Minimax Estimation of Quadratic Fourier Functionals

Abstract

We study estimation of (semi-)inner products between two nonparametric probability distributions, given IID samples from each distribution. These products include relatively well-studied classical L 2 and Sobolev inner products, as well as those induced by translation-invariant reproducing kernels, for which we believe our results are the first. We first propose estimators for these quantities, and the induced (semi)norms and (pseudo)metrics. We then prove non-asymptotic upper bounds on their mean squared error, in terms of weights both of the inner product and of the two distributions, in the Fourier basis. Finally, we prove minimax lower bounds that imply rate-optimality of the proposed estimators over Fourier ellipsoids. Singh, Sriperumbudur, and Póczosas well as the squared seminorm P 2 a := P, P a and squared pseudometric P − Q 2 a , using n i.i.d. samples X 1 , ..., X n i.i.d.∼ P and Y 1 , ..., Y n i.i.d.∼ Q from each distribution. Specifically, we assume that P and Q lie in a smaller subspace H b ⊆ H a parameterized by a Z-indexed real family b = {b z } z∈Z . In this setting, we study the minimax L 2 error M (a, b) of estimating P, Q a , over P and Q lying in a (unit) ellipsoid with respect to• b ; that is, the quantity M (a, b