SlowGuess/NeurIPS_2023_OCR huggingface dataset:
**Implementations.** We implement our Patch Diffusion on top of the current state-of-the-art Unet-based diffusion model EDM-DDPM++ [22] and EDM-ADM [22]. EDM-DDPM++ is our default backbone model for training low-resolution  $(64\times 64)$  datasets, while EDM-ADM coupling with Stable Diffusion [45] latent en/decoders is our backbone model for training high-resolution  $(256\times 256)$  datasets. We inherit the hyperparameter settings and the UNet architecture from Karras et al. [22]. We implement random cropping independently on each data point in the same batch, and the corresponding pixel coordinates are concatenated with the image channels to form the input for the UNet denoiser. When computing the diffusion loss at any  $\sigma$  level, we ignore the reconstructed pixel coordinates. We adopt the EDM-Sampling strategy with 50 deterministic reverse steps in the inference stage, for both the baseline model and ours.
We provide ablation studies to investigate what may impact the performance of Patch Diffusion.
Impact of full-size images. We first study the main effect of  $p$ , the ratio of full-size images during training, regarding the quality of generated examples and the training cost. We conduct experiments on a  $p$  gird, [0.0, 0.1, 0.25, 0.5, 0.75, 1.0]. Note here when  $p = 0.0$ , only patch images and their pixel coordinates are available during the training process, and when  $p = 1.0$ , patch training becomes the same as the standard training procedure of diffusion models. We train all models on 16 Nvidia V100 GPUs with a batch size of 512 for a duration of 200 million images.
We show the results in Figures 2 and 3. In the extreme case  $p = 0.0$ , the conditional score function is only trained on local patches, e.g.,  $16 \times 16$  and  $32 \times 32$  image patches, without the knowledge of how the full-size images look like. The FID is reasonably unsatisfactory while we still observe some coherently sampled images, such as the third and fourth faces shown in the 1st row of Figure 3. This observation validates our idea that using a mixture of large and small patch sizes could help the conditional score function to capture cross-region dependency. The learned small patch scores are guided by large patch score matching to form a coherent score mapping. Then, when  $p$  is greater than 0, even if  $p$  is as small as 0.1, the generation quality is dramatically improved in terms of FID. We reason this as that the involvement of full-size images provides the global score for patch-wise score matching, even though only a small ratio of the global score could guide the score function to learn towards it. As the  $p$  increases, we could see the FID also becomes better, and we hypothesize this is because the conditional score function converges better towards the local minimum due to more global score guidance. However, the improvement of FID gained from increasing  $p$  is not unlimited. We also observe that when  $p$  is large enough, such as  $p = 0.75$ , the FID values converge
Figure 2: FID results on CelebA-64×64 with different  $p$  values.
Metric	p=0.0	p=0.1	p=0.25	p=0.5	p=0.75	p=1.0
FID	14.51	3.05	2.10	1.77	1.65	1.66
Training Time (in hrs)	13.6	20.1	22.5	24.6	42.7	48.5
Figure 3: Ablation study on dataset CelebA-64×64 for the ratio of full-size images  $p$ . We generate from the trained models with different  $p$ , and show the generated images for a fixed set of seeds in the  $p$  increasing order from top to bottom.
to the minimum level, which indicates that sparsely using full-size images during training could be sufficient to guide the conditional score-matching to converge.
On the other hand, larger  $p$  means more training cost and longer training time. Hence, we pick the sweet point shown on the line plot,  $p = 0.5$ , which provides a good trade-off between generation quality and training efficiency, for our following experiments.
Patch size scheduling. Further, we investigate the impact of patch size scheduling, a stochastic or progressive way. Note usually the training of diffusion models applies learning rate decay. This may hurt the performance of progressive scheduling, since the training in the front stage is totally based on local patches while it has a large learning rate. We report FID comparison 1.66 (stochastic) v.s. 2.05 (progressive) on CelebA-64×64 and 3.11 (stochastic) v.s. 3.85 (progressive) on FFHQ-64×64. Therefore, unless otherwise specified, we employ stochastic patch size scheduling, as discussed in Section 3.2, for the following experiments.
4.2 Experiments on Large-scale Dataset
In this section, we aim to compare our Patch Diffusion Model (PDM) with other state-of-the-art diffusion model baselines in terms of both generation quality and training efficiency. As shown in Table 1, Patch Diffusion generally works well in terms of FID while having significantly reduced training time. Note, the FID results of our model and backbone are based on 50 deterministic reverse steps while the other baselines need a much larger number of reverse steps and are much slower in sampling.
We also combine our method with the Latent Diffusion [45] framework to achieve dramatic reduction in training time cost for high-resolution image synthesis, such as LSUN-Bedroom/Church-256×256 [63] and ImageNet-256×256 [9], and we denote it as Latent Patch Diffusion Model (LPDM). We borrow the pretrained image encoder and decoder from Stable Diffusion [45], encode the original images to a smaller latent space and then apply our patch diffusion training. Note the pretrained auto-encoder is not specifically trained for LSUN/ImageNet, which may limit the performance. We use the same latent diffusion model but without patch diffusion training under our codebase as a baseline, and we denote the implementation as LDM-ADM.
For LSUN-Bedroom/Church datasets, we train unconditional sampling, while for ImageNet dataset, we train conditional sampling. Following the classifier-free diffusion guidance (CFG) [14], during training, we randomly drop  $10\%$  of the given class labels, and during sampling, we use strength 1.3 for applying CFG,  $\mathrm{cfg} = 1.3$ .
Figure 4: Randomly generated images from Patch Diffusion (EDM-DDPM++ backbone) trained on CelebA-64×64 and FFHQ-64×64, and Latent Patch Diffusion (EDM-ADM backbone) trained on ImageNet-256×256.
Figure 5: Finetuning Results of ControlNet with Patch Diffusion Training. We finetune a ControlNet on the HED map to image generation task from Stable Diffusion checkpoints with 20k steps.
Figure 6: Randomly generated images from Patch Diffusion (EDM-DDPM++ backbone) trained on small datasets, AFHQv2-Cat-64×64, AFHQv2-Dog-64×64, and AFHQv2-Wild-64×64.
We present the FID and training cost in Table 2 and Table 3 for LSUN and ImageNet datasets, respectively. Patch Diffusion notably surpasses both the baseline and previous state-of-the-art Unet-based diffusion models in both generation quality and training efficiency. We show a number of uncurated image samples generated by Patch Diffusion in Figure 4 and Appendix B. Qualitatively, the images generated by Patch Diffusion exhibit photo-realism and are rich in variety.
Table 1: We report the FID and T-Time (training time in hours) on both CelebA-64×64 and FFHQ-64×64 dataset. For our model and our backbone model EDM [22], we train them on 16xV100 GPUs with batch size 512.
Algorithm	NFE↓	CelebA-64×64	FFHQ-64×64	Train-Time
DDIM [51]	1000	3.51	-	~48h
DDPM [15]	1000	3.26	-	~48h
NCSN++ [54]	1000	3.25	-	~48h
PNDM [33]	1000	2.71	-	~48h
DDPM++ [23]	1000	1.90	-	~48h
Soft Diffusion [8]	300	1.85	-	~48h
EDM-DDPM++ [22]	50	1.66	2.60	~48h
PDM-DDPM++ (ours)	50	1.77	3.11	~24h
Table 2: We report the FID and Train-Time (training time in hours) on LSUN-Bedroom-256×256 and LSUN-Church-256×256 datasets.
Algorithm	NFE↓	LSUN-Bedroom	LSUN-Church	Train-Time
DDIM [51]	50	10.58	6.62	~ 14 days
DDPM [15]	1000	7.89	4.90	~ 14 days
LDM-ADM	50	4.32	4.66	~ 8 Days
LPDM-ADM (ours)	50	2.75	2.66	~ 4 days
Table 3: We report the FID, sFID, IS, Precision and Recall on ImageNet-256×256 datasets. We also report the GFLOPs as a measure of compute cost for each algorithm. Each metric is presented in two columns, one without classifier-free guidance and one with, marked with  $\times$  and  $\checkmark$ , respectively. All methods use NFE 250 steps for reversing the learned diffusion process.
Evaluation Metric	FID↓	sFID↓	IS↑	Prec.↑	Rec.↑	Cost
Classifier-free guidance	X	✓	X	✓	X	✓	X	✓	X	✓	GFLOPs
ADM [10]	10.94	4.59	6.02	5.25	100.98	186.70	0.69	0.82	0.63	0.52	1120
ADM-Upsampled[10]	7.49	3.94	5.13	6.14	127.49	215.84	0.72	0.83	0.63	0.53	742
LDM-8 [45]	15.51	7.76	-	-	79.03	209.52	0.65	0.71	0.63	0.62	53
LDM-4 [45]	10.56	3.60	-	-	103.49	247.67	0.84	0.87	0.35	0.48	103
LPDM-ADM (ours, CFG=1.3)	7.64	2.72	5.36	4.86	130.23	243.25	0.73	0.84	0.63	0.57	78
4.3 Experiments on Finetuning
We evaluate Patch Diffusion in finetuning large-scale pretrained diffusion models. We plug it into ControlNet [64] as one example. We use the data proposed by Brooks et al. [3] as our base datasets and extract the HED maps by the HED boundary detector [62] as the input controls. We then finetune the ControlNet on the HED map to image generation task from the Stable Diffusion 'v1-5' checkpoint with 20k steps. We show the qualitative generation from HED maps to images in Figure 5. We observe that patch diffusion can be effectively applied to fine-tuning without compromising performance, while also enhancing training efficiency by approximately two times.
4.4 Experiments on Limited-size Dataset
We further investigate whether coordinate-guided score matching could improve the data efficiency of diffusion models. Note by doing random cropping with different patch sizes, even one single image could be extended to thousands of patch samples, which helps overcome the overfitting issue.
Specifically, we conduct experiments on three popular small datasets, AFHQv2-Cat, -Dog, and -Wild, each with as few as around 5k images [7]. We train our Patch Diffusion and the baseline approach EDM-DDPM++ from scratch for a duration of 75 million images. We compare the training cost and generation quality between different methods. As shown in Table 4, we observe that by using patch score matching, our model consistently outperforms the baseline model across all three datasets in terms of FID, while achieving  $\geq 2\times$  faster training at the same time. We show a number of uncurated
Table 4: Results reported on the limited-size dataset (AFHQ-v2). For all models, we train them on 16xV100 GPUs with batch size 512 for a duration of 75 million images.
Data(∼5k images)	Algorithm	FID ↓	NFE	Train-Time
AFHQv2-Cat	EDM-DDPM++ [22]	4.60	50	~18h
Patch Diffusion (ours)	3.11	50	~9h
AFHQv2-Dog	EDM-DDPM++ [22]	4.94	50	~18h
Patch Diffusion (ours)	4.80	50	~9h
AFHQv2-Wild	EDM-DDPM++ [22]	2.59	50	~18h
Patch Diffusion (ours)	1.93	50	~9h
image samples generated by Patch Diffusion in Figure 6. This experiment demonstrates that patch diffusion training could help improve the data efficiency of diffusion models.
4.5 Experiments on Image Extrapolation
In this section, we evaluate the out-painting performance of Patch Diffusion on the LSUN-Bedroom dataset. We initially enlarge the coordinate system to a higher resolution, maintaining the range [-1, 1]. The reference image is positioned at the center of the expanded coordinate system and remains static throughout the Patch Diffusion reversal process. In Figure 7, we extend the pixel manifold to dimensions of  $384 \times 384$ , even though our Patch Diffusion model is trained on  $256 \times 256$  image samples. The model effectively generates new content beyond the original boundary. Additionally, we present the extrapolation to a  $512 \times 512$  resolution in Appendix D.
Figure 7: Extrapolation Results. Patch Diffusion could generate beyond the boundary by extrapolating the learned coordinate manifold. For each pair of images, the left panel is the reference image in resolution  $256 \times 256$  and it is fixed in the center during the reverse process of Patch Diffusion, while the right panel shows the generated sample in resolution  $384 \times 384$ , where the out-of-boundary region is regenerated. Note our model is trained only on  $256 \times 256$  images.
We present Patch Diffusion, a novel patch-level training framework that trains diffusion models via coordinate conditioned score matching. We also propose to diversify the patch sizes for score matching in a progressive or stochastic schedule during training, to capture the cross-region dependency at multiple scales. Sampling with our method is as easy as in the original diffusion model. Patch Diffusion could significantly reduce the training time costs, e.g.,  $2 \times$  faster training, while improving the data efficiency of diffusion models, e.g., improving the performance of diffusion models trained on relatively small datasets. Going forward, the current coordinate system could be further improved by advanced positional embeddings, such as periodic one [38], to better incorporate the position information. We also leave the theoretical proof of the convergence of patch-wise score matching in general cases as our future work.
Z. Wang, H. Zheng, and M. Zhou acknowledge the support of NSF-IIS 2212418, NIH-R37 CA271186, and the NSF AI Institute for Foundations of Machine Learning (IFML).
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A Theoretical Interpretations
In this section, we provide mathematical intuitions for our patch diffusion from two perspectives.
Markov Random Field. Markov Random Field (MRF) has been widely used to represent image distributions [28, 29] due to its compactness and expressiveness in modeling dependence. Usually, images are modeled as an undirected regular graph (pixel grids)  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  in MRF, where each pixel is considered as a graph vertex, and each vertex will connect to its neighboring pixels on the image. Therefore, we can adopt the clique factorization form to represent the PDF defined over images:
$$
p (\boldsymbol {x}) = \frac {1}{Z} \prod_ {v \in \mathcal {V}} \phi_ {v} \left(\boldsymbol {x} _ {v}\right) \prod_ {e \in \mathcal {E}} \phi_ {e} \left(\boldsymbol {x} _ {e}\right), \tag {6}
$$
where  $\phi_v$  and  $\phi_e$  are node and edge potential functions,  $Z$  is a normalization term,  $\mathbf{x}_v$  and  $\mathbf{x}_e$  index the random variables via the subscripts  $v$  or  $e$ . One can see  $\phi_v$  is a univariate function and  $\phi_e$  is a pairwise function.
Next, we point out that the score function of the MRF parameterization can be even neat:
$$
\nabla \log p (\boldsymbol {x}) = \sum_ {v \in \mathcal {V}} \nabla \log \phi_ {v} \left(\boldsymbol {x} _ {v}\right) + \sum_ {e \in \mathcal {E}} \nabla \log \phi_ {e} \left(\boldsymbol {x} _ {e}\right), \tag {7}
$$
where  $Z$  is eliminated as it is irrelevant to  $x$ . As suggested by Equation 7, the score function can be eventually decomposed into independent pieces. That being said, we can separately learn each score function  $\nabla \log \phi_v$  and  $\nabla \log \phi_e$  first, and then average them up to approximate the entire score function during the inference. Our training procedure can be viewed as: each time we sample patches which contains subsets of  $\mathcal{V}$  and  $\mathcal{E}$ , and we conduct score matching on the corresponding  $\nabla \log \phi_v$  and  $\nabla \log \phi_e$ . Note that  $\nabla \log \phi_v$  and  $\nabla \log \phi_e$  is not necessarily spatial-invariant. We condition the network on the coordinates to model the location dependence for  $\nabla \log \phi_v$  and  $\nabla \log \phi_e$ .
Linear Regression. We also provide an alternative interpretation for patch-wise score matching through the lens of least square. We consider a multivariate Gaussian distribution as the demonstration example. Suppose we parameterize our target distribution as  $p_{\mu}(\boldsymbol{x}) = \mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu},\boldsymbol{\Sigma})$  where  $\boldsymbol{\mu}$  is the optimizee. Then its score function is written as:  $\nabla \log p_{\boldsymbol{\mu},\boldsymbol{\Sigma}}(\boldsymbol{x}) = -\boldsymbol{\Sigma}(\boldsymbol{x} - \boldsymbol{\mu})$  [18]. The original score matching with full-size image is equivalent to the following least square problem:
$$
\mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x})} \mathbb {E} _ {\boldsymbol {\epsilon} \sim \mathcal {N} \left(\boldsymbol {0}, \sigma_ {t} ^ {2} \boldsymbol {I}\right)} | | \boldsymbol {\Sigma} \boldsymbol {\mu} - \left(\boldsymbol {\Sigma} \boldsymbol {x} - \boldsymbol {\epsilon} / \sigma_ {t} ^ {2}\right) | | _ {2} ^ {2} \tag {8}
$$
On the other hand, the patch distribution can be considered as the marginal distribution on a subset of random variables. The transformation between patch distribution and full-size image distribution is a marginalization integral, which is as simple as a linear operator (more precisely an orthogonal projection). In our Gaussian example, the patch distribution has a closed form:  $p_{\mu}(\boldsymbol{x}_S) = \mathcal{N}(\boldsymbol{x}|\boldsymbol{P}_S\boldsymbol{\mu}, \boldsymbol{P}_S\Sigma \boldsymbol{P}_S^\top)$ , where  $S$  indices a set of pixels within an image patch identified by the location and size, and  $\boldsymbol{P}_S$  is a selection matrix associated with  $S$ . Then patch-wise score matching can be written as:
$$
\mathbb {E} _ {S} \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x})} \mathbb {E} _ {\boldsymbol {\epsilon} \sim \mathcal {N} \left(\boldsymbol {0}, \sigma_ {t} ^ {2} \boldsymbol {I}\right)} \| P _ {S} \boldsymbol {\Sigma} \boldsymbol {\mu} - \left(P _ {S} \boldsymbol {\Sigma} \boldsymbol {x} - \epsilon / \sigma_ {t} ^ {2}\right) \| _ {2} ^ {2}. \tag {9}
$$
From the linear regression perspective, the only difference between patch-based and full-size score matching is the measurement matrix. At first glance, patch diffusion trades the computation cost off the well-posedness. However, we argue that due to well-known redundancy and symmetry in image distributions, recovering the whole image distribution under limited observation can be factually feasible. This guarantees our patch diffusion can also converge to the true distribution. We consider a general argument beyond Gaussian examples as a promising future exploration.
B More Generated Images.
More generated images from Patch Diffusion are listed below.
Figure 8: Randomly generated images from Patch Diffusion (EDM-DDPM++ backbone) trained on LSUN-Bedroom/Church-256×256.
Figure 9: More generated images from Latent Patch Diffusion (EDM-ADM backbone) trained on ImageNet-256×256..
Figure 10: More Finetuning Results of ControlNet with patch diffusion training.
C Potential Social Implications