arXiv subjects

Mrinal Kanti Roychowdhury

Publications and source records attributed to Mrinal Kanti Roychowdhury.

At least 19 recordsLinked to original sources

arXiv2026

Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system

For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ on ${\mathbb R}^d$ represents the speed how fast the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. If $D_r(μ)$ does not exists, we call $\underline D_r(μ)$ and $\overline D_r(μ)$, the lower and upper quantization dimensions of $μ$ of order $r$. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems $(\{f_i\}_{i=1}^N, (p_i)_{i=0}^N, ν)$, where the mappings $f_i$ are bi-Lipschitz and the measure $ν$ is an image measure of an ergodic measure with bounded distortion supported on a conformal set. In addition, we determine the optimal quantization for an infinite discrete distribution, and give an example which shows that the quantization dimension of a Borel probability measure can be positive with zero quantization coefficient.

math.DS↗

arXiv2026

Linear stability analysis of the Lloyd algorithm on a circle

Lloyd algorithm is the standard iterative method for computing quantizers and codebooks in source coding and vector quantization. In this article, we study the dynamical and stability properties of the Lloyd map on the unit circle $\mathbb S^1$ using von Mises distributions. We construct the Lloyd iteration as a discrete dynamical system on the configuration space of ordered point sets modulo rotational symmetry. Also, we study the rotational equivarience of the Lloyd map. Further, we derive an explicit representation of the Jacobian matrix and prove that it possesses a circulant structure for the equally spaced configuration. Also, we study the bifurcation characteristics based on Lloyd map analysis. In the end, we provide the numerical algorithms for stability diagrams, Lyapunov spectrum estimation, and residue analysis, purely for empirical visualization. Our results provide a dynamical systems framework for Lloyd quantization on $\mathbb S^1$ for studying stability properties.

math.DS↗

arXiv2026

Discrete Quantization on Spherical Geometries: Explicit Models, Computations, and Didactic Exposition

We present an analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric, focusing on highly symmetric configurations on the unit sphere $\mathbb S^2$. Three discrete uniform models are analyzed and closed-form expressions for optimal quantizers and mean-square errors are derived. (I) For $N$ equally spaced points on the equator, exact error formulas are obtained for both divisible and non-divisible cases, showing that optimal Voronoi cells form contiguous arcs with midpoint representatives. (II) For two antipodally symmetric small circles at latitudes $\pmϕ_0$, each with $M$ longitudes, we establish a no-cross-circle Voronoi phenomenon, symmetry-preserving optimality, and finite-sum error formulas with curvature-dependent bounds and asymptotics. (III) For a single small circle at latitude $ϕ_0$, analogous formulas are proved and curvature is shown to reduce distortion by a factor $\cos^2ϕ_0$ while preserving the $n^{-2}$ decay rate. Across all models we rigorously formulate the block-midpoint principle: optimal Voronoi cells are contiguous azimuthal blocks whose representatives are azimuthal midpoints. These explicit benchmark models clarify curvature effects and support further developments in quantization on curved manifolds.

math.OC↗

arXiv2026

Optimal Quantization for Nonuniform Densities on Spherical Curves

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 point-density formula. We further quantify the asymptotic error formula for the nonuniform densities. We apply these theorems to the von Mises distributions and characterize the optimal condition. We also provide applications using the high-resolution asymptotic and its corresponding error formula. Our results can be used in geometric probability theory and quantization theory of spherical curves.

math.PR↗

arXiv2026

Constrained quantization for probability distributions

In this work, we extend the classical framework of quantization for Borel probability measures defined on normed spaces $\mathbb{R}^k$ by introducing and analyzing the notions of the $n$th constrained quantization error, constrained quantization dimension, and constrained quantization coefficient. These concepts generalize the well-established $n$th quantization error, quantization dimension, and quantization coefficient, traditionally considered in the unconstrained setting, and thereby broaden the scope of quantization theory. A key distinction between the unconstrained and constrained frameworks lies in the structural properties of optimal quantizers. In the unconstrained setting, if the support of $P$ contains at least $n$ elements, then the elements of an optimal set of $n$-points coincide with the conditional expectations over their respective Voronoi regions; this characterization does not, in general, persist under constraints. Moreover, it is known that if the support of $P$ contains at least $n$ elements, then any optimal set of $n$-points in the unconstrained case consists of exactly $n$ distinct elements. This property, however, may fail to hold in the constrained context. Further differences emerge in asymptotic behaviors. For absolutely continuous probability measures, the unconstrained quantization dimension is known to exist and equals the Euclidean dimension of the underlying space. In contrast, we show that this equivalence does not necessarily extend to the constrained setting. Additionally, while the unconstrained quantization coefficient exists and assumes a unique, finite, and positive value for absolutely continuous measures, we establish that the constrained quantization coefficient can exhibit significant variability and may attain any nonnegative value, depending critically on the specific nature of the constraint applied to the quantization process.

math.PR↗

arXiv2026

Optimal Quantization of Finite Uniform Data on the Sphere

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 geodesic subsets such as circles or arcs. We first establish the existence of optimal sets of $n$-means and characterize them through centroidal spherical Voronoi tessellations. Three fundamental structural results are obtained. First, a cluster - purity theorem shows that when the support consists of well-separated components, each optimal Voronoi region remains confined to a single component. Second, a ring - allocation (discrete water - filling) theorem provides an explicit rule describing how optimal representatives are distributed across multiple latitudinal rings, together with closed-form distortion formulas. Third, a Lipschitz - type stability theorem quantifies the robustness of optimal configurations under small geodesic perturbations of the support. In addition, a spherical analogue of Lloyd's algorithm is presented, in which intrinsic (Karcher) means replace Euclidean centroids for iterative refinement. These results collectively provide a unified and transparent framework for understanding the geometric and algorithmic structure of optimal quantization on $\mathbb S^2$.

math.OC↗

arXiv2025

Optimal Quantization on Spherical Surfaces: Continuous and Discrete Models -- A Beginner-Friendly Expository Study

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 first present a detailed geometric and analytical foundation for intrinsic quantization on the unit sphere, including definitions of great and small circles, spherical triangles, geodesic distance, Slerp interpolation, the Frechet mean, spherical Voronoi regions, centroid conditions, and quantization dimensions. Building upon this framework, we develop explicit continuous and discrete quantization models on spherical curves, namely great circles, small circles, and great circular arcs supported by rigorous derivations and pedagogical exposition. For uniform continuous distributions, we compute optimal sets of $n$-means and the associated quantization errors on these curves; for discrete distributions, we analyze antipodal, equatorial, tetrahedral, and finite uniform configurations, illustrating convergence to the continuous model. The central conclusion is that for a uniform probability distribution supported on a one-dimensional geodesic subset of total length $L$, the optimal $n$-means form a uniform partition and the quantization error satisfies $V_n = L^2/(12n^2)$. The exposition emphasizes geometric intuition, detailed derivations, and clear step-by-step reasoning, making it accessible to beginning graduate students and researchers entering the study of quantization on manifolds. This article is intended as an expository and tutorial contribution, with the main emphasis on geometric reformulation and pedagogical clarity of intrinsic quantization on spherical curves, rather than on the development of new asymptotic quantization theory.

math.OC↗

arXiv2025

Optimal Sets and Quantization Errors under Geometric Constraints for Discrete Distributions

This paper presents a detailed study of constrained quantization for both finite and infinite discrete probability distributions supported on subsets of the real line. Under specific geometric constraints - namely, a semicircular arc and the union of two sides of an equilateral triangle - we compute constrained optimal sets of $n$-points and the corresponding $n$th constrained quantization errors. For finite discrete distributions, we consider both uniform and nonuniform cases with support on $\{-3, -2, \dots, 3\}$. For infinite discrete distributions, two cases are analyzed: one supported on $\left\{ \frac{1}{n} : n \in \mathbb{N} \right\}$ and the other on the set of natural numbers $\mathbb{N}$. Explicit constructions and numerical computations of optimal quantizers and errors are provided. Furthermore, for the infinite discrete distribution supported on $\mathbb{N}$, we develop a general framework for constrained quantization under the linear constraint $y = mx + c$ and prove that the constrained quantization dimension in this setting is zero. Our results highlight how geometric constraints influence the structure and existence of optimal quantizers and pave the way for further investigations into constrained quantization theory.

math.PR↗

arXiv2025

Quantization for the mixtures of overlap probability distributions

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 distributions, we have examined the structure of optimal sets of $n$-means and the corresponding $n$th quantization errors for all positive integers $n$. Initially, we explicitly determined the optimal sets and quantization errors for $1 \leq n \leq 6$. Subsequently, we established several key lemmas and propositions and proposed an algorithm that facilitates the computation of optimal $n$-means and quantization errors for all $n \geq 5$. Numerical results are also presented to illustrate the application of the algorithm in deriving these quantities. The findings of this study offer valuable insight and serve as a foundation for further research on quantization in the context of mixed distributions with overlapping supports.

math.PR↗

arXiv2025

Quantization for the mixtures of uniform distributions on connected and disconnected line segments

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 distributions, we have determined the optimal sets of $n$-means and the corresponding $n$th quantization errors for all positive integers $n $. The methods developed in this paper can be applied more generally to investigate optimal quantization for any mixed distribution $P := pP_1 + (1 - p)P_2,$ where $P_1$ and $P_2$ are arbitrary probability distributions supported on either connected or disconnected line segments, and $(p, 1 - p)$ is any probability vector with $0 < p < 1$.

math.PR↗

arXiv2025

Conditional Optimal Sets and the Quantization Coefficients for Some Uniform Distributions

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 as a finite positive number. By giving different examples, in this paper, we have shown that the quantization coefficients for absolutely continuous probability measures defined on the same Euclidean space can be different. We have taken uniform distribution as a prototype of an absolutely continuous probability measure. In addition, we have also calculated the conditional optimal sets of $n$-points and the $n$th conditional quantization errors for the uniform distributions in constrained and unconstrained scenarios.

math.PR↗

arXiv2025

Optimal quantization for nonuniform discrete distributions

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 important in fields such as data compression and signal processing. We begin by examining two specific nonuniform distributions over a finite set of values and identify the best representative points for different levels of approximation. We then extend our analysis to two infinite discrete distributions: one supported on the reciprocals of natural numbers and another on the natural numbers themselves. For these distributions, we compute the optimal sets of representatives and assess how well they approximate the original distributions. Finally, we address the reverse problem-determining the underlying distribution when the optimal sets are known. Our results provide both theoretical insights and computational techniques useful in information theory and data analysis.

math.PR↗

arXiv2025

Conditional constrained and unconstrained quantization for probability distributions

In this paper, we introduce and develop the concept of conditional quantization for Borel probability measures on $\mathbb{R}^k,$ considering both constrained and unconstrained frameworks. For each setting, we define the associated quantization errors, dimensions, and coefficients, and provide explicit computations for specific classes of probability distributions. A key result in the unconstrained case is that the union of all optimal sets of $ n$-means is dense in the support of the measure. Furthermore, we demonstrate that in conditional constrained quantization, if the conditional set is contained within the union of the constraint family, then the lower and upper quantization dimensions, as well as the corresponding coefficients, remain unaffected by the conditional set for any Borel probability measure. In contrast, if the conditional set is not contained within this union, these properties may no longer hold, as illustrated through various examples.

math.PR↗

arXiv2025

Optimal quantization for a probability measure on a nonuniform stretched Sierpiński triangle

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. In this paper, we have considered a Borel probability measure $P$ on $\mathbb R^2$, which has support a nonuniform stretched Sierpiński triangle generated by a set of three contractive similarity mappings on $\mathbb R^2$. For this probability measure, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$.

cs.IT↗

arXiv2025

Quantization for a condensation system

For a given $r \in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, represents the rate at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. If $D_r(μ)$ does not exist, we define $\underline{D}_r(μ)$ and $\overline{D}_r(μ)$ as the lower and the upper quantization dimensions of $μ$ of order $r$, respectively. In this paper, we investigate the quantization dimension of the condensation measure $μ$ associated with a condensation system $(\{S_j\}_{j=1}^N, (p_j)_{j=0}^N, ν).$ We provide two examples: one where $ν$ is an infinite discrete distribution on $\mathbb{R}$, and one where $ν$ is a uniform distribution on $\mathbb{R}$. For both the discrete and uniform distributions $ν$, we determine the optimal sets of $n$-means, and calculate the quantization dimensions of condensation measures $μ$, and show that the $D_r(μ)$-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive.

math.DS↗

arXiv2025

Conditional quantization for uniform distributions on line segments and regular polygons

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 preselected, then the quantization is called a conditional quantization. In this paper, we investigate the conditional quantization for the uniform distributions defined on the unit line segments and $m$-sided regular polygons, where $m\geq 3$, inscribed in a unit circle.

math.PR↗

arXiv2024

Constrained quantization for a uniform distribution

Constrained quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with a finite number of supporting points lying on a specific set. The specific set is known as the constraint of the constrained quantization. A quantization without a constraint is known as an unconstrained quantization, which traditionally in the literature is known as quantization. Constrained quantization has recently been introduced by Pandey and Roychowdhury. In this paper, for a uniform distribution with support lying on a side of an equilateral triangle, and the constraint as the union of the other two sides, we obtain the optimal sets of $n$-points and the $n$th constrained quantization errors for all positive integers $n$. We also calculate the constrained quantization dimension and the constrained quantization coefficient.

math.PR↗

arXiv2024

Constrained quantization for the Cantor distribution with a family of constraints

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 constrained quantization dimension and the constrained quantization coefficient, and see that the constrained quantization dimension $D(P)$ exists as a finite positive number, but the $D(P)$-dimensional constrained quantization coefficient does not exist.

math.DS↗