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- W4321320499 abstract "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 $11Mtimes11M$ kernel matrix in 2 minutes on 3,072 x86 Haswell cores and a $4.5Mtimes4.5M$ matrix in 1 minute using 4,352 Knights Landing cores." @default.
- W4321320499 created "2023-02-19" @default.
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- W4321320499 date "2017-01-09" @default.
- W4321320499 modified "2023-09-29" @default.
- W4321320499 title "An $N log N$ Parallel Fast Direct Solver for Kernel Matrices" @default.
- W4321320499 doi "https://doi.org/10.48550/arxiv.1701.02324" @default.
- W4321320499 hasPublicationYear "2017" @default.
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