We propose a unified approximate message passing solution to bilinear matrix
recovery with applications to dictionary learning, low-rank matrix completion,
and robust PCA. In particular, we consider recovering the matrices D, X, and S
from Y = P(D*X) + S + W, where D*X is a low-rank product, S models sparse
disturbances, and W accounts for dense disturbances.  Here, P(.) is a known
element-wise mask, and X would be sparse for dictionary-learning problems.
We dub our approach Bilinear Generalized Approximate Message Passing (BiG-AMP),
a la Rangan. A salient feature of our approach is that it avoids calculating
matrix SVD's and hence is extremely efficient. We show empirically that its
performance compares favorably to other state-of-the-art matrix recovery
methods.