Related papers: Gaussian-width gradient complexity, reverse log-So…
Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $\mu$ with the property that $$ \int_{X} d(x, y) d\mu(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist,…
An important theme in recent work in asymptotic geometric analysis is that many classical implications between different types of geometric or functional inequalities can be reversed in the presence of convexity assumptions. In this note,…
The BBM equation is a Hamiltonian PDE which revealed to be a very interesting test-model to study the transformation property of Gaussian measures along the flow. In this paper we study the BBM equation with critical dispersion (which is a…
Motivated by problems on random differences in Szemer\'{e}di's theorem and on large deviations for arithmetic progressions in random sets, we prove upper bounds on the Gaussian width of point sets that are formed by the image of the…
In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of…
The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising from nonlinear regression problems is considered. Extending investigations from arXiv:2009.05298, local and global…
We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous…
In this paper, we study the quantitative convergence of shallow neural networks trained via gradient descent to their associated Gaussian processes in the infinite-width limit. While previous work has established qualitative convergence…
Quantum metrology pursues high-precision measurements of physical quantities by using quantum resources. However, the decoherence generally hinders its performance. Previous work found that the metrological error tends to diverge in the…
We study the behavior of the Wasserstein-$2$ distance between discrete measures $\mu$ and $\nu$ in $\mathbb{R}^d$ when both measures are smoothed by small amounts of Gaussian noise. This procedure, known as Gaussian-smoothed optimal…
This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous…
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we establish an optimal regularity result for…
A method is introduced for approximate marginal likelihood inference via adaptive Gaussian quadrature in mixed models with a single grouping factor. The core technical contribution is an algorithm for computing the exact gradient of the…
In object detection, a well-defined similarity metric can significantly enhance model performance. Currently, the IoU-based similarity metric is the most commonly preferred choice for detectors. However, detectors using IoU as a similarity…
In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we…
We use nonstandard analysis to study the problem of expressing a Gaussian integral in terms of the limiting behavior of a sequence of spherical integrals. Peterson and Sengupta proved that if a Gaussian measure $\mu$ has full support on a…
In this paper, we show that the empirical measure of mean-field model satisfies the large deviation principle with respect to the weak convergence topology or the stronger Wasserstein metric, under the strong exponential integrability…
Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown that it provides performances similar to its non-smoothed…
The mean-field theory for two-layer neural networks considers infinitely wide networks that are linearly parameterized by a probability measure over the parameter space. This nonparametric perspective has significantly advanced both the…
Identifying low-dimensional structure in high-dimensional probability measures is an essential pre-processing step for efficient sampling. We introduce a method for identifying and approximating a target measure $\pi$ as a perturbation of a…