Random Quadratic Form with random forcing: Metastable synchronization by noise
2026-08-17 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a mathematical model called the Random Quadratic Form (RQF) with added random noise (Brownian forcing) on a sphere. They find that even a tiny amount of this noise changes how the system behaves over time, causing it to go from only partly synchronized to fully synchronized by breaking symmetries. Their work focuses on small noise amounts and explains the step-by-step process where initially two opposite clusters form, then eventually merge due to this noise. This helps understand behaviors in continuous-time machine learning models like Neural ODEs and Transformers, especially regarding initialization choices.
Random Quadratic FormBrownian forcingsynchronizationsymmetry breakingmultiscale behaviorNeural ODEscontinuous-time transformersanti-polar configurationmachine learning initialization
Authors
Anna Shalova
Abstract
We study the Random Quadratic Form (RQF) on a sphere in the presence of random Brownian forcing. We show that the forcing does not effectively change the law of the process but affects the synchronization properties of the system. While the RQF without forcing exhibits partial synchronization due to the intrinsic symmetries, the introduction of an arbitrarily small forcing results in long-term symmetry breaking and leads to full synchronization. In this work we focus on the small forcing regime and recover the multiscale behavior of the two-point process. We show that in the first stage the model converges to an anti-polar configuration due to the symmetries of the RQF and in the second stage the two clusters meet due to the symmetry breaking phenomenon. The model is motivated by continuous-time machine learning models such as Neural ODEs and continuous-time formulations of transformers. In particular, the results of this work explain the role of the bias and the scale of its initialization.