This work considers recovery of signals that are sparse over two bases. For
instance, a signal might be sparse in both time and frequency, or a matrix can
be low rank and sparse simultaneously. To facilitate recovery, we consider
minimizing the sum of the ℓ1-norms that correspond to each basis, which
is a tractable convex approach. We find novel optimality conditions which
indicates a gain over traditional approaches where ℓ1 minimization is
done over only one basis. Next, we analyze these optimality conditions for the
particular case of time-frequency bases. Denoting sparsity in the first and
second bases by k1,k2 respectively, we show that, for a general class of
signals, using this approach, one requires as small as
O(max{k1,k2}loglogn) measurements for successful recovery hence
overcoming the classical requirement of
Θ(min{k1,k2}log(min{k1,k2}n)) for ℓ1
minimization when k1≈k2. Extensive simulations show that, our
analysis is approximately tight.Comment: 8 pages, 1 figure, submitted to ISIT 201