Even Perfect Diffusion Models Drift When Retrained on Their Own Output

Early stopping in reverse diffusion - a numerical necessity, not a modeling failure - alone drives model collapse even when score estimation is perfect.

The Real Mechanism Behind Model Collapse

Existing work bounded finite-round error but left two questions open: what distribution does the recursion converge to, and how fast? This paper answers both. The authors prove that recursive training converges geometrically to a unique limiting distribution. That limit takes a closed form: an infinite mixture of increasingly Gaussian-smoothed versions of the original data distribution. The culprit is the early truncation of the reverse diffusion process, required for numerical stability. Even with exact sampling and perfect scores, that truncation progressively drifts the model away from the true data.

A Spectral View: Recursive Training as a Low-Pass Filter

The paper uses a Hermite spectral decomposition to characterize the limiting distribution. Recursive training acts as a low-pass filter: higher-order modes, which encode fine non-Gaussian structure, are attenuated much more strongly than coarse modes. This explains why model collapse disproportionately loses high-frequency details like texture and sharp edges. The spectral picture also motivates a fix.

The Fix: Annealed Truncation Schedules

The authors propose annealed truncation schedules that progressively shrink truncation times across retraining rounds. Any schedule that converges to zero asymptotically eliminates recursive compounding. They also show robustness: with discretization and score estimation errors, the learned distribution stays within a Wasserstein-2 ball around the ideal limit, with mode-dependent contraction rates that suppress high-order errors faster than low-order ones. Validation on synthetic Gaussian mixtures and CIFAR-10 confirms the theory.

This work gives practitioners a concrete knob - truncation time scheduling - to prevent the silent erosion of data fidelity across generations of diffusion models.

Source: Recursively Trained Diffusion Models: Limiting Collapse Distribution and Spectral Characterization
Domain: arxiv.org