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Register a new userEmpirical measures: regularity is a counter-curse to dimensionality
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Authors
Beno^it Kloeckner
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We propose a decomposition method to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables, the convergence is much faster than for, say, merely Lipschitz observables. Actually, assuming $s$ derivatives with $s < d/2$ ($d$ the dimension) ensures an optimal rate of convergence of $1/sqrt{n}$ ($n$ the number of samples). The method is flexible enough to apply to Markov chains which satisfy a geometric contraction hypothesis, assuming neither stationarity nor reversibility, with the same convergence speed up to a power of logarithm factor. Our results are stated as controls of the expected distance between the empirical measure and its limit, but we explain briefly how the classical method of bounded difference can be used to deduce concentration estimates.
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This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimensional analysis and stands in contrast to classical metrics ({it e.g.}, the Wasserstein distance). The proposed metrics originate from the maximum mean discrepancy, which we generalize by proposing specific criteria for selecting test function spaces to guarantee the property of being free of CoD. Therefore, we call this class of metrics the generalized maximum mean discrepancy (GMMD). Examples of the selected test function spaces include the reproducing kernel Hilbert space, Barron space, and flow-induced function spaces. Three applications of the proposed metrics are presented: 1. The convergence of empirical measure in the case of random variables; 2. The convergence of $n$-particle system to the solution to McKean-Vlasov stochastic differential equation; 3. The construction of an $varepsilon$-Nash equilibrium for a homogeneous $n$-player game by its mean-field limit. As a byproduct, we prove that, given a distribution close to the target distribution measured by GMMD and a certain representation of the target distribution, we can generate a distribution close to the target one in terms of the Wasserstein distance and relative entropy. Overall, we show that the proposed class of metrics is a powerful tool to analyze the convergence of empirical measures in high dimensions without CoD.
In this paper we study the regularity properties of linear and polynomial images of Skorohod differentiable measures. Firstly, we obtain estimates for the Skorohod derivative norm of a projection of a product of Scorohod differentiable measures. In the second part of the paper we prove Nikolskii--Besov regularity of a polynomial image of a Skorohod differentiable measure on $mathbb{R}^n$.
Let $X_t$ be the (reflecting) diffusion process generated by $L:=Delta+ abla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $partial M$, where $Vin C^1(M)$ such that $mu(d x):= e^{V(x)}d x$ is a probability measure. We estimate the convergence rate for the empirical measure $mu_t:=frac 1 t int_0^t delta_{X_sd s$ under the Wasserstein distance. As a typical example, when $M=mathbb R^d$ and $V(x)= c_1- c_2 |x|^p$ for some constants $c_1in mathbb R, c_2>0$ and $p>1$, the explicit upper and lower bounds are present for the convergence rate, which are of sharp order when either $d<frac{4(p-1)}p$ or $dge 4$ and $ptoinfty$.
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Mingzhou Xu
, Kun Cheng (School of Information Engineering
, n Jingdezhen Ceramic Institute
2020
We prove that moderate deviations for empirical measures for countable nonhomogeneous Markov chains hold under the assumption of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chains in Ces`aro sense.
The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.
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