Sparse linear regression -- CLuP achieves the ideal \emph{exact} ML

TL;DR

This paper introduces a novel CLuP mechanism based on Random Duality Theory that achieves the ideal maximum-likelihood performance in sparse linear regression, outperforming previous algorithms like LASSO and SOCP, especially in large-scale problems.

Contribution

The paper presents a new CLuP algorithm that attains exact ML performance in sparse regression and demonstrates its scalability and superiority over existing methods.

Findings

CLuP achieves the ideal ML performance in sparse regression.

CLuP outperforms LASSO and SOCP variants in accuracy.

The algorithm is effective for problems with thousands of unknowns.

Abstract

In this paper we revisit one of the classical statistical problems, the so-called sparse maximum-likelihood (ML) linear regression. As a way of attacking this type of regression, we present a novel CLuP mechanism that to a degree relies on the \bl{\textbf{Random Duality Theory (RDT)}} based algorithmic machinery that we recently introduced in \cite{Stojnicclupint19,Stojnicclupcmpl19,Stojnicclupplt19,Stojniccluplargesc20,Stojniccluprephased20}. After the initial success that the CLuP exhibited in achieving the exact ML performance while maintaining excellent computational complexity related properties in MIMO ML detection in \cite{Stojnicclupint19,Stojnicclupcmpl19,Stojnicclupplt19}, one would naturally expect that a similar type of success can be achieved in other ML considerations. The results that we present here confirm that such an expectation is indeed reasonable. In particular, within the sparse regression context, the introduced CLuP mechanism indeed turns out to be able to \bl{\textbf{\emph{achieve the ideal ML performance}}}. Moreover, it can substantially outperform some of the most prominent earlier state of the art algorithmic concepts, among them even the variants of the famous LASSO and SOCP from \cite{StojnicPrDepSocp10,StojnicGenLasso10,StojnicGenSocp10}. Also, our recent results presented in \cite{Stojniccluplargesc20,Stojniccluprephased20} showed that the CLuP has excellent \bl{\textbf{\emph{large-scale}}} and the so-called \bl{\textbf{\emph{rephasing}}} abilities. Since such large-scale algorithmic features are possibly even more desirable within the sparse regression context we here also demonstrate that the basic CLuP ideas can be reformulated to enable solving with a relative ease the regression problems with \bl{\textbf{\emph{several thousands}}} of unknowns.