Discriminative reconstruction via simultaneous dense and sparse coding

TL;DR

This paper introduces a combined dense and sparse coding model that enhances reconstruction and classification by integrating both representation and discriminative features, validated through theoretical analysis and neural network experiments.

Contribution

The paper proposes a novel dense and sparse coding framework with theoretical guarantees and a tailored autoencoder, improving reconstruction and discriminative capabilities over traditional sparse coding models.

Findings

The model guarantees unique solutions under geometric conditions.

The autoencoder (DenSaE) outperforms traditional sparse coding in image denoising.

A balance between discriminative power and representation is achieved with DenSaE.

Abstract

Discriminative features extracted from the sparse coding model have been shown to perform well for classification. Recent deep learning architectures have further improved reconstruction in inverse problems by considering new dense priors learned from data. We propose a novel dense and sparse coding model that integrates both representation capability and discriminative features. The model studies the problem of recovering a dense vector $\mathbf{x}$ and a sparse vector $\mathbf{u}$ given measurements of the form $\mathbf{y} = \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$. Our first analysis relies on a geometric condition, specifically the minimal angle between the spanning subspaces of matrices $\mathbf{A}$ and $\mathbf{B}$, which ensures a unique solution to the model. The second analysis shows that, under some conditions on $\mathbf{A}$ and $\mathbf{B}$, a convex program recovers the dense and sparse components. We validate the effectiveness of the model on simulated data and propose a dense and sparse autoencoder (DenSaE) tailored to learning the dictionaries from the dense and sparse model. We demonstrate that (i) DenSaE denoises natural images better than architectures derived from the sparse coding model ($\mathbf{B}\mathbf{u}$), (ii) in the presence of noise, training the biases in the latter amounts to implicitly learning the $\mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$ model, (iii) $\mathbf{A}$ and $\mathbf{B}$ capture low- and high-frequency contents, respectively, and (iv) compared to the sparse coding model, DenSaE offers a balance between discriminative power and representation.