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Fast matrix multiplication algorithms may be useful, provided that theirrunning time is good in practice. Particularly, the leading coefficient oftheir arithmetic complexity needs to be small. Many sub-cubic algorithms havelarge leading coefficients, rendering them impractical. Karstadt and Schwartz(SPAA'17, JACM'20) demonstrated how to reduce these coefficients by sparsifyingan algorithm's bilinear operator. Unfortunately, the problem of finding optimalsparsifications is NP-Hard.  We obtain three new methods to this end, and apply them to existing fastmatrix multiplication algorithms, thus improving their leading coefficients.These methods have an exponential worst case running time, but run fast inpractice and improve the performance of many fast matrix multiplicationalgorithms. Two of the methods are guaranteed to produce leading coefficientsthat, under some assumptions, are optimal.

Sparsifying the Operators of Fast Matrix Multiplication Algorithms