Diffeomorphisms invariance is a proxy of performance in deep neural networks

Understanding why deep nets can classify data in large dimensions remains a challenge. It has been proposed that they do so by becoming stable to diffeomorphisms, yet existing empirical measurements support that it is often not the case. We revisit this question by defining a maximum-entropy distribution on diffeomorphisms, that allows to study typical diffeomorphisms of a given norm. We confirm that stability toward diffeomorphisms does not strongly correlate to performance on benchmark data sets of images. By contrast, we find that the stability toward diffeomorphisms relative to that of generic transformations R_f correlates remarkably with the test error ε_t. It is of order unity at initialization but decreases by several decades during training for state-of-the-art architectures. For CIFAR10 and 15 known architectures, we find ε_t ∼ 0.2 √R_f, suggesting that obtaining a small R_f is important to achieve good performance.