Fast Non-Asymptotic Testing And Support Recovery For Large Sparse Toeplitz Covariance Matrices

TL;DR

This paper introduces fast, non-asymptotic methods for testing and support recovery of large sparse Toeplitz covariance matrices in high-dimensional Gaussian data, with proven theoretical guarantees and practical efficiency.

Contribution

It develops computationally efficient tests and support selectors for sparse Toeplitz covariance matrices with non-asymptotic guarantees, extending to nearly Toeplitz and sub-Gaussian cases.

Findings

Test procedures perform well in high dimensions.

Support selectors effectively identify significant lags.

Rates improve as dimension increases.

Abstract

We consider $n$ independent $p$-dimensional Gaussian vectors with covariance matrix having Toeplitz structure. We test that these vectors have independent components against a stationary distribution with sparse Toeplitz covariance matrix, and also select the support of non-zero entries. We assume that the non-zero values can occur in the recent past (time-lag less than $p/2$). We build test procedures that combine a sum and a scan-type procedures, but are computationally fast, and show their non-asymptotic behaviour in both one-sided (only positive correlations) and two-sided alternatives, respectively. We also exhibit a selector of significant lags and bound the Hamming-loss risk of the estimated support. These results can be extended to the case of nearly Toeplitz covariance structure and to sub-Gaussian vectors. Numerical results illustrate the excellent behaviour of both test procedures and support selectors - larger the dimension $p$, faster are the rates.