7 research outputs found
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
Constrained quantization for the Cantor distribution
In this paper, we generalize the notion of unconstrained quantization of the
classical Cantor distribution to constrained quantization and give a general
definition of constrained quantization. Toward this, we calculate the optimal
sets of -points, th constrained quantization errors, the constrained
quantization dimensions, and the constrained quantization coefficients taking
different families of constraints for all . 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.Comment: arXiv admin note: text overlap with arXiv:2305.1111
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 -sided polygons inscribed in a circle. The result shows that the
quantization coefficient for the uniform distribution on the boundary of a
regular -sided polygon inscribed in a circle is an increasing function of
, and approaches to the quantization coefficient for the uniform
distribution on the circle as tends to infinity
Constrained quantization for probability distributions
In this paper, for a Borel probability measure on a Euclidean space
, we extend the definitions of th unconstrained quantization
error, unconstrained quantization dimension, and unconstrained quantization
coefficient, which traditionally in the literature known as th quantization
error, quantization dimension, and quantization coefficient, to the definitions
of th constrained quantization error, constrained quantization dimension,
and constrained quantization coefficient. The work in this paper extends the
theory of quantization and opens a new area of research. In unconstrained
quantization, the elements in an optimal set are the conditional expectations
in their own Voronoi regions, and it is not true in constrained quantization.
In unconstrained quantization, if the support of contains infinitely many
elements, then an optimal set of -means always contains exactly
elements, and it is not true in constrained quantization. It is known that the
unconstrained quantization dimension for an absolutely continuous probability
measure equals the Euclidean dimension of the underlying space. In this paper,
we show that this fact is not true as well for the constrained quantization
dimension. It is known that the unconstrained quantization coefficient for an
absolutely continuous probability measure exists as a unique finite positive
number. From work in this paper, it can be seen that the constrained
quantization coefficient for an absolutely continuous probability measure can
be any positive number depending on the constraint that occurs in the
definition of th constrained quantization error