Spectral Amplitude Purification in Distribution Matching for Diffusion Distillation

Zhenyu Zhou; Can Wang; Chun Chen; Zeyu Zheng; Defang Chen

cs.CVSep 24, 2026

Spectral Amplitude Purification in Distribution Matching for Diffusion Distillation

Authors: Zhenyu Zhou, Can Wang, Chun Chen, Zeyu Zheng, Defang Chen

Organizations: Zhejiang University · University of California, Berkeley

Abstract

Distribution Matching Distillation (DMD) enables high-quality diffusion sampling in only a few steps, but its optimization dynamics remain dominated by coarse, low-frequency signals, delaying the recovery of fine-grained details. We identify a pronounced concentration of spectral amplitudes at low frequencies in the DMD directional error, where dominant low-frequency components overwhelm weaker mid- and high-frequency signals. To address this issue, we propose Spectral Amplitude Purification for Distribution Matching Distillation (SAP-DMD), a plug-and-play approach that adaptively modulates the amplitude spectrum of the DMD directional field. By suppressing the dominant tail of the amplitude spectrum, SAP-DMD reduces low-frequency dominance and promotes more effective recovery of fine structures and textures. Experiments on PixArt- $α$ , SD3, and SD3.5 demonstrate that SAP-DMD accelerates training convergence and improves generation quality under both 2-step and 4-step sampling.

Figures & tables

Panel of Figure 1 — Figure 1: Performance overview on SD3.5 Medium. (a) Training evolution of DMD2 and SAP-DMD. SAP-DMD recovers fine details earlier and improves DrawBench ( Saharia et al., 2022 ) evaluation metrics faster on all 200 prompts using 2-step generation without GAN loss. (b) 2-step samples generated by SAP-DMD, which show sharp structures and rich details across diverse prompts.

Panel of Figure 2 — Figure 2: Frequency-domain diagnostics of the DMD directional error $\mathbf{\Delta}_{t}$ on SD3.5 Medium. Statistics are computed at $t=0.5$ from a DMD2 checkpoint trained for 200 steps and averaged over 1,000 samples. (a) Radial amplitude profiles before and after purification; the dotted line denotes the mean channel-wise threshold $\bar{\mathcal{S}}$ . (b) Spatial visualization of $\mathbf{\Delta}_{t}$ before and after purification (top), with the corresponding log-amplitude distributions and channel-wise thresholds $\mathcal{S}_{c}$ (bottom). (c) Relative spectral energy shares of low-, mid-, and high-frequency components, defined by $\|\mathbf{u}\|_{2}<16$ , $16\leq\|\mathbf{u}\|_{2}<32$ , and $32\leq\|\mathbf{u}\|_{2}\leq 64$ , respectively.

Method	NFE	GAN	MS-COCO			HPSv2.1
Method	NFE	GAN	IR	PS	CS	Anime	Concept	Painting	Photo	Avg.
PixArt- $\alpha$ $(512\times 512)$
PixArt- $\alpha$	25 $\times$ 2		0.7893	22.4067	30.47	30.87	29.04	28.69	28.43	29.26
YOSO	4	✓	0.7730	21.7805	27.87	30.43	30.44	30.38	27.42	29.67
DMD2 $\dagger$	4		0.7945	22.2972	30.53	32.20	31.27	31.27	28.68	30.86
DMD2 $\dagger$	4	✓	0.8047	22.4519	30.82	32.35	31.33	31.35	29.03	31.02

Table 1: Quantitative results of 4-step generation on MS-COCO and HPSv2.1.

\dagger

: results from our re-implementation.

\times 2

indicates classifier-free guidance ( Ho and Salimans, 2022 ) requiring two NFEs per sampling step. GAN: with adversarial training enabled. Gray rows: multi-step base models. Best results are in bold and second best underlined.

Method	NFE	GAN	MS-COCO			HPSv2.1
Method	NFE	GAN	IR	PS	CS	Anime	Concept	Painting	Photo	Avg.
SDXL	25 $\times$ 2		0.7586	22.4153	32.76	30.32	28.55	28.15	26.95	28.49
SD3 Medium	25 $\times$ 2		1.0026	22.5014	32.19	30.63	29.91	30.29	27.62	29.61
SD3.5 Medium	25 $\times$ 2		1.0208	22.4234	32.78	31.32	30.49	30.48	27.66	29.99
FLUX.1-dev	25		1.0653	23.0421	31.50	32.08	30.38	30.88	29.40	30.68
SDXL-Lightning	2	✓	0.6511	22.3895	31.28	30.86	29.53	29.49	27.23	29.28

Table 2: Cross-model comparison of 2-step generation on MS-COCO and HPSv2.1.

\dagger

: results from our re-implementation. GAN: with adversarial training enabled. Gray rows: multi-step base models. Best results are in bold and second best underlined.

Table 5

Panel of Figure 3 — Figure 3: Spectral evolution throughout SAP-DMD training on SD3.5 Medium. Statistics are averaged over 256 samples at $t=0.5$ with a checkpoint interval of 100 steps. (a) Low-frequency dominance $R_{\mathrm{LF}}$ , defined as the ratio of mean amplitudes over $\|\mathbf{u}\|_{2}<16$ and $16\leq\|\mathbf{u}\|_{2}\leq 64$ , remains consistently reduced after purification. (b) Relative amplitude suppression, measured as $1-\bar{\mathcal{A}}^{\mathrm{SAP}}(\|\mathbf{u}\|_{2})/\bar{\mathcal{A}}(\|\mathbf{u}\|_{2})$ , where $\bar{\mathcal{A}}$ denotes the sample-averaged radial amplitude.

Figure 4: Qualitative comparison of 4-step (top) and 2-step (bottom) generation.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: Radial amplitude profiles across representative diffusion timesteps on SD3.5 Medium. The DMD directional field consistently exhibits pronounced low-frequency amplitude concentration, while SAP-DMD suppresses dominant amplitudes and largely preserves weaker components.

Panel of Figure 6 — Figure 6: Relative spectral energy shares across representative diffusion timesteps on SD3.5 Medium. Low-frequency energy becomes increasingly dominant in the raw DMD directional field as $t$ increases. Across all timesteps, SAP-DMD substantially reduces this relative low-frequency dominance and increases the energy shares of mid- and high-frequency components.

Panel of Figure 7 — Figure 7: Standardized log-amplitude distributions across representative diffusion timesteps on SD3.5 Medium. Statistics are computed at each timestep from a DMD2 trained for 200 steps and averaged over 128 samples, with each distribution standardized independently before aggregation. The dashed curve denotes a standard Gaussian for reference. The log-amplitude distributions are substantially more symmetric than the raw amplitude distributions and remain bell-shaped across timesteps, supporting the log-domain $k$ -sigma boundary used by SAP-DMD.

Figure 8: Relative contributions of different source frequency bands to the parameter gradient on SD3.5. SAP-DMD consistently reduces the low-frequency contribution and increases the high-frequency contribution after backpropagation through the generator Jacobian.

t=0.98

is used because the generator Jacobian vanishes at the terminal timestep

t=1.0

Batch	TTUR	GAN	MS-COCO			HPSv2.1					A100
size	TTUR	GAN	IR	PS	CS	Anime	Concept	Painting	Photo	Avg.	hours
4	1		1.0867	22.4414	31.87	34.16	33.17	33.35	29.99	32.67	10.56
4	5		1.0761	22.4248	31.92	34.01	32.96	33.09	29.78	32.46	28.82
4	1	✓	1.1104	22.5945	32.14	33.58	32.92	33.04	29.97	32.38	+11.09
32	1		1.1245	22.5483	32.00	34.28	33.29	33.32	30.50	32.85	90.34
32	1	✓	1.1444	22.6080	31.98	34.45	33.48	33.63	30.77	33.08	+86.39

Appendix

Table 5: Sensitivity of 2-step SAP-DMD on SD3.5 Medium to the training recipe. GAN: with GAN loss in the second training stage;

+

: additional A100 hours of this stage.

Method	MS-COCO			HPSv2.1
Method	IR	PS	CS	Anime	Concept	Painting	Photo	Avg.
DMD2	$1.0693_{\pm.0246}$	$22.5196_{\pm.0359}$	$31.4974_{\pm.0727}$	$33.71_{\pm.40}$	$33.11_{\pm.31}$	$33.33_{\pm.30}$	$30.17_{\pm.29}$	$32.58_{\pm.32}$
SAP-DMD	$\mathbf{1.1125_{\pm.0009}}$	$\mathbf{22.6240_{\pm.0227}}$	$\mathbf{31.6910_{\pm.1253}}$	$\mathbf{34.50_{\pm.10}}$	$\mathbf{33.80_{\pm.10}}$	$\mathbf{33.98_{\pm.13}}$	$\mathbf{30.85_{\pm.24}}$	$\mathbf{33.28_{\pm.13}}$

Appendix

Table 6: Results of 4-step DMD2 and SAP-DMD on SD3.5 Medium without GAN loss over three matched training seeds, reported as mean

\pm

standard deviation.

Backbone	DMD2	SAP-DMD	$\Delta$
PixArt- $\alpha$	0.568 $\pm$ 0.007	0.566 $\pm$ 0.006	$-0.002$
SD3	0.598 $\pm$ 0.009	0.617 $\pm$ 0.007	$+0.019$
SD3.5	0.656 $\pm$ 0.008	0.672 $\pm$ 0.005	$+0.016$

Appendix

Table 7: Sample diversity measured by average pairwise LPIPS (higher is more diverse) for 4-step generation. Results are reported as mean

\pm

standard error over 128 prompts.

Figure 9: Early-stage high-frequency recovery on SD3.5, SD3, and PixArt-

\alpha

. We report the normalized AUC of the high-frequency log-power discrepancy relative to paired 25-step teacher outputs over the first

25\%

of training. Bars show the mean over 128 prompts, and error bars denote

95\%

bootstrap confidence intervals.

Figure 10: Fine-detail recovery during training on PixArt- $\alpha$ with 4-step generation. Columns: training steps; both methods share the prompt and initial noise.

Figure 11: Fine-detail recovery during training on SD3 with 4-step generation. Columns: training steps; both methods share the prompt and initial noise.

Figure 12: Fine-detail recovery during training on SD3.5 with 4-step generation. Columns: training steps; both methods share the prompt and initial noise.

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Step distillation has become a leading technique for accelerating diffusion models, among which Distribution Matching Distillation (DMD) and Consistency Distillation are two representative paradigms. While consistency methods enforce self-consistency along the full PF-ODE trajectory to steer it toward the clean data manifold, vanilla DMD relies on sparse supervision at a few predefined discrete timesteps. This restricted discrete-time formulation and mode-seeking nature of the reverse KL divergence tends to exhibit visual artifacts and over-smoothed outputs, often necessitating complex auxiliary modules -- such as GANs or reward models -- to restore visual fidelity. In this work, we introduce Continuous-Time Distribution Matching (CDM), migrating the DMD framework from discrete anchoring to continuous optimization for the first time. CDM achieves this through two continuous-time designs. First, we replace the fixed discrete schedule with a dynamic continuous schedule of random length, so that distribution matching is enforced at arbitrary points along sampling trajectories rather than only at a few fixed anchors. Second, we propose a continuous-time alignment objective that performs active off-trajectory matching on latents extrapolated via the student's velocity field, improving generalization and preserving fine visual details. Extensive experiments on different architectures, including SD3-Medium and Longcat-Image, demonstrate that CDM provides highly competitive visual fidelity for few-step image generation without relying on complex auxiliary objectives. Code is available at https://github.com/byliutao/cdm.

Abstract

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