A Concentration Inequality on Riemannian Path Space

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

Download Full-text

Antialiasing Complex Global Illumination Effects in Path-Space

ACM Transactions on Graphics ◽

10.1145/3072959.2990495 ◽

2017 ◽

Vol 36 (4) ◽

pp. 1 ◽

Cited By ~ 3

Author(s):

Laurent Belcour ◽

Ling-Qi Yan ◽

Ravi Ramamoorthi ◽

Derek Nowrouzezahrai

Keyword(s):

Global Illumination ◽

Path Space

Download Full-text

$$L_2$$-norm sampling discretization and recovery of functions from RKHS with finite trace

Sampling Theory, Signal Processing, and Data Analysis ◽

10.1007/s43670-021-00013-3 ◽

2021 ◽

Vol 19 (2) ◽

Author(s):

Moritz Moeller ◽

Tino Ullrich

Keyword(s):

Random Function ◽

Additional Assumption ◽

Main Tool ◽

Singular Values ◽

Rates Of Convergence ◽

Concentration Inequality ◽

Worst Case ◽

Mercer Kernels ◽

Finite Trace ◽

Complex Valued

AbstractIn this paper we study $$L_2$$ L 2 -norm sampling discretization and sampling recovery of complex-valued functions in RKHS on $$D \subset \mathbb {R}^d$$ D ⊂ R d based on random function samples. We only assume the finite trace of the kernel (Hilbert–Schmidt embedding into $$L_2$$ L 2 ) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomially in n, the number of samples. Under the mild additional assumption of separability we observe improved rates of convergence related to the decay of the singular values. Our main tool is a spectral norm concentration inequality for infinite complex random matrices with independent rows complementing earlier results by Rudelson, Mendelson, Pajor, Oliveira and Rauhut.

Download Full-text