# Minimax Estimation of Quadratic Fourier Functionals

arXiv: https://arxiv.org/abs/1803.11451 · HTML: https://mlsanity.com/paper/1803.11451/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Mar 2018.

## Built on by (most cited first, 1 of 1)

- [The Fundamental Limits of Structure-Agnostic Functional Estimation](https://mlsanity.com/paper/2305.04116/) (2023)

## Abstract (arXiv)

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 L2 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.