Sparse change detection in high-dimensional linear regression

TL;DR

This paper presents 'charcoal', a novel method for detecting sparse changes in high-dimensional linear regression coefficients using sketching techniques, with strong theoretical guarantees and practical effectiveness demonstrated through simulations and real data.

Contribution

The paper introduces 'charcoal', a new sketching-based approach for change detection in high-dimensional regression without requiring individual coefficient sparsity, along with theoretical analysis and empirical validation.

Findings

Method performs well in extensive simulations

Strong theoretical guarantees on estimation accuracy

Effective in real-world single-cell RNA sequencing data

Abstract

We introduce a new methodology 'charcoal' for estimating the location of sparse changes in high-dimensional linear regression coefficients, without assuming that those coefficients are individually sparse. The procedure works by constructing different sketches (projections) of the design matrix at each time point, where consecutive projection matrices differ in sign in exactly one column. The sequence of sketched design matrices is then compared against a single sketched response vector to form a sequence of test statistics whose behaviour shows a surprising link to the well-known CUSUM statistics of univariate changepoint analysis. The procedure is computationally attractive, and strong theoretical guarantees are derived for its estimation accuracy. Simulations confirm that our methods perform well in extensive settings, and a real-world application to a large single-cell RNA sequencing dataset showcases the practical relevance.