Efficient Nonparametric Smoothness Estimation
This work addresses a practical bottleneck for statisticians and data scientists by providing efficient estimators for Sobolev quantities, enabling broader application in nonparametric statistics, though it is incremental as it builds on existing theory.
The authors tackled the problem of estimating Sobolev quantities of probability density functions, which are important in nonparametric statistics but lacked practical estimators, by proposing a family of estimators that are minimax rate-optimal and computationally tractable, with empirical validation on synthetic data.
Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, partly due to a lack of practical estimators. They also include, as special cases, $L^2$ quantities which are used in many applications. We propose and analyze a family of estimators for Sobolev quantities of unknown probability density functions. We bound the bias and variance of our estimators over finite samples, finding that they are generally minimax rate-optimal. Our estimators are significantly more computationally tractable than previous estimators, and exhibit a statistical/computational trade-off allowing them to adapt to computational constraints. We also draw theoretical connections to recent work on fast two-sample testing. Finally, we empirically validate our estimators on synthetic data.