2016/05/19 by Shashank Singh, Simon S. Du, Singh, Shashank +3
Computer Science · Mathematics · #Advanced Statistical Methods and Models #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Machine Learning (stat.ML) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1605.05785
openalex publication_date 2016/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, L2 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.