相关论文: Computational Concentration of Measure: Optimal Bo…

A metric probability space $(\Omega,d)$ obeys the ${\it concentration\; of\; measure\; phenomenon}$ if subsets of measure $1/2$ enlarge to subsets of measure close to 1 as a transition parameter $\epsilon$ approaches a limit. In this paper…

Many modern machine learning classifiers are shown to be vulnerable to adversarial perturbations of the instances. Despite a massive amount of work focusing on making classifiers robust, the task seems quite challenging. In this work,…

This article investigates the probabilistic relationship between quantum classification of Boolean functions and their Hamming distance. By integrating concepts from quantum computing, information theory, and combinatorics, we explore how…

We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned…

We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from [DPMRF22]; (ii) injective matrix product states; (iii) output states of dense…

We study optimal transport between two high-dimensional distributions $\mu,\nu$ in $R^n$ from an algorithmic perspective: given $x \sim \mu$, find a close $y \sim \nu$ in $poly(n)$ time, where $n$ is the dimension of $x,y$. Thus, running…

Using geometric techniques like projection and dimensionality reduction, we show that there exists a randomized sub-linear time algorithm that can estimate the Hamming distance between two matrices. Consider two matrices ${\bf A}$ and ${\bf…

Accurate approximation of probability measures is essential in numerical applications. This paper explores the quantization of probability measures using the maximum mean discrepancy (MMD) distance as a guiding metric. We first investigate…

Concentration of measure is a phenomenon in which a random variable that depends in a smooth way on a large number of independent random variables is essentially constant. The random variable will "concentrate" around its median or…

Machine learning models with inputs in a Euclidean space $\mathbb{R}^d$, when implemented on digital computers, generalize, and their generalization gap converges to $0$ at a rate of $c/N^{1/2}$ concerning the sample size $N$. However, the…

Given a random subspace $H_n$ chosen uniformly in a tensor product of Hilbert spaces $V_n\otimes W$, we consider the collection $K_n$ of all singular values of all norm one elements of $H_n$ with respect to the tensor structure. A law of…

We study the Closest Pair Problem in Hamming metric, which asks to find the pair with the smallest Hamming distance in a collection of binary vectors. We give a new randomized algorithm for the problem on uniformly random input…

We study the computational complexity of approximating general constrained Markov decision processes. Our primary contribution is the design of a polynomial time $(0,\epsilon)$-additive bicriteria approximation algorithm for finding optimal…

We show how to compute any symmetric Boolean function on $n$ variables over any field (as well as the integers) with a probabilistic polynomial of degree $O(\sqrt{n \log(1/\epsilon)})$ and error at most $\epsilon$. The degree dependence on…

We introduce a new class of objectives for optimal transport computations of datasets in high-dimensional Euclidean spaces. The new objectives are parametrized by $\rho \geq 1$, and provide a metric space $\mathcal{R}_{\rho}(\cdot, \cdot)$…

Constrained counting and sampling are two fundamental problems in Computer Science with numerous applications, including network reliability, privacy, probabilistic reasoning, and constrained-random verification. In constrained counting,…

We present a new finite-sample analysis of M-estimators of locations in $\mathbb{R}^d$ using the tool of the influence function. In particular, we show that the deviations of an M-estimator can be controlled thanks to its influence function…

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…

Choosing decision variables deterministically (deterministic decision-making) can be regarded as a particular case of choosing decision variables probabilistically (probabilistic decision-making). It is necessary to investigate whether…

The most widely used internal measure for clustering evaluation is the silhouette coefficient, whose naive computation requires a quadratic number of distance calculations, which is clearly unfeasible for massive datasets. Surprisingly,…