School of Mathematics Colloquium
Optimal points for a probability distribution on a Cantor set
by Prof. Mrinal Kanti RoyChowdhury (University of Texas-Pan American, USA)
Thursday, July 21, 2011
from
to
(Asia/Kolkata)
at Colaba Campus ( AG-69 )
at Colaba Campus ( AG-69 )
TIFR
Description |
Given a probability measure $P$ on a compact subset of ${\mathbb R}^d$ and a natural number $n$, the {\it{$n$th quantization error of $P$} is defined to be $$V_n=\inf_{\ga} \int \min_{a\in\ga} \|x-a\|^2 dP(x),$$ where the infimum is taken over all subsets $\alpha$ of ${\mathbb R}^d$ with card $\alpha\leq n$, and $\| \cdot\|$ denotes the Euclidean norm on ${\mathbb R}^d$. A set $\alpha$ for which the infimum is achieved is called a {\it {$n$-optimal set}. The {\it {Quantization dimension} for the probability measure $P$ is defined by $$D(P)=\lim_{n\to \infty} \frac{2\log n}{-\log V_n},$$ and corresponds to the rate how fast $V_n$ goes to zero as $n$ tends to infinity. In this talk, we consider the Cantor set equipped with the natural homogeneous probability measure on it, and discuss the quantization errors of the measure and $n$-optimal sets for $n \geq 1$, and the quantization dimension. Some open problems in the area will be pointed out. |