Open Access
2024 Minimax Analysis for Inverse Risk in Nonparametric Planer Invertible Regression
Akifumi Okuno, Masaaki Imaizumi
Author Affiliations +
Electron. J. Statist. 18(1): 355-394 (2024). DOI: 10.1214/23-EJS2202


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 L2-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.

Funding Statement

A. Okuno is supported by JSPS KAKENHI (21K17718, 22H05106) and JST CREST (JPMJCR21N3). M. Imaizumi is supported by JSPS KAKENHI (18K18114) and JST Presto (JPMJPR1852).


We would like to thank Keisuke Yano for the helpful discussion.


Download Citation

Akifumi Okuno. Masaaki Imaizumi. "Minimax Analysis for Inverse Risk in Nonparametric Planer Invertible Regression." Electron. J. Statist. 18 (1) 355 - 394, 2024.


Received: 1 August 2022; Published: 2024
First available in Project Euclid: 2 February 2024

Digital Object Identifier: 10.1214/23-EJS2202

Primary: 62G08
Secondary: 54C56 , 62C20

Keywords: Invertible regression , minimax optimal rate , shape restriction

Vol.18 • No. 1 • 2024
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