Equivalent Polyadic Decompositions of Matrix Multiplication Tensors

Abstract

Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de Groote, who showed that for the multiplication of 2 × 2 matrices with 7 active multiplications, all algorithms are essentially equivalent to Strassen's algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices, (e.g., 2 × 3 by 3 × 2 or 3 × 3 by 3 × 3 matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors.

Characterization of Decomposition of Matrix Multiplication Tensors

“…It was observed in [5] that the matrices H r defined in (31) are similar to H ′ r computed for A ′ , B ′ , C ′ , see (35). The similarity of the two matrices implies that the matrices have the same characteristic polynomial [25].…”

“…It was also noted in [5] that if an integer-valued decomposition exists, its generalized signature should be integer-valued as well. However, we observed that many times, the matrices H r have a low rank, and the higher elements of the generalized signature are zeros, and do not help to distinguish different decompositions.…”

“…The lower bound is 19 and the upper bound is 23. A lot of effort has been exerted to improve the algorithm of Laderman [27] (decomposition with rank 23), there are many algorithms with the same complexity [18], [16], [31], [23], [5], but the proof that a decomposition to a rank smaller than 23 is impossible is not known yet. For more introductory texts to the subject see, e.g., [7], [27], [8], [19].…”

“…Once one decomposition of the MM tensor is found, there is the whole group of De Groote transformations that lead to a whole class of equivalent solutions [22]. An algorithm for determining wether two particular decompositions of the MM tensor are equivalent in the sense of the De Groote group has been proposed in [5]. In Section IV of this paper, we extend the results of [5] and suggest several characteristics that allow to identify equivalent decompositions and, hopefully, to construct new decompositions.…”

Topology in cyber research

“…Notwithstanding the problems of defining algorithms and the undecidability of gauging equality (much less computing a principled similarity) of algorithms in general, Dowker homology can capture salient information about straight-line or basic block algorithms. 8 In this restricted setting, it is not hard to identify various narrow classes of algorithms that admit reasonable definitions.…”