Supervised Learning of Analysis-Sparsity Priors With Automatic Differentiation

Hashem Ghanem, Joseph Salmon, Nicolas Keriven, Samuel Vaiter

Abstract: 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 unknown, 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w.r.t.</i> the dictionary have no closed-form expression. However, reconstructions can be iteratively computed using the Forward-Backward splitting (FB) algorithm. In this letter, we approximate reconstructions by the output of the FB algorithm, up to a linear transformation. Then, we leverage automatic differentiation to evaluate the gradient of this output <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w.r.t.</i> 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.