arXiv subjects

Mrinal Kanti Roychowdhury

Publications and source records attributed to Mrinal Kanti Roychowdhury.

At least 37 records · Page 2Linked to original sources

arXiv2023

Optimal quantizers for a nonuniform distribution on a Sierpinski carpet

The purpose of quantization for a probability distribution is to estimate the probability by a discrete probability with finite support. In this paper, a nonuniform probability measure $P$ on $\mathbb R^2$ which has support the Sierpiński carpet generated by a set of four contractive similarity mappings with equal similarity ratios has been considered. For this probability measure, the optimal sets of $n$-means and the $n$th quantization errors are investigated for all $n\geq 2$.

cs.IT↗

arXiv2022

Quantization dimension for self-similar measures of overlapping construction

Quantization dimension has been computed for many invariant measures of dynamically defined fractals having well separated cylinders, that is, in the cases when the so-called Open Set Condition (OSC) holds. To attack the same problem in case of heavy overlaps between the cylinders, we consider a family of self-similar measures, for which the underlying Iterated Function System satisfies the so-called Weak Separation Property (WSP) but does not satisfy the OSC since complete overlaps occur in between the cylinders. The work in this paper also shows that the quantization dimension determined for the set of overlap self-similar construction satisfying the WSP has a relationship with the temperature function of the thermodynamic formalism.

math.DS↗

arXiv2022

Optimal quantization for some triadic uniform Cantor distributions with exact bounds

Let $\{S_j : 1\leq j\leq 3\}$ be a set of three contractive similarity mappings such that $S_j(x)=rx+\frac {j-1}{2}(1-r)$ for all $x\in \mathbb R$, and $1\leq j\leq 3$, where $0<r<\frac 1 3$. Let $P=\sum_{j=1}^3 \frac 13 P\circ S_j^{-1}$. Then, $P$ is a unique Borel probability measure on $\mathbb R$ such that $P$ has support the Cantor set generated by the similarity mappings $S_j$ for $1\leq j\leq 3$. Let $r_0=0.1622776602$, and $r_1=0.2317626315$ (which are ten digit rational approximations of two real numbers). In this paper, for $0<r\leq r_0$, we give a general formula to determine the optimal sets of $n$-means and the $n$th quantization errors for the triadic uniform Cantor distribution $P$ for all positive integers $n\geq 2$. Previously, Roychowdhury gave an exact formula to determine the optimal sets of $n$-means and the $n$th quantization errors for the standard triadic Cantor distribution, i.e., when $r=\frac 15$. In this paper, we further show that $r=r_0$ is the greatest lower bound, and $r=r_1$ is the least upper bound of the range of $r$-values to which Roychowdhury formula extends. In addition, we show that for $0<r\leq r_1$ the quantization coefficient does not exist though the quantization dimension exists.

math.DS↗

arXiv2022

Uniform distributions on curves and quantization

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 to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of $n$-means and the $n$th quantization errors for different values of $n$ with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise (1982), which says that for a Borel probability measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number

math.PR↗

arXiv2022

Quantization for a probability distribution generated by an infinite iterated function system

Quantization for probability distributions concerns the best approximation of a $d$-dimensional probability distribution $P$ by a discrete probability with a given number $n$ of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on $\mathbb R$. For such a probability measure $P$, an induction formula to determine the optimal sets of $n$-means and the $n$th quantization error for every natural number $n$ is given. In addition, using the induction formula we give some results and observations about the optimal sets of $n$-means for all $n\geq 2$.

math.DS↗

arXiv2022

Canonical sequences of optimal quantization for condensation measures

We consider condensation measures of the form $P:=\frac 13 P\circ S_1^{-1}+ \frac 13 P\circ S_2^{-1}+ \frac 13 ν$ associated with the system $(\mathcal{S}, (\frac 13, \frac 13, \frac 13), ν) , $ where $\mathcal{S}=\{S_i\}_{i=1}^2 $ are contractions and $ ν$ is a Borel probability measure on $\mathbb R$ with compact support. Let $D(μ)$ denote the quantization dimension of a measure $μ$ if it exists. In this paper, we study self-similar measures $ν$ satisfying $D(ν)>κ$, $D(ν)<κ$, and $D(ν)=κ, $ respectively, where $κ$ is the unique number satisfying $[\frac13 (\frac{1}{5})^2]^{\fracκ{2+κ}}=\frac 12. $ For each case we construct two sequences $a(n)$ and $F(n)$, which are utilized in determining the optimal sets of $F(n)$-means and the $F(n)$th quantization errors for $P. $ We also show that for each measure $ν$ the quantization dimension $D(P)$ of $P$ exists and satisfies $D(P)=\max\{κ, D(ν)\}. $ Moreover, we show that for $D(ν)>κ$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for $D(ν)\leq κ$, the $D(P)$-dimensional lower quantization coefficient is infinity.

math.DS↗

arXiv2022

Quantization for infinite affine transformations

Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite set. In this article, we consider a probability distribution generated by an infinite system of affine transformations $\{S_{ij}\}$ on $\mathbb R^2$ with associated probabilities $\{p_{ij}\}$ such that $p_{ij}>0$ for all $i, j\in \mathbb N$ and $\sum_{i, j=1}^\infty p_{ij}=1$. For such a probability measure $P$, the optimal sets of $n$-means and the $n$th quantization error are calculated for every natural number $n$. It is shown that the distribution of such a probability measure is the same as that of the direct product of the Cantor distribution. In addition, it is proved that the quantization dimension $D(P)$ exists and is finite; whereas, the $D(P)$-dimensional quantization coefficient does not exist, and the $D(P)$-dimensional lower and the upper quantization coefficients lie in the closed interval $[\frac{1}{12}, \frac{5}{4}]$.

math.DS↗

arXiv2021

Optimal Quantization For Mixed Distributions

math.PR↗

arXiv2020

Quantization for uniform distributions on hexagonal, semicircular, and elliptical curves

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 to determine them, if $n$ is of the form $n=6k$ for some positive integer $k$. We further calculate the quantization dimension, the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number. Then, we define a mixture of two uniform distributions on the boundary of a semicircular disc, and obtain a sequence and an algorithm, with the help of which we determine the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$ with respect to the mixed distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$.

math.PR↗

arXiv2020

Local dimensions and quantization dimensions in dynamical systems

Let $μ$ be a Borel probability measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of $\mathbb R^k$. We study the Hausdorff and the packing dimensions, and the quantization dimensions of $μ$ with respect to the geometric mean error. The results establish the connections with various dimensions of the measure $μ$, and generalize many known results about local dimensions and quantization dimensions of measures.

math.DS↗

arXiv2020

Quantization for a mixture of uniform distributions associated with probability vectors

math.PR↗

arXiv2020

Optimal Quantization via Dynamics

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 stationary processes arising in dynamical systems, followed by a discussion of the special cases of stationary processes: random processes and Diophantine processes. We are interested in how close stationary process can be to giving optimal $n$-means and $n^{th}$ optimal mean distortion errors. We also consider different ways of measuring the degree of approximation by quantization, and their advantages and disadvantages in these different contexts.

math.DS↗

arXiv2019

Geometric and Measure-Theoretic Shrinking Targets in Dynamical Systems

We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.

math.DS↗

arXiv2019

The quantization of the standard triadic Cantor distribution

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. For a given $k\geq 2$, let $\{S_j : 1\leq j\leq k\}$ be a set of $k$ contractive similarity mappings such that $S_j(x)=\frac 1 {2k-1} x +\frac{2 (j-1)} {2k-1}$ for all $x\in \mathbb R$, and let $P= \frac 1 k \sum_{j=1}^kP\circ S_j^{-1}$. Then, $P$ is a unique Borel probability measure on $\mathbb R$ such that $P$ has support the Cantor set generated by the similarity mappings $S_j$ for $1\leq j\leq k$. In this paper, for the probability measure $P$, when $k=3$, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. We further show that the quantization coefficient does not exist though the quantization dimension exists.

math.DS↗

arXiv2019

Quantization for uniform distributions of Cantor dusts on $\mathbb{R}^2$

Let $P$ be a Borel probability measure on $\mathbb R^2$ supported by the Cantor dusts generated by a set of $4^u,\ u\geq 1$, contractive similarity mappings satisfying the strong separation condition. For this probability measure, we determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. In addition, it is shown that though the quantization dimension of the measure $P$ is known, the quantization coefficient for $P$ does not exist.

math.DS↗

arXiv2019

Center of mass and the optimal quantizers for some continuous and discrete uniform distributions

In this paper, we first consider a flat plate (called a lamina) with uniform density $ρ$ that occupies a region $\mathfrak R$ of the plane. We show that the location of the center of mass, also known as the centroid, of the region equals the expected vector of a bivariate continuous random variable with a uniform probability distribution taking values on the region $\mathfrak R$. Using this property, we prove that the Voronoi regions of an optimal set of two-means with respect to the uniform distribution defined on a disc partition the disc into two regions bounded by the semicircles. Besides, we show that if an isosceles triangle is partitioned into an isosceles triangle and an isosceles trapezoid in the Golden ratio, then their centers of mass form a centroidal Voronoi tessellation of the triangle. In addition, using the properties of center of mass we determine the optimal sets of two-means and the corresponding quantization error for a uniform distribution defined on a region with uniform density bounded by a rhombus. Further, we determine the optimal sets of $n$-means, and the $n$th quantization errors for two different discrete uniform distributions for some positive integers $n\leq \text{card(supp}(P))$.

math.PR↗

arXiv2019

Quantization for uniform distributions on stretched Sierpiński triangles

In this paper, we have considered a uniform probability distribution supported by a stretched Sierpiński triangle. For this probability measure, the optimal sets of $n$-means and the $n$th quantization errors are determined for all $n\geq 2$. In addition, it is shown that the quantization coefficient for such a measure does not exist though the quantization dimension exists.

math.DS↗

arXiv2019

An algorithm to compute CVTs for finitely generated Cantor distributions

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions with respect to a given probability measure. CVT is a fundamental notion that has a wide spectrum of applications in computational science and engineering. In this paper, an algorithm is given to obtain the CVTs with $n$-generators to level $m$, for any positive integers $m$ and $n$, of any Cantor set generated by a pair of self-similar mappings given by $S_1(x)=r_1x$ and $S_2(x)=r_2x+(1-r_2)$ for $x\in \mathbb R$, where $r_1, r_2>0$ and $r_1+r_2<1$, with respect to any probability distribution $P$ such that $P=p_1 P\circ S_1^{-1}+p_2 P\circ S_2^{-1}$, where $p_1, p_2>0$ and $p_1+p_2=1$.

cs.IT↗