The Average Number of Critical Rank-One Approximations to a Tensor

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Counts the average number of real critical rank‑one approximations to a random tensor and gives an explicit formula: the expected count equals a constant times the expected absolute determinant of a relatively small random symmetric matrix C whose blocks are (ni−1)×(nj−1), diagonal blocks scalar multiples of the identity, and off‑diagonals iid Gaussians. Methods use the ED correspondence and a birational parametrization to reduce a high‑dimensional integral to this determinant integral. Recovers SVD/Eckart‑Young for matrices, matches Friedland–Ottaviani’s complex ED degree (coefficient formula), yields numeric tables and observed stabilization when the largest mode ≥ sum(ni−1); asymptotics and the real jump hypersurface remain open.

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" chapters (18)

1. 0:00Introduction to tensors
2. 1:43Challenges with tensor rank
3. 4:03Understanding border rank
4. 5:46Approximation and inner products
5. 8:17Analyzing critical points
6. 10:33Matrix rank approximation
7. 14:56Average critical points
8. 18:22Structure of random matrices
9. 20:53Euclidean distance degree
10. 23:40Degrees for rank-one tensors
11. 26:07Computational analysis
12. 28:20Stabilization phenomena
13. 30:22ED correspondence
14. 34:00Proof ideas
15. 37:23Proof of the formula
16. 41:57Double counting technique
17. 47:31Symmetric tensors
18. 50:42Closing and questions