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Comparing metric measure spaces (i.e. a metric space endowed with aprobability distribution) is at the heart of many machine learning problems. The most popular distance between such metric measure spaces is theGromov-Wasserstein (GW)…

We present a metric-space approach to quantify the performance of density-functional approximations for interacting many-body systems and to explore the validity of the Hohenberg-Kohn-type theorem on fermionic lattices. This theorem…

Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance…

The empirical distribution function assigns mass $1/n$ to each of the $n$ observations in a sample. As these are highly variable, estimation error may be reduced by replacing them with estimated observations that are asymptotically less…

Data sets of multivariate normal distributions abound in many scientific areas like diffusion tensor imaging, structure tensor computer vision, radar signal processing, machine learning, just to name a few. In order to process those normal…

We revisit extending the Kolmogorov-Smirnov distance between probability distributions to the multidimensional setting and make new arguments about the proper way to approach this generalization. Our proposed formulation maximizes the…

The object of study in this paper is the expected $2$-Wasserstein distance between the empirical measures of several point processes and their respective limit. For this, the main tool developed is a smoothing procedure in Euclidean spaces…

We seek an analog for the quantum permutation group $S_n^+$ of the normalized Hamming distance for permutations. We define three distances on the tracial state space of $C(S_n^+)$ that generalize the $L^1$-Wasserstein distance of…

The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. They are particularly useful in areas such as…

Finite mixture models have long been used across a variety of fields in engineering and sciences. Recently there has been a great deal of interest in quantifying the convergence behavior of the \emph{mixing measure}, a fundamental object…

We investigate whether neighbor-density-weighted marked correlation functions (MCFs) can extract cosmological information beyond the standard redshift-space two-point correlation function (2PCF). Using the Kun suite of 129 $w_0w_a$CDM$+\sum…

Motivated by the statistical and computational challenges of computing Wasserstein distances in high-dimensional contexts, machine learning researchers have defined modified Wasserstein distances based on computing distances between…

We present measurements of the neutral atomic hydrogen (HI) mass function (HIMF) and cosmic HI density ($\Omega_{\rm HI}$) at $0 \leq z \leq 0.088$ from the Looking at the Distant Universe with MeerKAT Array (LADUMA) survey. Using LADUMA…

Statistical distances (SDs), which quantify the dissimilarity between probability distributions, are central to machine learning and statistics. A modern method for estimating such distances from data relies on parametrizing a variational…

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…

Despite the high accuracy of photometric redshifts (zphot) derived using Machine Learning (ML) methods, the quantification of errors through reliable and accurate Probability Density Functions (PDFs) is still an open problem. First, because…

This article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference…

We derive quantitative bounds on the rate of convergence in $L^1$ Wasserstein distance of general M-estimators, with an almost sharp (up to a logarithmic term) behavior in the number of observations. We focus on situations where the…

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order…

We present a simple method to approximate Rao's distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating Rao's distances between successive nearby normal distributions on…