Minimax Estimation of Quadratic Fourier Functionals

Understand

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 $\mathcal{L}^2$ 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.