Recursively Trained Diffusion Models: Limiting Collapse Distribution and Spectral Characterization

2026-06-11

AI Summary

This work addresses the persistent distributional drift and eventual model collapse in recursive self-training diffusion models caused by premature truncation of the backward process. The study establishes, for the first time, that under this mechanism the model converges geometrically to a unique limiting distribution, which admits a closed-form expression as a Gaussian-smoothed mixture of the data distribution. Leveraging Hermite spectral decomposition, the authors provide a low-pass filtering interpretation, identifying the accumulation of high-frequency errors as the root cause of drift. To mitigate this, they propose a progressive annealing truncation strategy. Combining Wasserstein-2 stability analysis with Gaussian mixture modeling, both theoretical and empirical results—validated on synthetic Gaussian mixtures and CIFAR-10—demonstrate that higher-order mode errors decay more rapidly and confirm the efficacy of the proposed approach.

Abstract

Recursive training of generative models on their own outputs can lead to model collapse, a compounding drift away from the true data distribution. Existing theoretical works bound finite-round error accumulation in the context of diffusion models, but two questions remain open:~what distribution does the recursion converge to, and how fast? We answer both, isolating a mechanism distinct from imperfect learning: even with perfect score estimation and exact sampling, the early stopping of the reverse diffusion (required for numerical stability) drives a progressive drift away from the data distribution. We prove that this recursion converges geometrically to a unique limiting distribution, which admits a closed-form characterization as an infinite mixture of increasingly Gaussian-smoothed versions of the data distribution. A Hermite spectral decomposition of this limit reveals that 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 spectral picture motivates annealed truncation schedules that progressively shrink truncation times across retraining rounds; we prove that any schedule converging to $0$ asymptotically eliminates recursive compounding. Finally, we show our idealized characterization is robust: in the presence of discretization and score estimation errors, the learned distribution remains in a Wasserstein-2 ball around the ideal limit, with mode-dependent contraction rates that contract high-order errors faster than low-order ones. We validate the theory on synthetic Gaussian mixtures and CIFAR-10.

Problem

• model collapse
• diffusion models
• recursive training
• distribution drift
• spectral characterization

Innovation

• recursive training
• model collapse
• diffusion models
• spectral characterization
• annealed truncation