The matricizedtensor times KhatriRao product (MTTKRP) computation is the typical bottleneck in algorithms for computing a CP decomposition of a tensor. In order to develop high performance sequential and parallel algorithms, we establish communication lower bounds that identify how much data movement is required for this computation in the case of dense tensors. We also present sequential and parallel algorithms that attain the lower bounds and are therefore communication optimal. In particular, we show that the structure of the computation allows for less communication than the straightforward approach of casting the computation as a matrix multiplication operation.
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Parallel MemoryIndependent Communication Bounds for SYRK
In this paper, we focus on the parallel communication cost of multiplying a matrix with its transpose, known as a symmetric rankk update (SYRK). SYRK requires half the computation of general matrix multiplication because of the symmetry of the output matrix. Recent work (Beaumont et al., SPAA '22) has demonstrated that the sequential I/O complexity of SYRK is also a constant factor smaller than that of general matrix multiplication. Inspired by this progress, we establish memoryindependent parallel communication lower bounds for SYRK with smaller constants than general matrix multiplication, and we show that these constants are tight by presenting communicationoptimal algorithms. The crux of the lower bound proof relies on extending a key geometric inequality to symmetric computations and analytically solving a constrained nonlinear optimization problem. The optimal algorithms use a triangular blocking scheme for parallel distribution of the symmetric output matrix and corresponding computation.
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 NSFPAR ID:
 10425091
 Date Published:
 Journal Name:
 Proceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures
 Page Range / eLocation ID:
 391 to 401
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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