arXiv subjects

Antoine Commaret

Publications and source records attributed to Antoine Commaret.

4 recordsLinked to original sources

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↗

arXiv2025

Modeling high dimensional point clouds with the spherical cluster model

A parametric cluster model is a statistical model providing geometric insights onto the points defining a cluster. The {\em spherical cluster model} (SC) approximates a finite point set $P\subset \mathbb{R}^d$ by a sphere $S(c,r)$ as follows. Taking $r$ as a fraction $\eta\in(0,1)$ (hyper-parameter) of the std deviation of distances between the center $c$ and the data points, the cost of the SC model is the sum over all data points lying outside the sphere $S$ of their power distance with respect to $S$. The center $c$ of the SC model is the point minimizing this cost. Note that $\eta=0$ yields the celebrated center of mass used in KMeans clustering. We make three contributions. First, we show fitting a spherical cluster yields a strictly convex but not smooth combinatorial optimization problem. Second, we present an exact solver using the Clarke gradient on a suitable stratified cell complex defined from an arrangement of hyper-spheres. Finally, we present experiments on a variety of datasets ranging in dimension from $d=9$ to $d=10,000$, with two main observations. First, the exact algorithm is orders of magnitude faster than BFGS based heuristics for datasets of small/intermediate dimension and small values of $\eta$, and for high dimensional datasets (say $d>100$) whatever the value of $\eta$. Second, the center of the SC model behave as a parameterized high-dimensional median. The SC model is of direct interest for high dimensional multivariate data analysis, and the application to the design of mixtures of SC will be reported in a companion paper.

stat.ME↗

arXiv2024

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↗

arXiv2024

Generalized Morse theory for tubular neighborhoods

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.

math.GT↗