Recovery of Sparse Matrices via Matrix Sketching

TL;DR

This paper introduces efficient matrix-based algorithms for recovering sparse matrices from compressed measurements, avoiding computationally intensive Kronecker operations and demonstrating significant efficiency gains.

Contribution

It extends FISTA and OMP algorithms to matrix form, providing a more computationally efficient approach for sparse matrix recovery without using Kronecker products.

Findings

FISTA with matrix inputs is more efficient than its vector form.

OMP with matrix inputs performs similarly to its vector counterpart.

The proposed methods reduce computational complexity for large matrices.

Abstract

In this paper, we consider the problem of recovering an unknown sparse matrix X from the matrix sketch Y = AX B^T. The dimension of Y is less than that of X, and A and B are known matrices. This problem can be solved using standard compressive sensing (CS) theory after converting it to vector form using the Kronecker operation. In this case, the measurement matrix assumes a Kronecker product structure. However, as the matrix dimension increases the associated computational complexity makes its use prohibitive. We extend two algorithms, fast iterative shrinkage threshold algorithm (FISTA) and orthogonal matching pursuit (OMP) to solve this problem in matrix form without employing the Kronecker product. While both FISTA and OMP with matrix inputs are shown to be equivalent in performance to their vector counterparts with the Kronecker product, solving them in matrix form is shown to be computationally more efficient. We show that the computational gain achieved by FISTA with matrix inputs over its vector form is more significant compared to that achieved by OMP.