Related papers: Optimal quantization for a probability measure on …
For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(\mu)$, of a Borel probability measure $\mu$ on ${\mathbb R}^d$ represents the speed how fast the $n$th quantization error of order $r$…
The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability…
Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite set. In this article, we consider a probability distribution generated by an infinite system of…
We provide a complete picture of the upper quantization dimension in terms of the R\'enyi dimension by proving that the upper quantization dimension $\bar{D}_{r}(\nu)$ of order $r>0$ for an arbitrary compactly supported Borel probability…
We present an analysis of optimal quantization of probability measures with nonuniform densities on spherical curves. We begin by deriving the centroid condition, followed by a high-resolution asymptotic analysis to establish the…
Quantization of a probability measure means representing it with a finite set of Dirac masses that approximates the input distribution well enough (in some metric space of probability measures). Various methods exists to do so, but the…
We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is…
The polarization measure is the probability that among 3 individuals chosen at random from a finite population exactly 2 come from the same class. This index is maximum at the midpoints of the edges of the probability simplex. We compute…
Characterizations of surjective isometries with respect to the Kuiper distance on three classes of Borel probability measures of $\mathbb{R}$ (or equivalently, probability distribution functions) are presented here. These classes are the…
We describe a measure quantization procedure i.e., an algorithm which finds the best approximation of a target probability law (and more generally signed finite variation measure) by a sum of $Q$ Dirac masses ($Q$ being the quantization…
Let $E$ be a Moran set on $\mathbb{R}^1$ associated with a closed interval $J$ and two sequences $(n_k)_{k=1}^\infty$ and $(\mathcal{C}_k=(c_{k,j})_{j=1}^{n_k})_{k\geq1}$. Let $\mu$ be the infinite product measure (Moran measure) on $E$…
A sequential measurement of two non-commuting quantum observables results in a joint probability distribution for all output combinations that can be explained in terms of an initial joint quasi-probability of the non-commuting observables,…
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…
We approximate the uniform measure on an equilateral triangle by a measure supported on $n$ points. We find the optimal sets of points ($n$-means) and corresponding approximation (quantization) error for $n\leq4$, give numerical…
In this article, we study the approximation of a probability measure $\mu$ on $\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment…
A common way to discretize a probability measure is to use an empirical measure as a discrete approximation. But how far from being optimal is this approximation in the p-Wasserstein distance? In this paper, we study this question in two…
The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. In this paper, first we state and prove a…
We consider statistical learning problems in which data are observed as a set of probability measures. Optimal transport (OT) is a popular tool to compare and manipulate such objects, but its computational cost becomes prohibitive when the…
When measuring phase of quantum states of light, the optimal single-shot measurement implements projection on the un-physical phase states. If we want to improve the precision further we need to accept a reduced probability of success,…
We study the quantization errors for the doubling probability measures $\mu$ which are supported on a class of Moran sets $E\subset\mathbb{R}^q$. For each $n\geq 1$, let $\alpha_n$ be an arbitrary $n$-optimal set for $\mu$ of order $r$ and…