# Minimax Analysis for Inverse Risk in Nonparametric Planer Invertible Regression

arXiv: https://arxiv.org/abs/2112.00213 · HTML: https://mlsanity.com/paper/2112.00213/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 38 references that are in the map.
- First posted on arXiv 1 Dec 2021.

## Builds on (most cited first, 14 of 14)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Block Neural Autoregressive Flow](https://mlsanity.com/paper/1904.04676/) (2019)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [The Expressive Power of a Class of Normalizing Flow Models](https://mlsanity.com/paper/2006.00392/) (2020)
- [Frequentist coverage and sup-norm convergence rate in Gaussian process regression](https://mlsanity.com/paper/1708.04753/) (2017)

## Built on by (most cited first, 1 of 1)

- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)

## Abstract (arXiv)

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. % the minimax rate, similar to other shape constraints.