Supervised learning of analysis-sparsity priors with automatic differentiation
This addresses the challenge of learning dictionaries for analysis priors in signal processing, offering a method for cases where closed-form solutions are unavailable, though it is incremental as it builds on existing optimization techniques.
The paper tackles the problem of learning analysis-sparsity priors, such as dictionaries for denoising and image reconstruction, by approximating reconstructions with the Forward-Backward algorithm and using automatic differentiation to compute gradients for learning via projected gradient descent. It successfully learns the 1D Total Variation dictionary from piecewise constant signals and improves stability by constraining dictionaries to 0-centered columns.
Sparsity priors are commonly used in denoising and image reconstruction. For analysis-type priors, a dictionary defines a representation of signals that is likely to be sparse. In most situations, this dictionary is not known, and is to be recovered from pairs of ground-truth signals and measurements, by minimizing the reconstruction error. This defines a hierarchical optimization problem, which can be cast as a bi-level optimization. Yet, this problem is unsolvable, as reconstructions and their derivative wrt the dictionary have no closed-form expression. However, reconstructions can be iteratively computed using the Forward-Backward splitting (FB) algorithm. In this paper, we approximate reconstructions by the output of the aforementioned FB algorithm. Then, we leverage automatic differentiation to evaluate the gradient of this output wrt the dictionary, which we learn with projected gradient descent. Experiments show that our algorithm successfully learns the 1D Total Variation (TV) dictionary from piecewise constant signals. For the same case study, we propose to constrain our search to dictionaries of 0-centered columns, which removes undesired local minima and improves numerical stability.