant: Hugo Henneuse.

To our knowledge, the analysis of convergence rates for persistence diagram estimation from noisy signals had predominantly relied on lifting signal estimation results through sup norm (or other functional norm) stability theorems. We believe that moving forward from this approach can lead to considerable gains. In 55, we illustrate it in the setting of Gaussian white noise model. We examine from a minimax perspective, the inference of persistence diagram (for sublevel sets filtration). We show that for piecewise Hölder-continuous functions, with control over the reach of the discontinuities set, taking the persistence diagram coming from a simple histogram estimator of the signal, permit to achieve the minimax rates known for Hölder-continuous functions.

8.2.12 Persistence-based Modes Inference

Participant: Hugo Henneuse.

In 56, we address the problem of estimating multiple modes of a multivariate density, using persistent homology, a key tool from Topological Data Analysis. We propose a procedure, based on a preliminary estimation of the $H_0$-persistence diagram, to estimate the number of modes, their locations, and the associated local maxima. For large classes of piecewise-continuous functions, we show that these estimators are consistent and achieve nearly minimax rates. These classes involve geometric control over the set of discontinuities of the density, and differ from commonly considered function classes in mode(s) inference. Interestingly, we do not suppose regularity or even continuity in any neighborhood of the modes.

8.2.13 Transductive conformal inference with adaptive scores

Participant: Gilles Blanchard.

In collaboration with Ulysse Gazin (U. Paris-Sorbonne), Etienne Roquain (U. Paris-Sorbonne)

Conformal inference is a fundamental and versatile tool that provides distribution-free guarantees for many machine learning tasks. In 30, we consider the transductive setting, where decisions are made on a test sample of $m$ new points, giving rise to $m$ conformal $p$-values. While classical results only concern their marginal distribution, we show that their joint distribution follows a Pólya urn model, and establish a concentration inequality for their empirical distribution function. The results hold for arbitrary exchangeable scores, including adaptive ones that can use the covariates of the test+calibration samples at training stage for increased accuracy. We demonstrate the usefulness of these theoretical results through uniform, in-probability guarantees for two machine learning tasks of current interest: interval prediction for transductive transfer learning and novelty detection based on two-class classification.

8.2.14 Moment inequalities for sums of weakly dependent random fields

Participant: Gilles Blanchard.

In collaboration A. Carpentier (U. Potsdam), O. Zadorozhnyi (TU München)

In 8, we derive both Azuma-Hoeffding and Burkholder-type inequalities for partial sums over a rectangular grid of dimension $d$ of a random field satisfying a weak dependency assumption of projective type: the difference between the expectation of an element of the random field and its conditional expectation given the rest of the field at a distance more than $\delta$ is bounded, in $L_p$ distance, by a known decreasing function of $\delta$. The analysis is based on the combination of a multi-scale approximation of random sums by martingale difference sequences, and of a careful decomposition of the domain. The obtained results extend previously known bounds under comparable hypotheses, and do not use the assumption of commuting filtrations.

8.3.1 Diffeomorphic interpolation for efficient persistence-based topological optimization

Participant: Mathieu Carrière.

In collaboration with Marc Theveneau (McGill University, Canada) and Théo Lacombe (Université Gustave Eiffel)

Topological Data Analysis (TDA) provides a pipeline to extract quantitative topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to what extent a given object exhibits some topological properties. These losses can then be used to perform topological optimizationvia gradient descent routines. While theoretically sounded, topological optimization faces an important challenge: gradients tend to be extremely sparse, in the sense that the loss function typically depends on only very few coordinates of the input object, yielding dramatically slow optimization schemes in this http URL on the central case of topological optimization for point clouds, we propose in this work to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space, with quantifiable Lipschitz constants. In particular, we show that our approach combines efficiently with subsampling techniques routinely used in TDA, as the diffeomorphism derived from the gradient computed on a subsample can be used to update the coordinates of the full input object, allowing us to perform topological optimization on point clouds at an unprecedented scale. Finally, we also showcase the relevance of our approach for black-box autoencoder (AE) regularization, where we aim at enforcing topological priors on the latent spaces associated to fixed, pre-trained, black-box AE models, and where we show thatlearning a diffeomorphic flow can be done once and then re-applied to new data in linear time (while vanilla topological optimization has to be re-run from scratch). Moreover, reverting the flow allows us to generate data by sampling the topologically-optimized latent space directly, yielding better interpretability of the model.

8.3.2 Differentiability and optimization of multiparameter persistent homology

Participant: Mathieu Carrière, David Loiseaux.

In collaboration with Siddharth Setlur (University of Edinburgh), Luis Scoccola (U. Sherbrooke) and Steve Oudot (Geomerix, Inria Saclay)

Real-valued functions on geometric data – such as node attributes on a graph – can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

8.3.3 Differentiable Mapper for topological optimization of data representation

Participant: Mathieu Carrière.

In collaboration with Ziyad Oulhaj, Bertrand Michel (Ecole Centrale de Nantes)

Unsupervised data representation and visualization using tools from topology is an active and growing field of Topological Data Analysis (TDA) and data science. Its most prominent line of work is based on the so-called Mapper graph, which is a combinatorial graph whose topological structures (connected components, branches, loops) are in correspondence with those of the data itself. While highly generic and applicable, its use has been hampered so far by the manual tuning of its many parameters-among these, a crucial one is the so-called filter: it is a continuous function whose variations on the data set are the main ingredient for both building the Mapper representation and assessing the presence and sizes of its topological structures. However, while a few parameter tuning methods have already been investigated for the other Mapper parameters (i.e., resolution, gain, clustering), there is currently no method for tuning the filter itself. In this work, we build on a recently proposed optimization framework incorporating topology to provide the first filter optimization scheme for Mapper graphs. In order to achieve this, we propose a relaxed and more general version of the Mapper graph, whose convergence properties are investigated. Finally, we demonstrate the usefulness of our approach by optimizing Mapper graph representations on several datasets, and showcasing the superiority of the optimized representation over arbitrary ones.

8.3.4 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 10, we study both theoretically and through simulations how to select this parameter.

8.3.5 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 14, 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.

8.3.6 Topological phase estimation method for reparameterized periodic functions

Participant: Frédéric Chazal, Wojciech reise.

In collaboration with Thomas Bonis (Univ. G. Eiffel) and Bertrand Michel (Ecole Centrale Nantes).

In 9, we consider a signal composed of several periods of a periodic function, of which we observe a noisy reparametrisation. The phase estimation problem consists of finding that reparametrisation, and, in particular, the number of observed periods. Existing methods are well-suited to the setting where the periodic function is known, or at least, simple. We consider the case when it is unknown and we propose an estimation method based on the shape of the signal. We use the persistent homology of sublevel sets of the signal to capture the temporal structure of its local extrema. We infer the number of periods in the signal by counting points in the persistence diagram and their multiplicities. Using the estimated number of periods, we construct an estimator of the reparametrisation. It is based on counting the number of sufficiently prominent local minima in the signal. This work is motivated by a vehicle positioning problem, on which we evaluated the proposed method.

8.3.7 ML Model Coverage Assessment by Topological Data Analysis Exploration

Participant: Martin Royer.

In collaboration with Ayman Fakhouri (IRT SystemX), Faouzi Adjed (IRT SystemX), Martin Gonzalez (IRT SystemX).

The increasing complexity of deep learning models necessitates advanced methods for model coverage assessment, a critical factor for their reliable deployment. In 29, we introduce a novel approach leveraging topological data analysis to evaluate the coverage of a couple dataset & classification model. By using tools from topological data analysis, our method identifies underrepresented regions within the data, thereby enhancing the understanding of both model performances and data completeness. This approach simultaneously evaluates the dataset and the model, highlighting areas of potential risk. We report experimental evidence demonstrating the effectiveness of this topological framework in providing a comprehensive and interpretable coverage assessment. As such, we aim to open new avenues for improving the reliability and trustworthiness of classification models, laying the groundwork for future research in this domain.

8.3.8 Euler Characteristic Tools for Topological Data Analysis

[Added by Gilles]

Participant: Olympio Hacquard, Vadim Lebovici.

Note: the authors are no longer members of DATASHAPE as of 2024, but this work is the outcome of work done during their Ph.D. in DATASHAPE.

In 13, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic profile. We show that this simple descriptor achieves state-of-the-art performance in supervised tasks at a meagre computational cost. Inspired by signal analysis, we compute hybrid transforms of Euler characteristic profiles. These integral transforms mix Euler characteristic techniques with Lebesgue integration to provide highly efficient compressors of topological signals. As a consequence, they show remarkable performances in unsupervised settings. On the qualitative side, we provide numerous heuristics on the topological and geometric information captured by Euler profiles and their hybrid transforms. Finally, we prove stability results for these descriptors as well as asymptotic guarantees in random settings.

8.4.1 Supervised Contamination Detection, with Flow Cytometry Application

Participant: Gilles Blanchard, Frédéric Chazal, Solenne Gaucher.

In 52, The contamination detection problem aims to determine whether a set of observations has been contaminated, i.e. whether it contains points drawn from a distribution different from the reference distribution. Here, we consider a supervised problem, where labeled samples drawn from both the reference distribution and the contamination distribution are available at training time. This problem is motivated by the detection of rare cells in flow cytometry. Compared to novelty detection problems or two-sample testing, where only samples from the reference distribution are available, the challenge lies in efficiently leveraging the observations from the contamination detection to design more powerful tests. In this article, we introduce a test for the supervised contamination detection problem. We provide non-asymptotic guarantees on its Type I error, and characterize its detection rate. The test relies on estimating reference and contamination densities using histograms, and its power depends strongly on the choice of the corresponding partition. We present an algorithm for judiciously choosing the partition that results in a powerful test. Simulations illustrate the good empirical performances of our partition selection algorithm and the efficiency of our test. Finally, we showcase our method and apply it to a real flow cytometry dataset.

8.4.2 Iteration Head: A Mechanistic Study of Chain-of-Thought.

Participant: Charles Arnal.

In collaboration with Vivien Cabannes (FAIR, Meta AI), Wassim Bouaziz (FAIR, Meta AI), Alice Yang (FAIR, Meta AI), Francois Charton (FAIR, Meta AI), Julia Kempe (Courant University and Center for Data Science, NYU & FAIR, Meta AI).

Chain-of-Thought (CoT) reasoning is known to improve Large Language Models both empirically and in terms of theoretical approximation power. However, our understanding of the inner workings and conditions of apparition of CoT capabilities remains limited. 25 helps fill this gap by demonstrating how CoT reasoning emerges in transformers in a controlled and interpretable setting. In particular, we observe the appearance of a specialized attention mechanism dedicated to iterative reasoning, which we coined "iteration heads". We track both the emergence and the precise working of these iteration heads down to the attention level, and measure the transferability of the CoT skills to which they give rise between tasks.

8.4.3 Touring sampling with pushforward maps

Participant: Charles Arnal.

In collaboration with Vivien Cabannes (FAIR, Meta AI).

The number of sampling methods could be daunting for a practitioner looking to cast powerful machine learning methods to their specific problem. 26 takes a theoretical stance to review and organize many sampling approaches in the “generative modeling” setting, where one wants to generate new data that are similar to some training examples. By revealing links between existing methods, it might prove useful to overcome some of the current challenges in sampling with diffusion models, such as long inference time due to diffusion simulation, or the lack of diversity in generated samples.

8.4.4 Prompt Selection Matters: Enhancing Text Annotations for Social Sciences with Large Language Models

Participant: Charles Arnal.

In collaboration with Louis Abraham (Université Paris 1 Panthéon-Sorbonne) and Antoine Marie (Institut Jean Nicod, Ecole Normale Supérieure, PSL-EHESS-CNRS).

Large Language Models have recently been applied to text annotation tasks from social sciences, equalling or surpassing the performance of human workers at a fraction of the cost. However, no inquiry has yet been made on the impact of prompt selection on labelling accuracy. In 33, we show that performance greatly varies between prompts, and we apply the method of automatic prompt optimization to systematically craft high quality prompts. We also provide the community with a simple, browser-based implementation of the method at github.

8.4.5 Scaling Laws with Hidden Structure

Participant: Charles Arnal.

In collaboration with Clement Berenfeld (Postdam Univ.), Simon Rosenberg (C3IA) and Vivien Cabannes (FAIR, Meta AI).

Statistical learning in high-dimensional spaces is challenging without a strong underlying data structure. Recent advances with foundational models suggest that text and image data contain such hidden structures, which help mitigate the curse of dimensionality. Inspired by results from nonparametric statistics, we hypothesize that this phenomenon can be partially explained in terms of decomposition of complex tasks into simpler subtasks. In 34, we present a controlled experimental framework to test whether neural networks can indeed exploit such “hidden factorial structures.” We find that they do leverage these latent patterns to learn discrete distributions more efficiently, and derive scaling laws linking model sizes, hidden factorizations, and accuracy. We also study the interplay between our structural assumptions and the models’ capacity for generalization.

8.4.6 Mode Estimation with Partial Feedback

Participant: Charles Arnal.

In collaboration with Vivien Cabannes (FAIR, Meta AI) and Vianney Perchet (Center for Research in Economics and Statistics (CREST), ENSAE, Palaiseau, France).

The combination of lightly supervised pre-training and online finetuning has played a key role in recent AI developments. These new learning pipelines call for new theoretical frameworks. In 35, we formalize core aspects of weakly supervised and active learning with a simple problem: the estimation of the mode of a distribution using partial feedback. We show how entropy coding allows for optimal information acquisition from partial feedback, develop coarse sufficient statistics for mode identification, and adapt bandit algorithms to our new setting. Finally, we combine those contributions into a statistically and computationally efficient solution to our problem.

8.4.7 False discovery proportion envelopes with $m$-consistency

Participant: Gilles Blanchard.

In collaboration with Iqraa Meah (U. Paris-Cité and INSERM), Etienne Roquain (U. Paris-Sorbonne)

In 17 We provide new nonasymptotic false discovery proportion (FDP) confidence envelopes in several multiple testing settings relevant for modern high dimensional-data methods. We revisit the multiple testing scenarios considered in the recent work of Katsevich and Ramdas (2020): top-k, preordered (including knockoffs), online. Our emphasis is on obtaining FDP confidence bounds that both have nonasymptotical coverage and are asymptotically accurate in a specific sense, as the number m of tested hypotheses grows. Namely, we introduce and study the property (which we call m-consistency) that the confidence bound converges to or below the desired level $\alpha$ when applied to a specific reference $\alpha$-level false discovery rate (FDR) controlling procedure. In this perspective, we derive new bounds that provide improvements over existing ones, both theoretically and practically, and are suitable for situations where at least a moderate number of rejections is expected. These improvements are illustrated with numerical experiments and real data examples. In particular, the improvement is significant in the knockoffs setting, which shows the impact of the method for a practical use. As side results, we introduce a new confidence envelope for the empirical cumulative distribution function of i.i.d. uniform variables and we provide new power results in sparse cases, both being of independent interest.

10.1.1 Inria associate team not involved in an IIL or an international program

10.2.1 Visits of international scientists

10.2.2 Visits to international teams

Extended visit

Participants: Corentin Lunel, Clément Maria.

- Duration : 1 years from Septembre 2024

- Coordinator : Clément Maria

- Location : Institut Montpelliérain Alexandre Grothendieck (IMAG) - Université de Montpellier

The visit consists of federating mathematicians from IMAG working on low dimensional and quantum topology together with computer scientists from Datashape, to work at the interface of the two fields.

10.3.1 ANR

10.3.2 Collaboration with other national research institutes

11.1.1 Scientific events: organisation

• Blanche Buet was a member of the organisation committee of the conference Geometric Sciences in Action: from Geometric Statistics to Shape Analysis (CIRM, may 2024).
• Nina Otter was a co-organiser of an American Mathematical Society Mathematics Research Community (AMS MRC) on “Climate science at the interface between Topological Data Analysis and Dynamical Systems Theory”, held in Buffalo, NY, in June 2024.
• Clément Maria was co-organizer of the Geometry & Computing conference held at CIRM, Marseille, October 2024.
• Clément Maria was co-organizer of the QuantAzur Days, Nice, October 2024.
• DataShape seminar: the seminar is held in person either at the LMO or at Inria Sophia Antipolis; all talks are recorded and can be accessed at: bbb2.imo.universite-paris-saclay.fr.

11.1.2 Scientific events: selection

11.1.3 Journal

11.1.4 Invited talks

• Gilles Blanchard gave an invited talk at the International Conference on Statistics and Data Science (ICSDS) organized by the IMS, Nice, Dec. 2024.
• Gilles Blanchard gave an invited talk at the International Symposium on Nonparametric Statistics (ISNP) organized by the IMS in Braga, Portugal, June 2024.
• Vincent Rouvreau gave a presentation and a practical session of the GUDHI library at the Jeunes Chercheur.e.s en Géométrie (jcgeo24) organized by the GDR IG-RV and the GDR IFM in ESIEE Paris, Université Gustave Eiffel, June 2024.
• Clément Maria gave an invited keynote talk at the ComPerWorkshop 2024, Graz, Austria.
• Nina Otter was a plenary speaker at the Spires 2024 Conference, University of Oxford, UK.

11.1.5 Leadership within the scientific community

• Frédéric Chazal has been the Scientific Director of the DATAIA Institute at Université Paris-Saclay until Sept.1, 2024.
• Frédéric Chazal is a member of the board of directors of the DIM project AI4IDF of the Région Ile-de-France until Sept.1, 2024.
• Frédéric Chazal is a member of the Scientific Advisory Board of the Centre for Topological Data Analysis of the Mathematical Institute at Oxford.

11.1.6 Scientific expertise

• Frédéric Chazal is a member of the “commission prospective de l’I2M” (Institut de Mathématiques de Marseille).
• Clément Maria was a jury member for the UCA-DS4H PhD grant allocation scheme for 2024.

11.1.7 Research administration

• Frédéric Chazal is co-responsible of the “programme Mathématiques et IA” of the Fondation Mathématique Jacques Hadamard, Paris-Saclay Univ.
• Frédéric Chazal is a member of the council of the Graduate School in Mathematics, Paris-Saclay Univ.
• Blanche Buet is member of the laboratoire council and CCUPS of LMO, Paris-Saclay Univ.
• Gilles Blanchard is part of the Sustainable Development Commission at the LMO, U. Paris-Saclay.
• Clément Maria is co-responsible of the CNRS-Groupe de Travail GeoAlgo.
• Clément Maria is a member of the steering committee of the QuantAzur federative institute.
• Clément Maria is a member of the Steering Committee of the Année de la Géométrie du GdR IFM (Informatique Fondamentale et ses Mathématiques).
• Marc Glisse is president of the CDT at Inria Saclay.

11.2.1 Teaching

• Master: Frédéric Chazal, Analyse Topologique des Données, 30h eq-TD, Université Paris-Sud, France.
• Master: Marc Glisse, 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 Data Analysis, with Applications in Biology and Medecine, 24h eq-TD, M2, Université Côte d'Azur, France.
• Master: Mathieu Carrière, Basic Algebra for Data Analysis, 15h eq-TD, M1, 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: Gilles Blanchard, Kernel and operator-theoretic methods in machine learning, 32h eq-TD, 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, 44h eq-TD, M1, École Polytechnique, France.
• Master: Nina Otter, Méthodes de modellisation statistique, ENSTA Paris, 25.5h eq-TD. (2024)
• Master: Mathijs Wintraecken, Geometrie et topologie (colles), 16h eq-TD, M1 Mathématiques, Laboratoire Jean Alexandre Dieudonné, Nice, France.

11.2.2 Supervision

• PhD: Bastien Dussap, comparaison de données cytométriques. Defended in October 2024. Gilles Blanchard and Marc Glisse
• PhD: Jérémie Capitao-Miniconi, deconvolution for singular measures with geometric support. Defended in October 2024. Frédéric Chazal and Elisabeth Gassiat.
• PhD: Jean-Baptiste Fermanian, Estimation de Kernel Mean Embedding et tests multiples en grande dimension. Defended in November 2024. Gilles Blanchard and Magalie Fromont-Renoir.
• PhD: David Loiseaux, Multivariate topological data analysis for statistical machine learning. Defended in December 2024. Mathieu Carrière and Frédéric Cazals.
• PhD: Antoine Commaret, Persistent Geometry. Defended in December 2024. David Cohen-Steiner and Indira Chatterji.
• PhD in progress: Myriam Frikha, Domain adaptation for temporal data. Started in Nov. 2024. Frédéric Chazal.
• PhD in progress: Hannah Marienwald (TU Berlin), Transfer learning in high dimension. Started September 2019. Gilles Blanchard and Klaus-Robert Müller (TU Berlin).
• 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: 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: António Leitão, started on November 2024. Nina Otter and Fosca Gianotti (Scuola Normale Superiore di Pisa).
• PhD in progress: Henrique Lovisi Ennes, started October 2023. Topological approach to Quantum Complexity. Clément Maria and Nicolas Nisse (INRIA UniCA, EPI COATI).

11.2.3 Juries

• Nina Otter was an external jury member for Renata Turkeš's' PhD defense at the University of Antwerp.

11.3.1 Others science outreach relevant activities

• Clément Maria gave a talk on quantum computing at the INRIA UniCA C@fé-In, March 2024
• Clément Maria gave a talk “Portrait de chercheur” at the C@fé ADSTIC Sophia Tech.

International journals

• 8 articleG.Gilles Blanchard, A.Alexandra Carpentier and O.Oleksandr Zadorozhnyi. Moment inequalities for sums of weakly dependent random fields.Bernoulli303August 2024HALDOIback to text
• 9 articleT.Thomas Bonis, F.Frédéric Chazal, B.Bertrand Michel and W.Wojciech Reise. Topological phase estimation method for reparameterized periodic functions.Advances in Computational Mathematics504July 2024, 66HALDOIback to text
• 10 articleF.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.Transactions on Machine Learning Research Journal2024, 2835-8856HALDOIback to text
• 11 articleF.Frédéric Chazal, C.Clément Levrard and M.Martin Royer. Topological Analysis for Detecting Anomalies (TADA) in dependent sequences: application to Time Series.Journal of Machine Learning Research25December 2024, 1-49HALback to text
• 12 articleH.Herbert Edelsbrunner, A.Alexey Garber, M.Mohadese Ghafari, T.Teresa Heiss, M.Morteza Saghafian and M.Mathijs Wintraecken. Brillouin Zones of Integer Lattices and Their Perturbations.SIAM Journal on Discrete Mathematics382June 2024, 1784-1807HALDOIback to text
• 13 articleO.Olympio Hacquard and V.Vadim Lebovici. Euler Characteristic Tools For Topological Data Analysis.Journal of Machine Learning Research25July 2024HALback to text
• 14 articleF.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.Transactions on Machine Learning Research JournalMay 2024HALback to text
• 15 articleS.Shuang Liang, R.Renata Turkes, J.Jiayi Li, N.Nina Otter and G.Guido Montufar. Pull-back Geometry of Persistent Homology Encodings.Transactions on Machine Learning Research Journal2024HAL
• 16 articleD.David Loiseaux and H.Hannah Schreiber. multipers: Multiparameter Persistence for Machine Learning.Journal of Open Source Software9103November 2024, 6773HALDOI
• 17 articleI.Iqraa Meah, G.Gilles Blanchard and E.Etienne Roquain. False discovery proportion envelopes with m-consistency.Journal of Machine Learning Research25270October 2024, 1-52HALback to text
• 18 articleM.Mona Michaud, A.Alexandre Guérin, M.Marguerite Dejean de la Bâtie, L.Léopold Bancel, L.Laurent Oudre and A.Alexis Tricot. The Analytical Validity of Stride Detection and Gait Parameters Reconstruction Using the Ankle-Mounted Inertial Measurement Unit Syde®.Sensors248April 2024, 2413HALDOI
• 19 articleW.Wojciech Reise, B.Bertrand Michel and F.Frédéric Chazal. Topological signatures of periodic-like signals.Bernoulli2024. In press. HALback to text

International peer-reviewed conferences

• 20 inproceedingsC.Charles Arnal, D.David Cohen-Steiner and V.Vincent Divol. Wasserstein convergence of Čech persistence diagrams for samplings of submanifolds.NeurIPS 2024Vancouver (Canada), CanadaDecember 2024HAL
• 21 inproceedingsD.Dominique Attali, H.Hana Dal Poz Kouřimská, C.Christopher Fillmore, I.Ishika Ghosh, A.André Lieutier, E.Elizabeth Stephenson and M.Mathijs Wintraecken. The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).Leibniz International Proceedings in InformaticsSoCG 2024 - 40th International Symposium on Computational GeometryAthènes, GreeceSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024, 87:1–87:6HALDOIback to text
• 22 inproceedingsD.Dominique Attali, H.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.https://drops.dagstuhl.de/entities/volume/LIPIcs-volume-293SoCG 2024 - 40th International Symposium on Computational Geometry293Athènes, GreeceSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024HALDOIback to textback to text
• 23 inproceedingsJ.-D.Jean-Daniel Boissonnat and K.Kunal Dutta. A Euclidean Embedding for Computing Persistent Homology with Gaussian Kernels.Leibniz International Proceedings in InformaticsESA 2024 - European Symposium on AlgorithmsLIPIcs-30832nd Annual European Symposium on Algorithms (ESA 2024)Egham, United KingdomSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024, 29:1-29:18HALDOI
• 24 inproceedingsJ.-D.Jean-Daniel Boissonnat, K.Kunal Dutta, S.Soumik Dutta and S.Siddharth Pritam. On Edge Collapse of Random Simplicial Complexes.Leibniz International Proceedings in InformaticsSoCG 2024 - 40th International Symposium on Computational GeometryLIPIcs-29340th International Symposium on Computational Geometry (SoCG 2024)Athens, GreeceSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024, 21:1--21:16HALDOI
• 25 inproceedingsV.Vivien Cabannes, C.Charles Arnal, W.Wassim Bouaziz, A.Alice Yang, F.Francois Charton and J.Julia Kempe. Iteration Head: A Mechanistic Study of Chain-of-Thought.NeurIPS 2024Vancouver (Canada), CanadaJune 2024HALback to text
• 26 inproceedingsV.Vivien Cabannes and C.Charles Arnal. Touring sampling with pushforward maps.ICASSP 2024 - International Conference on Acoustics, Speech, and Signal ProcessingSeoul, South KoreaApril 2024HALDOIback to text
• 27 inproceedingsM.Mathieu Carriere, M.Marc Theveneau and T.Théo Lacombe. Diffeomorphic interpolation for efficient persistence-based topological optimization.Advances in Neural Information Processing Systems 37 (NeurIPS)Vancouver, CanadaDecember 2024HAL
• 28 inproceedingsH.Hana Dal Poz Kouřimská, A.André Lieutier and M.Mathijs Wintraecken. The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.Leibniz International Proceedings in InformaticsSoCG 2024 - 40th International Symposium on Computational GeometryLIPIcs-29340th International Symposium on Computational GeometryAthens, GreeceSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024HALDOIback to text
• 29 inproceedingsA.Ayman Fakhouri, F.Faouzi Adjed, M.Martin Gonzalez and M.Martin Royer. ML Model Coverage Assessment by Topological Data Analysis Exploration.ATRACC workshop 2024 - AI Trustworthiness and Risk Assessment for Challenged Contexts / AAAI 2024 Fall SymposiumArlington (VA), United StatesNovember 2024HALback to text
• 30 inproceedingsU.Ulysse Gazin, G.Gilles Blanchard and E.Etienne Roquain. Transductive conformal inference with adaptive scores.Proceedings of Machine Learning ResearchProceedings of The 27th International Conference on Artificial Intelligence and Statistics238Proceedings of Machine Learning Research (PMLR)Valencia, SpainPMLR2024, 1504-1512HALback to text
• 31 inproceedingsZ.Ziyad Oulhaj, M.Mathieu Carrière and B.Bertrand Michel. Differentiable mapper for topological optimization of data representation.The Forty-first International Conference on Machine Learning (ICML 2024)Wien, AustriaFebruary 2024HAL
• 32 inproceedingsL.Luis Scoccola, S.Siddharth Setlur, D.David Loiseaux, M.Mathieu Carrière and S.Steve Oudot. Differentiability and Optimization of Multiparameter Persistent Homology.ICML 2024 - The Forty-first International Conference on Machine LearningWien, AustriaJune 2024HAL

Reports & preprints

• 33 miscL.Louis Abraham, C.Charles Arnal and A.Antoine Marie. Prompt Selection Matters: Enhancing Text Annotations for Social Sciences with Large Language Models.July 2024HALback to text
• 34 miscC.Charles Arnal, C.Clement Berenfeld, S.Simon Rosenberg and V.Vivien Cabannes. Scaling Laws with Hidden Structure.November 2024HALback to text
• 35 miscC.Charles Arnal, V.Vivien Cabannes and V.Vianney Perchet. Mode Estimation with Partial Feedback.2024HALDOIback to text
• 36 miscC.Charles Arnal, D.David Cohen-Steiner and V.Vincent Divol. Critical points of the distance function to a generic submanifold.2024HALDOI
• 37 miscC.Charles Arnal. The distance function to a finite set is a topological Morse function.July 2024HALback to text
• 38 miscD.Dominique Attali, H.Hana Dal Poz Kouřimská, C.Christopher Fillmore, I.Ishika Ghosh, A.André Lieutier, E.Elizabeth Stephenson and M.Mathijs Wintraecken. Supplementary material: The ultimate frontier: An optimality construction for homotopy inference.2024HAL
• 39 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.March 2024HAL
• 40 miscA.Annalisa Baldi, B.Bruno Franchi and P.Pierre Pansu. Continuous primitives for higher degree differential forms in Euclidean spaces, Heisenberg groups and applications.March 2024HAL
• 41 miscG.Gilles Blanchard, G.Guillermo Durand, A.Ariane Marandon-Carlhian and R.Romain Périer. FDR control and FDP bounds for conformal link prediction.April 2024HAL
• 42 miscG.Gilles Blanchard, J.-B.Jean-Baptiste Fermanian and H.Hannah Marienwald. Estimation of multiple mean vectors in high dimension.March 2024HAL
• 43 miscA. J.Andrew J. Blumberg, M.Mathieu Carriere, J. H.Jun Hou Fung and M. A.Michael A. Mandell. Resampling and averaging coordinates on data.August 2024HAL
• 44 miscA. J.Andrew J. Blumberg, M.Mathieu Carrière, J. H.Jun Hou Fung and M. A.Michael A. Mandell. Subsampling, aligning, and averaging to find circular coordinates in recurrent time series.December 2024HAL
• 45 miscJ.Jeremie Capitao-Miniconi, É.Élisabeth Gassiat and L.Luc Lehéricy. Deconvolution of repeated measurements corrupted by unknown noise.September 2024HALback to text
• 46 miscJ.Jeremie Capitao-Miniconi, É.Élisabeth Gassiat and L.Luc Lehéricy. Support and distribution inference from noisy data.May 2024HALback to text
• 47 miscM.Mathieu Carriere, S.Seunghyun Kim and W.Woojin Kim. Sparsification of the Generalized Persistence Diagrams for Scalability through Gradient Descent.December 2024HAL
• 48 miscD.David Cohen-Steiner and A.Antoine Commaret. Persistent intrinsic volumes.July 2024HAL
• 49 miscA.Antoine Commaret. Generalized Morse theory for tubular neighborhoods.2024HAL
• 50 miscH.Hana Dal Poz Kouřimská, A.André Lieutier and M.Mathijs Wintraecken. A free lunch: manifolds of positive reach can be smoothed without decreasing the reach.December 2024HAL
• 51 miscJ. H.Jun Hou Fung, M.Mathieu Carrière and A.Andrew Blumberg. Statistical estimation of sparsity and efficiency for molecular codes.August 2024HALDOI
• 52 miscS.Solenne Gaucher, G.Gilles Blanchard and F.Frédéric Chazal. Supervised Contamination Detection, with Flow Cytometry Application.April 2024HALback to text
• 53 miscH.Hugo Henneuse. A Geometric Approach for Multivariate Jumps Detection.October 2024HALback to text
• 54 miscH.Hugo Henneuse. Persistence Diagram Estimation : Beyond Plug-in Approaches.May 2024HALback to text
• 55 miscH.Hugo Henneuse. Persistence Diagram Estimation of Multivariate Piecewise Hölder-continuous Signals.March 2024HALback to text
• 56 miscH.Hugo Henneuse. Persistence-based Modes Inference.July 2024HALback to text
• 57 miscE. K.Elena K Kandror, A.Anqi Wang, M.Mathieu Carriere, A.Alexis Peterson, W.Will Liao, A.Andreas Tjarnberg, J. H.Jun Hou Fung, K. T.Krishnaa T Mahbubani, J.Jackson Loper, W.William Pangburn, Y.Yuchen Xu, K.Kourosh Saeb-Parsy, R.Raul Rabadan, T.Tom Maniatis and A. H.Abbas H Rizvi. Enhancer Dynamics and Spatial Organization Drive Anatomically Restricted Cellular States in the Human Spinal Cord.January 2025HALDOI
• 58 miscA.Antonio Leitao and N.Nina Otter. Time-optimal persistent homology representatives for univariate time series.2024HALback to text
• 59 miscA.André Lieutier and M.Mathijs Wintraecken. Manifolds of positive reach, differentiability, tangent variation, and attaining the reach.December 2024HAL
• 60 miscA.Adam Onus, N.Nina Otter and R.Renata Turkeš. Shoving tubes through shapes gives a sufficient and efficient shape statistic.2024HALback to text
• 61 miscE. M.El Mehdi Saad, G.Gilles Blanchard and S.Sylvain Arlot. Online Orthogonal Matching Pursuit.October 2024HAL
• 62 miscD.David Williams and M.Mathijs Wintraecken. Quasi-optimal interpolation of gradients and vector-fields on protected Delaunay meshes in ${}^d$.2024HAL

Other scientific publications

• 63 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. The Ultimate Frontier: An Optimality Construction for Homotopy Inference.2024HALback to textback to text

Scientific popularization

• 64 articleD.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.Notices of the American Mathematical Society71February 2024, 267-271HALDOI