Efficient Sparse Coding Using Hierarchical Riemannian Pursuit

Sparse coding is a class of unsupervised methods for learning a sparse representation of the input data in the form of a linear combination of a dictionary and a sparse code. This learning framework has led to state-of-the-art results in various signal processing tasks. However, classical methods learn the dictionary and the sparse code based on alternating optimizations, usually without theoretical guarantees for either optimality or convergence due to the non-convexity of the problem. Recent works on sparse coding with a complete dictionary provide strong theoretical guarantees thanks to the development of non-convex optimization. However, initial non-convex approaches learned the dictionary in the sparse coding problem sequentially in an atom-by-atom manner, which led to a long execution time. More recent works  have sought to directly learn the entire dictionary at once, which substantially reduces the execution time. However, the associated recovery performance is degraded with a finite number of data samples. In this paper, we propose an efficient sparse coding scheme with a two-stage optimization. The proposed scheme leverages the global and local Riemannian geometry of the two-stage optimization problem and facilitates fast implementation for superb dictionary recovery performance by a finite number of samples. We further prove that, with high probability, the proposed scheme can exactly recover any atom in the target dictionary with a finite number of samples. Experiments on both synthetic and real-world data verify the efficiency and robustness of the proposed scheme.¹