Understanding Trainable Sparse Coding via Matrix Factorization

Abstract: Sparse coding is a core building block in many data analysis and machine learning pipelines. Typically it is solved by relying on generic optimization techniques, that are optimal in the class of first-order methods for non-smooth, convex functions, such as the Iterative Soft Thresholding Algorithm and its accelerated version (ISTA, FISTA). However, these methods don't exploit the particular structure of the problem at hand nor the input data distribution. An acceleration using neural networks was proposed in \cite{Gregor10}, coined LISTA, which showed empirically that one could achieve high quality estimates with few iterations by modifying the parameters of the proximal splitting appropriately. In this paper we study the reasons for such acceleration. Our mathematical analysis reveals that it is related to a specific matrix factorization of the Gram kernel of the dictionary, which attempts to nearly diagonalise the kernel with a basis that produces a small perturbation of the $\ell_1$ ball. When this factorization succeeds, we prove that the resulting splitting algorithm enjoys an improved convergence bound with respect to the non-adaptive version. Moreover, our analysis also shows that conditions for acceleration occur mostly at the beginning of the iterative process, consistent with numerical experiments. We further validate our analysis by showing that on dictionaries where this factorization does not exist, adaptive acceleration fails.