Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the isoperimetric constant, covariance inequalities and $L_p$-Poincar{e} inequalities in dimension one
73
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheegers inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincar{e} inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincar{e} inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheegers inequality, which is a $L_{p}$-Poincar{e} inequality for $p=2$, to any real $pgeq 1$.
rate research
Read More
The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{e} constant. First we revisit E. Milmans result [40] on the link between weak (Poincar{e} or concentration) inequalities and Cheegers inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincar{e} (dimensional) bound in the inconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein). A mollification procedure is also introduced enabling, in the logconcave case, to reduce to the case of the Poincar{e} inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz,...).
We prove that if ${(P_x)}_{xin mathscr X}$ is a family of probability measures which satisfy the log-Sobolev inequality and whose pairwise chi-squared divergences are uniformly bounded, and $mu$ is any mixing distribution on $mathscr X$, then the mixture $int P_x , mathrm{d} mu(x)$ satisfies a log-Sobolev inequality. In various settings of interest, the resulting log-Sobolev constant is dimension-free. In particular, our result implies a conjecture of Zimmermann and Bardet et al. that Gaussian convolutions of measures with bounded support enjoy dimension-free log-Sobolev inequalities.
In this paper we establish some explicit and sharp estimates of the spectral gap and the log-Sobolev constant for mean field particles system, uniform in the number of particles, when the confinement potential have many local minimums. Our uniform log-Sobolev inequality, based on Zegarlinskis theorem for Gibbs measures, allows us to obtain the exponential convergence in entropy of the McKean-Vlasov equation with an explicit rate constant, generalizing the result of [10] by means of the displacement convexity approach, or [19, 20] by Bakry-Emery technique or the recent [9] by dissipation of the Wasserstein distance.
We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.
If Poincar{e} inequality has been studied by Bobkov for radial measures, few is known about the logarithmic Sobolev inequalty in the radial case. We try to fill this gap here using different methods: Bobkovs argument and super-Poincar{e} inequalities, direct approach via L1-logarithmic Sobolev inequalities. We also give various examples where the obtained bounds are quite sharp. Recent bounds obtained by Lee-Vempala in the logconcave bounded case are refined for radial measures.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
