Robust Testing in High-Dimensional Sparse Models

TL;DR

This paper establishes tight sample complexity bounds for robustly testing the norm of high-dimensional sparse signals under two observation models, highlighting increased complexity due to robustness constraints.

Contribution

It provides the first tight bounds on sample complexity for robust testing of sparse signals in high dimensions across two common models.

Findings

Sample complexity for robust testing is rac{s \, ext{log}(d/s)}{} in the first model.

Sample complexity in the linear regression model depends on rac{s \, ext{log} d}{} and rac{1}{\u03b3^4}.

Robustness significantly increases the difficulty of testing in high-dimensional sparse models.

Abstract

We consider the problem of robustly testing the norm of a high-dimensional sparse signal vector under two different observation models. In the first model, we are given $n$ i.i.d. samples from the distribution $\mathcal{N}\left(\theta,I_d\right)$ (with unknown $\theta$), of which a small fraction has been arbitrarily corrupted. Under the promise that $\|\theta\|_0\le s$, we want to correctly distinguish whether $\|\theta\|_2=0$ or $\|\theta\|_2>\gamma$, for some input parameter $\gamma>0$. We show that any algorithm for this task requires $n=\Omega\left(s\log\frac{ed}{s}\right)$ samples, which is tight up to logarithmic factors. We also extend our results to other common notions of sparsity, namely, $\|\theta\|_q\le s$ for any $0 < q < 2$. In the second observation model that we consider, the data is generated according to a sparse linear regression model, where the covariates are i.i.d. Gaussian and the regression coefficient (signal) is known to be $s$-sparse. Here too we assume that an $\epsilon$-fraction of the data is arbitrarily corrupted. We show that any algorithm that reliably tests the norm of the regression coefficient requires at least $n=\Omega\left(\min(s\log d,{1}/{\gamma^4})\right)$ samples. Our results show that the complexity of testing in these two settings significantly increases under robustness constraints. This is in line with the recent observations made in robust mean testing and robust covariance testing.