Communication Lower Bounds and Optimal Algorithms for Symmetric Matrix Computations
<|reference_start|>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.<|reference_end|>
@article{daas2024communication,
    title={Communication Lower Bounds and Optimal Algorithms for Symmetric Matrix
  Computations},
    author={Hussam Al Daas (STFC, Scientific Computing Department, Rutherford
  Appleton Laboratory, Didcot, UK) and Grey Ballard (Wake Forest University,
  Computer Science Department, Winston-Salem, NC, USA) and Laura Grigori (EPFL,
  Institute of Mathematics, Lausanne, Switzerland and PSI, Center for
  Scientific Computing, Theory and Data, Villigen, Switzerland) and Suraj Kumar
  (Institut national de recherche en sciences et technologies du num'erique,
  Lyon, France) and Kathryn Rouse (Inmar Intelligence, Winston-Salem, NC, USA)
  and Mathieu Verite (EPFL, Institute of Mathematics, Lausanne, Switzerland)},
    journal={arXiv preprint arXiv:2409.11304},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.11304},
    primaryClass={cs.DC}
}