  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.
