Tensor decomposition of higher-order correlations by nonlinear Hebbian plasticity

TL;DR

This paper introduces generalized nonlinear Hebbian learning rules enabling neurons to learn tensor decompositions of higher-order input correlations, revealing how nonlinearities influence neural encoding of complex input structures.

Contribution

It presents a family of nonlinear Hebbian rules and proves their ability to learn tensor eigenvectors, expanding understanding of synaptic plasticity beyond classic models.

Findings

Neurons can learn eigenvectors of higher-order input correlation tensors.

Basins of attraction for tensor eigenvectors are characterized and largest for dominant eigenvectors.

Any rule with a finite Taylor expansion admits stable equilibria at tensor eigenvectors.

Abstract

Biological synaptic plasticity exhibits nonlinearities that are not accounted for by classic Hebbian learning rules. Here, we introduce a simple family of generalized nonlinear Hebbian learning rules. We study the computations implemented by their dynamics in the simple setting of a neuron receiving feedforward inputs. These nonlinear Hebbian rules allow a neuron to learn tensor decompositions of its higher-order input correlations. The particular input correlation decomposed and the form of the decomposition depend on the location of nonlinearities in the plasticity rule. For simple, biologically motivated parameters, the neuron learns eigenvectors of higher-order input correlation tensors. We prove that tensor eigenvectors are attractors and determine their basins of attraction. We calculate the volume of those basins, showing that the dominant eigenvector has the largest basin of attraction. We then study arbitrary learning rules and find that any learning rule that admits a finite Taylor expansion into the neural input and output also has stable equilibria at generalized eigenvectors of higher-order input correlation tensors. Nonlinearities in synaptic plasticity thus allow a neuron to encode higher-order input correlations in a simple fashion.