Eigenvalues of block structured asymmetric random matrices

We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into $D$ partitions with arbitrary size (linear in $N$). The parameters of the model are the variances of elements in each block, summarized in $g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by studying the matrix-valued Stieltjes transform we show that these matrices have a circularly symmetric spectrum, we give an explicit formula for their spectral radius and a set of implicit equations for the full density function. We discuss applications of this model to neural networks

A Random Matrix Approach to Echo-State Neural Networks

Abstract Recurrent neural networks, especially in their linear version, have provided many qualitative insights on their performance under different configurations. This article provides, through a novel random matrix framework, the quantitative counterpart of these performance results, specifically in the case of echo-state networks. Beyond mere insights, our approach conveys a deeper understanding on the core mechanism under play for both training and testing

Analysis and Approximate Inference of Large Random Kronecker Graphs

Random graph models are playing an increasingly important role in various fields ranging from social networks, telecommunication systems, to physiologic and biological networks. Within this landscape, the random Kronecker graph model, emerges as a prominent framework for scrutinizing intricate real-world networks. In this paper, we investigate large random Kronecker graphs, i.e., the number of graph vertices $N$ is large. Built upon recent advances in random matrix theory (RMT) and high-dimensional statistics, we prove that the adjacency of a large random Kronecker graph can be decomposed, in a spectral norm sense, into two parts: a small-rank (of rank $O(\log N)$) signal matrix that is linear in the graph parameters and a zero-mean random noise matrix. Based on this result, we propose a ``denoise-and-solve'' approach to infer the key graph parameters, with significantly reduced computational complexity. Experiments on both graph inference and classification are presented to evaluate the our proposed method. In both tasks, the proposed approach yields comparable or advantageous performance, than widely-used graph inference (e.g., KronFit) and graph neural net baselines, at a time cost that scales linearly as the graph size $N$.Comment: 27 pages, 5 figures, 2 table

Bayesian Deep Net GLM and GLMM

Deep feedforward neural networks (DFNNs) are a powerful tool for functional approximation. We describe flexible versions of generalized linear and generalized linear mixed models incorporating basis functions formed by a DFNN. The consideration of neural networks with random effects is not widely used in the literature, perhaps because of the computational challenges of incorporating subject specific parameters into already complex models. Efficient computational methods for high-dimensional Bayesian inference are developed using Gaussian variational approximation, with a parsimonious but flexible factor parametrization of the covariance matrix. We implement natural gradient methods for the optimization, exploiting the factor structure of the variational covariance matrix in computation of the natural gradient. Our flexible DFNN models and Bayesian inference approach lead to a regression and classification method that has a high prediction accuracy, and is able to quantify the prediction uncertainty in a principled and convenient way. We also describe how to perform variable selection in our deep learning method. The proposed methods are illustrated in a wide range of simulated and real-data examples, and the results compare favourably to a state of the art flexible regression and classification method in the statistical literature, the Bayesian additive regression trees (BART) method. User-friendly software packages in Matlab, R and Python implementing the proposed methods are available at https://github.com/VBayesLabComment: 35 pages, 7 figure, 10 table