  We extend an adaptive partially matrix-free Hierarchically Semi-Separable
(HSS) matrix construction algorithm by Gorman et al. [SIAM J. Sci. Comput.
41(5), 2019] which uses 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 discuss the implementation details
of applying SJLT efficiently and 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
run time with similar HSS construction accuracy.
