  In this paper we provide a complementary set of results to those we present
in our companion work \cite{Stojnicl1HidParasymldp} regarding the behavior of
the so-called partial $\ell_1$ (a variant of the standard $\ell_1$ heuristic
often employed for solving under-determined systems of linear equations). As is
well known through our earlier works
\cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13}, the partial $\ell_1$
also exhibits the phase-transition (PT) phenomenon, discovered and well
understood in the context of the standard $\ell_1$ through Donoho's and our own
works \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}.
\cite{Stojnicl1HidParasymldp} goes much further though and, in addition to the
determination of the partial $\ell_1$'s phase-transition curves (PT curves)
(which had already been done in
\cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13}), provides a
substantially deeper understanding of the PT phenomena through a study of the
underlying large deviations principles (LDPs). As the PT and LDP phenomena are
by their definitions related to large dimensional settings, both sets of our
works, \cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13} and
\cite{Stojnicl1HidParasymldp}, consider what is typically called the asymptotic
regime. In this paper we move things in a different direction and consider
finite dimensional scenarios. Basically, we provide explicit performance
characterizations for any given collection of systems/parameters dimensions. We
do so for two different variants of the partial $\ell_1$, one that we call
exactly the partial $\ell_1$ and another one, possibly a bit more practical,
that we call the hidden partial $\ell_1$.
