Quantization Dimension Research Articles

Fractal dimensions of fractal transformations and quantization dimensions for bi-Lipschitz mappings

In this paper, we study the fractal dimension of the graph of a fractal transformation and also determine the quantization dimension of a probability measure supported on the graph of the fractal transformation. Moreover, we estimate the quantization dimension of the invariant measure corresponding to a weighted iterated function system consisting of bi-Lipschitz mappings under the strong open set condition.

Quantization dimensions for the bi-Lipschitz recurrent iterated function systems

In this paper, the quantization dimensions of the Borel probability measures supported on the limit sets of the bi-Lipschitz recurrent iterated function systems under the strong open set condition in terms of the spectral radius have been estimated.

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.

On the Quantization Dimension of Maximal Linked Systems

On the Quantization Dimension of Maximal Linked Systems

On the Intermediate Values of the Lower Quantization Dimension

On the Intermediate Values of the Lower Quantization Dimension

Quantization dimension of probability measures

The quantization dimension of a probability measure defined on a metric compact space $X$ is known not to exceed the box dimension of its support. It is proved that on any metric compact space of box dimension $\dim_BX=a\leq\infty$, for arbitrary two numbers $b\in[0,a]$ and $c\in[b,a]$ there is a probability measure such that its lower quantization dimension is $b$ and its upper quantization dimension is $c$. Bibliography: 6 titles.

Asymptotics of the quantization errors for some Markov-type measures with complete overlaps

Let G be a directed graph with 2N vertices 1,2,…,2N. Let T=(Ti,j)(i,j)∈G be a family of contractive similitudes on Rq. We assume that Ti˜,j˜=Ti,j for every (i˜,j˜) in {(i,j),(i,j+N),(i+N,j),(i+N,j+N)}∩G. We denote by K the Mauldin-Williams fractal determined by T. Let χ be a positive probability vector and P a 2N×2N row-stochastic matrix. We denote by ν the Markov-type measure associated with χ and P. Let μ be the image measure of ν under the natural projection, which is supported on K. We consider the following two cases: 1. G has two strongly connected components consisting of N vertices; 2. G is strongly connected. With some assumptions for G and T, for case 1, we determine the quantization dimension for μ in terms of the spectral radius of a related matrix; we prove that the lower quantization coefficient is always positive and establish a necessary and sufficient condition for the upper one to be finite. For case 2, we express the quantization dimension in terms of a pressure-like function and prove that the upper and lower quantization coefficient are always positive and finite.

Quantization dimensions of compactly supported probability measures via Rényi dimensions

We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension of order r > 0 r>0 for an arbitrary compactly supported Borel probability measure ν \nu is given by its Rényi dimension at the point q r q_{r} where the L q L^{q} -spectrum of ν \nu and the line through the origin with slope r r intersect. In particular, this proves the continuity of r ↦ D ¯ r ( \nu ) r\mapsto \overline {D}_{r}(\text {\nu )} as conjectured by Lindsay [Quantization dimension for probability distributions, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of North Texas]. This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the L q L^{q} -spectrum restricted to ( 0 , 1 ) \left (0,1\right ) . We give sufficient conditions in terms of the L q L^{q} -spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a C 1 \mathcal {C}^{1} -self-conformal iterated function system on R d \mathbb {R}^{d} without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved.

On metrization of the idempotent measures functor and quantization dimensions

On metrization of the idempotent measures functor and quantization dimensions

Optimal Quantization for Some Triadic Uniform Cantor Distributions with Exact Bounds

Let $$\{S_j: 1\le j\le 3\}$$ 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 $$1\le j\le 3$$ , where $$0<r<\frac{1}{3}$$ . Let $$P=\sum _{j=1}^3 \frac{1}{3} P\circ S_j^{-1}$$ . 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 $$1\le j\le 3$$ . Let $$r_0=0.1622776602$$ , and $$r_1=0.2317626315$$ (which are ten digit rational approximations of two real numbers). In this paper, for $$0<r\le r_0$$ , we give a general formula to determine the optimal sets of n-means and the nth quantization errors for the triadic uniform Cantor distribution P for all positive integers $$n\ge 2$$ . Previously, Roychowdhury gave an exact formula to determine the optimal sets of n-means and the nth quantization errors for the standard triadic Cantor distribution, i.e., when $$r=\frac{1}{5}$$ . 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 $$0<r\le r_1$$ the quantization coefficient does not exist though the quantization dimension exists.

Canonical Sequences of Optimal Quantization for Condensation Measures

We consider condensation measures of the form \(P:=\frac{1}{3} P\circ S_1^{-1}+ \frac{1}{3} P\circ S_2^{-1}+ \frac{1}{3} \nu \) associated with the system \((\mathcal {S}, (\frac{1}{3}, \frac{1}{3}, \frac{1}{3}), \nu ) , \) where \(\mathcal {S}=\{S_i\}_{i=1}^2 \) are contractions and \( \nu \) is a Borel probability measure on \(\mathbb R\) with compact support. Let \(D(\mu )\) denote the quantization dimension of a measure \(\mu \) if it exists. In this paper, we study self-similar measures \(\nu \) satisfying \(D(\nu )>\kappa \), \(D(\nu )<\kappa \), and \(D(\nu )=\kappa , \) respectively, where \(\kappa \) is the unique number satisfying \([\frac{1}{3} (\frac{1}{5})^2]^{\frac{\kappa }{2+\kappa }}=\frac{1}{2}\). 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 \(\nu \) the quantization dimension D(P) of P exists and satisfies \(D(P)=\max \{\kappa , D(\nu )\}\). Moreover, we show that for \(D(\nu )>\kappa \), the D(P)-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for \(D(\nu )\le \kappa \), the D(P)-dimensional lower quantization coefficient is infinity.

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 {Sij} on R2 with associated probabilities {pij} such that pij&gt;0 for all i,j∈N and ∑i,j=1∞pij=1. For such a probability measure P, the optimal sets of n-means and the nth 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 [112,54].

On quantization dimensions of idempotent probability measures

The concept of quantization dimensions (upper and lower) of idempotent probability measures on a metric compactum is introduced. These dimensions, which are defined using a general functorial scheme, are analogous to the quantization dimensions of classical probability distributions. It is proved that the quantization dimensions of an idempotent measure do not exceed the corresponding box dimensions of its support. At the same time, for any non-negative number b not exceeding the upper box dimension of a metric compactum X, there exists an idempotent probability measure μb with support equal to X and the upper quantization dimension equal to b. For the lower quantization dimension, a similar intermediate value theorem is valid under some additional restrictions on the compactum X.

OPTIMAL QUANTIZATION FOR MIXED DISTRIBUTIONS

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 distribution by a discrete distribution. Mixed distributions are an exciting new area for optimal quantization. In this paper, we have determined the optimal sets of n-means, the nth quantization errors, and the quantization dimensions of different mixed distributions. Besides, we have discussed whether the quantization coefficients for the mixed distributions exist. The results in this paper will give a motivation and insight into more general problems in quantization for mixed distributions.

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 investigate the optimal sets of n-means and the nth 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 nth 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 nth quantization errors for all positive integers n.

Local Dimensions and Quantization Dimensions in Dynamical Systems

Let $$\mu $$ 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 $$\mu $$ with respect to the geometric mean error. The results establish the connections with various dimensions of the measure $$\mu $$ and generalize many known results about local dimensions and quantization dimensions of measures.

On probability measures with a maximum of quantization dimension

On probability measures with a maximum of quantization dimension

Grassmannian Product Codebooks for Limited Feedback Massive MIMO With Two-Tier Precoding

In frequency-division duplex full-dimension massive MIMO systems, one of the main challenges is to obtain sufficiently accurate channel state information (CSI) at the transmitter to enable efficient multi-user MIMO transmission. In this paper, we propose a novel dual stage Grassmannian product quantization approach that is suitable for high-dimensional CSI quantization and feedback. We apply the proposed method for outer-tier CSI feedback in two-tier precoding architectures, which employ channel-subspace based outer-tier precoding strategies, such as maximum eigenmode transmission. The proposed method is especially effective if the channel can be decomposed in the angular domain, such that DFT codebooks enable an efficient CSI compression. Our dual stage product codebook quantization approach mitigates the well-known inefficiency of oversampled DFT codebooks for growing codebook sizes, by varying the intermediate quantization dimension of the proposed quantizer.

Quantization for uniform distributions on stretched Sierpiński triangles

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

The upper and lower quantization coefficient for Markov-type measures

We study the asymptotic quantization error of order $r$ for Markov-type measures $\mu$ on a class of ratio-specified graph directed fractals. We show that the quantization dimension of $\mu$ exists and determine its exact value $s_{r}$ in terms of spectral radius of a related matrix. We prove that the $s_{r}$-dimensional lower quantization coefficient of $\mu$ is always positive. Moreover, inspired by Mauldin-Williams's work on the Hausdorff measure of graph directed fractals, we establish a necessary and sufficient condition for the $s_{r}$-dimensional upper quantization coefficient of $\mu$ to be finite.