On Multi-Layer Basis Pursuit, Efficient Algorithms and Convolutional Neural Networks

Jeremias Sulam, Aviad Aberdam, Michael Elad

Abstract:

Parsimonious representations are ubiquitous in modeling and processing information. Motivated by the recent Multi-Layer
Convolutional Sparse Coding (ML-CSC) model, we herein generalize the traditional Basis Pursuit problem to a multi-layer setting,
introducing similar sparse enforcing penalties at different representation layers in a symbiotic relation between synthesis and analysis
sparse priors. We explore different iterative methods to solve this new problem in practice, and we propose a new Multi-Layer Iterative
Soft Thresholding Algorithm (ML-ISTA), as well as a fast version (ML-FISTA). We show that these nested first order algorithms
converge, in the sense that the function value of near-fixed points can get arbitrarily close to the solution of the original problem.
We further show how these algorithms effectively implement particular recurrent convolutional neural networks (CNNs) that generalize
feed-forward ones without introducing any parameters. We present and analyze different architectures resulting unfolding the iterations
of the proposed pursuit algorithms, including a new Learned ML-ISTA, providing a principled way to construct deep recurrent CNNs.
Unlike other similar constructions, these architectures unfold a global pursuit holistically for the entire network. We demonstrate the
emerging constructions in a supervised learning setting, consistently improving the performance of classical CNNs while maintaining
the number of parameters constant.