Related papers: Optimal quantization for nonuniform Cantor distrib…
In this paper, for a given family of constraints and the classical Cantor distribution we determine the constrained optimal sets of $n$-points, $n$th constrained quantization errors for all positive integers $n$. We also calculate the…
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:…
In this paper, we have studied various mixed distributions generated by two uniform distributions: first, where the supports are two connected line segments, and second, where the supports are two disconnected line segments. For these mixed…
This paper explores the process of optimal quantization for several types of discrete probability distributions. Quantization is a technique used to approximate a complex distribution with a smaller set of representative points, which is…
A method to compute the optimal success probability of discrimination of N arbitrary quantum states is presented, based on the decomposition of any N-outcome measurement into sequences of nested two-outcome ones. In this way the…
Let $S= (s_1<s_2<\dots)$ be a strictly increasing sequence of positive integers and denote $\mathbf{e}(\beta)=\mathrm{e}^{2\pi i \beta}$. We say $S$ is good if for every real $\alpha$ the limit $\lim_N \frac1N\sum_{n\le N}…
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…
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 $E$…
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…
This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We…
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…
The comparison of benchmark error sets is an essential tool for the evaluation of theories in computational chemistry. The standard ranking of methods by their Mean Unsigned Error is unsatisfactory for several reasons linked to 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 $r$…
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…
We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.
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…
The theory of constrained quantization has been recently introduced by Pandey and Roychowdhury. In this paper, they have further generalized their previous definition of constrained quantization and studied the constrained quantization for…
In this paper, we first consider a family of constraints given by straight lines. For a uniform probability distribution, we determine the constrained optimal sets of $n$-points and the corresponding $n$th constrained quantization errors…
We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics.…
Estimating the expectation value of an operator corresponding to an observable is a fundamental task in quantum computation. It is often impossible to obtain such estimates directly, as the computer is restricted to measuring in a fixed…