Related papers: The quantization of the standard triadic Cantor di…

Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using…

We investigate quantization coefficients for self-similar probability measures \mu on limit sets which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization…

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are…

Let $P:=\frac 1 3 P\circ S_1^{-1}+\frac 13 P\circ S_2^{-1}+\frac 13\nu$, where $S_1(x)=\frac 15 x$, $S_2(x)=\frac 1 5 x+\frac 45$ for all $x\in \mathbb R$, and $\nu$ be a Borel probability measure on $\mathbb R$ with compact support. Such a…

Optimal quantization for mixed distributions has emerged as a compelling area of study. In this work, we have focused on a mixed distribution formed from two uniform distributions with partially overlapping supports. For this class of…

The representation of a given quantity with less information is often referred to as `quantization' and it is an important subject in information theory. In this paper, we have considered absolutely continuous probability measures on unit…

This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…

Let $\nu$ be a Borel probability measure on a $d$-dimensional Euclidean space $\mathbb{R}^d$, $d\geq 1$, with a compact support, and let $(p_0, p_1, p_2, \ldots, p_N)$ be a probability vector with $p_j>0$ for $0\leq j\leq N$. Let $\{S_j:…

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…

Bucklew and Wise (1982) showed that the quantization dimension of an absolutely continuous probability measure on a given Euclidean space is constant and equals the Euclidean dimension of the space, and the quantization coefficient exists…

We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $\mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic…

Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly…

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…

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 $r$…

Quantum Clustering is a powerful method to detect clusters in data with mixed density. However, it is very sensitive to a length parameter that is inherent to the Schr\"odinger equation. In addition, linking data points into clusters…

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…

This paper develops a systematic and geometric theory of optimal quantization on the unit sphere $\mathbb S^2$, focusing on finite uniform probability distributions supported on the spherical surface - rather than on lower-dimensional…

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…

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…

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…