Use of Johnson–Lindenstrauss (JL) sketching operators in faster hierarchical matrix construction

We present an adaptive, partially matrix-free, hierarchical matrix construction framework using a broader class of Johnson–Lindenstrauss (JL) sketching operators. On the theoretical side, 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 discuss the implementation details of applying SJLT and SRHT efficiently. Then we demonstrate experimentally that using SJLT or SRHT instead of Gaussian sketching operators leads to up to 2.5 times speedups of the serial HSS construction implementation in the STRUMPACK C++ library. Additionally, we discuss the implementation of a parallel distributed HSS construction that leverages Gaussian or SJLT sketching operators. We observe a performance improvement of up to 35 times when using SJLT sketching operators over Gaussian sketching operators. 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.

This seminar is part of the Recent Progress and Open Directions in Matrix Computations series.