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In this paper, we propose a uniformly dithered 1-bit quantization scheme for high-dimensional statistical estimation. The scheme contains truncation, dithering, and quantization as typical steps. As canonical examples, the quantization…
In recent decades, statisticians have been increasingly encountering spatial data that exhibit non-Gaussian behaviors such as asymmetry and heavy-tailedness. As a result, the assumptions of symmetry and fixed tail weight in Gaussian…
As well known, cumulant expansion is an alternative way to moment expansion to fully characterize probability distributions provided all the moments exist. If this is not the case, the so called escort mean values (or q-moments) have been…
We consider the problem of probabilistic quantification of dynamical systems that have heavy-tailed characteristics. These heavy-tailed features are associated with rare transient responses due to the occurrence of internal instabilities.…
Gaussian universality results assert that the properties of many estimators remain unchanged when the input data are replaced by Gaussians. Such results have gained popularity in high-dimensional statistics and machine learning, as…
We study polynomial time algorithms for estimating the mean of a heavy-tailed multivariate random vector. We assume only that the random vector $X$ has finite mean and covariance. In this setting, the radius of confidence intervals achieved…
In this study, a numerical quadrature for the generalized inverse Gaussian distribution is derived from the Gauss-Hermite quadrature by exploiting its relationship with the normal distribution. The proposed quadrature is not Gaussian, but…
By considering a nonlinear combination of the probabilities of a system, a physical interpretation of Tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed. The escort probability is…
Studying the dynamics of open quantum systems can enable breakthroughs both in fundamental physics and applications to quantum engineering and quantum computation. Since the density matrix $\rho$, which is the fundamental description for…
Given a finite family of functions, the goal of model selection aggregation is to construct a procedure that mimics the function from this family that is the closest to an unknown regression function. More precisely, we consider a general…
In this article, we propose a communication-efficient decentralized machine learning (ML) algorithm, coined quantized group ADMM (Q-GADMM). To reduce the number of communication links, every worker in Q-GADMM communicates only with two…
Analyzing neural network dynamics via stochastic gradient descent (SGD) is crucial to building theoretical foundations for deep learning. Previous work has analyzed structured inputs within the \textit{hidden manifold model}, often under…
Multivariate generalized Gamma convolutions are distributions defined by a convolutional semi-parametric structure. Their flexible dependence structures, the marginal possibilities and their useful convolutional expression make them…
Modern risk modelling approaches deal with vectors of multiple components. The components could be, for example, returns of financial instruments or losses within an insurance portfolio concerning different lines of business. One of the…
Score-based model research in the last few years has produced state of the art generative models by employing Gaussian denoising score-matching (DSM). However, the Gaussian noise assumption has several high-dimensional limitations,…
We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called 'normex' approach from a univariate to a multivariate framework. We…
The multivariate generalised Gaussian distribution (MGGD) is commonly used to model high-dimensional vectors with non-Gaussian radial behaviour, ranging from sharp-peaked to heavy-tailed profiles. However, because many classical…
The heavy-tailed mutation operator, proposed by Doerr, Le, Makhmara, and Nguyen (2017) for evolutionary algorithms, is based on the power-law assumption of mutation rate distribution. Here we generalize the power-law assumption using a…
We introduce a q-weighted version of the Robinson-Schensted (column insertion) algorithm which is closely connected to q-Whittaker functions (or Macdonald polynomials with t=0) and reduces to the usual Robinson-Schensted algorithm when q=0.…
Riemannian Gaussian distributions were initially introduced as basic building blocks for learning models which aim to capture the intrinsic structure of statistical populations of positive-definite matrices (here called covariance…