The Geometry of the Identifiability of Tensors
summary
Uses projective geometry and Terracini's lemma to turn identifiability into concrete tests on secant varieties and tangent spans, implements a linear-algebra stacked matrix test so computations scale to tens of thousands of factors, and checks smoothness of points on sigma_k. Produces exact nonidentifiable cases: P1^5 (2x2x2x2x2) general tensors have exactly two decompositions via an elliptic normal curve, 4x4x4 rank-6 also gives two decompositions, and 3x6x6 rank-8 shows six decompositions; symmetric cubic rank-9 yields another elliptic-driven counterexample.
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chapters (23)
- 0:00Intro
- 0:54Simple tensors
- 1:34The complexity of a tensor
- 2:21The geometric point of view
- 4:05Problems on the Geometry of tensors
- 5:23Projective Geometry: secant varieties
- 6:57Geometric interpretation
- 8:03Geometric translation
- 9:21Local method: Terracini's Lemma
- 11:14Consequences of Terracini's Lemma
- 16:14Using degenerate subvarieties
- 23:26Maximal rank for identifiability
- 26:30General results
- 30:07Initial step
- 31:39Conjecture
- 34:15Unbalanced tensors of type a xbxc
- 34:53Other examples
- 37:33Some computational aspect
- 39:04Stacked Hessians
- 39:42Specific tensors
- 41:18Symmetric tensors
- 42:08Cubics
- 42:54Other perspectives
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