  Sparse linear regression -- finding an unknown vector from linear
measurements -- is now known to be possible with fewer samples than variables,
via methods like the LASSO. We consider the multiple sparse linear regression
problem, where several related vectors -- with partially shared support sets --
have to be recovered. A natural question in this setting is whether one can use
the sharing to further decrease the overall number of samples required. A line
of recent research has studied the use of \ell_1/\ell_q norm
block-regularizations with q>1 for such problems; however these could actually
perform worse in sample complexity -- vis a vis solving each problem separately
ignoring sharing -- depending on the level of sharing.
  We present a new method for multiple sparse linear regression that can
leverage support and parameter overlap when it exists, but not pay a penalty
when it does not. A very simple idea: we decompose the parameters into two
components and regularize these differently. We show both theoretically and
empirically, our method strictly and noticeably outperforms both \ell_1 or
\ell_1/\ell_q methods, over the entire range of possible overlaps (except at
boundary cases, where we match the best method). We also provide theoretical
guarantees that the method performs well under high-dimensional scaling.
