Related papers: High order tensor moments of random vectors
Observables in random tensor theory are polynomials in the entries of a tensor of rank $d$ which are invariant under $U(N)^d$. It is notoriously difficult to evaluate the expectations of such polynomials, even in the Gaussian distribution.…
The factor analysis model is a statistical model where a certain number of hidden random variables, called factors, affect linearly the behaviour of another set of observed random variables, with additional random noise. The main assumption…
As in random matrix theories, eigenvector/value distributions are important quantities of random tensors in their applications. Recently, real eigenvector/value distributions of Gaussian random tensors have been explicitly computed by…
The main purpose of this paper is to introduce the random tensor with normal distribution, which promotes the matrix normal distribution to a higher order case. Some basic knowledge on tensors are introduced before we focus on the random…
The factorial moments of the standard Poisson distribution are well known and are simple, but the raw moments are considered to be more complicated (Touchard polynomials). The present note presents a recurrence relation and an explicit…
In this paper, explicit error bounds are derived in the approximation of rank $k$ projections of certain $n$-dimensional random vectors by standard $k$-dimensional Gaussian random vectors. The bounds are given in terms of $k$, $n$, and a…
The points of a moment variety are the vectors of all moments up to some order of a family of probability distributions. We study this variety for mixtures of Gaussians. Following up on Pearson's classical work from 1894, we apply current…
We study the problem of estimating the mean of a random vector $X$ given a sample of $N$ independent, identically distributed points. We introduce a new estimator that achieves a purely sub-Gaussian performance under the only condition that…
We provide moment bounds for expressions of the type $(X^{(1)} \otimes \dots \otimes X^{(d)})^T A (X^{(1)} \otimes \dots \otimes X^{(d)})$ where $\otimes$ denotes the Kronecker product and $X^{(1)}, \dots, X^{(d)}$ are random vectors with…
The article derives multivariate Generalized Gram-Charlier (GGC) series that expands an unknown joint probability density function (\textit{pdf}) of a random vector in terms of the differentiations of the joint \textit{pdf} of a reference…
The projected normal distribution, also known as the angular Gaussian distribution, is obtained by dividing a multivariate normal random variable $\mathbf{x}$ by its norm $\sqrt{\mathbf{x}^T \mathbf{x}}$. The resulting random variable…
Given a subset $A$ of $\mathbb{R}^n$, we define \begin{align*} \mathrm{conv}_k(A) := \left\{ \lambda_1 s_1 + \cdots + \lambda_k s_k : \lambda_i \in [0,1], \sum_{i=1}^k \lambda_i = 1 , s_i \in A \right\} \end{align*} to be the set of vectors…
Gaussian mixture models (GMMs) are fundamental tools in statistical and data sciences. We study the moments of multivariate Gaussians and GMMs. The $d$-th moment of an $n$-dimensional random variable is a symmetric $d$-way tensor of size…
The sub-Gaussian stable distribution is a heavy-tailed elliptically contoured law which has interesting applications in signal processing and financial mathematics. This work addresses the problem of feasible estimation of distributions. We…
The factorial moments of the standard Poisson distribution are well known. The present note presents an explicit combinatorial sum for the factorial moments of the Poisson distribution of order $k$. Unlike the standard Poisson distribution…
Random tensors are the natural generalization of random matrices to higher order objects. They provide generating functions for random geometries and, assuming some familiarity with random matrix theory and quantum field theory, we discuss…
We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some…
In this article, we define the matricization of a tensor and we present some properties of the matricization. After that, we define the determinant of a tensor and we present some properties of the determinant. We define the covariance…
Relying on random matrix theory (RMT), this paper studies asymmetric order-$d$ spiked tensor models with Gaussian noise. Using the variational definition of the singular vectors and values of (Lim, 2005), we show that the analysis of the…
Computing the distribution of permanents of random matrices has been an outstanding open problem for several decades. In quantum computing, "anti-concentration" of this distribution is an unproven input for the proof of hardness of the task…