On singular values of data matrices with general independent columns

TL;DR

This paper analyzes the asymptotic behavior of singular values of high-dimensional data matrices with independent, potentially differently distributed columns, under a unifying diagonalizability assumption, with applications to covariance estimation and auto-regressive models.

Contribution

It introduces a general framework for the limiting spectral distribution of singular values in high-dimensional matrices with independent columns, extending existing models and applications.

Findings

Established a limiting distribution for singular values as dimensions grow large.

Applied results to realized covariance matrices in diffusion processes.

Derived spectral distributions for matrix auto-regressive models.

Abstract

In this paper, we analyse singular values of a large $p\times n$ data matrix $\mathbf{X}_n= (\mathbf{x}_{n1},\ldots,\mathbf{x}_{nn})$ where the column $\mathbf{x}_{nj}$'s are independent $p$-dimensional vectors, possibly with different distributions. Such data matrices are common in high-dimensional statistics. Under a key assumption that the covariance matrices $\mathbf{\Sigma}_{nj}=\text{Cov}(\mathbf{x}_{nj})$ can be asymptotically simultaneously diagonalizable, and appropriate convergence of their spectra, we establish a limiting distribution for the singular values of $\mathbf{X}_n$ when both dimension $p$ and $n$ grow to infinity in a comparable magnitude. The matrix model goes beyond and includes many existing works on different types of sample covariance matrices, including the weighted sample covariance matrix, the Gram matrix model and the sample covariance matrix of linear times series models. Furthermore, we develop two applications of our general approach. First, we obtain the existence and uniqueness of a new limiting spectral distribution of realized covariance matrices for a multi-dimensional diffusion process with anisotropic time-varying co-volatility processes. Secondly, we derive the limiting spectral distribution for singular values of the data matrix for a recent matrix-valued auto-regressive model. Finally, for a generalized finite mixture model, the limiting spectral distribution for singular values of the data matrix is obtained.