Related papers: The deepest point for distributions in infinite di…
Collections of probability distributions arise in a variety of applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions can be defined over diverse domain types including finite…
For a given metric measure space $(X,d,\mu)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS)…
A novel, non-trivial, probabilistic upper bound on the entropy of an unknown one-dimensional distribution, given the support of the distribution and a sample from that distribution, is presented. No knowledge beyond the support of the…
Intrinsic dimensionality (ID) is one of the most fundamental characteristics of multi-dimensional data point clouds. Knowing ID is crucial to choose the appropriate machine learning approach as well as to understand its behavior and…
The problem of distributed function computation is studied, where functions to be computed is not necessarily symbol-wise. A new method to derive a converse bound for distributed computing is proposed; from the structure of functions to be…
Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain…
Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to clustering, exponential families, parameter estimation and…
Robust estimation under Huber's $\epsilon$-contamination model has become an important topic in statistics and theoretical computer science. Statistically optimal procedures such as Tukey's median and other estimators based on depth…
We give examples of different multivariate probability distributions whose halfspace depths coincide at all points of the sample space.
Tukey's halfspace depth can be seen as a stochastic program and as such it is not guarded against optimizer's curse, so that a limited training sample may easily result in a poor out-of-sample performance. We propose a generalized halfspace…
While efficient distribution learning is no doubt behind the groundbreaking success of diffusion modeling, its theoretical guarantees are quite limited. In this paper, we provide the first rigorous analysis on approximation and…
Inference of the marginal probability distribution is defined as the calculation of the probability of a subset of the variables and is relevant for handling missing data and hidden variables. While inference of the marginal probability…
Quantization for probability distributions concerns the best approximation of a $d$-dimensional probability distribution $P$ by a discrete probability with a given number $n$ of supporting points. In this paper, we have considered a…
Understanding the tail behavior of distributions is crucial in statistical theory. For instance, the tail of a distribution plays a ubiquitous role in extreme value statistics, where it is associated with the likelihood of extreme events.…
We introduce a novel projection depth for data lying in a general Hilbert space, called the regularized projection depth, with a focus on functional data. By regularizing projection directions, the proposed depth does not suffer from the…
Notions of depth in regression have been introduced and studied in the literature. The most famous example is Regression Depth (RD), which is a direct extension of location depth to regression. The projection regression depth (PRD) is the…
Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant functions on unstructured meshes satisfying an orthogonality…
This paper studies the universal approximation property of deep neural networks for representing probability distributions. Given a target distribution $\pi$ and a source distribution $p_z$ both defined on $\mathbb{R}^d$, we prove under…
The halfspace depth of a $d$-dimensional point $x$ with respect to a finite (or probability) Borel measure $\mu$ in $\mathbb{R}^d$ is defined as the infimum of the $\mu$-masses of all closed halfspaces containing $x$. A natural question is…
We present a novel approach to estimating discrete distributions with (potentially) infinite support in the total variation metric. In a departure from the established paradigm, we make no structural assumptions whatsoever on the sampling…