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In this paper, we focus on the parallel communication cost of multiplying a matrix with its transpose, known as a symmetric rank-k 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 memory-independent parallel communication lower bounds for SYRK with smaller constants than general matrix multiplication, and we show that these constants are tight by presenting communication-optimal 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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This content will become publicly available on April 10, 2026
Communication Lower Bounds and Optimal Algorithms for Symmetric Matrix Computations
In this article, we focus on the communication costs of three symmetric matrix computations: i) multiplying a matrix with its transpose, known as a symmetric rank-k update (SYRK) ii) adding the result of the multiplication of a matrix with the transpose of another matrix and the transpose of that result, known as a symmetric rank-2k update (SYR2K) iii) performing matrix multiplication with a symmetric input matrix (SYMM). All three computations appear in the Level 3 Basic Linear Algebra Subroutines (BLAS) and have wide use in applications involving symmetric matrices. We establish communication lower bounds for these kernels using sequential and distributed-memory parallel computational models, and we show that our bounds are tight by presenting communication-optimal algorithms for each setting. Our lower bound proofs rely on applying a geometric inequality for symmetric computations and analytically solving constrained nonlinear optimization problems. The symmetric matrix and its corresponding computations are accessed and performed according to a triangular block partitioning scheme in the optimal algorithms.
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- PAR ID:
- 10583125
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- ACM Transactions on Parallel Computing
- ISSN:
- 2329-4949
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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