arXiv subjects

David Cohen-Steiner

Publications and source records attributed to David Cohen-Steiner.

9 recordsLinked to original sources

arXiv2026

Persistent Intrinsic Volumes

We develop a new method to estimate the area, and more generally the intrinsic volumes, of a compact subset $X$ of $\mathbb{R}^d$ 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.

math.MG↗

arXiv2026

Continuity of the normal cycle with respect to $C^0$ topologies

We show that the generalized curvatures and more specifically the normal cycle of a compact subset of $\mathbb{R}^d$ are continuous under weak notions of convergence, assuming an a priori mass bound. In particular, provided its mass remains bounded, the normal cycle behaves continuously when the subset is perturbed by a homeomorphism that is $C^0$-close to the identity. Our approach relies on a combination of persistent homology with the geometric measure theory framework classically used in the study of curvatures of singular sets. As an application, we prove that every compact definable set in an o-minimal structure admits a normal cycle by showing that any such set is a limit, in the above sense, of a family of smooth sets whose normal cycles have uniformly bounded mass. We also show that WDC sets are limits of nested smooth sets with uniformly bounded normal cycle masses.

math.MG↗

arXiv2024

Wasserstein convergence of Čech persistence diagrams for samplings of submanifolds

Čech Persistence diagrams (PDs) are topological descriptors routinely used to capture the geometry of complex datasets. They are commonly compared using the Wasserstein distances $OT_{p}$; however, the extent to which PDs are stable with respect to these metrics remains poorly understood. We partially close this gap by focusing on the case where datasets are sampled on an $m$-dimensional submanifold of $\mathbb{R}^{d}$. Under this manifold hypothesis, we show that convergence with respect to the $OT_{p}$ metric happens exactly when $p\gt m$. We also provide improvements upon the bottleneck stability theorem in this case and prove new laws of large numbers for the total $α$-persistence of PDs. Finally, we show how these theoretical findings shed new light on the behavior of the feature maps on the space of PDs that are used in ML-oriented applications of Topological Data Analysis.

cs.CG↗

arXiv2024

Critical points of the distance function to a generic submanifold

In general, the critical points of the distance function $d_{\mathsf{M}}$ to a compact submanifold $\mathsf{M} \subset \mathbb{R}^D$ can be poorly behaved. In this article, we show that this is generically not the case by listing regularity conditions on the critical and $μ$-critical points of a submanifold and by proving that they are generically satisfied and stable with respect to small $C^2$ perturbations. More specifically, for any compact abstract manifold $M$, the set of embeddings $i:M\rightarrow \mathbb{R}^D$ such that the submanifold $i(M)$ satisfies those conditions is open and dense in the Whitney $C^2$-topology. When those regularity conditions are fulfilled, we prove that the distance function to $i(M)$ satisfies Morse-like conditions and that the critical points of the distance function to an $\varepsilon$-dense subset of the submanifold (e.g., obtained via some sampling process) are well-behaved. We also provide many examples that showcase how the absence of these conditions allows for pathological situations.

math.DG↗

arXiv2020

Spectral Properties of Radial Kernels and Clustering in High Dimensions

In this paper, we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in $\mathbb{R}^n$. Our approach focuses on high dimensions and relies solely on the concentration properties of the components in the mixture. We give several results describing of the structure of kernel matrices for a sample drawn from such a mixture. Based on these results, we analyze the ability of kernel PCA to cluster high dimensional mixtures. In particular, we exhibit a specific kernel leading to a simple spectral algorithm for clustering mixtures with possibly common means but different covariance matrices. We show that the minimum angular separation between the covariance matrices that is required for the algorithm to succeed tends to $0$ as $n$ goes to infinity.

stat.ML↗

arXiv2017

Approximating the Spectrum of a Graph

The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of approximating the spectrum $λ= (λ_1,\dots,λ_{|V|})$, $0 \le λ_1,\le \dots, \le λ_{|V|}\le 2$ of $G$ in the regime where the graph is too large to explicitly calculate the spectrum. We present a sublinear time algorithm that, given the ability to query a random node in the graph and select a random neighbor of a given node, computes a succinct representation of an approximation $\widetilde λ= (\widetilde λ_1,\dots,\widetilde λ_{|V|})$, $0 \le \widetilde λ_1,\le \dots, \le \widetilde λ_{|V|}\le 2$ such that $\|\widetilde λ- λ\|_1 \le ε|V|$. Our algorithm has query complexity and running time $exp(O(1/ε))$, independent of the size of the graph, $|V|$. We demonstrate the practical viability of our algorithm on 15 different real-world graphs from the Stanford Large Network Dataset Collection, including social networks, academic collaboration graphs, and road networks. For the smallest of these graphs, we are able to validate the accuracy of our algorithm by explicitly calculating the true spectrum; for the larger graphs, such a calculation is computationally prohibitive. In addition we study the implications of our algorithm to property testing in the bounded degree graph model.

cs.DS↗

arXiv2016

A transfer principle and applications to eigenvalue estimates for graphs

In this paper, we prove a variant of the Burger-Brooks transfer principle which, combined with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a function of their genus. More precisely, we show the existence of a universal constants $C$ such that the $k$-th eigenvalue $λ_k^{nr}$ of the normalized Laplacian of a graph $G$ of (geometric) genus $g$ on $n$ vertices satisfies $$λ_k^{nr}(G) \leq C \frac{d_{\max}(g+k)}{n},$$ where $d_{\max}$ denotes the maximum valence of vertices of the graph. This result is tight up to a change in the value of the constant $C$, and improves recent results of Kelner, Lee, Price and Teng on bounded genus graphs. To show that the transfer theorem might be of independent interest, we relate eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple graph models, and discuss an application to the mesh partitioning problem, extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang to arbitrary meshes.

math.MG↗

arXiv2008

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive $μ$-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive $μ$-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample. These results can also be interpreted as a framework for an effective and robust notion of curvature.

cs.CG↗

arXiv2007

Stability of boundary measures

We introduce the boundary measure at scale r of a compact subset of the n-dimensional Euclidean space. We show how it can be computed for point clouds and suggest these measures can be used for feature detection. The main contribution of this work is the proof a quantitative stability theorem for boundary measures using tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer's curvature measures of a compact, allowing to compute them from point-cloud approximations of the compact.

cs.CG↗