Efficient Matrix Multiplication: The Sparse Power-of-2 Factorization

Abstract

We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer powers of two utilizing the principles of sparse recovery. While classical low resolution quantization achieves an accuracy of 6 dB per bit, our method can achieve many times more than that for large matrices. Numerical and analytical evidence suggests that the improvement actually grows unboundedly with matrix size. Due to sparsity, the algorithm even allows for quantization levels below 1 bit per matrix entry while achieving highly accurate approximations for large matrices. Applications include, but are not limited to, neural networks, as well as fully digital beam-forming for massive MIMO and millimeter wave applications.