Emergence of Quantised Representations Isolated to Anisotropic Functions

TL;DR

This paper introduces a new method to analyze how discrete representations emerge in autoencoders, revealing that certain symmetries in activation functions induce quantization, which impacts interpretability and reconstruction quality.

Contribution

The study demonstrates how specific algebraic symmetries in activation functions lead to quantized representations, providing a novel tool and causal understanding of representation discretization.

Findings

Discrete symmetries induce quantization in representations

Quantization correlates with increased reconstruction error

Symmetries influence the emergence of interpretable discrete codes

Abstract

Presented is a novel methodology for determining representational structure, which builds upon the existing Spotlight Resonance method. This new tool is used to gain insight into how discrete representations can emerge and organise in autoencoder models, through a controlled ablation study that alters only the activation function. Using this technique, the validity of whether function-driven symmetries can act as implicit inductive biases on representations is determined. Representations are found to tend to discretise when the activation functions are defined through a discrete algebraic permutation-equivariant symmetry. In contrast, they remain continuous under a continuous algebraic orthogonal-equivariant definition. This confirms the hypothesis that the symmetries of network primitives can carry unintended inductive biases, leading to task-independent artefactual structures in representations. The discrete symmetry of contemporary forms is shown to be a strong predictor for the production of symmetry-organised discrete representations emerging from otherwise continuous distributions -- a quantisation effect. This motivates further reassessment of functional forms in common usage due to such unintended consequences. Moreover, this supports a general causal model for a mode in which discrete representations may form, and could constitute a prerequisite for downstream interpretability phenomena, including grandmother neurons, discrete coding schemes, general linear features and a type of Superposition. Hence, this tool and proposed mechanism for the influence of functional form on representations may provide insights into interpretability research. Finally, preliminary results indicate that quantisation of representations correlates with a measurable increase in reconstruction error, reinforcing previous conjectures that this collapse can be detrimental.