Bayesian one- and two-sided inference on the local effective dimension

It is a challenge to manage infinite-or high-dimensional data in situations where storage, transmission, or computation resources are constrained. In the simplest scenario when the data consists of a noisy infinite-dimensional signal, we introduce the notion of local effective dimension as the smallest number of coordinates needed to effectively approximate the signal, formulate and study the problem of its recovery on the basis of noisy data. This problem can be associated to the problems of adaptive quantization, data compression, oracle estimation. We apply a Bayesian approach and study frequentist properties of the resulting posterior, a purely frequentist version of the results is also proposed. We derive upper and lower bounds for the local effective dimension which show that only the so called one-sided inference on the local effective dimension can be ensured whereas the two-sided inference, on the other hand, is in general impossible. We establish the minimal conditions under which two-sided inference can be made, and in passing derive an oracle estimation result. Finally, connection to the problem of smoothness estimation for some traditional smoothness scales is considered, also in the context of one-and two-sided inference on smoothness. In particular, folklore ‘it is impossible to estimate smoothness’ is formalized in a mathematically precise manner.