Geometry-oblivious FMM for compressing dense SPD matrices

We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, or "compression," of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate matrix-vector multiplication in N log N or eve...

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Bibliographic Details
Published in:International Conference for High Performance Computing, Networking, Storage and Analysis (Online) pp. 1 - 14
Main Authors: Yu, Chenhan D., Levitt, James, Reiz, Severin, Biros, George
Format: Conference Proceeding
Language:English
Published: New York, NY, USA ACM 12.11.2017
Series:ACM Conferences
Subjects:
Computer architecture
Computer systems organization > Architectures > Other architectures > Heterogeneous (hybrid) systems
Computer systems organization > Architectures > Parallel architectures > Multicore architectures
Computing methodologies > Machine learning > Machine learning approaches > Kernel methods
Computing methodologies > Parallel computing methodologies > Parallel algorithms
Computing methodologies > Symbolic and algebraic manipulation > Symbolic and algebraic algorithms > Linear algebra algorithms
Dynamic scheduling
Fast Matrix Multiplication
Fast Multipole Methods
Geometry-oblivious
Heterogeneous Computing
Hierarchical Matrices
Mathematics of computing > Mathematical software > Mathematical software performance
Mathematics of computing > Probability and statistics > Probabilistic representations > Nonparametric representations > Kernel density estimators
Runtime
Scientific computing
Software algorithms
Software performance
Symmetric matrices
Theory of computation > Design and analysis of algorithms > Approximation algorithms analysis > Numeric approximation algorithms
Theory of computation > Design and analysis of algorithms > Streaming, sublinear and near linear time algorithms > Sketching and sampling
heterogeneous computing
hierarchical matrices
fast multipole methods
geometry-oblivious
fast matrix multiplication
ISBN:9781450351140, 145035114X
ISSN:2167-4337
Online Access:Get full text
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Summary:We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, or "compression," of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate matrix-vector multiplication in N log N or even N time, where N is the matrix size. Compression requires N log N storage and work. In general, our scheme belongs to the family of hierarchical matrix approximation methods. In particular, it generalizes the fast multipole method (FMM) to a purely algebraic setting by only requiring the ability to sample matrix entries. Neither geometric information (i.e., point coordinates) nor knowledge of how the matrix entries have been generated is required, thus the term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme for hierarchical matrix computations that reduces synchronization barriers. We present results on the Intel Knights Landing and Haswell architectures, and on the NVIDIA Pascal architecture for a variety of matrices.
ISBN:9781450351140
145035114X
ISSN:2167-4337
DOI:10.1145/3126908.3126921