High-Dimensional Statistics I

on YouTube ↗01h 30m

" summary

High-dimensional settings need non-asymptotic guarantees: classification with D comparable to N shows a Kolmogorov-style alpha = N/D offset in error rates; covariance spectra follow Marchenko–Pastur so raw sample covariances are inconsistent. L1 methods (lasso) recover S-sparse signals when design obeys a restricted eigenvalue or RIP-like bound; random designs give sample complexity roughly N ~ s log D (not s^2 log D), and oracle inequalities yield prediction and parameter-error rates ~s log D / N.

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" chapters (17)

1. 0:00Start
2. 5:28Vignette I: Linear discriminant analysis
3. 6:45Classical vs. high-dimensional asymptotics
4. 10:19Vignette Il: Covariance estimation
5. 19:57Low-dimensional structure: Gaussian graphical models
6. 21:14Gauss-Markov models with hidden variables
7. 23:26Introduction
8. 24:06Outline
9. 25:16Noiseless linear models and basis pursuit
10. 27:30Noiseless recovery: Unrescaled sample size
11. 28:16Noiseless recovery: Rescaled
12. 29:00Restricted nullspace: necessary and sufficient
13. 36:58Illustration of restricted nullspace property
14. 40:10Some sufficient conditions
15. 45:29Violating matrix incoherence (elementwise/RIP)
16. 49:36Direct result for restricted nullspace/eigenvalues
17. 56:50Easy verification of restricted nullspace

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