Steering Dynamical Regimes of Diffusion Models by Breaking Detailed Balance

Abstract

We show that deliberately breaking detailed balance in generative diffusion processes can accelerate the reverse process without changing the stationary distribution. Considering the Ornstein--Uhlenbeck process, we decompose the dynamics into a symmetric component and a non-reversible anti-symmetric component that generates rotational probability currents. We then construct an exponentially optimal non-reversible perturbation that improves the long-time relaxation rate while preserving the stationary target. We analyze how such non-reversible control reshapes the macroscopic dynamical regimes of the phase transitions recently identified in generative diffusion models. We derive a general criterion for the speciation time and show that suitable non-reversible perturbations can accelerate speciation. In contrast, the collapse transition is governed by a trace-controlled phase-space contraction mechanism that is fixed by the symmetric component, and the corresponding collapse time remains unchanged under anti-symmetric perturbations. Numerical experiments on Gaussian mixture models support these findings.

Figures (3)

• Figure 1: Acceleration of speciation by non-reversible drift in Gaussian mixture. Left: Dynamics under Lelièvre's exponentially optimal drift. The labels $\mathbf{Q}_1$, $\mathbf{Q}_2$, and $\mathbf{Q}_3$ correspond to different choices of the auxiliary spectrum used in the construction algorithm. $\mathbf{Q}_1$ uses a standard linear spectrum with $\lambda \in [1, d]$, $\mathbf{Q}_2$ uses a shifted high-frequency spectrum with $\lambda \in [d+1, 2d]$, and $\mathbf{Q}_3$ uses a geometric spectrum. Right: Dynamics under a simple $\mathbf{Q}$ strategy defined as a dense anti-symmetric matrix where all upper-triangular elements are set to a constant magnitude $q$, specifically $Q_{ij} = q$ for $i < j$. The theoretical speciation times $t_S$ in legends are computed by solving $\lambda_{\min}(\widetilde{\mathbf{M}}(t_S))=0$. Notably, the simple strategy achieves faster speciation than the "exponentially optimal" designs. This arises because Lelièvre's optimality applies to the asymptotic decay rate when $t \to \infty$, whereas the speciation event occurs at short times where transient non-normal effects dominate the dynamics.
• Figure 2: Validation of the theoretical scaling for speciation dynamics. The probability $\phi(t)$ is plotted against the rescaled time $t/t_S$ using the same experimental configuration and drift matrices defined in Figure 1. Left: Results for Lelièvre's exponentially optimal drift. Right: Results for the simple $\mathbf{Q}$ strategy. The theoretical $t_S$ matches perfectly with the numerical results obtained from the Gaussian mixture simulations.
• Figure 3: Invariance of the collapse time to non-reversible perturbations. The normalized excess entropy density is plotted against time for different values of $\mathbf{Q}$. The collapse time $t_C$ is defined as the time at which the curves lift off from zero. Left: For non-diagonal $\mathbf{U}$, while the shapes of the transition curves differ, their onset point remains the same regardless of the value of $\mathbf{Q}$, validating that the collapse time is robust to non-reversible perturbations. Right:For diagonal $\mathbf{U}$, the entropy curves for different $\mathbf{Q}$ values are indistinguishable, showing that both the collapse time and subsequent dynamics are independent of the non-reversible component.

Theorems & Definitions (1)

• Proposition 1: Maximal spectral gap lelievre2013optimal