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Mathieu Carrière

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10 papers

2 author rows

NeurIPS Conference 2024 Conference Paper

Diffeomorphic interpolation for efficient persistence-based topological optimization

Mathieu Carrière
Marc Theveneau
Théo Lacombe

Topological Data Analysis (TDA) provides a pipeline to extract quantitative and powerful topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to which extent a given object exhibits some topological properties. One can then use these losses to perform topological optimization via 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 (locally) on only very few coordinates of the input object, yielding dramatically slow optimization schemes in practice. In this work, focusing on the central case of topological optimization for point clouds, we propose to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space. In particular, this approach combines efficiently with subsampling techniques routinely used in TDA, as the diffeomorphism derived from the gradient computed on the subsample can be used to update the coordinates of the full and possibly large input object. We then illustrate the usefulness of our approach on black-box autoencoder (AE) regularization, where we aim at applying some topological priors on the latent spaces associated to fixed, black-box AE models without modifying their (unknown) architectures and parameters. We empirically show that, while vanilla topological optimization has to be re-run every time that new data comes out of the black-box models, learning a diffeomorphic flow can be done once and then re-applied to new data in linear time. Moreover, reverting the flow allows us to generate data by sampling the topologically-optimized latent space directly, allowing for better interpretability of the model.

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ICML Conference 2024 Conference Paper

Differentiability and Optimization of Multiparameter Persistent Homology

Luis Scoccola
Siddharth Setlur
David Loiseaux
Mathieu Carrière
Steve Y. Oudot

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.

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ICML Conference 2024 Conference Paper

Differentiable Mapper for Topological Optimization of Data Representation

Ziyad Oulhaj
Mathieu Carrière
Bertrand Michel

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.

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TMLR Journal 2024 Journal Article

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

Charles Arnal
Felix Hensel
Mathieu Carrière
Théo Lacombe
Hiroaki Kurihara
Yuichi Ike
Frederic Chazal

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 this article, 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.

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NeurIPS Conference 2023 Conference Paper

A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions

David Loiseaux
Mathieu Carrière
Andrew Blumberg

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

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NeurIPS Conference 2023 Conference Paper

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures

David Loiseaux
Luis Scoccola
Mathieu Carrière
Magnus Bakke Botnan
Steve OUDOT

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

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ICML Conference 2021 Conference Paper

Optimizing persistent homology based functions

Mathieu Carrière
Frédéric Chazal
Marc Glisse
Yuichi Ike
Hariprasad Kannan
Yuhei Umeda

Solving optimization tasks based on functions and losses with a topological flavor is a very active and 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. 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.

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IJCAI Conference 2021 Conference Paper

Topological Uncertainty: Monitoring Trained Neural Networks through Persistence of Activation Graphs

Théo Lacombe
Yuichi Ike
Mathieu Carrière
Frédéric Chazal
Marc Glisse
Yuhei Umeda

Although neural networks are capable of reaching astonishing performance 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 paper, 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.

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JMLR Journal 2018 Journal Article

Statistical Analysis and Parameter Selection for Mapper

Mathieu Carrière
Bertrand Michel
Steve OUDOT

In this article, we study the question of the statistical convergence of the 1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that the Mapper is an optimal estimator of the Reeb graph, which gives, as a byproduct, a method to automatically tune its parameters and compute confidence regions on its topological features, such as its loops and flares. This allows to circumvent the issue of testing a large grid of parameters and keeping the most stable ones in the brute-force setting, which is widely used in visualization, clustering and feature selection with the Mapper. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

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ICML Conference 2017 Conference Paper

Sliced Wasserstein Kernel for Persistence Diagrams

Mathieu Carrière
Marco Cuturi
Steve Y. Oudot

Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe succinctly complex topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however, live in a space naturally endowed with a Hilbert structure and are usually compared with specific distances, such as the bottleneck distance. To incorporate PDs in a learning pipeline, several kernels have been proposed for PDs with a strong emphasis on the stability of the RKHS distance w. r. t. perturbations of the PDs. In this article, we use the Sliced Wasserstein approximation of the Wasserstein distance to define a new kernel for PDs, which is not only provably stable but also provably discriminative w. r. t. the Wasserstein distance $W^1_\infty$ between PDs. We also demonstrate its practicality, by developing an approximation technique to reduce kernel computation time, and show that our proposal compares favorably to existing kernels for PDs on several benchmarks.

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