Geometric and Information Compression of Representations in Deep Learning

Abstract

Deep neural networks transform input data into latent representations that support a wide range of downstream tasks. These representations can be characterized along information-theoretic and geometric dimensions, but their relationship remains poorly understood. A central open question is whether low mutual information (MI) between inputs and representations necessarily implies geometrically compressed latent spaces and vice versa. We investigate this question using class-wise clustering as a measure of geometric compression and theoretically sound MI estimation in conditional entropy bottleneck (CEB) networks and continuous dropout networks. We evaluate the interplay between MI, geometric compression, and generalization on classification tasks under controlled noise injection schemes. Our findings show that low MI does not reliably correspond to geometric compression, and that the connection between the two is more nuanced than often assumed. Indeed, our experiments reveal a negative and nonlinear relationship that can reverse when varying training setup. Our results put forward a hypothesis that generalization acts as a potential confounder in this connection rather than being their direct consequence.