Gaussian measure
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Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces Open
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization …
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Gradient-based dimension reduction of multivariate vector-valued functions Open
Multivariate functions encountered in high-dimensional uncertainty quantification problems often vary most strongly along a few dominant directions in the input parameter space. We propose a gradient-based method for detecting these direct…
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Monte Carlo with determinantal point processes Open
We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical $N^{-1/2}$, where $N$ is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures …
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Gaussian mixtures: Entropy and geometric inequalities Open
A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixture…
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The Gram-Schmidt walk: a cure for the Banaszczyk blues Open
An important result in discrepancy due to Banaszczyk states that for any set of n vectors in Rm of ℓ2 norm at most 1 and any convex body K in Rm of Gaussian measure at least half, there exists a ±1 combination of these vectors which lies i…
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Stein's method for normal approximation in Wasserstein distances with\n application to the multivariate Central Limit Theorem Open
We use Stein's method to bound the Wasserstein distance of order $2$ between\na measure $\\nu$ and the Gaussian measure using a stochastic process $(X_t)_{t\n\\geq 0}$ such that $X_t$ is drawn from $\\nu$ for any $t > 0$. If the stochastic…
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Regularization under diffusion and anticoncentration of the information content Open
Under the Ornstein-Uhlenbeck semigroup $\\{U_t\\}$, any non-negative measurable\n$f : \\mathbb R^n \\to \\mathbb R_+$ exhibits a uniform tail bound better than\nthat implied by Markov's inequality and conservation of mass: For every $\\alp…
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On locally essentially bounded divergence measure fields and sets of locally finite perimeter Open
Chen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter usi…
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On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates Open
We show that, for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (sub-Gaussian estimates) impl…
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An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities Open
The optimal transport map between the standard Gaussian measure and an -strongly log-concave probability measure is -Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequ…
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Computable genuine multimode entanglement measure: Gaussian versus non-Gaussian Open
Genuine multimode entanglement in continuous variable systems can be\nquantified by exploring the geometry of the state-space, namely via the\ngeneralized geometric measure (GGM) which is defined as the shortest distance\nof a given multim…
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The Φ34 measure via Girsanov’s theorem Open
We construct the Φ34 measure on a periodic three dimensional box as an absolutely continuous perturbation of a random translation of the Gaussian free field. The shifted measure is constructed via Girsanov's theorem and the relevant filtra…
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The Gaussian Double-Bubble Conjecture Open
We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of th…
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The Gaussian Multi-Bubble Conjecture Open
We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a cluster, obtained from …
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Quantitative logarithmic Sobolev inequalities and stability estimates Open
We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré inequality. The result implies a lower bound on the deficit in terms of the…
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Hessian formulas and estimates for parabolic Schrödinger operators Open
We study the Hessian of the fundamental solution to the parabolic problem for weighted Schrödinger operators of the form $\frac 12 Δ+\nabla h-V$ proving a second order Feynman-Kac formula and obtaining Hessian estimates. For manifolds with…
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Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance Open
We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the o…
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Moment measures and stability for Gaussian inequalities Open
Let $γ$ be the standard Gaussian measure on $\mathbb{R}^n$ and let $\mathcal{P}_γ$ be the space of probability measures that are absolutely continuous with respect to $γ$. We study lower bounds for the functional $\mathcal{F}_γ(μ) = {\rm E…
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General measure extensions of projection bodies Open
International audience
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functions on open domains: the Wiener case and a Fomin differentiable case Open
We provide three different characterizations of the space BV (O, gamma) of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure gamma on open domains O in Wiener spaces. Throughout these different ch…
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Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations Open
We prove structure theorems for measures on the discrete cube and on Gaussian space, which provide sufficient conditions for mean-field behavior. These conditions rely on a new notion of complexity for such measures, namely the Gaussian-wi…
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Wasserstein Stability of the Entropy Power Inequality for Log-Concave Densities Open
We establish quantitative stability results for the entropy power inequality (EPI). Specifically, we show that if uniformly log-concave densities nearly saturate the EPI, then they must be close to Gaussian densities in the quadratic Wasse…
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Linear dynamical systems on Hilbert spaces: typical properties and explicit examples Open
We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. - …
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Efficient algorithms for discrepancy minimization in convex sets Open
A result of Spencer states that every collection of n sets over a universe of size n has a coloring of the ground set with of discrepancy . A geometric generalization of this result was given by Gluskin (see also Giannopoulos) who showed t…
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Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise Open
In this article, we consider the stochastic wave equation on $\\mathbb{R} _{+} \\times \\mathbb{R} $, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structures are given by loc…
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Gromov-Wasserstein Distances between Gaussian Distributions Open
The Gromov-Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measure…
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The $\Phi^4_3$ measure via Girsanov's theorem Open
We construct the Phi(4)(3) measure on a periodic three dimensional box as an absolutely continuous perturbation of a random translation of the Gaussian free field. The shifted measure is constructed via Girsanov's theorem and the relevant …
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The Deficit in the Gaussian Log-Sobolev Inequality and Inverse Santaló Inequalities Open
We establish dual equivalent forms involving relative entropy, Fisher information, and optimal transport costs of inverse Santaló inequalities. We show in particular that the Mahler conjecture is equivalent to some dimensional lower bound …
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On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness Open
We study the numerical approximation of integrals over $\mathbb{R}^s$ with respect to the standard Gaussian measure for integrands which lie in certain Hermite spaces of functions. The decay rate of the associated sequence is specified by …
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Higher-order Stein kernels for Gaussian approximation Open
We introduce higher-order Stein kernels relative to the standard Gaussian measure, which generalize the usual Stein kernels by involving higher-order derivatives of test functions. We relate the associated discrepancies to various metrics …