16914

An N log N Parallel Fast Direct Solver for Kernel Matrices

Chenhan D. Yu, William B. March, George Biros
Department of Computer Science, The University of Texas at Austin, Austin, Texas, USA
arXiv:1701.02324 [cs.DC], (9 Jan 2017)

@article{yu2017parallel,

   title={An $N log N$ Parallel Fast Direct Solver for Kernel Matrices},

   author={Yu, Chenhan D. and March, William B. and Biros, George},

   year={2017},

   month={jan},

   archivePrefix={"arXiv"},

   primaryClass={cs.DC}

}

Kernel matrices appear in machine learning and non-parametric statistics. Given N points in d dimensions and a kernel function that requires $mathcal{O}(d)$ work to evaluate, we present an $mathcal{O}(dNlog N)$-work algorithm for the approximate factorization of a regularized kernel matrix, a common computational bottleneck in the training phase of a learning task. With this factorization, solving a linear system with a kernel matrix can be done with $mathcal{O}(Nlog N)$ work. Our algorithm only requires kernel evaluations and does not require that the kernel matrix admits an efficient global low rank approximation. Instead our factorization only assumes low-rank properties for the off-diagonal blocks under an appropriate row and column ordering. We also present a hybrid method that, when the factorization is prohibitively expensive, combines a partial factorization with iterative methods. As a highlight, we are able to approximately factorize a dense 11M*11M kernel matrix in 2 minutes on 3,072 x86 "Haswell" cores and a 4.5M*4.5M matrix in 1 minute using 4,352 "Knights Landing" cores.
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