# Reproduction: Quantifying Error Propagation and Model Collapse in Diffusion Models

## Pages

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| [Executive summary](#/executive-summary) |
| [Claim 1: Under finite drift energy assumptions, the intra-generation KL divergence between the generated distribution and the training target is bounded above by (1/2)ε̂ᵢ², the learned-path score error energy (Proposition 3.1).](#/claim-1-under-finite-drift-energy-assumptions-the-intra-generation-kl-divergence-between-the-generated-distribution-and-the-training-target-is-bounded-above-by-1-2-the-learned-path-score-error-energy-proposition-3-1) |
| [Claim 2: A matching lower bound on the chi-squared divergence, χ²(p̂^(i+1) ∥ qᵢ) ≥ (1/4)ηᵢε²⋆,ᵢ − C·ε⁴⋆,ᵢ, holds for small score errors (ε²⋆,ᵢ ≤ 1), where ηᵢ ∈ [0,1] is the observability coefficient (Proposition 3.3, Definition 3.2).](#/claim-2-a-matching-lower-bound-on-the-chi-squared-divergence-p-i-1-q-1-4-c-holds-for-small-score-errors-1-where-0-1-is-the-observability-coefficient-proposition-3-3-definition-3-2) |
| [Claim 3: Combining the upper and lower bounds yields a two-sided equivalence χ²(p̂^(i+1) ∥ qᵢ) ≍ ε²⋆,ᵢ in the perturbative regime (Theorem 3.4).](#/claim-3-combining-the-upper-and-lower-bounds-yields-a-two-sided-equivalence-p-i-1-q-in-the-perturbative-regime-theorem-3-4) |
| [Claim 4: When the score-error series ∑ᵢ ε²⋆,ᵢ diverges, the accumulated divergence across generations cannot vanish and has a non-zero floor limsup Dᵢ ≥ αη̄ε̄ / [16(1+(1-α)²)] (Proposition 4.1).](#/claim-4-when-the-score-error-series-diverges-the-accumulated-divergence-across-generations-cannot-vanish-and-has-a-non-zero-floor-limsup-d-16-1-1-proposition-4-1) |
| [Claim 5: When ∑ᵢ ε²⋆,ᵢ converges, accumulated divergence across generations satisfies D_{N+1} + C_bias ≍ Σᵢ(1-α)^{2(N-i)}ε²⋆,ᵢ + (1-α)^{2(N+1-i₀)}D_{i₀}, decaying geometrically at rate (1-α)² per generation, where α is the fraction of fresh data mixed in each round (Theorem 4.2).](#/claim-5-when-converges-accumulated-divergence-across-generations-satisfies-d-n-1-c-bias-1-2-n-i-1-2-n-1-i-d-i-decaying-geometrically-at-rate-1-per-generation-where-is-the-fraction-of-fresh-data-mixed-in-each-round-theorem-4-2) |
| [Claim 6: The α-dependent tradeoff between drift and stability is empirically confirmed on a 10D Gaussian mixture and on Fashion-MNIST/CIFAR-10, with low α producing collapse-driven drift and high α maintaining stability (Figure 1).](#/claim-6-the-dependent-tradeoff-between-drift-and-stability-is-empirically-confirmed-on-a-10d-gaussian-mixture-and-on-fashion-mnist-cifar-10-with-low-producing-collapse-driven-drift-and-high-maintaining-stability-figure-1) |
| [Conclusion](#/conclusion) |