Related papers: When Random Tensors meet Random Matrices
We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let $v$ be the top eigenvector of an $N\times N$ sparse random symmetric matrix with an average of $d$ non-zero centered entries per row. We…
We consider the problem of estimating a rank-one nonsymmetric matrix under additive white Gaussian noise. The matrix to estimate can be written as the outer product of two vectors and we look at the special case in which both vectors are…
Random contractions (sub-unitary random matrices) appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. We analyze statistical properties of complex…
This work introduces an asymptotic study of Hotelling-type tensor deflation in the presence of noise, in the regime of large tensor dimensions. Specifically, we consider a low-rank asymmetric tensor model of the form $\sum_{i=1}^r…
It is described how one comes to the Wigner-Dyson random matrix theory (RMT) starting from a model of a disordered metal. The lectures start with a historical introduction where basic ideas of the RMT and theory of disordered metals are…
The largest eigenvalue of random tensors is an important feature of systems involving disorder, equivalent to the ground state energy of glassy systems or to the injective norm of quantum states. For symmetric Gaussian random tensors of…
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.…
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…
We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. For general non-Gaussian noise, assuming that the signal is drawn from the Rademacher prior, we prove that the log likelihood ratio (LR) of the…
This work analyzes singular-value spectra of weight matrices in pretrained transformer models to understand how information is stored at both ends of the spectrum. Using Random Matrix Theory (RMT) as a zero information hypothesis, we…
This paper studies the deflation algorithm when applied to estimate a low-rank symmetric spike contained in a large tensor corrupted by additive Gaussian noise. Specifically, we provide a precise characterization of the large-dimensional…
Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions. Many statistics in one can be interpreted in terms of quantities of the other; for example, zeros of…
Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and…
In this paper, we focus on the fixed TT-rank and precision problems of finding an approximation of the tensor train (TT) decomposition of a tensor. Note that the TT-SVD and TT-cross are two well-known algorithms for these two problems.…
Computing the first few singular vectors of a large matrix is a problem that frequently comes up in statistics and numerical analysis. Given the presence of noise, exact calculation is hard to achieve, and the following problem is of…
The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…
In this paper we discuss the notion of singular vector tuples of a complex valued $d$-mode tensor of dimension m_1 x ... x m_d. We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the…
In this paper, we focus on developing randomized algorithms for the computation of low multilinear rank approximations of tensors based on the random projection and the singular value decomposition. Following the theory of the singular…
Across many disciplines from neuroscience and genomics to machine learning, atmospheric science and finance, the problems of denoising large data matrices to recover signals obscured by noise, and of estimating the structure of these…
Let $A\in\mathbb{R}^{m\times n}$ be a matrix of rank $r$ with singular value decomposition (SVD) $A=\sum_{k=1}^r\sigma_k (u_k\otimes v_k),$ where $\{\sigma_k, k=1,\ldots,r\}$ are singular values of $A$ (arranged in a non-increasing order)…