  Given an overcomplete dictionary $A$ and a signal $b$ that is a linear
combination of a few linearly independent columns of $A$, classical sparse
recovery theory deals with the problem of recovering the unique sparse
representation $x$ such that $b = A x$. It is known that under certain
conditions on $A$, $x$ can be recovered by the Basis Pursuit (BP) and the
Orthogonal Matching Pursuit (OMP) algorithms. In this work, we consider the
more general case where $b$ lies in a low-dimensional subspace spanned by some
columns of $A$, which are possibly linearly dependent. In this case, the
sparsest solution $x$ is generally not unique, and we study the problem that
the representation $x$ identifies the subspace, i.e. the nonzero entries of $x$
correspond to dictionary atoms that are in the subspace. Such a representation
$x$ is called subspace-sparse. We present sufficient conditions for
guaranteeing subspace-sparse recovery, which have clear geometric
interpretations and explain properties of subspace-sparse recovery. We also
show that the sufficient conditions can be satisfied under a randomized model.
Our results are applicable to the traditional sparse recovery problem and we
get conditions for sparse recovery that are less restrictive than the canonical
mutual coherent condition. We also use the results to analyze the sparse
representation based classification (SRC) method, for which we get conditions
to show its correctness.
