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Mrinal K. Roychowdhury

Publications and source records attributed to Mrinal K. Roychowdhury.

9 recordsLinked to original sources
arXiv2025

Constrained Quantization for Uniform Distributions with Two Constraint Families

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 for all positive integers $n$. In addition, we calculate the constrained quantization dimension and the constrained quantization coefficient with respect to this family of constraints. Next, we turn to another family of constraints, consisting of concentric circles. For the same probability distribution, we present a methodology to compute the constrained optimal sets of $n$-points and the corresponding $n$th constrained quantization errors for all positive integers $n$. Finally, we conclude the paper with a summary of the results and a discussion of future research directions.

math.PR↗
arXiv2025

Conditional constrained and unconstrained quantization for uniform distributions on regular polygons

In this paper, we have considered a uniform distribution on a regular polygon with $k$-sides for some $k\geq 3$ and the set of all its $k$ vertices as a conditional set. For the uniform distribution under the conditional set first, for all positive integers $n\geq k$, we obtain the conditional optimal sets of $n$-points and the $n$th conditional quantization errors, and then we calculate the conditional quantization dimension and the conditional quantization coefficient in the unconstrained scenario. Then, for the uniform distribution on the polygon taking the same conditional set, we investigate the conditional constrained optimal sets of $n$-points and the conditional constrained quantization errors for all $n \geq 6$, taking the constraint as the circumcircle, incircle, and then the different diagonals of the polygon.

math.PR↗
arXiv2025

Quantization dimensions for inhomogeneous bi-Lipschitz Iterated Function Systems

Let $ν$ 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 $\{S_j: 1\leq j\leq N\}$ be a set of contractive mappings on $\mathbb{R}^d$. Then, a Borel probability measure $μ$ on $\mathbb R^d$ such that $μ=\sum_{j=1}^N p_jμ\circ S_j^{-1}+p_0ν$ is called an inhomogeneous measure, also known as a condensation measure on $\mathbb{R}^d$. 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 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. In this paper, we investigate the quantization dimension for such a condensation measure.

math.PR↗
arXiv2024

Constrained quantization for the Cantor distribution

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 the classical Cantor distribution. Toward this, they have calculated the optimal sets of $n$-points, $n$th constrained quantization errors, the constrained quantization dimensions, and the constrained quantization coefficients, taking different families of constraints for all $n\in \mathbb N$. The results in this paper show that both the constrained quantization dimension and the constrained quantization coefficient for the Cantor distribution depend on the underlying constraints. It also shows that the constrained quantization coefficient for the Cantor distribution can exist and be equal to the constrained quantization dimension. These facts are not true in the unconstrained quantization for the Cantor distribution.

math.DS↗
arXiv2023

Quantization for a set of discrete distributions on the set of natural numbers

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 theorem, and then give a conjecture. We verify the conjecture by a few examples. Assuming that the conjecture is true, for a set of discrete distributions on the set of natural numbers we have calculated the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. In addition, the quantization dimension is also calculated.

math.PR↗
arXiv2021

Quantization coefficients for uniform distributions on the boundaries of regular polygons

In this paper, we give a general formula to determine the quantization coefficients for uniform distributions defined on the boundaries of different regular $m$-sided polygons inscribed in a circle. The result shows that the quantization coefficient for the uniform distribution on the boundary of a regular $m$-sided polygon inscribed in a circle is an increasing function of $m$, and approaches to the quantization coefficient for the uniform distribution on the circle as $m$ tends to infinity.

math.DS↗
arXiv2020

Quantization dimension and stability for infinite self-similar measures with respect to geometric mean error

Let $μ$ be a Borel probability measure associated with an iterated function system consisting of a countably infinite number of contracting similarities and an infinite probability vector. In this paper, we study the quantization dimension of the measure $μ$ with respect to the geometric mean error. The quantization for infinite systems is different from the well-known finite case investigated by Graf and Luschgy. That is, many tools which are used in the finite setting, for example, existence of finite maximal antichains, fail in the infinite case. We prove that the quantization dimension of the measure $μ$ is equal to its Hausdorff dimension which extends a well-known result of Graf and Luschgy for the finite case to an infinite setting. In the last section, we discuss the stability of quantization dimension for infinite systems.

math.DS↗
arXiv2020

A study on Quantization Dimension in complete metric spaces

The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a complete metric space. This generalizes the known results on quantization dimension of self-similar measures in the Euclidean space to a complete metric space. In the last part, continuity of quantization dimension is discussed.

math.DS↗