On the sparsity of LASSO minimizers in sparse data recovery

TL;DR

This paper analyzes the sparsity and error bounds of LASSO minimizers in sparse data recovery, showing support size remains proportional to the true sparsity under RIP conditions and providing new error bounds.

Contribution

It provides a detailed analysis of LASSO support size and error bounds in sparse recovery, with explicit dependence on RIP constants and parameters.

Findings

Support size of LASSO minimizer is proportional to true sparsity.

Derived new $ ext{ell}_2/ ext{ell}_1$ error bounds with explicit parameter dependence.

Support size bounds improve with smaller RIP constant $\delta$.

Abstract

We present a detailed analysis of the unconstrained $\ell_1$-weighted LASSO method for recovery of sparse data from its observation by randomly generated matrices, satisfying the Restricted Isometry Property (RIP) with constant $\delta<1$, and subject to negligible measurement and compressibility errors. We prove that if the data is $k$-sparse, then the size of support of the LASSO minimizer, $s$, maintains a comparable sparsity, $s\leq C_\delta k$. For example, if $\delta=0.7$ then $s< 11k$ and a slightly smaller $\delta=0.4$ yields $s< 4k$. We also derive new $\ell_2/\ell_1$ error bounds which highlight precise dependence on $k$ and on the LASSO parameter $\lambda$, before the error is driven below the scale of negligible measurement/ and compressiblity errors.