Tightness of the O(n^2) lower bound for matrix multiplication

Determine whether the O(n^2) lower bound on the time complexity of matrix multiplication is tight, i.e., establish whether the matrix multiplication exponent ω equals 2 (up to polylogarithmic factors), which would set the ultimate limit for speedups based on fast matrix multiplication in non-ellipsoid-specific fitting and related linear algebra routines.

Background

In assessing computational limits for ellipsoid fitting and related linear algebra tasks, the authors note that faster methods based on matrix multiplication (e.g., Strassen’s algorithm) can reduce complexity in non-ellipsoid-specific settings but cannot bypass an O(n2) barrier due to fundamental output-size constraints.

They explicitly point out that whether this O(n2) barrier is tight remains a broader open question in computer science, and thus directly impacts the best achievable asymptotics for any approach that relies on fast matrix multiplication to accelerate the fitting or solving steps discussed in the paper.

References

Faster methods, such as those based on Strassen's algorithm (≈ O(n{2.807})), are possible in the non-ellipsoid-specific case, but the absolute lower bound for such speedups is O(n2) (whether this bound is tight is an open question in computer science), which still leaves us with a complexity at least O(d4).

— Every Language Model Has a Forgery-Resistant Signature  (2510.14086 - Finlayson et al., 15 Oct 2025) in Section 3.3 (Ellipsoid fitting takes sextic time)

The algebraic complexity of this operation is $\mathcal{O}(Np)$ for some $2\le p<2.372$; cf., and it remains open whether $p=2$.

— A Closed-Form Formula for Consistent Lipschitz Regression on Metric Spaces with Sparse Neural Network Realizations  (2609.03129 - Hong et al., 2 Sep 2026) in Section 3, subsection “Comparison: Complexity of Our Formula Vs. Exact KRR”