Related papers: Higher Order Concentration of Measure
In this article we shall study the following elliptic system with coefficients: \begin{equation}\notag \left\{\begin{aligned} -\epsilon^2\Delta u +c(x)u=b(x)|v|^{q-1}v, &\text{ and } -\epsilon^2\Delta v +c(x)v=a(x) |u|^{p-1}u &&\text{in }…
We prove density of smooth functions in subspaces of Sobolev- and higher order $BV$-spaces of kind $W^{m,p}(\Omega)\cap L^q(\Omega-D)$ and $BV^m(\Omega)\cap L^q(\Omega-D)$, respectively, where $\Omega\subset\mathbb{R}^n$ ($n\in\mathbb{N}$)…
The study of finite approximations of probability measures has a long history. In (Xu and Berger, 2017), the authors focus on constrained finite approximations and, in particular, uniform ones in dimension $d=1$. The present paper gives an…
We present new concentration of measure inequalities for Markov chains, generalising results for chains that are contracting in Wasserstein distance. These are particularly suited to establishing the cut-off phenomenon for suitable chains.…
As context windows in large language models continue to expand, it is essential to characterize how attention behaves at extreme sequence lengths. We introduce token-sample complexity: the rate at which attention computed on $n$ tokens…
During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity…
We show an extension of Sanov's theorem on large deviations, controlling the tail probabilities of i.i.d. random variables with matching concentration and anti-concentration bounds. This result has a general scope, applies to samples of any…
Viewing a two time scale stochastic approximation scheme as a noisy discretization of a singularly perturbed differential equation, we obtain a concentration bound for its iterates that captures its behavior with quantifiable high…
Motivated by problems in high-dimensional statistics such as mixture modeling for classification and clustering, we consider the behavior of radial densities as the dimension increases. We establish a form of concentration of measure, and…
Suppose (standardized) measurements or statistics are monitored to raise an alarm when a threshold is exceeded. Often, the underlying population is heterogenous with respect to important discrete variables and thus samples may consist of…
We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for…
This note is concerned with weakly interacting stochastic particle systems with possibly singular pairwise interactions. In this setting, we observe a connection between entropic propagation of chaos and exponential concentration bounds for…
Probabilistic submeasures generalizing the classical (numerical) submeasures are introduced and discussed in connection with some classes of aggregation functions. A special attention is paid to triangular norm-based probabilistic…
We investigate the relation between moments and tails of heavy-tailed (in particular, Pareto-type) distributions. We also discuss the sharpness of our results in a number of examples under certain regularity conditions like log-convexity.…
The strength function of a single state $|d>$ is studied using the deformed Gaussian orthogonal ensemble. In particular we study the dependence of the spreading width of $|d>$ on the degree of mixing.
This paper establishes sharp dimension-free concentration inequalities and expectation bounds for the deviation of the sum of simple random tensors from its expectation. As part of our analysis, we use generic chaining techniques to obtain…
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,…
In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative $1$-Lipschitz functions with Dirichlet boundary condition around zero, which is called…
We propose a novel approach to the analysis of covariance operators making use of concentration inequalities. First, non-asymptotic confidence sets are constructed for such operators. Then, subsequent applications including a k sample test…
We consider divergence form uniformly parabolic SPDEs with bounded and measurable leading coefficients and possibly growing lower-order coefficients in the deterministic part of the equations. We look for solutions which are summable to the…