Powers of Tensors and Fast Matrix Multiplication

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Claims a tiny improvement to the best known upper bound on the matrix multiplication exponent omega by applying the laser method (version 2.3) to high powers of the Coppersmith–Winograd tensor and computing the analysis in polynomial time. The work implements a convex optimization formulation of the laser entropy/value method with an explicit penalty (gamma) term, runs it on powers 16 and 32, and produces numerical bounds slightly better than the prior ~2.3725, demonstrating the procedure is effective.

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1. 0:00Intro
2. 0:20Algebraic Complexity of Matrix Multiplication
3. 2:21Applications of our method
4. 5:04The tensor of matrix multiplication
5. 5:19General 3-tensors
6. 8:17The "laser method"
7. 14:51The first CW construction: Analysis
8. 16:24Idea behind the proof
9. 29:37The laser method: general formulation
10. 32:52Example: The first CW construction
11. 35:00The second CW construction: laser method
12. 35:47Analysis of the second construction
13. 37:21Analysis of the fourth power
14. 41:55The laser method: computing the bound
15. 45:18Conclusion