Learning reveals invisible structure in low-rank RNNs

TL;DR

This paper develops a theoretical framework for understanding learning dynamics in low-rank RNNs, revealing how hidden overlaps influence learning and memory, with implications for biological neural systems.

Contribution

It extends the low-rank RNN framework to include learning dynamics through a reduced ODE system, distinguishing between loss-visible and loss-invisible overlaps.

Findings

Learning acts as a perturbation revealing connectivity differences.

Loss-invisible overlaps encode training history as memory variables.

The theory makes testable predictions for biological learning experiments.

Abstract

Learning in neural systems arises from synaptic changes that reshape the representations underlying behavior. While low-rank recurrent neural networks (RNNs) have emerged as a powerful framework for linking connectivity to function, a theoretical understanding of their learning process remains elusive. Here, we extend the low-rank framework from activity to learning by deriving gradient-descent dynamics directly in a reduced overlap space. We formulate a closed-form, low-dimensional system of ODEs that governs learning in this space, exact for linear RNNs and asymptotically exact for nonlinear RNNs in the large-$N$ Gaussian limit. Central to our analysis is a distinction between two classes of overlaps: loss-visible overlaps, which fully determine network activity, output, and loss, and loss-invisible overlaps, which do not affect function but are required to describe learning. We illustrate the consequences of this decomposition through two phenomena. First, we show that learning can serve as a perturbation that exposes differences in connectivity between functionally equivalent networks. Second, we show that loss-invisible overlaps can act as memory variables that encode training history, and characterize the conditions under which this occurs. Finally, we present several testable predictions for biological learning experiments derived from our theory.