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
Proves the Poincaré inequality on the discrete cube using an explicit semi-group Pt (a product formula with biased Bernoullis), computes D_t Pt = ΔPt and integration-by-parts identities, then asks how vector-valued norms behave and introduces a linear-to-nonlinear principle. States Pisier/PCA’s Gaussian inequality: E||F−E F||^p ≤ (π/2)^p E||Σ G′_i ∂_iF(G)||^p, proved by rotating the Gaussian pair G,G′ and interpolating the function values; applies it to matrices/tensors (noncommutative Khinchin/Rudelson) and notes the π/2 constant is sharp.
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chapters (17)
- 0:00Start
- 0:25Poker Inequality
- 0:59The Semi Group Ptf
- 5:45The Product Rule
- 10:50Matrix Vector Formulas
- 13:26Formulas for Laplacian
- 26:16L1 Norm
- 29:40Linear to Non-Linear Principle
- 33:38Linear Non-Linear Principle
- 35:32The Triangle Inequality
- 36:33Bonus Space Theory
- 47:51The Non-Commutative Kitchen Inequality
- 48:09Operator Norm
- 1:01:42Rotation and Variance
- 1:05:36Expectation of the Absolute Value of F
- 1:06:31Indicator Functions
- 1:12:05Isoparametric Inequality