Geometry-oblivious FMM for compressing dense SPD matrices

Yu, Chih‐Wei; Levitt, James; Reiz, Severin; Biros, George

doi:10.1145/3126908.3126921

Cited by 18 publications

(29 citation statements)

References 31 publications

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“…Following [1], GOFMM comprises two phases: compression and evaluation. In the compression phase, an SPD matrix K is compressed to K recursively using a binary tree such that…”

Section: A Geometric-oblivious Fmm Reviewmentioning

confidence: 99%

“…GOFMM uses nearest neighborbased importance sampling to avoid the O(N 2 ) compression cost of K Iα using standard algorithms (say QR). Using neighbors as row samples, as opposed to uniform sampled rows, produces more accurate skeletonizations [1], [16].…”

Section: A Geometric-oblivious Fmm Reviewmentioning

confidence: 99%

“…If K ∈ R N ×N is a dense SPD matrix it requires O(N 2 ) storage and work for a matrix-vector multiplication (hereby "matvec"). GOFMM [1] (reviewed in §II) constructs a matrix K (hereby "compression") using O(N log N ) entries from K so that K − K ≤ K (for a user-defined > 0) and a matvec with K requires O(N log N ) work. The constants in the compression and multiplication complexity estimates depend on .…”

Section: Introductionmentioning

confidence: 99%

“…• We describe and implement interfaces for: (1) in memory evaluation of K, (2) out-of-core memory evaluation of K. We also define an interface that allows the user to redistribute any data related to the evaluation of K ij ( §IV, §V, and Appendix VII-A).…”

Section: Introductionmentioning

confidence: 99%

“…Related work: The literature on H-matrix methods is vast. For a review, see [1]. To our knowledge, there is no distributedmemory algorithm that (1) supports permutations to expose low-rank structure and (2) supports sparse direct evaluations (FMM-like interactions).…”

Section: Introductionmentioning

confidence: 99%

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Distributed-Memory Hierarchical Compression of Dense SPD Matrices

Reiz

Biros

2018

SC18: International Conference for High Performance Computing, Networking, Storage and Analysis

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We present a distributed memory algorithm for the hierarchical compression of symmetric positive definite (SPD) matrices. Our method is based on GOFMM, an algorithm that appeared in doi:10.1145/3126908.3126921. For many SPD matrices, GOFMM enables compression and approximate matrix-vector multiplication that for many matrices can reach N log N time-as opposed to N 2 required for a dense matrix. But GOFMM supports only shared memory parallelism. In this paper, we use the message passing interface (MPI) and extend the ideas of GOFMM to the distributed memory setting. We also propose and implement an asynchronous algorithm for faster multiplication. We present different usage scenarios on a selection of SPD matrices that are related to graphs, neural-networks, and covariance operators. We present results on the Texas Advanced Computing Center's "Stampede 2" system. We also compare with the STRUMPACK software package, which, to our knowledge, is the only other available software that can compress arbitrary SPD matrices in parallel. In our largest run, we were able to compress a 67M-by-67M matrix in less than three minutes and perform a multiplication with 512 vectors within 5 seconds on 6,144 Intel "Skylake" cores.

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“…Following [1], GOFMM comprises two phases: compression and evaluation. In the compression phase, an SPD matrix K is compressed to K recursively using a binary tree such that…”

Section: A Geometric-oblivious Fmm Reviewmentioning

confidence: 99%

Section: A Geometric-oblivious Fmm Reviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Distributed-Memory Hierarchical Compression of Dense SPD Matrices

Reiz

Biros

2018

SC18: International Conference for High Performance Computing, Networking, Storage and Analysis

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A literature survey of matrix methods for data science

Stoll

2020

GAMM-Mitteilungen

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Efficient numerical linear algebra is a core ingredient in many applications across almost all scientific and industrial disciplines. With this survey we want to illustrate that numerical linear algebra has played and is playing a crucial role in enabling and improving data science computations with many new developments being fueled by the availability of data and computing resources. We highlight the role of various different factorizations and the power of changing the representation of the data as well as discussing topics such as randomized algorithms, functions of matrices, and high‐dimensional problems. We briefly touch upon the role of techniques from numerical linear algebra used within deep learning.

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