# The All-or-Nothing Phenomenon in Sparse Linear Regression

## TL;DR

This paper characterizes a sharp phase transition in sparse linear regression, showing when exact recovery and hypothesis testing are possible based on sample size relative to a critical threshold.

## Contribution

It establishes the precise information-theoretic threshold for recovery and detection in sparse linear regression, resolving a conjecture and introducing new proof techniques.

## Key findings

- Recovery is almost perfect above the threshold
- Detection becomes impossible below the threshold
- New methods for bounding KL divergence and MMSE are developed

## Abstract

We study the problem of recovering a hidden binary $k$-sparse $p$-dimensional vector $\beta$ from $n$ noisy linear observations $Y=X\beta+W$ where $X_{ij}$ are i.i.d. $\mathcal{N}(0,1)$ and $W_i$ are i.i.d. $\mathcal{N}(0,\sigma^2)$. A closely related hypothesis testing problem is to distinguish the pair $(X,Y)$ generated from this structured model from a corresponding null model where $(X,Y)$ consist of purely independent Gaussian entries. In the low sparsity $k=o(p)$ and high signal to noise ratio $k/\sigma^2=\Omega\left(1\right)$ regime, we establish an `All-or-Nothing' information-theoretic phase transition at a critical sample size $n^*=2 k\log \left(p/k\right) /\log \left(1+k/\sigma^2\right)$, resolving a conjecture of \cite{gamarnikzadik}. Specifically, we show that if $\liminf_{p\to \infty} n/n^*>1$, then the maximum likelihood estimator almost perfectly recovers the hidden vector with high probability and moreover the true hypothesis can be detected with a vanishing error probability. Conversely, if $\limsup_{p\to \infty} n/n^*<1$, then it becomes information-theoretically impossible even to recover an arbitrarily small but fixed fraction of the hidden vector support, or to test hypotheses strictly better than random guess. Our proof of the impossibility result builds upon two key techniques, which could be of independent interest. First, we use a conditional second moment method to upper bound the Kullback-Leibler (KL) divergence between the structured and the null model. Second, inspired by the celebrated area theorem, we establish a lower bound to the minimum mean squared estimation error of the hidden vector in terms of the KL divergence between the two models.