  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.
