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+# $\beta$ -DARTS: Beta-Decay Regularization for Differentiable Architecture Search
+
+Peng Ye $^{1*}$ , Baopu Li $^{2}$ , Yikang Li $^{3}$ , Tao Chen $^{1\dagger}$ , Jiayuan Fan $^{1}$ , Wanli Ouyang $^{4}$
+
+$^{1}$ Fudan University,  $^{2}$ BAIDU USA LLC,  $^{3}$ Shanghai AI Laboratory,
+
+4The University of Sydney, SenseTime Computer Vision Group, Australia
+
+yepeng20@fudan.edu.cn
+
+# Abstract
+
+Neural Architecture Search (NAS) has attracted increasingly more attention in recent years because of its capability to design deep neural network automatically. Among them, differential NAS approaches such as DARTS, have gained popularity for the search efficiency. However, they suffer from two main issues, the weak robustness to the performance collapse and the poor generalization ability of the searched architectures. To solve these two problems, a simple-but-efficient regularization method, termed as Beta-Decay, is proposed to regularize the DARTS-based NAS searching process. Specifically, Beta-Decay regularization can impose constraints to keep the value and variance of activated architecture parameters from too large. Furthermore, we provide in-depth theoretical analysis on how it works and why it works. Experimental results on NAS-Bench-201 show that our proposed method can help to stabilize the searching process and makes the searched network more transferable across different datasets. In addition, our search scheme shows an outstanding property of being less dependent on training time and data. Comprehensive experiments on a variety of search spaces and datasets validate the effectiveness of the proposed method. The code is available at https://github.com/Sunshine-Ye/Beta-DARTS.
+
+# 1. Introduction
+
+Neural architecture search (NAS) has attracted lots of interests for its potential to automatize the process of architecture design. Previous reinforcement learning [26, 33] and evolutionary algorithm [25] based methods usually incur massive computation overheads, which hinder their practical applications. To reduce the search cost, a variety of approaches are proposed, including performance estimation [17], network morphisms [2] and one-shot architecture search [13, 21]. In particular, one-shot methods resort to
+
+![](images/04ec9ff44c0b9cbf44ffcab85ab8d80a81577effe48d5b8da3e89eab1badb240.jpg)  
+Figure 1. Schematic illustration about (a) DARTS [21] and our proposed  $\beta$ -DARTS, (b) DARTS- [5]. DARTS- adds an auxiliary skip connection with a decay rate  $\beta_{skip}$  to alleviate the performance collapse problem.  $\beta$ -DARTS introduces the Beta-Decay regularization to improve both the robustness of the searching process and the generalization ability of the searched architecture.
+
+weight sharing technique, which only needs to train a supernet covering all candidate sub-networks once. Based on this weight sharing strategy, differentiable architecture search [21] (namely DARTS, as shown in Fig. 1) relaxes the discrete operation selection problem to learn differentiable architecture parameters, which further improves the search efficiency by alternately optimizing supernet weights and architecture parameters.
+
+Although differentiable method has the advantages of simplicity and computational efficiency, its robustness and architecture generalization challenges still needs to be fully resolved. Firstly, lots of studies have shown that DARTS frequently suffers from performance collapse, that is the searched architecture tends to accumulate parameter-free operations especially for skip connection, leading to the performance degradation [1, 5]. To handle this robustness challenge, lots of instructive works are proposed: directly restricting the number of skip connections [4, 20]; exploiting or regularizing relevant indicators such as the norm of Hessian regarding the architecture parameters [1, 3]; changing the searching and/or discretization process [5, 6, 12]; implicitly regularizing the learned architecture parameters [1]. However, the explicit regularization of architecture parameters optimization receives little attention, as previous works (including above methods) adopt L2 or
+
+weight decay regularization by default on learnable architecture parameters (i.e.,  $\alpha$ ), without exploring solution along this direction. Secondly, several works have pointed out that the optimal architecture obtained on the specific dataset cannot guarantee its good performance on another dataset [19,22], namely the architecture generalization challenge. To improve the generalization of searched model, AdaptNAS [19] explicitly minimizes the generalization gap of architectures between domains via the idea of cross domain, MixSearch [22] searches a generalizable architecture by mixing multiple datasets of different domains and tasks. However, both methods solve this issue by leveraging larger datasets, while how to use a single dataset to learn a generalized architecture remains challenging.
+
+This paper is dedicated to simultaneously solve the above-mentioned two challenges in an efficient way. Inspired by the widely-used L2 [7] or weight decay regularization [18] approaches, we intend to design a customized regularization for DARTS-based methods, which can explicitly regularize the optimizing process of architecture parameters. However, different from the regularization on the learnable architecture parameter set,  $\alpha$  (before the nonlinear activation of softmax), commonly used in standard DARTS and its subsequent variants, we propose a novel and generic Beta-Decay regularization, imposing regularization on the activated architecture parameters  $\beta$  (after softmax), where  $\beta_{k} = \frac{\exp(\alpha_{k})}{\sum_{k^{\prime} = 1}^{|\mathcal{O}|}\exp(\alpha_{k^{\prime}})}$ . On one hand, the proposed Beta-Decay regularization is very simple to implement, achieved with only additional one line of PyTorch code in DARTS (Alg 1). On the other hand, this simple implementation is grounded by in-depth theoretical support. We provide theoretical analysis to show that, Beta-decay regularization not only mitigates unfair competition advantage among operations and solve the domination problem of parameter-free operations, but also minimizes the Lipschitz constraint defined by architecture parameters and make sure the generalization ability of searched architecture. In addition, we mathematically and experimentally demonstrate that, commonly-used L2 or weight decay regularization on  $\alpha$  may not be effective or even counterproductive for improving robustness and generalization of DARTS.
+
+Algorithm 1 PyTorch Implementation in DARTS  
+1:  $\mathcal{L}_{Beta} =$  torch.mean(torch.logsumexp( self.model._arch_parameters, dim=-1))   
+2: loss  $=$  self._val_loss(self.model, input_valid, target_valid)  $+\lambda \mathcal{L}_{Beta}$
+
+DARTS with Beta-Decay regularization ( $\beta$ -DARTS) is illustrated in Fig. 1. Extensive experiments on various search spaces (i.e. NAS-Bench-201, DARTS, NAS-Bench1Shot1) and datasets (i.e. CIFAR-10, CIFAR-100, ImageNet) verify the effectiveness of our method. Besides, our search scheme shows the following outstanding properties:
+
+- The search trajectories on NAS-Bench-201 and NAS-Bench-1Shot1 show that, the found architecture has continuously rising performance, and the search process can reach its optimal point at an early epoch.  
+- We only need to search once on the proxy dataset (i.e., CIFAR-10), but the searched architecture can obtain promising performance on various datasets (i.e., CIFAR-10, CIFAR-100 and ImageNet).
+
+# 2. Related Works
+
+# 2.1. Robustness of DARTS
+
+As DARTS is known to chronically suffer from the performance collapse issue caused by the domination of parameter-free operators, lots of works have dedicated to resolving it. P-DARTS [4] and DARTS+ [20] directly limit the number of skip connections. Such handcrafted rules are somewhat suspicious and may mistakenly reject good networks. R-DARTS [1] finds that the Hessian eigenvalues can be regarded as an indicator for the collapse, and employs stronger regularization or augmentation on the training of supernet weights to reduce this value. Then SDARTS [3] implicitly regularizes this indicator by adding perturbations to architecture parameters via random smoothing or adversarial attack. Both methods are indirect solutions and rely heavily on the quality of the indicator. FairDARTS [6] avoids operation competition by weighting each operation via independent sigmoid function, which will be pushed to zero or one by an MSE loss. DropNAS [15] proposes a grouped operation dropout for the co-adaption problem and matthew effect. DOTS [12] further uses the group operation search scheme to decouple the operation and topology search. DARTS- [5] factors out the optimization advantage of skip connection by adding an auxiliary one. However, these methods circumvent the domination effect of parameter-free operations by changing the searching and/or discretization process or adding extra parameters. Different from these works, we explore a more generic solution by explicitly regularizing the architecture parameters optimization, making original DARTS great again.
+
+# 2.2. Generalization of DARTS
+
+Improving the generalization ability of deep model has always been the focus of deep learning research. Recent works provide guarantee on model generalization by minimizing loss value and loss sharpness simultaneously [11]. However, the model generalization is not only related to the network weights, but also determined by its architecture. To this end, several methods attempt to improve the generalization of architectures in the field of NAS. AdaptNAS [19] incorporates the idea of domain adaptation into the search process of DARTS, which can minimize the generalization gap of neural architectures between domains.
+
+MixSearch [22] uses a composited multi-domain multi-task dataset to search a generalizable architecture in a differentiable manner. On one hand, both above methods are built on the assumption of having multiple datasets, while our method is not built on multiple datasets. On the other hand, our focus is on regularizing architecture parameters, which is not investigated in AdaptNAS and MixSearch.
+
+# 3. Proposed method
+
+# 3.1. Formulation of DARTS
+
+Following [34], DARTS searches the structure of normal cell and reduction cell to stack the full network. Typically, a cell is defined as a directed acyclic graph (DAG) with  $N$  nodes, where each node denotes a latent representation and the information between every two nodes is transformed by an edge. Each edge  $(i,j)$  contains several candidate operations, and DARTS applies continuous relaxation via the learnable architecture parameters set  $\alpha$  to mix the outputs of different operations, converting the discrete operation selection into a differentiable parameter optimization problem,
+
+$$
+\bar {O} ^ {(i, j)} (x) = \sum_ {k = 1} ^ {| \mathcal {O} |} \beta_ {k} ^ {(i, j)} O _ {k} (x), \quad \beta_ {k} ^ {(i, j)} = \frac {\exp \left(\alpha_ {k} ^ {(i , j)}\right)}{\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \exp \left(\alpha_ {k ^ {\prime}} ^ {(i , j)}\right)} \tag {1}
+$$
+
+where  $x$  and  $\bar{O}$  are the input and mixed output of an edge,  $\mathcal{O}$  is the candidate operation set, and  $\beta$  denotes the softmax-activated architecture parameter set. In this way, we can perform architecture search in a differentiable manner by solving following bi-level optimization objective,
+
+$$
+\min  _ {\alpha} \mathcal {L} _ {v a l} (w ^ {*} (\alpha), \alpha) \tag {2}
+$$
+
+$$
+\text {s . t .} w ^ {*} (\alpha) = \arg \min  _ {w} \mathcal {L} _ {\text {t r a i n}} (w, \alpha)
+$$
+
+In practice, architecture parameters  $\alpha$  and network weights  $w$  are alternately updated on the validation and training datasets via gradient descent, and  $w^{*}$  is approximated by one-step forward or current  $w$  [21].
+
+# 3.2. Commonly-used Regularization
+
+In this paper, we intend to improve the robustness and architecture generalization of DARTS by explicitly regularizing the optimizing process of architecture parameters. Thus, we begin with the default settings of previous methods, namely L2 or weight decay regularization on architecture parameters,  $\alpha$ . For convenience of analysing, we consider the single-step update of the architecture parameters,
+
+$$
+\alpha_ {k} ^ {t + 1} \leftarrow \alpha_ {k} ^ {t} - \eta_ {\alpha} \cdot \nabla_ {\alpha_ {k}} \mathcal {L} _ {v a l} \tag {3}
+$$
+
+where  $\eta_{\alpha}$  and  $\mathcal{L}_{val}$  are the learning rate of architecture parameters and the corresponding loss respectively. For multi-step updates, it can be transformed into a single-step update problem through step-wise recursive analysis.
+
+In standard DARTS and its subsequent variants, Adam optimizer with L2 regularization is commonly used for architecture parameters optimization. However, for adaptive gradient algorithms, the gradients of L2 regularization are normalized  $(\mathcal{N})$  by their summed magnitudes, thus the penalty for each element is relatively even, which partly offsets the effect of L2 regularization [23], defined as
+
+$$
+\bar {\alpha} _ {k} ^ {t + 1} \leftarrow \alpha_ {k} ^ {t} - \eta_ {\alpha} \cdot \nabla_ {\alpha_ {k}} \mathcal {L} _ {v a l} - \eta_ {\alpha} \lambda \mathcal {N} \left(\alpha_ {k} ^ {t}\right) \tag {4}
+$$
+
+Considering that L2 regularization may not be effective in adaptive gradient algorithms and is not identical to weight decay regularization [23], without loss of generality, we also include architecture parameters optimization with weight decay regularization [14] for comparison, defined as
+
+$$
+\bar {\alpha} _ {k} ^ {t + 1} \leftarrow \alpha_ {k} ^ {t} - \eta_ {\alpha} \cdot \nabla_ {\alpha_ {k}} \mathcal {L} _ {v a l} - \eta_ {\alpha} \lambda \alpha_ {k} ^ {t} \tag {5}
+$$
+
+# 3.3. Beta-Decay Regularization
+
+Since the searching and discretization process of DARTS actually utilize softmax-activated architecture parameter set,  $\beta$ , to represent the importance of each operator, we shall pay more attention to the explicit regularization on  $\beta$ . As shown in Subsection 3.4, Beta regularization has the ability to improve the robustness and architecture generalization of DARTS, which further denotes its significance. Although important, Beta regularization is typically ignored by previous works. This paper is devoted to filling this gap. Similar to the idea of most regularization methods, the core purpose of Beta regularization is to constrain the value of Beta from changing too much, formulated as
+
+$$
+\bar {\beta} _ {k} ^ {t + 1} = \theta_ {k} ^ {t + 1} \left(\alpha_ {k} ^ {t}\right) \beta_ {k} ^ {t + 1} \tag {6}
+$$
+
+For simplicity, we use a  $\theta$  function with  $\alpha$  as the independent variable to express the total influence of Beta regularization here. To realize above Beta regularization similar to weight decay through  $\alpha$ , we firstly study the influence of  $\alpha$  regularization on  $\beta$ . Recalling Eq. (4) and Eq. (5), we can conclude a unified formula as:
+
+$$
+\bar {\alpha} _ {k} ^ {t + 1} \leftarrow \alpha_ {k} ^ {t} - \eta_ {\alpha} \nabla_ {\alpha_ {k}} \mathcal {L} _ {v a l} - \eta_ {\alpha} \lambda F \left(\alpha_ {k} ^ {t}\right) \tag {7}
+$$
+
+Further, we substitute Eq. (7) and Eq. (3) into Eq. (1) to get  $\bar{\beta}_k^{t + 1}$  and  $\beta_k^{t + 1}$ , and then divide the former by the latter.
+
+$$
+\frac {\bar {\beta} _ {k} ^ {t + 1}}{\beta_ {k} ^ {t + 1}} = \frac {\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \exp \left(\alpha_ {k ^ {\prime}} ^ {t + 1}\right)}{\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \left[ \exp \left(F \left(\alpha_ {k} ^ {t}\right) - F \left(\alpha_ {k ^ {\prime}} ^ {t}\right)\right) \right] ^ {\lambda \eta_ {\alpha}} \exp \left(\alpha_ {k ^ {\prime}} ^ {t + 1}\right)} \tag {8}
+$$
+
+As we can see in Eq. (8), the mapping function  $F$  determines the influence of  $\alpha$  on  $\beta$ . Thus, all we need is to look for a suitable mapping function,  $F$ . Intuitively, a satisfactory  $F$  should meet the following two points: (1)  $F$  is
+
+![](images/4d7ef0fa3beaf85d49adf171c97a4230f2d48f74f232ed29a9ad0d0098c7f70a.jpg)  
+(a) CIFAR10
+
+![](images/38d1f73f8ce398cd9b3887a34a4661024ce676ff0fc20e0dfc60ee6e5c1f84e8.jpg)  
+(b) CIFAR100
+
+![](images/001668c7bfd32de0b5670b29435759648700e70fab96a0278149edd846ec40cc.jpg)  
+(c) ImageNet16
+
+not affected by the amplitude of  $\alpha$  (to avoid invalid regularization and optimization difficulties). (2)  $F$  can reflect the relative amplitude of  $\alpha$  (to impose more penalty on larger amplitude). To satisfy the two requirements, we adopt the softmax to normalize  $\alpha$ ,
+
+$$
+F \left(\alpha_ {k}\right) = \frac {\exp \left(\alpha_ {k}\right)}{\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \exp \left(\alpha_ {k ^ {\prime}}\right)} \tag {9}
+$$
+
+Then we introduce our proposed Beta-Decay regularization loss, whose gradients with respective to  $\alpha$  equals to  $F(\alpha)$
+
+$$
+\mathcal {L} _ {B e t a} = \log \left(\sum_ {k = 1} ^ {| \mathcal {O} |} e ^ {\alpha_ {k}}\right) = \text {s m o o t h m a x} (\{\alpha_ {k} \}) \tag {10}
+$$
+
+After that, substituting Eq. (9) into Eq. (8), we can obtain following equation, which further accounts for the effect of Beta-Decay regularization,
+
+$$
+\theta_ {k} ^ {t + 1} \left(\alpha_ {k} ^ {t}\right) = \frac {\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \exp \left(\alpha_ {k ^ {\prime}} ^ {t + 1}\right)}{\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \left[ \exp \left(\frac {\exp \left(\alpha_ {k} ^ {t}\right) - \exp \left(\alpha_ {k ^ {\prime}} ^ {t}\right)}{\sum_ {k ^ {\prime \prime} = 1} ^ {| \mathcal {O} |} \exp \left(\alpha_ {k ^ {\prime \prime}} ^ {t}\right)}\right) \right] ^ {\lambda \eta_ {\alpha}} \exp \left(\alpha_ {k ^ {\prime}} ^ {t + 1}\right)} \tag {11}
+$$
+
+Observing the above formula, we can get following conclusions: (1) When  $\alpha$  is the largest,  $\theta$  is the smallest and less than 1; when  $\alpha$  is the smallest,  $\theta$  is the largest and greater than 1; and when  $\alpha$  is equal,  $\theta = 1$ . (2) In current iteration,  $\theta$  decreases as  $\alpha$  increases. (3)  $\theta$  is smaller when  $\alpha$  is larger, and  $\theta$  is larger when  $\alpha$  is smaller. As a result, the variance of  $\beta$  is constrained to be smaller, and the value of  $\beta$  is constrained to be closer to its mean, achieving the effect similar to weight decay, thus called Beta-Decay regularization.
+
+# 3.4. Theoretical Analysis
+
+Stronger Robustness. According to the theorem revealed by recent work [32], the convergence of network weights  $w$  can heavily rely on  $\beta_{skip}$  in the supernet. In details, supposing that there are three operations (convolution, skip connection and none) in the search space and the training loss is MSE, when fixing architecture parameters to optimize network weights via gradient descent, at one step the training loss can be reduced by ratio  $(1 - \eta_w\varphi /4)$  with a probability
+
+![](images/dcdc2931feec223f924cb8927aa5f6489b6f00469ec249cce71a211d9a82382f.jpg)  
+Figure 2. Accuracy of different datasets of DARTS with L2, Weight Decay (WD) and Beta-Decay (BD) regularization on NAS-Bench-201 benchmark. The curve is smoothed with a coefficient of 0.5. Note that we only search once on CIFAR-10 dataset.  
+(a) DARTS, total std=3.1  
+Figure 3. The beta distribution of normal cell learned by DARTS and  $\beta$ -DARTS, on the original search space in CIFAR-10. The operator indexes  $1/2/3$  mean the max pool/avg pool/skip connect, while others are the parametric operators. The total std is calculated by the sum of the standard deviation of all edges under the edge independence assumption.
+
+![](images/bdbbb027c1273a5c925449083b24fe4d6a2a8f60297e8423a4d4fed2ebf34c78.jpg)  
+(b)  $\beta$ -DARTS, total std=0.63
+
+of at least  $1 - \sigma$ , where  $\eta_w$  is the corresponding learning rate and will be bounded by  $\sigma$ , and  $\varphi$  obeys
+
+$$
+\varphi \propto \sum_ {i = 0} ^ {h - 2} \left[ \left(\beta_ {\text {c o n v}} ^ {(i, h - 1)}\right) ^ {2} \prod_ {t = 0} ^ {i - 1} \left(\beta_ {\text {s k i p}} ^ {(t, i)}\right) ^ {2} \right] \tag {12}
+$$
+
+where  $h$  is the number of supernet layers. From Eq. (12), we can see that  $\varphi$  depends more on  $\beta_{skip}$  than  $\beta_{conv}$ , which demonstrates that the supernet weights can converge much faster with large  $\beta_{skip}$ . However, by imposing Beta-Decay regularization, we can redefine Eq. (12) as follows
+
+$$
+\varphi \propto \sum_ {i = 0} ^ {h - 2} \left[ \left(\theta_ {\text {c o n v}} ^ {(i, h - 1)} \beta_ {\text {c o n v}} ^ {(i, h - 1)}\right) ^ {2} \prod_ {t = 0} ^ {i - 1} \left(\theta_ {\text {s k i p}} ^ {(i, h - 1)} \beta_ {\text {s k i p}} ^ {(t, i)}\right) ^ {2} \right] \tag {13}
+$$
+
+As mentioned before,  $\theta$  becomes smaller when  $\beta$  is larger and  $\theta$  becomes larger when  $\beta$  is smaller, which makes the convergence of network weights rely more on  $\beta_{conv}$  and less on  $\beta_{skip}$ . From the perspective of convergence theorem [32], the Beta-Decay regularization constrains the privilege of  $\beta_{skip}$  and ensures the fair competition among architecture parameters. As shown in Fig. 2, DARTS with L2 or weight decay regularization suffers from the performance collapse issue, while DARTS with Beta-Decay regularization has a stable search process. As shown in Fig. 3, original
+
+![](images/091e7cb995965f58e4f08b17ac39f401d0239baa03d190fb29ee8301be14457a.jpg)  
+(a) L2 regularization
+
+![](images/6ab4afe4107434c737de7b4d977bf169d85fe117628513c79849a8a80dafb6c3.jpg)  
+(b) weight decay regularization
+
+![](images/4077674bdc701b3ef0b99dd93c156846464fb36e1a2f565f23959f4ac576b604.jpg)  
+(c) Beta-Decay regularization  
+Figure 4. The alpha statistical characteristics (i.e. mean, median and standard deviation) of different edges of each epoch when searching on NAS-Bench-201 benchmark with (a) L2 regularization, (b) weight decay regularization, and (c) Beta-Decay regularization.
+
+DARTS is dominated by skip connections while  $\beta$ -DARTS tends to favor parametric operators.
+
+Stronger Generalization. Referring to [24] and [10], Lipschitz constraint is commonly used to measure and improve the generalization ability of trained deep model. Specifically, suppose the function fitted by a deep model is  $f_{w}(x)$  where  $x$  is the input, when  $\| x_1 - x_2\|$  is very small, a well-trained model should meet the following constraint.
+
+$$
+\left\| f _ {w} \left(x _ {1}\right) - f _ {w} \left(x _ {2}\right) \right\| \leq C (w) \cdot \| x _ {1} - x _ {2} \| \tag {14}
+$$
+
+where  $C(w)$  is the Lipschitz constant. The smaller the constant is, the trained model will be less sensitive to input disturbances and have better generalization ability.
+
+Furthermore, we can extend this theory to differentiable architecture search. For simplicity, we consider a single-layer neural network, and multi-layer neural network can be solved through step-wise recursive analysis. Suppose the single-layer network is mixed by the operation set  $F(x) = (f_{1}(x), f_{2}(x), f_{3}(x))$ , with the corresponding architecture parameters  $\beta = (\beta_{1}, \beta_{2}, \beta_{3})$ . According to the Cauchy's inequality, we can get the following inequality.
+
+$$
+\left\| \beta F ^ {\mathrm {T}} \left(x _ {1}\right) - \beta F ^ {\mathrm {T}} \left(x _ {2}\right) \right\| \leq \| \beta \| \left\| F ^ {\mathrm {T}} \left(x _ {1}\right) - F ^ {\mathrm {T}} \left(x _ {2}\right) \right\| \tag {15}
+$$
+
+where  $\| \beta \| = \sqrt{\sum \beta_i^2}$  can be regarded as Lipschitz constant and  $\sum \beta_{i} = 1$ . As a result, the smaller the measure  $\| \beta \|$  is, the supernet will be less sensitive to the impact of input on the operation set, and the searched architecture will have better generalization ability. As shown in Fig. 2, the model searched by  $\beta$ -DARTS on CIFAR-10 can well generalize to the CIFAR-100 and ImageNet16 datasets and achieve excellent results. As shown in Fig. 4 and Fig. 3, the architecture parameter distribution learned by  $\beta$ -DARTS maintains a relative small standard deviation, making sure the generalization ability of the searched model.
+
+# 3.5. Commonly-used Regularization May Not Work
+
+When using L2 regularization on  $\alpha$ , we can obtain its effect on  $\beta$  according to Eq. (4) and Eq. (8), defined as
+
+$$
+\frac {\bar {\beta} _ {k} ^ {t + 1}}{\beta_ {k} ^ {t + 1}} = \frac {\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \exp \left(\alpha_ {k ^ {\prime}} ^ {t + 1}\right)}{\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \left[ \exp \left(\mathcal {N} \left(\alpha_ {k} ^ {t}\right) - \mathcal {N} \left(\alpha_ {k ^ {\prime}} ^ {t}\right)\right) \right] ^ {\lambda \eta_ {\alpha}} \exp \left(\alpha_ {k ^ {\prime}} ^ {t + 1}\right)} \tag {16}
+$$
+
+Similarly, when using weight decay on  $\alpha$ , we can obtain its effect on  $\beta$  according to Eq. (5) and Eq. (8), as follows
+
+$$
+\frac {\bar {\beta} _ {k} ^ {t + 1}}{\beta_ {k} ^ {t + 1}} = \frac {\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \exp \left(\alpha_ {k ^ {\prime}} ^ {t + 1}\right)}{\sum_ {k ^ {\prime} = 1} ^ {| \mathcal {O} |} \left[ \exp \left(\alpha_ {k} ^ {t} - \alpha_ {k ^ {\prime}} ^ {t}\right) \right] ^ {\lambda \eta_ {\alpha}} \exp \left(\alpha_ {k ^ {\prime}} ^ {t + 1}\right)} \tag {17}
+$$
+
+From Eq. (16) and Eq. (17), we can find that: (1) When the values in  $\alpha$  are all around 0, achieving the purpose of L2 and weight decay regularization, Alpha regularization has little effect on Beta; while when the values in  $\alpha$  are all not near 0, it means that both regularization do not work. (2) For L2 and weight decay regularization on  $\alpha$ , only when the median of  $\alpha$  is equal to 0, Alpha regularization has the same and correct effect as Beta regularization. (3) A large variance of  $\alpha$  is undesirable, which conflicts with the purpose of L2 and weight decay regularization, and makes the optimization process more sensitive to the hyperparameters  $\lambda$  and  $\eta_{\alpha}$ . In addition, we show the alpha statistical characteristics when searching with different regularization in Fig. 4, we can see that for L2 and weight decay regularization: (1) The mean and median of  $\alpha$  continue to decrease and gradually move away from 0. (2) The standard deviation of  $\alpha$  increases monotonically. These mathematical and experimental results show that L2 or weight decay regularization commonly used in existing gradient-based methods are not identical to Beta regularization, and may not be effective or even counterproductive. As a comparison, with our proposed Beta-Decay regularization: (1) The mean and median
+
+Table 1. Performance comparison on NAS-Bench-201 benchmark [9]. Note that  $\beta$ -DARTS only searches on CIFAR-10 dataset, but can robustly achieve new SOTA on CIFAR-10, CIFAR-100 and ImageNet16-120. Averaged on 4 independent runs of searching.  
+
+<table><tr><td rowspan="2">Methods</td><td rowspan="2">Cost (hours)</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td><td colspan="2">ImageNet16-120</td></tr><tr><td>valid</td><td>test</td><td>valid</td><td>test</td><td>valid</td><td>test</td></tr><tr><td>DARTS(1st) [21]</td><td>3.2</td><td>39.77±0.00</td><td>54.30±0.00</td><td>15.03±0.00</td><td>15.61±0.00</td><td>16.43±0.00</td><td>16.32±0.00</td></tr><tr><td>DARTS(2nd) [21]</td><td>10.2</td><td>39.77±0.00</td><td>54.30±0.00</td><td>15.03±0.00</td><td>15.61±0.00</td><td>16.43±0.00</td><td>16.32±0.00</td></tr><tr><td>GDAS [8]</td><td>8.7</td><td>89.89±0.08</td><td>93.61±0.09</td><td>71.34±0.04</td><td>70.70±0.30</td><td>41.59±1.33</td><td>41.71±0.98</td></tr><tr><td>SNAS [28]</td><td>-</td><td>90.10±1.04</td><td>92.77±0.83</td><td>69.69±2.39</td><td>69.34±1.98</td><td>42.84±1.79</td><td>43.16±2.64</td></tr><tr><td>DSNAS [16]</td><td>-</td><td>89.66±0.29</td><td>93.08±0.13</td><td>30.87±16.40</td><td>31.01±16.38</td><td>40.61±0.09</td><td>41.07±0.09</td></tr><tr><td>PC-DARTS [29]</td><td>-</td><td>89.96±0.15</td><td>93.41±0.30</td><td>67.12±0.39</td><td>67.48±0.89</td><td>40.83±0.08</td><td>41.31±0.22</td></tr><tr><td>iDARTS [31]</td><td>-</td><td>89.86±0.60</td><td>93.58±0.32</td><td>70.57±0.24</td><td>70.83±0.48</td><td>40.38±0.59</td><td>40.89±0.68</td></tr><tr><td>DARTS- [5]</td><td>3.2</td><td>91.03±0.44</td><td>93.80±0.40</td><td>71.36±1.51</td><td>71.53±1.51</td><td>44.87±1.46</td><td>45.12±0.82</td></tr><tr><td>β-DARTS</td><td>3.2</td><td>91.55±0.00</td><td>94.36±0.00</td><td>73.49±0.00</td><td>73.51±0.00</td><td>46.37±0.00</td><td>46.34±0.00</td></tr><tr><td>optimal</td><td>-</td><td>91.61</td><td>94.37</td><td>73.49</td><td>73.51</td><td>46.77</td><td>47.31</td></tr></table>
+
+of  $\alpha$  are basically equal. (2) When the standard deviation of  $\alpha$  increases to a certain extent, it will remain unchanged.
+
+# 4. Experiments
+
+In this section, we conduct extensive experiments on various search spaces (i.e. NAS-Bench-201, DARTS, NAS-Bench-1Shot1) and datasets (i.e. CIFAR-10, CIFAR-100, ImageNet) to verify the robustness and generalization of  $\beta$ -DARTS, and we further give some experimental insights about DARTS' dependence on training and data. The overall process of  $\beta$ -DARTS is summarized in Alg 2.
+
+# Algorithm 2  $\beta$ -DARTS
+
+# Require:
+
+Architecture parameters  $\alpha$ ; Network weights  $w$ ; Number of search epochs  $E$ ; Regularization coefficient adjustment scheme  $\lambda_{e}, e \in \{1,2,\dots,E\}$ .
+
+1: Construct a supernet and initialize architecture parameters  $\alpha$  and supernet weights  $w$  
+2: For each  $e \in [1, E]$  do  
+3: Update architecture parameters  $\alpha$  by descending  $\nabla_{\alpha}\mathcal{L}_{val} + \lambda_{e}\mathcal{L}_{Beta}$  
+4: Update network weights w by descending  $\nabla_w\mathcal{L}_{train}$  
+5: Derive the final architecture based on the learned  $\alpha$ .
+
+# 4.1. Results on NAS-Bench-201 Search Space
+
+Settings. NAS-Bench-201 [9] is the most widely used NAS benchmark analyzing various NAS methods. NAS-Bench-201 provides a DARTS-like search space, containing 4 internal nodes with 5 associated operations. The search space consists of 15,625 architectures, with the ground truth performance of CIFAR-10, CIFAR-100 and ImageNet16-120 of each architecture provided. On NAS-Bench-201, the searching settings are kept the same as DARTS on [9].
+
+Results. The comparison results are shown in Table 1. We only search on CIFAR-10 and use the found genotype to query the performance of various datasets. For robustness, our 4 runs of searching under different random seeds always find the same optimal solution, which is very close
+
+to the optimal performance of NAS-Bench-201. Moreover, as shown in Fig. 2, the performance collapse issue is well solved and  $\beta$ -DARTS has a stable search process. For generalization ability, we can see that the architecture found on CIFAR-10 achieves consistent new SOTA on CIFAR-10, CIFAR-100 and ImageNet. For dependency on training and data, as shown in Fig. 2, the search process reaches its optimal point at an early stage (i.e., before 20 epochs), on different datasets. Such results validate that  $\beta$ -DARTS has the ability to find the optimal architecture rapidly. More interestingly, we find that the search process of different datasets reach the optimal point in different epochs, although they belong to the same run of searching on CIFAR-10. More similar results can be found in Appendix A.1.
+
+# 4.2. Results on DARTS Search Space
+
+Settings. Common DARTS search space [21] is also popular for evaluating NAS methods. The search space consists of normal cell and reduction cell. Each cell has 4 intermediate nodes with 14 edges, and each edge is associated with 8 candidate operations. On DARTS search space, all the search settings are kept the same as DARTS since our method only introduces the simple regularization. For evaluation settings, the evaluation on CIFAR-10/100 follows DARTS [21] and the evaluation on ImageNet-1K follows P-DARTS [4] and PC-DARTS [29].
+
+Results. The comparison results are shown in Table 2. We search on CIFAR-10 or CIFAR-100 while evaluating the inferred architecture on CIFAR-10, CIFAR-100 and ImageNet. For robustness, the average results of multiple independent runs of  $\beta$ -DARTS achieve the SOTA performance on both CIFAR-10 and CIFAR-100, namely  $97.47 \pm 0.08\%$  and  $83.48 \pm 0.03\%$ , without extra changes or any cost. For generalization ability, architectures found on CIFAR-100 can still yield a SOTA result of  $97.49 \pm 0.07\%$  on CIFAR-10, and models found on CIFAR-10 obtain a new SOTA of  $83.76 \pm 0.22\%$  on CIFAR-100, and networks found on both CIFAR-10 and CIFAR-100 datasets can achieve comparable results on ImageNet with those of directly searching on
+
+Table 2. Comparison of SOTA models on CIFAR-10/100(left) and ImageNet-1K(right). For CIFAR-10/100, results in the top block are obtained by training the best searched model while the bottom block shows the average results of multiple runs of searching.  $\ddagger$  denotes the results of independently searching 3 times on CIFAR-100 and evaluating on both CIFAR-10 and CIFAR-100, while  $\dagger$  denotes the results on CIFAR-10. Because of the difference on classifiers, the network parameters on CIFAR-100 is slightly more than that of CIFAR-10 (about 0.05M). For ImageNet, the top block denotes networks are directly searched on ImageNet (Img.), the middle block indicates architectures are searched via the idea of Cross Domain (CD.) using CIFAR-10 and part of ImageNet, models in the bottom block are transferred from the searching results of CIFAR-10 (C10) or CIFAR-100 (C100). \* denotes the model is obtained on a different search space.
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">GPU (Days)</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>Params(M)</td><td>Acc(%)</td><td>Params(M)</td><td>Acc(%)</td></tr><tr><td>NASNet-A [33]</td><td>2000</td><td>3.3</td><td>97.35</td><td>3.3</td><td>83.18</td></tr><tr><td>DARTS(1st) [21]</td><td>0.4</td><td>3.4</td><td>97.00±0.14</td><td>3.4</td><td>82.46</td></tr><tr><td>DARTS(2nd) [21]</td><td>1</td><td>3.3</td><td>97.24±0.09</td><td>-</td><td>-</td></tr><tr><td>SNAS [28]</td><td>1.5</td><td>2.8</td><td>97.15±0.02</td><td>2.8</td><td>82.45</td></tr><tr><td>GDAS [8]</td><td>0.2</td><td>3.4</td><td>97.07</td><td>3.4</td><td>81.62</td></tr><tr><td>P-DARTS [4]</td><td>0.3</td><td>3.4</td><td>97.50</td><td>3.6</td><td>82.51</td></tr><tr><td>PC-DARTS [29]</td><td>0.1</td><td>3.6</td><td>97.43±0.07</td><td>3.6</td><td>83.10</td></tr><tr><td>P-DARTS [4]</td><td>0.3</td><td>3.3±0.21</td><td>97.19±0.14</td><td>-</td><td>-</td></tr><tr><td>R-DARTS(L2) [1]</td><td>1.6</td><td>-</td><td>97.05±0.21</td><td>-</td><td>81.99±0.26</td></tr><tr><td>SDARTS-ADV [3]</td><td>1.3</td><td>3.3</td><td>97.39±0.02</td><td>-</td><td>-</td></tr><tr><td>DOTS [12]</td><td>0.3</td><td>3.5</td><td>97.51±0.06</td><td>4.1</td><td>83.52±0.13</td></tr><tr><td>DARTS+PT [27]</td><td>0.8</td><td>3.0</td><td>97.39±0.08</td><td>-</td><td>-</td></tr><tr><td>DARTS- [5]</td><td>0.4</td><td>3.5±0.13</td><td>97.41±0.08</td><td>3.4</td><td>82.49±0.25</td></tr><tr><td>β-DARTS†</td><td>0.4</td><td>3.78±0.08</td><td>97.49±0.07</td><td>3.83±0.08</td><td>83.48±0.03</td></tr><tr><td>β-DARTS†</td><td>0.4</td><td>3.75±0.15</td><td>97.47±0.08</td><td>3.80±0.15</td><td>83.76±0.22</td></tr></table>
+
+<table><tr><td>Method</td><td>GPU (Days)</td><td>Params (M)</td><td>FLOPs (M)</td><td>Top1 (%)</td><td>Top5 (%)</td></tr><tr><td>MnasNet-92*(Img.) [26]</td><td>1667</td><td>4.4</td><td>388</td><td>74.8</td><td>92.0</td></tr><tr><td>FairDARTS*(Img.) [6]</td><td>3</td><td>4.3</td><td>440</td><td>75.6</td><td>92.6</td></tr><tr><td>PC-DARTS(Img.) [29]</td><td>3.8</td><td>5.3</td><td>597</td><td>75.8</td><td>92.7</td></tr><tr><td>DOTS(Img.) [12]</td><td>1.3</td><td>5.3</td><td>596</td><td>76.0</td><td>92.8</td></tr><tr><td>DARTS-*(Img.) [5]</td><td>4.5</td><td>4.9</td><td>467</td><td>76.2</td><td>93.0</td></tr><tr><td>AdaptNAS-S(CD.) [19]</td><td>1.8</td><td>5.0</td><td>552</td><td>74.7</td><td>92.2</td></tr><tr><td>AdaptNAS-C(CD.) [19]</td><td>2.0</td><td>5.3</td><td>583</td><td>75.8</td><td>92.6</td></tr><tr><td>AmoebaNet-C(C10) [25]</td><td>3150</td><td>6.4</td><td>570</td><td>75.7</td><td>92.4</td></tr><tr><td>SNAS(C10) [28]</td><td>1.5</td><td>4.3</td><td>522</td><td>72.7</td><td>90.8</td></tr><tr><td>P-DARTS(C100) [4]</td><td>0.3</td><td>5.1</td><td>577</td><td>75.3</td><td>92.5</td></tr><tr><td>SDARTS-ADV(C10) [3]</td><td>1.3</td><td>5.4</td><td>594</td><td>74.8</td><td>92.2</td></tr><tr><td>DOTS(C10) [12]</td><td>0.3</td><td>5.2</td><td>581</td><td>75.7</td><td>92.6</td></tr><tr><td>DARTS+PT(C10) [27]</td><td>0.8</td><td>4.6</td><td>-</td><td>74.5</td><td>92.0</td></tr><tr><td>β-DARTS(C100)</td><td>0.4</td><td>5.4</td><td>597</td><td>75.8</td><td>92.9</td></tr><tr><td>β-DARTS(C10)</td><td>0.4</td><td>5.5</td><td>609</td><td>76.1</td><td>93.0</td></tr></table>
+
+Table 3. Influence of different weighting schemes on  $\beta$  -DARTS.  
+
+<table><tr><td>Weighting Scheme</td><td>CIFAR-10 valid</td><td>CIFAR-10 test</td></tr><tr><td>0-15/25/50/100</td><td>91.21/91.55/91.55/91.55</td><td>93.83/94.36/94.36/94.36</td></tr><tr><td>5/10/15/25</td><td>84.96/90.59/91.55/90.59</td><td>88.02/93.31/94.36/93.31</td></tr><tr><td>25-15/10/5/0</td><td>90.59/87.30/73.58/39.77</td><td>93.31/90.65/76.88/54.30</td></tr></table>
+
+![](images/22a50d4a8d5487200851dbc408463c46ed9e81ecd74556ce6635b9c6ea2aa345.jpg)  
+(a)  $\beta$ -DARTS on NAS-Bench-201
+
+![](images/1f68c91f37b1bbccc9bbdc8cb20f2a7a9baa1d9509b3ac2806ae20ed0fc8ad31.jpg)  
+(b)  $\beta$ -SDARTS-RS on NAS-Bench-1Shot1  
+Figure 5. The effect of different max weight of linear increased weighting schemes on the searching results.
+
+ImageNet or using cross domain method.
+
+# 4.3. Ablation Study
+
+Importance of Increased Weighting Scheme. We firstly explore the influence of different weighting schemes on  $\beta$ -DARTS, including linear increased weighting scheme, constant weighting scheme and linear decay weighting scheme. The results are shown in Table 3. As we can see, linear decay weighting scheme impedes the effect of regularization, constant weighting scheme is sensitive to the hyperparameter, while linear increased weighting scheme is not only effective but also insensitive to hyperparameter. Besides, combining with the results of Fig. 2 that the performance on CIFAR-10, CIFAR-100, andImagenet in single run of searching reach the optimal point in order, we conclude that linear increased regularization coefficient can further maximize the generalization ability of inferred model after the
+
+searching performance on current data is maximized, as evidenced by Eq. (11) and Eq. (15).
+
+Wide Range of The Optimal Weight. We further investigate the optimal max weight of linear increased weighting scheme. The results on CIFAR-10 of NAS-Bench-201 and search space 1 of NAS-Bench-1Shot1 are provided in Fig. 5. We can see that the best performance is achieved in a wide range of max weights, namely about 25-100 and 1-13 for  $\beta$ -DARTS on NAS-Bench-201 and  $\beta$ -SDARTS-RS on NAS-Bench-1Shot1 respectively. There are similar results on CIFAR-10 and CIFAR-100 in common DARTS search space, as shown in Appendix A.2. If not mentioned specially, the default values of max weight for NAS-Bench-201, NAS-Bench-1Shot1, CIFAR-10 and CIFAR-100 are set to 50, 7, 0.5, 5 respectively in all our experiments. Furthermore, comparing Eq. (11) with Eq. (16) and Eq. (17), we find that the normalized values of  $\alpha$  in Eq. (11) has the ability to make sure that the optimization process is not sensitive to the hyperparameter of  $\lambda$ .
+
+# 4.4. Discussions
+
+Non-uniqueness. Actually, the idea of Beta regularization is what really matters, and the way to realize it is nonunique. Here, we show two kinds of variants of Beta regularization loss. Recalling Eq. (10), we can naturally figure out an alternative, using the smoothmax of all architecture parameters on the entire supernet, namely Beta-Global loss.
+
+$$
+\begin{array}{l} \mathcal {L} _ {B e t a - G l o b a l} = \operatorname {s m o o t h m a x} \left( \begin{array}{c} \alpha_ {1} ^ {1}, \dots , \alpha_ {| \mathcal {O} |} ^ {L} \end{array} \right) \\ = \log \left(\sum_ {l = 1} ^ {L} \sum_ {k = 1} ^ {| \mathcal {O} |} e ^ {\alpha_ {k} ^ {l}}\right) \tag {18} \\ \end{array}
+$$
+
+Table 4. The results of different Beta regularization loss with different weighting schemes on NAS-Bench-201 benchmark. Note that we only search on CIFAR-10 dataset, and perform 2 runs of searching under different random seeds.  
+
+<table><tr><td rowspan="2">Methods</td><td rowspan="2">Weighting Scheme</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td><td colspan="2">ImageNet16-120</td></tr><tr><td>valid</td><td>test</td><td>valid</td><td>test</td><td>valid</td><td>test</td></tr><tr><td>DARTS(1st) [21]</td><td>3.2</td><td>39.77±0.00</td><td>54.30±0.00</td><td>15.03±0.00</td><td>15.61±0.00</td><td>16.43±0.00</td><td>16.32±0.00</td></tr><tr><td>Beta-Global</td><td>0-25</td><td>91.55/91.55</td><td>94.36/94.36</td><td>73.49/73.49</td><td>73.51/73.51</td><td>46.37/46.37</td><td>46.34/46.34</td></tr><tr><td>Beta-Global</td><td>0-50</td><td>91.55/91.55</td><td>94.36/94.36</td><td>73.49/73.49</td><td>73.51/73.51</td><td>46.37/46.37</td><td>46.34/46.34</td></tr><tr><td>Beta-Global</td><td>0-75</td><td>91.55/91.55</td><td>94.36/94.36</td><td>73.49/73.49</td><td>73.51/73.51</td><td>46.37/46.37</td><td>46.34/46.34</td></tr><tr><td>Beta-Global</td><td>0-100</td><td>91.21/91.55</td><td>93.83/94.36</td><td>71.60/73.49</td><td>71.88/73.51</td><td>45.75/46.37</td><td>44.65/46.34</td></tr><tr><td>Beta-Zero</td><td>0-25</td><td>91.21/90.97</td><td>93.83/93.91</td><td>71.60/70.41</td><td>71.88/70.78</td><td>45.75/43.77</td><td>44.65/44.78</td></tr><tr><td>Beta-Zero</td><td>0-50</td><td>91.55/91.21</td><td>94.36/93.83</td><td>73.49/71.60</td><td>73.51/71.88</td><td>46.37/45.74</td><td>46.34/44.65</td></tr><tr><td>Beta-Zero</td><td>0-75</td><td>91.61/91.05</td><td>94.37/93.66</td><td>72.75/71.02</td><td>73.22/71.38</td><td>45.56/45.23</td><td>46.71/44.70</td></tr><tr><td>Beta-Zero</td><td>0-100</td><td>91.21/91.21</td><td>93.83/93.83</td><td>71.60/71.60</td><td>71.88/71.88</td><td>45.75/45.75</td><td>44.65/44.65</td></tr></table>
+
+![](images/1a80e6e631585c362becfa00ee07871cff239fab552cc1d79b58e54d9cd6766e.jpg)  
+(a) Search Space 1
+
+![](images/c5040ab34b5a40b287a2ad1ec0f98cc1d4f3134750f6e8b0be2d9539d6647352.jpg)  
+(b) Search Space 2  
+Figure 6. Error of SDARTS-RS and  $\beta$ -SDARTS-RS on 3 search spaces of NAS-Bench-1Shot1 [30]. The curve is smoothed with 0.5.
+
+![](images/aa27a8f5a13a87c66557fd07ed42d7b3ea24203c9cba7bcf432f36e48358dc25.jpg)  
+(c) Search Space 3
+
+In addition, by introducing a threshold, we can get the smoothmax between the threshold and each architecture parameter. We simply set the threshold as 0 in this paper, namely Beta-Zero loss.
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {B e t a - Z e r o}} = \operatorname {s m o o t h m a x} \left(0, \alpha_ {k} ^ {l}\right) \\ = - \log \left(1 + e ^ {- \alpha_ {k} ^ {l}}\right) \tag {19} \\ \end{array}
+$$
+
+The results of DARTS with Beta-Global and Beta-Zero regularization loss are shown in Table. 4. As we can see, both loss can promote original DARTS by a large margin, while Beta-Global loss that takes the same effect with Beta-Decay loss, can more stably obtain better results than Beta-Zero loss under different weighting schemes. Such results validate that regularizing  $\beta$  is important, while the way to achieve it has a lot of room for exploration.
+
+Generality. Moreover, we utilize NAS-Bench-1Shot1 [30] benchmark and SDARTS-RS [3] baseline to demonstrate the generality of Beta-Decay regularization. NAS-Bench-1Shot1 contains 3 search spaces, which consist of 6,240, 29,160 and 363,648 architectures with the CIFAR-10 performance separately. On NAS-Bench-1Shot1, both the operator of each edge and the topology of the cell need to be determined. We show the search trajectory in Fig. 6. On one hand,  $\beta$ -SDARTS-RS can yield much lower test/Validation error than SDARTS-RS across different search spaces. On the other hand, the error of  $\beta$ -SDARTS-RS keeps decreasing while the error of SDARTS-RS increases first and then decreases, validating the more stable search process of  $\beta$ -SDARTS-RS. Besides, the search process of  $\beta$ -SDARTS
+
+RS also reaches its optimal point at an early stage (i.e., around 10 epochs), on different search spaces.
+
+# 5. Conclusion
+
+In this paper, we investigate the explicit regularization on the optimization of architecture parameters of DARTS in depth, which is typically ignored by previous works. Firstly, we identify that L2 or weight decay regularization on alpha commonly used by DARTS and its variants may not be effective or even counterproductive. Then, we propose a novel and generic Beta-Decay regularization loss, for improving DARTS-based methods without extra changes or cost. In addition, we theoretically and experimentally show Beta-Decay regularization can improve both the robustness and the generalization of DARTS. Besides, we find that the proposed search scheme is less dependent on training time and data. Extensive experiments on various search spaces and datasets validate the superiority of our method.
+
+# Acknowledgements
+
+This work was supported by National Natural Science Foundation of China (No.62071127), Shanghai Municipal Science and Technology Major Project (No. 2021SHZDZX0103), and Zhejiang Lab Project (No. 2021KH0AB05). Wanli Ouyang was supported by the Australian Research Council Grant (DP200103223), Australian Medical Research Future Fund (MRFAI000085), and CRC-P Smart Material Recovery Facility SMRF-Curby Soft Plastics.
+
+# References
+
+[1] Thomas Elsken Arber Zela, Tonmoy Saikia, Yassine Marrakchi, Thomas Brox, and Frank Hutter. Understanding and robustifying differentiable architecture search. In International Conference on Learning Representations, volume 3, page 7, 2020.  
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+# $\phi$ -SfT: Shape-from-Template with a Physics-Based Deformation Model
+
+Navami Kairanda $^{1,2}$  Edith Tretschk $^{1}$  Mohamed Elgharib $^{1}$  Christian Theobalt $^{1}$  Vladislav Golyanik $^{1}$ $^{1}$ Max Planck Institute for Informatics, SIC  
+ $^{2}$ Saarland University, SIC
+
+# Abstract
+
+Shape-from-Template (SfT) methods estimate 3D surface deformations from a single monocular RGB camera while assuming a 3D state known in advance (a template). This is an important yet challenging problem due to the under-constrained nature of the monocular setting. Existing SfT techniques predominantly use geometric and simplified deformation models, which often limits their reconstruction abilities. In contrast to previous works, this paper proposes a new SfT approach explaining 2D observations through physical simulations accounting for forces and material properties. Our differentiable physics simulator regularises the surface evolution and optimises the material elastic properties such as bending coefficients, stretching stiffness and density. We use a differentiable renderer to minimise the dense reprojection error between the estimated 3D states and the input images and recover the deformation parameters using an adaptive gradient-based optimisation. For the evaluation, we record with an RGB-D camera challenging real surfaces exposed to physical forces with various material properties and textures. Our approach significantly reduces the 3D reconstruction error compared to multiple competing methods. For the source code and data, see https://4dqv.mpi-inf.mpg.de/phi-SfT/.
+
+# 1. Introduction
+
+Reconstructing general deformable, temporally-coherent surfaces in 3D from monocular RGB videos is a longstanding, challenging and ill-posed problem. It was studied under different assumptions, and methods addressing it can be roughly classified into (template-free) non-rigid structure from motion (NRSfM) [8, 14], (template-based) shape-from-template (SfT) [34, 40], and neural 3D mesh regression [27]. The objective of SfT is: Given a known initial 3D state (a template) of an observed deformable scene or an object, reconstruct all its 3D states observed in the entire image sequence [45]. Recent learning-based SfT methods encode prior knowledge in neural network weights [13, 47]. This offers multiple advantages over a vast body of previ
+
+![](images/4140e2c384e9b724e5a196cefa9cc84efb75b2a4f03058eb09e5cdade71e86a2.jpg)  
+Figure 1. Our  $\phi$ -SfT approach uses a physics simulator to reconstruct challenging deforming 3D surfaces observed in a monocular RGB video. Compared to existing methods, our estimates are significantly more accurate and physically plausible.
+
+ous, model-based works [34, 36, 38, 40, 45, 46, 61], such as the ability to handle larger deformations, a broader spectrum of supported types of motions and deformations (including highly nonlinear ones), and real-time operation. One of the pivotal limitations of both classical and neural SfT methods is that they capture general 3D states well but not fine local surface deformations. This is a consequence of the non-awareness of the physical fold formation process attributable to the elastic properties of the materials and forces acting on them. As a result, existing methods can only reconstruct predominantly global deformations.
+
+This paper proposes  $\phi$ -SfT (from Greek  $\phi \upsilon \sigma \iota \kappa \eta$  meaning physics): A new analysis-by-synthesis SfT method which addresses several limitations of the current state of the art and improves the accuracy of monocular non-rigid 3D reconstruction by a significant margin; see Fig. 1 for an overview. Our approach explicitly models the physical fold formation process, and its parameters are physically meaningful.  $\phi$ -SfT does not require training data. We enable gradient-based optimisation by employing two components: A differentiable renderer and a differentiable physics simulator. Our core idea is to use the latter as a regulariser during the optimisation of our objective function. Next, the differentiable renderer ensures that the reprojections of the recovered 3D states accurately match the observed images. In contrast to earlier photometric terms used for SfT [61],
+
+differentiable rendering allows us for the first time to define the reprojection error densely per pixel and not only per vertex. We can thus exploit the information present in the texture regardless of the mesh resolution.  $\phi$ -SfT is significantly more accurate than related methods and supports finer-scale local folds, which is shown on a wide spectrum of deformations in extensive experiments (Sec. 4). In summary, this paper has the following technical contributions:
+
+- A new optimisation-based SfT approach with a physics-based deformation model that ensures high physical fidelity and realism for the surface evolution, outputting temporally smooth 3D shape sequences that are aware of the forces acting on the object (Sec. 3).  
+- Differentiable rendering for SfT to encourage the 2D projections of the reconstructed 3D structure to match the observed images. The differentiability allows optimising for the deformation parameters by minimising a dense per-pixel photometric energy (Sec. 3.2).  
+- A new dataset of real deforming surfaces recorded in a way to facilitate the quantitative evaluation of reconstruction methods against reference depth maps (Sec. 4.1). The dataset contains surfaces of various textures and materials, exposed to different external forces. We release our dataset and source code to encourage future research.
+
+# 2. Related Work
+
+The methods for monocular non-rigid 3D reconstruction differ in the assumptions they make about the input, available prior knowledge and how they model the deformations. This section reviews methods that can be classified into non-rigid structure from motion (NRSfM), shape from template (SfT) and monocular 3D mesh reconstruction.
+
+NRSfM operates on 2D point tracks over the input monocular views. Earlier NRSfM methods were designed for sparse measurements and modelled deformations with linear subspaces along with various priors [2, 8, 12, 37, 53]. More recent techniques [3, 14, 17, 26] allow reconstructing dense image points observed in a reference frame. They impose constraints on spatial point locations to infer smooth and continuous deforming surfaces. A sparse NRSfM method relying on principles of continuum mechanics represents a deformable object using an estimated (not simulated) elastic model and a low-rank force field acting on it [1]. Even though the force prior has a direct physical interpretation, this model still shares most limitations with other methods. Diff-NRSfM [39] assumes the observed structure preserves its differentiable structure and infinitesimal planarity. This method produces impressive results for smooth surfaces but struggles to reconstruct fine-scale folds, unlike our  $\phi$ -SfT.
+
+Recently, neural NRSfM approaches both for sparse [35, 55] and dense [44, 50] cases were proposed in the literature. Some of them need to be trained for each object
+
+category [35, 44], whereas N-NRSfM [50] and PAUL [55] run on unknown data. Some 2D keypoint lifting approaches for 3D human pose estimation, such as Chen et al. [10], require only 2D data for supervision and share similarities with neural sparse NRSfM.
+
+SfT algorithms operate directly on images and assume a known 3D surface prior as input [34, 40, 45, 61]. These methods minimise the 3D-2D reprojection error and impose geometric constraints such as surface inextensibility [40, 45] or isometry [5, 34, 61]. Recent neural SfT methods [13, 18, 41, 47] predict 3D surfaces from monocular images relying on datasets with different template states. Our  $\phi$ -SfT contrasts with other SfT methods in that it uses temporal information and a differentiable physics simulator as a regulariser for high-fidelity 3D surface tracking instead of approximating the underlying physical properties via geometric constraints. Moreover, none of these methods uses a per-pixel differentiable photometric loss which ensures that 3D estimates accurately reproject into the 2D images.
+
+Physics-based priors in 3D human performance capture is an emerging field, although there is some early work on it relying on multi-view data [51]. Rempe et al.'s [43] method and PhysCap [48, 49] show that integrating physics laws into an objective for sparse 3D human motion capture improves the accuracy of the 3D estimates. The proposed constraints reduce the artefacts arising from the monocular setting (e.g., unnatural jitter and body leaning, foot sliding and foot-floor penetration). Several methods for 3D human performance capture include clothes deformations, such as Guo et al. [19] and Li et al. [28]. The method of Guo et al. operates on point clouds and optimises the states of the simulated clothes so that they match the inputs. The cloth motion is expressed through a combination of skin friction, gravity and forces attributed to the material (elasticity). Thus, their focus is cloth state recovery from sparse point cloud measurements, which provide a strong 3D shape cues, whereas we assume a single 3D template and operate on monocular videos; this inverse problem is much more ill-posed. Li et al. generate training data with a physics-based simulator on-the-fly and use it to train a neural network for 3D human performance capture, including clothes deformations. Thus, they do not impose hard physics-based constraints as we do with the differentiable physics simulator. Liang et al. [29] use 3D supervision for physics-based cloth simulation. The work by Weiss et al. [58] recovers material parameters of a physics simulator in an analysis-by-synthesis policy to solve an inverse elasticity problem. In contrast to our method, they additionally require depth inputs for a strong 3D cue and do not recover local surface deformations. The idea of combining a differentiable physics and a differentiable graphics engine has been recently explored in contexts different from ours, i.e., estimation of material properties and visuomotor control [21, 23, 31].
+
+Monocular 3D mesh reconstruction approaches can be trained on extensive collections of unstructured views in the desired object category. Some works [6, 11, 57] require 3D supervision, while others, similar to ours, do not: In an early work, Cashman et al. [9] show that almost-rigid object categories, like dolphins, can be reconstructed from image collections. Kanazawa et al. [22] relax the need for input annotations. Li et al. [27] extend [22] to video input and estimate a temporally consistent coarse mesh reconstruction for weakly articulated objects. LASR [60] further relaxes the need for an initial coarse template. We differ from these by the usage of a physics-based deformation model, and we focus on recovering local surface deformations.
+
+# 3. Our Approach
+
+We propose  $\phi$ -SfT, a new method for the 3D reconstruction of a deforming surface (such as cloth) from a monocular RGB video  $\{\mathbf{I}_t\}_{t \in [1, \dots, T]}$  with known intrinsics. As is common for SfT methods [34, 61], we assume that the camera is static and take as input a flat rest shape of the target deformable surface  $\mathbf{S}_1$  for  $t = 1$  with a corresponding texture map  $\mathbf{T}$ . We also assume that a segmentation mask separating a foreground object and background is available. To encourage physically plausible deformations, we use a full physical model, described in Sec. 3.1, that explicitly models forces acting on the surface and the underlying elastic properties of the material. Sec. 3.2 presents the objective function we employ to relate the 2D observations to the estimated reconstructions. We then describe how we optimise the objective function for the physical parameters in Sec. 3.3 and provide implementation details in Sec. 3.4. See Fig. 2 for an overview of  $\phi$ -SfT.
+
+Physics-based simulators are widely used in computer graphics for 3D simulations [4, 33], and differentiable versions exist [29]. Our idea is to use a differentiable physics simulator as a regulariser (deformation model prior) in monocular non-rigid 3D reconstruction. Its usage in SfT is, unfortunately, not straightforward. To integrate the physics-based simulator into our framework, we have to make several crucial improvements to it. First, the idealised assumptions of simulated environments cannot account for the variety of forces and effects causing surface deformations in the real world (e.g., wind turbulence). We take inspiration from the recent work on physically plausible 3D human motion capture [49]. This approach uses a virtual force acting on the root joint of a human skeleton to account for the effects the physics model does not consider. We, therefore, introduce corrective forces accounting for mismatched assumptions about the natural scene (Sec. 3.1). Second, while most simulators used in computer graphics are designed to create simulations following 3D reference motions, it now has to be driven by the 2D input images, and the gradients need to be backpropagated from the image-based losses. Hence,
+
+our energy function includes a dense photometric loss and a silhouette loss (Sec. 3.2). Lastly, the optimisation strategy suitable for 3D simulations is not the best choice for our analysis-by-synthesis  $\phi$ -SfT approach—optimising for deformed surface states, material properties and forces—and we propose an adaptive training strategy instead (Sec. 3.3).
+
+# 3.1. Deformation Model
+
+We seek to reconstruct a sequence of deforming surfaces as 2D manifold meshes in 3D space with fixed topology (edges  $\mathbf{E}$ ), thereby providing temporal correspondences. A surface in this sequence can be parameterised as a triangular mesh  $\mathbf{S}_t = \{\mathbf{V}_t, \mathbf{E}\}$  where the state of the  $i$ -th vertex in  $\mathbf{V}_t$  comprises its 3D position  $\mathbf{x}_t^i \in \mathbb{R}^3$  and its velocity  $\mathbf{v}_t^i \in \mathbb{R}^3$ .
+
+Surface Parametrisation At the core of our method, we model deformations of the non-rigid surface as a physical process, i.e., as elastic deformations resulting from internal stretching and bending forces as well as external forces acting on the surface. We thus do not treat the mesh states  $\mathbf{S}_t$  as parameters but instead use a physical parametrisation.
+
+We initialise the differentiable cloth simulator from the template  $\mathbf{S}_1$  and generate  $\mathbf{S}_t$  with physics simulation  $PS$  according to the material parameters and external forces:
+
+$$
+\mathbf {S} _ {t} = P S \left(\mathbf {S} _ {t - 1}; \phi_ {t - 1}\right), \tag {1}
+$$
+
+where  $\phi_{t - 1}$  are the estimated physics parameters, i.e.,
+
+$$
+\phi_ {t - 1} = \{d, \mathcal {S}, \mathcal {B}, w, \mathcal {F} _ {t - 1} \}. \tag {2}
+$$
+
+Here, material density  $d\in \mathbb{R}$ , stretching stiffness  $\mathcal{S}\in \mathbb{R}^{24}$  (resistance to stretching), and bending stiffness  $\mathcal{B}\in \mathbb{R}^{15}$  (resistance to bending and folding) all together describe the elastic properties of the material and, hence, determine the cloth's internal forces. We also optimise for the wind force  $w\in \mathbb{R}^3$ . However, the wind model is not sufficient to fully describe all the external forces in the scene, such as hand contacts and wind turbulence. We seek to correct for these model insufficiencies by additionally defining a set of corrective forces  $\mathcal{F} = \{\mathcal{F}_t\in \mathbb{R}^{|\mathbf{V}_t|\times 3}\}_{t\in [1,\dots,T - 1]}$ . Note that these vary across vertices and across time. They can, in principle, account for any physical force that the simulator does not explicitly model. In the following, we use the shorthand  $\phi = \{d,S,\mathcal{B},w,\mathcal{F}\}$ .
+
+Physics Simulator For  $PS$ , we follow the cloth simulation pipeline introduced by Narain et al. [4, 32, 33]. In the continuous domain, physics-based simulation can be formulated as a time-varying partial differential equation [4]:
+
+$$
+\frac {\partial^ {2} \mathbf {x}}{\partial t} = \mathbf {M} ^ {- 1} \mathbf {f} (\mathbf {x}, \mathbf {v}), \tag {3}
+$$
+
+![](images/f161b3aefd0121733a83df3012a50fd5948c5c340dc67feb86036bcb9f6848ad.jpg)  
+Figure 2. Given a sequence of monocular input images  $\{\mathbf{I}_t\}_t$ , a template at the rest position  $\mathbf{S}_1$  and the corresponding texture map  $\mathbf{T}$ , our technique solves for the unknown physical parameters  $\phi$  that describe the deforming 3D surface  $\{\mathbf{S}_t\}_t$ . We optimise for the per-sequence physical parameters of  $\{d, S, B, w\}$  as well as the per-frame corrective forces  $\{\mathcal{F}_t\}_t$  in a gradient-based manner. We utilise (1) a physics-based differentiable simulator  $PS$  for reconstructing meshes with a physical deformation model and (2) a differentiable renderer  $R$  for projecting the reconstructions into image space, which allows us to define a reprojection error over all pixels (instead of vertices) during optimisation. The differentiable nature of both components enables us to back-propagate the gradients of the total energy  $E$  all the way back to the unknown physics parameters. Note that the gradients are calculated automatically and provided here for completeness.
+
+where  $(\mathbf{x},\mathbf{v})$  is the vertex state, and  $\mathbf{M}$  is a diagonal matrix of the mass distribution derived from the material density  $d$  and surface area.  $\mathbf{f}(\cdot)$  are the forces, i.e., internal forces which are a function of cloth elastic properties  $\mathcal{S}$  and  $\mathcal{B}$  as well as external forces such as wind  $w$  and corrective forces  $\mathcal{F}_t$ . We follow the elastic model for cloth material of Wang et al. [56] for describing the effects of  $d$ ,  $\mathcal{S}$  and  $\mathcal{B}$ . This model approximates various nonlinear and anisotropic behaviours seen in cloths when subject to external forces.
+
+In practice, we are given the known position  $\mathbf{x}_{t - 1}$  and velocity  $\mathbf{v}_{t - 1}$  of the system at time  $t - 1$ . Our goal is to determine the new position  $\mathbf{x}_t = \mathbf{x}_{t - 1} + \Delta \mathbf{x}$  and velocity  $\mathbf{v}_t = \mathbf{v}_{t - 1} + \Delta \mathbf{v}$  at time  $t$  with a time step size  $h = 1$ . To that end, (3) can be transformed into a first-order differential equation and solved for  $\Delta \mathbf{x}$  and  $\Delta \mathbf{v}$  with the implicit, backward Euler method [4]:
+
+$$
+\binom {\Delta \mathbf {x}} {\Delta \mathbf {v}} = h \binom {\mathbf {v} _ {t}} {\mathbf {M} ^ {- 1} \mathbf {f} \left(\mathbf {x} _ {t}, \mathbf {v} _ {t}\right)}, \tag {4}
+$$
+
+which is nonlinear due to  $\mathbf{f}$ . To turn (4) into a linear system,  $\mathbf{f}$  can be approximated via linearisation:
+
+$$
+\mathbf {f} \left(\mathbf {x} _ {t}, \mathbf {v} _ {t}\right) = \mathbf {f} _ {t - 1} + \frac {\partial \mathbf {f}}{\partial \mathbf {x}} h \left(\mathbf {v} _ {t - 1} + \Delta \mathbf {v}\right) + \frac {\partial \mathbf {f}}{\partial \mathbf {v}} \Delta \mathbf {v}, \tag {5}
+$$
+
+where the Jacobians  $\frac{\partial\mathbf{f}}{\partial\mathbf{x}}$  and  $\frac{\partial\mathbf{f}}{\partial\mathbf{v}}$  are evaluated at  $\mathbf{f}_{t - 1}$ . Thus, a simple cloth simulation process involves solving for  $\mathbf{v}_t$
+
+using (4) and (5), and computing the subsequent simulation state as  $\mathbf{x}_t = \mathbf{x}_{t - 1} + h\mathbf{v}_t$
+
+There can additionally be self-collisions and collisions with a dynamic obstacle mesh  $(\mathbf{x}_t^{obs},\mathbf{v}_t^{obs})$  during simulation. Harmon et al. [20] determine the collision response at the impact zones to update the vertex positions accordingly:
+
+$$
+\mathbf {x} _ {t} = \mathbf {x} _ {t} + \text {c o l l i s i o n - r e s p o n s e} \left(\mathbf {x} _ {t}, \mathbf {v} _ {t}, \mathbf {x} _ {t} ^ {\text {o b s}}, \mathbf {v} _ {t} ^ {\text {o b s}}\right). \tag {6}
+$$
+
+Since we want to use end-to-end gradient-based optimisation, we need to backpropagate gradients through this extra step. However, due to the high dimensionality of the dynamical system when modelling cloth, a naive gradient computation for the general system (4) and (6) (the collision response) can become impractical. Liang et al. [29] propose a solution for this problem, and we proceed with their approach. Specifically, they use implicit differentiation for (4) and (6), where the gradient of the latter is approximated via QR decomposition of a much smaller constraint matrix. For more details on the backward pass, please refer to [29]. We next describe the objective function we use to optimise for the parameters  $\phi$  of the differentiable simulator.
+
+# 3.2. Our Objective Function
+
+We now have a physical deformation model that is parametrised by  $\phi$ , and that outputs a 3D mesh  $\mathbf{S}_t$  for time
+
+$t$ . We solve for the optimal parameters  $\phi^{*}$  by minimising the objective function  $E = E_{p} + \lambda E_{s}$  (with  $\lambda \in \mathbb{R}$ ):
+
+$$
+\phi^ {*} = \underset {\phi} {\arg \min } E (\phi). \tag {7}
+$$
+
+Since we are only given RGB images for time  $t > 1$ , we define a photometric energy term  $E_{p}$  in the image space. Specifically,  $E_{p}$  is an  $\ell_{1}$  RGB data term to encourage photometric consistency between the reconstructed surface rendered into 2D and the input frames:
+
+$$
+E _ {p} = \sum_ {t = 2} ^ {T} \| R (\mathbf {S} _ {t}, \mathbf {T}) - \mathbf {I} _ {t} \| _ {1}, \tag {8}
+$$
+
+where  $R(\cdot)$  is a differentiable renderer outputting a perspective projection of the input mesh (textured with  $\mathbf{T}$ ) onto the image plane with known intrinsic. We implement  $R$  with Soft Rasterizer [30], which introduces useful gradients by composing probability maps of rendered triangles into the final image. This allows us 1) to define  $E_{p}$  densely over all pixels, instead of just the vertices; 2) use the information in the high-resolution  $\mathbf{T}$  that would have been ignored if we had used a photometric term only at the vertices.
+
+While the photometric energy term works well for local corrections, it does not provide a signal for mismatches that are farther apart in the image space. To get a signal even for larger, coarser errors, we add a silhouette energy term that encourages consistency between the foreground segmentation mask of the input frames and the rendered 2D surface:
+
+$$
+E _ {s} = \sum_ {t = 2} ^ {T} \| G (B _ {t}; \sigma) - G (\hat {B} _ {t}; \sigma) \| _ {1}. \tag {9}
+$$
+
+Here,  $B$  and  $\hat{B}$  are the foreground binary segmentation masks of the reconstructed and the input images, respectively.  $G(\cdot, \sigma)$  represents a Gaussian filter of standard deviation  $\sigma$ . The Gaussian filter smooths the binary masks, extending the spatial area where informative gradients are obtainable. Without it, non-zero gradients would be obtained only at pixels located immediately next to the silhouettes of both binary masks. Thus, if the silhouettes do not match almost perfectly at a pixel, the gradient would be zero there, providing no signal to the network as to the target direction to move the mesh's triangles. Importantly, both the groundtruth mask  $B$  and the rendered mask  $\hat{B}$  are processed in the same way, ensuring that  $E_{s}$  is well-behaved.
+
+Given our model and the objective function, we next look at how we solve the optimisation problem.
+
+# 3.3. Optimisation
+
+Our goal is to obtain the optimal physical parameters  $\phi^{*}$  via (7). We use iterative, gradient-based optimisation to that end. Since both the simulator  $PS$  and the renderer  $R$  are
+
+differentiable, we can back-propagate gradients from the objective function  $E$  through the rendering to  $\mathbf{S}_t$  and from there further through the physics simulation to the physical parameters  $\phi$  and further to all earlier meshes (see Fig. 2).
+
+Initialisation To obtain a sufficiently accurate initial guess for the elastic properties  $d$ ,  $\mathcal{S}$ , and  $\mathcal{B}$ , we set them to the average values of ten different real fabrics [56]. The wind and corrective forces  $\mathcal{F}$  are initialised to  $\mathbf{0}$ , i.e., a zero vector. Note that this initialisation leads  $PS$  to initially generate surfaces  $\{\mathbf{S}_t\}_t$  that are identical to the template  $\mathbf{S}_1$ .
+
+![](images/d52692ed63eb7710d250b2cd46e80eb05437ef5886070454c334e3cd3741e502.jpg)  
+Figure 3. Our adaptive optimisation scheme.
+
+Adaptive Optimisation Scheme Since simulation is a temporal process,  $\phi$  for the early frames directly influence the reconstructions of the later frames. In addition, later frames are initially reconstructed at lower fidelity than earlier ones. Therefore, similar to
+
+tracking a surface, we exploit the temporal frame order and do not optimise for all  $t \geq 1$  from the start. Instead, we adaptively grow the active temporal window, i.e., the latest time  $t_a$  up to which all earlier frames  $t \leq t_a$  participate in the optimisation. We start with the first five frames as active and optimise  $E(7)$  for them. Once the energy of the latest frame decreases below a threshold  $b$ , we add the next frame to the optimisation; see Fig. 3.  $b$  is set to the energy that the fifth frame has when the sixth frame is added. We assume the fifth frame to be well-reconstructed since it is early in the sequence. (In case the optimisation cannot reach the threshold, we set a maximum number of iterations, after which we progress regardless.) This adaptive scheme speeds up optimisation by converging to a reasonable guess for  $\phi$  before later frames become active. Moreover, it allocates more iterations to frames with more challenging deformations. We evaluate the adaptive optimisation scheme experimentally in Sec. 4.3.
+
+# 3.4. Implementation Details
+
+We implement  $\phi$ -SfT in Pytorch [42]. (7) is solved with the Adam optimiser [25] with learning rate  $10^{-3}$ . The adaptive optimisation scheme leads to several hundred iterations in most cases, which takes 16-24 hours on an Nvidia RTX 8000 GPU. Due to the sequential nature of the simulator, we compute the objective function for all active times  $t$  for each optimisation iteration. We set  $\sigma = 7px$ ,  $\lambda = 0.5$  for  $E_{s}$  and apply the corrective forces  $\mathcal{F}$  by modifying the velocity of vertex  $i$  at time  $t$ :  $v_{t}^{i} = v_{t}^{i} + \mathcal{F}_{t}^{i}$  (because both mass and time steps are constant). We keep the wind air density fixed at  $1\mathrm{kg / m^3}$  and optimise only for the wind velocity.
+
+The images in our real scenes have resolution  $1920 \times 1080$  pixels. We also pre-process the real scenes (recorded with an RGB-D camera): We first segment out the background from the captured images and point clouds by depth thresholding. We next use Poisson surface reconstruction [24] on the template (at  $t = 1$ ), which yields  $\sim 300$  vertices on average. We then determine the initial rigid pose relative to a flat sheet (which is required by the simulator) using iterative closest point (ICP) [7], and initialise the simulator with it. We obtain the texture map  $\mathbf{T}$  by projecting the vertices of the template mesh  $\mathbf{S}_1$  onto the image space of the first image  $\mathbf{I}_1$  with known camera intrinsics.
+
+# 4. Experimental Results
+
+We evaluate our technique on real and synthetic data. We recorded the  $\phi$ -SfT real dataset (Sec. 4.1) of natural sequences using a monocular RGB camera and the depth camera Azure Kinect. The latter is used to obtain pseudoground-truth segmentations and deformations. Qualitative and quantitative results on this dataset in Sec. 4.2 show our technique clearly outperforms the state of the art by capturing a wider variety of deformations and local folds. In addition, we generate a new  $\phi$ -SfT synthetic dataset of four monocular RGB sequences of naturally deforming surfaces using physics-based simulation; see our supplement for further details and evaluations, likewise demonstrating the superior performance of  $\phi$ -SfT. We also perform an ablation study in Sec. 4.3 that demonstrates the importance of corrective forces and other design choices.
+
+# 4.1. The  $\phi$ -SfT Real Dataset
+
+We have recorded a new dataset of deforming surfaces to allow quantitative evaluations of reconstruction methods on real data against pseudo ground truth. The dataset includes nine sequences of various surface shapes and textures, including differing material properties due to differences in the cloths' fabric and weaving (e.g., there are more and less elastic and more and less dense materials). The texture pattern varies from fine-grained and regular to more global and irregular patterns; the cloth size ranges from  $55 \times 55 \mathrm{~cm}$  to  $95 \times 95 \mathrm{~cm}$ . The surfaces are exposed to external forces, i.e., gravity, wind, and hand contacts. Each sequence is simultaneously recorded using a monocular RGB and depth camera and has a length of about 40 frames, such that they focus on challenging folds. To obtain the texture map  $\mathbf{T}$  under the same lighting conditions as the recorded sequence, we start with a flat rest state for  $t = 1$  and obtain  $\mathbf{T}$  from it. Then, pseudo-ground-truth deformations for each frame are reconstructed as point clouds through backprojection, using the depth images and known camera intrinsics. The point clouds are provided in absolute distance units (meters). Fig. 4 shows an overview of the recorded sequences.
+
+![](images/b5921067abc1bfd9b404bbd1568fc6fa2e96d4facc6f57d08ad8ebfbadd25662.jpg)  
+Figure 4. We record a new  $\phi$ -SfT real dataset of nine sequences with reference depth data to facilitate quantitative comparisons of monocular 3D surface reconstruction methods. Our setup consists of a synchronised RGB camera and an Azure Kinect. The depth camera provides depth maps serving as pseudo ground truth.
+
+# 4.2. Comparison with Existing Methods
+
+We compare our technique to SfT methods, namely Yu et al.'s Direct, Dense, Deformable (DDD) [61], Ngo et al.'s Ngo2015 [34] and Shimada et al.'s IsMo-GAN [47], as well as Sidhu et al.'s Neural NRSfM (N-NRSfM) [50] and Parashar et al.'s Diff-NRSfM [39].
+
+Since NRSfM methods accept 2D point correspondences, we track 2D points densely across the input images with multi-frame subspace flow (MFSF) [15, 16], as suggested in [50]. The first frame of the sequence is selected as a keyframe for tracking. We provide DDD with the required hierarchy of coarse-to-fine templates and Ngo2015 with the same template as ours. To demonstrate the necessity for the physical simulation based on the internal stretching and bending forces, we show results of the "Only  $\mathcal{F}$ " baseline where the only forces are the correctives  $\{\mathcal{F}_t\}_t$ . As other forces and parameters are omitted, we implement this baseline as optimisation of per-vertex offsets over time.
+
+Due to the monocular scale and depth ambiguities, we align reconstructions of all methods to the ground truth in a rigid-body fashion. For the first frame, we determine the transformation matrix using Procrustes alignment [54], which is further refined with rigid ICP [7] for later frames.
+
+For quantitative evaluation, we compute the Chamfer distance between the pseudo-ground-truth point cloud from Kinect  $G = \{\mathbf{g}_i\in \mathbb{R}^3\}_{i}$  and points  $M = \{\mathbf{m}_j\in \mathbb{R}^3\}_{j}$  sampled from the reconstructed mesh:
+
+$$
+C h _ {G, M} = \frac {1}{| G |} \sum_ {\mathbf {g} \in G} \min  _ {\mathbf {m} \in M} \| \mathbf {g} - \mathbf {m} \| _ {2} ^ {2} + \frac {1}{| M |} \sum_ {\mathbf {m} \in M} \min  _ {\mathbf {g} \in G} \| \mathbf {m} - \mathbf {g} \| _ {2} ^ {2}. \tag {10}
+$$
+
+We report average Chamfer distance per sequence  $\tilde{C} h_{G,M}$ . Figs. 1 and 6 show that  $\phi$ -SfT substantially outperforms all tested methods qualitatively; see the supplement for more visualisations of our results. Tab. 1 shows that  $\tilde{C} h_{G,M}$  of  $\phi$ -SfT is on average lower than  $\tilde{C} h_{G,M}$  of competing methods.
+
+Our results confirm that SfT and NRSfM, both relying on simple geometric prior assumptions (such as surface smoothness, isometry or small local deformations), cannot cope with the elaborate fold patterns present in our dataset. The results of N-NRSfM follow the silhouettes of the input images better than DDD and IsMO-GAN, thanks to 2D point tracking, even though its  $\tilde{C} h_{G,M}$  is the highest (due to rigidity assumption for the initialisation [50, 52]). Its 3D surfaces are somewhat rugged, and the surface pattern allows to recognise the observed texture in the second row. Moreover, as expected, more fine-grained textures result in more accurate 2D point tracking by MFSF. DDD does not track large deformations and silhouettes well in our tests. IsMo-GAN, trained on rather smooth surfaces, cannot reproduce local folds and barely captures the contours. Diff-NRSfM produces reasonable reconstructions in smooth regions owing to its differentiable structure-preserving formulation. However, it is sensitive to noise in 2D measurements, which leads to visual artefacts in the regions with challenging folds. Ngo2015 failed on three scenes (partially on S7 and entirely on S8 and S9). Ngo2015 mostly produces physically implausible results but achieves lower  $\tilde{C} h_{G,M}$  on a few scenes; qualitative observations, however, remain the same for all sequences (Figs. 1 and 6). This suggests that an isometry prior [34] is less effective compared to our physics-based elastic model, which can even express non-isometric deformations (depending on the parameters). When removing all forces from the model except for the correctives, the results degrade in quality and average error increases  $>50\%$ , see "Only  $\mathcal{F}$ " in Fig. 6 and Tab. 1. This experiment demonstrates that the accuracy of the full  $\phi$ -SfT model does not rely solely on the corrective forces. In contrast,  $\phi$ -SfT estimates temporally coherent surfaces and captures all significant folds while missing only small nuances. As seen in Fig. 1, our physics-based model provides a reasonable prior for self-occluded surface parts. Moreover, our method is not sensitive to initialisation. We empirically find no issue with always initialising with our default material parameters.
+
+See Fig. 5 for depth maps reconstructed by  $\phi$ -SfT. This alternative way to visualise the results allows us to study and perceive even smaller surface details. We also strongly encourage the readers to watch our supplementary video.
+
+# 4.3. Ablative Study
+
+We conduct an ablative study on the various design choices to integrate the physics simulator into the SfT setting. We test the following modes: 1) Operation without corrective forces  $\mathcal{F}$ , 2) Influence of the adaptive training by considering all frames from the start (Sec. 3.3), and 3) Disabling the silhouette term (9). On average across all sequences, we obtain  $\tilde{C} h_{G,M} = 10.69$ , 6.21, 5.33, and 4.48 for w/o  $\mathcal{F}$ , w/o adaptive, w/o  $E_{s}$ , and the full model, re
+
+![](images/72b128ae252d5ba474e625ed20b287fdfb90e92751780b55b3bbfe8b0342b13d.jpg)  
+Figure 5. We show qualitative results as colour-coded depth maps. For (a) the given RGB image, (b) the ground-truth depth map exhibits similar features as (c) our reconstructed depth. Both the coarse shape and local folds are well captured.
+
+<table><tr><td>Seq.</td><td>IsMo-GAN</td><td>N-NRSfM</td><td>DDD</td><td>Diff-NRSfM</td><td>Ngo2015</td><td>Only F</td><td>φ-SFT</td></tr><tr><td>S1</td><td>19.69</td><td>8.25</td><td>2.95</td><td>17.14</td><td>2.19</td><td>2.59</td><td>0.79</td></tr><tr><td>S2</td><td>22.18</td><td>33.62</td><td>1.69</td><td>4.46</td><td>1.51</td><td>1.60</td><td>2.75</td></tr><tr><td>S3</td><td>33.54</td><td>104.60</td><td>3.80</td><td>4.40</td><td>2.17</td><td>3.23</td><td>3.54</td></tr><tr><td>S4</td><td>90.30</td><td>77.02</td><td>25.73</td><td>41.37</td><td>15.90</td><td>14.95</td><td>7.60</td></tr><tr><td>S5</td><td>92.78</td><td>72.66</td><td>10.46</td><td>26.92</td><td>10.72</td><td>21.32</td><td>6.15</td></tr><tr><td>S6</td><td>57.62</td><td>8.73</td><td>6.97</td><td>14.02</td><td>3.01</td><td>3.08</td><td>3.14</td></tr><tr><td>S7</td><td>49.27</td><td>129.44</td><td>15.64</td><td>12.49</td><td>7.95*</td><td>6.03</td><td>4.73</td></tr><tr><td>S8</td><td>24.45</td><td>38.06</td><td>7.61</td><td>9.91</td><td>fail</td><td>3.78</td><td>2.52</td></tr><tr><td>S9</td><td>53.12</td><td>19.81</td><td>11.77</td><td>5.29</td><td>fail</td><td>4.39</td><td>2.36</td></tr><tr><td>Avg.</td><td>49.22</td><td>54.69</td><td>10.87</td><td>15.11</td><td>5.92*</td><td>6.77</td><td>3.93</td></tr></table>
+
+Table 1. We quantitatively compare  $\phi$ -SfT to the state of the art on the real dataset. The average Chamfer distance  $\bar{C} h_{G,M}$  is multiplied by  $10^{4}$  for readability. \*\*: Ngo2015 failed on the last few frames of S7, which we exclude from the error computation.
+
+spectively. Omitting  $\mathcal{F}$  always leads to a significant error increase, and abandoning our adaptive training policy increases it by  $39\%$ . Fig. 7 shows qualitative results confirming the statistics over all sequences, i.e., the largest errors are present in the colour-coded error maps when  $\mathcal{F}$  is disabled. The second most crucial component of  $\phi$ -SfT is the adaptive training strategy. Note that  $E_{s}$  helps when the structure deforms and significantly changes its shape in the re-projection. See our supplement for the complete report.
+
+# 4.4. Discussion and Future Directions
+
+We use PyTorch3D [42] for rendering; it only supports a single light source, which can differ from the illumination during the recording. There is, therefore, a systematic mismatch between the rendered reconstructions during training and the input images. Still,  $\phi$ -SfT achieves high accuracy, as our differentiable rendering loss is empirically robust to small noise and mismatches in colour and shading. The runtime of our approach is comparably high (i.e., twice as long as N-NRSfM [50]). This is due to single-threading
+
+![](images/a6b6cb95cff1b3866a138e2963097471f5233b268ef28c0fa7ae2e1095a0d05c.jpg)  
+Figure 6. Qualitative comparisons of several tested methods [34, 39, 47, 50, 61] and  $\phi$ -SfT for a representative frame of the S1, S3 and S4 sequences. Our results are significantly more accurate and, unlike the other methods, capture the folds well (especially for S3).
+
+![](images/864714b4d204f0c13f109642b8d3a2a8f7cf7a721ee96342992ddb075d9a219e.jpg)  
+Figure 7. In the ablative study, we remove corrective forces, the adaptive scheme, or the silhouette energy term. The corrective forces are the most crucial component to make our method work.
+
+![](images/0ba65cd70fc8d78d84214711d3f2f7d63f46c433ffedd46c8a2ea9401e8a40aa.jpg)  
+Figure 8. Limitation: Due to the high runtime of the simulator, we use a lower-resolution mesh, which suffices to capture local folds but limits the reconstruction of very fine wrinkles (dashed area).
+
+when resolving collisions which can be improved through parallelisation. Next, we use lower-resolution meshes for all sequences, with  $\sim 300$  vertices, as optimising model parameters with higher resolutions is not feasible. This limits the reconstruction accuracy as we cannot capture very fine wrinkles, as shown in Fig. 8. However, note that our photometric energy is defined densely over all pixels, instead of just the vertices, and thus uses the information in the high-resolution texture map. Therefore, the lower number of vertices does not impede  $\phi$ -SfT's ability to capture folds. Moreover, even more sophisticated physics simula
+
+tors could be implemented, taking into account more physical laws (e.g., wind turbulence or electrostatic forces). Such requirements depend on downstream applications such as game and movie production or industrial quality control.
+
+In this first work of its kind, we focus on investigating how a physics-based deformation model can be leveraged in classical SfT. Accurate inference of deformation forces and materials remains a difficult problem that deserves more investigation in the future. Nonetheless,  $\phi$ -SfT infers them well enough to enable intuitive editing; see the supplement. While we do not target complex objects or new scenarios—such as the separate field of virtual garment and dress simulations [19, 59]—this can be an interesting future avenue.
+
+# 5. Conclusion
+
+We introduced  $\phi$ -SfT, a new optimisation-based SfT method that models deformations with a physical simulator and uses differentiable rendering to define a reprojection energy term over all pixels, exploiting texture information independent of the mesh resolution. Experiments on the new dataset demonstrate that our approach improves the reconstructions qualitatively and quantitatively by a significant margin over competing techniques of several method classes. Especially remarkable is  $\phi$ -SfT's accuracy in folded surface regions. We believe that the proposed technique has a high potential for future research, and we hope to see more improvements on physically principled methods for monocular non-rigid 3D reconstruction.
+
+Acknowledgements. We thank Shaifali Parashar for her help with the source code of Diff-NRSfM [39]. This work was supported by the ERC Consolidator Grant 4DRepLy (770784).
+
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+# gDNA: Towards Generative Detailed Neural Avatars
+
+Xu Chen $^{1,3}$  Tianjian Jiang $^{1}$  Jie Song $^{1}$  Jinlong Yang $^{3}$
+
+Michael J. Black<sup>3</sup> Andreas Geiger<sup>2,3</sup> Otmar Hilliges<sup>1</sup>
+
+$^{1}$ ETH Zürich, Department of Computer Science  $^{2}$ University of Tübingen
+
+3Max Planck Institute for Intelligent Systems, Tübingen
+
+https://xuchen-ethz.github.io/gdna
+
+![](images/a72d48fe4481bbe04d57ac1757d8ec7ae430b892ea7e7ccd4cf1073ee426080f.jpg)  
+Figure 1. Generative Detailed Neural Avatars. We propose a method to generate 1) a diverse set of 3D virtual humans of 2) varied identity, gender and shapes, appearing in 3) different clothing styles and poses, with 4) realistic and stochastic details such as wrinkles in garments. Our multi-subject method learns shape, articulation and clothing details from few posed scans without requiring skinning weight supervision. The method is able to synthesize novel identities that are not in the training set and generalizes to unseen poses.
+
+# Abstract
+
+To make 3D human avatars widely available, we must be able to generate a variety of 3D virtual humans with varied identities and shapes in arbitrary poses. This task is challenging due to the diversity of clothed body shapes, their complex articulations, and the resulting rich, yet stochastic geometric detail in clothing. Hence, current methods that represent 3D people do not provide a full generative model of people in clothing. In this paper, we propose a novel method that learns to generate detailed 3D shapes of people in a variety of garments with corresponding skinning weights. Specifically, we devise a multi-subject forward skinning module that is learned from only a few posed, un-rigged scans per subject. To capture the stochastic nature of high-frequency details in garments, we leverage an adversarial loss formulation that encourages the model to capture the underlying statistics. We provide empirical evidence that this leads to realistic generation of local details such as wrinkles. We show that our model is able to generate natural human avatars wearing diverse and detailed clothing. Furthermore, we show that our method can be used on the task of fitting human models to raw scans, outperforming the previous state-of-the-art.
+
+# 1. Introduction
+
+The ability to easily create diverse high-quality virtual humans with full control over their pose has many applications in movie production, games, VR/AR, architecture, and computer vision. While modern computer graphics techniques achieve photorealism, they typically require a lot of expertise and extensive manual effort. Our goal is to make 3D human avatars widely accessible by learning a generative model of people. Towards this goal, we propose the first method that can generate 1) diverse 3D virtual humans with 2) various identities and shapes, appearing in 3) different clothing styles and poses, with 4) realistic and stochastic high-frequency details such as wrinkles in garments.
+
+Generative modeling of 3D rigid objects has recently seen rapid progress, fueled by continuous and resolution-independent neural 3D representations [13, 40, 43, 48, 54]. However, modeling clothed humans and their articulation is more difficult due to the complex interaction of garments, their topology, and pose-driven deformations. Recent work leverages neural implicit surfaces to learn high-quality articulated avatars for a single subject [12, 15, 52, 58] but these methods are not generative, i.e. they cannot synthesize novel human identities and shapes. Generative models of clothing exist that augment SMPL by predicting displacements from the body mesh (CAPE [36]), or by drap
+
+ing an implicit garment representation on a T-posed body (SMPLicit [14]), and relying on SMPL's learned skinning for reposing. We show empirically that holistic modeling of identity, shape, articulation and clothing leads to higher fidelity generation and animation of virtual humans and to higher accuracy in fitting to 3D scans.
+
+Taking a step towards fully generative modeling of detailed neural avatars, we propose gDNA, a method that synthesizes 3D surfaces of novel human shapes, with control over the clothing style and pose, and that produces realistic high-frequency details of the garments. To leverage raw (posed) 3D scans, we build a multi-subject implicit generative representation. We build upon SNARF [12], a recent method for learning single-subject articulation-dependent effects that has been shown to generalize well to unseen poses. SNARF [12] requires many poses of a single subject for training. In contrast, our multi-subject method can be learned from very few posed scans (1-3) of many different subjects. This is achieved via the addition of a latent space for the conditional generation of shape and skinning weights for clothed humans. Furthermore, a learned warping field yields accurate deformations, using the same skinning field, independent of body size.
+
+Clothing wrinkles are produced by an underlying stochastic process. To capture these effects, we propose a method that learns the underlying statistics of 3D clothing details via an adversarial loss. Previous mesh-based approaches formulate this in UV-space [28], which is not directly applicable to implicit surfaces due to the lack of mesh connectivity. To learn high-frequency details, we first predict a 3D normal field, conditioned on the coarse shape features. To backpropagate the adversarial loss to the 3D normal field we establish 3D-2D correspondences by augmenting forward skinning with an implicit surface renderer. We show that adversarial training leads to significantly improved fidelity of 3D geometric details, see Fig. 9.
+
+Trained from posed scans only, we demonstrate the first method that can generate a large variety of 3D clothed human shapes with detailed wrinkles under pose control. The generated samples can be reposed via the learned skinning weights. We evaluate gDNA quantitatively, qualitatively, and through a perceptual study; gDNA strongly outperforms baselines. Furthermore, we show that gDNA can be used for fitting and re-animation of 3D scans, outperforming the state of the art (SOTA). In summary, we contribute:
+
+- The first method to generate a large variety of animatable 3D human shapes in detailed garments; that  
+- learns from raw posed 3D scans without requiring canonical shapes, detailed surface registration, or manually defined skinning weights, and  
+- a technique to significantly improve the geometric detail in clothing deformation, based on recovering the underlying statistics of cloth deformation.
+
+# 2. Related Work
+
+2D and 3D Generative Models: Most modern methods for synthesizing natural images leverage generative adversarial networks (GANs) [18] or variational auto-encoders (VAEs) [27]. These methods have achieved a high level of photorealism [24-26] and can yield impressive results on the task of synthesizing 2D images of humans [4,11,19,29, 34,53]. However, such methods reason in 2D and hence 3D consistency cannot be guaranteed [11,29,34] nor is extracting 3D geometry from such approaches straightforward.
+
+Several methods for the task of learning rigid 3D shapes exist. Early methods rely on voxel [61] or point cloud [3] representations. More recently, several methods represent object shapes by learning an implicit function using neural networks [13, 40, 48]. Such representations have also been proposed for the task of generative modeling of 3D shapes [10, 13, 16, 40, 42, 43, 48, 54]. However, these methods are typically not easily extended to non-rigid clothed humans. In this paper, we study the problem of 3D implicit generative modeling of non-rigid human shape.
+
+3D Human Models: Parametric 3D human body models [7,23,33,46,64] can synthesize 3D human shapes from a set of low-dimensional control parameters by deforming a template mesh. This idea has also been extended to model clothed humans [5, 36]. However, geometric expressivity is limited due to the fixed mesh topology and the bounded resolution of the template mesh.
+
+To overcome the topology and resolution limitations of meshes, other representations, including point clouds [35, 37,65], implicit surfaces [12,47,52,55,58,60], and radiance fields [32, 45, 49, 56, 63], have been explored. In particular, neural implicit surface representations have emerged as a powerful tool to model 3D (clothed) human shapes [6, 15, 17, 20, 21, 30, 41, 50, 51, 62, 66, 67] due to their topological flexibility and resolution independence. Recent work [12, 52, 58] uses implicit surfaces to learn human avatars for a single subject, wearing a specific garment. These methods model clothing details such as wrinkles as a deterministic function of the body poses. However, due to hysteresis and complex material properties, garment folds and wrinkles are stochastic and existing methods struggle to capture these effects. In contrast, we propose a multi-subject generative model of 3D humans that provides separate control over poses, garments and can synthesize realistic geometric details.
+
+CAPE [36] and SMPLicit [14] are generative models of clothing only, based on meshes and implicit surfaces respectively. Both methods are purely additive, that is they drape an implicit garment over the SMPL body [14] or predict the displacement parameters of a SMPL+D template mesh [36]. We experimentally show that this leads to lower fidelity in generated samples and higher error when fitting
+
+![](images/57948e5cbf8524429be9b6d39b50c8626ea92af795e4a422bbeaa252ee9aca5d.jpg)  
+Figure 2. Method Overview. We represent clothed humans using coarse shapes and detailed normals in a pose- and body-size-independent canonical space. This canonical representation can then be deformed to target body poses and sizes via a multi-subject forward skinning module. The deformed shapes are compared to raw posed scans via a 3D reconstruction loss to learn canonical shapes and skinning. To improve surface details, we augment the forward skinning module with an implicit surface renderer to generate 2D normal maps and learn detailed 3D normal fields by applying a 2D adversarial loss formulation.
+
+to 3D scans. NPMs [47] provide a latent space of multiple subjects for fitting to RGB-D depth maps or 3D scans.
+
+A common problem of all aforementioned approaches is the specific training data requirements these models impose: They either require synthetic data in canonical space [14, 47], or precise registration of a template mesh to posed scans [36, 47]. The former are rare and suffer from a domain gap, while the latter is challenging to attain. Our method overcomes this issue by requiring only a few training samples of each subject in posed space. We show that our method learns complex shape and clothing details and models realistic deformation even from such limited data.
+
+Adversarial Training of Clothing Details: Adversarial loss formulations have been used to learn detailed cloth wrinkles by optimizing 2D representations such as UV normal maps [28] or depth images [59]. It is noteworthy that implicit surfaces lack a notion of connectivity and therefore, incorporating 2D representations that have been designed to augment explicitly parametrized meshes is not straightforward. In contrast, we propose a formulation that leverages a 2D adversarial loss computed with posed images to optimize a 3D implicit representation in canonical space. Finally, our focus is the generation of human shapes appearing in varied clothing styles and diverse identities while previous methods focus on reconstruction [59] or single garment pose-dependent wrinkle enhancement [28].
+
+# 3. Method
+
+Our goal is to build a model that generates diverse 3D clothed humans with varying identities and fine-grained geometric details in arbitrary poses. Our model is learned from a sparse set of static scans without assuming surface correspondences. Our method is summarized Fig. 2.
+
+First, we formulate a pose- and body-size-independent canonical representation of clothed human shapes (Section 3.1). Second, to learn the canonical shape and defor
+
+mation properties from very few posed scans of each of the subjects, we extend a single-subject differentiable forward skinning method [12] to multiple subjects via a latent space of shape, articulation and garment (Section 3.2). Finally, to learn rich yet stochastic geometric details, we learn a detailed 3D normal field via a 2D adversarial loss formulation. To achieve this, we augment the forward skinning module with an implicit surface renderer (Section 3.3). Training details are discussed in Section 3.4.
+
+# 3.1. Canonical Representation
+
+Our method is based on neural implicit representations, leveraging their topological flexibility and resolution independence. We model the clothed human shape and geometric clothing details jointly.
+
+Coarse Shape: We model the shape in canonical space as the  $\tau = 0.5$  level set of a neural occupancy function:
+
+$$
+\mathcal {S} (\mathbf {z} _ {\text {s h a p e}}) = \{\mathbf {x} \mid \mathcal {O} (\mathbf {x}, \mathbf {z} _ {\text {s h a p e}}) = \tau \}, \tag {1}
+$$
+
+where  $\mathcal{O}$  is a neural network that predicts the occupancy probability  $o$  for any 3D point  $\mathbf{x}$  in canonical space. The prediction is conditioned on a shape code  $\mathbf{z}_{\mathrm{shape}} \in \mathbb{R}^{L_{\mathrm{shape}}}$ :
+
+$$
+\mathcal {O}: \mathbb {R} ^ {3} \times \mathbb {R} ^ {L _ {\text {s h a p e}}} \rightarrow [ 0, 1 ] \times \mathbb {R} ^ {L _ {\mathbf {f}}} \tag {2}
+$$
+
+$$
+\left(\mathbf {x}, \mathbf {z} _ {\text {s h a p e}}\right) \mapsto (o, \mathbf {f})
+$$
+
+This occupancy network also outputs a feature vector  $\mathbf{f}$  of dimension  $L_{\mathbf{f}}$  for each surface point. This feature carries coarse shape information and is used to predict fine details.
+
+We combine a 3D CNN-based feature generator and a locally conditioned MLP to model  $\mathcal{O}$ . A 3D style-based [25, 42] generator, illustrated in Fig. 3 first produces a 3D feature volume conditioned on  $\mathbf{z}_{\mathrm{shape}}$  via adaptive instance normalization [22]. The final occupancy is obtained via trilinear sampling of the feature volume and by feeding the feature and the 3D coordinate into an MLP.
+
+![](images/25d6dfec0464a39d08c00c54abaf2581e145916176d6fd43d28cb3b246cd4291.jpg)  
+Figure 3. Canonical Implicit Model. Given latent codes  $\mathbf{z}_{\mathrm{shape}}$  and  $\mathbf{z}_{\mathrm{detail}}$ , this module predicts the occupancy probability  $o$  and the normals  $\mathbf{n}$  for 3D points in canonical space. To predict  $o$ , a 3D style-based generator first produces a 3D feature volume conditioned on  $\mathbf{z}_{\mathrm{shape}}$ . The final occupancy of a 3D point is obtained via trilinear sampling of the feature volume and by feeding the feature and the 3D coordinate into an MLP. To predict normals  $\mathbf{n}$ , we use an MLP conditioned on feature  $\mathbf{f}$  and latent code  $\mathbf{z}_{\mathrm{detail}}$ .
+
+Detailed Surface Normals: Learning an occupancy field for multiple subjects and garment types with accurate and detailed normals is challenging and we empirically show that a naive implementation leads to artifacts on the surface (cf. Fig. 9). Analogous to normal mapping for polygon meshes [8, 28], we model surface details via normals in canonical 3D space. Such surface normals can be represented by the gradient of the implicit function, but this results in considerable computational complexity. Therefore, we use an MLP to predict surface normals similar to [58]. However, because implicit surfaces have no notion of connectivity, we propose a geometry-aware approach to link coarse geometry and the detailed normal field. More specifically, we condition the surface normal prediction on the underlying shape, leveraging the feature  $\mathbf{f}$  from the occupancy network. We further condition the field on a latent  $\mathbf{z}_{\mathrm{detail}} \in \mathbb{R}^{L_{\mathrm{detail}}}$  to enable generation of controllable details for the same coarse shape:
+
+$$
+\mathcal {N}: \mathbb {R} ^ {3} \times \mathbb {R} ^ {L _ {\text {d e t a i l}}} \times \mathbb {R} ^ {L _ {f}} \rightarrow \mathbb {R} ^ {3} \tag {3}
+$$
+
+$$
+\left(\mathbf {x}, \mathbf {z} _ {\text {d e t a i l}}, \mathbf {f}\right) \mapsto \mathbf {n}
+$$
+
+# 3.2. Multi-Subject Forward Skinning
+
+We additionally model the deformation properties and define the body size  $(\beta)$  and pose  $(\theta)$  parameters to be consistent with SMPL, enabling use of existing datasets (e.g. AMASS [38]) for animation. The body size parameter  $\beta$  is a 10-dimensional vector, and the body pose parameter  $\theta$  represents the joint angles of SMPL's skeleton.
+
+Single-Subject Skinned Representation: To animate implicit human shapes in controllable body poses  $\theta$ , recent work [12, 41, 52, 58] generalizes mesh-based linear blend skinning algorithms to neural implicit surfaces. The skeletal deformation of each 3D point is modeled as the weighted average of a set of bone transformations, with weights at each point predicted by an MLP. A key difference is whether this skinning weight field is defined in canonical space or in
+
+![](images/e5e00b1dcd494d9cf76b1d1710eace239cbe89db5e2915a7a39d179724408d55.jpg)  
+Figure 4. Multi-subject Forward Skinning. This module deforms the canonical occupancy and normal field to the target pose  $\theta$  and size  $\beta$  by establishing correspondences from posed space  $\mathbf{x}'$  to canonical space  $\mathbf{x}^*$ . First, given a deformed point  $\mathbf{x}'$ , we find its correspondence in resized canonical space  $\hat{\mathbf{x}}^*$  by iteratively finding the root of (9). Subsequently, we map  $\hat{\mathbf{x}}^*$  to  $\mathbf{x}^*$  in size-neutral canonical space using the warping field.
+
+posed space. We follow Chen et al. [12] who define the skinning field in canonical space:
+
+$$
+\mathcal {W}: \mathbb {R} ^ {3} \rightarrow \mathbb {R} ^ {n _ {b}} \tag {4}
+$$
+
+$$
+\mathbf {x} \mapsto \mathbf {w},
+$$
+
+where  $n_b$  denotes the number of bones and the weights  $\mathbf{w} = \{w_1, \dots, w_{n_b}\}$  of each point  $\mathbf{x}$  are enforced to satisfy  $w_i \geq 0$  and  $\sum_i w_i = 1$  by a softmax activation function. As shown in [12], defining the skinning weights field in canonical space is desirable because the skinning weights are then pose-independent, thus easier to learn and enabling generalization to out-of-distribution poses.
+
+Multi-Subject Skinned Representation: We extend this forward skinning idea to multiple subjects. Since the skinning weight field is defined in canonical space, the model can aggregate information over multiple training instances. Importantly, this enables us to learn skinning from one or a few poses of multiple subjects, instead of requiring many poses of the same subject.
+
+To achieve this, we decouple the effects originating from the body size variation  $\beta$  and the clothed human shape  $\mathbf{z}_{\mathrm{shape}}$ . We model the skinning field in a body-size-neutral space, analogously to the canonical surface representations. To capture diverse clothed human shapes, we condition the field on the latent shape code  $\mathbf{z}_{\mathrm{shape}}$ :
+
+$$
+\mathcal {W}: \mathbb {R} ^ {3} \times \mathbb {R} ^ {L _ {\text {s h a p e}}} \rightarrow \mathbb {R} ^ {n _ {b}} \tag {5}
+$$
+
+$$
+\left(\mathbf {x}, \mathbf {z} _ {\text {s h a p e}}\right) \mapsto \mathbf {w}
+$$
+
+We then model body size change with an additional warping field. Given a point  $\hat{\mathbf{x}}$  in  $\beta$ -size space, the warping field maps it back to the mean size by predicting its canonical correspondence  $\mathbf{x}$  (see Fig. 4):
+
+$$
+\mathcal {M}: \mathbb {R} ^ {3} \times \mathbb {R} ^ {L _ {\beta}} \rightarrow \mathbb {R} ^ {3} \tag {6}
+$$
+
+$$
+(\hat {\mathbf {x}}, \boldsymbol {\beta}) \mapsto \mathbf {x}
+$$
+
+In this formulation,  $\beta$  captures body shape variations analogously to SMPL, e.g. body height. Therefore, the canonical shape network only needs to model the remaining shape variations beyond SMPL, e.g. clothing and hair, controlled by  $\mathbf{z}_{\mathrm{shape}}$ . The final resized canonical surface is defined by:
+
+$$
+\hat {S} (\mathbf {z} _ {\text {s h a p e}}, \boldsymbol {\beta}) = \left\{\hat {\mathbf {x}} \mid \mathcal {O} \left(\mathcal {M} (\hat {\mathbf {x}}, \boldsymbol {\beta}), \mathbf {z} _ {\text {s h a p e}}\right) = \tau \right\} \tag {7}
+$$
+
+Given the target body pose  $\theta$ , a point  $\hat{\mathbf{x}}$  in  $\beta$ -size space is transformed to posed space  $\mathbf{x}'$  via
+
+$$
+\begin{array}{l} \mathbf {x} ^ {\prime} = \mathbf {d} (\mathbf {x}, \boldsymbol {\beta}, \boldsymbol {\theta}, \mathbf {z} _ {\text {s h a p e}}) \\ = \sum_ {i = 1} ^ {n _ {\mathrm {b}}} \mathcal {W} _ {i} (\mathcal {M} (\hat {\mathbf {x}}, \boldsymbol {\beta}), \mathbf {z} _ {\text {s h a p e}}) \cdot \boldsymbol {B} _ {i} (\boldsymbol {\beta}, \boldsymbol {\theta}) \cdot \hat {\mathbf {x}}, \tag {8} \\ \end{array}
+$$
+
+where  $B_{i}(\beta, \theta)$  are the bone transformation matrices obtained from the parametric skeleton of SMPL.
+
+Implicit Differentiable Forward Skinning: While our model learns a canonical representation, its supervision is provided in posed space. Given a point  $\mathbf{x}'$  in posed space we need to determine its correspondence in canonical space  $\mathbf{x}$  to compare the predicted occupancy and normals to groundtruth. We first find the correspondence  $\hat{\mathbf{x}}^*$  of  $\mathbf{x}'$  in resized canonical space and then map  $\hat{\mathbf{x}}^*$  to canonical space  $\mathbf{x}^*$ . An overview is provided in Fig. 4. While the goal is to determine  $\mathbf{x}' \mapsto \hat{\mathbf{x}}$ , we only have direct access to the inverse mapping defined by forward skinning Eq. (8), which is not invertible. Following [12], we determine the correspondence numerically by finding the root of the equation:
+
+$$
+\mathbf {d} (\hat {\mathbf {x}}, \boldsymbol {\beta}, \boldsymbol {\theta}, \mathbf {z} _ {\text {s h a p e}}) - \mathbf {x} ^ {\prime} = \mathbf {0}, \tag {9}
+$$
+
+using Broyden's method [9]. Subsequently, the canonical correspondence  $\mathbf{x}^*$  is given by:
+
+$$
+\mathbf {x} ^ {*} = \mathcal {M} (\hat {\mathbf {x}} ^ {*}, \boldsymbol {\beta}) \tag {10}
+$$
+
+We can now determine the occupancy at  $\mathbf{x}'$  as  $o' = \mathcal{O}(\mathbf{x}^*,\mathbf{z}_{\mathrm{shape}})$  and the normal  $\mathbf{n}'$  as
+
+$$
+\mathbf {n} ^ {\prime} = \left(\sum_ {i = 1} ^ {n _ {\mathrm {b}}} \mathcal {W} _ {i} \left(\mathbf {x} ^ {*}, \mathbf {z} _ {\text {s h a p e}}\right) \cdot \mathbf {R} _ {i}\right) ^ {- T} \mathcal {N} \left(\mathbf {x} ^ {*}, \mathbf {f}, \mathbf {z} _ {\text {d e t a i l}}\right) \tag {11}
+$$
+
+where  $\mathbf{R}_i$  denotes the rotational component of  $B_{i}$
+
+For convenient future reference, we define the occupancy field  $\mathcal{O}'$  and normal function  $\mathcal{N}'$  in posed space as:
+
+$$
+\mathcal {O} ^ {\prime}: \left(\mathbf {x} ^ {\prime}, \mathbf {z} _ {\text {s h a p e}}, \boldsymbol {\beta}, \boldsymbol {\theta}\right) \mapsto o ^ {\prime}, \mathbf {f} \tag {12}
+$$
+
+$$
+\mathcal {N} ^ {\prime}: \left(\mathbf {x} ^ {\prime}, \mathbf {z} _ {\text {d e t a i l}}, \mathbf {f}, \beta , \theta\right) \mapsto \mathbf {n} ^ {\prime} \tag {13}
+$$
+
+# 3.3. Implicit Surface Rendering
+
+Geometric clothing details are challenging to learn due to their stochastic nature. In 2D image generation tasks, GANs have achieved impressive results on learning high fidelity local textures. We propose to learn better geometric details  $\mathcal{N}$  using an adversarial loss. Towards this goal, we
+
+augment the forward skinning module with an implicit renderer to establish direct correspondences between 2D projections of 3D points in posed space and corresponding 3D points in canonical space, enabling end-to-end training.
+
+Implicit Rendering with Skinning: Given a pixel  $\mathbf{p}$  in the 2D posed normal map, its correspondence in deformed 3D space  $\mathbf{x}'$  can be determined by the intersection between the ray through  $\mathbf{p}$  and the forward skinned surface:
+
+$$
+\mathcal {O} ^ {\prime} \left(\mathbf {x} ^ {\prime}, \mathbf {z} _ {\text {s h a p e}}, \boldsymbol {\beta}, \boldsymbol {\theta}\right) = \tau , \text {w i t h} \mathbf {x} ^ {\prime} = \mathbf {r} _ {c} + t \cdot \mathbf {r} _ {d} \tag {14}
+$$
+
+where  $\mathbf{r}_d$  and  $\mathbf{r}_c$  denote the ray direction and origin, and  $t$  is the scalar distance along the ray. Following [44], we determine the intersection point  $\mathbf{x}'$  by finding the first change of occupancy  $\mathcal{O}'$  along the ray using the Secant method. We also obtain the canonical correspondence point  $\mathbf{x}$  of  $\mathbf{p}$  via forward skinning. Solving the 3D canonical correspondence for each pixel, yields the 2D normal map  $I$ :
+
+$$
+I _ {\mathbf {p}} = \mathcal {N} ^ {\prime} \left(\mathbf {x} ^ {\prime}, \mathbf {z} _ {\text {d e t a i l}}, \mathbf {f}, \beta , \boldsymbol {\theta}\right) \tag {15}
+$$
+
+# 3.4. Training
+
+We train our method via a set of posed scans and their corresponding SMPL parameters  $\theta, \beta$ . We follow the autodecoding framework of [48], and assign one shape code  $\mathbf{z}_{\mathrm{shape}}$  and one details code  $\mathbf{z}_{\mathrm{detail}}$  to each training sample. These are initialized to be zero and optimized jointly with the network weights. To enable sampling, we fit a Gaussian distribution to the latent codes after training.
+
+We split training into two stages: We first train the coarse shape, skinning, and warping networks and then train the normal network. This two-stage training is essential. Otherwise, the normal supervision will be back-propagated to wrong locations in canonical space due to wrong correspondences before training of shape and skinning converges.
+
+For the first stage, we use the binary cross entropy loss  $\mathcal{L}_{\mathrm{BCE}}$  between predicted occupancy  $\mathcal{O}'(\mathbf{x}', \mathbf{z}_{\mathrm{shape}}, \boldsymbol{\beta}, \boldsymbol{\theta})$  and ground-truth  $o_{\mathrm{gt}}$ . Following [12], we add auxiliary losses  $\mathcal{L}_{\mathrm{bone}}$  and  $\mathcal{L}_{\mathrm{joint}}$  to guide learning during early iterations:
+
+$$
+\mathcal {L} _ {\text {b o n e}} = B C E \left(\mathcal {O} \left(\mathbf {x} _ {\text {b o n e}}, \mathbf {z} _ {\text {s h a p e}}\right), 1\right) \tag {16}
+$$
+
+$$
+\mathcal {L} _ {\text {j o i n t}} = \left\| \mathbf {w} \left(\mathbf {x} _ {\text {j o i n t}}, \mathbf {z} _ {\text {s h a p e}}\right) - \mathbf {w} _ {\text {j o i n t , t a r g e t}} \right\| _ {2} ^ {2} \tag {17}
+$$
+
+where  $\mathbf{x}_{\mathrm{bone}}$  are randomly sampled points on canonical bones,  $\mathbf{x}_{\mathrm{joint}}$  are randomly sampled canonical joints, and  $\mathbf{w}_{\mathrm{joint},\mathrm{target}}$  is a vector that is 0.5 for the neighboring bones and 0 elsewhere (for details see Sup. Mat.). To ensure that the warping field changes body size consistently, we enforce the warping field to warp SMPL vertices  $\mathbf{v}(\beta)$  to the corresponding location in neutral shape  $\mathbf{v}(\beta_0)$ :
+
+$$
+\mathcal {L} _ {\text {w a r p}} = \left\| \mathcal {M} (\mathbf {v} (\boldsymbol {\beta}), \boldsymbol {\beta}) - \mathbf {v} \left(\boldsymbol {\beta} _ {0}\right) \right\| _ {2} ^ {2} \tag {18}
+$$
+
+Finally, we regularize the latent code to be close to the origin of the latent space via  $\mathcal{L}_{\mathrm{reg,shape}} = \| \mathbf{z}_{\mathrm{shape}}\| _2^2$
+
+![](images/bac003eec446e99acd255f30f4463e40236a9682ea97a175e10e4af925f04a92.jpg)  
+Figure 5. Clothed Human Generation. We show randomly sampled clothed human shapes generated by our method. Top: generated canonical shapes with detailed normals and skinning weights. Bottom: generated shapes reposed (2 poses each). Shapes are visualized as normal maps to highlight the detailed geometry synthesized by our method. For shaded results, see other figures, e.g. Fig. 1.
+
+The normal prediction network is trained subsequently. Here we penalize differences between the predicted and GT normal  $\mathbf{n}_{\mathrm{gt}}^{\prime}$  for randomly sampled surface points:
+
+$$
+\mathcal {L} _ {\text {n o r m}} = 1 - \mathbf {n} _ {\mathrm {g t}} ^ {\prime} ^ {T} \cdot \mathcal {N} ^ {\prime} \left(\mathbf {x} ^ {\prime}, \mathbf {z} _ {\text {d e t a i l}}, \mathbf {f}, \boldsymbol {\beta}, \boldsymbol {\theta}\right) \tag {19}
+$$
+
+In addition, we apply non-saturating adversarial losses [18]  $\mathcal{L}_{\mathrm{adv}} = -\log (1 + \exp (D(I)))$  with  $R_{1}$  gradient penalty [39] on the predicted 2D normal maps  $I$  and the real normal maps rendered from the posed scans  $I_{\mathrm{real}}$ .  $D$  is a jointly trained discriminator (see Sup. Mat. for details). We further regularize  $\mathbf{z}_{\mathrm{detail}}$  with  $\mathcal{L}_{\mathrm{reg,detail}} = \| \mathbf{z}_{\mathrm{shape}}\| _2^2$ .
+
+# 3.5. Inference
+
+We generate human avatars by randomly sampling  $\mathbf{z}_{\mathrm{shape}}$  and  $\mathbf{z}_{\mathrm{detail}}$  from the estimated Gaussian distribution. We then extract meshes in resized canonical space using MISE [40] from the implicit representation  $\hat{S} (\mathbf{z}_{\mathrm{shape}},\beta)$  and predict the vertex normal with our normal field. Finally, we pose the meshes to desired poses  $\pmb{\theta}$  following Eq. (8).
+
+# 4. Experiments
+
+Our main goal is to generate 3D human avatars. Since we are the first to tackle this problem setting, we compare our method to carefully designed ablative baselines, enabling analysis of each component of our method. We also evaluate the expressiveness of our model by fitting it to unseen scans and compare the accuracy to SOTA 3D human shape modeling methods. We outline the evaluation protocols in the following and refer the readers to Sup. Mat. for details.
+
+# Datasets:
+
+3D Scans: We train our model on commercial scans [1, 2].
+
+SIZER: Following [14], we use the SIZER dataset [57] to evaluate fitting. This dataset contains 3D scans of humans in 21 garments, including shirts, T-shirts, coats and pants.
+
+# Metrics:
+
+Fréchet Inception Distance (FID): To evaluate generation quality, we compute FID between 2D normal maps of training scans and those of randomly generated 3D shapes.
+
+User Preference: We conduct a perceptual study among 44 subjects and report how often participants preferred a particular method over ours.
+
+Surface Distance: To evaluate fitting accuracy, we measure the one-directional Chamfer distance between predicted surfaces and the target scans, following SMPLicit [14].
+
+# Baselines:
+
+NPMs [47]: NPMs learn the latent space of human shapes and deformation from ground-truth canonical shapes and vertex displacements, obtained from synthetic 3D animations [31] and real scans with registered surfaces [36].
+
+SMPLicit [14]: SMPLicit learns a generative model of 3D garments, and drapes these over the SMPL T-pose. This model is trained with a collection of 3D synthetic garments.
+
+# 4.1. Quality of Generated Samples
+
+Random Generation of Canonical Shapes: We show random samples generated by our method in Fig. 5 (top). While trained with posed scans only, our method learns plausible canonical shapes with surface details.
+
+Disentangled Pose and Shape: The generated shapes can be reposed as desired, even to poses far beyond the training pose distribution (cf. Fig. 5 bottom and Fig. 1).
+
+Interpolation: Interpolating the shape and details codes, yields smooth transitions of shapes and details between two very different samples, as shown in Fig. 6.
+
+![](images/8cf91a3922215ff588f6f61e3d5fcbe3f5479c3ba58643c10225e8e034d02569.jpg)  
+Figure 6. Interpolation. We interpolate the pose and shape code and the detail code between the leftmost and rightmost sample.
+
+![](images/39b396b98878d39b433e3baefda96cfa0fb1e27f1f2b523e859a3e2810688d71.jpg)  
+Figure 7. Disentangled Generation of Shape and Details. We generate samples with the same coarse shape and different detail codes. Note that the details appear noticeably different from each other while all match the underlying coarse shape.
+
+![](images/7c71922675aabfeaa3f1c612a5ff9e23fbb3380a266e0ba545618f9f0fbe6b1b.jpg)  
+Figure 8. Nearest Training Samples. Note that the pose and clothing of the generated shape is noticeably different from that of the nearest training sample.
+
+Disentangled Shape and Details: Our disentangled formulation allows us to generate diverse clothing details for the same coarse shape. Fig. 7 shows results with the same coarse shape  $\mathbf{z}_{\mathrm{shape}}$  but different details codes  $\mathbf{z}_{\mathrm{detail}}$ . While the coarse shape remains the same, gDNA generates varied plausible wrinkles that match the underlying coarse shape.
+
+Extrapolation Beyond Training Distribution: To further illustrate generalization, we show the training samples with the most similar pose and latent code to the generated sample in Fig. 8. The nearest neighbors are noticeably different from our generation, demonstrating that our method generalizes and is able to generate novel shapes in novel poses.
+
+# 4.2. Ablation Study
+
+We now ablate our design choices. The results are summarized in Tab. 1 and Fig. 9.
+
+Canonical Space Modeling: We verify the necessity to model shapes in canonical space and joint learning of skinn- ing weights. Towards this goal, we implement a baseline that generates posed shapes directly given the latent code and the body pose as input. As shown in Fig. 9 (first row), the individual samples lack details as the baseline must capture a large shape space caused by the pose change. Since the method does not reason about articulation, the sampled shapes suffer from invalid pose configurations, leading to high FID values as shown in Tab. 1 (Pose ONet).
+
+Adversarial Learning: The adversarial loss plays an important role in improving the perceptual realism of the generated samples, as evidenced by the FID improvement from Detailed Normal (w/o Adversarial) to Ours in Tab. 1. The normals estimated directly from the occupancy field suffer from artifacts on the surface (Fig. 9 (second row)). Training without adversary leads to overly smooth geometry (Fig. 9 (third row)), as the reconstruction loss induces a bias that averages out details. In contrast, our method produces realistic high-frequency details (Fig. 9 (bottom)). Notably, in  $21.3\%$  of the cases, users consider our generated shapes to be even more realistic than real scans.
+
+# 4.3. Comparison with SOTA on Model Fitting
+
+While our main goal is to generate clothed human shapes, our model can be fit to raw observations, just like existing 3D parametric human or clothing models. We consider two recent SOTA methods, i.e. NPMs [47] and SMPLicit [14]. We follow SMPLicit [14] and fit ours and the baselines to scans from the SIZER dataset.
+
+Accuracy: While not designed for fitting, our method achieves better accuracy than previous special purpose methods, as demonstrated in Tab. 2. Our method captures the person identity and clothing shapes more faithfully than NPMs and SMPLlicit, and our results exhibit more details such as wrinkles (Fig. 10 top). Since the model is trained directly from posed scans, disentangling pose and shape, it learns about real clothing details and can reproduce them.
+
+<table><tr><td>Method</td><td>FID↓</td><td>User Preference↑</td></tr><tr><td>Pose ONet</td><td>43.80</td><td>8.11%</td></tr><tr><td>Coarse Shape</td><td>29.34</td><td>26.1%</td></tr><tr><td>Detailed Normal (w/o Adv.)</td><td>42.18</td><td>15.4%</td></tr><tr><td>Ours</td><td>11.54</td><td>-</td></tr><tr><td>Ground-truth Scans</td><td>N/A</td><td>78.7%</td></tr></table>
+
+![](images/53db5b9d7281837b8d5c274b855c13a6de5ca2d10bf685026cda1e687d088c9f.jpg)  
+Figure 9. Generation Comparison. We show random samples from ablative baselines and our method. Without adversarial loss, the generated shapes appear either bumpy (Coarse Shape) or oversmooth (Detailed Normal w/o Adv.).
+
+Reposing Scans: During fitting we also recover skinning weights. This enables reposing of the shape as demonstrated in Fig. 10 bottom.
+
+# 5. Conclusion
+
+We propose gDNA, a generative model of 3D clothed humans that can produce a large variety of clothed people with detailed wrinkles and explicit pose control. Using implicit multi-subject forward skinning enables learning from only a few posed scans per subject. To model the stochastic details of garments, we exploit a 2D adversarial loss to update a 3D normal field. We demonstrate that gDNA can be used in various applications such as animation and 3D fitting, outperforming state-of-the-art methods.
+
+Table 1. Ablation Study. We report FID and user preference. The user preference score indicates how often participants of our user study preferred a particular method over ours.  
+
+<table><tr><td>Method</td><td>Pred-to-Scan ↓</td><td>Scan-to-Pred ↓</td></tr><tr><td>SMPLicit [14]</td><td>N/A</td><td>0.0240</td></tr><tr><td>NPMs [47]</td><td>0.0156</td><td>0.0215</td></tr><tr><td>Ours</td><td>0.0134</td><td>0.0123</td></tr></table>
+
+Table 2. Fitting Comparison. We report the distance between the target scan and 3D shapes fit by SOTA methods and ours. Pred-to-Scan metric does not apply to multi-layer surfaces from SMPLicit.
+
+![](images/d92ff418a18136181881d5e72ef829a839a70182d55982c1c4d1db96f847bb40.jpg)  
+Figure 10. Fitting and Reposing. We compare model fitting results on the SIZER dataset with SMPLicit [14] and NPMs [47]. We also show the fitted shapes reposed into target poses. As NPMs do not allow specifying target pose, random poses are shown.
+
+Limitations: Learning loose clothing (e.g. skirts) from deformed observations remains challenging due to the topology ambiguity and the large pose-dependent non-linear cloth deformation. Please refer to Sup. Mat. for more discussions about limitations and societal impact.
+
+Acknowledgements: Xu Chen was supported by the Max Planck ETH Center for Learning Systems. Andreas Geiger was supported by the DFG EXC number 2064/1 - project number 390727645. We thank Alex Zicong Fan, Marcel C. Bühler, Priyanka Patel, Qianli Ma, Sai Kumar Dwivedi, Thomas Langerak and Yuliang Xiu for their feedback, Garvita Tiwari for her suggestions about the SIZER dataset, and Tsvetelina Alexiadis for her help with the user study.
+
+Disclosure: MJB has received research gift funds from Adobe, Intel, Nvidia, Meta/Facebook, and Amazon. MJB has financial interests in Amazon, Datagen Technologies, and Meshcapade GmbH. MJB's research was performed solely at, and funded solely by, the Max Planck.
+
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+# iFS-RCNN: An Incremental Few-shot Instance Segmenter
+
+Khoi Nguyen  
+VinAI Research, Ha Noi, Vietnam  
+ducminhkhoi@gmail.com
+
+Sinisa Todorovic  
+Oregon State University, Oregon, USA  
+sinisa@oregonstate.edu
+
+# Abstract
+
+This paper addresses incremental few-shot instance segmentation, where a few examples of new object classes arrive when access to training examples of old classes is not available anymore, and the goal is to perform well on both old and new classes. We make two contributions by extending the common Mask-RCNN framework in its second stage – namely, we specify a new object class classifier based on the probit function and a new uncertainty-guided bounding-box predictor. The former leverages Bayesian learning to address a paucity of training examples of new classes. The latter learns not only to predict object bounding boxes but also to estimate the uncertainty of the prediction as a guidance for bounding box refinement. We also specify two new loss functions in terms of the estimated object-class distribution and bounding-box uncertainty. Our contributions produce significant performance gains on the COCO dataset over the state of the art – specifically, the gain of +6 on the new classes and +16 on the old classes in the AP instance segmentation metric. Furthermore, we are the first to evaluate the incremental few-shot setting on the more challenging LVIS dataset.
+
+# 1. Introduction
+
+This paper addresses the two related problems of incremental few-shot object detection (iFSOD) and instance segmentation (iFSIS). Initially, we are given a large training set of base object classes, which can be used for pre-training an instance segmenter. After this pre-training, access to training examples of the base classes becomes unavailable. With an arrival of a few training examples of new classes, the goal is to achieve successful object detection and instance segmentation on both new and base classes. Our key challenges include: how to address a paucity of data for new classes, and how to train on the new classes such that the base classes are not "forgotten".
+
+Tab. 1 compares iFSOD and iFSIS with other related problems. iFSOD and iFSIS are important problems arising in many applications, where access to old training data
+
+![](images/faff38a87248ea73f1a101f4087d60335661ce12d441eb8af36df909849ac779.jpg)
+
+![](images/c1667023e324c33d45c6ce5acab3b3e9e044814c4ecb93c01a0a955fb3177662.jpg)  
+Figure 1. Our iFS-RCNN is first pre-trained on abundant examples of base classes, and then fine-tuned on a few examples of new classes. iFS-RCNN modifies the classification head of Mask-RCNN by estimating the class-weight distribution via Bayesian learning. iFS-RCNN modifies the bounding-box head of Mask-RCNN by computing uncertainty of predicting the left (L), right (R), top (T), and bottom (B) sides of the bounding boxes. For testing, the last layers learned on the new classes (diagonal stripes) are concatenated with the corresponding ones learned on the base classes (solid color).
+
+becomes unavailable, due to, e.g., privacy and security issues or new legal regulations of data access. Also, they are critical in applications where limited time budgets prohibit retraining on both base and new classes.
+
+There is scant work on iFSOD and iFSIS. Following recent FSIS approaches, we use Mask-RCNN [13], and modify its prediction heads, as shown in Fig. 1. Mask-RCNN is first pre-trained with abundant examples of base classes, and then fine-tuned on new classes by "freezing" all mod
+
+<table><tr><td>Settings</td><td>Pretrained on</td><td>Fine-tuned on</td><td>Tested on</td></tr><tr><td>FSOD - FSIS</td><td>base</td><td>new</td><td>new</td></tr><tr><td>gFSOD - gFSIS</td><td>base</td><td>base + new</td><td>base + new</td></tr><tr><td>CL</td><td>base</td><td>new</td><td>base + new</td></tr><tr><td>iFSOD - iFSIS</td><td>base</td><td>new</td><td>base + new</td></tr></table>
+
+Table 1. A comparison of related problems. The blue and red indicate abundant and a few examples, respectively, of base and new classes. FSOD (FSIS): few-shot object detection (instance segmentation), gFSOD (gFSIS): generalized FSOD (FSIS), CL: continual learning, iFSOD (iFSIS): incremental FSOD (FSIS). iFSOD (iFSIS) are more challenging than: FSOD (FSIS), since we test on both classes; gFSOD (gFSIS), since our training cannot access the base classes; CL, since they work with more examples.
+
+ules except for the classification head, bounding-box head, and segmentation-mask head. Finally, for testing on both base and new classes, weights learned on the new classes are concatenated with weights learned on the base classes to make the corresponding last layers in the classification, bounding-box, and segmentation-mask heads.
+
+As shown in Fig. 1 and depicted in more detail in Fig. 2, we make two contributions aimed at addressing overfitting of Mask-RCNN in few-shot fine-tuning and improving its generalization to query images with large appearance-shape-scale variations of both base and new classes.
+
+Inspired by deep Bayesian learning [2], our first contribution is about learning a distribution of the classification head's weights on the new classes, and using the estimated distribution for regularization of the fine-tuning. Instead of using the standard Monte Carlo sampling of weights for this Bayesian learning, our key technical novelty is in casting the weight-distribution learning as a Bayesian logistic regression problem, and specifying an efficient approximation to this intractable problem using the probit function. From our ablation studies, our probit-based approximation gives a significantly better performance than the Monte Carlo sampling, even under a reasonable training-time budget.
+
+Our second contribution is about estimating the uncertainty of bounding-box localization on the new classes, and using the estimated uncertainty for two purposes – to refine the bounding-box prediction and to appropriately weight the loss of bounding-box prediction. As shown in Fig. 2, we use the estimated uncertainty along with the ROI-aligned-pooled feature map as input to a new bounding-box refinement module. The refined bounding box is subsequently input to the segmentation head. Also, we define a new loss between the ground truth and predicted bounding box, such that the loss becomes smaller for highly uncertain predictions, i.e., our fine-tuning stronger penalizes errors on training examples with highly certain bounding-box predictions.
+
+It is worth noting that we neither use Bayesian learning nor explicitly estimate uncertainty for fine-tuning of the segmentation head of Mask-RCNN. This is because fine
+
+tuning of the segmentation head on a few examples of the new classes does not face the common challenges of few-shot learning. Recall that the segmentation head predicts pixel labels independently with a  $1 \times 1$  convolution. Consequently, every (pixel, label) pair is an independent training example, giving rise to a sufficiently large training set for fine-tuning of the segmentation mask.
+
+Our extension of Mask-RCNN for incremental few-shot setting gives the name to our approach - iFS-RCNN. iFS-RCNN is evaluated on the COCO dataset [24] for few-shot object detection and instance segmentation with the iFSOD, iFSIS, FSOD, and FSIS tasks. iFS-RCNN significantly outperforms a recent approach [11]. In comparison with the standard Mask-RCNN trained in the gFSOD and gFSIS settings, iFS-RCNN shows a considerable improvement on the new classes while retaining the same performance on the base classes. On iFSOD and iFSIS, we also achieve the higher COCO AP rates by  $+6$  and  $+16$  for the new and base classes, respectively, relative to the state of the art.
+
+In addition, we are the first to report the results of iFSOD, FSIS, and iFSIS on the more challenging LVIS dataset [12] having significantly more classes and long-tailed class distributions.
+
+In the following, Sec. 2 reviews prior work; Sec. 3 specifies iFS-RCNN; and Sec. 4 presents our implementation details and experimental results.
+
+# 2. Related Work
+
+This section reviews closely related work.
+
+FSOD approaches [6, 8, 9, 16, 17, 19, 32, 33, 35, 36, 38] typically adapt Faster-RCNN [29], YOLO [28], or DETR [5] for standard object detection to few-shot setting. These approaches can be divided into two subgroups based on: episodic training [8, 16, 35, 36, 38] and fine-tuning [6, 9, 32, 33]. The former uses episodic training to mimic the setting of few-shot learning, by limiting access to a few annotated support images for each base class. The latter fine-tunes the weights of some layers while freezing the rest to preserve knowledge learned on the base classes.
+
+FSIS approaches typically use Mask-RCNN [13] as their backbone network. As in FSOD, these methods either train episodically [10, 25, 26, 37] or pretrain on the base classes then fine-tune the last layers of each head [32] on the new classes. Our iFS-RCNN follows the second training strategy and applies it to all three heads – the classification, bounding-box, and segmentation heads – of Mask-RCNN.
+
+iFSOD approaches [20, 21, 27] typically use Faster-RCNN as their backbone network. In [20], knowledge distillation ensures the prediction of the base classes, after finetuning on the new classes, to match its pretrained prediction. In [27], weights of the box head are generated on the fly, based on a class-specific code extracted from examples of the target classes. In this way, each class has a distinct box
+
+![](images/08199517f042255fa3206de9915033c70ebb8f0ec0daeaadbbb89227a159c4fe.jpg)  
+Figure 2. iFS-RCNN extends Mask-RCNN with two contributions: probit classifier and uncertainty-guided box predictor. The former uses Bayesian learning to estimate a probability distribution of the classifier head's weights (red diagonal stripes). Our contribution 1 is the efficient, analytical formulation of this Bayesian learning using the probit function. The latter explicitly estimates uncertainty of predicting bounding boxes (blue diagonal stripes), and uses the estimated uncertainty (violet) as input features along with the ROI-align-pooled features (dark yellow) for refining the boxes. The colored stripes depict the last layers of the classification, box, and segmentation heads learned in the few-shot setting during fine-tuning of the new classes.
+
+head for object detection. Our iFS-RCNN also uses for each class a distinct set of the classification, bounding-box, and segmentation heads of Mask-RCNN, where the difference between the respective heads is in the last layer fine-tuned for each new class separately.
+
+iFSIS: A recent approach to iFSIS [11] replaces the standard fully-connected classifier in Mask-RCNN with the cosine-similarity classifier. Unlike our iFS-RCNN, they do not use Bayesian learning and do not estimate bounding-box uncertainty. They convert activations of the classification head to a softmax distribution, and train the head with cross-entropy loss. In this way, the activation scores of all classes compete with each other to determine the class of the bounding box. As training examples of the new classes are scarce, their activation scores are likely to be smaller than those of the base classes in the softmax function. Consequently, their classifier is likely to be biased toward favoring the base classes. They address this bias by using the cosine-similarity classifier, where both the box feature and class weights are normalized to have unit length before the dot product for reducing a statistical difference between the base and new classes. In contrast, we directly use sigmoid activation of the classification head for predicting the class
+
+of the bounding box, and train our fully-connected classifier with the focal loss [23]. Our iFS-RCNN uses the sigmoid activation to predict each class independently, and thus alleviate the aforementioned bias.
+
+# 3. Specification of iFS-RCNN
+
+# 3.1. Problem Statement
+
+We address N-way K-shot iFSOD and iFSIS, where abundant training examples of  $N_{b}$  base classes are provided for pre-training. After that, access to training examples of the base classes is not available. When a few training examples  $K$  of additional  $N_{n}$  new classes are arbitrarily provided, our goal is to detect and segment all object instances belonging to all  $N = N_{b} + N_{n}$  classes in a query image.
+
+In the following, we specify our two contributions – the probit classifier and uncertainty-guided box predictor.
+
+# 3.2. The Probit Classifier
+
+As mentioned in Sec. 2, our iFS-RCNN uses sigmoid activation instead of softmax activation for predicting the class of a bounding box. With sigmoid activation, the scores for all classes are independently predicted, and thus our classi
+
+fication head effectively addresses the statistical difference between the base and new classes. However, in the FSOD and FSIS settings, our experiments (see Tab. 2) demonstrate that the classification head with the sigmoid activation usually gives a lower performance than that with the softmax activation, when weights of the classifier head (also referred to as class weights) are learned as point estimation.
+
+To address this problem, we resort to Bayesian learning of a distribution of the class weights, and adopt the common variational framework. Formally, the class weights  $w$  are characterized by the normal distribution,  $w \sim \mathcal{N}(\mu, \Sigma)$ , with mean  $\mu \in \mathbb{R}^D$  and diagonal covariance matrix  $\Sigma \in \mathbb{R}_{+}^{D \times D}$ . Our goal is to learn  $\mu$  and  $\Sigma$  by minimizing the following variational objective:
+
+$$
+L _ {c} = l _ {d} \left(p \left(c | f, \mu , \Sigma\right), c ^ {*}\right) + \mathrm {K L} \left(\mathcal {N} (\mu , \Sigma) | | \mathcal {N} (\mathbf {0}, \mathbf {1})\right), \tag {1}
+$$
+
+where  $f \in \mathbb{R}^D$  is a feature extracted from the bounding box;  $c$  and  $c^*$  are the predicted and ground-truth classes;  $l_d$  is the incurred sigmoid focal loss [23] for predicting  $c$ ; KL stands for the Kullback-Leibler divergence; and  $p(c|f, \mu, \Sigma)$  is the posterior predictive distribution defined as
+
+$$
+p (c | f, \mu , \Sigma) = \int \sigma (f ^ {\top} w) \mathcal {N} (w | \mu , \Sigma) d w, \tag {2}
+$$
+
+where  $\sigma (\cdot)$  denotes the sigmoid activation.
+
+Once  $\mu$  and  $\Sigma$  are learned, class prediction amounts to the MAP problem:  $c = \arg \max_{c'} p(c'|f, \mu, \Sigma)$ . However, the integral in (2) is intractable. Prior work typically approximates the MAP problem with Monte Carlo sampling:  $p(c|f, \mu, \Sigma) \approx \frac{1}{T} \sum_{w_t \sim \mathcal{N}(\mu, \Sigma)} \sigma(f^\top w_t)$  where  $T$  is the number of Monte-Carlo samples. The Monte Carlo approximation, however, has a poor trade off between efficiency and accuracy, where using a sufficiently large  $T$  would make our training and testing prohibitively slow.
+
+Instead of the stochastic Monte Carlo sampling, we specify a more efficient deterministic approximation of the posterior predictive distribution. We first observe that our fine-tuning of the classification head's last layer using Bayesian learning is equivalent to learning a Bayesian logistic regression (BLR). Conveniently, the well-known probit function  $\Phi(x)$  [30], [1, p. 219] provides a deterministic approximation to BLR. The probit function approximates the sigmoid function as  $\sigma(x) \approx \Phi(\lambda x) = \frac{1}{2}\left[1 + \mathrm{erf}\left(\frac{\lambda x}{\sqrt{2}}\right)\right]$ , where  $\mathrm{erf}(\cdot)$  is the error function, and  $\lambda^2 = \pi/8$  ensures that the two functions have the same slope at the origin. An important property of the probit function is that its convolution with a Gaussian function can be expressed analytically. Let  $a = f^\top w \in \mathbb{R}$  be a random variable whose expectation and variance can be expressed as  $\mathbb{E}[a] = \mu_a = f^\top \mu \in \mathbb{R}$  and  $\mathbb{V}[a] = \Sigma_a = f^\top \Sigma f \in \mathbb{R}$ . Then the posterior predictive distribution given by (2) can be efficiently approximated as
+
+$$
+p (c \mid f, \mu , \Sigma) = \int \sigma (a) \mathcal {N} (a \mid \mu_ {a}, \Sigma_ {a}) d a, \tag {3}
+$$
+
+$$
+\approx \int \Phi (\lambda a) \mathcal {N} (a | \mu_ {a}, \Sigma_ {a}) d a = \Phi \left(\frac {\lambda \mu_ {a}}{\left(1 + \lambda^ {2} \Sigma_ {a}\right) ^ {\frac {1}{2}}}\right), \tag {4}
+$$
+
+$$
+\approx \sigma \left(\frac {f ^ {\top} \mu}{\left(1 + \frac {\pi}{8} f ^ {\top} \Sigma f\right) ^ {\frac {1}{2}}}\right). \tag {5}
+$$
+
+As our classifier head uses the probit function for the MAP class prediction, we call it the probit classifier. It is suitable for iFSOD and iFSIS for two reasons. It leverages Bayesian learning to address the paucity of training data. Also, it predicts a score for each class independently to address the incremental-learning setting.
+
+# 3.3. Uncertainty-Guided Bounding Box Predictor
+
+Object appearances, shapes, and scales in test images may significantly differ from a few training examples available. Also, target objects in query images may be subject to partial occlusion. All this gives rise to uncertainty in bounding box prediction. We seek to explicitly model this uncertainty when predicting four offset values  $\{m_k\}_{k = 1..4} \in \mathbb{R}^4$  that initially identify the location of bounding boxes. Specifically, as shown in Fig. 2, our box predictor additionally estimates four uncertainty values  $\{u_k\}_{k = 1..4} \in \mathbb{R}_+^4$  of the bounding box prediction, one for each of the  $\{m_k\}$  predictions. The estimated uncertainty  $\{u_k\}$  is then used as input along with the ROI-align-pooled features - extracted from the initially predicted box  $m$  - to the box refinement module for the final offset bounding-box prediction  $\{b_k\}_{k = 1..4} \in \mathbb{R}^4$ .
+
+To learn how to predict uncertainty  $\{u_k\}$ , initial bounding-box  $\{m_k\}$ , and refined bounding-box  $\{b_k\}$  on a few training examples, we specify the following box loss:
+
+$$
+L _ {b} = L _ {\mathrm {u}} + L _ {\text {r e f i n e}}, \tag {6}
+$$
+
+where  $L_{\mathrm{u}}$  is our new uncertainty-weighted box loss and  $L_{\mathrm{refine}}$  is loss incurred by the box refinement module.
+
+We define  $L_{\mathrm{u}}$  as
+
+$$
+L _ {\mathrm {u}} = \sum_ {k = 1} ^ {4} \frac {1}{2} \left(\frac {\left(m _ {k} - b _ {k} ^ {*}\right) ^ {2}}{u _ {k} ^ {2}} + u _ {k} ^ {2}\right), \tag {7}
+$$
+
+where  $b^{*}$  is the ground-truth box. The first term in (7) is aimed at minimizing a weighted difference between the ground-truth and predicted boxes. The weighting is inversely proportional to the predicted uncertainty  $u_{k}$  such that the lower loss is incurred for box predictions with high uncertainty. The second term in (7) is aimed at minimizing uncertainty values such that the network incurs a penalty for predicting high uncertainty when trying to reduce the first term in (7).
+
+When the box refinement module makes the final prediction  $b$ , it incurs the following loss:
+
+$$
+L _ {\text {r e f i n e}} = \sum_ {k = 1} ^ {4} \text {s m o o t h L 1} \left(b _ {k}, b _ {k} ^ {*}\right). \tag {8}
+$$
+
+It is worth noting that our loss formulation fundamentally differs from other recent approaches aimed at estimating uncertainty in object detection. For example, recent approaches [15, 18] make the assumption that the bounding-box location and its uncertainty are governed by a Gaussian distribution. In contrast, we do not explicitly specify any probability distribution of box locations. iFS-RCNN appears related to Cascade-RCNN [3, 4] which also refines the initial box. However, these approaches do not explicitly predict uncertainty and thus cannot use uncertainty as an input feature for the box refinement as we do. By contrast, our uncertainty estimation in training serves to transfer "knowledge" from the base classes to the new testing classes. In experiments, these approaches show worse performance than our uncertainty-guided box refinement module.
+
+# 3.4. Our Training and Testing Strategies
+
+# Training on the base classes
+
+1. Obtain a variant of Mask-RCNN named Mask+Sigmoid by replacing the standard softmax classifier with the sigmoid classifier. Train Mask+Sigmoid with the following loss functions: sigmoid focal loss,  $L_{\mathrm{refine}}$  in (8), and mask-BCE loss.  
+2. Obtain a variant of Mask+Sigmoid named Mask+Sigmoid+Uncertainty by replacing the box predictor with the uncertainty-guided box predictor (contribution 2 in Fig. 2). While freezing other modules, train the uncertainty-guided box predictor of Mask+Sigmoid+Uncertainty with loss  $L_{b}$  in (6).  
+3. Store the class weights  $\mu_{b}$  of the base classes for the sigmoid classifier of  $\mathrm{Mask + Sigmoid + Uncertainty}$ .
+
+# Fine-tuning on the new classes
+
+1. Obtain a variant of Mask+Sigmoid+Uncertainty named iFS-RCNN by replacing the sigmoid classifier with the probit classifier (contribution 1 in Fig. 2).  
+2. While freezing other modules, train the probit classifier with loss  $L_{c}$  in (1) to obtain the class weights  $\mu_{n}, \Sigma_{n}$  of the new classes. Also, train the last layer of the box predictor with  $L_{b}$  in (6); and the last layer of the segmentation head with the mask-BCE loss.
+
+# Testing on the base and new classes
+
+1. Set  $\mu = [\mu_b; \mu_n]$  and  $\Sigma = [\mathbf{0}; \Sigma_n]$  for the probit classifier, where  $[\cdot; \cdot]$  is a concatenation. Also, concatenate the weights estimated for the base and new classes to obtain the box and segmentation-mask heads.  
+2. Run iFS-RCNN on query images.
+
+# 4. Experimental Results
+
+Datasets & Metrics: We evaluate iFS-RCNN on the modified version of the COCO 2014 dataset [24] introduced by [16] for FSIS and FSOD. Also, we are the first to evaluate iFSOD, FSIS, and iFSIS on a new split of the LVIS dataset [12] introduced by [32] for FSOD. We report the common COCO-style evaluation metrics of both object detection and instance segmentation – namely, the average precision (AP) at multiple intersection-over-union (IoU) thresholds ranging from 0.5 to 0.95.
+
+For COCO, the 20 categories shared with PASCAL VOC [7] are used as new classes while the remaining 60 classes are used as the base classes. We vary the number of examples for the new classes, i.e.  $K = \{1,2,3,5,10,30\}$ , and report average results with  $95\%$  confidence interval over 10 runs with different sets of few-shot examples for each  $K$ . This paper reports results for  $K = \{1,5,10\}$  for brevity. Other results are in the supplementary material.
+
+LVIS has 1230 classes where some have a large number of examples and some other, called rare classes, have only a few examples (less than 10 examples per class). Hence, the number of images for each class in LVIS has a long-tail distribution. We take the frequent (appearing in more than 100 images) classes and the common classes (10-100 images) in LVIS as the base classes, while the 454 rare classes (appearing less than 10 images) as the new classes. Due to the small number of training examples for the rare classes in LVIS, we cannot have multiple runs as in COCO, thus we follow [32] on their split with  $K \leq 10$ .
+
+# 4.1. Implementation Details
+
+Our backbone CNN is ResNet-50 [14] with the FPN of [22], as in recent work on FSIS. All variants of iFSRCNN are implemented using the detectron2 toolbox [34] with the codebase of [32]. All variants of our approach, specified in Sec. 4.2, are trained using SGD and a batch size of 16 on 8 NVIDIA GPUs V100s, with two images per GPU. The learning rate is set to 0.02 and 0.01 for the pre-training and fine-tuning stages respectively. The number of iterations for the pre-training stage is 110000 with two weight decay steps with the rate of 10 at 80000 and 100000 iterations. The number of iterations for fine-tuning stage depends on the number of examples ranging from 500 iterations (with  $K = 1$ ) to 6000 iterations (with  $K = 30$ ). The hyper-parameters of the focal loss are  $\gamma = 0.25$ ,  $\alpha = 2$ .
+
+The threshold for filtering predictions before the NMS is 0.05. To select proposals for training the uncertainty-guided box predictor, we choose the predicted boxes with IoU larger than 0.7 with their closest ground-truth boxes. The threshold for deciding foreground and background in the segmentation mask head is 0.5. We use the softplus function  $f(x) = \ln (1 + \exp (x))$  to ensure the class variances  $\Sigma$  and box uncertainty  $u$  are non-negative.
+
+<table><tr><td></td><td colspan="6">Object Detection</td><td colspan="6">Instance Segmentation</td></tr><tr><td>Number of shots</td><td>1</td><td>2</td><td>3</td><td>5</td><td>10</td><td>30</td><td>1</td><td>2</td><td>3</td><td>5</td><td>10</td><td>30</td></tr><tr><td>Mask-RCNN</td><td>3.58</td><td>5.07</td><td>5.79</td><td>7.81</td><td>8.59</td><td>12.68</td><td>3.71</td><td>5.24</td><td>5.29</td><td>7.66</td><td>8.46</td><td>11.09</td></tr><tr><td>Mask+Cosine</td><td>3.39</td><td>4.87</td><td>5.19</td><td>6.87</td><td>7.96</td><td>12.52</td><td>3.40</td><td>5.00</td><td>4.75</td><td>6.68</td><td>7.72</td><td>11.03</td></tr><tr><td>Mask+Sigmoid</td><td>3.60</td><td>4.49</td><td>5.63</td><td>7.06</td><td>7.68</td><td>11.40</td><td>3.92</td><td>4.63</td><td>5.63</td><td>7.15</td><td>7.67</td><td>10.94</td></tr><tr><td>Mask+Probit</td><td>5.18</td><td>5.90</td><td>7.82</td><td>9.45</td><td>10.43</td><td>13.48</td><td>5.15</td><td>6.03</td><td>7.67</td><td>9.34</td><td>9.52</td><td>12.07</td></tr><tr><td>Mask+MC</td><td>4.52</td><td>5.45</td><td>7.02</td><td>8.85</td><td>9.57</td><td>13.05</td><td>4.54</td><td>5.19</td><td>6.91</td><td>8.26</td><td>8.98</td><td>11.55</td></tr><tr><td>Mask+Sig+Uncert</td><td>4.48</td><td>5.32</td><td>6.76</td><td>8.49</td><td>9.16</td><td>12.85</td><td>4.84</td><td>5.88</td><td>7.00</td><td>8.62</td><td>9.22</td><td>11.98</td></tr><tr><td>Mask+Sig+Gauss</td><td>3.74</td><td>4.77</td><td>5.89</td><td>7.33</td><td>7.86</td><td>11.65</td><td>3.94</td><td>4.72</td><td>5.87</td><td>7.24</td><td>7.76</td><td>11.02</td></tr><tr><td>Mask+Sig+Refine</td><td>3.87</td><td>4.57</td><td>5.78</td><td>7.48</td><td>8.23</td><td>11.95</td><td>3.99</td><td>4.77</td><td>5.68</td><td>7.40</td><td>7.87</td><td>11.01</td></tr><tr><td>iFS-RCNN</td><td>6.34</td><td>6.93</td><td>8.93</td><td>10.53</td><td>11.27</td><td>14.66</td><td>5.54</td><td>6.33</td><td>7.80</td><td>9.41</td><td>10.23</td><td>13.08</td></tr></table>
+
+Table 2. Our ablation study on FSOD and FSIS with different  $K = \{ 1,2,3,5,{10},{30}\}$  on COCO. The best results are in red, second-best results are in blue. Mask-RCNN and Mask+Cosine use the softmax activation with cross-entropy loss for training. The remaining ablations use the sigmoid activation with the focal loss for training.  
+
+<table><tr><td></td><td colspan="8">Object Detection</td><td colspan="7">Instance Segmentation</td></tr><tr><td>Tested on</td><td colspan="3">New classes</td><td colspan="3">Base classes</td><td colspan="2">All classes</td><td colspan="3">New classes</td><td colspan="3">Base classes</td><td>All classes</td></tr><tr><td>Number of shots</td><td>1</td><td>5</td><td>10</td><td>1</td><td>5</td><td>10</td><td>1</td><td>5</td><td>10</td><td>1</td><td>5</td><td>10</td><td>1</td><td>5</td><td>10</td></tr><tr><td>TFA [32]</td><td>2.90</td><td>7.00</td><td>9.10</td><td>31.90</td><td>32.30</td><td>32.40</td><td>3.60</td><td>11.50</td><td>14.20</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FSDetView [35]</td><td>3.35</td><td>8.53</td><td>12.50</td><td>25.75</td><td>25.05</td><td>24.82</td><td>20.15</td><td>20.92</td><td>21.74</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>GIFSOD [21]</td><td>-</td><td>-</td><td>8.50</td><td>-</td><td>-</td><td>28.10</td><td>-</td><td>-</td><td>23.20</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ONCE [27]</td><td>0.70</td><td>1.00</td><td>1.20</td><td>17.90</td><td>17.90</td><td>17.90</td><td>13.60</td><td>13.70</td><td>13.70</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>LEAST [20]</td><td>4.40</td><td>9.40</td><td>12.50</td><td>24.60</td><td>25.20</td><td>23.10</td><td>7.50</td><td>13.70</td><td>16.20</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>iMTFA [11]</td><td>3.23</td><td>6.07</td><td>6.97</td><td>27.81</td><td>24.13</td><td>23.36</td><td>21.67</td><td>19.62</td><td>19.26</td><td>2.81</td><td>5.19</td><td>5.88</td><td>25.90</td><td>22.56</td><td>21.87</td></tr><tr><td>Mask+Sigmoid</td><td>2.85</td><td>6.34</td><td>8.04</td><td>38.55</td><td>38.53</td><td>38.53</td><td>29.62</td><td>30.49</td><td>30.91</td><td>3.06</td><td>6.52</td><td>8.00</td><td>35.70</td><td>35.69</td><td>35.69</td></tr><tr><td>iFS-RCNN</td><td>4.54</td><td>9.91</td><td>12.55</td><td>40.08</td><td>40.06</td><td>40.05</td><td>31.19</td><td>32.52</td><td>33.02</td><td>3.95</td><td>8.80</td><td>10.06</td><td>36.35</td><td>36.33</td><td>36.32</td></tr></table>
+
+Table 3. iFSOD and iFSIS results on COCO with different  $K = \{ 1,5,{10}\} .{}^{\prime } - {}^{\prime }$  indicates no results are reported. Best results are in bold.
+
+Our segmentation head is the same as in Mask-RCNN [13]. It is trained with the binary cross-entropy (BCE) loss. Our box refinement module has the same architecture as the box head. The final layers of the uncertainty-guided box predictor and segmentation head are class-specific, and obtained by concatenating the class weights learned for the base and new classes.
+
+# 4.2. Ablation Study
+
+The following ablations are evaluated on the first run of COCO for studying how each component of iFS-RCNN affects the final performance.
+
+- Mask-RCNN: the original Mask-RCNN [13] with the softmax classifier.  
+- Mask+Cosine: replace the dot product with the cosine-similarity classifier in Mask-RCNN as in [11].  
+- Mask+Sigmoid: replace the softmax classifier with the sigmoid classifier in Mask-RCNN (our strong baseline).  
+- Mask+Probit: replace the sigmoid classifier with the probit classifier in Mask+Sigmoid, our contribution 1 in Fig. 2.  
+- Mask+MC: replace the probit approximation with the Monte Carlo (MC) sampling in  $\mathrm{Mask + Probit}$ , where the number of samplings is  $T = 10$ .
+
+- Mask+Sig+Uncert: replace the box predictor in Mask+Sigmoid with the uncertainty-guided box predictor, our contribution 2 in Fig. 2.  
+- Mask+Sig+Gauss: a variant of Mask+Sigmoid which additionally predicts box uncertainty (similar to [15]) with Gaussian distribution assumption.  
+- Mask+Sig+Refine: similar to Cascade RCNN [3] as it does not explicitly predict uncertainty when refining the initial box prediction  
+- Our iFS-RCNN in Fig. 2.
+
+Tab. 2 shows our evaluation of the above ablations on FSOD and FSIS (see Tab. 1). From Tab. 2, Mask-RCNN with the softmax activation outperforms other pointestimation-based ablations with the sigmoid activation. However, Mask-RCNN gives worse performance than the ablations which use the sigmoid activation with Bayesian learning for the Probit or MC. This justifies our choice of Bayesian-based classifiers with the sigmoid activation. Also, our probit classifier outperforms the MC classifier, suggesting that MC requires longer stochastic sampling  $T \gg 10$ , which would be prohibitively slow in practice. Importantly, our modeling of uncertainty in  $\text{Mask} + \text{Sig} + \text{Uncert}$  gives a substantial performance gain over  $\text{Mask} + \text{Sigmoid}$ ,  $\text{Mask} + \text{Sig} + \text{Gauss}$ , and  $\text{Mask} + \text{Sig} + \text{Refine}$ . This justifies our contribution 2. Finally, our full approach, iFS-RCNN yields
+
+<table><tr><td>Settings</td><td>FSOD</td><td>FSIS</td><td colspan="4">Object Detection</td><td colspan="4">Instance Segmentation</td></tr><tr><td>Tested on</td><td>New</td><td>New</td><td>New</td><td>Base-c</td><td>Base-f</td><td>All</td><td>New</td><td>Base-c</td><td>Base-f</td><td>All</td></tr><tr><td>TFA [32] (gFSOD)</td><td>18.35</td><td>-</td><td>16.90</td><td>24.30</td><td>27.90</td><td>24.40</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Mask-RCNN [13] (gFSOD &amp; gFSIS)</td><td>16.50</td><td>18.31</td><td>12.11</td><td>24.54</td><td>28.59</td><td>24.04</td><td>12.75</td><td>25.35</td><td>27.75</td><td>24.36</td></tr><tr><td>Mask+Sigmoid (iFSOD &amp; iFSIS)</td><td>16.93</td><td>19.18</td><td>15.02</td><td>23.33</td><td>27.23</td><td>23.55</td><td>17.39</td><td>25.26</td><td>27.05</td><td>24.75</td></tr><tr><td>iFS-RCNN (iFSOD &amp; iFSIS)</td><td>20.76</td><td>21.06</td><td>18.38</td><td>26.11</td><td>30.12</td><td>26.46</td><td>18.26</td><td>26.29</td><td>28.46</td><td>25.90</td></tr></table>
+
+Table 4. Object detection and instance segmentation results with AP metric for FSOD, FSIS, iFSOD, and iFSIS tasks on LVIS. The best results are in bold. TFA and Mask-RCNN are trained with more supervision than Mask+Sigmoid and iFS-RCNN as described in Tab. 1. Base-c and Base-f indicate the common classes (≥ 100 images) and frequent classes (10-100 images) among the base classes.
+
+the best performance, about +2.5 performance gain over the strong baseline Mask+Sigmoid and +2 over Mask-RCNN. In the following experiments, we choose Mask+Sigmoid and iFS-RCNN to compare with prior work.
+
+# 4.3. Comparison with Prior Work on COCO
+
+Tab. 3 compares our results on COCO in iFSOD and iFSIS with the strong baseline Mask+Sigmoid, approaches designed for FSOD and FSIS (TFA [32], FSDetView [35]), and approaches designed for iFSOD and iFSIS (GIFSOD [21], ONCE [27], LEAST [20], MTFA, and iMTFA [11]).
+
+TFA (a fine-tuning-based approach) and FSDetView (an episodic-training-based approach) are adapted to the iFSOD setting as follows. TFA is first trained on the base classes, resulting in model 1. Then, TFA is fine-tuned on the new classes, resulting in model 2. Finally, we run the models 1 and 2 on the same test images and select the top 100 predictions, as in the COCO evaluation protocol. For FSDetView, after pre-training the model on training examples of the base classes, we use the pretrained model to estimate the prototype for each base class, and then run the pretrained model on the new classes to extract their prototypes. FSDetView uses the prototypes of both new and base classes for object detection in test images. As TFA and FSDetView were not originally designed for iFSOD, their aforementioned adaptation from FSOD to iFSOD has two limitations: (1) Storage and running of two distinct models/prototypes for predicting the base and new classes; (2) The two distinct models may independently each yield a high score on a base and new class in the test image, which is then hard to resolve.
+
+Among the approaches aimed at iFSOD, ONCE is based on YOLO [28], and LEAST and GIFSOD are based on Faster-RCNN [29]. iMTFA is a variant of Mask+Cosine, where the weight  $\mu_{n}$  is set to the box feature  $f$  extracted from an example of the new class.
+
+For the comparison in Tab. 3, results are averaged over 10 runs with different sets of few-shot examples. From this table, for the iFSOD setting, iFS-RCNN outperforms state of the art (SOTA) approaches slightly on new classes  $(+0.05)$  while significantly on the base classes  $(+12)$  for  $K = 10$ . For the iFSIS setting, iFS-RCNN outperforms the SOTA iMTFA with significant margins. Specifically, for  $K = 10$ , our performance gains are about  $+6$  for the new
+
+classes and +16 for the base classes on segmentation. Our performance gains are very large, especially for the challenging COCO dataset.
+
+# 4.4. Results on LVIS
+
+Tab. 4 reports our results on LVIS with  $K \leq 10$  on iFSIS and FSIS. Although Mask+Sigmoid uses less supervision than Mask-RCNN and TFA, they give comparable results. iFS-RCNN significantly outperforms Mask-RCNN with the gains of +3 on FSIS and +6 on iFSIS on the new classes. These results demonstrate the effectiveness of our iFS-RCNN on the more challenging dataset LVIS.
+
+# 4.5. Qualitative Evaluation
+
+Fig. 3 illustrates some of our results. The top two rows show success cases while the bottom row shows failure cases. For the failure cases from left to right: the train is misclassified as bus due to very similar appearance, the bird detection has a large bounding box due to the occlusion, the human leg is segmented as a part of the motorbike due to similar appearance, and the small boats far behind are not detected. Fig. 4 shows our box refinement results. As can be seen, high uncertainty about the box prediction gives a considerable box refinement.
+
+# 5. Conclusion and Discussion
+
+We have specified iFS-RCNN to address N-way K-shot incremental few-shot object detection (iFSOD) and instance segmentation (iFSIS). iFS-RCNN leverages Mask-RCNN, but modifies the standard softmax classifier and bounding-box prediction head. The softmax classifier has been replaced with the sigmoid classifier for alleviating a statistical imbalance of the base and new classes. The sigmoid classifier has been further improved via Bayesian learning to robustly estimate a distribution of the classifier head's weights on a few examples of the new classes. For this Bayesian learning, we have proposed an analytical approximation using the probit function. Also, the standard box predictor of Mask-RCNN has been extended to explicitly predict uncertainty of box prediction, and use the estimated uncertainty as input to a bounding-box refinement module. iFS-RCNN significantly outperforms iMTFA – the state of
+
+![](images/22e770d1f503147f7a48f03afaa9b6371b49b926d3cfa0532bf6775afd0c3170.jpg)
+
+![](images/4819c6bf50fa96a72deab1fbe2c62ede079b9c180848b55db20fb76b3d343a0c.jpg)
+
+![](images/8e3732362065f5de74e60e9c76bf43ad638ebd4661bbd7a769e594d634469560.jpg)
+
+![](images/7c869f6dae5dfeba21784b65cfba62103da3ce27e4c5cff7750d638977bcb24c.jpg)
+
+![](images/039c20ea7b18f8df53fe5b746e264456c150e4331355280cf5c5edbf4e79a491.jpg)
+
+![](images/40e24b45f4b6c8d19bd11fa73317566b77c32c585ef952a2de0d772dc137a1ad.jpg)
+
+![](images/2739a54fead598d8d07efcce255d7155a2a005a22809ff8def623094a136ee99.jpg)
+
+![](images/a5f8a48d02c38c4f746d733426ed6f0a2c3d7e14a660251d3b324a192ef5018a.jpg)
+
+![](images/14f94e9d2b09c002fab05baffdb7788fa8c81c8e669b5386a9bd86df2f64ca4f.jpg)  
+Figure 3. Representative results of iFS-RCNN on COCO-new with  $K = 30$ . The top two rows show success cases while the bottom row shows failure cases. In the last row, from left to right: the train is detected as bus, bonding-box is too large for the bird, the human leg is segmented as a part of the motorbike, the small boats far behind are not detected.
+
+![](images/c18972a0b751cf70a2ba71b56a2261ef0c36441cad140a30398b34f0efcafd35.jpg)
+
+![](images/996e61372dac99ddf5f608d9ee69835cd0dc1e1113a25a552a73dbffa49ab709.jpg)
+
+![](images/1d6a6eac26cc816e6426ef1e14626cf6b1deee5802c4fd687f3af97250575b6e.jpg)
+
+![](images/e7956e4734498575ac01cc1f958dce8fdfc7b0e502b7c43150bba6b6bd8b1a8a.jpg)
+
+![](images/e4a1c49ba67fd690caf558ede811459a57096f14080ad818542cd395d6606d24.jpg)
+
+![](images/01d237d0e366e181f573aa81689845cb40a959f9d2a7c90247b3fc8a194f6967.jpg)  
+Figure 4. Bounding-box refinement with our uncertainty-guided box predictor on COCO-new with  $K = 1$ . For each pair, left: initial box, right: refined box. The box labels show left (L), top (T), right (R), and bottom (B) side uncertainties; red text indicates high uncertainty ( $\geq 10$ ) that lead to large refinements. Yellow arrows indicate the refined direction of each box side.
+
+![](images/b0dda1ca990fe7d63a85ef80be031d40667fa30777762e790a35c99847481a40.jpg)
+
+![](images/f4b9c198a0d3e4630511b2e4700000b03da16c572268ab416d6735da3f8aa685.jpg)
+
+the art in iFSIS - with very large performance gains on the COCO dataset,  $+6$  on the new classes and  $+16$  on the base classes with 10 training examples. Moreover, we are
+
+the first to report the results on the LVIS dataset for the iFSOD, FSIS, and iFSIS problems, where we outperform strong baselines.
+
+As for potential limitations, our contribution 1, in general, could be applied to any object detector, including RPN-based ones like Faster-RCNN [29] and Mask-RCNN [13]), point-based ones like FCOS [31] and Center-Net [39]), and transformer-based ones like DETR [5]. However, in our experiments, we have succeeded in applying it only to the RPN-based detector. This might be due to the pre-selected balanced training examples of background and foreground classes in the second stage of Faster-RCNN, which is absent in the other two detector frameworks. Also, while our estimation of uncertainty of box prediction and our new uncertainty-weighted box loss have enabled significant performance gains, our contribution 2 lacks a theoretical underpinning as to why our formulation outperforms a Gaussian-based uncertainty estimation.
+
+As any system for object detection and segmentation, ours could be weaponized and misused for malicious violations of privacy, as well as used for efficiently discovering and fighting against such misuses due to its few-shot learning capabilities.
+
+Acknowledgement. This work was supported in part by DARPA MCS Award N66001-19-2-4035.
+
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+proposal networks. In Advances in neural information processing systems, pages 91-99, 2015.  
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+# iPLAN: Interactive and Procedural Layout Planning
+
+Feixiang He  
+University of Leeds, UK  
+scfh@leeds.ac.uk
+
+Yanlong Huang  
+University of Leeds, UK  
+y.l.huang@leeds.ac.uk
+
+He Wang *  
+University of Leeds, UK  
+h.e.wang@leeds.ac.uk
+
+# Abstract
+
+Layout design is ubiquitous in many applications, e.g. architecture/urban planning, etc, which involves a lengthy iterative design process. Recently, deep learning has been leveraged to automatically generate layouts via image generation, showing a huge potential to free designers from laborious routines. While automatic generation can greatly boost productivity, designer input is undoubtedly crucial. An ideal AI-aided design tool should automate repetitive routines, and meanwhile accept human guidance and provide smart/proactive suggestions. However, the capability of involving humans into the loop has been largely ignored in existing methods which are mostly end-to-end approaches. To this end, we propose a new human-in-the-loop generative model, iPLAN, which is capable of automatically generating layouts, but also interacting with designers throughout the whole procedure, enabling humans and AI to co-evolve a sketchy idea gradually into the final design. iPLAN is evaluated on diverse datasets and compared with existing methods. The results show that iPLAN has high fidelity in producing similar layouts to those from human designers, great flexibility in accepting designer inputs and providing design suggestions accordingly, and strong generalizability when facing unseen design tasks and limited training data.
+
+# 1. Introduction
+
+Layout generation has recently spiked research interests in computer vision/graphics, aiming to automate the design process and boost the productivity. The traditional design process follows a diagram of iteratively adjusting/finalizing details from coarse to fine and global to local, which imposes repetitive and laborious routines on designers. Very recently, it has been shown that automatic image generation of such designs (with minimal human input) is possible through learning from data [12, 26, 27, 36]. This new line of research combines deep learning with design and has
+
+demonstrated a new avenue for AI-aided design.
+
+To achieve full automation, current research tends to learn from existing designs in an end-to-end fashion, and then to generate new ones with qualitative similarity and sufficient diversity. Taking floorplan as an example, automated generation can be based on simple human input, such as the boundary of the floor space [36], the relations among rooms [26, 27], or both [12]. While fully automated generation is important, design is in nature a procedural process, which involves alternations between repetitive routines and creative thinking at multiple intermediate stages [29]. Therefore, an ideal AI-aided system should automate the routine part while allowing the designer to impart creativity. This requires the system to be able to interact with the designer, in the sense that it should accept designer's guidance, then actively suggest possible solutions accordingly, completing a feed-back loop. So far, the human-in-the-loop element is largely missing, which prevents a closer integration of AI and existing design practice.
+
+Designing such an AI model faces several intrinsic challenges. In practice, learning how to interact with the designer requires a full observation of the decisions made at every intermediate stage. However, existing datasets, such as RPLAN [36] and LIFULL [22], usually only include the final designs, without the stage-to-stage design process. One potential solution to overcome this issue is to reverse-engineer intermediate stages from final designs, which however leads to another difficulty: the order of stages depends on the specific task/goal and could vary dramatically even for the same final design. Further, the order uncertainty is exacerbated by the strong personal styles and preferences of designers. Thus, how to design an AI system that can account for the above factors is a key research question, which is under-explored to date.
+
+In this paper, we propose a new human-in-the-loop generative model for layout generation, which is referred to as interactive planning (iPLAN). Unlike previous work, iPLAN is equipped with a user-friendly interaction mechanism, which is achieved by letting the AI model learn the multi-stage design process, aiming to accommodate freeform user interactions and propose design suggestions at
+
+every stage. This allows designer inputs at different stages across a wide range of levels of detail, while offering the capability of fully automated generation. To address the challenge of missing procedural design data, we reverse-engineer the final design to obtain the stage-to-stage process, based on principles that are widely adopted by professional designers [29]. This enables us to design a Markov chain model to capture the full design procedure. Since there is more than one way to reverse-engineer the final designs (i.e., the stage order can vary), our model is designed with the capacity of accepting inputs with an arbitrary order, and consequently can learn the style variations implicitly from the data.
+
+While iPLAN is general, we focus on floorplan design in this paper. iPLAN has been validated on two large-scale benchmark datasets, i.e., RPLAN [36] and LIFULL [22], under diverse scenarios. The experiments show that our model is versatile in accepting designer inputs at various levels of detail, from minimal input and automatic generation, to stage-to-stage human guidance and interactive design. By learning from designs augmented by reverse-engineered processes, our model exhibits high fidelity in generating new designs with close style similarity and sufficient diversity. Finally, our model is highly flexible and generalizable when trained on varying amounts of data and facing unseen spaces and design requirements that are categorically different from the training data.
+
+Contributions: (i) We propose a novel human-in-the-loop generative model iPLAN which respects design principles and mimics the design styles of professional designers implicitly. (ii) We demonstrate a successful fine-grained stage-to-stage generative model for floorplan, as opposed to existing end-to-end approaches. (iii) We show a variety of design scenarios, including fully automated generation, interactive planning with user instructions, and generalization for unseen tasks; (iv) We conduct extensive evaluations on diverse benchmark datasets and demonstrate that iPLAN outperforms the state of the art under multiple metrics.
+
+# 2. Related Work
+
+Layout generation has been an active research area in computer vision, e.g., indoor scene synthesis [6, 24, 37, 39, 40] and floorplan generation [2, 9, 12, 25, 28, 36], image composition [1, 15], etc. Existing approaches can be generally grouped into two categories: handcrafted rule-based methods and data-driven methods. We mainly review the latter as they are closely related to our research.
+
+Indoor scene synthesis. The synthesis of indoor scenes typically involves the placement of furniture models from an existing database into a given room. Convolutional neural networks can be trained to iteratively insert one object at a time into a room for indoor scene generation [31, 35]. High-level scene semantics can also be employed,
+
+e.g., scene graphs, as a prior for a more controlled generation [34]. The biggest difference between indoor scene synthesis and floorplan generation is their requirements on the space partitioning. While indoor scene synthesis places objects in a room where the room itself does not need to be divided, floorplan normally requires explicit space division for different functionalities.
+
+Image composition from scene graphs. Another related field is image composition from scene graphs, where the task is to derive the scene from a layout graph that describes the locations and features of the objects. Such generation can be achieved by Generative Adversarial Networks (GANs) based on graph convolution [15]. Further improvements can be obtained by separating the layout from the appearance of the objects [1]. For more controllability, Li et al. [21] synthesize images from a scene graph and the corresponding image crops. In contrast to the floorplan, the challenge in image composition is how to compose different objects into an image rather than partitioning the space.
+
+Floorplan generation Floorplan generation can be formulated as an image synthesis problem which is one active research area in computer vision. Due to the surge of deep learning, the most promising approaches are GANs [3,7,16-19,38]. Image-based GANs have been proven effective in floorplan generation [4,13,14,30,41-43]. Graph-based GANs can also produce floorplans by only taking spacial constraints, such as room connections, room types, in the form of graph [26,27]. However, all these methods are end-to-end approaches, and therefore provide limited interactivity to the designer.
+
+More recently, some human-in-the-loop approaches are proposed. Wu et al. [36] propose a two-phase approach to produce floorplans of residential buildings. The model successively predicts locations of rooms and walls given a building boundary, and converts the predicted layout into a vector graphics format. Graph2Plan [12] combines the topology information in the form of graphs with spatial constraints, to instantiate rooms accordingly. These methods enable human interactions at certain stages, e.g., modifying room locations and retrieving the graph. Different from the existing methods, we propose a fine-grained generative model which enables interactions with the designer at different levels, from providing high-level design requirements to low-level instructions at a step.
+
+# 3. Methodology
+
+The overview of iPLAN is presented in Fig. 1. Without loss of generality, our model takes the space boundary as input, and decomposes the design procedure into: acquiring room types, locating rooms, and finalizing room partitions. Such workflow aims to mimic human designers [29] and accept designer inputs at any stage. The workflow is modeled as a joint probability distribution of all the afore
+
+![](images/37584f6b8e345164c61e064344d6ab5d28b4e6686977a205c1b078390d18edfb.jpg)  
+Figure 1. Overview of our framework iPLAN. Room types are predicted at a time, while room locations and partitions are predicted iteratively.
+
+mentioned factors, which is then factorized into stages and later formulated as a Markov chain (Sec. 3.1). Next, each factorized distribution provides a flexible entry point to incorporate user input or can be used for automatic generation (Sec. 3.2–Sec. 3.3).
+
+# 3.1. Problem Formulation
+
+A dataset of  $H$  layouts is denoted by  $\pmb{D} = \{\pmb{D}_i\}_{i=1}^H$  with the  $i$ -th layout  $\pmb{D}_i = (\pmb{B}_i, \pmb{R}_i, \pmb{T}_i, N_i, \pmb{C}_i)$ .  $\pmb{B}_i \in \mathbb{R}^{128 \times 128}$  is the boundary,  $N_i$  denotes the total number of rooms. We use  $j$  to index a specific room.  $\pmb{R}_i = \{\pmb{r}_{i,j}\}_{j=1}^{N_i}$  denotes the room regions with  $\pmb{r}_{i,j} \in \mathbb{R}^4$  indicating the top-left and bottom-right corners of the bounding box of the room.  $\pmb{T}_i = \{t_{i,j}\}_{j=1}^{N_i}$  is a set of room types  $t_{i,j} \in \mathbb{Z}^+$ .  $\pmb{C}_i = \{\pmb{c}_{i,j}\}_{j=1}^{N_i}$  is a set of room centers  $c_{i,j} \in \mathbb{R}^2$ . Given  $\pmb{D}$ , we aim to design a generative model for  $\mathcal{P}(\pmb{D}) = \prod_{i=1}^H \mathcal{P}(\pmb{D}_i)$ .
+
+Similarly to existing methods, we also formulate floor-plan design as an image generation problem, but with our focus on proposing a fine-grained generative model to enable human-in-the-loop interaction, through decomposing  $\mathcal{P}(D)$  appropriately. From a mathematical perspective, there are many ways to decompose  $\mathcal{P}(D)$ . In order to allow human inputs across various levels of detail, we rely on design principles and widely adopted practice [29] to mimic the human design workflow, and naturally divide a floorplan design procedure into several stages. First, the desired number of rooms and their types are determined. Next, locations of areas with specific functionality (e.g., living rooms, bedrooms) are roughly estimated. Finally, room partitioning is conducted to finalize the design. This design diagram serves as a strong inductive bias in our model, following which  $\mathcal{P}(D)$  is decomposed into:
+
+$$
+\begin{array}{l} \mathcal {P} (\boldsymbol {D}) = \prod_ {i = 1} ^ {H} \mathcal {P} (\boldsymbol {D} _ {i}) = \prod_ {i = 1} ^ {H} \mathcal {P} (\boldsymbol {R} _ {i}, \boldsymbol {C} _ {i}, \boldsymbol {T} _ {i}, N _ {i}, \boldsymbol {B} _ {i}) \\ = \prod_ {i = 1} ^ {H} \mathcal {P} \left(\boldsymbol {R} _ {i} \mid \boldsymbol {C} _ {i}, \boldsymbol {T} _ {i}, N _ {i}, \boldsymbol {B} _ {i}\right) \mathcal {P} \left(\boldsymbol {C} _ {i} \mid \boldsymbol {T} _ {i}, N _ {i}, \boldsymbol {B} _ {i}\right) \tag {1} \\ \mathcal {P} (\boldsymbol {T} _ {i}, N _ {i} | \boldsymbol {B} _ {i}) \mathcal {P} (\boldsymbol {B} _ {i}) \\ \end{array}
+$$
+
+where  $\mathcal{P}(\pmb {B}_i)$  accounts for the boundary known a priori;  $\mathcal{P}(\pmb {T}_i,N_i|\pmb {B}_i)$  is to infer the desired number of rooms and
+
+room types;  $\mathcal{P}(\pmb {C}_i|\pmb {T}_i,N_i,\pmb {B}_i)$  and  $\mathcal{P}(\pmb {R}_i|\pmb {C}_i,\pmb {T}_i,N_i,\pmb {B}_i)$  respectively correspond to the coarse and fine designs of the layout, where the former estimates the room locations while the latter predicts the exact partitions. A visualization of Eq. (1) is provided in Figure 1, where the second, third and fourth blocks correspond to  $\mathcal{P}(\pmb {T}_i,N_i|\pmb {B}_i)$ $\mathcal{P}(\pmb {C}_i|\pmb {T}_i,N_i,\pmb {B}_i)$  and  $\mathcal{P}(\pmb {R}_i|\pmb {C}_i,\pmb {T}_i,N_i,\pmb {B}_i)$ , respectively.
+
+Inspired by indoor scene synthesis where objects are placed iteratively [31, 35], we assume rooms are designed one by one. Formally, we model  $P(\pmb{R}_i | \pmb{C}_i, \pmb{T}_i, N_i, \pmb{B}_i)$  and  $\mathcal{P}(\pmb{C}_i | \pmb{T}_i, N_i, \pmb{B}_i)$  as Markov chains, i.e., designs are conducted in a step-wise manner and early decisions will affect later ones, which allows a designer to focus on one room at a time and give guidance at any step:
+
+$$
+\begin{array}{l} \mathcal {P} \left(\boldsymbol {C} _ {i} \mid \boldsymbol {T} _ {i}, N _ {i}, \boldsymbol {B} _ {i}\right) = \prod_ {j = 1} ^ {N _ {i}} \mathcal {P} \left(\boldsymbol {c} _ {i, j} \mid \boldsymbol {c} _ {i, <   j}, t _ {i, j}, N _ {i}, \boldsymbol {B} _ {i}\right) (2) \\ \mathcal {P} \left(\boldsymbol {R} _ {i} \mid \boldsymbol {C} _ {i}, \boldsymbol {T} _ {i}, N _ {i}, \boldsymbol {B} _ {i}\right) = \prod_ {j = 1} ^ {N _ {i}} \mathcal {P} \left(\boldsymbol {r} _ {i, j} \mid \boldsymbol {r} _ {i, <   j}, \boldsymbol {c} _ {i, j}, t _ {i, j}, N _ {i}, \boldsymbol {B} _ {i}\right) (3) \\ \end{array}
+$$
+
+where  $\pmb{r}_{i, < j} = \{\pmb{r}_{i,1}, \dots, \pmb{r}_{i,j-1}\}$  and  $\pmb{c}_{i, < j} = \{\pmb{c}_{i,1}, \dots, \pmb{c}_{i,j-1}\}$  denote the set of allocated room partitioning and centers before the  $j$ -th room, respectively.
+
+Connections to existing research. Eq. (1)-(3) is a generalization of existing methods and is more fine-grained. Graph2Plan [12] simultaneously determines  $C_i$ ,  $T_i$  and  $N_i$  given  $B_i$ , and then  $\mathcal{P}(R_i|C_i, T_i, N_i, B_i)$  is predicted. RPLAN [36] estimates  $\mathcal{P}(C_i, T_i, N_i|B_i)$  via predicting  $\mathcal{P}(c_{i,j}, t_{i,j}|c_{i,j+1}, t_{i,j+1}, B_i)$  consecutively, and then indirectly estimates room areas  $R_i$  by locating walls. In contrast, we further decompose  $\mathcal{P}(C_i, T_i, N_i|B_i)$  into  $\mathcal{P}(T_i, N_i|B_i)$  and  $\mathcal{P}(C_i|T, N_i, B_i)$ , and further decompose the latter using Eq. (2). Moreover,  $\mathcal{P}(R_i|C_i, T_i, N_i, B_i)$  is also decomposed into multiple steps by Eq. (3). Our decompositions bring a more fine-grained procedural generative model that allows for user interactions at arbitrary steps. This enables more flexible and closer human-AI interactions. For the sake of brevity, we omit the subscripts  $i$  of  $T_i, N_i, B_i$  and  $C_i$  in the following sections.
+
+# 3.2. The Number and Types of Rooms
+
+$\pmb{T}$  and  $N$  are normally given beforehand. However, given a specific  $\pmb{B}$ , the design might not be unique, e.g., the same space can be designed as a 2-bed or 3-bed flat. In other words, there exists a distribution of possible designs. Thus, we propose to learn their distributions  $\mathcal{P}(\pmb{T}, N|\pmb{B})$  from real designs by professional designers to enable automatic exploration.  $\{\pmb{T}, N\}$  can be replaced by a random variable  $\pmb{Q} = \{q_k\}_{k=1}^K$ , where  $K$  denotes the number of room types in  $\pmb{D}$  and  $q_k$  corresponds to the number of rooms
+
+![](images/4961a483f9ae1fb89b21844359426462000af9fb8ae6b6d720d0a8de611c5f53.jpg)  
+Figure 2. Prediction of a room center at one step
+
+under the  $k$ -th type. So, we model  $P(\mathbf{Q}|\mathbf{B})$  instead.
+
+We propose a boundary-conditioned Variational Autoencoder (BCVAE) based on VAE [32] where  $B$  serves as a condition. The model consists of an embedding module  $\mathcal{F}_{ed}$ , an encoder  $\mathcal{F}_{en}$  and a decoder  $\mathcal{F}_{de}$ . By feeding  $\mathcal{F}_{ed}$  with  $B$ , it outputs an embedded vector  $\boldsymbol{\gamma} \in \mathbb{R}^{128}$ . Also,  $\{\pmb{\mu}, \pmb{\Sigma}\} = \mathcal{F}_{en}(\pmb{Q}, \boldsymbol{\gamma})$  where  $\pmb{\mu} \in \mathbb{R}^{32}$  and  $\pmb{\Sigma} \in \mathbb{R}^{32 \times 32}$  are the mean and covariance of a Gaussian distribution. Furthermore, a latent variable  $z$  is sampled from  $\mathcal{N}(\pmb{\mu}, \pmb{\Sigma})$  using the reparameterization trick [20]. Given  $z$  and  $\boldsymbol{\gamma}$ , we reconstruct  $\pmb{Q}$  by  $\hat{\pmb{Q}} = \mathcal{F}_{de}(z, \boldsymbol{\gamma})$ . We employ a standard VAE loss for training. The detailed network architecture and the training loss are discussed in the supplementary material.
+
+During inference, for a new boundary  $\pmb{B}$ , we sample  $\pmb{z}$  from  $\mathcal{N}(\mathbf{0},\pmb{I})$  and predict  $\hat{\pmb{Q}}$  by  $\hat{\pmb{Q}} = \mathcal{F}_{de}(\pmb{z},\mathcal{F}_{ed}(\pmb{B}))$  which is then used to recover  $\hat{T}$  and  $\hat{N}$ .
+
+# 3.3. Locating Rooms
+
+Locating room regions  $\mathcal{P}(\pmb {C}|\pmb {T},\pmb {N},\pmb {B})$  plays an essential role in the design process. Similarly to [36], we model room region prediction as a step-wise classification task. At each step, given a multi-channel image representation of the current design state and the next desired room type  $t_j$ , we predict the center of the next room. The multi-channel image encodes the boundary  $\pmb{B}$  and all previously predicted room centers  $c_{< j}$ . The image consists of  $K + 4$  binary channels, three of which label  $\pmb{B}$ , i.e., the boundary, the front door and the interior area pixels.  $K$  channels represent the predicted room centers, with each channel corresponding to a room type. For each room, its center is represented by a  $9\times 9$  square of pixels with value 1. We also use a channel to summarize the centers of all predicted rooms. Regarding the desired room type  $t_j$ , we convert it into a one-hot vector.
+
+We feed the multi-channel representation to a Resnet-18 [10] network and the one-hot vector of the desired room type to an embedding network to extract features. The embedding network has three fully-connected layers, followed by four convolution blocks, a Batch Normalization Layer and a LeakyReLU Layer. The extracted features are concatenated and subsequently fed to a decoder module that contains 4 atrous spatial pyramid pooling (ASPP) [5], a Convolutional block and a Deconvolutional block. The output is  $\mathbf{O} \in \mathbb{R}^{(K + 3)\times 128\times 128}$ , giving a probability vector of
+
+![](images/a20d9f50019a6de6656860b6f67d301c41321dbf907c9180ab1e41774c5eaf85.jpg)  
+Figure 3. Procedural room shape generation
+
+all pixels. For each pixel, we predict  $K + 3$  labels, i.e.,  $K$  room types and EXISTING, FREE, and OUTSIDE, where EXISTING, FREE and OUTSIDE indicate whether a pixel belongs to an existing room, a free space (in the interior of the boundary) and the exterior, respectively. The whole model is shown in Figure 2.
+
+For training, we predict one room at a time, by decomposing the final design into a series of design states with one room added at each stage. However, there are multiple possible sequences for a given final design and the ground-truth step-to-step decisions are unavailable. Therefore, we employ a stochastic training process to learn all possible sequences by randomly removing rooms from a design. We propose a pixel-wise cross-entropy loss:
+
+$$
+\mathcal {L} = \sum_ {h = 1} ^ {1 2 8} \sum_ {w = 1} ^ {1 2 8} - \omega_ {y} \log \frac {\exp \left(O _ {y , h , w}\right)}{\sum_ {k = 1} ^ {K + 3} \exp \left(O _ {k , h , w}\right)}, \tag {4}
+$$
+
+where  $y$  is the ground-truth class index for the pixel located at  $(h, w)$ .  $\omega_y$  represents the weight of  $y$ -th label, we set it to be 2 for  $K$  room types and 1.25 for the other three labels.
+
+During the inference phase, for a sequence of room types  $\pmb{T} = \{t_1,t_2,\dots ,t_N\}$  we first predict the room center for  $t_1$ , i.e.,  $\mathcal{P}(c_{1}|t_{1},N,B)$ , then we update the current design state using this predicted center  $\hat{c}_1$  and continue to predict the next room center for  $t_2$ , i.e.,  $\mathcal{P}(c_2|\hat{c}_1,t_2,N,B)$ . This procedure is repeated until all centers  $\hat{C}$  are determined. Note that when the order of elements in  $T$  varies, its corresponding layout can also change, thus providing design diversity. Also, at any intermediate step, user input (e.g., adjust a room center) can be incorporated into our model before it predicts the next room center.
+
+# 3.4. Predicting Room Partitioning
+
+After predicting the room centers  $\hat{C}$ , we are ready for detailed room partitioning  $\mathcal{P}(\boldsymbol{R}|\hat{\boldsymbol{C}},\hat{\boldsymbol{T}},\hat{\boldsymbol{N}},\boldsymbol{B})$ , where we need to consider the room size and the room shape. The room size is directly related to the specific functionality of the room, e.g., living rooms are normally larger than bathrooms for social interactions. The shape is affected by the functionality too but also strongly affected by the boundary geometry and the inner walls. Further, there is usually no 'optimal' solution but a distribution of near-optimal solutions
+
+when looking at the final designs. Since the distribution can be arbitrary, we propose a new Generative Adversarial Network (GAN) to model the step-wise prediction of room partitioning. The general model is shown in Figure 3. Instead of directly predicting the shape of each room, we first predict its bounding box. This allows the bounding box predicted at a later stage to grab the space that is already allocated. We find this is a straightforward yet effective way to generate non-box rooms.
+
+Generator: The generator begins with a bounding-box regressor  $\mathcal{F}_b$  that outputs the top-left and bottom-right corners of the box.  $\mathcal{F}_b$  consists of six convolutional units and two fully connected layers, each unit with a Convolution layer, a Layer-Normalization layer and a ReLU layer.  $\mathcal{F}_b$  enables the designer to interact with the system easily, e.g., relocating a room, modifying a room bounding box. However, only predicting the corners of bounding boxes is not convenient. We want to label all the pixels within the box so that we can work in the image space consistently. So we design a tailor-made masking module  $\mathcal{F}_m$  to map each room bounding box into a room mask image.  $\mathcal{F}_m$  contains six convolution layers with  $3\times 3$  kernel, followed by a Sigmoid activation function.
+
+For a given sequence of room centers and types  $\{(c_1,t_1),(c_2,t_2),\ldots ,(c_N,t_N)\}$  , we use  $\mathcal{F}_b$  to predict the coordinates  $\hat{\pmb{r}}_1$  of the bounding box of the first room specified by  $c_{1}$  and  $t_1$  , where  $\hat{r}_1 = \mathcal{F}_b(S_0,c_1,t_1)$  with  $S_0 =$ $\pmb{B}$  . Furthermore, we obtain the room mask using  $\hat{M}_{1} =$ $\mathcal{F}_m(\hat{r}_1)\in \mathbb{R}^{128\times 128}$  . The output of the generator at the first step is computed by
+
+$$
+\hat {\boldsymbol {S}} _ {1} = \boldsymbol {S} _ {0} \times (\mathbf {1} - \hat {\boldsymbol {M}} _ {1}) + \hat {\boldsymbol {M}} _ {1} \times t _ {1}, \tag {5}
+$$
+
+where  $\mathbf{1} \in \mathbb{R}^{128 \times 128}$  represents a matrix with all elements being 1. During the prediction in the second step, the generator takes as input  $\hat{\pmb{S}}_1$ ,  $\pmb{c}_2$ ,  $t_2$  and outputs  $\hat{\pmb{S}}_2 = \hat{\pmb{S}}_1 \times (\mathbf{1} - \hat{\pmb{M}}_2) + \hat{\pmb{M}}_2 \times t_2$  with  $\hat{\pmb{M}}_2 = \mathcal{F}_m(\hat{\pmb{r}}_2) = \mathcal{F}_m\big(\mathcal{F}_b(\hat{\pmb{S}}_1,\pmb{c}_2,t_2)\big)$ . The same procedure is iterated (for totally  $\hat{N}$  steps) until the entire design is accomplished.
+
+Discriminator: The discriminator retains the similar backbone as  $\mathcal{F}_b$  in the generator, except that the last two convolution layers are dropped. The discriminator is employed to distinguish whether a sequence of design states comes from the generator or the ground truth. Specifically, a sequence of predicted design states  $\hat{\pmb{S}} = \{\hat{\pmb{S}}_1,\hat{\pmb{S}}_1,\dots ,\hat{\pmb{S}}_{\hat{N}}\}$  should be recognized as 'FALSE', while a sequence from the ground truth with the same order of room types (i.e.,  $\{t_1,t_2,\ldots ,t_{\hat{N}}\}$ ) should be predicted as 'TRUE'.
+
+For training, we define a loss on top of WGAN-GP [8] loss, i.e.,
+
+$$
+\begin{array}{l} \mathcal {L} = \mathbb {E} _ {\hat {\boldsymbol {S}} \sim \mathbb {P} _ {g}} [ D (\hat {\boldsymbol {S}}) ] - \mathbb {E} _ {\boldsymbol {S} \sim \mathbb {P} _ {r}} [ D (\boldsymbol {S}) ] \tag {6} \\ + \lambda_ {1} \mathbb {E} _ {\bar {S} \sim \mathbb {P} _ {\bar {S}}} (| | \bigtriangledown_ {\bar {S}} D (\bar {S}) | | _ {2} - 1) ^ {2} + \lambda_ {2} \mathcal {L} _ {s}, \\ \end{array}
+$$
+
+where  $\mathbb{P}_g$  and  $\mathbb{P}_r$  represent the generated and real data distribution, respectively.  $\bar{\boldsymbol{S}}\sim \mathbb{P}_{\bar{\boldsymbol{S}}}$  denotes a random interpolation between  $\hat{\boldsymbol{S}}$  and  $S$ , and is used for a gradient penalty with  $\lambda_1 = 10$ .  $||\cdot ||_2$  is the  $\ell_2$ -norm. The last term is introduced to explicitly regularize the bounding boxes with  $\lambda_{2} = 100$ :
+
+$$
+\mathcal {L} _ {s} = \sum_ {j = 1} ^ {N} l _ {j}, \quad l _ {j} = \left\{ \begin{array}{c c} 0. 5 \left(\boldsymbol {r} _ {j} - \hat {\boldsymbol {r}} _ {j}\right), & \text {i f} \| \boldsymbol {r} _ {j} - \hat {\boldsymbol {r}} _ {j} \| _ {1} <   1 \\ \| \boldsymbol {r} _ {j} - \hat {\boldsymbol {r}} _ {j} \| _ {1} - 0. 5, & \text {o t h e r w i s e} \end{array} \right. \tag {7}
+$$
+
+where  $\boldsymbol{r}_j$  is the ground truth and  $||\cdot||_1$  is the  $\ell_1$ -norm.
+
+At the inference stage, our procedural model predicts room areas in a step-wise manner, which hence allows user input at an arbitrary intermediate step, such as modifying room types, changing room centers or even the already predicted room areas. Gaps may exist between the predicted rooms. Thus, we employ a simple post-processing step as in [33] and detail it in the supplementary material.
+
+# 4. Experiments
+
+# 4.1. Datasets
+
+We conduct experiments on two commonly used datasets, RPLAN [36] and LIFULL [22]. RPLAN is collected from the real-world residential buildings in the Asian real estate market, which contains over 80k floorplans and 13 types of rooms<sup>1</sup>. All floorplans in RPLAN are axis-aligned and pre-processed to the same scale. The training-validation-test split of the dataset is  $70\% - 15\% - 15\%$  [12]. LIFULL HOME's dataset offers approximately five million apartment floorplans from the Japanese house market. The original dataset is given in the form of images, but a subset has been parsed into vector format by [23]. We select a subset with 4–10 rooms. This specific dataset consists of approximately 54k floorplans and 9 room types<sup>2</sup>, among which  $85\%$  (randomly sampled) of the data serves as the training set while the remaining for testing.
+
+# 4.2. Metrics
+
+Although each floorplan only contains one design (i.e., one boundary corresponds to one design), our model can predict the distribution of plausible designs, offering design diversity and alternative choices. To evaluate the predicted distribution, we employ the Fréchet Inception Distance (FID) [11] to calculate the distance between two distributions, which was also used in floorplan generation [26,27]. Built on FID, we introduce three metrics  $FID_{img}$ ,  $FID_{area}$  and  $FID_{type}$ : (i)  $FID_{img}$ : computed on rendered images to evaluate the distributional differences of
+
+<table><tr><td>Dataset</td><td>Method</td><td>FIDimg</td><td>FIDarea</td><td>FIDtype</td></tr><tr><td rowspan="7">RPLAN</td><td>HouseGAN++</td><td>51.33</td><td>1.36 × 108</td><td>0.038</td></tr><tr><td>Rplan</td><td>4.1</td><td>2.29 × 105</td><td>0.58</td></tr><tr><td>OurIII</td><td>1.22</td><td>3.13 × 104</td><td>0.05</td></tr><tr><td>OurII</td><td>0.72</td><td>1.09 × 104</td><td>0.03</td></tr><tr><td>Rplan*</td><td>0.11</td><td>8.62 × 103</td><td>6.4 × 10-3</td></tr><tr><td>Graph2Plan</td><td>0.62</td><td>8.82 × 103</td><td>2.70 × 10-4</td></tr><tr><td>OurI</td><td>0.16</td><td>4.89 × 102</td><td>4.44 × 10-6</td></tr><tr><td rowspan="6">LIFULL</td><td>Rplan</td><td>50.19</td><td>4.29 × 106</td><td>5.15</td></tr><tr><td>OurIII</td><td>37.35</td><td>9.81 × 105</td><td>2.52</td></tr><tr><td>OurII</td><td>32.65</td><td>7.75 × 105</td><td>2.14</td></tr><tr><td>Rplan*</td><td>1.43</td><td>5.62 × 105</td><td>0.064</td></tr><tr><td>Graph2Plan</td><td>0.64</td><td>2.87 × 103</td><td>2.63 × 10-5</td></tr><tr><td>OurI</td><td>0.38</td><td>2.07 × 103</td><td>3.59 × 10-6</td></tr></table>
+
+Table 1. FID-based metrics on RPLAN and LIFULL.
+
+the generated and true images. (ii)  $FID_{area}$ : to evaluate the distributional differences of room areas. Each layout is represented by a  $1 \times K$  vector  $area_i$ , with its  $k$ -th element  $area_{i,k}$  representing the average area of the  $k$ -th type of rooms in the  $i$ -th floorplan.  $K$  denotes the number of room types. (iii)  $FID_{type}$ : to calculate the distributional differences of room numbers against room types. Each layout is represented by a  $1 \times K$  vector  $type_i$ , whose element  $type_{i,k}$  represents the number of rooms under the  $k$ -th type in the  $i$ -th floorplan. It is worthwhile to mention that the above three metrics are not biased towards our method as they are not involved in the training process at all.
+
+# 4.3. Baselines
+
+We choose three methods as baselines. Rplan<sup>3</sup> [36] is an image-based method, which outputs a floorplan given a boundary. We further use a variant of Rplan, named Rplan*, which also takes as input the room types and centers. HouseGAN++ [27] is a graph-constrained approach, which treats rooms as nodes and requires node types and connectivity as input. It generates mask images for all nodes according to the graph, then blends them to form the layout. Graph2plan [12] requires a boundary and a room relation graph as input, where the graph includes information on room size, room center, room type and their connections. All floorplans are rendered in the same way as [12], including door and window placements.
+
+# 4.4. Quantitative Evaluations
+
+Ablation Study. iPLAN is capable of fully automated generation while allowing for user interactions at different stages. Based on the amount of information received from the designer, three variants are evaluated: (v1)  $\text{Our}_I$  takes as input the boundary, room types, and room centers; (v2)  $\text{Our}_{II}$  takes as input the boundary and room
+
+types; (v3)  $\text{Our}_{II}$  takes as input only the boundary. The variants show different levels of automation/user interactivity, demonstrating iPLAN's flexibility when the depth of human involvement varies, from little human input (e.g., iPLAN freely explores designs), to step-to-step guidance (e.g., where to put each room). The implementation details are in the supplementary document. Table 1 shows the results. iPLAN achieves good results on all three settings, among which  $\text{Our}_I$  achieves the best results and  $\text{Our}_{II}$  is better than  $\text{Our}_{III}$ . This is natural as more prior information helps for better prediction. Considering RPLAN and LIFULL are significantly different in terms of the overall shapes of boundaries and rooms, iPLAN can indeed handle different data distributions well.
+
+Comparisons on RPLAN iPLAN in general outperforms all baseline methods by large margins shown in Table 1. Looking closely, HouseGAN++ achieves worse results, except  $FID_{type}$  with a small margin, than other methods. The reason is that its layouts are not bounded by boundaries. By taking the boundary as input, both Rplan and Our $_{III}$  predict room centers and regions successively, but Our $_{III}$  outperforms Rplan on all three metrics. Furthermore, when both the boundary and room types are given, Our $_{II}$  outperforms the first three methods. Finally, Rplan*, Graph2Plan and Our $_{I}$  are given the full information (boundary, room types and centers). Our $_{I}$  further improves on all metrics, where Rplan* achieves a slightly better score on  $FID_{img}$ . We observe that the step-wise prediction in iPLAN is less likely to generate partitioning ambiguity (allocating a space to multiple rooms), while is observed in Rplan*. Also, learning conditional probabilities (as iPLAN does) as oppose to a joint probability of multiple factors from final designs (as Rplan* and Graph2Plan do) is easier, where the decomposition of final designs can be seen as a data augmentation strategy.
+
+Comparisons on LIFULL Similar comparisons have been done on LIFULL (Table 1). We skip HouseGAN++ because it needs door locations in the input but there is no such data in LIFULL. Compared with Rplan*, Our $_I$  and Graph2Plan, room centers are not provided as priors for Rplan, Our $_{III}$  and Our $_{II}$ , leading to a large performance gap between these two groups. It is understandable because LIFULL is more challenging/heterogeneous with multi-scale samples and nested rooms. Next, Our $_{II}$  outperforms Our $_{III}$  on all three metrics because more prior knowledge (i.e., room types) is given, and both are better than Rplan. Furthermore, when the room centers are provided, inaccurate wall prediction affects Rplan*. Different from the results on RPLAN, Graph2Plan outperforms Rplan* on LIFULL. This is because Graph2Plan takes spacial room relations as input which is more robust for such a multi-scale dataset. Our $_I$  is still better than Rplan* and Graph2Plan. Therefore, we can conclude that, when the
+
+![](images/044cd633abd665c037af10f5b086ac1ac619a10064ab4c7c124651d4abeec9fe.jpg)  
+Figure 4. Qualitative comparisons. Shaded areas indicate design choices that are questionable and unobserved in data.
+
+same amount of prior knowledge is given, iPLAN achieves the best performance.
+
+# 4.5. Qualitative Evaluation
+
+To show intuitive results, qualitative comparisons are given in Figure 4. Each color represents one room type (see color indication in Fig. 5). The shaded boxes indicate questionable design areas. Specifically, in the first row, Rplan designs an isolated bathroom at a corner in the first image, which is a strange design and not observed in the data. In the second row, the first image by Rplan allocates an unreasonable space (shaded) to the public area, which however should be assigned to the neighboring bedroom. Similarly, the layout produced by Graph2Plan consists of 3 bedrooms (second row), but the bathroom is embedded among them (shaded), which means people need to pass through one bedroom to go to the toilet. Besides, the bathroom is too small. As for the last row, the room area is not properly predicted by Graph2plan because the region (shaded) at the bottom-left corner is too narrow.
+
+By comparing  $\mathrm{Our}_I$ ,  $\mathrm{Our}_{II}$  and  $\mathrm{Our}_{III}$ , it is apparent that  $\mathrm{Our}_I$  is closest to GT as a result of the strong input constraints. In contrast, when less constraints are imposed, the diversity is significant, as depicted in  $\mathrm{Our}_{II}$  and  $\mathrm{Our}_{III}$ . More results and analysis are shown in the supplementary material.
+
+# 4.6. User Interactions
+
+The performance of iPLAN under different levels of human interactions is showcased in Figure 5. Figure  $5(a)$  is fully automated generation without human input other than a boundary. iPLAN can generate diverse layouts, with a varying number of rooms and different room types/areas. Figure  $5(b)$  shows the user can decide the room types, where the final layouts are different when the order of room types varies, showing the flexibility of iPLAN. We further provide iPLAN with more priors in Fig.  $5(c)$ , i.e., the bound-
+
+<table><tr><td>Dataset</td><td>Method</td><td>FIDimg</td><td>FIDarea</td><td>FIDtype</td></tr><tr><td rowspan="4">RPLAN</td><td>Our8I</td><td>1.2</td><td>1.36 × 104</td><td>0.06</td></tr><tr><td>Our8I</td><td>0.2</td><td>2.69 × 102</td><td>1.25 × 10-5</td></tr><tr><td>Our8I~8</td><td>2.28</td><td>4.87 × 104</td><td>0.18</td></tr><tr><td>Our8I~8</td><td>0.24</td><td>6.89 × 102</td><td>1.54 × 10-5</td></tr></table>
+
+Table 2. Generalization results on RPLAN.
+
+ary, room types and centers, leading to a planned layout that is nearly the same as the ground truth. We also evaluate iPLAN by introducing a user input at an intermediate step (Fig. 5(d)), where the balcony is moved to the left and consequently a different layout is obtained. Figure 5 demonstrates the interactive ability of iPLAN, which is crucial in settings where the designer leads the design.
+
+# 4.7. Generalizability
+
+Quantitative results To further push iPLAN, we quantitatively evaluate the generalizability of iPLAN by setting up two groups of experiments on RPLAN. In the first group, we select 51322 layouts consisting of  $4 \sim 7$  rooms for training and randomly choose 12,000 8-room layouts for testing. In the second group, we consider a more challenging task by randomly selecting 26574 layouts containing  $4 \sim 6$  rooms for training and 12000 layouts containing  $7 \sim 8$  rooms for testing. Note that in both experiments, the layouts for test include more rooms than the ones used for training.
+
+Quantitative results are reported in Table 2. Again, we can observe that  $\mathrm{Our}_I$  outperforms  $\mathrm{Our}_{II}$ , which is consistent with our intuition since more prior knowledge is provided to  $\mathrm{Our}_I$ . Further, the results in first group are better than those in the second group, which is reasonable because the generalization setting in the second group is more challenging. However, both groups achieve comparable performance to Table 1 and still outperform the baselines.
+
+Qualitative Results An interesting test is to see if iPLAN trained on RPLAN can be generalized to unseen types of boundaries. The qualitative results are also pro
+
+![](images/4d5a1c3f8a727cd9a4f2bdfba095abfb38a3bc79ae1aca06a6109d62531d22da.jpg)  
+Input
+
+GT  
+![](images/145811a11126359456102d7d6acdf3613c9b22c46271ccab074ea4aa061381b5.jpg)  
+: Human input
+
+![](images/e3daa94d0107ef83c95ef85eafb455929c566d840bb5ae088959fc199bce4bc3.jpg)
+
+![](images/8b2883f6b78c480a7ea5a89b55cae49754c2d99704814ccd5f599496df3a7292.jpg)
+
+![](images/676d243be4f2d581ebe51f4e75de43e98b75b884515d4406484e05d7446c519a.jpg)
+
+![](images/a86f8bcfdf66c57d40794ab19e02b93fc1502c8e3f0935cfe60b8a05d7f6b46f.jpg)
+
+![](images/bb532270882e5052abd62deba3f7795622212905e55a36256ccc854a02244f5b.jpg)
+
+![](images/89da12fd151930aa1de4906073841f211f4e58bdaa796f19591706fb6f5ab9dd.jpg)  
+Public Area
+
+![](images/69dde06f1b07307519c8fc6112621621719486f75779a0df671c501acf9ab00a.jpg)
+
+![](images/462f4444365c75cf507575c0a62ce1b972de5822013f7dcb115bd60dd6cddcb7.jpg)
+
+Figure 5. User interactions. (a)-(c) indicate interactions via providing different levels of human input; (d) shows fine-grained interaction in a step-wise generation.  
+![](images/b24f995b59a761e1471806cd609a36fe78ef21cc718b7585eb7089562b6266e1.jpg)  
+$\odot$  Bedroom  
+Kitchen  
+$\odot$  Bathroom  
+$\odot$  Balcony  
+Storage
+
+![](images/c832f3a99ebff10e314afb49073c83da7de50a3da794e419df674a60c8cca92a.jpg)  
+Figure 6. Layouts with non-axis aligned edges. Left: successful examples. Right: Rplan fails but our model succeeds.
+
+vided to verify the generalizability of iPLAN. Note that RPLAN only contains layouts with straight axis-aligned edges in the boundaries, so we consider the boundaries with non-axis aligned edges and curves. Since HouseGAN++ cannot design a layout for a specific building boundary and Graph2Plan is restricted to boundaries with axis-aligned edges, we only compare iPLAN with Rplan.
+
+Figure 6 shows several examples of non-axis aligned design produced by Rplan and iPLAN. The left lists four successful cases on both approaches and the right shows two examples where Rplan fails but iPLAN succeeds. Even when successful, Rplan often predicts isolated rooms which divide the public areas into strange shapes and reduce their usability (the first two examples), while iPLAN utilizes the space better. As for the failures of Rplan, we give two examples of room center estimation and wall prediction, where it fails to reasonably fill and partition the space, a problem from which iPLAN does not suffer. Please refer to the supplementary document for more examples.
+
+# 5. Limitation & Conclusion
+
+While iPLAN is able to handle some irregular boundaries, it cannot cope with extremely irregular ones. One important future direction would be to handle complex environments, e.g., non-axis aligned boundaries and compound and nested rooms, which is crucial to generalize iPLAN to public spaces such as train stations and shopping malls.
+
+In this paper, we proposed a novel human-in-the-loop generative model iPLAN to learn professional designs in a stage-to-stage fashion while respecting design principles. While being capable of fully automated generation, iPLAN allows close interactions with humans by accepting user guidance at every stage and automatically suggesting possible designs accordingly. Comprehensive evaluations on two benchmark datasets RPLAN and LIFULL show that iPLAN outperforms the state-of-the-art methods, both quantitatively and qualitatively. Importantly, iPLAN has exhibited strong generalization capability to unseen design tasks and boundary inputs.
+
+Acknowledgements: In this paper, we used "LIFULL HOME'S Snapshot Data of Rentals" provided by LIFULL Co., Ltd. via IDR Dataset Service of National Institute of Informatics. We thank Jing Li for her input on the design practice. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 899739 CrowdDNA and the Marie Skłodowska-Curie grant agreement No 101018395. Feixiang He has been supported by UKRI PhD studentship [EP/R513258/1, 2218576].
+
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+# vCLIMB: A Novel Video Class Incremental Learning Benchmark
+
+Andrés Villa $^{1,2*}$ , Kumail Alhamoud $^{2}$ , Victor Escorcia $^{3}$ , Fabian Caba Heilbron $^{4}$ , Juan León Alcazar $^{2}$ , Bernard Ghanem $^{2}$
+
+<sup>1</sup>Pontificia Universidad Católica de Chile, <sup>2</sup>King Abdullah University of Science and Technology (KAUST),
+
+$^{3}$ Samsung AI Center Cambridge,  $^{4}$ Adobe Research
+
+afvilla@uc.cl, kumail.hamoud@kaust.edu.sa, v.castillo@samsung.com
+
+caba@adobe.com, {juancarlo.alcazar, bernard.ghanem}@kaust.edu.sa
+
+# Abstract
+
+Continual learning (CL) is under-explored in the video domain. The few existing works contain splits with imbalanced class distributions over the tasks, or study the problem in unsuitable datasets. We introduce vCLIMB, a novel video continual learning benchmark. vCLIMB is a standardized test-bed to analyze catastrophic forgetting of deep models in video continual learning. In contrast to previous work, we focus on class incremental continual learning with models trained on a sequence of disjoint tasks, and distribute the number of classes uniformly across the tasks. We perform in-depth evaluations of existing CL methods in vCLIMB, and observe two unique challenges in video data. The selection of instances to store in episodic memory is performed at the frame level. Second, untrimmed training data influences the effectiveness of frame sampling strategies. We address these two challenges by proposing a temporal consistency regularization that can be applied on top of memory-based continual learning methods. Our approach significantly improves the baseline, by up to  $24\%$  on the untrimmed continual learning task. The code of our benchmark can be found at: https://vclimb.netlity.app/.
+
+# 1. Introduction
+
+Deep neural networks rely on large-scale datasets to achieve state-of-the-art performance on modern computer vision tasks [6, 10, 18, 20]. A significant result of such pretraining is enabling feature reuse [12, 41], by means of finetuning the learned weights for smaller downstream tasks [3, 22, 28]. Due to legal or technical constraints, and the fact that labeling data is expensive and time-consuming [34], real-world deep-learning pipelines would rarely involve a single fine-tuning stage. Instead, these pipelines could require the sequential fine-tuning of large models in a set
+
+of independent tasks that are learned sequentially. Under these conditions, deep neural networks suffer from what is known as catastrophic forgetting [13], where the fine-tuning on novel tasks significantly reduces the performance of the model in a previously learned task, and drift [29], where unseen training data does not fit the previously estimated class distribution. Continual learning [11] directly models such a scenario, by adapting a neural network model into a sequential series of tasks. We focus on a special case of CL: class incremental learning (CIL), where the labels and data are mutually exclusive between tasks, training data is available only for the current task, and there are no tasks ids.
+
+With 500 hours of video from diverse categories uploaded to YouTube every minute and a billion people actively using TikTok every month [4, 36], video content of varying quality is available at an unprecedented scale. With such large volumes of data, it is important to develop models that can effectively learn from continuous streams of untrimmed video data. Remarkably, few research efforts have addressed continual learning with video [24, 26, 44]. Despite these current works, video continual learning methods still show a large variability in their experimental protocols, making direct comparisons hard to establish. These protocols present the following limitations. (1) They are not publicly available. (2) They do not explore a realistic setup with untrimmed videos. (3) Most of them leverage a large pretraining step, which warms up the model by learning a sample of classes from the same distribution and is not always available in a real continual learning scenario.
+
+We directly address these limitations and propose vCLIMB (video CLass IncreMental Learning Benchmark), a novel benchmark devised for the evaluation of continual learning in video. Our test-bed defines a fixed task split on the original training and validation sets of three well known video datasets: UCF101 [33], ActivityNet [5] and Kinetics [6]. vCLIMB follows the standard class incremental continual learning scenario, but includes some modifications to better fit the nature of video data in human ac
+
+tion recognition tasks. First, to achieve fair comparisons between video CIL methods that use memory, we re-define memory size [27, 38] to be the total number of frames, instead of the total number of video instances, that the memory can store. We report this as Memory Frame Capacity in our tables to avoid confusion with the memory size defined in image CIL. This means we are not only concerned about selecting the best videos to store in memory, but we also want to identify the best set of frames to keep in memory. Second, since fine-grained temporal video annotations are expensive (especially for long videos), we analyze the effect of using trimmed and untrimmed video data in continual learning. To the best of our knowledge, this is the first work to explore continual learning with untrimmed videos.
+
+Using vCLIMB's data splits, we establish an initial set of baselines by adapting well-known continual learning methods [1, 17, 27, 38] from the image recognition domain into the activity recognition domain. After benchmarking these baseline methods, we propose a novel strategy for video continual learning that leverages the inherent temporal consistency in video data to better approach the continual action recognition problem. We believe that vCLIMB's standardized approach to prototyping and evaluating video continual learning will enable future works in this area.
+
+Contributions. This paper proposes vCLIMB, a novel benchmark for continual learning in video, which focuses on the activity classification task. Our work brings the following contributions: (1) A standardized benchmark for continual learning in video action recognition, which defines the training protocols and associated metrics for three video datasets; this novel continual learning setup includes a more realistic combination of trimmed and untrimmed videos. (2) We re-purpose and evaluate four baseline methods from the image domain into the video domain. (3) We present a novel strategy based on consistency regularization that can be built on top of memory-based methods to reduce memory consumption while improving performance.
+
+# 2. Related Work
+
+Continual Learning (CL) studies methods that can learn novel concepts continually without forgetting previous knowledge. In the class incremental learning (CIL) setting, models are trained sequentially on a set of tasks. Each incremental task consists of labeled data from novel classes, which are learned individually without considering the data from previous tasks. CIL approaches can be grouped into two broad categories: regularization-based and memory-based methods. While regularization-based methods penalize abrupt changes in the most relevant parameters learned from previous tasks, memory-based methods mitigate catastrophic forgetting by retaining a limited amount of training instances from previous tasks. Although there is extensive literature on continual learning
+
+[1, 7, 8, 15, 17, 21, 25, 27, 31, 42], we restrict our review to a sample of the most widely adopted baselines.
+
+Regularization-Based Methods. Regularization techniques try to keep constant the weights that are important for the previous tasks. They differ in how they estimate the importance value of the model parameters learned in previous tasks. These values are commonly updated and stored in an importance matrix at the end of each task. Elastic Weight Consolidation (EWC) [17] uses the Fisher Information Matrix to make that estimation. Memory Aware Synapses (MAS) [1] estimates the importance of each parameter in a self-supervised manner by measuring how small changes in the parameters affect the output of the model. MAS is of particular interest in the video domain because it can handle weakly labeled instances. These methods include a regularization factor  $\lambda_{reg}$ , which controls how relevant the previously learned tasks are. For a large factor, the model will prioritize the previous tasks over the current one.
+
+Memory-Based Methods. Memory-based methods select and store samples into a memory buffer for future replay. This memory has a limited size and is available when learning a new task. While a naive baseline randomly chooses instances from previously learned classes, current strategies attempt to select the best subset of training samples per class to move into the buffer. We focus on two representative methods that have been shown to work well with high-resolution image datasets. Incremental Classifier and Representation Learning (iCaRL) [27] combines rehearsal and distillation strategies. During training, it selects the most representative instances per class following a nearest-mean-of-exemplars rule. The Bias Correction method (BiC) [38] follows iCaRL's instance sampling approach, but augments the classification layer with a bias correction layer. Bias correction mitigates the imbalance between the large amount of data from the new task and the relatively scarce data from previous tasks, which is only available in memory.
+
+Consistency Regularization. Consistency regularization techniques are used to ensure that a model's output is invariant to various augmentations, which is common in semisupervised learning [2, 19, 30, 32, 39]. For example, such methods have been shown to improve the generation of images from a few samples [43, 45]. Our work investigates catastrophic forgetting in video continual learning and proposes a consistency loss to help memory-replay methods significantly alleviate this impeding nuisance.
+
+Continual Learning in Video. Despite the growing interest in CL in the image domain, the first three works to report results on video data were only recently published. Zhao et al. [44] proposed a spatio-temporal knowledge transfer strategy to mitigate catastrophic forgetting. A concurrent work [26] estimated the subset of feature channels that contributes the most to the predictions of the previous tasks,
+
+<table><tr><td rowspan="2">Set</td><td rowspan="2">In-distribution Pretraining</td><td rowspan="2">Tasks</td><td colspan="3">Videos Per Task</td><td rowspan="2">Classes Per Task</td><td rowspan="2">Avg. Frames Per Video</td><td rowspan="2">Untrimmed Video</td></tr><tr><td>Train</td><td>Val</td><td>Test</td></tr><tr><td>i-Something-Something-B0 [44]</td><td>None</td><td>4</td><td>-</td><td>-</td><td>-</td><td>10</td><td>-</td><td>X</td></tr><tr><td>i-Something-Something-B20 [44]</td><td>20 classes</td><td>5</td><td>-</td><td>-</td><td>-</td><td>4</td><td>-</td><td>X</td></tr><tr><td>i-Kinetics-B0 [44]</td><td>None</td><td>4</td><td>-</td><td>-</td><td>-</td><td>10</td><td>-</td><td>X</td></tr><tr><td>i-Kinetics-B20 [44]</td><td>20 classes</td><td>5</td><td>-</td><td>-</td><td>-</td><td>4</td><td>-</td><td>X</td></tr><tr><td>UCF101-50 [26]</td><td>51 classes</td><td>5/10/25</td><td>-</td><td>-</td><td>-</td><td>10/5/2</td><td>-</td><td>X</td></tr><tr><td>HMDB51-25 [26]</td><td>26 classes</td><td>5/25</td><td>-</td><td>-</td><td>-</td><td>5/1</td><td>-</td><td>X</td></tr><tr><td>Something-Something V2-90 [26]</td><td>84 classes</td><td>9/18</td><td>-</td><td>-</td><td>-</td><td>10/5</td><td>-</td><td>X</td></tr><tr><td>vCLIMB UCF101</td><td>None</td><td>10</td><td>928</td><td>131</td><td>272</td><td>10</td><td>183</td><td>X</td></tr><tr><td>vCLIMB UCF101</td><td>None</td><td>20</td><td>464</td><td>65</td><td>136</td><td>5</td><td>183</td><td>X</td></tr><tr><td>vCLIMB Kinetics</td><td>None</td><td>10</td><td>24628</td><td>1988</td><td>3977</td><td>40</td><td>250</td><td>X</td></tr><tr><td>vCLIMB Kinetics</td><td>None</td><td>20</td><td>12314</td><td>994</td><td>1988</td><td>20</td><td>250</td><td>X</td></tr><tr><td>vCLIMB ActivityNet-Untrim.</td><td>None</td><td>10</td><td>1001</td><td>492</td><td>-</td><td>20</td><td>3542</td><td>✓</td></tr><tr><td>vCLIMB ActivityNet-Untrim.</td><td>None</td><td>20</td><td>500</td><td>246</td><td>-</td><td>10</td><td>3542</td><td>✓</td></tr><tr><td>vCLIMB ActivityNet-Trim.</td><td>None</td><td>10</td><td>1541</td><td>765</td><td>-</td><td>20</td><td>3879</td><td>X</td></tr><tr><td>vCLIMB ActivityNet-Trim.</td><td>None</td><td>20</td><td>770</td><td>383</td><td>-</td><td>10</td><td>3879</td><td>X</td></tr></table>
+
+Table 1. CIL Benchmark Statistics. In vCLIMB we provide 8 splits for class incremental learning, each split contains 10 or 20 tasks. Our Kinetics and ActivityNet setups contain large-scale video data and provide long sequences of tasks with no video pretraining. This makes our splits more suitable for measuring forgetting. We highlight that ActivityNet-Untrim provides a realistic challenge to test a model's ability to continually learn from weakly labeled video streams.
+
+and introduced a temporal mask that keeps this subset stable while learning a new task. It also included a distillation loss, which allows only the least relevant feature maps to be updated while learning a new task. Finally, Ma et al. [24] also approached the class incremental video classification problem by regularizing the feature space in consecutive tasks.
+
+Although existing works bootstrapped the study of continual learning in video data, all of them use different evaluation protocols, making direct comparisons between methods difficult. Moreover, these works propose to pretrain the model with a large set of classes of the same data distribution (up to half of the total), as shown in Table 1. Such an arrangement is unnatural for continual learning, as it makes it difficult to disentangle the effects of catastrophic forgetting. In contrast, our benchmark presents a more challenging and realistic setup. Our splits contain up to 20 tasks, with a balanced number of classes per task. Furthermore, vCLIMB includes three video datasets, which enables the study of the continual learning problem in more diverse scenarios. Finally, previous works do not provide a detailed analysis of the proposed video continual learning setups. In our benchmark, we provide extensive empirical evaluations to analyze individual splits and identify unique properties of continual learning in video, including the memory size and the adoption of untrimmed video configurations.
+
+# 3. vCLIMB: A Video Class Incremental Learning Benchmark
+
+Notation & Problem Definition Similar to the image domain, we train a single neural network  $(F_{\omega})$  with parameter set  $(\omega)$  over a sequence of tasks  $(R_{n})$ . Each task in the sequence contains its own training data  $R_{i} =$
+
+$\{(X_0,Y_0),(X_1,Y_1)\dots (X_n,Y_n)\}$ , with  $X_{n}$  the input data and  $Y_{n}$  its corresponding ground truth. We optimize the parameter set  $\omega$  in  $F$  sequentially over  $R_{n}$ , searching for  $\bar{F}_{\omega}$  that maximizes the average accuracy over  $R_{n}$ .
+
+Datasets and Tasks. The bottom half of Table 1 summarizes the main attributes of vCLIMB. We design vCLIMB on top of three well-known datasets for action recognition: (1) UCF101 [33], which has 13.3K videos from 101 classes, (2) Kinetics [6], a large-scale video dataset with over 300K short clips distributed over 400 action classes, and (3) ActivityNet [5], which can be used for trimmed and untrimmed activity classification and contains 20K videos from 200 activity classes. We utilize the diversity of videos in ActivityNet and provide two subsets: ActivityNet-Trim, in which every part of the videos when an action happens is considered as an independent video, and ActivityNet-Untrim, which is more challenging and labels a whole video with its most representative action class. We create two different CIL sequences of tasks for each dataset. The first sequence contains ten tasks, and the second sequence contains twenty tasks. Statistics about the number of classes per task, the number of videos per task, and the average number of frames in each split are provided in Table 1.
+
+Metrics for Video Continual Learning (CL). In vCLIMB, we use the standard CL metrics: Final Average Accuracy (Acc) and Backward Forgetting (BWF). Acc is the average classification accuracy of the model evaluated on all learned tasks, including the last task it was trained on [23, 37]. This metric is essential to show how the average performance of the model degrades as it learns new tasks. BWF complements Acc and measures the influence of the learned task  $i$
+
+![](images/6f22481846e23a9f9b8aece716c8f372d6da552a9eba1923a790ee6c896f1199.jpg)  
+Figure 1. Outline of our Memory-based CIL setting. In video CIL, a model sequentially learns a set of video classification tasks. Memory-based methods define a limited episodic memory to store a few temporally down-sampled examples of previous tasks. We propose a consistency regularization that helps the model to better remember previous tasks from down-sampled examples in memory.
+
+in the performance of the previous tasks [23], as:
+
+$$
+B W F _ {i} = \frac {1}{N _ {i} - 1} \sum_ {j = 1} ^ {N _ {i} - 1} \left(R _ {j, j} - R _ {N _ {i}, j}\right) \tag {1}
+$$
+
+where  $N_{i}$  is the number of learned tasks after learning the task  $i$ , and  $R_{j,j}$  and  $R_{N_i,j}$  represents the accuracy on the task  $j$  after learning the task  $j$ , and learning a new task the task  $i$ , respectively. Given a total of  $N$  tasks, we report the final Backward Forgetting  $BWF = BWF_{N}$  in our tables.
+
+# 3.1. Unique Challenges of Video CL
+
+Video CIL comes with unique challenges. (1) Memory-based methods developed in the image domain are not scalable to store full-resolution videos, so novel methods are needed to select representative frames to store in memory. (2) Untrimmed videos have background frames that contain less helpful information, thus making the selection process more challenging. (3) The temporal information is unique to video data, and both memory-based and regularization-based methods need to mitigate forgetting while also integrating key information from this temporal dimension. These challenges informed the design choices of vCLIMB.
+
+Re-defining Memory Size. Unlike image benchmarks for class incremental learning (CIL), our video instances contain a temporal dimension whose size could show large variability. To favor fair comparisons between methods, we define the working memory size in terms of stored frames. This design choice avoids a scenario where longer videos are preferred (or even trivially selected) to maximize the amount of data stored in the working memory. Moreover, it creates a new unique aspect of CIL in video data, as methods must decide first what subset of frames should be selected, and then decide what video to store according to the sub-sampled videos. We follow the same instance per class ratio as [27], which is 20 when the model has learned all training classes. Therefore, we define a memory that can save at most 8000, 4000, and 2020 videos for Kinetics, ActivityNet, and UCF101, respectively. If we store videos in memory without down-sampling them, this would correspond to saving  $3.25\%$ ,  $25.95\%$ , and  $21.76\%$  of the total frames in Kinetics, ActivityNet, and UCF101. As we
+
+show later, these storage requirements can be significantly reduced using temporal consistency regularization.
+
+Untrimmed Video Data for CIL. In the untrimmed classification setting, the action of interest may occur at any time in the video and its boundaries are not known. This problem formulation has no direct counterpart in the image classification domain. The annotation scheme of ActivityNet allows us to analyze this scenario for class incremental learning. In ActivityNet, videos contain one or multiple temporal segments defining the occurrence of an action instance, while unlabeled segments constitute a background set where no relevant action takes place [5].
+
+We leverage this unique property of video data and define two independent setups for video CIL. In the trimmed setup, we only use frames that belong to a labeled action segment. In the untrimmed setup, we freely sample frames from the whole video, regardless of whether they belong to the main action or the background. For consistency in the untrimmed scenario, we give every frame in the video the same label. We select a primary label as the action with the longest temporal support in the video, and discard any video that contains instances of 2 or more different labels. We empirically find that this assignment only discards  $0.15\%$  of the ActivityNet dataset. This untrimmed learning task more closely resembles the real world scenario of CIL, where a model learns from a continuous stream of diverse videos. Given the scale of current video services and the costly nature of fine-grained labels, real models would likely be learning from a stream with weak video-level annotations.
+
+Baselines We implement and evaluate these four continual learning methods to serve as baselines [1, 17, 27, 38] because they are widely used and easily scalable to the video domain. We also compare with a naive memory-based strategy, which selects samples randomly for memory creation. We provide the implementations for these methods as part of our video CIL benchmark.
+
+# 3.2. A Stronger Baseline with Temporal-consistency Regularization
+
+We present a novel strategy for CIL in the video domain. Our approach relies on one of the unique characteristics of
+
+![](images/87e8bb28cf799fe6b81770db50ee906e5d8c67315e066e2d3c5257c0441a69ba.jpg)  
+Figure 2. Temporal Consistency Loss. Our consistency loss encourages learning representations that are robust to frame sampling, enabling the CIL model to remember old tasks by looking only at down-sampled versions of those videos.
+
+video data, temporal resolution consistency. Figure 1 illustrates our pipeline for memory-based video CIL methods. After completing the training on videos from the first task, the model has the option to select a few temporally subsampled examples to store in the episodic memory. When learning the second task, the model is trained on both the new task videos and a few past-task examples it retained in memory. We introduce a regularization loss, represented by the dashed arrow in Figure 1, during the fine-tuning phase of every new task. This loss constrains the network to estimate similar feature representations for the original video clip and a temporally down-sampled version of it.
+
+This constraint aims to optimize the effectiveness of the data stored in the episodic memory, by enforcing a similar representation between the original clip and its temporally down-sampled version. As shown in Figure 2, the most relevant aspects of our loss is that it reduces the drift between the samples stored in memory and the original samples used at training time. While simple, our constraint directly addresses one of the key aspects of CIL in video: its episodic memory is composed of frame sets instead of full-resolution video clips. In practice, we enforce similar representations by adding a regularization term to the loss function.
+
+Temporal Consistency Loss. When training on a new task, each video  $(X)$  will have an augmented version  $(X^{d})$  by means of temporal down-sampling. We use pairs of  $X$  and  $X^{d}$  to calculate a consistency loss  $L_{c}$  over a single forward pass of the network  $F$ :
+
+$$
+L _ {c} = (1 - \lambda) L _ {c l s} (F (X), Y) + \lambda L _ {c l s} (F (X ^ {d}), Y), \tag {2}
+$$
+
+where  $L_{cls}$  is the cross-entropy loss,  $Y$  is the ground truth label of  $X$ , and  $\lambda$  is the consistency regularization factor.
+
+Our consistency regularization strategy is model agnostic and thus can be adapted to any backbone and CIL memory-replay strategy. Since we use the same set of weights  $(F)$  for both loss terms, our method only introduces a linear time increase in the total number of FLOPs.
+
+# 4. Experimental Evaluation
+
+We now proceed with the experimental assessment of the class incremental continual learning task in vCLIMB. In this section, we first discuss the details of the reimplementation of image methods for continual learning in the video domain. Then, we proceed with the empirical assessment of these baseline methods on the three datasets included in the vCLIMB benchmark: UFC101 [33], Kinetics [6] and ActivityNet [5]. For the ActivityNet dataset, we also evaluate the CIL task in the presence of trimmed and untrimmed video annotations. Moreover, we assess the effectiveness of the proposed regularization method in all the previous scenarios and datasets.
+
+Implementation Details. The memory-based [27,38] and regularization-based [1, 17] baselines are trained for 50 and 20 epochs, respectively, following a fully-supervised setup on each CIL task. We use TSN [35] with a ResNet-34 backbone pretrained on ImageNet. We follow the same temporal data augmentation proposed in [35], using  $N = 8$  segments per video. We optimize our model using Adam [16] with a learning rate of  $1 \times 10^{-3}$ . For the temporal consistency loss factor, we use  $\lambda = 0.5$ . We run MAS [1] with a regularization factor  $\lambda_{reg}$  of  $3 \times 10^{5}$ . We found EWC [17] to be more challenging to tune for video CIL. The best results are obtained by picking a different regularization factor  $\lambda_{reg}$  for EWC on each individual dataset:  $3 \times 10^{3}$  for UCF101,  $5 \times 10^{2}$  for Kinetics, and  $3 \times 10^{5}$  for ActivityNet.
+
+# 4.1. Baselines for Video CIL
+
+As outlined in 3.1 the datasets in vCLIMB differ in scale, so we set a different working memory limit for each dataset according to their total number of frames. For each dataset, we perform experiments on two different splits: a 10-task split and a 20-task split. In Table 2, we report the results obtained using the best hyper-parameters for each method.
+
+Consistent with image class incremental learning [14], the regularization-based methods EWC [17] and MAS [1] lag significantly behind replay-based methods, regardless of the difficulty of the dataset or the number of tasks. This is because regularization methods only penalize changes to the model parameters, and thus experience an unavoidable trade-off between learning new tasks and forgetting older tasks. If a larger regularization parameter is used to emphasize learning new tasks, forgetting increases and the average accuracy on old tasks is compromised. If a smaller regularization parameter is used to emphasize remembering old tasks, the accuracy on newer tasks is impaired. Given the difficulty of video CIL, this limitation highlights the importance of future exploration of more sophisticated strategies of memory-free approaches.
+
+Surprisingly, no memory-based approach is superior across all video datasets. While the naive baseline, which
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Num. Task</td><td colspan="4">Kinetics</td><td colspan="4">ActivityNet-Trim</td><td colspan="4">UCF101</td></tr><tr><td>Mem. Video Instances</td><td>Mem. Frame Capacity</td><td>Acc ↑</td><td>BWF ↓</td><td>Mem. Video Instances</td><td>Mem. Frame Capacity</td><td>Acc ↑</td><td>BWF ↓</td><td>Mem. Frame Capacity</td><td>Mem. Frame Capacity</td><td>Acc ↑</td><td>BWF ↓</td></tr><tr><td>EWC</td><td>10</td><td>None</td><td>None</td><td>5.81%</td><td>16.05%</td><td>None</td><td>None</td><td>4.02%</td><td>5.32%</td><td>None</td><td>None</td><td>9.51%</td><td>98.94%</td></tr><tr><td>MAS</td><td>10</td><td>None</td><td>None</td><td>7.81%</td><td>10.12%</td><td>None</td><td>None</td><td>8.11%</td><td>0.18%</td><td>None</td><td>None</td><td>10.89%</td><td>11.11%</td></tr><tr><td>EWC</td><td>20</td><td>None</td><td>None</td><td>2.95%</td><td>32.70%</td><td>None</td><td>None</td><td>1.28%</td><td>3.77%</td><td>None</td><td>None</td><td>4.71%</td><td>92.12%</td></tr><tr><td>MAS</td><td>20</td><td>None</td><td>None</td><td>4.25%</td><td>5.54%</td><td>None</td><td>None</td><td>4.61%</td><td>0.1%</td><td>None</td><td>None</td><td>5.90%</td><td>5.31%</td></tr><tr><td>Naive</td><td>10</td><td>8000</td><td>2 × 10^6</td><td>30.14%</td><td>41.30%</td><td>4000</td><td>15.5 × 10^6</td><td>47.20%</td><td>20.64%</td><td>2020</td><td>3.69 × 10^5</td><td>91.42%</td><td>7.43%</td></tr><tr><td>iCaRL</td><td>10</td><td>8000</td><td>2 × 10^6</td><td>32.04%</td><td>38.74%</td><td>4000</td><td>15.5 × 10^6</td><td>48.53%</td><td>19.72%</td><td>2020</td><td>3.69 × 10^5</td><td>80.97%</td><td>18.11%</td></tr><tr><td>BiC</td><td>10</td><td>8000</td><td>2 × 10^6</td><td>27.90%</td><td>51.96%</td><td>4000</td><td>15.5 × 10^6</td><td>51.96%</td><td>24.27%</td><td>2020</td><td>3.69 × 10^5</td><td>78.16%</td><td>18.49%</td></tr><tr><td>Naive</td><td>20</td><td>8000</td><td>2 × 10^6</td><td>23.47%</td><td>48.05%</td><td>4000</td><td>15.5 × 10^6</td><td>40.78%</td><td>23.18%</td><td>2020</td><td>3.69 × 10^5</td><td>87.40%</td><td>10.96%</td></tr><tr><td>iCaRL</td><td>20</td><td>8000</td><td>2 × 10^6</td><td>26.73%</td><td>42.25%</td><td>4000</td><td>15.5 × 10^6</td><td>43.33%</td><td>21.57%</td><td>2020</td><td>3.69 × 10^5</td><td>76.59%</td><td>21.83%</td></tr><tr><td>BiC</td><td>20</td><td>8000</td><td>2 × 10^6</td><td>23.06%</td><td>58.97%</td><td>4000</td><td>15.5 × 10^6</td><td>46.53%</td><td>15.95%</td><td>2020</td><td>3.69 × 10^5</td><td>70.69%</td><td>24.90%</td></tr></table>
+
+Table 2. Baseline Video CIL Results. We report the average accuracy (Acc) and the backward forgetting (BWF) at 10 and 20 tasks from three action recognition benchmarks. Regularization-based methods (at the top of the table), under-perform memory-based methods, shown at the bottom. Consistent with results in the image domain, the longer 20-task sequences are more challenging for all methods across all dataset. We highlight that the reported improvements in the image domain do not directly translate to the video domain, as no single method obtains the best performance in every setup.
+
+randomly samples instances for memory, outperforms the other baselines on UCF101, iCaRL and BiC are better on Kinetics and ActivityNet. We believe that fixing this discrepancy by finding memory-based methods designed specifically for video is an important research direction.
+
+Number of Tasks and Forgetting. We observe that the longer (20-task) sequence is a more challenging setup, where the average accuracy is always lower for any method on any of the three datasets. Following this trend, all memory-based methods forget more as the number of tasks in the sequence increases. Similar to the image domain, evaluating on long sequences of tasks accentuates the shortcomings of CIL methods, and is thus suitable for the study of strategies that mitigate forgetting. We pose closing the forgetting gap between the 10-task and 20-task scenarios as an important research direction.
+
+Relevance of Memory Size in Video. UCF101 stands out in Table 2, since memory-based methods perform surprisingly well in this dataset. In fact, the  $91.42\%$  accuracy obtained by the naive rehearsal baseline on the UCF101 10-task CIL is almost on par with the  $94.9\%$  accuracy reported for training TSN [35], which is the backbone in our experiments, on all UCF101 classes simultaneously. This creates a sharp contrast with Kinetics and ActivityNet results, where the best CIL baseline achieve  $32.04\%$  and  $51.96\%$  respectively. In comparison, TSN trained on all the action classes of the whole dataset at once achieves  $73.9\%$  on Kinetics [40] and  $88\%$  on ActivityNet [35].
+
+Park et al. [26] conducted experiments on UCF101 and made the observation that storing a subset of frames in memory results in a similar performance to storing the whole video [26]. Our hypothesis is that such good performance and apparent invariance to frame sampling is an artifact specific to the UCF101 dataset, and that it is not the norm in video class incremental learning. Our experi
+
+ments on Kinetics and ActivityNet in Table 3 show that this is indeed not the case for more challenging video datasets. In particular for ActivityNet, storing four frames per video in memory, results in an overall decrease of accuracy by  $27\%$  compared to storing all frames. We hypothesize that the temporal dependencies in Kinetics and ActivityNet are more complex and more relevant for the task, thus models with naive sampling strategies struggle to remember older tasks from severely down-sampled videos.
+
+# 4.2. Remembering from Down-sampled Videos.
+
+To avoid incurring large memory requirements, it is favorable for video CIL models to down-sample videos before storing them for future replay. This means the model will learn a new task with full-resolution videos, but its memory will be composed of temporally sub-sampled versions of videos belonging to older tasks. Unfortunately, the distribution shift from the original training data to the modified stored exemplars causes a decline in the accuracy, which is evident from the Kinetics and ActivityNet results in Table 3.
+
+To mitigate forgetting, continual video action recognition models must learn robust action embeddings, which are invariant to the temporal resolution of the video. We use our temporal consistency loss to train the CIL model jointly on both full-resolution videos and temporally down-sampled videos. Our proposed strategy helps achieve this desirable invariance. Table 3 reports the results of using our consistency loss, which is explained in Section 3.2, to mitigate forgetting in models that remember from down-sampled videos. Since no single memory-based method consistently outperforms the other methods across all datasets, we choose the more established baseline iCaRL [27] to perform the sub-sampled memory experiments.
+
+Consistency Regularization on Kinetics. The last three rows of Table 3 summarize the effect of applying our temporal regularization on limited size memories. Temporal
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Frames per video</td><td colspan="3">Kinetics</td><td colspan="3">ActivityNet-Trim</td><td colspan="3">UCF101</td></tr><tr><td>Mem. Frame Capacity</td><td>Acc ↑</td><td>BWF ↓</td><td>Frame Capacity</td><td>Acc ↑</td><td>BWF ↓</td><td>Mem. Frame Capacity</td><td>Acc ↑</td><td>BWF ↓</td></tr><tr><td>iCaRL</td><td>4</td><td>3.2 × 104</td><td>30.73%</td><td>40.36%</td><td>1.6 × 104</td><td>21.63%</td><td>36.98%</td><td>8.08 × 103</td><td>80.32%</td><td>17.13%</td></tr><tr><td>iCaRL</td><td>8</td><td>6.4 × 104</td><td>32.04%</td><td>38.48%</td><td>3.2 × 104</td><td>21.54%</td><td>33.41%</td><td>16.16 × 103</td><td>81.12%</td><td>18.25%</td></tr><tr><td>iCaRL</td><td>16</td><td>12.8 × 104</td><td>31.36%</td><td>38.74%</td><td>6.4 × 104</td><td>25.27%</td><td>29.71%</td><td>32.32 × 103</td><td>81.06%</td><td>18.23%</td></tr><tr><td>iCaRL</td><td>ALL</td><td>2 × 106</td><td>32.04%</td><td>38.74%</td><td>15.5 × 106</td><td>48.53%</td><td>19.72%</td><td>3.69 × 105</td><td>80.97%</td><td>18.11%</td></tr><tr><td>iCaRL+TC</td><td>4</td><td>3.2 × 104</td><td>35.32%</td><td>34.07%</td><td>1.6 × 104</td><td>42.99%</td><td>23.82%</td><td>8.08 × 103</td><td>73.85%</td><td>26.35%</td></tr><tr><td>iCaRL+TC</td><td>8</td><td>6.4 × 104</td><td>36.24%</td><td>33.83%</td><td>3.2 × 104</td><td>45.73%</td><td>18.90%</td><td>16.16 × 103</td><td>74.25%</td><td>25.27%</td></tr><tr><td>iCaRL+TC</td><td>16</td><td>12.8 × 104</td><td>36.54%</td><td>33.53%</td><td>6.4 × 104</td><td>44.04%</td><td>22.82%</td><td>32.32 × 103</td><td>75.84%</td><td>23.23%</td></tr></table>
+
+Table 3. Ablation study results with different memory sizes. We compare iCaRL [27] with and without Temporal Consistency (iCaRL+TC) on the 10-task trimmed action recognition setups. iCaRL experiences a loss of accuracy as the memory size decreases in the challenging Kinetics and ActivityNet-Trim setups. Applying TC allows us to reduce the memory size by 2 orders of magnitude while retaining the performance in ActivityNet-Trim, and it even outperforms the version of iCaRL that uses all frames in Kinetics.
+
+consistency regularization (labeled TC in the table) reduces forgetting on the 10-task Kinetics split regardless of how many frames per video are stored. In particular, our best baseline (iCaRL) tested on a memory of 16, 8, or 4 frames per video is significantly improved when our temporal consistency term is added to iCaRL's loss objective. For example, the accuracy is improved by more than  $4.5\%$  in the scenario where 4 frames per video are stored. It is worth highlighting that adding temporal consistency enables us to store as few as 4 frames per video and yet achieve more than  $3\%$  accuracy improvement over storing full videos, which requires about 100 times greater memory frame capacity.
+
+Consistency Regularization on Trimmed ActivityNet. We observe a similar trend on ActivityNet with temporal consistency resulting in even more sizable improvements. Specifically, adding the regularization term with a model that has access to a memory consisting of 8 frames per video results in a massive  $24\%$  improvement. Moreover, with 4 frames per video stored in memory, our temporally consistent model comes close to achieving similar performance to the model that uses full-resolution videos. In fact, this significantly closes the  $27\%$  accuracy gap between storing full-resolution videos in memory and storing 4 frames per video when our regularization is not used. Our results show that our method is most relevant for datasets that require more sophisticated temporal reasoning like ActivityNet.
+
+How Many Frames Should be Stored in Memory? A major challenge for adapting continual learning methods to video is that videos consume significantly more memory than images due to the added temporal dimension. Our experiments in Table 3 show that a consistency-based training framework makes it possible to down-sample videos before storing them for future replay, resulting in remarkably small memories and minor performance degradation. In particular on the challenging dataset Kinetics, iCaRL without TC trained with full-resolution memory achieves an average accuracy of  $32.04\%$ . Yet, adding the TC loss and training on
+
+memory of only 8 frames per video exemplar results in an even better average performance of  $36.24\%$ . This is impressive, since the average number of frames per video in Kinetics is 250, which translates to only  $3.2\%$  of the average Kinetics video being stored in memory.
+
+Our experiments on ActivityNet-Trim also show a similar trend towards alleviating large memory requirements. Since 8 frames represent  $0.21\%$  of the average ActivityNet video, naively storing 8-frame exemplars with no temporal consistency in training results in a  $27\%$  drop in accuracy. Using consistency regularization and 8 frames per memory sample, we are able to reduce the difference in accuracy from  $27\%$  to only  $3\%$ .
+
+Frame Sampling in UCF101. We revisit our assertion that remembering old tasks in UCF101 is unaffected by the number of frames stored in memory. We vary the number of frames per video used: from storing all frames to storing 16, 8, and 4 frames. The results reported in Table 3 clearly show that the performance of iCaRL is almost the same in all these different scenarios, validating the claim that UCF101 is not a prototypical dataset to evaluate CIL methods. For completeness, we also run the same set of experiments using the proposed consistency regularization scheme. We notice that it does not help increase the performance. This is unsurprising for two reasons. (i) Class incremental learning on UCF101 is already not that much more challenging than training on the whole dataset in one task, as we showed in Section 4.1. (ii) As the first half of Table 3 shows, class incremental learning with a memory buffer on UCF101 is invariant to the number of frames per video stored in memory. This can be explained by the strong scene bias exhibited in UCF101 [9]. Thus, incorporating the TC loss in this dataset, which has a very small accuracy gap between CIL training and fully-supervised training, is not expected to improve accuracy.
+
+UCF101 for video CIL. In conclusion, we still recommend using the UCF101 splits for prototyping video CIL methods. However, due to its simplicity, we encourage the
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Frames per video</td><td rowspan="2">Mem. Frame Capacity</td><td colspan="2">ActivityNet-Untrim</td><td colspan="2">ActivityNet-Trim</td></tr><tr><td>Acc ↑</td><td>BWF ↓</td><td>Acc ↑</td><td>BWF ↓</td></tr><tr><td>iCaRL</td><td>4</td><td>1.6 × 104</td><td>16.28%</td><td>32.75%</td><td>21.63%</td><td>36.98%</td></tr><tr><td>iCaRL</td><td>8</td><td>3.2 × 104</td><td>16.67%</td><td>31.96%</td><td>21.54%</td><td>33.41%</td></tr><tr><td>iCaRL</td><td>16</td><td>6.4 × 104</td><td>21.27%</td><td>28.94%</td><td>25.27%</td><td>29.71%</td></tr><tr><td>iCaRL+TC</td><td>4</td><td>1.6 × 104</td><td>36.07%</td><td>22.39%</td><td>42.99%</td><td>23.82%</td></tr><tr><td>iCaRL+TC</td><td>8</td><td>3.2 × 104</td><td>40.29%</td><td>20.80%</td><td>45.73%</td><td>18.90%</td></tr><tr><td>iCaRL+TC</td><td>16</td><td>6.4 × 104</td><td>40.45%</td><td>21.21%</td><td>44.04%</td><td>22.82%</td></tr></table>
+
+Table 4. Ablation study results with trimmed and untrimmed videos. All the experiments involve sequentially training on 10 tasks. ActivityNet-Untrim provides a more realistic and challenging setup to evaluate CIL models. We impose the strict resource constraint of 4, 8, and 16 frames per video stored in memory. Our temporal consistency approach significantly improves the accuracy of the baseline method trained on both ActivityNet setups with limited memory. The best performing setting for each data split is highlighted in the table.
+
+community to evaluate new video CIL methods on the more challenging splits from Kinetics and ActivityNet.
+
+Class Incremental Learning from Untrimmed Videos. As observed in Table 4, we perform a set of experiments to evaluate the realistic class incremental learning scenario with untrimmed videos and make a few interesting observations. First, ActivityNet-Untrim is more challenging than ActivityNet-Trim. iCaRL baseline [27] achieves a better performance on ActivityNet-Trim regardless of the number of frames per video stored in memory. Second, our temporal consistency regularization improves [27] by large margins in both ActivityNet setups. The progressive performance of iCaRL on the trimmed and untrimmed setups in the 10-task case, where 8 frames are used to represent a memory video, are plotted in Figure 3. Our regularization loss enhances the performance by  $24\%$  in both cases.
+
+Why Does Consistency Regularization Work? We hypothesize that the large accuracy gains garnered by adding the consistency regularization loss are due to two reasons. First, regularization may make the training more stable. When training the model on augmented examples, we expect the model to learn more robust representations that can make learning new tasks easier. Second, consistency regularization forces the backbone to learn action embeddings that are invariant to the number of frames used to represent the video. Current video CIL methods store down-sampled videos in the episodic memory. However, without applying the consistency loss, the model naturally learns completely different features for densely sampled videos and sparsely sampled videos. Thus, the model struggles to remember the old action representations from this memory of down-sampled videos. This distribution shift between videos of different temporal resolution is especially apparent in datasets with more complicated temporal dependencies like ActivityNet. This points to why our approach improves the performance by large margins on this dataset and
+
+suggests TC loss does not skew the models towards spurious scene features, rather it manages to retain meaningful temporal features.
+
+![](images/2430802446918b67ad1ef9da4ce67c3e04abe927ab170982b83dae0686e33100.jpg)  
+Figure 3. Average Accuracy in the Validation Set. We train iCaRL sequentially on 10 tasks and enforce a memory constraint of 8 frames per video. Our temporal consistency (TC) loss significantly improves iCaRL's performance on ActivityNet-Trim and ActivityNet-Untrim
+
+# 5. Conclusion and Limitations
+
+In this paper, we propose and analyze vCLIMB, a continual learning benchmark for video action recognition. We expose and tackle the unstudied accuracy drop, which is experienced by memory-based video CIL models and is caused by frame sub-sampling. In our experiments, we sample frames uniformly and leverage a consistency loss to significantly alleviate that accuracy drop, by up to  $24\%$  in CIL untrimmed video classification. We think that exploring non-uniform sampling strategies is another promising direction, but we leave that exploration for future work.
+
+Acknowledgments. This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. OSR-CRG2021-4648. Authors also thank Centro Nacional de Inteligencia Artificial CENIA, FB210017, BASAL, ANID.
+
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+# Voxel Field Fusion for 3D Object Detection
+
+Yanwei Li $^{1}$  Xiaojuan Qi $^{2}$  Yukang Chen $^{1}$  Liwei Wang $^{1}$  Zeming Li $^{3}$  Jian Sun $^{3}$  Jiaya Jia $^{1,4}$
+
+The Chinese University of Hong Kong<sup>1</sup> The University of Hong Kong<sup>2</sup>  
+MEGVII Technology<sup>3</sup> SmartMore<sup>4</sup>
+
+# Abstract
+
+In this work, we present a conceptually simple yet effective framework for cross-modality 3D object detection, named voxel field fusion. The proposed approach aims to maintain cross-modality consistency by representing and fusing augmented image features as a ray in the voxel field. To this end, the learnable sampler is first designed to sample vital features from the image plane that are projected to the voxel grid in a point-to-ray manner, which maintains the consistency in feature representation with spatial context. In addition, ray-wise fusion is conducted to fuse features with the supplemental context in the constructed voxel field. We further develop mixed augmentor to align feature-variant transformations, which bridges the modality gap in data augmentation. The proposed framework is demonstrated to achieve consistent gains in various benchmarks and outperforms previous fusion-based methods on KITTI and nuScenes datasets. Code is made available at https://github.com/dvlab-research/VFF.
+
+# 1. Introduction
+
+Object detection in 3D scenes is regarded as a vital task to provide accurate perception for real-world applications. Over the past decades, research attentions [15,27,38,40,51] have been dedicated to 3D object detection from raw point clouds. Due to the inherent properties of LiDAR sensors, the captured point clouds are usually sparse and cannot provide sufficient context to distinguish among hard cases in distant or occluded regions, which consequently yields inferior performance in such scenarios. However, in safety-critical applications like autonomous driving, the frequently occurred miss-detection is unacceptable.
+
+To address this issue, previous studies introduce image features in the cross-modality fusion [13, 32, 33]. The main challenge is to maintain the cross-modality consistency in this process that might be damaged in feature representation.
+
+![](images/eded8e72ac91c26741194ee0125774e46f6ddfece2a07a384c75869ff60bc2f0.jpg)  
+(a) Point-to-point manner
+
+![](images/8ff250ba4d63aa6c0c79237451bdf5e3ff39d5c820c82a0c40eb71acdb676cdd.jpg)  
+(b) Point-to-ray manner  
+Figure 1. Compared with previous work [17,50] in 1a that projects feature from raw image to voxel and represents in a point-to-point manner, the proposed method in 1b projects feature from augmented image to voxel field and represents in a point-to-ray manner. Dotted and solid arrow denote point- and ray-level projection.
+
+tion considering context deficiency and density variance, and data augmentation for cross-modality misalignment. In particular, previous work represents image features in a point-to-point manner in Figure 1a, which conducts fusion in each single point, constrained by the sparsity of point clouds. In this case, the rich context cues from images cannot be well utilized because adjacency in the image plane cannot be guaranteed in 3D space. Meanwhile, given augmented point clouds, traditional approaches [17, 50] usually keep raw images unchanged and reverse transformations in point clouds for pairwise correspondence. However, because of the flipping and scaling variance in 2D convolutions, the asynchronous augmentation brings cross-modality misalignment and instability.
+
+In this paper, we propose a new cross-modality framework, called voxel field fusion (VFF). Mixed augmentation for both modalities is first applied for data-level preprocessing. As briefly illustrated in Figure 1b, VFF projects augmented image features to the voxel grid and represents it in a point-to-ray manner, called voxel field similar to that in neural rendering [19, 20]. In this way, representations of both modalities are well aligned, and surrounding spatial context is replenished in voxel field. In short, the key idea of VFF is to maintain modality consistency by representing and fusing augmented image feature as a ray in voxel field.
+
+The image-to-voxel dense rendering is usually resource-intensive or requires extra models for depth prediction [25, 47]. To facilitate this process, we draw inspiration from recent advances in neural rendering [19, 20, 48] and propose a learnable sampler and ray-wise fusion for efficient ray construction and cross-modality fusion. In particular, instead of random sampling [20], learnable sampler is designed to select image features for interaction within the activated area with high responses, where features are represented in a point-to-ray manner as aforementioned. Then, ray-wise fusion is conducted in the voxel field according to the predicted score of each voxel along the ray. For the misalignment in augmentation, the mixed augmentor is further proposed to bridge this gap by aligning feature-variant augmentation (flipping and scaling) in image level.
+
+With the above designs, the cross-modality consistency can be maintained from the aspect of feature representation and data augmentation in an end-to-end manner. Generally, the proposed VFF is distinguished from two aspects. First, it projects image features in a point-to-ray manner and represents, as well as fuses them, in the voxel field, which eliminates the modality gap and provides accurate 3D context to detect hard cases. Second, it efficiently samples high-responded features from augmented images, which enables the network to construct each ray on the fly.
+
+The overall framework, called voxel field fusion, can be easily instantiated with various voxel-based backbones for 3D object detection, which is fully elaborated in Section 3. Extensive empirical studies of the designed workflow are conducted in Section 4 to reveal the effect of each component. We further report experimental results on two widely-adopted datasets, namely KITTI [10] and nuScenes [2]. The proposed VFF is proved to achieve consistent increases over various benchmarks and attains significant gain with  $2.2\%$  AP over strong baselines on hard cases of KITTI test set. Meanwhile, it surpasses previous fusion-based methods by a large margin and achieves leading performance on nuScenes test set with  $68.4\%$  mAP and  $72.4\%$  NDS.
+
+# 2. Related Work
+
+LiDAR-based 3D Detection. Given point clouds as input, traditional LiDAR-based approaches are usually distinguished by their representation for irregular data, e.g., grid and point. Grid-based methods project the point clouds to regular grids and process them by 2D or 3D networks. The methods with 2D networks usually construct 2D bird-view grids [6, 39, 40] or pseudo image [15] and generating 3D bonding boxes on top of it. Meanwhile, the approaches with 3D networks construct 3D voxels from the divided point clouds and predict boxes with detection heads [8, 38, 44]. Point-based methods directly handle raw point cloud with set abstraction [24] and generate 3D proposals on top of it [21, 22, 28, 42]. Given the sparse property of point cloud,
+
+the recognition ability is limited due to lack of texture features, especially in real scenes with multiple categories like nuScenes [2] dataset.
+
+Image-based 3D Detection. Previous image-based methods construct the network and extract features from pure monocular or multiple images for 3D box prediction. Given a single image, several monocular-based approaches [1, 31, 35] try to regress and predict 3D boxes directly, while others propose to construct middle-level representation and perform detection on top of it [36, 47]. Because of the depth requirement in 3D detection, previous work also tries to enhance the ability from depth estimation [5, 25, 30]. Another stream for relatively accurate depth is utilizing stereo or multi-view images to construct 3D geometry volume [4, 7, 43] and conduct object detection on top of it. Although depth estimated from multi-view is much better than that from a single image, it still lags behind the accurate point cloud from LiDAR.
+
+Cross-modality Fusion. With inherent limitation of every single modality, there are several methods to combine the strength of image and LiDAR with cross-modality fusion. In particular, point- and proposal-level fusion are introduced to combine features from different modalities. Point-level fusion [13, 17, 33] is usually applied in early stage of the network, while the proposal-based manner [6, 14, 46] is often adopted in the late stage for instance-level fusion. There are also methods that combine these two fusion manners, such as MVX-Net [32]. Compared with proposal-level fusion, the point-level one is a more subtle manner, which is also adopted in our method for deep fusion. Previous point-level fusion approaches [13, 33] usually enhance the point feature from image semantics in a point-to-point manner, ignoring the surrounding context in 3D space. Different from them, the proposed voxel field fusion represents the augmented image feature in a point-to-ray manner in the voxel field, which makes further use of merits from both modalities with sufficient context.
+
+# 3. Voxel Field Fusion
+
+The overall framework is conceptually simple: mixed augmentor is designed to align data augmentation across modalities; learnable sampler is introduced to efficiently select key features for interaction; and ray-wise fusion is proposed to fuse and combine features along the ray.
+
+# 3.1. Mixed Augmentor
+
+Given inputs captured from the camera and LiDAR, we first process the data with correspondence, as presented in Figure 2. To this end, a joint strategy, called mixed augmentor, is proposed to handle the aforementioned augmentation misalignment during training from two aspects, namely sample-added and sample-static augmentation.
+
+![](images/c134d46344e8ed036e64cd2b642ef1edebb595d7cd74a53cf71e2e48d1836419.jpg)  
+Figure 2. The framework for 3D object detection with voxel field fusion. In particular, inputs with different modalities are firstly processed with the mixed augmentor that is adopted for training only. Then, features of both modalities are respectively extracted in feature encoder, where the correspondence is established in ray constructor. In voxel field fusion, the vital image feature for interaction is selected with the designed sampler. And the ray-wise fusion is then conducted with the high-responded feature along each ray. With the fused and newly generated feature in the voxel field, the following detection backbone and head are applied to predict the final 3D proposals.
+
+Table 1. Corresponding operations in the mixed augmentor.  
+
+<table><tr><td>type</td><td>Point operation</td><td>Image operation</td></tr><tr><td>Sample-added</td><td>GT-sampling</td><td>Copy-paste</td></tr><tr><td rowspan="3">Sample-static</td><td>Flip</td><td>Image-flip</td></tr><tr><td>Rescale</td><td>Image-rescale</td></tr><tr><td>Rotate</td><td>Reproject</td></tr></table>
+
+Sample-added. The sample-added augmentation is defined to increase samples in each scene from the whole database, i.e., GT-sampling [38]. In this case, we supplement the RGB data of sampled 3D objects in a copy-paste manner [9, 49]. That means for each sampled object, we crop data within the projected 2D box and paste it onto the input image, where the crops are reorganized according to the actual depth or the cropping order. In this process, the occluded points covered by near samples are filtered to avoid the cross-modality ambiguity in nuScenes dataset, similar to that of [34].
+
+Sample-static. The sample-static augmentation includes a set of transformations without new sample added, e.g., flipping, rescaling, and rotation. Different from previous work [34, 50], which uses reprojection to find the pairwise correspondence across modalities, we utilize image-level operations for augmentations that affect pretrained 2D convolutions, as summarized in Table 1. Specifically, due to inherent properties like flipping and scaling variance of convolution, the asynchronous augmentation across modalities brings the misalignment. For example, if a flipping operation is applied to point cloud  $\mathcal{C}$  but not corresponding images  $\mathcal{P}$ , the left-right context of point  $p_i$  projected from point  $c_i$  would be malposed. We further validate effectiveness of the proposed workflow in Tables 2 and 3.
+
+# 3.2. Voxel Field Construction
+
+With the above designed augmentor, we have input image  $\mathbf{X}^I\in \mathbb{R}^{H\times W\times 3}$  and voxelized point cloud  $\mathbf{X}^V\in$ $\mathbb{R}^{X\times Y\times Z\times 4}$ , as depicted in Figure 2. The feature encoder with stacked convolutions is applied to extract feature  $\mathbf{F}_l^I$  and  $\mathbf{F}_l^V$  for image and voxel in the  $l$ -th stage, where  $l$  is set to 1 by default and further investigated in Table 7. With the ray constructor, the correspondence between voxel bin  $v_{i}$  and image pixel  $p_i$  is established in voxel field with the given projection matrix  $\mathbf{T}_{\mathrm{Voxel}}\to \mathrm{Image}$ .
+
+Voxel Field. In voxel representation, point clouds of a scene are captured within a voxel space  $\mathcal{V}$  that contains several bins. Every single voxel  $v$  in it can be represented by the function  $\mathcal{F}$ , called voxel field. Specifically, for the feature  $\mathrm{F}_{l,v}^{V}$  in voxel bin  $v$  with coordinates  $(x,y,z)$ , we have  $\mathrm{F}_{l,v}^{V} = \mathcal{F}(x,y,z)$ . In the voxel field, the ray  $\mathcal{R}_i\in \mathcal{V}$  from the point  $p_i$  through the voxel space  $\mathcal{V}$  in a fixed direction is constructed with
+
+$$
+p _ {i} = v _ {j} \mathbf {T} _ {\mathrm {V o x e l} \rightarrow \mathrm {I m a g e}} ^ {T}, \forall v _ {j} \in \mathcal {R} _ {i}. \tag {1}
+$$
+
+This means all voxel bins  $v_{j}$  with the same projected point  $p_i$  are marked to be located in the  $i$ -th ray set  $\mathcal{R}_i$ . Theoretically, the number of the whole set  $\mathcal{R}$  can be up to  $W \times H$  if without constraint, which brings huge computational cost that linearly increases with the number of sampled points.
+
+Learnable Sampler. To reduce the computational burden, learnable sampler is proposed to select  $n$  points from the image plane for ray interaction, which brings  $n$  rays in total. In particular, we first split the image to several non-overlap windows with size  $w \times w$  and filter out the empty windows with no projected point  $p_i$ , where  $w$  is set to 64 by default. Different from heuristic approaches for sampling, we adopt a learnable strategy to select key features by importance, as illustrated in Figure 3. For heuristic manner,  $n$  features from the image plane are randomly sampled
+
+![](images/f13698a867862433267549673813545ee12a5d84c01c1c27de256f33079acdf5.jpg)  
+(a) Sample by uniformity
+
+![](images/a69c035a72daa88cbaee93131a63958e2d8a0b01d79c4218c35db9e596bb0faa.jpg)  
+(b) Sample by density
+
+![](images/87c43779856950738058b2569aca051402f9a0b3e1df5cd1f02a126060fa74ce.jpg)  
+(c) Sample by sparsity
+
+![](images/3a0c7a4f047429850d88c23b9d93d510f5a436e760350dcc47ec5eea3612b937.jpg)  
+(d) Sample by importance
+
+according to uniformity, density, and sparsity of projected LiDAR points. Although the heuristic sampling method reduces the cost for ray construction, it still introduces several useless points that could increase computation in this process. To facilitate the efficiency, a learnable sampler  $\mathcal{S}$  is further proposed, which only conducts the sampling procedure from the predicted important sub-region with high responses, as depicted in Figure 3d. Therefore, a set of sampled pixels  $\widehat{\mathcal{P}}$  is achieved by
+
+$$
+\mathcal {S} (\mathcal {P}) = \mathcal {U} \left(\left\{p _ {i}: \mathbb {1} \left(\delta \left(f \left(\mathrm {F} _ {l, i} ^ {I}\right)\right)\right) = 1 \right\}\right), \tag {2}
+$$
+
+where  $f, \delta$ , and  $\mathcal{U}$  denote stacked convolutions, sigmoid activation, and uniform sampler, respectively. The indicator  $\mathbb{1}$  is set to 1 if the activated response in  $p_i$  surpasses the threshold 0.5. Considering the importance of foreground instance in 3D detection, here we set the Gaussian region inside each 2D object box to be the positive area for supervision, which is further explained in Section 3.4. In this way, the amount of sampled pixels  $\widehat{\mathcal{P}}$ , as well as the cost, are further reduced. Meanwhile, the high accuracy is still retained thanks to the proposed learnable sampler, as compared in Table 4.
+
+# 3.3. Ray-voxel Interaction
+
+Recall from Equation (1), the ray  $\mathcal{R}_i$  is constructed with the pixel  $\hat{p}_i \in \widehat{\mathcal{P}}$  from the above designed sampler  $\mathcal{S}$ . Thus, the cross-modality fusion can be conducted with the ray  $\mathcal{R}_i$  in the voxel field. Previous research [17, 33, 34] performs fusion with sparse points from LiDAR sensor only, with no consideration of surrounding 3D context, as presented in Figure 4a. Therefore, a naive solution is to extend the perception region to include points that are located in the nearby region with radius  $r$  of the voxel space  $\mathcal{V}$ , named local fusion. In this section, we first introduce the basic local fusion and then improve over it to formulate the designed ray-wise fusion.
+
+![](images/d2265541b9bdc056f2a65a0ee1de5015802f0817737e74c29493e23a246f6d81.jpg)  
+(a) Single fusion with each point
+
+![](images/2b04f66efc516a4c3adb04acf1dbde202dca663bd6d645e589652dcd610996b1.jpg)  
+(b) Local fusion with aggregation
+
+![](images/90c99ee854a4fd7adad81fc9bb1e70716c0b665943c46fa6c8830cefb23f462f.jpg)  
+Figure 3. Toy examples of different sampling methods. The blue points denote the sampled pixels for ray interaction. Compared with heuristic manners, our proposed learnable sampler in 3d only considers the important green area with high responses.  
+Figure 4. Toy examples of different fusion methods. The red point  $\hat{v}_j$  denotes the anchor voxel with LiDAR points. Green points in  $\mathcal{R}_i$  indicate voxel bins  $v_{j}$  within the ball with radius  $r$ . The dotted line in 4d represents the assigned existence probability for training.
+
+![](images/6fd4dd7d1cd1259581bf419455246af77cf6dc00014777e5582b0a8992613b82.jpg)  
+(c) Local fusion with propagation  
+(d) Ray-wise fusion along the ray
+
+Local Fusion. In this case, two types of local fusion are designed as our benchmark. Given the prior that features along the ray are more likely to be located near LiDAR points and the closer features usually contribute more, local fusion is conducted inside each Gaussian ball and ignores outside features along ray  $\mathcal{R}_i$ . Here, we call the voxels that contains LiDAR points as anchor voxels. The local fusion is divided into aggregation and propagation according to specific operations. The aggregation manner in Figure 4b aggregates image features to the anchor voxel  $\hat{v}_j$  with Gaussian weight  $\omega$ , while the propagation one in Figure 4c propagates features in  $\hat{v}_j$  to each voxel with weight  $\omega$ .
+
+Ray-wise Fusion. Although the designed local fusion extends the perception region from a single voxel to near area, it still sets a hard boundary in this process. And such an approach cannot fully release the potential of ray-wise representation, especially in most of the voxels without LiDAR points. Thus, besides the anchor voxel  $\hat{v}_j$ , we further extend the operation region to the whole ray, called ray-wise fusion. Compared with single fusion in Figure 4a, ours is more robust to sensor jitter due to a larger fault-tolerant space brought by Gaussian ball. Different from local fusion, the ray-wise manner in Figure 4d takes the above local prior to label assign in the training stage only, as elaborated in Section 3.4. Specifically, given the voxel  $v_j$  along the ray  $\mathcal{R}_i$ , its probability  $\omega_{j}$  is calculated by
+
+$$
+\omega_ {j} = \delta \left(\left\langle \mathbf {F} _ {l, i} ^ {I}, \mathbf {F} _ {l, v _ {j}} ^ {\prime} \right\rangle\right), \tag {3}
+$$
+
+where  $\delta$  denotes sigmoid activation, and the voxel feature  $\mathbf{F}_{l,v_j}^{\prime} = \mathrm{MLP}([x_j,y_j,z_j])$  is transformed from the coordinate  $(x_{j},y_{j},z_{j})$  of  $v_{j}$  with multi-layer perceptron. Here,  $\omega_{j}$  can be viewed as the response of image feature  $\mathbf{F}_{l,i}^{I}$  to the position of  $v_{j}$ . From another perspective, this operation converts the monocular depth estimation with  $\mathbf{F}_{l,i}^{I}$  from a
+
+regression problem to classification in a single ray  $\mathcal{R}_i$ . It actually reduces the solution space to a ray. With the predicted score  $\omega_{j}$ , we conduct the fusion in voxel  $v_{j}$  with
+
+$$
+\widehat {\mathcal {F}} (x _ {j}, y _ {j}, z _ {j}) = \mathcal {F} (x _ {j}, y _ {j}, z _ {j}) + \omega_ {j} f ([ \mathbf {F} _ {l, i} ^ {I}, \mathbf {F} _ {l, v _ {j}} ^ {\prime} ]). \quad (4)
+$$
+
+Here,  $f$  denotes the convolution.  $\widehat{\mathcal{F}}(x_j, y_j, z_j)$  indicates the new generated feature in voxel  $v_j$ . In this process,  $\mathcal{F}(x_j, y_j, z_j)$  is set to 0 if the original voxel  $v_j$  is empty, which can be viewed as completion in empty voxels of  $\mathcal{V}$ . For network efficiency, only the voxels with top predicted scores  $\omega$  are selected for fusion, which accounts for a quarter of original non-empty voxels in total. As presented in Figure 2, the fused feature  $\widehat{F}_l^V$  is obtained for the next stage in the 3D detection backbone and the following detection head. This framework can be instantiated with various voxel-based backbones for experiments in Section 4, e.g., PV-RCNN [27], Voxel R-CNN [8], and CenterPoint [44].
+
+# 3.4. Optimization Objectives
+
+Recall from Equations (2) and (3), there are two learnable factors required supervision. For learnable sampler  $S$ , given the prior that foreground objects are usually more important, we draw Gaussian distribution inside each box with
+
+$$
+\mathbf {Y} _ {l, u, v} = \exp \left(- \frac {\left(u - \hat {u} _ {i}\right) ^ {2} + \left(v - \hat {v} _ {i}\right) ^ {2}}{2 \sigma_ {i} ^ {2}}\right), \tag {5}
+$$
+
+where  $(\hat{u}_i,\hat{v}_i)$  denotes coordinate of the  $i$  -th object center, and  $\sigma_{i}$  indicates the object size-adaptive standard deviation. For activated probability  $\omega_{j}$  , the Gaussian-like supervision  $\hat{\omega}_{j}$  of voxel  $v_{j}$  is formulated as
+
+$$
+\hat {\omega} _ {j} = \exp \left(- \frac {(x - \hat {x} _ {j}) ^ {2} + (y - \hat {y} _ {j}) ^ {2} + (z - \hat {z} _ {j}) ^ {2}}{2 \sigma_ {j} ^ {2}}\right), \tag {6}
+$$
+
+where  $(\hat{x}_j,\hat{y}_j,\hat{z}_j)$  denotes the location of anchor voxel  $\hat{v}_j$  in Figure 4, and  $\sigma_{j}$  indicates the size-adaptive standard deviation. And the voxels with Euclidean distance larger than radius  $r$  are assigned as 0, which is investigated in Table 6. In this way, positions of LiDAR points can be utilized to provide supervision for feature selection in the ray. Finally, the objective function of voxel field fusion is defined as
+
+$$
+\mathcal {L} _ {\mathrm {v f f}} = \lambda_ {\mathrm {s}} \operatorname {B C E} \left(f \left(\mathrm {F} _ {l} ^ {I}\right), \mathbf {Y} _ {l}\right) + \lambda_ {\mathrm {r}} \sum^ {m} \operatorname {F L} (\omega , \hat {\omega}) / m, \tag {7}
+$$
+
+where  $f(\mathrm{F}_l^I)$  represents the convolved feature in Equation (2), and  $m$  denotes the size of ray set  $\mathcal{R}$ . BCE and FL indicate the binary cross-entropy loss and focal loss [18], respectively.  $\lambda_{\mathrm{s}}$  and  $\lambda_{\mathrm{r}}$  denote the balanced loss factor for sampler and ray-wise fusion. The optimization target for the whole network is the summation of raw detection loss  $\mathcal{L}_{\mathrm{det}}$  and voxel field fusion loss  $\mathcal{L}_{\mathrm{vff}}$ .
+
+# 4. Experiments
+
+In this section, the experimental setup is first introduced. Then, we give the analysis of each component on KITTI val set with PV-RCNN [27] as the backbone. Comparisons with previous work on nuScenes [2] and KITTI [10] datasets are reported in the end.
+
+# 4.1. Experimental Setup
+
+Dataset. KITTI dataset [10] is a widely-adopted multimodality benchmark for 3D object detection, which provides synchronized LiDAR points and front-view camera images. It contains 7,481 samples for training and 7,518 samples for testing, where the training samples are usually split to train set with 3,712 samples and val set with 3,769 samples. nuScenes dataset [2] is a large-scale benchmark for autonomous driving with 1,000 scenes, which are divided into 700, 150, 150 scenes in the train set, val set, and test set, respectively. Here, we use the synchronized data with 10 object categories that are collected from a 32-beam LiDAR and six cameras in a 360-degree field of view.
+
+Implementation Details. In this work, three different backbones are adopted to validate the proposed framework, i.e., PV-RCNN [27] and Voxel R-CNN [8] on KITTI dataset, and CenterPoint [44] on nuScenes dataset. We follow corresponding architecture and training settings in each network. In the proposed VFF, three convolutions and MLPs are applied to the learnable sampler in Equation (2) and feature transformation in Equation (3), where individual MLP is used for each camera view. For optimization,  $\lambda_{\mathrm{s}}$  and  $\lambda_{\mathrm{r}}$  in Equation (7) is set to 2 and 5 in all our experiments. Besides the non-empty voxels, we select features along each ray with probability  $\omega$  bigger than 0.05 in the inference stage.
+
+# 4.2. Component-wise Analysis
+
+Aligned Augmentation. As elaborated in Section 3.1, aligned data plays a vital role in cross-modality consistency. In Table 2, we compare the joint strategy for the sample-static augmentation listed in Table 1, where PV-RCNN and basic single fusion are adopted. As compared in Table 2, the designed image-level transformation contributes significantly to feature-variant augmentations, namely Flip and Rescale. If flip augmentation is adopted only, the performance gain is up to  $1.65\%$  AP on moderate cases.
+
+Mixed Augmentor. The proposed mixed augmentor in Section 3.1 maintains consistency in augmentation from sample-added and sample-static strategies. Here, we investigate the augmentor with VFF in Table 3. In the mixed augmentor, sample-added strategy yields notable gain with  $6.28\%$  AP. With the sample-static manner, the network with VFF is further improved to  $85.51\%$  AP on moderate cases. For clear comparisons, the following ablation studies are conducted with sample-added strategy only by default.
+
+Table 2. Comparisons on different augmentations on the KITTI val set. aug type and align type denote the adopted sample-static augmentation and type of cross-modality alignment, respectively.  
+
+<table><tr><td rowspan="2">aug type</td><td rowspan="2">align type</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>None</td><td>-</td><td>88.69</td><td>81.82</td><td>79.91</td></tr><tr><td rowspan="2">+ Flip</td><td>reproject</td><td>89.26</td><td>82.50</td><td>82.27</td></tr><tr><td>ours</td><td>91.34</td><td>84.15</td><td>82.48</td></tr><tr><td rowspan="2">+ Rescale</td><td>reproject</td><td>91.78</td><td>84.25</td><td>82.58</td></tr><tr><td>ours</td><td>91.68</td><td>84.53</td><td>82.59</td></tr><tr><td rowspan="2">+ Rotate</td><td>reproject</td><td>91.30</td><td>84.45</td><td>82.64</td></tr><tr><td>ours</td><td>91.43</td><td>84.57</td><td>82.61</td></tr></table>
+
+Table 3. Comparisons on different strategies in mixed augmentor on the KITTI val set. strategy and fusion denote the strategy in mixed augmentor and the usage of voxel field fusion, respectively.  
+
+<table><tr><td rowspan="2">strategy</td><td rowspan="2">fusion</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td rowspan="2">Original</td><td>×</td><td>86.40</td><td>75.47</td><td>71.32</td></tr><tr><td>✓</td><td>87.08</td><td>76.25</td><td>72.03</td></tr><tr><td rowspan="2">+ Added</td><td>×</td><td>87.77</td><td>79.19</td><td>76.71</td></tr><tr><td>✓</td><td>89.56</td><td>82.53</td><td>80.00</td></tr><tr><td rowspan="2">+ Static</td><td>×</td><td>91.53</td><td>84.36</td><td>82.29</td></tr><tr><td>✓</td><td>92.31</td><td>85.51</td><td>82.92</td></tr></table>
+
+Table 4. Comparisons on different sampling types on the KITTI val set. sample type and learn denote the adopted sampler and learnable manner for feature selection in Section 3.2, respectively.  
+
+<table><tr><td rowspan="2">sample type</td><td rowspan="2">learn</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>Uniformity</td><td>X</td><td>88.69</td><td>81.90</td><td>79.82</td></tr><tr><td>Sparsity</td><td>X</td><td>89.09</td><td>81.68</td><td>79.85</td></tr><tr><td>Density</td><td>X</td><td>88.72</td><td>82.01</td><td>81.42</td></tr><tr><td>Importance</td><td>✓</td><td>89.11</td><td>82.10</td><td>79.98</td></tr></table>
+
+Learnable Sampler. To facilitate voxel field construction, learnable sampler is proposed in Section 3.2. In Table 4, different sampling methods are compared, which are divided into heuristic and learnable sets. As presented in Figure 3, the heuristic sampler contains the type of uniformity, sparsity, and density. And the network with heuristic sampler achieves peak performance with  $82.01\% \mathrm{AP}$  when sampling based on density. As for the learnable sampler, the proposed sampling with importance in Figure 3d attains a superior result than the heuristic one with  $82.10\% \mathrm{AP}$ .
+
+Table 5. Comparisons on different fusion strategies on the KITTI val set. range and operation denote the fusion range and specific operation, respectively. Single fusion is a subset of the others.  
+
+<table><tr><td rowspan="2">range</td><td rowspan="2">operation</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>Single</td><td>-</td><td>88.69</td><td>81.82</td><td>79.91</td></tr><tr><td rowspan="2">Local</td><td>aggregate</td><td>89.00</td><td>82.12</td><td>81.52</td></tr><tr><td>propagate</td><td>89.16</td><td>82.07</td><td>81.17</td></tr><tr><td>Ray-wise</td><td>generate</td><td>89.56</td><td>82.53</td><td>80.00</td></tr></table>
+
+Table 6. Comparisons on different types of supervision on the KITTI val set. radius and gaussian denote the adopted radius  $r$  and the usage of Gaussian distribution in Section 3.3, respectively.  
+
+<table><tr><td rowspan="2">radius</td><td rowspan="2">gaussian</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>0</td><td>×</td><td>89.11</td><td>82.10</td><td>79.98</td></tr><tr><td>1</td><td>✓</td><td>89.56</td><td>82.53</td><td>80.00</td></tr><tr><td>2</td><td>✓</td><td>88.33</td><td>81.75</td><td>79.54</td></tr></table>
+
+Table 7. Comparisons on different fusion positions on the KITTI val set. stage and fusion denote the specific fusion stage in the network and the usage of voxel field fusion, respectively.  
+
+<table><tr><td rowspan="2">stage</td><td rowspan="2">fusion</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>-</td><td>X</td><td>87.77</td><td>79.19</td><td>76.71</td></tr><tr><td>Stage-1</td><td>✓</td><td>89.56</td><td>82.53</td><td>80.00</td></tr><tr><td>Stage-2</td><td>✓</td><td>89.12</td><td>80.43</td><td>79.70</td></tr><tr><td>Stage-3</td><td>✓</td><td>89.64</td><td>80.27</td><td>77.93</td></tr><tr><td>Stage-4</td><td>✓</td><td>88.69</td><td>80.22</td><td>78.10</td></tr></table>
+
+Ray-voxel Interaction. The interaction between ray and voxel is a core operation of the proposed framework in Section 3.3. Here, we compare different fusion methods presented in Figure 4. As shown in Table 5, the performance improves with fusion range increases from a single point to the whole ray. Compared with single fusion and local fusion, the ray-wise strategy surpasses them with a significant gap, which proves the effectiveness of ray-wise fusion.
+
+Ray-wise Supervision. Considering noise from sensor jitter or other issues, the Gaussian-like assignment is proposed to provide supervision in Figure 4d and Equation (6). Different supervision strategies are compared in Table 6. It is clear that the Gaussian distribution with radius 1 results in the best performance  $82.53\%$  AP. And extra region provides wrong guidance to the ray that goes through each Gaussian ball, which harms feature localization in each ray.
+
+Table 8. Comparisons on different methods with a single model on the KITTI val set. * denotes our result from official source code.  
+
+<table><tr><td rowspan="2">method</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td><td colspan="3">AP3D@Car-R11 (IoU=0.7)</td><td colspan="3">APBEV@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td><td>Easy</td><td>Moderate</td><td>Hard</td><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td colspan="10">LiDAR-based</td></tr><tr><td>SECOND [38]</td><td>-</td><td>-</td><td>-</td><td>87.43</td><td>76.48</td><td>69.10</td><td>-</td><td>-</td><td>-</td></tr><tr><td>PointRCNN [28]</td><td>-</td><td>-</td><td>-</td><td>88.88</td><td>78.63</td><td>77.38</td><td>-</td><td>-</td><td>-</td></tr><tr><td>STD [42]</td><td>-</td><td>-</td><td>-</td><td>89.70</td><td>79.80</td><td>79.30</td><td>-</td><td>-</td><td>-</td></tr><tr><td>PV-RCNN [27]</td><td>92.57</td><td>84.83</td><td>82.69</td><td>89.35</td><td>83.69</td><td>78.70</td><td>95.76</td><td>91.11</td><td>88.93</td></tr><tr><td>Voxel R-CNN [8]</td><td>92.38</td><td>85.29</td><td>82.86</td><td>89.41</td><td>84.52</td><td>78.93</td><td>95.52</td><td>91.25</td><td>88.99</td></tr><tr><td colspan="10">LiDAR+RGB</td></tr><tr><td>UberATG-MMF [16]</td><td>-</td><td>-</td><td>-</td><td>88.40</td><td>77.43</td><td>70.22</td><td>-</td><td>-</td><td>-</td></tr><tr><td>3D-CVF [46]</td><td>89.67</td><td>79.88</td><td>78.47</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>EPNet [13]</td><td>92.28</td><td>82.59</td><td>80.14</td><td>-</td><td>-</td><td>-</td><td>95.51</td><td>88.76</td><td>88.36</td></tr><tr><td>PV-RCNN*</td><td>91.53</td><td>84.36</td><td>82.29</td><td>88.95</td><td>83.51</td><td>78.72</td><td>92.82</td><td>90.43</td><td>88.41</td></tr><tr><td>+ VFF</td><td>92.31</td><td>85.51</td><td>82.92</td><td>89.45</td><td>84.21</td><td>79.13</td><td>95.43</td><td>91.40</td><td>90.66</td></tr><tr><td>Voxel R-CNN*</td><td>92.27</td><td>84.88</td><td>82.50</td><td>89.46</td><td>83.61</td><td>78.80</td><td>95.51</td><td>91.13</td><td>88.85</td></tr><tr><td>+ VFF</td><td>92.47</td><td>85.65</td><td>83.38</td><td>89.51</td><td>84.76</td><td>79.21</td><td>95.65</td><td>91.75</td><td>91.39</td></tr></table>
+
+Table 9. Comparisons on cross-modality fusion of different categories on the KITTI val set. We report results of car, pedestrian, and cyclist with IoU=0.7, 0.5, and 0.5, respectively.  
+
+<table><tr><td rowspan="2">category</td><td rowspan="2">fusion</td><td colspan="3">AP3D-R40</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td rowspan="2">Car</td><td>X</td><td>91.53</td><td>84.36</td><td>82.29</td></tr><tr><td>✓</td><td>92.31</td><td>85.51</td><td>82.92</td></tr><tr><td rowspan="2">Pedestrian</td><td>X</td><td>66.04</td><td>59.19</td><td>54.15</td></tr><tr><td>✓</td><td>73.26</td><td>65.11</td><td>60.03</td></tr><tr><td rowspan="2">Cyclist</td><td>X</td><td>91.31</td><td>72.18</td><td>67.60</td></tr><tr><td>✓</td><td>89.40</td><td>73.12</td><td>69.86</td></tr></table>
+
+Table 10. Comparisons on different types of pretrained 2D backbone on the KITTI val set. model denotes the adopted pretrained model in feature encoder with corresponding task.  
+
+<table><tr><td rowspan="2">model</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>Cls: ResNet [12]</td><td>91.96</td><td>85.33</td><td>84.24</td></tr><tr><td>Det: Faster R-CNN [26]</td><td>91.98</td><td>85.11</td><td>82.52</td></tr><tr><td>Seg: DeepLabV3 [3]</td><td>92.31</td><td>85.51</td><td>82.92</td></tr></table>
+
+Fusion Stage. We further investigate the fusion stage of the proposed VFF in Table 7. Compared with the baseline, the designed fusion yields superior performance in each stage. The early-stage fusion contributes more, which surpasses the baseline with  $3.34\%$  AP. Meanwhile, fusion in the later stage brings less gain, which could be attributed to the low
+
+Table 11. Comparisons on cross-modality fusion of different data on the KITTI val set. beam num and fusion denote LiDAR beam number and the usage of voxel field fusion, respectively.  
+
+<table><tr><td rowspan="2">beam num</td><td rowspan="2">fusion</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td rowspan="2">Beam-64</td><td>X</td><td>91.53</td><td>84.36</td><td>82.29</td></tr><tr><td>✓</td><td>92.31</td><td>85.51</td><td>82.92</td></tr><tr><td rowspan="2">Beam-32</td><td>X</td><td>91.14</td><td>79.51</td><td>76.54</td></tr><tr><td>✓</td><td>92.20</td><td>82.36</td><td>79.81</td></tr></table>
+
+resolution for key feature selection and insufficient fusion.
+
+Different Categories. The ray-wise representation of image features in VFF introduces sufficient context for ambiguous examples. In Table 9, we report comparisons with VFF on various categories. It is clear that the performance of each class has been improved especially for pedestrian, which is up to nearly  $6\%$  AP on cases of all difficulties.
+
+Pretrained Network. In Table 10, we analyze the pretrained 2D backbone, which provides the feature  $\mathbf{F}_l^I$  in the feature encoder of Figure 2. Here, we adopt all the augmentations and ResNet-50 based models for different tasks in Table 10, i.e., classification, detection, and semantic segmentation. Compared with other tasks, the network [3] provides better features if pretrained with semantic setting.
+
+Sparse LiDAR. To verify the effectiveness of VFF with different LiDAR sparsity, we downsample the LiDAR points on the KITTI dataset to 32-beam following [47]. As pre
+
+Table 12. Comparisons on different methods with single model on the nuScenes test set.  
+
+<table><tr><td>method</td><td>mAP</td><td>NDS</td><td>Car</td><td>Truck</td><td>Bus</td><td>Trailer</td><td>C.V.</td><td>Ped.</td><td>Motor.</td><td>Bicycle</td><td>T.C.</td><td>Barrier</td></tr><tr><td colspan="13">LiDAR-based</td></tr><tr><td>PointPillars [15]</td><td>30.5</td><td>45.3</td><td>68.4</td><td>23.0</td><td>28.2</td><td>23.4</td><td>4.1</td><td>59.7</td><td>27.4</td><td>1.1</td><td>30.8</td><td>38.9</td></tr><tr><td>3DSSD [41]</td><td>42.6</td><td>56.4</td><td>81.2</td><td>47.2</td><td>61.4</td><td>30.5</td><td>12.6</td><td>70.2</td><td>36.0</td><td>8.6</td><td>31.1</td><td>47.9</td></tr><tr><td>CBGS [52]</td><td>52.8</td><td>63.3</td><td>81.1</td><td>48.5</td><td>54.9</td><td>42.9</td><td>10.5</td><td>80.1</td><td>51.5</td><td>22.3</td><td>70.9</td><td>65.7</td></tr><tr><td>CenterPoint [44]</td><td>60.3</td><td>67.3</td><td>85.2</td><td>53.5</td><td>63.6</td><td>56.0</td><td>20.0</td><td>84.6</td><td>59.5</td><td>30.7</td><td>78.4</td><td>71.1</td></tr><tr><td colspan="13">LiDAR+RGB</td></tr><tr><td>PointPainting [33]</td><td>46.4</td><td>58.1</td><td>77.9</td><td>35.8</td><td>36.2</td><td>37.3</td><td>15.8</td><td>73.3</td><td>41.5</td><td>24.1</td><td>62.4</td><td>60.2</td></tr><tr><td>FusionPainting [37]</td><td>66.3</td><td>70.4</td><td>86.3</td><td>58.5</td><td>66.8</td><td>59.4</td><td>27.7</td><td>87.5</td><td>71.2</td><td>51.7</td><td>84.2</td><td>70.2</td></tr><tr><td>MVP [45]</td><td>66.4</td><td>70.5</td><td>86.8</td><td>58.5</td><td>67.4</td><td>57.3</td><td>26.1</td><td>89.1</td><td>70.0</td><td>49.3</td><td>85.0</td><td>74.8</td></tr><tr><td>PointAugmenting [34]</td><td>66.8</td><td>71.0</td><td>87.5</td><td>57.3</td><td>65.2</td><td>60.7</td><td>28.0</td><td>87.9</td><td>74.3</td><td>50.9</td><td>83.6</td><td>72.6</td></tr><tr><td>VFF + CenterPoint</td><td>68.4</td><td>72.4</td><td>86.8</td><td>58.1</td><td>70.2</td><td>61.0</td><td>32.1</td><td>87.1</td><td>78.5</td><td>52.9</td><td>83.8</td><td>73.9</td></tr></table>
+
+Table 13. Comparisons on different methods on KITTI test set.  
+
+<table><tr><td rowspan="2">method</td><td colspan="3">AP3D@Car-R40 (IoU=0.7)</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td colspan="4">LiDAR-based</td></tr><tr><td>PointPillars [15]</td><td>82.58</td><td>74.31</td><td>68.99</td></tr><tr><td>PointRCNN [28]</td><td>86.96</td><td>75.64</td><td>70.70</td></tr><tr><td>Part-A2[29]</td><td>87.81</td><td>78.49</td><td>73.51</td></tr><tr><td>STD [42]</td><td>87.95</td><td>79.71</td><td>75.09</td></tr><tr><td>SA-SSD [11]</td><td>88.75</td><td>79.79</td><td>74.16</td></tr><tr><td>PV-RCNN [27]</td><td>90.25</td><td>81.43</td><td>76.82</td></tr><tr><td>Voxel R-CNN [8]</td><td>90.90</td><td>81.62</td><td>77.06</td></tr><tr><td colspan="4">LiDAR+RGB</td></tr><tr><td>MV3D [6]</td><td>74.97</td><td>63.63</td><td>54.00</td></tr><tr><td>F-PointNet [23]</td><td>82.19</td><td>69.79</td><td>60.59</td></tr><tr><td>AVOD [14]</td><td>83.07</td><td>71.76</td><td>65.73</td></tr><tr><td>UberATG-MMF [16]</td><td>88.40</td><td>77.43</td><td>70.22</td></tr><tr><td>EPNet [13]</td><td>89.81</td><td>79.28</td><td>74.59</td></tr><tr><td>3D-CVF [46]</td><td>89.20</td><td>80.05</td><td>73.11</td></tr><tr><td>VFF + PV-RCNN</td><td>89.58</td><td>81.97</td><td>79.17</td></tr><tr><td>VFF + Voxel R-CNN</td><td>89.50</td><td>82.09</td><td>79.29</td></tr></table>
+
+sented in Table 11, the proposed VFF achieves significant gain with  $2.85\%$  AP over the baseline. For hard cases, the gap is enlarged to  $3.27\%$  AP. It could attribute to the replenishment of empty voxels that lack LiDAR points.
+
+# 4.3. Main Results
+
+nuScenes. We further report results on the large scale nuScenes test set. As shown in Table 12, the proposed method surpasses all previous approaches with  $68.4\%$  mAP and  $72.4\%$  NDS. Compared with our strong backbone CenterPoint [44], the performance gain brought by VFF is up to
+
+8.1% mAP and 5.1% NDS. As for ambiguous classes like motorcycle and bicycle, the gain is even up to 19% AP.
+
+KITTI. In Table 8, we carry out experiments on KITTI val set. Compared with the baseline, our proposed VFF achieves consistent gain on various evaluation metrics and attains  $85.51\%$  and  $85.65\%$  AP with PV-RCNN [27] and Voxel R-CNN [8], respectively. The result on KITTI test set is reported on Table 13. Compared with previous fusion-based methods, the proposed VFF pushes the top performance to  $81.97\%$  AP and  $82.09\%$  AP with PV-RCNN and Voxel R-CNN as the backbone, respectively. Thanks to the designed fusion manner, our method outperforms all previous models on hard cases with  $79.29\%$  AP, which attains  $2.2\%$  AP improvement over the baseline.
+
+# 5. Conclusion
+
+We have presented the voxel field fusion, a conceptually simple yet effective framework for cross-modality fusion in 3D object detection. The key difference from prior works lies in that we maintain modality consistency by representing and fusing augmented image features as a ray in the voxel field. In particular, inconsistency in feature representation for multi-modalities is eliminated with learnable sampler and ray-wise fusion. Meanwhile, the mixed aug-mentor is developed to bridge the gap in cross-modality data augmentation. Experiments on KITTI and nuScenes dataset prove the effectiveness of the proposed framework, which achieves consistent gains in various benchmarks and surpasses previous fusion-based models on both datasets.
+
+# 6. Acknowledgments
+
+We thank Yilun Chen and Tao Hu for their suggestions. This work is supported by The National Key Research and Development Program of China (No. 2017YFA0700800) and Beijing Academy of Artificial Intelligence (BAAI).
+
+# References
+
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+[2] Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. In CVPR, 2020.  
+[3] Liang-Chieh Chen, George Papandreou, Florian Schroff, and Hartwig Adam. Rethinking atrous convolution for semantic image segmentation. arXiv:1706.05587, 2017.  
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+# Voxel Set Transformer: A Set-to-Set Approach to 3D Object Detection from Point Clouds
+
+Chenhang He, Ruihuang Li, Shuai Li, Lei Zhang* The Hong Kong Polytechnic University
+
+{csche, csrhli, csshuali, cshzeng}@comp.polyu.edu.hk
+
+# Abstract
+
+Transformer has demonstrated promising performance in many 2D vision tasks. However, it is cumbersome to compute the self-attention on large-scale point cloud data because point cloud is a long sequence and unevenly distributed in 3D space. To solve this issue, existing methods usually compute self-attention locally by grouping the points into clusters of the same size, or perform convolutional self-attention on a discretized representation. However, the former results in stochastic point dropout, while the latter typically has narrow attention fields. In this paper, we propose a novel voxel-based architecture, namely Voxel Set Transformer (VoxSeT), to detect 3D objects from point clouds by means of set-to-set translation. VoxSeT is built upon a voxel-based set attention (VSA) module, which reduces the self-attention in each voxel by two cross-attention and models features in a hidden space induced by a group of latent codes. With the VSA module, VoxSeT can manage voxelized point clusters with arbitrary size in a wide range, and process them in parallel with linear complexity. The proposed VoxSeT integrates the high performance of transformer with the efficiency of voxel-based model, which can be used as a good alternative to the convolutional and point-based backbones. VoxSeT reports competitive results on the KITTI and Waymo detection benchmarks. The source codes can be found at https://github.com/skyhehe123/VoxSeT.
+
+# 1. Introduction
+
+Object detection from 3D point cloud has been receiving extensive attention as it empowers many applications like autonomous driving, robotics and virtual reality. Unlike 2D images, 3D point clouds are naturally sparse and unevenly distributed in continuous space, impeding the CNN layers from being directly applied. To resolve this issue, some approaches [5, 10, 42, 49, 51] first transform the point cloud
+
+![](images/75497b0b8dc70c1b96b91e553beca6126402b9aa92fd4139695e96e8de076ad7.jpg)  
+(a)
+
+![](images/0ae7a92ed1f8098e739115f7e00ef04991c476ba946440ac00ea81ed991f6408.jpg)  
+(b)
+
+![](images/f0e68a2bae6f6f84994cc24cbb06cb2b5f8e22be8f6c1726836d71ed0c3f0505.jpg)  
+(c)  
+Figure 1. The illustrations of (a) grouping-based, (b) convolutional-based and (c) our proposed induced set-based attention mechanisms.
+
+into a discrete representation and then apply CNN models to extract high dimensional features. Another class of approaches [27, 33, 41, 44, 45] model the point cloud in continuous space, where the multi-scale features are extracted through interleaved grouping and aggregation steps.
+
+Beyond the above two schemes, transformer-based models [9, 21-23, 31, 47] have recently attracted great interest in processing point cloud data as the self-attention used in transformers is invariant to permutation and cardinality of the input components, which makes transformer an appropriate choice for point cloud processing. The main limitation of transformer models, however, lies in that the self-attention computation is quadratic. Each token has to be updated by using all the other tokens from previous layers, making self-attention intractable for long sequence point clouds. Point Transformer [47] builds transformers upon a PointNet [28] architecture, which hierarchically groups the point cloud data into different clusters and computes self-attention in each cluster. CT3D [31] presents a two-stage point cloud detector, where 3D RoIs are extracted to group the raw points in the first stage and transformers are applied to the grouped points in the second stage.
+
+However, since the distribution of point clouds is extremely uneven, the number of points in each cluster varies a lot. To enable the self-attention to run in parallel, cur
+
+rent approaches [23, 31, 47] balance the token number in each cluster by stochastically dropping points or padding dummy points (see Figure 1(a)). This results in unstable detection results and redundant computations. Besides, each operation of grouping  $n$  points to  $m$  clusters will cost  $\mathcal{O}(nm)$  complexity, which is relatively intensive. Alternatively, Voxel Transformer [21] performs self-attention on a discrete voxel grid, as depicted in Figure 1(b). It computes self-attention in a convolutional manner and hence is as efficient as sparse convolution with  $\mathcal{O}(n)$  complexity. However, since convolutional attention is a point-wise operation, to save the memory, the attention field of the convolutional kernel is typically small, thus hindering the voxel transformer to model long-range dependencies. It is worth mentioning that though Group-free [20] and 3DETR [22] present a promising solution by computing self-attention on a reduced set of seed points, this solution is only applicable to indoor scenes, where the point clouds are relatively dense and concentrated. Considering that the point clouds of outdoor scenes are typically sparse, large-scale ( $e.g., > 20k$ ), and unevenly distributed, the scale and coverage of seed points remain an issue.
+
+To address the above issues, we introduce a voxel-based set attention (VSA) module. For each VSA, we divide the whole scene into non-overlapping 3D voxels and compute the voxel indices of the input point with instant efficiency. We use these voxels to determine the attentive region which is analogous to the window attention in SwinTransformer [19]. Unlike image, LiDAR has irregular structures, and the resulting attention groups have different lengths, which hinders the parallelization of the model.
+
+Inspired by the induced set transformer [13], we assign a group of trainable "latent codes" to each voxel. These latent codes build a fixed-length bottleneck for the point cloud, through which the information from input points within the voxel can be compressed to a static hidden space. This formulation is based on the key observation that the self-attention matrix is typically low-rank, and hence we can decompose an intensive full self-attention into two consecutive cross-attention modules. As shown in Figure 1(c), VSA first transforms the latent codes, which serve as queries, to a hidden space by attending to the projected features, i.e., keys and values, from the input points. The transformed hidden features, which encode the context information of the input points in each voxel, are enriched by a convolutional feed-forward network, in which the features across voxels exchange their information in spatial domain. After that, the hidden features are attentively fused with input, producing output features of the input resolution. By leveraging the latent codes, the cross-attention performed in all voxels can be vectorized, making VSA a highly parallel module. Given  $n$ $d$ -dimensional input features and  $k$  latent codes, VSA has a complexity of  $\mathcal{O}(nkd)$  and it can
+
+be implemented with general matrix multiplications.
+
+With VSA, we propose a Voxel Set Transformer (VoxSeT) to detect 3D objects by learning point cloud features in a set-to-set translation process. VoxSeT is composed of VSA modules, MLP layers and a shallow CNN for Birds-Eye-View (BEV) feature extraction. To verify the effectiveness of the proposed model, we conduct experiments on two 3D detection benchmarks, KITTI and Waymo open dataset. VoxSeT achieves competitive performance with current state-of-the-arts. In addition, the proposed VSA module can be seamlessly adopted into point-based detectors such as PointRCNN [33], and demonstrates advantages over the set abstraction module.
+
+In summary, in this work we first invent a voxel-based set attention module, which can model long-range dependencies from the token cluster of arbitrary size, bypassing the limitation of current grouped-based and convolution-based attention modules. We then present a Voxel Set Transformer to learn point cloud features effectively by leveraging the superiority of transformer on large-scale sequential data. Our work provides a novel alternative to the current convolutional and point-based backbones for 3D point cloud data processing.
+
+# 2. Related work
+
+# 2.1. 3D object detection from point clouds
+
+Early approaches on 3D object detection from point cloud can be categorized into two classes. The first class of methods transform the point cloud into more compact representations, e.g., Birds-Eye-View (BEV) images [3,11,34], frontal-view range images [2,7,18], and volumetric features [14,43,51]. Yan et al. [42] developed a sparse convolutional backbone to efficiently process the point clouds by encoding the point clouds into a 3D sparse tensor. Lang et al. [12] further accelerated the detection rate by stacking the voxel features as a "pillar" and using 2D CNN to process. Another class of methods [25,27,33,44,45] process the point cloud in a continuous space by employing a PointNet [29] architecture. The point-wise features in multi-scales are extracted in stages with interleaved grouping and sampling operations. Shi et al. [33] and Yang et al. [45] proposed to generate 3D RoIs from PointNet outputs and apply the RoIs to group point-wise features for further refinement. Qi et al. [26] proposed a deep voting method to cluster the points from objects' surface to detect the object with insufficient points. Unlike compact representations, point-wise features preserve more details and fine-grained structures of original point clouds. Based on this fact, some approaches [10,32] employ a hybrid representation in both point and voxel spaces to achieve more reliable detection outputs.
+
+Our proposed architecture is largely motivated by voxel
+
+based approaches. We partition the point cloud into voxel grid and execute self-attention locally, endowing our model with inductive bias and computational efficiency.
+
+# 2.2. Transformer in point cloud analysis
+
+Recently, Transformer [38] has demonstrated its great success in many computer vision tasks such as image classification [6, 37], 2D object detection [1, 19], and other dense prediction tasks [30, 48]. For point cloud analysis, Zhao et al. [47] proposed a novel subtraction attention based operator for point cloud classification and segmentation. Guo et al. [9] investigated a dual attention to process the point clouds in feature and edge space. Misra et al. [22] and Liu et al. [20] used transformer to process point clouds as sequential data, preventing the models from stacking hierarchical grouping and sampling modules. Miao et al. [21] embedded self-attention into a sparse convolutional kernel. Sheng et al. [31] built the transformer on top of a two-stage detector and operated attention on the points grouped by RoIs.
+
+Unlike the above approaches that perform self-attention on a fixed-size token cluster, our proposed Voxel Set Transformer leverages the idea of induced set transformer [13] to decompose self-attention into two cross-attentions, making it possible to perform self-attention on the token clusters of arbitrary size.
+
+# 3. Methodology
+
+# 3.1. Preliminary
+
+It is prohibitive to directly apply self-attention on point cloud data due to its quadratic computational complexity. To bypass the issue, an induced set attention block was proposed in [13], where the full self-attention in a set was approximated by two reduced cross-attentions induced by a group of latent codes. Given an input set  $X \in \mathcal{R}^{n \times d}$  of size  $n$  with dimension  $d$  and  $k$  latent codes  $L \in \mathcal{R}^{k \times d}$ , the output set  $O \in \mathcal{R}^{n \times d}$  from the induced set attention block can be formulated as
+
+$$
+H = \operatorname {C r o s s A t t e n t i o n} (L, X) \in \mathcal {R} ^ {k \times d}, \tag {1}
+$$
+
+$$
+\hat {H} = \operatorname {F F N} (H) \in \mathcal {R} ^ {k \times d}, \tag {2}
+$$
+
+$$
+O = \operatorname {C r o s s A t t e n t i o n} (X, \hat {H}) \in \mathcal {R} ^ {n \times d}. \tag {3}
+$$
+
+The first cross attention transforms the latent features  $L$  into hidden features  $H$  by attending to the input set. This step costs  $\mathcal{O}(nkd)$  complexity, which is linear to  $n$  as the number of latent codes  $k$  is fixed and usually very small. The transformed hidden features contain information about the input set  $X$  and then they are updated by a pointwise feed-forward network (FFN). This point-wise operation costs  $\mathcal{O}(k)$  complexity and it learns highly semantic features from the input set. The second cross attention attends the input set to the resulting hidden features, which
+
+costs  $\mathcal{O}(nkd)$  complexity, producing an output set of length  $n$ . The induced set attention is based on the assumption that the self-attention can be approximated with low-rank projections, thus the self-attention can be regarded as performing a  $k$ -clustering on the inputs where the latent codes serve as cluster centers. This is also analogous to the clustered attention [39] and Linformer [40], where the input set is explicitly reduced with linear projection.
+
+# 3.2. Voxel-based Set Attention (VSA)
+
+Unlike images, point clouds are widely distributed and have weak semantic associations in scene level, while they have strong structural details in the local region. Instead of compressing all the input points into a hidden space, we modify the above induced set attention to be performed locally. Specifically, we partition the scene into a voxel grid and assign a set of latent codes to each voxel. We refer to the module as Voxel-based Set Attention (VSA).
+
+Scatter kernel function. As mentioned before, VSA is a highly parallel module, where the operations across voxels can be vectorized. This vectorization can be achieved by the scatter function<sup>1</sup>, which is a CUDA kernel library that performs symmetric reduction, e.g., sum, max and mean, on different segments of a matrix. In our case, we regard the input set as a single matrix, each row of which corresponds to a point-wise feature, and its belonging voxel can be indexed by a table of voxel coordinates.
+
+Let  $\{p_i = (x_i,y_i,z_i):i = 1,\dots n\}$  denote the coordinates of point cloud and  $[d_x,d_y,d_z]$  be the voxel size in three dimensions. The voxel coordinates  $\mathcal{V}$  can be computed by  $\mathcal{V} = \{\mathcal{V}_i = (\lfloor \frac{x_i}{d_x}\rfloor ,\lfloor \frac{y_i}{d_y}\rfloor ,\lfloor \frac{z_i}{d_z}\rfloor):i = 1,\ldots ,n\}$ , where  $\lfloor \cdot \rfloor$  is the floor function. Hence, given point-wise input features  $\{X_{i}:1 = 1,\dots n\}$ , their reduced voxel-wise form  $\{Y_j:j = 1,\dots,m\}$  after a symmetric function  $\mathrm{F}(\cdot)$  can be represented as:
+
+$$
+Y = \left\{\mathrm {F} \left(\left\{X _ {i}: \mathcal {V} _ {i} = j \right\}\right): j = 1, \dots , m \right\} \tag {4}
+$$
+
+where  $m$  is the number of non-empty voxels. With scatter function  $\mathbf{F}_{scatter}$ , the above equation can be written in a vectorized form, i.e.,
+
+$$
+Y = \mathrm {F} _ {\text {s c a t t e r}} (X, \mathcal {V}). \tag {5}
+$$
+
+By deploying VSA, we do not need to stochastically drop or pad the points in each voxel and the complexity of the model is linear.
+
+In Figure 2, we illustrate the VSA in a matrix-multiplication form for ease of comprehension. As can be seen, the module is analogous to an encoder-decoder architecture, where the input set is encoded to a hidden space, then the hidden features are refined through a ConvFFN and finally decoded to produce the output set.
+
+![](images/eed3e2ff6d0d30af933baca0de8b4122d8c4d4d880522599d059ec67eeca11ea.jpg)  
+Figure 2. The matrix multiplication used in implementing the voxel-based set attention (VSA) module. The pseudo codes in PyTorch-stype of each step are presented bellow the diagram.
+
+Encoder. In the encoder, we first project the input features from the previous module to the key  $K \in \mathcal{R}^{n \times d}$  and the value  $V \in \mathcal{R}^{n \times d}$  with linear projections, respectively. Next, we perform cross attention between the key and the latent codes (query)  $L \in \mathcal{R}^{k \times d}$ , producing the attention matrix  $\mathcal{A} \in \mathcal{R}^{n \times k \times d}$ . The attention matrix  $\mathcal{A}$  is then normalized voxel-wisely to obtain  $\tilde{\mathcal{A}}$ , and multiplied with the value, producing hidden features  $\tilde{H}$ . The calculation of  $\tilde{H}$  can be formulated as:
+
+$$
+\tilde {\mathcal {A}} = \operatorname {S o f t m a x} _ {\text {s c a t t e r}} (\mathcal {A}, \mathcal {V}), \mathcal {A} = K L ^ {T}, \tag {6}
+$$
+
+$$
+H = \tilde {A} ^ {T} V. \tag {7}
+$$
+
+After that, we perform voxel-wise reduction on the hidden features based on the voxel indices  $\nu$ :
+
+$$
+H _ {r} = \operatorname {S u m} _ {\text {s c a t t e r}} (H, \mathcal {V}). \tag {8}
+$$
+
+The overall computations in the encoder include two GEMMs and two scatter operations. The overall complexity of the encoder is  $\mathcal{O}(2n(k + 1)d)$ .
+
+It is worth mentioning that the cross-attention based encoding scheme can be viewed as an extension of voxel feature encoding (VFE) used in [12, 42, 51]. The difference is that VFE encodes the points within a voxel into a single feature vector, while our scheme encodes the points based on a codebook consisting of the latent features. Owing to the high expressive power of VSA, we can use a relatively large voxel size to capture the features in a wide range.
+
+Convolutional feed-forward network. The core idea of VSA is to encode the region-wise features into a hidden space using latent codes. The hidden features work as a bottleneck, through which we apply a ConvFFN to achieve more flexible and complex information update. Unlike conventional FFN that only performs point-wise token update, ConvFFN enables the information exchange across voxels, which is especially important for dense prediction. To adaptively integrate voxel features with global dependencies, we
+
+scatter the reduced hidden features into a 3D sparse tensor based on their voxel coordinates  $C_r$ , and then conduct two depth-wise convolutions (DwConv) on them to enforce the feature interaction in spatial domain. Given convolutional weights  $W_1$  and  $W_2$ , the enriched hidden features  $\hat{H}_r \in \mathcal{R}^{m \times k \times d}$  from ConvFFN can be written as:
+
+$$
+\hat {H} _ {r} = \operatorname {D w C o n v} \left(\sigma \left(\operatorname {D w C o n v} \left(\mathcal {T} \left(H _ {r}, C _ {r}\right); W _ {1}\right)\right); W _ {2}\right), \tag {9}
+$$
+
+where  $\sigma$  denotes the non-linear activation,  $\mathcal{T}$  denotes the formulation of sparse tensor, and the number of groups in DwConv equals to that of latent features. This operation costs  $\mathcal{O}(\frac{HWDkd}{d_x d_y d_z})$  complexity, where  $H, W, D$  refer to the point cloud range in three directions, respectively, and  $[d_x, d_y, d_z]$  specifies the voxel size. The ConvFFN plays an important role in VSA as it introduces desirable inductive bias and global context to the module. More studies on ConvFNN will be discussed in Sec 4.5.
+
+Decoder. The decoder reconstructs the output set from the enriched hidden features  $\hat{H}_r$ . Specifically, we first broadcast the hidden features based on the voxel indices  $\mathcal{V}$ , producing  $\hat{H} \in \mathcal{R}^{n \times k \times d}$  which has the same length as the input set. Then we generate the query, key-value pair from the input set and the hidden features, respectively, with linear projections. Given matrices of query  $Q \in \mathcal{R}^{n \times d}$ , key  $K \in \mathcal{R}^{n \times k \times d}$  and value  $V \in \mathcal{R}^{n \times k \times d}$ , the decoder output  $O$  can be calculated as:
+
+$$
+\mathcal {A} = \left[ \mathcal {A} _ {1}, \dots , \mathcal {A} _ {n} \right] = \left[ K _ {1} Q _ {1} ^ {T}, \dots , K _ {n} Q _ {n} ^ {T} \right], \tag {10}
+$$
+
+$$
+\tilde {\mathcal {A}} = \left[ \operatorname {S o f t m a x} \left(\mathcal {A} _ {1}\right), \dots , \operatorname {S o f t m a x} \left(\mathcal {A} _ {n}\right) \right], \tag {11}
+$$
+
+$$
+O = \left[ O _ {1}, \dots , O _ {n} \right] = \left[ \tilde {\mathcal {A}} _ {1} ^ {T} V _ {1}, \dots , \tilde {\mathcal {A}} _ {n} ^ {T} V _ {n} \right] \tag {12}
+$$
+
+The overall computational complexity of the decoder is  $\mathcal{O}(2nkd)$ . Owing to the flexibility of cross-attention mechanism, VSA formulates the point cloud processing as a set-to-set translation problem.
+
+Relative position embedding. As discussed in [23, 29], preserving the local structure of point cloud is crucial to
+
+![](images/bc1ae233982b3623084061b1a337054b0ff5cbfb1a878a0f964253cd67da88a2.jpg)  
+Figure 3. The overall architecture of the proposed Voxel Set Transformer (VoxSeT).
+
+improve performance. Therefore, we introduce a Positional Embedding (PE) module to encode the local coordinates of the point clouds within a voxel to a high dimensional feature and inject them into each VSA module. Specifically, the PE module applies the Fourier parameterization to take values  $[sin(f_k\pi x),\cos (f_k\pi ,x)]$ , given the normalized local coordinates  $x\in [0,1]$  and the  $k^{th}$  frequency  $f_{k}$  with bandwidth  $L$ . The resulted Fourier embedding, which has a dimension of  $3L$ , is further mapped to the input dimension of the first MLP module through a learnable linear layer.
+
+# 3.3. Voxel Set Transformer (VoxSeT)
+
+The overall architecture of VoxSeT is illustrated in Figure 3. Following the traditional transformer paradigm, the VoxSeT backbone is composed of inter-connected multilayer perception (MLP) and VSA modules. We use batch norm as the normalization layer and wrap each VSA module into a residual block for optimal gradient flow.
+
+Unlike grouping-based approaches [25, 29] that progressively downsample and aggregate point-wise features for context extraction, our backbone extracts point cloud features as a set-to-set translation process. The semantic level of the features is controlled by the size of voxels in the VSA module. We empirically found that applying large voxels can learn richer context information, and demonstrate better understanding of the objects with sparse points, especially pedestrian and cyclist instances. We will present the settings of VSA modules in Sec. 4.1.
+
+Birds-eye-view feature encoding. In point cloud detection, a common phenomenon is that models using dense birds-eye-view (BEV) features [4, 5, 42] generally achieve higher recall than those using sparse point-wise features [33, 45]. In this regard, we encode the point-wise features from the backbone into a BEV representation and apply a shallow CNN to increase the feature density. The CNN has only two strides, each involving three convolutions. The convolutional features of two strides are finally concatenated and passed to the detection head for bounding-box prediction. To generate BEV features, we aggregate the point-wise features within a pillar of size  $0.36\mathrm{m} \times 0.36\mathrm{m}$  and apply a "soft-pooling" operation to produce features in BEV. Given point-wise output features  $X^{j} \in \mathcal{R}^{k \times d}$  in the
+
+$j^{th}$  pillar, the pillar-wise features after pooling  $F^j$  can be formulated as:
+
+$$
+F ^ {j} = \sum_ {m = 1} ^ {k} X _ {m} ^ {j} * w _ {m} ^ {j}, \quad w _ {m} ^ {j} = \frac {e ^ {X _ {m} ^ {j}}}{\sum_ {m = 1} ^ {k} e ^ {X _ {m} ^ {j}}}. \tag {13}
+$$
+
+Detection head and training objectives. To enhance the expressiveness of VoxSeT backbone, we follow PointR-CNN [33] to apply the foreground segmentation loss  $\mathcal{L}_{seg}$  to the output features. This forces VoxSeT to capture contextual information for generating accurate bounding-boxes. The detection head follows the traditional anchor-based design [12, 42]. The final loss then becomes:
+
+$$
+\mathcal {L} = \mathcal {L} _ {s e g} + \frac {1}{N _ {p}} \left(\mathcal {L} _ {c l s} + \mathcal {L} _ {r e g}\right) + \mathcal {L} _ {d i r}), \tag {14}
+$$
+
+where  $N_{p}$  is the number of positive samples whose IoU with anchors lies between  $[\sigma_1,\sigma_2]$ .  $\mathcal{L}_{\mathrm{cls}}$  is the focal loss for bounding-box classification and  $\mathcal{L}_{\mathrm{reg}}$  is the Smooth- $L_{l}$  loss for bounding-box offsets regression.  $\mathcal{L}_{dir}$  is a binary entropy loss for bounding-box orientation prediction. The readers are referred to [33] and [42] for details.
+
+Two-stage model. It is worth noting that VoxSeT can be extended to a two-stage detector, in which we employ the efficient RoI head from LiDAR-RCNN [15] as our second-stage module. To more clearly illustrate our contribution, we also report the performance of a single-stage detector, and our VoxSeT demonstrates superior performance over the current single-stage baselines.
+
+# 4. Experiments
+
+In this section, we evaluate our proposed VoxSeT on two public detection datasets, KITTI [8] and Waymo [35]. We first introduce the training details of VoxSeT and the evaluation settings, and then compare our models with state-of-the-art detection models. Finally, we conduct an in-depth analysis of each component of VoxSeT.
+
+# 4.1. Implementation details
+
+Model setup. On the KITTI dataset, we select the LiDAR points that fall into the ranges [0m, 70.4m], [-40m,
+
+Table 1. Performance comparison with state-of-the-art methods on the Waymo dataset with 202 validation sequences ( $\sim$ 40k samples) for vehicle detection.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Backbone</td><td colspan="4">3D mAP</td><td colspan="4">BEV mAP</td></tr><tr><td>Overall</td><td>0-30m</td><td>30-50m</td><td>50m-inf</td><td>Overall</td><td>0-30m</td><td>30-50m</td><td>50m-inf</td></tr><tr><td>LEVEL1 (IoU=0.7):</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PointPillar [12] (CVPR19)</td><td>CNN</td><td>56.62</td><td>81.01</td><td>51.75</td><td>27.94</td><td>75.57</td><td>92.10</td><td>74.06</td><td>55.47</td></tr><tr><td>MVF [50] (CoRL20)</td><td>CNN</td><td>62.93</td><td>86.30</td><td>60.02</td><td>36.02</td><td>80.40</td><td>93.59</td><td>79.21</td><td>63.09</td></tr><tr><td>PV-RCNN [32] (CVPR20)</td><td>SpCNN</td><td>70.30</td><td>91.92</td><td>69.21</td><td>42.17</td><td>82.96</td><td>97.35</td><td>82.99</td><td>64.97</td></tr><tr><td>Voxel-RCNN [5] (AAAI21)</td><td>SpCNN</td><td>75.59</td><td>92.49</td><td>74.09</td><td>53.15</td><td>88.19</td><td>97.62</td><td>87.34</td><td>77.70</td></tr><tr><td>VoTR-TSD [21] (ICCV21)</td><td>Transformer</td><td>74.95</td><td>92.28</td><td>73.36</td><td>51.09</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CT3D [31] (ICCV21)</td><td>SpCNN</td><td>76.30</td><td>92.51</td><td>75.07</td><td>55.36</td><td>90.50</td><td>97.64</td><td>88.06</td><td>78.89</td></tr><tr><td>VoxSeT (ours)</td><td>Transformer</td><td>76.02</td><td>91.13</td><td>75.75</td><td>54.23</td><td>89.12</td><td>95.12</td><td>87.36</td><td>77.78</td></tr><tr><td>VoxSeT + CT3D (RoI head)</td><td>Transformer</td><td>77.82</td><td>92.78</td><td>77.21</td><td>54.41</td><td>90.31</td><td>96.11</td><td>88.12</td><td>77.98</td></tr><tr><td>LEVEL2 (IoU=0.7):</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PV-RCNN [32] (CVPR20)</td><td>SpCNN</td><td>65.36</td><td>91.58</td><td>65.13</td><td>36.46</td><td>77.45</td><td>94.64</td><td>80.39</td><td>55.39</td></tr><tr><td>Voxel-RCNN [5] (AAAI21)</td><td>SpCNN</td><td>66.59</td><td>91.74</td><td>67.89</td><td>40.80</td><td>81.07</td><td>96.99</td><td>81.37</td><td>63.26</td></tr><tr><td>VoTR-TSD [21] (ICCV21)</td><td>Transformer</td><td>65.91</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CT3D [31] (ICCV21)</td><td>SpCNN</td><td>69.04</td><td>91.76</td><td>68.93</td><td>42.60</td><td>81.74</td><td>97.05</td><td>82.22</td><td>64.34</td></tr><tr><td>VoxSeT (ours)</td><td>Transformer</td><td>68.16</td><td>91.03</td><td>67.13</td><td>42.23</td><td>76.13</td><td>94.13</td><td>81.78</td><td>58.13</td></tr><tr><td>VoxSeT + CT3D (RoI head)</td><td>Transformer</td><td>70.21</td><td>92.05</td><td>70.10</td><td>43.20</td><td>80.56</td><td>96.79</td><td>80.44</td><td>62.37</td></tr></table>
+
+Table 2. Performance comparison with traditional single-stage baseline models on the KITTI validation set. The results are reported by the mAP with 11 recall points.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Vehicle</td><td colspan="3">Pedestrian</td><td colspan="3">Cyclist</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td><td>Easy</td><td>Moderate</td><td>Hard</td><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>SECOND [42]</td><td>88.61</td><td>78.62</td><td>77.22</td><td>56.55</td><td>52.98</td><td>47.73</td><td>80.58</td><td>67.15</td><td>63.10</td></tr><tr><td>PointPillars [12]</td><td>86.46</td><td>77.28</td><td>74.65</td><td>57.75</td><td>52.29</td><td>47.90</td><td>80.04</td><td>62.61</td><td>59.52</td></tr><tr><td>VoxSeT (single-stage)</td><td>88.45</td><td>78.48</td><td>77.07</td><td>60.62</td><td>54.74</td><td>50.39</td><td>84.07</td><td>68.11</td><td>65.14</td></tr><tr><td>Improvements</td><td>-0.16</td><td>-0.14</td><td>-0.15</td><td>2.87</td><td>1.76</td><td>2.49</td><td>3.49</td><td>0.96</td><td>2.04</td></tr></table>
+
+$40\mathrm{m}$ ,  $[-3\mathrm{m}, 1\mathrm{m}]$  along X, Y, Z axes, respectively, and abandon those points with the frontal view projections out of image. On the Waymo dataset, the points that lie between  $[-75.2\mathrm{m}, 75\mathrm{m}]$  in the X and Y axes, and  $[-2\mathrm{m}, 4\mathrm{m}]$  in the Z axis are selected. The voxel size of the first VSA layer is  $[0.32\mathrm{m}, 0.32\mathrm{m}, 4\mathrm{m}]$  on KITTI and  $[0.32\mathrm{m}, 0.32\mathrm{m}, 6\mathrm{m}]$  on Waymo. The voxel size is doubled along the X and Y axes in the next VSA block. The feature dimensions of the four VSA blocks are 16, 32, 64 and 128, respectively. The number of latent codes in each VSA block is 8 and the bandwidth  $L$  of the Positional Embedding (PE) module is 64.
+
+Training and inference. The network is trained end-to-end on four RTX Quodra 8000 GPUs for 100 epochs with the Adam optimizer. The batch size, learning rate, and weight decay are set to 4, 0.003 and 0.01, respectively. The learning rate is decayed with the onecycle policy, where the momentum has a damping range of  $[85\%, 95\%]$ .
+
+We apply the anchor settings in SECOND [42] in our single stage model. For the two-stage model, we sample 512 RoIs in training and 128 RoIs in inference. In the post-processing phase, the bounding-boxes are filtered by NMS with an IoU threshold of 0.1, and those having confidence over 0.3 are selected as final predictions. Data augmenta
+
+tions [4,42,51] are applied to improve the model generalization performance. For other default settings, the readers are referred to the OpenPCDet toolbox [36] used in this work.
+
+# 4.2. Dataset and evaluation metrics
+
+KITTI dataset [8]. KITTI contains 7,481 training samples and 7,518 testing samples. Following the common protocol [3], we split the labeled data into a training set with 3,712 samples and a validation set with 3,769 samples. We conduct experiments on the commonly used car category whose detection IoU threshold is 0.7, and report the results on three difficulty levels (easy, moderate and hard) according to the object size, occlusion state and truncation level.
+
+Waymo open dataset [35]. This dataset consists of 798 training sequences and 202 validation sequences, where there are 158,361 samples and 40,077 samples, respectively. The evaluation metrics used are 3D mean Average Precision (mAP) with IoU threshold of 0.7 on the vehicle category. The measures are reported based on the distances from objects to sensor, i.e.,  $0 - 30\mathrm{m}$ ,  $30 - 50\mathrm{m}$  and  $>50\mathrm{m}$ , respectively. Two difficulty levels, LEVEL 1 (boxes with more than five LiDAR points) and LEVEL 2 (boxes with at least one LiDAR point) are considered.
+
+Table 3. Performance comparison with state-of-the-art methods on the KITTI test set. The results are reported by the mAP with 0.7 IoU threshold and 40 recall points.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">3D</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>LiDAR + RGB:</td><td></td><td></td><td></td></tr><tr><td>MV3D [3] (CVPR17)</td><td>74.97</td><td>63.63</td><td>54.00</td></tr><tr><td>ContFuse [17] (ECCV18)</td><td>83.68</td><td>68.78</td><td>61.67</td></tr><tr><td>AVOD-FPN [11] (IROS18)</td><td>83.07</td><td>71.76</td><td>65.73</td></tr><tr><td>F-PointNet [27] (CVPR18)</td><td>82.19</td><td>69.79</td><td>60.59</td></tr><tr><td>MMF [16] (CVPR19)</td><td>88.40</td><td>77.43</td><td>70.22</td></tr><tr><td>3D-CVF [46] (ECCV20)</td><td>89.20</td><td>80.05</td><td>73.11</td></tr><tr><td>CLOCs [24] (IROS20)</td><td>88.94</td><td>80.67</td><td>77.15</td></tr><tr><td>LiDAR only:</td><td></td><td></td><td></td></tr><tr><td>VoxelNet [51] (CVPR18)</td><td>77.47</td><td>65.11</td><td>57.73</td></tr><tr><td>SECOND [42] (Sensor18)</td><td>83.34</td><td>72.55</td><td>65.82</td></tr><tr><td>PointPillars [12] (CVPR19)</td><td>82.58</td><td>74.31</td><td>68.99</td></tr><tr><td>STD [45] (ICCV19)</td><td>87.95</td><td>79.71</td><td>75.09</td></tr><tr><td>PointRCNN [33] (CVPR19)</td><td>86.96</td><td>75.64</td><td>70.70</td></tr><tr><td>SA-SSD [10] (CVPR20)</td><td>88.75</td><td>79.79</td><td>74.16</td></tr><tr><td>3DSSD [45] (CVPR20)</td><td>88.36</td><td>79.57</td><td>74.55</td></tr><tr><td>PV-RCNN [45] (CVPR20)</td><td>90.25</td><td>81.43</td><td>76.82</td></tr><tr><td>Voxel-RCNN [45] (AAAI21)</td><td>87.95</td><td>79.71</td><td>75.09</td></tr><tr><td>CT3D [31] (ICCV21)</td><td>87.83</td><td>81.77</td><td>77.16</td></tr><tr><td>VoTR-TSD [21] (ICCV21)</td><td>89.90</td><td>82.09</td><td>79.14</td></tr><tr><td>VoxSeT (ours)</td><td>88.53</td><td>82.06</td><td>77.46</td></tr></table>
+
+# 4.3. Results on the Waymo open dataset
+
+We first evaluate the performance of VoxSeT on the Waymo open dataset. The results are summarized in Table 1. VoxSeT outperforms most of CNN-based models, leading PV-RCNN [32] by  $5\%$  LEVEL_1 mAP and VoxelRCNN [5] by  $2.4\%$  LEVEL_2 mAP. As one of the few transformer based models, our VoxSeT achieves better performance than its transformer-based competitor VoTR-TSD [21], which brings  $0.9\%$  and  $1.4\%$  improvements on LEVEL_1 and LEVEL_2 mAP, respectively. VoxSeT achieves comparable performance to the state-of-the-art method CT3D [31]. It should be noted that, however, CT3D actually employs a heavy transformer based RoI head, which has three self-attention encoding layers. By adopting this RoI head into VoxSeT, our model achieves better results, outperforming CT3D by  $1.5\%$  LEVEL_1 mAP and  $1.2\%$  LEVEL_2 mAP. This demonstrates that as a new transformer based backbone network, VoxSeT surpasses Sparse CNN based networks. VoxSeT works especially well in the range of  $30 - 50\mathrm{m}$ , which indicates that transformer modules are better in capturing the context information in long-range areas.
+
+# 4.4. Results on the KTTI Dataset
+
+We then conduct experiments on the KITTI dataset to evaluate the performance of VoxSeT as a single-stage detec
+
+Table 4. Performance comparison with state-of-the-art methods on the KITTI validation set. The results are reported by the mAP with 0.7 IoU threshold and 11 recall points.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">3D</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>LiDAR + RGB:</td><td></td><td></td><td></td></tr><tr><td>MV3D [3] (CVPR17)</td><td>71.29</td><td>62.68</td><td>56.56</td></tr><tr><td>F-PointNet [27] (CVPR18)</td><td>83.76</td><td>70.92</td><td>63.65</td></tr><tr><td>3D-CVF [46] (ECCV20)</td><td>89.67</td><td>79.88</td><td>78.47</td></tr><tr><td>LiDAR only:</td><td></td><td></td><td></td></tr><tr><td>SECOND [42] (Sensor18)</td><td>88.61</td><td>78.62</td><td>77.22</td></tr><tr><td>PointPillars [12] (CVPR19)</td><td>86.62</td><td>76.06</td><td>68.91</td></tr><tr><td>STD [45] (ICCV19)</td><td>89.70</td><td>79.80</td><td>79.30</td></tr><tr><td>PointRCNN [33] (CVPR19)</td><td>88.88</td><td>78.63</td><td>77.38</td></tr><tr><td>SA-SSD [10] (CVPR20)</td><td>90.15</td><td>79.91</td><td>78.78</td></tr><tr><td>3DSSD [45] (CVPR20)</td><td>89.71</td><td>79.45</td><td>78.67</td></tr><tr><td>PV-RCNN [45] (CVPR20)</td><td>89.35</td><td>83.69</td><td>78.70</td></tr><tr><td>Voxel-RCNN [45] (AAAI21)</td><td>89.41</td><td>84.52</td><td>78.93</td></tr><tr><td>CT3D [31] (ICCV21)</td><td>89.54</td><td>86.06</td><td>78.99</td></tr><tr><td>VoTR-TSD [21] (ICCV21)</td><td>89.04</td><td>84.04</td><td>78.68</td></tr><tr><td>VoxSeT (ours)</td><td>89.21</td><td>86.71</td><td>78.56</td></tr></table>
+
+tion model. Our competitors are SECOND [42] and Point-Pillars [12], which represent two widely used baseline feature extractors. All the three methods use the same detection head and hyper-parameters in training. As shown in Table 2, VoxSeT achieves comparable performance to SECOND on vehicle class, but much better performance on Pedestrian and Cyclist classes. We believe this is because VoxSeT has a wider effective receptive field through the VSA conditioned on the large voxel, which is crucial to detecting the objects with sparse points.
+
+As a two-stage detection model, VoxSeT achieves better performance than CT3D by  $0.7\%$  (Easy),  $0.4\%$  (Moderate) and  $0.3\%$  (Hard) mAP, respectively, as shown in Table 3. Compared with VoTR-TSD which relies on multiscale backbone features, VoxSeT can still achieve comparable performance by using only singular point-wise features. We also evaluate VoxSeT on KITTI val. One can see that VoxSeT achieves leading accuracy on "Moderate" level but slightly lower accuracy on "Easy" and "Hard" levels. We believe this is because KITTI has a long-tailed distribution, and hence the "Moderate" samples dominate the hidden space of the VSA module.
+
+It should be pointed out that both CT3D and VoTR employ convolutional architectures, and their performances on KITTI and Waymo datasets are not consistent. Specifically, CT3D works better on Waymo but its performance drops much on KITTI, while VoTR works better on KITTI but its performance on Waymo is much worse. In contrast, our VoxSeT exhibits consistently superior performance on both datasets, demonstrating its good generalization capacity.
+
+![](images/5b30e1eea9053c4fc5a5867e0a0a5cff03856cf537b018808618f3b6aed3225a.jpg)  
+Figure 4. Visualization of the spatial attention maps induced by different latent codes in the last VSA module.
+
+Table 5. Effects of enabling token interactions by convolutional feed-forward network. The mAP (11 recall points) of RPN on KITTI val are reported.  
+
+<table><tr><td>Settings</td><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>default</td><td>88.31</td><td>79.56</td><td>77.84</td></tr><tr><td>ConvFFN→FFN</td><td>70.12</td><td>69.54</td><td>54.23</td></tr></table>
+
+Table 6. The mAP (11 recall points) of RPN on KITTI val by using different number of latent codes (LC) in four VSA modules.  
+
+<table><tr><td>No. of latent codes</td><td>4</td><td>8</td><td>16</td></tr><tr><td>mAP</td><td>76.63</td><td>78.25</td><td>78.74</td></tr></table>
+
+Table 7. Comparison of PointRCNN [33] detectors with VoxSeT and PointNet++ backbones. The mAP (11 recall points) on KITTI val are reported.  
+
+<table><tr><td>Settings</td><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>PointRCNN</td><td>88.52</td><td>78.95</td><td>77.81</td></tr><tr><td>VSA-PointRCNN</td><td>89.61</td><td>80.14</td><td>78.69</td></tr></table>
+
+Table 8. The latency and runtime memory on the KITTI dataset, tested by NVIDIA 2080Ti GPU.  
+
+<table><tr><td>Models</td><td>Latency</td><td>Memory (runtime)</td></tr><tr><td>SECOND [42]</td><td>48 ms</td><td>6093MB</td></tr><tr><td>PointPillars [12]</td><td>22ms</td><td>1508 MB</td></tr><tr><td>VoxSeT (single-stage)</td><td>34 ms</td><td>2381 MB</td></tr></table>
+
+# 4.5. Ablation study
+
+We conduct a series of ablation experiments to comprehend the roles of different components in VoxSeT.
+
+Convolutional feed-forward network. Table 5 shows that replacing the proposed ConvFFN with the conventional FFN significantly degrades the accuracy, indicating that the local connectivity is crucial to the detection performance.
+
+Effects of number of latent codes. In Table 6, we investigate the number of latent codes used in four VSA modules. We see that more latent codes can encode more context information of point cloud, and enhance the modeling capacity of VoxSeT.
+
+Comparison with PointNet++ backbone. We train a PointRCNN [33] variant by replacing its PointNet++ back
+
+bone [33] with our VoxSeT backbone. From Table [29], one can observe obvious performance improvements. We believe this is because VSA module has better modeling power in terms of dynamic learning and large receptive field than the set abstraction (SA) module in PointNet++.
+
+Latency and runtime memory. Table 8 shows that VoxSeT is faster and has less memory consumption compared to the sparse 3D CNN (SECOND). The higher performance than PointPillars and the acceptable runtime cost suggest that VoxSeT can be a good alternative to PointPillars in real-time applications.
+
+Visualization of attention weights. Figure 4 visualizes the spatial attention maps for the latent codes in the last VSA module. We show the attention maps for 4 out of a total of 8 latent codes. One can observe that the VSA module focuses more on the object region and different latent codes encode different contexts of the objects, indicating the high expressiveness of VSA for point cloud data.
+
+# 5. Conclusion and discussions
+
+We proposed VoxSeT, a novel transformer-based framework for 3D object detection from LiDAR point clouds. In contrast to previous 3D LiDAR detectors, which use sparse CNN and PointNet backbones to learn point cloud features, we made the first attempt to model point cloud processing as set-to-set translation, which preserves the full resolution of raw point cloud at every step of feature extraction. We presented a voxel-based set attention module that performs self-attention on voxel clusters of arbitrary size and encodes point features with more discriminative context information from a large receptive field. Experimental results on the Waymo and KITTI datasets demonstrated that our VoxSeT can achieve competitive performance, making it a good alternative for point cloud modeling.
+
+It should be noted that in VoxSeT, we only explored one possible formulation of liner attention based on the induced latent codes. This limits the expressive power of VoxSeT to represent different point cloud structures and their correlations. By using stronger attention mechanisms, the performance of VoxSeT can be further improved, which will be our future research direction.
+
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