Minimax Analysis for Inverse Risk in Nonparametric Planer Invertible Regression

TL;DR

This paper analyzes the minimax risk of estimating invertible functions on a plane, establishing rates for inverse function estimation and introducing an estimator that nearly attains these rates, demonstrating that invertibility does not increase estimation complexity.

Contribution

It derives minimax rates for estimating invertible bi-Lipschitz functions and proposes an estimator that nearly achieves these rates, advancing understanding of invertible function estimation.

Findings

Minimax rates for inverse function estimation match non-invertible cases.

An asymptotically almost everywhere invertible estimator attains the minimax rate.

Invertibility does not increase the estimation complexity in terms of rate.

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 bi-Lipschitz 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 level-sets. 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 non-invertible bi-Lipschitz 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.