Adaptive detection of a high-variable function

A major difficulty arising in statistics of multi-variable functions is “the curse of dimensionality:” Rates of accuracy in estimation and separation rates in detection problems behave poorly as the number of variables increases. This difficulty arises for most popular functional classes such as Sobolev or Hölder balls.

In 2007, the authors considered functional classes of a new type, which were first introduced by Sloan and Woźniakowski. These classes are balls SW_σ,s in weighed tensor product spaces, which are characterized by two parameters: σ> 0 is a “smoothness” parameter, and s> 0 determines the weight sequence that characterizes the “importance” of variables. In particular, it was shown that under the white Gaussian noise model, the log-asymptotics of separation rates in detection are similar to those for one-variable functions of smoothness σ^* = min (s, σ) independently of the original problem dimensions; thus, the curse of dimensionality is “lifted.” However, the test procedure depends on parameters (σ, s), which are typically unknown.

In the present paper, we propose a test procedure which does not depend on parameters (σ, s) and provides the same log-asymptotics of separation rates uniformly over any compact set of parameters (σ, s). In addition, we give an independent simple proof of log-asymptotics of separation rates in the problem. Bibliography: 16 titles.