Asymptotic Spectra: Old and New Insights
summary
Introduces the asymptotic spectrum of tensors as equivalence classes in a commutative semiring, links it to the matrix multiplication exponent ω (Strassen recursion gives ω ≤ 2.81), and formalizes asymptotic restriction via an archimedean positive cone (a Stone pre-order) so the spectrum embeds in a compact space of continuous functions. Proposes a stronger preorder called disjoint copy restriction that yields a parametrized exponent inequality involving ε, sketches how algorithms become inequalities on the spectrum, and outlines where concrete bounds remain unknown.
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chapters (17)
- 0:00Start
- 4:16Strassen's algorithm (2)
- 4:46Tensor rank
- 8:38Strassen's algorithm and tensors
- 11:46Equivalence classes of tensors
- 19:22The ring B
- 23:53Archimedian orders
- 26:39A counter example
- 32:12Asymptotic restriction
- 34:21Asymptotic cone
- 44:50Imbedded asymptotic spectra (2)
- 46:09Example: Strassen's algorithm
- 48:21Virtual tensors
- 48:31The spectrum of matrix multiplication, once again
- 50:25Example: Group theoretic approach
- 51:54Why is it useful?
- 53:18Again, why is this useful?