Related papers: High order tensor moments of random vectors
A tensor is a multidimensional array of numbers that can be used to store data, encode a computational relation and represent quantum entanglement. In this sense a tensor can be viewed as valuable resource whose transformation can lead to…
Based on the canonical correlation analysis we derive series representations of the probability density function (PDF) and the cumulative distribution function (CDF) of the information density of arbitrary Gaussian random vectors as well as…
To address the common problem of high dimensionality in tensor regressions, we introduce a generalized tensor random projection method that embeds high-dimensional tensor-valued covariates into low-dimensional subspaces with minimal loss of…
Dimensionality reduction for high-order tensors is a challenging problem. In conventional approaches, higher order tensors are `vectorized` via Tucker decomposition to obtain lower order tensors. This will destroy the inherent high-order…
Motivated by the many potential applications of low-rank multi-way tensor approximations, we set out to count the rank-one tensors that are critical points of the distance function to a general tensor v. As this count depends on v, we…
The general linear model is a universally accepted method to conduct and test multiple linear regression models. Using this model one has the ability to simultaneously regress covariates among different groups of data. Moreover, there are…
Suppose X is a random vector, that is distributed uniformly in some n-dimensional convex set. It was conjectured that when the dimension n is very large, there exists a non-zero vector u, such that the distribution of the real random…
We consider the following natural question. Given a matrix $A$ with i.i.d. random entries, what are the moments of the determinant of $A$? In other words, what is $\mathbb{E}[\det(A)^k]$? While there is a general expression for…
Granville and Soundararajan showed that the $k$th moment in the Erd\H{o}s--Kac theorem is equal to the $k$th moment of the standard Gaussian distribution in the range $k=o((\log \log x)^{1/3})$, up to a negligible error term. We show that…
The spectral form factor of random matrix theory plays a key role in the description of disordered and chaotic quantum systems. While its moments are known to be approximately Gaussian, corrections subleading in the matrix dimension, $D$,…
Current work in lexical distributed representations maps each word to a point vector in low-dimensional space. Mapping instead to a density provides many interesting advantages, including better capturing uncertainty about a representation…
We describe an elementary method to get non-asymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. As the basic example, we consider the GUE matrices. Immediate…
The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential…
In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.
In this paper we show that the closure of a random sample for a k-dimensional random vector is almost surely a deterministic set of all heavy points of the distribution. A heavy point is defined to be a point for which all its neighborhoods…
The problem of estimating a complex measure made up by a linear combination of Dirac distributions centered on points of the complex plane from a finite number of its complex moments affected by additive i.i.d. Gaussian noise is considered.…
We propose an estimator for the mean of a random vector in $\mathbb{R}^d$ that can be computed in time $O(n^4+n^2d)$ for $n$ i.i.d.~samples and that has error bounds matching the sub-Gaussian case. The only assumptions we make about the…
Perturbation of the GUE are known in physics to be related to enumeration of graphs on surfaces. We investigate this idea and show that for a small convex perturbation, we can perform a genus expansion: the moments of the empirical measure…
The aim of this expository note is to prove that any $1$-subgaussian random vector is dominated in the convex ordering by a universal constant times a standard Gaussian vector. This strengthens Talagrand's celebrated subgaussian comparison…
Although quantum systems are generally described by quantum state vectors, we show that in certain cases their measurement processes can be reformulated as probabilistic equations expressed in terms of probabilistic state vectors. These…