Sharp quantization for Lane–Emden problems in dimension two

Quantization dimension via quantization numbers

We give a characterization of the quantization dimension of Borel probability measures on R in terms of ε-quantization numbers. Using this concept, we show that the upper rate distortion dimension is not greater than the upper quantization dimension of order one. We also prove that the upper quantization dimension of a product measure is not greater than the sum of that of its marginals. Finall...

Two Open Problems on Effective Dimension

Effective fractal dimension was defined by Lutz [13] in order to quantitatively analyze the structure of complexity classes. The dimension of a class X inside a base class C is a real number in [0,1] corresponding to the relative size of X ∩ C inside C. Basic properties include monotonicity, so dimension 1 classes are maximal and dimension 0 ones are minimal, and the fact that dimension is defi...

Sharp quadratic majorization in one dimension

Majorization methods solve minimization problems by replacing a complicated problem by a sequence of simpler problems. Solving the sequence of simple optimization problems guarantees convergence to a solution of the complicated original problem. Convergence is guaranteed by requiring that the approximating functions majorize the original function at the current solution. The leading examples of...

Quantization Dimension for Conformal Iterated Function Systems

Fractal dimension and vector quantization

We show that the performance of a vector quantizer for a self-similar data set is related to the intrinsic (“fractal”) dimension of the data set. We derive a formula for predicting the error-rate, given the fractal dimension and discuss how we can use our result for evaluating the performance of vector quantizers quickly.