SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Guided diffusion model and retrieval augmentation. Guided diffusion is a technique applied to diffusion models for conditional generation. Classifier guidance [47] is an cost-effective method leveraging the gradient of a classifier to steer the generative process of a trained diffusion model. On the other hand, Classifier-free guidance [48] learns the conditional distribution during training for improved generation quality. This approach also allows for the use of continuous guidance information, such as embedding vectors, rather than being limited to discrete labels. Classification and Regression Diffusion Models (CARD) [27] formulates classification and regression as a conditional generation task that generates labels or target variables conditioned on images. Our approach follows the same paradigm, and leverages the multi-modal coverage ability of diffusion models to learn the label distribution within the neighborhood. SS-DDPM [49] proposes a diffusion model with arbitrary noising distributions defined on constrained manifold, e.g., the probabilistic simplex for label generation. Retrieval-augmented diffusion models [30] used retrieved neighbors from an external database as conditional information to train diffusion models for image synthesis. Retrieval Augmented Classification [31] used retrieval-augmentation to train classification model using class-imbalanced training data. Our approach differs from theirs by retrieving labels instead of data to reduce label noise in training rather than increasing the training data. In addition, our model does not require an external database.
We first evaluate the performance of our method on datasets with various types synthetic noises. Then, we perform experiments on four real-world datasets. To better understand the performance gain sources, we conduct ablation studies to measure the impacts of conditional diffusion and different pseudo-label construction strategies. All experiments were done on four NVIDIA Titan V GPUs. Comprehensive implementation details and hyper-parameters are provided in the Supplementary C.
Table 1: Classification accuracy (%) on CIFAR-10 and CIFAR-100 datasets under PMD noises and hybrid noises, combining PMD noise with Uniform (U) and Asymmetric (A) noise.
Methods	CIFAR-10
35% PMD	70% PMD	35% PMD + 30% U	35% PMD + 60% U	35% PMD + 30% A
Standard	78.11 ± 0.74	41.98 ± 1.96	75.26 ± 0.32	64.25 ± 0.78	75.21 ± 0.64
Co-teaching+ [50]	79.97 ± 0.15	40.69 ± 1.99	78.72 ± 0.53	55.49 ± 2.11	75.43 ± 2.96
GCE [35]	80.65 ± 0.39	36.52 ± 1.62	78.08 ± 0.66	67.43 ± 1.43	76.91 ± 0.56
SL [36]	79.76 ± 0.72	36.29 ± 0.66	77.79 ± 0.46	67.63 ± 1.36	77.14 ± 0.70
LRT [14]	80.98 ± 0.80	41.52 ± 4.53	75.97 ± 0.27	59.22 ± 0.74	76.96 ± 0.45
CC [44]	81.23 ± 0.78	42.43 ± 1.56	79.6 ± 0.44	70.71 ± 0.34	78.66 ± 0.66
PLC [15]	82.80 ± 0.27	42.74 ± 2.14	79.04 ± 0.50	72.21 ± 2.92	78.31 ± 0.41
SimCLR KNN	83.71	29.45	78.25	54.82	75.37
C2D + SimCLR [46]	83.84 ± 0.13	34.23 ± 0.45	85.61 ± 0.29	81.39 ± 0.68	83.06 ± 0.57
LRA-diffusion (SimCLR)	88.76 ± 0.24	42.63 ± 1.97	88.41 ± 0.37	84.43 ± 0.82	85.64 ± 0.23
CLIP KNN	91.80	30.66	84.67	57.03	81.76
LRA-diffusion (CLIP)	96.54 ± 0.13	44.62 ± 0.18	95.71 ± 0.17	87.21 ± 0.71	93.65 ± 0.40
Methods	CIFAR-100
35% PMD	70% PMD	35% PMD + 30% U	35% PMD + 60% U	35% PMD + 30% A
Standard	57.68 ± 0.29	39.32 ± 0.43	48.86 ± 0.56	35.97 ± 1.12	45.85 ± 0.93
Co-teaching+	56.70 ± 0.71	39.53 ± 0.28	52.33 ± 0.64	27.17 ± 1.66	51.21 ± 0.31
GCE	58.37 ± 0.18	40.01 ± 0.71	52.90 ± 0.53	38.62 ± 1.65	52.69 ± 1.14
SL	55.20 ± 0.33	40.02 ± 0.85	51.34 ± 0.64	37.57 ± 0.43	50.18 ± 0.97
LRT	56.74 ± 0.34	45.29 ± 0.43	45.66 ± 1.60	23.37 ± 0.72	52.04 ± 0.15
CC	59.44 ± 0.33	42.79 ± 1.21	56.58 ± 0.45	43.64 ± 1.71	54.45 ± 1.22
PLC	60.01 ± 0.43	45.92 ± 0.61	60.09 ± 0.15	51.68 ± 0.10	56.40 ± 0.34
SimCLR KNN	54.22	39.25	51.87	41.73	46.50
C2D + SimCLR	69.28 ± 0.31	51.63 ± 0.53	59.87 ± 0.66	64.45 ± 1.41	65.65 ± 0.93
LRA-diffusion (SimCLR)	61.39 ± 0.15	53.37 ± 0.81	60.52 ± 0.28	55.79 ± 0.31	59.28 ± 0.11
CLIP KNN	79.58	52.55	69.66	50.91	61.19
LRA-diffusion (CLIP)	81.91 ± 0.10	74.52 ± 0.12	82.80 ± 0.11	81.10 ± 0.09	81.78 ± 0.15
5.1 Results on Synthetic Noisy Datasets
We conduct simulation experiments on the CIFAR-10 and CIFAR-100 datasets [51] to evaluate our method's performance under various noise types. Specifically, following [15], we test with polynomial margin diminishing (PMD) noise, a novel instance-dependent noise, at two noise levels and three hybrid noise types by adding independent and identically distributed (i.i.d) noises on top of instance-dependent noise.
For instance-dependent noise, we adopt the recently proposed polynomial margin diminishing (PMD) noise [15]. Following the original paper, we train a classifier  $\pmb{\eta}(x)$  using clean labels to approximate the probability mass function of the posterior distribution  $p(y|\mathbf{x})$ . Images are initially labeled as their most likely class  $u_{\mathbf{x}}$  according to the predictions of  $\pmb{\eta}(x)$ . Then, we randomly alter the labels to the second most likely class  $s_{\mathbf{x}}$  for each image with probability:  $p_{u_{\mathbf{x}}, s_{\mathbf{x}}} = -\frac{c}{2} \left[ \pmb{\eta}_{u_{\mathbf{x}}}(\mathbf{x}) - \pmb{\eta}_{s_{\mathbf{x}}}(\mathbf{x}) \right]^{2} + \frac{c}{2}$ , where  $c$  is a constant noise factor that controls the final percentage of noisy labels. Since corrupting labels to the second most likely class can confuse the "clean" classifier the most, it is expected to have the most negative impact on the performance of models learned with noisy labels. For PMD noise, we simulate two noise levels where  $35\%$  and  $70\%$  of the labels are corrupted.
For i.i.d noise, following [52, 15], we use a transition probability matrix  $\mathbf{T}$  to generate noisy labels. Specifically, we corrupt the label of the  $i$ -th class to the  $j$ -th class with probability  $T_{ij}$ . We adopt two types of i.i.d noise in this study: (1) Uniform noise, where samples are incorrectly labeled as one of the other  $(n - 1)$  classes with a uniform probability  $T_{ij} = \tau / (n - 1)$  and  $T_{ii} = 1 - \tau$ , with  $\tau$  the pre-defined noise level; (2) Asymmetric noise: we carefully design the transition probability matrix such that for each class  $i$ , the label can only be mislabeled as one specific class  $j$  or remain unchanged with probability  $T_{ij} = \tau$  and  $T_{ii} = 1 - \tau$ . In our experiment, we generated three types of hybrid noise by adding  $30\%$ ,  $60\%$  uniform, and  $30\%$  asymmetric noise on top of  $35\%$  PMD noise.
We test our proposed label-retrieval-augmented diffusion model using two pre-trained encoders: (1) SimCLR [21]: We trained two encoders using the ResNet50 [53] architecture on the CIFAR-10 and CIFAR-100 datasets through contrastive learning; (2) CLIP [24]: the model is pre-trained on a large dataset comprising 400 million image-text pairs. Specifically, we used the vision transformer [54] encoder (ViT-L/14) with pre-trained weights, the best-performing architecture in CLIP. For
simplification, we refer to these configurations as LRA-diffusion (SimCLR) and LRA-diffusion (CLIP). We also investigated the performance of the KNN algorithm within the feature space defined by the SimCLR and CLIP encoders, denoted as SimCLR KNN and CLIP KNN respectively.
Table 1 lists the performance of the Standard method (train classifier using noisy labels), our method, and baseline methods for learning from noisy labels. The results in white rows are borrowed directly from [15]. We can see that using the SimCLR encoder in the LRA-diffusion method results in superior test accuracy on both CIFAR-10 and CIFAR-100 datasets compared to other baselines, without the need for additional training data. This is because the SimCLR encoder is trained in an unsupervised manner, making it immune to label noise, and it can effectively extract categorical features for accurate image retrieval. Therefore, when the correct labels dominate the label distribution in the neighborhood, training with the labels of the retrieved neighbor images allows the model to learn with more correct labels. C2D [46] also utilizes a pre-trained SimCLR encoder for initialization, but label noise may still affect the feature space during training. In contrast, our method freezes the feature encoder, shielding the pre-trained features from noise. Results on CIFAR-10 demonstrate that when the pre-trained feature has high KNN accuracy, our method performs better. On CIFAR-100, where the SimCLR feature has lower KNN accuracy, C2D is more effective, as it can refine the feature space through training. Moreover, freezing the feature encoder allows for efficient integration of large pre-trained encoders like CLIP, saving us from the prohibitive computational cost of fine-tuning. Notably, incorporating the CLIP encoder into our method significantly improves test accuracy over our LRA-diffusion (SimCLR) due to its excellent representation capabilities. In fact, by performing KNN in the CLIP feature space alone was able to achieve accuracy surpassing all competing methods in most experiments. This allows for the use of more clean labels during training, thus result in even higher accuracy.
To evaluate the contribution of diffusion and the pre-trained features, we conducted ablation experiments using CARD [27], SS-DDPM [49], and linear probing to incorporate pre-trained models. It is worth noting that the SimCLR model was trained on the same training set without access to external data. Results are given in Table 2.
Table 2: Classification accuracy (%) on CIFAR-10 and CIFAR-100 datasets with PMD noises using different combinations of model, pre-trained feature, and label.
Methods	Feature space	Label	CIFAR-10	CIFAR-100
35% PMD	70% PMD	35% PMD	70% PMD
Linear probing	SimCLR	noisy	86.9	38.93	56.18	51.87
Linear probing	SimCLR	sample	63.8	35.84	53.34	52
Linear probing	SimCLR	mean	86.27	39.94	55.95	52.61
ResNet+Linear	SimCLR	sample	86.55	38.06	56.57	51.36
CARD	SimCLR	sample	75.08	34.35	52.03	32.67
SS-DDPM (Dirichlet)	SimCLR	sample	88.13	40.11	59.3	51.67
LRA-diffusion (ours)	SimCLR	sample	88.96	42.63	61.38	53.57
Linear probing	CLIP	noisy	85.35	37.4	65.02	53.21
Linear probing	CLIP	sample	95.61	40.17	63.98	58.53
Linear probing	CLIP	mean	96.19	35.19	69.05	62.76
ResNet+Linear	CLIP	sample	88.72	43.26	59.78	51.47
CARD	CLIP	sample	79.72	33.57	47.1	23.45
SS-DDPM (Dirichlet)	CLIP	sample	96.03	42.03	80.72	72.24
LRA-diffusion (ours)	CLIP	sample	96.55	44.51	81.92	74.58
Linear probing with sampled labels yielded lower accuracy than using noisy labels or the mean of neighboring labels. This difference may be due to the linear layer's inability to yield stochastic outputs from a multimodal distribution. During training, conflicting gradient directions may arise if the model tries to predict different labels across gradient steps, which can impede learning. However, due to the mode coverage ability of the diffusion model, our method can effectively learn from retrieval-augmented labels to generate different labels with different probabilities. We also test a baseline using an additional ResNet encoder along with the linear layer to mimic our model architecture shown in Figure C.1. The results are comparable with linear probing with sampled labels. Our model
significantly outperforms CARD, mainly due to the more informative  $f_{p}$  encoder. We use the same model architecture for SS-DDPM, which uses Dirichlet distributions for noisy states. SS-DDPM performed slightly worse than our method, indicating that constraining noisy states within probability simplex may not benefit our task.
Additional ablation studies are included in the Supplementary D. The results demonstrate that in the absence of a pre-trained encoder, our model can leverage the features of a noisy classifier to enhance its accuracy. We also include results of ablation experiments using different approaches for incorporating the CLIP model. The results indicate that the superior performance of our method is not solely attributable to the strength of the CLIP features. In conclusion, our LRA-diffusion model provides an efficient approach for incorporating pre-trained encoders for learning from noisy labels.
5.3 Results on Real-world Noisy Datasets
We further evaluate the performance of our proposed method on real-world label noise. Following previous work [42, 15, 44, 43], we conducted experiments on four image datasets, i.e., WebVision [55], ImageNet ILSVRC12 [56], Food-101N [57], and Clothing1M [58]. For experiments on Webvision, ILSVRC12, and Food-101N datasets, we use the CLIP image encoder as the  $f_{p}$  encoder to train LRA-diffusion models. Comprehensive dataset description and implementation details can be found in the Supplementary C. We evaluated the performance of our method against a group of state-of-the-art (SOTA) methods. The results are presented in Table 3 and Table 4. Our approach significantly outperforms all the previous methods in terms of classification accuracy. It is important to highlight that EPL [59] incorporates the most powerful CLIP and ConvNext-XL [60] encoders and cooperates with other SOTA methods such as ELR [61], DivideMix [42], and UNICON [62]. However, our method outperforms EPL by achieving ~6% higher accuracy on WebVision and ILSVRC12 datasets. This improvement over EPL demonstrates that developing better ways to incorporate pre-trained models to facilitate learning from noisy labels is a non-trivial task, highlighting the valuable contribution of our approach.
Table 3: Classification accuracies (%) on WebVision, ILSVRC2012 datasets.
Dataset	DivideMix	ELR	UNICON	EPL	LongReMix	C2D	CC	NCR	LRA-diffusion
WebVision	77.32	77.78	77.60	78.77	78.92	79.42	79.36	80.5	84.16
ILSVRC2012	75.20	70.29	75.29	76.51	-	78.57	76.08	-	82.56
Table 4: Classification accuracies (%) on the Food-101N dataset.
Standard	CleanNet [57]	BARE [63]	DeepSelf [64]	PLC	LongReMix	LRA-diffusion
81.67	83.95	84.12	85.10	85.28	87.39	93.42
For experiments on the Clothing1M dataset, we found that LRA-diffusion conditioned on the CLIP image encoder did not achieve the SOTA accuracy. A potential explanation is that the CLIP feature is too general for this domain specific task for categorizing fashion styles. However, our method is orthogonal to most traditional learning with noisy label approaches. As shown in the additional ablation study in Supplementary D.1, our method can collaborate with a trained classifier by conditioning on its feature encoder to achieve improved performance. We first use the CC [44] method to select clean samples and train a ResNet50 classifier, which achieved  $75.32\%$  accuracy (refer to as  $\mathrm{CC}^*$ ). Then, we condition on its feature before the classification head to train our LRA-diffusion model on the selected samples, which achieved  $75.70\%$  accuracy. As Table 5 shows, our method achieved a  $0.38\%$  improvement based on  $\mathrm{CC}^*$  and beat all SOTA methods.
Table 5: Classification accuracies (%) on Clothing1M
Standard	BARE	PLC	LongReMix	DeepSelf	C2D	NCR	CleanNet
68.94	72.28	74.02	74.38	74.45	74.58	74.60	74.69
DivideMix	ELR	UNICON	EPL	CC*	CC	SANM [65]	LRA-diffusion
74.76	74.81	74.98	75.21	75.32	75.40	75.63	75.70
5.4 Inference Efficiency Analysis
In order to test the efficiency of our model, we perform experiments assessing the runtime on CIFAR-10 dataset and compare our method with a standard classifier that uses ResNet50. It's worth noting that our SimCLR encoder is also built on the ResNet50. Thus, the standard method's runtime also reflects the linear probing runtime on SimCLR. Table 6 shows the results.
Table 6: Inference time (s) of standard classifier and LRA-diffusion models on CIFAR-10 images.
Number of images	Standard (ResNet50)	LRA-diffusion
SimCLR (ResNet50)	CLIP (ViT-B/32)	CLIP (ViT-B/16)
10000	3.96	9.52	9.13	17.12
50000	20.77	41.31	39.82	92.34
We can see, the computation bottleneck lies on the large pre-trained encoder but not the diffusion model itself. In general, our method takes twice as long as a standard classifier (ResNet50) when using SimCLR (ResNet50) and CLIP (ViT-B/32) pre-trained encoders. Larger CLIP encoders can increase the time further. However, it can be further accelerated if the features can be pre-computed in advance or be computed in parallel (as they are only required to be computed once and can be reused later).
Our method, while being effective in many scenarios, does have certain limitations that we acknowledge. Its performance enhancement can be compromised if a pre-trained  $f_{p}$  feature encoder isn't available or is inadequately trained. Additionally, the diffusion model introduces Gaussian noise in the forward process, leading to latent label vectors not being confined within the probability simplex, which could increase training time. Lastly, our method's performance becomes less effective when label noise levels surpass  $50\%$ . However, supervised learning can not be the optimal choice in such situations.
In this paper, by viewing the noisy labeling process as a conditional generative process, we leverage diffusion models to denoise the labels and accurately capture label uncertainty. A label-retrieval-augmented diffusion model was proposed to effectively learn from noisy label data by incorporating the principle of neighbor consistency. Additionally, by incorporating auxiliary conditional information from large pre-trained models such as CLIP, we are able to significantly boost the model performance. The proposed model is tested on several benchmark datasets, including CIFAR-10, CIFAR-100, Food101N, and Clothing1M, achieving state-of-the-art results in most experiments. Future work could extend our model to multi-label settings, as it does not require a one-hot representation for labels. It is also promising to use semantic segmentation to guide the generation, potentially enhancing our model's interpretability and performance.
Acknowledgement: This work is partially supported by NSF AI Institute-2229873, NSF RI2223292, an Amazon research award, and an Adobe gift fund. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, the Institute of Education Sciences, or the U.S. Department of Education.
[1] Yu Yao, Tongliang Liu, Mingming Gong, Bo Han, Gang Niu, and Kun Zhang. Instance-dependent label-noise learning under a structural causal model. Advances in Neural Information Processing Systems, 34:4409-4420, 2021.
[2] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning (still) requires rethinking generalization. Communications of the ACM, 64(3):107-115, 2021.
A Denoising Diffusion Implicit Model with non-zero mean latent space
The forward process of a diffusion process with non-zero mean latent distribution  $\mathbf{y}_T\sim \mathcal{N}(f(\mathbf{x}),\mathbf{I}))$  has a closed form representation:
$$
q \left(\mathbf {y} _ {t} \mid \mathbf {y} _ {0}, f\right) = \mathcal {N} \left(\mathbf {y} _ {t}; \sqrt {\bar {\alpha} _ {t}} \mathbf {y} _ {0} + (1 - \sqrt {\bar {\alpha} _ {t}}) f (\mathbf {x}), (1 - \bar {\alpha} _ {t}) \mathbf {I}, \right. \tag {A.1}
$$
which could be reparameterized as:
$$
\mathbf {y} _ {t} = \sqrt {\bar {\alpha} _ {t}} \mathbf {y} _ {0} + (1 - \sqrt {\bar {\alpha} _ {t}}) f (\mathbf {x}) + \sqrt {1 - \bar {\alpha} _ {t}} \boldsymbol {\epsilon} \tag {A.2}
$$
Then, similar to the DDIM, we define a Non-Markovian forward process with  $\sigma_t \geq 0$ ,  $t = 1:T$ .
$$
q _ {\sigma} \left(\mathbf {y} _ {1: T} \mid \mathbf {y} _ {0}, f\right) := q _ {\sigma} \left(\mathbf {y} _ {T} \mid \mathbf {y} _ {0}, f\right) \prod_ {t = 2} ^ {T} q _ {\sigma} \left(\mathbf {y} _ {t - 1} \mid \mathbf {y} _ {t}, \mathbf {y} _ {0}, f\right) \tag {A.3}
$$
Where  $q_{\sigma}(\mathbf{y}_T|\mathbf{y}_0,f) = \mathcal{N}(\mathbf{y}_T;\sqrt{\bar{\alpha}_T}\mathbf{y}_0 + (1 - \sqrt{\bar{\alpha}_T})f(\mathbf{x}),(1 - \bar{\alpha}_T)\mathbf{I})$  and for all  $t > 1$
$$
q _ {\sigma} \left(\mathbf {y} _ {t - 1} \mid \mathbf {y} _ {t}, \mathbf {y} _ {0}, f\right) = \mathcal {N} \left(\mathbf {y} _ {t - 1}; \sqrt {\bar {\alpha} _ {t - 1}} \mathbf {y} _ {0} + \left(1 - \sqrt {\bar {\alpha} _ {t - 1}}\right) f (\mathbf {x}) + \sqrt {1 - \bar {\alpha} _ {t - 1} - \sigma_ {t} ^ {2}} \cdot \tilde {\epsilon}, \sigma_ {t} ^ {2} \mathbf {I}\right), \tag {A.4}
$$
$$
\tilde {\boldsymbol {\epsilon}} = \frac {1}{\sqrt {1 - \bar {\alpha} _ {t}}} \cdot \left(\mathbf {y} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {y} _ {0} - (1 - \sqrt {\bar {\alpha} _ {t}}) f (\mathbf {x})\right)
$$
We can prove that the sampling process defined by Eq. (A.3) and Eq. (A.4) has the same marginal distribution as the closed-form sampling process in Eq. (A.1) by the following Lemma:
Lemma A.1. For  $q_{\sigma}(\mathbf{y}_{1:T}|\mathbf{y}_0,f)$  defined in Eq. (A.3) and  $q_{\sigma}(\mathbf{y}_{t - 1}|\mathbf{y}_t,\mathbf{y}_0,f)$  defined in Eq. (A.4), we have:
$$
q _ {\sigma} \left(\mathbf {y} _ {t} \mid \mathbf {y} _ {0}, f\right) = \mathcal {N} \left(\mathbf {y} _ {t}; \sqrt {\bar {\alpha} _ {t}} \mathbf {y} _ {0} + (1 - \sqrt {\bar {\alpha} _ {t}}) f (\mathbf {x}), (1 - \bar {\alpha} _ {t}) \mathbf {I}\right) \tag {A.5}
$$
Proof. Assume for any  $t \leq T$ , if Eq. (A.5) is true, the following is also true:
$$
q _ {\sigma} \left(\mathbf {y} _ {t - 1} \mid \mathbf {y} _ {0}, f\right) = \mathcal {N} \left(\mathbf {y} _ {t - 1}; \sqrt {\bar {\alpha} _ {t - 1}} \mathbf {y} _ {0} + \left(1 - \sqrt {\bar {\alpha} _ {t - 1}}\right) f (\mathbf {x}), \left(1 - \bar {\alpha} _ {t - 1}\right) \mathbf {I}\right), \tag {A.6}
$$
then we can prove the statement with an induction argument for  $t$  from  $T$  to 1, since the base case ( $t = T$ ) already holds.
First, we have that:
$$
q _ {\sigma} \left(\mathbf {y} _ {t - 1} \mid \mathbf {y} _ {0}\right) := \int_ {\mathbf {y} _ {t}} q _ {\sigma} \left(\mathbf {y} _ {t} \mid \mathbf {y} _ {0}, f\right) q _ {\sigma} \left(\mathbf {y} _ {t - 1} \mid \mathbf {y} _ {t}, \mathbf {y} _ {0}, f\right) d \mathbf {y} _ {t}, \tag {A.7}
$$
$$
q _ {\sigma} \left(\mathbf {y} _ {t} \mid \mathbf {y} _ {0}, f\right) = \mathcal {N} \left(\mathbf {y} _ {t}; \sqrt {\bar {\alpha} _ {t}} \mathbf {y} _ {0} + \left(1 - \sqrt {\bar {\alpha} _ {t}}\right) f (\mathbf {x}), \left(1 - \bar {\alpha} _ {t}\right) \mathbf {I}\right) \tag {A.8}
$$
$$
q _ {\sigma} \left(\mathbf {y} _ {t - 1} \mid \mathbf {y} _ {t}, \mathbf {y} _ {0}, f\right) = \mathcal {N} \left(\mathbf {y} _ {t - 1}; \sqrt {\bar {\alpha} _ {t - 1}} \mathbf {y} _ {0} + \left(1 - \sqrt {\bar {\alpha} _ {t - 1}}\right) f (\mathbf {x}) + \sqrt {1 - \bar {\alpha} _ {t - 1} - \sigma_ {t} ^ {2}} \cdot \tilde {\boldsymbol {\epsilon}}, \sigma_ {t} ^ {2} \mathbf {I}\right), \tag {A.9}
$$
$$
\tilde {\boldsymbol {\epsilon}} = \frac {1}{\sqrt {1 - \bar {\alpha} _ {t}}} \cdot \left(\mathbf {y} _ {t} - \sqrt {\bar {\alpha} _ {t}} \mathbf {y} _ {0} - (1 - \sqrt {\bar {\alpha} _ {t}}) f (\mathbf {x})\right).
$$
According to [66] Eq. (2.115), we have that  $q_{\sigma}(\mathbf{y}_{t - 1}|\mathbf{y}_t,\mathbf{y}_0,f)$  is Gaussian, with mean  $\pmb{\mu}_{t - 1}$  and co-variance  $\pmb{\Sigma}_{t - 1}$ :
$$
\begin{array}{l} \boldsymbol {\mu} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \mathbf {y} _ {0} + (1 - \sqrt {\bar {\alpha} _ {t - 1}}) f (\mathbf {x}) (A.10) \\ + \sqrt {1 - \bar {\alpha} _ {t - 1} - \sigma_ {t} ^ {2}} \left(\frac {\sqrt {\bar {\alpha} _ {t}} \mathbf {y} _ {0} + (1 - \sqrt {\bar {\alpha} _ {t}}) f (\mathbf {x}) - \sqrt {\bar {\alpha} _ {t}} \mathbf {y} _ {0} - (1 - \sqrt {\bar {\alpha} _ {t}}) f (\mathbf {x})}{\sqrt {1 - \bar {\alpha} _ {t}}}\right) \\ = \sqrt {\bar {\alpha} _ {t - 1}} \mathbf {y} _ {0} + (1 - \sqrt {\bar {\alpha} _ {t - 1}}) f (\mathbf {x}). (A.11) \\ \end{array}
$$
$$
\boldsymbol {\Sigma} _ {t - 1} = \sigma_ {t} ^ {2} \mathbf {I} + \frac {1 - \bar {\alpha} _ {t - 1} - \sigma_ {t} ^ {2}}{1 - \bar {\alpha} _ {t}} (1 - \bar {\alpha} _ {t}) \mathbf {I} = (1 - \bar {\alpha} _ {t}) \mathbf {I}. \tag {A.12}
$$
Therefore, Eq. (A.6) holds. Following the induction, the lemma is proved.