Related papers: Constrained quantization for a uniform distributio…
We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order $r$, including negative values of $r$. To this end, we use the concept of partition…
New measures for the quantization of systems with constraints are discussed and applied to several examples, in particular, examples of alternative but equivalent formulations of given first-class constraints, as well as a comparison of…
Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum state space may thus be thought…
The Born rule provides a probability vector (distribution) with a quantum state for a measurement setting. For two settings, we have a pair of vectors from the same quantum state. Each pair forms a combined-probability vector that obeys…
For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and…
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…
A quantization scheme based on the extension of phase space with application of constrained quantization technic is considered. The obtained method is similar to the geometric quantization. For constrained systems the problem of scalar…
We discuss in detail the uniform discretization approach to the quantization of totally constrained theories. This approach allows to construct the continuum theory of interest as a well defined, controlled, limit of well behaved discrete…
In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon, and then investigated the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. We give an exact formula…
This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The…
If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present…
Quantization summarizes continuous distributions by calculating a discrete approximation. Among the widely adopted methods for data quantization is Lloyd's algorithm, which partitions the space into Vorono\"i cells, that can be seen as…
We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of measures. The limit is (not the normal distribution and is)…
A theory of measurement uncertainty is presented, which, since it is based exclusively on the Bayesian approach and on the subjective concept of conditional probability, is applicable in the most general cases. The recent International…
Let $P$ be a Borel probability measure on $\mathbb R$ generated by an infinite system of similarity mappings $\{S_j : j\in \mathbb N\}$ such that $P=\sum_{j=1}^\infty \frac 1{2^j} P\circ S_j^{-1}$, where for each $j\in \mathbb N$ and $x\in…
We have previously reported a Bayesian algorithm for determining the coordinates of points in three-dimensional space from uncertain constraints. This method is useful in the determination of biological molecular structure. It is limited,…
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 study the problem of quantization of discrete probability distributions, arising in universal coding, as well as other applications. We show, that in many situations this problem can be reduced to the covering problem for the unit…
Measuring the concentration of random variables is a fundamental concept in probability and statistics. Here, we explore a type of concentration measure for continuous random variables with bounded support and use it to provide a notion of…
The concept of randomized measurements on individual particles has proven to be useful for analyzing quantum systems and is central for methods like shadow tomography of quantum states. We introduce $\textit{collective}$ randomized…