arXiv subjects

Steve Oudot

Publications and source records attributed to Steve Oudot.

At least 19 recordsLinked to original sources

arXiv2026

IdEst: Assessing Self-Supervised Learning Representations via Intrinsic Dimension

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning meaningful representations from unlabeled data. However, the standard protocol for evaluating these representations, linear probing, is computationally expensive, sensitive to hyperparameters, and provides limited insight into the geometric structure of the representation space. In this work, motivated by connections between neural network generalization and intrinsic dimension (ID) we propose IdEst, a method for estimating the ID of SSL representations via the Minimum Spanning Tree dimension estimator ($\mathrm{dim}_\mathrm{MST}$). Across diverse datasets, architectures, and SSL pretraining objectives, we show that IdEst strongly correlates with downstream linear probe performances. Furthermore, we demonstrate that IdEst enables efficient hyperparameter selection, significantly reducing the computational cost compared to supervised alternatives. Our results highlight intrinsic dimensionality as a principled geometric proxy for assessing SSL representations, complementing standard supervised probing protocols.

cs.LG↗

arXiv2026

Function-Rips complexes in persistent homotopy theory: Stability and persistent Latschev theorems

Classical results of Hausmann and Latschev show that Vietoris-Rips complexes can recover the homotopy type of a manifold, even from finite metric spaces that are nearby in Gromov-Hausdorff distance. We prove persistent homotopical versions of these theorems for metric spaces equipped with filtration functions. The central object of study is the so-called persistent homotopy type of the function-Rips complex, a filtered simplicial complex that combines a fixed Rips scale with the filtration data on the underlying space. Using techniques from CAT($\kappa$)-geometry and persistent simplicial homotopy theory, we generalize Latschev's and Hausmann's theorems to the setting of spaces with filtration functions and homotopical interleavings. A fundamental ingredient is a new homotopical stability theorem. The fixed-scale function-Rips construction is known not to be globally stable with respect to function Gromov-Hausdorff distance and homotopical interleaving distance. Here, we show that it is nevertheless stable for appropriate choices of the Rips parameter at such pairs $(M,f)$ for which $M$ is a complete metric space of curvature bounded above, and $f$ is a Lipschitz continuous multivariate function.

math.AT↗

arXiv2025

T-REGS: Minimum Spanning Tree Regularization for Self-Supervised Learning

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data, often by enforcing invariance to input transformations such as rotations or blurring. Recent studies have highlighted two pivotal properties for effective representations: (i) avoiding dimensional collapse-where the learned features occupy only a low-dimensional subspace, and (ii) enhancing uniformity of the induced distribution. In this work, we introduce T-REGS, a simple regularization framework for SSL based on the length of the Minimum Spanning Tree (MST) over the learned representation. We provide theoretical analysis demonstrating that T-REGS simultaneously mitigates dimensional collapse and promotes distribution uniformity on arbitrary compact Riemannian manifolds. Several experiments on synthetic data and on classical SSL benchmarks validate the effectiveness of our approach at enhancing representation quality.

cs.LG↗

arXiv2025

Counts and end-curves in two-parameter persistence

Given a finite dimensional, bigraded module over the polynomial ring in two variables, we define its two-parameter count, a natural number, and its end-curves, a set of plane curves. These are two-dimensional analogues of the notions of bar-count and endpoints of singly-graded modules over the polynomial ring in one variable, from persistence theory. We show that our count is the unique one satisfying certain natural conditions; as a consequence, several inclusion-exclusion formulas in two-parameter persistence yield the same positive number, which equals our count, and which in turn equals the number of end-curves, giving geometric meaning to this count. We show that the end-curves determine the classical Betti tables by showing that they interpolate between generators, relations, and syzygies. Using the band representations of a certain string algebra, we show that the set of end-curves admits a canonical partition, where each part forms a closed curve on the plane; we call this the boundary of the module. As an invariant, the boundary is neither weaker nor stronger than the rank invariant, but, in contrast to the rank invariant, it is a complete invariant on the set of spread-decomposable representations. Our results connect several lines of work in multiparameter persistence, and their extension to modules over the real-exponent polynomial ring in two variables relates to two-dimensional Morse theory.

math.RT↗

arXiv2025

D-GRIL: End-to-End Topological Learning with 2-parameter Persistence

End-to-end topological learning using 1-parameter persistence is well-known. We show that the framework can be enhanced using 2-parameter persistence by adopting a recently introduced 2-parameter persistence based vectorization technique called GRIL. We establish a theoretical foundation of differentiating GRIL producing D-GRIL. We show that D-GRIL can be used to learn a bifiltration function on standard benchmark graph datasets. Further, we exhibit that this framework can be applied in the context of bio-activity prediction in drug discovery.

cs.LG↗

arXiv2024

Local characterization of block-decomposability for multiparameter persistence modules

Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a prominent role in levelset persistence. Here we generalize the local condition for decomposability into block modules to the n-parameter setting, and prove a corresponding structure theorem. Our result holds in the generality of pointwise finite-dimensional modules over finite products of arbitrary totally ordered sets. Our proof extends the one by Botnan and Crawley-Boevey from 2 to n parameters, which requires some crucial adaptations at places where their proof is fundamentally tied to the 2-parameter setting.

math.RT↗

arXiv2024

Estimating the persistent homology of $\mathbb{R}^n$-valued functions using function-geometric multifiltrations

Given an unknown $\mathbb{R}^n$-valued function $f$ on a metric space $X$, can we approximate the persistent homology of $f$ from a finite sampling of $X$ with known pairwise distances and function values? This question has been answered in the case $n=1$, assuming $f$ is Lipschitz continuous and $X$ is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary $n$, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach. From a practical standpoint, we show that our approximation results are robust to input noise, and that function-geometric multifiltrations have good statistical convergence properties. We also provide an algorithm to compute our estimators, and we use its implementation to conduct extensive experiments, on both synthetic and real biological data, in order to validate our theoretical results and to assess the practicality of our approach.

math.AT↗

arXiv2024

Differential Calculus and Optimization in Persistence Module Categories

Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other areas of mathematics, including symplectic topology, complex analysis, and topological data analysis, where they arise from filtrations of topological spaces by the sublevel sets of real-valued functions. Two fundamental properties of persistence modules make them useful in such contexts: (1) the fact that they are stable under perturbations of the originating functions, and (2) the fact that they can be approximated, in the sense of relative homological algebra, by classes of indecomposable modules with an elementary structure. In this text we give an introduction to the theory of persistence modules, then we explain how the above properties can be leveraged to build a framework for differential calculus and optimization with convergence guarantees in persistence module categories.

math.AT↗

arXiv2024

Differentiability and Optimization of Multiparameter Persistent Homology

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.

cs.CG↗

arXiv2024

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

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

math.AT↗

arXiv2024

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

In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode decomposes the rank invariant as a $\Z$-linear combination of rank invariants of indicator modules supported on segments in the poset. We develop the theory behind these decompositions, both for the usual rank invariant and for its generalizations, showing under what conditions they exist and are unique. We also show that, like its unsigned counterpart, the signed barcode reflects in part the algebraic structure of the module: specifically, it derives from the terms in the minimal rank-exact resolution of the module, i.e., its minimal projective resolution relative to the class of short exact sequences on which the rank invariant is additive. To complete the picture, we show some experimental results that illustrate the contribution of the signed barcode in the exploration of multi-parameter persistence modules.

math.AT↗

arXiv2024

Computation of $γ$-linear projected barcodes for multiparameter persistence

The $γ$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along linear forms in the polar of some fixed cone $γ$. So far, the computation of the $γ$-linear projected barcode has only been studied in the functional setting, in which persistence modules come from the persistent cohomology of $\mathbb{R}^n$-valued functions. Here we develop a method that works in the algebraic setting directly, for any multiparameter persistence module over $\mathbb{R}^n$ that is given via a finite free resolution. Our approach is similar to that of RIVET: first, it pre-processes the resolution to build an arrangement in the dual of $\mathbb{R}^n$ and a barcode template in each face of the arrangement; second, given any query linear form $u$ in the polar of $γ$, it locates $u$ within the arrangement to produce the corresponding barcode efficiently. While our theoretical complexity bounds are similar to the ones of RIVET, our arrangement turns out to be simpler thanks to the linear structure of the space of linear forms. Our theoretical analysis combines sheaf-theoretic and module-theoretic techniques, showing that multiparameter persistence modules can be converted into a special type of complexes of sheaves on vector spaces called conic-complexes, whose derived pushforwards by linear forms have predictable barcodes.

math.AT↗

arXiv2024

ADAPT: Multimodal Learning for Detecting Physiological Changes under Missing Modalities

Multimodality has recently gained attention in the medical domain, where imaging or video modalities may be integrated with biomedical signals or health records. Yet, two challenges remain: balancing the contributions of modalities, especially in cases with a limited amount of data available, and tackling missing modalities. To address both issues, in this paper, we introduce the AnchoreD multimodAl Physiological Transformer (ADAPT), a multimodal, scalable framework with two key components: (i) aligning all modalities in the space of the strongest, richest modality (called anchor) to learn a joint embedding space, and (ii) a Masked Multimodal Transformer, leveraging both inter- and intra-modality correlations while handling missing modalities. We focus on detecting physiological changes in two real-life scenarios: stress in individuals induced by specific triggers and fighter pilots' loss of consciousness induced by $g$-forces. We validate the generalizability of ADAPT through extensive experiments on two datasets for these tasks, where we set the new state of the art while demonstrating its robustness across various modality scenarios and its high potential for real-life applications.

cs.CV↗

arXiv2024

Efficient computation of topological integral transforms

Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric information on weighted polytopal complexes. While some implementations exist, they only enable discretized representations of the transforms, and they do not handle weighted complexes (such as for instance images). Moreover, recent hybrid transforms lack an implementation. In this paper, we introduce Eucalc, a novel implementation of three topological integral transforms -- the Euler characteristic transform, the Radon transform, and hybrid transforms -- for weighted cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the algorithms significantly reduce computational complexity by focusing on critical points. Our software provides exact representations of transforms, handles both binary and grayscale images, and supports multi-core processing. It is publicly available as a C++ library with a Python wrapper. We present mathematical foundations, implementation details, and experimental evaluations, demonstrating Eucalc's efficiency.

cs.CG↗

arXiv2024

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures

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

cs.LG↗

arXiv2024

On the stability of multigraded Betti numbers and Hilbert functions

Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free modules in its minimal free resolution -- as well as in practice -- it is easy to visualize and it is one of the main outputs of current multiparameter persistent homology software, such as RIVET. However, to the best of our knowledge, no bottleneck stability result with respect to the interleaving distance has been established for this invariant so far, and this potential lack of stability limits its practical applications. We prove a stability result for multigraded Betti numbers, using an efficiently computable bottleneck-type dissimilarity function we introduce. Our notion of matching is inspired by recent work on signed barcodes, and allows matching bars of the same module in homological degrees of different parity, in addition to matchings bars of different modules in homological degrees of the same parity. Our stability result is a combination of Hilbert's syzygy theorem, Bjerkevik's bottleneck stability for free modules, and a novel stability result for projective resolutions. We also prove, in the $2$-parameter case, a $1$-Wasserstein stability result for Hilbert functions with respect to the $1$-presentation distance of Bjerkevik and Lesnick.

math.AT↗

arXiv2022

Local characterizations for decomposability of 2-parameter persistence modules

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

math.RT↗

arXiv2022

On rectangle-decomposable 2-parameter persistence modules

This paper addresses two questions: (a) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (b) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: on the one hand, a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; on the other hand, algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local characterization is key to the efficiency of our algorithms, and it generalizes previous conditions derived for the smaller class of block-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. By contrast, we show that general interval-decomposability does not admit such a local characterization, even when locality is understood in a broad sense. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work.

math.AT↗