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QUANTIZATION DIMENSION FUNCTIONS AND GIBBS MEASURES MRINAL KANTI ROYCHOWDHURY
 

Summary: QUANTIZATION DIMENSION FUNCTIONS AND GIBBS MEASURES
MRINAL KANTI ROYCHOWDHURY
DEPT OF MATHEMATICS
THE UNIVERSITY OF TEXAS-PAN AMERICAN
Abstract: The term "quantization" in the title originates in the theory or signal pro-
cessing. It was used by electrical engineers starting in the late 40's. Mathematicians started
working in this area almost in the last two decades. As a mathematical topic quantization
for probability distributions concerns the best approximation of a d-dimensional probability
distribution P by a discrete probability with a given number of n-supporting points or in
other words, the best approximation of a d-dimensional random vector X with distribution
P by a random vector Y with at most n values in its image. A given random vector X
can be approximated by a discrete random vector Y in infinitely many different ways. The
random vector Y which gives the error minimum is called the optimal quantizer of the ran-
dom vector X and the corresponding error is called the optimal error. The image set of the
optimal quantizer is called the optimal set.
Goals of Quantization Theory:
Two main goals are as follows:
(1) Finding the exact configuration of the so called n-optimal sets,
(2) Estimating the rate called Quantization dimension at which the specified measure
of the error goes to zero as n increases.

  

Source: Assani, Idris - Department of Mathematics, University of North Carolina at Chapel Hill

 

Collections: Mathematics