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

The paper studies the robustness properties of discrete-time stochastic optimal control under Wasserstein model approximation for both discounted-cost and average-cost criteria. Specifically, we study the performance loss when applying an…

We study the average $p-$Wasserstein distance between a finite sample of an infinite hyperuniform point process on $\mathbb{R}^2$ and its mean for any $p\geq 1$. The average Wasserstein transport cost is shown to be bounded from above and…

We prove a number of results relating the concepts of Keisler measures, generic stability, randomizations, and NIP formulas. Among other things, we do the following: (1) We introduce the notion of a Keisler-Morley measure, which plays the…

Our purpose in this paper is to study isometries and isometric embeddings of the $p$-Wasserstein space $\mathcal{W}_p(\mathbb{H}^n)$ over the Heisenberg group $\mathbb{H}^n$ for all $p>1$ and for all $n\geq 1$. First, we create a link…

We establish upper bounds for the expected $p$-th power of the Gaussian-smoothed $p$-Wasserstein distance between a probability measure $\mu$ and the corresponding empirical measure $\mu_N$, whenever $\mu$ has finite $q$-th moment for some…

We study universal aspects of polymer conformations and transverse fluctuations for a single swollen chain characterized by a contour length $L$ and a persistence length $\ell_p$ in two dimensions (2D) and in three dimensions (3D) in the…

The concept of edit distance, which dates back to the 1960s in the context of comparing word strings, has since found numerous applications with various adaptations in computer science, computational biology, and applied topology. By…

In this paper we study the isometric rigidity of certain classes of metric spaces with respect to the $p$-Wasserstein space. We prove that spaces that split a separable Hilbert space are not isometrically rigid with respect to…

We consider the problem of learning the interaction strength between the nodes of a network based on dependent binary observations residing on these nodes, generated from a Markov Random Field (MRF). Since these observations can possibly be…

In this note, we propose an extension of the Wasserstein 1-metric ($W_1$) for matrix probability densities, matrix-valued density measures, and an unbalanced interpretation of mass transport. The key is using duality theory, in particular,…

Stability and robustness are critical for deploying Transformers in safety-sensitive settings. A principled way to enforce such behavior is to constrain the model's Lipschitz constant. However, approximation-theoretic guarantees for…

Distance measures play an important role in shape classification and data analysis problems. Topological distances based on Reeb graphs and persistence diagrams have been employed to obtain effective algorithms in shape matching and scalar…

We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…

Let $V$ be an inner product space, and $x, y \in V$; the conjecture is made that, for any $p \in [1, \infty]$, the function $d_p(x, y):=\|x-y\|/(\|x\|^p+ \|y\|^p)^{1/p}$ is a distance on $V$.

We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $\mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic…

We propose a methodology for intercomparing climate models and evaluating their performance against benchmarks based on the use of the Wasserstein distance (WD). This distance provides a rigorous way to measure quantitatively the difference…

In this paper, we prove that the Euclidean distance between two independent random vectors uniformly distributed on $l_p^n$-balls $(1 \leq p \leq \infty)$ or on its boundary satisfies a central limit theorem as $n$ tends to $\infty$. Also,…

We consider structured approximation of measures in Wasserstein space $\mathrm{W}_p(\mathbb{R}^d)$ for $p\in[1,\infty)$ using general measure approximants compactly supported on Voronoi regions derived from a scaled Voronoi partition of…

The maximum mean discrepancy and Wasserstein distance are popular distance measures between distributions and play important roles in many machine learning problems such as metric learning, generative modeling, domain adaption, and…

Persistence diagrams do not admit an inner product structure compatible with any Wasserstein metric. Hence, when applying kernel methods to persistence diagrams, the underlying feature map necessarily causes distortion. We prove persistence…