Quantized Nonparametric Estimation

We present an extension to Pinsker's theorem for nonparametric estimation over Sobolev ellipsoids when estimation is carried out under storage or communication constraints. Placing limits on the number of bits used to encode any estimator, we give tight lower and upper bounds on the excess risk due to quantization in terms of the number of bits, the signal size, and the noise level. This establishes the Pareto optimal minimax tradeoff between storage and risk under quantization constraints for Sobolev spaces. Our results and proof techniques combine elements of rate distortion theory and minimax analysis.