Formulas for Riesz Kernels

on YouTube ↗51m

" summary

Asks when P^{-α} (P a hyperbolic homogeneous polynomial positive on its hyperbolicity cone C) is completely monotone and constructs the Riesz kernel Q_α via the inverse Laplace transform; shows Q_α ≥ 0 is equivalent to complete monotonicity. Gives exact exponent classification for determinant polynomials: α ∈ {0, 1/2, 1, …, (n−1)/2} or α > (n−1)/2, with the kernel realized by the Wishart distribution. For product-of-linears Q_α is an explicit Gelfand–Aomoto hypergeometric function. Conjecture: Q_α ≥ 0 for all sufficiently large α remains open.

AI-written summary · about

" chapters (4)

1. 0:00Start
2. 12:42Inverse Laplace Transform
3. 17:27Determinant of Polynomial
4. 29:40Product of Linear Form

" related talks

• 12mDeep Neural Tangent Kernel and Laplace Kernel Have the Same RKHSLin Chen · Meet the Fellows Welcome Event
• 01h 22mGeneralization IVPeter Bartlett, Sasha Rakhlin · Deep Learning Boot Camp 2019
• 54mNegative Dependence and Lorentzian PolynomialsPetter Brändén · Deterministic Counting, Probability, and Zeros of Partition Functions (Simons Institute workshop, 2019)
• 01h 00mA Theoretical Framework of Convolutional Kernels on Image DatasetsSong Mei · Deep Learning Theory Workshop and Summer School 2022
• 45mTractability And Overparametrization In Some Simple ModelsAndrea Montanari · Algorithmic Advances for Statistical Inference with Combinatorial Structure
• 35mTesting Positive Semidefiniteness and Eigenvalue ApproximationDavid P. Woodruff · Sketching and Algorithm Design 2023
• 29mComputational Barriers to Estimation from Low-Degree PolynomialsTselil Schramm · Rigorous Evidence for Information-Computation Trade-offs
• 01h 14mApproximation PowerMatus Telgarsky · Deep Learning Boot Camp 2019