The article presents a note on the 6 well known matrix compositions. He states that all of them have cubic complexity, but practical algorithms with better exponents exist for all of them.
(of course, it's not so clear cut, since robustness varies, not all methods are applicable, and the benchmark is for solving the system, not computing the decomposition, but overall, knowledge of which decomposition is fast and which is not is absolutely crucial to practitioners)
[1]: https://eigen.tuxfamily.org/dox/group__DenseDecompositionBen...
I'm aware of randomized algorithms with better complexity, which come at the cost of only giving approximate results (though the approximation may be perfectly good for practical purposes). See e.g. [1]. Are there other approaches?
For example, the SVD (Golub-Kahan-Reinsch) method generally involves bidiagonalisation of the matrix and then implicit QR steps, which are often achieved with Householder reflections and Givens rotations. Sure, all of those are conceptually matrix multiplications, but they're not implemented as O(N^3) matrix multiplications; rather, their special structure is exploited so that they're faster. Yet, the entire algorithm is still cubic. So not sure Strassen would accelerate things.