Towards improving discriminative reconstruction via simultaneous dense and sparse coding.

Discriminative features extracted from the sparse coding model have been shown to perform well for classification and reconstruction. 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 considers 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 proposes a natural geometric condition based on the minimal angle between spanning subspaces corresponding to the measurement matrices $\mathbf{A}$ and $\mathbf{B}$ to establish the uniqueness of solutions to the linear system. The second analysis shows that, under mild assumptions, a convex program recovers the dense and sparse components. We validate the effectiveness of the proposed 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 a) DenSaE denoises natural images better than architectures derived from the sparse coding model ($\mathbf{B}\mathbf{u}$), b) 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, c) $\mathbf{A}$ and $\mathbf{B}$ capture low- and high-frequency contents, respectively, and d) compared to the sparse coding model, DenSaE offers a balance between discriminative power and representation.