Related papers: On Random Matrices Arising in Deep Neural Networks…
We show that the operatorial framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called…
We review methods to calculate eigenvalue distributions of products of large random matrices. We discuss a generalization of the law of free multiplication to non-Hermitian matrices and give a couple of examples illustrating how to use…
We introduce a theory of probability in $\lambda$-rings designed to efficiently describe random variables valued in multisets of complex numbers, varieties over a field, or other similar enriched settings. A key role is played by the…
We briefly review the random matrix theory for large N by N matrices viewed as free random variables in a context of stochastic diffusion. We establish a surprising link between the spectral properties of matrix-valued multiplicative…
Random matrix models consisting of normal matrices, defined by the sole constraint $[N^{\dag},N]=0$, will be explored. It is shown that cubic eigenvalue repulsion in the complex plane is universal with respect to the probability…
In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian…
Situations in many fields of research, such as digital communications, nuclear physics and mathematical finance, can be modelled with random matrices. When the matrices get large, free probability theory is an invaluable tool for describing…
We study random matrices with independent subgaussian columns. Assuming each column has a fixed Euclidean norm, we establish conditions under which such matrices act as near-isometries when restricted to a given subset of their domain. We…
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 compute analytically the probability distribution and moments of the sum and product of the non-zero eigenvalues and singular values of random matrices with (i) non-negative entries, (ii) fixed rank, and (iii) prescribed sums of the…
We establish the universality of the singular numbers in random matrix products over $\mathrm{GL}_n(\mathbb{Q}_p)$ as the number of products approaches infinity, with a fixed $n\ge 1$. We demonstrate that, under a broad class of…
Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus,…
This paper considers several aspects of random matrix universality in deep neural networks. Motivated by recent experimental work, we use universal properties of random matrices related to local statistics to derive practical implications…
Given a collection $\{\lambda_1, \dots, \lambda_n\} $ of real numbers, there is a canonical probability distribution on the set of real symmetric or complex Hermitian matrices with eigenvalues $\lambda_1,\ldots,\lambda_n$. In this paper, we…
We study the eigenvalue distribution of a random matrix, at a transition where a new connected component of the eigenvalue density support appears away from other connected components. Unlike previously studied critical points, which…
We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…
It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this…
We study universal traits which emerge both in real-world complex datasets, as well as in artificially generated ones. Our approach is to analogize data to a physical system and employ tools from statistical physics and Random Matrix Theory…
This review article provides an overview of random matrix theory (RMT) with a focus on its growing impact on the formulation and inference of statistical models and methodologies. Emphasizing applications within high-dimensional statistics,…
Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of…