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.

翻译：从稀疏编码模型中提取的判别性特征已被证明在分类任务中表现良好。近期深度学习架构通过考虑从数据中学到的新型稠密先验，进一步改善了逆问题重建。我们提出了一种融合表示能力与判别性特征的新型稠密-稀疏编码模型。该模型研究在观测数据形式为$\mathbf{y} = \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$时，恢复稠密向量$\mathbf{x}$与稀疏向量$\mathbf{u}$的问题。我们的第一项分析依赖一个几何条件——具体为矩阵$\mathbf{A}$与$\mathbf{B}$张成子空间间的最小夹角——该条件确保模型解的唯一性。第二项分析表明，在关于$\mathbf{A}$和$\mathbf{B}$的特定条件下，凸规划能够恢复稠密与稀疏分量。我们通过模拟数据验证了模型的有效性，并提出了一种适用于从稠密-稀疏模型中学习字典的稠密-稀疏自编码器（DenSaE）。我们证明：（i）在自然图像去噪任务中，DenSaE优于基于稀疏编码模型（$\mathbf{B}\mathbf{u}$）的架构；（ii）当存在噪声时，训练后者偏置项等价于隐式学习$\mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$模型；（iii）$\mathbf{A}$与$\mathbf{B}$分别捕获低频与高频内容；（iv）与稀疏编码模型相比，DenSaE在判别能力与表示能力之间实现了平衡。