Multi-Layer Sparse Coding: The Holistic Way

TL;DR

This paper introduces a holistic multi-layer sparse coding framework that unifies synthesis and analysis models, proposes an integrated pursuit algorithm, and demonstrates its advantages over previous methods through theoretical analysis and numerical experiments.

Contribution

It generalizes multi-layer sparse models to include fully connected networks, and develops a holistic pursuit algorithm with provable improvements.

Findings

The holistic approach outperforms previous pursuit methods.

The integrated algorithm effectively estimates all representations simultaneously.

Numerical results confirm practical advantages of the proposed method.

Abstract

The recently proposed multi-layer sparse model has raised insightful connections between sparse representations and convolutional neural networks (CNN). In its original conception, this model was restricted to a cascade of convolutional synthesis representations. In this paper, we start by addressing a more general model, revealing interesting ties to fully connected networks. We then show that this multi-layer construction admits a brand new interpretation in a unique symbiosis between synthesis and analysis models: while the deepest layer indeed provides a synthesis representation, the mid-layers decompositions provide an analysis counterpart. This new perspective exposes the suboptimality of previously proposed pursuit approaches, as they do not fully leverage all the information comprised in the model constraints. Armed with this understanding, we address fundamental theoretical issues, revisiting previous analysis and expanding it. Motivated by the limitations of previous algorithms, we then propose an integrated - holistic - alternative that estimates all representations in the model simultaneously, and analyze all these different schemes under stochastic noise assumptions. Inspired by the synthesis-analysis duality, we further present a Holistic Pursuit algorithm, which alternates between synthesis and analysis sparse coding steps, eventually solving for the entire model as a whole, with provable improved performance. Finally, we present numerical results that demonstrate the practical advantages of our approach.