Using alternative tensor identities for sparse output computation

Determine whether alternative rank or border-rank identities, including identities associated with the Coppersmith–Winograd tensor, can be adapted to compute sparse sets of entries of thin matrix products and thereby extend the paper’s savings to larger inner dimensions.

Background

The presented method relies on specific sparsity and sharing properties of Schönhage’s identity. The paper suggests that identities with better inner-product dimensions might extend the method, but states that it is unclear how to exploit other tensors and whether suitable rank or border-rank expressions exist.

References

It is not immediately clear how to make use of other tensors, since our analysis uses specific properties of Sch\"onhage's identity, such as its sparsity and the way the leaves of its recursion are shared among the output entries (Section~\ref{sec:sparse}).

— Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs  (2610.06783 - Alman et al., 5 Oct 2026) in Section 8, Section “Conclusion,” paragraph “Algebraic algorithm design”