Spectral analysis of the Gram matrix of mixture models

Abstract

This text is devoted to the asymptotic study of some spectral properties of the Gram matrix \(W^{\sf T} W\) built upon a collection \(w_1, \ldots, w_n\in \mathbb{R}^p\) of random vectors (the columns of \(W\)), as both the number \(n\) of observations and the dimension \(p\) of the observations tend to infinity and are of similar order of magnitude. The random vectors \(w_1, \ldots, w_n\) are independent observations, each of them belonging to one of \(k\) classes \(\mathcal{C}_1,\ldots, \mathcal{C}_k\). The observations of each class \(\mathcal{C}_a\) (\(1\le a\le k\)) are characterized by their distribution \(\mathcal{N}(0, p^{-1}C_a)\), where \(C_1, \ldots, C_k\) are some non negative definite \(p\times p\) matrices. The cardinality \(n_a\) of class \(\mathcal{C}_a\) and the dimension \(p\) of the observations are such that \(\frac{n_a}{n}\) (\(1\le a\le k\)) and \(\frac{p}{n}\) stay bounded away from \(0\) and \(+\infty\). We provide deterministic equivalents to the empirical spectral distribution of \(W^{\sf T}W\) and to the matrix entries of its resolvent (as well as of the resolvent of \(WW^{\sf T}\)). These deterministic equivalents are defined thanks to the solutions of a fixed-point system. Besides, we prove that \(W^{\sf T} W\) has asymptotically no eigenvalues outside the bulk of its spectrum, defined thanks to these deterministic equivalents. These results are directly used in our companion paper "Kernel spectral clustering of large dimensional data", which is devoted to the analysis of the spectral clustering algorithm in large dimensions. They also find applications in various other fields such as wireless communications where functionals of the aforementioned resolvents allow one to assess the communication performance across multi-user multi-antenna channels.