Tensor Decomposition II

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" summary

Claims that low border rank is certified by specific minors and by two families of equations (flattenings plus Strassen-type polynomials), and gives a concrete decomposition workflow: compute the Catalecticant matrix, if that fails build an 18×18 Koszul-twisted flattening, find its kernel (dimension 4 for a general plane quintic), extract seven projective eigenvectors of tensors from a general kernel element, and solve a small linear system to recover the seven linear forms. Implementation exists in Macaulay2 and decodes a general plane quintic in ~0.5s (vs Bertini minutes on many cores).

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" chapters (13)

1. 0:00Identifying tensor rank
2. 1:58Secant varieties equations
3. 4:57Wearing decomposition
4. 7:03Solving decomposition systems
5. 10:45Dimension and rank
6. 15:18Uniqueness of decomposition
7. 19:58Sylvester's algorithm
8. 29:00Example of decomposition
9. 37:12Algorithm success conditions
10. 46:46Eigenvectors of tensors
11. 51:59Eigenvector counting
12. 55:48Cozul algorithm
13. 1:02:04Applications and discussion