Remarks on the Discrete Cube
Talkat Probability, Geometry, and Computation in High Dimensions Boot Camp (Simons Institute, Fall 2020) · 2020-08-19..08-28
summary
Shows the naive Gaussian→cube replacement fails: Talagrand’s L∞ counterexample forces a different formulation. Proves a precise linear→nonlinear principle on the discrete cube that replaces IID signs by an average over biased Rademachers (a T‑dependent reweighting) and yields a bound with a universal π/2 constant inside an explicit μ(T) integral. Recovers Pisier’s Gaussian result by the CLT, explains why L1 Poincaré is sharp for majority, and how symmetrization handles applications.
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chapters (32)
- 0:00Introduction to lecture
- 1:37The discrete cube conjecture
- 3:38Counterexample to the conjecture
- 5:26Revisiting the conjecture
- 7:33UMD spaces and harmonic analysis
- 8:43Alternative proof methods
- 10:49Quantum methods for proof
- 13:10Classical vs quantum methods
- 14:24The correct discrete principle
- 15:49Interpolation and the semi group
- 19:36The punch line formula
- 22:11Proof of the lemma
- 26:19Smoothing interpretation
- 30:42Final theorem formulation
- 33:37The correct principle explained
- 36:10Application of the principle
- 38:46Q&A on biased signs
- 40:04Gaussian case and finite type
- 42:19Q&A on constants and moments
- 44:57Transition to boolean functions
- 45:32Boolean function properties
- 47:34Majority function analysis
- 52:34Boolean properties and variance
- 56:16L1 version of Poincare
- 58:25Proof of the L1 inequality
- 1:01:02Q&A and majority function check
- 1:04:03Gaussian case versus discrete
- 1:07:20Q&A on rotational invariance
- 1:09:07Independence in PCA proof
- 1:11:27Reparameterization and insights
- 1:13:54Conjectures and conclusions
- 1:15:43Q&A and final remarks