Multi-Layer Convolutional Sparse Modeling: Pursuit and Dictionary Learning

TL;DR

This paper introduces a pursuit algorithm for Multi-Layer Convolutional Sparse Coding (ML-CSC), providing theoretical stability bounds, an online training method for filters, and demonstrating its effectiveness in unsupervised applications, bridging sparse modeling and neural networks.

Contribution

It proposes a new pursuit algorithm for ML-CSC, offers stability analysis, and develops an online filter learning method, advancing understanding and practical implementation of convolutional sparse models.

Findings

Proposed a projection-based pursuit algorithm for ML-CSC.

Derived improved stability bounds for the pursuit solution.

Demonstrated competitive results in unsupervised applications.

Abstract

The recently proposed Multi-Layer Convolutional Sparse Coding (ML-CSC) model, consisting of a cascade of convolutional sparse layers, provides a new interpretation of Convolutional Neural Networks (CNNs). Under this framework, the computation of the forward pass in a CNN is equivalent to a pursuit algorithm aiming to estimate the nested sparse representation vectors -- or feature maps -- from a given input signal. Despite having served as a pivotal connection between CNNs and sparse modeling, a deeper understanding of the ML-CSC is still lacking: there are no pursuit algorithms that can serve this model exactly, nor are there conditions to guarantee a non-empty model. While one can easily obtain signals that approximately satisfy the ML-CSC constraints, it remains unclear how to simply sample from the model and, more importantly, how one can train the convolutional filters from real data. In this work, we propose a sound pursuit algorithm for the ML-CSC model by adopting a projection approach. We provide new and improved bounds on the stability of the solution of such pursuit and we analyze different practical alternatives to implement this in practice. We show that the training of the filters is essential to allow for non-trivial signals in the model, and we derive an online algorithm to learn the dictionaries from real data, effectively resulting in cascaded sparse convolutional layers. Last, but not least, we demonstrate the applicability of the ML-CSC model for several applications in an unsupervised setting, providing competitive results. Our work represents a bridge between matrix factorization, sparse dictionary learning and sparse auto-encoders, and we analyze these connections in detail.