Exact Equivariance, Disentanglement and Invariance of Transformations

Abstract

Invariance, equivariance and disentanglement of transformations are important topics in the field of representation learning. Previous models like Variational Autoencoder [1] and Generative Adversarial Networks [2] attempted to learn disentangled representations from data with different levels of successes. Convolutional Neural Networks are approximately equivariant and invariant (if pooling is performed) to input translations. In this report, we argue that the recently proposed Object-Oriented Learning framework [3] offers a new solution to the problem of Equivariance, Invariance and Disentanglement: it systematically factors out common transformations like translation and rotation in inputs and achieves “exact equivariance” to these transformations — that is, when the input is translated and/or rotated by some amount, the output and all intermediate representations of the network are also translated and rotated by exactly the same amount. The transformations are “exactly disentangled” in the sense that the translations and rotations can be read out directly from a few known variables of the system without any approximation. Invariance can be achieved by reading other variables that are known not to be affected by the transformations. No learning is needed to achieve these properties. Exact equivariance and disentanglement are useful properties that augment the expressive power of neural networks. We believe it will enable new applications including but not limited to precise visual localization of objects and measuring of motion and angles.