Stability of Floating-Point Least-Squares Approximate Matrix Multiplication

Prove numerical stability for an approximate matrix multiplication algorithm obtained from a floating-point least-squares solution to the matrix-multiplication tensor decomposition problem.

Background

The paper establishes stability results for APA algorithms whose symbolic parameterized decompositions are analyzed explicitly. It distinguishes these schemes from approximate matrix multiplication algorithms generated by floating-point least-squares optimization of tensor decompositions. Although several such algorithms exist, their numerical stability remains unresolved, and the paper identifies proving stability for them as an open problem.

References

In particular, it is still open to prove numerical stability for an approximate matrix multiplication algorithm obtained from a floating-point, least-squares solution to the tensor decomposition problem eqn: MM_tensor; several such algorithms exist in the literature and should, like APA matrix multiplication, be considered for practical use.

eqn: MM_tensor:

M⟨m,k,n⟩≔∑i1=1m∑i2=1k∑i3=1nEi1i2(m,k)⊗Ei2i3(k,n)⊗Ei3i1(n,m).{\mathcal M}_{\langle m,k,n \rangle} \coloneqq \sum_{i_1 = 1}^m \sum_{i_2 = 1}^k \sum_{i_3=1}^n E_{i_1i_2}^{(m,k)} \otimes E_{i_2i_3}^{(k,n)} \otimes E_{i_3i_1}^{(n,m)}.

— Fast and Stable Arbitrary Precision Approximate Matrix Multiplication  (2610.13905 - Dumitriu et al., 9 Oct 2026) in Section 7, Conclusion