Recent Progress in High-Dimensional Learning
Talkat Probability, Geometry, and Computation in High Dimensions Boot Camp (Simons Institute, Fall 2020) · 2020-08-19..08-28·
summary
Argues that low‑rank tensor decompositions uniquely recover latent factors under mild linear‑independence conditions and gives an explicit spectral method: form two matrix „flattenings” T( x ), T( y ), multiply one by the inverse of the other and diagonalize to extract factors — Generick's algorithm. Shows concrete applications: phylogenetic trees and hidden Markov models via a third‑moment tensor and mixtures of identical spherical Gaussians where T = sum w_i μ_i⊗μ_i⊗μ_i and empirical moment corrections recover parameters in polynomial time under full‑rank and eigengap assumptions. Notes harder continuous cases (orbit tensor decomposition) and dependence on conditioning for noise robustness.
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chapters (13)
- 0:00High-dimensional learning intro
- 0:22Tensor decompositions
- 5:21Rotation problem with tensors
- 9:12Generick’s algorithm
- 17:28Tensor decompositions in phylogeny
- 26:31HMM learning via tensors
- 34:04Conditional independence in tensors
- 35:49Mixture of spherical Gaussians
- 41:15Q&A
- 48:55Orbit retrieval and latent models
- 54:40Tensor networks and spectral methods
- 55:26Future directions
- 57:21Q&A on tensor rank
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