Volume Doubling Condition and a Local Poincaré Inequality on Unweighted Random Geometric Graphs
The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider $\varepsilon$-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of $\mathbb{R}^K$. We show that a volume doubling condition (VD) and local Poincar\'e inequality (LPI) hold for the random geometric graph (with high probability, and uniformly over all shortest path distance balls in a certain radius range) under suitable regularity conditions of the underlying submanifold and the sampling distribution.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Conditionally Resampled Sliding-Window Count Kernels: Spectral-Gap Bounds and Poincaré Inequalities
We study the conditionally resampled sliding-window count kernel associated with the empirical counts of length-$n$ windows from a stationary finite-state reversible Markov chain. Although the resulting count process is …
Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials
In this paper, we prove that optimizability of any F using Gradient Flow from all initializations implies a Poincar\'e Inequality for Gibbs measures mu_{beta} = e^{-\beta F}/Z at low temperature. In particular, under mil…
Diffusion annealed Langevin dynamics: a theoretical study
In this work we study the diffusion annealed Langevin dynamics, a score-based diffusion process recently introduced in the theory of generative models and which is an alternative to the classical overdamped Langevin diff…
Poincaré Inequality for Local Log-Polyak-Lojasiewicz Measures : Non-asymptotic Analysis in Low-temperature Regime
Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynami…
Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev
Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $\pi$ under the sole assumption that $\pi$ satisfies a Poincar\'e inequality. Using this fact to provide gua…