Orthogonal Tensor Decomposition

on YouTube ↗48m

" summary

Characterizes which symmetric tensors admit an orthogonal decomposition, proves the orthonormal factors are exactly the robust eigenvectors that attract almost every random initialization under the tensor power map θ ↦ T·θ^{d−1} (iterate, normalize, subtract rank‑1 term, repeat), and shows the ML topic‑model tensor can be turned into an ODECO tensor and efficiently decomposed to recover parameters. Gives exact eigen/singular counts (example: 7 isolated solutions; generic counts from Sturmfels/Cartwright and Friedland–Ottaviani) and conjectures explicit polynomial equations for the ODECO variety.

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" chapters (15)

1. 0:00Intro
2. 1:25Symmetric Tensors and Polynomials
3. 4:40Orthogonal Tensor Decomposition
4. 6:36An Application: Exchangeable Single Topic Models
5. 11:29Eigenvectors of Symmetric Tensors
6. 16:09Robust Eigenvectors
7. 20:41The Number of Eigenvectors of a Tensor
8. 21:17Eigenvectors of Odeco Tensors
9. 23:34The Set of Odeco Tensors
10. 25:12The Odeco Variety
11. 27:26Nonsymmetric Tensor Decomposition
12. 28:24Nonsymmetric Odeco Tensors
13. 30:46Examples
14. 34:26The number of singular vector tuples of a tensor
15. 34:57Singular Vectors of Odeco Tensors

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