Matrix Multiplication, And The Asymptotic Spectrum Of Tensors

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" summary

Survey revives Strassen's asymptotic spectra theory, frames tensors as a preordered semiring and proves a strong duality: asymptotic rank equals the maximum over monotone semiring homomorphisms (the asymptotic spectrum). It uses real algebraic geometry–style duality to rederive Schönhage’s asymptotic sum (tau) inequality that turns bounds like rank(MM2) ≤ 7 into ω ≤ log2 7 ≈ 2.81 and explains how Schönhage’s direct‑sum analyses yield ω ≤ 2.55. The work also proves star‑convexity of tensor spectra and extends the framework to tensor networks, plus a detailed modern survey and proofs.

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" chapters (12)

1. 0:00Introduction and motivation
2. 2:13Overview of the framework
3. 4:25Asymptotics and complexity
4. 11:19Matrix multiplication tensors
5. 18:24Asymptotic rank definition
6. 21:46Symmetry and order on tensors
7. 28:34Semi-ring structure
8. 32:14The duality theorem
9. 37:37Structure of asymptotic spectra
10. 44:24Application to matrix exponents
11. 50:53Connectivity and conclusions
12. 58:00Q&A session