  We consider the problem of sparse coding, where each sample consists of a
sparse linear combination of a set of dictionary atoms, and the task is to
learn both the dictionary elements and the mixing coefficients. Alternating
minimization is a popular heuristic for sparse coding, where the dictionary and
the coefficients are estimated in alternate steps, keeping the other fixed.
Typically, the coefficients are estimated via $\ell_1$ minimization, keeping
the dictionary fixed, and the dictionary is estimated through least squares,
keeping the coefficients fixed. In this paper, we establish local linear
convergence for this variant of alternating minimization and establish that the
basin of attraction for the global optimum (corresponding to the true
dictionary and the coefficients) is $\order{1/s^2}$, where $s$ is the sparsity
level in each sample and the dictionary satisfies RIP. Combined with the recent
results of approximate dictionary estimation, this yields provable guarantees
for exact recovery of both the dictionary elements and the coefficients, when
the dictionary elements are incoherent.
