Convolutional Sparse Coding with Overlapping Group Norms
This is an incremental study for researchers in sparse coding and image processing, showing that more complex norms may not outperform basic methods without careful tuning.
The paper tackled the limitation of homogeneous sparsity control in convolutional sparse coding by replacing the ℓ₁ penalty with mixed group norms, but found that these norms performed poorly in denoising, and while weighting improved them, it also benefited the simpler ℓ₁ norm.
The most widely used form of convolutional sparse coding uses an $\ell_1$ regularization term. While this approach has been successful in a variety of applications, a limitation of the $\ell_1$ penalty is that it is homogeneous across the spatial and filter index dimensions of the sparse representation array, so that sparsity cannot be separately controlled across these dimensions. The present paper considers the consequences of replacing the $\ell_1$ penalty with a mixed group norm, motivated by recent theoretical results for convolutional sparse representations. Algorithms are developed for solving the resulting problems, which are quite challenging, and the impact on the performance of the denoising problem is evaluated. The mixed group norms are found to perform very poorly in this application. While their performance is greatly improved by introducing a weighting strategy, such a strategy also improves the performance obtained from the much simpler and computationally cheaper $\ell_1$ norm.