Related papers: $\ell^p$-Distances on Multiparameter Persistence M…

Machine learning image classifiers are susceptible to adversarial and corruption perturbations. Adding imperceptible noise to images can lead to severe misclassifications of the machine learning model. Using $L_p$-norms for measuring the…

We consider metrics which are preserved under a $p$-Wasserstein transport map, up to a possible contraction. In the case $p=1$ this corresponds to a metric which is uniformly curved in the sense of coarse Ricci curvature. We investigate the…

A limit theorem for the largest interpoint distance of $p$ independent and identically distributed points in $\mathbb{R}^n$ to the Gumbel distribution is proved, where the number of points $p=p_n$ tends to infinity as the dimension of the…

Techniques from metric geometry have become fundamental tools in modern mathematical data science, providing principled methods for comparing datasets modeled as finite metric spaces. Two of the central tools in this area are the…

The central limit theorem is one of the most fundamental results in probability and has been successfully extended to locally dependent data and strongly-mixing random fields. In this paper, we establish its rate of convergence for…

We study the Wasserstein distance of order 1 between the empirical distribution and the marginal distribution of stationary $\alpha$-dependent sequences. We prove some moments inequalities of order p for any p $\ge$ 1, and we give some…

In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance $W_2$ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance…

Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…

We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than $d$. The achievability argument involves an iterative construction of a set of radius-$d$ balls, each centered at a…

We study $p-$Wasserstein spaces $ \mathcal{W}_p(\mathbb{R}^n, d_N)$ over $\mathbb{R}^n$ equipped with a norm metric $d_N$. We show that, if the norm is smooth enough, then the Wasserstein space is isometrically rigid whenever $p \neq 2$. We…

We provide a definition of ephemeral multi-persistent modules and prove that the quotient of persistent modules by the ephemeral ones is equivalent to the category of $\gamma$-sheaves. In the case of one-dimensional persistence, our…

The analysis of various models in statistical physics relies on the existence of decompositions of measures into mixtures of product-like components, where the goal is to attain a decomposition into measures whose entropy is close to that…

We establish stability estimates for the $k$-plane transform on positive Radon measures, with particular emphasis on Fourier and Wasserstein metrics. We first introduce a metric on $k$-plane data and prove a bi-Lipschitz stability estimate…

We introduce the {\em $\mu$-topological stability}. This is a type of stability depending on the measure $\mu$ different from the set-valued approach \cite{lm}. We prove that the map $f$ is $m_p$-topologically stable if and only if $p$ is a…

In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, we prove: the m.m.e. is Bernoulli, and the periodic points with Lyapunov exponents bounded away from…

We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence…

We establish the well-posedness of stationary solutions for a class of SPDEs with locally monotone coefficients, and prove the Freidlin--Wentzell large deviation principle (LDP) for these stationary solutions. The LDP for the associated…

We study the discretization of generalized Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of…

The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space.…

We show that the observable category of q-tame multiparameter persistence modules satisfies good metric and algebraic properties: it forms a complete metric space with respect to the interleaving distance, and it is Krull--Schmidt in the…