Correlations in high dimensional or asymmetric data sets: Hebbian neuronal processing

The Hebbian neural learning algorithm that implements Principal Component Analysis (PCA) can be extended for the analysis of more realistic forms of neural data by including higher than two-channel correlations and non-Euclidean 1p metrics. Maximizing a dth rank tensor form which correlates d channels is equivalent to raising the exponential order of variance correlation from 2 to d in the algorithm that implements PCA. Simulations suggest that a generalized version of Oja's PCA neuron can detect such a dth order principal component. Arguments from biology and pattern recognition suggest that neural data in general is not symmetric about its mean; performing PCA with an implicit 1l metric rather than the Euclidean metric weights exponentially distributed vectors according to their probability, as does a highly nonlinear Hebb rule. The correlation order d and the 1p metric exponent p were each roughly constant for each of several Hebb rules simulated. High-order correlation analysis may prove increasingly useful as data from large networks of cells engaged in information processing becomes available.