Not all solutions are created equal: An analytical dissociation of functional and representational similarity in deep linear neural networks

2026-09-30

AI Summary

This study addresses the frequent conflation of representation and function in connectionist systems through analytical derivations in two-layer linear networks, numerical simulations of nonlinear networks, and multivariate pattern analysis. It provides the first quantitative demonstration that representational similarity is separable from functional similarity. Furthermore, this work reveals a distinct mechanism whereby robustness to parameter noise compels networks to adopt specific representations, whereas input noise or generalization error imposes no such constraint. It also establishes that representational alignment reflects computational advantages beyond mere functional alignment. Collectively, this research offers a novel theoretical foundation for interpreting and comparing internal representations in artificial neural networks.

Abstract

A foundational principle of connectionism is that perception, action, and cognition emerge from parallel computations among simple, interconnected units that generate and rely on neural representations. Accordingly, researchers employ multivariate pattern analysis to decode and compare the neural codes of artificial and biological networks, aiming to uncover their functions. However, there is limited analytical understanding of how a network's representation and function relate, despite this being essential to any quantitative notion of underlying function or functional similarity. We address this question using analysable two-layer linear networks and numerical simulations in non-linear networks. We find that function and representation are dissociated, allowing representational similarity without functional similarity and vice versa. Further, we show that neither robustness to input noise nor the level of generalization error constrain representations to the task. In contrast, networks robust to parameter noise have limited representational flexibility and must employ task-specific representations. Our findings suggest that representational alignment reflects computational advantages beyond functional alignment alone, with significant implications for interpreting and comparing the representations of connectionist systems.

Problem

• representational similarity
• functional similarity
• deep linear neural networks
• connectionism
• neural representations

Innovation

• representational similarity
• functional similarity
• deep linear neural networks
• parameter noise robustness
• representational alignment