Group-Theoretic Partial Matrix Multiplication

A generalization of recent group-theoretic matrix multiplication algorithms to an analogue of the theory of partial matrix multiplication is presented. We demonstrate that the added flexibility of this approach can in some cases improve upper bounds on the exponent of matrix multiplication yielded by group-theoretic full matrix multiplication. The group theory behind our partial matrix multiplication algorithms leads to the problem of maximizing a quantity representing the “fullness” of a given partial matrix pattern. This problem is shown to be NP-hard, and two algorithms, one optimal and another non-optimal but polynomial-time, are given for solving it.

💡 Research Summary

The paper extends the Cohn‑Umans group‑theoretic framework for fast matrix multiplication by allowing the underlying subsets of a group to violate the strict triple‑product property (TPP). In the original Cohn‑Umans approach, three ordered subsets S, T, U of a finite group G must satisfy TPP so that each product term in the group algebra corresponds uniquely to a single entry of the matrix product. The authors relax this requirement and introduce the notion of “aliasing”: when TPP fails, extra terms appear in the convolution, contaminating the output matrix. They observe that if any of the input entries that contribute to an aliasing term are set to zero, the erroneous term disappears. Consequently, they define an “aliasing set” A consisting of triples ((i,j),(j′,k),(i′,k′)) that describe which pairs of input entries generate a spurious output entry.

To control aliasing they introduce covering sets I ⊆ indices of the left factor and J ⊆ indices of the right factor. If every aliasing triple has at least one of its two input positions in I∪J, then zeroing those positions eliminates all spurious terms and the Cohn‑Umans algorithm computes the correct partial matrix product. The quality of a covering pair (I,J) is measured by a function

 f(I,J) = Σ_ℓ k_ℓ·n_ℓ,

where k_ℓ counts non‑zero entries in column ℓ of the left matrix that are not forced to zero, and n_ℓ counts non‑zero entries in row ℓ of the right matrix that are not forced to zero. The maximal value over all coverings, f(A), quantifies how much genuine computation the partial product performs. When A is empty (i.e., TPP holds) we have f(A)=|S||T||U|, the full amount of work.

The authors prove a new upper bound on the matrix‑multiplication exponent ω that incorporates f(A):

 ω ≤ 3·log(∏_i d_i)·log f(A),

where {d_i} are the character degrees of G. This is a direct analogue of the Cohn‑Umans bound, but now the “information content” of the multiplication is captured by f(A). Hence, by allowing aliasing and carefully selecting which entries to zero, one can increase f(A) and potentially obtain a tighter bound on ω.

The central combinatorial problem is: given A, find a covering (I,J) that maximizes f(I,J). The paper shows this problem is NP‑hard by reduction from a known hard covering problem. Consequently, exact optimization requires exponential time in the worst case. Nevertheless, the authors present two algorithms:

1. Polynomial‑time non‑optimal algorithm – They model the aliasing triples as edges of a bipartite graph whose left vertices correspond to entries of the left matrix and right vertices to entries of the right matrix. Finding a minimum vertex cover (equivalently, a maximum matching via König’s theorem) yields a covering that zeroes the fewest entries. This does not guarantee maximal f, but provides a “good enough” covering quickly.

2. Exact exponential‑time algorithm – A branch‑and‑bound search explores, for each aliasing triple, whether to zero the left or the right entry. The current f(I,J) is computed incrementally; if it falls below the best value found so far, the subtree is pruned. This algorithm returns the optimal f(A) but runs in time exponential in |A|.

To demonstrate the practical impact, the authors revisit the construction from Cohn et al. (2005) based on the group G = (C_n × C_n × C_n) ⋊ S_2. In the original setting, subsets S, T, U each have size 2n(n‑1) and satisfy TPP, giving f = 8n³(n‑1)³ and an exponent bound ω ≤ 2.9088 for n = 17. The paper augments each subset with the identity element (e_H, e_H), forming S₀, T₀, U₀. This introduces three categories of aliasing (bottom, top‑easy, top‑hard). By zeroing only the newly added rows/columns that participate in aliasing, they obtain a covering that leaves the original entries untouched while still eliminating all spurious terms. The resulting f(A) is shown to be at least

 (2n(n‑1))³ + (2n(n‑1))² + 2n(n‑1) − (n‑1)² + 1,

which for n = 17 yields a slightly larger information count and improves the exponent bound to ω ≤ 2.9084. Although the improvement is modest, it validates the central thesis: partial matrix multiplication with controlled aliasing can surpass the best known bounds derived from full‑matrix constructions.

In conclusion, the paper introduces a novel dimension to group‑theoretic matrix multiplication by allowing partial products and explicitly handling aliasing through combinatorial coverings. It establishes the computational hardness of the optimal covering problem, supplies both practical and exact algorithms, and demonstrates a concrete improvement on the exponent ω. The work opens several avenues for future research, such as exploring larger or non‑abelian groups, designing better heuristics for the covering problem, or integrating the partial‑product perspective with other algebraic techniques (e.g., laser method, tensor surgery) to push ω ever closer to the conjectured optimal value of 2.