Construction of Hierarchically SemiSeparable Mat. Representation

Join us for ASE-60, where we celebrate the life and the career of Professor Alan Stuart Edelman, on the occasion of his 60th birthday: https://math.mit.edu/events/ase60celebration/

We extend our early work on adaptive partially matrix-free Hierarchically Semi-Separable (HSS) matrix construction algorithm using Gaussian sketching operators to a broader class of Johnson-Lindenstrauss (JL) sketching operators. We present theoretical work which justifies this extension. In particular, we extend the earlier concentration bounds to all JL sketching operators and examine this bound for specific classes of such operators including the original Gaussian sketching operators, subsampled randomized Hadamard transform (SRHT) and the sparse Johnson-Lindenstrauss transform (SJLT). We demonstrate experimentally that using SJLT instead of Gaussian sketching operators leads to 1.5 - 2.5X speedups of the HSS construction implementation in the STRUMPACK C++ library. The generalized algorithm allows users to select their own JL sketching operators with theoretical lower bounds on the size of the operators which may lead to faster runtime with similar HSS construction accuracy.