Divisive Normalization: Justification and Effectiveness as Efficient Coding Transform

Divisive normalization (DN) has been advocated as an effective nonlinear {\em efficient coding} transform for natural sensory signals with applications in biology and engineering. In this work, we aim to establish a connection between the DN transform and the statistical properties of natural sensory signals. Our analysis is based on the use of multivariate {\em t} model to capture some important statistical properties of natural sensory signals. The multivariate {\em t} model justifies DN as an approximation to the transform that completely eliminates its statistical dependency. Furthermore, using the multivariate {\em t} model and measuring statistical dependency with multi-information, we can precisely quantify the statistical dependency that is reduced by the DN transform. We compare this with the actual performance of the DN transform in reducing statistical dependencies of natural sensory signals. Our theoretical analysis and quantitative evaluations confirm DN as an effective efficient coding transform for natural sensory signals. On the other hand, we also observe a previously unreported phenomenon that DN may increase statistical dependencies when the size of pooling is small.