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 the invertible estimator is used in many domains, such as statistics, econometrics, and machine learning. Although the consistency and universality of invertible estimators have been well investigated, 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 an inverse $L^2$-risk to evaluate an estimator which preserves invertibility. Then, we derive lower and upper rates for a minimax inverse risk by exploiting a representation of invertible functions using level-sets. To obtain an upper bound, 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 rejects the expectation of whether invertibility improves the minimax rate, similar to other shape constraints.
翻译:我们研究的是在平面上估计反向函数的微小风险,而保持偏差也是不可忽略的。 从数据中学习不可视性,在统计、计量经济学和机器学习等许多领域都使用不可逆估计值。虽然对不可逆估计值的一致性和普遍性进行了很好地调查,但这些方法的效率仍在发展之中。在这项研究中,我们研究的是估算在2美元平面方方形上不可逆的双利普西茨函数的微小风险。我们首先引入了逆值 $2$2 的风险来评价一个能保持不可逆性的估值。然后,我们通过利用水平设置的不可逆函数的表示,得出低和高的微反向风险率。为了获得上界,我们开发了一个几乎不可逆的估算值,其风险几乎在所有地方都可视性地达到从微偏差中得出的低至对正数系数的微度值。我们测出的微缩缩率相当于非可逆性负负值的预期值,即其他不可逆性负向的负值。