Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems

Ann Dooms

cs.LGMar 13, 2026

Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems

Authors: Ann Dooms

Organizations: Department of Mathematics & Data Science Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels, BELGIUM

Abstract

What is a diffusion model actually doing when it turns noise into a photograph? We show that the deterministic DDIM reverse chain operates as a Partitioned Iterated Function System (PIFS) and that this framework serves as a unified design language for denoising diffusion model schedules, architectures, and training objectives. From the PIFS structure we derive three computable geometric quantities: a per-step contraction threshold $L^*_t$ , a diagonal expansion function $f_t(λ)$ and a global expansion threshold $λ^{**}$ . These quantities require no model evaluation and fully characterize the denoising dynamics. They structurally explain the two-regime behavior of diffusion models: global context assembly at high noise via diffuse cross-patch attention and fine-detail synthesis at low noise via patch-by-patch suppression release in strict variance order. Self-attention emerges as the natural primitive for PIFS contraction. The Kaplan-Yorke dimension of the PIFS attractor is determined analytically through a discrete Moran equation on the Lyapunov spectrum. Through the study of the fractal geometry of the PIFS, we derive three optimal design criteria and show that four prominent empirical design choices (the cosine schedule offset, resolution-dependent logSNR shift, Min-SNR loss weighting, and Align Your Steps sampling) each arise as approximate solutions to our explicit geometric optimization problems tuning theory into practice.

Figures & tables

Schedule	$N$	Mean $L_{t}^{*}$	CV	$L_{t}^{*}$ at finest step	finest $t$
Linear (DDPM)	1000	0.805	0.341	0.00500	1
Cosine (orig., $s_{\mathrm{off}}=0$ )	1000	0.637	0.483	0.00079	1
Cosine (impr., $s_{\mathrm{off}}=0.008$ )	1000	0.641	0.474	0.00321	1
50-step DDIM (cosine)	50	0.637	0.483	0.01571	20

Table 1: Schedule comparison: statistics of the per-step contraction threshold

L_{t}^{*}

for four schedules (

T=1000

; 50-step DDIM subsamples the cosine schedule at stride 20). Lower CV indicates a more equalised schedule; higher mean indicates a stronger average contraction margin. No schedule dominates on both criteria. The final column reports

L_{t}^{*}

at the finest step actually executed by each chain.

Schedule	$\mathrm{CV}(\mathrm{IG}_{t})$	$\mathrm{CV}(\|\Delta d_{t}\|)$	Spearman $\rho$	Ratio/theory
Linear (DDPM)	1.107	0.836	0.9999	$1.04\times$
Cosine ( $s_{\mathrm{off}}=0.008$ )	0.867	0.570	0.9998	$1.02\times$
Ratio/theory $=\overline{\mathrm{IG}_{t}/\Delta d_{t}^{2}}\cdot N^{++}$ ; equals 1 at CS equality.

Table 2: Information gain equalisation: coefficient of variation of

\mathrm{IG}_{t}

and

|\Delta d_{t}|

across 1000 training steps, computed analytically from CIFAR-10 patch variances (

n_{k}=192

M=16

, all patches above

\lambda^{**}

). Lower CV indicates a more information-equalised schedule. Spearman

\rho(\mathrm{IG}_{t},|\Delta d_{t}|)\geq 0.9999

for both.

$t$	Regime	$\rho$	$p$ -value	sig
1	II	$+0.118$	$0.664$	n.s. ( $S_{k,t}\sim 10^{-4}$ , near-uniform)
101	II	$+0.009$	$0.974$	n.s.
201	II	$+0.459$	$0.074$	n.s.
381	I/II	$+0.594$	$0.015$	$*$
461	I	$+0.647$	$0.007$	$**$
601	I	$+0.953$	$1.2\times 10^{-8}$	$**$ (peak)

Table 3: Spearman rank-correlation

\rho(\lambda_{k},S_{k,t})

across

M=16

patches at selected timesteps (

N_{\mathrm{seeds}}=5

). Positive

\rho

confirms (MM): higher-variance patches carry larger suppression. Significance levels:

*\,p<0.05

**\,p<0.01

DDIM step	$t$ (train)	Mean $\hat{\kappa}_{t}$	Std	Regime
1 of 50	980	1.0011	0.0019	high-noise
5 of 50	900	1.0002	0.0001	high-noise
10 of 50	800	1.0005	0.0004	high-noise
20 of 50	600	1.0248	0.0177	transition
30 of 50	400	1.1108	0.0247	mid-noise
35 of 50	300	1.1408	0.0859	mid-noise

Table 4: Global per-step spectral norm for DDPM CIFAR-10 (

n=3{,}072

, 50-step DDIM, 7 seeds).

$t$	$\hat{\kappa}_{t}^{\mathrm{diag}}$	$\pm$ std	$\hat{\delta}_{t}^{\mathrm{cross}}$	Global $\hat{\kappa}_{t}$	Regime
980	1.0004	0.0001	0.0007	1.0011	high-noise
800	1.0002	0.0001	0.0008	1.0010	high-noise
600	1.0000	0.0015	0.0853	1.0853	denoising
400	1.0026	0.0153	0.1273	1.1300	denoising
280	1.0070	0.0135	0.1038	1.1107	denoising/fine-detail
220	1.0325	0.0432	0.0768	1.1092	fine-detail

Table 5: Block-Jacobian decomposition for DDPM CIFAR-10 (

8\times 8

patches,

M=16

N_{\mathrm{seeds}}=4

). Selected representative steps.

$t$	Regime	$H(A_{t})$ (nats)	$\hat{\delta}_{t}^{\mathrm{cross}}$	$\pm$ std
980	high-noise	4.963	0.00946	0.00018
840	high-noise	4.930	0.01511	0.00076
700	I	4.794	0.04109	0.00074
560	I	4.662	0.09463	0.01117
440	I	4.656	0.14824	0.02529
300	I	4.631	0.19204	0.00187

Table 6: Cross-patch coupling

\hat{\delta}_{t}^{\mathrm{cross}}

and mean attention entropy

H(A_{t})

(nats) at selected DDIM timesteps.

\hat{\delta}_{t}^{\mathrm{cross}}

grows

218\times

from the high-noise limit of Regime I to Regime II; Spearman

\rho(H,\hat{\delta}^{\mathrm{cross}})=-1.000

(

p\approx 0.00025

, exact permutation probability for a perfect ranking on 8 points).

Distance $d$	$\bar{C}$ (mean)	std
1 (adjacent)	0.0362	0.0789
2 (next-nearest)	0.0123	0.0203
3 (diagonal)	0.0117	0.0193

Table 7: Mean coupling

\bar{C}_{kj}=\tfrac{1}{2}(\|[J]_{kj}\|_{F}+\|[J]_{jk}\|_{F})

vs. Chebyshev patch distance, averaged across all timesteps and trajectories.

$t$	Regime	Mean slack	Min slack	Frac. violated
980	high-noise	$-0.000524$	$-0.000650$	$16/16$
800	high-noise	$-0.001927$	$-0.002632$	$16/16$
700	I	$-0.003942$	$-0.005122$	$16/16$
600	I	$-0.004958$	$-0.006310$	$16/16$
280	I	$-0.003904$	$-0.004638$	$16/16$
200	I/II	$-0.000304$	$-0.000717$	$14/16$

Table 8: Per-patch (PC) margin slack

\delta^{\mathrm{loc}}=(1-\hat{\kappa}_{k,t}^{\mathrm{diag}})-\mathcal{C}_{k,t}(x_{t})

for DDPM CIFAR-10 (

M=16

patches,

8\times 8

N_{\mathrm{seeds}}=5

K_{\mathrm{PI}}=30

). Positive slack means (PC) is satisfied.

$t$	$\rho$	$p$ -value	sig	Note
260	$-0.668$	0.005	**	transition zone: learned ordering dominates
240	$-0.503$	0.047		transition zone
220	$-0.391$	0.134	n.s.
200	$-0.268$	0.316	n.s.
180	$-0.344$	0.192	n.s.
160	$-0.165$	0.542	n.s.

Table 9: Spearman rank-correlation

\rho(\hat{\lambda}_{k},\hat{\kappa}_{t}^{\mathrm{diag}})

across 16 patches at Regime II steps (

N=16

8\times 8

patches). Negative

\rho

confirms that low-variance patches release suppression first, as predicted by Theorem 22 .

DDIM step	$t$	$\sqrt{\bar{\alpha}_{t}}$	$\mathbb{E}[\\|\widehat{\Delta}_{t}\\|_{2}]$	Regime
1 of 50	999	0.0064	$0.18\pm 0.04$	high-noise
(extra)	990	0.0070	$0.19\pm 0.05$	high-noise
(extra)	980	0.0078	$0.21\pm 0.05$	high-noise
2 of 50	960	0.0094	$0.26\pm 0.06$	high-noise
3 of 50	940	0.0114	$0.31\pm 0.07$	high-noise
4 of 50	920	0.0138	$0.38\pm 0.09$	high-noise

Table 10: Paired-difference proxy

\mathbb{E}[\|\widehat{\Delta}_{t}\|_{2}]

at selected timesteps for DDPM CIFAR-10 (

N=200

images, mean

\pm

std). High-noise points (

t\geq 800

) are used for OLS log-log fit; at these timesteps

\widehat{\Delta}_{t}\approx\Delta_{t}

Figure 1: Paired-difference proxy $\mathbb{E}[\|\widehat{\Delta}_{t}\|_{2}]$ for DDPM CIFAR-10 ( $N=200$ images); $\widehat{\Delta}_{t}\approx\Delta_{t}$ at high noise and overestimates $\Delta_{t}$ at lower noise (see text). Left : proxy vs. timestep $t$ ; the dashed line marks the high-noise boundary at $t=800$ . Right : log-log plot with OLS fit over the high-noise region ( $t\geq 800$ , dark markers); slope $0.95$ is consistent with $O(\sqrt{\bar{\alpha}_{t}})$ scaling of $\|\Delta_{t}\|_{2}$ (Theorem 16 ). Mid-noise points (light markers) lie above the extrapolated line.

patch $R_{k}$	$\lambda_{k}$	$\Lambda_{\mathrm{pred}}$	$\hat{\Lambda}_{k}$	std
0	231.7	$-0.0552$	$-0.00496$	$0.0067$
3	181.9	$-0.0572$	$-0.00269$	$0.0068$
12	184.8	$-0.0587$	$-0.00399$	$0.0064$
4	176.1	$-0.0656$	$-0.00809$	$0.0067$
8	174.1	$-0.0639$	$-0.00666$	$0.0087$
15	156.7	$-0.0625$	$-0.00419$	$0.0068$

Table 11: Suppression-corrected Lyapunov exponents for CelebA-HQ ( google/ddpm-celebahq-256 ,

64\times 64

16\times 16

patches,

N_{\mathrm{traj}}=8

T_{\mathrm{DDIM}}=49

\Lambda_{\mathrm{pred}}

is the predicted exponent from the suppression-corrected formula;

\hat{\Lambda}_{k}

is the empirical estimate (mean

\pm

std over trajectories). All 16 patches agree in sign: both predicted and empirical exponents are negative, so

d_{\mathrm{KY}}^{\mathrm{eff}}=0

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Forward process and schedule
Symbol	Meaning
$x_{0}\in\mathbb{R}^{n}$	Clean image
$x_{t}\in\mathbb{R}^{n}$	Noisy image at timestep $t$
$\bar{\alpha}_{t}$	Cumulative noise schedule coefficient
$v_{t}=1-\bar{\alpha}_{t}$	Residual noise variance at step $t$
$\mathrm{SNR}_{t}=\bar{\alpha}_{t}/v_{t}$	Signal-to-noise ratio at step $t$

Appendix

Table 13

Attractor geometry
$\ell_{i}$	Lyapunov exponents of $\Phi$ at fixed point ( 43 )
$\Lambda(\mu)=\frac{1}{T}\sum_{t}\log f_{t}(\mu)$	Mean log-expansion rate at spectral value $\mu$
$\lambda^{**}$	Global Moran expansion threshold ( $\prod_{t}f_{t}(\lambda^{**})=1$ ) ( 45 )
$d_{\mathrm{KY}}(\mathcal{A})$	Kaplan-Yorke dimension ( 44 )
$g_{k,t}=f_{t}(\lambda_{k})-S_{k,t}(x_{t})$	Suppression-corrected diagonal factor, evaluated along $x_{t}\sim q(x_{t})$ ( 48 )
$\Lambda_{\mathrm{eff},k}=\frac{1}{T}\sum_{t}\mathbb{E}_{x_{t}}[\log g_{k,t}]$	Suppression-corrected Lyapunov exponent ( 49 )

Appendix

Table 14

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