A note on compressed sensing and the complexity of matrix multiplication

We consider the conjectured O(N2+ ) time complexity of multiplying any two N × N matrices A and B. Our main result is a deterministic Compressed Sensing (CS) algorithm that both rapidly and accurately computes A · B provided that the resulting matrix product is sparse/compressible. As a consequence of our main result we increase the class of matrices A, for any given N × N matrix B, which allows the exact computation of A · B to be carried out using the conjectured O(N2+ ) operations. Additionally, in the process of developing our matrix multiplication procedure, we present a modified version of Indyk’s recently proposed extractor-based CS algorithm [12] which is resilient to noise.