Block Subsampled Randomized Hadamard Transform for Nyström Approximation on Distributed Architectures

This article introduces a novel structured random matrix composed blockwise from subsampled randomized Hadamard transforms (SRHTs). The block SRHT is expected to outperform well-known dimension reduction maps, including SRHT and Gaussian matrices, on distributed architectures with not too many cores compared to the dimension. We prove that a block SRHT with enough rows is an oblivious subspace embedding, i.e., an approximate isometry for an arbitrary low-dimensional subspace with high probability. Our estimate of the required number of rows is similar to that of the standard SRHT. This suggests that the two transforms should provide the same accuracy of approximation in the algorithms. The block SRHT can be readily incorporated into randomized methods, for instance to compute a low-rank approximation of a large-scale matrix. For completeness, we revisit some common randomized approaches for this problem such as Randomized Singular Value Decomposition and Nyström approximation, with a discussion of their accuracy and implementation on distributed architectures.