Mrinal Kanti Roychowdhury
University of Texas-Pan American, USA
July 21, 2011
Optimal points for a probability distribution on a Cantor set: Given a probability measure $P$ on a compact subset of ${\mathbb R}^d$ and a natural number $n$, the {$n$th quantization error of $P$} is defined to be $$V_n=\inf_{\gamma} \int \min_{a\in\gamma} \|x-a\|^2 dP(x), $$ 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.
University of Texas-Pan American, USA
July 21, 2011
Optimal points for a probability distribution on a Cantor set: Given a probability measure $P$ on a compact subset of ${\mathbb R}^d$ and a natural number $n$, the {$n$th quantization error of $P$} is defined to be $$V_n=\inf_{\gamma} \int \min_{a\in\gamma} \|x-a\|^2 dP(x), $$ 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.