Parameter symmetries explain geometry of neural network representations
Parameter symmetries determine representational geometry in overparameterized nonlinear networks
Machine Learning
Summary
Neural networks that have more parameters than needed can produce many different internal ways of representing information that still perform the same task. The authors show that these different representations come from three simple operations: adding, copying, or scaling features. They explain exactly how these operations shape what the network’s internal representations look like and how overparameterization increases this variety. They also find that certain ways of choosing among these possibilities help pick out unique, meaningful representations that reveal how the network is computing.
What this means in practice
- •For machine learning engineers: Design neural network architectures with awareness of representational ambiguities caused by parameter symmetries, improving interpretability efforts.
- •For computational neuroscientists: Disentangle when observed neural representations can reveal underlying computations despite parameter redundancies in models of brain function.
A theory result. No direct application yet.
Authors
Marvin Theiss, Lukas Braun, Andrew M. Saxe, Erin Grant
Abstract
Representations are routinely used across machine learning, psychology, and neuroscience to draw inferences about the computations of biological and artificial systems. Such inferences presume a meaningful link between representational geometry and the computation being performed. For artificial neural networks, however, the extent to which function constrains representation remains unclear. One key obstacle is that these networks admit parameter symmetries: changes in parameterization that preserve function exactly while reshaping representational geometry. Here, we show that a broad class of parameter symmetries acts on representations through just three primitive feature transformations: addition, duplication, and scaling. This feature-level characterization yields a closed-form decomposition of representational geometry into essential and auxiliary components, which makes precise how degeneracy in representational geometry can grow with overparameterization even when function is held fixed. Finally, we show that implementation-level selection rules can resolve this degeneracy, yielding identifiable geometries in which features are weighted according to their contributions to the network's function. Together, our results delineate when representations can support inferences about computation, and when they cannot.