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+# Revisiting the Transferability of Supervised Pretraining: an MLP Perspective
+
+Yizhou Wang $^{1,3*†}$ , Shixiang Tang $^{2†}$ , Feng Zhu $^{3}$ , Lei Bai $^{2‡}$ , Rui Zhao $^{3,4}$ , Donglian Qi $^{1}$ , Wanli Ouyang $^{2}$ $^{1}$ Zhejiang University,  $^{2}$ The University of Sydney,  $^{3}$ SenseTime Research,  $^{4}$ Qing Yuan Research Institute, Shanghai Jiao Tong University, Shanghai, China
+
+{yizhouwang, qidl}@zju.edu.cn, stan3906@uni.sydney.edu.au, baisanshi@gmail.com, {zhufeng, zhaorui}@sensetime.com, wanli.ouyang@sydney.edu.au
+
+# Abstract
+
+The pretrain-finetune paradigm is a classical pipeline in visual learning. Recent progress on unsupervised pretraining methods shows superior transfer performance to their supervised counterparts. This paper revisits this phenomenon and sheds new light on understanding the transferability gap between unsupervised and supervised pretraining from a multilayer perceptron (MLP) perspective. While previous works [6, 8, 17] focus on the effectiveness of MLP on unsupervised image classification where pretraining and evaluation are conducted on the same dataset, we reveal that the MLP projector is also the key factor to better transferability of unsupervised pretraining methods than supervised pretraining methods. Based on this observation, we attempt to close the transferability gap between supervised and unsupervised pretraining by adding an MLP projector before the classifier in supervised pretraining. Our analysis indicates that the MLP projector can help retain intra-class variation of visual features, decrease the feature distribution distance between pretraining and evaluation datasets, and reduce feature redundancy. Extensive experiments on public benchmarks demonstrate that the added MLP projector significantly boosts the transferability of supervised pretraining, e.g.  $+7.2\%$  top-1 accuracy on the concept generalization task,  $+5.8\%$  top-1 accuracy for linear evaluation on 12-domain classification tasks, and  $+0.8\%$  AP on COCO object detection task, making supervised pretraining comparable or even better than unsupervised pretraining.
+
+# 1. Introduction
+
+While Supervised Learning with the cross-entropy loss<sup>1</sup> (SL) were the de facto pretraining paradigm in computer vision [14, 20, 26, 39] for a long period, recent unsupervised
+
+learning methods [3-9, 15, 17, 18, 49, 52] show better transfer learning performance on various visual tasks [17, 22, 53]. This raised the question of why unsupervised pretraining surpasses supervised pretraining even though supervised pretraining uses annotations with rich semantic information.
+
+Several works have attempted to explain the better transferability of unsupervised pretraining than supervised pretraining by the following two reasons: (1) Learning without semantic information in annotations [16, 37, 45, 53], which makes the backbone less overfit to semantic labels to preserve instance-specific information which may be useful in transfer tasks, and (2) Special design of the contrastive loss [22, 23, 53], which helps the learned features to contain more low/mid-level information for effective transfer to downstream tasks. Starting from the perspective of supervision and loss design, these works provide intuitive explanations for better transferability.
+
+In this paper, we shed new light on understanding transferability by considering the multilayer perception (MLP) projector. While previous works [6, 8, 17] verified its effectiveness on the unsupervised image classification task: unsupervised training and evaluating the model on the same ImagNet-1K dataset, they did not explore its effectiveness on transfer tasks thoroughly and rigorously. It is not straightforward to extend the effectiveness of MLP on the unsupervised image classification task to downstream tasks if not supported by rigorous experiments or theoretical analysis, because the performance on the pretraining task is not always predictive of the performance on transfer tasks when there exists a large semantic gap [16, 35, 43]. To our best knowledge, we are the first to identify the MLP projector as the core factor for the transferability with deep empirical and theoretical analysis. With this new viewpoint, we find that a simple yet effective method, adding an MLP projector, can promote the transferability of the conventional
+
+supervised pretraining methods with the cross-entropy loss (SL) to be comparable or even better than representative unsupervised pretraining methods.
+
+Specifically, we use the concept generalization task [37] on ImageNet-1K, where the pretraining and the evaluation datasets have a large semantic distance, as a probe to analyze the transferability of different models. Our experimental results and corresponding analysis indicate that the MLP projector in unsupervised pretraining methods is important for their better transferability. Motivated by this observation, we insert an MLP projector before the classifier in SL, forming SL-MLP. The added MLP can improve the transferability of supervised pretraining, making supervised pretraining comparable or even better than unsupervised pretraining. Experimental results on SL and SL-MLP show three interesting findings: 1) The added MLP preserves the intra-class variation on the pretraining dataset. 2) The added MLP decreases the feature distribution distance between the pretraining and the evaluation dataset; 3) The added MLP decreases the feature redundancy in the pretraining dataset. We also provide theoretical analysis on how the preserved intra-class variation and the decreased feature distribution distance improve the performance on the target dataset, by adding an MLP projector.
+
+Extensive experimental results confirm that adding an MLP projector into the supervised pretraining method (SL) can consistently improve the transferability of the model on various downstream tasks. Specifically, on the concept generalization task [37], SL-MLP boosts the top-1 accuracy compared to SL  $(55.9\% \rightarrow 63.1\%)$  by  $+7.2\%$ . It also achieves better performance  $(64.1\%)$  than Byol  $(62.3\%)$  by  $+1.8\%$  on the 300-epochs pretraining setting. In classification tasks on 12 cross-domain datasets [22], SL-MLP improves SL by  $+5.8\%$  accuracy on average. Moreover, SL-MLP shows better transferability than SL on COCO object detection [25] by  $+0.8\%$  AP. These improvements brought by the MLP projector can largely bridge the transferability gap between supervised and unsupervised pretraining as detailed in Sec. 5.2.
+
+The main contributions of our paper are three-fold. (1) We reveal that the MLP projector is the main factor for the transferability gap between existing unsupervised and supervised learning methods. (2) We empirically demonstrate that, by adding an MLP projector, supervised pretraining methods can have comparable or even better transferability than representative unsupervised pretraining methods. (3) We theoretically prove that the MLP projector can improve transferability of pretrained models by preserving intra-class feature variation.
+
+# 2. Related Works
+
+MLP in unsupervised learning methods. Adding a multi-layer perceptron (MLP) projector after the encoder was first
+
+introduced in SimCLR [6] and followed by recent unsupervised learning frameworks [3, 7-9, 17, 49]. SimCLR claims that the MLP can reduce the loss of information caused by the contrastive loss, and various works [6, 8] have verified that the MLP projector can enhance the discriminative ability of unsupervised models on the unsupervised image classification task, where unsupervised training and evaluation are conducted on the same dataset. However, the relation between the MLP and the transferability of unsupervised learning methods is under-explored. In this paper, we reveal that the MLP projector is also important for the desirable transferability of unsupervised learning.
+
+MLP in supervised learning methods. The typical supervised learning method (SL) only uses the cross-entropy loss and shows inferior performance on various transfer tasks than recent unsupervised learning methods. Inspired by [15, 48], recent works [22, 23] introduced the contrastive loss equipped with an MLP projector into SL to improve its transferability. Nonetheless, those works ignored the ablation on the MLP and attributed the better transfer performance to the contrastive mechanism in the loss. In this paper, we propose that the MLP is important for the improved transferability of recent supervised learning methods [22, 23] with empirical and theoretical analysis.
+
+Transferability gap between supervised and unsupervised learning. Previous works attributed the superior transferability of unsupervised learning to lack of annotation [16, 37, 45, 53] or special design of the contrastive loss [22, 23, 40, 53]. Different from both reasons, we explain the transferability gap by considering the architectural difference between the supervised and the unsupervised learning frameworks. From this perspective, we analyze the role of the MLP projector in both supervised and unsupervised learning methods, and are the first to identify its key importance to model transferability to the best of our knowledge.
+
+# 3. Transferability Analysis of the Unsupervised and Supervised Pretraining Methods
+
+# 3.1. The Concept Generalization Task
+
+We use the concept generalization task [37] to analyze the transferability gap between the unsupervised and supervised pretraining methods.
+
+Data preparation. Sariyildiz et al. [37] evaluated the transferability of methods when the pretraining and evaluation dataset have semantic distance. Their experimental results show that larger semantic distance will lead to more accuracy differences among different pretraining methods. Therefore, we enlarge the semantic gap between the pretraining and the evaluation dataset to help us compare different pretraining methods. Sariyildiz et al. [37] use the hierarchy in WordNet [29] and divide ImageNet-21K [13] into six class-exclusive datasets with different semantic distance
+
+![](images/f3cb62352ca205d0b839843e8a7948b3a6e763a98171d40eb29b95f7b0e28275.jpg)  
+Figure 1. Schematic illustration of stage-wise evaluation. We flatten intermediate feature maps from different stages and then use them to train stage-wise classifiers. Top-1 accuracy is reported by evaluating images in eval-D with the stage-wise classifiers.
+
+- one for pretraining, and others for evaluation. Without loss of generality, we construct a smaller pretraining dataset (pre-D) and evaluation dataset (eval-D) based on ImageNet1K [36] to reduce the experimental burden. Pre-D contains 652 classes mostly of organisms, and eval-D contains the other 348 classes of instrumentality.
+
+Transferability evaluation. Following [37], to assess the transferability, we freeze all parameters in the pretrained backbone  ${}^{2}$  ,and finetune the classifier with the ImageNet- 1K training samples in eval-D for reporting top-1 accuracy on ImageNet-1K validation samples in eval-D.
+
+# 3.2. Stage-wise Evaluation on Existing Methods
+
+Motivated by works [22, 50, 53] we make a thorough stage-wise investigation of the conventional supervised pretraining method (SL) and the existing representative unsupervised pretraining methods (Mocov1 [18], Mocov2 [8], Byol [17]) by evaluating the transferability of intermediate feature maps (Fig. 1). After pretraining the model on pre-D, we freeze all model parameters and use the extracted intermediate feature maps of images in eval-D to finetune a stage-wise classifier for a stage-wise linear evaluation.
+
+The evaluation results of these existing methods are depicted in Fig. 2 (underlined on the legend). Our stage-wise evaluation shows two new findings that have not been reported by existing works. First, on stage-wise evaluation from stage 1 to stage 4, SL is consistently higher than Byol, Mocov1, and Mocov2, which suggests that the semantic information in annotations can benefit the transferability of low/middle-level feature maps. Second, on stage-wise evaluation from stage 4 to stage 5, the performance of Byol and Mocov2 still increase while SL and Mocov1 have a transferability drop. By carefully inspecting these methods, we notice an architectural difference between SL, Mocov1, Mocov2, and Byol after stage 5: An MLP projector is in-
+
+![](images/a811f4dc15d1d1cac11a471de2abadaf8216290bbd1bdf1b48e5a4eb324a9f6c.jpg)  
+Figure 2. Top-1 accuracy of stage-wise evaluation. All methods use ResNet50 as their backbones and are trained by 300 epochs with the setting in original papers. The results of linear evaluation of layer4-pooled-features (see Fig. 1) are reported in the legend.
+
+serted after stage 5 in Byol and Mocov2, which does not exist in SL and Mocov1. Such difference, together with the experimental results in Fig. 2, leads to a new hypothesis that the MLP projector might be the core factor of the desirable transferability of unsupervised pretraining.
+
+# 3.3. MLP Improves the Transferability of Unsupervised Pretraining Methods
+
+To confirm our hypothesis of the effectiveness on unsupervised pretraining methods, we ablate the MLP projectors on existing unsupervised methods, using stage-wise evaluation. Specifically, we remove the MLP projector in Byol and Mocov2 as Byol w/o MLP and Mocov2 w/o MLP, and add an MLP projector in Mocov1 as Mocov1 w/ MLP. The stage-wise evaluation results of these ablations are summarized in Fig. 2. We use solid lines for methods that have an MLP projector and dash lines for those that do not.
+
+These ablation results offer us two observations. First, when evaluating the layer4-pooled-features (depicted in the legend), unsupervised learning methods with an MLP projector achieve better transferability than their variants without the MLP projector, e.g., Byol, Mocov1 w/ MLP, Mocov2 achieve higher accuracy than Byol w/o MLP, Mocov1, and Mocov2 w/o MLP by  $+23.3\%$ ,  $+5.1\%$  and  $+3.7\%$ , respectively. Second, on stage-wise evaluation from stage 4 to stage 5, the MLP projector can help unsupervised learning methods without the MLP projector to avoid the transferability drop. These consistent improvements by adding an MLP projector empirically show that the MLP projector is important for the transferability of unsupervised pretraining. While there might exist some other non-linear structures that can boost the transferability, we only explore from an MLP perspective in this paper because of its simplicity and demonstrated effectiveness.
+
+![](images/5316cdf3662e43153ab9499920599233607a989b40c22ae93ecb89f76ec2a037.jpg)  
+Figure 3. The difference between SL and SL-MLP. Our SL-MLP adds an MLP before the classifier compared to SL. Only the encoders in both methods are utilized for downstream tasks.
+
+![](images/a8cb2e34d41aff78826586be8422f55552468fb9371a597c47399a9228c97587.jpg)
+
+# 4. MLP Can Enhance Supervised Pretraining
+
+# 4.1. SL-MLP: Adding an MLP Projector to SL
+
+Motivated by the empirical results in Sec. 3, an interesting question is whether the MLP projector can also promote the transferability of supervised pretraining? We attempt to insert an MLP projector before the classifier on SL for better transferability. We denote this supervised pretraining method as SL-MLP (see Fig. 3 for their comparison). Specifically, SL-MLP includes a feature extractor  $f(\cdot)$ , an MLP projector  $g(\cdot)$ , and a classifier  $\mathbf{W}$ . Given an input image  $\mathbf{x}$ , the feature extractor outputs a feature  $\mathbf{f} = f(\mathbf{x})$ . For example,  $f(\mathbf{x})$  transforms an image  $\mathbf{x}$  to a 2048 dimensional feature  $\mathbf{f}$  when using the ResNet-50 backbone. The MLP projector maps  $\mathbf{f}$  into a projection vector  $\mathbf{g} = g(\mathbf{f})$ . Following Byol, the MLP projector consists of two fully connected layers, a batch normalization layer, and a ReLU layer, which can be mathematically formulated as  $g(\mathbf{f}) = fc_{2}(ReLU(BN(fc_{1}(\mathbf{f})))\in \mathbb{R}^{D_{g}}$ , where  $fc_{1}$  and  $fc_{2}$  are fully connected layers, the hidden feature dimension in the MLP projector is set to 4096, and  $D_{g}$  is set to 256. Given the label denoted by  $y$  for image  $\mathbf{x}$ , the objective function for SL-MLP can be formulated as
+
+$$
+\mathcal {L} (\mathbf {x}) = \operatorname {C E} \left(\mathbf {W} \cdot g (f (\mathbf {x})), y\right), \tag {1}
+$$
+
+where  $\mathrm{CE}(\cdot)$  is the cross-entropy loss. Same as SL, only the learned feature extractor  $f(\cdot)$  is utilized in downstream transfer tasks after supervised pretraining.
+
+# 4.2. Empirical Findings of MLP in SL-MLP
+
+MLP projector avoids transferability drop at stage 5 in supervised pretraining. We conduct stage-wise evaluation as Sec. 3.2 again to see whether the transferability drop from stage 4 to stage 5 exists in SL-MLP. In Fig. 6(a), the transferability of SL-MLP continuously increases from stage 1 to 5, avoiding a decrease at stage 5 as SL. Besides, we observe that the transferability of SL-MLP is higher than that of Byol from stage 1 to 4, indicating that annotations benefit the transferability of intermediate feature maps.
+
+MLP projector enlarges the intra-class variation of features. According to [22, 53], features with large intra-class variation can preserve more instance discriminative information, which is beneficial for transfer learning. We reveal
+
+![](images/8aa03bb729e3c8063a71031ac58850d496f6295da629fb4aee9a4178481db558.jpg)  
+(i) SL
+
+![](images/9040b6a88926816e953b376b32e669cf0b97579403147c20a746f7f2d7af8842.jpg)  
+(iii) SupCon w/o MLP
+
+![](images/fc4121af51463738e2750944c03f7c5038228a5a127e09d7337cdf38e0a31663.jpg)  
+(v) Byol w/o MLP
+
+![](images/6ca4cbd30495d11f0de7cb0355d404fcd7e984c19c9e51dc0d0da358a3a7aaac.jpg)  
+(ii) SL-MLP
+
+![](images/7ff2d633be1e80cc57dde1bdab68af3c8ae750828205c1d22cfc44556d207677.jpg)  
+(iv) SupCon
+
+![](images/00b06132f603875002c5d9b16dfcfdfb9f4317d36e339f2f74b93eefe53c6af8.jpg)  
+(vi) Byol  
+Figure 4. Visualization of different methods with 10 randomly selected classes on pre-D. Different colors denote different classes. Features extracted by pretrained models without an MLP projector (top row) have less intra-class variation than those extracted by pretrained models with an MLP projector (bottom row).
+
+![](images/997f3f4ff14022a2f445ef9af65973dd374ba9ea49d7b003ef4656a01c642605.jpg)  
+(i) SL
+
+![](images/24593a43168ba872cd4d323425cef0c60822e19f538f2928f2d68a64e352c998.jpg)  
+(ii) SL-MLP
+
+![](images/2846e74625dc3bf4d38e251fe01ef83ebec29f2225a0eb92c43ddca36f05895f.jpg)  
+(iii) Byol  
+Figure 5. Visualization of Feature Mixtuneness between pre-D and eval-D. Cold colors denote features from 5 classes that are randomly selected from pre-D, and warm colors denote features from 5 classes that are randomly selected from eval-D.
+
+that adding an MLP projector also can enlarge the intra-class variation. We compare two supervised pretraining methods, i.e., SL, SupCon [23], and one unsupervised pretraining method, i.e., Byol, with their variants with/without MLP. Qualitatively, we visualize their features learned on pre-D by t-SNE in Fig. 4. The intra-class variation of features from SL-MLP, SupCon, and Byol are larger than that from SL, SupCon w/o MLP, and Byol w/o MLP, respectively. Quantitatively, following LDA [1], we utilize a discriminative ratio  $\phi(I^{pre})$  to measure intra-class variation on pre-D, where  $I^{pre}$  denotes pre-D (mathematically defined in Sec. 4.3). Smaller discriminative ratio  $\phi$  usually means larger intra-class variation $^{4}$ . Comparing Fig. 6(c) with Fig. 6(b), we can see Byol and SL-MLP have smaller  $\phi(I^{pre})$  but higher accuracy on eval-D than SL, which shows larger intra-class variation can benefit transferability. Furthermore, when inspecting SL only, we can see a process where the accuracy on eval-D first rises and then descends (after 210 epochs) along with  $\phi(I^{pre})$  increasing. This phenomenon can be theoretically explained in Sec. 4.3. We additionally provide the visualization of intra-class variation on different pretraining epochs in Supplementary B.1. MLP projector reduces feature distribution distance be
+
+![](images/3c267f0df44d7bddc0194fdf7d5d4d3fcea09ae4ccc48843fd09c1c19f448349.jpg)  
+(a)
+
+![](images/cdf0f817254a5532000e3d0a25ac0ff2c97fe0b5e45e713cef4a323d5554e94d.jpg)  
+(b)
+
+![](images/c03ae9a9dcf009d29823a52f1e06428268eb47644e48377d2f4ed097b1358772.jpg)  
+(c)
+
+![](images/b657a8e3547de5b8d2e3ed5db70b97e05e654a6bc9be5d39400658ca230cc522.jpg)  
+Figure 6. (a) Stage-wise evaluation on eval-D. (b) Linear evaluation accuracy on eval-D. (c) Discriminative ratio of features on pre-D. Following [17, 19], we pretrain SL, SL-MLP, and Byol for 300 epochs.  
+(a)  
+Figure 7. (a) Feature Mixtuneness between pre-D and eval-D. (b) Redundancy  $\mathcal{R}$  of pretrained features during different epochs. Following [17, 19], we pretrain SL, SL-MLP, and Byol for 300 epochs.
+
+![](images/b3d0c78b221e76c1b6a2e200ebe540371fb2abd945d351c3420c3f8bce8786f3.jpg)  
+(b)
+
+<table><tr><td>Method</td><td>EP</td><td>Top-1(↑)</td><td>R(↓)</td></tr><tr><td>SL</td><td>100</td><td>55.9</td><td>0.078</td></tr><tr><td>SL-MLP</td><td>100</td><td>63.1</td><td>0.035</td></tr><tr><td>SL</td><td>300</td><td>54.4</td><td>0.087</td></tr><tr><td>SL-MLP</td><td>300</td><td>64.1</td><td>0.034</td></tr><tr><td>Byol w/o MLP</td><td>300</td><td>39.0</td><td>0.247</td></tr><tr><td>Byol</td><td>300</td><td>62.3</td><td>0.037</td></tr><tr><td>Mocov1</td><td>300</td><td>54.1</td><td>0.069</td></tr><tr><td>Mocov1 w/ MLP</td><td>300</td><td>59.2</td><td>0.058</td></tr></table>
+
+Table 1. Redundancy  $\mathcal{R}$  of pretrained features. Methods with an MLP obtain lower channel redundancy and transfer better.
+
+tween pre-D and eval-D. According to [2, 27], decreasing the feature distribution distance between pre-D and eval-D in the representation space can benefit transfer learning. Intuitively, the distribution distance between two sets of features is small when features are well mixed (visualization provided in Supplementary A). Therefore, we compare the mixtuneness of features in pre-D and eval-D to indicate the feature distribution distance between SL and SL-MLP. Graphically, we visualize features from pre-D and eval-D by t-SNE in Fig. 5. We observe that features from pre-D and eval-D are more mixed comparing SL and SL-MLP, indicating that MLP projector can mitigate the distribution distance between pre-D and eval-D. Quantitatively, we define Feature Mixtuneness II in the feature space as
+
+$$
+\Pi = 1 - \frac {1}{C} \sum_ {i = 1} ^ {C} \left| \frac {\operatorname {t o p} _ {k} ^ {e v a l} (i)}{k} - \frac {C ^ {e v a l}}{C} \right|, \tag {2}
+$$
+
+where  $C = 1000$  is total number of classes in ImageNet-1K,  $C^{eval}$  represents the number of classes in eval-D, and  $top_{k}^{eval}(i)$  represents the number of classes in eval-D found by top  $k$  neighbors search of any class  $i \in C$ . Since the percentage of finding a sample from eval-D in  $k$  nearest neighbors is  $C^{eval} / C$  when pre-D and eval-D are uniformly mixed, Feature MixTUREness measures the similarity of the current and the uniformly mixed distribution between pre-D and eval-D in the feature space. We examine Feature MixTUREness of SL, SL-MLP, and Byol during different pretrain
+
+ing epochs in Fig. 7(a). Feature MixTURENess of SL gradually decreases, which indicates that SL will enlarge the distribution difference between pre-D and eval-D. In contrast, SL-MLP and Byol show consistently high Feature MixTUREness, indicating that the MLP projector can reduce the distribution distance between pre-D and eval-D. We visualize the evolution of Feature MixTUREness in Supplementary B.2. MLP projector reduces feature redundancy. Inspired by [52], high channel redundancy limits the capability of feature expression in deep learning. Mathematically, we compute Pearson correlation coefficient among feature channels to evaluate feature redundancy  $\mathcal{R}$ , i.e,
+
+$$
+\mathcal {R} = \frac {1}{d ^ {2}} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} | \rho (i, j) |, \quad \rho (i, j) = \frac {\sum_ {n = 1} ^ {N} \mathbf {f} _ {n , i} \cdot \mathbf {f} _ {n , j}}{\sqrt {\sum_ {n = 1} ^ {N} | | \mathbf {f} _ {n , i} | | ^ {2}} \sqrt {\sum_ {n = 1} ^ {N} | | \mathbf {f} _ {n , j} | | ^ {2}}}
+$$
+
+where  $d = 2048$  is the feature dimension,  $\rho(i,j)$  is Pearson correlation coefficient of feature channel  $i$  and  $j$ . As shown in Fig. 7(b), SL-MLP has lower feature redundancy than SL, which indicates that the MLP projector can reduce feature redundancy. In Tab. 1, we further confirm that the MLP projector can reduce the feature redundancy and thus increase the accuracy on eval-D by ablating the MLP projector on various pretraining methods.
+
+# 4.3. Theoretical Analysis for Empirical Findings
+
+In this section, we provide a theoretical analysis to reveal that: 1) maximizing the discriminative ratio  $\phi(I^{pre})$  of
+
+![](images/3032868e0e96e9f033f07fe30d8686d001b8be7fc876aa04cbf8698ec6ee7a6a.jpg)  
+Figure 9. Insights for transferability.  $\phi(I^{pre})$  and  $\phi(I^{eval})$  are the discriminative ratios (Eq. 4) on the pretraining and evaluation datasets. Higher  $\phi(I^{eval})$  indicates better model transferability. Green and Blue line show the performance curve on the evaluation dataset with small and large semantic gap, respectively.
+
+a model on the pretraining dataset above a certain threshold will lead to a transferability decrease (shown by blue/green lines in Fig. 9); 2) the threshold is smaller when the semantic gap between the pretraining and evaluaion dataset is larger ( $t_{l} < t_{s}$  in Fig. 9).
+
+Mathematically, the discriminative ratio  $\phi(I)$  on the dataset  $I$  can be defined by LDA [1] as
+
+$$
+\phi (I) = D _ {i n t e r} (I) / D _ {i n t r a} (I), \tag {4}
+$$
+
+where  $D_{inter}(I) = \frac{1}{C(C - 1)}\sum_{j = 1}^{C}\sum_{k = 1,k\neq j}^{C}||\mu (I_j) - \mu (I_k)||^2$  is the inter-class distance,  $D_{intra}(I) = \frac{1}{C}\sum_{j = 1}^{C}(\frac{1}{|I_j|}\sum_{(x_i,y_i)\in I_j}||\mathbf{f}_i - \mu (I_j)||^2)$  is the intra-class distance, and  $C$  is the number of classes.  $\mu (I_j) = \frac{1}{I_j}\sum_{(x_i,y_i)\in I_j}\mathbf{f}_i$  is the center of features in class  $I_{j}$ , and  $\mathbf{f}$  is the feature in Sec. 4.1. Higher discriminative ratio  $\phi$  indicates higher classification accuracy. Inspired by [26], we analyze the relation between  $\phi (I^{pre})$  and  $\phi (I^{eval})$  in Theorem 1 (Supplementary C).
+
+Theorem 1 Given  $\phi_1(I^{pre}) < \phi_2(I^{pre}), \phi_1(I^{eval}) > \phi_2(I^{eval})$  when  $\phi_1(I^{pre}) > t$ , where  $t$  is a threshold that is negatively related to the feature distribution distance.
+
+Insights for the intra-class variation. Theorem 1 reveals that continuously minimizing the intra-class variation (maximizing the discriminative ratio) on the pretraining dataset will decrease the transferability of the model when the discriminative ratio  $\phi(I^{pre})$  is larger than  $t$ . It explains the observation in Fig. 6(b) and Fig. 6(c) that training with more than 210 epochs leads to better performance on pre-D, but a worse transferability on eval-D. This insight suggests that we should not make the intra-class variation on the pretraining dataset too small when designing the objective function or network architecture (e.g., adding an MLP projector).
+
+Insights for the relation between the feature distribution distance and threshold  $t$ . When the feature distribution distance between the pretraining and evaluation dataset is large, the threshold  $t$  is small, in which case it is easier to have the undesirable effect of increasing the discriminative
+
+ratio  $\phi(I^{pre})$  on pre-D leading to decreasing the discriminative ratio  $\phi(I^{eval})$  on eval-D (and thus the accuracy on the evaluation data). This insight suggests that we should maintain more intra-class variation on the pretraining dataset when transferring the model to a target dataset which has a larger semantic distance from the pretraining dataset.
+
+# 5. Experiment
+
+# 5.1. Experimental Setup
+
+Datasets. For backbone analysis, we keep using the concept generalization setting described in Sec. 3.1. For generalization to other classification tasks, we follow the setup in [22], which pretrains the models on the whole ImageNet-1K dataset and then evaluates the transferability on 12 classification datasets [10-12, 21, 24, 31-34, 41, 42, 44] from different domains. Furthermore, the COCO [25] dataset is used to evaluate the performance of SL-MLP pretrained by ImageNet-1K [36] on object detection task.
+
+Details. For SL and SL-MLP pretraining, the cross-entropy is deployed as the loss function. The MLP projector deployed in SL-MLP is described in Sec. 4.1. Following [20], we use the SGD optimizer with a cosine decay learning rate of 0.4 to optimize SL and SL-MLP, and set the batch size to 1024. Byol is used as a representative method for comparisons in backbone analysis and object detection. Following [17], we use LARS optimizer [51] with a cosine decay learning rate schedule and a warm-up of 10 epochs to pretrain the network. The initial learning rate is set to 4.8. We set the batch size to 4096 and the initial exponential moving average parameter  $\tau$  to 0.99. Except for the backbone analysis, we use ResNet50 as the default backbone. More detailed pretraining setups of different backbones and different methods are provided in Supplementary H.1.
+
+# 5.2. Experimental Results
+
+Generalize to unseen concepts with diverse backbones. We verify the effectiveness of the added MLP projector on SL using concept generalization task with different backbones. Following evaluation method mentioned in Sec. 3.1, we train a linear classifier with the frozen backbone for 100 epochs, and report the top-1 accuracy on eval-D in Tab. 2. Firstly, SL-MLP obtains better performance than SL among different backbones. Specifically, with ResNet50, SL-MLP improves SL to  $63.1(+7.2\%)$  when we pretrain the model by only 100 epochs, which bridges the performance gap between SL and Byol. In 300 epochs setting, SL has a transferability drop compared to 100 epochs setting  $(55.9\% \rightarrow 54.4\%)$ , but the transferability of SL-MLP continue to increase  $(63.1\% \rightarrow 64.1\%)$ . Secondly, SL-MLP  $(64.1\%)$  performs better than Byol  $(62.3\%)$  when both are trained by 300 epochs. Experimental results in Tab. 2 also confirm that SL-MLP can consistently improve the trans
+
+Table 2. Concept generalization task. We report Top-1 accuracy on eval-D of SL-MLP, Byol, and SL on various backbones. SL-MLP and Byol share the same MLP projector.  
+
+<table><tr><td>Method</td><td>Architecture</td><td>Labels</td><td>MLP</td><td>Epochs</td><td>Top-1(↑)</td></tr><tr><td>SL</td><td>ResNet50</td><td>✓</td><td></td><td>100</td><td>55.9</td></tr><tr><td>SL-MLP</td><td>ResNet50</td><td>✓</td><td>✓</td><td>100</td><td>63.1</td></tr><tr><td>Byol</td><td>ResNet50</td><td></td><td>✓</td><td>300</td><td>62.3</td></tr><tr><td>SL</td><td>ResNet50</td><td>✓</td><td></td><td>300</td><td>54.4</td></tr><tr><td>SL-MLP</td><td>ResNet50</td><td>✓</td><td>✓</td><td>300</td><td>64.1</td></tr><tr><td>SL</td><td>ResNet34</td><td>✓</td><td></td><td>100</td><td>50.1</td></tr><tr><td>SL-MLP</td><td>ResNet34</td><td>✓</td><td>✓</td><td>100</td><td>55.0</td></tr><tr><td>Byol</td><td>ResNet34</td><td></td><td>✓</td><td>300</td><td>54.8</td></tr><tr><td>SL</td><td>ResNet34</td><td>✓</td><td></td><td>300</td><td>50.2</td></tr><tr><td>SL-MLP</td><td>ResNet34</td><td>✓</td><td>✓</td><td>300</td><td>55.8</td></tr><tr><td>SL</td><td>ResNet101</td><td>✓</td><td></td><td>100</td><td>56.0</td></tr><tr><td>SL-MLP</td><td>ResNet101</td><td>✓</td><td>✓</td><td>100</td><td>63.6</td></tr><tr><td>SL</td><td>ResNet101</td><td>✓</td><td></td><td>300</td><td>53.9</td></tr><tr><td>SL-MLP</td><td>ResNet101</td><td>✓</td><td>✓</td><td>300</td><td>64.7</td></tr><tr><td>SL</td><td>Swin-tiny</td><td>✓</td><td></td><td>100</td><td>58.9</td></tr><tr><td>SL-MLP</td><td>Swin-tiny</td><td>✓</td><td>✓</td><td>100</td><td>60.6</td></tr><tr><td>SL</td><td>EfficientNetb2</td><td>✓</td><td></td><td>100</td><td>57.6</td></tr><tr><td>SL-MLP</td><td>EfficientNetb2</td><td>✓</td><td>✓</td><td>100</td><td>64.2</td></tr></table>
+
+Table 3. Object detection results. All methods are pretrained on ImagNet-1K, then finetuned on COCO using Mask-RCNN (R50-FPN) based on Detectron2 [46]. Sup. and Unsup. are short for supervised learning and unsupervised learning, respectively. Results of methods† are from [48].  
+
+<table><tr><td>Method</td><td>Sup.</td><td>Unsup.</td><td>Epoch</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>SL</td><td>✓</td><td></td><td>100</td><td>38.9</td><td>59.6</td><td>42.7</td></tr><tr><td>SL-MLP</td><td>✓</td><td></td><td>100</td><td>39.7</td><td>60.4</td><td>43.1</td></tr><tr><td>InsDis† [47]</td><td></td><td>✓</td><td>200</td><td>37.4</td><td>57.6</td><td>40.6</td></tr><tr><td>PIRL† [30]</td><td></td><td>✓</td><td>200</td><td>37.5</td><td>57.6</td><td>41.0</td></tr><tr><td>SwAV† [3]</td><td></td><td>✓</td><td>200</td><td>38.5</td><td>60.4</td><td>41.4</td></tr><tr><td>Mocov2† [8]</td><td></td><td>✓</td><td>200</td><td>38.9</td><td>59.4</td><td>42.4</td></tr><tr><td>Byol [17]</td><td></td><td>✓</td><td>300</td><td>39.4</td><td>60.4</td><td>43.2</td></tr><tr><td>SL</td><td>✓</td><td></td><td>300</td><td>40.1</td><td>61.1</td><td>43.8</td></tr><tr><td>SL-MLP</td><td>✓</td><td></td><td>300</td><td>40.7</td><td>61.8</td><td>44.2</td></tr></table>
+
+ferability of SL on various backbones, e.g. ResNet101 [20], Swin-tiny [28], and EfficientNetb2 [38]. Swin-tiny achieves relatively smaller gain  $(+1.7\%)$  due to its good Feature MixTURENess (0.86), which is close to SL-MLP in Fig. 7(a).
+
+Generalize to other classification tasks. To evaluate if the added MLP can help SL to transfer better on cross-domain tasks, following [22], we pretrain the model on ImageNet-1K, and evaluate the transferability on 12 classification datasets from different domains. As illustrated in Tab. 4, supervised pretraining methods with the MLP projector, i.e., SL-MLP and SupCon [23], outperform their no MLP counterparts, i.e., SL and SupCon w/o MLP on linear evaluation, by  $5.79\%$ ,  $13.71\%$  on the averaged Top-1 accuracy, respectively. Consistent results can be observed on finetuning and few-shot learning settings. More results are
+
+provided in Supplementary I. Besides, by comparing SupCon, SL-MLP and SupCon w/o MLP, SL, we conclude that the MLP projector instead of the contrastive loss plays the key role in increasing transferability. Our conclusion contrasts with previous works [22, 53] because they ignore the MLP projector before the contrastive loss.
+
+Generalize to object detection. We assess the transferability improvement by the MLP projector on COCO object detection task. We follow the settings in [18] to finetune the whole network with  $1 \times$  schedule. In Tab. 3, we report results using Mask-RCNN (R50-FPN), as detailed in Supplementary H. When both are pretrained over 100 epochs, SL-MLP performs better than SL (without MLP) on object detection by +0.8 AP. If MLP is used by both supervised and unsupervised pretraining, SL-MLP pretrained by 100 epochs achieves better performance than unsupervised pretraining (e.g. SwAV and Mocov2) which are pretrained with 200 epochs. When both pretrained over 300 epochs, SL-MLP shows better performance than Byol with +1.3 AP.
+
+# 5.3. Ablation Study
+
+Effectiveness of different components in MLP. In this part, we investigate the influence of different components in the MLP projector. We set the hidden units and output dimension of MLP to be 2048 to retain the dimension of output features the same as SL. Variants are constructed by adding the components incrementally: (a) no MLP projector; (b) only Input FC; (c) Input FC+BN+output FC; (d) Input FC+ReLU+output FC; (e) BN+ReLU. All experiments are pretrained on pre-D over 100 epochs. As shown in Tab. 5, SL-MLP achieves the best accuracy among all variants. We analyze the influence of different components on discriminative ratio  $\phi$  on pre-D, Feature MixTUREN II, feature redundancy  $\mathcal{R}$  qualitatively and quantitatively in Supplementary D.3. Besides, we also observe an interesting phenomenon on Tab. 5(e) that only inserting a lightweight BN-ReLU also achieves good transfer performance. As this is not our main focus, we will investigate it in future works.
+
+Epochs and layers. Fig. 10(a) shows that adding one MLP projector achieves the optimal transferability. In addition, larger pretraining epochs benefit the transferability of SL-MLP when one MLP projector is added, but it has little influence when more MLP projectors are used.
+
+SL-MLP is less sensitive to batch size. Most unsupervised methods depend on big mini-batches to train a representation with strong transferability. To investigate the sensitivity of SL-MLP to batch size, we do experiments with batch size from 256 to 4096 for Byol and to 1024 for SL-MLP over 300 epochs. As shown in Fig. 10(b), the transferability of Byol drops when the batch size decreases. In contrast, the transferability of SL-MLP retains when batch size changes.
+
+SL-MLP is less sensitive to augmentation. Unsupervised methods benefit from a broader space of colors and more
+
+Table 4. Linear evaluation on fixed backbone, full network finetuning, and few-shot learning performance on 12 classification datasets in terms of top-1 accuracy. All models are pretrained for 300 epochs with the same code base except for SelfSupCon† (Mocov2) which pretrained for 400 epochs using the results illustrated in [22]. Average results style: best, second best.  
+
+<table><tr><td>Method</td><td>ChestX</td><td>CropDisease</td><td>DeepWeeds</td><td>DTD</td><td>EuroSAT</td><td>Flowers102</td><td>Kaokore</td><td>Omniglot</td><td>Resisc45</td><td>Sketch</td><td>SVHN</td><td>ISIC</td><td>Average</td></tr><tr><td colspan="14">linear evaluation</td></tr><tr><td>SL</td><td>45.45</td><td>96.80</td><td>84.02</td><td>66.22</td><td>95.07</td><td>83.69</td><td>75.40</td><td>64.14</td><td>85.36</td><td>67.82</td><td>67.13</td><td>79.58</td><td>75.89</td></tr><tr><td>SL-MLP</td><td>49.89</td><td>99.02</td><td>87.86</td><td>72.61</td><td>96.63</td><td>93.46</td><td>81.12</td><td>76.73</td><td>91.66</td><td>74.51</td><td>75.16</td><td>81.53</td><td>81.68</td></tr><tr><td>SupCon w/o MLP</td><td>41.38</td><td>91.52</td><td>73.16</td><td>62.93</td><td>89.84</td><td>73.23</td><td>66.38</td><td>44.54</td><td>76.55</td><td>55.21</td><td>61.45</td><td>68.54</td><td>67.06</td></tr><tr><td>SupCon</td><td>47.71</td><td>98.79</td><td>85.66</td><td>74.20</td><td>95.83</td><td>92.24</td><td>79.42</td><td>73.42</td><td>91.14</td><td>76.80</td><td>74.26</td><td>79.78</td><td>80.77</td></tr><tr><td>SelfSupCon†</td><td>48.08</td><td>99.06</td><td>87.88</td><td>72.71</td><td>96.97</td><td>89.62</td><td>81.67</td><td>69.66</td><td>90.88</td><td>69.12</td><td>69.95</td><td>81.51</td><td>79.70</td></tr><tr><td colspan="14">finetuned with 1000 training samples</td></tr><tr><td>SL</td><td>40.86</td><td>94.31</td><td>86.95</td><td>62.12</td><td>94.05</td><td>88.94</td><td>78.22</td><td>46.16</td><td>80.32</td><td>14.17</td><td>82.16</td><td>78.28</td><td>70.54</td></tr><tr><td>SL-MLP</td><td>42.34</td><td>94.48</td><td>89.64</td><td>63.90</td><td>95.30</td><td>90.20</td><td>77.98</td><td>46.66</td><td>83.13</td><td>17.32</td><td>80.19</td><td>78.82</td><td>71.66</td></tr><tr><td>SupCon w/o MLP</td><td>41.72</td><td>93.52</td><td>84.95</td><td>58.09</td><td>95.15</td><td>88.23</td><td>78.95</td><td>45.68</td><td>80.63</td><td>14.39</td><td>82.25</td><td>77.96</td><td>70.12</td></tr><tr><td>SupCon</td><td>41.84</td><td>93.46</td><td>88.70</td><td>61.81</td><td>94.54</td><td>91.28</td><td>78.35</td><td>46.02</td><td>81.62</td><td>15.84</td><td>81.85</td><td>78.51</td><td>71.15</td></tr><tr><td>SelfSupCon†</td><td>43.09</td><td>93.95</td><td>88.10</td><td>62.95</td><td>95.47</td><td>88.92</td><td>79.41</td><td>45.33</td><td>81.14</td><td>10.57</td><td>82.37</td><td>78.27</td><td>70.88</td></tr><tr><td colspan="14">5-ways 5-shots few-shot classification</td></tr><tr><td>SL</td><td>25.64</td><td>89.07</td><td>54.32</td><td>78.58</td><td>82.96</td><td>93.14</td><td>46.14</td><td>92.82</td><td>84.17</td><td>87.06</td><td>38.03</td><td>41.22</td><td>67.76</td></tr><tr><td>SL-MLP</td><td>26.89</td><td>93.45</td><td>59.08</td><td>83.04</td><td>87.16</td><td>96.88</td><td>50.77</td><td>95.73</td><td>89.00</td><td>89.84</td><td>41.96</td><td>46.76</td><td>71.71</td></tr><tr><td>SupCon w/o MLP</td><td>23.62</td><td>75.64</td><td>49.34</td><td>73.04</td><td>73.90</td><td>82.16</td><td>38.10</td><td>67.87</td><td>75.18</td><td>81.01</td><td>34.92</td><td>35.16</td><td>59.16</td></tr><tr><td>SupCon</td><td>26.18</td><td>94.09</td><td>59.36</td><td>85.02</td><td>87.97</td><td>96.55</td><td>51.02</td><td>94.49</td><td>89.01</td><td>89.75</td><td>41.67</td><td>43.48</td><td>71.55</td></tr><tr><td colspan="14">Top-1 Accuracy vs Epochs &amp; MLPs</td></tr><tr><td colspan="14">Top-1Accuracy on evalD</td></tr><tr><td colspan="14">Top-1 accuracy on evalD</td></tr><tr><td colspan="14">Top-1 accuracy on evalD</td></tr><tr><td colspan="14">Top-1 accuracy on evalD</td></tr><tr><td colspan="14">Top-1 accuracy on evalD</td></tr><tr><td colspan="5">Top-1 accuracy on evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy on evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy on evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy on evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy on evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td colspan="9"></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td colspan="9"></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td colspan="9"></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td colspan="9"></td></tr><tr><td colspan="5">Top-1 accuracy in evalD</td><td colspan="9"></td></tr><tr><td colspan="5">Top-1 accuracy is evalD</td><td colspan="9"></td></tr></table>
+
+Figure 10. (Left to right) (a) Top-1 accuracy with different pretraining epochs and number of MLP projectors. (b) Top-1 accuracy with different batch sizes. (c) Top-1 accuracy with different pretraining augmentations.  
+Table 5. Empirical analysis of architectural design of the MLP projector. We pretrain models over 100 epochs and set the output dimension to 2048. Top-1 accuracy on eval-D is reported.  
+
+<table><tr><td rowspan="2">Exp</td><td colspan="4">Components</td><td rowspan="2">+Params</td><td rowspan="2">Top-1</td></tr><tr><td>Input FC</td><td>BN</td><td>ReLU</td><td>Output FC</td></tr><tr><td>(a)</td><td></td><td></td><td></td><td></td><td>/</td><td>55.9</td></tr><tr><td>(b)</td><td>✓</td><td></td><td></td><td></td><td>4.196M</td><td>56.6</td></tr><tr><td>(c)</td><td>✓</td><td>✓</td><td></td><td>✓</td><td>8.395M</td><td>61.0</td></tr><tr><td>(d)</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>8.391M</td><td>60.1</td></tr><tr><td>(e)</td><td></td><td>✓</td><td>✓</td><td></td><td>0.004M</td><td>60.5</td></tr><tr><td>SL-MLP</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>8.395M</td><td>62.5</td></tr></table>
+
+intensive augmentations during pretraining, which always lead to undesirable degradation when some augmentations are missing. Supervised models trained only with horizontal flipping may perform well [53]. We set Byol's augmentations as a baseline setting for both SL-MLP and Byol. We then compare the robustness on augmentation between SL-MLP and Byol by removing augmentation step by step. Experiments of SL-MLP and Byol are all constructed on their default condition with only augmentations changed. The results are illustrated on Fig. 10(c). We find that SL-MLP inherits the robustness of SL and shows a little accuracy drop with simple augmentations.
+
+# 6. Limitations and Conclusions
+
+In this paper, we study the transferability gap between supervised and unsupervised pretraining. Based on empirical results, we identify that the MLP projector is a key factor for the good transferability of unsupervised pretraining methods. By adding an MLP projector into supervised pretraining methods, we close the gap between supervised and unsupervised pretraining and even make supervised pretraining better. Our finding is confirmed with extensive experiments on diverse backbone networks and various downstream tasks, including the concept generalization tasks, cross-domain image classifications, and objection detection. While the MLP is a simple design for better transferability, there might exist some straightforward designs on the objective function, which we leave for future work.
+
+# 7. Acknowledgment
+
+This work is supported by the Key R&D Project of Zhejiang Province (No.2022C01056), Natural Science Foundation of Zhejiang Province (No.LQ21F030017), the National Natural Science Foundation of China (No.62127803) and Hetao Shenzhen-Hong Kong Science and Technology Innovation Cooperation Zone (HZQB-KCZYZ-2021045).
+
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+# RigidFlow: Self-Supervised Scene Flow Learning on Point Clouds by Local Rigidity Prior
+
+Ruibo Li $^{1,2}$ , Chi Zhang $^{2}$ , Guosheng Lin $^{1,2*}$ , Zhe Wang $^{3}$ , Chunhua Shen $^{4}$ $^{1}$ S-Lab for Advanced Intelligence, Nanyang Technological University  
+ $^{2}$ School of Computer Science and Engineering, Nanyang Technological University  
+ $^{3}$ SenseTime Research  $^{4}$ Zhejiang University  
+E-mail: ruibo001@e.ntu.edu.sg, gslin@ntu.edu.sg
+
+# Abstract
+
+In this work, we focus on scene flow learning on point clouds in a self-supervised manner. A real-world scene can be well modeled as a collection of rigidly moving parts, therefore its scene flow can be represented as a combination of rigid motion of each part. Inspired by this observation, we propose to generate pseudo scene flow for self-supervised learning based on piecewise rigid motion estimation, in which the source point cloud is decomposed into a set of local regions and each region is treated as rigid. By rigidly aligning each region with its potential counterpart in the target point cloud, we obtain a region-specific rigid transformation to represent the flow, which together constitutes the pseudo scene flow labels of the entire scene to enable network training. Compared with most existing approaches relying on point-wise similarities for scene flow approximation, our method explicitly enforces region-wise rigid alignments, yielding locally rigid pseudo scene flow labels. We demonstrate the effectiveness of our self-supervised learning method on FlyingThings3D and KITTI datasets. Comprehensive experiments show that our method achieves new state-of-the-art performance in self-supervised scene flow learning, without any ground truth scene flow for supervision, even outperforming some supervised counterparts.
+
+# 1. Introduction
+
+Scene flow [35] is a 3D motion field to describe the motion of every point between two time steps. As an essential representation of dynamics, scene flow can be applied in numerous downstream applications, such as robotics and autonomous driving, for dynamic scene understanding. Due to the wide applications of 3D sensors, scene flow estimation with point clouds as input has attracted broad inter
+
+![](images/005e811b5eeab757e048273367cf7d04e25af2bf103bef29f5d00a4be9449e7e.jpg)
+
+![](images/57b53d7976cc0e1ae5dbfb594f774bd86cabad10b743b91de46abd4c45f78321.jpg)
+
+![](images/7c442d4af4f794710bd3c35517a012ae537093a0b95c7a8a4f8b21063538041a.jpg)  
+Figure 1. Comparison of the pseudo scene flow labels produced by nearest neighbor search and our proposed method. (a) Input point clouds from two consecutive frames; (b) Pseudo labels generated by the point matching with per-point nearness as a measure; (c) Pseudo labels generated by our proposed method. Green line represents the correct pseudo label with absolute error less than  $0.1m$  or relative error less than  $10\%$ . Red line represents the incorrect pseudo label.
+
+est. However, scene flow data is difficult to collect [25], which makes supervised learning approaches suffer from a shortage of real-world training samples. To overcome the reliance on scene flow data, scene flow learning in a self-supervised manner is proposed as a promising solution, which is the focus of our work here.
+
+To enable deep network training under the self-supervised setting, it is crucial to approximate the scene flow labels given unlabelled point cloud data. To do so,
+
+most previous approaches [1, 13, 16, 27, 29, 34, 42] perform point matching between two consecutive point clouds to establish point correspondences and treat the 3D coordinate difference between each matching pair as the pseudo scene flow label. However, the point matching strategies applied in these approaches mainly consider point-wise similarities, but often fail to capture potential structured motions of points, leading to inconsistent pseudo scene flow labels. Fig. 1 (b) shows an example of the estimated pseudo scene flow labels based on point matching with per-point nearness as a measure.
+
+For a real-world scene, most of the structures in this scene are rigid or almost so [23]. This allows us to decompose a non-rigid scene into a collection of rigidly moving parts, such that the entire scene flow can be approximated by estimating the rigid motion of individual parts. Based on this intuition, in this work, we propose to generate the pseudo scene flows from point clouds by piecewise rigid motion estimations.
+
+To achieve this goal, an over-segmentation approach is employed to decompose the source point cloud into supervoxels and these supervoxels are treated as rigid during the pseudo label generation. By solving an independent rigid registration for each supervoxel, we find a rigid transformation that rigidly aligns this supervoxel with its potential counterpart in the target point cloud. Based on the rigid transformation estimate, we produce the rigid flow of each supervoxel, thereby generating the entire scene flow approximation as pseudo labels for self-supervised learning. Different from most existing self-supervised approaches that neglect potential structured motions of points, our method explicitly enforces region-wise rigid alignments and generates locally rigid pseudo scene flow labels, where the pseudo labels of points in the same supervoxel obey the same rigid motion pattern.
+
+Main contributions of this paper are listed as follows:
+
+- We present a new self-supervised scene flow learning approach (RigidFlow) that solves the pseudo scene flow label generation as a piecewise rigid motion estimation task;  
+- By decomposing the source point cloud into a set of local regions, we propose a piecewise pseudo label generation module that explicitly enforces region-wise rigid alignments and produces rigid pseudo flow for each local region.  
+- Our proposed RigidFlow achieves state-of-the-art performance in self-supervised scene flow learning, without any ground truth scene flow for supervision, even outperforming some supervised counterparts.
+
+# 2. Related Work
+
+Scene flow estimation on images. Scene flow estimation [35] aims to predict a 3D field in a scene to represent the motion of each point. Nowadays, the local rigidity assumption has been widely used in numerous advanced approaches [2, 9-11, 14, 18, 21, 22, 25, 33, 36-38] to estimate scene flow from images. Especially, UnRigidFlow [18] and EffiScene [11] address unsupervised scene flow estimation from images by jointly learning rigidity masks to constrain scene flow predictions. Unlike these methods that use highly regular 2D images as input and employ photometric error as the major loss function, we focus on scene flow estimation from sparse 3D point clouds and explore the application of the local rigidity to pseudo scene flow label generation, so that self-supervised scene flow learning can be achieved by any supervised loss functions with our generated pseudo labels.
+
+Supervised scene flow estimation on point clouds. With the development of 3D sensors, estimating scene flow from point clouds has been drawing a lot of attention. Various approaches [3, 6-8, 15, 19, 20, 30, 41] have been proposed to estimate scene flow from point clouds by supervised learning. Particularly, the local rigidity has also been applied in some of them [3, 6, 15]. PointFlowNet [3] and Rigid3DSceneFlow [6] propose to directly estimate rigid motion for each 3D object. HCRF-Flow [15] employs the local rigidity to refine scene flow predictions. Different from these fully-supervised approaches learning to refine or constrain predicted flow by the local rigidity, without any ground truth data, we explore how to produce pseudo labels with the guidance of the local rigidity assumption to achieve self-supervised scene flow learning.
+
+Self-supervised scene flow estimation on point clouds. In self-supervised scene flow learning, without ground truth scene flow, most previous methods [1, 13, 16, 27, 29, 34, 42] propose to establish point correspondences between source and target point clouds by point matching, and treat the 3D coordinate difference between each matching pair as a pseudo scene flow label. Specifically, [1, 27, 34] adopt a nearest neighbor loss and [13, 29, 42] adopt a Chamfer loss for self-supervision. In the two loss functions, point correspondences are built by the nearest search. And, [16] builds point correspondences by solving an optimal transport. However, the two point matching strategies mainly consider point-wise similarities but fail to capture potential structured motions of points, leading to unsatisfactory scene flow approximation. Motivated by the local rigidity assumption, we propose to generate pseudo labels by a piecewise rigid motion estimation. By explicitly enforcing region-wise rigid alignments between the source and the target, our method generate locally rigid pseudo scene flow labels, where the pseudo labels of points in the same supervoxel obey the same rigid motion pattern. Although
+
+the local rigidity has been considered in some recent self-supervised works [1, 29], the clue of rigidity is limited to smooth or constrain the flow predictions as a regularization. Its ability to improve pseudo label generation is far from being explored.
+
+# 3. Preliminaries: Rigid registration and ICP
+
+As a crucial task in computer vision, point cloud registration has been well studied in the literature [4, 5, 31, 32, 39, 40]. Given two point clouds,  $\mathbf{X} = \{\pmb{x}_i \in \mathbb{R}^3\}_{i=1}^{N_x}$  and  $\mathbf{Y} = \{\pmb{y}_i \in \mathbb{R}^3\}_{i=1}^{N_y}$ , point cloud rigid registration aims to predict a rigid transformation that aligns  $\mathbf{X}$  to  $\mathbf{Y}$ . The rigid transformation can be written as  $[\pmb{R}_{XY}, \pmb{t}_{XY}]$ , where the rotation matrix  $\pmb{R}_{XY} \in \mathrm{SO}(3)$  and the translation vector  $\pmb{t}_{XY} \in \mathbb{R}^3$ . The objective function of point cloud registration can be expressed as:
+
+$$
+E \left(\boldsymbol {R} _ {X Y}, \boldsymbol {t} _ {X Y}; \boldsymbol {X}, \boldsymbol {Y}\right) = \frac {1}{n} \sum_ {i = 1} ^ {n} \| \boldsymbol {R} _ {X Y} \boldsymbol {x} _ {i} + \boldsymbol {t} _ {X Y} - \boldsymbol {y} _ {m (i)} \| _ {2} ^ {2}, \tag {1}
+$$
+
+where  $m$  is the point mapping from points in  $X$  to their corresponding points in  $Y$ .
+
+Since the point mapping  $m$  is unknown, the iterative closest point (ICP) [4] is widely employed to address this problem by alternating between estimating the rigid transformation and finding the point mapping. In each iteration, based on the previous point mapping estimate, the current rigid transformation is updated by solving the least-square problem in Eq. (1). And then, based on the current rigid transformation estimate, the point mapping of each point is updated to its closest match in another point cloud:
+
+$$
+m (i) = \arg \min  _ {j} \| \boldsymbol {R} _ {X Y} \boldsymbol {x} _ {i} + \boldsymbol {t} _ {X Y} - \boldsymbol {y} _ {j} \| _ {2} ^ {2}. \tag {2}
+$$
+
+Although the ICP is efficient, the performance depends heavily on the initialization of rigid transformation and point matching.
+
+# 4. Method
+
+Given a temporal sequence of point clouds,  $\pmb{P} = \{\pmb{p}_i \in \mathbb{R}^3\}_{i=1}^N$  at frame  $t$  and  $\pmb{Q} = \{\pmb{q}_i \in \mathbb{R}^3\}_{i=1}^N$  at frame  $t + 1$ , scene flow estimation aims to estimate the 3D motion field  $\pmb{F} = \{\pmb{f}_i \in \mathbb{R}^3\}_{i=1}^N$  in a scene. In this paper, we target at self-supervised point cloud scene flow estimation, where no ground truth scene flow annotations  $\pmb{D}$  are provided. To enable network training without ground truth labels, we focus on effective pseudo scene flow label generation, such that the scene flow learning can be achieved using the pseudo labels  $\widehat{D}$  as supervision.
+
+In this section, we first introduce how to produce pseudo scene flow for a real-world scene by a piecewise rigid motion estimation. After that, we present the details of our algorithm and explain how to use the generated pseudo labels to achieve self-supervised training. Fig. 2 is the overview of our method.
+
+# 4.1. Generating Pseudo Labels by Piecewise Rigid Motion Estimation
+
+Scene flow is the 3D motion field of objects in a scene. If the scene contains only a rigidly moving object, the scene flow from  $\pmb{P}_{\text{rigid}}$  to  $\pmb{Q}_{\text{rigid}}$  follows the rigid transformation  $[R_{PQ}, t_{PQ}]$  between the two point clouds:
+
+$$
+\boldsymbol {D} = \boldsymbol {R} _ {P Q} \boldsymbol {P} _ {\text {r i g i d}} + \boldsymbol {t} _ {P Q} - \boldsymbol {P} _ {\text {r i g i d}}. \tag {3}
+$$
+
+Thus, for a rigidly moving object, when the ground truth scene flow is unavailable, we can produce the scene flow by finding its optimal rigid transformation between  $P_{rigid}$  and  $Q_{rigid}$ .
+
+For a complex real-world scene, although it is not rigid, most of the structures in this scene are rigid or almost rigid, which makes it possible to approximate a non-rigid scene into a set of rigidly moving regions. Therefore, we can estimate the flow of each rigidly moving region by finding its optimal rigid motion, thereby generating the entire scene flow. In other words, we can generate pseudo scene flows by a piecewise rigid motion estimation.
+
+Decomposing the point cloud  $\pmb{P}$  into  $K$  rigid regions  $\{P^{(1)}, P^{(2)}, \dots, P^{(K)}\}$ , the piecewise rigid motion estimation from  $\pmb{P}$  to  $\pmb{Q}$  for the region  $P^{(k)}$  can be considered as an independent rigid body registration from  $P^{(k)}$  to  $Q$ :
+
+$$
+\left[ \boldsymbol {R} _ {k} ^ {*}, \boldsymbol {t} _ {k} ^ {*} \right] = \arg \min  _ {\left[ \boldsymbol {R} _ {k}, \boldsymbol {t} _ {k} \right]} E \left(\boldsymbol {R} _ {k}, \boldsymbol {t} _ {k}; \boldsymbol {P} ^ {(k)}, \boldsymbol {Q}\right), \tag {4}
+$$
+
+$$
+E \left(\boldsymbol {R} _ {k}, \boldsymbol {t} _ {k}; \boldsymbol {P} ^ {(k)}, \boldsymbol {Q}\right) = \frac {1}{N _ {k}} \sum_ {i = 1} ^ {N _ {k}} \left\| \boldsymbol {R} _ {k} \boldsymbol {p} _ {i} ^ {(k)} + \boldsymbol {t} _ {k} - \boldsymbol {q} _ {m ^ {(k)} (i)} \right\| _ {2} ^ {2}, \tag {5}
+$$
+
+where  $N_{k}$  is the point number in the rigid region  $\pmb{P}^{(k)}$ ,  $m^{(k)}(i)$  is the point mapping from the  $i$ -th point in  $\pmb{P}^{(k)}$  to its correspondence in  $\pmb{Q}$ , and  $[\pmb{R}_k^*,\pmb{t}_k^*]$  is the optimal rigid transformation for  $\pmb{P}^{(k)}$ .
+
+Following Eq. (3), with  $\left[R_k^*, t_k^*\right]$ , the pseudo rigid scene flow estimate  $\widehat{D}^{(k)}$  for this region can be computed by:
+
+$$
+\hat {\boldsymbol {D}} ^ {(k)} = \boldsymbol {R} _ {k} ^ {*} \boldsymbol {P} ^ {(k)} + \boldsymbol {t} _ {k} ^ {*} - \boldsymbol {P} ^ {(k)}. \tag {6}
+$$
+
+Combining the pseudo rigid scene flow estimates for all  $K$  rigid regions  $\{\widehat{D}^{(1)},\widehat{D}^{(2)},\dots,\widehat{D}^{(K)}\}$ , we obtain the final pseudo scene flow labels  $\widehat{D}$  for self-supervised training.
+
+# 4.2. Piecewise pseudo label generation module
+
+In order to convert pseudo label generation into a piecewise rigid motion estimation, we first employ an oversegmentation method [17] to split the point cloud  $P$  into supervoxels and treat these supervoxels as rigid moving regions. After obtaining the supervoxels, we employ a piecewise pseudo label generation module to generate scene flow labels for each supervoxel via finding its rigid transformation from  $P^{(k)}$  to  $Q$ . In this module, we follow the principle of ICP algorithm [4], i.e., alternately estimating point mapping and rigid transformation, such
+
+![](images/dbbcd52d15141e1a87d46f6d02a34a3f5f45ee3a8faddb2968393a751f0ac78d.jpg)  
+Figure 2. Overview of our pseudo scene flow label generation approach. In our approach, we employ an optimization-based oversegmentation method [17] to split the point cloud in the first frame into a set of supervoxels and estimate the rigid transformation of each supervoxel. Based on the rigid transformation estimate, we generate the pseudo labels for each supervoxel, thereby obtaining the pseudo labels for the entire scene.
+
+Algorithm 1 Pseudo scene flow label generation by piecewise rigid motion estimation
+
+Input: Source point cloud  $P$ , Target point cloud  $Q$ , Predicted scene flow from NNs being trained  $F$ ;
+
+Output: Pseudo scene flow labels  $\hat{D}$
+
+# Procedure:
+
+1: Split  $\pmb{P}$  into a set of supervoxels  $\{\pmb{P}^{(1)},\dots,\pmb{P}^{(K)}\}$ ;
+
+$\triangleright$  Oversegmentation
+
+2: for  $k = 1, \dots, K$  do
+
+3:  $m^{(k)}(i)\gets \arg \min_{j}\| \pmb{p}_{i}^{(k)} + \pmb{f}_{i}^{(k)} - \pmb{q}_{j}\|_{2}^{2}$
+
+- Initializing by predicted flow
+
+4: while not converged do
+
+5: Find the matches  $\mathbf{Q}^{(k)}$  for  $P^{(k)}$  based on  $\mathbf{m}^{(k)}$ .
+
+6:  $\pmb{H}^{(k)}\gets \sum_{i = 1}^{N_k}(\pmb {p}_i^{(k)} - \overline{\pmb{p}}^{(k)})(\pmb {q}_i^{(k)} - \overline{\pmb{q}}^{(k)})^\top$
+
+7:  $\pmb{R}_k\gets \pmb{V}^{(k)}\pmb{U}^{(k)^{\top}},\pmb{t}_k\gets -\pmb{R}_k\overline{\pmb{p}}^{(k)} + \overline{\pmb{q}}^{(k)}$
+
+$\triangleright$  Updating rigid transformation
+
+8:  $m^{(k)}(i)\gets \arg \min_{j}\| R_{k}\pmb{p}_{i}^{(k)} + \pmb{t}_{k} - \pmb{q}_{j}\|_{2}^{2}$
+
+$\triangleright$  Updating point mapping
+
+9: end while
+
+10:  $\widehat{\pmb{D}}^{(k)}\gets \pmb{R}_k^*\pmb{P}^{(k)} + \pmb{t}_k^* -\pmb{P}^{(k)}$
+
+$\triangleright$  Generating pseudo labels
+
+11: end for  
+12:  $\hat{\pmb{D}}\gets \{\hat{\pmb{D}}^{(1)},\dots,\hat{\pmb{D}}^{(K)}\}$
+
+that the region-wise rigid alignment between this region with its counterpart is explicitly enforced. Next, we present the details of this module.
+
+Initializing point mapping by predicted flow. The performance of ICP relies greatly on the initialization of rigid transformation and point mapping. When solving the registration from  $\pmb{P}^{(k)}$  to  $\pmb{Q}$ , for each point  $\pmb{p}_i^{(k)}$  in  $\pmb{P}^{(k)}$ , a
+
+straightforward way of initialization is to set its closest point in  $Q$  as the initial correspondence. Inspired by [16], we establish the initial point mapping based on the predicted scene flow  $F$  from neural networks being trained. Specifically, we warp the point  $p_i^{(k)}$  by its predicted flow  $f_i^{(k)}$ , and then take the closest point of this warped point as the initial match:
+
+$$
+m _ {i n i t a l} ^ {(k)} (i) = \arg \min  _ {j} \| \boldsymbol {p} _ {i} ^ {(k)} + \boldsymbol {f} _ {i} ^ {(k)} - \boldsymbol {q} _ {j} \| _ {2} ^ {2}. \tag {7}
+$$
+
+As the training progresses, the accuracy of the predicted scene flow will be gradually improved, making the closest search of the warped points more likely to find the correct matches and establish good initial point correspondences.
+
+Updating rigid transformation estimate. Based on the previous point mapping estimate, we update the rigid transformation for each supervoxel by solving the least-square problem shown in Eq. (4) and Eq. (5) with the point mapping fixed. Specifically, following [4, 39, 40], we apply the singular value decomposition (SVD) to it.
+
+For the points in supervoxel  $\pmb{P}^{(k)}$ , we first select their matches  $\pmb{Q}^{(k)}$  from  $\pmb{Q}$  according to the point mapping estimate  $\pmb{m}^{(k)}$ . The centers of  $\pmb{P}^{(k)}$  and  $\pmb{Q}^{(k)}$  are defined as  $\overline{\pmb{p}}^{(k)} = \frac{1}{N_k}\sum_{i=1}^{N_k}\pmb{p}_i^{(k)}$  and  $\overline{\pmb{q}}^{(k)} = \frac{1}{N_k}\sum_{i=1}^{N_k}\pmb{q}_i^{(k)}$ . The cross-covariance matrix for supervoxel  $\pmb{P}^{(k)}$  can be written as:
+
+$$
+\boldsymbol {H} ^ {(k)} = \sum_ {i = 1} ^ {N _ {k}} \left(\boldsymbol {p} _ {i} ^ {(k)} - \overline {{\boldsymbol {p}}} ^ {(k)}\right) \left(\boldsymbol {q} _ {i} ^ {(k)} - \overline {{\boldsymbol {q}}} ^ {(k)}\right) ^ {\top}. \tag {8}
+$$
+
+Using SVD to decompose  $H^{(k)}$ , we have  $H^{(k)} = U^{(k)}S^{(k)}V^{(k)^{\top}}$ . The rigid transformation estimate for supervoxel  $P^{(k)}$  can be updated in closed-form as:
+
+$$
+\boldsymbol {R} _ {k} = \boldsymbol {V} ^ {(k)} \boldsymbol {U} ^ {(k) ^ {\top}}, \tag {9}
+$$
+
+$$
+\boldsymbol {t} _ {k} = - \boldsymbol {R} _ {k} \overline {{\boldsymbol {p}}} ^ {(k)} + \overline {{\boldsymbol {q}}} ^ {(k)}.
+$$
+
+Updating point mapping estimate. Based on the current rigid transformation estimate, the point mapping of each point in supervoxel  $P^{(k)}$  is updated to its closest point in  $Q$ :
+
+$$
+m ^ {(k)} (i) = \arg \min  _ {j} \| \boldsymbol {R} _ {k} \boldsymbol {p} _ {i} ^ {(k)} + \boldsymbol {t} _ {k} - \boldsymbol {q} _ {j} \| _ {2} ^ {2}. \tag {10}
+$$
+
+Generating pseudo labels for each supervoxel. After several alternating iterations between Eq. (9) and Eq. (10), we obtain the final rigid transformation estimates for each supervoxel as the optimal rigid transformation. Following Eq. (6), we generate pseudo rigid scene flow for each supervoxel. The method of our pseudo label generation is sketched in Algorithm 1.
+
+# 4.3. Self-supervised training with pseudo labels
+
+Using the generated pseudo labels for supervision, we can achieve the self-supervised training of scene flow estimation networks with supervised loss functions. In this paper, we choose FLOT [30] as the default estimation network and use the  $l_{1}$ -norm loss function for self-supervised learning:
+
+$$
+L = \frac {1}{3 N} \sum_ {i = 1} ^ {N} \| \boldsymbol {f} _ {i} - \widehat {\boldsymbol {d}} _ {i} \| _ {1}, \tag {11}
+$$
+
+where  $\pmb{f}_i$  is the predicted scene flow for point  $i$  and  $\widehat{d}_i$  is our generated pseudo label.
+
+# 5. Experiment
+
+To validate the effectiveness of our self-supervised learning method, we compare our method with the state-of-the-art fully-supervised and self-supervised approaches. Then, we conduct various ablation experiments to analyze the contribution of different components. Finally, we design some quantitative and qualitative experiments to evaluate the generated pseudo labels for further analysis. All experiments are performed on the large-scale synthetic FlyingThings3D [24] dataset and the real-world KITTI 2015 [25, 26] dataset.
+
+Datasets. 3D data are not directly provided by the two original datasets, thus the point clouds need to be extracted from the original data. Following [30], we denote the two point cloud datasets prepared by HPLFlowNet [7] as FT3Ds and KITTIs, respectively. For FT3Ds and KITTIs, there are no occluded points in the processed point clouds. We denote the two datasets prepared by FlowNet3D [19] as FT3Do and KITTIo, respectively, where occluded points are preserved. Specially, FlowNet3D [19] also splits the KITTIo data to use the first 100 pairs for finetuning and the rest 50 pairs for testing. Here, we denote the finetuning part as KITTIf and the rest testing data as KITTIt. Following the raw data sampling strategy used in [16], we extract some raw point clouds from KITTI dataset as training samples (6026 pairs) and denote them as KITTIr. There is no overlap between KITTIr and KITTIo.
+
+Implementation details. During evaluation, we conduct two main experiments, one for point clouds without occlusions and one for point clouds with occlusions.
+
+For the experiment on point clouds without occlusions, we follow the experimental setting in [7, 13, 42]. Specifically, we train a FLOT [30] model by our self-supervised approach on  $\mathrm{FT3D_s}$  training set and test it on  $\mathrm{FT3D_s}$  testing set and KITTIs. For a pair of point clouds, we randomly sample 8192 points in each point cloud as input. In the pseudo label generation phase, we decompose the source point cloud into 40 supervoxels with an over-segmentation method [17] and set the iteration number in our piecewise pseudo label generation module to 4. We set the batchsize to 1 and use Adam optimizer [12] with an initial learning rate of 0.001.
+
+For the experiment on point clouds with occlusions, we train a FLOT model on  $\mathrm{KITTI_r}$  using our self-supervised method and evaluate it on  $\mathrm{KITTI_o}$  and  $\mathrm{KITTI_t}$ . For each training sample, we randomly sample 2048 points in each point cloud as input. In the pseudo label generation phase, we split the source point cloud into 40 supervoxels and set the iteration number to 2. We set the batchsize to 4 and use Adam with an initial learning rate of 0.001.
+
+Our code is implemented based on PyTorch [28], FLOT [30] and S3DPC [17].
+
+Evaluation metrics. We evaluate our method on four evaluation metrics adopted in [7, 30]. We denote the ground truth scene flow and predicted scene flow as  $D$  and  $F$ , respectively. The metrics are defined as follows: EPE(m):  $\| D - F \|_2$ , end point error, averaged over all points; AS(%): the ratio of points with EPE < 0.05m or relative error < 5%; AR(%): the ratio of points with EPE < 0.1m or relative error < 10%; Out(%): the ratio of points with EPE > 0.3m or relative error > 10%.
+
+# 5.1. Comparison with State-of-the-art Methods
+
+# 5.1.1 Results on FT3Ds and KITTIs
+
+We evaluate our self-supervised learning method on  $\mathrm{FT3D_s}$  testing set and  $\mathrm{KITTI_s}$  data, following the experimental setting in [7, 13, 42]. We compare our method with five advanced self-supervised approaches: Ego-motion [34], PointPWC-Net [42], SLIM [1], Self-Point-Flow [16] and FlowStep3D [13]. In Table 1, our approach achieves the best performance on EPE, AS, and AR, and obtains the second best performance on Out, which demonstrates the effectiveness and the generalization ability of our self-supervised learning algorithm. Especially, our method is the only self-supervised approach that achieves an EPE metric below  $7cm$ . For this metric, our method outperforms the recent FlowStep3D by  $17.6\%$  and  $38.3\%$  on  $\mathrm{FT3D_s}$  and  $\mathrm{KITTI_s}$  respectively, despite the model capacity of our used FLOT is significantly less than that of their FlowStep3D  $(0.11\mathrm{M}v.s.0.68\mathrm{M})$ .
+
+Table 1. Quantitative results on FT3Ds testing set and KITTIs. Full. represents fully-supervised training, Self. represents self-supervised training. Compared with self-supervised approaches, our approach achieve the state-of-the-art performance on three metrics. Especially, our approach is the only self-supervised one that achieves an EPE metric below  $7cm$ . Without any ground truth scene flow for supervision, our approach even outperforms some supervised ones.  
+
+<table><tr><td>Data</td><td>Method</td><td>Sup.</td><td>EPE ↓</td><td>AS ↑</td><td>AR ↑</td><td>Out ↓</td></tr><tr><td rowspan="10">FT3Ds</td><td>FlowNet3D [19]</td><td>Full.</td><td>0.0864</td><td>47.89</td><td>83.99</td><td>54.64</td></tr><tr><td>HPLFlowNet [7]</td><td>Full.</td><td>0.0804</td><td>61.44</td><td>85.55</td><td>42.87</td></tr><tr><td>PointPWC-Net [42]</td><td>Full.</td><td>0.0588</td><td>73.79</td><td>92.76</td><td>34.24</td></tr><tr><td>FLOT [30]</td><td>Full.</td><td>0.0520</td><td>73.20</td><td>92.70</td><td>35.70</td></tr><tr><td>FlowStep3D [13]</td><td>Full.</td><td>0.0455</td><td>81.62</td><td>96.14</td><td>21.65</td></tr><tr><td>Ego-motion [34]</td><td>Self.</td><td>0.1696</td><td>25.32</td><td>55.01</td><td>80.46</td></tr><tr><td>PointPWC-Net [42]</td><td>Self.</td><td>0.1213</td><td>32.39</td><td>67.42</td><td>68.78</td></tr><tr><td>Self-Point-Flow [16]</td><td>Self.</td><td>0.1009</td><td>42.31</td><td>77.47</td><td>60.58</td></tr><tr><td>FlowStep3D [13]</td><td>Self.</td><td>0.0852</td><td>53.63</td><td>82.62</td><td>41.98</td></tr><tr><td>Ours</td><td>Self.</td><td>0.0692</td><td>59.62</td><td>87.10</td><td>46.42</td></tr><tr><td rowspan="11">KITTI_s</td><td>FlowNet3D [19]</td><td>Full.</td><td>0.1064</td><td>50.65</td><td>80.11</td><td>40.03</td></tr><tr><td>HPLFlowNet [7]</td><td>Full.</td><td>0.1169</td><td>47.83</td><td>77.76</td><td>41.03</td></tr><tr><td>PointPWC-Net [42]</td><td>Full.</td><td>0.0694</td><td>72.81</td><td>88.84</td><td>26.48</td></tr><tr><td>FLOT [30]</td><td>Full.</td><td>0.0560</td><td>75.50</td><td>90.80</td><td>24.20</td></tr><tr><td>FlowStep3D [13]</td><td>Full.</td><td>0.0546</td><td>80.51</td><td>92.54</td><td>14.92</td></tr><tr><td>Ego-motion [34]</td><td>Self.</td><td>0.4154</td><td>22.09</td><td>37.21</td><td>80.96</td></tr><tr><td>PointPWC-Net [42]</td><td>Self.</td><td>0.2549</td><td>23.79</td><td>49.57</td><td>68.63</td></tr><tr><td>SLIM(8192 point) [1]</td><td>Self.</td><td>0.1207</td><td>51.78</td><td>79.56</td><td>40.24</td></tr><tr><td>Self-Point-Flow [16]</td><td>Self.</td><td>0.1120</td><td>52.76</td><td>79.36</td><td>40.86</td></tr><tr><td>FlowStep3D [13]</td><td>Self.</td><td>0.1021</td><td>70.80</td><td>83.94</td><td>24.53</td></tr><tr><td>Ours</td><td>Self.</td><td>0.0619</td><td>72.37</td><td>89.23</td><td>26.18</td></tr></table>
+
+We also compare our self-supervised method with some supervised approaches that are trained on  $\mathrm{FT3D_s}$  training set. As shown in Table 1, without any ground truth for supervision, our self-supervised method outperforms FlowNet3D [19] and performs on par with HPLFlowNet [7] on  $\mathrm{FT3D_s}$ . Evaluated on KITTI $_s$  without fine-tuning, our self-supervised method has better generalization ability than FlowNet3D and HPLFlowNet. Qualitative results on  $\mathrm{FT3D_s}$  are shown in Fig. 3.
+
+# 5.1.2 Results on KITTI $_0$  and KITTI $_t$
+
+We evaluate our self-supervised learning method on  $\mathrm{KITTI_o}$  and  $\mathrm{KITTI_r}$ . In  $\mathrm{KITTI_o}$  data, following [30], we remove all points with depth larger than  $35m$  for evaluation. As shown in Table 2, our method outperforms Self-Point-Flow [16] on all metrics. The FLOT trained on  $\mathrm{KITTI_r}$  via our self-supervised method achieves better performance than the fully-supervised FLOT [30] trained on  $\mathrm{FT3D_o}$ , which demonstrates the advantage of our self-supervised learning strategy. That is, compared with learning from labeled synthetic data, the proposed self-supervised learning allows
+
+Table 2. Quantitative results on KITTI<sub>0</sub>. Without any ground truth scene flow for supervision, our method outperforms Self-Point-Flow [16], and even performs better than supervised FlowNet3D [19] and FLOT [30].  
+
+<table><tr><td>Method</td><td>Sup.</td><td>Train. data</td><td>EPE ↓</td><td>AS ↑</td><td>AR ↑</td><td>Out ↓</td></tr><tr><td>FlowNet3D [19]</td><td>Full.</td><td>FT3D0</td><td>0.173</td><td>27.6</td><td>60.9</td><td>64.9</td></tr><tr><td>FLOT [30]</td><td>Full.</td><td>FT3D0</td><td>0.107</td><td>45.1</td><td>74.0</td><td>46.3</td></tr><tr><td>Self-Point-Flow [16]</td><td>Self.</td><td>KITTIr</td><td>0.105</td><td>41.7</td><td>72.5</td><td>50.1</td></tr><tr><td>Ours</td><td>Self.</td><td>KITTIr</td><td>0.102</td><td>48.4</td><td>75.6</td><td>44.2</td></tr></table>
+
+Table 3. Quantitative results on  $\mathrm{KITTI_t}$  following the test settings in [29]. Without pre-trained on  $\mathrm{FT3D_o}$  with supervision, our self-supervised FLOT and FlowNet3D still achieve better performance.  
+
+<table><tr><td>Method</td><td>Pre-train</td><td>Training data</td><td>EPE ↓ AS ↑ AR ↑</td></tr><tr><td>JGF [27]</td><td>✓</td><td>FT3Do + KITTIf</td><td>0.218 10.17 34.38</td></tr><tr><td>WWL [29]</td><td>✓</td><td>FT3Do + KITTIf</td><td>0.169 21.71 47.75</td></tr><tr><td>Ours (FlowNet3D)</td><td></td><td>KITTIr</td><td>0.152 30.17 61.14</td></tr><tr><td>Ours (FLOT)</td><td></td><td>KITTIr</td><td>0.117 38.75 69.73</td></tr></table>
+
+Table 4. Ablation study for piecewise pseudo label generation module (PPLG). NN: nearest neighbor search. RCA: region-wise center alignment. RRA: region-wise rigid alignment.  $\Delta$  denotes the difference in metric with respect to the baseline method.  
+
+<table><tr><td>Method</td><td>NN</td><td>RCA</td><td>RRA</td><td>EPE ↓</td><td>ΔEPE</td></tr><tr><td>nearest neighbor search</td><td>✓</td><td></td><td></td><td>0.217</td><td>0.000</td></tr><tr><td>+ region-wise center alignment</td><td>✓</td><td>✓</td><td></td><td>0.089</td><td>-0.128</td></tr><tr><td>+ region-wise rigid align. (Ours)</td><td>✓</td><td></td><td>✓</td><td>0.071</td><td>-0.146</td></tr></table>
+
+models to learn better representations from unannotated realistic data directly. Some qualitative results on  $\mathrm{KITTI_o}$  are shown in Fig. 3.
+
+Then, we compare the performance of our method with WWL [29] and JGF [27] in a self-supervised learning manner. Following WWL [29], the experiment is conducted on  $\mathrm{KITTI_t}$ . As shown in Table 3, without pre-trained on  $\mathrm{FT3D_o}$  with full-supervision, the FLOT and the FlowNet3D trained via our proposed self-supervised method still perform better than theirs.
+
+# 5.2. Ablation study
+
+We validate the effectiveness of each component in our method. Models are trained on  $\mathrm{FT3D_s}$  training set and tested on  $\mathrm{FT3D_s}$  testing set.
+
+Piecewise pseudo label generation module. At the core of our design is the piecewise pseudo label generation module (PPLG). For each supervoxel, this module generates pseudo labels by alternately estimating point mapping and rigid transformation to explicitly enforce region-wise rigid alignments. The module contains three key steps: point mapping initialization, rigid transformation update, and point mapping update.
+
+![](images/79e6426f2d044757981c688947a516dbe2b48fb27dd44b79ac7147fa59baa451.jpg)
+
+![](images/405dd1600e3b8d8cd73286d0d10ebce5029d796c7131389823a4bed24e7501ae.jpg)
+
+![](images/166ec9acd31322986eeaefd10d992c8deb3066b8e72359a3a103f1577e1ceb4e.jpg)
+
+![](images/8d01ac8f02d2913e974c623fa8433adb1a3b36cb638edda94f5fab11d4f08fab.jpg)  
+Figure 3. Qualitative results on FT3Ds (top) and KITTIo (bottom). Bule points represent the source point cloud. Green points represent the points translated by the correct scene flow predictions. Red points represent the points translated by the incorrect predictions. The scene flow predictions are measured by AR.
+
+![](images/b86aac8d7e497cf19bae59e1148bd4023b269f4197a9f63770a6254f832d0a22.jpg)
+
+![](images/48b8421363bf048bf2f43cc6a055a8c98b4bdb4fb8926afdbcca20a4721331f9.jpg)
+
+Table 5. Comparison of different strategies in pseudo label generation module. IwoF: point mapping initialization without predicted flow. IwF: point mapping initialization with predicted flow. GC: label generation from correspondence. GR: label generation from rigid motion.  
+
+<table><tr><td>Method</td><td>IwoF</td><td>IwF</td><td>GC</td><td>GR</td><td>EPE ↓</td></tr><tr><td>Strategy I</td><td>✓</td><td></td><td></td><td>✓</td><td>0.356</td></tr><tr><td>Strategy II</td><td></td><td>✓</td><td>✓</td><td></td><td>0.080</td></tr><tr><td>Ours</td><td></td><td>✓</td><td></td><td>✓</td><td>0.071</td></tr></table>
+
+To demonstrate the advantage of region-wise rigid alignment in pseudo scene flow generation, we design two baseline methods for comparison: nearest neighbor search, denoted by NN, and region-wise center alignment, denoted by RCA. In nearest neighbor search, for each point, we directly treat the initial match derived from our point mapping initialization (Eq. 7) as the corresponding point to produce a pseudo label without considering any region-wise constraints. As shown in Table 4, compared with the method of nearest neighbor search (NN), by enforcing region-wise alignments, our module significantly reduces the EPE metric by  $14.6\text{cm}$ . For the method of region-wise center alignment, when updating transformation, we only encourage the center of each supervoxel to coincide with that of its counterpart rather than enforcing region-wise rigid alignments. Specifically, we fix the rotation matrix  $\pmb{R}_k$  in Eq. (6), Eq. (9) and Eq. (10) to an identity matrix. As shown in Table 4, by enforcing region-wise rigid alignments, our module outperforms the method of region-wise center alignment (RCA) by  $22.8\%$  on EPE metric, which demonstrates the effectiveness of our region-wise rigid alignment approach.
+
+We next study the initialization strategy and the label generation strategy in PPLG. In our module, we initialize the point mapping by the predicted flow from neural networks being trained, denoted as  $\mathbf{I}\mathbf{w}\mathbf{F}$ , and produce pseudo rigid scene flow from the optimal rigid transformation,
+
+Table 6. The impact of different update iteration numbers.  
+
+<table><tr><td>Iteration No.</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>EPE ↓</td><td>0.077</td><td>0.073</td><td>0.071</td><td>0.069</td></tr></table>
+
+Table 7. The impact of different supervoxel numbers.  
+
+<table><tr><td>Desired supervoxel No.</td><td>80</td><td>60</td><td>40</td><td>20</td></tr><tr><td>EPE ↓</td><td>0.074</td><td>0.073</td><td>0.071</td><td>0.077</td></tr></table>
+
+Table 8. Time consumption of our method to generate pseudo labels for a training sample with 8,192 points in each point cloud.  
+
+<table><tr><td>Component</td><td>Over-segmentation</td><td>Label generation</td><td>Total</td></tr><tr><td>Time (ms)</td><td>76.1</td><td>96.4</td><td>172.5</td></tr></table>
+
+denoted as GR. To verify the advantages of our design, we include two strategies for comparison: Strategy I initializes the point mapping without using predicted flow, denoted as IwoF, and Strategy II produces pseudo scene flow based on the coordinate difference between each optimal correspondence, denoted as GC. As shown in Table 5, our module outperforms Strategy I and Strategy II by around  $80\%$  and  $14\%$  on the EPE metric, respectively, which demonstrates the effectiveness of our design. And Table 6 shows the impact of different update iteration numbers on our method.
+
+Impact of supervoxel number. When generating pseudo labels, we decompose a scene into a set of supervoxels and find the rigid motion of each supervoxel. As shown in Table 7, the model achieves the best performance when our self-supervised method decomposes each scene into 40 supervoxels for label generation.
+
+Time consumption. We evaluate the running time of our pseudo label generation method for a training sample with 8192 points in each point cloud. In Table 8, the total time
+
+![](images/d7535055dc30fc042cde3d147b46d479131f12d1bf94df9f9e54da42ffbe9fdb.jpg)  
+Figure 4. The end point error (EPE, the lower, the better) of model predictions (Blue curve) and generated pseudo labels (Orange curve) on some training samples. During the training, the accuracy of pseudo labels is consistently higher than that of predicted flow. This allows us to apply the pseudo labels as supervision.
+
+consumption for a training sample is  $172.5\mathrm{ms}$  on a single 1080ti GPU.
+
+# 5.3. Analysis on pseudo labels
+
+We conduct some quantitative and qualitative experiments to evaluate the generated pseudo labels for further analysis. In Fig. 4, using part of training samples (197 samples) as testing data, we compare the error of our generated pseudo labels and the predictions of the trained network. We can make the following observations from Fig. 4. (1) During the training, the quality of our generated pseudo labels is gradually improved. (2) The accuracy of pseudo labels is consistently higher than that of predicted flow. This allows us to apply the pseudo labels as supervision. (3) The performance gap between pseudo labels and network predictions is gradually reduced. As illustrated in Fig. 5 (c)-(e), the quality of our generated pseudo labels for the airplane and the chair are gradually improved along with training iterations, which demonstrates the effectiveness of our pseudo label generation method. More quantitative and qualitative results are in supplementary.
+
+# 6. Conclusion
+
+In this paper, we propose to produce pseudo scene flow labels by a piecewise rigid motion estimation. By solving an independent rigid registration for each region, we find a region-specific rigid transformation to represent the flow, thereby generating the entire scene flow approximation as pseudo labels for self-supervised learning. Comprehensive experiments on FlyingThings3D and KITTI datasets demonstrate that our proposed approach achieves state-of-the-art performance in self-supervised scene flow learning, without any ground truth scene flow for supervision, even outperforming some supervised counterparts.
+
+![](images/36ad1854babbdf12a682aee7f1b63cce679f35b778944cc0d60028b78d04ea02.jpg)  
+(a) Input point clouds
+
+![](images/8a1c7a91c362211e51be242e3aae1d07fae4465723184377d553669e92f90b85.jpg)  
+(b)Supervoxels
+
+![](images/23c5a6ef1744138d17912e6aea8db20f977c4be327819888e8e7bea1f33b25fd.jpg)  
+(c) Pseudo labels after 2e3 iter.
+
+![](images/a394d9c70dc160f5e070a541cc21d788704d7a6bfc2c5acaadac44ffe0291aa7.jpg)
+
+![](images/295553dc66725e5dc5f33e6b633386314fed2eb25bb8923b2716b72cc241dfa9.jpg)  
+(d) Pseudo labels after 2e4 iter.  
+(e) Pseudo labels after 2e5 iter.
+
+![](images/8ffcfa76e4d3949ca315c3833a257b18ded06e88b9196b4495f704bd559fea31.jpg)  
+(f) Ground truth scene flow  
+Figure 5. Our generated pseudo labels after different training iterations for the same scene. (a) Input point clouds. Blue points represent the source point cloud and gray points represent the target. (b) Supervoxel results of the source point cloud. Different colors indicate different supervoxels. (c) - (e) Our generated pseudo labels after different training iterations. Green line indicates the correct pseudo label measured by AR. Red line indicates the incorrect pseudo label. (f) Ground truth scene flow. The quality of our generated pseudo labels for the airplane and the chair are gradually improved along with training iterations.
+
+# 7. Limitations
+
+Although achieving remarkable results, our method still has some limitations. Firstly, our pseudo label generation method is based on the local rigidity assumption. For scenes with strongly non-rigid motion, the local rigidity assumption may be not suitable. In practice, these scenes are not common. Secondly, our method generates pseudo labels by enforcing region-wise alignments. Occluded regions without valid counterparts will affect our method. To evaluate the impact of occlusion issues on our method, in Sec. 5.1.2, we conduct experiments on real-world point cloud data with occluded regions. As shown in Table 2 and Table 3, compared with existing approaches, our method achieve the best results. Dealing with non-rigid motions and occlusion issues is worthy of in-depth exploration in our future work.
+
+# Acknowledgments
+
+This study is supported under the RIE2020 Industry Alignment Fund - Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s). This research is also supported by the National Research Foundation, Singapore, under its AI Singapore Programme (AISG Award No: AISG-RP-2018-003), and the Ministry of Education, Singapore, under its Academic Research Fund (Tier-2 research grant: MOE-T2EP20220-0007, Tier-1 research grant: RG95/20).
+
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+# RigNeRF: Fully Controllable Neural 3D Portraits
+
+ShahRukh Athar*  
+Stony Brook University  
+sathar@cs.stonybrook.edu
+
+Zexiang Xu  
+Adobe Research  
+zexu@adobe.com
+
+Kalyan Sunkavalli  
+Adobe Research  
+sunkaval@adobe.com
+
+Eli Shechtman Adobe Research elishe@adobe.com
+
+Zhixin Shu  
+Adobe Research  
+zshu@adobe.com
+
+![](images/d3976fdf437b60dd85e736c3b9894d5d469c1e1dc1fff6636a6980a6704d55a1.jpg)  
+(a) Pose Control  
+Figure 1. Our method, RigNeRF, enables full control of head-posed and facial expression of a human portrait using a 3DMM-guided deformable neural radiance field. With a short portrait video for training, RigNeRF can reanimate the subject with arbitrary (a) head-posed and (b) facial expressions. It also allows controllable viewpoints of the 3D scene (c). Naturally RigNeRF can be used to transfer facial animation from a driving video sequence to the 3D portraits faithfully (d).
+
+![](images/975a64a758ed3fcf9d34226c5cc4e2025cb03377479fa7f544c61b49eb74b60e.jpg)  
+(b) Expression Control
+
+![](images/55d9d498679dc4e803b428644b5d5456708c7bc271d3bafeb91d668907f865d6.jpg)  
+(c) Novel View Synthesis
+
+![](images/ba2a5899864a2982432624f6055beaa2ee2e38105d3c932604a5a95c0be564c7.jpg)  
+(d) Video based reanimation with novel view synthesis
+
+![](images/002136ca38adbdc6cf48230f1f2b5f2585d70b3a4209fae97b9bcda93c5483e3.jpg)
+
+![](images/7da1707e41626e50dfcaf6d9382bf868233f0fcc3cd8a066a640a4e32c32060a.jpg)
+
+# Abstract
+
+Volumetric neural rendering methods, such as neural radiance fields (NeRFs), have enabled photo-realistic novel view synthesis. However, in their standard form, NeRFs do not support the editing of objects, such as a human head, within a scene. In this work, we propose RigNeRF, a system that goes beyond just novel view synthesis and enables full control of head pose and facial expressions learned from a single portrait video. We model changes in head pose and facial expressions using a deformation field that is guided by a 3D morphable face model (3DMM). The 3DMM effectively acts as a prior for RigNeRF that learns to predict only residuals to the 3DMM deformations and allows us to render novel (rigid) poses and (non-rigid) expressions that were not present in the input sequence. Using only a smartphone-captured short video of a subject for training, we demonstrate the effectiveness of our method on free view synthesis of a portrait scene with explicit head pose and
+
+expression controls.
+
+# 1. Introduction
+
+Photo-realistic editing of human portraits is a longstanding topic in the computer graphics and computer vision community. It is desirable to be able to control certain attributes of a portrait, such as 3D viewpoint, lighting, head pose, and even facial expression, after capturing. It also has great potential in AR/VR applications where a 3D immersive experience is valuable. However, it is a challenging task: modeling and rendering a realistic human portrait with complete control over 3D viewpoint, facial expressions, and head pose in natural scenes remains elusive, despite the longtime interest and recently increased research.
+
+3D Morphable Face Models (3DMMs) [4] were among the earliest attempts towards a fully controllable 3D human
+
+head model. 3DMMs use a PCA-based linear subspace to control face shape, facial expressions, and appearance independently. A face model of desired properties can be rendered in any view using standard graphics-based rendering techniques such as rasterization or ray-tracing. However, directly rendering 3DMMs [4], which only models face region, is not ideal for photo-realistic applications as its lacks essential elements of the human head such as hair, skin details, and accessories such as glasses. Therefore, it is better employed as an intermediate 3D representation [21, 47, 48] due to its natural disentanglement of face attributes such as shape, texture, and expression, which makes 3DMMs an appealing representation for gaining control of face synthesis.
+
+On the other hand, recent advances on neural rendering and novel view synthesis [3,6,13,14,28,29,32,33,35,39,50, 52,53] have demonstrated impressive image-based rendering of complex scenes and objects. Despite that, existing works are unable to simultaneously generate high quality novel views of a given natural scene and control the objects within it, including that of the human face and its various attributes. In this work, we would like to introduce a system to model a fully controllable portrait scene: with camera view control, head pose control, as well as facial expression control.
+
+Control of head-pose and facial expressions can be enabled in NeRFs via a deformation module as done in [35, 36, 39]. However, since those deformations are learnt in a latent space, they cannot be explicitly controlled. A natural way to add control to head-pose and facial expressions, via deformations, is by parameterizing the deformation field using the 3DMM head-pose and facial expression space. However, as shown in Fig 7, such naive implementation of a deformation field leads to artefacts during the reanimation due to the loss of rigidity and incorrect modelling of facial expressions. To address these issues, we introduce RigNeRF, a method that leverages a 3DMM to generate a coarse deformation field which is then refined by corrective residual predicted by an MLP to account for the non-rigid dynamics, hair and accessories. Beyond giving us a controllable deformation field, the 3DMM acts as an inductive bias allowing our network to generalize to novel head poses and expressions that were not observed in the input video.
+
+Our model is designed to be trained on a short video captured using a mobile device. Once trained, RigNeRF allows for explicit control of head pose, facial expression and camera viewpoint. Our results capture rich details of the scene along with details of the human head such as the hair, beard, teeth and accessories. Videos reanimated using our method maintain high fidelity to both the driving morphable model in terms of facial expression and head-pose and the original captured scene and human head.
+
+In summary, our contributions in this paper are as follows: 1) We propose a neural radiance field capable of full control of the human head along with simultaneously modelling the
+
+full 3D scene it is in. 2) We experimentally demonstrate the loss of rigidity when dynamic neural radiance fields are reanimated. 3) We introduce a deformation prior that ensures rigidity of the human head during reanimation thus significantly improves its quality.
+
+# 2. Related works
+
+RigNeRF is a method for full control of head pose, facial expressions, and novel view synthesis of 3D portrait scene. It is closely related to recent work on neural rendering, novel view synthesis, 3D face modeling, and controllable face generation.
+
+Neural Scene Representations and Novel View Synthesis. RigNeRF is related to recent work in neural rendering and novel view synthesis [3,6,13,14,25,28,29,31-36,39,40,45, 49-53]. Neural Radiance Fields (NeRF) use a Multi-Layer Perceptron (MLP),  $F$ , to learn a volumetric representation of a scene. For every 3D point and the direction from which the point is being viewed,  $F$  predicts its color and volume density. For any given camera pose,  $F$  is first evaluated densely enough throughout the scene using hierarchical volume sampling [33], then volume rendering is used to render the final image.  $F$  is trained by minimizing the error between the predicted color of a pixel and its ground truth value. While NeRFs are able to generate photo-realistic images for novel view synthesis, it is only designed for a static scene and is unable to represent scene dynamics. Specifically designed for dynamic portrait video synthesis, our approach not only models the dynamics of human faces, but also allows specific controls on the facial animation.
+
+Dynamic Neural Scene Representations. Although NeRF [33] is designed for a static scene, several works have attempted to extend it to model dynamic objects or scene. There is a line of work [27, 28, 39, 50] that extend NeRF to dynamic scenes by providing as input a time component and along with it imposing temporal constraints either by using scene flow [28, 50] or by using a canonical frame [39]. Similarly, Nerfies [35] too work with dynamic scenes by mapping to a canonical frame, however it assumes that the movement is small. In [36], authors build upon [35] and use an ambient dimension in order to model topological changes in the deformation field. The deformation fields in these approaches are conditioned on learnt latent codes without specific physical or semantic meaning, and therefore not controllable in an intuitive manner. RigNeRF, similar to [35, 36], models the portrait video by mapping to a canonical frame but in addition also enables full parameterized control of head pose and facial expression.
+
+Controllable Face Generation. Recent breakthroughs in Generative Adversarial Networks(GANs) [15, 17-20, 55]
+
+have enabled high-quality image generation and manipulation. They also inspired a large collection of work [2, 7-9, 24, 38, 42, 43, 46, 47] focusing on face image manipulation and editing. However, majority of these work are intrinsically image-based and lack explicit 3D representation. Therefore it is challenging to enable high-quality view synthesis and 3D controls of the portraits such as large pose changes or extreme facial expressions. Another line of work [1, 10, 22, 23] made use of 3D Morphable Model as intermediate 3D face representation to reanimate face images/videos. While being able to model head poses with great detail, thanks to the disentangled representation in 3DMM, they often unable to perform novel view synthesis as they focus on face region but neglect the geometry or appearance of the scene. Similarly, NerFACE [13] uses neural radiance fields to model a 4D face avatar and allows pose/expression control on the head. However, they assume a static background and fixed camera, thus cannot perform view synthesis of the person or the scene. In contrast, our method RigNeRF provides full control over the head pose and facial expressions of the person captured in the portrait video while simultaneously being able to synthesize novel views of the 3D portrait scene.
+
+Hybrid Representations. The photorealism of volumetric and implicit representations have encouraged works that combine them with classical representations in order to improve reconstruction [5] or lend control over foreground [12, 30, 37]. In [30], authors learn a deformation field along with a texture mapping to reanimate human bodies. Similarly, [37] learns a 3D skinning field to accurately deform points according to the target pose. Neither [30] nor [37] model the full 3D scene. In contrast, RigNeRF models the whole 3D scene with full control of head-pose, facial expressions and viewing directions.
+
+# 3. RigNeRF
+
+In this section, we describe our method, RigNeRF, that enables novel view synthesis of 3D portrait scenes and arbitrary control of head pose and facial expressions. A Neural Radiance Field (NeRF) [33] with a per-point deformation is used to control the head pose and facial expressions of the subject. The deformation field deforms the rays of each frame to a canonical space, defined by a the 3DMM in the frontal head-pose and neutral expression, where the colors are sampled. In order to model deformations due to both head-pose (a rigid deformation) and facial expressions (a non-rigid deformation) and to correctly deform facial details such as hair and glasses, the deformation field is defined as the sum of the 3DMM deformation field and a residual deformation predicted by a deformation MLP.
+
+# 3.1. Deformable Neural Radiance Fields
+
+A neural radiance field (NeRF) is defined as a continuous function  $F: (\gamma_{m}(\mathbf{x}), \gamma_{n}(\mathbf{d})) \to (\mathbf{c}(\mathbf{x}, \mathbf{d}), \sigma(\mathbf{x}))$ , that, given the position of a point in the scene  $\mathbf{x}$  and the direction it is being viewed in,  $\mathbf{d}$ , outputs the color  $\mathbf{c} = (r, g, b)$  and the density  $\sigma$ .  $F$  is usually represented as a multi-layer perceptron (MLP) and  $\gamma_{m}: \mathbb{R}^{3} \to \mathbb{R}^{3 + 6m}$  is the positional encoding [33] defined as  $\gamma_{m}(\mathbf{x}) = (\mathbf{x}, \dots, \sin(2^{k}\mathbf{x}), \cos(2^{k}\mathbf{x}), \dots)$  where  $m$  is the total number of frequency bands and  $k \in \{0, \dots, m - 1\}$ . The expected color of the pixel through which a camera ray passes is calculated via volume rendering. The parameters of  $F$  are trained to minimize the L2 distance between the expected color and the ground-truth.
+
+NeRFs, as defined above, are designed for static scenes and offer no control over the objects within the scene. In order to model a dynamic scene, NeRFs are extended by additionally learning a deformation field to map each 3D point of the scene to a canonical space, where the volumetric rendering takes place [35, 36, 39]. The deformation field is also represented by an MLP  $D_{i}:\mathbf{x}\rightarrow \mathbf{x}_{\mathrm{can}}$  where  $D_{i}$  is defined as  $D(\mathbf{x},\omega_{i}) = \mathbf{x}_{\mathrm{can}}$  and  $\omega_{i}$  is a per-frame latent deformation code. In addition to a deformation code,  $\omega_{i}$ , a per-frame appearance code is also used [35, 36, 39],  $\phi_{i}$ , thus the final radiance field for the  $i$ -th frame is as follows:
+
+$$
+(\mathbf {c} (\mathbf {x}, \mathbf {d}), \sigma (\mathbf {x})) = F (\boldsymbol {\gamma} (D (\mathbf {x}, \omega_ {i})), \boldsymbol {\gamma} (\mathbf {d}), \phi_ {i}) \tag {1}
+$$
+
+In addition to the parameters of  $F$ , each  $\omega_{i}$  and  $\phi_{i}$  are also optimized through stochastic gradient descent. While the aforementioned modifications are able generate novel views [35, 39] of dynamic videos and handle small movement of objects in the scene [35], the deformations they estimate are conditioned on learnt deformation codes that can be arbitrary. Instead, we seek intuitive deformation controls that explicitly disentangles and controls facial appearance based on camera viewpoint, head pose and expression.
+
+# 3.2. A 3DMM-guided deformation field
+
+RigNeRF enables novel view synthesis of dynamic portrait scene and arbitrary control of head pose and facial expressions. For each frame  $i$ , we first extract its head-pose and expression parameters  $\{\beta_{i,\mathrm{exp}},\beta_{i,\mathrm{pose}}\}$  using DECA [11] and landmark fitting [16]. Next, we shoot rays through each pixel,  $p$ , of the frame and deform each point on the ray,  $\mathbf{x}$ , to a position in the canonical space,  $\mathbf{x}_{\mathrm{can}} = (x^{\prime},y^{\prime},z^{\prime})$  where its color is computed. A natural way to parameterize this canonical space, and any deviations from it, is using 3DMMs [4,26]. Thus, RigNeRF's canonical space is defined as the one where the head has zero head-pose and a neutral facial expression.
+
+Unfortunately, a 3DMM is only defined accurately for a subset of points on the head—3DMM fitting is often not perfect and they do not model hair, glasses, etc.—and is
+
+![](images/1acca458b1bcbd2b18e3551fc66ad5071cdb80d22194f7e4bae10ebd8b487a5d.jpg)  
+Figure 2. Overview of RigNeRF. RigNeRF is a deformable NeRF architecture that consists of two learnable MLPs: a deformation MLP  $D$  and a color MLP  $F$ . Given an image, we shoot rays through each of its pixels. For every ray, we deform each point on it according to a 3DMM-guided deformation field. This deformation field is the sum of the 3DMM deformation field (see Sect 3.2) and the residual predicted by the deformation MLP,  $D$ . Next, the deformed point is given as input to the color MLP,  $F$ , which additionally takes as input the pose and expression parameters  $\{\beta_{\mathrm{exp}}, \beta_{\mathrm{pose}}\}$ , the viewing direction  $\mathbf{d}$  and an appearance embedding  $\phi$  to predict the color and density. The final color of the pixel is calculated via volume rendering.
+
+undefined for point in the rest of 3D space. Hence, a deformation MLP  $D_{i}:\mathbf{x}\rightarrow \mathbf{x}_{\mathrm{can}}$  is still necessary to perform the transformation to the canonical space. However, as detailed in Sect 4.3, we find that directly predicting the deformation to the canonical space gives rise to artefacts during reanimation. The artefacts arise due to the inability of  $D$  to 1) maintain the rigidity of the head and 2) to model facial expressions correctly.
+
+![](images/2f501c7035111aa16fccbfade0a42a0a144ba1a4dbbf1db25ee0e5049e20b9a0.jpg)  
+Figure 3. The 3DMM deformation field. The 3DMM deformation field at any point in space,  $\mathbf{x}$ , is equal to the deformation of its closest neighbor on the mesh,  $\hat{\mathbf{x}}$ , weighted by the inverse of the exponential of the distance between  $\mathbf{x}$  and  $\hat{\mathbf{x}}$ .
+
+In order to fix this and ensure RigNeRF is able to handle both rigid deformations due to head-pose changes and non-rigid deformations due to changes in facial expressions, we use a deformation field prior derived using the 3DMM. For expression and head-pose parameters,  $\{\beta_{\mathrm{exp}},\beta_{\mathrm{pose}}\}$ , the value of the 3DMM deformation field at any point  $\mathbf{x} = (x,y,z)$  is:
+
+$$
+3 \mathrm {D M M D e f} (\mathbf {x}, \beta_ {\exp}, \beta_ {\text {p o s e}}) = \frac {3 \mathrm {D M M D e f} (\hat {\mathbf {x}} , \beta_ {\exp} , \beta_ {\text {p o s e}})}{\exp (\operatorname {D i s t T o M e s h} (\mathbf {x}))} \tag {2}
+$$
+
+where, 3DMMDef(x) is the value of the 3DMM deformation field,  $\hat{\mathbf{x}} = (\hat{x},\hat{y},\hat{z})$  is the closest point to  $(x,y,z)$  on the mesh and DistToMesh  $= ||\mathbf{x} - \hat{\mathbf{x}} ||$  is the distance between  $\mathbf{x}$  and  $\hat{\mathbf{x}}$ . The 3DMM deformation of any point on the mesh,  $\hat{\mathbf{x}}$ , is given by the difference between its position in the canonical space (i.e when the mesh had a zero head-pose and neutral facial expression) and its current articulation, as follows:
+
+$$
+3 \mathrm {D M M D e f} (\hat {\mathbf {x}}, \beta_ {\exp}, \beta_ {\text {p o s e}}) = \hat {\mathbf {x}} _ {\mathrm {F L A M E} (\mathbf {0}, \mathbf {0})} - \hat {\mathbf {x}} _ {\mathrm {F L A M E} \left(\beta_ {\exp}, \beta_ {\text {p o s e}}\right)} \tag {3}
+$$
+
+where,  $\hat{\mathbf{x}}_{\mathrm{FLAME}}(\mathbf{0},\mathbf{0})$  is the position of  $\mathbf{x}$  in the canonical space and  $\hat{\mathbf{x}}_{\mathrm{FLAME}}(\beta_{\mathrm{exp}},\beta_{\mathrm{pose}})$  is its position with head pose and facial expression parameters  $\{\beta_{\mathrm{exp}},\beta_{\mathrm{pose}}\}$ .
+
+The RigNeRF deformation field can now be defined as the sum of the 3DMM deformation field and the residual predicted by  $D$ , as follows
+
+$$
+\begin{array}{l} \hat {D} (\mathbf {x}) = 3 \mathrm {D M M D e f} (\mathbf {x}, \beta_ {\mathrm {i , e x p}}, \beta_ {\mathrm {i , p o s e}}) \\ + D \left(\boldsymbol {\gamma} _ {a} (\mathbf {x}), \boldsymbol {\gamma} _ {b} \left(3 \mathrm {D M M D e f} \left(\mathbf {x}, \beta_ {\mathrm {i , e x p}}, \beta_ {\mathrm {i , p o s e}}\right), \omega_ {i}\right) \quad (4) \right. \\ \end{array}
+$$
+
+$$
+\mathbf {x} _ {\text {c a n}} = \mathbf {x} + \hat {D} (\mathbf {x})
+$$
+
+where,  $\hat{D} (\mathbf{x})$  is the value of the RigNeRF deformation field at  $\mathbf{x}$ ,  $\{\gamma_{a},\gamma_{b}\}$  is the positional embedding on  $\mathbf{x}$  and 3DMMDef(x,...) respectively and  $\omega_{i}$  is the deformation embedding for current frame. We use  $\omega_{i}$  to model deformations that cannot be accounted for by head-pose and expression changes. Experimentally, we find that conditioning  $D$  directly on the expression and pose parameters,  $\{\beta_{\mathrm{i,exp}},\beta_{\mathrm{i,pose}}\}$ , leads to overfitting and poor generalization. This is likely due to the high dimensionality of the code (59), that makes it prone to overfitting. Instead, we condition  $D$  on the 3DMM
+
+deformation of the point  $\mathbf{x}$ , 3DMMDef $(\mathbf{x}, \beta_{\mathrm{i,exp}}, \beta_{\mathrm{i,pose}})$ . Since 3DMMDef $(\mathbf{x}, \beta_{\mathrm{i,exp}}, \beta_{\mathrm{i,pose}}) \in \mathbb{R}^3$ , it is itself relatively low dimensional, and it can be pushed into higher dimensions by adjusting the number of frequencies of its positional embedding,  $\gamma_b$ . We find that using  $b = 2$  frequencies in  $\gamma_b$  for the 3DMM deformation, 3DMMDef $(\mathbf{x}, \beta_{\mathrm{i,exp}}, \beta_{\mathrm{i,pose}})$ , works the best. In Fig 4, we show renders of both the output of  $D$  and the RigNeRF deformation field,  $\hat{D}$ , as described in Eq. (4). In Fig 4(c), we see that  $D$  generates the accurate deformation around the glasses, which the 3DMM deformation cannot do, for both head-poses. In Fig 4(d), we see that the  $\hat{D}$  is only concentrated on the head as it should be.
+
+![](images/5fbce531f14cfdc8c7f1a264e050e9f160d57b93f6aa67c00ad70c44e304f5b1.jpg)  
+Figure 4. Visualize learnt depth and deformation in RigNeRF. Here we show the depth, the magnitude of the output of the Deformation MLP  $D$ , and magnitude of the  $\hat{D}$  i.e the sum of the 3DMM Deformation and  $D$ . In (b), we can see that despite large changes in head-pose, the depth remains consistent. Next, in (c), we see  $D$  generates a deformation around the glasses for both poses so that it can be accurately deformed along with the head. Finally, in the last column we see how  $\hat{D}$  is only concentrated on the head.
+
+# 3.3. 3DMM-conditioned Appearance
+
+In order to accurately model expression and head-pose based textures, such as teeth, we condition  $F$  on both expression and head-pose parameters and on features extracted from the penultimate layer of the deformation MLP  $D(\gamma_{a}(\mathbf{x}),\ldots)$ . We find that using these features as input improves the overall quality of the render, please check the supplementary for details. Thus, once a point  $\mathbf{x}$  has been deformed to its location in the canonical space,  $\mathbf{x}_{\mathrm{can}}$ , using Eq. (4), its color is calculated as follows:
+
+$$
+\mathbf {c} (\mathbf {x}, \mathbf {d}), \sigma (\mathbf {x}) = F \left(\gamma_ {c} \left(\mathbf {x} _ {\text {c a n}}\right), \gamma_ {d} (\mathbf {d}), \phi_ {i}, D _ {F, i} \left(\mathbf {x} _ {\text {c a n}}\right), \beta_ {\mathrm {i , e x p}}, \beta_ {\mathrm {i , p o s e}}\right) \tag {5}
+$$
+
+where,  $\mathbf{d}$  is the viewing direction,  $\gamma_{c},\gamma_{d}$  is the positional embedding on  $\mathbf{x}_{\mathrm{can}}$  and  $\mathbf{d}$ , and  $D_{F,i}(\mathbf{x}_{\mathrm{can}})$  are features from the penultimate layer of the deformation MLP  $D(\pmb{\gamma}_a(\mathbf{x}),\dots)$ . The pixel color for  $p$  is then calculated using volume rendering and parameters of RigNeRF are minimized w.r.t to the ground truth color of  $p$ . The full architecture is shown in Fig 2.
+
+# 4. Results
+
+In this section, we show results of head-pose control, facial expression control, and novel view synthesis using RigNeRF. For each scene, the model is trained on a short portrait video captured using a consumer smartphone.
+
+Baseline approaches. To the best of our knowledge, RigNeRF is the first method that enables dynamic control of head-pose, facial expressions along with the ability to synthesize novel views of full portrait scenes. Thus, there is no existing work for an apple-to/apple comparison. We qualitatively and quantitatively compare our method to three other methods that perform closely related tasks: (1) HyperNeRF [36]: a state-of-the-art method using NeRFs for novel view synthesis of dynamic portrait scenes, without any control (2) NerFACE [12], a state-of-the-art method using NeRF for face dynamics control without modeling camera viewpoint and the entire scene, and (3) First Order Motion Model (FOMM) [44], a general-purpose image reanimation pipeline.
+
+When generating renanimation videos, RigNeRF, HyperNeRF [36] and NerFACE [12] require an appearance code for rendering; we use the appearance code of the first frame here. Similarly, RigNeRF and Nerfies [35] require a deformation code and we use the deformation code from the first frame. Full videos of the reanimation can be found in the supplementary material. We strongly urge the readers to refer to the videos to evaluate the quality of the results.
+
+Training Data Capture and Training details. The training and validation data was captured using an iPhone XR or iPhone 12 for all the experiments in the paper. In the first half of the capture, we ask the subject to enact a wide range of expressions and speech while trying to keep their head still as the camera is panned around them. In the next half, the camera is fixed at head-level and the subject is asked to rotate their head as they enact a wide range of expressions. Camera parameters are calculated using COLMAP [41]. We calculate the expression and shape parameters of each frame in the videos using DECA [11] and further optimize them using via standard landmark fitting using the landmarks predicted by [16] and camera parameters given by COLMAP [41]. All training videos are between 40-70 seconds long ( $\sim$  1200-2100 frames). Due to compute restrictions, the video is down-sampled and the models are trained at  $256 \times 256$  resolution. We use coarse-to-fine and vertex deformation regularization [35] to train the deformation network  $D(\mathbf{x}, \omega_i)$ . Please find full details of each experiment in the supplementary.
+
+# 4.1. Evaluation on Test Data
+
+We evaluate RigNeRF, HyperNeRF [36], NerFACE [12] and FOMM [44] on held out images on the captured video
+
+![](images/82e7f881b6be3b3431fec3d685c03bae06e8834c3e7cc396c32af26e88f69c87.jpg)  
+(a) Fixed view with varying expression and pose
+
+![](images/74739177844a431f52271b806910492d058a76ea1ba8fd16086f1d53061bfd04.jpg)  
+(b) Varying expression, pose and view  
+Figure 5. Qualitative comparison by reanimation with novel facial expression, head-pose, and camera view parameters. Here we reanimate RigNeRF, HyperNeRF [36] and NerFACE [12] using facial expression and head-pose derived from source images (top-row). We observe that while HyperNeRF [36] is able to generate realistic looking images of a portrait, it is unable to control head pose or facial expression in the result. On the other hand, NerFACE [12] attempts to render the correct pose and expression, but is unable to generate plausible face regions. Since, NerFACE [12] lacks an explicit deformation module it is unable to model deformation due to head-pose and facial expression changes. In contrast, our approach RigNeRF can effectively control the head pose, facial expression, and camera view, generating high quality facial appearance.
+
+<table><tr><td colspan="4">Subject 1</td><td colspan="3">Subject 2</td><td colspan="3">Subject 3</td><td colspan="3">Subject 4</td></tr><tr><td>Models</td><td>PSNR ↑</td><td>LPIPS ↓</td><td>FaceMSE ↓</td><td>PSNR ↑</td><td>LPIPS ↓</td><td>FaceMSE ↓</td><td>PSNR ↑</td><td>LPIPS ↓</td><td>FaceMSE ↓</td><td>PSNR ↑</td><td>LPIPS ↓</td><td>FaceMSE ↓</td></tr><tr><td>RigNeRF (Ours)</td><td>29.55</td><td>0.136</td><td>9.6e-5</td><td>29.36</td><td>0.102</td><td>1e-4</td><td>28.39</td><td>0.109</td><td>8e-5</td><td>27.0</td><td>0.092</td><td>2.3e-4</td></tr><tr><td>HyperNeRF [36]</td><td>24.58</td><td>0.22</td><td>8.14e-4</td><td>22.55</td><td>0.1546</td><td>9.48e-4</td><td>19.29</td><td>0.26</td><td>2.74e-3</td><td>21.19</td><td>0.182</td><td>1.58e-3</td></tr><tr><td>NerFACE [13]</td><td>24.2</td><td>0.217</td><td>7.84e-4</td><td>24.57</td><td>0.174</td><td>6.7e-4</td><td>28.00</td><td>0.1292</td><td>1.2e-4</td><td>28.47</td><td>0.134</td><td>2.7e-4</td></tr><tr><td>FOMM [44]</td><td>11.45</td><td>0.432</td><td>7.65e-3</td><td>12.7</td><td>0.582</td><td>6.31e-3</td><td>10.17</td><td>0.601</td><td>1.7e-2</td><td>11.17</td><td>0.529</td><td>6.8e-3</td></tr></table>
+
+Table 1. Quantitative results of Subject 1,2,3 and 4 on test data. Our results are better than HyperNeRF [36], NerFACE [12] and FOMM [44] on most metrics across all subjects.
+
+sequences. We use the camera view, pose and expression parameters of these images. Since RigNeRF and HyperNeRF [36] use a per-frame deformation  $\omega_{i}$ , we can't use the first frame (which is what we use as default for reanimation) to perform a direct comparison with the ground truth image as it may have a different deformation to the canonical space than the first frame. Therefore, we first optimize for the deformation code,  $\omega_{v}$ , of a given validation image by minimizing rendering error wrt that frame as follows:
+
+$$
+\omega_ {v} = \min  _ {\omega} | | C _ {p} (\omega ; \mathbf {x}, \mathbf {d}, \theta , \phi_ {0}, \beta_ {\mathrm {i}, \exp}, \beta_ {\mathrm {i}, \text {p o s e}}) - C _ {p} ^ {G T} | | \tag {6}
+$$
+
+where,  $C_p(\omega; \mathbf{x}, \mathbf{d}, \theta, \phi_0)$  is the predicted color at pixel  $p$  generated using Eq. (5) and volume-rendering,  $\phi_0$  is the appearance code of the first frame,  $\theta$  are the parameters of  $F$  as defined in Eq. (5) and  $C_p^{GT}$  is the ground-truth pixel
+
+value. Note, we only optimize  $\omega$ , all other parameters of the radiance field are kept fixed. We optimize Eq. (6) for 200 epochs which we observe to be more than enough to find the loss plateau. Once the optimization finishes, we report the final MSE, PSNR, LPIPS and Face MSE i.e the MSE only over the face region. We use no such optimization with NerFACE [12] since it does not have a deformation module and on FOMM [44] since it is a image-based method.
+
+As can be seen in Table 1, our method outperforms HyperNeRF [36], NerFACE [12] and FOMM [44] on held-out test images. RigNeRF, HyperNeRF [36] and NerFACE [12] are trained on dynamic portrait videos with changing headpose and facial expressions, HyperNeRF [36], lacking any head- pose and facial expression control, is unable to generate the head-poses and facial expressions seen in the held-out
+
+![](images/f0a5a6fd2f64f12ece49645afa9639c3c8b6e6c3ed9cadf74c9b279b86e62ab5.jpg)  
+(a) Image-driven Reanimation
+
+![](images/321d8731330059801cd7478ab3bffe1a35b5ba2004a3926ecb731bcfe04dd915.jpg)  
+(b) Novel-View Synthesis
+
+![](images/18b0f19c607bba8d39b217635506b8babb8966ef7644d94e63a445682daf3f55.jpg)
+
+![](images/2e22cda35e41d4f191010f94fbab42863348e84e6ceeba97328a301796fad35d.jpg)  
+(c) 3DMM-based control of 3D Portraits  
+Figure 6. Applications of RigNeRF. RigNeRF allows for full control head-pose, facial expressions and viewing direction of 3D portrait scenes. This enables application like (a) Image-driven Reanimation, (b) Novel View Synthesis and (c) 3DMM-based control of 3D Portraits. In (a)-top row, we show images of "driving sequences" from which we extract pose and expression parameters; the results from 4 subjects are shown in (a)-bottom where we synthesize realistic portrait frames that closely matches the driving pose and expression. We show in (b) a set of view synthesis results where we fix the head pose and facial expression, rendering results with varying camera positions, in which we show high-quality results with dramatic view changes. In (c), we demonstrate the application of controlling the portrait appearance with explicit 3DMM parameters. Within each row, (c)-column 1 and (c)-column 2 have the same pose but different expressions; (c)-column 2 and (c)-column 3 have the different poses but share the same expression. The inset shows the input 3DMM pose and expression, both of which are faithfully rendered in the corresponding results. Please find more results in the supplemental document and video.
+
+test set. Even as we optimized the deformation code of HyperNeRF [36], we found it hard to fit the unseen test images. NerFACE [12], on the other hand, lacking a deformation module, therefore is unable to model the dynamism of head-pose changes solely by concatenating pose and expression parameters as input the NeRF MLP. As a result, NerFACE [12] generates significant artefacts on the face regions (see the third row of Fig 5(a) and Fig 5(b)). FOMM [44], being an image based method, is unable to model novel views. Qualitative results of FOMM [44] can be found in the supplementary. In contrast to other methods, RigNeRF, thanks to the use of a 3DMM-guided deformation module, is able to model head-pose, facial expressions and the full 3D portrait scene with high fidelity, thus giving better reconstructions with sharp details.
+
+# 4.2. Reanimation with pose and expression control
+
+In this section we show results of reanimating a portrait video using both RigNeRF, HyperNeRF [36] and Ner-
+
+FACE [12] using expression and head-pose parameters as the driving parameters. Per-frame expression and head-pose parameters from the driving video are extracted using DECA [11] + Landmark fitting [16] and are given as input to RigNeRF in Eq. (4) and Eq. (5). Since HyperNeRF [36] does not take as input head-pose or expression parameters, it's forward pass remains unchanged.
+
+First, in Fig 5(a) we show the results of changing head- pose and expression using a driving video while keeping view constant. As one can see, RigNeRF captures the driving head- pose and facial expressions with high fidelity without compromising the reconstruction of the entire 3D scene. In contrast, we see that HyperNeRF [36] is unable to change facial expressions or head- pose due to the lack of controls, while NerFACE [12] generates significant artefacts on the face, especially when head- pose is changed. In Fig 5(b), we show the results of novel view synthesis along with changing head- pose and facial expressions. Again, we see that RigNeRF reanimates the subject with the accurate head- pose and
+
+![](images/eed3b41d3519fd929b46eaaeb5fd2863ce06425acc0101f5c19cf94806a15000.jpg)  
+Figure 7. A qualitative comparison between RigNeRF and HyperNeRF+E/P. Here we show a qualitative comparison between RigNeRF and HyperNeRF+E/P when reanimated using source images. We see that HyperNeRF+E/P generates many artefacts during reanimation due to its inability to model deformations correctly. In constraint, RigNeRF generates realistic reanimations with high fidelity to both the head-pose and facial expressions.
+
+facial expressions and is able to do so regardless of viewing direction and without compromising the reconstruction of the background 3D scene. Again, HyperNeRF is unable to change the head-pose and facial expressions due to the aforementioned reason while NerFACE [12] generates significant artefacts during reanimation.
+
+In Fig 6, we show more qualitative results of RigNeRF. We use three different applications of RigNeRF to demonstrate its flexibility and full controllability of a portrait scene. In Fig 6-(a), we show additional results of image-driven animation. The result frames (Fig 6-(a)-bottom) closely reproduce the head pose and facial expression shown in the driving frames (Fig 6-(a)-top). In Fig 6-(b), we show results of varying camera views while fixing (an arbitrary) facial expression and head pose. We demonstrate robust view synthesis performance with dramatic view changes. In Fig 6-(c), we show that RigNeRF can take user-specified 3DMM parameters as input to generate high-quality portrait images: each frame faithfully reproduces the face and expression provided by a set of 3DMM parameters shown in the inset.
+
+# 4.3. Comparison with HyperNeRF+E/P
+
+In this section we compare against HyperNeRF [36] with added pose and expression control, which we named HyperNeRF+E/P. The forward pass of this model is as follows
+
+$$
+\begin{array}{l} \mathbf {x} _ {\text {c a n}} = D \left(\boldsymbol {\gamma} _ {a} (\mathbf {x}), \beta_ {\mathrm {i}, \exp}, \beta_ {\mathrm {i}, \text {p o s e}}, \omega_ {i}\right) \\ \mathbf {w} = H \left(\boldsymbol {\gamma} _ {l} (\mathbf {x}), \omega_ {i}\right) \tag {7} \\ \mathbf {c} (\mathbf {x}, \mathbf {d}), \sigma (\mathbf {x}) = F (\boldsymbol {\gamma} _ {c} (\mathbf {x} _ {\text {c a n}}), \boldsymbol {\gamma} _ {d} (\mathbf {d}), \phi_ {i}, \mathbf {w}, \beta_ {i, \exp}, \beta_ {i, \text {p o s e}}) \\ \end{array}
+$$
+
+where,  $H$  is the ambient MLP and  $\mathbf{w}$  are the ambient coordinates [36]. In Table 2, we show a quantitative comparison between RigNeRF and HyperNeRF+E/P. We see that RigN-
+
+<table><tr><td colspan="4">Subject 1</td><td colspan="3">Subject 2</td></tr><tr><td>Models</td><td>PSNR ↑</td><td>LPIPS ↓</td><td>FaceMSE ↓</td><td>PSNR ↑</td><td>LPIPS ↓</td><td>FaceMSE ↓</td></tr><tr><td>RigNeRF (Ours)</td><td>29.55</td><td>0.136</td><td>9.6e-5</td><td>29.36</td><td>0.102</td><td>1e-4</td></tr><tr><td>HyperNeRF+Exp</td><td>31.3</td><td>0.161</td><td>1.3e-4</td><td>30</td><td>0.116</td><td>1.9e-4</td></tr></table>
+
+Table 2. Quantitative comparison between RigNeRF and HyperNeRF+E/P. We see that RigNeRF generates better reconstruction of the face with lower perceptual distance to the ground-truth as compared to HyperNeRF+E/P.
+
+eRF is able to generate better reconstructions of the face and generates images that are perceptually closer to the ground truth, as measured by LPIPS [54], than those generated by HyperNeRF+E/P. In Fig 7 we show a qualitative comparison when both methods are reanimated using source images, where the head-pose and expression significantly differ from the training set. As can be seen, while HyperNeRF+E/P is able to turn the head, it is unable to do so rigidly (see row 3, column 2 and row 3, column 4 of Fig 7). Further, it is also unable to model facial expressions accurately and generates artefacts on the face region. This further demonstrates the benefit of our 3DMM-guided deformation learning.
+
+# 5. Limitations and Conclusion
+
+Our method has certain limitations. First, it is subject specific and trains an individual model for each scene. Due to the need to capture sufficient expression and head-pose variations for training, our method currently requires a training sequences ranging from 40-70 seconds. Additionally, like all other NeRF-based methods, the quality of camera-registration affects the quality of the results. Being a method that allows photorealistic facial reanimation, RigNeRF may have potentially negative societal impact if misused. We discuss this further in the supplementary.
+
+In conclusion, we present RigNeRF, a volumetric neural rendering model for fully controllable human portraits. Once trained, it allows full control of head-pose, facial expression, and viewing direction. In order to ensure generalization to novel head-pose and facial expression we use 3DMM-guided deformation field. This deformation field allows us to effectively model and control both the rigid deformations caused by head-pose change and non-rigid deformations of changes in facial expressions. Training with a short portrait video, RigNeRF enables applications includes image-based face reanimation, portrait novel-view synthesis, and 3DMM-based control of 3D portraits.
+
+# 6. Acknowledgements
+
+We would like to thank the anonymous CVPR reviewers for taking the time to review and suggest improvements to the paper. ShahRukh Athar is supported by a gift from Adobe, Partner University Fund 4DVision Project, and the SUNY2020 Infrastructure Transportation Security Center.
+
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+# Robust and Accurate Superquadric Recovery: a Probabilistic Approach
+
+Weixiao Liu $^{1,2}$  Yuwei Wu $^{1}$  Sipu Ruan $^{1}$  Gregory S. Chirikjian $^{1*}$ $^{1}$ National University of Singapore  $^{2}$ Johns Hopkins University  
+{mpewxl, yw.wu, ruansp, mpegre}@nus.edu.sg
+
+# Abstract
+
+Interpreting objects with basic geometric primitives has long been studied in computer vision. Among geometric primitives, superquadrics are well known for their ability to represent a wide range of shapes with few parameters. However, as the first and foremost step, recovering superquadrics accurately and robustly from 3D data still remains challenging. The existing methods are subject to local optima and sensitive to noise and outliers in real-world scenarios, resulting in frequent failure in capturing geometric shapes. In this paper, we propose the first probabilistic method to recover superquadrics from point clouds. Our method builds a Gaussian-uniform mixture model (GUM) on the parametric surface of a superquadric, which explicitly models the generation of outliers and noise. The superquadric recovery is formulated as a Maximum Likelihood Estimation (MLE) problem. We propose an algorithm, Expectation, Maximization, and Switching (EMS), to solve this problem, where: (1) outliers are predicted from the posterior perspective; (2) the superquadric parameter is optimized by the trust-region reflective algorithm; and (3) local optima are avoided by globally searching and switching among parameters encoding similar superquadrics. We show that our method can be extended to the multi-superquadrics recovery for complex objects. The proposed method outperforms the state-of-the-art in terms of accuracy, efficiency, and robustness on both synthetic and real-world datasets. The code is at http://github.com/bmlklwx/EMS-superquadric_fitting.git.
+
+# 1. Introduction
+
+Understanding 3D environments (and the objects therein) has been a crucial task for computer vision and intelligent systems [24]. With the recent breakthroughs in 3D vision and the advancement of computing power, modern computer vision systems are able to reconstruct and reason about scenes with low-level representations, such as point
+
+![](images/83ff591b3b303596a77de59f871745bde469aa643dce81196a39d7b20c06ee71.jpg)  
+Figure 1. (a) Shape vocabulary of convex superquadrics. We focus on the interpretation of point clouds with convex superquadrics. (b) Superquadric representations inferred by our method.
+
+![](images/73cedcca019f9c9058bc45a77c95567c121bee2bd5265cc2b7c1d91fb9eae485.jpg)
+
+clouds [11, 25, 26], meshes [18] and voxels [9, 38]. In contrast, the human visual system favors abstracting scenes into canonical parts for a better perceptual understanding [3, 15]. Human vision seems to work well with basic geometric structures of the scene, and not rely on detailed point-wise models [24]. Therefore, understanding the geometric structure of visual inputs appears to be a promising approach for intelligent systems to achieve high-level goals such as physical reasoning, decision making, planning and interaction with the environment. Inspired by this idea, researchers turned to exploring the possibility of using basic volumetric primitives for object description, e.g., cuboids [20, 33, 39]. However, due to the limited expressiveness of cuboids, objects can only be describe in a highly abstracted way.
+
+Superquadrics are a family of geometric primitives with a rich shape vocabulary, including cuboids, cylinders, ellipsoids, octahedra and their intermediates (Fig. 1), but only encoded by 5 parameters. In the recent few years, superquadrics have raised considerable attention in the community and are widely applied in robotics and computer vision tasks, e.g., object modeling [5, 22, 23], collision detection and motion planning [28, 29], pose estimation [4], and grasping [27, 36, 37]. A single superquadric is already expressive enough to reasonably model many everyday objects [14, 30]. The pioneering work on single superquadric recovery includes [1, 12, 32], where the recovery is formulated as a Least-Square (LSQ) problem. In [6, 16], the methods are extended to model complex objects with multiple superquadrics. However, the existing methods are vulnerable
+
+ble to noise and outliers, and the inherent deficiency of the local optimizer makes them sensitive to initialization.
+
+In this paper, we focus on the single superquadric recovery problem and propose the first probabilistic method to recover superquadrics from noisy point clouds. We build a probabilistic model to simulate how observations (point cloud) are possibly generated from a given parameterized superquadric surface. GUM is adopted to accommodate the generation of noise and outliers. The superquadric recovery is thus formulated as a MLE problem (Sec. 3.1). Expectation and Maximization (EM) algorithm is most commonly used to solve MLE problems with latent variables [10]. However, the EM algorithm typically converge to a local optimum, resulting in an inaccurate shape approximation. To solve this problem, we propose a novel algorithm, Expectation, Maximization and Switch (EMS), which takes advantage of the geometric features of superquadrics to avoid local optima (Sec. 3.2). In the E-step, the probability of a point being an outlier is inferred from a posterior perspective (Sec. 3.3). In the M-step, the parameter of the superquadric is updated given the current estimation of the latent variables in the E-step (Sec. 3.4). The E and M-step alternate until converged to a local optimum, where we introduce the S-step: globally search for candidate parameters encoding similar superquadrics in terms of shape and pose, and then switch to the one from which we can further increase the likelihood (Sec. 3.5). This strategy is made possible by investigating the geometric symmetry and algebraic ambiguity of superquadrics. Furthermore, we show our probabilistic formulation can be extended to multi-superquadrics recovery tasks (Sec. 3.6).
+
+The motivations and key features of this paper come from the two following questions. (1) Can we design an algorithm robust, accurate and efficient enough to abstract an optimal superquadric primitive from a noisy point cloud whose underlying shape is indeed within the vocabulary of superquadrics? This question is answered in Sec. 4.1 and Sec. 4.2. (2) If the point cloud is too complex to be approximate by a single superquadric, can the algorithm figure out a major part which can be approximated by a superquadric? This is a challenging goal which cannot be achieved by any existing single superquadric recovery algorithm. We demonstrate qualitatively that the proposed method is capable of doing this in Sec. 4.3.
+
+# 2. Related work
+
+# 2.1. Preliminary: Superquadrics
+
+Superquadrics are a family of geometric primitives (Fig. 1), which can be defined by the implicit function [2]:
+
+$$
+F (\mathbf {x}) \doteq \left(\left(\frac {x}{a _ {x}}\right) ^ {\frac {2}{\varepsilon_ {2}}} + \left(\frac {y}{a _ {y}}\right) ^ {\frac {2}{\varepsilon_ {2}}}\right) ^ {\frac {\varepsilon_ {2}}{\varepsilon_ {1}}} + \left(\frac {z}{a _ {z}}\right) ^ {\frac {2}{\varepsilon_ {1}}} \tag {1}
+$$
+
+where  $\mathbf{x} \doteq [x, y, z]^T \in \mathbb{R}^3$  is a point defined in the superquadric frame;  $a_x$ ,  $a_y$  and  $a_z \in \mathbb{R}_{>0}$  are the scale parameters corresponding to the x, y, z-axis;  $\epsilon_1$  and  $\epsilon_2 \in \mathbb{R}_{\geq 0}$  are the shape parameters.  $\epsilon_1$  governs the shape along the z-axis and  $\epsilon_2$  controls the shape orthogonal to the z-axis. A superquadric is convex, if both  $\epsilon_1$  and  $\epsilon_2$  are within (0,2]. Eq. (1) is also called the 'inside-outside' function, since  $\mathbf{x}$  locates on the surface if  $F(\mathbf{x}) = 1$ , inside if  $F(\mathbf{x}) < 1$  and outside otherwise. We are able to fully parameterize a superquadric of a general pose with  $\theta \doteq \{\epsilon_1, \epsilon_2, a_x, a_y, a_z, g\}$ , where  $g = [\mathbf{R} \in SO(3), \mathbf{t} \in \mathbb{R}^3] \in SE(3)$  is the Euclidean transformation.
+
+# 2.2. Superquadric Recovery
+
+The single superquadric recovery plays an important role in all superquadric recovery problems. In the previous works [1, 12, 13, 32, 35], it is formulated as a LSQ problem and solved with the Levenberg-Marquardt (LM) algorithm [17]. Recently, researchers have also explored the possibility of deep learning approaches [21, 31]. Solina et al. [1,32] propose a cost function based on the implicit function of superquadrics. Gross et al. [12] modifies the cost function with the radial distance between points and the superquadric surface. In [13, 34], the robustness of the algorithms is investigated, and outliers are randomly rejected by a customized heuristic function. All the above methods suffer from numerical instability when either  $\epsilon_{1}$  or  $\epsilon_{2}$  approaches 0. As a consequence, they compromise by constraining the lower bounds of  $\epsilon_{1}$  and  $\epsilon_{2}$  to 0.1, resulting in less accuracy when modeling shapes with sharp edges, e.g., cuboids and cylinders. Recently, this problem is revisited and solved by approximating the implicit function with an auxiliary function in the unstable region [35]. Also in [35], the authors point out that the methods are sensitive to the initialization, and suggest to try 3 different initial guesses obtained by performing the Principal Component Analysis (PCA) to the point cloud. However, all the methods are still subject to local optima, especially when the point cloud is partially sampled on the object surface.
+
+For the multi-superquadrics recovery, Leonardis [16] Borges [5], and Chevalier et al. [6] try to interpret complex objects with multiple superquadrics by first segmenting the point cloud into parts and fitting a superquadric to each of them. In contrast, our multi-superquadrics recovery extension works in a hierarchical way. Other than a multi-superquadrics representation, we can also obtain the hierarchical relationships among the superquadrics (Fig. 3).
+
+Another line of the multi-superquadrics recovery is based on deep learning [22, 23], where superquadrics can be obtained from meshes or even an RGB image. However, their approaches are data-driven with limitations in generalization, while we focus on the recovery of unseen real-world objects based on a case by case probabilistic reasoning.
+
+# 3. Our Method
+
+# 3.1. Probabilistic Formulation
+
+In this section, we demonstrate how to formulate the superquadric recovery as a MLE problem. Our model is built as follows. First, on the superquadric surface  $S_{\theta} \subset \mathbb{R}^3$  parameterized by  $\theta$ , a Gaussian centroid  $\mu \in S_{\theta}$  is sampled randomly according to the uniform density function
+
+$$
+p (\boldsymbol {\mu}) = \frac {1}{A _ {\boldsymbol {\theta}}}, \quad A _ {\boldsymbol {\theta}} = \int_ {S _ {\boldsymbol {\theta}}} 1 d S \tag {2}
+$$
+
+where  $A_{\theta}$  is the area of the superquadric surface. Subsequently, an observation  $\mathbf{x} \in \mathbb{R}^3$  is generated from a Gaussian-uniform model (GUM), the probability density function of which is given as
+
+$$
+p (\mathbf {x} \mid \boldsymbol {\mu}) = w _ {o} p _ {o} (\mathbf {x}) + (1 - w _ {o}) \mathcal {N} (\mathbf {x} \mid \boldsymbol {\mu}, \boldsymbol {\Sigma}) \tag {3}
+$$
+
+where  $\mathcal{N}(\cdot |\boldsymbol {\mu},\boldsymbol {\Sigma})$  represents the density function of a Gaussian distribution with mean  $\pmb{\mu}$  and covariance  $\pmb{\Sigma}$ . We assume the noise to be isotropic, i.e.,  $\pmb {\Sigma} = \sigma^2\mathbf{I}$ , where  $\mathbf{I}\in \mathbb{R}_{3\times 3}$  is the identity matrix. A uniform outlier component is introduced to model the generation of outliers.  $w_{o}\in [0,1]$  is the probability of a point being sampled from the outlier component. To make Eq. (3) a proper probability density function (integrates to 1 over  $\mathbb{R}^3$ ), we introduce a working space  $\mathbb{V}$  encapsulating the point set, whose volume equals  $V$ . The probability density of the outlier component  $p_o(\mathbf{x}) = \frac{1}{V}$  if  $\mathbf{x}\in \mathbb{V}$  and  $p_o(\mathbf{x}) = 0$  otherwise.
+
+For the convenience of the upcoming derivation, we reformulate Eq. (3) into an equivalent form by introducing a latent random variable  $z \in \{0,1\}$  encoding the membership of  $\mathbf{x}$ . When  $z = 0$ ,  $\mathbf{x}$  is sampled from the uniform outlier component. On the contrary,  $z = 1$  indicates that  $\mathbf{x}$  is generated from the Gaussian inlier component. We assume the membership  $z$  is independent of the mean  $\mu$  of the Gaussian component. Therefore, Eq. (3) is equivalent to
+
+$$
+\begin{array}{l} p (\mathbf {x} | \boldsymbol {\mu}, z) = p _ {o} (\mathbf {x}) ^ {1 - z} \cdot \mathcal {N} (\mathbf {x} | \boldsymbol {\mu}, \boldsymbol {\Sigma}) ^ {z} \tag {4} \\ z \sim p (z) = \mathrm {B e r n o u l l i} (1 - w _ {o}) \\ \end{array}
+$$
+
+Given a set of points  $\mathbf{X} = \{\mathbf{x}_i\in \mathbb{R}^3|i = 1,2,\dots,N\}$ , the parameters of the superquadric surface can be estimated by maximizing the following likelihood function
+
+$$
+L (\boldsymbol {\theta}, \sigma^ {2}) = \prod_ {i = 1} ^ {N} p \left(\mathbf {x} _ {i} \mid \boldsymbol {\mu} _ {i}, z _ {i}\right) p \left(\boldsymbol {\mu} _ {i}\right) p \left(z _ {i}\right) \tag {5}
+$$
+
+or equivalently, minimizing the negative log-likelihood function
+
+$$
+l \left(\boldsymbol {\theta}, \sigma^ {2}\right) = \sum_ {i = 1} ^ {N} z _ {i} \left(\frac {\left\| \mathbf {x} _ {i} - \boldsymbol {\mu} _ {i} \right\| _ {2} ^ {2}}{2 \sigma^ {2}} - \log c\right) + N \log \left(A _ {\boldsymbol {\theta}}\right) \tag {6}
+$$
+
+where  $c$  is the normalizing constant of the Gaussian distribution defined in Eq. (3). Note that the terms independent of  $\theta$  and  $\sigma^2$  have been omitted for simplicity.
+
+# 3.2. Expectation, Maximization and Switching
+
+Eq. (6) is intractable to solve directly because of the existence of a set of continuous latent variables  $\pmb{\mu}_i\in S_\theta ,i = 1,2,\dots,N$  and discrete latent variables  $z_{i}\in \{0,1\} ,i = 1,2,\dots,N$  . The EM algorithm [10] is most commonly used to solve MLE with latent variables. However, in our scenario, it suffers from multiple local optima and has difficulties in calculating the expectation of the latent variables in an exact form. Therefore, we propose a novel algorithm called EMS, which is adapted from the EM algorithm by utilizing the geometric properties of superquadrics. The algorithm first estimates the latent variables from a posterior perspective given the current estimation of the superquadric parameter (E-step). Next, the parameters are inferred and updated by minimizing the negative log-likelihood function (M-step). These two steps alternate until converged to a local minimum. Then, we globally search for candidate parameters which encode superquadrics geometrically similar to the current estimation but have the potential of further decreasing the negative log-likelihood function (S-step). If a valid candidate is found, we reinitialize the EM-phase with this candidate; else, we declare a termination and return the final result. Detailed motivations and derivations of each step are discussed in the following sections.
+
+# 3.3. E-step: Latent Variables Estimation
+
+Given the current superquadric parameter  $\theta$ , we estimate  $\pmb{\mu}_i$  via maximum a posteriori (MAP)
+
+$$
+\hat {\boldsymbol {\mu}} _ {i} = \underset {\boldsymbol {\mu} _ {i} \in \mathcal {S} _ {\boldsymbol {\theta}}} {\arg \min } p (\boldsymbol {\mu} _ {i} | \mathbf {x} _ {i}) \tag {7}
+$$
+
+which guarantees to minimize Eq. (6). By the Bayes' rule, we show that it is equivalent to find the nearest point on the superquadric surface to  $\mathbf{x}_i$
+
+$$
+\hat {\boldsymbol {\mu}} _ {i} = \underset {\boldsymbol {\mu} _ {i} \in \mathcal {S} _ {\boldsymbol {\theta}}} {\arg \min } \| \mathbf {x} _ {i} - \boldsymbol {\mu} _ {i} \| _ {2} \tag {8}
+$$
+
+As far as we know, there is no analytical solution to Eq. (8), and numerical optimization is prohibitively time consuming for a large number of observations. Therefore, we approximate  $\hat{\mu}_i$  with a sub-optimal solution
+
+$$
+\hat {\boldsymbol {\mu}} _ {i} ^ {s} = \mathbf {x} _ {i} - \left(1 - F ^ {- \frac {\epsilon_ {1}}{2}} \left(g ^ {- 1} \circ \mathbf {x} _ {i}\right)\right) \left(g ^ {- 1} \circ \mathbf {x} _ {i}\right) \tag {9}
+$$
+
+which is the intersection of the vector  $\mathbf{x}_i$  (defined in the superquadric frame) and the superquadric surface  $S_{\theta}$ .  $F(\cdot)$  is the implicit function of the superquadric. This idea was first introduced by [12] to establish the radial distance measure for superquadrics fitting.
+
+Given the current estimation of  $\pmb{\mu}_i$ , the expectation of the posterior probability of  $\mathbf{x}_i$  being an inlier is inferred via the Bayes' rule
+
+$$
+E \left(z _ {i} = 1 \mid \mathbf {x} _ {i}, \hat {\boldsymbol {\mu}} _ {i} ^ {s}\right) = \frac {\mathcal {N} \left(\mathbf {x} _ {i} \mid \hat {\boldsymbol {\mu}} _ {i} ^ {s} , \sigma^ {2} \mathbf {I}\right)}{\mathcal {N} \left(\mathbf {x} _ {i} \mid \hat {\boldsymbol {\mu}} _ {i} ^ {s} , \sigma^ {2} \mathbf {I}\right) + \frac {w _ {o} p _ {o}}{1 - w _ {o}}} \tag {10}
+$$
+
+For simplicity, we denote the value of the expectation as  $\hat{z}_i$ .
+
+# 3.4. M-step: Parameter Optimization
+
+Substituting  $\pmb{\mu}_i$  and  $z_{i}$  in Eq. (6) with the their posterior estimations  $\hat{\pmb{\mu}}_i^s$  and  $\hat{z}_i$ , we obtain an updated negative log-likelihood function
+
+$$
+\hat {l} (\boldsymbol {\theta}, \sigma^ {2}) = \sum_ {i = 1} ^ {N} \hat {z} _ {i} \left(\frac {\| \mathbf {x} _ {i} - \hat {\boldsymbol {\mu}} _ {i} ^ {s} \| _ {2} ^ {2}}{2 \sigma^ {2}} - \log c\right) + N \log (A _ {\boldsymbol {\theta}}) \tag {11}
+$$
+
+To the best of our knowledge,  $A_{\theta}$  cannot be expressed in closed-form with respect to  $\theta$ . For efficiency,  $A_{\theta}$  is approximated by a bi-linear interpolation along  $\epsilon_{1}$  and  $\epsilon_{2}$ . Details are presented in the Supplementary Material.  $N\log (A_{\theta})$  indicates a preference for a small superquadric. To avoid local optima, it is preferable to delay the introduction of this term until  $\sigma^2$  falls below a threshold, i.e., the shape is roughly captured. We apply the trust-region-reflective [8] to optimize  $\theta$ . The main reason we do not use the LM solver lies in its numerical instability when approaching the lower bound of the shape parameters. In contrast, the reflective transformation [7], empirically, always keeps the optimization numerically stable. More discussion can be found in the Supplementary Material. After  $\theta$  is solved,  $\sigma^2$  is updated by setting the corresponding partial derivative of Eq. (11) to zero.
+
+# 3.5. S-step: Geometric Local Optimum Avoidance
+
+Superquadrics hold many geometric and algebraic features, e.g., symmetry and parametric ambiguity. Trivial cases can be raised where two superquadrics possess a similar or even identical shape and pose but are distant in the parameter space. The likelihood function Eq. (11) quantifies how well an underlying geometric primitive fits the observations, and thus two similar (or identical) superquadrics must possess similar (or identical) likelihoods as well. Inspired by this idea, we propose a geometry-guided local optima avoidance strategy: globally search for candidate parameters encoding similar shapes, and switch to the one which has the potential of further increasing the likelihood. The switch action is discontinuous in the parameter space, while the change in terms of the underlying geometric shape is almost smooth. We investigate the conditions when two parameters distant to each other represent similar superquadrics, and summarize the cases into two categories. The idea of switch and examples of similarities are visualized in Fig. 2.
+
+![](images/a190f6ef1b78eeb153851aee22d6cc8acae2814f07c7aac42f39fed1704c56e9.jpg)  
+Figure 2. Visualization of the S-step and similarities. The EM steps change the shape and pose of the superquadric towards a local optimum, while the S-steps tries to preserve the current shape but jump between distant parameters to avoid the local optimum.
+
+Axis-mismatch similarity: According to the definition in Sec. 2.1, the x and y-axis are of an equivalent class in terms for interchangeability, i.e., if we interchange the value of  $a_{x}$  and  $a_{y}$ , we can obtain an identical superquadric by rotating 90 degrees about the z-axis. However, the z-axis distinguishes itself because it is generally not interchangeable with them. We thus name the z-axis as the principal axis. Now let us consider a special case when the shape parameters  $\epsilon_{1} = \epsilon_{2}$ . In this scenario, the implicit function (Eq. (1)) degenerates and makes all the three axes equivalent. From another perspective, the 'degree' of interchangeability decrease as the difference between  $\epsilon_{1}$  and  $\epsilon_{2}$  increases. Therefore, when  $\epsilon_{1}$  and  $\epsilon_{2}$  are close to each other, a similar superquadric can be obtained by reassigning the principal axis to either the x or y-axis and then applying a corresponding rotation. We call this axis-mismatch, because it helps to avoid the local optimum caused by the incorrect assignment of the principal axis, which prohibit the shape from further varying to the global optimum.
+
+Duality similarity: This case is derived from the shape ambiguity regarding  $\epsilon_{2}$ . When  $a_{x} = a_{y}$  and  $\epsilon_{2} = 0$ , we can always represent the same superquadric (in terms of shape and pose) by scaling  $a_{x}$  and  $a_{y}$  with  $1 / \sqrt{2}$ , setting  $\epsilon_{2} = 2$ , and rotating 45 degrees about the principal axis. This is because  $\epsilon_{2} = 0$  and  $\epsilon_{2} = 2$  define a pair of squares dual to each other in the subspace orthogonal to the principal axis. Utilizing this property, when  $a_{x} \approx a_{y}$ , a similar superquadric can be constructed by setting  $\epsilon_{2} = 2 - \epsilon_{2}$ , rescaling and rotating. The duality similarity is of great importance to local optimum avoidance, since it connects two extreme ends of the parameter space. Local optimizer tends to converge monotonically towards one of the ends, while the duality similarities glue the ends together, allowing the algorithm to check and explore the counterpart.
+
+Similar superquadrics can also be resulted by a combination of the two similarities. Detailed mathematical formulation about the candidates generation and switching strategy
+
+can be found in the Supplementary Material.
+
+# 3.6. Extension: Multi-superquadrics Recovery
+
+Although our method is designed for single superquadric recovery, it can be easily extended to understand the shape and structure of a complex object with multiple superquadrics. One of the key features of our single superquadric recovery method is that it can capture the major superquadric-like part of a point cloud and identify the unfitted points as 'outliers'. Therefore, from a point cloud  $\mathbf{X}$  of a complex object, we can first recover a superquadric capturing the major superquadric structure along with a set of unfitted 'outlier' clusters  $\mathcal{C}$ . Then, we feed  $\mathcal{C}$  back to the algorithm as the input to recover a new set of superquadrics along with an updated  $\mathcal{C}$ . The process (Algorithm 1) is repeated hierarchically until  $\mathcal{C}$  is empty or a maximum depth of layer is reached. In this way, not only the point cloud is represented with the union of the acquired superquadrics, but also the hierarchical relationship between the superquadrics is obtained, as illustrated in Fig. 3.
+
+Algorithm 1 A layer of hierarchical superquadric recovery  
+1: Input:  $\{\pmb{X}_1, \pmb{X}_2, \dots, \pmb{X}_n\}$   
+2: Output:  $\{\pmb{\theta}_1, \pmb{\theta}_2, \dots, \pmb{\theta}_n\}, \mathcal{C} = \{\bar{X}_1, \bar{X}_2, \dots, \bar{X}_m\}$   
+3:  $\mathcal{C} \gets \{\}$   
+4: for  $i = 1, \dots, n$  do  
+5:  $(\pmb{\theta}_i, \text{outliers}) \gets EMS(\pmb{X}_i)$   
+6:  $\{\bar{X}_{1_i}, \bar{X}_{2_i}, \dots, \bar{X}_{\gamma_i}\} \gets \text{Clustering(outliers)}$   
+7:  $\triangleright$  clustering the outliers by Euclidean distance [19]  
+8:  $\mathcal{C} \gets \mathcal{C} \cup \{\bar{X}_{1_i}, \bar{X}_{2_i}, \dots, \bar{X}_{\gamma_i}\}$   
+9:  $\triangleright$  clusters with points less than a threshold are pruned  
+10: end for
+
+![](images/7d4678ea4e67edbe4ae91cf2b87ea2eb583896990f9991453a59548ce259cc0b.jpg)
+
+![](images/f6a9975fd0fd63eba4eac2de18fa2a252069f221b4000a349afdee8367fedc7c.jpg)
+
+![](images/c9e725dd6871579c05796c08a40f3c9d2660c09cecac9fa0fd525c37111a10ee.jpg)
+
+![](images/97871976783002be2ccf4c4fbf91b34ee6608c6866284f0281b190eb30be68db.jpg)
+
+![](images/2305e0298a07185d04c65a0bcf23d2258ffa5b412de40f47dd139d836d0ee0ec.jpg)  
+Figure 3. Hierarchical process of the multi-superquadrics recovery. We demonstrate with a Seal from the KIT ObjectModels. (a) The original point cloud. (b-d) Steps of the hierarchical process and the generation of the structure graph. (e) The recovered superquadric model. (f) Segmentation of the original point cloud based on the superquadric model.
+
+![](images/8dd5bc43acddafc50c7be02e431a793fb9661b22372db08aec34903603610d76.jpg)
+
+# 4. Experiments
+
+We conduct experiments on both synthetic and real-world point cloud datasets to verify the accuracy, efficiency and robustness of our method. All the algorithms and experiments are implemented in MATLAB, on a computer running Intel Core i9-9900K(3.6GHz). In all the single superquadric recovery experiments, we compare with 4 baseline methods: Implicit-LSQ [32], Radial-LSQ [12], Robust-fitting [13] and Numerical Stable method [35].
+
+Initialization: Following [35], if not specified, the baseline methods are initialized with 3 different superquadric parameters obtained via PCA to alleviate the affect of local optima. To show the effectiveness of the proposed local optimum avoidance strategy, our method is only initialized once with the first one among the three. Implementation details can be found in the Supplementary Material.
+
+# 4.1. Recovery on Synthetic Datasets
+
+In this section, we evaluate the performance of the algorithms on recovering the underlying geometric primitives given points sampled on a superquadric. Three experiments are designed to evaluate the robustness to partial data, outliers, and noise, respectively. To cover the whole parameter space of superquadrics in convex region, we randomly generate 500 superquadrics under the following strategy: the shape parameter  $[\epsilon_1,\epsilon_2]$  is uniformly sampled on  $(0,2]^2\subset \mathbb{R}^2$ ; the scale parameter  $[a_x,a_y,a_z]$  is uniformly sampled on  $[0.5,3]^3\subset \mathbb{R}^3$ ; the rotation  $\mathbf{R}\in SO(3)$  is generated by first sampling a rotation axis on the unit sphere and then applying a random rotation in  $[0,2\pi ]$  about the axis; the translation  $\mathbf{t}$  is uniformly sampled on  $[-1,1]^3\subset \mathbb{R}^3$ . Point clouds are generated by sampling evenly on the surface of the random superquadrics with an equal interval of 0.2. In [35], a recursive algorithm is proposed to realize this task. However, the points sampled by the algorithm get denser when approaching the poles along the z-axis. Therefore, we further improve their algorithm by adaptively re-scaling the sampling intervals at different latitudes to achieve an overall equal-distance sampling (see Supplementary Material).
+
+The most natural metric to evaluate the goodness of fitting is the average point-to-surface orthogonal distance. However, given an arbitrary point, there is no analytical solution of the corresponding closest point on a superquadric surface. Therefore, we approximate the point-to-surface metric with
+
+$$
+e r r o r (\mathbf {X}, \mathbf {S}) \doteq \frac {1}{N} \sum_ {i = 1} ^ {N} \min  _ {\mathbf {s} _ {j} \in \mathbf {S}} \| \mathbf {x} _ {i} - \mathbf {s} _ {j} \| _ {2} \tag {12}
+$$
+
+where  $\mathbf{S} = \{\mathbf{s}_j\in S_\theta |j = 1,2,\dots,M\}$  is a set of points
+
+densely and evenly sampled (with an interval of 0.02 in the synthetic experiments) on the superquadric surface  $S_{\theta}$ .
+
+![](images/f0c99154d6340b0b2cdaeb411cdd32f771178d16b6c145291b2b775fecb12bac.jpg)
+
+![](images/f8f922d9c33a8f73965b557b37aaedfc9c606c8273c6bdfc388a7923a2f65708.jpg)
+
+![](images/b6df4e2dc1cb0a6e200561b2f2708c6cc4a0af1a8fad52cd1521185b9397b9fe.jpg)
+
+![](images/588f242524dd6cef9dac0c997bb788baa7049af34b2acdf66f68603cd84d5d34.jpg)
+
+<table><tr><td colspan="4">(e) Partial ratio</td><td colspan="2">Partial ratio</td></tr><tr><td>Partial Ratio</td><td>1.0</td><td>0.8</td><td>0.6</td><td>0.4</td><td>0.2</td></tr><tr><td>Radial (once)</td><td>33.2%</td><td>28.6%</td><td>30.8%</td><td>29.4%</td><td>34.4%</td></tr><tr><td>Radial (3 times)</td><td>99.0%</td><td>78.0%</td><td>70.4%</td><td>68.8%</td><td>65.6%</td></tr><tr><td>EMS</td><td>100%</td><td>100%</td><td>99.8%</td><td>97.4%</td><td>86.4%</td></tr></table>
+
+![](images/32ca762e1d96f56bdc3a74860c3959b95e96747a6fd240648f24b4f1dccfa183.jpg)  
+(f)  
+Numerical Stable  
+Figure 4. Results of recovery from partial data. (a) Average error (the baselines are initialized 3 times). (b) Average error (the baselines are initialized once). (c) Ablation study. (d) Average runtime. (e) Global Optimality rate (declared when the fitting error is less than 0.01) (f) Examples at partial ratio 0.4, when the baselines get stuck in local optima, while ours have managed to jump out.
+
+![](images/169cc03c6db3a8f9e215653b8455514a36013820e75ae88c0927b20855259bb7.jpg)  
+Radial-LSQ
+
+![](images/ea991d0eaf46acc1dc22feea2214a8da2a1ca9529b3779e83f0c525584920a9c.jpg)  
+EMS (Ours)
+
+![](images/419ede9b2db7cf072da6bd95305af969ea742238fe71a7c3df079c76c5889771.jpg)  
+Ground Truth
+
+Recovery on partial data: This experiment simulates the cases where points are sampled from a partial area of a superquadric in the sense that sometimes we can only obtain a partial view of an object. For each point cloud of the 500 superquadrics, a point is randomly selected and a certain percentage (partial ratio) of the points which are the closest to the random point are kept. We test on 5 different levels of partial ratio ranging from 1.0 (complete point cloud) to 0.2, resulting in totally 2500 point clouds. Results are shown in Fig. 4. Our method significantly outperforms the baseline methods at all levels of partial ratios. All the methods perform well when a complete point cloud of a superquadric is available. However, with the decreasing of the partial ratio, it becomes harder to make a reasonable initial guess of the parameter. Therefore, the baseline methods become more likely to get stuck at a local optimum, even with multiple trials. While, our method continuously avoids the local optima with the geometry-guided switching strategy, and thus maintains a high success rate in converging to the global optimum. We also perform ablation studies about the S-step. The results verify its effectiveness (Fig. 4(c)). Although our algorithm only guarantees local optimality, the
+
+empirical results indicate that our method is very likely to achieve the global optimum in practice (Fig. 4(e)).
+
+Robustness to outliers: We add different levels of Gaussian outliers to the 2500 point clouds. Outlier ratio (the number of outliers / the number of inliers) is used to quantify the level of outliers added. Results are shown in Fig. 5. Our method shows superior robustness to outliers and is able to recover the underlying superquadrics even from severely corrupted point clouds. The Implicit-LSQ, RadialLSQ and Numerical Stable are vulnerable to outliers, since they are not designed to tolerate outliers. The Robust-fitting performs better, however, is limited by its heuristic outlier rejection policy, which also imposes an undesirable negative effect on clean point clouds. In contrast, our method infers outliers in a posterior probabilistic approach and thus maintains high robustness among all levels of outliers.
+
+![](images/fef1e3e4d31bc09e2a1d7c3ff15d1f0432e792e6777b00b07503f2f863ad474a.jpg)  
+(c)
+
+![](images/f497d4d063a57eec986187f4ff1c76ee53f131c5816aa4988751b69de4b84207.jpg)  
+Implicit-LSQ
+
+![](images/9a85b505bb0a3b712f6a73d9ed5b313d7cf239e423e620c34ca0624d917b1d63.jpg)
+
+![](images/2ec0091bf1af7ab35f4808d995fc743c069597c19d579b0d03cdb6ea8b68ed7a.jpg)  
+Numerical Stable
+
+![](images/5003a064d9139deac0d83ac0955cf815df6f40037def5ed50b2556b9bf349b64.jpg)  
+Radial-LSQ
+
+![](images/72db99defbfb93427f307facaf600001598e43e4f58b046d112e4cc6dfd78184.jpg)  
+Robust-fitting  
+Figure 5. Outlier results. (a) Average error at different outlier ratios. (b) Average runtime. Runtime of Radial, Implicit and NS has no significant meaning since they cannot recover the superquadric correctly in most cases. (c) Examples at outlier ratio 0.4.
+
+![](images/1c64f2192247bd9adecb3e78975fb672951e8ec080912097f61b1104c0df2dc0.jpg)  
+EMS (Ours)
+
+![](images/3f71a1ec11717be281d126a417fef9956eb25fd6d1c2aca1e77d7f250d5a828f.jpg)  
+Ground Truth
+
+Robustness to noise: In this experiment, we corrupt each point in the point clouds with Gaussian noise of different variance. Results are shown in Fig. 6. It can be observed that the fitting error grows with the increase of noise level for all the methods, including ours. This is because, unlike outliers, the noise inherently corrupts the geometric information. But still, our method shows advantages in robustness and efficiency when dealing with noisy point clouds.
+
+# 4.2. Recovery on Real-world Datasets
+
+In this section, we evaluate the single superquadric recovery on two public datasets: the KIT ObjectModel [14] and BigBIRD [30]. Both of the datasets consist of various daily household objects, and are widely used as benchmarks
+
+![](images/2862e31bed0fc04bfaf9e98cbb060a79707d70f97d07abdc2e7ec3886779eeca.jpg)
+
+![](images/986f279550d043ba01d59407d4cfa66c1b39959bf27f6f1f3d6e888a6b63a154.jpg)
+
+![](images/717092e35a0174e129c693fb34fea7106a8f8d093e573e289f1147c7980e23e2.jpg)  
+Figure 6. Noise results (a) Average errors at different noise levels (evaluated on uncorrupted point clouds). (b) Average runtime. (c) Examples at  $\sigma^2 = 0.01$  Gaussian noise level.
+
+![](images/875f60cf5a01da52ff24757f04a18c9c5eb799c454001d3e28fc93549adee635.jpg)
+
+![](images/eab6faaa55f120c25d4eb2e1606fbf6c8982b4212581a998b509427f19dcd046.jpg)
+
+in 3D-object reconstruction, recognition, and robot grasping. KIT is scanned by a high-accuracy laser scanner (Minolta VI-900), while BigBIRD is captured by a commercial-grade RGB-D camera (PrimeSense Carmine 1.08).
+
+KIT ObjectModels: We follow the setting in [35]. The dataset includes 145 objects, among which [35] selects 105 items that can be reasonably approximated by a single superquadric. However, we decide to shrink our scope to 97 items, since some of the objects (e.g., bottles) would be better recovered with multiple superquadrics. Therefore, we defer the discussion of those cases to Sec. 4.3. We down-sample the point clouds with voxelized grids to about 1500 points. Quantitative results are shown in Fig. 7(c). Our method outperforms all the baseline methods in accuracy and efficiency. Our method also demonstrates some interesting features which can not be shown quantitatively. As shown in Fig. 7(a), some objects have minor local deformations and/or parts deviating from the main structure, which affect the major shape capturing. Our probabilistic method is able to reduce the impact of those parts (inferred as outliers) and achieves an overall better fitting. Also in Fig. 7(b), the box-like objects are usually filled with content inside, resulting in a shape slightly extended in the middle. This shape is within the vocabulary of superquadrics, however, is so close to cuboids that the other algorithms fail in telling them apart, i.e., get stuck at a local optimum. In contrast, our method is able to switch and check among similar shapes and recover the optimal superquadric.
+
+BigBIRD: Compared with laser scanners, low-cost RGB-D cameras are inferior in measurement accuracy, resulting in noisy point clouds with outliers. Therefore, the BigBIRD is an ideal dataset for evaluating the robustness of an algorithm in daily operation. The dataset consists of 125 object instances, and we test on 91 of them which can be reasonably approximated with single superquadrics.
+
+![](images/605ea94243902ea503382df4b28cff98b29bf3192e8e014ac9ebf9fb62b1447e.jpg)
+
+![](images/8137026b58ce2837d7582db793ba8a8fe415963b22482e46af4246318c521d8b.jpg)
+
+![](images/d081bb7c5687181734ff7a9b95ec6e8ce908009fcb9051078bf2deee64041e09.jpg)
+
+![](images/9ec2f5fdd2067359f712d731ab7ee368e81bf89062e6ce539b1575fddc939aad.jpg)
+
+(c)  
+
+<table><tr><td>implicit</td><td>NS</td></tr><tr><td>0.647</td><td>0.638</td></tr><tr><td>249.7</td><td>256.0</td></tr></table>
+
+<table><tr><td>Robust</td><td>EMS (Ours)</td></tr><tr><td>0.748</td><td>0.584</td></tr><tr><td>1065.0</td><td>91.1</td></tr></table>
+
+![](images/310c2aa409e1085d47920a17e0a00da85eeb8325322ad96cf519f441e59b6d13.jpg)  
+Figure 7. KIT results. (a) and (b) are the two cross-section views of fitting results between Radial-LSQ and the proposed method. (c) Quantitative results of average error (Eq. (12) with sampling interval as dense as  $0.1\mathrm{mm}$ ) and runtime.  
+(a) Implicit
+
+![](images/2828da05a4212efca4b4523d534242117b836e5a3e92d4f417d430b4ccd4620d.jpg)  
+NS
+
+![](images/7dc2cd07a748dca4cf1286c735becc2b5a87780d70b53b328e5a759c6b62168f.jpg)  
+Radial
+
+![](images/2da65d77f9e5fab57429dd0105153c82acea960e8280856e7ae65638b6aa4648.jpg)  
+Robust
+
+![](images/507c869d4200a7e9876849d7cd0aa33090178cf8a8fabf9b4e20dabc0be67e2c.jpg)  
+EMS (Ours)
+
+![](images/0686174a8885fa716237b621a14760ee17f74001202c3666bd737d47804a5e7d.jpg)
+
+![](images/68fbfb76e71a65f250bbfd8fae6d76fc4a7b0c014f9a939f2cdc1afd2c1b42b9.jpg)  
+Figure 8. BigBIRD results. (a) A heatmap showing the local fitting error between the recovered superquadric and the original point cloud. (b) Quantitative results of the average error (Eq. (12) with sampling interval as dense as  $0.1\mathrm{mm}$ ) and runtime.
+
+<table><tr><td>Method</td><td>Error (mm)</td><td>Time (ms)</td></tr><tr><td>Radial</td><td>0.821</td><td>175.0</td></tr><tr><td>Implicit</td><td>0.826</td><td>139.0</td></tr><tr><td>NS</td><td>0.847</td><td>133.9</td></tr><tr><td>Robust</td><td>0.807</td><td>834.9</td></tr><tr><td>EMS (Ours)</td><td>0.753</td><td>78.9</td></tr></table>
+
+(b)
+
+The point clouds are downsampled with voxelized grids to around 1000 points. Results are shown in Fig. 8. Among the baselines, the Robust-fitting shows better robustness but takes a much longer time to converge. Our method can achieve the best accuracy with the least time consumption.
+
+# 4.3. Multi-superquadrics Recovery
+
+In this section, we show the qualitative results of our extension to the multi-superquadrics recovery (Sec. 3.6). We use the complex objects which cannot be properly approximated by a single superquadric from the KIT Object-Models. We downsample the point clouds to around 5000 points, so as to keep enough details of the complex shapes. We set the maximum hierarchical layers to 3 and the prin
+
+![](images/85525fea747ba3ea4ef9d5b1980ea7a1c82690076b083ab91cb27bdb2f44913d.jpg)  
+(a)
+
+![](images/20dba65b7d1c687cb9f0d5c130a3acb3f6874784aa122354951e39709c8ecaa5.jpg)
+
+![](images/cc9ea7bf05427a6270a1dc4136ceac30a77411aae04487b2f059b2682eca6c16.jpg)
+
+![](images/462a9ca123eee5e18df868b6f523881947d56ee8347bbcaaa9e96ca879d25b13.jpg)
+
+![](images/be5931c9833e6adb1c04d50bbe1ad22280aac40368a8d100839ef2ca7aab78c5.jpg)
+
+![](images/2ff2f7cd38bf2b180b2ea7190693d1cabcce10fa5db3ee5305ae89d404b206dc.jpg)
+
+![](images/ca8a381022e5158f23c7b3a1458b2546ee9c262ad4e383f6c2b2e53dc1a43ad6.jpg)
+
+![](images/9734adfa3592f8777406df3acc6b7153f1e508d2e341937c311aa81866ca8249.jpg)
+
+![](images/dae23470eb2df0c7c8f32acab4d19d540efd7ccd44fa9a1c472752ca9791ca46.jpg)  
+(b)
+
+![](images/867cb644ade016d92e1342c03fdbf7b37912fd3c7156305520f214538fb69d0b.jpg)
+
+![](images/d0ddc198d3aa2fc774037db3b293ec6bd7e3bd20e7b14bc7ce7860caa4a02c16.jpg)
+
+![](images/b2c2dfdefc89a2fe2f5e3501046136662e50b382c40783d24543c60394d94a72.jpg)
+
+![](images/f249f4b8b2cfbe11e82c803a452a45006ced495803c9b4b6da422feff1e6f3ef.jpg)
+
+![](images/3b07b842c3b01d90219dbd51476a55ae388430c04e868dc7bb1bab7e2b9e97e4.jpg)
+
+![](images/38faed1ebb5adc22bc773f65bcd6a98578dfba522eba598d1b5232eb974122d2.jpg)
+
+![](images/35eda26e31ef2d9d845522c3f0c8cad186e0b900accf0b21196a00d1073503a2.jpg)
+
+![](images/54b8a023e3e4ae2b26939c249e76930aa5cfae95efd446139d5e41ee6e575a80.jpg)  
+(c)
+
+![](images/2800ce224f106e3d66581e784d4e6d94465d314491a5fd8223258ca355fbd094.jpg)  
+Figure 9. Examples of multi-superquadrics recovery. (a) The original point clouds. (b) The recovered multi-superquadrics representations with the proposed method. (c) Segmentation of the original point clouds based on the multi-superquadrics representations.
+
+![](images/07b83526323072a212fe46f4c329a928a024e42655c871ab726861a515ca713f.jpg)
+
+![](images/5c3611bd2ce3e1a80ea4ca63abd047119be02c25910e9f63a7dda2a16d985789.jpg)
+
+![](images/60b85f2beecd4441ad1dee543659e51eb02ea7eafd9c36a79702024a2d0e22a0.jpg)
+
+![](images/06b89b52bdd336ef4ebf6aa79dba44e6efa43e9725b3d8df48467db7c56f794f.jpg)
+
+![](images/5b4824f6e3b4a9e61d22aca74b9f4cf4f0c42e9637f04ae0f758d71718429b16.jpg)
+
+![](images/85a34da13bed27bd27feae6178e0af5c82f1d5fa0d8a9c58cc8860136501644f.jpg)
+
+ing threshold to 60 points. As shown in Fig. 10, the baseline single superquadric method fails in capturing the major superquadric-like part of a bottle as a cylinder. Our method succeeds in doing this and identifies the points around the cap of the bottle as outliers, from which the bottle is recovered hierarchically. Other results on more complex objects are shown in Fig. 9. Our method is able to represent the objects vividly with superquadrics. Some detailed parts such as ears and mouths of animals can also be recovered.
+
+![](images/cdef18899fef5d53d98422ca2a51f05ed1f5e06902bdbfb7a7b4f6e61befb006.jpg)  
+(b)
+
+![](images/2c89a2111687f0a89fd799eee1a023f9f38da4cf49a6006778b93f5704a29f6a.jpg)
+
+![](images/3eea63266ab1949d7c89469cbc4cc34b79b825a68f6e3a26cc7b7f62f4ac0cb0.jpg)  
+Figure 10. Qualitative multi-superquadrics recovery on bottles from KIT. (a) Comparison of single superquadric recovery between Radial-LSQ and EMS (ours). The proposed method is able to recover the major superquadric shape (a cylinder) from the point cloud. (b) Other results of the proposed method.
+
+![](images/3274db0b8b077649fb77eaee8c1fdffb210ef122539beccfe76677a70f597c9f.jpg)
+
+![](images/5b236f98ecee401339988f1f0c2435280754866e10142336be3593f696f00b3c.jpg)
+
+![](images/a35e76fec422d4ec5cc834ab7fc6b4742b624f8c0361dde6a160cea448a6f4bd.jpg)
+
+# 5. Discussion & Limitations
+
+Various experiments show that our method is robust, accurate and efficient. As shown in Sec. 4.1 and Sec. 4.2, the baseline methods are sensitive to initialization, and are easily trapped into local optima. Therefore, to obtain a satisfying result, they usually require multiple trials starting from different initial guesses. In comparison, our method is able to avoid local optima on the fly and achieve a better recovery accuracy with one shot. Also, our method is robust to outliers and noise. This feature is brought in by our probabilistic formulation of the problem, where we explicitly model
+
+the possible sources of outliers and noise with a uniform distribution and a Gaussian distribution, respectively. This feature also forms the basis for our multi-superquadrics extension. As shown in Sec. 4.3, our method is able to capture the major superquadric-like part from a point cloud and hierarchically generate subordinate superquadrics from the 'outliers' identified at the upper level.
+
+However, there is a limitation in our multi-superquadrics extension: our approach requires the object being recovered possesses an inherent hierarchical geometric structure. For more general cases, a preliminary point cloud segmentation is preferred [25, 26]. Another limitation is due to the expressiveness of superquadrics. It is hard to well capture the shape of a thin-walled non-convex object such as a cup. In future works, we plan to adapt deformations and primitive subtractions to extend the vocabulary of superquadrics.
+
+# 6. Conclusion
+
+We present the first probabilistic method to recover superquadrics from point clouds. We formulate the recovering process as an MLE problem and propose a novel geometry-guided approach, EMS, to infer an optimal superquadric. The proposed method outperforms the state-of-the-art in terms of accuracy, robustness, and efficiency, which is substantiated by extensive experiments on both synthetic and real-world datasets. We also extend our method to represent a complex object with a hierarchy of superquadrics. Such representation can reveal the geometric and hierarchical structure of an object, which, we believe, has great potential in facilitating tasks such as object classification, segmentation and grasping.
+
+Acknowledgements This research is supported by the National Research Foundation, Singapore, under its Medium Sized Centre Programme - Centre for Advanced Robotics Technology Innovation (CARTIN) R-261-521-002-592, and JHU internal funds.
+
+# References
+
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+# Robust Combination of Distributed Gradients Under Adversarial Perturbations
+
+Kwang In Kim UNIST
+
+# Abstract
+
+We consider distributed (gradient descent-based) learning scenarios where the server combines the gradients of learning objectives gathered from local clients. As individual data collection and learning environments can vary, some clients could transfer erroneous gradients e.g. due to adversarial data or gradient perturbations. Further, for data privacy and security, the identities of such affected clients are often unknown to the server. In such cases, naively aggregating the resulting gradients can mislead the learning process. We propose a new server-side learning algorithm that robustly combines gradients. Our algorithm embeds the local gradients into the manifold of normalized gradients and refines their combinations via simulating a diffusion process therein. The resulting algorithm is instantiated as a computationally simple and efficient weighted gradient averaging algorithm. In the experiments with five classification and three regression benchmark datasets, our algorithm demonstrated significant performance improvements over existing robust gradient combination algorithms as well as the baseline uniform gradient averaging algorithm.
+
+# 1. Introduction
+
+The success of deep learning relies on the abundance of training data and computational capability to process such data. When computational resources are limited at centralized servers, distributed learning can be employed, where data is distributed across multiple clients and part of the learning process is performed within each client. To enhance data privacy and security, and reduce the required communication cost, often in such environments, details of the localized data are hidden to the server and other clients, and the server aggregates only encapsulated information communicated from clients.
+
+We consider distributed gradient descent-based optimization as an instance of such learning problems: The main objective is to minimize a differentiable energy function presented as the sum of the client energies defined based on the respective local data (e.g. the average of squared errors or cross-entry losses over clients' datasets) [23, 27, 41]. Then each client iteratively calculates and transfers the gradient of
+
+the local energy with respect to the learner parameters while the server maintains and updates these parameters using the aggregated local gradients.
+
+The nature of individual clients and local data therein can vary significantly. For instance, a client could be a data center securing professionally annotated data while another client might be an edge device containing data labeled by inexperienced annotators. In this case, some clients could exhibit abnormal behavior e.g. due to incorrect labeling or adversarial attacks, and the corresponding local gradients can be unreliable. This can distract the gradient aggregation process at the server leading to suboptimal learning.
+
+Robust distributed learning with such affected clients has only recently come to recognition and existing approaches are limited in that they require a separate training set at the server (to train a detector of affected clients) [34], information on the number of affected clients [32], or access to local training data [12] (see Sec. 2 for details).
+
+We present a new approach that combines local gradients when some clients undergo adversarial perturbation. Our algorithm takes the naive uniform average of gradients as a noisy observation of an underlying ground-truth combination, and iteratively improves this during the gradient descent steps of the learner parameters. Specifically, our algorithm represents the combined gradient as a convex combination of all local gradients. The corresponding combination weights are uniformly initialized. As the optimization proceeds, the genuine (unaffected) local gradients are automatically identified by adjusting the combination weights via simulating the diffusion of the combined gradient. Our approach does not require a separate training set at the server, and it offers the advantage of refining the gradient combination without having to know the number and identities of affected clients as well as the nature of perturbations therein.
+
+In the experiments with five classification and three regression benchmark datasets, and two different scenarios of client perturbations, our approach demonstrated significant performance improvements over the standard uniform gradient averaging algorithm and existing robust gradient combination approaches.
+
+# 2. Related work
+
+Privacy-enhancing and communication-efficient distributed learning. Since the introduction of the federated averaging (FedAvg) algorithm [23], federated learning has been established as a standard problem and method for communication-efficient and cross-device distributed learning [17]. The main challenges in this scenario include minimizing the communication cost (e.g. via gradient compression [21] and quantization [19]), optimizing the topology of the underlying communication networks [43], data protection and encapsulation, personalization [7], and theoretical convergence guarantee [20, 35]. Recent work also focuses on robustness against non-independent and identically distributed (non-i.i.d.) data generation processes [24] and the extensions of FedAvg to e.g. multi-task learning [27] and meta-learning [18].
+
+Robust distributed learning. In privacy and security-enhancing distributed learning, the server has only access to the gradients or updated learner parameters provided by the individual clients, and therefore, it is susceptible to corrupted updates [8, 39].
+
+Learning approaches that are robust against such updates have only been recently introduced. Sun et al.'s gradient clipping approach assumes that the gradients generated from the affected clients tend to have large norms and it excludes the gradients whose norms are larger than a given threshold  $T$  in the combination [29]. This algorithm is tailored to attacks that focus on specific sub-tasks in multi-task environments, and it is not directly applicable to perturbations that we consider in this paper. Furthermore, in general adversarial perturbation scenarios, the threshold  $T$  might not be known in advance. Hu et al.'s gradient normalization approach builds on a similar idea [15]: Assuming that the affected gradients exhibit larger norms, they normalized the gradients to suppress their contributions. In our preliminary experiments (with random labeling and random gradient perturbations; see Sec. 4) we observed that the perturbed gradients do not necessarily exhibit larger norms than genuine gradients, and in such cases, gradient normalization or clipping are not directly applicable.
+
+In Han and Zhang's robust federated learning algorithm, the clients take the role of (partially corrupted) teachers while the server acts as a student [12]. Each teacher then provides a few known genuine data instances, verified by domain experts, to guide the teaching process. This approach requires access to local data of individual clients and known identities of genuine data instances therein, and therefore its application is limited when the server is agnostic to the client internals.
+
+The local gradients can be considered as abstract high-dimensional vectors and in this respect, combining perturbed local gradients can be seen as the problem of robustly estimating their mean (see Eq. 3). In general, when observed data are noisy, potentially including outliers, the central statistical tendency can be better captured by their medians as robust es
+
+timators than the means. Pillutla et al.'s gradient aggregation algorithm uses geometric median as another robust alternative to the mean [25]. In the experiments, we demonstrate that our approach outperforms the algorithms that replace the gradient means with their medians or geometric medians [25].
+
+Turan et al.'s gradient combination algorithm provides an alternative approach to exploit the median estimates [32]: For each optimization step, the median of the local gradients are calculated. Then, the genuine gradients are selected as  $K$ -nearest neighbors of the median. While this algorithm demonstrated significant improvements over federated averaging, it requires determining  $K$  which might vary across the problems and be difficult to estimate. In the experiments, we show that this algorithm achieves robustness when the numbers of affected clients are limited and when  $K$  is known explicitly. However, when more clients are affected, the median estimates were distracted by the perturbed gradients and the performance degraded accordingly. Our algorithm demonstrated improvements over [32] even without having to tune  $K$ , especially when large numbers of clients are affected. Other robust mean alternatives can be found e.g. in gradient filtering via low-rank signal recovery [28] and trimmed mean [38, 42].
+
+Some existing approaches build upon additional assumptions or trade data privacy and security with robustness: For example, Xie et al.'s Zeno algorithm detects affected clients based on the accuracy of each local model assessed based on labeled datasets at the server [40]. Alistarh et al.'s Byzantine stochastic gradient descent approach provides a theoretical guarantee on the near-optimality of the final solution, but it requires the convexity of training objectives [1]. Wan and Chen's robust federated learning algorithm detects the genuine gradients via training a separate learner that estimates the probability of each local gradient being genuine [34]. While this approach achieved a significant performance gain over uniform gradient averaging, its application domain is limited as it requires a separate training set at the server.
+
+# 3. Robust gradient combination against adversarial perturbations
+
+Problem formulation. Our goal is to learn a function  $f$  with domain  $\mathcal{X}$  and range  $\mathcal{Y}$ . We will focus on deep neural network learners  $f$  parameterized by weight vectors  $\mathbf{w}$  while our methods can be applied to other types of learners that can be trained using gradient descent techniques. For classification, we will use one-hot encoding where  $\mathcal{Y} \subset \mathbb{R}^c$  with  $c$  being the number of classes. For regression,  $\mathcal{Y} \subset \mathbb{R}$ . In standard supervised learning, one is provided with a single training set  $Z = \{(\mathbf{x}_1,\mathbf{y}_1),\dots,(\mathbf{x}_N,\mathbf{y}_N)\} \subset \mathcal{X} \times \mathcal{Y}$  and the prediction function  $f$  is constructed by minimizing the sum of losses:
+
+$$
+\mathcal {E} (\mathbf {w}; Z) = \sum_ {j = 1} ^ {N} l (f (\mathbf {x} _ {j}), \mathbf {y} _ {j}).
+$$
+
+For classification, we will employ the standard cross-entropy loss:  $l(\mathbf{a},\mathbf{b}) = -\sum_{j=1}^{c}[\mathbf{a}]_j \log ([\mathbf{b}]_j)$  with  $[\mathbf{a}]_j$  being the  $j$ -th element of the vector  $\mathbf{a}$  while for regression, the squared error will be used:  $l(\mathbf{a},\mathbf{b}) = (\mathbf{a} - \mathbf{b})^2$ . Applying the standard gradient descent, one obtains the final learner parameter (weight vector)  $\mathbf{w}^*$  by iteratively updating  $\mathbf{w}$
+
+$$
+\mathbf {w} (t + 1) = \mathbf {w} (t) - \eta \nabla_ {\mathbf {w}} \mathcal {E} (\mathbf {w}; Z) \tag {1}
+$$
+
+with a step size  $\eta > 0$ .
+
+In distributed learning, the training set  $Z$  is distributed across  $K$  different clients  $\{C_k\}_{k=1}^K$  where  $C_k$  is provided with a subset  $Z_k$  of  $Z$ , and it calculates the update of  $w$  based only on  $Z_k$ . For instance, McMahan et al.'s federated averaging (FedAvg) framework [23] replaces the centralized update in Eq. 1 with an aggregation of local gradients:
+
+$$
+\mathbf {w} (t + 1) = \mathbf {w} (t) - \eta \sum_ {k = 1} ^ {K} \mathbf {g} _ {k} (t), \tag {2}
+$$
+
+where  $\mathbf{g}_k(t)$  is the gradient  $\nabla_{\mathbf{w}}\mathcal{E}(\mathbf{w}(t);Z_k)$  calculated by the client  $C_k$  based on the learner parameter  $\mathbf{w}(t)$ . For stochastic gradient descent, only a small portion (called mini-batch) of  $Z_k$  contributes to the gradient evaluation  $\nabla_{\mathbf{w}}\mathcal{E}$ : All gradient combination algorithms considered in this work can be straightforwardly applied to this case. At iteration  $t$ , a server  $S$  distributes the previous learner parameter  $\mathbf{w}(t)$ , collects the resulting local gradients  $\{\mathbf{g}_k\}_{k=1}^K$ , and performs the parameter update in Eq. 2. To enhance security and privacy, the local datasets  $\{Z_k\}$  are not shared with the server or other clients. Therefore, the only information that is communicated from  $C_k$  to  $S$  is the gradient  $\mathbf{g}_k(t)$ .
+
+When each client provides a genuine gradient of  $\mathcal{E}$ , the FedAvg update  $\mathbf{w}(t + 1)$  in Eq. 2 is equivalent to the centralized update (Eq. 1) and they will lead to identical solutions. However, when some clients undergo adversarial perturbation and accordingly transfer erroneous gradients, the final solution of FedAvg process can be inferior to the centralized counterpart. The nature of the perturbations (e.g. the labels  $\{\mathbf{y}_j\}_{j = 1}^{|Z_k|}$  in  $Z_{k}$  are randomly changed or the  $k$  th client gradient  $\mathbf{g}_k(t)$  is replaced by a random vector; see Sec. 4) and the number and identities of the affected clients are not known to the server where the parameter update (Eq. 2) is performed.
+
+Robust estimation of gradient update using diffusion on the gradient manifold. We employ a modified parameter
+
+update rule:
+
+$$
+\mathbf {w} (t + 1) = \mathbf {w} (t) - \eta K \mathbf {q} (t), \tag {3}
+$$
+
+$$
+\mathbf {q} (t) = \sum_ {k = 1} ^ {K} [ \mathbf {a} (t) ] _ {k} \mathbf {g} _ {k} (t), \tag {4}
+$$
+
+where  $\mathbf{a}(t)$ $(\| \mathbf{a}(t)\| _1 = 1$  and  $[\mathbf{a}(t)]_k\geq 0$  for  $1\le k\le K$  regulates the contributions of the local gradients  $\{\mathbf{g}_k(t)\}$ . When  $\mathbf{a}(t)$  is uniform (i.e.  $[\mathbf{a}(t)]_k = 1 / K$ ), Eq. 3 becomes the standard FedAvg (Eq. 2) while general non-uniform  $\mathbf{a}(t)$  can instantiate different gradient combination strategies. Our algorithm automatically determines  $\mathbf{a}(t)$  via simulating a diffusion process on the manifold  $M$  of normalized gradients. Diffusion on data manifolds. The diffusion of a function  $f$  on a manifold  $M$  induced by its Laplace-Beltrami operator  $\Delta_M$  is described as a partial differential equation [16, 26]:
+
+$$
+\frac {\partial h}{\partial t} = \Delta_ {M} h. \tag {5}
+$$
+
+When  $h$  is a noisy observation of an underlying ground-truth function  $\overline{h}$
+
+$$
+h (\mathbf {r}) = \bar {h} (\mathbf {r}) + \varepsilon , \tag {6}
+$$
+
+with  $\varepsilon$  being Gaussian noise, it can be shown that the diffusion (Eq. 5) performs denoising of  $h$  depending on the curvature of  $M$  [14, 36]. In particular, when  $h$  is presented as a noisy embedding of  $M$  into an ambient Euclidean space  $\mathbb{R}^m$ , applying the diffusion in Eq. 5 tends to recover the underlying true embedding  $\{\overline{h}(\mathbf{r}) : \mathbf{r} \in M\} \subset \mathbb{R}^m$  by pushing along  $M$  the noisy observations  $\{h(\mathbf{r}) : \mathbf{r} \in M\} \subset \mathbb{R}^m$  towards them.
+
+As shown shortly, we will take the local client gradients  $\{\mathbf{g}_k\}$  and the combined gradient  $\mathbf{q}$  (Eq. 4) as a sample from  $M$  and denoise  $\mathbf{q}$  via simulating diffusion. For this, we first spatially discretize the original diffusion process: Suppose that we have a set of data instances  $\overline{H} = \{\overline{\mathbf{h}}_1,\dots,\overline{\mathbf{h}}_S\}$  sampled from a manifold  $M$ . The dataset  $\overline{H}$  is not directly observed but instead presented as a noisy embedding  $H = \{\mathbf{h}_1,\dots,\mathbf{h}_S\}$  within  $\mathbb{R}^m$ . The normalized graph Laplacian  $\mathbf{L}$  on  $H$  is defined as
+
+$$
+\mathbf {L} = \mathbf {I} - \mathbf {D} ^ {- 1} \mathbf {A}, \text {w h e r e} \tag {7}
+$$
+
+$$
+[ \mathbf {A} ] _ {i j} = \kappa (\mathbf {h} _ {i}, \mathbf {h} _ {j}; \sigma^ {2}) := \exp \left(- \frac {\| \mathbf {h} _ {i} - \mathbf {h} _ {j} \| ^ {2}}{\sigma^ {2}}\right), \tag {8}
+$$
+
+for  $i \neq j$  and  $[\mathbf{A}]_{ij} = 0$  for  $i = j$ . The  $i, j$ -th kernel matrix entry  $[\mathbf{A}]_{ij}$  measures the similarity of data instance  $\mathbf{h}_i$  and  $\mathbf{h}_j$  controlled by the parameter  $\sigma^2$ .  $\mathbf{D}$  is a diagonal matrix of the column-sums of  $\mathbf{A}$ :  $[\mathbf{D}]_{ii} = \sum_j[\mathbf{A}]_{ij}$ , which makes the corresponding diffusion process (presented below) probabilistically normalized. With graph Laplacian  $\mathbf{L}$  as an approximation of  $-\Delta_M$ , a spatial discretization of Eq. 5
+
+is formulated as a time evolution of  $H$  (see [6, 13] for convergence analysis of  $\mathbf{L}$  to  $-\Delta_M$ ):
+
+$$
+\frac {\partial \mathbf {H}}{\partial t} = - \mathbf {L} \mathbf {H} \tag {9}
+$$
+
+with  $\mathbf{H} = [\mathbf{h}_1^\top, \dots, \mathbf{h}_S^\top]^\top$ . This diffuses the rows of  $\mathbf{H}$  (equivalently,  $H$ ) along the manifold  $M$ . It should be noted that our diffusion is nonlinear as its generator  $\mathbf{L}$  depends on  $H$  which is being evolved: Initially,  $\mathbf{L}$  is given as a noisy discretization of  $-\Delta_M$  (as it is constructed based on  $H$ , instead of  $\overline{H}$ ), and  $\mathbf{L}$  and  $\mathbf{H}$  are jointly denoised during the diffusion process.
+
+Denoising combined gradients via diffusion. We consider a manifold  $M$  consisting of normalized gradients of  $\mathcal{E}$ :  $\| \mathbf{g}\| _2 = 1$ ,  $\sum_{j}[\mathbf{g}]_{j} = 0$  for  $\mathbf{g}\in h(M)$ . The corresponding graph Laplacian is constructed using the local gradients  $\{\mathbf{g}_k\}_{k = 1}^K$  and the combined gradient  $\mathbf{q}$ : The data matrix  $\mathbf{H}$  is obtained by assigning  $\mathbf{g}_k$  to  $\mathbf{h}_k$  for  $1\le k\le K$  and  $\mathbf{q} = \mathbf{h}_S$  with  $S = K + 1$ .
+
+Now applying the explicit Euler time-discretization to the continuous evolution of Eq. 9, we obtain to the following iterative algorithm
+
+$$
+\begin{array}{l} \frac {\mathbf {H} (t + 1) - \mathbf {H} (t)}{\delta} = - \mathbf {L} \mathbf {H} (t) \\ \Leftrightarrow \mathbf {H} (t + 1) = (\mathbf {I} - \delta \mathbf {L}) \mathbf {H} (t), \tag {10} \\ \end{array}
+$$
+
+where  $\delta$  is the discretization step.
+
+This diffusion process simultaneously denoises all gradients, which can be computationally demanding. Since our goal is to robustify the update process of  $\mathbf{w}$  (Eq. 3) via the gradient combination  $\mathbf{q}$  (Eq. 4), we focus on only diffusing  $\mathbf{q}$  given the local gradients  $\{\mathbf{g}_k\}$ : The evolution of  $\mathbf{q}$  is accordingly derived from Eqs. 7 and 10 as
+
+$$
+\mathbf {q} (t + 1) = (1 - \delta) \mathbf {q} (t) + \delta \sum_ {k = 1} ^ {K} [ \mathbf {a} (t) ] _ {k} \mathbf {g} _ {k} (t), \tag {11}
+$$
+
+$$
+\mathbf {a} (t) = \frac {\tilde {\mathbf {a}} (t)}{\| \tilde {\mathbf {a}} (t) \| _ {1}}, [ \tilde {\mathbf {a}} (t) ] _ {k} = \kappa (\mathbf {g} _ {k} (t - 1), \mathbf {q} (t); \sigma^ {2}).
+$$
+
+Finally, our learning algorithm is obtained by first determining the initial solution  $\mathbf{q}(1)$  as a uniform average of the local gradients:  $\mathbf{q}(1) = \sum_{k=1}^{K}[\mathbf{a}(1)]_k\mathbf{g}_i(1)$  with  $[\tilde{\mathbf{a}}(1)]_k = 1 / K$  and incorporating the resulting diffusion step of  $\mathbf{q}$  (Eq. 11) in the update of  $\mathbf{w}$  in Eq. 3. By noting that  $\mathbf{q}(t)$  is uniquely determined by  $\mathbf{a}(t)$  given the local gradients  $\{\mathbf{g}_k(t)\}_{k=1}^K$ , this casts the  $\mathbf{q}$ -update step of Eq. 11 into an update rule of the gradient combination weights  $\mathbf{a}$ :
+
+$$
+[ \mathbf {a} (t + 1) ] _ {k} = (1 - \delta) [ \mathbf {a} (t - 1) ] _ {k} + \delta [ \mathbf {a} (t) ] _ {k}. \tag {12}
+$$
+
+As  $\mathbf{a}(t + 1)$  is not automatically normalized, we explicitly normalize it before the update of  $\mathbf{w}$  (Eq. 3). Algorithm 1 summarizes the training process.
+
+Hyperparameters and complexity. The hyperparameters of our algorithm include the diffusion step  $\delta$  (Eq. 11) and the
+
+Algorithm 1: Server execution of our distributed learning framework: The client  $C_k$  is equipped with a dataset  $X_k$  and it provides the corresponding loss gradient  $\mathbf{g}_k(t)$  at step  $t$ . An unknown subset of the total clients  $\{C_k\}_{k=1}^K$  communicate erroneous gradients.
+
+Input: Diffusion step size  $\delta$  (Eq. 10) and kernel width  $\sigma^2$  of graph Laplacian (Eq. 8).
+
+Output: Optimized learner parameter  $\mathbf{w}^*$ .
+
+Initialize the learner parameters  $\mathbf{w}(1)$  randomly; Initialize combination coefficients  $\mathbf{a}(1)$  uniformly;
+
+for each round  $t = 1,\dots$  do
+
+Collects the local gradients  $\{\mathbf{g}_k(t)\}_{k = 1}^K$
+
+Normalize the gradients  $\mathbf{g}_k(t) = \frac{\mathbf{g}_k(t)}{\|\mathbf{g}_k(t)\|_2}$ ;
+
+Calculate the weighted
+
+average gradient  $\mathbf{q}(t) = \sum_{k=1}^{K}[\mathbf{a}(t)]_k\mathbf{g}_k(t)$ ;
+
+Calculate the kernel evaluations  $\tilde{\mathbf{a}}(t)$  (Eq. 11);
+
+Calculate
+
+$\mathbf{w}(t + 1)$  (Eq. 3) and distribute it to clients;
+
+Calculate and normalize  $\mathbf{a}(t + 1)$  (Eq. 12);
+
+end
+
+$$
+\mathbf {w} ^ {*} = \mathbf {w} (t);
+$$
+
+kernel width  $\sigma^2$  (Eq. 8). The parameter  $\sigma^2$  controls the uniformity of the combination coefficients  $\mathbf{a}$ : For small  $\sigma^2$  values, only a small portion of local gradients  $\{\mathbf{g}_k\}$  contributes to the combination  $\mathbf{q}$  helping suppress noisy gradients, at the expense of potentially neglecting some genuine gradients. Large  $\sigma^2$  values lead to close to uniform coefficients making more gradients contribute to the combination, possibly including affected gradients. We fix  $\sigma^2$  as a conservative value of 2 leading to close to uniform  $\mathbf{a}$  values for the first 50 iterations of optimization. Thereafter, it is determined as 0.2 which is roughly half the squared mean distance of the pairs of the initial gradients of CIFAR10 dataset (see Sec. 4). As the explicit Euler scheme in Eq. 11 is not stable for large  $\delta$  values, we fix  $\delta$  at a small value of 0.1. This offered fast enough  $\mathbf{q}$ -update while keeping the evolution stable. For large  $\delta$  values, when  $\mathbf{a}$  is not normalized,  $\mathbf{q}$  can quickly diverge to infinity or converge to zero: As  $\mathbf{a}$  is normalized in our algorithm,  $\mathbf{a}$  actually converges to a vector consisting of zeros except for an entry of value one. In general, tuning these hyperparameters per problem and dataset can improve performance. However, this might require separate validation sets at the server, which can limit the application domain of distributed learning.
+
+Each step of our algorithm requires updating the gradient combination weights  $\mathbf{a}$  (Eq. 12), and calculating the similarity of each local gradient and the combined gradient  $\mathbf{q}$  (Eq. 11). Given the local gradients  $\{\mathbf{g}_k(t)\}_{k=1}^K$ , the time complexity of each step of our algorithm depends linearly on the number  $K$  of clients and the size  $m$  of a gradient vector (equivalently, the number of weights of the learner  $f$ ) incur
+
+![](images/720ed535cb1e9d67d0126d8640fed58a51d5daa10bd4dfdd727b2d42bc9600be.jpg)
+
+![](images/3b4968436215ad69807c56aec28d499498206fffffd9e39147cbed5f8da7d28f.jpg)
+
+![](images/0d194fffb57497b51046658347698a28d5046532b51f6c78b63c4a3ac0c41e67.jpg)  
+(a) Ground truth
+
+![](images/7913170780e9e9cb3bfd98f11991b59a6227fcd4d7b24563c81438dd00a6874d.jpg)  
+(b) Training data
+
+![](images/c16c6c058a26bdfe28d5d45c5706ef9a23a42271dba08dbdba566ee71c374333.jpg)  
+(c) FedAvg  
+(d) Ours  
+(e) Distribution of the finalized combination weights a.  
+Figure 1. Our algorithm and FedAvg applied to a simple 2D example where the last 40 (out of 125) clients are affected. Our algorithm successfully suppressed the gradients from the affected clients leading to  $11.5\%$  accuracy improvements from FedAvg.
+
+ring the total complexity of  $O(K \times m)$ . We terminate the weight diffusion (Eq. 12) when the diversity of local gradients  $D_{t} = \sum_{k=1}^{K} \| \mathbf{g}_{k}(t-1) - \mathbf{q}(t) \|$  decreases rapidly. For CI-FAR100, each gradient combination step takes less than a second in a machine with i7-10700K CPU and RTX2080Ti GPU.
+
+An illustrative example. Figure 1 illustrates the learning behavior of our algorithm using a two-dimensional toy example: A training set (Fig. 1(b)) of size 500 is generated as a noisy sample from a ground-truth class map (Fig. 1(a)), which is distributed across 125 clients. At each training iteration, the gradients of 40 clients are replaced by Gaussian random vectors simulating adversarial attacks. Our algorithm successfully suppressed the perturbed gradients as indicated by the small a values assigned for these affected clients (Fig. 1(e)) and achieved the testing accuracy of  $90.81\%$  (Fig. 1(d)).
+
+The FedAvg was distracted by these random gradients significantly degrading the performance (81.38%; Fig. 1(c)).
+
+Discussion. The gradient combination framework in Eq. 3 is general and it includes other algorithms including Wan and Chen's robust federated learning approach [34] as a special case. In [34], the combination weight  $[\mathbf{a}]_k$  is constructed based on the probability of the gradient  $\mathbf{g}_k$  being genuine, which is estimated by a separate attention mechanism trained on an annotated dataset (of genuine and affected gradients). This requires a separate dataset at the server and incurs the overhead of transmitting such data to clients to collect the example gradients. Our algorithm can be considered as an alternative that can be applied even when such labeled datasets are not available at the server and when high data communication costs cannot be afforded.
+
+Diffusion processes have been widely used in semi-supervised learning [30, 44], image and surface enhancement [3, 36], clustering and data embedding [5, 33]. Most closely related to our approach is Hein and Maier's manifold denoising algorithm [14] which denoises a given point cloud sampled from a manifold. Unlike [14], our approach employs a probabilistic normalization (Eq. 8) leading to a convex combination of local gradients (Eq. 11) and focuses on diffusing a single combined gradient vector rendering the update rule for the combination weights a (Eq. 12). Further, our approach uses the explicit Euler time discretization. This leads to a computationally more affordable solution (Eq. 10) than the implicit Euler scheme of [14], as the former does not require solving a large-scale optimization problem per diffusion step. However, this comes at the expense that our update rule is not uniformly stable for any values of the diffusion step  $\delta > 0$  (Eqs. 11 and 12) and therefore, we keep it at a small value.
+
+Our algorithm was conceived by simulating diffusion on a data manifold. While this offers an advantage of theoretical rigor, inheriting the convergence analysis of diffusion processes and their denoising properties [14, 36], a more intuitive interpretation exists: At step  $t$ , the new combined gradient vector  $\mathbf{q}(t + 1)$  (Eq. 11) can be seen as the minimizer of the following convex energy
+
+$$
+\mathcal {E} ^ {W} (\mathbf {q}) = \frac {1}{2} \sum_ {k = 1} ^ {S} [ \mathbf {c} (t) ] _ {k} \| \mathbf {h} _ {k} (t) - \mathbf {q} \| ^ {2}, \tag {13}
+$$
+
+where for  $1 \leq k \leq S - 1$ ,  $\mathbf{h}_k(t) = \mathbf{g}_k(t)$ ,  $[\tilde{\mathbf{c}}(t)]_k = \delta[\mathbf{a}(t)]_k$ , and  $\tilde{\mathbf{c}}(t) = \frac{\mathbf{c}(t)}{\|\mathbf{c}(t)\|}$ , and  $\mathbf{h}_S(t) = \mathbf{q}(t)$  and  $[\tilde{\mathbf{c}}(t)]_S = 1 - \delta$ . This penalizes the deviation of  $\mathbf{q}(t + 1)$  from the previous solution  $\mathbf{q}(t)$  and the current local gradients  $\{\mathbf{g}_k(t)\}$ . The contribution  $[\mathbf{c}(t)]_k$  of  $\mathbf{g}_k(t)$  is controlled based on the error  $\|\mathbf{q}(t) - \mathbf{g}_k(t - 1)\|^2$  incurred at the previous step: The iterative minimization of  $\mathcal{E}^W$  tends to assign smaller weights  $\{[\mathbf{c}(t)]_k\}$  to the gradients  $\{\mathbf{g}_k\}$  that exhibit large errors. This introduces robustness in the gradient aggregation of Eq. 11 as the outliers are gradually ignored during optimization.
+
+Achieving robustness via reweighting the contributions of individual error terms has been studied in robust linear regression and compressed sensing [10].
+
+# 4. Experiments
+
+We evaluated our robust gradient combination algorithm on five classification and three regression datasets.
+
+# 4.1. Classification
+
+Setting. The Caltech256 dataset contains 30,607 color images from 257 object and background classes [11]. The CIFAR10 and CIFAR100 datasets consist of 60,000 images of 10 and 100 classes, respectively [22]. The FashionMNIST dataset consists of 70,000 grayscale images in 10 categories [37]. The CINIC10 dataset contains 270,000 images selected from ImageNet and CIFAR10 [9].
+
+We simulated random labeling perturbations where the ground-truth training labels of the affected clients are replaced by random class labels. In the accompanying supplemental document, we also report the results of experiments with random gradient perturbations where the local gradient vectors are replaced by i.i.d. Gaussian random vectors. As our algorithm operates at the server, it is agnostic to the nature of the local datasets and perturbations, and the only information that our algorithm received from each client is the output gradient.
+
+For all datasets, we performed experiments with varying portions of affected clients: Ten to sixty percent of the total clients were affected. For all datasets and portions of affected clients, we ran experiments 10 times with different distributions of training data across clients and averaged the results. For Caltech256, CIFAR10, CIFAR100, FashionMNIST, for each experimental run, 20,000 data instances were randomly selected and equally distributed to 100 clients while 10,000 points were used for testing. For CINIC10, 180,000 training instances (the union of the training and validation sets proposed by the authors of [9]) were randomly distributed to 1,000 clients, and the remaining 90,000 points were reserved for testing. Here, the sizes of the local datasets across clients varied: The largest local dataset was around 10 times larger than the smallest set.
+
+We used the learner that combines a fixed ResNet101 pretrained on ImageNet and three fully-connected layers.
+
+Baselines. We compared with 1) the naive application of FedAvg which uniformly averages all local gradients. As our algorithm combines these gradients by taking weighted averages instead (Eq. 4), it can be considered as an instance of robust alternatives to the mean. In this context, we compared with 2) the median of local gradients as a popular robust estimator of central statistical tendency (denoted as Median) and 3) Pillutla et al.'s geometric median-based approach as another robust average replacement (GeoMed) [25]. GeoMed
+
+requires solving an optimization problem at each gradient combination step (taking around 10 seconds per step). Also, we compared with 4) Turan et al.'s robust gradient-based optimization strategy [32] (MedTh). This selects local gradients whose distance from the median gradient is within a threshold. As it is challenging to determine such a threshold value without having access to separate validation sets, we set it in the way that the number of selected gradients becomes the same as the ground-truth number of unaffected clients; Our algorithm did not use this additional information. Lastly, we compared with FedAvg applied to only genuine clients (GTC). This will serve as the ideal performance upper bounds of all gradient combination algorithms.
+
+Results. Table 1 summarizes the results. Note that the performance of GTC degrades with increasing portions of affected clients since the numbers of total genuine training instances accordingly decrease. Federated averaging (FedAvg) demonstrated robustness when only moderate numbers (around  $10 - 20\%$ ) of clients are affected. However, its performance degraded rapidly as more clients were affected. This tendency was consistently observed across all datasets. While the median is a well-established robust alternative to the mean, combining local gradients based on their median (Median) turned out to be only comparable to, or sometimes even significantly worse than the naive mean (FedAvg). GeoMed was more effective than Median, yet it was noticeably inferior to FedAvg in a few cases despite the much higher computational cost. MedTh demonstrated competitive performance when affected clients were fewer than  $50\%$  of the total clients. However, for larger numbers of affected clients, it also delivered notably worse results than FedAvg. Our algorithm demonstrated graceful performance degradation even in these cases, and overall, it yielded (statistically) significantly better results than FedAvg on 28 out of 30 cases. Importantly, it was never significantly worse than FedAvg.
+
+Throughout the entire experiments on random gradient perturbations (see supplemental document), both MedTh and ours significantly outperformed FedAvg, constantly achieving the best performance.
+
+Combining MedTh and Ours. Our original algorithm estimates the continuous gradient combination weights  $\mathbf{a}$ . This helps suppress the perturbed gradients but it does not completely rule out their contributions. In the accompanying supplemental document, we demonstrate that when additional validation sets are available, we can achieve this goal and improve performance by combining our algorithm with the thresholding idea of MedTh.
+
+# 4.2. Regression
+
+Setting. We used three benchmark datasets. The MOCAP dataset provides 50,000 motion capture data instances [2]. Each instance provides the 3D locations of 62 skeletal
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Method</td><td colspan="6">Accuracy</td></tr><tr><td>10</td><td>20</td><td>30</td><td>40</td><td>50</td><td>60</td></tr><tr><td rowspan="6">Caltech256</td><td>GTC</td><td>76.54 (0.34)</td><td>75.96 (0.20)</td><td>75.47 (0.19)</td><td>74.84 (0.41)</td><td>73.83 (0.16)</td><td>72.26 (0.50)</td></tr><tr><td>FedAvg</td><td>76.29 (0.32)</td><td>75.16 (0.35)</td><td>74.14 (0.45)</td><td>72.29 (0.33)</td><td>70.50 (0.27)</td><td>66.75 (0.44)</td></tr><tr><td>Median</td><td>64.80 (0.47)</td><td>63.58 (0.97)</td><td>61.82 (0.74)</td><td>59.15 (0.98)</td><td>55.57 (0.85)</td><td>48.15 (0.99)</td></tr><tr><td>GeoMed</td><td>76.29 (0.36)</td><td>75.68 (0.41)</td><td>74.69 (0.44)</td><td>71.70 (0.53)</td><td>66.49 (0.36)</td><td>60.77 (0.82)</td></tr><tr><td>MedTh</td><td>76.28 (0.27)</td><td>75.59 (0.38)</td><td>74.70 (0.85)</td><td>72.51 (1.17)</td><td>69.74 (1.04)</td><td>46.31 (5.19)</td></tr><tr><td>Ours</td><td>76.39 (0.20)</td><td>75.95 (0.29)</td><td>75.06 (0.26)</td><td>73.72 (0.31)</td><td>71.90 (0.50)</td><td>69.25 (0.47)</td></tr><tr><td rowspan="6">CIFAR10</td><td>GTC</td><td>89.57 (0.11)</td><td>89.49 (0.15)</td><td>89.44 (0.15)</td><td>89.55 (0.20)</td><td>89.36 (0.16)</td><td>89.22 (0.10)</td></tr><tr><td>FedAvg</td><td>89.23 (0.10)</td><td>87.34 (0.23)</td><td>86.58 (0.15)</td><td>85.98 (0.37)</td><td>85.24 (0.41)</td><td>84.83 (0.37)</td></tr><tr><td>Median</td><td>87.21 (0.25)</td><td>86.89 (0.26)</td><td>86.74 (0.82)</td><td>86.30 (1.37)</td><td>84.65 (1.19)</td><td>82.01 (0.62)</td></tr><tr><td>GeoMed</td><td>87.38 (0.47)</td><td>87.49 (1.06)</td><td>87.14 (1.07)</td><td>86.89 (1.83)</td><td>84.99 (0.85)</td><td>82.46 (0.32)</td></tr><tr><td>MedTh</td><td>89.39 (0.18)</td><td>89.27 (0.10)</td><td>89.01 (0.19)</td><td>88.57 (0.25)</td><td>78.10 (19.90)</td><td>31.40 (18.43)</td></tr><tr><td>Ours</td><td>89.51 (0.06)</td><td>89.43 (0.17)</td><td>89.22 (0.19)</td><td>89.14 (0.13)</td><td>88.95 (0.16)</td><td>88.85 (0.20)</td></tr><tr><td rowspan="6">CIFAR100</td><td>GTC</td><td>66.86 (0.23)</td><td>66.50 (0.21)</td><td>66.10 (0.33)</td><td>65.81 (0.30)</td><td>65.14 (0.42)</td><td>64.41 (0.45)</td></tr><tr><td>FedAvg</td><td>66.50 (0.40)</td><td>65.95 (0.54)</td><td>64.67 (0.28)</td><td>63.25 (0.40)</td><td>61.99 (0.32)</td><td>60.53 (0.33)</td></tr><tr><td>Median</td><td>65.76 (0.23)</td><td>64.53 (0.29)</td><td>62.73 (0.31)</td><td>59.51 (0.75)</td><td>53.51 (0.85)</td><td>47.89 (0.65)</td></tr><tr><td>GeoMed</td><td>67.11 (0.40)</td><td>66.80 (0.63)</td><td>65.75 (0.69)</td><td>65.20 (0.23)</td><td>58.63 (0.57)</td><td>49.79 (1.05)</td></tr><tr><td>MedTh</td><td>66.80 (0.14)</td><td>66.35 (0.29)</td><td>65.78 (0.24)</td><td>64.11 (0.33)</td><td>38.34 (1.43)</td><td>36.93 (2.52)</td></tr><tr><td>Ours</td><td>66.80 (0.23)</td><td>66.41 (0.38)</td><td>65.87 (0.38)</td><td>65.44 (0.27)</td><td>64.80 (0.20)</td><td>63.56 (0.62)</td></tr><tr><td rowspan="6">FashionMNIST</td><td>GTC</td><td>86.96 (0.18)</td><td>86.84 (0.23)</td><td>86.84 (0.11)</td><td>86.92 (0.19)</td><td>86.77 (0.19)</td><td>86.77 (0.20)</td></tr><tr><td>FedAvg</td><td>86.48 (0.19)</td><td>83.12 (0.24)</td><td>81.93 (0.35)</td><td>81.30 (0.31)</td><td>80.80 (0.50)</td><td>80.47 (0.26)</td></tr><tr><td>Median</td><td>83.23 (1.44)</td><td>82.77 (1.30)</td><td>82.68 (1.71)</td><td>82.02 (1.57)</td><td>80.44 (1.05)</td><td>78.40 (0.44)</td></tr><tr><td>GeoMed</td><td>83.29 (1.65)</td><td>82.37 (0.40)</td><td>82.42 (1.72)</td><td>81.20 (0.36)</td><td>80.41 (0.45)</td><td>78.37 (0.30)</td></tr><tr><td>MedTh</td><td>86.75 (0.11)</td><td>86.55 (0.22)</td><td>86.27 (0.22)</td><td>83.57 (4.56)</td><td>76.16 (14.07)</td><td>52.65 (23.84)</td></tr><tr><td>Ours</td><td>87.01 (0.23)</td><td>86.73 (0.16)</td><td>86.63 (0.27)</td><td>86.40 (0.51)</td><td>86.24 (0.16)</td><td>86.41 (0.27)</td></tr><tr><td rowspan="6">CINIC10</td><td>GTC</td><td>79.99 (0.07)</td><td>79.96 (0.11)</td><td>79.75 (0.12)</td><td>79.70 (0.08)</td><td>79.66 (0.12)</td><td>79.60 (0.15)</td></tr><tr><td>FedAvg</td><td>78.20 (0.11)</td><td>76.88 (0.21)</td><td>76.04 (0.16)</td><td>75.41 (0.29)</td><td>74.81 (0.27)</td><td>74.38 (0.13)</td></tr><tr><td>Median</td><td>77.16 (1.59)</td><td>76.91 (1.61)</td><td>77.03 (1.95)</td><td>76.84 (1.33)</td><td>73.83 (1.44)</td><td>71.04 (0.72)</td></tr><tr><td>GeoMed</td><td>78.13 (1.95)</td><td>79.01 (1.86)</td><td>77.49 (2.61)</td><td>78.92 (1.20)</td><td>75.64 (1.28)</td><td>71.51 (0.75)</td></tr><tr><td>MedTh</td><td>79.48 (0.11)</td><td>79.31 (0.15)</td><td>79.06 (0.10)</td><td>78.57 (0.15)</td><td>57.10 (23.87)</td><td>42.19 (20.96)</td></tr><tr><td>Ours</td><td>80.03 (0.13)</td><td>79.95 (0.17)</td><td>79.85 (0.16)</td><td>79.64 (0.32)</td><td>79.19 (0.50)</td><td>78.69 (0.52)</td></tr></table>
+
+Table 1. Results of classification under random labeling perturbations: Mean accuracy and standard deviation (in parenthesis) are shown in % (higher is better). The best and second-best results excluding GTC are highlighted with bold and italic, respectively. For Median, GeoMed, MedTh, and ours, the results of statistical significance test with respect to FedAvg (t-test with  $\alpha = 0.95$ ) are shown: Blue and orange respectively represents significantly better and worse results than FedAvg.
+
+joints. A body surface mesh model is fitted to these 62 joints and the locations of five end effectors (two hands, two feet, and head) are recorded, constituting 15-dimensional coarse representations. The goal is to restore the  $186(62 \times 3)$ -dimensional joint locations from the input 15-dimensional coarse representations. We focused on estimating the first 12 attributes (4 out of 62 joints). Our learners consisted of five fully-connected layers. Experiments were repeated 10 times, and for each experimental run, a randomly selected subset of size 40,000 was used for training while the remaining data instances were used for testing. The training sets were distributed across 100 clients where  $40\%$  were affected: Each output instance is contaminated with a Gaussian
+
+noise whose standard deviation per dimension is twice the standard deviation of the corresponding dimension. The School dataset provides exam scores of 15,362 students in 139 UK secondary schools in the years 1985, 1986, and 1987 [4]. Our goal is to predict these exam scores using 27 input features including the year of the exam, gender, and ethnic group. A small subset was selected from each school, constituting a combined test set of size 5,362. The remaining 10,000 instances were distributed to the clients based on the corresponding schools. The ICVL hand dataset contains 17,604 depth frames of size  $320 \times 240$  capturing human hands and the corresponding 3D annotations of 16 joints [31]. Among them, 16,008 frames were used for training while the
+
+remaining 1,596 frames were assigned for testing. For each frame, features were extracted using the ImageNet-pretrained ResNet101. Similarly to MOCAP, for School and ICVL,  $40\%$  of total local gradients were contaminated by Gaussian noise.
+
+Results. Table 2 summarizes the results. For all datasets, the three robust gradient combination algorithms demonstrated performance gains over FedAvg. For MOCAP, GeoMed was marginally better than FedAvg while MedTh and ours achieved further significant improvements. For School, ours ranked the best while the other two robust algorithms also exhibited marked performance gain. For ICVL, GeoMed ranked first followed by ours.
+
+# 5. Conclusions
+
+We have considered the problem of robust distributed learning under adversarial perturbation. Our approach embeds the gradients calculated from local clients into the manifold of normalized vectors and performs diffusion therein. The resulting algorithm is instantiated as a robust average estimator against outliers and it improves the combined gradient without having to require additional labeled datasets at the server or known number of affected clients. On five classification and three regression benchmark datasets, and two scenarios of adversarial perturbations, our algorithm demonstrated significant improvements over baseline federated averaging [23], robust median and geometric median estimators [25], and median-based thresholding approach [32].
+
+Limitations. The application of our approach (as well as the other algorithms considered in our experiments) is limited to cases where individual clients are either entirely affected or not: As our original algorithm is agnostic to data distributions and operations performed within clients, it does not exploit the possibility that only part of each local dataset is perturbed. When access to local datasets is guaranteed, such a scenario could be approached by applying our main gradient update step (Eqs. 11 and 12) in two levels: Within each client, it can suppress perturbed data instances by taking each data instance and the corresponding error gradient as a sub-client. The resulting locally combined gradients could then be combined at the server as in our main experiments.
+
+# Future work and potential negative societal impacts.
+
+Our approach contributes to promoting distributed learning, in particular over devices where data acquisition and annotation processes might not be responsibly moderated. This can potentially facilitate the transfer of bias (e.g., age, gender, and ethnicity) present in individual clients to the entire learning process. Therefore, future work should investigate detecting and mitigating the effect of bias in local datasets. Further, future work should assess the performance of our algorithm under different adversarial perturbation scenarios, e.g. when the input images (instead of the labels as in our experiments)
+
+<table><tr><td>Attr.</td><td>FedAvg</td><td>GeoMed</td><td>MedTh</td><td>Ours</td></tr><tr><td colspan="5">MOCAP</td></tr><tr><td>1</td><td>2.05 (0.17)</td><td>2.01 (0.06)</td><td>1.75 (0.13)</td><td>1.74 (0.09)</td></tr><tr><td>2</td><td>5.67 (0.05)</td><td>5.66 (0.02)</td><td>3.48 (0.10)</td><td>2.69 (0.08)</td></tr><tr><td>3</td><td>18.76 (0.05)</td><td>18.62 (0.07)</td><td>9.39 (0.52)</td><td>4.40 (0.30)</td></tr><tr><td>4</td><td>3.43 (0.05)</td><td>3.36 (0.05)</td><td>2.75 (0.05)</td><td>2.55 (0.06)</td></tr><tr><td>5</td><td>18.95 (0.08)</td><td>18.86 (0.08)</td><td>9.48 (0.53)</td><td>4.31 (0.35)</td></tr><tr><td>6</td><td>1.93 (0.13)</td><td>1.84 (0.05)</td><td>1.58 (0.10)</td><td>1.62 (0.14)</td></tr><tr><td>7</td><td>3.96 (0.05)</td><td>3.96 (0.05)</td><td>2.89 (0.05)</td><td>2.46 (0.06)</td></tr><tr><td>8</td><td>18.83 (0.09)</td><td>18.73 (0.05)</td><td>9.25 (0.49)</td><td>3.87 (0.30)</td></tr><tr><td>9</td><td>2.02 (0.11)</td><td>1.92 (0.08)</td><td>1.69 (0.10)</td><td>1.68 (0.12)</td></tr><tr><td>10</td><td>8.78 (0.07)</td><td>8.69 (0.06)</td><td>4.41 (0.24)</td><td>2.47 (0.09)</td></tr><tr><td>11</td><td>19.43 (0.05)</td><td>19.37 (0.04)</td><td>9.23 (0.56)</td><td>3.60 (0.21)</td></tr><tr><td>12</td><td>2.63 (0.13)</td><td>2.55 (0.04)</td><td>2.32 (0.09)</td><td>2.23 (0.08)</td></tr><tr><td colspan="5">School</td></tr><tr><td>1</td><td>15.89 (0.36)</td><td>13.36 (0.43)</td><td>13.87 (0.63)</td><td>12.38 (0.41)</td></tr><tr><td colspan="5">ICVL</td></tr><tr><td>1</td><td>3.61 (0.04)</td><td>3.62 (0.11)</td><td>3.63 (0.06)</td><td>3.57 (0.05)</td></tr><tr><td>2</td><td>3.75 (0.09)</td><td>2.73 (0.09)</td><td>3.17 (0.14)</td><td>3.00 (0.19)</td></tr><tr><td>3</td><td>16.40 (0.08)</td><td>4.53 (0.08)</td><td>9.09 (0.67)</td><td>7.02 (1.60)</td></tr><tr><td>4</td><td>3.46 (0.02)</td><td>3.56 (0.08)</td><td>3.43 (0.09)</td><td>3.49 (0.04)</td></tr><tr><td>5</td><td>4.03 (0.04)</td><td>2.95 (0.12)</td><td>3.58 (0.11)</td><td>3.36 (0.14)</td></tr><tr><td>6</td><td>16.19 (0.04)</td><td>4.50 (0.10)</td><td>8.98 (0.60)</td><td>6.90 (1.60)</td></tr><tr><td>7</td><td>3.77 (0.02)</td><td>3.94 (0.11)</td><td>3.70 (0.04)</td><td>3.79 (0.08)</td></tr><tr><td>8</td><td>3.91 (0.10)</td><td>3.32 (0.25)</td><td>3.56 (0.11)</td><td>3.46 (0.09)</td></tr><tr><td>9</td><td>15.33 (0.08)</td><td>4.56 (0.10)</td><td>8.54 (0.63)</td><td>6.65 (1.43)</td></tr><tr><td>10</td><td>4.52 (0.05)</td><td>4.50 (0.05)</td><td>4.39 (0.03)</td><td>4.42 (0.10)</td></tr><tr><td>11</td><td>3.69 (0.05)</td><td>3.64 (0.29)</td><td>3.73 (0.12)</td><td>3.69 (0.06)</td></tr><tr><td>12</td><td>15.18 (0.09)</td><td>5.02 (0.04)</td><td>8.74 (0.56)</td><td>7.01 (1.33)</td></tr></table>
+
+Table 2. Regression results: Mean error rate and standard deviation in parenthesis (lower is better). The best and second-best results are highlighted with **bold** and **italic**, respectively. Statistically significantly better and worse results than **FedAvg** are highlighted in blue and **orange**, respectively.
+
+are randomly contaminated by noise or systematically altered, and adversarial perturbations are coordinated (e.g. by injecting noise correlated across clients). We experimentally demonstrated that our algorithm offers the capability of suppressing affected gradients. Future work should also conduct a theoretical analysis, e.g. on the deviation between the gradients combined by our algorithm and the corresponding ground-truth combinations (generated by  $GTC$ ).
+
+# Acknowledgments
+
+This work was supported by the National Research Foundation of Korea (NRF) grant (No. 2021R1A2C2012195), Institute of Information & Communications Technology Planning & Evaluation (IITP) grant (2021-0-00537, Visual Common Sense Through Self-supervised Learning for Restoration of Invisible Parts in Images), and IITP grant (2020-0-01336, Artificial Intelligence Graduate School Program, UNIST), all funded by the Korea government (MSIT).
+
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+# Robust Contrastive Learning against Noisy Views
+
+Ching-Yao Chuang† R Devon Hjelm‡ Xin Wang‡ Vibhav Vineet‡  
+Neel Joshi‡ Antonio Torralba† Stefanie Jegelka† Yale Song‡  
+†MIT CSAIL ‡ Microsoft Research
+
+https://github.com/chingyaoc/RINCE
+
+![](images/22643eb6673f78270006c87c5ac373d2d69f72c364d74733b094049b2a66c405.jpg)  
+ct
+
+an assumption that positat share certain underlye, e.g., patches of an imsignals of a video. Whatrature suggests thatmal representationsfalse positive pairsn this work, we provat is robust againstretical justificationsetric losses for noisyg a new contrastive...zation based on the
+
+mated Noise Level  
+masseteriastance measure. The proposed loss is completely modality-agnostic and a simple drop-in replacement for the InfoNCE loss, which makes it easy to apply to existing contrastive frameworks. We show that our approach provides consistent improvements over the state-of-the-art on image, video, and graph contrastive learning benchmarks that exhibit a variety of real-world noise patterns.
+
+# 1. Introduction
+
+Contrastive learning [1, 2, 3] has become one of the most prominent self-supervised approaches to learn representations of high-dimensional signals, producing impressive results with image [4, 5, 6, 7, 8, 9], text [10, 11, 12, 13], audio [14, 15, 16], and video [17, 18, 19]. The central idea is to learn representations that capture the underlying information shared between different "views" of data [3, 20]. For images, the views are typically constructed by applying common data augmentation techniques, such as jittering, cropping, resizing and rotation [6], and for video the views are often chosen as adjacent frames [21] or co-occurring multimodal signals, such as video and the corresponding optical flow [22], audio [19] and transcribed speech [17].
+
+Designing the right contrasting views has shown to be
+
+![](images/2e21565380bf21c81bce2f89909d863dd98a2021c10c129992c42379a663f222.jpg)  
+Figure 1. Noisy views can deteriorate contrastive learning. We propose a new contrastive loss function (RINCE) that rescales the sample importance in the gradient space based on an estimated noise level. With a simple turn of a knob  $(q \in (0,1])$ , we can upweight or downweight sample pairs with low shared information.
+
+a key ingredient of contrastive learning [6, 23]. This often requires domain knowledge, intuition, trial-and-error (and luck!). What would happen if the views are wrongly chosen and do not provide meaningful shared information? Prior work has reported deteriorating effects of such noisy views in contrastive learning under various scenarios, e.g., unrelated image patches due to extreme augmentation [20], irrelevant video-audio pairs due to overdubbing [24], and misaligned video-caption pairs [17]. The major issue with noisy views is that representations of different views are forced to align with each other even if there is no meaningful shared information. This often leads to suboptimal representations that merely capture spurious correlations [25] or make them collapse to a trivial solution [26]. Worse yet, when we attempt to learn from large-scale unlabeled data – i.e., the scenario where self-supervised learning is particularly expected to shine – the issue is only aggravated
+
+because of the increased noise in the real-world data [27], hindering the ultimate success of contrastive learning.
+
+Consequently, a few attempts have been made to design contrastive approaches that are noise-tolerant. For example, Morgado et al. [24] optimize a soft instance discrimination loss to weaken the impact of noisy views. Miech et al. [17] address the misalignment between video and captions by aligning multiple neighboring segments of a video. However, existing approaches are often tied to specific modalities or make assumptions that may not hold for general scenarios, e.g., MIL-NCE [17] is not designed to address the issues of irrelevant audio-visual signals.
+
+In this work, we develop a principled approach to make contrastive learning robust against noisy views. We start by making connections between contrastive learning and the classical noisy binary classification in supervised learning [28, 29]. This allows us to explore the wealth of literature on learning with noisy labels [30, 31, 32]. In particular, we focus on a family of robust loss functions that has the symmetric property [29], which provides strong theoretical guarantees against noisy labels in binary classification. We then show a functional form of contrastive learning that can satisfy the symmetry condition if given a proper symmetric loss function, motivating the design of new contrastive loss functions that provide similar theoretical guarantees.
+
+This leads us to propose Robust InfoNCE (RINCE), a contrastive loss function that satisfies the symmetry condition. RINCE can be understood as a generalized form of the contrastive objective that is robust against noisy views. Intuitively, its symmetric property provides an implicit means to reweight sample importance in the gradient space without requiring an explicit form of noise estimator. It also provides a simple "knob" (a real-valued scalar  $q \in (0,1]$ ) that controls the behavior of the loss function balancing the exploration-exploitation trade-off (i.e., from being conservative to playing adventures on potentially noisy samples).
+
+We also provide a theoretical analysis of the proposed RINCE objective and show that it extends the analyses by Ghosh et al. [29] to the self-supervised contrastive learning regime. Furthermore, we relate the proposed loss function to dependency measurement. Analogous to InfoNCE loss, which is a lower bound of mutual information between two views [3], we show that RINCE is a lower bound of Wasserstein Dependency Measure (WDM) [33], even in the noisy setting. By replacing the KL divergence in the mutual information estimator with the Wasserstein distance, WDM is able to capture the geometry of the representation space via the equipped metric space and provides robustness against noisy views better than the KL divergence, both in theory and practice. In particular, the features learned with RINCE achieve better class-wise separation, which is proved to be crucial to improve generalization [34].
+
+Despite its rigorous theoretical background, implement
+
+ing RINCE requires only a few lines code and can be a simple drop-in replacement for the InfoNCE loss to make contrastive learning robust against noisy views. Since InfoNCE sets the basis for many modern contrastive methods such as SimCLR [6] and MoCo-v1/v2/v3 [5, 7, 35], our construction can be easily applied to many existing frameworks.
+
+Finally, we provide strong empirical evidence demonstrating the robustness of RINCE against noisy views under various scenarios with different modalities and noise types. We show that RINCE improves over the state-of-the-art in image [36, 37], video [27, 38] and graph [39] self-supervised learning benchmarks, demonstrating its generality across different modalities. We also show that RINCE exhibits strong robustness against different types of noise such augmentation noise [20, 40], label noise [28, 41], and noisy audio-visual correspondence [24]. The improvement is consistently observed across different dataset scales and training epochs, demonstrating the scalability and computational efficiency. In short, our main contributions are:
+
+- We propose RINCE, a new contrastive learning objective that is robust against noisy views of data;  
+- We provide a theoretical analysis to relate the proposed loss to symmetric losses and dependency measurement;  
+- We demonstrate our approach on real-world scenarios of image, video, and graph contrastive learning.
+
+# 2. Related Work
+
+Contrastive Learning Contrastive approaches have become prominent in unsupervised representation learning [1, 2, 42]. InfoNCE [3] and its variants [5, 6, 8, 9] achieve state-of-the-art across different modalities [10, 14, 17, 18, 19]. Modern approaches improve upon InfoNCE from different directions. One line of work focuses on modifying training mechanisms, e.g., appending projection head [6], momentum encoder with dynamic dictionary update [5, 7], siamese networks with stop gradient trick [8, 43], and online cluster assignment [44]. Another line of work refines the loss function itself to make it more effective, e.g., upweight hard negatives [12, 45], correct false negatives [11], and alleviate feature suppression [46]. Along this second line of work, we propose a new contrastive loss function robust against noisy views. Some of prior work in this direction [17, 24] was demonstrated on limited modalities only; we demonstrate its generality on image [37], video [27, 38], and graph [40, 47, 48] contrastive learning scenarios. Our approach is orthogonal to the first line of work; we show that RINCE can easily be applied to some of existing training mechanisms such as SimCLR [6] and MoCo-v1/v2/v3 [5, 7, 35].
+
+Robust Loss against Noisy Labels Learning with noisy labels has been actively explored in recent years [28, 29, 31, 32, 49, 50, 51, 52, 53, 54, 55]. One line of work
+
+attempts to develop robust loss functions that are noisetolerant [29, 30, 56, 57]. Ghosh et al. [29] prove that symmetric loss functions are robust against noisy labels, e.g., Mean Absolute Error (MAE) [30], while commonly used Cross Entropy (CE) loss is not. Based on this idea, Zhang and Sabuncu [56] propose the generalized cross entropy loss to combine MAE and CE loss functions. A similar idea is adopted in [57] by combining the reversed cross entropy loss with CE loss. In the next section, we relate noisy views to noisy labels by interpreting contrastive learning as binary classification, and developed a robust symmetric contrastive loss that enjoys the similar theoretical guarantees.
+
+# 3. From Noisy Labels to Noisy Views
+
+We start by connecting two seemingly different but related frameworks: supervised binary classification with noisy labels and self-supervised contrastive learning with noisy views. We then introduce a family of symmetric loss functions that is noise-tolerant and show how we can transform contrastive objectives to a symmetric form.
+
+# 3.1. Symmetric Losses for Noisy Labels
+
+Denoting the input space by  $\mathcal{X}$  and the binary output space by  $\mathcal{Y} = \{-1,1\}$ , let  $S = \{x_{i},y_{i}\}_{i = 1}^{m}$  be the unobserved clean dataset that is drawn i.i.d. from the data distribution  $\mathcal{D}$ . In the noisy setting, the learner obtains a noisy dataset  $S_{\eta} = \{x_i,\hat{y}_i\}_{i = 1}^m$ , where  $\hat{y}_i = y_i$  with probability  $1 - \eta_{x_i}$  and  $\hat{y}_i = -y_i$  with probability  $\eta_{x_i}$ . Note that the noise rate  $\eta_{x}$  is data point-dependent. For a classifier  $f\in \mathcal{F}:\mathcal{X}\to \mathbb{R}$ , the expected risk under the noise-free scenario is  $R_{\ell}(f) = \mathbb{E}_{\mathcal{D}}[\ell (f(x),y)]$  where  $\ell :\mathbb{R}\times \mathcal{V}\rightarrow \mathbb{R}$  is a binary classification loss function. When the noise exists, the learner minimizes the noisy expected risk  $R_{\ell}^{\eta}(f) = \mathbb{E}_{\mathcal{D}_n}[\ell (f(x),\hat{y})]$ .
+
+Ghosh et al. [29] show that symmetric loss functions are robust against noisy labels in binary classification. In particular, a loss function  $\ell$  is symmetric if it sums to a constant:
+
+$$
+\ell (s, 1) + \ell (s, - 1) = c, \quad \forall s \in \mathbb {R}, \tag {1}
+$$
+
+where  $s$  is the prediction score from  $f$ . Note that the symmetry condition should also hold with the gradients w.r.t.  $s$ . They show that if the noise rate is  $\eta_x \leq \eta_{\max} < 0.5, \forall x \in \mathcal{X}$  and if the loss is symmetric and non-negative, the minimizer of the noisy risk  $f_{\eta}^{*} = \arg \inf_{f \in \mathcal{F}} R^{\eta}(f)$  approximately minimizes the clean risk:
+
+$$
+R (f _ {\eta} ^ {*}) \leq \epsilon / (1 - 2 \eta_ {\max}),
+$$
+
+where  $\epsilon = \inf_{f\in \mathcal{F}}R(f)$  is the optimal clean risk. This implies that the noisy risk under symmetric loss is a good surrogate of the clean risk. In Appendix A.2, we further relax the non-negative constraint on the loss with a corollary.
+
+# 3.2. Towards Symmetric Contrastive Objectives
+
+The results above suggest that we can achieve robustness against noisy views if a contrastive objective can be expressed in a form that satisfies the symmetry condition in the binary classification framework. To this end, we first relate contrastive learning to binary classification, and then express it in a form where symmetry can be achieved.
+
+Contrastive learning as binary classification. Given two views  $X$  and  $V$ , we can interpret contrastive learning as noisy binary classification operating over pairs of samples  $(x, v)$  with a label 1 if it is sampled from the joint distribution,  $(x, v) \sim P_{XV}$ , and -1 if it comes from the product of marginals,  $(x, v') \sim P_{X}P_{V}$ . In the presence of noisy views, some negative pairs  $(x, v') \sim P_{X}P_{V}$  could be mislabeled as positive, introducing noisy labels.
+
+To see this more concretely, let us consider the InfoNCE loss [3], one of the most widely adopted contrastive objectives [4, 6, 11, 58]. It minimizes the following loss function:
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {I n f o N C E}} (\mathbf {s}) = - \log \frac {e ^ {s ^ {+}}}{e ^ {s ^ {+}} + \sum_ {i = 1} ^ {K} e ^ {s _ {i} ^ {-}}} \\ := - \log \frac {e ^ {f (x) ^ {T} g (v) / t}}{e ^ {f (x) ^ {T} g (v) / t} + \sum_ {i = 1} ^ {K} e ^ {f (x) ^ {T} g \left(v _ {i}\right) / t}}, \tag {2} \\ \end{array}
+$$
+
+where  $\mathbf{s} = \{s^{+},\{s_{i}^{-}\}_{i = 1}^{K}\}$ ,  $s^+$  and  $s_i^-$  are the scores of related (positive) and unrelated (negative) pairs and  $t$  is the temperature parameter introduced to avoid gradient saturation. The expectation of the loss is taken over  $(x,v)\sim P_{XV}$  and  $K$  independent samples  $v_{i}\sim P_{V}$ , where  $P_{XV}$  denotes the joint distribution over pairs of views such as transformations of the same image or co-occurring multimodal signals. Although InfoNCE has a functional form of the  $(K + 1)$ -way softmax cross entropy loss, the model ultimately learns to classify whether a pair  $(x,v)$  is positive or negative by maximizing/minimizing the positive score  $s^+ /$  negative scores  $s_i^-$ . Therefore, InfoNCE under noisy views can be seen as binary classification with noisy labels. We acknowledge that similar interpretations have been made in prior works under different contexts [4, 59, 60].
+
+Symmetric form of contrastive learning. Now we turn to a functional form of contrastive learning that can achieve the symmetric property. Assume that we have a noisetolerant loss function  $\ell$  that satisfies the symmetry condition of equation 1. We say a contrastive learning objective is symmetric if it accepts the following form
+
+$$
+\mathcal {L} (\mathbf {s}) = \underbrace {\ell \left(s ^ {+} , 1\right)} _ {\text {P o s i t i v e P a i r}} + \lambda \underbrace {\sum_ {i = 1} ^ {K} \ell \left(s _ {i} ^ {-} , - 1\right)} _ {K \text {N e g a t i v e P a i r s}} \tag {3}
+$$
+
+which consists of a collection of  $(K + 1)$  binary classification losses;  $\lambda >0$  is a density weighting term controlling
+
+```txt
+pos: exponent for positive example   
+# neg: sum of exponents for negative examples   
+# q, lam: hyperparameters of RINCE   
+info_nce_loss  $=$  -log(pos/(pos+neg))   
+rince_loss  $=$  -pos\*\*q/q + (lam\*(pos+neg))\*\*q/q
+```
+
+the ratio between classes 1 (positive pairs) and  $-1$  (negative pairs). Reducing  $\lambda$  places more weight on the positive score  $s^+$ , while setting  $\lambda$  to zero recovers the negative-pair-free contrastive loss such as BYOL [8].
+
+Contrastive objectives that satisfy the symmetric form enjoy strong theoretical guarantees against noisy labels as described in Ghosh et al. [29], as long as we plug in the right contrastive loss function  $\ell$  that satisfies the symmetry condition. Unfortunately, the InfoNCE loss [3] does not satisfy the symmetry condition in the gradients w.r.t.  $s^{+/ - }$  (we provide the full derivations in Appendix A.5). This motivates us to develop a new contrastive loss function that satisfies the symmetry condition, described next.
+
+# 4. Robust InfoNCE Loss
+
+Based on the idea of robust symmetric classification loss, we present the following Robust InfoNCE (RINCE) loss:
+
+$$
+\mathcal {L} _ {\mathrm {R I N C E}} ^ {\lambda , q} (\mathbf {s}) = \frac {- e ^ {q \cdot s ^ {+}}}{q} + \frac {(\lambda \cdot (e ^ {s ^ {+}} + \sum_ {i = 1} ^ {K} e ^ {s _ {i} ^ {-}})) ^ {q}}{q},
+$$
+
+where  $q, \lambda \in (0,1]$ . Figure 2 shows the pseudo-code of RINCE; it is simple to implement. When  $q = 1$ , RINCE becomes a contrastive loss that fully satisfies the symmetry property in the form of equation 3 with  $\ell(s,y) = -ye^s$ :
+
+$$
+\mathcal {L} _ {\mathrm {R I N C E}} ^ {\lambda , q = 1} (\mathbf {s}) = - (1 - \lambda) e ^ {s ^ {+}} + \lambda \sum_ {i = 1} ^ {K} e ^ {s _ {i} ^ {-}}.
+$$
+
+Notice that the exponential loss  $-ye^{s}$  satisfies the symmetric condition defined in equation 1 with  $c = 0$ . Therefore, when  $q \to 1$ , we achieve robustness against noisy views in the same manner as binary classification with noisy labels.
+
+In the limit of  $q \to 0$ , RINCE becomes asymptotically equivalent to InfoNCE, as the following lemma describes:
+
+Lemma 1. For any  $\lambda >0$ , it holds that
+
+$$
+\lim  _ {q \rightarrow 0} \mathcal {L} _ {\text {R I N C E}} ^ {\lambda , q} (s) = \mathcal {L} _ {\text {I n f o N C E}} (s) + \log (\lambda);
+$$
+
+$$
+\lim  _ {q \rightarrow 0} \frac {\partial}{\partial s} \mathcal {L} _ {\text {R I N C E}} ^ {\lambda , q} (s) = \frac {\partial}{\partial s} \mathcal {L} _ {\text {I n f o N C E}} (s).
+$$
+
+We defer the proofs to Appendix A. Note that the convergence also holds for the derivatives: optimizing RINCE in the limit of  $q\to 0$  is mathematically equivalent to optimizing InfoNCE. Therefore, by controlling  $q\in (0,1]$  we smoothly interpolate between the InfoNCE loss  $(q\rightarrow 0)$  and the RINCE loss in its fully symmetric form  $(q\rightarrow 1)$ .
+
+![](images/1944dd187288cb7cf4c40b6f91c34ee6e9173db3eb359f83f061c92b7a011168.jpg)  
+Figure 2. Pseudocode for RINCE. The implementation only requires a small modification to the InfoNCE code.  
+Figure 3. Loss Visualization. We visualize the (a) loss value and the (b) gradient scale with respect to the positive score  $s^+$  for different  $q$  while setting  $\lambda = 0.5$ . The gradient scale of InfoNCE ( $q \to 0$ ) is larger when the positive score is smaller (hard positive pair). In contrast, for fully symmetric RINCE ( $q = 1$ ), the gradient is larger when positive score is large (easy positive pair).
+
+# 4.1. Intuition behind RINCE
+
+We now analyze the behavior of RINCE through the lens of exploration-exploitation trade-off. In particular, we reveal an implicit easy/hard positive mining scheme by inspecting the gradients of RINCE under different  $q$  values, and show that we achieve stronger robustness (more exploitation) with larger  $q$  at the cost of potentially useful clean hard positive samples (less exploration).
+
+To simplify the analysis, we consider InfoNCE and RINCE with a single negative pair  $(K = 1)$ :
+
+$$
+\mathcal {L} _ {\text {I n f o N C E}} (\mathbf {s}) = - \log \left(e ^ {s ^ {+}} / \left(e ^ {s ^ {+}} + e ^ {s ^ {-}}\right)\right);
+$$
+
+$$
+\mathcal {L} _ {\mathrm {R I N C E}} ^ {\lambda , q} (\mathbf {s}) = \frac {- e ^ {q \cdot s ^ {+}}}{q} + \frac {(\lambda \cdot (e ^ {s ^ {+}} + e ^ {s ^ {-}})) ^ {q}}{q}.
+$$
+
+We visualize the loss and the scale of the gradients with respect to positive scores  $s^+$  in Figure 3. Although the loss values are different for each  $q$ , they follow the same principle: The loss achieves its minimum when the positive score  $s^+$  is maximized and the negative score  $s^-$  is minimized.
+
+The interesting bit lies in the gradients. The InfoNCE loss  $(q\rightarrow 0)$  places more emphasis on hard positive pairs, i.e., the pairs with low positive scores  $s^+$  (the left-most part in the plot). In contrast, the fully symmetric RINCE loss  $(q = 1)$  places more weights on easy positive pairs (the right-most part). This reveals an implicit trade-off between exploration (convergence) and exploitation (robustness). When  $q\to 0$  , the loss performs hard positive mining, providing faster convergence in the noise-free setting. But in the presence of noise, exploration is harmful; it wrongly puts higher weights to false positive pairs because noisy samples tend to induce larger losses [24, 55, 61, 62], and this could hinder convergence. In contrast, when  $q\rightarrow 1$  we perform easy positive mining. This provides robustness
+
+especially against false positives; but this is done at the cost of exploration with clean hard positives. An important aspect here is that RINCE does not require an explicit form of noise estimator: the scores  $s^{+}$  and  $s^{-}$ , and the relationship between the two (which is what the loss function measures) act as noise estimates. In practice, we set  $q \in [0.1, 0.5]$  to strike the balance between exploration and exploitation.
+
+Note that both  $q \to 0$  and  $q \to 1$  naturally perform hard negative mining; both their derivatives put exponentially more weights on hard negative pairs. In Appendix C.1 we provide extended analysis of the gradients with respect to both  $s^{+}$  and  $s^{-}$  and show that RINCE performs easy positive mining and hard negative mining.
+
+# 4.2. Theoretical Underpinnings
+
+Next, we provide an information-theoretic explanation on what makes RINCE robust against noisy views. In particular, we show that RINCE is a contrastive lower-bound of mutual information (MI) expressed in Wasserstein dependency measure (WDM) [33], which provides superior robustness against sample noise compared to the Kullback-Leibler (KL) divergence thanks to strong geometric properties of the Wasserstein metric. We further show that, even in the presence of noise, RINCE is a lower bound of clean WDM, indicating its robustness against noisy views.
+
+Limitations of KL divergence in MI estimation. Without loss of generality, let  $f = g$  and consider  $f = f' \circ \phi$ , where  $\phi$  is a representation encoder and  $f'$  is a projection head [6]. Also, let  $P^{\phi} = \phi_{\#} P$  be the pushforward measure of  $P$  with respect to  $\phi$ . It has been shown that InfoNCE is a variational lower-bound of MI in the representation space expressed with KL-divergence [20, 63]:
+
+$$
+\begin{array}{l} - \mathbb {E} \left[ \mathcal {L} _ {\text {I n f o N C E}} (\mathbf {s}) \right] + \log (K) \leq I (\phi (X), \phi (V)) \\ = D _ {\mathrm {K L}} (P _ {X V} ^ {\phi}, P _ {X} ^ {\phi} P _ {V} ^ {\phi}). \\ \end{array}
+$$
+
+Intuitively, maximizing MI can be interpreted as maximizing the discrepancy between positive and negative pairs. However, prior works [33, 64] have identified theoretical limitations of maximizing MI using the KL divergence: Because KL divergence is not a metric, it is sensitive to small differences in data samples regardless of the geometry of the underlying data distributions. Therefore, the encoder  $\phi$  can capture limited information shared between  $X$  and  $V$  as long as the differences are sufficient to maximize the KL divergence. Note that this can be especially detrimental in the presence of noisy views, as the learner can quickly settle on spurious correlations in false positive pairs due to the absence of the actual shared information.
+
+RINCE is a lower bound of WDM. We now establish RINCE as a lower bound of WDM [33], which is proposed as a replacement for the KL divergence in MI estimation.
+
+WDM is based on the Wasserstein distance, a distance metric between probability distributions defined via an optimal transport cost. Letting  $\mu$  and  $\nu \in \mathrm{Prob}(\mathbb{R}^d\times \mathbb{R}^d)$  be two probability measures, we define the Wasserstein-1 distance with a Euclidean cost function as
+
+$$
+\mathcal{W}(\mu ,\nu) = \inf_{\pi \in \Pi (\mu ,\nu)}\mathbb{E}_{\substack{(X,V)\\ (X^{\prime},V^{\prime})\sim \pi}}\left[\| X - X^{\prime}\| +\| V - V^{\prime}\|\right]
+$$
+
+where  $\Pi (\mu ,\nu)$  denotes the set of measure couplings whose marginals are  $\mu$  and  $\nu$ , respectively. By virtue of symmetry when  $q = 1$ , if  $\lambda >1 / (K + 1)$ , the Kantorovich-Rubinstein duality [65] implies that (full theorem in Appendix A.3):
+
+$$
+\begin{array}{l} - \mathbb {E} \left[ \mathcal {L} _ {\text {R I N C E}} ^ {\lambda , q = 1} (\mathbf {s}) \right] \leq L \cdot I _ {\mathcal {W}} (\phi (X), \phi (V)) \\ := L \cdot \mathcal {W} \left(P _ {X V} ^ {\phi}, P _ {X} ^ {\phi} P _ {V} ^ {\phi}\right), \tag {4} \\ \end{array}
+$$
+
+where  $I_{\mathcal{W}}(\phi(X), \phi(V))$  is the WDM defined in [33] and  $L$  is a constant that depends on  $t, \lambda$ , and the Lispchitz constant of the projection head  $f$ . Note that we are not aware of any work that showed it is possible to establish a similar bound with WDM for the InfoNCE loss.
+
+This provides another explanation of what makes RINCE robust against noisy views. Unlike InfoNCE which maximizes the KL divergence, optimizing RINCE is equivalent to maximizing the WDM with a Lipschitz function. Equipped with a proper metric, this allows RINCE to measure the divergence between two distributions  $P_{XV}^{\phi}$  and  $P_X^\phi P_V^\phi$  without being overly sensitive to individual sample noise, as long as the noise does not alter the geometry of the distributions. This also allows the encoder  $\phi$  to learn more complete representations, as maximizing the Wasserstein distance requires the encoder to not only model the density ratio between the two distributions but also the optimal cost of transporting one distribution to another.
+
+RINCE is still a lower bound of WDM even with noise. Finally, we show that RINCE still maximizes the noise-less WDM under additive noise, corroborating the robustness of RINCE. Let's consider a simple mixture noise model:
+
+$$
+P _ {X V} ^ {\eta} = (1 - \eta) P _ {X V} + \eta P _ {X} P _ {V},
+$$
+
+where  $\eta$  is the noise rate and the noisy joint distribution  $P_{XV}^{\eta}$  is a weighted sum between the noise-less positive distribution  $P_{XV}$  and negative distribution  $P_{X}P_{V}$ . Note that the marginals of  $P_{XV}^{\eta}$  are still  $P_{X}$  and  $P_{V}$  by construction. The intuition behind mixture noise model is that when we draw positive pairs from  $P_{XV}^{\eta}$ , we obtain false positives from  $P_{X}P_{V}$  with probability  $\eta$ . Via the symmetry of the contrastive loss, we can extend bound (4) as follows (proof in Appendix A.4):
+
+$$
+- \mathbb {E} _ {P _ {X V} ^ {\eta}} \left[ \mathcal {L} _ {\text {R I N C E}} ^ {\lambda , q = 1} (\mathbf {s}) \right] \leq (1 - \eta) \cdot L \cdot I _ {\mathcal {W}} (\phi (X), \phi (V)).
+$$
+
+![](images/eb8da3ba65802b75c068c51ab0444b005b6e6af58560d9fcb5a5f1d5b6d37412.jpg)  
+(a) Label Noise
+
+![](images/6739378aa5fd7b73bbaf30413455c0ea9be9292e1ce513d4914bf1e72649eacb.jpg)  
+(b) Augmentation Noise
+
+![](images/a20996e6915b209e35ddbb555f4da95c4a930049e7547641c2823a2712004dcf.jpg)  
+Figure 4. Noisy CIFAR-10. We show the top-1 accuracy of RINCE with different values of  $q$  across different noise rate  $\eta$ . Large  $q$  ( $q = 0.5, 1$ ) leads to better robustness, while smaller  $q$  ( $q = 0.01$ ) performs similar to InfoNCE ( $q \to 0$ ).
+
+![](images/78bf22324ba47a3e2c9d4b225fe72ffa54a828fa46ab1fdd9f20a1c9aa55cb89.jpg)
+
+![](images/e285d1206643d515d582eb55759a60ce4964208c7b83ac561738dc32fbf1ecde.jpg)
+
+![](images/5296a1836b6aa8fea805ef06401eb63b2b21744cb0e03170d3a30a896b5fb4f0.jpg)  
+Figure 5. t-SNE Visualization on CIFAR-10 with label noise. Colors indicate classes. RINCE leads to better class-wise separation than the InfoNCE loss in both noise-less and noisy cases.
+
+![](images/892ed67cd09e5cb8b8e9a3d1831cc7b70ba30ee154f81e046ccb82c02dc72784.jpg)
+
+![](images/6cb9c37657d0c7abb0471c24369ae3d97e5b24160dbd6c61124bc421e84ca3ee.jpg)
+
+Comparing to the bound (4), the right hand side is reweighted with  $(1 - \eta)$ . This implies that minimizing RINCE with noisy views still maximizes a lower bound of noise-less WDM. Despite the simplicity of the analysis, it intuitively relates dependency measures and the noisy views with interpretable bounds. It would be an interesting future direction to extend the analysis to more complicated noise models, e.g.,  $P_{XV}^{\eta} = (1 - \eta)P_{XV} + \eta Q_{XV}$ , where  $Q$  is an unknown perturbation on positive distribution.
+
+# 5. Experiments
+
+We evaluate RINCE on various contrastive learning scenarios involving images (CIFAR-10 [36], ImageNet [37]), videos (ACAV100M [27], Kinetics400 [38]) and graphs (TUDataset [39]). Empirically, we find that RINCE is insensitive to the choice of  $\lambda$ ; we simply set  $\lambda = 0.01$  for all vision experiments and  $\lambda = 0.025$  for graph experiments.
+
+# 5.1. Noisy CIFAR-10
+
+We begin with controlled experiments on CIFAR-10 to verify the robustness of RINCE against synthetic noise by controlling the noise rate  $\eta$ . We consider two noise types:
+
+<table><tr><td>Method</td><td>Δ to SimCLR [6]</td><td>Top 1</td><td>Top 5</td></tr><tr><td>Supervised [67]</td><td>N/A</td><td>76.5</td><td>-</td></tr><tr><td>SimSiam [43]</td><td>No negative pairs</td><td>71.3</td><td>-</td></tr><tr><td>BYOL [8]</td><td>No negative pairs</td><td>74.3</td><td>91.6</td></tr><tr><td>Barlow Twins [9]</td><td>Redundancy reduction</td><td>73.2</td><td>91.0</td></tr><tr><td>SwAV [44]</td><td>Cluster discrimination</td><td>75.3</td><td>-</td></tr><tr><td>SimCLR [6]</td><td>None</td><td>69.3</td><td>89.0</td></tr><tr><td>+RINCE (Ours)</td><td>Symmetry controller q</td><td>70.0</td><td>89.8</td></tr><tr><td>MoCo [5]</td><td>Momentum encoder</td><td>60.6</td><td>-</td></tr><tr><td>MoCo-v2 [7]</td><td>Momentum encoder</td><td>71.1</td><td>90.1</td></tr><tr><td>MoCo-v3 [35]</td><td>Momentum encoder</td><td>73.8</td><td>-</td></tr><tr><td>+RINCE (Ours)</td><td>Symmetry controller q</td><td>74.2</td><td>91.8</td></tr></table>
+
+Table 1. Linear Evaluation on ImageNet. All the methods use ResNet-50 [67] as backbone architecture with 24M parameters.
+
+Label noise. We start with the case of supervised contrastive learning [41] where positive pairs are different images of the same label. This allows us to control noise in the traditional sense, i.e., learning with noisy labels. Similar to [56], we flip the true labels to semantically related ones, e.g.,  $\mathrm{CAT} \leftrightarrow \mathrm{DOG}$  with probability  $\eta / 2$ . This is commonly referred to as class-dependent noise [55, 56, 57].
+
+Augmentation noise. We consider the self-supervised learning scenario and vary the crop size during data augmentation similar to [20], i.e., after applying all the transformations as in SimCLR [6], images are further cropped into  $1/5$  of their original size with probability  $\eta$ . This effectively controls the noise rate as cropped patches will most likely be too small to contain any shared information.
+
+Figure 4 shows the results of SimCLR trained with InfoNCE and RINCE with different choices of  $q$  and  $\lambda$ . When the augmentation noise is present, e.g.,  $\eta = 0.4$ , the accuracy of InfoNCE drops from  $91.14\%$  to  $87.33\%$ . In contrast, the robustness of RINCE is enhanced by increasing  $q$ , achieving  $89.01\%$  when  $q = 1.0$ . InfoNCE also fails to address label noise and suffers from significant performance drop (93.38% → 87.11% when  $\eta = 0.8$ ). In comparison, RINCE retains the performance even when the noise rate is large (91.59% for  $q = 1.0$ ). In both cases, reducing the value of  $q$  makes the performance of RINCE closer to InfoNCE, verifying our analysis in Lemma 1.
+
+Figure 5 shows t-SNE visualization [66] of representations learned with InfoNCE and RINCE ( $q = 1.0$ ) under different label noise. As the noise rate increases, representations of different classes start to tangle up for InfoNCE, while RINCE still achieves decent class-wise separation.
+
+# 5.2. Image Contrastive Learning
+
+We verify our approach on the well-established ImageNet benchmark [37]. We adopt the same training proto
+
+<table><tr><td>Method</td><td>Backbone</td><td>Finetune Input Size</td><td>HMDB</td><td>UCF</td></tr><tr><td>3D-RotNet [68]</td><td>R3D-18</td><td>16 × 1122</td><td>33.7</td><td>62.9</td></tr><tr><td>ClipOrder [69]</td><td>R3D-18</td><td>16 × 1122</td><td>30.9</td><td>72.4</td></tr><tr><td>DPC [70]</td><td>R3D-18</td><td>25 × 1282</td><td>35.7</td><td>75.7</td></tr><tr><td>CBT [71]</td><td>S3D</td><td>16 × 1122</td><td>44.6</td><td>79.5</td></tr><tr><td>AVTS [72]</td><td>MC3-18</td><td>25 × 2242</td><td>56.9</td><td>85.8</td></tr><tr><td>SeLaVi [73]</td><td>R(2+1)D-18</td><td>32 × 1122</td><td>47.1</td><td>83.1</td></tr><tr><td>XDC [74]</td><td>R(2+1)D-18</td><td>32 × 2242</td><td>52.6</td><td>86.8</td></tr><tr><td>Robust-xID [24]</td><td>R(2+1)D-18</td><td>32 × 2242</td><td>55.0</td><td>85.6</td></tr><tr><td>Cross-AVID [19]</td><td>R(2+1)D-18</td><td>32 × 2242</td><td>59.9</td><td>86.9</td></tr><tr><td>AVID+CMA [19]</td><td>R(2+1)D-18</td><td>32 × 2242</td><td>60.8</td><td>87.5</td></tr><tr><td>InfoNCE (Ours)</td><td>R(2+1)D-18</td><td>32 × 2242</td><td>57.8</td><td>88.6</td></tr><tr><td>RINCE (Ours)</td><td>R(2+1)D-18</td><td>32 × 2242</td><td>61.6</td><td>88.8</td></tr><tr><td>GDT [23]</td><td>R(2+1)D-18</td><td>30 × 1122</td><td>62.3*</td><td>90.9*</td></tr></table>
+
+* Based on an advanced hierarchical data augmentation during pretraining.
+
+Table 2. Kinetics400-pretrained performance on UCF101 and HMDB51 (top-1 accuracy). Ours use the same data augmentation approach as Cross-AVID and AVID+CMA, while GDT uses a hierarchical sampling process to obtain good performance.
+
+col and hyperparameter settings of SimCLR [6] and MoCo-v3 [35] and simply replace the InfoNCE with our RINCE loss ( $q = 0.1$  and  $q = 0.6$ , respectively) as shown in Figure 2. Table 1 shows that RINCE improves InfoNCE (SimCLR and MoCo-v3) by a non-trivial margin. We also include results from the SOTA baselines, where they improve SimCLR by introducing dynamic dictionary plus momentum encoder (MoCo-v1/v2/v3 [5, 7, 35]), removing negative pairs plus the stop-gradient trick (SimSiam [43], BYOL [8]), or online cluster assignment (SwAV [44]). In comparison, our work is orthogonal to the recent developments, and the existing tricks can be applied along with RINCE.
+
+Figure 6 shows the positive pairs from SimCLR augmentations and the corresponding positive scores  $s^{+} = f(x)^{T}g(v)$  output by trained RINCE model. Examples with lower positive scores contain pairs that is less informative to each other, while semantically meaningful pairs often have higher scores. This implies that positive scores are good noise detectors, and down-weighting the samples with lower positive score brings robustness during training, verifying our analysis in section 4.1.
+
+# 5.3. Video Contrastive Learning
+
+We examine our approach in the audio-visual learning scenario using two video datasets: Kinetics400 [38] and ACAV100M [27]. Here, we find that simple  $q$ -warmup improves the stability of RINCE, i.e.,  $q$  starts at 0.01 and linearly increases to 0.4 until the last epoch. We apply this to all RINCE models in this section. As we show below, RINCE outperforms SOTA noise-robust contrastive methods [19, 24] on Kinetics400, while also providing scalability and computational efficiency compared to InfoNCE.
+
+![](images/ffd827ed5db19978017d432b63fa829fb128a407333d49df38767eff6fbba613.jpg)
+
+![](images/bba44ce272f95d7d6d17c967cd6bd8753be616d0bc7064e0cc51832614362e53.jpg)
+
+![](images/adfd567fec294ca081fc0bbf0780f5b924520e5ce4a8f7c7048e29a98bf55671.jpg)
+
+![](images/05a6212c9d4f8b6f84c7311fa011e6ee98b37fd4d817fab1b2e279a93f651bdf.jpg)  
+$s^+$  < 0.75  
+Figure 6. Positive pairs and their scores. The positive scores  $s^+ \in [-1,1]$  are output by the trained RINCE model (temperature = 1). Pairs that have lower scores are visually noisy, while informative pairs often have higher scores.
+
+![](images/0f1b087bb407608899af95dc1bfcb2a9d06a0bc1b12c3225cf2a68732700c32f.jpg)
+
+![](images/6904c5a38da69a697de19737dd6fdeee31648a4fbc393548034be238cf5633f5.jpg)
+
+![](images/edecd7e93b0c891dad50b1982cab05af1f2f8baac2ee5a32af64843be4899428.jpg)
+
+![](images/a224a93f7cbf67376391fbfc9f6d571e6ef55efa3052c0ad39d532721b52c86c.jpg)
+
+![](images/1b02e76383725be5c0488ec0546dfafd93e4094dd5388673df80d931b098b51f.jpg)  
+$0.8 <   s^{+} <   0.9$
+
+![](images/e34ac211b485366eb8718bd0ec4e34e1300ca561d6b9535bd9767999bde8aa8e.jpg)
+
+![](images/1bc43e279f491ae435f67f8c4f60776122e40a821a3a1fe3f049f48f4df45122.jpg)
+
+![](images/79160e7e2a3d6cf32b494a79031b005516afb523a1561a94ea13cc4ebd2b9e82.jpg)
+
+![](images/b6d3b367829703105c4480e108f23232c2cde1dff83de7e9fba512705d5df912.jpg)
+
+![](images/4dd64278c362d85efc350a107de0f32f58893fbfd9e43b99aecc679c9c1fd0ed.jpg)  
+$s^+ > 0.95$
+
+Kinetics400 For fair comparison to SOTA, we follow the same experimental protocol and hyperparameter settings of [19] and simply replace their InfoNCE loss functions with RINCE as shown in Figure 2. We use the same network architecture, i.e., 18-layer  $\mathrm{R}(2 + 1)\mathrm{D}$  video encoder [75], 9-layer VGG-like audio encoder, and 3-layer MLP projection head producing 128-dim embeddings. We use the ADAM optimizer [76] for 400 epochs with 4,096 batch size, 1e-4 learning rate and 1e-5 weight decay. The pretrained encoders are finetuned on UCF-101 [77] and HMDB-51 [78] with clips composed of 32 frames of size  $224 \times 224$ . We defer the full experimental details to Appendix B.
+
+Table 2 shows RINCE outperforming most of the baselines, including Robust-xID [24] and AVID+CMA [19] which are recent InfoNCE-based SOTA methods proposed to address the noisy view issues in audio-visual contrastive learning. Considering the only change required is the simple replacement of the InfoNCE with our RINCE loss, the results clearly show the effectiveness of our approach. The simplicity means we can easily apply RINCE to a variety of InfoNCE-based approaches, such as GDT [23] that uses advanced data augmentation to achieve SOTA results.
+
+ACAV100M We conduct an in-depth analysis of RINCE on ACAV100M [27], a recent large-scale video dataset for self-supervised learning. Compared to Kinetics400 which is limited to human actions, ACAV100M contains videos "in the wild" exhibiting a wide variety of audio-visual patterns. The unconstrained nature of the dataset makes it a good benchmark to investigate the robustness of RINCE to various types of real-world noise, e.g., background music, overdubbed audio, studio narrations, etc.
+
+We focus on evaluating the (a) scalability and (b) conver
+
+![](images/d5c05f6129f8512b9f1109f7f86c0309ffc555d5f6fcd2ff9f5d86a6288d118b.jpg)  
+(a) Scalability
+
+![](images/e222b2c3da6f480f97d6d3a6b5c877f5f2ab853c89d4bbd6b7d00f55c1eb81d9.jpg)  
+(b) Convergence  
+Figure 7. RINCE outperforms InfoNCE with fewer epochs across different scales The results are based on ACAV100M-pretrained models transferred to UCF-101.
+
+gence rate of RINCE, thereby answering the question: Will it retrain its edge over InfoNCE (a) even in the large-scale regime and (b) with a longer training time? We follow the same experimental setup as described above, but reduce the batch size to 512 and report the results only on the first split of UCF-101 to make our experiments tractable.
+
+Figure 7 (a) shows the top-1 accuracy of RINCE and InfoNCE across different data scales and training epochs. RINCE outperforms InfoNCE by a large margin at every data scale. In terms of the convergence rate, RINCE is comparable to or even outperforms fully-trained (200 epochs) InfoNCE models with only 100 or fewer epochs. Figure 7 (b) gives a closer look at the convergence at  $50\mathrm{K}$  and  $200\mathrm{K}$  scales. Interestingly, InfoNCE saturates and even degenerates after epoch 150, while RINCE keeps improving. This verifies our analysis in section 4.1: InfoNCE can overfit noisy samples due to its exploration property, while RINCE downweights them and continue to obtain the learning signal from clean ones, achieving robustness against noise.
+
+# 5.4. Graph Contrastive Learning
+
+To see whether the modality-agnostic nature of RINCE applies beyond image and video data, we examine our approach on TUDataset [39], a popular benchmark suite for graph inference on molecules (BZR, NCI1), bioinformatics (PROTEINS), and social network (RDT-B, IMDB-B). Unlike vision datasets, data augmentation for graphs requires careful engineering with domain knowledge, limiting the applicability of InfoNCE-type contrastive objectives.
+
+For fair comparison, we follow the protocol of [40] and train graph isomorphism networks [79] with four types of data augmentation: node dropout, edge perturbation, attribute masking, and subgraph sampling. We train models using ADAM [76] for 20 epochs with a learning rate 0.01 and report mean and standard deviation over 5 independent trials. We set  $q = 0.1$  for all the experiments in this section.
+
+Table 3 shows that RINCE outperforms SOTA InfoNCE-
+
+<table><tr><td>Methods</td><td>RDT-B</td><td>NCI1</td><td>PROTEINS</td><td>DD</td></tr><tr><td>node2vec [80]</td><td>-</td><td>54.9±1.6</td><td>57.5±3.6</td><td>-</td></tr><tr><td>sub2vec [81]</td><td>71.5±0.4</td><td>52.8±1.5</td><td>53.0±5.6</td><td>-</td></tr><tr><td>graph2vec [82]</td><td>75.8±1.0</td><td>73.2±1.8</td><td>73.3±2.1</td><td>-</td></tr><tr><td>InfoGraph [47]</td><td>82.5±1.4</td><td>76.2±1.1</td><td>74.4±0.3</td><td>72.9±1.8</td></tr><tr><td>GraphCL [40]</td><td>89.5±0.8</td><td>77.9±0.4</td><td>74.4±0.5</td><td>78.6±0.4</td></tr><tr><td>JOAO [83]</td><td>85.3±1.4</td><td>78.1±0.5</td><td>74.6±0.4</td><td>77.3±0.5</td></tr><tr><td>JOAOv2 [83]</td><td>86.4±1.5</td><td>78.4±0.5</td><td>74.1±1.1</td><td>77.4±1.2</td></tr><tr><td>InfoNCE* (Ours)</td><td>89.9±0.4</td><td>78.2±0.8</td><td>74.4±0.5</td><td>78.6±0.8</td></tr><tr><td>RINCE (Ours)</td><td>90.9±0.6</td><td>78.6±0.4</td><td>74.7±0.8</td><td>78.7±0.4</td></tr></table>
+
+* GraphCL [40] but uses the same data augmentation as RINCE.
+
+![](images/8432f25897adb5159c9a2aadbb261647f8314e9c97021bc7411cac8907a41449.jpg)  
+Table 3. Self-supervised representation learning on TUDataset: The baseline results are excerpted from the published papers.  
+Figure 8. Performance vs. Perturbation Rate: We increase the perturbation rate of node dropping, edge perturbation, and attribute masking from  $10\%$  to  $60\%$ . RINCE outperforms InfoNCE in terms of accuracy and variance when perturbation enhances.
+
+![](images/2ef20ff5e003aa92048944c16e7ec3f0fe695d9b2af643810c4c0ee58b6dabd4.jpg)
+
+based contrastive methods, GraphCL and JOAO /JOAOv2, setting the new records on all four datasets. GraphCL applies different augmentations for different datasets, while JOAO/JOAOv2 require solving bi-level optimization to choose optimal augmentation per dataset. In contrast, we apply the same augmentation across all four datasets and achieve competitive performance, demonstrating its generality and robustness. In Figure 8, we control perturbation rate by applying three augmentation types (node dropout, edge perturbation, attribute masking) to different  $\%$  of nodes/edges. We show results on two datasets most sensitive to augmentation. Again, RINCE consistently outperform InfoNCE and has relatively smaller variances when the noise rate increases.
+
+# 6. Conclusion
+
+We presented Robust InfoNCE (RINCE) as a simple drop-in replacement for the InfoNCE loss in contrastive learning. Despite its simplicity, it comes with strong theoretical justifications and guarantees against noisy views. Empirically, we provided extensive results across image, video, and graph contrastive learning scenarios demonstrating its robustness against a variety of realistic noise patterns.
+
+Acknowledgement This work was in part supported by NSF Convergence Award 6944221 and ONR MURI 6942251.
+
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+[81] B. Adhikari, Y. Zhang, N. Ramakrishnan, and B. A. Prakash, "Sub2vec: Feature learning for subgraphs," in Pacific-Asia Conference on Knowledge Discovery and Data Mining. Springer, 2018, pp. 170–182. 8  
+[82] A. Narayanan, M. Chandramohan, R. Venkatesan, L. Chen, Y. Liu, and S. Jaiswal, “graph2vec: Learning distributed representations of graphs,” arXiv preprint arXiv:1707.05005, 2017. 8  
+[83] Y. You, T. Chen, Y. Shen, and Z. Wang, “Graph contrastive learning automated,” arXiv preprint arXiv:2106.07594, 2021.8  
+[84] W. Falcon and K. Cho, “A framework for contrastive self-supervised learning and designing a new approach,” arXiv preprint arXiv:2009.00104, 2020. 17

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+# Robust Cross-Modal Representation Learning with Progressive Self-Distillation
+
+Alex Andonian* MIT CSAIL
+
+andonian@mit.edu
+
+Shixing Chen, Raffay Hamid  
+Amazon Prime Video
+
+{shixic, raffay}@amazon.com
+
+# Abstract
+
+The learning objective of vision-language approach of CLIP [63] does not effectively account for the noisy many-to-many correspondences found in web-harvested image captioning datasets, which contributes to its compute and data inefficiency. To address this challenge, we introduce a novel training framework based on cross-modal contrastive learning that uses progressive self-distillation and soft image-text alignments to more efficiently learn robust representations from noisy data. Our model distills its own knowledge to dynamically generate soft-alignment targets for a subset of images and captions in every minibatch, which are then used to update its parameters. Extensive evaluation across 14 benchmark datasets shows that our method consistently outperforms its CLIP counterpart in multiple settings, including: (a) zero-shot classification, (b) linear probe transfer, and (c) image-text retrieval, without incurring extra computational cost. Analysis using an ImageNet-based robustness test-bed [70] reveals that our method offers better effective robustness to natural distribution shifts compared to both ImageNet-trained models and CLIP itself. Lastly, pretraining with datasets spanning two orders of magnitude in size shows that our improvements over CLIP tend to scale with number of training examples.
+
+# 1. Introduction
+
+The convergence of self-supervised pretraining techniques in natural language processing and computer vision have brought about a renaissance of cross-modal representation learning methods [1, 19, 30, 39, 52, 63, 68, 75] where largescale weakly correlated multimodal data (e.g., image-text pairs) is used to learn cross-modal representations using contrastive learning techniques. In particular, the recently proposed CLIP [63] model has garnered significant attention due to its impressive zero-shot recognition ability and excellent transfer performance on downstream tasks.
+
+However, despite their recent success, multimodal pretraining methods like CLIP [63] are data and compute in-
+
+![](images/c656f8ccdfaf0185de409a80f9fb3ba66c758057f13fe4292937c98f595c867a.jpg)
+
+![](images/59c3e61d2993000306e719c3a9d9c9937333b0a08f514272bfa0d08fdfed74cd.jpg)  
+test image ground truth: goldfish  
+an out-of-distributionimage from ImageNet-R dataset showing a goldfish rendered in astained glass pane  
+pretrained models
+
+![](images/ab5baf01ae243ae5ab12a83afe84f89bd8d4b71663f06c9dd960ae8672ced2b2.jpg)
+
+![](images/d7e6b9f3fd0c538de5bb3c191a2eb60f4fc891c6969259bd385936e106bb76e3.jpg)  
+test image  
+model outputs  
+Figure 1. Illustrative Comparison with CLIP - CLIP [63] learns a joint vision-language embedding space by bringing corresponding image-text representation together (green links), while repelling unpaired instances away from each other (red links). This formulation does not account for potential semantic similarity between negative samples. We address this issue by learning to predict a distribution of soft-alignment targets (dotted blue edges) in a given minibatch, thereby allowing our model to learn more robust representations. This robustness is evident when comparing predicted distributions on an out-of-distribution image from ImageNet-R dataset [25], where unlike CLIP, our method can correctly classify a goldfish rendered in a stained glass pane.
+
+efficient. Much of CLIP's success can be attributed to its voracious appetite for training data, utilizing 400M image-text pairs and an estimated 3,584 GPU days for pretraining. As the scale of data increases, pretraining requirements of these methods become increasingly expensive, thereby limiting their widespread adoption in a sustainable manner.
+
+This data and compute inefficiency of CLIP [63] can be partially attributed to the underlying assumptions it makes about the web-harvested data it uses for training. Several mainstream vision-language datasets utilize the alt-text HTML attribute of images scraped from archived web pages [9, 67, 71] where captions can often have words unrelated to their corresponding image-content [67]. However, CLIP [63] models the caption for each image to be accurately and exclusively related to only that image (see Figure 1). Moreover, when using larger batch sizes (32K used for CLIP), the likelihood of observing negatives with
+
+high semantic similarity increases which can further degrade the learned representations especially those associated with shared semantics between faulty negatives [2].
+
+To address this challenge, we propose to model the many-to-many relationships between images of webharvested datasets and their corresponding captions more accurately using soft probabilities rather than hard pairing labels. Specifically, we propose a simple yet effective framework for robust contrastive language-image pretraining that uses progressive self-distillation and soft image-text alignment targets to more efficiently learn from noisy data. Instead of explicitly finding, correcting or even pruning noisy correspondences [75, 88], our joint student-teacher model dynamically generates a new set of soft-alignments for a random subset of images and captions in every minibatch. This enables our method to model many-to-many relationships while simultaneously re-calibrating potentially poorly matched instances without needing to identify them. Over the course of training, our network generates soft-alignments for increasingly large subsets of a minibatch, effectively becoming its own teacher. We identify several key elements that allow the student network to predict its targets without representation collapse or reinforcing its mistakes.
+
+We use multiple pretraining datasets to extensively compare our approach to CLIP [63] evaluated on 14 benchmark datasets, where our approach consistently outperforms CLIP under multiple settings. Analysis using an ImageNet-based robustness test-bed [70] shows that our method offers better effective robustness to natural distribution shifts compared to both ImageNet-trained models as well as CLIP. Pretraining with datasets spanning two orders of magnitude in size shows that our improvements over CLIP tend to scale with number of training examples. Lastly, the simplicity of our approach allows it to be readily incorporated into existing and future methods.
+
+# 2. Related Work
+
+a. Self-Supervised Representation Learning: Self-supervised learning (SSL) approaches use a pretext task to automatically generate a supervision signal from the data itself, thereby eliminating the dependence on expensive manual data-labeling [31]. Pretext tasks in computer vision include spatial reasoning [15, 20, 32, 54, 59], temporal context [23, 32, 36, 48, 49], and other visual properties such as hue [13, 35, 81, 82], brightness [29, 72] or optical flow [29, 62, 73, 79], reconstruction of modified inputs [59, 74, 81], and classifying inputs with pseudo-labels [15, 16, 58] or pseudo-clusters [7, 8, 86, 87]. A promising subset of SSL methods uses a variant of instance discrimination framework [17, 77] which learns to align augmented versions of features while distinguishing them from features of other instances using a contrastive loss [8, 11, 24, 47].
+
+b. Vision-Language Pretraining: Joint vision-language pretraining (VLP) is an active research area [1,19,39,63,68] where the availability of large-scale image-text datasets e.g., YFCC100M [71] and Conceptual Captions [9, 67] has played a key role in its progress. Although multiple concurrent works are being proposed to further improve VLP models [75], our work is different from them in a few important ways. Specifically, unlike EfficientCLIP [75] that proposes an ensemble approach to obtain a less noisy data subset for cross-modal training, our method attempts to sidestep this problem altogether by re-purposing as opposed to completely removing noisy data. Similarly, DeCLIP [39] improves on the data-efficiency of CLIP [63] by leveraging intra-model contrastive learning along with a nearest-neighbor feature bank to augment negatives. However, incorporating these supervision sources can be computationally expensive. In contrast, our approach offers a simple yet effective way to improve the data-efficiency of CLIP [63] without incurring additional computational cost.
+
+c. Learning from Noisy Data: Several techniques have been developed to increase the robustness of label noise especially in the supervised context [22, 40, 60, 65, 83]. These techniques include loss functions that reduce impact of outliers [76, 83], metalearning procedures that learn how to correct sources of label noise [1, 37, 69, 84], loss correction approaches that model label noise [50, 51, 65], regularization techniques aimed at lowering the impact of noise [61], and noise filtering processes that iteratively refine dataset labels and retrain models to obtain a more robust final model [55]. However, these works investigate noise-robust methods in the context of common object detection and classification tasks and cannot directly be applied effectively to cross-modal pretraining tasks. As it stands, noise-robust VLP pretraining methods are still a relatively unexplored topic.
+
+d. Knowledge Distillation: Approaches for knowledge distillation (KD) [27] aim to transfer knowledge from one model (i.e., a teacher) to another (i.e., a student). While KD techniques are often motivated by certain performance and efficiency goals [6, 10, 38, 66], researchers have also found that KD methods serve as an effective regularization technique that can reduce model overfitting and improve generalization capabilities [14, 40, 43, 53]. Our approach is motivated by the recent success of self-knowledge distillation approaches [21, 78, 80] that use the student network as a teacher under supervised settings to achieve high accuracy with reduced computational cost. To the best of our knowledge, we are among the first to investigate progressive self-distillation in the context of vision-language pretraining.
+
+# 3. Methods
+
+Unlike CLIP [63], we view the problem of learning aligned vision-language representations from web-scale weakly-annotated data as the challenge of learning many-to-many
+
+relationships from noisy image-text correspondences. To address this challenge, we propose a novel vision-language pretraining method that progressively distills a model's own knowledge to soften its initially hard-target alignments, thereby enabling it to learn more transferable representations from the same amount of training data (see Figure 2). In the following, we first establish cross-modal contrastive learning objective and identify some of its limitations. We then introduce our novel progressive self-distillation approach and explain how it addresses these limitations.
+
+# 3.1. Preliminaries
+
+We consider a batch of N semantically paired image-text tuples  $\{(v_i, t_i)\}_{i=1:N}$  drawn from a cross-modal dataset. The goal of cross-modal contrastive pretraining is to learn encoders  $f_v$  for image data and  $f_t$  for text data such that for a given semantically related instance  $(v_i, t_i)$ , the encoded  $\ell_2$ -normalized embeddings  $\mathbf{v}_i = f_v(v_i)$  and  $\mathbf{t}_i = f_t(t_i)$  with  $\mathbf{v}_i, \mathbf{t}_i \in \mathbb{R}^d$  are close together (i.e., "aligned") under some distance metric, while the unpaired image and text embeddings are farther apart (i.e., "unaligned").
+
+# 3.2. Contrastive Learning with InfoNCE Loss
+
+Recall that CLIP [63] trains these image and text encoders with a contrastive loss by minimizing the InfoNCE [56] loss  $\mathcal{L}_{\mathrm{InfoNCE}} = \mathcal{L}_v + \mathcal{L}_t$ , where  $\mathcal{L}_v$  is the loss for aligning images to text and  $\mathcal{L}_t$  is the loss for aligning text to images. Specifically,  $\mathcal{L}_v$  is defined as:
+
+$$
+\mathcal {L} _ {v} = - \frac {1}{N} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} I _ {i j} \log P _ {v} \left(\mathbf {v} _ {i}, \mathbf {t} _ {j}; \tau\right) \tag {1}
+$$
+
+$$
+\mathbf {P} _ {v} \left(\mathbf {v} _ {i}, \mathbf {t} _ {j}; \tau\right) = \frac {\exp \left(\operatorname {s i m} \left(\mathbf {v} _ {i} , \mathbf {t} _ {j}\right) / \tau\right)}{\sum_ {k = 1} ^ {N} \exp \left(\operatorname {s i m} \left(\mathbf {v} _ {i} , \mathbf {t} _ {k}\right) / \tau\right)} \tag {2}
+$$
+
+where  $\mathrm{sim}(\mathbf{v}_i,\mathbf{t}_j) = \mathbf{v}_i^T\mathbf{t}_j$  is typically chosen to be the dot product (cosine similarity),  $\tau$  is a learnable softmax temperature parameter and  $\mathbf{I}_{ij}$  is an element from the identity matrix  $\mathbf{I}_{\mathrm{N}}$ . Since InfoNCE is symmetric,  $\mathcal{L}_t$  and  $\mathrm{P}_t$  are defined in a similar manner.
+
+For convenience, let  $\mathbf{V},\mathbf{T}\in \mathbb{R}^{N\times d}$  be matrices that contain a batch of image and text embeddings whose rows are populated with  $\mathbf{v}_{1:\mathrm{N}}$  and  $\mathbf{t}_{1:\mathrm{N}}$ , respectively. Then, the InfoNCE loss can be re-written compactly in matrix form as:
+
+$$
+\mathcal {L} _ {\text {I n f o N C E}} = \mathcal {H} \left(\mathbf {I} _ {\mathrm {N}}, \rho \left(\mathbf {V T} ^ {\top}\right)\right) + \mathcal {H} \left(\mathbf {I} _ {\mathrm {N}}, \rho \left(\mathbf {T V} ^ {\top}\right)\right), \tag {3}
+$$
+
+where  $\mathcal{H}$  is the batched (row-wise) cross-entropy function with mean reduction and  $\rho$  is the standard softmax function applied row-wise such that each row sums to one.
+
+Equation 3 shows that InfoNCE loss is simply the cross entropy between a one-hot distribution  $\mathrm{I}_{ij}$  and estimated probability  $\mathrm{P}_v(\mathbf{v}_i, \mathbf{t}_j; \tau)$ . It enforces the strict assumption that an image  $v_i$  selected from a batch should be paired exclusively with text  $t_i$  while being repelled from all other  $t_j$ .
+
+However, this assumption generally does not hold for two important reasons. First, it is likely that a given image would be aligned to several text captions to different degrees especially under large batch-size settings. Second, the ground truth pairings in large-scale weakly-annotated datasets may be simply incorrect or describe loose correlations between images and their corresponding texts.
+
+# 3.3. Distillation through Soft-Alignments
+
+To address the aforementioned limitations of using the InfoNCE loss to train on noisy cross-modal data, we propose to adopt a knowledge distillation framework where the predictive probabilities produced by a teacher network are used as soft target distributions to train a student network.
+
+Our framework offers two key advantages. First, in the process of generating target image-text alignments, a well-trained teacher can combat poorly captioned images by re-pairing them with stronger semantic matches from the batch, thereby providing a cleaner learning signal for the student network. Second, by providing soft targets, the teacher can convey many-to-many relationships in a batch.
+
+Specifically, to estimate the correspondence between image  $v_{i}$  and text  $t_{i}$ , our teacher model employs image and text encoders  $\tilde{f}_{v}$  and  $\tilde{f}_{t}$  to compute  $\ell_{2}$ -normalized teacher embeddings  $\tilde{\mathbf{v}}_{i}$  and  $\tilde{\mathbf{t}}_{j}$  which are similarly batched as rows in the matrices  $\tilde{\mathbf{V}}$  and  $\tilde{\mathbf{T}}$  respectively. Our method uses a swapped prediction strategy to produce soft target distributions  $\mathbf{A}^{v}$  and  $\mathbf{A}^{t}$  to supervise student training. These target distributions are defined as:
+
+$$
+\mathbf {A} ^ {v} = \rho (\tilde {\mathbf {T}} \tilde {\mathbf {V}} ^ {\top}; \tilde {\tau}) \text {a n d} \mathbf {A} ^ {t} = \rho (\tilde {\mathbf {V}} \tilde {\mathbf {T}} ^ {\top}; \tilde {\tau}) \tag {4}
+$$
+
+where  $\rho$  is the standard softmax function now using a secondary teacher temperature  $\tilde{\tau}$  that transforms and re-scales raw logits into probabilities.
+
+Swapped prediction improves on the well established forward bootstrapping approach [65] by using predictions from opposite modality. Unlike bootstrapping, swapped prediction computes the image alignment scores  $\mathbf{A}^v$  from the text encoder posterior probabilities and vice-versa, thus aggregating information over all other instances from the opposite modality. Intuitively, the strength of alignment from image  $v_i$  to text  $t_j$  is based on the probability that text  $t_j$  should be matched with image  $v_i$  compared to all other  $v_j$ . This strategy has shown promise in related contrastive learning settings [50], which is consistent with our results.
+
+Replacing the  $\mathrm{I_N}$  targets in Equation 3 with estimated soft-alignment probabilities  $\mathbf{A}^v$  and  $\mathbf{A}^t$  allows the teacher to re-calibrate the attractive and repulsive forces between image and text embeddings in the representation space based on its estimated similarity between instances. For instance, a faulty negative pair  $(v_i,t_j)$ , which may have high semantic similarity is assigned a similarity score of zero by
+
+![](images/0d20c8e0e43191d020441de79a73ad54102ec3401eb6087de4143c6a1a0129b1.jpg)  
+Figure 2. Approach Overview - We partition a minibatch into aligned instances (rows above the dotted red line) and unaligned instances (rows below the dotted red line). Our teacher network is trained on the aligned instances using the standard InfoNCE loss. By normalizing the predictions from the opposite modality (columns to the right of the dotted yellow line), the teacher provides the estimated soft-alignments for the unaligned data to supervise the student. Over the course of training, the proportion of soft-alignments provided to the student increases as the teacher's representations become more reliable.
+
+the InfoNCE loss, whereas our method provides  $\mathbf{A}_{ij}$  as a target which should be larger given a well trained teacher.
+
+# 3.4. Progressive Self-Distillation
+
+We now explain how to start such a teacher network and how its contribution to learning process evolves over time.
+
+# 3.4.1 Teacher Network Selection
+
+Conventional KD and SSL methods offer numerous potential teacher choices, e.g., larger but static pretrained teacher networks [27], or networks that share the same model architecture but use weights from a previous epoch [42], or as an exponential moving average [24]. The primary drawback to these approaches is reduced computational and memory efficiency as they require a secondary inference stage using additional model weights that must be kept in memory.
+
+To circumvent these issues, we adopt a self-distillation strategy where the student network acts as its own teacher (i.e.,  $f_v = \tilde{f}_v, f_t = \tilde{f}_t$ ). The idea here is to update the targets of the student contrastive objective using the current state of the model. Intuitively, as the learning improves over time, its representation can be trusted to make more accurate predictions. This mitigates the negative effects of noisy pairings as incorrect pairs are increasingly likely to be inconsistent with the consensus learned from the rest of the data as training progresses. By refining inconsistent alignments, the model can develop more coherent representations, which further improves its ability to evaluate the consistency of noisy image-text pairs.
+
+# 3.4.2 Progressing from Student to Teacher
+
+As our objective relies on some basic level of alignment between corresponding image and text representations, we introduce a novel procedure that progressively increases the
+
+contribution of self-distillation to the contrastive learning process over the course of training. Our model therefore dynamically evolves into its own teacher as training progresses, which differs from the standard knowledge distillation setting where the teacher is often static and separate.
+
+We achieve this dynamic progression by randomly partitioning a batch of N image-text pairs into  $\mathrm{N}^a = \lfloor \alpha \mathrm{N}\rfloor$  "aligned" instances and  $\mathrm{N}^u = \mathrm{N} - \lfloor \alpha \mathrm{N}\rfloor$  "unaligned" instances where  $\alpha \in [0,1]$  determines their relative proportions. The aligned subset is used to train the teacher network using the hard ground truth pairings and the standard InfoNCE loss. The teacher network then employs our aforementioned swapped prediction strategy to estimate soft-alignments on the unaligned instances to supervise the student. We refer to this random minibatch partitioning as dynamic as opposed to static as the global partitioning of instances into aligned and unaligned subsets is refreshed for each training epoch.
+
+To increase the strength of the teacher's influence on learning, we decrease the value of  $\alpha$  gradually, in the same way that the learning rate can be scheduled. While there are several strategies to decrease  $\alpha$  as a function of the training iteration, e.g., step-wise, linear, etc., we use a cosine-annealing schedule [38] specified by a start and end value.
+
+To summarize our overall learning procedure, we first compute the batched student-teacher embedding with  $\mathbf{V} = \tilde{\mathbf{V}}$  and  $\mathbf{T} = \tilde{\mathbf{T}}$ . Next, we extract the first  $\mathrm{N}^a$  rows to form aligned subset of teacher embeddings  $\tilde{\mathbf{V}}^a$ ,  $\tilde{\mathbf{T}}^a$ , and the last  $\mathrm{N}^u$  rows for the unaligned student embeddings  $\mathbf{V}^u$ ,  $\mathbf{T}^u$ . Altogether, our final objective function is defined as:
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {I n f o N C E}} ^ {\text {P S D}} = \alpha \left[ \mathcal {H} \left(\mathbf {I} _ {\mathrm {N} _ {a}}, \rho (\tilde {\mathbf {V}} ^ {a} \tilde {\mathbf {T}} ^ {\top})\right) + \mathcal {H} \left(\mathbf {I} _ {\mathrm {N} _ {a}}, \rho (\tilde {\mathbf {T}} ^ {a} \tilde {\mathbf {V}} ^ {\top})\right) \right] + \\ (1 - \alpha) [ \mathcal {H} (\mathbf {A} ^ {v}, \rho (\mathbf {V} ^ {u} \mathbf {T} ^ {\top})) + \mathcal {H} (\mathbf {A} ^ {t}, \rho (\mathbf {T} ^ {u} \mathbf {V} ^ {\top})) ] \\ \end{array}
+$$
+
+where  $\mathbf{I}_{\mathrm{N}_a} \in \mathbb{R}^{\mathrm{N}_a \times \mathrm{N}}$  is the zero-padded identity matrix, while  $\mathbf{A}^v, \mathbf{A}^t$  are indexed to match the unaligned student embeddings.
+
+# 4. Experiments
+
+We start by describing our experimental setup which aims to match CLIP [63] as closely as possible for fair comparison. We then demonstrate the advantages of our method over CLIP [63] for: (a) zero-shot classification, (b) finetuning (i.e., linear probe), and (c) image-text retrieval.
+
+# 4.1. Pretraining Datasets
+
+We apply our pretraining method to three image-text datasets varying in scale, scope and noise:
+
+a. MS COCO Captions [41] - A widely used standard image captioning benchmark dataset with approximately 118K images, each labeled with 5 human evaluated captions, and a testing set of 5K images.  
+b. Conceptual Captions 3M (CC3M) [67] - A collection of over 3M images and their raw descriptions harvested from the alt-text HTML attribute associated with the web-scraped images, therefore representing a wider variety of content styles. After downloading and preprocessing, we utilized about 2.9M image-text pairs in our experiments.  
+c. Conceptual Captions 12M (CC12M) [9] - By relaxing multiple image and text filters used in CC3M [67], CC12M is a less precise but  $4 \times$  larger set of image-text pairs that covers a wider range of visual concepts. Due to unavailable URLs, we utilize about 10M examples from this dataset.
+
+# 4.2. Pretraining Details
+
+In the following experiments, the image encoders follow ViT-B/32 vision transformer architecture proposed in CLIP [63], while the text encoder's transformer-based architecture follows modification proposed in [63]. Image and text features are projected to a shared 512-D space and L2 normalized before participating in contrastive loss.
+
+Models are trained from scratch for 100 epochs using the Adam optimizer [33] with weight decay and a cosine annealing learning rate schedule with warmup [44]. As done in [63], the learnable temperature parameter  $\tau$  is initialized to 0.07 and clamped to values less than 100. Automatic mixed-precision [45] training is used to save on memory and achieve minibatch sizes of 4096. Input images are randomly cropped and resized to  $224 \times 224$  resolution during pretraining and the maximum length of the text is limited to 77 tokens via random sub-sequence sampling similar to [63]. Training is conducted on as many as 8 Nvidia A-100 GPUs with the longest experiments spanning up to several days. The partitioning factor  $\alpha$  is decayed from 0.8 to 0.2 over the course of training using cosine annealing. De
+
+tails regarding hyperparameter values for different datasets and models are provided in the supplementary material.
+
+We compare our approach to a re-implementation of CLIP based on the methods and pseudo-code described in the original paper as well as open source implementations such as OpenCLIP [28]. The performance of our re-implementation is consistent with that achieved by OpenCLIP and unreleased CLIP models trained on reduced subsets of the 400M dataset; thus, we retrain a baseline CLIP model for each pretraining dataset to serve as a proxy.
+
+# 4.3. Evaluation Details
+
+a. Evaluation Protocol - We measure zero-shot and linear probe classification performance using Top-1 accuracy. For the linear probe experiments, we train the linear classifier with the L-BFGS optimizer on extracted visual features as described in [63]. We use standard retrieval metrics: recall at rank K (R@K, higher is better) and mean rank (MnR, lower is better), to evaluate the retrieval performance of our model. R@K (Recall at K) calculates the percentage of test samples for which the correct result is found in the top-K retrieved points to the query sample.  
+b. Benchmark Datasets - We evaluate the zero-shot and linear probe classification performance of proposed approach on a suite of benchmark evaluation datasets, including ImageNet [12], Places365 [85], ObjectNet [3], and several recent variants of ImageNet aimed at evaluating the robustness of trained models to natural (as opposed to synthetic) distribution shifts [25, 26, 64].  
+c. Prompt Ensembling with Templates - Consistent with previous work, we find that our approach benefits from prompt ensembling to augment the original class label for downstream tasks. For fair comparison, we use the same set of prompt templates published in CLIP [63], which generally take on the form of "a photo of a {label}."
+
+# 4.4. Zero-Shot Image Classification
+
+Following the pretraining stage, we evaluate our method on zero-shot image classification achieved through natural language input, and compare to its CLIP counterpart trained under identical settings.
+
+Table 1 lists the top-1 accuracy  $(\%)$  of zero-shot image classification across a suite of benchmark datasets. Given a fixed amount of pretraining data, our method considerably outperforms its CLIP counterparts in terms of average top-1 accuracy over all datasets, achieving as much as a  $6.19\%$  absolute improvement in the CC3M pretraining regime. Notably, our method surpasses CLIP on ImageNet and all of the ImageNet variants. Our method also achieves substantial performance gains on ImageNet-Renditions (ImageNet-R) [25], a dataset specifically aimed at evaluating out-of-distribution robustness, with an average
+
+<table><tr><td>Pretraining Dataset</td><td>Method</td><td>Cifar10 [34]</td><td>Cifar100 [34]</td><td>Caltech101 [18]</td><td>Places365 [85]</td><td>ObjectNet [3]</td><td>ImageNet-R [25]</td><td>ImageNet-O [26]</td><td>ImageNet-A [26]</td><td>ImageNet-V2 [64]</td><td>ImageNet [12]</td><td>Average</td></tr><tr><td rowspan="2">COCO</td><td>CLIP</td><td>64.14</td><td>19.57</td><td>32.88</td><td>12.78</td><td>4.98</td><td>8.27</td><td>8.0</td><td>3.32</td><td>7.41</td><td>8.18</td><td>16.87</td></tr><tr><td>Ours</td><td>66.74</td><td>24.49</td><td>34.26</td><td>14.15</td><td>6.18</td><td>11.25</td><td>9.85</td><td>5.32</td><td>8.99</td><td>9.49</td><td>19.07+2.22</td></tr><tr><td rowspan="2">CC3M</td><td>CLIP</td><td>73.90</td><td>30.60</td><td>54.07</td><td>24.54</td><td>4.49</td><td>28.33</td><td>18.5</td><td>7.827</td><td>21.43</td><td>23.56</td><td>28.73</td></tr><tr><td>Ours</td><td>80.15</td><td>38.27</td><td>64.45</td><td>28.07</td><td>9.21</td><td>37.31</td><td>26.2</td><td>10.81</td><td>26.70</td><td>27.96</td><td>34.91+6.19</td></tr><tr><td rowspan="2">CC12M</td><td>CLIP</td><td>75.29</td><td>41.94</td><td>75.86</td><td>31.29</td><td>12.50</td><td>51.34</td><td>31.90</td><td>13.25</td><td>34.89</td><td>37.87</td><td>40.61</td></tr><tr><td>Ours</td><td>84.84</td><td>51.34</td><td>80.00</td><td>34.08</td><td>15.24</td><td>59.29</td><td>33.0</td><td>18.85</td><td>39.16</td><td>42.24</td><td>45.85+5.23</td></tr></table>
+
+Table 1. Zero-Shot Image Classification Comparison - Zero-shot top-1 accuracy (%) of our method compared to baseline CLIP on numerous ImageNet-based benchmark datasets using different pretraining datasets varying in scale.
+
+![](images/7568af630a0c0d3e9583cf57cc6db03a8400d0737568fefe1ed57254cfea9343.jpg)  
+Figure 3. Pretraining Data Size and Relative Performance - Average zero-shot classification performance of our method compared to CLIP as number of pretraining examples from the CC12M dataset grows from 600K to 12M. Our method consistently learns representations more amenable to zero-shot classification than CLIP given same amount of pretraining data.
+
+performance gain of nearly  $8.5\%$  when considering Conceptual Captions datasets [9, 67]. Figure 1 shows how our method is able to provide more accurate predictions for a goldfish rendered in a stained glass pane.
+
+To better understand the relationship between the amount of pretraining data and relative differences in downstream performance between our method and CLIP, we perform a sweep of experiments that restrict pretraining to 5, 10, 25, 50, and  $100\%$  of the CC12M dataset. Figure 3 plots the average zero-shot classification score for both methods, and shows that our method maintains higher performance throughout this range. Interestingly, our method appears to provide the largest performance increases at the boundaries of the plot, suggesting that our method can improve data-efficiency in data-limited regimes, in addition to providing larger performance gains as the dataset size increases.
+
+![](images/f9740d0b900a8f3f3d95e44587d9d4faa1a0a6f2dd70c55292b8575f26980ebb.jpg)  
+Figure 4. Effective Robustness Evaluation - Our method produces features more robust to natural distribution shift compared to CLIP features. When comparing models with similar ImageNet performance, our method produces representations that offer better performance on naturally shifted distributions. Mean Transfer Accuracy is computed over the ImageNet-R/O/A/V2 test sets. Best-fit trend lines suggests that the effective robustness of our method (blue) outpaces CLIP (red) as ImageNet accuracy increases.
+
+# 4.5. Evaluation of Effective Robustness
+
+CLIP models [63, 70] have been found to be more robust to natural distribution shifts when compared to standard models trained on ImageNet. This phenomena is illustrated in Figure 4 with ImageNet accuracy on the x-axis and mean accuracy on ImageNet-A, ImageNet-O, ImageNet-R, and ImageNetV2, on the y-axis. Previous work [46, 70] has found that the in-distribution and out-of-distribution accuracies of ImageNet models follow a predictable linear trend (plotted in green), and CLIP models established a trend (plotted in red) of improved effective robustness. Note that the slope of linear-fit to our model is higher than that for the CLIP model, suggesting that our effective robustness improves over CLIP with increasing scale.
+
+<table><tr><td>Pretraining Dataset</td><td>Method</td><td>Food[1] [5]</td><td>OxfordPeds [57]</td><td>Bindsnap [4]</td><td>ImageNet [12]</td></tr><tr><td rowspan="2">COCO</td><td>CLIP</td><td>53.04</td><td>76.86</td><td>37.17</td><td>52.66</td></tr><tr><td>Ours</td><td>53.60</td><td>80.59</td><td>42.85</td><td>56.21</td></tr><tr><td rowspan="2">CC3M</td><td>CLIP</td><td>53.33</td><td>78.11</td><td>37.75</td><td>56.28</td></tr><tr><td>Ours</td><td>60.69</td><td>80.40</td><td>43.17</td><td>61.00</td></tr><tr><td rowspan="2">CC12M</td><td>CLIP</td><td>67.41</td><td>85.17</td><td>41.06</td><td>59.42</td></tr><tr><td>Ours</td><td>71.87</td><td>86.32</td><td>47.16</td><td>65.33</td></tr></table>
+
+# 4.6. Linear Probe Performance
+
+We report our linear probe performance on 4 downstream datasets in Table 2. Our method outperforms its CLIP counterpart in every case, suggesting that our learned visual features alone (i.e., considering within-modal alignment only) are more transferable than CLIP.
+
+# 4.7. Image-Text Retrieval
+
+Given the cross-modal nature of models under investigation, image-text retrieval consists of two sub-tasks: image-to-text and text-to-image. We evaluate our method on COCO test set with 5K images, each with 5 unique captions.
+
+Table 3 shows our zero-shot retrieval performance relative to baseline CLIP under COCO, CC3M and CC12M pretraining datasets. Relative to CLIP, our method consistently outperforms in both text-to-image and image-to-text subtasks, with both methods generally achieving slightly higher performance in text-to-image sub-task. Note that, due to data domain differences, pretraining directly on COCO produces models with higher downstream performance than pretraining on CC12M despite the  $100 \times$  increase in data size. The last four rows show that, as expected, finetuning Conceptual Captions pretrained models on COCO produces retrieval performance that exceeds corresponding baselines and the performance from training on COCO alone.
+
+# 4.8. Ablation Study
+
+We now investigate the contributions of individual components of our method, specifically the swapped prediction strategy for estimating soft-alignments over the forward bootstrapping strategy, dynamic minibatch partitioning, and progressive self-distillation to downstream performance. Following pretraining on COCO, we measure models' performance on COCO zero-shot image-text retrieval and ImageNet zero-shot classification.
+
+Table 2. Linear Probe Performance – Linear probe top-1 accuracy (%) of our method compared to baseline CLIP on four benchmark datasets is given. Our method learns visual features that consistently achieve improved finetuning performance, suggesting that our loss helps to improve within-modal feature alignment.  
+
+<table><tr><td colspan="3">Method</td><td colspan="2">Performance</td></tr><tr><td>Prediction Mode</td><td>Dynamic Partitioning</td><td>Progressive Distillation</td><td>IN Top-1</td><td>MS R1</td></tr><tr><td>Baseline</td><td>-</td><td>-</td><td>8.18</td><td>27.58</td></tr><tr><td>Forward</td><td>X</td><td>X</td><td>6.23</td><td>20.40</td></tr><tr><td>Swapped</td><td>X</td><td>X</td><td>8.46</td><td>23.51</td></tr><tr><td>Forward</td><td>✓</td><td>X</td><td>8.37</td><td>25.64</td></tr><tr><td>Swapped</td><td>✓</td><td>X</td><td>8.81</td><td>26.24</td></tr><tr><td>Forward</td><td>✓</td><td>✓</td><td>8.87</td><td>26.18</td></tr><tr><td>Swapped</td><td>✓</td><td>✓</td><td>9.51</td><td>28.48</td></tr></table>
+
+Table 4. Abalation Study - Our results using COCO for pretraining. IN Top-1 is zero-shot ImageNet accuracy and MS R1 is mean recall at rank 1 across image-to-text and text-to-image subtasks.
+
+Table 4 shows that directly using the model's posterior (bootstrapping) to assign alignments to a static subset of inputs leads to a degradation of learned representations and decreased performance compared to the baseline. Here the teacher network is likely reinforcing its own mistakes without penalty from the loss, leading to partial representation collapse for the unaligned examples. Utilizing the swapped prediction strategy restores zero-shot performance to slightly above baseline suggesting effective method for addressing this issue; however, retrieval performance is still lower than the baseline. By incorporating a dynamic partitioning strategy, our method exceeds baseline performance.
+
+# 4.9. Qualitative Analysis
+
+a. Distribution of Similarity Scores: Our method is largely motivated by the observation that previous VLP approaches neglect potential semantic similarity between negative samples and that accounting for this phenomenon can improve learned representations. In Figure 5, we plot the distribution of similarity scores for both positive and negative samples drawn from the COCO testing set. The left subplot reveals that our method consistently yields larger similarity scores for positive pairs compared to its CLIP counterpart and the OpenAI pretrained CLIP. Interestingly, the histogram of negative similarity scores shows that our method also assigns higher similarity scores to negative pairs. While it may seem counter intuitive to assign greater similarity scores to negative samples, we argue that doing so is the very reason our method captures greater similarity between positive pairs. By allowing some degree of alignment between the right set of negative examples, our method is able to minimize the inconsistencies between shared context of related positives and negatives. This in turn allows us to learn an overall more coherent representation space, resulting in increased robustness and downstream performance.
+
+b. Visualizing Text-Image Retrievals: In Figure 6, we show the comparative list of top ten retrieved images for 5 example text queries. Overall, these retrievals suggest that
+
+<table><tr><td rowspan="2">Pretraining Dataset</td><td rowspan="2">Method</td><td colspan="4">Text-to-Image</td><td colspan="4">Image-to-Text</td></tr><tr><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>MnR↓</td><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>MnR↓</td></tr><tr><td rowspan="2">COCO</td><td>CLIP</td><td>27.76</td><td>57.34</td><td>70.70</td><td>23.10</td><td>27.40</td><td>56.31</td><td>68.65</td><td>20.11</td></tr><tr><td>Ours</td><td>28.42</td><td>57.14</td><td>68.86</td><td>26.11</td><td>28.53</td><td>56.75</td><td>68.10</td><td>22.01</td></tr><tr><td rowspan="2">CC3M</td><td>CLIP</td><td>12.50</td><td>29.76</td><td>40.92</td><td>91.04</td><td>9.88</td><td>24.86</td><td>35.30</td><td>106.94</td></tr><tr><td>Ours</td><td>16.98</td><td>37.12</td><td>48.28</td><td>63.80</td><td>13.19</td><td>31.54</td><td>43.00</td><td>72.92</td></tr><tr><td rowspan="2">CC12M</td><td>CLIP</td><td>19.64</td><td>40.66</td><td>51.72</td><td>55.23</td><td>17.63</td><td>39.67</td><td>50.77</td><td>55.29</td></tr><tr><td>Ours</td><td>22.94</td><td>46.60</td><td>57.82</td><td>46.24</td><td>22.79</td><td>45.95</td><td>56.81</td><td>43.41</td></tr><tr><td rowspan="2">CC3M</td><td>CLIP*</td><td>31.30</td><td>59.54</td><td>71.80</td><td>21.15</td><td>29.18</td><td>58.66</td><td>70.23</td><td>17.02</td></tr><tr><td>Ours*</td><td>33.26</td><td>66.70</td><td>76.92</td><td>18.77</td><td>32.20</td><td>60.14</td><td>71.96</td><td>17.05</td></tr><tr><td rowspan="2">CC12M</td><td>CLIP*</td><td>35.22</td><td>62.74</td><td>73.46</td><td>17.70</td><td>33.36</td><td>61.15</td><td>73.36</td><td>15.76</td></tr><tr><td>Ours*</td><td>38.66</td><td>66.74</td><td>77.10</td><td>13.35</td><td>38.15</td><td>65.85</td><td>77.02</td><td>12.54</td></tr></table>
+
+Table 3. Image-Text Retrieval - Zero-shot retrieval performance on the COCO 5K testing set as measured by recall at 1,5,10 (higher is better) and mean rank (lower is better). The last four rows (marked with *) report the zero-shot retrieval results of the same models further finetuned on the COCO captions training set.
+
+![](images/bd59496439884e9328e2aaae16e014305f8996ac53c0087b897acfd9dccddeab.jpg)  
+Figure 5. Similarity Scores - Similarity scores distribution for positive and negative pairs in joint cross-modal space after training with original CLIP loss and our proposed loss is provided. Compared to our baseline (COCO) pretrained CLIP and OpenAI's pretrained CLIP model, our method yields similarity scores with higher mean and lower variance for positive pairs. While COCO pretrained CLIP model concentrates negatives' similarity scores around zero, our method concentrates them at higher levels as it allows for some degree of semantic similarity between negatives.
+
+our learned features more comprehensively capture all potential similarities between a text snippet and image. In contrast, baseline CLIP features tend to more narrowly focus on one specific commonality while neglecting others. For example, in the last row of Figure 6, CLIP strongly targets "living room" but misses the cat. Our approach on the other hand is successfully able to extract all the key aspects of the query including the cat, the chair and the living room. Similar trends are found in other examples.
+
+# 5. Conclusions
+
+We proposed a novel cross-modal contrastive learning framework with progressive self-distillation and soft image-text alignments. Our approach distills its own knowledge to dynamically generate soft-alignment targets for a subset of samples in every minibatch, which enables it to efficiently learn robust representations from noisy data. Exten
+
+![](images/05269cac1ad05e3cd184253cdb0d5bd9825db3b1e768e363b70e3eb7a04e541d.jpg)  
+Figure 6. Example Text-Image Retrievals - Given a text query, we display the top ten most semantically related images (ranked left to right) retrieved by CLIP and our method. Compared to CLIP, our method continues to retrieve images that more holistically match the text description, even after the ground truth image has appeared in the ranking.
+
+sive evaluations across 14 benchmark datasets showed that our method consistently outperforms its CLIP counterpart in multiple settings. Moreover, our method provides better effective robustness to natural distribution shifts compared to existing state-of-the-art methods. Going forward, we plan to further improve the efficiency of our approach by investigating data redundancy, network architectures and optimization algorithms during pretraining, so that we can generalize to even larger scale with fewer resources.
+
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+# Robust Egocentric Photo-realistic Facial Expression Transfer for Virtual Reality
+
+Amin Jourabloo* Fernando De la Torre† Jason Saragih Shih-En Wei Stephen Lombardi  
+Te-Li Wang Danielle Belko Autumn Trimble Hernan Badino  
+Facebook Reality Labs, Pittsburgh, PA  
+†Robotics Institute, Carnegie Mellon University
+
+# Abstract
+
+Social presence, the feeling of being there with a "real" person, will fuel the next generation of communication systems driven by digital humans in virtual reality (VR). The best 3D video-realistic VR avatars that minimize the uncanny effect rely on person-specific (PS) models. However, these PS models are time-consuming to build and are typically trained with limited data variability, which results in poor generalization and robustness. Major sources of variability that affects the accuracy of facial expression transfer algorithms include using different VR headsets (e.g., camera configuration, slop of the headset), facial appearance changes over time (e.g., beard, make-up), and environmental factors (e.g., lighting, backgrounds). This is a major drawback for the scalability of these models in VR.
+
+This paper makes progress in overcoming these limitations by proposing an end-to-end multi-identity architecture (MIA) trained with specialized augmentation strategies. MIA drives the shape component of the avatar from three cameras in the VR headset (two eyes, one mouth), in untrained subjects, using minimal personalized information (i.e., neutral 3D mesh shape). Similarly, if the PS texture decoder is available, MIA is able to drive the full avatar (shape+texture) robustly outperforming PS models in challenging scenarios. Our key contribution to improve robustness and generalization, is that our method implicitly decouples, in an unsupervised manner, the facial expression from nuisance factors (e.g., headset, environment, facial appearance). We demonstrate the superior performance and robustness of the proposed method versus state-of-the-art PS approaches in a variety of experiments.
+
+# 1. Introduction
+
+Our experience with communication systems is two-dimensional, mostly via video teleconferencing (e.g., Messenger), that includes both audio and video transmissions. Recent studies on videoconferencing have shown that the
+
+![](images/7569d9c3dbe90391fad7d50e0aa208ec47d1d60fc3cd05636847114e53fa2545.jpg)  
+Figure 1. 3D Photo-realistic avatar driven by three headset-mounted camera (HMC) images in a VR headset. This paper presents a system to drive photo-realistic avatars robustly with variability in headsets, lighting, environmental background, head pose, and facial appearance.
+
+more closely technology can simulate a face-to-face interaction, the more participants are able to focus, engage, and retain information [43]. A more advanced level of communication with virtual reality (VR) via telepresence [5,8,12, 19,30,34,44,45] will allow virtual presence at a distant location and a more authentic interaction. If successful, this new form of face-to-face interaction can reduce the time and financial commitments of travel, make sales meetings or family meetings more immersive, with a huge impact for the environment and use of personal time.
+
+Today most real-time systems for avatars in AR/VR are cartoon-like (e.g., Hyprsense, Loom AI); on the other hand, Hollywood has animated nearly uncanny digital humans as virtual avatars using advanced computer graphics technology and person-specific models (e.g., Siren). While some of these avatars can be driven in real-time from cameras, building the PS model is an extremely time-consuming and hand-tuned process that prevents democratization of this technology. This paper makes progress in this direction by generating video-realistic avatars by transferring subtle facial expressions from the headset mounted cameras (HMC) images in a VR headset to a 3D talking head (see Fig. 1).
+
+We build on recent work on codec avatars (CA) [26] that learn a PS model from a Plenoptic study. Recall that driving an avatar from HMC cameras is typically more
+
+![](images/9558fed977ca780d5b5a843b1aa9b91aee539562bfa5ae8956ee0c6815414da0.jpg)  
+(a)
+
+![](images/35efc944a6abecfd0eae3ad73fa6fceb0e1d7c5babe0bc1597a23dcdd2fe4edd.jpg)  
+Headset Environment  
+(b)  
+Headset Environment
+
+![](images/1b4ff2847abaa0a8103c4103e34a1f5433e2564cc6e30c6a84ad7aadf8bf4356.jpg)  
+(c)  
+adset  
+Environment  
+Facial appearance Facial appearance Facial appearance  
+Figure 2. Comparing the HMC images of a subject in multiple HMC captures with variations in headset, environment and facial appearance. (a) The training HMC capture, (b-e) the testing HMC captures. The blue bold font shows the variations respect to the training capture (a). The red circles in (f) show the locations of the cameras inside the headset.
+
+![](images/211414fb6d1ce74fec725190a73c36aa591078770429d3b025b8a7e95aaf7572.jpg)  
+Facial appearance Facial appearance  
+(d)  
+Headset  
+Environment
+
+![](images/af41a186c88a528fefb7f9d264b56b666806955791dbafefe8a08c3cf212121a.jpg)
+
+![](images/430276ececfbeea8be897b3384fe8e28667a1cc7562c9ad412da6602d76cfe2c.jpg)
+
+challenging than driving it from regular cameras (e.g., iPhone) [25, 37, 52], due to the domain difference between IR cameras and the texture/shape of the avatar, variability in HMC images due to headset variability (e.g., camera location, IR LED illumination), high-distortion introduced by the near-camera views, and partial visibility of the face (Fig. 2). Wei et al. [49] proposed an end-to-end deep learning network for learning the mapping between the HMC images and the parameterized avatar. First, this model solves the unknown correspondence between HMC images and the avatar parameters in an unsupervised manner using an eleven-view HMC headset. Second, to animate the CA in real time from three HMC images(i.e., inference), Wei et al. [49] learns an encoder network to regress from 3-view HMC images to CA's parameters (Fig. 3).
+
+While previous work has reported compelling photorealistic facial expression transfer results, the existing method has limitations due to the PS nature of the approach. It is time-consuming, expensive and error-prone to capture sufficient statistical variability when collecting PS samples to learn a robust model. It will typically require recording several sessions with variability across lighting, headsets and iconic changes (e.g., makeup, beard), which limits its scalability. To build generic models (conditioned to the neutral shape), the most import contribution of this paper is to propose multi-identity architecture (MIA), an architecture that factorizes nuisance parameters such as camera parameters, facial aesthetic changes (e.g., beard, makeup) and environmental factors (e.g., lighting) from
+
+the facial motion (i.e., facial expression). This is critical because the encoder is able to extract from the HMC images only the information that is relevant to the final task, which is transferring subtle facial expressions, and it is able to marginalize information that is not relevant (headset, facial appearance, environment). Implicitly, this results in an algorithm that aligns facial expressions (3D shape + texture) across users in an unsupervised manner. Recall that is a very difficult problem to align subtle facial behavior (using both 3D shape + texture) across users in a supervised or unsupervised manner. That is, how can we find the correspondence of expression across subjects? Even if done manually, it is an extremely challenging problem and MIA (to the best of our knowledge) is the first algorithm that solves this problem in an unsupervised and discriminative manner (see subsection 4.3). MIA results in an algorithm for facial expression transfer for VR, that improves upon PS models in realistic scenarios.
+
+# 2. Prior Work
+
+# 2.1. Animating Stylized and Codec Avatars
+
+Animating stylized avatars from video has a long history, for instance [7] fits a generic 3DMM to the face and use it to retarget the facial motion to a 3D character. To improve the accuracy, Chaudhuri et al. [6] proposed to learn person-specific expression blendshapes and dynamic albedo maps from the input video of subjects. In [42], facial action unit intensity is estimated in a self-supervise manner by utilizing a differentiable rendering layer for fitting the expression and
+
+![](images/0de867379e23cf0608b8815c14c99ac4f184d530789233939bc7fc3ae1540eb2.jpg)  
+Figure 3. Training and testing pipeline for animating the face codec avatar. In the data collection stage, we perform a face capture to generate the codec avatar of the subject [26] and a HMC capture. We utilize [49] to find the correspondence between the avatar and the HMC capture. Finally, we can train a model to animate the codec avatar (CA) from the HMC images in real-time.
+
+![](images/c36533a46828efebc7d82fd5a6117f7ae6b766da6d9aaa8d87eb6a09530bcec9.jpg)  
+Face Capture and Face Modeling
+
+![](images/897b1c0c9255f3f1e9c93f153c8d18594e19885d6db380446a7c00a6e737f54a.jpg)
+
+![](images/72a645f5dc5c8ec79d27cac1a3da0ea7f07414fa87f88fbe711c837884131145.jpg)
+
+![](images/8d2e729bd830c9b39e7b86d2437a355c0c875018f6a0dcb7f69fdbee0e2be9fc.jpg)
+
+![](images/b9fab1c15344f3e04acad16dc80625b3f8f7dfcf371a143354f8fe0a2f6cb54e.jpg)  
+Figure 4. The proposed multi-identity architecture (MIA). It consists of three main parts: the backbone network  $\mathbf{B}$ , the 3D shape network  $\mathbf{G}$ , and the texture networks  $\mathbf{F}_i$ . The identity selector module pass the features to the corresponding texture network.
+
+to retarget the expression to the character. In contrast, expression transfer from a VR headset [12, 18, 27, 33] is more challenging due to partial visibility of face in HMC images, the specific hardware, and limited existing data.
+
+CAs animate avatars by estimating the parameters of a PS shape and texture model from HMC [9, 26, 39, 49], see Fig. 3. In [26], combination of real and synthetic HMC images are utilized for reducing the domain gap between real HMC images in IR spectrum and rendered images for training encoder and reducing the HMC-avatar domain gap. Wei et al. [49] utilize a cycle-GAN to achieve accurate cycle consistency between 11-view HMC images and CA. Then, they train a person specific regressor from 3-view HMC images to the CA's parameter. Chu et al. [9] propose to use modular CA to have more freedom for animating the eyes and mouth. In a different approach, Richard et al. [36] animate the CA based on the gaze direction and audio inputs. The aforementioned methods rely on PS models, are typically not robust to variations in headsets and environments.
+
+# 2.2. 3D Shape Estimation
+
+Early approaches for model-based shape and texture estimation are based on active shape model [11] (ASM) and Active Appearance Model [10, 28] (AAM). The AAM methods learn a joint holistic model of shape and appearance. 3D Morphable Model (3DMM) provides a dense 3D representation for faces e.g. the Basel Face Model [35] and the FaceWarehouse [3]. In [16, 21], 3DMM is incorporated in an end-to-end CNN training to dicriminatively estimate the 3D shape of faces given single input image. Tran et al. [47] propose to learn a nonlinear 3DMM via deep neural network from in-the-wild images, and in this way 3DMMs are capable of representing non-linear facial expressions. The proposed method in [13] can extract expression-dependent details of the 3D shape from a single image. [14] proposed to use the GAN generator
+
+for 3DMM fitting and estimating high-fidelity UV texture. Similarly, [20] proposed to utilize the volumetric representation of face instead of using 3DMM. An unsupervised method proposed in [15] for, identity 3DMM fitting, regressing the 3D shape and texture. Also, in [38] the identity constraints utilized among the images of the same subject. Similar to [47] we learn a non-linear discriminative 3DMM, but we extend it to learn the model from HMC images given a neutral 3D shape, and align the expressions across subjects in an unsupervised manner. To the best of our knowledge, this is the first work that solves the correspondence of expression across subjects in unsupervised manner.
+
+# 3. Multi-Identity Model
+
+This section describes the proposed multi-identity architecture (MIA) and augmentation techniques to robustify and generalize existing encoder models for driving CAs.
+
+# 3.1. Multi-Identity Architecture (MIA)
+
+Given 3-view HMC images of the eyes and mouth (see Fig. 1), our goal is to estimate the facial expression of a CA (shape+texture), and render it in an arbitrary view in VR. The MIA has three main parts (see Fig. 4): the backbone network, the 3D shape network, and the texture branch.
+
+Backbone network: The backbone network,  $\mathbf{B}_{\psi}$  in Fig. 4, is shared among subjects. Its goal is to factorize the expression from other nuisance factors such as lighting, background, or camera views, and build an internal representation that is invariant to those factors. As we will show in the experiments section, MIA naturally finds that the best way to encode HMC images across subjects, is by marginalizing out person-specific factors in addition to the nuisance factors mentioned. This results in learning an embedding that only preserves expression without the need of solving for correspondence across expression among subjects.
+
+3D shape network: MIA assumes that the neutral shape
+
+![](images/75fb4c08147b6ac0c634ac4ecc523c724a89b6780a34408d70ad0d714e9f5f48.jpg)  
+Figure 5. Examples of applying 3D augmentation layer to HMC images. First row: Real HMC images, second row: Augmented images by changing 3D pose, focal length and background.
+
+of the test subject,  $\mathbf{S}^N\in \mathbb{R}^{7306\times 3}$ , in given<sup>1</sup>. This is the only information MIA needs to generalize the shape component of the network to untrained subjects. A network  $\mathbf{G}_{\gamma}$  is trained to estimate 3D shape,  $\hat{\mathbf{S}}\in \mathbb{R}^{7306\times 3}$  from HMC images. The network  $\mathbf{G}_{\gamma}$  takes both of the output of the backbone network  $\mathbf{B}_{\psi}$  and  $\mathbf{S}^N$  to estimate the person-specific 3D shape expression residual. The neutral 3D shape is used to re-inject person-specific information that was factored out in  $\mathbf{B}_{\psi}$ . For instance, eye openness, which varies across identities, can be extracted from the neutral 3D shape  $\mathbf{S}^N$  of each subject. With this, we reconstruct the 3D shape of subject  $i$  as:
+
+$$
+\hat {\mathbf {S}} _ {i} = \mathbf {S} _ {i} ^ {N} + \mathbf {G} _ {\gamma} \left(\mathbf {B} _ {\psi} \left(\mathbf {H} _ {i} ^ {0}, \mathbf {H} _ {i} ^ {1}, \mathbf {H} _ {i} ^ {2}\right), \mathbf {S} _ {i} ^ {N}\right). \tag {1}
+$$
+
+The network  $\mathbf{G}_{\gamma}$  is trained by minimizing the Euclidean distance between the target  $\mathbf{S}_i$  and estimated  $\hat{\mathbf{S}}_i$  3D shapes,
+
+$$
+\mathbf {L} _ {S} ^ {i} = \left\| \mathbf {W} ^ {S} \odot \left(\mathbf {S} _ {i} - \hat {\mathbf {S}} _ {i}\right) \right\| _ {2} ^ {2}, \tag {2}
+$$
+
+where  $\mathbf{W}^S$  is the weight mask for the visible areas.
+
+Texture network: When the pre-trained PS texture decoder is available for each identity, our goal is to be able to animate the CA from HMC images robustly and with minimal adaptation effort. In this paper, we presume pre-trained decoders  $\mathbf{D}_{\phi}$  from [26] are available, but our work can be similarly applied to other PS models as well (e.g. [24, 46]). The network  $\mathbf{D}_{\phi}$  takes, as input, an expression parameter  $\mathbf{z} \in \mathbb{R}^{256}$  and a view vector  $\mathbf{v} \in \mathbb{R}^3$ , and generates person-specific and view-specific texture  $\mathbf{T}^{\mathbf{v}} \in \mathbb{R}^{1024 \times 1024 \times 3}$  that, together with shape, can be used to render the avatar,
+
+$$
+\mathbf {T} ^ {\mathbf {v}} = \mathbf {D} _ {\phi} (\mathbf {z}, \mathbf {v}). \tag {3}
+$$
+
+However, since each PS model is trained independently of all others, the structure of the latent space,  $\mathbf{z}$ , is not consistent across identities. We would like to utilize the shared
+
+backbone encoder  $\mathbf{B}_{\psi}$  across identities to encourage robustness via joint training. Inspired by multi-task learning techniques [4, 32], we additionally learn person-specific adaptation layers,  $\mathbf{F}_{\theta}$ , that transform the identity-consistent expression embedding produced by  $\mathbf{B}_{\psi}$  to each identity's personalized latent space. Finally, to eliminate unnecessary dimensions in  $\mathbf{z}$ , non-informative dimensions, we apply PCA dimensionality reduction, denoted  $\mathbf{P} \in \mathbb{R}^{256 \times 80}$  to each identity's latent space and fix it during training. Together, these components are used to generate PS expression parameters as follows:
+
+$$
+\hat {\mathbf {z}} _ {i} = \mathbf {P} _ {i} \left(\mathbf {F} _ {\theta_ {i}} \left(\mathbf {B} _ {\psi} \left(\mathbf {H} _ {i} ^ {0}, \mathbf {H} _ {i} ^ {1}, \mathbf {H} _ {i} ^ {2}\right)\right)\right) + \bar {\mathbf {z}} _ {i}, \tag {4}
+$$
+
+where  $i$  is subject index, and  $\overline{\mathbf{z}}_i$  is the average expression parameter for subject  $i$ . Then, we use Eqn. 3 to generate the estimated texture  $\mathbf{T}_i^{\mathbf{v}}$  from view  $\mathbf{v}$ . To guide the network, we minimize the Euclidean loss between the estimated and the target expression parameters and textures:
+
+$$
+\mathbf {L} _ {T} ^ {i} = \left\| \mathbf {z} _ {i} - \hat {\mathbf {z}} _ {i} \right\| _ {2} ^ {2} + \lambda_ {\mathbf {T}} \left\| \mathbf {W} ^ {T} \odot \left(\mathbf {T} _ {i} ^ {\mathbf {v}} - \hat {\mathbf {T}} _ {i} ^ {\mathbf {v}}\right) \right\| _ {2} ^ {2}, \tag {5}
+$$
+
+where  $\mathbf{W}^T$  is the weight mask for the visible areas from the HMC images and  $\lambda_T$  is the weight for the texture loss.
+
+Total Loss: The entire MIA network is trained end-to-end to optimize the networks' parameters by minimizing:
+
+$$
+\underset {\psi , \gamma , \left\{\theta_ {i} \right\} _ {i = 0} ^ {K}} {\text {m i n i m i z e}} \sum_ {i = 0} ^ {K} \mathbf {L} _ {T} ^ {i} + \lambda_ {\mathbf {S}} \mathbf {L} _ {S} ^ {i}, \tag {6}
+$$
+
+where  $K$  is the number of subjects and  $\lambda_{\mathbf{S}}$  is the weight for the shape loss.
+
+# 3.2. Augmentation
+
+Data augmentation is a wildly practiced heuristic in many deep learning tasks. The main goal is to make the distribution of variations in training data more similar to those in the test set. Most common data augmentations techniques include scaling [41], color augmentation [23], simple geometric transformations [40], and utilizing synthetic data [22, 29]. However, a major source of variability in our task stems from headset factors, such as variations in camera placement and focus, as well as the slop of the headset relative to the face which varies during usage. These variations are not easily modeled using standard augmentation techniques that do not take the 3D shape of the face into account. In this paper, we simulate headset-based variations by perturbing the 3D rotation and translation of the face shape in the training set, and use it to re-render augmented views of each HMC image on random backgrounds. Some examples are shown in Fig. 5. As demonstrated in the experiments section below, this simple augmentation technique substantially improves the robustness of our method to real world variations.
+
+![](images/6f1c1c83a1b1a9fe20629a72b8c506a715a1aaff0cd1f24b54a0bc626441f350.jpg)  
+Figure 6. The testing results for estimating the 3D shapes for six untrained subjects and their ground truth based on 11-view results in [49].
+
+![](images/639ebd3d4f41c07a910c676525569d95e680aa46e9dd91be237f3d1e20b05cc1.jpg)
+
+![](images/31212ea7d60215a0222b85b09a3170b6c95a48d48f4bb97cef96989bb072e2ac.jpg)
+
+# 4. Experimental Results
+
+This section reports experimental results and analysis on MIA. The first experiment shows how MIA can estimate accurate 3D shapes directly from HMC images of untrained subjects. In the second experiment, we evaluate the quality of MIA's texture prediction for identities with pre-trained avatars under challenging testing scenarios. In the third experiment, we show how MIA can incorporate new subjects with minimal training. In addition, we also present further analysis about what MIA learns prior and during adaptation.
+
+Data: We used 120 HMC captures of different subjects for training and 21 HMC captures for testing. Training and testing HMC captures do not overlap. Each HMC capture is a 45 minutes long video (30fps) of 11-views HMC images, and contains 73 peak expressions, two sets of continuous range-of-motion, recitation of 50 sentences and 5-10 minutes of conversion. The HMC images are in the IR spectrum with a resolution of  $480 \times 640$ . During testing only 3-views are available. For each subject, we have a pre-trained decoder to generate PS texture for various expressions from arbitrary views. For more information, of how to build the PS decoder see [26] and Eqn. 3.
+
+Ground Truth: We utilize the result of the method in [49], that solves for the correspondence between 11-views HMC images and the CA parameters as the ground truth. Recall that the training data is captured with 11-views to achieve more precise results in the correspondence between HMC and CA, while the testing data has only 3-views.
+
+**Baseline method:** We compare MIA with the person specific (PS) encoder in [49]. The PS encoder is trained with one HMC capture (3-view images) and uses a CNN architecture with the same number of parameters as ours.
+
+Evaluation metrics: We report the average Euclidean error for the eyes, mouth and face areas separately for both 3D shape and the texture. The 3D shape errors are measured in millimeters and the texture errors in raw intensity values (i.e. 0-255). We report the localized error metrics to analyze failure modes better. For example, the 3D shape error
+
+Table 1. Test results for 3D shape estimation in untrained subjects.  
+
+<table><tr><td rowspan="2">Subject</td><td colspan="3">3D Shape Error (mm)</td></tr><tr><td>Face</td><td>Eyes</td><td>Mouth</td></tr><tr><td>person 1</td><td>1.68</td><td>1.08</td><td>2.90</td></tr><tr><td>person 2</td><td>1.51</td><td>1.21</td><td>2.32</td></tr><tr><td>person 3</td><td>1.07</td><td>0.74</td><td>1.82</td></tr><tr><td>person 4</td><td>1.84</td><td>1.20</td><td>2.92</td></tr><tr><td>person 5</td><td>1.45</td><td>0.89</td><td>2.23</td></tr><tr><td>person 6</td><td>1.47</td><td>0.97</td><td>2.56</td></tr><tr><td>person 7</td><td>1.73</td><td>1.20</td><td>2.98</td></tr><tr><td>person 8</td><td>1.57</td><td>0.89</td><td>2.52</td></tr><tr><td>person 9</td><td>1.21</td><td>0.91</td><td>1.84</td></tr><tr><td>person 10</td><td>1.58</td><td>1.00</td><td>2.68</td></tr><tr><td>overall</td><td>1.51 ± 0.23</td><td>1.00 ± 0.16</td><td>2.47 ± 0.42</td></tr></table>
+
+in the eyes capture openness and blinking errors, while in the mouth, they capture deviations in lip shapes important for visual-speech. Similarly, texture error in the eyes is typically due to the errors in gaze direction, and in mouth, it corresponds to incorrect teeth and tongue estimation.
+
+Implementation details: In training, we use the Adam optimizer, setting the batch size to 32 and the initial learning rate to  $1\mathrm{e} - 3$ . We decrease learning rate by  $8\mathrm{e} - 1$  after each 25K iterations. In total, we train the encoder for 250K iterations, and set both of  $\lambda_{\mathrm{T}}$  and  $\lambda_{\mathrm{S}}$  to 100. We crop and resize the HMC images to  $192\times 192$  to focus on the face areas.
+
+The backbone network,  $\mathbf{B}_{\psi}$ , consists of two residual networks [17], one for eye images  $\left[\mathbf{H}^{0}, \mathbf{H}^{1}\right] \in \mathbb{R}^{192 \times 192 \times 2}$  and another for the mouth  $\mathbf{H}^{2} \in \mathbb{R}^{192 \times 192}$ . Each network consists of a Res-Net head module, five BottleNeck blocks and a 64-way fully connected layer. Each BottleNeck block consists of ten convolutional layers with  $3 \times 3$  and  $1 \times 1$  filters. We add shortcut connections among the convolutional layers, and each layer is followed by ReLU [31] and Instance normalization [48] layers. To extract the final identity invariant features, we apply a global average pooling and a 64-way fully connected layer to the activations of the last BottleNeck block. The architecture of the 3D shape network,  $\mathbf{G}_{\gamma}$ , consists of four fully connected layers where each one is followed by a leaky ReLU [51] layer with negative slope of 0.2. We normalize the extracted features
+
+![](images/99317f44d7a6b640abe16b96ff8a1500e9e49c973afbb0ce420e20c31200f9d9.jpg)  
+Figure 7. Test results to estimate the 3D shape from HMC images of an untrained subject for a wide range of expressions.
+
+from the HMC images and the neutral 3D shape, to account for their different domains, by employing group normalization [50] after concatenating the features. Finally, for the texture network  $\mathbf{F}_{\theta}$ , we utilize the combination of a ReLU layer and a fully connected layer without bias.
+
+# 4.1. Quantitative Evaluations
+
+This section quantifies the performance of MIA using three experiments: (1) driving the 3D shape of untrained subjects. (2) robustness of shape and texture estimation for subjects with trained PS models. (3) generalization of learned features on new subjects.
+
+Driving 3D Shape: Inputs to the shape generation network,  $\mathbf{G}_{\gamma}$ , are the HMC images and the corresponding identity's neutral 3D shape. We train the network with 120 subjects using the loss function in Eqn. 2 as guidance. Fig. 6 shows the estimated 3D shape for extreme expression examples from six untrained subjects along with their ground truth. Our 3D shape estimator captures subtle details in expressions necessary for inferring social signal. Table 1 shows the 3D shape errors for face, eyes and mouth areas of the whole sequence for ten untrained subjects. The error is less than  $2\mathrm{mm}$  in the face/eyes and  $3\mathrm{mm}$  in the mouth. Recall that MIA does not use any sample from the test subject other than the neutral shape and has never seen any HMC images for these subjects during training. Fig. 7 shows testing results of one untrained subject for a wide range of expressions. Note that PS [49] is not able to estimate the 3D
+
+shape for untrained subjects.
+
+Comparing Table 1 with PS's results for the 3D shape error in Table 2 (different capture), we find that MIA outperforms PS, despite PS having access to subject-specific HMC images, and their target shapes, during training. We suspect the reason for this is that MIA learns to marginalize the extrinsic variability of the problem (i.e. environment, headset) from the 120 subjects that is trained on, while the PS tends to overfit to the specific HMC capture session used for training. More comparative results can be found in the video in the supplementary material.
+
+Driving Full Avatars: In this experiment, we evaluate the ability of MIA to generate both shape and texture and its robustness against extrinsic factors such as headset, environment and facial appearance variations. Here, data for test subject is available during training, but from a different HMC capture. The selected subject was captured on five different dates; examples of the HMC images are shown in Fig. 2. These samples show large appearance variations due to facial hair, pose changes in the headset slop, and camera assembly differences across headsets; it also contains background variation due to changing environment and overall lighting differences. We use one HMC capture (Fig. 2(a)) of the subject with 119 HMC captures of other subjects for training, and test on the remaining four HMC captures of that subject. Table 2 compares the testing errors of MIA against PS [49]. On test capture 1, which is very similar to the training capture, PS [49] performs better than MIA. But,
+
+Table 2. Testing results for a subject with multiple testing HMC captures with different variations. One HMC capture of the subject is inside the training set. The 3D shape errors are in mm and the texture errors are in intensity.  
+
+<table><tr><td rowspan="2">Test Capture</td><td rowspan="2">Sample Image</td><td rowspan="2">Variations</td><td rowspan="2">Method</td><td colspan="3">3D Shape Error</td><td colspan="3">Texture Error</td></tr><tr><td>Face</td><td>Eyes</td><td>Mouth</td><td>Face</td><td>Eyes</td><td>Mouth</td></tr><tr><td rowspan="2">1</td><td rowspan="2">Fig. 2.(b)</td><td rowspan="2">Headset</td><td>PS [49]</td><td>0.85</td><td>0.65</td><td>1.33</td><td>1.13</td><td>1.93</td><td>1.50</td></tr><tr><td>MIA</td><td>1.20</td><td>0.85</td><td>1.90</td><td>1.33</td><td>2.34</td><td>1.79</td></tr><tr><td rowspan="2">2</td><td rowspan="2">Fig. 2.(c)</td><td rowspan="2">Headset Facial appearance</td><td>PS [49]</td><td>2.04</td><td>0.77</td><td>4.71</td><td>1.84</td><td>2.47</td><td>3.51</td></tr><tr><td>MIA</td><td>1.28</td><td>0.79</td><td>2.22</td><td>1.49</td><td>2.52</td><td>2.00</td></tr><tr><td rowspan="2">3</td><td rowspan="2">Fig. 2.(d)</td><td rowspan="2">Headset Facial appearance</td><td>PS [49]</td><td>1.90</td><td>0.98</td><td>3.68</td><td>1.65</td><td>2.73</td><td>2.84</td></tr><tr><td>MIA</td><td>1.26</td><td>0.86</td><td>2.23</td><td>1.32</td><td>2.52</td><td>2.03</td></tr><tr><td rowspan="2">4</td><td rowspan="2">Fig. 2.(e)</td><td rowspan="2">Environment Facial appearance</td><td>PS [49]</td><td>2.21</td><td>0.86</td><td>4.92</td><td>1.92</td><td>2.33</td><td>3.39</td></tr><tr><td>MIA</td><td>1.14</td><td>0.73</td><td>1.94</td><td>1.45</td><td>2.11</td><td>2.05</td></tr><tr><td rowspan="2" colspan="2"></td><td rowspan="2">overall</td><td>PS [49]</td><td>1.75 ± 0.61</td><td>0.81 ± 0.13</td><td>3.66 ± 1.64</td><td>1.63 ± 0.35</td><td>2.36±0.33</td><td>2.81 ± 0.92</td></tr><tr><td>MIA</td><td>1.22±0.06</td><td>0.80±0.06</td><td>2.07±0.17</td><td>1.39±0.08</td><td>2.37 ± 0.19</td><td>1.96±0.12</td></tr></table>
+
+Table 3. Testing results for training and testing on new subjects with pre-trained fixed backbone network.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">3D Shape Error</td><td colspan="3">Texture Error</td></tr><tr><td>Face</td><td>Eyes</td><td>Mouth</td><td>Face</td><td>Eyes</td><td>Mouth</td></tr><tr><td>PS [49]</td><td>1.12 ± 0.26</td><td>0.74 ± 0.11</td><td>1.98 ± 0.65</td><td>2.22±0.61</td><td>2.90 ± 0.95</td><td>2.77 ± 0.83</td></tr><tr><td>MIA</td><td>1.05±0.19</td><td>0.74±0.09</td><td>1.71±0.37</td><td>2.22 ± 0.62</td><td>2.88±0.81</td><td>2.65±0.78</td></tr></table>
+
+its performance declines significantly when testing on the other captures, where variations in environment and facial appearance are more extreme. Note that the overall errors for MIA, for all areas of 3D shape and texture, are more stable and are similarly low across all test captures. The first two rows of Fig. 10 shows visual comparison of methods on the test HMC captures, where a significant reduction in expressive detail is noticeable in results for PS [49]. We refer the reader to the supplementary material for more results.
+
+Adaptation to New Identities: We evaluated the generalization of MIA's feature extraction to new subjects on HMC captures of 6 subjects that are not trained in MIA. Each of the 6 subjects has more than one HMC capture exhibiting variations in extrinsic factors. We used the pre-trained MIA network with 120 subjects (excluding the test 6 subjects), and fix the shape generation network,  $\mathbf{G}$ , and backbone network,  $\mathbf{B}_{\psi}$ . For each new subject, we trained a new small texture network  $\mathbf{F}_{\theta}$ . During the testing on HMC captures with variations, we used the newly trained texture estimation branch for estimating the texture parameters, and decode both the texture and the 3D shape by utilizing Eqn. 3. Table 3 shows the overall errors for 3D shape and texture for different areas of 7 testing HMC captures of the 6 subjects. MIA achieves lower errors for all areas with smaller variability, demonstrating the effectiveness of the features extracted from the fixed backbone network. The last three rows of Fig. 10 show visualizations of this case for.
+
+# 4.2. Ablation Study
+
+3D augmentation layer: To analyze the advantage of using the 3D augmentation layer, we compare the errors of the PS [49] model, MIA with 3D augmentation trained with 1 subject, MIA without 3D augmentation (3D Aug) trained with 30 subjects, and MIA with 3D Aug trained with 30 subjects. Fig. 8 shows the average errors for the four test captures in Table 2. It shows that even using the 3D Aug layer with 1 subject reduces errors slightly in comparison to PS [49]. However, there is huge drop in errors by using the 3D Aug layer with 30 subjects. This reduction of er
+
+![](images/cb582c77e73bd9fa2285abc8c8d7380413c8eb0a19312a002a027c4c2e4edc85.jpg)  
+Figure 8. The advantage of the 3D augmentation layer, the errors drop significantly by using both of 3D augmentation and MIA.
+
+![](images/2afe26640f276a645b6e465fd11e8311cc68c3e16a2b3c7cabb56a7e743370f0.jpg)
+
+![](images/a8715660d3a95f736170b497b0ac4e1a291a2d3aaa3e3c57a20b52155ef85039.jpg)  
+Figure 9. The influence of number of training subjects. The shape errors are decreasing by increasing number of training subjects.
+
+ror is more significant in the mouth area. It shows that the combination of MIA and 3D Aug is effective.
+
+Influence of number of subjects: We evaluate the influence of number of training subjects in the performance of MIA during testing. We train MIA with 30, 60, 100 and 120 subjects, and test them on ten untrained subjects for estimating the 3D shapes. Fig. 9 shows that by increasing number of training subjects the 3D shape errors are decreasing, especially for the mouth area.
+
+# 4.3. Unsupervised Expression Correspondence
+
+The MIA implicitly learns to solve for correspondence across expressions in order to marginalize nuisance parameters (e.g., lighting). It naturally discovers that the best way to encode HMC multi-identity data is finding a latent space that only contains expression information. Fig. 11 illustrates how MIA learns to solve for correspondence across expressions. The first column shows the input HMC images and the second column is the CA of the subject in the first column. The remaining columns are the CAs of other subjects driven from the HMC images in first column, that is, the same extracted features from HMC images are utilized to estimate (by using the corresponding  $\mathbf{F}_{\theta}$ ) a new expression parameter (with the same facial expression meaning),  $\mathbf{z}$ , in the latent space of each of the remaining subjects. As we can observe, MIA is able to align the expression across all of the subjects in an unsupervised manner, and creating a common expression-only space. Please pay attention to the mouth area in the second row of Fig. 11 that shows the same expression with different mouth interior.
+
+# 5. Conclusion and Future Directions
+
+This paper proposes MIA to robustify and generalize existing PS methods for driving CAs. MIA learns to extract identity invariant features related to facial expression while marginalizing nuisance factors (headset, environment, facial expression) in an unsupervised manner. We show that MIA is able to drive the shape component in untrained subjects, and if the PS texture decoder is available, with a minimum training, MIA can drive CAs for new subjects. For future directions, first, we will design new loss function based on the closeness of 3D surfaces to model the lips and eyes closure. Second, we will work on texture-conditional decoders to make the texture part of the method generalizable for new subjects without pre-trained decoders.
+
+![](images/639b458ccd5244ef7070d39579a1b57744b78afc55b62fe4996e3cb5307e0712.jpg)
+
+![](images/53880a77b893aaf79613e7bbdcd30c00f3e009a8447271ef19b792076c79874e.jpg)
+
+![](images/7b4fee094c508b9036d738a44bc67f758b594e2bdbb33f59d34bc68b5b48ad02.jpg)
+
+![](images/dfd81b2883591857bd235d85c8c97d3e1240fdc5856670728d0b4d55aa781e1e.jpg)
+
+![](images/1383eb80c1ea32165fc082d1ec498c5524f6c9aa303271afd18c836bea73646f.jpg)
+
+![](images/ab02575f44caf7e191e155355a3058984f09a4405065806cf97f6f349eaf67b5.jpg)
+
+![](images/213b5d3a0f7304c46b454b0896cf21bf1ff8cf6e4c89179494b7df40df1efc05.jpg)
+
+![](images/ad6ab6b34642756aff1eaca733532d6dd8b1209a62228c4082538bce31b27cfd.jpg)
+
+![](images/593c1eb20cab6e65159d7510c23aecd22a91dd1b7343bf1cd1176309b37fa31d.jpg)
+
+![](images/607d106a2c47be3cba40df9811a4d6436d8d46a9dd553415c4a8e5b6edde0d3d.jpg)
+
+![](images/96af7eda862cb828ce35fad62d6dd2ee85efa90022659592395ecf292f53fb40.jpg)
+
+![](images/79a78f6dac93d91c012ee7b56afc24f157c60becf06515060c2acc32f1d0beac.jpg)
+
+![](images/757f337235fbd919cb2e6748c45e3e2b915dab580b6236e9f5fdbd63ee82857c.jpg)
+
+![](images/1d07c818b97a71bc4e433daaabcb4d9b59e017032cf8698e3549831282c2c5c4.jpg)
+
+![](images/e41d72891a4a097de38e7e5ef2eb0afbe0ae6e30f844f739e545201561de58b5.jpg)
+
+![](images/6c851fe6995eb858f43883c3249b0dc28d002fed5bbbec6a310a451b5e40bb50.jpg)
+
+![](images/95a76657bde14d2c5626f40861a53ccaa16458d4b3edfa30269b06031e615eb6.jpg)
+
+![](images/82afbd8ca900378b8b97f907a7fd054beb5eac6d87162d571dda4c7c24ded016.jpg)
+
+![](images/6eba8c5979d0c3430acd8d523c811ff108a549a4460bd3d3729f2f7ec2217115.jpg)
+
+![](images/3a41109b5420da3550e01e0f6673dabfd8c7d12482ee9eff6e17eec5e1146b4b.jpg)
+
+![](images/d6149613e9e2fc0302bb5c6d994885301b56b00c4e7e880a7c4967908d66abc3.jpg)
+
+![](images/dfc5213f0a20aeaf69d9cb0f9d5060fe387b8d6cd93f3471463fc04d65d93ac0.jpg)
+
+![](images/0f7e4d7c949b48988aa5810635fa77020b0bd17b341384d82443dc1d2396cc5e.jpg)
+
+![](images/b04bc398401b1744f1b7fa39192d002dde8545040a4c1c78896326aee3feddee.jpg)
+
+![](images/bdfa6178c87fd90e6135a064fcfcca3a8d0d8f15985f9a00c33a24dd5c4cbde9.jpg)
+
+![](images/87f041b442fe4345bf3f25baaeb397df2e9005c3f770c49245efd1ed7d446bf6.jpg)
+
+![](images/51f6ef87aaf1dc91bc97ac27f7a0532edc985566910ef8575e0e3edb33909091.jpg)
+
+![](images/27730e310285be9b0cb5616d30216c06c59d294bfe337136b82400fff930b4d3.jpg)
+
+![](images/1d1cfc39d219f92392cfc200179623e4d279019c6b699dcb1251c6f017ad7452.jpg)
+
+![](images/d15aa44afe966974d9b10e1a748e62bb1ecd42c16be89824ce041e6829b1cffc.jpg)
+
+![](images/b2da4e040b6d82aa797b88b28cba5a1ed86c0b97e66f07fbce18cea2aae0ff8d.jpg)
+
+![](images/925a7a3b06ac866c82a129f12cbff007e0a67f377c9e37bee02b6b4b8a1b4d97.jpg)
+
+![](images/cd402cbeb8292b1ba1b4911f0ca3b069f2736d34d4cdfc373f8186213429e676.jpg)  
+HMC
+
+![](images/6b40112f480b1b18ea3fab99a54126af02fbfa54a4081c8c2c3924a7b1d4c19b.jpg)  
+PS [49]
+
+![](images/f800031a13fdb81cf13f9eb00238a3b017da9c907d047cdff1df2993b2d71d38.jpg)  
+MIA
+
+![](images/6af2cd8bd625dd04b5514f7f7564ae053d01abace260453b43cb9ee01a6246d9.jpg)  
+GT
+
+![](images/7238e6b59b1a853c231b0e317284d7741e1ab9756715221a0eda4cb1ba7e94a9.jpg)  
+HMC
+
+![](images/d048ce80bf588f37901890ec61ae1b1826994363408027b85119e73af04337de.jpg)  
+PS [49]
+
+![](images/586b143b029aa15cf2887efb2da57c67b63dab42faf4f6a94284e5406b67c71b.jpg)  
+MIA
+
+![](images/76327ab6661f7fb2514924bcc50117942889d12af11cf4a6de5c6807b6b12e54.jpg)  
+GT
+
+![](images/ff8b998110aad31aa797e88d959c8029cafde9511335ab8d9eace61ae256f291.jpg)  
+Figure 10. The comparison of PS [49] and MIA methods for animating the codec avatar from HMC images. The MIA can estimate more expressive and accurate expressions.
+
+![](images/5504878c6ecb8f1d78407744d9b5b03fd53d1102b2c502b97f5b344eef780ffb.jpg)
+
+![](images/54ef51345e6a2ecf499845d11c23bd1403b74986173596a8c50fd019a1571e93.jpg)
+
+![](images/1215a72dccdfaca2788c9c680a4fb93a77b3a202f838d388a5d4446bed4d6bd4.jpg)
+
+![](images/a6076f361fd8dc8b88d444c9896f022a494d43f5f3d1d29038f0d646af380767.jpg)
+
+![](images/f27525a8a47fdf7224e3a96417c72c1c542d91f59d616773f6862bf33b1358dc.jpg)
+
+![](images/6fd42095b648656ddfba001406687ed8e539a22345adb7d1b301c0cc7a7f7c59.jpg)
+
+![](images/f5af64da274b764f318e7d72a02b87809f7343fcddcfe2d37fbec453d8140450.jpg)
+
+![](images/a878e87c446d8059e6c08ba4b5bd6192c72926ef71528648fa5d40ce8feb866c.jpg)
+
+![](images/9815929d009d9c55821cc239383cb209c62cab36348da742cb8faa2e196bf623.jpg)  
+Figure 11. The examples of synchronized expressions of training subjects by using the same input HMC images.
+
+![](images/5aee55ed7418636948155c31887c9fcfa2b92b9c6b6e9d2536d0c694b7d496f4.jpg)
+
+![](images/5d66fbcde73e7ff4df087ea9b151783ef58469fd6c48fe3d6c636b18ca47d2a2.jpg)
+
+![](images/2b45d80d65f20baa7a73bca5ff8101895e2214934bf09b5d8faf92fcf2d16489.jpg)
+
+![](images/7f1eb4984a84addd4c9c87c094802ef2aa39e2f13e9c787107523e10dcc391ce.jpg)
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+![](images/1450e8854933b42773058dd00f03bff3d499137abb17e20025c0b2607c7c70af.jpg)
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+![](images/e4f6f16c3982e4133147410ae9496848154832d058f7121a6e5fd1af396937f2.jpg)
+
+# References
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