Related papers: Cheeger's isoperimetric problem for Gaussian mixtu…

Mixture models, such as Gaussian mixture models, are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the…

We define a family {$\gamma(P)$} of generalized Euler constants indexed by finite sets of primes $P$ and study their distribution. These arise from partial sums of reciprocals of integers not divisible by any prime in $P$. An apparent…

Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be calculated…

We introduce notions of Cheeger constants for graphons and graphings. We prove Cheeger and Buser inequalities for these. On the way we prove co-area formulae for graphons and graphings.

Given a group $\Gamma$, we establish a connection between the unitarisability of its uniformly bounded representations and the asymptotic behaviour of the isoperimetric constants of Cayley graphs of $\Gamma$ for increasingly large…

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk…

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if…

We consider the inverse problem of estimating an unknown function $u$ from noisy measurements $y$ of a known, possibly nonlinear, map $\mathcal{G}$ applied to $u$. We adopt a Bayesian approach to the problem and work in a setting where the…

By exploiting the well-known observation that size-biasing or zero-biasing an infinitely divisible random variable may be achieved by adding an independent increment, combined with tools from Stein's method for compound Poisson and Gaussian…

The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse…

We focus on Bayesian inverse problems with Gaussian likelihood, linear forward model, and priors that can be formulated as a Gaussian mixture. Such a mixture is expressed as an integral of Gaussian density functions weighted by a mixing…

We study Cheeger and $p$-eigenvalue partition problems depending on a given evaluation function $\Phi$ for $p\in[1,\infty)$. We prove existence and regularity of minima, relations among the problems, convergence, and stability with respect…

Let $(X,d,\mu)$ be a complete metric measure space, with $\mu$ a locally doubling measure, that supports a local weak $L^2$-Poincar\'e inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic…

Let M be a bounded domain of a Euclidian space with smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over…

The method of moments is a classical statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding…

Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the nonlocal version of the…

The classical Gaussian mixture model assumes homogeneity within clusters, an assumption that often fails in real-world data where observations naturally exhibit varying scales or intensities. To address this, we introduce the…

We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above .

We consider estimating the parameters of a Gaussian mixture density with a given number of components best representing a given set of weighted samples. We adopt a density interpretation of the samples by viewing them as a discrete Dirac…

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more…