# ResearchLearning and high-order statistics

How might brains learn the statistical regularities of the world?

Clearly these go well beyond second order. Here we show that a third-order statistic of edge distributions is reminiscent of curve and texture geometry:

- Third-Order Edge Statistics: Contour Continuation, Curvature, and Cortical ConnectionsAdvances in Neural Information Processing Systems 262013

The original paper relating endstopping to curvature is:

- Endstopped neurons in the visual cortex as a substrate for calculating curvatureNature3296138438 - 4411987

In general, it supports the hypothesis that differential geometry can serve as a surrogate for very high-order statistical relationships. As required, brains can learn them.

Here is a generalization of Hebbian learning:

- Feedforward learning of mixture modelsAdvances in Neural Information Processing Systems 272014

Of course, connections to diffusion geometry exist:

- Geometric diffusions as a tool for harmonic analysis and structure definition of data: Multiscale MethodsProc. Nat. Acad. Sci. (USA)102217432 - 74372005