Provably accurate double-sparse coding

Sparse coding is a crucial subroutine in algorithms for various signal processing, deep learning, and other machine learning applications. The central goal is to learn an overcomplete dictionary that can sparsely represent a given input dataset. However, a key challenge is that storage, transmission, and processing of the learned dictionary can be untenably high if the data dimension is high. In this paper, we consider the double-sparsity model introduced by Rubinstein et al. (2010b) where the dictionary itself is the product of a fixed, known basis and a data-adaptive sparse component. First, we introduce a simple algorithm for double-sparse coding that can be amenable to efficient implementation via neural architectures. Second, we theoretically analyze its performance and demonstrate asymptotic sample complexity and running time benefits over existing (provable) approaches for sparse coding. To our knowledge, our work introduces the first computationally efficient algorithm for double-sparse coding that enjoys rigorous statistical guarantees. Finally, we corroborate our theory with several numerical experiments on simulated data, suggesting that our method may be useful for problem sizes encountered in practice.