Hebbian learning of recurrent connections: a geometrical perspective

TL;DR

This paper demonstrates how a Hebbian learning rule in recurrent neural networks can uncover the geometric structure of input data, leading to the emergence of cortical-like maps and connectivity patterns.

Contribution

It introduces a geometrical perspective on Hebbian learning in recurrent networks, showing how input geometry influences network structure and function.

Findings

Networks extract input geometry through local interactions.

Emergence of convolutional and patchy connectivity patterns.

Patterns similar to cortical maps like ocular dominance and orientation columns.

Abstract

We show how a Hopfield network with modifiable recurrent connections undergoing slow Hebbian learning can extract the underlying geometry of an input space. First, we use a slow/fast analysis to derive an averaged system whose dynamics derives from an energy function and therefore always converges to equilibrium points. The equilibria reflect the correlation structure of the inputs, a global object extracted through local recurrent interactions only. Second, we use numerical methods to illustrate how learning extracts the hidden geometrical structure of the inputs. Indeed, multidimensional scaling methods make it possible to project the final connectivity matrix on to a distance matrix in a high-dimensional space, with the neurons labelled by spatial position within this space. The resulting network structure turns out to be roughly convolutional. The residual of the projection defines the non-convolutional part of the connectivity which is minimized in the process. Finally, we show how restricting the dimension of the space where the neurons live gives rise to patterns similar to cortical maps. We motivate this using an energy efficiency argument based on wire length minimization. Finally, we show how this approach leads to the emergence of ocular dominance or orientation columns in primary visual cortex. In addition, we establish that the non-convolutional (or long-range) connectivity is patchy, and is co-aligned in the case of orientation learning.