Related papers: Concentration Bounds for Discrete Distribution Est…
In this work we analyze convergence of solutions for the Laplace operator with Neumann boundary conditions in a two-dimensional highly oscillating domain which degenerates into a segment (thin domains) of the real line. We consider the case…
Kullback-Leibler (KL) divergence is a fundamental concept in information theory that quantifies the discrepancy between two probability distributions. In the context of Variational Autoencoders (VAEs), it serves as a central regularization…
We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$. We find that the $j$th derivative of the estimators at $x_0$ converges at the rate…
We show that the moment generating function of the Kullback-Leibler divergence (relative entropy) between the empirical distribution of $n$ independent samples from a distribution $P$ over a finite alphabet of size $k$ (i.e. a multinomial…
We initiate the study of quadratic discrepancy for finite point sets on the Heisenberg group $\mathbb H^n$ with respect to upper Ahlfors regular probability measures. For a natural family of test sets given by left translations and…
In many contemporary statistical and machine learning methods, one needs to optimize an objective function that depends on the discrepancy between two probability distributions. The discrepancy can be referred to as a metric for…
We study the $k$-median with discounts problem, wherein we are given clients with non-negative discounts and seek to open at most $k$ facilities. The goal is to minimize the sum of distances from each client to its nearest open facility…
We study the connections between spectral clustering and the problems of maximum margin clustering, and estimation of the components of level sets of a density function. Specifically, we obtain bounds on the eigenvectors of graph Laplacian…
We consider the closeness testing problem for discrete distributions. The goal is to distinguish whether two samples are drawn from the same unspecified distribution, or whether their respective distributions are separated in $L_1$-norm. In…
We find the best asymptotic lower bounds for the coefficient of the leading term of the $L_1$ norm of the two-dimensional (axis-parallel) discrepancy that can be obtained by K.Roth's orthogonal function method among a large class of test…
This paper addresses the statistical problem of estimating the infinite-norm deviation from the empirical mean to the distribution mean for high-dimensional distributions on $\{0,1\}^d$, potentially with $d=\infty$. Unlike traditional…
We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Dirac equation. To this end we develop basic scattering theory and establish a limiting absorption principle for discrete perturbed Dirac operators.
In this paper, we derive a new dimension-free non-asymptotic upper bound for the quadratic $k$-means excess risk related to the quantization of an i.i.d sample in a separable Hilbert space. We improve the bound of order $\mathcal{O} \bigl(…
Knowledge distillation (KD) has been widely adopted to compress large language models (LLMs). Existing KD methods investigate various divergence measures including the Kullback-Leibler (KL), reverse Kullback-Leibler (RKL), and…
We study the fundamental problems of identity testing (goodness of fit), and closeness testing (two sample test) of distributions over $k$ elements, under differential privacy. While the problems have a long history in statistics, finite…
A key limitation of sampling algorithms for approximate inference is that it is difficult to quantify their approximation error. Widely used sampling schemes, such as sequential importance sampling with resampling and Metropolis-Hastings,…
We extend the notion of combinatorial discrepancy to \emph{non-additive} functions. Our main result is an upper bound of $O(\sqrt{n \log(nk)})$ on the non-additive $k$-color discrepancy when $k$ is a prime power. We demonstrate two…
Discrete diffusion models have recently gained significant prominence in applications involving natural language and graph data. A key factor influencing their effectiveness is the efficiency of discretized samplers. Among these,…
To make inferences about the shape of a population distribution, the widely popular mean regression model, for example, is inadequate if the distribution is not approximately Gaussian (or symmetric). Compared to conventional mean regression…
We study $L^p$ bounds on spectral projections for the Laplace operator on compact Riemannian manifolds, restricted to small frequency dependent neighborhoods of submanifolds. In particular, if $\lambda$ is a frequency and the size of the…