# Average case analysis of Lasso under ultra-sparse conditions

## TL;DR

This paper provides an average-case analysis of Lasso in ultra-sparse linear models using a novel replica method approach, offering insights into support recovery and performance without restrictive assumptions.

## Contribution

It introduces a new analytical framework for Lasso's performance in ultra-sparse regimes, extending previous results to more general settings and noise conditions.

## Key findings

- Provides a lower bound on sample complexity for support recovery
- Generalizes previous bounds to non-Gaussian noise
- Supports analysis with extensive numerical experiments

## Abstract

We analyze the performance of the least absolute shrinkage and selection operator (Lasso) for the linear model when the number of regressors $N$ grows larger keeping the true support size $d$ finite, i.e., the ultra-sparse case. The result is based on a novel treatment of the non-rigorous replica method in statistical physics, which has been applied only to problem settings where $N$ ,$d$ and the number of observations $M$ tend to infinity at the same rate. Our analysis makes it possible to assess the average performance of Lasso with Gaussian sensing matrices without assumptions on the scaling of $N$ and $M$, the noise distribution, and the profile of the true signal. Under mild conditions on the noise distribution, the analysis also offers a lower bound on the sample complexity necessary for partial and perfect support recovery when $M$ diverges as $M = O(\log N)$. The obtained bound for perfect support recovery is a generalization of that given in previous literature, which only considers the case of Gaussian noise and diverging $d$. Extensive numerical experiments strongly support our analysis.