We present several experiments demonstrating the geometric structure of the distribution of edges in natural scenes. We look at both the third-order interaction statistics and a mixture model characterization of natural images. Motivated by the mixture model characterization, we demonstrate that a novel learning rule, triplet BCM, can learn selectivity for low-rank mixture models under a broad variety of noise conditions. This learning rule could be easily implemented with a neural ciruit using only local computations without feedback or recursion. Furthermore, this learning rule predicts a timing dependent synaptic modification rule that matches measured synaptic modification in cortex through all measured third order interactions. We believe this learning rule provides an explaination for the functional role of spike timing dependent plasticity in visual cortex.

We provide an analysis of this learning rule, demonstrating that it performs stochastic gradient ascent in a tensor objective fuction. This objective function is related to the cross-moment skewness, a common objective function in the tensor decomposition literature. We show convergence with probability one for several variants of our learning rule, and show numerically that the expected update of this learning rule will also approximately converge to a selective state for input data drawn from a hidden Markov model when the Markov chain has sufficiently lazy transitions.