Divisive normalization is an efficient code for multivariate Pareto-distributed environments

Divisive normalization is a ubiquitous computation commonly thought to be an implementation of the efficient coding principle. Despite empirical evidence that it reduces statistical redundancy present in naturalistic stimuli, making the relationship between this neural code and the statistics of a stimulus precise has remained elusive. This paper closes this gap by providing a necessary and sufficient condition for divisive normalization to generate an efficient code. The multivariate Pareto distribution found to be efficiently encoded exhibits many stylized features of naturalistic stimulus statistics and provides testable predictions. In an empirical analysis, we find that the Pareto distribution captures the statistics of natural images well, suggesting that divisive normalization may have evolved to efficiently represent stimuli from such distributions.