  Convolutional sparse coding (CSC) improves sparse coding by learning a
shift-invariant dictionary from the data. However, existing CSC algorithms
operate in the batch mode and are expensive, in terms of both space and time,
on large datasets. In this paper, we alleviate these problems by using online
learning. The key is a reformulation of the CSC objective so that convolution
can be handled easily in the frequency domain and much smaller history matrices
are needed. We use the alternating direction method of multipliers (ADMM) to
solve the resulting optimization problem and the ADMM subproblems have
efficient closed-form solutions. Theoretical analysis shows that the learned
dictionary converges to a stationary point of the optimization problem.
Extensive experiments show that convergence of the proposed method is much
faster and its reconstruction performance is also better. Moreover, while
existing CSC algorithms can only run on a small number of images, the proposed
method can handle at least ten times more images.
