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Random Quadratic Form with random forcing: Metastable synchronization by noise
Anna Shalova
TL;DR
The paper asks how arbitrarily small Brownian forcing changes synchronization in the Random Quadratic Form on a sphere. It analyzes the one- and two-point dynamics, finding metastable anti-polar clustering before eventual full synchronization, while preserving the single-point process law.
Problem
It remains unclear how arbitrarily small additive forcing changes the RQF’s symmetry-driven partial synchronization and multiscale two-point dynamics.
Method
The paper characterizes the forced RQF’s global attractor, transient clustering dynamics, convergence rates, and one-point motion in the small-forcing regime.
Results
For any nonzero forcing, trajectories eventually synchronize to a single point after approaching an anti-polar meta-attractor on the time scale t ∼ log |γ|^-1.
Takeaways & Limitations
Forcing changes synchronization at the two-point level while preserving the qualitative law of the one-point process.
Takeaways & Limitations
The paper conjectures analogous iterative convergence results only when k_i mod k_{i+1} = 0 across the harmonic cascade.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper studies a randomly forced RQF on the sphere, showing that arbitrarily small forcing breaks the anti-polar symmetry and produces metastable, multiscale synchronization in the two-point process while preserving the one-point motion up to rescaling. It also connects the forcing scale γ to bias-versus-weight initialization in continuous-time neural models.
- RQF synchronization: With γ = 0, shared-noise trajectories cluster in aligned or anti-polar configurations, with the anti-polar attractor arising from the RQF’s intrinsic symmetry.The non-forced system exhibits partial synchronization rather than convergence to a single point.
- RQF synchronization: Any nonzero forcing breaks this symmetry, making the anti-polar state unstable and driving all trajectories toward a single-valued global attractor.The forced dynamics retains the anti-polar set as an intermediate attractor but not as the long-term attractor.
- Metastable regime: As γ → 0, trajectories first approach the anti-polar ’meta’-attractor on the time scale t ∼ log |γ|−1, then converge to the global singleton attractor.This two-stage evolution is termed metastable synchronization and produces distinct transient and long-term convergence regimes.
- One-point versus two-point dynamics: The forcing changes the two-point process but preserves the qualitative one-point process: Xt/(1+γ^2) is an Sn−1-valued Brownian motion for every γ ∈ R.Thus, metastability and symmetry breaking appear at the level of multiple coupled trajectories rather than single-point motion.
- Machine-learning motivation: In the Neural ODE interpretation, the parameter γ encodes the relative scale of weight and bias initialization and determines the system’s long-time behavior.The RQF also provides a counterpart for multipoint feature dynamics in continuous-time transformers without self-attention interaction.
2 Notation and Preliminaries
This section establishes the probability-space, SDE, and random dynamical system notation used throughout the paper. It also introduces generators, attractors, and structural results connecting SDEs with Markov RDSs.
- Noise and probability spaces: The framework uses continuous-path spaces for matrix and vector noise, their Borel σ-algebras, Wiener measures, and the resulting product probability space.The spaces are ΩQ = C0(R, Rn×n) and ΩW = C0(R, Rn), with product space (Ω, F, P) = (ΩQ × ΩW, FQ × FW, PQ × PW).
- Noise and probability spaces: Time shifts on the product space form a Wiener-measure-preserving family, providing the noise dynamics for the random dynamical system.The shifts are defined on (Ω, F), and the family preserves P for measurable sets.
- SDE preliminaries: For manifold-valued Itô SDEs, pathwise solutions are defined as measurable maps from the Brownian probability space into continuous manifold-valued trajectories.Definition 2.1 fixes a two-sided Brownian motion and requires a map Xω : Ω→C(R, M) satisfying the SDE almost surely.
- SDE preliminaries: The infinitesimal generator tracks diffusion statistics and expected distances, while Dynkin’s formula supplies a key proof tool for later theorems.The generator and its adjoint also define backward and forward Kolmogorov evolutions, with the latter describing the law through the Fokker–Planck equation.
- Random dynamical systems: An RDS combines measure-preserving noise shifts with a measurable cocycle, and sufficiently regular SDEs admit a unique equivalent Markov RDS representation.The cocycle satisfies φ(0, ω, x) = x and φ(t + s, ω, x) = φ(t, θsω, φ(s, ω, x)); under the stated coefficient regularity, it yields pathwise solutions and a Markov RDS.
- Random attractors: The section defines random point attractors and records existence and sample-measure dichotomies for ergodic Markov RDSs on compact state spaces.A unique ergodic measure guarantees a minimal weak point attractor, while sample measures are either almost surely continuous or supported on finitely many atoms.
3 Main results
The RQF with forcing has the law of a rescaled spherical Brownian motion, with the uniform measure as its unique invariant distribution. Any nonzero forcing produces a singleton random attractor, while vanishing forcing yields metastable anti-polar behavior before synchronization.
- RQF with forcing: RQF is a rescaled spherical Brownian motion and is ergodic with the uniform measure on S^n−1 as its unique invariant distribution.Theorem 3.1 identifies the law with a rescaled heat flow on the sphere.
- Two-point dynamics: The two-point process characterizes anti-polar and synchronized configurations through the boundary values −1 and 1 of the inner product Z_t = ⟨X_t,Y_t⟩.Coincidence corresponds to Z_t = 1, while anti-alignment corresponds to Z_t = −1.
- Random attractor: For every nonzero forcing γ, the RQF has a singleton strong forward random attractor measurable with respect to the past.The two-point distance converges almost surely to zero, collapsing trajectories onto one random point.
- Small forcing and metastability: When γ = 0, the attractor consists of two anti-polar points, whereas small nonzero forcing generates metastable synchronization of the two-point motion.The small-forcing analysis establishes multiscale behavior and a metastable anti-polar attractor.
- Small forcing and metastability: The anti-polar meta-attractor remains attractive on a time scale of order log |γ|−1 as γ → 0.This estimate describes the first stage of the multiscale dynamics before the symmetry-breaking synchronization stage.
4 Multi-cluster and multiscale dynamics
The section extends the RQF framework to multicluster attractors and multiscale dynamics on the circle. Harmonic and multiharmonic models preserve Brownian-motion laws while producing structured random attractors and successive clustering scales.
- Multicluster dynamics: The k-harmonic circle model has a random attractor supported on at most k points because its angular dynamics reduce to a fully synchronizing Brownian motion.The rescaling ψ_t = kϕ_t preserves the Brownian-motion representation and implies almost-sure singleton synchronization in the transformed coordinate.
- Multicluster dynamics: Proposition 4.1 establishes the harmonic model’s random-attractor structure and extends the corresponding result to every k ∈ N.The supplied proposition statement introduces the harmonic model on S1 and its minimal weak random point attractor.
- Multiscale dynamics: The multiharmonic model is a time-rescaled Brownian motion, with the overall scale determined by the number of components and γ rather than the harmonic indices.Proposition 4.2 gives the rescaling w(γ,m), while the accompanying remark states that the Fokker–Planck time scale depends only on m and γ.
- Multiscale dynamics: The m-harmonic system has m − 1 meta-attractors and one global attractor, whose configurations are determined progressively by the harmonic structure.The structure of the j-th attractor depends on the harmonics (k_i)_{i≤j}.
- Multiscale dynamics: Iteratively truncating the harmonic system yields a cascade of random weak point attractors and recovers m distinct dynamical scales.The first attractor consists of exactly k_0 equidistant points, and subsequent attractors depend on progressively larger sets of harmonic numbers.