Random Quadratic Form on a Sphere: Synchronization by Common Noise

TL;DR

This paper introduces the Random Quadratic Form (RQF), a stochastic model on a sphere that exhibits synchronization phenomena driven by common noise, providing insights into transformer models and token clustering without self-attention.

Contribution

The study presents the RQF as a new stochastic differential equation model demonstrating synchronization by common noise, with analysis of invariant measures and attractors, and offers an alternative explanation for token clustering in transformers.

Findings

Two-point motion shows nontrivial synchronization behavior.

Tokens cluster even without self-attention in transformers.

Invariant measures and attractors characterize the system dynamics.

Abstract

We introduce the Random Quadratic Form (RQF): a stochastic differential equation which formally corresponds to the gradient flow of a random quadratic functional on a sphere. While the one-point dynamics of the system is a Brownian motion and thus has no preferred direction, the two-point motion exhibits nontrivial synchronizing behaviour. In this work we study synchronization of the RQF, namely we give both distributional and path-wise characterizations of the solutions by studying invariant measures and random attractors of the system. The RQF model is motivated by the study of the role of linear layers in transformers and illustrates the synchronization by common noise phenomena arising in the simplified models of transformers. In particular, we provide an alternative (independent of self-attention) explanation of the clustering behaviour in deep transformers and show that tokens cluster even in the absence of the self-attention mechanism.