Related papers: On Random Matrices Arising in Deep Neural Networks…
The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important…
Neural networks have been used successfully in a variety of fields, which has led to a great deal of interest in developing a theoretical understanding of how they store the information needed to perform a particular task. We study the…
Random matrix theory has played a major role in several areas of pure and applied mathematics, as well as statistics, physics, and computer science. This lecture aims to describe the intrinsic freeness phenomenon and how it provides new…
Free probability theory started in the 1980s has attracted much attention lately in signal processing and communications areas due to its applications in large size random matrices. However, it involves with massive mathematical concepts…
The paper deals with the distribution of singular values of the input-output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent rectangular…
Free probability and random matrix theory has shown to be a fruitful combination in many fields of research, such as digital communications, nuclear physics and mathematical finance. The link between free probability and eigenvalue…
Neural network models are one of the most successful approaches to machine learning, enjoying an enormous amount of development and research over recent years and finding concrete real-world applications in almost any conceivable area of…
We consider a random matrix of the form $D_n \odot X_n$ (known as a variance profile matrix), where $\odot$ denotes the Hadamard product of the two matrices, $D_n$ is a deterministic matrix, and $X_n$ is a random matrix. We call $D_n\odot…
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…
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,…
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…
We introduce a new class of large structured random matrices characterized by four fundamental properties which we discuss. We prove that this class is stable under matrix-valued and pointwise non-linear operations. We then formulate an…
This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.'s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common…
This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of…
These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition…
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 use techniques from finite free probability to analyze matrix processes related to eigenvalues, singular values, and generalized singular values of random matrices. The models we use are quite basic and the analysis consists entirely of…
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…
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…
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…