  The rapid developing area of compressed sensing suggests that a sparse vector
lying in an arbitrary high dimensional space can be accurately recovered from
only a small set of non-adaptive linear measurements. Under appropriate
conditions on the measurement matrix, the entire information about the original
sparse vector is captured in the measurements, and can be recovered using
efficient polynomial methods. The vector model has been extended to a finite
set of sparse vectors sharing a common non-zero location set. In this paper, we
treat a broader framework in which the goal is to recover a possibly infinite
set of jointly sparse vectors. Extending existing recovery methods to this
model is difficult due to the infinite structure of the sparse vector set.
Instead, we prove that the entire infinite set of sparse vectors can recovered
by solving a single, reduced-size finite-dimensional problem, corresponding to
recovery of a finite set of sparse vectors. We then show that the problem can
be further reduced to the basic recovery of a single sparse vector by randomly
combining the measurement vectors. Our approach results in exact recovery of
both countable and uncountable sets as it does not rely on discretization or
heuristic techniques. To efficiently recover the single sparse vector produced
by the last reduction step, we suggest an empirical boosting strategy that
improves the recovery ability of any given sub-optimal method for recovering a
sparse vector. Numerical experiments on random data demonstrate that when
applied to infinite sets our strategy outperforms discretization techniques in
terms of both run time and empirical recovery rate. In the finite model, our
boosting algorithm is characterized by fast run time and superior recovery rate
than known popular methods.
