Minimax Analysis for Inverse Risk in Nonparametric Planer Invertible Regression
Abstract
We study a minimax risk of estimating inverse functions on a plane, while keeping an estimator is also invertible. Learning invertibility from data and exploiting an invertible estimator are used in many domains, such as statistics, econometrics, and machine learning. Although the consistency and universality of invertible estimators have been well investigated, analysis of the efficiency of these methods is still under development. In this study, we study a minimax risk for estimating invertible biLipschitz functions on a square in a $2$dimensional plane. We first introduce two types of $L^2$risks to evaluate an estimator which preserves invertibility. Then, we derive lower and upper rates for minimax values for the risks associated with inverse functions. For the derivation, we exploit a representation of invertible functions using levelsets. Specifically, to obtain the upper rate, we develop an estimator asymptotically almost everywhere invertible, whose risk attains the derived minimax lower rate up to logarithmic factors. The derived minimax rate corresponds to that of the noninvertible biLipschitz function, which shows that the invertibility does not reduce the complexity of the estimation problem in terms of the rate. % the minimax rate, similar to other shape constraints.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 DOI:
 10.48550/arXiv.2112.00213
 arXiv:
 arXiv:2112.00213
 Bibcode:
 2021arXiv211200213O
 Keywords:

 Mathematics  Statistics Theory;
 Statistics  Machine Learning
 EPrint:
 34 pages, 34 figures, accepted to Electronic Journal of Statistics