Visual Scene Representations: Sufficiency, Minimality, Invariance and Deep Approximations

Stefano Soatto, Alessandro Chiuso

Abstract: An ideal visual representation would be a function of visual data that is a minimal sufficient statistics for the scene, and a maximal invariant to nuisance variability. We derive analytical expressions for such representations and show that, under certain assumptions underlying the Lambert-Ambient model, they are related to convolutional architectures. This link highlights the conditions under which they can be expected to perform well, and also suggests ways to improve and generalize them. This new interpretation draws connections to the classical theories of sampling, hypothesis testing and group invariance. We show that one layer of a convolutional architecture can approximate an optimal representation of one im- age, given sufficiently many receptive fields. We also show that stacking multiple layers, each of which is invariant to a small group transformation such as affine, achieves invariance to larger groups, all the way to planar diffeomorphisms, given sufficiently many layers and receptive fields.