Elementary Results on Maximal, Generic and Typical Ranks
summary
Defines rank relative to a projective variety and works over C and R, proves max rank ≤ 2·generic rank over C and over R max rank ≤ 2·minimal typical rank, and shows minimal typical rank = generic rank of complexification. Proof is geometric: draw a line through any point, intersect an open ball of generic/typical points and write the point as a sum of two generic pieces, giving the factor-two bound. Gives examples (symmetric and small-format tensors) and lists sharpness and border-rank as open questions.
AI-written summary · about
chapters (8)
- 0:00Introduction
- 4:39Max rank over complex
- 11:03Max rank over real
- 17:11Max rank proof in one picture
- 29:57Typical ranks resolved
- 38:57Typical rank proof with semi-algebraic sets
- 47:00Open questions on ranks
- 51:58Q&A
related talks
52mVariety Membership Testing, Algebraic Natural Proofs, and Geometric Complexity Theory
36mThe Rank of Sparse Random Matrices
43mGeometric Complexity Theory and Tensor Rank
33mOn the Problems Related to Waring Rank and Border Waring Rank
30mOn Approximation Guarantees for Greedy Low Rank Optimization
01h 03mHigh Dimensional Geometry and Concentration II
58mSign-rank vs Margin
01h 00mAverage-case Complexity for Polynomials, and All That