Invariance to geometric transformations is a highly desirable property of automatic classifiers in many image recognition tasks. Nevertheless, it is unclear to which extent state-of-the-art classifiers are invariant to basic transformations such as rotations and translations. This is mainly due to the lack of general methods that properly measure such an invariance. In this paper, we propose a rigorous and systematic approach for quantifying the invariance to geometric transformations of any classifier. Our key idea is to cast the problem of assessing a classifier's invariance as the computation of geodesics along the manifold of transformed images. We propose the Manitest method, built on the efficient Fast Marching algorithm to compute the invariance of classifiers. Our new method quantifies in particular the importance of data augmentation for learning invariance from data, and the increased invariance of convolutional neural networks with depth. We foresee that the proposed generic tool for measuring invariance to a large class of geometric transformations and arbitrary classifiers will have many applications for evaluating and comparing classifiers based on their invariance, and help improving the invariance of existing classifiers.

本文提出了一种严谨和系统的方法来量化任何分类器对几何变换的不变性，并使用该方法证明了数据增强对学习不变性的重要性，以及卷积神经网络深度增加时不变性的增加。该方法可应用于评估和比较分类器的不变性，并有助于改进现有分类器的不变性。

Manitest：分类器真的具有不变性吗？