基于自适应Johnson-Lindenstrauss抽样技术的分层半分离矩阵表示构造
Construction of Hierarchically Semi-Separable matrix Representation using Adaptive Johnson-Lindenstrauss Sketching
AI总结：
本文提出了一种基于自适应Johnson-Lindenstrauss抽样技术的HSS矩阵构造算法，通过扩展抽样算子类型，提升了HSS构造的效率和性能。
摘要：
我们提出了一种扩展，将Gorman等人[SIAM J. Sci. Comput. 41(5), 2019]提出的自适应、部分矩阵无关的分层半分离（HSS）矩阵构造算法从高斯抽样算子扩展到更广泛的Johnson-Lindenstrauss（JL）抽样算子。我们开发了理论工作，证明了这一扩展的正确性。特别是，我们将先前的集中界扩展到所有JL抽样算子，并检查了这一界在特定类别的此类算子中的应用，包括原始的高斯抽样算子、子采样随机Hadamard变换（SRHT）和稀疏Johnson-Lindenstrauss变换（SJLT）。我们讨论了高效应用SJLT和SRHT的实现细节。然后我们实验表明，使用SJLT或SRHT代替高斯抽样算子可以将STRUMPACK C++库中的串行HSS构造实现速度提高高达2.5倍。此外，我们讨论了利用高斯或SJLT抽样算子实现并行分布式HSS构造的实现。我们观察到，使用SJLT抽样算子相对于高斯抽样算子，性能提升可达35倍。通用算法允许用户选择自己的JL抽样算子，并具有理论下界上的算子大小，这可能导致在相似HSS构造精度下更快的运行时间。
英文摘要：
We present an extension of 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 develop 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 and SRHT efficiently. Then we demonstrate experimentally that using SJLT or SRHT instead of Gaussian sketching operators leads to up to 2.5x 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 35x 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 run time with similar HSS construction accuracy.
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