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.
Figures & tables
Figure 1: Ensemble of solutions of eq. ( 1 ) with different initial conditions driven by the same realization of the noisy processes Qt and Wt with the forcing parameter γ=0.1 . In the first stage (the first row), the particles cluster into an anti-polar configuration and follow the dynamics of the ’meta’-attractor AQ(ω) . In the second stage (the second row), the particles converge to the global random attractor A(ω) and continue moving as a single cluster.
Figure 2: On the left: the ensemble of solutions of Eq. ( 17 ) driven by the same noise from different initial conditions. The system gradually approaches each of the random attractors for k=7,3 and 1 . On the right: the dynamics of pairwise scalar products between the particles showing the formation and dissolvement of the clusters.