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On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces
Authors:S Dereich  F Fehringer  A Matoussi  M Scheutzow
Abstract:Let mgr be a centered Gaussian measure on a separable Banach space E and N a positive integer. We study the asymptotics as Nrarrinfin of the quantization error, i.e., the infimum over all subsets Escr of E of cardinality N of the average distance w.r.t. mgr to the closest point in the set Escr. We compare the quantization error with the average distance which is obtained when the set Escr is chosen by taking N i.i.d. copies of random elements with law mgr. Our approach is based on the study of the asymptotics of the measure of a small ball around 0. Under slight conditions on the regular variation of the small ball function, we get upper and lower bounds of the deterministic and random quantization error and are able to show that both are of the same order. Our conditions are typically satisfied in case the Banach space is infinite dimensional.
Keywords:High-resolution quantization  small ball probability  Gaussian process  isoperimetric inequality
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