Tensor Decomposition I

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Defines nonsymmetric tensors as multidimensional arrays or elements of A⊗B⊗C, emphasizes NP‑hardness but constant‑time solvability for fixed dimensions, and uses flattening (matrix slices) plus minors to test multilinear rank and membership in the subspace variety. Uses tangent‑space dimension and Terracini’s lemma to predict expected rank and gives the formula r ≳ mnp/(m+n+p−2), shows 2×2×2 is nondefective (σ2 fills P7) and cites the P2×P2×P2 σ4 exception (Strassen’s degree‑9 hypersurface).

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" chapters (21)

1. 0:00Introduction to tensors
2. 0:36Defining tensors
3. 2:47Tensor product properties
4. 5:26Concept of decomposition
5. 7:52Tensor rank defined
6. 10:55Tensors as matrices
7. 14:28Tensor flattening
8. 17:18Matrix slices and rank
9. 21:25Multilinear rank example
10. 24:37Dimensionality reduction
11. 26:50Subspace variety
12. 32:31Secant varieties
13. 34:23Border rank
14. 37:06Expected tensor rank
15. 43:53Tangent space dimensions
16. 48:41Terracini's Lemma
17. 50:42Dimension example
18. 54:25Typical rank over complexes
19. 57:49Exceptions to rank rules
20. 1:02:42Symmetric tensors
21. 1:09:04Conclusion and Q&A