# Optimal Sparsity Testing in Linear regression Model

## TL;DR

This paper investigates the fundamental limits of testing the sparsity level in high-dimensional linear regression, providing minimax separation distances for different scenarios with known and unknown parameters.

## Contribution

It precisely characterizes the minimax separation distances for sparsity testing in high-dimensional linear regression under various conditions, highlighting the influence of null and alternative sparsity levels.

## Key findings

- Minimax separation distances depend on null and alternative sparsity levels.
- Different scenarios show distinct separation distances based on knowledge of covariance and noise.
- Both null and alternative hypotheses' sizes are crucial in the testing problem.

## Abstract

We consider the problem of sparsity testing in the high-dimensional linear regression model. The problem is to test whether the number of non-zero components (aka the sparsity) of the regression parameter $\theta^*$ is less than or equal to $k_0$. We pinpoint the minimax separation distances for this problem, which amounts to quantifying how far a $k_1$-sparse vector $\theta^*$ has to be from the set of $k_0$-sparse vectors so that a test is able to reject the null hypothesis with high probability. Two scenarios are considered. In the independent scenario, the covariates are i.i.d. normally distributed and the noise level is known. In the general scenario, both the covariance matrix of the covariates and the noise level are unknown. Although the minimax separation distances differ in these two scenarios, both of them actually depend on $k_0$ and $k_1$ illustrating that for this composite-composite testing problem both the size of the null and of the alternative hypotheses play a key role.