Limiting distribution under recursive training of diffusion models

Determine whether the sequence of model distributions (\hat p^i) produced by recursively training a score-based diffusion model on mixed data q_i = \alpha\,data + (1-\alpha)\,\hat p^i converges to a limiting distribution as i \to \infty, and, if convergence occurs, characterize how the limiting distribution depends on the fresh–data proportion \alpha and on the true data distribution data.

Background

The paper analyzes recursive training of score-based diffusion models where, at each generation i, the training distribution is the mixture q_i = \alpha\,data + (1-\alpha)\,\hat pi of fresh samples from the true data distribution and synthetic samples from the current model. It establishes lower and upper bounds on intra-generation and accumulated divergences, showing how fresh data mitigates error accumulation and how score estimation errors propagate.

While the analysis provides finite-horizon bounds and discounted accumulation laws under small error regimes, it leaves open the asymptotic behavior of the recursion. Specifically, the existence and characterization of a limiting distribution for (\hat pi) under recursive training are not resolved, motivating a precise convergence question tied to \alpha and data.

References

Finally, a key open question is: is there is a limiting distribution to which the model converges when recursively trained, and if so, how does it depend on \alpha and \textit{data}?

— Error Propagation and Model Collapse in Diffusion Models: A Theoretical Study  (2602.16601 - Khelifa et al., 18 Feb 2026) in Conclusion and Future Work

Their real proportion stays constant across iterations rather than being driven to zero as in our fixed budget protocol, so their result bounds how little synthetic data suffices but leaves the long-horizon asymptotics open.

— Feature Selective Model Collapse in Diffusion Models: Total Replacement versus Fixed-Budget Training  (2610.01318 - Malet et al., 1 Oct 2026) in Appendix, Section "Related literature and positioning", paragraph discussing Bohacek et al.