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Discriminative features extracted from the sparse coding model have beenshown to perform well for classification. Recent deep learning architectureshave further improved reconstruction in inverse problems by considering newdense priors learned from data. We propose a novel dense and sparse codingmodel that integrates both representation capability and discriminativefeatures. 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 analysisproposes a geometric condition based on the minimal angle between spanningsubspaces corresponding to the matrices $\mathbf{A}$ and $\mathbf{B}$ thatguarantees unique solution to the model. The second analysis shows that, undermild assumptions, a convex program recovers the dense and sparse components. Wevalidate the effectiveness of the model on simulated data and propose a denseand sparse autoencoder (DenSaE) tailored to learning the dictionaries from thedense and sparse model. We demonstrate that (i) DenSaE denoises natural imagesbetter than architectures derived from the sparse coding model($\mathbf{B}\mathbf{u}$), (ii) in the presence of noise, training the biases inthe 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 sparsecoding model, DenSaE offers a balance between discriminative power andrepresentation.

Towards improving discriminative reconstruction via simultaneous dense and sparse coding