Risk of the Least Squares Minimum Norm Estimator under the Spike Covariance Model

TL;DR

This paper analyzes the risk of the minimum norm least squares estimator in high-dimensional settings with spike covariance matrices, showing risk vanishes compared to the null estimator and providing tighter bounds.

Contribution

It introduces new asymptotic and non-asymptotic risk bounds for the estimator under spike covariance models, improving upon previous analyses.

Findings

Risk of the estimator vanishes compared to null estimator

Provides tighter bounds on risk using spike model assumptions

Risk bounds hold when d/n approaches infinity

Abstract

We study risk of the minimum norm linear least squares estimator in when the number of parameters $d$ depends on $n$, and $\frac{d}{n} \rightarrow \infty$. We assume that data has an underlying low rank structure by restricting ourselves to spike covariance matrices, where a fixed finite number of eigenvalues grow with $n$ and are much larger than the rest of the eigenvalues, which are (asymptotically) in the same order. We show that in this setting risk of minimum norm least squares estimator vanishes in compare to risk of the null estimator. We give asymptotic and non asymptotic upper bounds for this risk, and also leverage the assumption of spike model to give an analysis of the bias that leads to tighter bounds in compare to previous works.