On the well-spread property and its relation to linear regression

TL;DR

This paper explores the importance of the well-spread property in design matrices for robust linear regression, showing its necessity for consistent recovery and analyzing the computational complexity of certifying this property.

Contribution

It demonstrates the existence of matrices lacking the well-spread property where recovery is impossible and analyzes the complexity of certifying well-spreadness in random matrices.

Findings

Existence of design matrices without well-spreadness where recovery fails

Efficient certification of well-spreadness when observations are quadratic in dimension

Computational hardness of certification when observations are fewer than quadratic in dimension

Abstract

We consider the robust linear regression model $\boldsymbol{y} = X\beta^* + \boldsymbol{\eta}$, where an adversary oblivious to the design $X \in \mathbb{R}^{n \times d}$ may choose $\boldsymbol{\eta}$ to corrupt all but a (possibly vanishing) fraction of the observations $\boldsymbol{y}$ in an arbitrary way. Recent work [dLN+21, dNS21] has introduced efficient algorithms for consistent recovery of the parameter vector. These algorithms crucially rely on the design matrix being well-spread (a matrix is well-spread if its column span is far from any sparse vector). In this paper, we show that there exists a family of design matrices lacking well-spreadness such that consistent recovery of the parameter vector in the above robust linear regression model is information-theoretically impossible. We further investigate the average-case time complexity of certifying well-spreadness of random matrices. We show that it is possible to efficiently certify whether a given $n$-by-$d$ Gaussian matrix is well-spread if the number of observations is quadratic in the ambient dimension. We complement this result by showing rigorous evidence -- in the form of a lower bound against low-degree polynomials -- of the computational hardness of this same certification problem when the number of observations is $o(d^2)$.