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+# Dataset Distillation using Neural Feature Regression
+
+Yongchao Zhou Department of Computer Science University of Toronto yongchao.zhou@mail.utoronto.ca
+
+Ehsan Nezhadarya Toronto AI Lab LG Electronics Canada ehsan.nezhadarya@lge.com
+
+Jimmy Ba Department of Computer Science University of Toronto jba@cs.toronto.edu
+
+# Abstract
+
+Dataset distillation aims to learn a small synthetic dataset that preserves most of the information from the original dataset. Dataset distillation can be formulated as a bi-level meta-learning problem where the outer loop optimizes the metadataset and the inner loop trains a model on the distilled data. Meta-gradient computation is one of the key challenges in this formulation, as differentiating through the inner loop learning procedure introduces significant computation and memory costs. In this paper, we address these challenges using neural Feature Regression with Pooling (FRePo), achieving the state-of-the-art performance with an order of magnitude less memory requirement and two orders of magnitude faster training than previous methods. The proposed algorithm is analogous to truncated backpropagation through time with a pool of models to alleviate various types of overfitting in dataset distillation. FRePo significantly outperforms the previous methods on CIFAR100, Tiny ImageNet, and ImageNet-1K. Furthermore, we show that high-quality distilled data can greatly improve various downstream applications, such as continual learning and membership inference defense. Please check out our webpage at https://sites.google.com/view/frepo.
+
+# 1 Introduction
+
+Knowledge distillation [1] is a technique in deep learning to compress knowledge for easy deployment. Most previous works focus on model distillation [2, 3] where the knowledge acquired by a large teacher model is transferred to a small student model. In contrast, dataset distillation [4, 5] aims to learn a small set of synthetic examples preserving most of the information from a large dataset such that a model trained on it can achieve similar test performance as one trained on the original dataset. Distilled data can accelerate model training and reduce the cost of storing and sharing a dataset. Moreover, its highly condensed and synthetic nature can also benefit various applications, such as continual learning [5–8], neural architecture search [5, 7], and privacy-preserving tasks [9, 10].
+
+Dataset distillation was first studied by Maclaurin et al. [11] in the context of gradient-based hyperparameter optimization and subsequently Wang et al. [4] formally proposed dataset distillation as a new task. Dataset distillation can be naturally formulated as a bi-level meta-learning problem. The inner loop optimizes the model parameters on the distilled data (meta-parameters), while the outer loop refines the distilled data with meta-gradient updates.
+
+One key challenge in dataset distillation is computing the meta-gradient. Several methods [4, 11–13] compute it by back-propagating through the unrolled computation graph, but they often suffer from huge compute and memory requirement [14], training instability [15, 16], and truncation bias [17]. To avoid unrolled optimization, surrogate objectives are used to derive the meta-gradient, such as gradient matching [5, 7, 18], feature alignment [8, 19], and training trajectory matching [20]. Nevertheless, a surrogate objective may introduce its own bias [19], and thus, may not accurately reflect the true objective. An alternative is using kernel methods, such as Neural Tangent Kernel (NTK) [21], to approximate the inner optimization [22, 23]. However, computing analytical NTK for modern neural network can be extremely expensive [22, 23].
+
+![](images/99c51e47398cb446d8e74bd29e225e4ce9dc4051310f93f7693b478569789644.jpg)  
+Figure 1: Example distilled images from $3 2 \mathbf { x } 3 2$ CIFAR100, 64x64 Tiny ImageNet, and $1 2 8 \mathrm { x } 1 2 8$ ImageNet Subset. The images look real and transfer well to different architectures. They can be used for various downstream applications, such as continual learning and membership inference defense.
+
+Even with an accurate meta-gradient, dataset distillation still suffers from various types of overfitting. For instance, the distilled data can easily overfit to a particular learning algorithm [4, 13, 20], a certain stage of optimization [13, 19], or a certain network architecture [5, 7, 20, 22, 23]. Meanwhile, the model can also overfit the distilled data during training, which is the most common cause of overfitting when we train on a small dataset. All these kinds of overfitting impose difficulties on the training and general-purpose use of the distilled data.
+
+We propose an efficient meta-gradient computation method and a “model pool” to address the overfitting problems. The bottleneck in meta-gradient computation arises due to the complexity of inner optimization, as we need to know how the inner parameters vary with the outer parameters [24]. However, the inner optimization can be pretty simple if we only train the last layer of a neural network to convergence while keeping the feature extractor fixed. In this case, computing the prediction on the real data using the model trained on the distilled data can be expressed as a kernel ridge regression (KRR) with respect to the conjugate kernel [25]. Hence, computing the meta-gradient is simply back-propagating through the kernel and a fixed feature extractor. To alleviate overfitting, we propose to maintain a diverse pool of models instead of periodically training and resetting a single model as in prior work [7, 13, 18]. Intuitively, our algorithm targets the following question: what is the best data to train the linear classifier given the current feature extractor? Due to the diverse feature extractors we use, the distilled data generalize well to a wide range of model distributions.
+
+# Summary of Contributions:
+
+• We propose an effective method for dataset distillation. Our method, named neural Feature Regression with Pooling (FRePo), achieves state-of-the-art results on various benchmark datasets with a $1 0 0 \mathrm { x }$ reduction in training time and a $1 0 \mathrm { x }$ reduction in GPU memory requirement. Our distilled data looks real (Figure 1) and transfers well to different architectures. We show that FRePo scales well to datasets with high-resolution images or complex label space. We achieve $7 . 5 \%$ top1 accuracy on ImageNet-1K [26] using only one image per class. The same classifier obtains only $1 . 1 \%$ accuracy from a random subset of real images. The previous methods struggle in this task due to large memory and compute requirements. • We demonstrate that high-quality distilled data can significantly improve various downstream applications, such as continual learning and membership inference defense.
+
+![](images/bd356bac9e8a26f60ac46a3e8c2299fdbc56eeb9580bd6cdb628ffbec6e98257.jpg)  
+Figure 2: Comparison of FRePo and Unrolled Optimization. $S$ , $X _ { s }$ , $Y _ { s }$ are the distilled dataset, images and labels. $\mathcal { L }$ is the meta-training loss and $\dot { \theta } ^ { ( k ) }$ , $g ^ { ( k ) }$ are the model parameter and gradient at step $k$ . $f ( X )$ is the feature for input $X$ and $K _ { X _ { t } X _ { s } } ^ { \theta }$ is the Gram matrix of $X _ { t }$ and $X _ { s }$ . FRePo is analogous to 1-step TBPTT as it computes the meta-gradient at each step while performing the online model update. However, instead of backpropagating through the inner optimization, FRePo computes the meta-gradient through a kernel and feature extractor.
+
+# 2 Method
+
+# 2.1 Dataset Distillation as Bi-level Optimization
+
+Suppose we have a large labeled dataset $\mathcal { T } = \left\{ \left( \mathbf { x } _ { 1 } , \mathbf { y } _ { 1 } \right) , \dotsc , \left( \mathbf { x } _ { | T | } , \mathbf { y } _ { | T | } \right) \right\}$ with $| \tau |$ image and label\` ˘( pairs. Dataset distillation aims to learn a small synthetic dataset $\mathcal { S } = \left\{ ( \mathbf { x } _ { 1 } , \mathbf { y } _ { 1 } ) , \dotsc , \left( \mathbf { x } _ { | S | } , \mathbf { y } _ { | S | } \right) \right\}$ that preserves most of the information in $\tau$ . We train several neural networks parameterized by $\theta$ on the dataset $s$ and then compute the validation loss $\mathcal { L } ( \mathcal { A } l g \left( \theta , \mathcal { S } \right) , \mathcal { T } )$ on the real dataset $\tau$ , where ${ \mathcal { A } } l g \left( \theta , S \right)$ is the neural network parameters optimized by a learning algorithm $\mathcal { A } g$ with the model initialization $\theta$ and distilled dataset $s$ as its inputs. The validation loss $\mathcal { L } ( \mathcal { A } l g \left( \theta , S \right) , \mathcal { T } )$ is a noisy objective with the stochasticity coming from random model initialization and inner learning algorithm. Thus, we are interested in minimizing the expected value of this loss, which we denote it as $F ( S )$ . We formulate the dataset distillation as the following bi-level optimization problem.
+
+$$
+\overbrace { \mathcal { S } ^ { * } : = \mathop { \mathrm { a r g m i n } } _ { \mathcal { S } } F ( \mathcal { S } ) } ^ { o u t e r - l e v e l } , \mathrm { w h e r e } F ( \mathcal { S } ) = \mathbb { E } _ { \theta \sim P _ { \theta } } \biggl [ \mathcal { L } \Bigl ( \overbrace { \mathcal { A } l g \left( \theta , \mathcal { S } \right) } ^ { i n n e r - l e v e l } , \ T \Bigr ) \biggr ] .
+$$
+
+In this bi-level setup, the outer loop optimizes the distilled data to minimize $F ( S )$ , while the inner loop trains a neural network using the learning algorithm, $\mathcal { A } g$ , to minimize the training loss on the distilled data $s$ . From the meta-learning perspective, the task is defined by the model initialization $\theta$ , and we want to learn a meta-parameter $s$ that generalizes well to different models sampled from the model distributions $P _ { \theta }$ . During learning, we optimize the meta-parameter $s$ by minimizing the meta-training loss $F ( S )$ . In contrast, at meta-test time, we train a new model from scratch on $s$ and evaluate the trained model on a held-out real dataset. This meta-test performance reflects the quality of the distilled data.
+
+# 2.2 Dataset Distillation using Neural Feature Regression with Pooling $\mathbf { ( F R e P 0 ) }$
+
+The outer-level problem can be solved using gradient-based methods of the form $\mathcal { S } \gets \mathcal { S } \ – \alpha \nabla _ { \mathcal { S } } F ( \mathcal { S } )$ , where $\alpha$ is the learning rate for the distilled data and $\nabla _ { S } F ( S )$ is the meta-gradient [27]. For a particular model $\theta$ , the meta-gradient can be expressed as $\nabla _ { \mathcal { S } } \hat { \mathcal { L } } \left( \mathcal { A } l g \left( \theta , \mathcal { S } \right) , \mathcal { T } \right)$ . Computing this meta-gradient requires differentiating through inner optimization. If $\mathcal { A } \boldsymbol { { l } } _ { g }$ is an iterative algorithm like gradient descent, then backpropagating through the unrolled computation graph [14] can be a solution. However, this type of unrolled optimization introduces significant computation and memory overhead, as the whole training trajectory needs to be stored in memory (Figure 2(b)).
+
+Traditionally, these issues are alleviated with truncated backpropagation through time (TBPTT) [28–30]. Instead of backpropagating through an entire unrolled sequence, TBPTT performs backpropagation for each subsequence separately. It is efficient because its time and memory complexity scale linearly with respect to the truncation steps. However, truncation may yield highly biased gradients that severely impact training. To mitigate this truncation bias [14], we consider training only the top layer of a network to convergence. The key insight is that the data helpful for training the output layer can also help train the whole network. Thus, we decompose the neural network into a feature
+
+Require: $\tau$ : a labeled dataset; $\alpha$ : the learning rate for the distilled data nitialization: Initialize a labeled distilled dataset $\boldsymbol { \mathcal { S } } = \left( \boldsymbol { X _ { s } } , \boldsymbol { Y _ { s } } \right)$ .
+
+Initialization: Initialize a model pool $\mathcal { M }$ with $m$ models $\left\{ \boldsymbol { \theta } _ { i } \right\} _ { i = 1 } ^ { m }$ randomly initialized from $P _ { \theta }$
+
+1: while not converged do   
+2: Ż Sample a model uniformly from the model pool: $\theta _ { i } \sim \mathcal { M }$ .   
+3: Ż Sample a target batch uniformly from the labeled dataset: $( X _ { t } , Y _ { t } ) \sim \tau$ .   
+4: $\triangleright$ Compute the meta-training loss $\mathcal { L }$ using Eq. 2   
+5: Ż Update the distilled data $s$ $\mathrm { : } ~ X _ { s } \gets X _ { s } - \alpha \nabla _ { X _ { s } } \mathcal { L }$ , and $Y _ { s } \gets Y _ { s } - \alpha \nabla _ { Y _ { s } } \mathcal { L }$   
+6: $\triangleright$ Train the model $\theta _ { i }$ on the current distilled data $s$ for one step.   
+7: $\triangleright$ Reinitialize the model $\theta _ { i } \sim P _ { \theta }$ if $\theta _ { i }$ has been updated more than $K$ steps.
+
+8: end while
+
+Output: Learned distilled dataset $\boldsymbol { \mathcal { S } } = \left( \boldsymbol { X _ { s } } , \boldsymbol { Y _ { s } } \right)$
+
+extractor and a linear classifier. We fix the feature extractor at each meta-gradient computation and train the linear classifier to convergence before updating $s$ . After that, we adjust the feature extractor by training the whole network on the updated distilled data. We note that similar two-phase procedure has been studied in the context of representation learning [31].
+
+Meta-Gradient Computation: If we consider the mean square error loss, then the optimal weights for the linear classifier have a closed-form solution. Moreover, since the feature dimension is typically larger than the number of distilled data, we can use kernel ridge regression (KRR) with a conjugate kernel [25] rather than solving the weights explicitly [12]. The resulting meta-training loss (Eq. 2) is similar to that used in KIP [22, 23], but we use a more flexible kernel rather than NTK.
+
+$$
+\mathcal { L } \left( \mathcal { A } l g \left( \theta , \mathcal { S } \right) , \mathcal { T } \right) = \frac { 1 } { 2 } | | Y _ { t } - K _ { X _ { t } X _ { s } } ^ { \theta } ( K _ { X _ { s } X _ { s } } ^ { \theta } + \lambda I ) ^ { - 1 } Y _ { s } | | _ { 2 } ^ { 2 } ,
+$$
+
+where $( X _ { t } , Y _ { t } )$ and $( X _ { s } , Y _ { s } )$ are the inputs and labels of the real data $\tau$ and distilled data $s$ respectively. The Gram matrix between real inputs and distilled inputs is denoted as $K _ { X _ { t } X _ { s } } ^ { \theta } \in \mathbb { R } ^ { | T | \times | S | }$ while the Gram matrix between distilled inputs is denoted as $K _ { X _ { s } X _ { s } } ^ { \theta } \in \mathbb { R } ^ { | S | \times | S | }$ . $\lambda$ t scontrols the regularization strength for KRR. Let us denote the neural network feature for a given input $X$ and model parameter $\theta$ as $\mathbf { \bar { \chi } } _ { f ( X , \theta ) } \in \mathbb { R } ^ { N \times d }$ , where $N$ is the number of input and $d$ is the feature dimension 1. The conjugate kernel is defined by the inner product of the neural network features. Thus, the two Gram matrices are computed as follows:
+
+$$
+K _ { X _ { t } X _ { s } } ^ { \theta } = f ( X _ { t } , \theta ) f ( X _ { s } , \theta ) ^ { \top } , \quad K _ { X _ { s } X _ { s } } ^ { \theta } = f ( X _ { s } , \theta ) f ( X _ { s } , \theta ) ^ { \top } ,
+$$
+
+Now, computing the meta-gradient $\nabla _ { \mathcal { S } } \mathcal { L } \left( \mathcal { A } l g \left( \theta , \mathcal { S } \right) , \mathcal { T } \right)$ is just back-propagating through the conjugate kernel and a fixed feature extractor, which is very efficient and takes even fewer operations than computing the gradient for the network’s weights. Moreover, we decouple the meta-gradient computation from the model online update. Hence, we can train the online model using any optimizer, and the distilled data will be agnostic to the specific learning algorithm choice. Our proposed method is similar to 1-step TBPTT in that we compute the meta-gradient at each step while performing the online model update. Unlike the conventional 1-step TBPTT, we compute the meta-gradient using a KRR output layer to mitigate truncation bias, illustrated in Figure 2(a).
+
+Model Pool: As discussed in Section 1, there are various types of overfitting in dataset distillation. Several techniques have been proposed to alleviate such problem, such as random initialization [4], periodic reset [13, 5, 7], and dynamic bi-level optimization [19]. These techniques share the same underlying principle: the model diversity matters. Thus, we propose to maintain a “model pool” filled with diverse set of parameters obtained from different number of training steps and different random initializations. Unlike the previous methods that periodically training and resetting a single model, FRePo randomly sample a model from the pool at each meta-gradient computation and update it using the current distilled data. However, if a model has been updated more than $K$ steps, we reinitialize it with a new random seed. From the meta-learning perspective, we maintain a diverse set of meta-tasks to sample from and avoid sampling very similar tasks at each consecutive gradient computation to avoid overfitting to a particular setup.
+
+Pool Diversity: We can increase the regularization strength by increasing the diversity of the model pool by setting a larger $K$ , using data augmentation when training the model on the distilled data, or using models with different architectures. To keep our method simple, we use the same architecture for all models in the pool and do not use any data augmentation when training the model on the distilled data. Thus, our model pool only contains models with different initialization, at different optimization stages, and trained at different time-step of the distilled data.
+
+# 3 Related Work
+
+Unrolling in Bi-Level Optimization: One way to compute the meta-gradient is to differentiate through the unrolled inner optimization [4, 11–13]. However, this approach inherits several difficulties of the unrolled optimization, such as: 1) large computation and memory cost [14]; 2) truncation bias with short unrolls [17]; 3) exploding or vanishing gradients with long unrolls [15]; 4) chaotic and poorly conditioned loss landscapes with long unrolls [16]. In contrast, our method considers approximating the inner optimization with kernel ridge regression instead of unrolled optimization.
+
+Surrogate Objective: To avoid unrolled optimization, several works turn to surrogate objectives. DC [5], DSA [7], and DCC [18] formulate the dataset distillation as a gradient matching problem between the gradients of neural network weights computed on the real and distilled data. In contrast, DM [8] and CAFE [19] consider the feature distribution alignment between the real and distilled data. Moreover, MTT [20] shows that knowledge from many expert training trajectories can be distilled to a dataset by using a training trajectory matching objective. Nevertheless, surrogate objectives may introduce new biases and thus, may not accurately reflect the true objective. For example, gradient matching approaches [5, 7, 18] only focus on short-range behavior and may easily overfit to a biased set of samples that produce dominant gradients [19, 20].
+
+Closed-form Approximation: An alternative way to circumvent unrolled optimization is to find a closed-form approximation to the inner optimization. Based on the correspondence between infinitelywide neural networks and kernel methods, KIP [22, 23] approximates the inner optimization with NTK [21]. In this case, the meta-gradient can be computed by back-propagating through the NTK. However, computing NTK for modern neural networks is extremely expensive. Thus, using NTK for dataset distillation requires thousands of GPU hours and sophisticated implementation of the distributed kernel computation framework [23]. Similar to ours, Bohdal et al. [12] also decomposes the neural network as a feature extractor and a linear classifier. However, they only learn the label and explicitly solve for the optimal classifier weights rather than perform KRR.
+
+# 4 Dataset Distillation
+
+# 4.1 Implementation Details
+
+We compare our method to four state-of-the-art dataset distillation methods [7, 8, 20, 23] on various benchmark datasets [26, 32–37]. We train the distilled data using Algorithm 1 with the same set of hyperparameters for all experiments except stated otherwise. Unlike prior work [7, 8, 20], we do not apply data augmentation during training. However, we apply the same data augmentation [7, 20] during evaluation for a fair comparison. We preprocess the data in a similar way as in previous works [20, 23] but use a wider architecture than previous works [7, 8, 20] because the KRR component does not behave well when the feature dimension is low, resulting in a significant performance drop for our method. Results on the original architecture are included in Appendix ??. We evaluate each distilled data using five random neural networks and report the mean and standard deviation. For the baseline method, we report the best of the reported value in the original paper and our reproducing results.
+
+For the sake of brevity, we provide implementation details about data preprocessing, distilled data initialization, and hyperparameters in Appendix ?? and various ablation studies regarding the model pool, batch size, distilled data initialization, label learning, and model architectures in Appendix ??. More distilled image visualizations can be found in Appendix ??. Our code is available at https://github.com/yongchao97/FRePo.
+
+Table 1: Test accuracies of models trained on the distilled data from scratch. : denotes performance better than the original reported performance. KRR preformance is shown in bracket. FRePo performs extremely well for one image per class setting on CIFAR100, Tiny ImageNet and CUB-200.   
+
+<table><tr><td></td><td>Img/Cls</td><td>DSA [7]</td><td>DM[8]</td><td>KIP [23]</td><td>MTT [20]</td><td>FRePo</td></tr><tr><td rowspan="3">MNIST</td><td>1</td><td>88.7±0.6</td><td>89.9 ± 0.8†</td><td>90.1± 0.1</td><td>91.4 ± 0.9†</td><td>93.0 ± 0.4 (92.6 ± 0.4)</td></tr><tr><td>10</td><td>97.9 ±0.1†</td><td>97.6 ± 0.1†</td><td>97.5 ± 0.0</td><td>97.3 ± 0.1+</td><td>98.6 ± 0.1 (98.6 ± 0.1)</td></tr><tr><td>50</td><td>99.2 ± 0.1</td><td>98.6 ± 0.1</td><td>98.3 ± 0.1</td><td>98.5±0.1+</td><td>99.2 ± 0.0 (99.2± 0.1)</td></tr><tr><td rowspan="3">F-MNIST</td><td>1</td><td>70.6 ± 0.6</td><td>71.5 ± 0.5†</td><td>73.5± 0.5</td><td>75.1 ± 0.9†</td><td>75.6 ± 0.3 (77.1 ± 0.2)</td></tr><tr><td>10</td><td>84.8±0.3t</td><td>83.6±0.2t</td><td>86.8±0.1</td><td>87.2± 0.3+</td><td>86.2 ± 0.2 (86.8 ± 0.1)</td></tr><tr><td>50</td><td>88.8±0.2t</td><td>88.2±0.1†</td><td>88.0±0.1</td><td>88.3± 0.1+</td><td>89.6 ± 0.1 (89.9 ± 0.1)</td></tr><tr><td rowspan="3">CIFAR10</td><td>1</td><td>36.7± 0.8†</td><td>31.0 ± 0.6†</td><td>49.9 ± 0.2</td><td>46.3 ± 0.8</td><td>46.8 ± 0.7 (47.9 ± 0.6)</td></tr><tr><td>10</td><td>53.2 ± 0.8†</td><td>49.2 ± 0.8†</td><td>62.7± 0.3</td><td>65.3 ± 0.7</td><td>65.5 ± 0.4 (68.0 ± 0.2)</td></tr><tr><td>50</td><td>66.8± 0.4†</td><td>63.7±0.5t</td><td>68.6± 0.2</td><td>71.6 ± 0.2</td><td>71.7 ± 0.2 (74.4 ± 0.1)</td></tr><tr><td rowspan="3">CIFAR100</td><td>1</td><td>16.8± 0.2†</td><td>12.2 ± 0.4†</td><td>15.7 ± 0.2</td><td>24.3 ± 0.3</td><td>28.7 ± 0.1 (32.3 ± 0.1)</td></tr><tr><td>10</td><td>32.3 ±0.3</td><td>29.7 ±0.3</td><td>28.3 ± 0.1</td><td>40.1 ± 0.4</td><td>42.5 ± 0.2 (44.9 ± 0.2)</td></tr><tr><td>50</td><td>42.8± 0.4</td><td>43.6 ± 0.4</td><td>1</td><td>47.7 ± 0.2</td><td>44.3 ± 0.2 (43.0 ± 0.3)</td></tr><tr><td rowspan="2">T-ImageNet</td><td>1</td><td>6.6± 0.2t</td><td>3.9± 0.2</td><td></td><td>8.8 ±0.3</td><td>15.4 ± 0.3 (19.1 ± 0.3)</td></tr><tr><td>10</td><td>一</td><td>12.9 ± 0.4</td><td></td><td>23.2 ± 0.2</td><td>25.4 ± 0.2 (26.5± 0.1)</td></tr><tr><td rowspan="2">CUB-200</td><td>1</td><td>1.3 ± 0.1†</td><td>1.6 ± 0.1†</td><td></td><td>2.2± 0.1†</td><td>12.4 ± 0.2 (13.7 ± 0.2)</td></tr><tr><td>10</td><td>4.5 ± 0.3†</td><td>4.4 ± 0.2†</td><td></td><td>1</td><td>16.8 ± 0.1 (16.1 ± 0.3)</td></tr></table>
+
+![](images/f1494e6fda90eb5674ceeec703ad75133081347b4f6471179e08a6c207b5ecd8.jpg)  
+Figure 3: (a,b) Training efficiency comparison when learning $1 \mathrm { I m g / C l s }$ on CIFAR100. (c,d) Time per iteration and peak memory usage as we increase the model size. FRePo is significantly more efficient than the previous methods, almost two orders of magnitude faster than the second-best method (i.e., MTT), with only 1/10 of the GPU memory requirement.
+
+# 4.2 Standard Benchmarks
+
+Distillation Performance: We first evaluate our method on six standard benchmark datasets. We learn 1, 10, and 50 images per class for datasets with only ten classes, while we learn 1 and 10 images per class for CIFAR100 [34] with 100 classes, Tiny ImageNet [35] with 200 classes, and CUB-200 [37] with 200 fine-grained classes. As shown in Table in 1, we achieve the state-of-the-art performance in most settings despite the hyperparameter may be suboptimal. Our method performs exceptionally well on datasets with a complex label space when learning few images per class. For example, we improve the CIFAR100, Tiny ImageNet, and CUB-200 in one image per class setting from $2 4 . 3 \%$ , $8 . 8 \%$ , and $2 . 2 \%$ to $2 8 . 7 \%$ , $1 5 . 4 \%$ , and $1 2 . 4 \%$ , respectively. Figure 4 shows that our distilled images look real and natural though we do not directly optimize for this objective. We observe a strong correlation between the test accuracy and image quality: the better the image quality, the higher the test accuracy. Our results suggest that a highly condensed dataset does not need to be very different from the real dataset as it may just reflect the most common pattern in a dataset. We also report the KRR predictor’s test accuracy using the feature extractor trained on the distilled data. When the dataset is as simple as MNIST [32], the KRR predictor achieves similar performance as the neural network predictor. In contrast, for more complex datasets, the KRR predictor consistently outperforms the neural network predictor, with the most significant gap being $3 . 7 \%$ for Tiny ImageNet in the one image per class setting.
+
+Table 2: Cross-architecture transfer performance on CIFAR10 with $1 0 \mathrm { I m g / C l s }$ . Despite being trained for a specific architecture, our distilled data transfer well to various architectures unseen during training. Conv is the default evaluation model used for each method. NN, DN, IN, and BN stand for no normalization, default normalization, Instance Normalization, Batch Normalization respectively.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Train Arch</td><td colspan="6">Evaluation Architecture</td></tr><tr><td>Conv</td><td>Conv-NN</td><td>ResNet-DN</td><td>ResNet-BN</td><td>VGG-BN</td><td>AlexNet</td></tr><tr><td>DSA [7]</td><td>Conv-IN</td><td>53.2 ± 0.8</td><td>36.4 ± 1.5</td><td>42.1 ± 0.7</td><td>34.1 ± 1.4</td><td>46.3 ± 1.3</td><td>34.0 ± 2.3</td></tr><tr><td>DM[8]</td><td>Conv-IN</td><td>49.2 ± 0.8</td><td>35.2 ± 0.5</td><td>36.8 ± 1.2</td><td>35.5 ± 1.3</td><td>41.2 ± 1.8</td><td>34.9 ± 1.1</td></tr><tr><td>MTT[20]</td><td>Conv-IN</td><td>64.4 ± 0.9</td><td>41.6 ± 1.3</td><td>49.2 ± 1.1</td><td>42.9 ± 1.5</td><td>46.6 ± 2.0</td><td>34.2 ± 2.6</td></tr><tr><td>KIP [23]</td><td>Conv-NTK</td><td>62.7 ± 0.3</td><td>58.2 ±0.4</td><td>49.0 ± 1.2</td><td>45.8 ± 1.4</td><td>30.1 ± 1.5</td><td>57.2 ± 0.4</td></tr><tr><td>FRePo</td><td>Conv-BN</td><td>65.5 ± 0.4</td><td>65.5 ± 0.4</td><td>58.1 ± 0.6</td><td>57.7 ± 0.7</td><td>59.4 ± 0.7</td><td>61.9 ± 0.7</td></tr></table>
+
+![](images/06d156480cb24f7de4d97dc51031b5fb9e76281f33da496484aeb238e841000d.jpg)  
+Figure 4: (a,b,c) Distilled 1 img/cls from CIFAR100 using FRePo, MTT, and DSA. High quality images also produce high test accuracy. (d) Three categories of learned labels. (Top) High confidence, large margin; (Middle) High confidence, small margin; (Bottom) Low confidence, small margin.
+
+Label Learning: A similar trend can also be observed for label learning. When the dataset is simple and has only a few classes, label learning may not be necessary. However, it becomes crucial for complex datasets with many labels, such as CIFAR100 and Tiny-ImageNet (See more details in Appendix ??). Similar to the teacher label in the knowledge distillation [1], we observe that the distilled label also encodes the class similarity. We identify three typical cases in Figure 4d. The first group consists of highly confident labels with a much higher value for one class than other classes (large margin), such as sunflower, bicycle, and chair. In contrast, the distilled labels in the second group are confident but may get confused with some closely-related classes (small margin). For instance, the learned label for "girl" has almost equally high values for the girl, woman, man, boy, and baby, suggesting that these classes are very similar and may be difficult for the model to distinguish them apart. The last group contains distilled labels with low values for all classes, such as bear, beaver, and squirrel. It is often hard for humans to recognize the distilled images in such a group, suggesting that they may be the challenging classes in a dataset.
+
+Training Cost Analysis: Figure 3a, 3b shows that our method is significantly more time-efficient than the previous methods. When learning one image per class on CIFAR100, FRePo reaches a similar test accuracy $( 2 3 . 4 \% )$ to the second-best method $( 2 4 . 0 \% )$ in 38 seconds, compared to 3805 seconds for MTT, which is roughly two orders of magnitude faster. Moreover, FRePo achieves $92 \%$ of its final test accuracy ( $2 6 . 4 \%$ out of $2 8 . 7 \%$ ) in only 385 seconds. As shown in Figure 3c, our algorithm takes much less time to perform one gradient step on the distilled data. Thus, we can perform more gradient steps in a fixed time. Furthermore, Figure 3d suggests that our algorithm has much less GPU memory requirement. Therefore, we can potentially use a much larger and more complex model to take advantage of the advancement in neural network architecture.
+
+Cross-Architecture Generalization: One desired property of our distilled data is that it generalizes well to architecture it has not seen during the training. Similar to previous works [5, 20], we evaluate the distilled data from CIFAR10 on a wide range of architectures which it has not seen during training, including AlexNet [38], VGG [39], and ResNet [40]. Table 2 shows that our method outperforms previous methods on all unseen architectures. Instance Normalization (IN) [41], as the vital ingredient in several methods (DSA, DM, MTT), seems to hurt the cross-architecture transfer. The performance degrades a lot when no normalization (NN) is applied (Conv-NN, AlexNet) or using a different normalization, like Batch Normalization (BN) [42]. It suggests that the distilled data generated by those methods encode the inductive bias of a particular training architecture. In contrast, our distilled data generalize well to various architectures, including those without normalization (Conv-NN, AlexNet). Note that Figure 1, 4 also indicate that our distilled data encode less architectural bias as the distilled images look natural and authentic. A simple idea to further alleviate the overfitting of a particular architecture is to include more architectures in the model pool. However, the training may not be stable as the meta-gradient computed by different architectures can be very different.
+
+Table 3: Distillation performance on higher resolution (128x128) dataset (i.e. ImageNette, ImageWoof) and medium resolution (64x64) dataset with a complex label space (i.e. ImageNet-1K). FRePo scales to high-resolution images and learns the discriminate feature of complex datasets.   
+
+<table><tr><td></td><td colspan="2">ImageNette (128x128)</td><td colspan="2">ImageWoof (128x128)</td><td colspan="2">ImageNet (64x64)</td></tr><tr><td>Img/Cls</td><td>1</td><td>10</td><td>1</td><td>10</td><td>1</td><td>2</td></tr><tr><td>Random Subset</td><td>23.5± 4.8</td><td>47.7 ± 2.4</td><td>14.2 ± 0.9</td><td>27.0± 1.9</td><td>1.1 ± 0.1</td><td>1.4 ± 0.1</td></tr><tr><td>MTT[20]</td><td>47.7± 0.9</td><td>63.0 ± 1.3</td><td>28.6 ± 0.8</td><td>35.8 ± 1.8</td><td>1</td><td>1</td></tr><tr><td>FRePo</td><td>48.1 ± 0.7</td><td>66.5 ± 0.8</td><td>29.7 ± 0.6</td><td>42.2 ± 0.9</td><td>7.5 ± 0.3</td><td>9.7 ± 0.2</td></tr></table>
+
+# 4.3 ImageNet
+
+High Resolution ImageNet Subset To understand how well our method performs on high-resolution images, we evaluate it on ImageNette and ImageWoof datasets [36] with a resolution of 128x128. We learn 1 and 10 images per class on both datasets and report the performance in Table 3 and visualize some distilled images in Figure 1. As shown in Table 3, we outperform MTT on all settings and achieve much better performance when we distill ten images per class on a more difficult dataset ImageWoof. It suggests that our distilled data is better at capturing the discriminative features for each class. Figure 1 shows that our distilled images look real and capture the distinguishable feature of different classes. For the easy dataset (i.e., ImageNette), all images have clear different structures, while for ImageWoof, the texture of each dog seems to be crucial.
+
+Resized ImageNet-1K: We also evaluate our method on a resized version of ILSVRC2012 [26] with a resolution of 64x64 to see how it performs on a complex label space. Surprisingly, we can achieve $7 . 5 \%$ and $9 . 7 \%$ Top1 accuracy using only 1k and $2 \mathrm { k }$ training examples, compared to $1 . 1 \%$ and $1 . 4 \%$ using an equally-sized real subset.
+
+# 5 Application
+
+# 5.1 Continual Learning
+
+Continual learning (CL) [43] aims to address the catastrophic forgetting problem [43–45] when a model learns sequentially from a stream of tasks. A commonly used strategy to recall past knowledge is based on a replay buffer, which stores representative samples from previous tasks [46–49]. Since sample selection is an important component of constructing an effective buffer [48–51], we believe distilled data can be a key ingredient for a continual learning algorithm due to its highly condensed nature. Several works [6–8, 52] have successfully applied the dataset distillation to the continual learning scenario. Our work shows that we can achieve much better results by using a better dataset distillation technique.
+
+We follow Zhao and Bilen [8] that sets up the baseline based on GDumb [49] which greedily stores class-balanced training examples in memory and train model from scratch on the latest memory only. In that case, the continual learning performance only depends on the quality of the replay buffer. We perform 5 and 10 step class-incremental learning [53] on CIFAR100 with an increasing buffer size of 20 images per class. Specifically, we distill 400 and 200 images at each step and put them into the replay buffer. We follow the same class split as Zhao and Bilen [8] and compare our method to random [49], herding [54, 55], DSA [7], and DM [8]. We use the default data preprocessing and default model for each method in this experiment as we find it gives the best performance for each method. We use the test accuracy on all observed classes as the performance measure [8, 48].
+
+Table 4: AUC of five attackers on models trained on the real and distilled MNIST data. The model trained on the real data is vulnerable to MIAs, while the model trained on the distilled data is robust to MIAs. Training on distilled data allows privacy preservation while retaining model performance.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Test Acc (%)</td><td colspan="5">Attack AUC</td></tr><tr><td>Threshold</td><td>LR</td><td>MLP</td><td>RF</td><td>KNN</td></tr><tr><td>Real</td><td>99.2 ± 0.1</td><td>0.99 ± 0.01</td><td>0.99 ± 0.00</td><td>1.00 ±0.00</td><td>1.00 ± 0.00</td><td>0.97 ±0.00</td></tr><tr><td>Subset</td><td>96.8± 0.2</td><td>0.52 ±0.00</td><td>0.50 ± 0.01</td><td>0.53 ± 0.01</td><td>0.55 ± 0.00</td><td>0.54 ±0.00</td></tr><tr><td>DSA</td><td>98.5 ± 0.1</td><td>0.50 ± 0.00</td><td>0.51 ± 0.00</td><td>0.54 ± 0.00</td><td>0.54 ± 0.01</td><td>0.54 ± 0.01</td></tr><tr><td>DM</td><td>98.3 ± 0.0</td><td>0.50 ± 0.00</td><td>0.51 ± 0.01</td><td>0.54 ± 0.01</td><td>0.54 ± 0.01</td><td>0.53 ± 0.01</td></tr><tr><td>FRePo</td><td>98.5± 0.1</td><td>0.52 ±0.00</td><td>0.51 ± 0.00</td><td>0.53 ± 0.01</td><td>0.52 ± 0.01</td><td>0.51 ± 0.01</td></tr></table>
+
+![](images/3425d57f4bfa2229b9a55a067323e25dd62e8fcc52fb803fdd882c3fcc7566d9.jpg)  
+Figure 5: (a,b) Multi-class accuracies across all classes observed up to a certain time point. We perform significantly better than other methods in both 5 and 10 step class-incremental continual learning. (c,d) Test accuracy and attack AUC as we increase the number of training steps. AUC keeps increasing when training a model on the real data for more steps. In contrast, AUC keeps low when training on distilled data.
+
+Figure 5 shows that our method performs significantly better than all previous methods. The final test accuracy for all classes for our method (FRePo) and the second-best method (DM) are $4 1 . 6 \%$ , $3 3 . 9 \%$ in 5-step learning, and $3 8 . 0 \%$ , $3 4 . 0 \%$ in 10-step learning. However, we notice that for FRePo, distilling 2000 images in a continual learning setup achieves a similar test accuracy $( 4 1 . 6 \% )$ as distilling only 1000 images from the whole dataset $( 4 1 . 3 \%$ from Table 1). In addition, performance drops as we perform more steps. It suggests that FRePo considers all available classes to derive the most condensed dataset. Splitting the data into multiple groups and performing independent distillation may generate redundant information or fail to capture the distinguishable features.
+
+# 5.2 Membership Inference Defense
+
+Membership inference attacks (MIA) aim to infer whether a given data point has been used to train the model or not [56–58]. Ideally, we want a model to learn from the data but not memorize it to preserve privacy. However, deep neural networks are well-known for their ability to memorize all the training examples, even on large and randomly labeled datasets [59]. Several methods have been proposed to defend against such attacks by either modifying the training procedure [60] or changing the inference workflow [61]. This section shows that the distilled data contain little information regarding sample presence in the original dataset. Thus, instead of training on the original datasets, training on distilled data allows privacy preservation while retaining model performance.
+
+We consider three distilled data generated by DSA [7], DM [8] and FRePo. We perform five popular "black box" MIA provided by Tensorflow Privacy [62] on models trained on the real data or the data distilled from it. The attack methods include a threshold attack and four model-based attacks using logistic regression (LR), multi-layer perceptron (MLP), random forest (RF) and K-nearest neighbor (KNN). The inputs to those attack methods are ground-truth labels, model predictions, and losses. To measure the privacy vulnerability of the trained model, we compute the area under the ROC curve (AUC) of an attack classifier. Following prior work, [56, 63], we keep a balanced set of training examples (member) and test examples (non-member) with 10K each to maximize the uncertainty of MIA. Thus, the random guessing strategy results in a $50 \%$ MIA accuracy. We conduct experiments on MNIST and FashionMNIST with a distillation size of 500. For space reasons, we provide more implementation details and results in appendix.
+
+As shown in Table 4, all models trained on the distilled data preserve privacy as their attack AUCs are closed to random guessing. However, we observe a small drop in test accuracy compared to the model trained on the full dataset, which is expected as we only distill 500 examples instead of 10,000 examples. Compared to the model trained on an equally sized subset of the original data, the model trained on distilled data results in much better test performance. Figure 5c, 5d demonstrate the trade-off between test accuracy and attack effectiveness as measured by ROC AUC. It shows that early stopping can be an effective technique to preserve privacy. However, we will still be under high MIA risk if we perform early stopping by monitoring the validation loss. In contrast, training a model on the distilled data does not have this problem as the attack AUCs keep at a very low level regardless of training steps.
+
+# 6 Conclusion
+
+We propose neural Feature Regression with Pooling (FRePo) to overcome two challenges in dataset distillation: meta-gradient computation and various types of overfitting in dataset distillation. We obtain state-of-the-art performance on various datasets with a $1 0 0 \mathrm { x }$ reduction in training time and a 10x reduction in GPU memory requirement. The distilled data generated by FRePo looks real and natural and generalizes well to a wide range of architectures. Furthermore, we demonstrate two applications that take advantage of the high-quality distilled data, namely, continual learning and membership inference defense.
+
+Broader Impact “Synthetic data”, in the broader sense of artificial data created by generative models, can help researchers understand how an otherwise opaque learning machine “sees” the world. There have been concerns regarding the risk of fake data. This paper explores a new research direction in generating synthetic data only for downstream classification tasks. We believe this work can provide additional interpretability and potentially address the common concerns in machine learning regarding training data privacy.
+
+# Acknowledgments and Disclosure of Funding
+
+We would like to thank Harris Chan, Andrew Jung, Michael Zhang, Philip Fradkin, Denny Wu, Chong Shao, Leo Lee, Alice Gao, Keiran Paster, and Lazar Atanackovic for their valuable feedback. Jimmy Ba was supported by NSERC Grant [2020-06904], CIFAR AI Chairs program, Google Research Scholar Program and Amazon Research Award. This project was supported by LG Electronics Canada. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute for Artificial Intelligence.
+
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+        "text": "Dataset Distillation using Neural Feature Regression ",
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+        "text": "Yongchao Zhou Department of Computer Science University of Toronto yongchao.zhou@mail.utoronto.ca ",
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+        "text": "Ehsan Nezhadarya Toronto AI Lab LG Electronics Canada ehsan.nezhadarya@lge.com ",
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+        "text": "Jimmy Ba Department of Computer Science University of Toronto jba@cs.toronto.edu ",
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+        "text": "Abstract ",
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+        "text": "Dataset distillation aims to learn a small synthetic dataset that preserves most of the information from the original dataset. Dataset distillation can be formulated as a bi-level meta-learning problem where the outer loop optimizes the metadataset and the inner loop trains a model on the distilled data. Meta-gradient computation is one of the key challenges in this formulation, as differentiating through the inner loop learning procedure introduces significant computation and memory costs. In this paper, we address these challenges using neural Feature Regression with Pooling (FRePo), achieving the state-of-the-art performance with an order of magnitude less memory requirement and two orders of magnitude faster training than previous methods. The proposed algorithm is analogous to truncated backpropagation through time with a pool of models to alleviate various types of overfitting in dataset distillation. FRePo significantly outperforms the previous methods on CIFAR100, Tiny ImageNet, and ImageNet-1K. Furthermore, we show that high-quality distilled data can greatly improve various downstream applications, such as continual learning and membership inference defense. Please check out our webpage at https://sites.google.com/view/frepo. ",
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+        "text": "1 Introduction ",
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+        "text": "Knowledge distillation [1] is a technique in deep learning to compress knowledge for easy deployment. Most previous works focus on model distillation [2, 3] where the knowledge acquired by a large teacher model is transferred to a small student model. In contrast, dataset distillation [4, 5] aims to learn a small set of synthetic examples preserving most of the information from a large dataset such that a model trained on it can achieve similar test performance as one trained on the original dataset. Distilled data can accelerate model training and reduce the cost of storing and sharing a dataset. Moreover, its highly condensed and synthetic nature can also benefit various applications, such as continual learning [5–8], neural architecture search [5, 7], and privacy-preserving tasks [9, 10]. ",
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+        "text": "Dataset distillation was first studied by Maclaurin et al. [11] in the context of gradient-based hyperparameter optimization and subsequently Wang et al. [4] formally proposed dataset distillation as a new task. Dataset distillation can be naturally formulated as a bi-level meta-learning problem. The inner loop optimizes the model parameters on the distilled data (meta-parameters), while the outer loop refines the distilled data with meta-gradient updates. ",
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+        "text": "One key challenge in dataset distillation is computing the meta-gradient. Several methods [4, 11–13] compute it by back-propagating through the unrolled computation graph, but they often suffer from huge compute and memory requirement [14], training instability [15, 16], and truncation bias [17]. To avoid unrolled optimization, surrogate objectives are used to derive the meta-gradient, such as gradient matching [5, 7, 18], feature alignment [8, 19], and training trajectory matching [20]. Nevertheless, a surrogate objective may introduce its own bias [19], and thus, may not accurately reflect the true objective. An alternative is using kernel methods, such as Neural Tangent Kernel (NTK) [21], to approximate the inner optimization [22, 23]. However, computing analytical NTK for modern neural network can be extremely expensive [22, 23]. ",
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+            "Figure 1: Example distilled images from $3 2 \\mathbf { x } 3 2$ CIFAR100, 64x64 Tiny ImageNet, and $1 2 8 \\mathrm { x } 1 2 8$ ImageNet Subset. The images look real and transfer well to different architectures. They can be used for various downstream applications, such as continual learning and membership inference defense. "
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+        "text": "Even with an accurate meta-gradient, dataset distillation still suffers from various types of overfitting. For instance, the distilled data can easily overfit to a particular learning algorithm [4, 13, 20], a certain stage of optimization [13, 19], or a certain network architecture [5, 7, 20, 22, 23]. Meanwhile, the model can also overfit the distilled data during training, which is the most common cause of overfitting when we train on a small dataset. All these kinds of overfitting impose difficulties on the training and general-purpose use of the distilled data. ",
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+        "text": "We propose an efficient meta-gradient computation method and a “model pool” to address the overfitting problems. The bottleneck in meta-gradient computation arises due to the complexity of inner optimization, as we need to know how the inner parameters vary with the outer parameters [24]. However, the inner optimization can be pretty simple if we only train the last layer of a neural network to convergence while keeping the feature extractor fixed. In this case, computing the prediction on the real data using the model trained on the distilled data can be expressed as a kernel ridge regression (KRR) with respect to the conjugate kernel [25]. Hence, computing the meta-gradient is simply back-propagating through the kernel and a fixed feature extractor. To alleviate overfitting, we propose to maintain a diverse pool of models instead of periodically training and resetting a single model as in prior work [7, 13, 18]. Intuitively, our algorithm targets the following question: what is the best data to train the linear classifier given the current feature extractor? Due to the diverse feature extractors we use, the distilled data generalize well to a wide range of model distributions. ",
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+        "type": "text",
+        "text": "Summary of Contributions: ",
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+        "text": "• We propose an effective method for dataset distillation. Our method, named neural Feature Regression with Pooling (FRePo), achieves state-of-the-art results on various benchmark datasets with a $1 0 0 \\mathrm { x }$ reduction in training time and a $1 0 \\mathrm { x }$ reduction in GPU memory requirement. Our distilled data looks real (Figure 1) and transfers well to different architectures. We show that FRePo scales well to datasets with high-resolution images or complex label space. We achieve $7 . 5 \\%$ top1 accuracy on ImageNet-1K [26] using only one image per class. The same classifier obtains only $1 . 1 \\%$ accuracy from a random subset of real images. The previous methods struggle in this task due to large memory and compute requirements. • We demonstrate that high-quality distilled data can significantly improve various downstream applications, such as continual learning and membership inference defense. ",
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+        "image_caption": [
+            "Figure 2: Comparison of FRePo and Unrolled Optimization. $S$ , $X _ { s }$ , $Y _ { s }$ are the distilled dataset, images and labels. $\\mathcal { L }$ is the meta-training loss and $\\dot { \\theta } ^ { ( k ) }$ , $g ^ { ( k ) }$ are the model parameter and gradient at step $k$ . $f ( X )$ is the feature for input $X$ and $K _ { X _ { t } X _ { s } } ^ { \\theta }$ is the Gram matrix of $X _ { t }$ and $X _ { s }$ . FRePo is analogous to 1-step TBPTT as it computes the meta-gradient at each step while performing the online model update. However, instead of backpropagating through the inner optimization, FRePo computes the meta-gradient through a kernel and feature extractor. "
+        ],
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+        "text": "2 Method ",
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+        "text": "2.1 Dataset Distillation as Bi-level Optimization ",
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+        "text": "Suppose we have a large labeled dataset $\\mathcal { T } = \\left\\{ \\left( \\mathbf { x } _ { 1 } , \\mathbf { y } _ { 1 } \\right) , \\dotsc , \\left( \\mathbf { x } _ { | T | } , \\mathbf { y } _ { | T | } \\right) \\right\\}$ with $| \\tau |$ image and label\\` ˘( pairs. Dataset distillation aims to learn a small synthetic dataset $\\mathcal { S } = \\left\\{ ( \\mathbf { x } _ { 1 } , \\mathbf { y } _ { 1 } ) , \\dotsc , \\left( \\mathbf { x } _ { | S | } , \\mathbf { y } _ { | S | } \\right) \\right\\}$ that preserves most of the information in $\\tau$ . We train several neural networks parameterized by $\\theta$ on the dataset $s$ and then compute the validation loss $\\mathcal { L } ( \\mathcal { A } l g \\left( \\theta , \\mathcal { S } \\right) , \\mathcal { T } )$ on the real dataset $\\tau$ , where ${ \\mathcal { A } } l g \\left( \\theta , S \\right)$ is the neural network parameters optimized by a learning algorithm $\\mathcal { A } g$ with the model initialization $\\theta$ and distilled dataset $s$ as its inputs. The validation loss $\\mathcal { L } ( \\mathcal { A } l g \\left( \\theta , S \\right) , \\mathcal { T } )$ is a noisy objective with the stochasticity coming from random model initialization and inner learning algorithm. Thus, we are interested in minimizing the expected value of this loss, which we denote it as $F ( S )$ . We formulate the dataset distillation as the following bi-level optimization problem. ",
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+        "text": "$$\n\\overbrace { \\mathcal { S } ^ { * } : = \\mathop { \\mathrm { a r g m i n } } _ { \\mathcal { S } } F ( \\mathcal { S } ) } ^ { o u t e r - l e v e l } , \\mathrm { w h e r e } F ( \\mathcal { S } ) = \\mathbb { E } _ { \\theta \\sim P _ { \\theta } } \\biggl [ \\mathcal { L } \\Bigl ( \\overbrace { \\mathcal { A } l g \\left( \\theta , \\mathcal { S } \\right) } ^ { i n n e r - l e v e l } , \\ T \\Bigr ) \\biggr ] .\n$$",
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+        "text": "In this bi-level setup, the outer loop optimizes the distilled data to minimize $F ( S )$ , while the inner loop trains a neural network using the learning algorithm, $\\mathcal { A } g$ , to minimize the training loss on the distilled data $s$ . From the meta-learning perspective, the task is defined by the model initialization $\\theta$ , and we want to learn a meta-parameter $s$ that generalizes well to different models sampled from the model distributions $P _ { \\theta }$ . During learning, we optimize the meta-parameter $s$ by minimizing the meta-training loss $F ( S )$ . In contrast, at meta-test time, we train a new model from scratch on $s$ and evaluate the trained model on a held-out real dataset. This meta-test performance reflects the quality of the distilled data. ",
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+        "text": "2.2 Dataset Distillation using Neural Feature Regression with Pooling $\\mathbf { ( F R e P 0 ) }$ ",
+        "text_level": 1,
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+        "text": "The outer-level problem can be solved using gradient-based methods of the form $\\mathcal { S } \\gets \\mathcal { S } \\ – \\alpha \\nabla _ { \\mathcal { S } } F ( \\mathcal { S } )$ , where $\\alpha$ is the learning rate for the distilled data and $\\nabla _ { S } F ( S )$ is the meta-gradient [27]. For a particular model $\\theta$ , the meta-gradient can be expressed as $\\nabla _ { \\mathcal { S } } \\hat { \\mathcal { L } } \\left( \\mathcal { A } l g \\left( \\theta , \\mathcal { S } \\right) , \\mathcal { T } \\right)$ . Computing this meta-gradient requires differentiating through inner optimization. If $\\mathcal { A } \\boldsymbol { { l } } _ { g }$ is an iterative algorithm like gradient descent, then backpropagating through the unrolled computation graph [14] can be a solution. However, this type of unrolled optimization introduces significant computation and memory overhead, as the whole training trajectory needs to be stored in memory (Figure 2(b)). ",
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+    {
+        "type": "text",
+        "text": "Traditionally, these issues are alleviated with truncated backpropagation through time (TBPTT) [28–30]. Instead of backpropagating through an entire unrolled sequence, TBPTT performs backpropagation for each subsequence separately. It is efficient because its time and memory complexity scale linearly with respect to the truncation steps. However, truncation may yield highly biased gradients that severely impact training. To mitigate this truncation bias [14], we consider training only the top layer of a network to convergence. The key insight is that the data helpful for training the output layer can also help train the whole network. Thus, we decompose the neural network into a feature ",
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+    {
+        "type": "text",
+        "text": "Require: $\\tau$ : a labeled dataset; $\\alpha$ : the learning rate for the distilled data nitialization: Initialize a labeled distilled dataset $\\boldsymbol { \\mathcal { S } } = \\left( \\boldsymbol { X _ { s } } , \\boldsymbol { Y _ { s } } \\right)$ . ",
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+        "type": "text",
+        "text": "Initialization: Initialize a model pool $\\mathcal { M }$ with $m$ models $\\left\\{ \\boldsymbol { \\theta } _ { i } \\right\\} _ { i = 1 } ^ { m }$ randomly initialized from $P _ { \\theta }$ ",
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+        "type": "text",
+        "text": "1: while not converged do   \n2: Ż Sample a model uniformly from the model pool: $\\theta _ { i } \\sim \\mathcal { M }$ .   \n3: Ż Sample a target batch uniformly from the labeled dataset: $( X _ { t } , Y _ { t } ) \\sim \\tau$ .   \n4: $\\triangleright$ Compute the meta-training loss $\\mathcal { L }$ using Eq. 2   \n5: Ż Update the distilled data $s$ $\\mathrm { : } ~ X _ { s } \\gets X _ { s } - \\alpha \\nabla _ { X _ { s } } \\mathcal { L }$ , and $Y _ { s } \\gets Y _ { s } - \\alpha \\nabla _ { Y _ { s } } \\mathcal { L }$   \n6: $\\triangleright$ Train the model $\\theta _ { i }$ on the current distilled data $s$ for one step.   \n7: $\\triangleright$ Reinitialize the model $\\theta _ { i } \\sim P _ { \\theta }$ if $\\theta _ { i }$ has been updated more than $K$ steps. ",
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+    {
+        "type": "text",
+        "text": "8: end while ",
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+    {
+        "type": "text",
+        "text": "Output: Learned distilled dataset $\\boldsymbol { \\mathcal { S } } = \\left( \\boldsymbol { X _ { s } } , \\boldsymbol { Y _ { s } } \\right)$ ",
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+    {
+        "type": "text",
+        "text": "extractor and a linear classifier. We fix the feature extractor at each meta-gradient computation and train the linear classifier to convergence before updating $s$ . After that, we adjust the feature extractor by training the whole network on the updated distilled data. We note that similar two-phase procedure has been studied in the context of representation learning [31]. ",
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+    {
+        "type": "text",
+        "text": "Meta-Gradient Computation: If we consider the mean square error loss, then the optimal weights for the linear classifier have a closed-form solution. Moreover, since the feature dimension is typically larger than the number of distilled data, we can use kernel ridge regression (KRR) with a conjugate kernel [25] rather than solving the weights explicitly [12]. The resulting meta-training loss (Eq. 2) is similar to that used in KIP [22, 23], but we use a more flexible kernel rather than NTK. ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/0cd9f2dca22b293f4e069930dff443abaa6142b03906a3c62fda7f6203ea2b36.jpg",
+        "text": "$$\n\\mathcal { L } \\left( \\mathcal { A } l g \\left( \\theta , \\mathcal { S } \\right) , \\mathcal { T } \\right) = \\frac { 1 } { 2 } | | Y _ { t } - K _ { X _ { t } X _ { s } } ^ { \\theta } ( K _ { X _ { s } X _ { s } } ^ { \\theta } + \\lambda I ) ^ { - 1 } Y _ { s } | | _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $( X _ { t } , Y _ { t } )$ and $( X _ { s } , Y _ { s } )$ are the inputs and labels of the real data $\\tau$ and distilled data $s$ respectively. The Gram matrix between real inputs and distilled inputs is denoted as $K _ { X _ { t } X _ { s } } ^ { \\theta } \\in \\mathbb { R } ^ { | T | \\times | S | }$ while the Gram matrix between distilled inputs is denoted as $K _ { X _ { s } X _ { s } } ^ { \\theta } \\in \\mathbb { R } ^ { | S | \\times | S | }$ . $\\lambda$ t scontrols the regularization strength for KRR. Let us denote the neural network feature for a given input $X$ and model parameter $\\theta$ as $\\mathbf { \\bar { \\chi } } _ { f ( X , \\theta ) } \\in \\mathbb { R } ^ { N \\times d }$ , where $N$ is the number of input and $d$ is the feature dimension 1. The conjugate kernel is defined by the inner product of the neural network features. Thus, the two Gram matrices are computed as follows: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/87a4499ed59b016b53bcda7f7527fcb6ea8d32afd3493c3f5288d6937be23d26.jpg",
+        "text": "$$\nK _ { X _ { t } X _ { s } } ^ { \\theta } = f ( X _ { t } , \\theta ) f ( X _ { s } , \\theta ) ^ { \\top } , \\quad K _ { X _ { s } X _ { s } } ^ { \\theta } = f ( X _ { s } , \\theta ) f ( X _ { s } , \\theta ) ^ { \\top } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Now, computing the meta-gradient $\\nabla _ { \\mathcal { S } } \\mathcal { L } \\left( \\mathcal { A } l g \\left( \\theta , \\mathcal { S } \\right) , \\mathcal { T } \\right)$ is just back-propagating through the conjugate kernel and a fixed feature extractor, which is very efficient and takes even fewer operations than computing the gradient for the network’s weights. Moreover, we decouple the meta-gradient computation from the model online update. Hence, we can train the online model using any optimizer, and the distilled data will be agnostic to the specific learning algorithm choice. Our proposed method is similar to 1-step TBPTT in that we compute the meta-gradient at each step while performing the online model update. Unlike the conventional 1-step TBPTT, we compute the meta-gradient using a KRR output layer to mitigate truncation bias, illustrated in Figure 2(a). ",
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+    {
+        "type": "text",
+        "text": "Model Pool: As discussed in Section 1, there are various types of overfitting in dataset distillation. Several techniques have been proposed to alleviate such problem, such as random initialization [4], periodic reset [13, 5, 7], and dynamic bi-level optimization [19]. These techniques share the same underlying principle: the model diversity matters. Thus, we propose to maintain a “model pool” filled with diverse set of parameters obtained from different number of training steps and different random initializations. Unlike the previous methods that periodically training and resetting a single model, FRePo randomly sample a model from the pool at each meta-gradient computation and update it using the current distilled data. However, if a model has been updated more than $K$ steps, we reinitialize it with a new random seed. From the meta-learning perspective, we maintain a diverse set of meta-tasks to sample from and avoid sampling very similar tasks at each consecutive gradient computation to avoid overfitting to a particular setup. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Pool Diversity: We can increase the regularization strength by increasing the diversity of the model pool by setting a larger $K$ , using data augmentation when training the model on the distilled data, or using models with different architectures. To keep our method simple, we use the same architecture for all models in the pool and do not use any data augmentation when training the model on the distilled data. Thus, our model pool only contains models with different initialization, at different optimization stages, and trained at different time-step of the distilled data. ",
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+    {
+        "type": "text",
+        "text": "3 Related Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Unrolling in Bi-Level Optimization: One way to compute the meta-gradient is to differentiate through the unrolled inner optimization [4, 11–13]. However, this approach inherits several difficulties of the unrolled optimization, such as: 1) large computation and memory cost [14]; 2) truncation bias with short unrolls [17]; 3) exploding or vanishing gradients with long unrolls [15]; 4) chaotic and poorly conditioned loss landscapes with long unrolls [16]. In contrast, our method considers approximating the inner optimization with kernel ridge regression instead of unrolled optimization. ",
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+    {
+        "type": "text",
+        "text": "Surrogate Objective: To avoid unrolled optimization, several works turn to surrogate objectives. DC [5], DSA [7], and DCC [18] formulate the dataset distillation as a gradient matching problem between the gradients of neural network weights computed on the real and distilled data. In contrast, DM [8] and CAFE [19] consider the feature distribution alignment between the real and distilled data. Moreover, MTT [20] shows that knowledge from many expert training trajectories can be distilled to a dataset by using a training trajectory matching objective. Nevertheless, surrogate objectives may introduce new biases and thus, may not accurately reflect the true objective. For example, gradient matching approaches [5, 7, 18] only focus on short-range behavior and may easily overfit to a biased set of samples that produce dominant gradients [19, 20]. ",
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+    {
+        "type": "text",
+        "text": "Closed-form Approximation: An alternative way to circumvent unrolled optimization is to find a closed-form approximation to the inner optimization. Based on the correspondence between infinitelywide neural networks and kernel methods, KIP [22, 23] approximates the inner optimization with NTK [21]. In this case, the meta-gradient can be computed by back-propagating through the NTK. However, computing NTK for modern neural networks is extremely expensive. Thus, using NTK for dataset distillation requires thousands of GPU hours and sophisticated implementation of the distributed kernel computation framework [23]. Similar to ours, Bohdal et al. [12] also decomposes the neural network as a feature extractor and a linear classifier. However, they only learn the label and explicitly solve for the optimal classifier weights rather than perform KRR. ",
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+    {
+        "type": "text",
+        "text": "4 Dataset Distillation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 Implementation Details ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We compare our method to four state-of-the-art dataset distillation methods [7, 8, 20, 23] on various benchmark datasets [26, 32–37]. We train the distilled data using Algorithm 1 with the same set of hyperparameters for all experiments except stated otherwise. Unlike prior work [7, 8, 20], we do not apply data augmentation during training. However, we apply the same data augmentation [7, 20] during evaluation for a fair comparison. We preprocess the data in a similar way as in previous works [20, 23] but use a wider architecture than previous works [7, 8, 20] because the KRR component does not behave well when the feature dimension is low, resulting in a significant performance drop for our method. Results on the original architecture are included in Appendix ??. We evaluate each distilled data using five random neural networks and report the mean and standard deviation. For the baseline method, we report the best of the reported value in the original paper and our reproducing results. ",
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+    {
+        "type": "text",
+        "text": "For the sake of brevity, we provide implementation details about data preprocessing, distilled data initialization, and hyperparameters in Appendix ?? and various ablation studies regarding the model pool, batch size, distilled data initialization, label learning, and model architectures in Appendix ??. More distilled image visualizations can be found in Appendix ??. Our code is available at https://github.com/yongchao97/FRePo. ",
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+    {
+        "type": "table",
+        "img_path": "images/a5d293177e310c0c210335bf5ab5229cfb17e497a341c9da5f6d75598dd1f5dd.jpg",
+        "table_caption": [
+            "Table 1: Test accuracies of models trained on the distilled data from scratch. : denotes performance better than the original reported performance. KRR preformance is shown in bracket. FRePo performs extremely well for one image per class setting on CIFAR100, Tiny ImageNet and CUB-200. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Img/Cls</td><td>DSA [7]</td><td>DM[8]</td><td>KIP [23]</td><td>MTT [20]</td><td>FRePo</td></tr><tr><td rowspan=\"3\">MNIST</td><td>1</td><td>88.7±0.6</td><td>89.9 ± 0.8†</td><td>90.1± 0.1</td><td>91.4 ± 0.9†</td><td>93.0 ± 0.4 (92.6 ± 0.4)</td></tr><tr><td>10</td><td>97.9 ±0.1†</td><td>97.6 ± 0.1†</td><td>97.5 ± 0.0</td><td>97.3 ± 0.1+</td><td>98.6 ± 0.1 (98.6 ± 0.1)</td></tr><tr><td>50</td><td>99.2 ± 0.1</td><td>98.6 ± 0.1</td><td>98.3 ± 0.1</td><td>98.5±0.1+</td><td>99.2 ± 0.0 (99.2± 0.1)</td></tr><tr><td rowspan=\"3\">F-MNIST</td><td>1</td><td>70.6 ± 0.6</td><td>71.5 ± 0.5†</td><td>73.5± 0.5</td><td>75.1 ± 0.9†</td><td>75.6 ± 0.3 (77.1 ± 0.2)</td></tr><tr><td>10</td><td>84.8±0.3t</td><td>83.6±0.2t</td><td>86.8±0.1</td><td>87.2± 0.3+</td><td>86.2 ± 0.2 (86.8 ± 0.1)</td></tr><tr><td>50</td><td>88.8±0.2t</td><td>88.2±0.1†</td><td>88.0±0.1</td><td>88.3± 0.1+</td><td>89.6 ± 0.1 (89.9 ± 0.1)</td></tr><tr><td rowspan=\"3\">CIFAR10</td><td>1</td><td>36.7± 0.8†</td><td>31.0 ± 0.6†</td><td>49.9 ± 0.2</td><td>46.3 ± 0.8</td><td>46.8 ± 0.7 (47.9 ± 0.6)</td></tr><tr><td>10</td><td>53.2 ± 0.8†</td><td>49.2 ± 0.8†</td><td>62.7± 0.3</td><td>65.3 ± 0.7</td><td>65.5 ± 0.4 (68.0 ± 0.2)</td></tr><tr><td>50</td><td>66.8± 0.4†</td><td>63.7±0.5t</td><td>68.6± 0.2</td><td>71.6 ± 0.2</td><td>71.7 ± 0.2 (74.4 ± 0.1)</td></tr><tr><td rowspan=\"3\">CIFAR100</td><td>1</td><td>16.8± 0.2†</td><td>12.2 ± 0.4†</td><td>15.7 ± 0.2</td><td>24.3 ± 0.3</td><td>28.7 ± 0.1 (32.3 ± 0.1)</td></tr><tr><td>10</td><td>32.3 ±0.3</td><td>29.7 ±0.3</td><td>28.3 ± 0.1</td><td>40.1 ± 0.4</td><td>42.5 ± 0.2 (44.9 ± 0.2)</td></tr><tr><td>50</td><td>42.8± 0.4</td><td>43.6 ± 0.4</td><td>1</td><td>47.7 ± 0.2</td><td>44.3 ± 0.2 (43.0 ± 0.3)</td></tr><tr><td rowspan=\"2\">T-ImageNet</td><td>1</td><td>6.6± 0.2t</td><td>3.9± 0.2</td><td></td><td>8.8 ±0.3</td><td>15.4 ± 0.3 (19.1 ± 0.3)</td></tr><tr><td>10</td><td>一</td><td>12.9 ± 0.4</td><td></td><td>23.2 ± 0.2</td><td>25.4 ± 0.2 (26.5± 0.1)</td></tr><tr><td rowspan=\"2\">CUB-200</td><td>1</td><td>1.3 ± 0.1†</td><td>1.6 ± 0.1†</td><td></td><td>2.2± 0.1†</td><td>12.4 ± 0.2 (13.7 ± 0.2)</td></tr><tr><td>10</td><td>4.5 ± 0.3†</td><td>4.4 ± 0.2†</td><td></td><td>1</td><td>16.8 ± 0.1 (16.1 ± 0.3)</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/f1494e6fda90eb5674ceeec703ad75133081347b4f6471179e08a6c207b5ecd8.jpg",
+        "image_caption": [
+            "Figure 3: (a,b) Training efficiency comparison when learning $1 \\mathrm { I m g / C l s }$ on CIFAR100. (c,d) Time per iteration and peak memory usage as we increase the model size. FRePo is significantly more efficient than the previous methods, almost two orders of magnitude faster than the second-best method (i.e., MTT), with only 1/10 of the GPU memory requirement. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.2 Standard Benchmarks ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Distillation Performance: We first evaluate our method on six standard benchmark datasets. We learn 1, 10, and 50 images per class for datasets with only ten classes, while we learn 1 and 10 images per class for CIFAR100 [34] with 100 classes, Tiny ImageNet [35] with 200 classes, and CUB-200 [37] with 200 fine-grained classes. As shown in Table in 1, we achieve the state-of-the-art performance in most settings despite the hyperparameter may be suboptimal. Our method performs exceptionally well on datasets with a complex label space when learning few images per class. For example, we improve the CIFAR100, Tiny ImageNet, and CUB-200 in one image per class setting from $2 4 . 3 \\%$ , $8 . 8 \\%$ , and $2 . 2 \\%$ to $2 8 . 7 \\%$ , $1 5 . 4 \\%$ , and $1 2 . 4 \\%$ , respectively. Figure 4 shows that our distilled images look real and natural though we do not directly optimize for this objective. We observe a strong correlation between the test accuracy and image quality: the better the image quality, the higher the test accuracy. Our results suggest that a highly condensed dataset does not need to be very different from the real dataset as it may just reflect the most common pattern in a dataset. We also report the KRR predictor’s test accuracy using the feature extractor trained on the distilled data. When the dataset is as simple as MNIST [32], the KRR predictor achieves similar performance as the neural network predictor. In contrast, for more complex datasets, the KRR predictor consistently outperforms the neural network predictor, with the most significant gap being $3 . 7 \\%$ for Tiny ImageNet in the one image per class setting. ",
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+    {
+        "type": "table",
+        "img_path": "images/4e44b3dde3c2a554e1627b4447e7df3e28b023aaff4e028f17f27dce37cbd8da.jpg",
+        "table_caption": [
+            "Table 2: Cross-architecture transfer performance on CIFAR10 with $1 0 \\mathrm { I m g / C l s }$ . Despite being trained for a specific architecture, our distilled data transfer well to various architectures unseen during training. Conv is the default evaluation model used for each method. NN, DN, IN, and BN stand for no normalization, default normalization, Instance Normalization, Batch Normalization respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Train Arch</td><td colspan=\"6\">Evaluation Architecture</td></tr><tr><td>Conv</td><td>Conv-NN</td><td>ResNet-DN</td><td>ResNet-BN</td><td>VGG-BN</td><td>AlexNet</td></tr><tr><td>DSA [7]</td><td>Conv-IN</td><td>53.2 ± 0.8</td><td>36.4 ± 1.5</td><td>42.1 ± 0.7</td><td>34.1 ± 1.4</td><td>46.3 ± 1.3</td><td>34.0 ± 2.3</td></tr><tr><td>DM[8]</td><td>Conv-IN</td><td>49.2 ± 0.8</td><td>35.2 ± 0.5</td><td>36.8 ± 1.2</td><td>35.5 ± 1.3</td><td>41.2 ± 1.8</td><td>34.9 ± 1.1</td></tr><tr><td>MTT[20]</td><td>Conv-IN</td><td>64.4 ± 0.9</td><td>41.6 ± 1.3</td><td>49.2 ± 1.1</td><td>42.9 ± 1.5</td><td>46.6 ± 2.0</td><td>34.2 ± 2.6</td></tr><tr><td>KIP [23]</td><td>Conv-NTK</td><td>62.7 ± 0.3</td><td>58.2 ±0.4</td><td>49.0 ± 1.2</td><td>45.8 ± 1.4</td><td>30.1 ± 1.5</td><td>57.2 ± 0.4</td></tr><tr><td>FRePo</td><td>Conv-BN</td><td>65.5 ± 0.4</td><td>65.5 ± 0.4</td><td>58.1 ± 0.6</td><td>57.7 ± 0.7</td><td>59.4 ± 0.7</td><td>61.9 ± 0.7</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/06d156480cb24f7de4d97dc51031b5fb9e76281f33da496484aeb238e841000d.jpg",
+        "image_caption": [
+            "Figure 4: (a,b,c) Distilled 1 img/cls from CIFAR100 using FRePo, MTT, and DSA. High quality images also produce high test accuracy. (d) Three categories of learned labels. (Top) High confidence, large margin; (Middle) High confidence, small margin; (Bottom) Low confidence, small margin. "
+        ],
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+    {
+        "type": "text",
+        "text": "Label Learning: A similar trend can also be observed for label learning. When the dataset is simple and has only a few classes, label learning may not be necessary. However, it becomes crucial for complex datasets with many labels, such as CIFAR100 and Tiny-ImageNet (See more details in Appendix ??). Similar to the teacher label in the knowledge distillation [1], we observe that the distilled label also encodes the class similarity. We identify three typical cases in Figure 4d. The first group consists of highly confident labels with a much higher value for one class than other classes (large margin), such as sunflower, bicycle, and chair. In contrast, the distilled labels in the second group are confident but may get confused with some closely-related classes (small margin). For instance, the learned label for \"girl\" has almost equally high values for the girl, woman, man, boy, and baby, suggesting that these classes are very similar and may be difficult for the model to distinguish them apart. The last group contains distilled labels with low values for all classes, such as bear, beaver, and squirrel. It is often hard for humans to recognize the distilled images in such a group, suggesting that they may be the challenging classes in a dataset. ",
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+    {
+        "type": "text",
+        "text": "Training Cost Analysis: Figure 3a, 3b shows that our method is significantly more time-efficient than the previous methods. When learning one image per class on CIFAR100, FRePo reaches a similar test accuracy $( 2 3 . 4 \\% )$ to the second-best method $( 2 4 . 0 \\% )$ in 38 seconds, compared to 3805 seconds for MTT, which is roughly two orders of magnitude faster. Moreover, FRePo achieves $92 \\%$ of its final test accuracy ( $2 6 . 4 \\%$ out of $2 8 . 7 \\%$ ) in only 385 seconds. As shown in Figure 3c, our algorithm takes much less time to perform one gradient step on the distilled data. Thus, we can perform more gradient steps in a fixed time. Furthermore, Figure 3d suggests that our algorithm has much less GPU memory requirement. Therefore, we can potentially use a much larger and more complex model to take advantage of the advancement in neural network architecture. ",
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+        "type": "text",
+        "text": "Cross-Architecture Generalization: One desired property of our distilled data is that it generalizes well to architecture it has not seen during the training. Similar to previous works [5, 20], we evaluate the distilled data from CIFAR10 on a wide range of architectures which it has not seen during training, including AlexNet [38], VGG [39], and ResNet [40]. Table 2 shows that our method outperforms previous methods on all unseen architectures. Instance Normalization (IN) [41], as the vital ingredient in several methods (DSA, DM, MTT), seems to hurt the cross-architecture transfer. The performance degrades a lot when no normalization (NN) is applied (Conv-NN, AlexNet) or using a different normalization, like Batch Normalization (BN) [42]. It suggests that the distilled data generated by those methods encode the inductive bias of a particular training architecture. In contrast, our distilled data generalize well to various architectures, including those without normalization (Conv-NN, AlexNet). Note that Figure 1, 4 also indicate that our distilled data encode less architectural bias as the distilled images look natural and authentic. A simple idea to further alleviate the overfitting of a particular architecture is to include more architectures in the model pool. However, the training may not be stable as the meta-gradient computed by different architectures can be very different. ",
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+        "table_caption": [
+            "Table 3: Distillation performance on higher resolution (128x128) dataset (i.e. ImageNette, ImageWoof) and medium resolution (64x64) dataset with a complex label space (i.e. ImageNet-1K). FRePo scales to high-resolution images and learns the discriminate feature of complex datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">ImageNette (128x128)</td><td colspan=\"2\">ImageWoof (128x128)</td><td colspan=\"2\">ImageNet (64x64)</td></tr><tr><td>Img/Cls</td><td>1</td><td>10</td><td>1</td><td>10</td><td>1</td><td>2</td></tr><tr><td>Random Subset</td><td>23.5± 4.8</td><td>47.7 ± 2.4</td><td>14.2 ± 0.9</td><td>27.0± 1.9</td><td>1.1 ± 0.1</td><td>1.4 ± 0.1</td></tr><tr><td>MTT[20]</td><td>47.7± 0.9</td><td>63.0 ± 1.3</td><td>28.6 ± 0.8</td><td>35.8 ± 1.8</td><td>1</td><td>1</td></tr><tr><td>FRePo</td><td>48.1 ± 0.7</td><td>66.5 ± 0.8</td><td>29.7 ± 0.6</td><td>42.2 ± 0.9</td><td>7.5 ± 0.3</td><td>9.7 ± 0.2</td></tr></table>",
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+        "text": "4.3 ImageNet ",
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+        "text": "High Resolution ImageNet Subset To understand how well our method performs on high-resolution images, we evaluate it on ImageNette and ImageWoof datasets [36] with a resolution of 128x128. We learn 1 and 10 images per class on both datasets and report the performance in Table 3 and visualize some distilled images in Figure 1. As shown in Table 3, we outperform MTT on all settings and achieve much better performance when we distill ten images per class on a more difficult dataset ImageWoof. It suggests that our distilled data is better at capturing the discriminative features for each class. Figure 1 shows that our distilled images look real and capture the distinguishable feature of different classes. For the easy dataset (i.e., ImageNette), all images have clear different structures, while for ImageWoof, the texture of each dog seems to be crucial. ",
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+        "text": "Resized ImageNet-1K: We also evaluate our method on a resized version of ILSVRC2012 [26] with a resolution of 64x64 to see how it performs on a complex label space. Surprisingly, we can achieve $7 . 5 \\%$ and $9 . 7 \\%$ Top1 accuracy using only 1k and $2 \\mathrm { k }$ training examples, compared to $1 . 1 \\%$ and $1 . 4 \\%$ using an equally-sized real subset. ",
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+        "type": "text",
+        "text": "5 Application ",
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+    {
+        "type": "text",
+        "text": "5.1 Continual Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Continual learning (CL) [43] aims to address the catastrophic forgetting problem [43–45] when a model learns sequentially from a stream of tasks. A commonly used strategy to recall past knowledge is based on a replay buffer, which stores representative samples from previous tasks [46–49]. Since sample selection is an important component of constructing an effective buffer [48–51], we believe distilled data can be a key ingredient for a continual learning algorithm due to its highly condensed nature. Several works [6–8, 52] have successfully applied the dataset distillation to the continual learning scenario. Our work shows that we can achieve much better results by using a better dataset distillation technique. ",
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+        "text": "We follow Zhao and Bilen [8] that sets up the baseline based on GDumb [49] which greedily stores class-balanced training examples in memory and train model from scratch on the latest memory only. In that case, the continual learning performance only depends on the quality of the replay buffer. We perform 5 and 10 step class-incremental learning [53] on CIFAR100 with an increasing buffer size of 20 images per class. Specifically, we distill 400 and 200 images at each step and put them into the replay buffer. We follow the same class split as Zhao and Bilen [8] and compare our method to random [49], herding [54, 55], DSA [7], and DM [8]. We use the default data preprocessing and default model for each method in this experiment as we find it gives the best performance for each method. We use the test accuracy on all observed classes as the performance measure [8, 48]. ",
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+        "table_caption": [
+            "Table 4: AUC of five attackers on models trained on the real and distilled MNIST data. The model trained on the real data is vulnerable to MIAs, while the model trained on the distilled data is robust to MIAs. Training on distilled data allows privacy preservation while retaining model performance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Test Acc (%)</td><td colspan=\"5\">Attack AUC</td></tr><tr><td>Threshold</td><td>LR</td><td>MLP</td><td>RF</td><td>KNN</td></tr><tr><td>Real</td><td>99.2 ± 0.1</td><td>0.99 ± 0.01</td><td>0.99 ± 0.00</td><td>1.00 ±0.00</td><td>1.00 ± 0.00</td><td>0.97 ±0.00</td></tr><tr><td>Subset</td><td>96.8± 0.2</td><td>0.52 ±0.00</td><td>0.50 ± 0.01</td><td>0.53 ± 0.01</td><td>0.55 ± 0.00</td><td>0.54 ±0.00</td></tr><tr><td>DSA</td><td>98.5 ± 0.1</td><td>0.50 ± 0.00</td><td>0.51 ± 0.00</td><td>0.54 ± 0.00</td><td>0.54 ± 0.01</td><td>0.54 ± 0.01</td></tr><tr><td>DM</td><td>98.3 ± 0.0</td><td>0.50 ± 0.00</td><td>0.51 ± 0.01</td><td>0.54 ± 0.01</td><td>0.54 ± 0.01</td><td>0.53 ± 0.01</td></tr><tr><td>FRePo</td><td>98.5± 0.1</td><td>0.52 ±0.00</td><td>0.51 ± 0.00</td><td>0.53 ± 0.01</td><td>0.52 ± 0.01</td><td>0.51 ± 0.01</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/3425d57f4bfa2229b9a55a067323e25dd62e8fcc52fb803fdd882c3fcc7566d9.jpg",
+        "image_caption": [
+            "Figure 5: (a,b) Multi-class accuracies across all classes observed up to a certain time point. We perform significantly better than other methods in both 5 and 10 step class-incremental continual learning. (c,d) Test accuracy and attack AUC as we increase the number of training steps. AUC keeps increasing when training a model on the real data for more steps. In contrast, AUC keeps low when training on distilled data. "
+        ],
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+    {
+        "type": "text",
+        "text": "Figure 5 shows that our method performs significantly better than all previous methods. The final test accuracy for all classes for our method (FRePo) and the second-best method (DM) are $4 1 . 6 \\%$ , $3 3 . 9 \\%$ in 5-step learning, and $3 8 . 0 \\%$ , $3 4 . 0 \\%$ in 10-step learning. However, we notice that for FRePo, distilling 2000 images in a continual learning setup achieves a similar test accuracy $( 4 1 . 6 \\% )$ as distilling only 1000 images from the whole dataset $( 4 1 . 3 \\%$ from Table 1). In addition, performance drops as we perform more steps. It suggests that FRePo considers all available classes to derive the most condensed dataset. Splitting the data into multiple groups and performing independent distillation may generate redundant information or fail to capture the distinguishable features. ",
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+    {
+        "type": "text",
+        "text": "5.2 Membership Inference Defense ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Membership inference attacks (MIA) aim to infer whether a given data point has been used to train the model or not [56–58]. Ideally, we want a model to learn from the data but not memorize it to preserve privacy. However, deep neural networks are well-known for their ability to memorize all the training examples, even on large and randomly labeled datasets [59]. Several methods have been proposed to defend against such attacks by either modifying the training procedure [60] or changing the inference workflow [61]. This section shows that the distilled data contain little information regarding sample presence in the original dataset. Thus, instead of training on the original datasets, training on distilled data allows privacy preservation while retaining model performance. ",
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+    {
+        "type": "text",
+        "text": "We consider three distilled data generated by DSA [7], DM [8] and FRePo. We perform five popular \"black box\" MIA provided by Tensorflow Privacy [62] on models trained on the real data or the data distilled from it. The attack methods include a threshold attack and four model-based attacks using logistic regression (LR), multi-layer perceptron (MLP), random forest (RF) and K-nearest neighbor (KNN). The inputs to those attack methods are ground-truth labels, model predictions, and losses. To measure the privacy vulnerability of the trained model, we compute the area under the ROC curve (AUC) of an attack classifier. Following prior work, [56, 63], we keep a balanced set of training examples (member) and test examples (non-member) with 10K each to maximize the uncertainty of MIA. Thus, the random guessing strategy results in a $50 \\%$ MIA accuracy. We conduct experiments on MNIST and FashionMNIST with a distillation size of 500. For space reasons, we provide more implementation details and results in appendix. ",
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+    {
+        "type": "text",
+        "text": "As shown in Table 4, all models trained on the distilled data preserve privacy as their attack AUCs are closed to random guessing. However, we observe a small drop in test accuracy compared to the model trained on the full dataset, which is expected as we only distill 500 examples instead of 10,000 examples. Compared to the model trained on an equally sized subset of the original data, the model trained on distilled data results in much better test performance. Figure 5c, 5d demonstrate the trade-off between test accuracy and attack effectiveness as measured by ROC AUC. It shows that early stopping can be an effective technique to preserve privacy. However, we will still be under high MIA risk if we perform early stopping by monitoring the validation loss. In contrast, training a model on the distilled data does not have this problem as the attack AUCs keep at a very low level regardless of training steps. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We propose neural Feature Regression with Pooling (FRePo) to overcome two challenges in dataset distillation: meta-gradient computation and various types of overfitting in dataset distillation. We obtain state-of-the-art performance on various datasets with a $1 0 0 \\mathrm { x }$ reduction in training time and a 10x reduction in GPU memory requirement. The distilled data generated by FRePo looks real and natural and generalizes well to a wide range of architectures. Furthermore, we demonstrate two applications that take advantage of the high-quality distilled data, namely, continual learning and membership inference defense. ",
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+    {
+        "type": "text",
+        "text": "Broader Impact “Synthetic data”, in the broader sense of artificial data created by generative models, can help researchers understand how an otherwise opaque learning machine “sees” the world. There have been concerns regarding the risk of fake data. This paper explores a new research direction in generating synthetic data only for downstream classification tasks. We believe this work can provide additional interpretability and potentially address the common concerns in machine learning regarding training data privacy. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We would like to thank Harris Chan, Andrew Jung, Michael Zhang, Philip Fradkin, Denny Wu, Chong Shao, Leo Lee, Alice Gao, Keiran Paster, and Lazar Atanackovic for their valuable feedback. Jimmy Ba was supported by NSERC Grant [2020-06904], CIFAR AI Chairs program, Google Research Scholar Program and Amazon Research Award. This project was supported by LG Electronics Canada. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute for Artificial Intelligence. ",
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+    {
+        "type": "text",
+        "text": "References ",
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parse/dev/Ix37FJYDkBp/Ix37FJYDkBp.md

@@ -0,0 +1,265 @@
+# SemMAE: Semantic-Guided Masking for Learning Masked Autoencoders
+
+Gang $\mathbf { L i } ^ { 1 , 2 }$ ∗, Heliang Zheng3, Daqing $\mathbf { L i u ^ { 3 } }$ , Chaoyue Wang3, Bing $\mathbf { S u ^ { 4 } }$ , Changwen Zheng1†
+
+Institute of Software, Chinese Academy of Sciences1, University of Chinese Academy of Sciences2, JD Explore Academy3, Renmin University of China4 ucasligang@gmail.com, {zhengheliang,liudaqing1,wangchaoyue9}@jd.com, bingsu@ruc.edu.cn, changwen@iscas.ac.cn
+
+# Abstract
+
+Recently, significant progress has been made in masked image modeling to catch up to masked language modeling. However, unlike words in NLP, the lack of semantic decomposition of images still makes masked autoencoding (MAE) different between vision and language. In this paper, we explore a potential visual analogue of words, i.e., semantic parts, and we integrate semantic information into the training process of MAE by proposing a Semantic-Guided Masking strategy. Compared to widely adopted random masking, our masking strategy can gradually guide the network to learn various information, i.e., from intra-part patterns to inter-part relations. In particular, we achieve this in two steps. 1) Semantic part learning: we design a self-supervised part learning method to obtain semantic parts by leveraging and refining the multi-head attention of a ViT-based encoder. 2) Semantic-guided MAE (SemMAE) training: we design a masking strategy that varies from masking a portion of patches in each part to masking a portion of (whole) parts in an image. Extensive experiments on various vision tasks show that SemMAE can learn better image representation by integrating semantic information. In particular, SemMAE achieves $8 4 . 5 \%$ fine-tuning accuracy on ImageNet-1k, which outperforms the vanilla MAE by $1 . 4 \%$ . In the semantic segmentation and fine-grained recognition tasks, SemMAE also brings significant improvements and yields the state-of-the-art performance. Our code is available at https://github.com/ucasligang/SemMAE.
+
+# 1 Introduction
+
+Together with transformers, masked language modeling (MLM) has revolutionized the field of self-supervised learning (SSL) in natural language processing (NLP), which enables training of generalizable NLP models containing over one hundred billion parameters [5]. The concept of MLM is quite intuitive, i.e., a portion of the data is removed and a model is trained to predict the removed content. Recently, significant progress has been made in masked image modeling to catch up to masked language modeling, where the masking mechanism is a key factor. Context encoder [28], an inpainting-based masked image modeling (MIM) pioneer, proposes to use a random and fix-shaped mask; SiT [2] and BEiT [3] use random “blockwise” masks, where patches in the local neighbourhood are masked together (also called GMML: group mask model learning); MAE [19] randomly masks out $7 5 \%$ patches of an image. Actually, masking mechanisms define the specific pretext task, i.e., what kind of information is to be exploited and what kind of information is to be predicted. Thus
+
+AttMask [21] studies the problem of which tokens to mask and proposes an attention-guided mask strategy to make informed decisions. ADIOS [30] takes one step further to “learn to mask” by adversarial training.
+
+Although promising performance has been achieved, there is still a large gap for masked autoencoding (MAE) between vision and language due to different signal natures. A sentence can be semantically decomposed into words, while the semantic decomposition of an image is not trivial to be obtained. To find a visual analogue of words, we investigate part-based image representation. Specifically, the real world is composed of objects, which consist of different parts. Therefore, part-based image representation is a fundamental image representation method that fits the inherent properties of objects [8, 15, 16, 18, 20]. For example, part-based Pictorial Stracture [16] dominated the image representation field for several years in the early days of computer vision, and Deformable Part Model (DPM) [15] was also a milestone in image recognition and detection. Moreover, GLOM [20] argues that the hierarchical representation with five levels (i.e., the lowest level, sub-part level, part level, object level, and scene level) would be a powerful image representation method in the future. To this end, we argue that semantic parts would be a potential visual analogue of words. With such visual analogue, more controllable hints can be built up to guide the learning of MAE, thus high-level visual representations can be well learned.
+
+In this paper, we first propose a self-supervised semantic part learning method to obtain semantic parts for each image. Our insight is that the spatial information to reconstruct an image is highly correlated to the position of semantic parts. In particular, our part learning model consists of a ViT-based encoder together with an attention module that generates a class token and multiple attention maps, and a StyleGAN-based decoder that reconstructs the original image. The attention maps are optimized to provide spatial information, and the class token is integrated into the decoder via AdaIN to provide texture information. We find that the optimized attention maps can indicate part positions, and we conduct an argmax operation to obtain part segmentation maps. After that, we study how semantic parts can facilitate the learning of MAE. We design a masking strategy that varies from masking a portion of patches in each part to masking a portion of (whole) parts in an image. Such a design can gradually guide the network to learn various information, i.e., from intra-part patterns to inter-part relations. Extensive experiments on various vision tasks (e.g., linear probing, fine-tuning, semantic segmentation, and fine-grained recognition) show that SemMAE can learn better image representation by integrating semantics.
+
+Our contributions include 1) designing a self-supervised semantic part learning method that can generate promising semantic parts on multi-class datasets, i.e., ImageNet, and 2) verifying that semantic parts can facilitate the learning of MAE by proposing a semantic-guided masking strategy. While more importantly, we hope our attempts can provide insights for the community to study the visual analogue of words and unified vision and language modeling.
+
+# 2 Related work
+
+Semantic part learning. Part-based image representation is a fundamental image representation method that fits the inherent properties of objects [8, 15, 16, 18, 20]. However, due to the tremendous cost of labeling parts, there are still no large-scale datasets containing part labels. Thus previous works are mainly two-fold, i.e., unsupervised/weakly-supervised part learning and few-shot part segmentation. Unsupervised/weakly-supervised part learning methods [10, 22, 39] propose to mine part information by leveraging spatial priors, the semantics of convolutional channels, or designing contrastive proxy tasks. Few-shot part segmentation methods [4, 29, 38] mainly learn an additional classifier over pre-trained features that are trained by GAN, self-supervised contrastive learning, or denoising diffusion probabilistic modeling. Although promising results have been obtained, these models are designed to deal with fine-grained datasets, where all images belong to a single super-class (e.g., birds, cars, or human faces). It is much more challenging to solve the problem of unsupervised part learning on multi-class datasets such as ImageNet. With the development of ViT and self-supervised learning (SSL), some recent works show a potential solution. In particular, DINO [6] and iBOT [42] have observed intuitive semantics in the ViT trained by their SSL methods, where the multi-head attention maps can somehow indicate different semantic parts of an object. Inspired by these works, we design a reconstruction-based method to further refine the attention maps learned by iBOT to obtain semantic parts on the ImageNet dataset.
+
+![](images/cbb8251d2beda93c0b0f022dbc58fb0ff4c31d7a644b9d6801d62c9b5eb8f663.jpg)  
+Figure 1: Comparison of different masking strategies. Detailed information for each compared model can be found in Section 2 Masked Image Modeling.
+
+Masked image modeling. Inspired by the success of Masked Language Modeling (MLM) [5, 12] in pre-training of the NLP field, Masked Image Modeling (MIM) has been proposed recently and exhibits promising potential for visual pre-training [3, 7, 19, 30]. Existing works mainly study the problem of MIM from two directions, i.e., regression targets and masking strategies. In terms of regression targets, BeiT [3], mc-BEiT [24], and PeCo [13] adopt tokens produced by VQ-VAE [32] or its variants. MaskFeat [35] studies a broad spectrum of feature types and proposes to regress Histograms of Oriented Gradients (HOG) features of the masked content. MAE [19] and SimMIM [37] argue that predicting RGB values of raw pixels by direct regression performs no worse than the patch classification approaches with complex designs. In this paper, we follow MAE [19] to adopt the most simple and intuitive raw pixels regression. In terms of masking strategies, SiT [2], MC-SSL0.0 [1] and BeiT [3] use a block-wise masking strategy, where a block of neighbouring tokens arranged spatially are masked. MAE [19] and SimMIM [37] use random masking with a large masked patch size or a large proportion of masked patches. MST [25] and AttMask [21] propose to use attention maps to guide the masking strategy, where the former proposes to mask the nonessential regions to preserve crucial patches while the latter proposes to learn image representations with challenging tasks by masking the most attended tokens. Moreover, ADIOS [30] proposes to learn an optimal mask by adversarial training. Compared to these works, our SemMAE takes one step further and explicitly learn semantic parts to build reasonable hints for masked image modeling. Figure 1 is an illustration of different masking strategies.
+
+# 3 Semantic-guided masked autoencoders
+
+We propose a Semantic-guided Masked Autoencoder (SemMAE) for self-supervised image representation learning with mask image modeling. The framework of SemMAE is shown in Figure 2, which consists of two key components, i.e., Semantic Part Learning (A) and SemanticGuided Masking (B). First, given an image in Figure 2 (a), we extract the class token in Figure 2 (b) and patch tokens in Figure 2 (c) by an iBOT-pretrained ViT. After that, we learn an embedding over the class token to obtain part tokens in Figure 2 (d). We calculate the correlation of each part token to patch tokens to obtain attention maps in Figure 2 (e), whose texture information is further removed by a large-kernel blur operation. The attention maps are optimized by a diversity constraint and a reconstruction task where the attention maps and the class token are fed into a StyleGAN-based decoder to control the spatial and texture information of the reconstructed image, respectively. Finally, we conduct argmax over the attention maps to obtain part segmentation maps in Figure 2 (f) and used them to guide the mask generation for MAE. Specifically, we design a masking strategy that varies from masking a portion of patches in each part to masking a portion of (whole) parts in an image. Such a design can gradually guide the network to learn various information, i.e., from intra-part patterns to inter-part relations.
+
+![](images/90a996e4f562df22d25af9c680404d44a9450d3443e9296a7ba446d8c69e9239.jpg)  
+Figure 2: An illustration of the proposed SemMAE. (A) Semantic Part Learning. A ViT-based encoder takes as input an image in (a) and produces a class token in (b) and patch tokens in (c). Our attention module first learns to embed the class token into part tokens in (d) and then generates an attention map for each part token by calculating the correlation between the part token and patch tokens. As an objective function of the attention maps, our StyleGAN-based decoder learns to reconstruct the original image from attention maps with texture information from the class token. (B) Semantic-Guided Masking. We conduct argmax over the attention maps to obtain part segmentations in (f), which are used to guide the mask generation. During the training of the MAE, the masks vary from a portion of patches in each part to a portion of (whole) parts in an image.
+
+# 3.1 Semantic part learning
+
+In this subsection, we introduce our self-supervised semantic part learning method. Previous unsupervised/weakly-supervised part learning methods are mainly designed to deal with singleclass datasets (i.e., fine-grained datasets where images belong to the same superclass). Few methods are able to solve this problem under a multi-class scenario (e.g., ImageNet). While some recent works (i.e., DINO [6] and iBOT [42]) on ViT-based self-supervised learning show that the multi-head attention maps in their model can somehow indicate different semantic parts of an object. In this work, we take advantage of semantics learned in iBOT and design a reconstruction task together with a diversity constraint to refine and obtain semantic parts.
+
+In particular, given an image I, we first use an iBOT-pretrained ViT to extract its features, i.e., a class token $\mathbf { F } _ { \mathrm { c } } ^ { - } \in \mathbb { R } ^ { C \times 1 }$ and patch tokens $\mathbf { F } \in \mathbb { R } ^ { C \times H \mathbf { \bar { W } } }$ . Then, we embed the class token into $N$ part tokens $\mathbf { F } _ { \mathrm { p } } \in \mathbb { R } ^ { C \times N }$ . The main idea of such embedding is to re-weight feature channels of the class token. As shown in previous methods [39], feature channels may be corresponding to specific semantics and channel re-weighting can group channels with similar semantics together to obtain semantic part features. Thus we can obtain part tokens by:
+
+$$
+\mathbf { F } _ { \mathrm { p } } ^ { ( i ) } = \mathbf { F } _ { \mathrm { c } } \circ \mathrm { s i g m o i d } ( \mathbf { W } _ { \mathrm { c 2 } } ^ { ( i ) } \operatorname { t a n h } ( \mathbf { W } _ { \mathrm { c 1 } } ^ { ( i ) } \mathbf { F } _ { \mathrm { c } } ) ) ,
+$$
+
+where $i \in [ 1 , 2 , . . . , N ]$ , $\mathbf { F } _ { \mathrm { p } } ^ { ( i ) } \in \mathbb { R } ^ { C \times 1 }$ is the $i ^ { t h }$ column vector of $\mathbf { F } _ { \mathrm { p } } \in \mathbb { R } ^ { C \times N }$ , $\circ$ indicates hadamard product, $\mathbf { W } _ { \mathrm { c 1 } } ^ { ( i ) }$ and $\mathbf { W } _ { \mathrm { c 2 } } ^ { ( i ) }$ are embedding weights, $\operatorname { t a n h } ( { \cdot } )$ and sigmoid $( \cdot )$ are activation functions.
+
+After that, we calculate the correlation of each part token to the patch token in each position, thus we can obtain attention maps, i.e., the possibility of a semantic part to appear in each position:
+
+$$
+\mathbf { M } = \mathbf { F } _ { \mathrm { p } } \otimes \mathbf { F } : = \mathrm { s o f t m a x } ( \mathbf { F } _ { \mathrm { p } } ^ { T } \mathbf { W } _ { \mathrm { p } } ^ { T } \mathbf { W } \mathbf { F } ) ,
+$$
+
+where $\textbf { M } \in \ \mathbb { R } ^ { N \times H W }$ denotes $N$ attention maps, $\otimes$ indicates correlation function, which is implemented by softmax $( \mathbf { F } _ { \mathrm { p } } ^ { T } \mathbf { W } _ { \mathrm { p } } ^ { T } \mathbf { W } \mathbf { F } )$ in our work. $\mathbf { W } _ { \mathrm { p } }$ and $\mathbf { W }$ are embedding matrixes.
+
+To learn such multi-attention maps (i.e., to optimize the parameters in Equation 1 and Equation 2), we propose a reconstruction task. Our insight is that the spatial information to reconstruct an image is highly correlated to the position of semantic parts. Thus, we adopt a StyleGAN-based decoder to reconstruct the original image based on the spatial information from the attention maps and the texture information from the class token. To ensure the attention maps learn spatial information, we 1) remove texture information from the attention maps by conducting a large-kernel blur operation and 2) further feed the blurred attention maps to stacked convolutional layers. To integrate the texture information from the class token into the decoder, we use Adaptive Instance Normalization (AdaIN) operation, which is widely used to integrate texture/style information:
+
+$$
+[ \mathbf { F } _ { \mathrm { d } } ] _ { i } = \mathrm { A d a l N } ( [ \mathrm { c o n v } ( \mathbf { M } ) ] _ { i } , \mathbf { F } _ { \mathrm { c } } ) : = [ \mathbf { W } _ { \mathrm { s } } \mathbf { F } _ { \mathrm { c } } ] _ { i } \frac { [ \mathrm { c o n v } ( \mathbf { M } ) ] _ { i } - \mu ( [ \mathrm { c o n v } ( \mathbf { M } ) ] _ { i } ) } { \sigma ( [ \mathrm { c o n v } ( \mathbf { M } ) ] _ { i } ) } + [ \mathbf { W } _ { \mathrm { b } } \mathbf { F } _ { \mathrm { c } } ] _ { i } ,
+$$
+
+where each feature channel $[ \mathrm { c o n v } ( \mathbf { M } ) ] _ { i }$ is normalized separately, and then scaled and biased using the corresponding scalar components from the embedded class token $\mathbf { F } _ { \mathrm { c } }$ . $\mathbf { F } _ { \mathrm { d } }$ denotes the convolutional feature in the decoder, ${ \bf W _ { s } }$ and $\mathbf { W } _ { \mathrm { b } }$ are embedding weights, $[ \cdot ] _ { i }$ denotes the $i ^ { t h }$ feature channel, $\mathrm { c o n v } ( \cdot )$ denotes convolutional layers, $\mu ( \cdot )$ and $\sigma ( \cdot )$ calculate the mean and variance values, respectively. The reconstructed image $\hat { \bf I }$ can be obtained by stacking convolutional and AdaIN layers:
+
+$$
+\hat { \bf I } = \mathrm { c o n v } ( \mathrm { A d a I N } ( \mathrm { c o n v } ( { \bf F } _ { \mathrm { d } } ) , { \bf F } _ { \mathrm { c } } ) ) .
+$$
+
+We use the Mean squared error (MSE) loss function to optimize such reconstruction task:
+
+$$
+\mathcal { L } _ { r e c } ( \mathbf { I } , \hat { \mathbf { I } } ) = \frac { 1 } { H W } \sum _ { i , j } ^ { H W } ( \mathbf { I } ( i , j ) - \hat { \mathbf { I } } ( i , j ) ) ^ { 2 } .
+$$
+
+Moreover, to obtain diverse multiple attention maps, we follow previous work [40] and add a diversity constraint over attention maps:
+
+$$
+\mathcal { L } _ { d i v } ( \mathbf { M } ) = \frac { 1 } { N ^ { 2 } } ( \sum _ { i \neq j } ( 0 - \frac { \mathbf { m } _ { i } \mathbf { m } _ { j } ^ { T } } { \| \mathbf { m } _ { i } \| _ { 2 } \| \mathbf { m } _ { j } \| _ { 2 } } ) ^ { 2 } + \sum _ { i = j } ( 1 - \frac { \mathbf { m } _ { i } \mathbf { m } _ { j } ^ { T } } { \| \mathbf { m } _ { i } \| _ { 2 } \| \mathbf { m } _ { j } \| _ { 2 } } ) ^ { 2 } ) ,
+$$
+
+where attention maps are optimized to be different from each other, $\mathbf { m } _ { i }$ and $\mathbf { m } _ { j }$ denotes the $i ^ { t h }$ and $j ^ { t h }$ attention map, respectively. The overall objective function can be denoted by:
+
+$$
+\mathcal { L } = \mathcal { L } _ { r e c } ( \mathbf { I } , \hat { \mathbf { I } } ) + \lambda \mathcal { L } _ { d i v } ( \mathbf { M } ) ,
+$$
+
+where $\lambda$ is the loss weight.
+
+# 3.2 Semantic-guided masking
+
+After finished semantic part learning, we move to the next stage, i.e., semantic-guided MAE training. Our informed masking strategy is based on the part information learned in Subsection 3.1. Specifically, we can obtain multiple attention maps by Equation 2, where each attention map $\mathbf { m } \in \mathbb { R } ^ { H \times \hat { W } }$ indicates the possibility of the corresponding semantic part appearing in $H \times W$ positions. Thus we conduct argmax $( \cdot )$ operation over attention maps to obtain part segmentation, where each patch is classified into a particular semantic part. The patches in the same semantic part compose a visual analogue of words, which are semantically meaningful. To leverage such visual analogue of words for MAE training, a most intuitive way is to mask a portion of semantic parts and learn to predict the masked semantic parts by other parts. However, due to the learned semantic parts being coarse-grained (e.g., 6 parts for each image), we experimentally find that such a masking strategy makes the task too hard to effectively learn image representations.
+
+To this end, we propose an easy-to-hard reconstruction task, which can provide reasonable hints (i.e., visible patches) for the model to predict the masked patches during the training process of the MAE. Specifically, at the beginning of the training process, we mask a portion of patches in each part, thus the masked patches can be predicted based on the visual patches that belong to the same semantic part. Such a design can facilitate the models to learn intra-part patterns. After that, we gradually mask all patches belonging to some parts and a portion of patches belong to the remaining parts. Finally, we mask all patches belonging to a portion of parts and predict the remaining patches belong to the other parts, where inter-part relations or visual reasoning ability can be learned.
+
+Algorithm 1 shows the details to obtain the number of masked patches for each semantic part. First, we define two masking settings, i.e., 1) mask a portion of patches in each part and 2) random select some parts to mask (the whole part). The number of masked patches for each semantic part can be calculated for these two settings. After that, we introduce an interpolation hyper-parameter $\alpha$ . A small $\alpha$ means the first setting dominates the masking strategy, and vice versa. $\alpha$ is adjusted based on training iterations and keeps increasing during the training process. Finally, we random mask a certain number of patches based on the calculated masking number.
+
+# Algorithm 1 Algorithm of Semantic-Guided Masking in a PyTorch-like style.
+
+Input: $L$ , $x$ , num_patches, mask_ratio, total_epoches, epoch # $L$ : the number of patches per image. # $x \in \mathbb { R } ^ { L \times C }$ : the token embeddings of image patches. # num_patches $\mathbf { \Psi } \in \mathbb { R } ^ { N \times 1 }$ : the patch number of each part, where $N$ is the number of parts. # mask_ratio: the ratio of masked patches. # total_epoches: the number of pre-training epochs. # epoch: current epoch number. # mask a portion of patches in each part   
+1: num_mask1 = mask_ratio \* num_patches # randomly select some parts to mask (with tricks to ensure a fixed mask ratio)   
+2: shuffle_num_patches $=$ shuffle_parts(num_patches)   
+3: marks $\ v { U } = \mathbf { L } \ v { \Sigma } ^ { * }$ mask_ratio-cumsum(shuffle_num_patches)+shuffle_num_patches   
+4: marks_remains $=$ where(marks $< 0$ , 0, marks)   
+5: num_mask2 $=$ where(marks_remains $<$ shuffle_num_patches, marks_remains, shuffle_num_patches)   
+6: num_mask2 $=$ unshuffle_parts(num_mask2) # adaptive masking by interpolating between num_mask1 and num_mask2   
+7: α = ( epochtotal_epoches )   
+8: $\ n u m \_ m a s k = ( 1 - \alpha ) * n u m \_ m a s k 1 + \alpha * n u m \_ m a s k 2$   
+9: return num_mask # $n u m \_ m a s k \in \mathbb { R } ^ { N \times 1 } ;$ : the number of patches to be masked in each part.
+
+# 4 Experiments
+
+# 4.1 Experiment setup
+
+Semantic part learning. As introduced in Section 3.1, we use ViT-small [14] as our part learning encoder, which is pre-trained by a self-supervised method iBOT [42]. We follow iBOT [42] to learn 6 semantic parts for each image, as the head number of the multi-head attention in ViT-Small is 6. The size of the blur kernel is experimentally set to be 7, and the loss weight $\lambda$ in Equation 7 is set to be 0.03. The experiment is performed on ImageNet-1k [11] dataset. The parameters of the ViT-based encoder are fixed, and we only optimize the attention module and the StyleGAN-based decoder. Our model converges fast, which only takes 2 hours on one A100 GPU card.
+
+Semantic-guided MAE training. We follow MAE [19] and adopt an encoder-decoder structure to perform MIM. Our method is general for ViT backbones, while most experiments are conducted with a relatively small version, i.e., the original ViT-Base [14], due to the limitation of computation resources. We follow the most comment setting to optimize our model by AdamW [27] with a learning rate of $2 . 4 \mathrm { e } { - 3 }$ . The batch size is set to be 4096, and the weight decay is set to be 0.05. We use a cosine learning rate strategy [26] with warmup [17]. The warmup number is set to be 40 epochs, and we pre-train our model for 800 epochs. For data augmentation, we only employ random horizontal flipping in our pre-training stage. The hyper-parameter $\gamma$ in Algorithm 1 is experimentally set to be 2. Our model is trained on 16 A-100 GPUs for 3 days, and more details can be found in our code, which is in the supplemental material and will be made publicly released.
+
+Table 1: Quantitative evaluation of the effectiveness of integrating semantic information for MAE.   
+
+<table><tr><td rowspan=2 colspan=1>Setting</td><td rowspan=1 colspan=2>16×16 patch size</td><td rowspan=1 colspan=2>8×8 patch size</td></tr><tr><td rowspan=1 colspan=1>MAE [19]</td><td rowspan=1 colspan=1>SemMAE</td><td rowspan=1 colspan=1>MAE [19]</td><td rowspan=1 colspan=1>SemMAE</td></tr><tr><td rowspan=1 colspan=1>Linear probing</td><td rowspan=1 colspan=1>63.7</td><td rowspan=1 colspan=1>65.0</td><td rowspan=1 colspan=1>66.8</td><td rowspan=1 colspan=1>68.7</td></tr></table>
+
+# 4.2 Semantic-guided MAE
+
+The effectiveness of integrating semantic information. We conduct experiments under two different settings (i.e., with a patch size of $1 6 \times 1 6$ and $8 \times 8$ ) to verify the effectiveness of integrating semantic information for training MAE. The results in Table 1 show that integrating semantic information can bring $1 . 3 \%$ and $1 . 9 \%$ accuracy gains for linear probing, respectively. As we use $8 \times 8$ patch size to learn semantic parts, the coarse-grained patch (i.e., large patch size) in the pre-training stage would cause imprecise part segment and suppresses the benefits of semantic parts. To further study the impact of patch size for masked image modeling, we conduct fine-tuning experiments in Table 2. It can be observed that in SimMIM and original MAE, a larger patch size performs better; while in our SemMAE, more precise semantic parts with $8 \times 8$ patch size can significantly improve the performance. Thus in the following experiments, we adopt $8 \times 8$ patch size for SemMAE. It is notable that although using a smaller patch size, our parameters and computational cost do not increase during pre-training and linear probing as only 1/4 patches in each image are used. Specifically, we leverage the learned attentions maps to remove $3 / 4$ patches that are most likely to be the background.
+
+Table 2: The optimal patch size for different models.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Patch size</td><td rowspan=1 colspan=1>Fine-tuning Acc.(%)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Patch size</td><td rowspan=1 colspan=1>Fine-tuning Acc.(%)</td></tr><tr><td rowspan=4 colspan=1>SimMIM [37]</td><td rowspan=1 colspan=1>32x32</td><td rowspan=1 colspan=1>82.8</td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=1>MAE [19]</td><td rowspan=1 colspan=1>16x16</td><td rowspan=1 colspan=1>83.26</td></tr><tr><td rowspan=1 colspan=1>16x16</td><td rowspan=1 colspan=1>82.7</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>8x8</td><td rowspan=1 colspan=1>83.10</td></tr><tr><td rowspan=1 colspan=1>8x8</td><td rowspan=1 colspan=1>82.1</td><td rowspan=2 colspan=2>SemMAE</td><td rowspan=1 colspan=1>16x16</td><td rowspan=1 colspan=1>83.34</td></tr><tr><td rowspan=1 colspan=1>4x4</td><td rowspan=1 colspan=1>82.0</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>8x8</td><td rowspan=1 colspan=1>84.50</td></tr></table>
+
+A detailed study on masking strategies. Once obtained semantic parts, a most intuitive way to leverage such visual analogue of words for MAE training is to mask a portion of semantic parts and make the model to predict the removed content. However, due to the learned semantic parts being coarse-grained (e.g., 6 parts for each image), we experimentally find that such a masking strategy makes the task too hard to effectively learn image representations. The results can be found in Table 3, where masking $7 5 \%$ parts cause $1 3 . 9 \%$ performance drops compared to random masking. Moreover, it can be observed that masking $7 5 \%$ patches per part achieves comparable results with random masking. The self-supervised learning task of masking $7 5 \%$ patches per part would encourage the model to learn local contexts/intra-part patterns, and masking $7 5 \%$ parts would encourage the model to learn inter-part relations. Interestingly, we find that the former task can enable the model to further learn better image representation in the latter task. The results in Table 3 show that our proposed adaptive masking strategy (i.e., varying from masking $7 5 \%$ patches per part to masking $7 5 \%$ parts gradually) with $\gamma = 2$ yields the best performance.
+
+![](images/b7d4598335ce52c0e42d64c8d328fb3696e4079b4b81264741277c0a31a90363.jpg)  
+Figure 3: The curves of $\alpha$ and $\gamma$
+
+Table 3: Quantitative evaluation of different masking strategies.   
+
+<table><tr><td rowspan=1 colspan=1>Mask strategy</td><td rowspan=1 colspan=1>a</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>Linear probing</td></tr><tr><td rowspan=1 colspan=1>Random masking</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>66.8</td></tr><tr><td rowspan=1 colspan=1>Mask 75% patches</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>66.5</td></tr><tr><td rowspan=1 colspan=1>Mask 75% parts</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>52.9</td></tr><tr><td rowspan=6 colspan=1>Adaptive masking</td><td rowspan=1 colspan=1>1→0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>63.3</td></tr><tr><td rowspan=5 colspan=1>0→1</td><td rowspan=1 colspan=1>1/3</td><td rowspan=1 colspan=1>66.2</td></tr><tr><td rowspan=1 colspan=1>1/2</td><td rowspan=1 colspan=1>67.3</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>67.9</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>68.7</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>68.6</td></tr></table>
+
+# 4.3 Semantic part learning
+
+We first evaluate the effectiveness of our proposed semantic part learning method. Both qualitative and quantitative experiments are conducted. Note that most of the previous part learning models are designed for single-class datasets and cannot be effectively applied to ImageNet. iBOT [42] is not proposed for part learning, while the multi-head attention in their model achieves the state-of-the-art part learning performance on ImageNet. Figure 4 shows the qualitative comparison of our model and iBOT [42], and it can be observed that our model can generate more complete semantic part segmentation maps where different parts and the background are better separated with less noise. Moreover, as there is no part segmentation ground truth, we conduct quantitatively evaluation in an indirect way by training SemMAE and analyzing ImageNet classification performance. The results in Table 4 show that the semantic parts obtained by iBOT are not able to benefit the learning of MAE, while our semantic part learning methods can generate more precise part segmentation maps, which are vital to learning a better image representation.
+
+![](images/523aa9018548f6cdc8c65ae729be27cf7503ce05af1ee7af051e233f591e0c0c.jpg)  
+Figure 4: Qualitative comparison of semantic part learning. Different color indicates different semantic parts, and it can be observed that our model can better separate different parts and the background with less noise.
+
+Table 4: Quantitative evaluation of semantic part learning in terms of classification accuracy $( \% )$ .   
+
+<table><tr><td>Semantic parts for masking Baseline (w/o parts） iBOT-initialized partsOur learned parts</td><td></td><td></td><td></td></tr><tr><td>Linear probing Acc. (%)</td><td>63.7</td><td>63.6</td><td>65.0</td></tr></table>
+
+Table 5: System-level comparison on ImageNet-1k in terms of classification accuracy using ViT-Base as the encoder. Note that we list the best performance in previous papers with $2 2 4 \times 2 2 4$ inputs, and some experiment settings (e.g., training epochs and patch size) may be different.   
+
+<table><tr><td>Method</td><td></td><td>Pre-train dataset Pre-train epoches Linear probing</td><td></td><td>gFintuning</td></tr><tr><td colspan="5">Traning from scratch</td></tr><tr><td colspan="2">ViT384 [14]</td><td></td><td></td><td>77.9</td></tr><tr><td colspan="2">DeiT[31]</td><td></td><td></td><td>81.8</td></tr><tr><td colspan="2">ViT[19]</td><td>=</td><td></td><td>82.3</td></tr><tr><td colspan="5">Contrastive-based SSL Pre-Training</td></tr><tr><td>AttMask [42]</td><td>ImageNet-1K</td><td>100</td><td>75.7</td><td>1</td></tr><tr><td>DINO [6]</td><td>ImageNet-1K</td><td>300</td><td>78.2</td><td>82.8</td></tr><tr><td>MoCo v3 [9]</td><td>ImageNet-1K</td><td>300</td><td>76.5</td><td>83.2</td></tr><tr><td>iBOT[42]</td><td>ImageNet-1K</td><td>1600</td><td>79.5</td><td>84.0</td></tr><tr><td colspan="5">MIM-based SSL Pre-Training</td></tr><tr><td>BeiT[3]</td><td>ImageNet-1K</td><td>800</td><td>56.7</td><td>83.2</td></tr><tr><td>MAE [19]</td><td>ImageNet-1K</td><td>1600</td><td>68.0</td><td>83.6</td></tr><tr><td>SimMIM [37]</td><td>ImageNet-1K</td><td>800</td><td>56.7</td><td>83.8</td></tr><tr><td> SemMAE</td><td>ImageNet-1K</td><td>800</td><td>68.7</td><td>84.5</td></tr></table>
+
+# 4.4 Compared with other methods on ImageNet
+
+Linear probing and fine-tuning on ImageNet-1K classification dataset is the most common setting to evaluate SSL methods. We collect all competitive methods that report their results on ImageNet-1K dataset. For example, we do not include the related work MST [25] and ADIOS [30] as they evaluate their model on other benchmarks. Table 5 shows the comparison of our model and previous models in terms of linear probing and fine-tuning. For a fair comparison, all experiments adopt the same input size, i.e., $2 2 4 \times 2 2 4$ unless specified otherwise. Compared with “training from scratch”, our SemMAE can significantly improve the performance for both linear probing and fine-tuning. For linear probing, our SemMAE outperforms the most competitive MIM-based methods by $0 . 8 \%$ even with fewer training epochs. For fine-tuning, our SemMAE achieves $8 4 . 5 \%$ top-1 classification accuracy, outperforming SimMIM[37] and MAE[19] by $0 . 9 \%$ and $0 . 7 \%$ respectively. Moreover, our SemMAE can also surpass previous contrastive learning-based methods[6, 9] for fine-tuning.
+
+# 4.5 Downstream tasks
+
+Fine-grained image classification. Table 6 shows our results of transfer learning on fine-grained datasets. Our model can surpass the most competitive MAE [19] with a clear margin, i.e., $0 . 3 \%$ , $0 . 6 \%$ , and $0 . 2 \%$ on the iNaturalists[33], CUB-Bird[34], and Stanford-Cars[23] dataset, respectively. These results show the promising transfer ability of our SemMAE for downstream classification tasks.
+
+Table 6: Fine-tuning results on fine-grained datasets.   
+
+<table><tr><td>Method</td><td>iNa19</td><td>CUB</td><td>Cars</td></tr><tr><td>BeiT[3]</td><td>79.2</td><td>-</td><td>94.2</td></tr><tr><td>DINO [3]</td><td>78.6</td><td></td><td>93.0</td></tr><tr><td>iBoT[19]</td><td>79.6</td><td>1</td><td>94.3</td></tr><tr><td>MAE [19]</td><td>81.8</td><td>86.5</td><td>94.2</td></tr><tr><td> SemMAE</td><td>82.1</td><td>87.1</td><td>94.4</td></tr></table>
+
+Semantic segmentation. Semantic segmentation aims to assign a label to each pixel of the input image. We evaluate our SemMAE on the widely used semantic segmentation dataset ADE20K [41], which contains 25K images and 150 semantic categories. We follow the most common setting to use the task layer in UPerNet [36] and fine-tune the pre-trained ViT-Base model. We use the standard setting that pre-train a ViT-Base model with a patch size of $1 6 \times 1 6$ and fine-tunes 160K iterations with a batch size of 16. Such an experiment can validate the transfer ability of our SemMAE for semantic segmentation. As shown in Table 7, SeMAE surpasses MAE by 0.2 (46.3 vs. 46.1) mIoU and outperforms the supervised pre-train model by 1.0 mIoU. These results show the promising transfer ability of our SemMAE for dense prediction visual tasks.
+
+Table 7: Semantic segmentation results on ADE-20K.   
+
+<table><tr><td>Method</td><td>mIoU</td></tr><tr><td>Supervised Pre-Training</td><td>45.3</td></tr><tr><td>Self-Supervised Pre-Training</td><td></td></tr><tr><td>BeiT</td><td>45.8</td></tr><tr><td>MAE(800 epochs)</td><td>46.1</td></tr><tr><td>SemMAE (Ours)</td><td>46.3</td></tr></table>
+
+# 5 Conclusion
+
+In this paper, we study the visual analogue of words and propose a semantic-guided masked autoencoder model to reduce the gap between masked language modeling and masked image modeling. Our proposed self-supervised semantic part learning method can generate promising semantic parts on ImageNet and we show that the learned semantic parts can facilitate the learning of MAE. Unlike the main-stream random masking strategy, our semantic-guided mask strategy can effectively integrate semantic information in the pre-training process. Extensive experiments with superior results show the effectiveness of our SemMAE.
+
+Limitations: due to the lack of part segmentation labels, the semantic part in our work is kind of coarse (e.g., 6 parts per image), making it not an ideal visual analogue of words yet. Moreover, using a small patch size increases the computational cost in the fine-tuning stage. In the future, we will 1) investigate finer-grained semantic parts (e.g., 20-30 parts per image) by few-shot part segmentation and 2) replace the widely obtained patch-based tokenization with part-based tokenization to further reduce the gap between vision and language modeling.
+
+# References
+
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+[3] Bao, H., Dong, L., Piao, S., Wei, F.: BEit: BERT pre-training of image transformers. In: ICLR (2022)   
+[4] Baranchuk, D., Rubachev, I., Voynov, A., Khrulkov, V., Babenko, A.: Label-efficient semantic segmentation with diffusion models. In: ICLR (2022)   
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+[14] Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., Houlsby, N.: An image is worth 16x16 words: Transformers for image recognition at scale. In: ICLR (2021)   
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+[16] Fischler, M.A., Elschlager, R.A.: The representation and matching of pictorial structures. TC 100(1), 67–92 (1973)   
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+[20] Hinton, G.: How to represent part-whole hierarchies in a neural network. arXiv preprint arXiv:2102.12627 (2021)   
+[21] Kakogeorgiou, I., Gidaris, S., Psomas, B., Avrithis, Y., Bursuc, A., Karantzalos, K., Komodakis, N.: What to hide from your students: Attention-guided masked image modeling. arXiv preprint arXiv:2203.12719 (2022)   
+[22] Krause, J., Jin, H., Yang, J., Fei-Fei, L.: Fine-grained recognition without part annotations. In: CVPR. pp. 5546–5555 (2015)   
+[23] Krause, J., Stark, M., Deng, J., Fei-Fei, L.: 3d object representations for fine-grained categorization. In: Proceedings of the IEEE international conference on computer vision workshops. pp. 554–561 (2013)   
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+[27] Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101 (2017)   
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+[30] Shi, Y., Siddharth, N., Torr, P.H., Kosiorek, A.R.: Adversarial masking for self-supervised learning. arXiv preprint arXiv:2201.13100 (2022)   
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+[36] Xiao, T., Liu, Y., Zhou, B., Jiang, Y., Sun, J.: Unified perceptual parsing for scene understanding. In: ECCV. pp. 418–434 (2018)   
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+
+# Checklist
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section ??.   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 5   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] No theoretical results. (b) Did you include complete proofs of all theoretical results? [N/A] No theoretical results.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] In the supplemental material   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4 Experiment Setup.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4 Experiment Setup.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes] In our code.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/LdVQGdXkkG/LdVQGdXkkG_content_list.json

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+        "type": "text",
+        "text": "MODULAR ACTION CONCEPT GROUNDINGIN SEMANTIC VIDEO PREDICTION",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        ],
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+    },
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+        "text": "Recent works in video prediction have mainly focused on passive forecasting and low-level action-conditional prediction, which sidesteps the learning of interaction between agents and objects. We introduce the task of semantic actionconditional video prediction, which uses semantic action labels to describe those interactions and can be regarded as an inverse problem of action recognition. The challenge of this new task primarily lies in how to effectively inform the model of semantic action information. Inspired by the idea of Mixture of Experts, we embody each abstract label by a structured combination of various visual concept learners and propose a novel video prediction model, Modular Action Concept Network (MAC). Our method is evaluated on two newly designed synthetic datasets, CLEVR-Building-Blocks and Sapien-Kitchen, and one real-world dataset called Tower-Creation. Extensive experiments demonstrate that MAC can correctly condition on given instructions and generate corresponding future frames without need of bounding boxes. We further show that the trained model can make out-of-distribution generalization, be quickly adapted to new object categories and exploit its learnt features for object detection, showing the progression towards higher-level cognitive abilities. More visualizations can be found at https://iclr-mac.github.io/MAC/. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            540,
+            336,
+            556
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, video prediction has drawn a lot of attention due to its ability to capture meaningful representations through self-supervision (Wang et al. (2018b); Yu et al. (2019)). Although modern video prediction methods have made significant progress in improving predictive accuracy, most of their applications are limited in the scenarios of passive forecasting (Villegas et al. (2017); Wang et al. (2018a); Byeon et al. (2018); Jin et al. (2020)), meaning models can only passively observe a short period of dynamics and accordingly make a short-term extrapolation. Such settings neglect the fact that the observer can also become an active participant in the environment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To model the movements of active manipulators, several low-level action-conditional video prediction models have been proposed in the community (Oh et al. (2015); Mathieu et al. (2015); Babaeizadeh et al. (2017); Ebert et al. (2017)). In this work, we go one step further by introducing the task of semantic action-conditional video prediction which emphasizes the modeling of interactions between agents and environment. Instead of using low-level single-entity actions such as action vectors of robot arms as done in prior works (Finn et al. (2016); Kurutach et al. (2018)), our new task provides semantic descriptions of interactive actions, e.g. \"Open the door\", and asks the model to imagine \"What if I open the door\" in the form of future frames. This task requires the model to recognize the object identity, assign correct affordances to objects and envision the long-term expectation by planning a reasonable trajectory toward the goal, which resembles how humans might imagine conditional futures. The ability to predict correct and semantically consistent future perceptual information is indicative of conceptual grounding of actions, in a manner similar to object grounding in image-based detection and generation tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The challenge of action-conditional video prediction primarily lies in how to correctly inform the model of more abstract semantic action information. Existing low-level counterparts usually achieve this by employing a naive concatenation (Finn et al. (2016); Babaeizadeh et al. (2017)) with action vector of each timestep. While this implementation might enable model to move the desired objects, it fails to produce consistent long-term predictions toward target locations in the multi-entity settings because it was originally designed to only encode the motion information of a single entity. If we take \"put A on $B \"$ as an example, it turns out to be difficult to make the model learn what and where $B$ is, because the main self-supervisory signals in the framework of video prediction are pixel changes and $B$ is not moving in this case. In order to distinguish and locate instances in the scene, other related works heavily rely on pre-trained object detectors or ground-truth bounding boxes (Bar et al. (2020); Ji et al. (2020); Huang et al. (2018); Wu et al. (2020)). However, we argue that utilizing a pre-trained detector actually simplifies the task since such a detector already solves the major difficulty by mapping high-dimension inputs to low-dimension groundings. Furthermore, bounding boxes cannot effectively describe complex visual changes including rotations and occlusions. Thus, a more flexible way of representing objects and actions is required. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/18a22a9ed4bce1235e535341f86718ebefdcca7ae56cf92f60f98c955783bfeb.jpg",
+        "image_caption": [
+            "Figure 1: Concept Grounding in Semantic Video Prediction. After observing the scene, an agent predicts future frames conditioned on a series of semantic actions describing agent-object interactions. Neither bounding boxes nor key points are provided. Conditioning on different action labels leads to Counterfactual generations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "We present a new video prediction model, MAC, short for Modular Action Concept Network. Inspired by the idea of Mixture of Experts, MAC embodies each semantic label by a structured combination of various concept slots, each of which encodes the spatial representation of a specific concept. Such design allows MAC to reuse and integrate the knowledge learnt from different scenarios so that it can perceive the locations of motionless objects and extrapolate to unseen cases, showing the progression towards higher-level cognitive abilities. The contributions of this work are summarized as follows: ",
+        "bbox": [
+            173,
+            496,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We introduce a new task, semantic action-conditional video prediction as illustrated in Fig 1, which can be viewed as an inverse problem of action recognition.   \n2. We create two new synthetic video datasets, CLEVR-Building-blocks and Sapien-Kitchen, and label one real-world dataset called Tower-Creation for evaluation.   \n3. We propose a novel video prediction model, Modular Action Concept Network, in which routing of visual concept slots is directly controlled by action labels. We show that MAC can successfully depict the long-term counterfactual evolution without need of bounding boxes.   \n4. We demonstrate that the trained MAC can make out-of-distribution generalization, be adapted for new object categories with a small number of samples and exploit its learnt features for detection. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 APPROACH",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+        "type": "text",
+        "text": "We begin with defining the task of semantic action-conditional video prediction. Given an initial frame $x _ { 0 }$ and a sequence of action labels $a _ { 1 : T }$ , the model is required to predict the corresponding future frames $x _ { 1 : T }$ . Each action label is a pre-defined semantic description of a spatiotemporal movement that involves multiple objects in a scene and spans over multiple frames such as \"take the yellow cup on the table\" from $t = 0$ to $t = 1 0$ . So technically, one can regard this task as an inverse problem of action recognition. It should also be pointed out that our semantic task is different from common dense video prediction and generation tasks in the sense that it focuses on predicting time-agnostic events. Hence, we design the corresponding datasets as videos capturing sufficient key frames of entire actions. In future practices, we can further apply video interpolation methods in CV or motion planner algorithms in RL to make up the intermediate process if needed. ",
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+        "img_path": "images/dfc5e76ab0af39d06f4ca0181dfcd70b8e217e161695cec07bf20e324f1972f5.jpg",
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+            "Figure 2: The pipeline of MAC in which the computation of concept slot module is elaborated (Better viewed in color). Feature maps extracted by encoder are mapped into the concept slot tensors. Concept slot module receives an action label that controls the collection of concept slot tensors and outputs representations encapsulating this action. A recurrent predictor updates representations before sending them to decoder to predict the next frame. "
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+        "text": "2.1 MOTIVATION ",
+        "text_level": 1,
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+        "text": "The design of our new task is necessary for studying compositional generalization as it detaches the definition of object from its specific location. However, it also requires a successful model to figure out where the desired object is through leveraging abstract labels. Our main idea is that we create a large number of small specialized learners called concept slots for each word in the dictionary of action labels to capture their corresponding spatial representations from observations. During training, action labels will be translated as constituency trees to control the activations of all related concept slots and to assemble the representations of given actions for next-frame prediction. As a result, this language-guided gating mechanism embeds the syntactic structures into the learning system and enables the proposed model to dynamically recombine its learnt concepts so that it can understand the combinatorial complexity of the world. In this paper, we demonstrate that our method possesses many key characteristics of system-2 learning (Goyal et al. (2019); Goyal & Bengio (2020)), including concept grounding, sample efficiency, counterfactual generations, out-of-distribution generalization and fast transfer. ",
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+        "text": "2.2 MODULAR ACTION CONCEPT NETWORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The MAC model is composed of 4 modules including encoder $\\mathcal { E }$ , decoder $\\mathcal { D }$ , concept slot module $\\mathcal { C }$ and recurrent predictor $\\mathcal { P }$ . The goal of our model is to learn the following mapping: ",
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+    {
+        "type": "equation",
+        "img_path": "images/e0ef079ae8b925a388a79ea3ed9d054070cc3b62d6d5cf4959e34f9e90480717.jpg",
+        "text": "$$\n\\hat { x } _ { t } = \\mathcal { D } ( \\mathcal { P } ( \\mathcal { C } ( \\mathcal { E } _ { t - 1 } ) | a _ { t } ) | h _ { t - 1 } ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $x _ { t } , a _ { t }$ and $h _ { t }$ are video frame, action labels and hidden states at time $t$ . The overall architecture of our method is illustrated in Fig 2. In the case of stochastic video generation, another two modules, prior $p ( z )$ and posterior $q ( z )$ , will be added to help estimate the latent distribution of trajectories. ",
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+        "text": "Encoder and Decoder: At each timestep $t - 1$ , the encoder $\\mathcal { E }$ receives visual input $x _ { t - 1 }$ and extracts a set of multi-scale feature maps. In the deterministic setting, we employ a convolutional neural network with an architecture similar to VGG16 (Simonyan $\\&$ Zisserman (2014)). The matching decoder $\\mathcal { D }$ is a mirrored version of the encoder with down-sampling operations replaced with spatial up-sampling and additional sigmoid output layer. It aggregates the updated latent representations produced by predictor and multi-scale feature maps from encoder to predict the next frame $\\hat { x } _ { t }$ . ",
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+        "type": "text",
+        "text": "In the stochastic setting, we use invertible autoencoder introduced in CrevNet (Yu et al. (2019)) instead as we find this information-preserving architecture can better preserve the attributes of randomly moving objects. The corresponding decoder is the backward pass, i.e. inverse computation, of the same network of the encoder. Readers can find more details about invertible autoencoder and coupling layer in Appendix B. ",
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+        "type": "text",
+        "text": "Concept Slot Module: The concept slot module $\\mathcal { C }$ is the core module of MAC. It resembles the mixture of experts as each slot focuses on only one concept in the space of action labels and will be activated and assembled to represent the given actions through the language-guided gating functions. ",
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+        "type": "text",
+        "text": "Each action label will first be decomposed into several constituents of sentence. A constituent is a verb or object phrase, like “pick” or \"large red bowl\". Since we are mostly dealing with manipulation videos, actions are usually divided into 3 constituents, verb, object1, object2. Each constituent will have its own dictionary recording all pre-defined words or concepts and gating functions can be derived based on these dictionaries to establish bottom-up connections from concept slots. The computation of concept slot module is given as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/1a8ee3b54ae98fc32dab1d4a9aae96bc6fe31dd84c89aa0b2b0b70370f23c3f2.jpg",
+        "text": "$$\n\\mathbf { w } ^ { i } = \\Psi ^ { i } ( \\mathbf { f } ) , \\qquad \\mathbf { c } ^ { j } = \\Phi ^ { j } ( \\mathrm { C o n c a t } ( \\{ \\mathbf { w } ^ { i } | \\forall i , \\delta ^ { j } ( i ) = 1 \\} ) )\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where w and c are concept and constituent representations and $\\delta ^ { j }$ is the indicator function for gating function of $j _ { \\mathrm { t h } }$ constituent. More specifically, after the feature maps f are extracted from the input image, they are fed into $\\kappa$ convolutional units $\\Psi ^ { i }$ , i.e. the concept slot layer, to create $\\kappa$ concept slot tensors of dimension $N _ { d }$ . Here, $\\kappa$ is the total number of possible concepts we pre-defined in the dictionary of action labels. Since verbs can be interpreted as spatiotemporal changes of relationships between objects, not only slots for objects but also slots for verbs, like ’take’ or ’put on’, are computed from the extracted feature maps. ",
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+        "type": "text",
+        "text": "Next, a gating function will collect all involved concept slot tensors and create an ensemble as input for each constituent. This assembly process simulates the formation of simplified constituency parse trees. Constituent slot layer $\\Phi ^ { j }$ can either be resolution-preserving or upsampling operators as spatial information is important for our new task. Finally, outputs of all constituent slots are concatenated pixel-wisely to obtain the representation of actions before sending them to predictor. It is worth noticing that MAC is allowed to have multiple concurrent actions in a scene at inference time. In this case, we copy additional groups of trained constituent slots to represent other actions. ",
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+        "type": "text",
+        "text": "Learned Prior: We leverage a technique called learned prior from SVG (Denton & Fergus (2018)) to model the stochastic movements in videos. In particular, we build two additional recurrent inference networks, prior and posterior respectively, to capture the randomness of motions. During training, the posterior inference network $q ( z )$ can access to the representations of target frames to estimate a true distribution of trajectory that we expect its prior counterpart $p ( z )$ to mimic at test time. Codes of motions $z _ { t }$ estimated by posterior during training (or by prior during testing) will then be concatenated with latent representations before sent to predictor. ",
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+        "type": "text",
+        "text": "Predictor: The recurrent predictor $\\mathcal { P }$ , implemented as a stack of residual ConvLSTM layers (Shi et al. (2015)), calculates the spatiotemporal evolution for each action label respectively. The memory mechanism of ConvLSTM is essential for MAC to remember its previous actions and to recover the occluded objects. To prevent interference between concurrent actions, hidden states are not shared between actions. The outputs of predictor for all action labels are added point-wisely. ",
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+        "text": "Training: In the deterministic setting, we train our model by minimizing the mean squared error the between the target frames and the predictions. In the stochastic setting, we optimize the following variational lower bound (ELBO) using re-parameterization trick (Kingma & Welling (2013)): ",
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+        "type": "equation",
+        "img_path": "images/2e5807e4581256b203aea64fc87c680cd2835818abc3becbe2344b4018b9bc4b.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\theta , \\phi , \\psi } ( x _ { 1 : T } ) = \\sum _ { t = 1 } ^ { T } [ \\mathbb { E } _ { q _ { \\phi } ( z _ { 1 : t } | x _ { 1 : t } ) } \\log p _ { \\theta } ( x _ { t } | z _ { 1 : t } , x _ { 1 : t - 1 } ) - \\beta D _ { K L } ( q _ { \\phi } ( z _ { t } | x _ { 1 : t } ) | | p _ { \\psi } ( z _ { t } | x _ { 1 : t - 1 } ) ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $p _ { \\theta }$ is the future frame generator, $z _ { t }$ represents the latent codes of motion, $p _ { \\psi } \\big ( z _ { t } | x _ { 1 : t - 1 } \\big )$ is the prior distribution, $q _ { \\phi } \\big ( { z } _ { t } | { x } _ { 1 : t } \\big )$ is the posterior distribution and $D _ { K L }$ denotes the Kullback–Leibler (KL) divergence which forces the posterior to approximate the prior distribution. Since $p _ { \\theta }$ is modeled by conditional Gaussian, the likelihood term reduces to MSE measure between the ground truth frames and the predictions. The full derivation of ELBO is provided in the Appendix A. ",
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+        "type": "text",
+        "text": "At the inference phase, the model will use its previous predictions as visual inputs instead except for the first pass. Hence, a training strategy called scheduled sampling (Bengio et al. (2015)) is adopted to alleviate the discrepancy between training and inference. ",
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+    {
+        "type": "text",
+        "text": "3 DATASETS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this study, we create two new synthetic datesets, CLEVR-Building-blocks and Sapien-Kitchen, and label one real-world dataset called Tower-Creation from Roboturk (Mandlekar et al. (2018)) ",
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+        "type": "text",
+        "text": "for evaluation. This is because most existing video datasets either don’t come with semantic action labels (Babaeizadeh et al. (2017)) or fail to provide necessary visual information in their first frames due to egomotions and occlusions (Hundt et al. (2018)). Although there are several candidate datasets like Penn Action (Zhang et al. (2013)), BAIR (Finn et al. (2016)) and KTH (Schuldt et al. (2004)) for multi-modal learning, they all adopt the same single-entity setting which actually indicates they can be solved by a much simpler model. To tackle the above issues, we design each video in our datatsets as a depiction of certain atomic action performed by an agent with objects which are observable in the starting frame. Furthermore, we add functions to generate bounding boxes of all objects for both synthetic datasets in order to train AG2Vid. It is worth noting that all three of these domains exhibit a key property named combinatorial explosion, resulting in factorial complexity growth in both spatial and temporal dimensions even with a small object set. For instance, a sequence with 6 (out of 32) objects and 6 actions can have 333,396,000 possibilities without considering any continuous factor. Hence, our model only sees a small fraction of these potential scenarios during training. ",
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+    {
+        "type": "text",
+        "text": "3.1 CLEVR-BUILDING-BLOCKS DATASET ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "CLEVR-Building-blocks dataset is built upon CLEVR environment (Johnson et al. (2017)). For each video, the data generator initializes the scene with 4 - 6 randomly positioned and visually different objects. There are totally 32 combinations of shapes, colors and materials of objects and at most one instance of each combination is allowed to appear in a video sequence. The agent can perform one of the following 8 actions on objects ${ \\mathcal { O } } _ { A }$ and $\\mathcal { O } _ { B }$ : Pick ${ \\mathcal { O } } _ { A }$ , Pick and Rotate ${ \\mathcal { O } } _ { A }$ transversely / longitudinally, Put ${ \\mathcal { O } } _ { A }$ on $\\mathcal { O } _ { B }$ , Put ${ \\mathcal { O } } _ { A }$ on the left / right side of $\\mathcal { O } _ { B }$ , Put ${ \\mathcal { O } } _ { A }$ in the front of / behind $\\mathcal { O } _ { B }$ . Each training sample contains a video of three consecutive Pick- and $P u t -$ action pairs and a sequence of semantic action labels of every frame. ",
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+    {
+        "type": "text",
+        "text": "3.2 SAPIEN-KITCHEN DATASET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Sapien-Kitchen Dataset describes a more complicated environment in the sense that: (a). It contains deformable actions like \"open\" and \"close\"; (b). The structures of different objects in the same category are highly diverse; (c). Objects can be initialized with randomly assigned relative positions like \"along the wall\" and \"on the dishwasher\". We collect totally 21 types of small movable objects in 3 categories, bottle, kettle and kitchen pot, and 19 types of large openable appliances in another 3 categories, oven, refrigerator and dishwasher, from Sapien engine (Xiang et al. (2020)). The agent can perform one of the following 6 atomic actions on small object $\\mathcal { O } _ { s }$ and large appliance $\\mathcal { O } _ { l }$ : Take $\\mathcal { O } _ { s }$ on $\\mathcal { O } _ { l }$ , Take $\\mathcal { O } _ { s }$ in $\\mathcal { O } _ { l }$ , Put $\\mathcal { O } _ { s }$ on $\\mathcal { O } _ { l }$ , Put $\\mathcal { O } _ { s }$ in $\\mathcal { O } _ { l }$ , Open $\\mathcal { O } _ { l }$ and Close $\\mathcal { O } _ { l }$ . Composite action sequences are defined as follows: \"Take_on–Put_on\", \"Take_on–Open–Put_in–Close\", \"Open–Take_in–Close\". ",
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+    {
+        "type": "text",
+        "text": "3.3 TOWER-CREATION DATASET ",
+        "text_level": 1,
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+        "text": "Each video in Tower-Creation Dataset depicts a robotic arm building a tower with flatware present on the table. We have labeled 524 videos in total since semantic descriptions are not provided and prodce 1867 samples consists of two actions: Pick ${ \\mathcal { O } } _ { A }$ and Put ${ \\mathcal { O } } _ { A }$ on $\\mathcal { O } _ { B }$ . We use 1536 video clips for training and 331 for evaluation. It should be pointed out that the size of Tower-Creation dataset is small compared with commonly used datasets such as BAIR (Finn et al. (2016)) which has $5 9 \\mathrm { k }$ videos in total. Thus, our experiments can also tell whether evaluated methods are data efficient. ",
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+        "type": "text",
+        "text": "4 EXPERIMENTAL EVALUATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 ACTION-CONDITONAL VIDEO PREDICTION ",
+        "text_level": 1,
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+        "text": "Baselines and setup: We evaluate the proposed model on CLEVR-Building-blocks and SapienKitchen Datasets. AG2Vid (Bar et al. (2020)) is re-implemented as the baseline model because it is the most related work. Unlike our method which only needs visual input and action sequence, AG2Vid also requires bounding boxes of all objects and progress meters of actions, i.e. clock edge, for training and testing. Furthermore, we conduct an ablation study by replacing concept slot module with the concatenation of features and tiled action vector, which is commonly used in low-level action-conditional video prediction (Finn et al. (2016)), to show the effectiveness of our module. ",
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+    {
+        "type": "text",
+        "text": "Metrics: To estimate the fidelity of action-conditional video prediction, MSE, SSIM (Wang et al. (2004)), PSNR and LPIPS (Zhang et al. (2018)) are calculated between the predictions and groundtruths. However, these metrics may not effectively tell if actions are successfully completed due to the small sizes of the moving objects. Hence, we also perform a human study to assess the accuracy of performing the correct action in generated videos for each model. The human judges annotate whether the model can identify the desired objects, perform actions specified by action labels and maintain the consistent visual appearances of all objects in its generations and only videos meeting all three criterions are scored as correct. ",
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+    {
+        "type": "image",
+        "img_path": "images/3038eadaf59d6f016ac8de99ed21432c8034e5f2bfb1414e7818098f737fcf0c.jpg",
+        "image_caption": [
+            "Figure 3: The qualitative comparison on CLEVR-Building-blocks and Sapien-Kitchen. The first row of each figure is the groundtruth sequence. The red, blue and green boxes highlight the quality of predictions by each method. In contrast to the success of MAC, concatenation-based method fails to find the correct destinations or to preserve attributes of moving objects. Also, bounding boxes used in AG2Vid cannot portray visual changes like rotations correctly. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/d537a98800aea288af65e98ccd6259c7be134266dd2b797d52be492b8e51cc2c.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Quantitative evaluation on CLEVR-Building-blocks and Sapien-Kitchen. All metrics are averaged frame-wisely except for accuracy. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"4\">CLEVR-Building-blocks</td><td colspan=\"4\">Sapien-Kitchen</td></tr><tr><td>SSIM↑</td><td>MSE↓</td><td>LPIPS↓</td><td>Accuracy↑</td><td>SSIM↑</td><td>MSE↓</td><td>LPIPS↓</td><td>Accuracy↑</td></tr><tr><td>Copy-First-Frame</td><td>0.962</td><td>251.38</td><td>0.1320</td><td>-</td><td>0.951</td><td>152.87</td><td>0.0393</td><td>1</td></tr><tr><td>Concatenation Baseline</td><td>0.961</td><td>226.53</td><td>0.1301</td><td>50.8%</td><td>0.962</td><td>23.13</td><td>0.0232</td><td>52.4%</td></tr><tr><td>AG2Vid</td><td>0.956</td><td>58.67</td><td>0.0399</td><td>78.8%</td><td>0.947</td><td>270.87</td><td>0.0684</td><td>5.2%</td></tr><tr><td>MAC</td><td>0.983</td><td>43.52</td><td>0.0303</td><td>95.2%</td><td>0.971</td><td>11.16</td><td>0.0178</td><td>86.4%</td></tr></table>",
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+        "text": "",
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+        "type": "text",
+        "text": "Results: The quantitative comparisons of all methods are summarized in Table 1. The MAC achieves the best scores on all metrics without access to additional information like bounding boxes, showing the superior performance of our concept slot module. The qualitative analysis in Fig 3 further reveals the drawbacks of other baselines. For CLEVR-Building-blocks, the concatenation-based variant fails to recognize the right objects due to its limited inductive bias. Although AG2Vid has no difficulty in identifying the desired objects, assumptions made by flow warping are too strong to handle rotation and occlusion. Consequently, the adversarial loss enforces AG2Vid to fix these errors by converting them to wrong poses or colors. These limitations of AG2Vid will be further amplified in a more complicated environment, i.e. Sapien-Kitchen. The same architecture used for CLEVR can only learn to remove the moving objects from their starting positions in Sapien-Kitchen because rotation and occlusion occur more often. The concatenation baseline performs better by showing correct generation of open and close actions on large appliance. Yet, it still fails to produce long-term consistent predictions as the visual appearances of moving objects are altered. On the contrary, MAC can authentically depict the correct actions specified by action labels on both datasets. ",
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+    {
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+        "img_path": "images/107061b0061d8ec09a04d6e8033e1a292090fa7f6e39534a41674c392584f842.jpg",
+        "image_caption": [
+            "Figure 4: Counterfactual video generation: Conditioning on the same initial frame and different action labels, MAC can produce high-quality imaginations of counterfactual futures. Various visual outcomes present in the final frames are highlighted with red boxes and enlarged in the final column. Top: Generative results on CLEVR-Building-blocks. 34 frames are generated. Bottom: Generative results on Sapien-Kitchen dataset. 35 frames are generated. "
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+        "image_caption": [
+            "Figure 5: Left: Visual comparison between sMAC and SVG-LP on Tower-Creation. The supposed completions of Pick and Put in the final frames are highlighted by red and yellow boxes while incorrect completions in SVG-LP generations are labelled by grey boxes. The last two rows are counterfactual generations in which models are given different action labels. Right: Quantitative comparison per-frame. Higher SSIM and PSNR indicate better performance. "
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+    {
+        "type": "text",
+        "text": "4.2 COUNTERFACTUAL GENERATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Counterfactual generation: The most intriguing application of MAC is counterfactual generation. More specifically, counterfactual generation means that our model will observe the same starting frame but receive different valid action labels to produce the corresponding future frames. ",
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+    {
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+        "img_path": "images/0466372b18118f224d4c7199030b2f35ad76a99ed819f8b9789e8c8b8973b942.jpg",
+        "image_caption": [
+            "Figure 6: Compositional generalization and feature reuse.Top: Unobserved scenarios. All red cubes are removed from the tranining data, but the trained model can still manipulate red cube at test time. Middle: Concurrent actions. Inputting two action sequences at the same time. Both actions are depicted correctly. Bottom Left: New-object adaptation. Even with a few training samples, MAC can be fast adapted for generation of new objects. Red arrows point to new objects present in images. Bottom Right: Object detection. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results: The visual results of counterfactual generations on each dataset are displayed in Fig 4. As we can see, our model successfully identifies the desired objects, plans correct trajectories toward the target places and generates high-quality imaginations of counterfactual futures. It is also worth noticing that all displayed generations are long-term generations , i.e. more than 30 frames are predicted for each sequence. Our recurrent predictor plays an very important role in sustaining the spatiotemporal consistency and in reconstructing the fully-occluded objects. ",
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+    {
+        "type": "text",
+        "text": "4.3 STOCHASTIC VIDEO GENERATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Baselines and setup: We continue to evaluate the stochastic version of MAC (sMAC) on TowerCreation dataset. SVG-LP was extended to action-conditional version in (Villegas et al. (2019)) so that we can adopt it as the baseline model to demonstrate the effectiveness of concept slot module. ",
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+    {
+        "type": "text",
+        "text": "Results: The qualitative and quantitative comparison between sMAC and action-conditional SVG-LP is provided in Fig 5. Although SVG-LP can partially understand the given action labels, it often fails to locate and manipulate the desired objects. Consequently, it will generate the moving object out of nowhere and often place it on a wrong target object. In contrast, sMAC can successfully simulate the trajectory of robotic arms and correctly animate the \"Pick\" and \"Put\" actions thanks to the concept slot module. Row 3 and 5 in Fig 5 show that sMAC is also capable of producing diverse future frames and predicting counterfactual results following different action instructions. ",
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+    {
+        "type": "text",
+        "text": "4.4 COMPOSITIONAL GENERALIZATION ",
+        "text_level": 1,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We further explore other interesting features of our MAC. We first demonstrate that MAC is capable of making out-of-distribution generalization by designing two experiments. We evaluate how quickly our model can be adapted to new objects. It turns out for each new object, the trained MAC only requires a few training video examples to generate decent results. Finally, to verify that our model encodes the spatial information, we add SSD (Liu et al. (2016)) head after the frozen encoder and concept slot layer to conduct object detection. ",
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+    {
+        "type": "text",
+        "text": "Unobserved scenarios: We design an interesting experiment where only a subset of CLEVRBuilding-blocks data are used for training and check what will happen if we input the unobserved action labels to the trained model. More precisely, we exclude all videos manipulating red cubes in the training sets and send the instructions involving red cubes at test time. The visualization of this experiment can be found in Fig 6. As we can see, MAC can still identify and manipulate red cubes correctly, showing its ability to recombine the learnt concept to comprehend new objects. ",
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+    {
+        "type": "text",
+        "text": "Concurrent actions: Concurrent actions means multiple action inputs at the same time. It can be considered as out-of-distribution generalization because our model only observes single-action videos during training. Generating concurrent-action videos needs to employ copied constituent slots and parallel hidden states. As illustrated in Fig 6, MAC can linearly integrate the action information in the latent space and correctly portray 2 concurrent actions in the same scene. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Adaptation: We add a new openable category \"safe\" and a new movable category \"dispenser\" into Sapien-Kitchen and generate 100 video sequences for each new object showing its interaction with other objects. Approximately, there are about 5 new sequences created for each new action pair between 2 objects. Blank concept slots for new categories are attached to trained MAC and we finetune it on this small new training set. Visualization in Fig 6 shows that even with a few training samples, MAC is accurately adapted for video generation of new objects. This is because, with the help of concept slots, MAC can disentangle actions into relatively independent grounded concepts. When it learns new concepts, MAC reuses and integrates prior knowledge learnt from different cases. ",
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+    {
+        "type": "text",
+        "text": "Object detection: The quantitative results of object detection and more visualizations can be found in Appendix D. We observe that the features learnt by MAC can be easily transferred for detection as our video prediction task is highly location-dependent. This result indicates that utilizing bounding boxes might be a little redundant for some video tasks because videos already provide rich motion information that can be used for salient object detection. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Video prediction: ConvLSTM (Shi et al. (2015)) was the first deep learning model that employed a hybrid of convolutional and recurrent units for passive video prediction. This architectural design was soon followed by studies looking at a similar problem (Kalchbrenner et al. (2017); Mathieu et al. (2015); Wang et al. (2017); Yu et al. (2019); Wang et al. (2018b)). However, the capability of passive video prediction framework is very limited as models usually don’t have sufficient information to predict the long-term future due to partial observation, egomotion and randomness. More importantly, this setting prevents models from interacting with environment. On the other hand, the low-level action-conditional video prediction task provides an action vector at each timestep as additional input to guide the prediction (Oh et al. (2015); Chiappa et al. (2017); Babaeizadeh et al. (2017); Wu et al. (2021)). CDNA (Finn et al. (2016)) is a representative of such models. In CDNA, the states and action vectors of the robotic manipulator are first spatially tiled and integrated into the model through concatenation. SVG (Denton & Fergus (2018)) was initially proposed for stochastic video generation but later was extended to action-conditional version in (Villegas et al. (2019)). It is worth noticing that SVG also used concatenation to incorporate action information. Such implementations are prevalent in low-level action-conditional video prediction because the action vector only encodes the spatial information of a single entity, usually a robotic manipulator (Finn et al. (2016)) or a human hand. A common failure case for such models is the presence of multiple affordable entities (Kim et al. (2019)), a scenario that our task definition and datasets focus on. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Modularity: Mixture of Experts refers to a classical machine learning technique where various learners are employed, each of which specializes in one particular function, and their output are aggregated through a gating function. This modular design makes each submodule relatively independent and thus leads to better generalization and robustness to compositional changes, which has been studied in several works (Goyal et al. (2019); Afshar et al. (2021); Sabour et al. (2017); Henaff et al. (2016)). In this work, we hypothesis that the underlying syntactic structures of semantic labels can tell how to aggregate the representations of individual concept learners. By translating labels into constituency trees, action graphs are embedded into the learning system to get the entire perspective of ongoing activities while each concept learner can focus on its specific subtask. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose the new task of semantic action-conditional video prediction and introduce 3 new datasets that are meant to bridge the gap towards a robust solution to this task in complex interactive scenarios. MAC, a novel video prediction model, was also designed by utilizing the idea of Mixture of Experts to ground action concept for video generation. Our proposed model can generate alternative futures without requiring additional auxiliary data such as bounding boxes, and is shown to be both quickly extendible and adaptable to novel scenarios and entities. It is our hope that our contributions will advance progress and understanding within this new task space, and that a model robust enough for real-world applications (i.e. in robotic systems) in perception and control will be eventually proposed as a descendant of this work. ",
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+    {
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+        "text": "Fanbo Xiang, Yuzhe Qin, Kaichun Mo, Yikuan Xia, Hao Zhu, Fangchen Liu, Minghua Liu, Hanxiao Jiang, Yifu Yuan, He Wang, et al. Sapien: A simulated part-based interactive environment. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11097–11107, 2020. ",
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+        "text": "Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 586–595, 2018. ",
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+        "bbox": [
+            174,
+            234,
+            825,
+            273
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A VARIATIONAL LOWER BOUND DERIVATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            575,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The original variational lower bound was derived in (Kingma & Welling (2013)). ",
+        "bbox": [
+            174,
+            122,
+            704,
+            138
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/576191b509435fbe389b7f978c0f09d994627dbfe229aaa0c87bb6e6cb172ad4.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\quad \\log \\gamma _ { \\mathbb { P } } ( \\mathbf { x } ) = \\log \\int _ { \\mathcal { R } } \\log \\left( \\mathbf { x } \\mid \\mathbf { x } _ { \\mathcal { P } } \\rho ( \\mathbf { x } ) \\mid \\mathbf { x } _ { \\mathcal { P } } \\rho ( \\mathbf { x } ) \\right. } \\\\ & { = \\log \\int _ { \\mathcal { R } } p ( \\mathbf { x } ) \\mid p ( \\mathbf { x } ) \\mid p ( \\mathbf { x } ) \\mid \\frac { \\log ( \\mathbf { x } ) \\mid \\exp ( \\mathbf { x } ) } { \\exp ( \\mathbf { x } ) } } \\\\ & { = \\log \\mathbb { E } _ { \\rho _ { \\xi } \\in \\{ \\mathbf { x } , \\mathbf { x } \\} } \\frac { p ( \\mathbf { x } ) \\exp ( \\mathbf { x } ) \\mid \\exp ( \\mathbf { x } ) } { \\exp ( \\mathbf { x } ) } } \\\\ & { \\geq \\mathbb { E } _ { \\mathbf { x } _ { \\mathbb { P } } ( \\mathbf { x } ) = \\mathbf { b } } \\frac { p ( \\mathbf { x } ) \\mid p ( \\mathbf { x } ) \\mid p ( \\mathbf { x } ) } { \\exp ( \\mathbf { x } ) } } \\\\ & { = \\mathbb { E } _ { \\mathbf { x } _ { \\mathbb { P } } ( \\mathbf { x } ) = \\mathbf { b } } \\frac { \\log ( \\mathbf { x } ) \\mid \\exp ( \\mathbf { x } ) } { \\exp ( \\mathbf { x } ) } } \\\\ & { = \\mathbb { E } _ { \\mathbf { x } _ { \\mathbb { P } } ( \\mathbf { x } ) = \\mathbf { b } } \\frac { \\log ( \\mathbf { x } ) \\mid \\exp ( \\mathbf { x } ) - \\mathbf { E } _ { \\mathbf { x } _ { \\mathbb { P } } ( \\mathbf { x } ) } \\mid \\log \\frac { \\log ( \\mathbf { x } \\mid \\mathbf { x } ) } { p ( \\mathbf { x } ) } } { \\exp ( \\mathbf { x } ) } } \\\\ &  = \\mathbb { E } _ { \\mathbf { x } _ { \\mathbb { P } } ( \\mathbf { x } ) \\mid \\mathbf { x } _ { \\mathcal { P } } \\rho ( \\mathbf { x } ) \\mid \\exp ( \\mathbf { x } ) - \\mathcal { P } _ { \\mathbf { x } _ { \\mathbb { P } } ( \\mathbf { x } _ { \\mathbb { P } } ( \\mathbf { x } ) \\mid \\rho ( \\mathbf { x } ) \\mid \\exp ( \\mathbf { x } ) } { \\exp ( \\mathbf { x } ) \\mid \\exp ( \\mathbf { x } \\mid \\mathbf { x } _ { \\mathbb { P } } \\rho ( \\mathbf { x } ) \\mid \\mathbf { x } ) } } \\\\ &  = \\sum _  \\mathbf { x } \\in \\{ \\mathbf { x } , \\mathbf { x } \\mid \\mathbf { x } _ { \\mathbb { P } } \\rho ( \\mathbf  x \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The final step (Denton & Fergus (2018)) is obtained through the factorization of reconstruction and KL-Divergence term into individual time steps due to the independence across time. ",
+        "bbox": [
+            169,
+            405,
+            823,
+            434
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B INVERTIBLE ARCHITECTURE AND COUPLING LAYER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            439,
+            650,
+            455
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The additive coupling layer was first introduced in (Dinh et al. (2014)). Following (Yu et al. (2019)), we use it as the building block to construct the invertible autoencoder. More specifically, the reshaped input $x$ is divided into two groups, denoted as $x ^ { 1 }$ and $x ^ { 2 }$ , channel-wisely. In its forward pass, one group, e.g. $x ^ { 1 }$ , passes through several convolutional layers and updates the other group, $x ^ { 2 }$ , through addition. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3681ad4ada7f73ff3bc09ff55c77fa84bf68ffb1e13ad40af7aabb0704be1c0d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { x } ^ { 2 } = x ^ { 2 } + \\mathcal { F } _ { 1 } ( x ^ { 1 } ) } \\\\ { \\hat { x } ^ { 1 } = x ^ { 1 } + \\mathcal { F } _ { 2 } ( \\hat { x } ^ { 2 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            536,
+            563,
+            578
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { F }$ is a composite non-linear transformation consisting of convolutions and activations, and ${ \\hat { x } } ^ { 1 }$ and ${ \\hat { x } } ^ { 2 }$ are the updated $x ^ { 1 }$ and $x ^ { 2 }$ . In its backward pass, we can retrieve $x ^ { 1 }$ and $x ^ { 2 }$ from ${ \\hat { x } } ^ { 2 }$ and ${ \\hat { x } } ^ { 1 }$ by the following inverse computation: ",
+        "bbox": [
+            174,
+            584,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/48e5ae1128506f15e43bb35b1f5c52d58cb88c4ea5a8bf39044ff371c17b7cd1.jpg",
+        "text": "$$\n\\begin{array} { r } { x ^ { 1 } = \\hat { x } ^ { 1 } - \\mathcal { F } _ { 2 } ( \\hat { x } ^ { 2 } ) } \\\\ { x ^ { 2 } = \\hat { x } ^ { 2 } - \\mathcal { F } _ { 1 } ( x ^ { 1 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            632,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Pixel shuffle layer (Shi et al. (2016)), a bijective downsampling, is also employed to change the shape of feature from $( w , h , c )$ to $( w / n , h / \\bar { n } , c \\times n ^ { 2 } )$ to enable the invertibility of the entire network. Stacking these building blocks and downsampling in an alternating fashion between two groups, we will obtain a two-way autoencoder. The property of invertibility ensures no information loss during feature extraction, which is better at preserving the attributes of moving objects. The same network can serve as both the encoder and the decoder by using its forward and backward pass respectively. ",
+        "bbox": [
+            174,
+            678,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C TRAINING SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            768,
+            354,
+            785
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the deterministic setting, MAC adopts VGG16 (Simonyan & Zisserman (2014)) and a mirrored network as encoder and decoder and 2 layers of residual ConvLSTM as predictor. In the stochastic setting, sMAC replaces its encoder with 24-layer invertible autoencoder and use its backward pass as decoder. Additionally, it also deploys two inference networks composed of 2 layers of ConvLSTM, named prior and posterior, to model conditionally Gaussian distribution of trajectories . ",
+        "bbox": [
+            174,
+            790,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use the Adam optimizer (Kingma & Ba (2014)) with a starting learning rate of $2 \\times 1 0 ^ { - 4 }$ to optimize the MAC and sMAC. The training process is stopped after $2 0 0 , 0 0 0$ iterations with the batch size of 4. 20,000 video clips of CLEVR-Building-Blocks and 30,000 of Sapien-Kitchen are generated for model training and additionally 5,000 videos are generated for each dataset for evaluation. Considering the size of Tower-Creation dataset, various traditional data augmentation methods are used and we also implement a new trick in which the neighbouring frames of key frames are sampled from Gaussian distributions to serve as small temporal variations. This trick can significantly improve the visual quality and diversity of stochastic generation for both sMAC and SVG-LP. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/51f6aa92cc2c5b0a060201d9781ad652e3b3bd6e0b1bf5fdad4f296e657b3e01.jpg",
+        "table_caption": [
+            "Table 2: Quantitative measures of object detection on Sapien-Kitchen in terms of average precision. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Oven</td><td rowspan=1 colspan=1>Fridge</td><td rowspan=1 colspan=2>DishwashetBottle</td><td rowspan=1 colspan=1>Kettle</td><td rowspan=1 colspan=1>Kitchenpot</td><td rowspan=1 colspan=1>mAP</td></tr><tr><td rowspan=1 colspan=1>MACencoder+SSD</td><td rowspan=1 colspan=1>92.75</td><td rowspan=1 colspan=1>94.56</td><td rowspan=1 colspan=1>90.89</td><td rowspan=1 colspan=1>83.25</td><td rowspan=1 colspan=1>77.18</td><td rowspan=1 colspan=1>81.32</td><td rowspan=1 colspan=1>86.66</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/d097e23d4c7e9f3da44e7b0d74273c8fdef42dca11cbfe59335262791a00725d.jpg",
+        "image_caption": [
+            "Figure 7: Visualization of 2D Object Detection on Sapien-Kitchen. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            223,
+            818,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D OBJECT DETECTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            816,
+            382,
+            832
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The quantitative results and visualization of object detection is provided in the Table 2 and Fig 7. SSD head was optimized following its protocol while the MAC encoder was frozen to demonstrate that features learnt through self-supervision can be directly transferred for detection because our video prediction task is highly location-dependent. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    }
+]

parse/dev/Qaw16njk6L/Qaw16njk6L_content_list.json

@@ -0,0 +1,1374 @@
+[
+    {
+        "type": "text",
+        "text": "NASVIT: NEURAL ARCHITECTURE SEARCH FOR EF-FICIENT VISION TRANSFORMERS WITH GRADIENTCONFLICT-AWARE SUPERNET TRAINING",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chengyue $\\mathbf { G o n g } ^ { 2 * }$ , Dilin Wang1, Meng $\\mathbf { L i ^ { 1 } }$ , Xinlei Chen1, Zhicheng $\\mathbf { Y a n } ^ { 1 }$ ,   \nYuandong Tian1, Qiang $\\mathbf { L i u } ^ { 2 }$ , Vikas Chandra1   \n1 Meta Reality Labs 2 University of Texas at Austin   \n{wdilin, meng.li, xinleic, zyan3, yuandong, vchandra} $@$ fb.com, {cygong, lqiang}@cs.utexas.edu ",
+        "bbox": [
+            186,
+            193,
+            763,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            271,
+            544,
+            286
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Designing accurate and efficient vision transformers (ViTs) is an important but challenging task. Supernet-based one-shot neural architecture search (NAS) enables fast architecture optimization and has achieved state-of-the-art results on convolutional neural networks (CNNs). However, directly applying the supernetbased NAS to optimize ViTs leads to poor performance - even worse compared to training single ViTs. In this work, we observe that the poor performance is due to a gradient conflict issue: the gradients of different sub-networks conflict with that of the supernet more severely in ViTs than CNNs, which leads to early saturation in training and inferior convergence. To alleviate this issue, we propose a series of techniques, including a gradient projection algorithm, a switchable layer scaling design, and a simplified data augmentation and regularization training recipe. The proposed techniques significantly improve the convergence and the performance of all sub-networks. Our discovered hybrid ViT model family, dubbed NASViT, achieves top-1 accuracy from $7 8 . 2 \\%$ to $8 1 . 8 \\%$ on ImageNet from 200M to 800M FLOPs, and outperforms all the prior art CNNs and ViTs, including AlphaNet and LeViT. When transferred to semantic segmentation tasks, NASViTs also outperform previous backbones on both Cityscape and ADE20K datasets, achieving $7 3 . 2 \\%$ and $3 7 . 9 \\%$ mIoU with only 5G FLOPs, respectively. Code is available at https://github.com/facebookresearch/NASViT. ",
+        "bbox": [
+            233,
+            305,
+            766,
+            568
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            602,
+            336,
+            618
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformers have recently been applied to various vision tasks, including image classification (Liu et al., 2021; Dong et al., 2021), object detection (Carion et al., 2020; Zhu et al., 2020), semantic segmentation (Xie et al., 2021; Cheng et al., 2021), video understanding (Bertasius et al., 2021; Fan et al., 2021), etc. Vision transformers (ViTs) benefit from high model capacity, large receptive field, and grouping effect, etc (Dosovitskiy et al., 2020), and demonstrate superior performance compared to convolutional neural networks (CNNs) especially with the scaling of the model size and training data size. For example, CoAtNet (Dai et al., 2021) achieves $9 0 . 8 8 \\%$ top-1 accuracy on Imagenet by scaling the model to 2586G FLOPs and pre-training the model on JFT-3B dataset (Sun et al., 2017). ",
+        "bbox": [
+            174,
+            637,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Though promising in the high computation budget regime, the performance of ViTs is still inferior to that of the CNN counterparts on small- or medium-sized architectures, especially compared to CNN architectures that are highly optimized by neural architecture search (NAS), e.g., AlphaNet (Wang et al., 2021a), FBNetV3 (Dai et al., 2020), etc. For example, the initial DeiT-Tiny (Touvron et al., 2020) only achieves $7 2 . 2 \\%$ top-1 accuracy with 1.2G FLOPs. The recently proposed LeViT (Graham et al., 2021) makes significant progress to achieve $7 6 . 6 \\%$ top-1 accuracy with 305M FLOPs with convolution/transformer hybrid architectures and a $3 \\mathbf { x }$ longer training schedule. In contrast, AlphaNet (Wang et al., 2021a) achieves $7 7 . 8 \\%$ top-1 accuracy with only 203M FLOPs. The large accuracy gap illustrated above raises a natural question: are transformer blocks that build large and dynamic receptive fields beneficial for small models? ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To answer the question above, in this work, we target at developing a family of efficient ViTs with FLOPs ranging from 200M to 800M . A natural approach is to leverage NAS, which has achieved state-of-the-art (SOTA) accuracy-efficiency trade-off for CNNs (Wang et al., 2021a; Dai et al., 2020; Cai et al., 2019). The recently proposed supernet-based NAS, e.g., BigNAS (Yu et al., 2020a) and AlphaNet (Wang et al., 2021a), builds a weight-sharing graph including all the sub-networks in the architecture search space. A sandwich sampling rule with inplace knowledge distillation (KD) (Yu et al., 2018) is leveraged to simultaneously optimize the supernet and sub-networks for each mini-batch, which stabilizes the training and improves the training convergence. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To leverage the supernet-based NAS, we first modify the LeViT model to build the architecture search space for ViTs and then jointly optimize the model architectures and parameters following AlphaNet. However, we find that directly applying AlphaNet achieves poor performance on the ViT search space, even worse compared to training single ViTs. To understand the root cause of the poor performance, we examine the supernet training procedure and observe that the gradients of the supernet and the different sub-networks conflict with each other during the sandwich sampling, which makes the training loss saturates much more quickly for ViTs, thus leading to slow convergence. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To alleviate the issue of conflicting gradients, we propose three different techniques to improve the supernet training. Firstly, instead of directly adding the gradients from different sub-networks together, we find it beneficial to prioritize the training of the sub-networks over the supernet, as our main purpose is to build efficient sub-networks. We achieve this with a projection gradient algorithm which removes the component of the supernet gradient that is conflict with the sub-network gradient. Secondly, to alleviate the gradient conflicts among different sub-networks, we propose to augment each transformer layer with switchable channel-wise scaling layers. The weights of different scaling layers are not shared among different transformer blocks to reduce gradient conflicts. Thirdly, we propose to use a weak data augmentation scheme and reduce the regularization in training to decrease the optimization difficulty and hence reduce gradient conflicts. ",
+        "bbox": [
+            174,
+            327,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our proposed techniques significantly alleviate the gradient conflict issue and empirically improve the convergence of supernet training. Compared to the baseline supernet training algorithm in AlphaNet, we can improve the top-1 accuracy to $7 8 . 2 \\%$ for the small model with 205M FLOPs and achieve $8 1 . 8 \\%$ for the large model with 757M FLOPs. Meanwhile, the resulting model family, NASViT, outperforms all the SOTA CNN and ViT models across a wide range of computation constraints. NASViT also demonstrates good performance on downstream tasks. When transferring to semantic segmentation tasks, NASViT backbones outperform previous CNN and ViT backbones on both Cityscape and ADE20K datasets, achieving $7 3 . 2 \\%$ and $3 7 . 9 \\%$ mIoU with 5G FLOPs, respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Related Works Recently, researchers have used supernet-based NAS to optimize the architecture for transformers. For example, HAT (Han et al., 2021) uses supernet for hardware-aware transformer optimization. HAT mainly focuses on NLP tasks and features a design space with heterogeneous transformer layers. AutoFormer (Chen et al., 2021a) and ViTAS (Su et al., 2021) leverages supernetbased NAS to optimize the ViT architecture. By searching the width, depth, K/Q/V dimension, MLP ratio, etc, better accuracy is achieved compared to the baseline DeiT models (Chen et al., 2021a). However, these works focus on large ViT models with more than 1G FLOPs and their accuracy is still inferior to the CNN backbones with similar compute, e.g., EfficientNet (Tan & Le, 2019). We refer readers to appendix for more discussions about related works. ",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "2 NAS FOR EFFICIENT TRANSFORMERS ",
+        "text_level": 1,
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+        "text": "Our goal is to design efficient small- and medium-sized ViTs in the FLOPs regime from 200M to 800M. We build our search space inspired by the recently proposed LeViT (Graham et al., 2021). LeViT is a family of efficient models leveraging a hybrid architecture of convolutions and transformers. In LeViT, the convolutions are introduced to handle high resolution inputs thanks to their efficiency from local computation while the transformers are leveraged for lower resolution features to extract global information. We closely follow LeViT to build our search space; see Figure 1 for an overview. ",
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+        "text": "Search Space We summarize the detailed search dimensions of our search space in Table 1. For each CNN block, we directly follow the design in AlphaNet (Wang et al., 2021a;b) and search for the optimal channel widths, block depths, expansion ratios and kernel sizes; for each transformer ",
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+        "img_path": "images/da1a4f953ba5b77a13ec2de1120b220d0b596e2e0020a1f8d55db5b3ee9b11fd.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of our ViT search space. MBConv refers to inverted residual blocks (Sandler et al., 2018). All CNN and transformer blocks contain a stack of dynamic layers with searchable architecture configurations. Additionally, we also search for the input resolutions. "
+        ],
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+        "table_body": "<table><tr><td>Block</td><td>Width</td><td>Depth</td><td></td><td></td><td></td><td></td><td>Kernel size Expansion ratio SE Stride NumberofWindows</td></tr><tr><td>Conv</td><td>{16,24}</td><td>-</td><td>3</td><td>=</td><td>-</td><td>2</td><td></td></tr><tr><td>MBConv-1</td><td>{16,24}</td><td>{1,2}</td><td>{3,5}</td><td>1</td><td>N</td><td>1</td><td></td></tr><tr><td>MBConv-2</td><td>{24,32}</td><td>{3,4,5}</td><td>{3,5}</td><td>{4,5,6}</td><td>N</td><td>2</td><td></td></tr><tr><td>MBConv-3</td><td>{32,40}</td><td>{3,4,5,6}</td><td>{3,5}</td><td>{4,5,6}</td><td>Y</td><td>2</td><td></td></tr><tr><td>Transformer-4</td><td>{64,72}</td><td>{3,4,5,6}</td><td>-</td><td>{1,2}</td><td>-</td><td>2</td><td>1</td></tr><tr><td>Transformer-5</td><td>{112,120,128}</td><td>{3,4,5,6,7,8}</td><td>=</td><td>{1,2}</td><td></td><td>2</td><td>1</td></tr><tr><td>Transformer-6</td><td>{160,168,176,184}</td><td>{3,4,5,6,7,8}</td><td></td><td>{1,2}</td><td></td><td>1</td><td>1</td></tr><tr><td>Transformer-7</td><td>{208,216,224}</td><td>{3,4,5,6}</td><td></td><td>{1,2}</td><td></td><td>2</td><td></td></tr><tr><td>MBPool</td><td>{1792,1984}</td><td>=</td><td>1</td><td>6</td><td></td><td></td><td></td></tr><tr><td>Input resolution</td><td colspan=\"7\">{192,224,256,288}</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: An illustration of our search space. MBConv refers to the inverted residual block (Sandler et al., 2018). MBPool denotes the efficient last stage (Howard et al., 2019). SE represents the squeeze and excite layer (Hu et al., 2018). Transformer stands for the transformer blocks (Vaswani et al., 2017). For MBConv blocks, the expansion ratio refers to the expansion ratio of the depth-wise convolution layer. For transformer layers, it refers to the MLP expansion ratio. For each transformer block, we use $3 \\times 3$ depth-wise convolution with stride 2 for down-sampling and the down-sampling layer is placed as the first layer for that block. ",
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+        "type": "text",
+        "text": "block, we search for the best number of windows, hidden feature dimensions (denoted as Width in Table 1)1, depths and MLP expansion ratios. Compared to CNN blocks, one special search dimension for transformer blocks is the number of windows $k$ . When the number of windows $k$ is greater than 1, we follow Swin transformer (Liu et al., 2021) and partition the input tokens into $k$ groups. We then compute the self-attention weights for each group separately to reduce computational cost. Standard global self-attention is a special case of $k = 1$ . In this work, we only search the number of windows for the first transformer block, as the input resolutions to the other transformer blocks are already small after 4 times of down-sampling. Similar to the search range of AlphaNet, the smallest sub-network in our search space has 190M FLOPs and the largest sub-network has FLOPs of 1,881M. we refer the reader to Appendix B for more description of our search space. ",
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+    {
+        "type": "text",
+        "text": "Naive supernet-based NAS fails to find accurate ViTs We first closely follow the previous best practices in AlphaNet (Wang et al., 2021a) for the supernet training. We train the supernet for 360 epochs on ImageNet (Deng et al., 2009). At each training step, we adopt the sandwich sampling rule (Yu et al., 2018) and sample four sub-networks: the smallest sub-network, the supernet (a.k.a. the largest sub-network), and two random sub-networks. All small sub-networks are supervised by the supernet with $\\alpha$ -divergence-based KD; see Algorithm 1 in Appendix C.1 for an overview of the supernet training procedure. Additionally, as our candidate networks contain transformer blocks, we further incorporate the best training recipe from LeViT (Graham et al., 2021) by replacing the SGD optimizer with Adam (Kingma & Ba, 2014) and leveraging an external pre-trained teacher model for the best accuracy. Specifically, we use the pre-trained teacher to supervise the supernet and still constrain all other small sub-networks to learn from the supernet. In this work, we always use an EfficientNet-B5 (Tan & Le, 2019) with $8 3 . 3 \\%$ top-1 accuracy on ImageNet as the teacher to train our ViT supernet unless otherwise specified. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We plot the training curves of the smallest sub-network and the largest sub-network in Figure 2. We find both the smallest sub-network and the largest sub-network from our search space converge poorly compared to the CNN baseline. Specifically, the validation accuracy of both the smallest and the largest sub-network is saturated at around the 250-th epoch, and the final accuracy is much worse than the CNN baselines. To understand the inferior model performance, we investigate the potential issues of our ViT supernet training from the following three directions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/d583c7b7b70b8e1371eac7ee76bea6ab3d53afe7a4946fa797a8428054d7f638.jpg",
+        "image_caption": [
+            "Figure 2: (a-b) show the training curves of the smallest sub-network and the largest sub-network (i.e., the supernet), respectively. Note that AlphaNet is trained without external teacher models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/f3b7f134a114539d1ff0aec67542cf10278e59d8a3de0b482d7fe89fdafb3e49.jpg",
+        "table_caption": [
+            "Table 2: ImageNet top-1 accuracy from subnetworks trained from scratch vs. results from sub-networks sampled from the supernet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>FLOPs (M)</td><td>190</td><td>208</td><td>309</td><td>591</td></tr><tr><td>Scratch</td><td>77.2</td><td>77.5</td><td>79.1</td><td>80.4</td></tr><tr><td>Supernet</td><td>76.4</td><td>76.6</td><td>78.5</td><td>80.6</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/84b38559dfcbc451de609cede0e8ffeb67a277c15d939a28d894f09ec4a40dfd.jpg",
+        "table_caption": [
+            "Table 3: ImageNet Top-1 accuracy from the smallest and the largest sub-network by using different training recipes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>AlphaNet</td><td>DeiT</td><td>LeViT</td></tr><tr><td>Smallest</td><td>77.0</td><td>76.6</td><td>76.8</td></tr><tr><td>Largest</td><td>82.4</td><td>82.2</td><td>82.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Investigation 1: Is our search space designed badly? We seek to understand if the performance gap is caused by a bad search space design. To verify, we randomly pick four sub-networks from the search space with computation cost ranging from 190M to 591M FLOPs. Then, we train these networks from scratch with the same data augmentation and regularization. As we can see from Table 2, the sub-networks trained from scratch outperform the sub-networks sampled from the supernet. Note that from previous works (e.g. Yu et al., 2020a), supernet often learns more accurate sub-networks compared to the training from scratch performance, by taking advantage of inplace knowledge KD and weight-sharing. Our observations in Table 2 indicate that the poor performance does not come from the search space but from the interference with the training of the supernet. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Investigation 2: Are the training settings suitable for ViTs? Our default training settings from AlphaNet are originally optimized for CNNs only. Compared with AlphaNet, recent ViT methods, e.g., DeiT and LeViT, suggest to use stronger data augmentation schemes (e.g., a combination of CutMix (Yun et al., 2019), Mixup (Zhang et al., 2017), randaugment (Cubuk et al., 2020), random erasing (Zhong et al., 2020), and repeated augmentation) and stronger regularization (e.g., large weight decay, large drop path probability) for training. We evaluate the effectiveness of these ViT specific training recipes and summarize our findings in Table 3. As we can see from Table 3, DeiT- or LeViT-based training recipe produces even worse accuracy compared to the results from AlphaNet-based training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Investigation 3: Saturated supernet training due to gradient conflicts? Compared to the standard single network training, a major difference of supernet training is that multiple networks are sampled and trained at each step. We hypothesize that the training loss from the supernet and that from the sub-networks may yield conflicting gradients due to the heterogeneous and complex structures of networks, and the conflict gradients may consequently lead to slow convergence and undesirable performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To verify this hypothesis, we compute the cosine similarity between the gradients from the supernet and the averaged gradients from the sub-networks. A negative cosine similarity indicates the supernet and sub-networks produce conflict gradients and tend to update model parameters in opposite directions. To quantitatively examine the gradient conflict issue, we go through the entire ImageNet training set and calculate the percentage of negative cosine similarity between the gradients of supernet and sub-networks among all training images at a per layer granularity. The gradients are computed under the same data augmentation and regularization as the supernet training stage. For AlphaNet, we train the model using its official code 2. As shown in Table 4, our ViT supernet suffers from more severe gradient conflicts compared to the CNN baseline. According to existing works in multi-task learning, large gradient conflict ratios may result in significant accuracy drop even for binary classification problems (see Figure 3 in Du et al. (2018) and Figure 4(b) in Yu et al. (2020b)). We hypothesize that the inferior performance of our ViT supernet is mainly caused by the large percentage of disagreements between the supernet gradients and the subnetworks gradients. ",
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+    {
+        "type": "table",
+        "img_path": "images/24bb1c1ab12b647254ecc12bf716d862e1aae05a20990fa35778856d799edd62.jpg",
+        "table_caption": [
+            "Table 4: An estimation of negative cosine similarity ratio (gradient conflict ratio) between the supernet gradient and the averaged gradient of the sub-networks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Epoch</td><td>1st</td><td>90th</td><td>180th</td><td>270th</td><td>360th</td></tr><tr><td>AlphaNet</td><td>27%</td><td>20%</td><td>21%</td><td>24%</td><td>28%</td></tr><tr><td>ViT</td><td>36%</td><td>27%</td><td>27%</td><td>32%</td><td>34%</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "3 GRADIENT CONFLICT AWARE SUPERNET TRAINING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We propose to improve the ViT supernet training by addressing the gradient conflict issue between the supernet and the sub-networks from three aspects: 1) manually resolving the gradient conflict by projecting the supernet gradients to the normal vector of the sub-networks gradients; 2) introducing switchable scaling layers to the search space to give more optimization freedom for sub-networks; 3) reducing data augmentation and regularization to provide easier training signals. ",
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+        "type": "text",
+        "text": "Gradient projection to prioritize sub-networks update Our first idea is to focus on training the sub-networks whenever gradients from the supernet and the gradients from the sub-networks conflicted with each other. As we are interested in the sub-networks from the 200M to 800M FLOPs range, we propose to prioritize the optimization of the sub-networks over the supernet when gradient conflicts are observed. Let $\\nabla \\ell _ { s u p }$ and $\\nabla \\ell _ { s u b }$ denote the gradients of the supernet and the sub-networks, respectively. To prioritize sub-networks training, we always project $\\nabla \\ell _ { s u p }$ to the normal vector of $\\nabla \\ell _ { s u b }$ to avoid gradient conflicts when the cosine similarity between $\\nabla \\ell _ { s u p }$ and $\\nabla \\ell _ { s u b }$ is negative. The overall accumulated gradient at each training iteration with projection can be written as follows, ",
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+    {
+        "type": "image",
+        "img_path": "images/6e1beb18a66bc7af0d7217a75f272d4ba9d9948a9673c971ed1bbba09287b7f6.jpg",
+        "image_caption": [
+            "Figure 3: A basic transformer layer with scaling. Activated components are the neurons selected in the forward path for one sub-network. ‘c1’ and $\\cdot _ { \\mathrm { c 2 } } ,$ represent the number of channels activated in a selfattention layer and MLP, respectively. "
+        ],
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+    {
+        "type": "equation",
+        "img_path": "images/be7b5992e435c04b8538c4b23ccac72496c66cf97504b38ea425c133ec1ac4e8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { g = \\nabla \\ell _ { s u b } + \\mathrm { p r o j } ( \\nabla \\ell _ { s u p } ) \\mathrm { w i t h } } \\\\ & { \\mathrm { p r o j } ( \\nabla \\ell _ { s u p } ) = \\left\\{ \\begin{array} { l l } { \\nabla \\ell _ { s u p } } & { \\mathrm { i f ~ } \\mathrm { c o s } ( \\nabla \\ell _ { s u p } , \\nabla \\ell _ { s u b } ) \\geq 0 , } \\\\ { \\nabla \\ell _ { s u p } - \\frac { \\nabla \\ell _ { s u p } ^ { \\top } \\nabla \\ell _ { s u b } } { \\| \\nabla \\ell _ { s u b } \\| ^ { 2 } } \\nabla \\ell _ { s u b } } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that $\\cos ( \\nabla \\ell _ { s u b } , \\ \\mathrm { p r o j } ( \\nabla \\ell _ { s u p } ) ) = 0$ if $\\cos ( \\nabla \\ell _ { s u p } , \\nabla \\ell _ { s u b } ) < 0$ , which ensures the gradient cosine similarity is non-negative. In sandwich sampling, since several sub-networks are sampled in each iteration, $\\ell _ { s u b }$ is computed as the summation of the gradients from all sub-networks. Note similar ideas of gradient projection have also been explored in multi-tasks learning, see e.g., Yu et al. (2020b); Du et al. (2018); Real et al. (2019); Dery et al. (2021). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While the gradient projection in Eqn. (1) eliminates the gradient conflicts, it may lead to slow convergence as the resulting gradients are biased. Hence, we also propose the follow two techniques to reduce the gradient conflicts from a search space design and training strategy refinement perspective. ",
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+        "type": "text",
+        "text": "Switchable scaling layer Motivated by Slimmable NN (Yu et al., 2018), we introduce additional switchable scaling layers to allow sub-networks with different layer widths and depths to re-scale their features in a privatized way. Specifically, for each transformer layer, a switchable scaling layer is introduced at the output of the self-attention (SA) and the MLP, respectively, as shown in Figure 3. Assume $\\pmb { x } _ { [ c , d ] } \\in \\mathbb { R } ^ { c }$ is a input feature of a scaling layer, with $c$ the feature dimension (i.e. the number of selected channels in the forward path) and $d$ the index of this layer in a transformer search block. The scaling layer transforms ${ \\pmb x } _ { [ c , d ] }$ as ${ \\pmb w } _ { [ c , d ] } \\odot { \\pmb x } _ { [ c , d ] }$ . Here $\\pmb { w } _ { [ c , d ] } \\in \\mathbb { R } ^ { c }$ are learnable parameters and $\\odot$ denotes element-wise multiplication. For each transformer block (see Table 1), each different configuration of $[ c , d ]$ will specify a set of independent switchable scaling layers. Following CaiT (Touvron et al., 2021), we initialize all scaling factors $\\pmb { w }$ to a small value (e.g. $1 { \\dot { 0 } } ^ { - 4 }$ ) for fast convergence and stable training. Intuitively, the switchable scaling layers effectively increase the model capacity of sub-networks and give the sub-networks more optimization flexibility. ",
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+    {
+        "type": "text",
+        "text": "Reduced data augmentation and regularization Furthermore, we observe that the supernet and the sub-networks are more likely to conflict with each other in the presence of stronger data augmentations and stronger regularization, e.g., large weight decay, large DropConnect (Wan et al., 2013). Hence, we simplify the AlphaNet training recipe and use a weaker data augmentation scheme - RandAugment (Cubuk et al., 2019) with both the number of augmentation transformations and the magnitude set to 1, and remove the regularization, e.g. DropConnect (Wan et al., 2013), dropout and weight decay, from the training; see Table 5 for a comparison. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7b9515097054bb610027d4d8bacecc722cc60c148a51f9c5e45f627afe848bcf.jpg",
+        "table_caption": [
+            "Table 5: An illustration of our simplified training settings, where $n$ is the number of augmentation transformations and $m$ the number of magnitudes in RandAugment. A typical setting of RandAugment is $n { = } 2$ and $\\scriptstyle m = 9$ for training a single network; see Cubuk et al. (2020); Liu et al. (2021). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Dataaugmentation</td><td>Weight decay</td><td>DropConnect</td><td>Dropout</td></tr><tr><td>AlphaNet</td><td>AutoAugment</td><td>10-5</td><td>0.2</td><td>0.2</td></tr><tr><td>Ours</td><td>RandAugment (n = 1,m = 1)</td><td>0</td><td>0</td><td>0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first retrain our ViT supernet with our proposed gradient conflict reduction techniques on ImageNet (Deng et al., 2009); we then conduct an evolutionary search on a subset of the ImageNet training dataset to search the accuracy vs. FLOPs Pareto following (Wang et al., 2021b). We refer the reader to Appendix C.1 for more details. Note all the models are directly sampled from the supernet without retraining or finetuning. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/b33ff03bfeab5dd2ef33dfa988ce54ce684b202c094e467af4bd14fb8446902f.jpg",
+        "image_caption": [
+            "Figure 4: Comparison with prior-art CNNs and ViTs on ImageNet. Here $\" + T \"$ indicates methods that are trained with external teacher models. Note that Mobile-Former (Chen et al., 2021c) and MNasNet (Tan et al., 2019) are trained without additional teacher models. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "We call our discovered as NASViT models and compare with state-of-the-art efficient CNNs and ViTs, including FBNetV3 (Dai et al., 2020), AlphaNet (Wang et al., 2021a), LeViT (Graham et al., 2021) and Segformer (Xie et al., 2021), on both image classification (e.g., ImageNet) and semantic segmentation benchmarks (e.g., Cityscapes and ADE20K). ",
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+    {
+        "type": "text",
+        "text": "4.1 IMAGENET ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We compare our NASViT models with state-of-the-art NAS-based CNNs, including AlphaNet (Wang et al., 2021a) and FBNetV3 (Dai et al., 2020), and recently-proposed efficient ViTs, e.g., LeViTs. ",
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+    },
+    {
+        "type": "text",
+        "text": "Settings Note that our ViT supernet is trained with a pretrained Efficient-B5 teacher model $( 8 3 . 3 \\%$ top-1) model. For fair comparison, we retrain AlphaNet with the same teacher. For FBNet-V3 and ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/d75d7ebc28b4fbc0cfbd9ccc7a8da89171bd6b2c33f16b72d3c5fe1ae0e09b1d.jpg",
+        "table_caption": [
+            "Table 6: Comparison with prior art efficient CNNs and ViTs on ImageNet. The reported AlphaNet models are trained with an external teacher model. The “\\*” indicates that the ViTs are trained without external teacher models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Group</td><td>Method</td><td>M FLOPs</td><td>Top-1 accuracy (%)</td></tr><tr><td rowspan=\"2\">200-300 (M)</td><td>AlphaNet-A0</td><td>203</td><td>77.9</td></tr><tr><td>NASViT-A0 (ours)</td><td>208</td><td>78.2</td></tr><tr><td rowspan=\"4\">300-400 (M)</td><td>LeViT (Graham et al.,2021)</td><td>300</td><td>76.6</td></tr><tr><td>NASViT-A1 (ours)</td><td>309</td><td>79.7</td></tr><tr><td>AlphaNet-A2</td><td>317</td><td>79.4</td></tr><tr><td>FBNetV3 (Dai et al.,2020)</td><td>357</td><td>79.6</td></tr><tr><td rowspan=\"3\">400-500 (M)</td><td>LeViT</td><td>406</td><td>78.6</td></tr><tr><td>NASViT-A2 (ours)</td><td>421</td><td>80.5</td></tr><tr><td>AlphaNet-A4</td><td>444</td><td>80.4</td></tr><tr><td rowspan=\"4\">500-600 (M)</td><td>NASViT-A3 (ours) FBNetV3</td><td>528</td><td>81.0</td></tr><tr><td>NASViT-A4 (ours)</td><td>557</td><td>80.8</td></tr><tr><td></td><td>591</td><td>81.4</td></tr><tr><td>AlphaNet</td><td>596</td><td>81.1</td></tr><tr><td rowspan=\"3\">600 - 1000 (M)</td><td>LeViT NASViT-A5 (ours)</td><td>658</td><td>80.0</td></tr><tr><td>FBNetV3</td><td>757</td><td>81.8</td></tr><tr><td>AutoFormer* (Chen et al.,2021a)</td><td>762</td><td>81.5</td></tr><tr><td rowspan=\"8\">&gt;1000 (M)</td><td>PiT-XS (Heo et al.,2021)</td><td>1,300 1,400</td><td>74.7</td></tr><tr><td>ViTAS-D* (Su et al., 2021)</td><td></td><td>79.1</td></tr><tr><td></td><td>1,600</td><td>76.2</td></tr><tr><td>NASViT (supernet) (ours)</td><td>1,881</td><td>82.9</td></tr><tr><td>CVT-13-NAS*(Wu et al., 2021)</td><td>4,100</td><td>82.2</td></tr><tr><td>Swin-Tiny* (Liu et al., 2021)</td><td>4,500</td><td>81.3</td></tr><tr><td>CVT-13* (Wu et al., 2021)</td><td>4,500</td><td>81.6</td></tr><tr><td>T2T-ViT-14* (Yuan et al., 2021a)</td><td>5,200</td><td>81.5</td></tr><tr><td></td><td>Deep ViT (Zhou et al.,2021)</td><td>6,200</td><td>82.3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "LeViT models, these models already use teachers with better performance than Efficient-B5 for training, and therefore we directly report their results following their papers. Specifically, FBNet-V3 use a RegNetY-32G with $8 4 . 5 \\%$ top-1 and LeViT use a RegNetY-16G with $8 3 . 6 \\%$ top-1 as the teacher model, respectively. ",
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+        "type": "text",
+        "text": "Results We summarize our results in both Table 6 and Figure 4. Our discovered NASViT models outperform all evaluated CNN and ViT baselines. Our models are the first models with transformers blocks that outperform state-of-the-art efficient CNNs with similar FLOPs on ImageNet. For example, with $< 6 0 0 \\mathrm { M }$ FLOPs, our NASViT-A4 achieves $8 1 . 4 \\%$ top-1 accuracy on ImageNet. As a reference point, a ResNet-50 model (4G FLOPs) achieves $8 1 . 5 \\%$ top-1 accuracy by distilling from a BiT (Kolesnikov et al., 2020) teacher $( 8 7 . 5 \\%$ top-1 accuracy) with 1200 epochs of training (Beyer et al., 2021). ",
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+        "img_path": "images/90ef698b21efb7472680c03fb25c20841793f890a160dfbeab03dde01b671d48.jpg",
+        "image_caption": [
+            "Figure 5: Results of our method and baselines on semantic segmentation. (a-b) show the results on the Cityscapes and ADE20K validation set, respectively. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.2 SEMANTIC SEGMENTATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate the transfer learning performance of our discovered NASViT models by fine-tuning them on downstream semantic segmentation tasks. In particular, we fine-tune NASViT-A1 to NASViT-A4 as backbones and we show that our NASViT models yield the best segmentation performance compared to the results from the previous efficient CNN backbones, e.g., AlphaNet and MobileNetV3 (Howard et al., 2019), as well as the recently proposed transformer-based Segformer (Xie et al., 2021). ",
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+        "type": "text",
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+        "type": "text",
+        "text": "Settings We evaluate on two benchmark datasets, Cityscapes (Cordts et al., 2016) and ADE20K (Zhou et al., 2017). To handle large input resolutions efficiently, for each transformer block, we set the number of windows to be the input feature map size divided by 7 instead of using our searched settings on ImageNet. We use the recent proposed light-weight Segformer head (Xie et al., 2021) as the decoder head for all the backbones, to achieve a better accuracy efficiency trade-off. ",
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+        "type": "text",
+        "text": "Results As shown in Figure 5, our models yield the best FLOPs vs. mIoU trade-offs. For example, our model achieves $7 6 . 1 \\%$ mIoU and $4 1 . 4 \\%$ mIOU with less than 30G FLOPs on the Cityscapes and the ADE20K validation set, respectively. ",
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+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDIES ON GRADIENT CONFLICT AWARE TRAINING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We conduct ablation studies on ImageNet to have a better understanding on the effectiveness of our proposed methods. We mainly study 1) how our techniques can mitigate the gradient conflict issues and improve the performance 2) whether CNN supernets can also benefit from our techniques. All the comparisons in this section are conducted on ImageNet. ",
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+        "type": "text",
+        "text": "On the effectiveness of our gradient conflict aware training techniques As demonstrated in Table 7, both weak data augmentation and regularization and switchable scaling layer and can significantly reduce the gradient conflict ratios and in the meantime, improve the top-1 accuracy of both the smallest sub-network and the supernet. By further applying gradient projection to prioritize the sub-networks update (denoted by Prioritize (sub)), the performance of both the smallest and largest sub-network is boosted by around $0 . 3 \\%$ on top-1 accuracy. ",
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+        "type": "text",
+        "text": "Prioritizing the supernet update Instead of focusing on training the sub-networks, we retrain our ViT supernet and prioritize the supernet update by moving the proj(·) term in Eqn. (1) to $\\nabla \\ell _ { s u b }$ . As demonstrated in the last column of Table 7, this training strategy (denoted by Prioritize (sup)) leads to a slightly improved supernet while resulting in less competitive performance on the smallest sub-network. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 7: Ablation study results on ImageNet. We show the top-1 validation accuracy of the smallest and largest sub-network, and the negative cosine similarity ratio for each case. Note that switchable scaling layer is applied on top of Weak $D A$ & Reg; and Prioritize (sub) is applied on top of both Weak $D A$ & reg and switchable scaling layer. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Baseline</td><td>Weak DA &amp;Reg</td><td>Switchable scaling</td><td>Prioritize (sub)</td><td>Prioritize (sup)</td></tr><tr><td>Top-1 (smallest)</td><td>76.6</td><td>77.4</td><td>77.6</td><td>78.1</td><td>77.9</td></tr><tr><td>Top-1 (supernet)</td><td>82.2</td><td>82.5</td><td>82.6</td><td>82.9</td><td>83.0</td></tr><tr><td>Negative Cosine Similarity Ratio</td><td>34%</td><td>30%</td><td>29%</td><td>0%</td><td>0%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Improving CNN-based supernets We verify the generalizability of our three techniques to the CNN supernets. In this setting, we applied all three techniques together to improve CNN-based supernets. We show in Table 8 that our method is especially helpful for AlphaNet trained with KL based KD (denoted by AlphaNet $( w / K L ) ,$ ). To further understand the large improvements on AlphaNet $( w / K L )$ , we follow ours studies in Table 4 and compute the gradient conflict ratio for AlphaNet $( w /$ $K L )$ at epoch 1st, 90th, 180th, 270th, and 360th, and the corresponding gradient conflicts ratio is $2 5 \\%$ , $18 \\%$ , $24 \\%$ , $28 \\%$ and $31 \\%$ , respectively. The gradient conflict issue is more severe for AlphaNet $^ Ḋ r Ḍ w Ḍ$ $\\scriptstyle \\alpha - K L ,$ compared with AlphaNet trained with $\\alpha$ -divergence based KD (AlphaNet $( w / \\alpha$ -div)). Our findings indicate that our techniques are not restricted to the ViT supernet training and might be beneficial for all supernets in which a large ratio of gradient conflicts presents. ",
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+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDIES ON SEARCH SPACE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this part, we provide additional ablation studies to support some key design choices of our ViT search space. For all the studies, we use a 250M sub-network that is randomly sub-sampled from our search space for consistency. ",
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+        "table_caption": [
+            "Table 8: Improving CNN-based supernets on ImageNet. AlphaNet (w/ KL) and AlphaNet (w/ $\\alpha$ -div) denote AlphaNets trained with KL and $\\alpha$ -divergence based knowledge distillation, respectively. A0 to A6 are the architectures reported in AlphaNet (Wang et al., 2021a). Note that the AlphaNet supernets here are trained without external teacher models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architectures (M FLOPs)</td><td colspan=\"7\">A0 (203M) A1(279M)A2(317M) A3(357M) A4(444M) A5 (491M) A6 (709M)</td></tr><tr><td>AlphaNet (w/ KL)</td><td>77.0</td><td>78.2</td><td>78.5</td><td>78.8</td><td>79.3</td><td>79.6</td><td>80.1</td></tr><tr><td>AlphaNet (w/ KL) + Ours</td><td>77.5</td><td>78.6</td><td>78.9</td><td>79.2</td><td>79.8</td><td>80.1</td><td>80.7</td></tr><tr><td>AlphaNet (w/ α-div)</td><td>77.8</td><td>78.9</td><td>79.2</td><td>79.4</td><td>80.0</td><td>80.3</td><td>80.8</td></tr><tr><td>AlphaNet (w/ α-div) + Ours</td><td>77.8</td><td>78.9</td><td>79.2</td><td>79.4</td><td>80.0</td><td>80.4</td><td>80.9</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Global attention vs. local attention In our search space (Table 1), we mainly use the global self-attention for the best representation learning capacity. However, with the more computationally efficient local and linear self-attention schemes, we would be able to use a slightly bigger model under similar FLOPs constraints with a sacrifice of global context modeling. To test this trade-off, we train the aforementioned model from scratch with different types of self-attention strategies. Specifically, in addition to the global self-attention, we further evaluate a number of local and linear self-attention mechanisms, including Swin (Liu et al., 2021), CSwin (Dong et al., 2021), VOLO (Yuan et al., 2021b) and LSH (Kitaev et al., 2020). We uniformly scale the width of the transformer blocks to ensure all models have similar compute FLOPs. In Table 9, we show that the standard global attention achieves the best accuracy compared to other faster local and linear self-attention methods. Additionally, from our evolutionary search results, we also notice that the sub-networks with all standard global self-attention layers often yield the best accuracy vs. FLOPs trade-offs. ",
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+    {
+        "type": "table",
+        "img_path": "images/9e65e1effab5058f323566b0875af3f14c198e00341f73326814c3ba53799cb4.jpg",
+        "table_caption": [
+            "Table 9: ImageNet top-1 accuracy with different types of self-attention mechanisms. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Global</td><td>Swin</td><td>CSwin</td><td>VOLO</td><td>LSH</td></tr><tr><td>78.5</td><td>78.0</td><td>77.9</td><td>78.1</td><td>78.0</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The placement of transformer blocks Our ViT supernet has a convolution stem with 3 downsampling strides. We further test the optimal choice of where to switch to transformer blocks. We uniformly scale the width of transformer layers of our baseline model to ensure similar FLOPs for different architecture designs. As shown in Table 10, we notice that our current design, a convolution stem with 3 strides, yields the best performance. ",
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+    {
+        "type": "table",
+        "img_path": "images/c8c3f4ae9a1db165e9643af746159f9043a85302bbb3f18aed2e308d033ff166.jpg",
+        "table_caption": [
+            "Table 10: Ablation studies on where to switch to transformer blocks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>#strides</td><td>2</td><td>3</td><td>4</td></tr><tr><td>Top-1 Accuracy</td><td>78.1</td><td>78.5</td><td>78.4</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 11: Ablation studies on the impact of head dimension. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HeadDimension</td><td>8</td><td>16</td><td>32</td></tr><tr><td>Top-1 Accuracy</td><td>78.5</td><td>78.3</td><td>78.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Head Dimension In previous ViT works (e.g Touvron et al., 2020; 2021; Zhou et al., 2021; Wu et al., 2021; Liu et al., 2021), the feature dimension of each self-attention head is usually set to be 32, 64, or larger. In this work, as shown in Table 11, we found a smaller head feature dimension (e.g., 8) yields better performance. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work, we identify one key issue of ViT supernet training that the supernet gradients and the sub-network gradients are likely to disagree with each other, and consequently leading to inferior NAS performance. We fix this gradient conflict issue by introducing a gradient projection method to prioritize the sub-networks update, designing switchable scaling layers to increase the model capacities of sub-networks and simplifying the training recipe to provide easier training signals. With our improved ViT supernet training techniques, our method finds a family of efficient models, called NASViT, that establishes a new state-of-the-art top-1 accuracy vs. FLOPs trade-offs on ImageNet. Our NASViT models are the first ViT variants that outperform prior-art efficient CNNs on the mobile FLOPs regime. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements Chengyue Gong and Qiang Liu are supported in part by CAREER-1846421, SenSE-2037267, EAGER-2041327, and Office of Navy Research, and NSF AI Institute for Foundations of Machine Learning (IFML). We would like to thank the anonymous reviewers and the area chair for their thoughtful comments and efforts towards improving our manuscript. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    },
+    {
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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1989–1998, 2020.   \nHanxiao Liu, Karen Simonyan, and Yiming Yang. Darts: Differentiable architecture search. arXiv preprint arXiv:1806.09055, 2018.   \nZe Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. Swin transformer: Hierarchical vision transformer using shifted windows. arXiv preprint arXiv:2103.14030, 2021.   \nYiping Lu, Zhuohan Li, Di He, Zhiqing Sun, Bin Dong, Tao Qin, Liwei Wang, and Tie-Yan Liu. Understanding and improving transformer from a multi-particle dynamic system point of view. arXiv preprint arXiv:1906.02762, 2019.   \nHouwen Peng, Hao Du, Hongyuan Yu, Qi Li, Jing Liao, and Jianlong Fu. Cream of the crop: Distilling prioritized paths for one-shot neural architecture search. arXiv preprint arXiv:2010.15821, 2020.   \nHieu Pham, Melody Y Guan, Barret Zoph, Quoc V Le, and Jeff Dean. Efficient neural architecture search via parameter sharing. arXiv preprint arXiv:1802.03268, 2018.   \nYongming Rao, Wenliang Zhao, Benlin Liu, Jiwen Lu, Jie Zhou, and Cho-Jui Hsieh. Dynamicvit: Efficient vision transformers with dynamic token sparsification. arXiv preprint arXiv:2106.02034, 2021.   \nEsteban Real, Sherry Moore, Andrew Selle, Saurabh Saxena, Yutaka Leon Suematsu, Jie Tan, Quoc V Le, and Alexey Kurakin. Large-scale evolution of image classifiers. In International Conference on Machine Learning, pp. 2902–2911. PMLR, 2017.   \nEsteban Real, Alok Aggarwal, Yanping Huang, and Quoc V Le. Regularized evolution for image classifier architecture search. In Proceedings of the aaai conference on artificial intelligence, volume 33, pp. 4780–4789, 2019.   \nGobinda Saha, Isha Garg, and Kaushik Roy. Gradient projection memory for continual learning. arXiv preprint arXiv:2103.09762, 2021.   \nMark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4510–4520, 2018.   \nPeter Shaw, Jakob Uszkoreit, and Ashish Vaswani. Self-attention with relative position representations. arXiv preprint arXiv:1803.02155, 2018.   \nNoam Shazeer, Zhenzhong Lan, Youlong Cheng, Nan Ding, and Le Hou. Talking-heads attention. arXiv preprint arXiv:2003.02436, 2020.   \nDimitrios Stamoulis, Ruizhou Ding, Di Wang, Dimitrios Lymberopoulos, Bodhi Priyantha, Jie Liu, and Diana Marculescu. Single-path nas: Designing hardware-efficient convnets in less than 4 hours. arXiv preprint arXiv:1904.02877, 2019.   \nXiu Su, Shan You, Jiyang Xie, Mingkai Zheng, Fei Wang, Chen Qian, Changshui Zhang, Xiaogang Wang, and Chang Xu. Vision transformer architecture search. arXiv preprint arXiv:2106.13700, 2021.   \nChen Sun, Abhinav Shrivastava, Saurabh Singh, and Abhinav Gupta. Revisiting unreasonable effectiveness of data in deep learning era. In Proceedings of the IEEE international conference on computer vision, pp. 843–852, 2017.   \nMingxing Tan and Quoc Le. Efficientnet: Rethinking model scaling for convolutional neural networks. In International Conference on Machine Learning, pp. 6105–6114. PMLR, 2019.   \nMingxing Tan, Bo Chen, Ruoming Pang, Vijay Vasudevan, Mark Sandler, Andrew Howard, and Quoc V Le. Mnasnet: Platform-aware neural architecture search for mobile. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2820–2828, 2019.   \nHugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Herve´ Jegou. Training data-efficient image transformers & distillation through attention. ´ arXiv preprint arXiv:2012.12877, 2020.   \nHugo Touvron, Matthieu Cord, Alexandre Sablayrolles, Gabriel Synnaeve, and Herve J ´ egou. 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Slimmable neural networks. arXiv preprint arXiv:1812.08928, 2018.   \nJiahui Yu, Pengchong Jin, Hanxiao Liu, Gabriel Bender, Pieter-Jan Kindermans, Mingxing Tan, Thomas Huang, Xiaodan Song, Ruoming Pang, and Quoc Le. Bignas: Scaling up neural architecture search with big single-stage models. In European Conference on Computer Vision, pp. 702–717. Springer, 2020a.   \nTianhe Yu, Saurabh Kumar, Abhishek Gupta, Sergey Levine, Karol Hausman, and Chelsea Finn. Gradient surgery for multi-task learning. arXiv preprint arXiv:2001.06782, 2020b.   \nLi Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zihang Jiang, Francis EH Tay, Jiashi Feng, and Shuicheng Yan. Tokens-to-token vit: Training vision transformers from scratch on imagenet. arXiv preprint arXiv:2101.11986, 2021a.   \nLi Yuan, Qibin Hou, Zihang Jiang, Jiashi Feng, and Shuicheng Yan. Volo: Vision outlooker for visual recognition. arXiv preprint arXiv:2106.13112, 2021b.   \nSangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 6023–6032, 2019.   \nHongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017.   \nZhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang. Random erasing data augmentation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp. 13001–13008, 2020.   \nAimin Zhou, Bo-Yang Qu, Hui Li, Shi-Zheng Zhao, Ponnuthurai Nagaratnam Suganthan, and Qingfu Zhang. Multiobjective evolutionary algorithms: A survey of the state of the art. 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In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8697–8710, 2018. ",
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+        "text": "A RELATED WORK ",
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+        "text": "ViTs ViT (Dosovitskiy et al., 2020) and its follow-ups (Wu et al., 2021; Liu et al., 2021; Zhou et al., 2021; Touvron et al., 2021) have been demonstrated to be an alternative choice to CNNs for challenging vision tasks, especially for image classification. In (Dosovitskiy et al., 2020), a large-scale ViT-Large model is trained on JFT-300 to obtain good performance. The follow-ups mainly focus on making the data size and model size smaller without loss of accuracy. A line of works introduce inductive bias or CNN layers to keep the good performance of ViTs while reducing the data and model sizes (e.g. Liu et al., 2021; Dong et al., 2021; Yuan et al., 2021b; Wu et al., 2021). For smaller data sizes, researchers successfully achieve good performance using ViTs without extra data. For example, VOLO (Yuan et al., 2021b) achieves $8 7 . 3 \\%$ top-1 accuracy on ImageNet without extra data. ",
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+        "text": "For smaller model sizes, however, ViTs have not achieved comparable results to efficient CNNS smaller than 1G FLOPs, even if additional CNN layers are introduced (Graham et al., 2021). Dynamic ViTs (Rao et al., 2021; Chen et al., 2021b) propose to dynamically filter the tokens to reduce the computation cost, and the efficiency is not comparable to efficient CNNs. LeViT (Graham et al., 2021), Xiao et al. (2021) and PiT (Heo et al., 2021) processes the high resolution inputs with early convolution layers or spatial-aware layers and also adopt more efficient self-attention designs. Mobileformer (Chen et al., 2021c) proposes a two-branch neural network: one is efficient CNN layers and the other is transformer layers with a small number of tokens (e.g. 6, 8). ",
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+        "text": "NAS NAS is a powerful tool for automating efficient neural architecture design. It often targets at searching for the best model in a search space under given efficiency-related constraints. Earlier NAS solutions often build on reinforcement learning (e.g. Zoph & Le, 2016; Zoph et al., 2018; Howard et al., 2019) and evolutionary algorithms (e.g. Real et al., 2019; 2017; Wan et al., 2020). More NAS practices have made the search more efficient through weight-sharing and search architectures with gradient-based methods (e.g. Liu et al., 2018; Pham et al., 2018; Stamoulis et al., 2019). This helps alleviate the heavy computational burden of training all candidate networks from scratch and accelerates the NAS process significantly. and researchers work on how to rank the model performance more accurately (Dong & Yang, 2020). Recently, training a large supernet without retraining candidate sub-networks with inplace KD is shown to be an effective mechanism that significantly improves the supernet performance (e.g. Yu et al., 2020a; Wang et al., 2021b;a). In addition to inplace KD, various of KD variants have also been investigated in the literature. For example, Peng et al. (2020) proposes to search a prioritized path as the teacher; Li et al. (2020) proposes to distill the feature level knowledge from an additional teacher model to improve the NAS performance. ",
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+        "text": "NAS for ViT Most recently, several related works, e.g., AutoFormer (Chen et al., 2021a), and ViTAS (Su et al., 2021), have been proposed to search for ViTs. AutoFormer is the first paper that leverages NAS for ViT optimization. A comprehensive search space for the ViT architecture is proposed and a weight-entanglement training strategy is developed to improve the NAS efficiency. ViTAS leverages a similar supernet-based NAS method compared to AutoFormer and introduces private class token and self-attention maps to cater for the variance of distinct ViT architectures. Both work demonstrate promising accuracy improvement compared to the baseline DeiT models for large models with more than 1G FLOPs. ",
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+        "text": "Gradient cosine similarity in multi-mask learning A line of prior approaches have observed that the difficult training with multiple objectives can be improved by using the cosine similarity between gradients (e.g. Du et al., 2018; Yu et al., 2020b; Real et al., 2019). The cosine similarity is used to as a regularization or an indicator. Real et al. (2019) adds a regularization term to force the cosine similarity between two different losses to be larger than zero. Du et al. (2018) and Dery et al. (2021) propose to use gradient cosine similarity to identify whether auxiliary tasks can benefit the main task. In Du et al. (2018), once the cosine similarity is negative (gradient conflict), the weight of the auxiliary task is set to be zero. Yu et al. (2020b) is most related to our projection method, which projects the gradient of every loss to achieve orthogonal gradients. To avoid negative gradient cosine similarity, we project the gradient of the supernet to prioritize the training of sub-networks, which have similar intuition as auxiliary losses. In continual learning, many works uses orthogonal gradient descent to restrict the direction of gradient updates of new tasks in order to avoid catastrophic forgetting (e.g. Farajtabar et al., 2020; Bennani et al., 2020; Saha et al., 2021). ",
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+        "text": "B SEARCH SPACE ",
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+        "img_path": "images/67202428f8dd91712368b350a25205c8163436478dcb77286b52fb694ac0a454.jpg",
+        "image_caption": [
+            "Figure 6: A demonstration of our self-attention module. ‘RPE’, ’Dw Conv, ‘Talking Head’, ‘Proj’ and ‘MLP’ refer to relative positional embedding, depth-wise convolutional layer, talking head attention, projection layer and MLP layer, respectively. "
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+        "text": "B.1 EFFICIENT TRANSFORMER BUILDING LAYER ",
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+        "text": "In this section, we give a detailed introduction about our transformer building layer with self-attention. In the literature, researchers have develop many variants of the standard self-attention with different focuses (e.g. efficiency, convergence, lone-term dependency, etc.). Motivated recent works, e.g., LeVit, SWIN-tranformer and VOLO (Graham et al., 2021; Liu et al., 2021; Yuan et al., 2021b), we develop a transformer layer for the purpose of efficiency and effectiveness in vision tasks. A demonstration of our transformer layer is shown in Figure 6. To enhance the learning capacity of our ViT models, we incorporate talking head (Shazeer et al., 2020) layers and depth-wise convolution layer in the self-attention module. Additionally, following LeViT (Graham et al., 2021), we expand the dimension of V matrix by expansion ratio 4 and introduce activation function after the projection matrix. Following Swin Transformer (Liu et al., 2021), we use relative positional embedding for the attention scores. For efficiency, we reduce the MLP expansion ratio to $\\{ 1 , 2 \\}$ and add one additional MLP layer to keep the model complexity following MacaronNet (Lu et al., 2019). ",
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+        "text": "Positional information The positional embedding in transformer architectures is location-dependent trainable parameters. Recent works propose absolute positional embedding, relative positional embedding or additional depth-wise convolution layers (Dong et al., 2021) to enhance the local information. We introduce two additional depth-wise convolution layers into a MHSA with relative positional embedding. For relative positional embedding, we directly follow the implementation in NLP (Shaw et al., 2018). For depth-wise convolution, we add one depth-wise convolution layer in the MLP layer and another depth-wise convolution layer after the linear transformations of V matrix. ",
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+        "text": "Expansion Ratio In the self-attention design space, researchers have explored whether expanding the channels can have good performance. LeViT proposes to expand the dimension of V. We follow LeViT’s design which expand the dimension of $\\mathrm { v }$ by an expansion ratio 4. researchers have explored how many layers (Lu et al., 2019) should we use for MLP in a self-attention block. We follow the strategy developed by (Lu et al., 2019) which adds one more MLP layer for each self-attention block, but reduce the MLP expansion ratio to $\\{ 1 , 2 \\}$ for efficiency. As displayed in Figure 6, we place an additional MLP layer after the first MLP layer. ",
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+        "text": "Normalization Layers and activation Many recent works apply additional batch normalization layers, layer normalization layers or activation functions to the network. Taking the computation cost of layer normalization layers into consideration, we do not introduce any new normalization layers to the basic self-attention layers. ",
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+        "text": "Talking-head attention and number of heads Most of the existing ViTs set the dimension of each head to be 24/32 (e.g Touvron et al., 2020; 2021; Zhou et al., 2021; Wu et al., 2021; Liu et al., 2021). However, for a model with few channels, a large head dimension leads to few number of heads. We set a smaller head dimension (e.g. 8, 16) to make the number of heads to be large, and further introduce the talking-head attention to improve the capacity of different heads. Talking-head attention (Shazeer et al., 2020) introduce two additional linear transformation between all the heads, one is before softmax and another is after softmax. ",
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+        "text": "Classification head Due to the use of depth-wise convolution layers and down-sampling, we remove the classification token for simplicity. While LeViT and DeiT use two heads for the teacher knowledge distillation and supervised labels, we use one head for all the training settings and replace the onelayer fully-connected layer head with MobilenetV3 (Howard et al., 2019) head so as to reducing computation cost. ",
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+        "text": "Scaling Factor To train very deep transformer models, Touvron et al. (2021) introduces additional learnable channel-wise scaling factors initialized with $1 0 ^ { - 4 }$ into the models. The channel-wise scaling factors are introduced to the output of each MLP and multi-head attention (MHA) layer in the model. Notice that many of sub-networks in our search space are very deep, we introduce architecture-dependent switchable scaling factors into the supernet (see Section 3). ",
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+        "text": "C IMPLEMENTATION DETAILS ON TRAINING AND SEARCHING ",
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+        "text": "C.1 TRAINING AND SEARCH ALGORITHM ",
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+        "text": "Training Consider a supernet with trainable parameter $\\theta$ and the candidate sub-networks set $\\mathcal { A }$ The goal of training a supernet is to learn model parameter $\\theta$ target at optimizing all the sub-networks in $\\mathcal { A }$ and simultaneously achieving good accuracy. Let $s \\sim A$ , $p ( x ; \\theta )$ and $q ( x ; \\theta _ { s } )$ denote the output probability of the supernet and the sub-network $s$ , we have the loss ",
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+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f06341a22fc18cd9b79604538a657deca3b6e4858ab3e83efed23ba25e588230.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { \\ell } ( \\boldsymbol { \\theta } ) + \\sum _ { i = 1 } ^ { k } \\ell _ { \\mathrm { K D } } ( \\theta _ { s _ { i } } , \\theta _ { d e t a c h } ) , \\ w h e r e \\ \\ell _ { \\mathrm { K D } } ( \\theta _ { s _ { i } } , \\theta _ { d e t a c h } ) = \\mathbb { D } \\bigg ( p ( \\boldsymbol { x } ; \\theta _ { d e t a c h } ) \\mid | \\mathbf { \\ell } q ( \\boldsymbol { x } ; \\theta _ { s _ { i } } ) \\bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            526,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\ell ( \\theta )$ denotes the loss of the supernet, $\\theta _ { s _ { i } }$ represents the parameters for a sampled sub-network $s _ { i }$ , and ${ \\dot { \\mathbb { D } } } ( p | | q )$ is a divergence that measures the difference between $p$ and $q$ . The $\\theta _ { d e t a c h }$ denotes a copy of $\\theta$ whose gradient is stopped during back-propagation. ",
+        "bbox": [
+            176,
+            570,
+            823,
+            612
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Algorithm: Supernet based NAS training ",
+        "text_level": 1,
+        "bbox": [
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+            626,
+            532,
+            640
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/32f0bce657699b0bd6631bbc49349dd05a98ea71bab6c11872ee631bf9c65498.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "5: end while "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">while not converged do</td></tr><tr><td>1: 2:</td><td>Sample a mini-batch data from dataset</td></tr><tr><td>3:</td><td>Sample the supernet (i.e., the biggest sub-network) from the search space and train the</td></tr><tr><td>4:</td><td>supernet with with ground truth labels (or with KD from an external teacher model) Sample k random sub-networks from the search space and train them with KD by using the supernet network as the teacher model</td></tr></table>",
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Search After training, a random forest based neural predictor is trained to fit the map from the architecture hyper-parameters to the model performance (e.g., accuracy). A number of sub-networks are sampled from the trained supernet to train the neural predictor. ",
+        "bbox": [
+            174,
+            763,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We then follow the strategy in previous works (e.g. Cai et al., 2019; Wang et al., 2021a) to do evolutionary search: 1) we randomly sample 1024 sub-networks from the supernet and estimate their accuracy on a sub-sampled subset of the ImageNet training set, which is never used during the supernet training; 2) we apply crossover and random mutation (see Zhou et al. (2011) for more details about evolutionary algorithms) on the best performing 128 sub-networks. We fix both the crossover size and mutation size to be 128, yielding 256 new sub-networks. We then evaluate the performance of these sub-networks; 3) We repeat the second step 40 times. The total number of sub-networks thus evaluated is around 10K. ",
+        "bbox": [
+            174,
+            811,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "1: Input: a pretrained supernet with fixed weights ",
+        "text_level": 1,
+        "bbox": [
+            186,
+            121,
+            534,
+            133
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "2: Randomly sample 1024 sub-networks and evaluate their performance on a withhold training set (which is not used during training).   \n3: Partition 1024 sub-networks into training and validation subset with equal size. Train a random forest regressor to predict sub-netowrk’s accuracy given the sub-network architecture hyperparameters as the input.   \n4: Run evolution algorithm to search the Pareto of sub-networks. The sub-network accuracy is given by the random forest based predictor. ",
+        "bbox": [
+            179,
+            135,
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+            233
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C.2 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            262,
+            356,
+            277
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Latency-aware Searching Note that the networks in the paper were optimized for the best FLOPs vs. accuracy trade-off. However, it is expected that the networks that achieve the best FLOPs vs. accuracy trade-off don’t necessary yield the best latency vs. accuracy trade-off in the same time. Therefore, to achieve the best latency vs. accuracy trade-off, we re-search three NASViTs (B0/B1/B2) that form better latency vs. accuracy trade-off compared to the results from NASViTs (A0/A1/A2). For latency comparison, we evaluate the latency of NASViTs-A0/A1/A2 and AlphaNet-A0/A2/A4 on Intel(R) Xeon CPUs with a batch size of 1. See the result in the table below. Here, we do not remove BN or LN layers when estimating latency. ",
+        "bbox": [
+            173,
+            289,
+            826,
+            401
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/8fb732f4d20a9eecf9a971098ae2cf953e803a35e1f574600992cc1e7a13bd85.jpg",
+        "table_caption": [
+            "Algorithm 2 Algorithm: Supernet based NAS searching ",
+            "Table 12: NASViT models searched with for better latency vs. accuracy trade-off. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Accuracy (%)</td><td>CPU latency (ms)</td></tr><tr><td>NASViT-B0</td><td>78.2</td><td>21.0±0.4</td></tr><tr><td>NASViT-A0</td><td>78.2</td><td>21.6±0.5</td></tr><tr><td>AlphaNet-A0</td><td>77.9</td><td>21.4±0.5</td></tr><tr><td>NASViT-B1</td><td>79.6</td><td>26.6±0.6</td></tr><tr><td>NASViT-A1</td><td>79.7</td><td>27.2±0.6</td></tr><tr><td>AlphaNet-A2</td><td>79.4</td><td>27.4±0.5</td></tr><tr><td>NASViT-B2</td><td>80.6</td><td>29.1±0.6</td></tr><tr><td>NASViT-A2</td><td>80.5</td><td>29.8±0.5</td></tr><tr><td>AlphaNet-A4</td><td>80.4</td><td>30.4±0.5</td></tr></table>",
+        "bbox": [
+            316,
+            412,
+            676,
+            556
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "D ARCHITECTURE VISUALIZATION OF NASVIT MODELS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            661,
+            118
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/20eadf1ba4d31f5b686ff8ddc93ccb41a2e97b44f21a880e77169d1e8667b608.jpg",
+        "table_caption": [
+            "Table 13: Here, ‘c’ denotes the number of output channels, ‘d’ denotes number of layers, ‘ks’ denotes kernel size, ‘e’ denotes expansion ratio, $\\mathbf { \\cdot } _ { \\mathbf { k } } ,$ denotes number of windows, ‘s’ denotes stride. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>NASViT-A1</td><td>NASViT-A2</td><td>NASViT-A3</td><td>NASViT-A4</td></tr><tr><td rowspan=\"4\">Conv</td><td>c: 16</td><td>c: 16</td><td>c: 16</td><td>c:16</td></tr><tr><td>d:1</td><td>d:1</td><td>d:1</td><td>d:1</td></tr><tr><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td></tr><tr><td>s: 2</td><td>s: 2</td><td>s: 2</td><td>s:2</td></tr><tr><td rowspan=\"5\">MBConv-1</td><td>c: 16</td><td>c: 16</td><td>c: 16</td><td>c: 16</td></tr><tr><td>d:1</td><td>d:1</td><td>d:1</td><td>d:1</td></tr><tr><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td></tr><tr><td>e:3</td><td>e:3</td><td>e:3</td><td>e:3</td></tr><tr><td>s:1</td><td>s:1</td><td>s:1</td><td>s:1</td></tr><tr><td rowspan=\"5\">MBConv-2</td><td>c: 24</td><td>c: 24</td><td>c: 24</td><td>c: 24</td></tr><tr><td>d:3</td><td>d:3</td><td>d:3</td><td>d:3</td></tr><tr><td>ks: 3</td><td>ks: 3</td><td>ks:3</td><td>ks: 3</td></tr><tr><td>e:4</td><td>e: 4</td><td>e:5</td><td>e:4</td></tr><tr><td>s:2</td><td>s:2</td><td>s:2</td><td>s:2</td></tr><tr><td rowspan=\"5\">MBConv-3</td><td>c: 32</td><td>c: 32</td><td>c: 32</td><td>c: 32</td></tr><tr><td>d: 3</td><td>d:3</td><td>d: 3</td><td>d: 3</td></tr><tr><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td></tr><tr><td>e:4</td><td>e: 6</td><td>e:5</td><td>e:6</td></tr><tr><td>s: 2</td><td>s: 2</td><td>s: 2</td><td>s: 2</td></tr><tr><td rowspan=\"5\">Transformer-4</td><td>c: 64</td><td>c: 64</td><td>c: 64</td><td>c: 64</td></tr><tr><td>d: 4</td><td>d: 4</td><td>d: 4</td><td>d: 4</td></tr><tr><td>k: 1</td><td>k:1</td><td>k:1</td><td>k: 1</td></tr><tr><td>e:1</td><td>e:1</td><td>e:1</td><td>e:1</td></tr><tr><td>s:2</td><td>s: 2</td><td>s: 2</td><td>s: 2</td></tr><tr><td rowspan=\"4\">Transformer-5</td><td>c: 112</td><td>c: 112</td><td>c: 112</td><td>c: 120</td></tr><tr><td>d: 3</td><td>d: 3</td><td>d: 4</td><td>d: 3</td></tr><tr><td>e:1</td><td>e:1</td><td>e:1</td><td>e:1</td></tr><tr><td>s: 2</td><td>s: 2</td><td>s: 2</td><td>s: 2</td></tr><tr><td rowspan=\"4\">Transformer-6</td><td>c: 160</td><td>c: 160</td><td>c: 160</td><td>c: 160</td></tr><tr><td>d: 3</td><td>d: 5</td><td>d: 7</td><td>d: 6</td></tr><tr><td>e:1</td><td>e:1</td><td>e: 1</td><td>e:1</td></tr><tr><td>s: 1</td><td>s:1</td><td>s: 1</td><td>s:1</td></tr><tr><td rowspan=\"4\">Transformer-7</td><td>c: 216</td><td>c: 208</td><td>c: 216</td><td>c: 216</td></tr><tr><td>d: 3</td><td>d: 4</td><td>d: 5</td><td>d: 6</td></tr><tr><td>e:1</td><td>e:1</td><td>e:1</td><td>e: 1</td></tr><tr><td>s:2</td><td>s: 2</td><td>s: 2</td><td>s:2</td></tr><tr><td>MBPool</td><td>c: 1792</td><td>c: 1792</td><td>c: 1984</td><td>c: 1984</td></tr><tr><td>Resolution</td><td>192</td><td>224</td><td>256</td><td>288</td></tr></table>",
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+    }
+]

parse/dev/aPXMGv7aeOn/aPXMGv7aeOn.md

@@ -0,0 +1,272 @@
+# Compressible-composable NeRF via Rank-residual Decomposition
+
+Jiaxiang Tang1, Xiaokang Chen1, Jingbo Wang2, Gang Zeng1,3
+
+1School of Intelligence Science and Technology, Peking University 2Chinese University of Hong Kong 3Intelligent Terminal Key Laboratory of SiChuan Province {tjx, pkucxk}@pku.edu.cn, wj020@ie.cuhk.edu.hk, zeng@pku.edu.cn
+
+# Abstract
+
+Neural Radiance Field (NeRF) has emerged as a compelling method to represent 3D objects and scenes for photo-realistic rendering. However, its implicit representation causes difficulty in manipulating the models like the explicit mesh representation. Several recent advances in NeRF manipulation are usually restricted by a shared renderer network, or suffer from large model size. To circumvent the hurdle, in this paper, we present a neural field representation that enables efficient and convenient manipulation of models. To achieve this goal, we learn a hybrid tensor rank decomposition of the scene without neural networks. Motivated by the low-rank approximation property of the SVD algorithm, we propose a rank-residual learning strategy to encourage the preservation of primary information in lower ranks. The model size can then be dynamically adjusted by rank truncation to control the levels of detail, achieving near-optimal compression without extra optimization. Furthermore, different models can be arbitrarily transformed and composed into one scene by concatenating along the rank dimension. The growth of storage cost can also be mitigated by compressing the unimportant objects in the composed scene. We demonstrate that our method is able to achieve comparable rendering quality to state-of-the-art methods, while enabling extra capability of compression and composition. Code is available at https://github.com/ashawkey/CCNeRF.
+
+# 1 Introduction
+
+Photo-realistic rendering and manipulation of 3D scenes have been long standing problems with numerous real-world applications, such as VR/AR, computer games, and video creation. Recently, the volumetric Neural Radiance Field (NeRF) representations [26, 1, 8, 27] show impressive progress in rendering photo-realistic images with rich details. However, due to this implicit representation of geometry and appearance, manipulating the underlying scenes encoded by NeRF still remains a challenging problem. To solve this problem, some works [21, 45, 19] introduce scene-specific features and scene agnostic rendering network, so that scenes trained with a shared rendering network can be composed together. However, the constrained and biased capability of these rendering networks causes difficulty in extending to various objects or scenes. New objects have to be trained with a fixed rendering network to be compatible with the old objects. Other works [36] discard the rendering network and adopt an no-neural-network NeRF representation, which is more convenient to manipulate the reconstructed scenes and is still able to render high-quality images. Nevertheless, the large storage requirement for each single model is detrimental to composing complex scenes with lots of objects.
+
+We present a novel approach that allows efficient and convenient manipulation of scenes represented with our model. Two aspects should be fulfilled to achieve this goal. The first is that we can dynamically adjust the model size to support different levels of detail (LOD) in different scenarios.
+
+![](images/bafcbe764de067e330aa7da67877ab6d1b420be5d0ac6176fbde4b4a1a8f969c.jpg)  
+Figure 1: Compressibility and Composability of our method. We present a tensor rank decomposition based neural field representation, which supports model compression through rank truncation, and arbitrary composition between different models through rank concatenation. Both of these operations require no extra optimization, or any constraints in training (e.g., a shared renderer).
+
+This functions similar to mipmaps in graphics and requires no extra optimization step. The second is that all models can be transformed and composed arbitrarily for manipulation with no constraints in training. This promises that our models are always reusable, and support the most basic operations in a 3D editor like blender [9]. We name these two properties as compressibility and composability.
+
+For the compressibility, we are motivated by the properties of Singular Value Decomposition (SVD) and High-order SVD (HOSVD) [10]. Our aim is to learn the decomposition of a 3D scene from only 2D observations like TensoRF [8], and further preserve the near-optimal low-rank approximation property. We propose a simple and flexible tensor rank decomposition based neural radiance field, and a rank-residual learning strategy. Each 3D scene is modeled by a 4D feature volume, which can be described with a set of rank components and a matrix storing the weights for each feature channel. The rank components are either vector- or matrix-based, corresponding to the CANDECOMP/PARAFAC (CP) decomposition [14, 4] and a less compact triple plane variant. We introduce a rank-residual learning strategy to encourage the lower ranks to preserve more important information of the whole scene. Combined with an empirical sort-and-truncate strategy, the proposed method achieves nearoptimal low-rank approximation at any targeted rank. Different LODs are represented with different low-rank truncations of the model, allowing dynamic trade-off between model size and rendering quality without retraining. Besides, our model contains no neural networks and thus naturally supports composability. Since there are no MLP renderers in our model, we can compose different objects by simply concatenating their rank components. A transformation matrix is recorded for each object to control its position and orientation in the scene.
+
+As demonstrated in Figure 1, we are able to control each model’s LOD and size in a flexible range, and perform arbitrary transformation and composition of different models. Furthermore, these two properties are connected together through the underlying concept of rank, and can be combined in practical use. For example, we can mitigate the growth of model size of a complex scene composed of multiple objects, by compressing the less important objects. Our contributions can be summarized as follows: (1) We propose a simple radiance field representation based on two types of tensor rank decomposition, which allows flexible control of model size and naturally supports transformation and composition of different models. (2) We design a rank-residual learning strategy to enable near-optimal low-rank approximation. After training, our model can be dynamically adjusted to trade off between performance and model size without retraining. (3) The proposed method reaches comparable rendering quality with state-of-the-arts, while additionally enabling both compressibility and composability.
+
+# 2 Related Work
+
+# 2.1 Scene Representation with NeRF
+
+3D scenes can be represented with various forms, including volumes, point clouds, meshes, and implicit representations [33, 34, 7, 30, 38, 25, 23, 28]. NeRF [26] proposes to use a 5D function to represent the scene and applies volumetric rendering for novel view synthesis, achieving photorealistic results and detailed geometry reconstruction. This powerful representation quickly receives attention and is extensively studied and applied in various fields [50, 24], such as generative settings [6, 37, 29, 5], dynamic scenes [20, 31], and texture mapping [44]. In particular, we categorize recent progress by the design of the underlying functions into three classes: neural network-based, hybrid and no-neural-network. neural network-based representations typically apply an MLP, as the implicit function to encode 3D scenes. The original NeRF [26] and most following works [1, 2, 51, 43, 46, 35] choose this representation for its simplicity. However, the training and inference speed of such a network is generally slow due to the relatively expensive MLP computation. Therefore, hybrid representations try to reduce the size of the MLP, by storing the 3D features in an explicit data structure. Since a dense 3D representation is unaffordable, different methods are explored. For example, NSVF [21] adopts sparse voxel grids, PlenOctrees [49] adopts octrees, instant-ngp [27] adopts a multi-scale hashmap, and TensoRF [8] factorizes the scene into lower-rank components. Querying such hybrid representation is much faster, thus reducing training and inference time and even reaching interactive FPS. Lastly, no-neural-network representations attempt to model the 3D scene without neural networks. Plenoxels [36] shows that only the explicit sparse voxels representation is enough to model complex 3D scenes. Our method also belongs to this representation, sharing the similar tensor rank decomposition idea to TensoRF [8], but we focus on two additional capabilities, i.e., compressibility and composability, which are important yet usually absent in previous work.
+
+# 2.2 Tensor Decomposition and Low-rank Approximation
+
+Decomposition of high-order tensors [18] can be considered as the generalizations of matrix singular value decomposition. The Tucker decomposition [40] decomposes a tensor into a core tensor multiplied by a matrix along each mode. The CANDECOMP/PARAFAC (CP) decomposition [14, 4] factorizes a tensor into a sum of component rank-one tensors, and can be viewed as a special case of Tucker where the core tensor is superdiagonal. The high-order singular value decomposition [10] provides a method to compute a specific Tucker decomposition with an all-orthogonal core tensor. Low-rank approximation is a common problem that applies tensor decomposition, and has found various applications such as image compression. Although the truncated HOSVD does not hold the optimal property contrary to the truncated SVD, it still results in a quasi-optimal solution [10, 41, 12], which is enough to yield a sufficiently good solution in practical uses. Tensor rank decomposition and its variants [10, 11] has been used in various vision and learning tasks [47, 8, 48]. Specifically, TensoRF [8] first leverages the CP decomposition and a Vertex-Matrix (VM) decomposition to factorize neural radiance fields, but its other designs (e.g., use of MLP) disturbs the property of tensor rank decomposition and prevents it from achieving compression or composition. Instead, we focus on modeling neural radiance fields only with tensor rank decomposition, and aim to preserve the low-rank approximation property, enabling the compression of a learned neural radiance field similar to the SVD compression of an image.
+
+# 2.3 Manipulation and Composition of NeRF
+
+Manipulation and Composition are important for a 3D representation’s practical usage. Explicit 3D representations, e.g., meshes, are natively editable and composable. However, neural network-based implicit representations like a vanilla NeRF is difficult to perform such operations. NSVF [21] can composite separate objects together, but these objects have to be trained together using a shared MLP, which limits its flexibility and potential usage. Later works [45, 29, 13, 19] learn object-compositional NeRF, but are usually scene-specific and do not allow cross-scene composition without retraining. Geometry and appearance editing [22, 42] of neural fields also requires an extra optimization step to modify the neural network-based representation. With the explicit sparse voxel representation, Plenoxels [36] naturally supports direct composition of different objects, but suffers from the large storage on the dense index matrix. Our method also supports arbitrary affine transformations and compositions without extra optimization. Further, we can efficiently mitigate the model size growth due to the compact tensor rank decomposition and the compressibility.
+
+![](images/e1b5dcf7a911e3f2e62b1fa494065264baf201f3c861c31390038eea63a4cad2.jpg)  
+Figure 2: Model structure. Our model is composed of a matrix storing rank weights for different feature channels, and a set of decomposed rank components. Each rank component can be either vectoror matrix-based, and the ratio can be controlled to trade off between model size and performance. To query any 3D coordinate, we first project it to the decomposed vectors or matrices as denoted by the black lines, and then perform weighted interpolation. $| |$ denotes concatenation along the rank dimension.
+
+# 3 Methodology
+
+# 3.1 Preliminaries on Neural Radiance Fields
+
+Neural Radiance Fields (NeRF) [26] represents a 3D volumetric scene with a 5D function $f _ { \Theta }$ that maps a 3D coordinate $\mathbf { x } = ( x , y , z )$ and a 2D viewing direction $\mathbf { d } = ( \theta , \phi )$ into a volume density $\sigma$ and an emitted color $\mathbf { c } = ( r , g , b )$ . Given a ray $\mathbf { r }$ originating at $\mathbf { o }$ with direction $\mathbf { d }$ , we query $f _ { \Theta }$ at points $\mathbf x _ { i } = \mathbf o + t _ { i } \mathbf d$ sequentially sampled along the ray to get densities $\{ \sigma _ { i } \}$ and colors $\{ \mathbf { c } _ { i } \}$ . The color of the pixel corresponding to the ray is then estimated by numerical quadrature:
+
+$$
+\hat { \mathbf { C } } ( \mathbf { r } ) = \sum _ { i } T _ { i } \alpha _ { i } \mathbf { c } _ { i } , T _ { i } = \prod _ { j < i } ( 1 - \alpha _ { j } ) , \alpha _ { i } = 1 - \exp ( - \sigma _ { i } \delta _ { i } ) , \delta _ { i } = t _ { i + 1 } - t _ { i }
+$$
+
+where $\delta _ { i }$ is the step size, $\alpha _ { i }$ is the opacity, and $T _ { i }$ is the transmittance. Since this volume rendering process is differentiable, NeRF can be optimized only from 2D image supervision by minimizing the L2 difference between each pixel’s predicted color $\hat { \mathbf { C } } ( \mathbf { r } )$ and the ground truth color $\mathbf { C } ( \mathbf { r } )$ from the image:
+
+$$
+\mathcal { L } _ { \mathrm { N e R F } } = \sum _ { \mathbf { r } } | | \mathbf { C } ( \mathbf { r } ) - \hat { \mathbf { C } } ( \mathbf { r } ) | | _ { 2 } ^ { 2 }
+$$
+
+# 3.2 Preliminaries on Tensor Decomposition
+
+For a 3D tensor $\mathcal { T } \in \mathbb { R } ^ { H \times W \times D }$ , each element $\mathcal { T } _ { i , j , k } \ \in \ \mathbb { R }$ can be represented via the Tucker decomposition [40] by:
+
+$$
+\mathcal { T } _ { i , j , k } = \sum _ { p = 1 } ^ { P } \sum _ { q = 1 } ^ { Q } \sum _ { r = 1 } ^ { R } S _ { p , q , r } \mathbf { U } _ { i , p } ^ { x } \mathbf { U } _ { j , q } ^ { y } \mathbf { U } _ { k , r } ^ { z }
+$$
+
+where $\pmb { S } \in \mathbb { R } ^ { P \times Q \times R }$ is the core tensor, $P , Q , R$ are the number of components along each axis, and ${ \bf U } ^ { x } \in \mathbb { R } ^ { H \times P } , { \bf U } ^ { y } \in \mathbb { R } ^ { W \times Q } , { \bf U } ^ { z } \in \bar { \mathbb { R } } ^ { \bar { D } \times R }$ are the factor matrices. The CP decomposition can be viewed as a special case of Tucker when $P = Q = R$ and $s$ is superdiagonal [18] (i.e., $S _ { i , j , k } \neq 0 \Longleftrightarrow i = j = k )$ ):
+
+$$
+\mathcal { T } _ { i , j , k } = \sum _ { r = 1 } ^ { R } \mathbf { s } _ { r } \mathbf { U } _ { i , r } ^ { x } \mathbf { U } _ { j , r } ^ { y } \mathbf { U } _ { k , r } ^ { z }
+$$
+
+where $\mathbf { s } = \mathrm { d i a g } ( \mathcal { S } ) \in \mathbb { R } ^ { R }$ is the reduced core tensor (or rank weights). Although s is usually absorbed into the factor matrices, we write it out for the convenience of later discussion.
+
+# 3.3 Decompose NeRF without MLP
+
+Hybrid Feature Volume Decomposition. We are interested in multi-feature volumetric encoded as a 4D tensor $\mathcal { T } \in \mathbb { R } ^ { C \times H \times W \times D }$ , where $C$ is the feature dimension (e.g., density, RGB values, or other features), and $( H , W , D )$ is the spatial resolution (usually $C \ll \operatorname* { m i n } ( H , W , D ) )$ . A straightforward way is to perform $C$ independent decompositions for each channel. However, different feature channels such as the RGB values are highly correlated in real 3D scenes. A more compact way is to share the factorized matrices like TensoRF [8], and only use different rank weights s for different channels. Therefore, we propose to represent $\tau$ through the CP decomposition by:
+
+$$
+\mathcal { T } _ { i , j , k } = \mathbf { S } \cdot ( \mathbf { U } _ { i } ^ { x } * \mathbf { U } _ { j } ^ { y } * \mathbf { U } _ { k } ^ { z } )
+$$
+
+where $\mathbf { S } \in \mathbb { R } ^ { C \times R }$ is the matrix of rank weights for $C$ channels, and $^ *$ denotes the Hadamard product. Since the above decomposition relies on rank-one vectors (1D tensors), it may require very high ranks to represent complex 3D scenes, which leads to expensive computation at each location. A less compact but more computation-friendly alternative is to adopt matrices (2D tensors) to factorize the 3D scene. This variant in the form of CP decomposition is given by:
+
+$$
+\mathcal { T } _ { i , j , k } = \mathbf { S } \cdot ( \mathcal { U } _ { i , j } ^ { x , y } * \mathcal { U } _ { j , k } ^ { y , z } * \mathcal { U } _ { i , k } ^ { x , z } )
+$$
+
+where $\mathcal { U } ^ { x , y } \in \mathbb { R } ^ { H \times W \times R } , \mathcal { U } ^ { y , z } \in \mathbb { R } ^ { W \times D \times R } , \mathcal { U } ^ { x , z } \in \mathbb { R } ^ { H \times D \times R }$ are the factorized matrices along three planes, each containing $R$ components. This variant can be comprehended by first slicing and tiling the original 3D space along each axis, and then learning a CP decomposition on $\mathbb { R } ^ { H W \times W D \times H D }$ Although this representation is less compact and takes more storage, recent works [8, 5] have shown that it is able to represent scenes with smaller $R$ and better quality. We denote this variant as the Triple Plane (TP) decomposition. Further, we notice that for each individual rank, the underlying vector- or matrix-based decomposition can be selected independently. Therefore, a hybrid variant (HY) that combines the above CP and TP decomposition is proposed. We can flexibly adjust the ratio of two decompositions by $R = R _ { \mathrm { v e c } } + R _ { \mathrm { m a t } }$ to trade off between model size and performance. The model structure is illustrated in Figure 2.
+
+Learning the Decomposition via Differentiable Rendering. For simplicity, we take the CP decomposition as an example. Our model only consists of four tensors to optimize, i.e., S $, \mathbf { U } ^ { x } , \mathbf { U } ^ { y } , \mathbf { U } ^ { z }$ . To represent scenes through neural radiance fields, we need to learn the volume density $\sigma$ and color c at ach location. As the volu with one feature channel only dependent on the 3D coordinate . However, the color is dependent on $\mathbf { x }$ , we can representth the 3D location $\mathcal { T } _ { i , j , k } ^ { \mathrm { d e n s i t y } } \in \mathbb { R }$ $\mathbf { x }$ and the 2D viewing direction $\mathbf { d }$ , which is a 5D function in total. To represent it within the 3D feature volumes, we adopt the spherical harmonics (SH) functions to approximate the additional 2D viewing directions dependency [49, 36]. In particular, for spherical harmonics of maximum degree $\ell _ { \mathrm { m a x } }$ , it takes $( \ell _ { \mathrm { m a x } } + 1 ) ^ { 2 }$ SH coefficients to model the view-dependent color per channel. We use $\mathcal { T } _ { i , j , k } ^ { \kappa } \in \mathbb { R } ^ { ( \ell _ { \operatorname* { m a x } } + 1 ) ^ { 2 } } , \kappa \in \{ r , g , b \}$ to represent these coefficients. The density and color at each location (coordinate indices are omitted for simplicity) can then be represented with:
+
+$$
+\sigma = \phi ( \mathcal { T } ^ { \mathrm { d e n s i t y } } ) ; c ^ { \kappa } = \psi ( \sum _ { \ell = 0 } ^ { \ell _ { \mathrm { m a x } } } \sum _ { m = - \ell } ^ { \ell } \mathcal { T } _ { \ell , m } ^ { \kappa } Y _ { \ell } ^ { m } ( \mathbf { d } ) ) , \kappa \in \{ r , g , b \}
+$$
+
+where $\phi ( \cdot )$ is the density activation, $\psi ( \cdot )$ is the color activation, $\mathbf { d }$ is the viewing direction, and $Y _ { \ell } ^ { m }$ are the SH functions. Our model can then be optimized through the standard RGB loss in Equation 2.
+
+# 3.4 Rank-residual Learning for Compressibility
+
+The idea of low-rank approximation is to only keep the most important rank components, where the importance of each rank component can be represented by the singular values in the SVD algorithm. Similarly, we can define the importance of each rank component in our decomposition by the rank weights $\mathbf { S }$ averaged on all feature channels, and multiplied with the magnitude of three factor matrices along the rank dimension. However, the CP decomposition doesn’t hold this low-rank approximation property [18]. If we directly sort the rank components by the rank importance and truncate the model for compression, the rendering quality drops sharply compared to the optimal model (i.e., model retrained with the same parameters), as illustrated by the baseline method in Figure 3.
+
+We propose a rank-residual learning strategy to close this performance gap and achieve near-optimal compression results. This strategy aims to simulate SVD’s low-rank approximation property, where the lower rank components contain more information and contribute more to the approximation. Since the rank components in our method can be flexibly adjusted, a direct solution is to train in a progressive way. Suppose the total number of ranks is $R$ , and the number of training stages is $M$ . The total $R$ rank components can be sequentially divided into $M$ non-empty groups, and we denote the accumulated number of ranks for each group by $R _ { m }$ where $m \in \{ 1 , 2 , \cdots , M \}$ . We start from training the first $R _ { 1 }$ rank components, and after its convergence, we fix them and append the next $R _ { 2 } - R _ { 1 }$ rank components to train in a new stage. In the last stage, all $R$ ranks are involved. However, this is inefficient since we need to make sure each stage is fully converged before increasing the number of ranks. A more efficient way is to train all stages in parallel. In particular, we simultaneously supervise the sequentially accumulated outputs from all $M$ groups. This can be viewed as supervising the outputs from $M$ truncations of the decomposition with a rank-residual loss:
+
+![](images/c5cea6b97e70b34d604315f259907174a09e2041fc8106a010ef6da1143187a5.jpg)  
+Figure 3: Compression at any rank. Combined with the empirical sort-and-truncate strategy, the proposed model achieves nearoptimal compression at any rank. We use the HY-S model on the LEGO dataset as an example, and the dashed lines indicate where we apply rank-residual supervision.
+
+![](images/0ecc8acb1ad7e105cb053cdc255af16b1c4713ce9161f1548c4826aafe7fb7d9.jpg)  
+Figure 4: Visualization of rank importance. Ranks are sorted column-wisely based on the averaged rank importance. The rank importance is more concentrated in the proposed method (right) compared to the baseline (left), which is crucial for truncationbased compression. We use the HY model on the LEGO dataset as an example, and the dashed lines indicate where we apply rank-residual supervision.
+
+$$
+\mathcal { L } _ { \mathrm { r e s i d u a l } } = \sum _ { \mathbf { r } \in \mathcal { R } } \sum _ { m = 1 } ^ { M } | | \mathbf { C } ( \mathbf { r } ) - \hat { \mathbf { C } } _ { m } ( \mathbf { r } ) | | _ { 2 } ^ { 2 }
+$$
+
+where $\hat { \mathbf { C } } _ { m } ( \mathbf { r } )$ is RGB values calculated from the truncated decomposition ${ \bf S } _ { m } , { \bf U } _ { m } ^ { x } , { \bf U } _ { m } ^ { y } , { \bf U } _ { m } ^ { z }$ which only keeps the first $R _ { m }$ rank components (taking the CP decomposition as an example). During training, each group is able to learn the residual error from previous groups, and eventually leads to the desired low-rank approximation property. Ideally, choosing $M = R$ groups assures this low-rank approximation property to hold at any targeted rank, but usually computationally unaffordable with a large $R$ . In practice we choose $M \ll R$ , so the property only holds at those dividing ranks $\{ R _ { m } \}$ . For any other rank $R ^ { \prime }$ , assuming $R _ { m } < R ^ { \prime } < R _ { m + 1 }$ , we first keep the fully covered $R _ { m }$ rank components, and then use the empirical sort-and-truncate strategy to select the remaining $R ^ { \prime } - R _ { m }$ ranks in the last group $\{ R _ { m } + 1 , \bar { R } _ { m } + 2 \cdots , R _ { m + 1 } \}$ . By selecting the top rank components with the largest average importance, it is enough to keep the near-optimal low-rank approximation property at any targeted ranks, as shown by the proposed method in Figure 3.
+
+The decomposition model trained with our rank-residual learning allows dynamic adjustment of model size and rendering quality with no extra optimization. In practice, we usually need different LODs to adapt to different cases, such as the texture mipmaps. While the other NeRF representations need to retrain for different LODs and store every model separately, we just train once and get a unified model for all LODs. Given a targeted storage upper bound or performance lower bound, we only need to select the targeted rank and truncate the decomposition with the simple slicing operation.
+
+# 3.5 Composability without Constraints
+
+As illustrated in Figure 5, any 3D object or scene represented by our model can be composed without the constraints in previous work [21, 45]. Since our model describes each 3D scene with a set of rank components, composability is naturally accomplished by concatenating along the rank dimension and summing up the number of ranks $\begin{array} { r } { R = \sum _ { n = 1 } ^ { N } R _ { n } } \end{array}$ , where $n$ denotes the object index. In practice, this is implemented by appending new rank components to a parameter list, so that models with different resolution and decomposition forms can be composed together. For each object, we record its number of ranks so we can still distinguish it from the whole scene. Arbitrary affine transformations of individual objects are supported by recording a transformation matrix $\dot { \mathbf { T } } _ { n } \in \mathbb { R } ^ { 4 \times 4 }$ for each object, which includes the translation $\mathbf { t } _ { n } \in \mathbb { R } ^ { 3 }$ , rotation ${ \bf R } _ { n } \in \mathrm { S O } ( 3 )$ and scale $\mathbf { s } _ { n } \in \mathbb { R } ^ { 3 }$ . We warp the ray into each object’s coordinate system before querying the density $\{ \sigma _ { i , n } \}$ and color $\{ \mathbf { c } _ { i , n } \}$ at the sampled points:
+
+![](images/3fb0d6c3726bc16a007586f8374d2c1c375e3deb063aec6e745877c75f87e36f.jpg)  
+Figure 5: Compressing a scene composed of multiple objects. For a scene composed of lots of different objects, we can compress the less important objects to achieve better efficiency and less storage with a little sacrifice of rendering quality.
+
+$$
+\sigma _ { i , n } , \mathbf { c } _ { i , n } = f _ { \boldsymbol \Theta } ( \mathbf { T } _ { n } \mathbf { x } _ { i } , \mathbf { R } _ { n } \mathbf { d } )
+$$
+
+Here we assume the coordinate $\mathbf { x } _ { i }$ is homogeneous, and the viewing direction $\mathbf { d }$ is represented by a unit vector. To correctly handle the occlusion between different objects, we still sample one ray per pixel, and perform the composition at each sample point by:
+
+$$
+\left\{ \begin{array} { l l } { \alpha _ { i } = 1 - \exp ( - \delta _ { i } \sum _ { n = 1 } ^ { N } \sigma _ { i , n } ) } \\ { \mathbf { c } _ { i } = \sum _ { n = 1 } ^ { N } \varphi _ { N } ( \sigma _ { i , n } ) \mathbf { c } _ { i , n } } \end{array} \right.
+$$
+
+We sum up the density from all objects to calculate opacity, and weight the color by the density after the softmax function $\varphi _ { N }$ . The rendering formula in Equation 1 can then be applied to calculate the pixel color. Although the model size and rendering time grows linearly with the total number of ranks (complexity of the scene), we show in experiments that the compression property can be applied to mitigate the growth and improve efficiency.
+
+# 4 Experiments
+
+# 4.1 Implementation Details
+
+The model is implemented with the PyTorch framework [32]. The degree for SH coefficients is 3, which equals 48 channels for the color feature. We use the Adam optimizer [16] with an initial learning rate of 0.02 for the factorized matrices, and 0.001 for the singular values. All the experiments are performed on one NVIDIA V100 GPU. The resolution of the feature grid is determined by the total number of voxels $N$ and the bounding box, where $N$ is increased from $1 2 8 ^ { 3 }$ to $3 0 0 ^ { 3 }$ for HY models and $5 0 0 ^ { 3 }$ for CP models in early training steps. To accelerate rendering, we adopt the binary occupancy mask pruning technique as in [8, 27], and use separate rank components for density and color to avoid unnecessary querying in empty space. This occupancy mask is also used to shrink the initial bounding box for more precise modeling. We mainly carry out experiments on the NeRF-synthetic dataset [26] (CC BY 3.0 license) and the Tanks and Temples dataset [17] (CC BY-NC-SA 3.0 license). Please check the supplementary materials for more details.
+
+# 4.2 Compression Results
+
+Firstly, We evaluate the compressibility of our model. Since the color components cost most of the total storage, we mainly focus on compressing the color components, and keep the density components fixed. Given the number of ranks $R$ , we first train two models with the original loss
+
+![](images/440fb15bcd764ce251c2e9118f22e9b583b824939d158ed24bfd4361e041371b.jpg)  
+Figure 6: Visualization of compression. The baseline method deteriorates significantly, while our proposed method remains high rendering quality.
+
+![](images/82d225ebc903aee716e3d89e9a9f41b632a5d290ccc78cf2027bb673c04a27e1.jpg)  
+Figure 7: Visualization of composition. We show composition between different models with our method. Our method can successfully handle occlusion and remain high-quality rendering.
+
+Table 1: Compression Results. We report the PSNR for different compression strategies. CP, $R _ { \mathrm { v e c } } ^ { \mathrm { d e n s i t y } } / R _ { \mathrm { m a t } } ^ { \mathrm { d e n s i t y } } { - } R _ { \mathrm { v e c } } ^ { \mathrm { c o l o r } } / R _ { \mathrm { m a t } } ^ { \mathrm { c o l o r } }$ model settings with different range of ranks. Ranks are denoted by. We emphasize how the proposed method improves over the truncate baseline, especially at highly compressed conditions.
+
+<table><tr><td>Model</td><td>Ranks</td><td>Resolution</td><td>Size (MB)</td><td>Optimal</td><td>Baseline</td><td>Proposed</td></tr><tr><td rowspan="4">CP</td><td>96/0-384/0</td><td>500</td><td>4.4</td><td>30.78</td><td>30.78</td><td>30.55 (-0.23)</td></tr><tr><td>96/0-288/0</td><td>500</td><td>3.8</td><td>30.68</td><td>28.78</td><td>30.46 (+1.68)</td></tr><tr><td>96/0-192/0</td><td>500</td><td>3.2</td><td>30.38</td><td>26.95</td><td>30.15 (+3.20)</td></tr><tr><td>96/0-96/0</td><td>500</td><td>2.7</td><td>29.78</td><td>24.97</td><td>29.53 (+4.56)</td></tr><tr><td rowspan="5">HY-S</td><td>96/0-96/64</td><td>300</td><td>68.9</td><td>31.54</td><td>31.54</td><td>31.22 (-0.32)</td></tr><tr><td>96/0-96/32</td><td>300</td><td>35.2</td><td>31.36</td><td>27.57</td><td>31.09 (+3.52)</td></tr><tr><td>96/0-96/16</td><td>300</td><td>18.4</td><td>31.04</td><td>25.66</td><td>30.83 (+5.17)</td></tr><tr><td>96/0-96/4</td><td>300</td><td>5.7</td><td>30.40</td><td>24.24</td><td>30.13 (+5.89)</td></tr><tr><td>96/0-96/0</td><td>300</td><td>1.5</td><td>29.49</td><td>23.54</td><td>29.30 (+5.76)</td></tr><tr><td rowspan="4">HY</td><td>64/16-256/64</td><td>300</td><td>88.0</td><td>32.43</td><td>32.43</td><td>32.36(-0.07)</td></tr><tr><td>64/16-192/48</td><td>300</td><td>70.8</td><td>32.42</td><td>30.63</td><td>32.35 (+1.72)</td></tr><tr><td>64/16-128/32</td><td>300</td><td>53.7</td><td>32.31</td><td>28.50</td><td>32.29 (+4.09)</td></tr><tr><td>64/16-64/16</td><td>300</td><td>36.5</td><td>31.96</td><td>26.30</td><td>31.94 (+5.64)</td></tr></table>
+
+$\mathcal { L } _ { \mathrm { N e R F } }$ in Equation 2 and our rank-residual loss $\mathcal { L } _ { \mathrm { r e s i d u a l } }$ in Equation 8. We denote them as $\mathcal { M } _ { R }$ and ${ \mathcal { M } } _ { R } ^ { \mathrm { o u r s } }$ , respectively. At any targeted rank $r \leq R$ to compress, we design three strategies to verify whether the proposed compression is near-optimal: (1) Retrain a model $\mathcal { M } _ { r }$ with $\mathcal { L } _ { \mathrm { N e R F } }$ at the given rank. This requires a retraining from scratch, and can be viewed as the optimal compression result at the given rank. (2) Sort and truncate the rank of the original model $\mathcal { M } _ { R }$ to ${ \mathcal { M } } _ { r } ^ { \mathrm { b a s e } }$ , which can be viewed as the baseline compression result. (3) Sort and truncate the rank of the proposed model ${ \mathcal { M } } _ { R } ^ { \mathrm { o u r s } }$ to $\mathcal { M } _ { r } ^ { \mathrm { o u r s } }$ . We denote these three settings as ‘Optimal’, ‘Baseline’, and ‘Proposed’ respectively.
+
+The quantitative results are listed in Table 1. We find that the performance of the baseline method degrades significantly compared to the optimal method, whereas our proposed method is comparable to the optimal method at all targeted ranks. The visualization in Figure 6 demonstrates how the baseline model gradually deteriorates compared to the proposed model. Note that at the the right-most column where the baseline model is not compressed and the same as the optimal model, the rendering quality of the proposed model is hard to discern from the optimal model. In Figure 3, we show that even though our model is only supervised at 5 discrete ranks, it can remain good compression quality at any other ranks. Figure 3 provides an explanation for the compressibility of our model. With the rank-residual learning, the rank importance is more concentrated to the lower ranks, which benefits the low-rank approximation.
+
+Table 2: Comparison with recent methods. Our method achieves comparable results while enabling both capability of compression and composition.   
+
+<table><tr><td></td><td></td><td colspan="2">Capability</td><td colspan="2">Synthetic-NeRF</td><td colspan="2">TanksTemples</td></tr><tr><td>Method</td><td>Size (MB)</td><td>Composable</td><td>Compressible</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>SRN [38]</td><td>-</td><td>X</td><td>X</td><td>22.26</td><td>0.846</td><td>24.10</td><td>0.847</td></tr><tr><td>NeRF[26]</td><td>5.0</td><td>×</td><td>X</td><td>31.01</td><td>0.947</td><td>25.78</td><td>0.864</td></tr><tr><td>NSVF [21]</td><td>1</td><td>√</td><td>X</td><td>31.75</td><td>0.953</td><td>28.48</td><td>0.901</td></tr><tr><td>SNeRG[15]</td><td>1771.5</td><td>X</td><td>X</td><td>30.38</td><td>0.950</td><td>1</td><td>-</td></tr><tr><td>PlenOctrees [49]</td><td>1976.3</td><td>√</td><td>X</td><td>31.71</td><td>1</td><td>27.99</td><td>0.917</td></tr><tr><td>Plenoxels [36]</td><td>778.1</td><td>√</td><td>X</td><td>31.71</td><td>-</td><td>27.43</td><td>0.906</td></tr><tr><td>DVGO [39]</td><td>612.1</td><td>X</td><td>X</td><td>31.95</td><td>0.975</td><td>28.41</td><td>0.911</td></tr><tr><td>TensoRF-CP-384 [8]</td><td>3.9</td><td>X</td><td>X</td><td>31.56</td><td>0.949</td><td>27.59</td><td>0.897</td></tr><tr><td>TensoRF-VM-192 [8]</td><td>71.8</td><td>X</td><td>X</td><td>33.14</td><td>0.963</td><td>28.56</td><td>0.920</td></tr><tr><td>Instant-NGP [27]</td><td>63.3</td><td>X</td><td>X</td><td>33.18</td><td>1</td><td>-</td><td>-</td></tr><tr><td>Ours-CP</td><td>4.4</td><td>√</td><td>√</td><td>30.55</td><td>0.935</td><td>27.01</td><td>0.878</td></tr><tr><td>Ours-HY-S</td><td>68.9</td><td>√</td><td>√</td><td>31.22</td><td>0.947</td><td>27.52</td><td>0.900</td></tr><tr><td>Ours-HY</td><td>88.0</td><td>√</td><td>√</td><td>32.37</td><td>0.955</td><td>28.08</td><td>0.913</td></tr></table>
+
+# 4.3 Composition Results
+
+In Figure 7, we demonstrate the composability of our method. Without extra optimization, we are able to perform affine transformation and composition of different models like composing meshes in a 3D editor. Note that our model can correctly handle the occlusion and collision between different objects. In Figure 5, we combine the compression capability of our model in scene composition. In a complex scene composed of lots of objects, we can compress the less important objects to make a trade-off between the model size and rendering quality.
+
+# 4.4 Comparisons and Discussion
+
+Rendering Quality. We compare our method with some recent works in Table 2. We focus on the extra capabilities to facilitate practical applications, but not boosting the rendering quality over the previous state-of-the-arts, since the vanilla NeRF [26] already reaches photo-realistic rendering in most cases. Although the performance of the proposed method is not the best, the simple model design with the extra compressibility and composability is unique and enables various applications.
+
+MLP Renderers. As discussed in [8, 27], the absence of a small MLP renderer network generally leads to worse performance, especially for the specular details. However, these renderers are trained separately and cannot be shared across scenes unless explicitly restricted, which causes inconvenience or limits the potency of composition. We therefore discard the MLP at the cost of a slightly worse rendering quality, but facilitates the composition and compression.
+
+Limitations. Although our method can correctly compose the geometry of multiple objects, we don’t consider the lighting conditions. Our model bakes lighting conditions into the color like the vanilla NeRF, and cannot perform re-lighting to achieve consistent lighting effect after composition. A future direction is to integrate the reflectance models [3, 52]. Besides, we only support modeling bounded objects for now. A background model [51] can be combined to simulate unbounded scenes.
+
+# 5 Conclusion
+
+In this work, we present a novel compressible and composable neural radiance field representation. Our model is designed to be simple and flexible, yet still being effective enough to render photorealistic images. A rank-residual learning strategy is proposed to enable near-optimal low-rank approximation, which allows dynamic adjustment of the model size to support different LODs in different scenarios. All models represented with our method can be arbitrarily transformed and composed together like in a 3D editor. Powered by these properties, we are able to efficiently and conveniently manipulate complex scenes with multiple objects. We believe our method will further facilitate the NeRF-based scene representation in real-world applications.
+
+Acknowledgements. This work is supported by the National Key Research and Development Program of China (2020YFB1708002), National Natural Science Foundation of China (61632003, 61375022, 61403005), Grant SCITLAB-20017 of Intelligent Terminal Key Laboratory of SiChuan Province, Beijing Advanced Innovation Center for Intelligent Robots and Systems (2018IRS11), and PEK-SenseTime Joint Laboratory of Machine Vision.
+
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+# SOLVING INVERSE PROBLEMS IN MEDICAL IMAGING WITH SCORE-BASED GENERATIVE MODELS
+
+Yang $\mathbf { S o n g ^ { * } }$ , Liyue Shen˚, Lei Xing & Stefano Ermon Stanford University {yangsong@cs,liyues@,lei@,ermon@cs}.stanford.edu
+
+# ABSTRACT
+
+Reconstructing medical images from partial measurements is an important inverse problem in Computed Tomography (CT) and Magnetic Resonance Imaging (MRI). Existing solutions based on machine learning typically train a model to directly map measurements to medical images, leveraging a training dataset of paired images and measurements. These measurements are typically synthesized from images using a fixed physical model of the measurement process, which hinders the generalization capability of models to unknown measurement processes. To address this issue, we propose a fully unsupervised technique for inverse problem solving, leveraging the recently introduced score-based generative models. Specifically, we first train a score-based generative model on medical images to capture their prior distribution. Given measurements and a physical model of the measurement process at test time, we introduce a sampling method to reconstruct an image consistent with both the prior and the observed measurements. Our method does not assume a fixed measurement process during training, and can thus be flexibly adapted to different measurement processes at test time. Empirically, we observe comparable or better performance to supervised learning techniques in several medical imaging tasks in CT and MRI, while demonstrating significantly better generalization to unknown measurement processes.
+
+# 1 INTRODUCTION
+
+Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are commonly used imaging tools for medical diagnosis. Reconstructing CT and MRI images from raw measurements (sinograms for CT and $\mathbf { k }$ -spaces for MRI) are well-known inverse problems. Specifically, measurements in CT are given by $\mathbf { X }$ -ray projections of an object from various directions, and measurements in MRI are obtained by inspecting the Fourier spectrum of an object with magnetic fields. However, since obtaining the full sinogram for CT causes excessive ionizing radiation for patients, and measuring the full k-space of MRI is very time-consuming, it has become important to reduce the number of measurements in CT and MRI. In many cases, only partial measurements, such as sparse-view sinograms and downsampled $\mathbf { k }$ -spaces, are available. Due to this loss of information, the inverse problems in CT and MRI are often ill-posed, making image reconstruction especially challenging.
+
+With the rise of machine learning, many methods (Zhu et al., 2018; Mardani et al., 2017; Shen et al., 2019; Würfl et al., 2018; Ghani & Karl, 2018; Wei et al., 2020) have been proposed for medical image reconstruction using a small number of measurements. Most of these methods are supervised learning techniques. They learn to directly map partial measurements to medical images, by training on a large dataset comprising pairs of CT/MRI images and measurements. These measurements need to be synthesized from medical images with a fixed physical model of the measurement process. However, when the measurement process changes, such as using a different number of CT projections or different downsampling ratio of MRI $\mathbf { k }$ -spaces, we have to re-collect the paired dataset with the new measurement process and re-train the model. This prevents models from generalizing effectively to new measurement processes, leading to counter-intuitive instabilities such as more measurements causing worse performance (Antun et al., 2020).
+
+In this work, we sidestep this difficulty completely by proposing unsupervised methods that do not require a paired dataset for training, and therefore are not restricted to a fixed measurement process. Our main idea is to learn the prior distribution of medical images with a generative model in order to infer the lost information due to partial measurements. Specifically, we propose to train a score-based generative model (Song & Ermon, 2019; 2020; Song et al., 2021) on medical images as the data prior, due to its strong performance in image generation (Ho et al., 2020; Dhariwal & Nichol, 2021). Given a trained score-based generative model, we provide a family of sampling algorithms to create image samples that are consistent with the observed measurements and the estimated data prior, leveraging the physical measurement process. Once our model is trained, it can be used to solve any inverse problem within the same image domain, as long as the mapping from images to measurements is linear, which holds for a large number of medical imaging applications.
+
+We evaluate the performance of our method on several tasks in CT and MRI. Empirically, we observe comparable or better performance compared to supervised learning counterparts, even when evaluated with the same measurement process in their training. In addition, we are able to uniformly surpass all baselines when changing the number of measurements, e.g., using a different number of projections in sparse-view CT or changing the k-space downsampling ratio in undersampled MRI. Moreover, we show that by plugging in a different measurement process, we can use a single model to perform both sparse-view CT reconstruction and metal artifact removal for CT imaging with metallic implants. To the best of our knowledge, this is the first time that generative models are reported successful on clinical CT data. Collectively, these empirical results indicate that our method is a competitive alternative to supervised techniques in medical image reconstruction and artifact removal, and has the potential to be a universal tool for solving many inverse problems within the same image domain.
+
+# 2 BACKGROUND
+
+# 2.1 LINEAR INVERSE PROBLEMS
+
+An inverse problem seeks to recover an unknown signal from a set of observed measurements. Specifically, suppose $\mathbf { x } \in \mathbb { R } ^ { n }$ is an unknown signal, and $\mathbf { y } \in \mathbb { R } ^ { m } = A \mathbf { x } + \epsilon$ is a noisy observation given by $m$ linear measurements, where the measurement acquisition process is represented by a linear operator $\pmb { A } \in \mathbb { R } ^ { m \times n }$ , and $\epsilon \in \mathbb { R } ^ { n }$ represents a noise vector. Solving a linear inverse problem amounts to recovering the signal $\mathbf { x }$ from its measurement y. Without further assumptions, the problem is ill-defined when $m < n$ , so we additionally assume that $\mathbf { x }$ is sampled from a prior distribution $p ( \mathbf { x } )$ . In this probabilistic formulation, the measurement and signal are connected through a measurement distribution $p ( \mathbf { y } \mid \mathbf { x } ) = q _ { \epsilon } ( \mathbf { y } - A \mathbf { x } )$ , where $q _ { \epsilon }$ denotes the noise distribution of $\epsilon$ . Given $p ( \mathbf { y } \mid \mathbf { x } )$ and $p ( \mathbf { x } )$ , we can solve the inverse problem by sampling from the posterior distribution $p ( \mathbf { x } \mid \mathbf { y } )$ .
+
+Examples of linear inverse problems in medical imaging include image reconstruction for CT and MRI. In both cases, the signal $\mathbf { x }$ is a medical image. The measurement y in CT is a sinogram formed by $\mathrm { X }$ -ray projections of the image from various angular directions (Buzug, 2011), while the measurement $\mathbf { y }$ in MRI consists of spatial frequencies in the Fourier space of the image (a.k.a. the $\mathbf { k }$ -space in the MRI community) (Vlaardingerbroek & Boer, 2013).
+
+# 2.2 SCORE-BASED GENERATIVE MODELS
+
+When solving inverse problems in medical imaging, we are given an observation $\mathbf { y }$ , the measurement distribution $\bar { p } ( \mathbf { y } \mid \mathbf { x } )$ and aim to sample from the posterior distribution $p ( \mathbf { x } \mid \mathbf { y } )$ . The prior distribution $p ( \mathbf { x } )$ is typically unknown, but we can train generative models on a dataset $\{ \mathbf { x } ^ { ( 1 ) } , \mathbf { x } ^ { ( 2 ) } , \cdot \cdot \cdot , \mathbf { x } ^ { ( N ) } \} \sim$ $p ( \mathbf { x } )$ to estimate this prior distribution. Given an estimate of $p ( \mathbf { x } )$ and the measurement distribution $p ( \mathbf { y } \mid \mathbf { x } )$ , the posterior distribution $p ( \mathbf { x } \mid \mathbf { y } )$ can be determined through Bayes’ rule.
+
+We propose to estimate the prior distribution of medical images using the recently introduced scorebased generative models (Song & Ermon, 2019; Ho et al., 2020; Song et al., 2021), whose iterative sampling procedure makes it especially easy for controllable generation conditioned on an observation y. Specifically, we adopt the formulation of score-based generative models in Song et al. (2021), where we leverage a Markovian diffusion process to progressively perturb data to noise, and then smoothly convert noise to samples of the data distribution by estimating and simulating its time reversal. We provide an illustration of this generative modeling framework in Fig. 1.
+
+![](images/b88e7401b989af752cb00b405dac997e7560fb8f85d563ef8d39e58b11a2057f.jpg)  
+Figure 1: We can smoothly perturb images to noise by following the trajectory of an SDE. By estimating the score function $\nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } )$ with neural networks (called score models), it is possible to approximate the reverse SDE and then solve it to generate image samples from noise.
+
+Perturbation process Suppose the dataset is sampled from an unknown data distribution $p ( \mathbf { x } )$ . We perturb datapoints with a stochastic process over a time horizon $[ 0 , 1 ]$ , governed by a linear stochastic differential equation (SDE) of the following form
+
+$$
+\mathrm { d } \mathbf { x } _ { t } = f ( t ) \mathbf { x } _ { t } \mathrm { d } t + g ( t ) \mathrm { d } \mathbf { w } _ { t } , \qquad t \in [ 0 , 1 ] ,
+$$
+
+where $f : [ 0 , 1 ] \to \mathbb { R }$ , $g : [ 0 , 1 ] \to \mathbb { R }$ , $\{ \mathbf { w } _ { t } \in \mathbb { R } ^ { n } \} _ { t \in [ 0 , 1 ] }$ denotes a standard Wiener process (a.k.a., Brownian motion), and $\{ \mathbf { x } _ { t } \in \mathbb { R } ^ { n } \} _ { t \in [ 0 , 1 ] }$ symbolizes the trajectory of random variables in the stochastic process. We further denote the marginal probability distribution of $\mathbf { x } _ { t }$ as $p _ { t } ( \mathbf { x } )$ , and the transition distribution from $\mathbf { x } _ { \mathrm { 0 } }$ to $\mathbf { x } _ { t }$ as $p _ { 0 t } ( \mathbf { x } _ { t } \mid \mathbf { x } _ { 0 } )$ . By definition, we clearly have $p _ { 0 } ( \mathbf { x } ) \equiv p ( \mathbf { x } )$ . Moreover, the functions $f ( t )$ and $g ( t )$ are specifically chosen such that for any initial distribution $p _ { 0 } ( \mathbf { x } )$ , the distribution at the end of the perturbation process, $p _ { 1 } ( \mathbf { x } )$ , is close to a pre-defined noise distribution $\pi ( \mathbf { x } )$ . In addition, the transition density $p _ { 0 t } ( \mathbf { x } _ { t } \mid \mathbf { \dot { x } } _ { 0 } )$ is always a conditional linear Gaussian distribution, taking the form $p _ { 0 t } ( \mathbf { x } _ { t } \mid \mathbf { x } _ { 0 } ) = \mathcal { N } ( \mathbf { x } _ { t } \mid \alpha ( t ) \mathbf { x } _ { 0 } , \beta ^ { 2 } ( t ) I )$ where $\alpha : [ 0 , 1 ]  \mathbb { R }$ and $\beta : [ 0 , 1 ]  \mathbb { R }$ can be derived analytically from $f ( t )$ and $g ( t )$ (Särkkä & Solin, 2019). Examples of such SDEs include Variance Exploding (VE), Variance Preserving (VP), and subVP SDEs proposed in Song et al. (2021). We found VE SDEs performed the best in our experiments.
+
+Reverse process By reversing the perturbation process in Eq. (1), we can start from a noise sample $\mathbf { x } _ { 1 } \sim p _ { 1 } ( \mathbf { x } )$ and gradually remove the noise therein to obtain a data sample $\mathbf { x } _ { 0 } \sim p _ { 0 } ( \mathbf { x } ) \equiv p ( \mathbf { x } )$ . Crucially, the time reversal of Eq. (1) is given by the following reverse-time SDE (Song et al., 2021)
+
+$$
+\mathrm { d } \mathbf { x } _ { t } = \left[ f ( t ) \mathbf { x } _ { t } - g ( t ) ^ { 2 } \nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } ) \right] \mathrm { d } t + g ( t ) \mathrm { d } \bar { \mathbf { w } } _ { t } , \qquad t \in [ 0 , 1 ] ,
+$$
+
+where $\{ \bar { \mathbf { w } } _ { t } \} _ { t \in [ 0 , 1 ] }$ denotes a standard Wiener process in the reverse-time direction, and $\mathrm { d } t$ represents an infinitesimal negative time step, since the above SDE must be solved backwards from $t = 1$ to $t = 0$ . The quantity $\nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } )$ is known as the score function of $p _ { t } ( \mathbf { x } _ { t } )$ . By the definition of time reversal, the trajectory of the reverse stochastic process given by Eq. (2) is $\{ \mathbf { x } _ { t } \} _ { t \in [ 0 , 1 ] }$ , same as the one from the forward SDE in Eq. (1).
+
+Sampling Given an initial sample from $p _ { 1 } ( \mathbf { x } )$ , as well as scores at each intermediate time step, $\nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } )$ , we can simulate the reverse-time SDE in Eq. (2) to obtain samples from the data distribution $p _ { 0 } ( \mathbf { x } ) \equiv p ( \mathbf { x } )$ . In practice, the initial sample is approximately drawn from $\pi ( \mathbf { x } )$ since $\pi ( \mathbf { x } ) \approx p _ { 1 } ( \mathbf { x } )$ , and the scores are estimated by training a neural network $s _ { \theta } ( \mathbf { x } , t )$ (named the score model) on a dataset $\{ \mathbf { x } ^ { ( 1 ) } , \mathbf { x } ^ { ( 2 ) } , \cdot \cdot \cdot , \mathbf { x } ^ { ( N ) } \} \sim p ( \mathbf { x } )$ with denoising score matching (Vincent, 2011; Song et al., 2021), i.e., solving the following objective
+
+$$
+\theta ^ { * } = \underset { \theta } { \operatorname { a r g m i n } } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathbb { E } _ { t \sim \mathcal { U } [ 0 , 1 ] } \mathbb { E } _ { \mathbf { x } _ { t } ^ { ( i ) } \sim p _ { 0 t } ( \mathbf { x } _ { t } ^ { ( i ) } | \mathbf { x } ^ { ( i ) } ) } \Big [ \left\| s _ { \theta } ( \mathbf { x } _ { t } ^ { ( i ) } , t ) - \nabla _ { \mathbf { x } _ { t } ^ { ( i ) } } \log p _ { 0 t } ( \mathbf { x } _ { t } ^ { ( i ) } | \mathbf { x } ^ { ( i ) } ) \right\| _ { 2 } ^ { 2 } \Big ] ,
+$$
+
+where $\mathcal { U } [ 0 , 1 ]$ denotes a uniform distribution over $[ 0 , 1 ]$ . The theory of denoising score matching ensures that $\bar { s } _ { \theta ^ { * } } ( \mathbf { x } , t ) \approx \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } )$ . After training this score model, we plug it into Eq. (2) and solve the resulting reverse-time SDE
+
+$$
+\mathrm { d } \mathbf { x } _ { t } = \left[ f ( t ) \mathbf { x } _ { t } - g ( t ) ^ { 2 } s _ { \theta ^ { \ast } } ( \mathbf { x } _ { t } , t ) \right] \mathrm { d } t + g ( t ) \mathrm { d } { \bar { \mathbf { w } } } _ { t } , \qquad t \in [ 0 , 1 ] ,
+$$
+
+for sample generation. One sampling method is to use the Euler-Maruyama discretization for solving Eq. (3), as given in Algorithm 1. Other sampling methods include annealed Langevin dynamics (ALD, Song & Ermon, 2019), probability flow ODE solvers (Song et al., 2021), and Predictor-Corrector samplers (Song et al., 2021).
+
+<table><tr><td>Algorithm 1 Unconditional sampling</td><td>Algorithm 2 Inverse problem solving</td></tr><tr><td>Require: N</td><td>Require: N,y, 入</td></tr><tr><td></td><td></td></tr><tr><td>2:fori=N-1to0do 3:t←</td><td>2:fori=N-1 to 0 do 3: t←</td></tr><tr><td>N</td><td>4: N yt ~ pot(yt |y)</td></tr><tr><td></td><td>5: xt ←T-1[λAp-1(△)yt+(1-λ)ATxt +</td></tr><tr><td>4: Xt-△t←xt-f(t)xt△t</td><td>(I-∧)Txt] 6: xt-△t←xt-f(t)xt△t</td></tr><tr><td>5: Xt-△t←Xt-△t+g(t)²sθ*(xt,t)△t</td><td>7: Xt-△t←Xt-△t+g(t)²sθ*(xt,t)△t</td></tr><tr><td>6: z ~ N(0,I)</td><td>8: z ~ N(0,1)</td></tr><tr><td>7: Xt-△t←xt-△t+g(t)√△tz</td><td>Xt-△t←Xt-△t+g(t)√△tz</td></tr><tr><td>8: return Xo</td><td>9: 10: return Xo</td></tr></table>
+
+# 3 SOLVING INVERSE PROBLEMS WITH SCORE-BASED GENERATIVE MODELS
+
+With score-based generative modeling, we can train a score model $s _ { \theta ^ { * } } ( \mathbf { x } , t )$ to generate unconditional samples from the the prior distribution of medical images $p ( \mathbf { x } )$ . To solve inverse problems however, we will need to sample from the posterior $p ( \mathbf { x } \mid \mathbf { y } )$ . This can be accomplished by conditioning the original stochastic process $\{ \mathbf { x } _ { t } \} _ { t \in [ 0 , 1 ] }$ on an observation $\mathbf { y }$ , yielding a conditional stochastic process $\{ \mathbf { x } _ { t } \ | \ \mathbf { y } \} _ { t \in [ 0 , 1 ] }$ . We denote the marginal distribution at $t$ as $p _ { t } ( \mathbf { x } _ { t } \mid \mathbf { y } )$ , and our goal is to sample from $p _ { 0 } ( \mathbf { x } _ { 0 } \mid \mathbf { y } )$ , the same distribution as $p ( \mathbf { x } \mid \mathbf { y } )$ by definition. Much like generating unconditional samples by solving the reverse-time SDE in Eq. (2), we can reverse the conditional stochastic process $\{ \mathbf { x } _ { t } \mid \mathbf { y } \} _ { t \in [ 0 , 1 ] }$ to sample from the posterior distribution $p _ { 0 } ( \mathbf { x } _ { 0 } \mid \mathbf { y } )$ by solving the following conditional reverse-time SDE (Song et al., 2021):
+
+$$
+\mathrm { d } \mathbf { x } _ { t } = \left[ f ( t ) \mathbf { x } _ { t } - g ( t ) ^ { 2 } \nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } \mid \mathbf { y } ) \right] \mathrm { d } t + g ( t ) \mathrm { d } \bar { \mathbf { w } } _ { t } , \qquad t \in [ 0 , 1 ] .
+$$
+
+The conditional score function $\nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } \mid \mathbf { y } )$ is a critical part of Eq. (4), yet it is non-trivial to compute. One solution is to estimate the score function by training a new score model $s _ { \theta ^ { * } } ( \mathbf x _ { t } , \mathbf y , t )$ that explicitly depends on $\mathbf { y }$ (Song et al., 2021; Dhariwal $\&$ Nichol, 2021), such that $s _ { \theta ^ { * } } ( \mathbf x _ { t } , \mathbf y , t ) \approx$ $\nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } \mid \mathbf { y } )$ . However, this requires paired data $\{ ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \} _ { i = 1 } ^ { N }$ for training and has the same drawbacks as supervised learning techniques. We do not consider this approach in this work.
+
+An unsupervised alternative is to approximate the conditional score function with an unconditionallytrained score model $s _ { \theta ^ { * } } ( \mathbf { x } _ { t } , t ) \approx \mathbf { \bar { \nabla } } \varphi _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } )$ and the measurement distribution $p ( \mathbf { y } \mid \mathbf { x } )$ . Many existing works (Song et al., 2021; Kawar et al., 2021; Kadkhodaie & Simoncelli, 2020; Jalal et al., 2021) have implemented this idea in different ways. However, the methods in Kawar et al. (2021) and Kadkhodaie & Simoncelli (2020) both require computing the singular value decomposition (SVD) of $\pmb { A } \in \mathbb { R } ^ { m \times n }$ , which can be difficult for many measurement processes in medical imaging. The method proposed in Jalal et al. (2021) is only designed for a specific sampling method called annealed Langevin dynamics (ALD, Song & Ermon, 2019), which proves to be inferior to more advanced sampling algorithms such as Predictor-Corrector methods (Song et al., 2021).
+
+In what follows, we propose a new conditional sampling approach for inverse problem solving with score-based generative models. Our method is computationally efficient for medical image reconstruction, and is applicable to a large family of iterative sampling methods for score-based generative models. At a high level, we first train an unconditional score model $s _ { \theta ^ { * } } ( \mathbf { x } , t )$ on medical images without assuming any measurement process. Given an observation $\mathbf { y }$ at test time, we form a stochastic process $\{ \mathbf { y } _ { t } \} _ { t \in [ 0 , 1 ] }$ by adding appropriate noise to y. We then discretize the reverse-time SDE in Eq. (3) with existing unconditional samplers for $s _ { \theta ^ { * } } ( \mathbf { x } , t )$ , while incorporating the conditional information from y with a proximal optimization step to generate intermediate samples that are consistent with tytutPr0,1s.
+
+# .1 A CONVENIENT FORM OF THE LINEAR MEASUREMENT PROCESS
+
+Many different measurement processes in medical imaging share same components of computation. For example, sparse-view CT reconstruction and metal artifact removal for CT both involve computing the same Radon transform. Similarly, MRI measurement processes require computing the same spatial Fourier transform regardless of different downsampling ratios. To rigorously characterize this structure of measurement processes, we propose a special formulation of $\pmb { A }$ that is efficient to obtain in medical imaging applications. Without loss of generality, we assume that the linear operator $\pmb { A }$ has full rank, i.e., $\operatorname { r a n k } ( A ) = \operatorname* { m i n } ( n , m ) = m$ . The result below gives the alternative formulation of $\pmb { A }$ :
+
+![](images/346b278b9fdf4594092ef31abf6fa3cf132d1289dadfc04da652c6ace75a75bd.jpg)  
+Figure 2: Linear measurement processes for sparse-view CT (left) and undersampled MRI (right).
+
+Proposition 1. $I f \operatorname { r a n k } ( A ) = m$ , then there exist an invertible matrix $\pmb { T } \in \mathbb { R } ^ { n \times n }$ , and a diagonal matrix $\pmb { \Lambda } \in \{ 0 , 1 \} ^ { n \times n }$ with $\operatorname { t r } ( \mathbf { \mathbf { \boldsymbol { \Lambda } } } ) = m$ , such that $A = \mathcal { P } ( \mathbf { \boldsymbol { \Lambda } } ) \mathbf { \boldsymbol { T } }$ . Here $\mathcal { P } ( \mathbf { A } ) \in \{ 0 , 1 \} ^ { m \times n }$ is an operator that, when multiplied with any vector $\mathbf { \pmb { a } } \in \mathbb { R } ^ { n }$ , reduces its dimensionality to m by removing each $i$ -th element of $\textbf { \em a }$ for $i = 1 , 2 , \cdots , n$ if $\mathbf { \Lambda } \Lambda _ { i i } = 0$ .
+
+We illustrate this decomposition for CT/MRI in Fig. 2. Many measurement processes in medical imaging share the same $_ { \mathbf { T } }$ , even if they correspond to different $\pmb { A }$ . For example, $_ { \mathbf { T } }$ corresponds to the Radon transform and Fourier transform in sparse-view CT and undersampled MRI respectively, regardless of the number of measurements, i.e., CT projections and $\mathbf { k }$ -space downsampling ratios. For both sparse-view CT reconstruction and metal artifact removal for CT images, the operator $_ { \mathbf { T } }$ is the Radon transform (see Fig. 8). Intuitively, $\mathrm { d i a g } ( \pmb { \Lambda } )$ can be viewed as a subsampling mask on the sinogram/k-space, and ${ \mathcal { P } } ( \Lambda )$ subsamples the sinogram $/ \mathrm { k }$ -space into an observation y with a smaller size according to this subsampling mask. In addition, we note that $\pmb { T } ^ { - 1 }$ can be efficiently implemented with the inverse Radon transform or the inverse Fourier transform in CT/MRI applications.
+
+# 3.2 INCORPORATING A GIVEN OBSERVATION INTO AN UNCONDITIONAL SAMPLING PROCESS
+
+In what follows, we show that the decomposition in Proposition 1 provides an efficient way to generate approximate samples from the conditional stochastic process $\{ \mathbf { x } _ { t } \mid \mathbf { y } \} _ { t \in [ 0 , 1 ] }$ with an unconditional score model $s _ { \theta ^ { * } } ( \mathbf { x } , t )$ . The basic idea is to “hijack” the unconditional sampling process of scorebased generative models to incorporate an observed measurement y.
+
+As we have already discussed, it is difficult to directly solve $\{ \mathbf { x } _ { t } \mid \mathbf { y } \} _ { t \in [ 0 , 1 ] }$ for sample generation. To bypass this difficulty, we first consider a related stochastic process that is much easier to sample from. Recall that $p _ { 0 t } ( \dot { \mathbf { x } } _ { t } \mid \mathbf { x } _ { 0 } ) = \mathcal { N } ( \mathbf { x } _ { t } \mid \alpha ( t ) \mathbf { x } _ { 0 } , \beta ^ { 2 } ( t ) \mathbf { I } )$ where $\alpha ( t )$ and $\beta ( t )$ can be derived from $f ( t )$ and $g ( t )$ (Song et al., 2021). Given the unconditional stochastic process $\{ \mathbf { x } _ { t } \} _ { t \in [ 0 , 1 ] }$ , we define $\{ \mathbf { y } _ { t } \} _ { t \in [ 0 , 1 ] }$ , where $\mathbf { y } _ { t } = A \mathbf { x } _ { t } + \alpha ( t ) \mathbf { \epsilon }$ . Unlike $\{ \mathbf { x } _ { t } \ | \ \mathbf { y } \} _ { t \in [ 0 , 1 ] }$ , the conditional stochastic process $\{ \mathbf { y } _ { t } \ | \ \mathbf { y } \} _ { t \in [ 0 , 1 ] }$ is fully tractable. First, we have ${ \bf y } _ { 0 } = \tilde { \bf A } { \bf x } _ { 0 } + \alpha ( 0 ) \epsilon = A { \bf x } _ { 0 } + \epsilon = { \bf y }$ . Since $p _ { 0 t } ( \mathbf { x } _ { t } \mid \mathbf { x } _ { 0 } ) = \mathcal { N } ( \mathbf { x } _ { t } \mid \alpha ( t ) \mathbf { x } _ { 0 } , \beta ^ { 2 } ( t ) \mathbf { I } )$ , we have $\mathbf { x } _ { t } = \alpha ( t ) \mathbf { x } _ { 0 } + \beta ( t ) \mathbf { z }$ , where $\mathbf { z } \in \mathbb { R } ^ { n } \sim \mathcal { N } ( \mathbf { 0 } , I )$ . By definition, $\mathbf { y } _ { t } = A \mathbf { x } _ { t } + \alpha ( t ) \boldsymbol { \epsilon }$ , so we have ${ \bf y } _ { t } = A ( \alpha ( t ) { \bf x } _ { 0 } + \beta ( t ) { \bf z } ) + \alpha ( t ) \epsilon = \alpha ( t ) ( { \bf y } - \epsilon ) +$ $\beta ( t ) A \mathbf { z } + \alpha ( t ) \boldsymbol { \epsilon } = \alpha ( t ) \mathbf { y } + \beta ( t ) A \mathbf { z }$ . Therefore, we can easily generate a sample $\hat { \mathbf { y } } _ { t } \sim p _ { t } ( \mathbf { y } _ { t } \mid \mathbf { y } )$ by first drawing $\mathbf { z } \sim \mathcal { N } ( \mathbf { 0 } , I )$ and then computing $\hat { \mathbf { y } } _ { t } = \alpha ( t ) \mathbf { y } + \beta ( t ) A \mathbf { z }$ .
+
+The key of our approach is to modify any existing iterative sampling algorithm designed for the unconditional stochastic process $\{ \mathbf { x } _ { t } \} _ { t \in [ 0 , 1 ] }$ so that the samples are consistent with $\{ \mathbf { y } _ { t } \mid \mathbf { \bar { y } } \} _ { t \in [ 0 , 1 ] }$ . In general, an iterative sampling process of score-based generative models selects a sequence of time steps $\left\{ 0 = t _ { 0 } < t _ { 1 } < \cdots < t _ { N } = 1 \right\}$ and iterates according to
+
+$$
+\begin{array} { r } { \hat { \mathbf { x } } _ { t _ { i - 1 } } = h \big ( \hat { \mathbf { x } } _ { t _ { i } } , \mathbf { z } _ { i } , s _ { \theta ^ { * } } \big ( \hat { \mathbf { x } } _ { t _ { i } } , t _ { i } \big ) \big ) , \quad i = N , N - 1 , \cdots , 1 , } \end{array}
+$$
+
+where $\hat { \mathbf { x } } _ { t _ { N } } \sim \pi ( \mathbf { x } )$ , $\mathbf { z } _ { i } \sim \mathcal { N } ( \mathbf { 0 } , I )$ , and $\pmb { \theta } ^ { * }$ denotes the parameters in an unconditional score model $s _ { \theta ^ { * } } ( \mathbf { x } , t )$ . Here the iteration function $^ { h }$ takes a noisy sample $\hat { \mathbf { x } } _ { t _ { i } }$ and reduces the noise therein to generate $\hat { \mathbf { x } } _ { t _ { i - 1 } }$ , using the unconditional score model $s _ { \theta ^ { * } } ( \mathbf { x } , t )$ . For example, for the Euler-Maruyama sampler detailed in Algorithm 1, this iteration function is given by
+
+$$
+h ( \hat { \bf x } _ { t _ { i } } , { \bf z } _ { i } , s _ { \theta ^ { * } } ( \hat { \bf x } _ { t _ { i } } , t _ { i } ) ) = \hat { \bf x } _ { t _ { i } } - f ( t _ { i } ) \hat { \bf x } _ { t _ { i } } / N + g ( t _ { i } ) ^ { 2 } s _ { \theta ^ { * } } ( \hat { \bf x } _ { t _ { i } } , t _ { i } ) / N + g ( t _ { i } ) { \bf z } _ { i } / \sqrt { N } .
+$$
+
+Samples obtained by this procedure $\{ \hat { \mathbf { x } } _ { t _ { i } } \} _ { i = 0 } ^ { N }$ constitute an approximation of $\{ \mathbf { x } _ { t } \} _ { t \in [ 0 , 1 ] }$ , where the last sample $\hat { \mathbf { x } } _ { t _ { 0 } }$ can be viewed as an approximate sample from $p _ { 0 } ( \mathbf { x } )$ . Most existing sampling
+
+![](images/c0af5c9ecf9519f112faae328016232d8434e2a0ad75f382a2514566b95319b5.jpg)  
+Figure 3: (Left) An overview of our method for solving inverse problems with score-based generative models. (Right) An illustration about how to combine $\hat { \mathbf { x } } _ { t _ { i } }$ and $\mathbf { y }$ to form $\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime }$ .
+
+methods for score-based generative models are instances of this iterative sampling paradigm, including Algorithm 1, ALD (Song & Ermon, 2019), probability flow ODEs (Song et al., 2021) and PredictorCorrector samplers (Song et al., 2021).
+
+To enforce the constraint implied by $\{ \mathbf { y } _ { t } \mid \mathbf { y } \} _ { t \in [ 0 , 1 ] }$ , we prepend an additional step to the iteration rule in Eq. (5), leading to
+
+$$
+\begin{array} { r } { \hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = k ( \hat { \mathbf { x } } _ { t _ { i } } , \hat { \mathbf { y } } _ { t _ { i } } , \lambda ) \qquad } \\ { \hat { \mathbf { x } } _ { t _ { i - 1 } } = h ( \hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } , \mathbf { z } _ { i } , s _ { \theta ^ { * } } ( \hat { \mathbf { x } } _ { t _ { i } } , t _ { i } ) ) , \quad i = N , N - 1 , \cdots , 1 , } \end{array}
+$$
+
+where $\hat { \mathbf { x } } _ { t _ { N } } \sim \pi ( \mathbf { x } ) , \hat { \mathbf { y } } _ { t _ { i } } \sim p _ { t _ { i } } ( \mathbf { y } _ { t _ { i } } \mid \mathbf { y } )$ , and $0 \leqslant \lambda \leqslant 1$ is a hyper-parameter. We provide an illustration of this process in Fig. 3. The iteration function $\pmb { k } ( \cdot , \hat { \mathbf { y } } _ { t _ { i } } , \lambda ) : \mathbb { R } ^ { n }  \mathbb { R } ^ { n }$ promotes data consistency by solving a proximal optimization step (Nesterov, 2003; Boyd et al., 2004; Hammernik et al., 2021) that simultaneously minimizes the distance between $\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime }$ and $\hat { \mathbf { x } } _ { t _ { i } }$ , and the distance between $\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime }$ and the hyperplane $\left\{ \pmb { x } \in \mathbb { R } ^ { n } \ \lvert \ A \pmb { x } = \hat { \mathbf { y } } _ { t _ { i } } \right\}$ , with a hyperparameter $0 \leqslant \lambda \leqslant 1$ balancing between the two:
+
+$$
+\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = \underset { z \in \mathbb { R } ^ { n } } { \arg \operatorname* { m i n } } \{ \left( 1 - \lambda \right) \| z - \hat { \mathbf { x } } _ { t _ { i } } \| _ { T } ^ { 2 } + \underset { u \in \mathbb { R } ^ { n } } { \operatorname* { m i n } } \lambda \| z - u \| _ { T } ^ { 2 } \} \quad s . t . \quad A u = \hat { \mathbf { y } } _ { t _ { i } } .
+$$
+
+Recall that $A = \mathcal { P } ( \mathbf { \boldsymbol { \Lambda } } ) \mathbf { \boldsymbol { T } }$ according to Proposition 1. In the equation above we choose the norm $\left\| \pmb { a } \right\| _ { T } ^ { 2 } : = \left\| \pmb { T } \pmb { a } \right\| _ { 2 } ^ { 2 }$ to simplify our theoretical analysis. The decomposition in Proposition 1 allows us to derive a closed-form solution to the optimization problem in Eq. (8), as given below:
+
+Theorem 1. The solution of Eq. (8) can be given by
+
+$$
+\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = \pmb { T } ^ { - 1 } [ \lambda \pmb { \Lambda } \mathcal { P } ^ { - 1 } ( \pmb { \Lambda } ) \hat { \mathbf { y } } _ { t _ { i } } + ( 1 - \lambda ) \pmb { \Lambda } \pmb { T } \hat { \mathbf { x } } _ { t _ { i } } + ( \pmb { I } - \pmb { \Lambda } ) \pmb { T } \hat { \mathbf { x } } _ { t _ { i } } ] ,
+$$
+
+where $\mathcal { P } ^ { - 1 } ( \mathbf { \Lambda } ) : \mathbb { R } ^ { m }  \mathbb { R } ^ { n }$ denotes any right inverse of ${ \mathcal { P } } ( \Lambda )$ .
+
+See Fig. 3 for an illustration of the function $\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = k ( \hat { \mathbf { x } } _ { t _ { i } } , \hat { \mathbf { y } } _ { t _ { i } } , \lambda )$ . The right inverse $\mathcal { P } ^ { - 1 } ( \pmb { \Lambda } )$ increases the dimensionality of a vector $\pmb { a } \in \mathbb { R } ^ { m }$ to $n$ by putting its entries on every index $i$ of an $n$ -dimensional vector where $\mathbf { \Lambda } \Lambda _ { i i } = 1$ . Recall that in sparse-view CT or undersampled MRI, $\mathrm { d i a g } ( \pmb { \Lambda } )$ represents a subsampling mask, and ${ \mathcal { P } } ( \Lambda )$ subsamples the full sinogram/k-space to generate the observation y. In this case, $\bar { \mathcal { P } } ^ { - 1 } ( \pmb { \Lambda } )$ pads the observation y so that it has the same size as the full sinogram $/ \mathrm { k }$ -space.
+
+When $\lambda = 0$ , $\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = k ( \hat { \mathbf { x } } _ { t _ { i } } , \hat { \mathbf { y } } _ { t _ { i } } , 0 ) = \hat { \mathbf { x } } _ { t _ { i } }$ completely ignores the constraint $A \hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = \hat { \mathbf { y } } _ { t _ { i } }$ , in which case our sampling method in Eq. (7) performs unconditional generation. On the other hand, when $\lambda = 1 , \hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = k ( \hat { \mathbf { x } } _ { t _ { i } } , \hat { \mathbf { y } } _ { t _ { i } } , 1 )$ satisfies $A \hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = \hat { \mathbf { y } } _ { t _ { i } }$ exactly. When the measurement is noisy, we choose $0 < \lambda < 1$ to allow slackness in the constraint $A \hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = \hat { \mathbf { y } } _ { t _ { i } }$ . The value of $\lambda$ is important for balancing between $\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } \approx \hat { \mathbf { x } } _ { t _ { i } }$ and $A \hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } \approx \hat { \mathbf { y } } _ { t _ { i } }$ . In practice, we use Bayesian optimization to tune this $\lambda$ automatically on a validation dataset. When the measurement process contains no noise, we replace $\hat { \mathbf { x } } _ { t _ { 0 } }$ with $k ( \hat { \mathbf x } _ { t _ { 0 } } , \mathbf y , 1 )$ at the last sampling step to guarantee $\boldsymbol { A } \hat { \mathbf x } _ { t _ { 0 } } = \mathbf y$ .
+
+In summary, our method given in Eq. (7) introduces minimal modifications to an existing iterative sampling method of score-based generative models. For example, we can convert the sampler in Algorithm 1 to an inverse problem solver in Algorithm 2 by adding/modifying just three lines of pseudo-code. Unlike the concurrent work Jalal et al. (2021), our method is not limited to annealed Langevin dynamics (ALD). As demonstrated in our experiments, we outperform Jalal et al. (2021) even with the same ALD sampler, and can widen the performance gap further by using more advanced approaches like the Predictor-Corrector sampler (Song et al., 2021). Unlike Kadkhodaie & Simoncelli (2020); Kawar et al. (2021), we rely on the efficient alternative representation of $\pmb { A }$ given in Section 3.1, and do not require expensive SVD computation.
+
+![](images/9bb6d3714bf650018a582b295662c97782f135a63fdf0565d501ed37743dae90.jpg)  
+Figure 4: Examples of sparse-view CT reconstruction results on LIDC $3 2 0 \times 3 2 0$ (Top row) and LDCT $5 1 2 \times 5 1 2$ (Bottom row), all with 23 projections. You may zoom in to view more details.
+
+# 4 EXPERIMENTS
+
+We aim to answer the following questions in this section: (1) Can we directly compete with best-inclass supervised learning techniques for the same measurement process used in their training, even though our approach is fully unsupervised? (2) Can our method generalize better to new measurement processes? (3) How do we fare against other unsupervised approaches? To study these questions, we experiment on several tasks in medical imaging, including sparse-view CT reconstruction, metal artifact removal (MAR) for CT, and undersampled MRI reconstruction. More experimental details are provided in Appendix B.
+
+Datasets We consider two datasets for CT experiments. The first is the Lung Image Database Consortium (LIDC) image collection dataset (Armato III et al., 2011; Clark et al., 2013) where we slice the original 3D CT volumes to obtain 130304 2D images of resolution $3 2 0 \times 3 2 0$ for training. The second is the Low Dose CT (LDCT) Image and Projection dataset (Moen et al., 2021) that contains CT scans of multiple anatomic sites, including head, chest, and abdomen, from which we generate 47006 2D image slices of resolution $5 1 2 \times 5 1 2$ for training. We simulate CT measurements (sinograms) with a parallel-beam geometry using projection angles equally distributed across 180 degrees. For MAR experiments, we follow Yu et al. (2020) to synthesize metal artifacts. For undersampled MRI experiments, we use the Brain Tumor Segmentation (BraTS) 2021 dataset (Menze et al., 2014; Bakas et al., 2017), where we slice 3D MRI volumes to get 297270 images of resolution $2 4 0 \times 2 4 0$ as the training dataset. We simulate MRI measurements with Fast Fourier Transform using a single-coil setup, and follow Zbontar et al. (2018); Knoll et al. (2020) to undersample the $\mathbf { k }$ -space with an equispaced Cartesian mask. The performance is measured on 1000 test images with peak signal-to-noise ratio (PSNR) and structural similarity (SSIM).
+
+Standard techniques in medical imaging We include two standard learning-free techniques as baselines for sparse-view CT reconstruction. The first is filtered back projection on sparse-view sinograms, which is denoted by “FBP”. The second is an iterative reconstruction method with total variation regularization called FISTA-TV (Beck & Teboulle, 2009). For MAR experiments, we include another learning-free baseline called linear interpolation (LI, Kalender et al., 1987).
+
+Table 1: Results for undersampled MRI reconstruction on BraTS. First two methods are supervised learning techniques trained with $8 \times$ acceleration. The others are unsupervised techniques.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">24×Acceleration</td><td colspan="2">8× Acceleration</td><td colspan="2">4× Acceleration</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>Cascade DenseNet</td><td>23.39±2.17</td><td>0.765±0.042</td><td>28.35±2.30</td><td>0.845±0.038</td><td>30.97±2.33</td><td>0.902±0.028</td></tr><tr><td>DuDoRNet</td><td>18.46±3.05</td><td>0.662±0.093</td><td>37.88±3.03</td><td>0.985±0.007</td><td>30.53±4.13</td><td>0.891±0.071</td></tr><tr><td>Score SDE</td><td>27.83±2.73</td><td>0.849±0.038</td><td>35.04±2.11</td><td>0.943±0.016</td><td>37.55±2.08</td><td>0.960±0.013</td></tr><tr><td>Langevin</td><td>28.80±3.21</td><td>0.873±0.039</td><td>36.44±2.28</td><td>0.952±0.016</td><td>38.76±2.32</td><td>0.966±0.012</td></tr><tr><td>Ours</td><td>29.42±3.03</td><td>0.880±0.035</td><td>37.63±2.70</td><td>0.958±0.015</td><td>39.91±2.67</td><td>0.965±0.013</td></tr></table>
+
+Table 2: Results for sparse-view CT reconstruction on LIDC and LDCT. FISTA-TV is a standard iterative reconstruction method that does not need training. cGAN, Neumann, and SIN- $_ { \mathrm { 4 c } }$ -PRN are supervised learning techniques trained with 23 projection angles.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Projections</td><td colspan="2">LIDC 320× 320</td><td colspan="2">LDCT 512 × 512</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>FBP</td><td>23</td><td>10.18±1.38</td><td>0.230±0.072</td><td>10.11±1.19</td><td>0.302±0.078</td></tr><tr><td>FISTA-TV</td><td>23</td><td>20.08±4.89</td><td>0.799±0.061</td><td>21.88±4.42</td><td>0.850±0.067</td></tr><tr><td>cGAN</td><td>23</td><td>19.83±3.07</td><td>0.479±0.103</td><td>19.90±2.52</td><td>0.545±0.065</td></tr><tr><td>Neumann</td><td>23</td><td>17.18±3.79</td><td>0.454±0.128</td><td>18.83±3.29</td><td>0.525±0.073</td></tr><tr><td>SIN-4c-PRN</td><td>23</td><td>30.48±3.99</td><td>0.895±0.047</td><td>34.82±3.55</td><td>0.877±0.116</td></tr><tr><td rowspan="3"> Ours</td><td>10</td><td>29.52±2.63</td><td>0.823±0.061</td><td>28.96±4.41</td><td>0.849±0.086</td></tr><tr><td>20</td><td>34.40±2.66</td><td>0.895±0.048</td><td>36.80±4.50</td><td>0.936±0.058</td></tr><tr><td>23</td><td>35.24±2.71</td><td>0.905±0.046</td><td> 37.41±4.62</td><td>0.941±0.057</td></tr></table>
+
+Supervised learning baselines For sparse-view CT on both LIDC and LDCT, we include cGAN (Ghani & Karl, 2018), Neumann (Gilton et al., 2019), and SIN- $_ \mathrm { 4 c }$ -PRN (Wei et al., 2020) as supervised learning baselines. We follow the settings in Wei et al. (2020) and train all methods with 23 projection angles. For MAR, we use cGANMAR (Wang et al., 2018) and SNMAR (Yu et al., 2020) as the baselines. For undersampled MRI on BraTS, we compare against Cascade DenseNet (Zheng et al., 2019) and DuDoRNet (Zhou & Zhou, 2020), which are both trained with a $8 \times$ acceleration factor by measuring only $1 / 8$ of the full $\mathbf { k }$ -space.
+
+Unsupervised learning baselines For unsupervised techniques, so far only score-based generative models have witnessed success on clinic data. We compare with several existing methods that apply score-based generative models to inverse problem solving. Specifically, we consider the “Langevin” approach proposed in Jalal et al. (2021), and the “Score SDE” method in Song et al. (2021), where the former is limited to annealed Langevin dynamics (ALD) sampling, and the latter was based on a crude approximation to the conditional score function $\nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } \mid \mathbf { y } )$ in Eq. (4), and was proposed as a theoretical possibility in Appendix I.4 of Song et al. (2021) without experiments. We only focus on undersampled MRI for these baselines, since it is the only medical imaging problem ever tackled with score-based generative models before our work. All methods share the same score models and only differ in terms of inference. We make sure all sampling algorithms have comparable number of iteration steps ( $N$ in Eqs. (5) and (7)).
+
+Competing with supervised learning approaches Thanks to the outstanding sample quality of score-based generative models, we can achieve comparable or better performance than best-in-class supervised learning methods even for the same measurement process used in their training. As shown in Table 2, we outperform the top supervised learning technique SIN- $_ { \mathrm { 4 c } }$ -PRN on sparse-view CT reconstruction by a significant margin, on both the LIDC and LDCT datasets. Our results with
+
+Table 3: MAR results on LIDC.   
+
+<table><tr><td>Method</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>LI</td><td>26.30±2.62</td><td>0.910±0.028</td></tr><tr><td>cGANMAR</td><td>27.27±1.96</td><td>0.927±0.060</td></tr><tr><td>SNMAR</td><td>27.28±1.43</td><td>0.937±0.048</td></tr><tr><td>Ours</td><td> 32.16±2.32</td><td>0.939±0.022</td></tr></table>
+
+20 measurements are even better than supervised learning counterparts with 23 measurements. In Fig. 4, we provide a visual comparison of the reconstruction quality for various methods, where it is clear to see that our method can recover more details faithfully. From results in Table 3, we also outperform the top supervised learning method SNMAR on metal artifact removal. As shown in Fig. 7, our method generates images with less artifacts and preserves the structure better. For undersampled MRI reconstruction results given in Tables 1 and 3, our method is ranked the 2nd for the case of $8 \times$ acceleration, with comparable performance to the top supervised method DuDoRNet.
+
+![](images/70209982aeff030c78d4431e06a6f3bef473450317e7fa79ef05888d70dd21da.jpg)  
+Figure 5: Performance vs. numbers of measurements. Shaded areas represent standard deviation. (Left) MRI on BraTS. (Center) CT on LIDC. (Right) Comparing score-based generative models for undersampled MRI reconstruction on BraTS.
+
+Generalizing to different number of measurements Since our approach is fully unsupervised, we can naturally apply the same score model to different measurement processes. We first consider changing the number of measurements at the test time, e.g., using different number of projection angles (resp. different acceleration factors) for sparse-view CT (resp. undersampled MRI) reconstruction. As shown in Table 1 and Fig. 5 (Left), we achieve the best performance on undersampled MRI for both $2 4 \times$ and $4 \times$ acceleration factors, whereas DuDoRNet fails to generalize when the acceleration factor changes. The other supervised learning approach Cascade DenseNet demonstrates limited adaptability by building a model architecture inspired by the physical measurement process of MRI, but fails to yield top-level performance. For sparse-view CT reconstruction, all supervised learning methods struggle to generalize to different projection angles, as shown in Fig. 5 (Center).
+
+Generalizing to different measurement processes in CT We can perform both sparse-view CT reconstruction and metal artifact removal (MAR) with a single score model trained on CT images. These two tasks are inverse problems in CT imaging with different measurement processes $\pmb { A }$ , but they share the same $\mathbf { T }$ in the decomposition of Proposition 1. We provide a visualization of the measurement process corresponding to MAR in Fig. 8. As shown in Table 3, we can outperform supervised learning techniques specifically designed and trained for MAR, while using the same score model used in sparse-view CT reconstruction on LIDC.
+
+Comparing against existing score-based methods We compare our method against Langevin (Jalal et al., 2021) and Score SDE (Song et al., 2021) for undersampled MRI reconstruction on BraTS. Two variants of our approach are considered, which respectively use annealed Langevin dynamics (ALD) and the Predictor-Corrector (PC) sampler for score-based generative models as the backend. We denote the former by $\mathrm { \ddot { \ s u } L D + O u r s { \vec { \nu } } }$ , and the latter by “PC $^ +$ Ours” (our default method for all other experiments). Recall that Langevin uses ALD as the sampler, same as $\mathrm { ^ { 6 6 } A L D + O u r s ^ { 3 7 } }$ . All results are provided in Fig. 5 (Right). We observe that “ALD $^ +$ Ours” uniformly outperform Langevin and Score SDE across all numbers of measurements in the experiment. Moreover, ${ } ^ { \mathrm { s } } \mathrm { P C } +$ Ours” can further improve “ALD $^ +$ Ours”, demonstrating the power of switching to more advanced sampling methods of score-based generative models in our proposed approach.
+
+# 5 CONCLUSION
+
+We propose a new method to solve linear inverse problems with score-based generative models. Our method is fully unsupervised, requires no paired data for training, can flexibly adapt to different measurement processes at test time, and only requires minimal modifications to a large number of existing sampling methods of score-based generative models. Empirical results demonstrate that our method can match or outperform existing supervised learning counterparts on image reconstruction for sparse-view CT and undersampled MRI, and has better generalization to new measurement processes, such as using a different number of projections or downsampling ratios in CT/MRI, and tackling both sparse-view CT reconstruction and metal artifact removal with a single model.
+
+# AUTHOR CONTRIBUTIONS
+
+Yang Song designed the project, wrote the paper, and ran all experiments for score-based generative models. Liyue Shen preprocessed data, ran all baseline experiments, and helped write the paper. Lei Xing and Stefano Ermon supervised the project, provided valuable feedback, and helped edit the paper.
+
+# ACKNOWLEDGMENTS
+
+YS is supported by the Apple PhD Fellowship in AI/ML. LS is supported by the Stanford Bio-X Graduate Student Fellowship. This research was supported by NSF (#1651565, #1522054, #1733686), ONR (N000141912145), AFOSR (FA95501910024), ARO (W911NF-21-1-0125), Sloan Fellowship, and Google TPU Research Cloud. This research was also supported by NIH/NCI (1R01 CA256890 and 1R01 CA227713).
+
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+
+# A PROOFS
+
+Proposition 1. $I f \operatorname { r a n k } ( A ) = m$ , then there exist an invertible matrix $\pmb { T } \in \mathbb { R } ^ { n \times n }$ , and a diagonal matrix $\pmb { \Lambda } \in \{ 0 , 1 \} ^ { n \times n }$ with $\operatorname { t r } ( \mathbf { \mathbf { \boldsymbol { \Lambda } } } ) = m$ , such that $A = \mathcal { P } ( \mathbf { \boldsymbol { \Lambda } } ) \mathbf { \boldsymbol { T } }$ . Here $\mathcal { P } ( \mathbf { A } ) \in \{ 0 , 1 \} ^ { m \times n }$ is an operator that, when multiplied with any vector $\mathbf { \pmb { a } } \in \mathbb { R } ^ { n }$ , reduces its dimensionality to m by removing each $i$ -th element of $\textbf { \em a }$ for $i = 1 , 2 , \cdots , n$ if $\mathbf { \Lambda } \Lambda _ { i i } = 0$ .
+
+Proof. Let $\textbf { \textit { A } } = \mathbf { \beta } \left( \pmb { a } _ { 1 } ^ { \mathsf { T } } , \pmb { a } _ { 2 } ^ { \mathsf { T } } , \cdot \cdot \cdot , \pmb { a } _ { m } ^ { \mathsf { T } } \right) \ \in \ \mathbb { R } ^ { m \times n }$ . Since $\pmb { A }$ has full rank, the row vectors $\{ a _ { 1 } , a _ { 2 } , \cdots , a _ { m } \}$ are linearly independent. We can therefore extend them to a total of $n$ linearly independent vectors, i.e., $\{ a _ { 1 } , \dotsc , a _ { 2 } , \dotsc , a _ { m } , b _ { 1 } , \dotsc , b _ { n - m } \}$ . Due to the linear independence, we know $\pmb { T } = ( \pmb { a } _ { 1 } ^ { \top } , \pmb { a } _ { 2 } ^ { \top } , \cdot \cdot \cdot , \pmb { a } _ { m } ^ { \top } , \pmb { b } _ { 1 } ^ { \top } , \cdot \cdot \cdot , \pmb { b } _ { n - m } ^ { \top } ) \in \mathbb { R } ^ { n \times n }$ has full rank and is invertible. Next, we define
+
+$$
+\pmb { \Lambda } = \mathrm { d i a g } ( \underbrace { 1 , 1 , \cdots , 1 } _ { m } , \underbrace { 0 , 0 , \cdots , 0 } _ { n - m } ) ,
+$$
+
+where diag converts a vector to a diagonal matrix. Clearly $\operatorname { t r } ( \mathbf { \mathbf { \boldsymbol { \Lambda } } } ) = m$ and $A = \mathcal { P } ( \mathbf { \boldsymbol { \Lambda } } ) \mathbf { \boldsymbol { T } }$ , which completes the proof. □
+
+Lemma 1. Let $\mathcal { P } ^ { - 1 } ( \pmb { \Lambda } ) : \mathbb { R } ^ { m }  \mathbb { R } ^ { n }$ be any right inverse of $\mathcal { P } ( \mathbf { A } ) : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ . For any $\pmb { u } \in \mathbb { R } ^ { n }$ and $\hat { \mathbf { y } } _ { t } \in \mathbb { R } ^ { m }$ , we have
+
+$$
+\mathcal { P } ( \Lambda ) \pmb { T } \pmb { u } = \hat { \mathbf { y } } _ { t } \iff \Lambda \pmb { T } \pmb { u } = \Lambda \mathcal { P } ^ { - 1 } ( \pmb { \Lambda } ) \hat { \mathbf { y } } _ { t }
+$$
+
+Proof. By the definition of ${ \mathcal { P } } ( \Lambda )$ , we have $\mathcal { P } ( \mathbf { \boldsymbol { \Lambda } } ) = \mathcal { P } ( \mathbf { \boldsymbol { \Lambda } } ) \mathbf { \boldsymbol { \Lambda } }$ , and
+
+$$
+\forall a \in \mathbb { R } ^ { n } , b \in \mathbb { R } ^ { n } : \quad \mathcal { P } ( \Lambda ) a = \mathcal { P } ( \Lambda ) b \iff \Lambda a = \Lambda b .
+$$
+
+To prove the “if” direction, we note that
+
+$$
+\begin{array} { r l } & { \Lambda T u = \Lambda \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t } \implies \mathcal { P } ( \Lambda ) \Lambda T u = \mathcal { P } ( \Lambda ) \Lambda \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t } } \\ & { \implies \mathcal { P } ( \Lambda ) T u = \mathcal { P } ( \Lambda ) \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t } } \\ & { \implies \mathcal { P } ( \Lambda ) T u = \hat { \mathbf { y } } _ { t } . } \end{array}
+$$
+
+To prove the “only if” direction, we have
+
+$$
+\begin{array} { r } { \begin{array} { r } { \mathcal { P } ( \Lambda ) \pmb { T } \pmb { u } = \hat { \mathbf { y } } _ { t } \implies \mathcal { P } ( \Lambda ) \pmb { T } \pmb { u } = \mathcal { P } ( \Lambda ) \mathcal { P } ^ { - 1 } ( \pmb { \Lambda } ) \hat { \mathbf { y } } _ { t } } \\ { \overset { ( i ) } { \implies } \pmb { \Lambda } \pmb { T } \pmb { u } = \pmb { \Lambda } \mathcal { P } ^ { - 1 } ( \pmb { \Lambda } ) \hat { \mathbf { y } } _ { t } , } \end{array} } \end{array}
+$$
+
+where (i) is due to the property in Eq. (10). This completes the proof for both directions.
+
+Theorem 1. The solution of Eq. (8) can be given by
+
+$$
+\hat { \mathbf { x } } _ { t _ { i } } ^ { \prime } = \pmb { T } ^ { - 1 } [ \lambda \pmb { \Lambda } \mathcal { P } ^ { - 1 } ( \pmb { \Lambda } ) \hat { \mathbf { y } } _ { t _ { i } } + ( 1 - \lambda ) \pmb { \Lambda } \pmb { T } \hat { \mathbf { x } } _ { t _ { i } } + ( \pmb { I } - \pmb { \Lambda } ) \pmb { T } \hat { \mathbf { x } } _ { t _ { i } } ] ,
+$$
+
+where $\mathcal { P } ^ { - 1 } ( \mathbf { \Lambda } ) : \mathbb { R } ^ { m }  \mathbb { R } ^ { n }$ denotes any right inverse of ${ \mathcal { P } } ( \Lambda )$ .
+
+Proof. The optimization objective function in Eq. (8) can be written as
+
+$$
+\begin{array} { r l } & { \quad ( 1 - \lambda ) \left\| z - \hat { \mathbf { x } } _ { t } \right\| _ { T } ^ { 2 } + \lambda \left\| z - u \right\| _ { T } ^ { 2 } } \\ & { = ( 1 - \lambda ) \left\| T z - T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| T z - T u \right\| _ { 2 } ^ { 2 } } \\ & { = ( 1 - \lambda ) \left\| T z - T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| \Lambda T ( z - u ) + ( I - \Lambda ) T ( z - u ) \right\| _ { 2 } ^ { 2 } } \\ & { = ( 1 - \lambda ) \left\| T z - T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| \Lambda T ( z - u ) \right\| _ { 2 } ^ { 2 } + \lambda \left\| ( I - \Lambda ) T ( z - u ) \right\| _ { 2 } ^ { 2 } } \\ & { = ( 1 - \lambda ) \left\| T z - T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| \Lambda T z - \Lambda \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| ( I - \Lambda ) T ( z - u ) \right\| _ { 2 } ^ { 2 } } \end{array}
+$$
+
+![](images/9a8115b2f58451217268eeeb35a5d6f7410b77885fd91f449da92be6afcbfdae.jpg)  
+Figure 6: SSIM vs. numbers of measurements. Shaded areas represent standard deviation. (Left) MRI on BraTS. (Center) CT on LIDC. (Right) Comparing score-based generative models for undersampled MRI reconstruction on BraTS.
+
+![](images/2bdfae454ea0dd7e1f8188a61e71a34848c1b6f31521152d564565d1d6621544.jpg)  
+Figure 7: Examples of metal artifact removal on LIDC. You may zoom in to view more details.
+
+Since $\mathbf { \nabla } A \mathbf { u } = \hat { \mathbf { y } } _ { t }$ , we have $\mathcal { P } ( \mathbf { A } ) \pmb { T } \pmb { u } = \hat { \mathbf { y } } _ { t }$ and equivalently $\Lambda T u = \Lambda \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t }$ due to Lemma 1. This constraint does not restrict the value of $( I - \Lambda ) T u$ . Therefore, when $\mathbf { \nabla } A \mathbf { u } = \hat { \mathbf { y } } _ { t }$ , we have
+
+$$
+\begin{array} { r l } & { \quad \left\| z - \dot { \mathbf { x } } _ { t } \right\| _ { T } ^ { 2 } + \operatorname* { m i n } ( 1 - \lambda ) \lambda \left\| z - u \right\| _ { T } ^ { 2 } } \\ & { = ( 1 - \lambda ) \left\| T z - T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } + \operatorname* { m i n } \lambda \left\| \Lambda T z - \Lambda \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| ( I - \Lambda ) T ( z - u ) \right\| _ { 2 } ^ { 2 } } \\ & { - ( 1 - \lambda ) \left\| T z - T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| \Lambda T z - \Lambda \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t } \right\| _ { 2 } ^ { 2 } } \\ & { = ( 1 - \lambda ) \left\| \Lambda T z - \Lambda T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| \Lambda T z - \Lambda \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t } \right\| _ { 2 } ^ { 2 } + ( 1 - \lambda ) \left\| ( I - \Lambda ) T z - ( I - \Lambda ) T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } } \end{array}
+$$
+
+This simplifies the optimization problem in Eq. (8) to
+
+$\operatorname* { n i n } _ { z } ( 1 - \lambda ) \left\| \Lambda T z - \Lambda T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } + \lambda \left\| \Lambda T z - \Lambda \mathcal { P } ^ { - 1 } ( \Lambda ) \hat { \mathbf { y } } _ { t } \right\| _ { 2 } ^ { 2 } + ( 1 - \lambda ) \left\| ( I - \Lambda ) T z - ( I - \Lambda ) T \hat { \mathbf { x } } _ { t } \right\| _ { 2 } ^ { 2 } ,$ k 22 , which is minimizing a quadratic function of $_ z$ . The optimal solution $z ^ { * }$ is thus in closed form:
+
+$$
+\begin{array} { r } { z ^ { * } = \pmb { T } ^ { - 1 } [ ( \pmb { I } - \pmb { \Lambda } ) \pmb { T } \hat { \mathbf { x } } _ { t } + ( 1 - \lambda ) \pmb { \Lambda } \pmb { T } \hat { \mathbf { x } } _ { t } + \lambda \pmb { \Lambda } \pmb { \mathcal { P } } ^ { - 1 } ( \pmb { \Lambda } ) \hat { \mathbf { y } } _ { t } ] . } \end{array}
+$$
+
+According to the definition, $\hat { \mathbf { x } } _ { t } ^ { \prime } = z ^ { * }$ , whereby the proof is completed.
+
+# B ADDITIONAL EXPERIMENTAL DETAILS
+
+# B.1 ADDITIONAL RESULTS
+
+In Fig. 6, we provide SSIM results versus the number of measurements for multiple methods and tasks. In general, the SSIM curves have very similar trends to the PSNR curves in Fig. 5. We additionally provide a visualization of metal artifact removal results in Fig. 7.
+
+# B.2 THE TASK OF METAL ARTIFACT REMOVAL
+
+Metallic implants in an object can cause strong metal artifacts in CT imaging. As shown in Fig. 8, the source of artifacts come from extremely bright regions in the sinogram, called metal traces. To reduce or ideally remove metal artifacts from a CT image, we remove metal traces from the sinogram and leverage the data prior to complete the sinogram. As a result, metal artifact removal can be viewed as an inverse problem, where the measurement process gives the full sinogram except for the metal trace region, and our goal is to reconstruct the full CT image using this partially known sinogram, which will be artifact-free assuming perfect inpainting of the sinogram.
+
+![](images/4f7fbf69277046619f0d9f5574bcbb471b9501e40ab3e9337e7c63761cba0910.jpg)  
+Figure 8: The linear measurement process of metal artifact removal.
+
+# B.3 DETAILS OF DATASETS
+
+CT datasets We conduct experiments of 2D CT image reconstruction on two datasets. First, the Lung Image Database Consortium image collection (LIDC) (Armato III et al., 2011; Clark et al., 2013) consists of diagnostic and lung cancer screening thoracic computed tomography (CT) scans for lung cancer detection and diagnosis, which contains 1018 cases. Second, the Low Dose CT Image and Projection dataset (LDCT) (Clark et al., 2013; Moen et al., 2021) involves CT images of multiple anatomic sites, including 99 head CT scans, 100 chest CT scans, and 100 abdomen CT scans. Note that for the LDCT dataset, we only use the full-dose CT images in our experiments. In CT image processing, we convert the Hounsfield units from dicom files to the attenuation coefficients and set the background pixels to zero. Then, 2D CT images are sliced from 3D CT volumes. The sinograms are simulated from 2D CT images based on parallel-beam geometry with different number of projection angles that are equally distributed across 180 degrees.
+
+MRI dataset The Brain Tumor Segmentation (BraTS) 2021 dataset (Menze et al., 2014; Bakas et al., 2017) collected for the image segmentation challenge contains 2000 cases (8000 MRI scans), where each case has four different MR contrasts: native (T1), post-contrast T1-weighted (T1Gd), T2-weighted (T2), and T2 Fluid Attenuated Inversion Recovery (T2-FLAIR). For each 3D MR volume, we extract 2D slices from 3D volumes and simulate k-space data by Fast Fourier Transform. To reconstruct MR images, we follow Knoll et al. (2020); Zbontar et al. (2018) to undersample $\mathbf { k }$ -space data with an equispaced Cartesian mask, where the center k-space is fully sampled while the left $\mathbf { k }$ -space is under-sampled by equispaced columns.
+
+# B.4 DETAILS OF SCORE-BASED GENERATIVE MODELS
+
+We use the ${ \mathrm { N C S N } } { + + }$ model architecture in Song et al. (2021), and perturb the data with the Variance Exploding (VE) SDE. Our training procedure follows that of Song et al. (2021). Instead of generating samples according to the numerical SDE solver in Algorithm 1, we use the Predictor-Corrector (PC) sampler as described in Song et al. (2021) since it generally has better performance for VE SDEs. In PC samplers, the predictor refers to a numerical solver for the reverse-time SDE while the corrector can be any Markov chain Monte Carlo (MCMC) method that only depends on the scores. One such MCMC method considered in this work is Langevin dynamics, whereby we transform any initial sample $\mathbf { x } ^ { ( 0 ) }$ to an approximate sample from $p _ { t } ( \mathbf { x } )$ via the following procedure:
+
+$$
+\begin{array} { r } { \mathbf { x } ^ { ( i + 1 ) } \gets \mathbf { x } ^ { ( i ) } + \epsilon \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } ^ { ( i ) } ) + \sqrt { 2 \epsilon } \mathbf { z } ^ { ( i ) } , \quad i = 0 , 1 , \cdots , N - 1 . } \end{array}
+$$
+
+Here $N \in \mathbb { N } _ { > 0 } , \epsilon > 0$ , and $\mathbf { z } ^ { ( i ) } \sim \mathcal { N } ( \mathbf { 0 } , I )$ . The theory of Langevin dynamics guarantees that in the limit of $N  \infty$ and $\epsilon \to 0 , \mathbf { x } ^ { ( N ) }$ is a sample from $p _ { t } ( \mathbf { x } )$ under some regularity conditions. Note that Langevin dynamics only requires the knowledge of $\nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } )$ , which can be approximated using the time-dependent score model $s \mathbf { \boldsymbol { \theta } } \ast \left( \mathbf { \boldsymbol { x } } , t \right)$ . In PC samplers, each predictor step immediately follows multiple consecutive corrector steps, all using the same $s _ { \theta ^ { * } } ( \mathbf { x } , t )$ evaluated at the same $t$ . This jointly ensures that our intermediate sample at $t$ is approximately distributed according to $p _ { t } ( \mathbf { x } )$ . As shown in Song et al. (2021), PC sampling often outperforms numerical solvers for the reverse-time SDE, especially when the forward SDE in Eq. (1) is a VE SDE. In order to use PC samplers for inverse problem solving, our modification is similar to the change made in Algorithm 2 for Algorithm 1. Specifically, we run line 4 & 5 in Algorithm 2 before every corrector or predictor step.
+
+When comparing our approach to previous methods with score-based generative models, we use the same score model to isolate the confounding factors in model training and architecture design. Moreover, we make sure the total cost of sampling is comparable across different methods. For the ALD sampler used in Jalal et al. (2021), we use 700 noise scales with 3 steps of Langevin dynamics per noise scale, resulting in a total of $7 0 0 \times 3 = 2 1 0 0$ steps that require score function evaluation. For the PC sampler, we use 1000 noise scales and 1 step of Langevin dynamics per noise scale, totalling $1 0 0 0 + 1 0 0 0 = 2 0 0 0$ steps of score model evaluation.
+
+For PC samplers, the step size $\epsilon$ in Langevin dynamics is determined by a signal-to-noise ratio $\eta$ . For all methods, we tune $\eta$ and $\lambda$ in Eq. (8) with 100 steps of Bayesian optimization on a validation dataset, and report the results on the test dataset with the optimal parameters. We use the $\mathsf { a x } - \mathsf { p } \bot$ atform toolkit for Bayesian optimization. The optimal parameters in our experiments are given by
+
+• Sparse-view CT on LIDC $3 2 0 \times 3 2 0$ : $\eta = 0 . 2 4 6$ , $\lambda = 0 . 8 4 1$ .   
+• Metal artifact removal on LIDC $3 2 0 \times 3 2 0$ : $\eta = 0 . 2 0 9$ , $\lambda = 0 . 2 2 7$ .   
+• Sparse-view CT on LDCT $5 1 2 \times 5 1 2$ : $\eta = 0 . 4 , \lambda = 0 . 7 2$ .   
+• Accelerated MRI on BraTS $2 4 0 \times 2 4 0$ : $\eta = 0 . 5 7 7$ , $\lambda = 0 . 9 8 2$ .
+
+B.5 TRAINING DETAILS OF BASELINE MODELS
+
+B.5.1 BASELINE MODELS FOR SPARSE-VIEW CT RECONSTRUCTION
+
+FBP Filtered back projection (FBP) is a standard way for CT image reconstruction, which simply put the projections (sinogram) back to the image space based on the corresponding projection angles and geometry to get an approximated estimation of the unknown image. Usually, a high-pass filter, ramp filter is used to eliminate the blurring during this process. In our experiments, we conduct FBP on sparse-view sinograms using the torch radon toolbox (Ronchetti, 2020).
+
+FISTA-TV FISTA-TV is a fast iterative shrinkage-thresholding algorithm (FISTA) for solving linear inverse problems in image processing (Beck & Teboulle, 2009). It adopts a total variation (TV) term as the regularization in the optimization procedure. Each optimization iteration involves a matrixvector multiplication followed by a shrinkage-threshold step. In experiments, FISTA is implemented using the tomobar toolbox (Kazantsev & Wadeson, 2020) with the regularization using the CCPi regularisation toolkit (Kazantsev et al., 2019). We run 300 iterations for reconstructing each CT image with regularization parameter 0.001. Considering the nature of iterative reconstruction in FISTA, it is quite natural to generalize this method to different number of projections for reconstructing CT images. In experiments of generalizing to different number of measurements, FISTA method takes as input the sinogram with different numbers of projections and the corresponding angles for these input projections for the iterative procedure.
+
+cGAN Conventional iterative CT reconstruction algorithms like FISTA are typically slow due to their iterative nature. Ghani & Karl (2018) proposed to cast sparse-view CT reconstruction as a sinogram inpainting problem. Specifically, it used a conditional generative adversarial network (cGAN) to first complete the sinogram data prior to reconstructing CT images, thereby avoiding the costly iterative tomographic processing. However, the imperfect sinogram inpainting may further cause image artifacts. Specifically, cGAN model takes zero-padded sparse-view sinogram with 23 projections as input and generates the completed full-angle sinogram with 180 projections. The cGAN model was implemented using PyTorch (Paszke et al., 2019) and trained using a batchsize of 64 and learning rate of 0.0001 with 50 epochs in total. In experiments of generalizing to different number of measurements, we deployed the trained cGAN model by zero-padding sparse-view sinogram with different numbers of projections to full-view sinogram as the input. After obtaining the output inpainted sinogram, we replace the corresponding projections in the output based on the ground truth projections in the input. Finally, the images were reconstructed from the overlayed sinogram. Note that we trained the model using 23 projections and tested it on other projection settings to evaluate the generalization.
+
+SIN- $\mathbf { 4 c }$ -PRN To further reduce the artifacts in both sinogram and image space, SIN- $. 4 \mathrm { c }$ -PRN (Wei et al., 2020) proposed a two-step sparse-view CT reconstruction model. It involves a sinogram inpainting network (SIN) to generate super-resolved sinograms with different number of projections, and then a post-processing refining network (PRN) to further remove image artifacts. Both networks are connected through a filtered back-projection operation (FBP). Specifically, SIN model takes 23- view sinogram as input to fistly upsample to full-view sinogram and then generate sinograms through network for 23, 45, 90, 180 projections respectively. FBP transforms these generated sinograms to image space, which was then concatenated and feed into PRN model for refinement. The framework was implemented using PyTorch (Paszke et al., 2019) while FBP operation was implemented using . SIN model was trained using a batchsize of 20 and learning rate of 0.0001, while PRN model was trained using a batchsize of 15 and learning rate of 0.0001. Considering that LIDC dataset is much larger than LDCT dataset, the SIN- $_ { \cdot 4 \mathrm { c } }$ -PRN model was trained for 30 epochs on LIDC dataset and 50 epochs on LDCT dataset. To deploy the trained SIN model to different numbers of measurements, the sinograms with various number of projections are taken as the input for SIN model to generate multi-view sinograms, which were also overlayed with corresponding ground truth projections in inputs. The generated multi-view sinograms are then used for PRN model inference. Since SIN- $_ \mathrm { 4 c }$ -PRN model involves the dual-domain learning in both sinogram and image spaces to remove artifacts, and generates multi-scale sinograms during sinogram inpainting, it shows a better generalization to different numbers of measurements compared with cGAN model as shown in Figure 5 and Figure 6.
+
+Neumann Meanwhile, in another parallel direction, researchers proposed to learn the regularizer used in optimization from training data, outperforming traditional regularizers. Specifically, Gilton et al. (2019) presented an end-to-end, data-driven method for learning a nonlinear regularizer for solving inverse problems inspired by the Neumann series, called Neumann network. Neumann network was implemented using PyTorch (Paszke et al., 2019). Due to GPU memory constraints, the model training used the batchsize of 5 on LIDC dataset and the batchsize of 2 on LDCT dataset. The initial learning rate was 0.00001 with an exponential learning rate decay. The network was trained with 15 training epochs on both datasets.
+
+# B.5.2 BASELINE MODELS FOR UNDERSAMPLED MRI RECONSTRUCTION
+
+DuDoRNet Zhou & Zhou (2020) proposed a dual domain recurrent network (DuDoRNet) to simultaneously recover k-space data and images for MRI reconstruction, in order to address aliasing artifacts in both frequency and image domains. The original model in Zhou & Zhou (2020) also embedded a deep T1 prior to make use of fully-sampled short protocol (T1) as complementary information. For a fair comparison with other supervised learning approaches, in our experiments, we do not include this additional information but train the DuDoRNet model without T1 prior. The DuDoRNet was trained using a batchsize of 6 and a learning rate of 0.0005 with 5 training epochs. In experiments of generalizing to different number of measurements, we trained the model with an acceleration factor of 8 and deployed the trained model to other acceleration factors during testing. Specifically, for inference, we use different Cartesian masking function corresponding to different acceleration factors or down-sampling ratios to sub-sample the $\mathbf { k }$ -space data for the network input with the corresponding initial reconstructed image with zero-padding $\mathbf { k }$ -space.
+
+Cascade DenseNet To reconstruct de-aliased MR images from under-sampled k-space data, Zheng et al. (2019) proposed a cascaded dilated dense network (CDDN) for MRI reconstruction, based on stacked dense blocks with residual connections while using the zero-filled MR image as inputs. Specifically, they used a two-step data consistency layer for k-space correction, and replaced corresponding phase-coding lines of the generated image with the original sampled k-space data after each block. In experiments, we trained the model using a batchsize of 8 and a learning rate of 0.0001, with 5 epochs on BraTS dataset. In experiments of generalizing to different number of measurements, we trained the model with an acceleration factor of 8 and deployed the trained model to other acceleration factors during testing. Similarly, different masking functions corresponding to different acceleration factors were used to sub-sample $\mathbf { k }$ -space data to get network inputs. From results, we observe that
+
+Cascaded DenseNet generalizes better to more measurements than DuDoRNet as shown in Figure 5 and Figure 6.
+
+# B.5.3 BASELINE MODELS FOR METAL ARTIFACT REMOVAL
+
+LI One straightforward way for reducing metal artifacts is to complete or inpaint the metal-affected missing regions in sinogram directly through linear interpolation (Kalender et al., 1987). This method does not need any network training. However, the imperfect completion of sinogram may introduce secondary artifacts to the reconstructed image. In our experiments setting, to fit for the practical applications in real world, we assume the ground truth metal trace and mask information are unknown, which can only be estimated by a rough thresholding in artifacts-affected images. We use the estimated metal mask and metal trace for linear interpolation baseline.
+
+cGANMAR Wang et al. (2018) proposed a conditional generative adversarial network (cGAN)- based approach for metal artifacts reduction (MAR) in CT. Specifically, cGANMAR network learns the mapping directly from the artifacts-affected CTs to artifacts-free CTs through refinement in image space. The cGANMAR model was implemented using PyTorch (Paszke et al., 2019) and was trained with the batchsize of 64 and the learning rate of 0.0001. The network was trained with 400 epochs.
+
+SNMAR Yu et al. (2020) proposed a sinogram completion neural network (SinoNet) to recover the metal-affected projections. Especially, it leveraged the learning in both sinogram domain and image domain by using a prior network to generate a good prior image to guide sinogram learning. Note that in original setting, SNMAR required linear interpolated sinogram and CT as inputs and used ground truth metal trace and mask information to generated them. But in our method, we assume the ground truth metal trace and mask information are unknown according to practical scenario and estimate it by a rough thresholding, which will introduce estimation errors. In SNMAR experiments, we still follow the original setting to guarantee the best performance of this baseline method for a strong comparison. We trained the SNMAR using the batchsize of 64 and the learning rate of 0.0001, with a total of 100 training epochs.

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+# The Curse of Low Task Diversity: On the Failure of Transfer Learning to Outperform MAML and their Empirical Equivalence
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Recently, it has been observed that a transfer learning solution might be all we need   
+2 to solve many few-shot learning benchmarks – thus raising important questions   
+3 about when and how meta-learning algorithms should be deployed. In this paper,   
+4 we seek to clarify these questions by 1. proposing a novel metric – the diversity   
+5 coefficient – to measure the diversity of tasks in a few-shot learning benchmark   
+6 and 2. by comparing MAML and transfer learning under fair conditions (same   
+7 architecture, same optimizer and all models trained to convergence). Using the   
+8 diversity coefficient, we show that the popular MiniImagenet and Cifar-fs few-shot   
+9 learning benchmarks have low diversity. This novel insight contextualizes claims   
+10 that transfer learning solutions are better than meta-learned solutions in the regime   
+11 of low diversity under a fair comparison. Specifically, we empirically find that a low   
+12 diversity coefficient correlates with a high similarity between transfer learning and   
+13 Model-Agnostic Meta-Learning (MAML) learned solutions in terms of accuracy   
+14 at meta-test time and classification layer similarity (using feature based distance   
+15 metrics like SVCCA, PWCCA, CKA, and OPD). To further support our claim,   
+16 we find this meta-test accuracy holds even as the model size changes. Therefore,   
+17 we conclude that in the low diversity regime, MAML and transfer learning have   
+18 equivalent meta-test performance when both are compared fairly. We also hope   
+19 our work inspires more thoughtful constructions and quantitative evaluations of   
+20 meta-learning benchmarks in the future.
+
+# 21 1 Introduction
+
+22 The success of deep learning in computer vision (1; 2), natural language processing (3; 4), game   
+23 playing $\textcircled { 5 } \textcircled { 6 } \textcircled { 7 }$ and more, keeps motivating a growing body of applications of deep learning on   
+24 an increasingly wide variety of domains. In particular, deep learning is now routinely applied to   
+25 few-shot learning – a research challenge that assesses a model’s ability to learn to adapt to new tasks,   
+26 new distributions, or new environments. This has been the main research area where meta-learning   
+27 algorithms have been applied – since such a strategy seems promising in a small data regime due to   
+28 its potential to learn to learn or learn to adapt. However, it was recently shown $\textcircled{8}$ that a transfer   
+29 learning model with a fixed embedding can match and outperform many modern sophisticated meta  
+30 learning algorithms on numerous few-shot learning benchmarks $\bigoplus \iiiint \bigoplus \iiiint \bigoplus \iiiint$ . This growing body of   
+31 evidence – coupled with these surprising results in meta-learning – raise the question if researchers are   
+32 applying meta-learning with the right inductive biases $\textcircled { 1 3 } ; \textcircled { 1 4 }$ and designing appropriate benchmarks   
+33 for meta-learning. Our evidence suggests this is not the case.   
+34 In this work, we show that when the task diversity – a novel measure of variability across tasks – is   
+35 low, then MAML (Model Agnostic Meta-Learning) $\textcircled{1 1 5 }$ learned solutions have the same accuracy   
+36 as transfer learning (i.e., a supervised learned model with a fine-tuned final linear layer). We want   
+37 to emphasize the importance of doing such an analysis fairly: with the same architecture, same   
+38 optimizer and all models trained to convergence. This empirical equivalence remained true even as   
+39 the model size changed – thus further suggesting this equivalence is more a property of the data than   
+40 of the model. Therefore, we suggest taking a problem-centric approach to meta-learning and suggest   
+41 applying Marr’s level of analysis $\textcircled { 1 6 } ; \textcircled { 1 7 } \textcircled { }$ to few-shot learning – to identify the family of problems   
+42 suitable for meta-learning. Marr emphasized the importance of understanding the computational   
+43 problem being solved and not only analyzing the algorithms or hardware that attempts to solve   
+44 them. An example given by Marr is marveling at the rich structure of bird feathers without also   
+45 understanding the problem they solve is flight. Similarly, there has been analysis of MAML solutions   
+46 and transfer learning without putting the problem such solutions should solve into perspective $( \overline { { 1 8 } } ; \overline { { 1 9 } } )$ .   
+47 Therefore, in this work, we hope to clarify some of these results by partially placing the current   
+48 state of affairs in meta-learning from a problem-centric view. In addition, the novelty of our analysis   
+49 compared to previous work is that we make analysis intrinsic of the data as a first class citizen.
+
+50 Our contributions summarized as follows:
+
+1. We propose a novel metric that quantifies the intrinsic diversity of the data of a few-shot learning benchmark. We call it the diversity coefficient. It enables analysis of meta-learning algorithms through a problem-centric framework. It also goes beyond counting the number of classes or number of data points or counting the number concatenated data sets – and instead quantifies the expected diversity/variability of tasks in a few-shot learning benchmark.
+
+2. We analyze the two most prominent few-shot learning benchmarks – MiniImagenet and Cifar-fs – and show that their diversity is low. These results are robust across different ways to measure the diversity coefficient, suggesting that our approach is robust.
+
+3. With this context, we partially clarify the surprising results from $\textcircled { 1 1 9 }$ by comparing their transfer learning method against models trained with MAML $\textcircled{1 1 5 }$ . In particular, when making a fair comparison, transfer learning method with a fixed feature extractor fails to outperform MAML. We define a fair comparison when the two methods are compared using the same architecture (backbone), same optimizer and all models trained to convergence. We also show that their final layer makes similar predictions according to neural network distance techniques like distance based Singular Value Canonical Correlation Analysis (SVCCA), Projection Weighted (PWCCA), Linear Centered Kernel Analysis (LINCKA) and Orthogonal Procrustes Distance (OPD). This equivalence holds even as the model size increases.
+
+4. Interestingly, we also find that even in the regime where task diversity is low (in MiniImagenet and Cifar-fs), the features extracted by supervised learning and MAML are different – implying that the mechanism by which they function is different despite the similarity of their final predictions.
+
+5. As an actionable conclusion, we provide a metric that can be used to analyze the intrinsic diversity of the data in a few-shot learning benchmarks and therefore build more thoughtful environments to drive research in meta-learning. In addition, our evidence suggests the following test to predict the empirical equivalence of MAML and transfer learning: if the task diversity is low, then transfer learned solutions might fail to outperform meta-learned solutions. This test is easy to run because our diversity coefficient can be done using the Task2Vec method $\textcircled { 1 2 0 }$ using pre-trained neural network. We also found that random networks were consistent with the results of pre-trained networks on Imagenet.
+
+81 We hope that this line of work inspires a problem-centric first approach to meta-learning – which   
+82 appears to be especially sensitive to the properties of the problem in question. Therefore, we hope   
+83 future work takes a more thoughtful and quantitative approach to benchmark creation – instead of   
+84 focusing only on making huge data sets.
+
+# 2 Background
+
+In this section, we provide a summary of the background needed to understand our main results.
+
+87 Model-Agnostic Meta-Learning (MAML): The MAML algorithm $\textcircled{1 1 5 }$ attempts to meta-learn   
+88 an initialization of parameters for a neural network so that it is primed for fast gradient descent
+
+adaptation. It consists of two main optimization loops: 1) an outer loop used to prime the parameters for fast adaptation, and 2) an inner loop that does the fast adaptation. During meta-testing, only the inner loop is used to adapt the representation learned by the outer loop.
+
+92 Transfer Learning with Union Supervised Learning (USL): Previous work $\textcircled { 1 9 }$ shows that   
+93 an initialization trained with supervised learning, on a union of all tasks, can outperform many   
+94 sophisticated methods in meta-learning. In particular, their method consists of two stages: 1) first   
+95 they use a union of all the labels in the few-shot learning benchmark during meta-training and train   
+96 with standard supervised learning (SL), then 2) during the meta-testing, they use an inference method   
+97 common in transfer learning: extract a fixed feature from the neural network and fully fine-tune the   
+98 final classification layer (i.e., the head). Note that our experiments only consider when the final layer   
+99 is regularized Logistic Regression trained with LBGFS.   
+100 Distances for Deep Neural Network Feature Analysis: To compute the distance between neural   
+101 networks we use the distance versions of Singular Value Canonical Correlation Analysis (SVCCA)   
+102 $\textcircled { 2 1 }$ , Projection Weighted Canonical Correlation (PWCCA) $\textcircled { 1 2 2 }$ , Linear Centered Kernel Analysis   
+103 (LINCKA) $\textcircled { 1 2 3 }$ and Orthogonal Procrustes Distance (OPD) $\textcircled { 1 2 4 }$ . These distances are in the interval   
+104 $[ 0 , 1 ]$ and are not necessarily a formal distance metric but are guaranteed to be zero when their   
+105 inputs are equal and nonzero otherwise. This is true because SVCCA, PWCCA, LINCKA are based   
+106 on similarity metrics and OPD is already a distance. Note that we use the formula $d ( X , Y ) =$   
+107 $1 - s i m ( X , Y )$ for our distance metrics where sim is one either SVCCA, PWCCA, LINCKA   
+108 similarity metric and $X , Y$ are matrices of activations (called layer matrices). The distance between   
+109 two models is computed by choosing a layer and then comparing the features/activations after   
+110 adaptation for that layer given a batch of tasks represented as a support and query set. A more   
+111 thorough overview of these metrics for the analysis of internal representations for convolutional   
+112 neural networks (CNNS) can be found in the appendix, section G.   
+113 Task2Vec Embeddings for Distances between Tasks: The diversity coefficient we propose is   
+114 the expectation of distance between tasks (explain in more detail in section $\textcircled { 3 }$ ). Therefore, it is   
+115 essential to define the distance between different pairs of tasks. We choose the cosine distance   
+116 between Task2Vec (vectorial) embeddings as in $\textcircled { 1 2 0 }$ . Therefore, we provide a summary of the   
+117 Task2Vec method to compute task embeddings. The vectorial representation of tasks provided by   
+118 Task2Vec $\textcircled { 1 2 0 }$ is the vector of diagonal entries of the Fisher Information Matrix (FIM) given a fix   
+119 neural network as a feature extractor – also called a probe network – after fine-tuning the final   
+120 classification layer to the task. The authors explain this is a good vectorial representation of tasks   
+121 because 1. It approximately indicates the most informative weights for solving the current task   
+122 (up to a second order approximation) 2. For rich probe networks like CNNs, the diagonal is more   
+123 computationally tractable. We choose Task2Vec because the original authors provide extensive   
+124 evidence that their embeddings correlate with semantic and taxonomic relations between different   
+125 visual classes – making it a convincing embedding for tasks $\textcircled { 1 2 0 }$ . The Task2Vec embedding of task $\tau$   
+126 is the diagonal of the following matrix:
+
+$$
+\hat { F } _ { D _ { \tau } , f _ { w } } = \hat { F } ( D _ { \tau } , f _ { w } ) = \mathbb { E } _ { \boldsymbol { x } , \boldsymbol { y } \sim \hat { p } ( \boldsymbol { x } | \tau ) p ( \boldsymbol { y } | \boldsymbol { x } , f _ { w } ) } [ \nabla _ { w } \log p ( \boldsymbol { y } \mid \boldsymbol { x } , f _ { w } ) \nabla _ { w } p ( \boldsymbol { y } \mid \boldsymbol { x } , f _ { w } ) ^ { \top } ]
+$$
+
+127 where $f _ { w }$ is the neural networks used as a feature extractor with architecture $f$ and weights $w$ ,   
+128 $\hat { p } ( x \mid \tau )$ is the empirical distribution defined by the training data $D _ { \tau } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { \bar { n } }$ for task $\tau$ , and   
+129 $p ( y \mid x , f _ { w } )$ is a deep neural network trained to approximate the (empirical) posterior $\hat { p } ( y \mid x , \tau )$ .   
+130 We’d like to emphasize that the there is a dependence on target label since Task2Vec fixes the   
+131 feature extractor (using $f _ { w , }$ ) and then fits the final layer (or “head") to approximate the task posterior   
+132 distribution $\hat { p } ( y \mid x , \tau )$ .
+
+# 133 3 Definition of the Diversity Coefficient
+
+134 The diversity coefficient aims to measure the intrinsic diversity (or variability) of tasks in a few-shot   
+135 learning benchmark. At a high level, the diversity coefficient is the expected distance between a   
+136 pair of different tasks given a fixed probe network. In this work, we choose the distance to be the   
+137 cosine distance between vectorial representations (i.e. embeddings) of tasks according to Task2Vec   
+138 as described in section $2 .$ Using a fixed probe networks is essential because: 1. Using a fixed probe   
+139 network means that the distances between different tasks are comparable, as discussed in the original   
+140 Task2Vec $\textcircled{20 }$ and 2. Since we are computing the distance between different tasks, we need to make   
+141 sure the difference comes from intrinsic properties of the data and not from a different source, e.g. if   
+142 one uses different models then this might confound the source of variability in our metric. We define   
+143 the diversity coefficient of a few-shot learning benchmark $B$ as follows:
+
+$$
+\begin{array} { r } { \hat { d i } v ( B ) = \mathbb { E } _ { \tau _ { 1 } \sim \hat { p } ( \tau | B ) , \tau _ { 2 } \sim \hat { p } ( \tau | B ) } \mathbb { E } _ { D _ { 1 } \sim \hat { p } ( x _ { 1 } , y _ { 1 } | \tau _ { 1 } ) , D _ { 2 } \sim \hat { p } ( x _ { 2 } , y _ { 2 } | \tau _ { 2 } ) } \left[ d ( \hat { F } _ { D _ { 1 } , f _ { w } } , \hat { F } _ { D _ { 2 } , f _ { w } } ) \right] } \end{array}
+$$
+
+144 where $f _ { w }$ is the neural networks used as a feature extractor with architecture $f$ and weights $w$ ,   
+145 $\hat { p } ( x \mid \tau )$ is the empirical distribution defined by the training data $D _ { \tau } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ for task $\tau$   
+146 $\tau _ { 1 } , \tau _ { 2 }$ are tasks sampled from the empirical distribution of tasks $\hat { p } ( \tau \mid B )$ for the current benchmark   
+147 $B$ (i.e. a batch of tasks with their data sets $\mathcal { D } = ( \tau _ { i } , D _ { \tau _ { i } } ) _ { i = 1 } ^ { N } )$ , a task $\tau _ { i }$ is the probability distribution   
+148 $p ( x , y \mid \tau )$ of the data, is a distance metric (for us cosine), $f _ { w }$ is the neural networks used as   
+149 a feature extractor with architecture $f$ and weights $w$ , and $\hat { p } ( x \mid \tau )$ is the empirical distribution   
+150 defined by the training data $D _ { \tau } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ for task $\tau$ . We’d also like to recall the reader that the   
+151 definition of a task in this setting is of a n-way, $\mathbf { k }$ -shot few-shot learning task. Therefore, each task has   
+152 n classes sampled with $\mathrm { k }$ examples used for the adaptation. We’d like to emphasize that the adaptation   
+153 here is only to fine-tune the final layer according to the Task2Vec method for the correct computation   
+154 of the FIM. Therefore, in this setting we combine the support and query set as the split is not relevant   
+155 for the computation of the task embedding using Task2Vec. Note that the above formulation can be   
+156 easily adapted to any distance function between tasks, and is not necessarily specific to using the   
+157 FIM or cosine distance. For example, given the true distributions for tasks one can use real distances   
+158 between probability distributions e.g. Hellinger distance. In addition, it is obvious one can use a   
+159 distance function besides the cosine distance – but choose it in accordance to the original work of   
+160 Task2Vec $\textcircled { 1 2 0 }$ .
+
+# 4 Experiments
+
+62 This section explains the experiments backing up our main results outlined in our list of contributions.   
+63 Experimental details are provided in the supplementary section $\bigstar$ and the learning curves displaying   
+64 the convergence for a fair comparison are in supplementary section B.
+
+# 165 4.1 The Diversity Coefficient of MiniImagenet and Cifar-fs
+
+66 To put our analysis into a problem-centric framework, we first analyze the problem they are trying   
+7 to solve through the diversity coefficient. Recall that the diversity coefficient aims to quantify the   
+8 intrinsic variation of tasks in a few-shot learning benchmark. We show that the diversity coefficient   
+69 of the popular MiniImagenet and Cifar-fs benchmarks are low with good confidence intervals using   
+70 four different probe networks in table 1.
+
+<table><tr><td>Probe Network</td><td>Diversity on MI</td><td>Diversity on Cifar-fs</td></tr><tr><td>Resnet18 (pt)</td><td>0.117 ± 2.098e-5</td><td>0.100 ± 2.18e-5</td></tr><tr><td>Resnet18 (rand)</td><td>0.0955 ± 1.29e-5</td><td>0.103 ± 1.05e-5</td></tr><tr><td>Resnet34 (pt)</td><td>0.0999 ± 1.95e-5</td><td>0.0847 ± 3.06e-5</td></tr><tr><td>Resnet34 (rand)</td><td>0.0620 ± 8.12e-6</td><td>0.0643 ± 9.64e-6</td></tr></table>
+
+Table 1: The diversity coefficient of MiniImagenet (MI) and Cifar-fs is low. The diversity coefficient was computed using the cosine distance between different standard n-way, $\mathbf { k }$ -shot classification tasks from the few-shot learning benchmark using the Task2Vec method described in section $3 .$ We used $\mathrm { n } { = } 5$ (number of classes) and ${ \bf k } = 2 0$ (number of examples per class) since we can use the whole task data to compute the diversity coefficient (no splitting of support and query set required). We used Resnet18 and Resnet34 networks as probe networks – both pre-trained on ImageNet (indicated as “pt" on table) and randomly initialized (indicated as “rand" on table). We observe that both type of networks and weights give similar diversity results. All confidence intervals were at $9 5 \%$ . To compute results, we used 500 few-shot learning tasks and only compare pairs of different tasks. This results in $( 5 0 0 ^ { 2 } - 5 0 0 ) / 2 = 1 2 4 ,$ 750 pair-wise distances used to compute the diversity coefficient.
+
+# 171 4.2 Low Diversity Correlates with Equivalence of MAML and Transfer Learning
+
+172 Now that we have placed ourselves in a problem-centric framework and shown the diversity coefficient   
+173 of the popular MiniImagenet and Cifar-fs benchmarks are low – we proceed to show the failure of   
+174 transfer learning (with USL) to outperform MAML. Crucially, the analysis was done using a fair   
+175 comparison: using the same model architecture, optimizer, and training all models to convergence   
+176 – details in section A. We used the five-layer CNN used in $\textcircled { 1 5 } \textcircled { 2 5 }$ and Resnet12 as in $\textcircled { 1 9 }$ . We   
+177 provide evidence that in the setting of low diversity:
+
+1. The accuracy of an adapted MAML meta-learner vs. an adapted USL pre-trained model are similar and statistically significant, except for one result where transfer learning with USL is worse. This is shown in table 2 and 1.   
+2. The distance for the classification layer decreases sharply according to four distance-based metrics – SVCCA, PWCCA, LINCKA, and OPD – as shown in figure $2 .$ This implies the predictions of the two are similar.
+
+184 For the first point, we emphasize that tables 1 and table 2 taken together support our central hypothesis:   
+185 that models trained with meta-learning are not inferior to transfer learning models (using USL) when   
+186 the diversity coefficient is low. Careful inspection reveals that the methods have the same meta-test   
+187 accuracy with intersecting confidence intervals – making the results statistically significant across   
+188 few-shot benchmarks and architectures. The one exception is the third set of bar plots, where transfer   
+189 learning with USL is in fact worse.   
+190 For the second point, refer to figure $\nsubseteq$ and observe that as the depth of the network increases, the   
+191 distance between the activation layers of a model trained with MAML vs USL increases until it   
+192 reaches the final classification layer – where all four metrics display a noticeable dip. In particular,   
+193 PWCCA considers the two prediction layers identical (approximately zero distance). This final point   
+194 is particularly interesting because PWCCA is weighted according to the CCA weights that stabilize   
+195 with the final predictions of the network. This means that the PWCCA distance value is reflective of   
+196 what the networked actually learned and gives a more reliable distance metric (for details, refer to the   
+197 appendix section G.5). This is important because this supports our main hypothesis: that at prediction   
+198 time there is an equivalence between transfer learning and MAML when the diversity coefficient is   
+199 low.
+
+![](images/a9b91dba46f4e1bb66a582521775272ad42f050a687fee3846c532e28aaf2892.jpg)  
+Figure 1: MAML trained models and union supervised trained (USL) models have statistically equivalent meta-test accuracy for MiniImagenet and Cifar-fs with Resnet12 and five layer CNNs. This holds for both the Resnet12 architecture used in $\textcircled { 1 9 }$ and the 5 layer CNN (indicated as “5CNN") in $\textcircled { 1 2 5 }$ . Results used a (meta) batch-size of 100 tasks and $9 5 \%$ confidence intervals. All MAML models were trained with 5 inner steps during meta-training. “MAML5" and “MAML10" in the bar plot indicates the adaptation method used at test time i.e. we used 5 inner steps and 10 inner steps at test time. MiniImagenet is abbreviated as “MI" in the figure.
+
+# 00 4.3 Is the Equivalence of MAML and Transfer Learning related to Model Size or Low 01 Diversity?
+
+An alternative hypothesis to explain the equivalence of transfer learning (with USL) and MAML could be due to the capabilities of large neural networks to be better meta-learners in general. Inspired by the impressive ability of large language models to be few-shot (or even zero-shot) learners $\textcircled { 1 4 } \textcircled { 2 7 } \textcircled { 2 8 } \textcircled { 3 } \textcircled { - }$ we hypothesized that perhaps the meta-learning capabilities of deep learning models is a function of the model size. If this were true, then we expected to see the difference in meta-test accuracy
+
+<table><tr><td>Meta-train Initialization </td><td>Adaptation at Inference</td><td>Meta-test Accuracy</td></tr><tr><td>Random</td><td>no adaptation</td><td>19.3 ± 0.80</td></tr><tr><td>MAML0</td><td>no adaptation</td><td>20.0 ± 0.00</td></tr><tr><td>USL</td><td>no adaptation</td><td>15.0 ± 0.26</td></tr><tr><td>Random</td><td>MAML5 adaptation</td><td>34.2 ± 1.16</td></tr><tr><td>MAML5</td><td>MAML5 adaptation</td><td>62.4 ± 1.64</td></tr><tr><td>USL</td><td>MAML5 adaptation</td><td>25.1 ± 0.98</td></tr><tr><td>Random</td><td>MAML10 adaptation</td><td>34.1 ± 1.23</td></tr><tr><td>MAML5</td><td>MAML10 adaptation</td><td>62.3 ± 1.50</td></tr><tr><td>USL</td><td>MAML10 adaptation</td><td>25.1 ± 0.97</td></tr><tr><td>Random</td><td>Adapt Head only (with LR)</td><td>40.2 ± 1.30</td></tr><tr><td>MAML5</td><td>Adapt Head only (with LR)</td><td>59.7 ± 1.37</td></tr><tr><td>USL</td><td>Adapt Head only (with LR)</td><td>60.1 ± 1.37</td></tr></table>
+
+Table 2: MAML trained representations and supervised trained representation have statistically equivalent meta-test accuracy on MiniImagenet – which has low diversity. The transfer model’s adaptation is labeled as “Adapted Head only (with LR)" – which stands for “Logistic Regression (LR)" used in $\textcircled{1 1 9 }$ . More precisely, we used Logistic Regression (LR) with LBFGS with the default value for the l2 regularization parameter given by Python’s Sklearn. Note that an increase in inner steps from 5 to 10 with the MAML5 trained model does not provide an additional meta-test accuracy boost, consistent with previous work $\textcircled { 1 2 6 }$ . Note that the fact that the MAML5 representation matches the USL representation when both use the same adaptation method is not surprising – given that: 1) previous work has shown that the distance between the body of an adapted MAML model is minimal compared to the unadapted MAML (which we reproduce in $\boxed { 5 }$ in the green line) and 2) the fact that a MAML5 adaptation is only 5 steps of MAML while LR fully converges the prediction layer. We want to highlight that only the MAML5 model achieved the maximum meta-test performance of 0.6 with the MAML5 adaptation – suggesting that the USL and MAML5 meta-learning algorithms might learn different representations. For USL to have a fair comparison during meta-test time when using the MAML adaptation, we provide the MAML final layer learned initialization parameters to the USL model (but any is fine due to convexity when using a fixed feature extractor). This is needed since during meta-training USL is trained with a union of all the labels (64) – so it does not even have the right output size of 5 for few-shot prediction. Meta-testing was done in the standard 5-way, 5-shot regime.
+
+207 of MAML and USL to be larger for smaller models and the difference to decrease as the model   
+208 size increased. Once the two models were, of the same size but large enough, we hypothesized that   
+209 the meta-test accuracy would be the same. We tested this to rule out that our observations were a   
+210 consequence of the model size. The results were negative and surprisingly the equivalence between   
+211 MAML and USL seems to hold even as the model increased – strengthening our hypothesis that the   
+212 low task diversity might be a bigger factor explaining our observations. We show this in figure $^ { 3 , }$   
+213 and we want to draw attention to the fact this statistical equivalence holds even when using only four   
+214 filters – the case where we expected the biggest difference.
+
+# 15 4.4 MAML learns a different base model compared to Union Supervised Learned models – 16 even in the presence of low task diversity
+
+The first four layers of figure 2 shows how large the distance is of a MAML representation compared to a SL representation. In particular, it is much larger than the distance value in the range $[ 0 , 0 . 1 ]$ from previous work that compared MAML vs. adapted MAML $\textcircled { 1 8 }$ . We reproduced that and indeed MAML vs. adapted MAML has a small difference (smaller for us) – supporting our observations that a MAML vs. a USL learned representations are different at the feature extractor layer even when the diversity is low. Results are statistically significant.
+
+![](images/faf7c973aba4c9d5f64586a69eae521cde521f73c78e55e91d2b0c7fe09fd7fe.jpg)  
+Figure 2: The classification layer of transfer learning and a MAML5 model decrease in distance – implying similar predictions. More precisely, an initialization trained with 5 inner steps (MAML5) has an increasingly similar head (classifier) after adaptation with MAML5 compared to the classifier layer of the union supervise learned (USL) model that has been adapted only at the final layer. In particular, the USL model has been adapted with Logistic Regression (LR) with LBFGS with the default value for the l2 regularization parameter given by Python’s Sklearn (as in $\textcircled { 1 1 9 }$ ). We showed this trend with four different distance metrics SVCCA, PWCCA, LICKA, and OPD referenced in section $\bigstar$ Observe that according to PWCCA the distance between the predictions is zero. This is true because the distance of classification layer (indicated as “head" in the figure) is zero. The architecture used here is a five layer CNN as in $\textcircled { 1 5 } , \textcircled { 2 5 } )$ with their same setup. The benchmark used for this analysis is MiniImagenet.
+
+![](images/08a586d62c82c32f160979b9c038cea04ff6c8519c3fb06b1a4cc273bbedcf98.jpg)  
+Figure 3: The meta-test accuracy of MAML and transfer learning using USL is similar in a statistically significant way – regardless of the model size. In this experiment, we used the MiniImagenet benchmark, the five layer CNN used in $\textcircled { 1 5 } \textcircled { 2 5 }$ , and only increased the filter size using sizes 4, 8, 16, and 32. We made sure the comparison was fair by using the same architecture, optimizer, and trained all models to convergence. During meta-training, the MAML model was trained using 5 inner steps. The legends indicating MAMl5 and MAML10 refer to the number of inner steps used at test time. We used a (meta) batch size of 100 tasks.
+
+In this section, we show the closeness of MAML and transfer learning (with USL) for synthetic experiments for low and high diversity regimes in Figure $\textcircled { 4 }$ In the low regime, the two methods are equivalent in a statistically significant way – which supports the main claims of our paper. As the diversity increases, however, the difference between USL and MAML increases (in favor of USL). This will be explored further in future work.
+
+The task is the usual n-way, $\mathbf { k }$ -shot tasks, but the data comes from a Gaussian and the meta-learners are tasked with classifying from which Gaussian the data points came from in a few-shot learning manner. Benchmarks are created by sampling a Gaussian distribution with means moving away from the origin as the benchmark changes. Therefore, the Gaussian benchmark with the highest diversity coefficient has Gaussians that are the furthest from the origin. We computed the diversity coefficient using a proper distance between distributions using the Hellinger distance eluded in section $3$ instead of the FIM distance. We can do this because we know the ground truth distribution in our synthetic experiments, and Gaussians have a closed form Hellinger distance. Details on the n-way Gaussian benchmark and diversity coefficient using the Hellinger distance can be found in supplementary section E and F.
+
+![](images/699c005e17187b208d460b65ed9d829c88d2ab9c6d3faa2d76636d5608875fd0.jpg)  
+Figure 4: The meta-test accuracy of MAML and transfer learning using USL is similar in a statistically equivalent way in the low diversity regime in the 5-way, 10-shot Gaussian Benchmarks. MAML models were trained with 5 inner steps. MAML5 and MAML10 indicate the adaptation procedure at test time. Results used a (meta) batch-size of 500 tasks and $9 5 \%$ confidence intervals. As the diversity of the benchmark increases, the Gaussian tasks are sampled further away from the origin. Note, as the diversity increases, the difference between USL and MAML increases (in favor of USL).
+
+# 240 5 Related Work
+
+241   
+242   
+243   
+244   
+245   
+246   
+247   
+248   
+249   
+250   
+251   
+252   
+253   
+254
+
+Our work proposes a problem-centric framework for the analysis of meta-learning algorithms inspired from previous puzzling results $\textcircled { 1 9 }$ . We propose to use a pair-wise distance between tasks and analyze how this metric might correlate with meta-learning. The closest line of work for this is the long line of work by $\textcircled { 1 2 0 }$ where they suggest methods to analyze the complexity of a task, propose unsymmetrical distance metrics for data sets, reachability of tasks with SGD, ways to embed entire data sets and more $( | 2 0 ; | 2 9 ; | 3 0 ; | 3 1 | )$ . We believe this line of work to be very fruitful and hope that more people adopt tools like the ones they suggest and we propose in this paper before researching or deploying meta-learning algorithms. We hope this helps meta-learning methods succeed in practice – since cognitive science suggests meta-learning is a powerful method humans use to learn $\textcircled{3 2 }$ . In the future, we hope to compare $\textcircled { 1 2 0 }$ ’s distance metrics between tasks with ours to provide a further unified understanding of meta-learning and transfer learning. A contrast between their work and ours is that we focus our analysis from a meta-learning perspective applied to few-shot learning – while their focus is understanding transfer learning methods between data sets.
+
+The use of a distance metric in our definition of the diversity coefficient is inspired by the analysis 255 done by $\textcircled { 1 8 }$ . They showed that MAML functions mainly via feature re-use than by rapid learning i.e., 256 that a model trained with MAML changes very little after the MAML adaptation. The main difference
+
+257 of their work with our is: 1) that we compare MAML trained models against union supervised learned   
+258 models (USL) instead of only comparing MAML against adapted MAML, and 2) that we explicitly   
+259 analyzed properties of the data sets. In addition, we use a large set of distance metrics for our analysis   
+260 including: SVCCA, PWCCA, LINCKA and OPD as proposed by (21; 22; 23; 24).
+
+Our work is most influenced by previous work suggesting modern meta-learning requires rethinking $\textcircled{1 1 9 }$ The main difference of our work with theirs is that we analyzed the internal representation of the meta-learning algorithms and contextualize these with quantifiable metrics of the problem being solved. Unlike their work, we focused on a fair comparison between meta-learning methods by ensuring the same neural network backbone was used. Another difference is that they gained further accuracy gains by using distillation – a method we did not analyze and leave for future work.
+
+267 A related line of work $\textcircled { 1 3 3 } \textcircled { 2 6 }$ first showed that there exist synthetic data sets that are capable of   
+268 exhibiting higher degrees of adaptation as compared to the original work by $\textcircled { 1 1 8 }$ . The difference is   
+269 that they did not compare MAML models against transfer learning methods like we did here. Instead,   
+270 they focused on comparing adapted MAML models vs. unadapted MAML models.   
+271 Another related line of work is the predictability of adversarial transferability and transfer learning.   
+272 They show this both theoretically and with extensive experiments $\textcircled{3 4 }$ . The main difference between   
+273 their work and ours is that they focus their analysis mainly on transfer learning, while we concentrated   
+274 on meta-learning for few-shot learning. In addition, we did not consider adversarial transferability –   
+275 while that was a central piece of their analysis. Further, related work is outlined in the supplementary   
+276 section I.
+
+# 77 6 Discussion and Future Work
+
+In this work, we presented a problem-centric framework when comparing transfer learning methods with meta-learning algorithms – using USL and MAML as the canonical representatives of transfer and meta-learning methods respectively. We showed the diversity coefficient of the popular MiniImagenet and Cifar-fs benchmark is low and that under a fair comparison – MAML is very similar to transfer learning with USL. This was also true even when decreasing the model size – removing the alternative hypothesis that the equivalence of MAML and transfer learning with USL held due to large models. Instead, this suggests strengthens our hypothesis that the diversity of the data might be the driving factor. The equivalence of MAML and USL also replicated in our synthetic experiments. Therefore, we challenge the suggestions from previous work $\textcircled { 1 1 9 }$ that only a good embedding can beat more effective than sophisticated meta-learning – especially in the low diversity regime. In addition, our synthetic experiments show a promising scenario where we can systematically differentiate meta-learning algorithms from transfer learning algorithms – which supports our actionable suggestion to use the diversity coefficient to effectively study meta-learning and transfer learning algorithms. We hope to study this in more depth in the future with real and synthetic data.
+
+We also have theoretical results from a statistical decision perspective in the supplementary section ?? that inspired this work and suggest that when the distance between tasks is zero – then the predictions of transfer learning, meta-learning and even a fixed model with no adaptation are all equivalent (with the l2 loss). The results are theoretically limited because we can only reason when the diversity is exactly zero, but regardless provided an interesting perspective to study and inspire empirical work.
+
+We hope this work inspires the community in meta-learning and machine learning to construct benchmarks from a problem-centric perspective – that go beyond large scale data sets – using have quantitative metrics.
+
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+376 [28] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell,   
+377 P. Mishkin, J. Clark, G. Krueger, and I. Sutskever, “Learning Transferable Visual Models From   
+378 Natural Language Supervision,” feb 2021.   
+379 [29] “The Dynamic Distance Between Learning Tasks: \* From Kolmogorov Complexity to Transfer   
+380 Learning via Quantum Physics and the Information Bottleneck of the Weights of Deep Networks,”   
+381 2018.   
+382 [30] A. Achille, G. B. Mbeng, and S. Soatto, “Dynamics and Reachability of Learning Tasks,” 2019.   
+383 [31] A. Achille, G. Paolini, G. Mbeng, and S. Soatto, “The Information Complexity of Learning   
+384 Tasks, their Structure and their Distance,” 2020.   
+385 [32] B. M. Lake, T. D. Ullman, J. B. Tenenbaum, and S. J. Gershman, “Building Machines That   
+386 Learn and Think Like People,” Behavioral and Brain Sciences, vol. 40, 2016.   
+387 [33] B. Miranda, “An empirical study of the properties of meta-learning - presentation,” Illinois   
+388 Digital Environment for Access to Learning and Scholarship (IDEALS), 2020.   
+389 [34] K. Liang, J. Y. Zhang, B. Wang, Z. Yang, O. Koyejo, and B. Li, “Uncovering the Connections   
+390 Between Adversarial Transferability and Knowledge Transferability,” 2021.
+
+# Checklist
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section ??. • Did you include the license to the code and datasets? [No] Code and data will be released if accepted.
+
+400 Please do not modify the questions and only use the provided macros for your answers. Note that the   
+401 Checklist section does not count towards the page limit. In your paper, please delete this instructions   
+402 block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [No]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [TODO]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [TODO]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [TODO]

parse/dev/x2WTG5bV977/x2WTG5bV977_content_list.json

@@ -0,0 +1,966 @@
+[
+    {
+        "type": "text",
+        "text": "The Curse of Low Task Diversity: On the Failure of Transfer Learning to Outperform MAML and their Empirical Equivalence ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Recently, it has been observed that a transfer learning solution might be all we need   \n2 to solve many few-shot learning benchmarks – thus raising important questions   \n3 about when and how meta-learning algorithms should be deployed. In this paper,   \n4 we seek to clarify these questions by 1. proposing a novel metric – the diversity   \n5 coefficient – to measure the diversity of tasks in a few-shot learning benchmark   \n6 and 2. by comparing MAML and transfer learning under fair conditions (same   \n7 architecture, same optimizer and all models trained to convergence). Using the   \n8 diversity coefficient, we show that the popular MiniImagenet and Cifar-fs few-shot   \n9 learning benchmarks have low diversity. This novel insight contextualizes claims   \n10 that transfer learning solutions are better than meta-learned solutions in the regime   \n11 of low diversity under a fair comparison. Specifically, we empirically find that a low   \n12 diversity coefficient correlates with a high similarity between transfer learning and   \n13 Model-Agnostic Meta-Learning (MAML) learned solutions in terms of accuracy   \n14 at meta-test time and classification layer similarity (using feature based distance   \n15 metrics like SVCCA, PWCCA, CKA, and OPD). To further support our claim,   \n16 we find this meta-test accuracy holds even as the model size changes. Therefore,   \n17 we conclude that in the low diversity regime, MAML and transfer learning have   \n18 equivalent meta-test performance when both are compared fairly. We also hope   \n19 our work inspires more thoughtful constructions and quantitative evaluations of   \n20 meta-learning benchmarks in the future. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "21 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
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+            312,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "22 The success of deep learning in computer vision (1; 2), natural language processing (3; 4), game   \n23 playing $\\textcircled { 5 } \\textcircled { 6 } \\textcircled { 7 }$ and more, keeps motivating a growing body of applications of deep learning on   \n24 an increasingly wide variety of domains. In particular, deep learning is now routinely applied to   \n25 few-shot learning – a research challenge that assesses a model’s ability to learn to adapt to new tasks,   \n26 new distributions, or new environments. This has been the main research area where meta-learning   \n27 algorithms have been applied – since such a strategy seems promising in a small data regime due to   \n28 its potential to learn to learn or learn to adapt. However, it was recently shown $\\textcircled{8}$ that a transfer   \n29 learning model with a fixed embedding can match and outperform many modern sophisticated meta  \n30 learning algorithms on numerous few-shot learning benchmarks $\\bigoplus \\iiiint \\bigoplus \\iiiint \\bigoplus \\iiiint$ . This growing body of   \n31 evidence – coupled with these surprising results in meta-learning – raise the question if researchers are   \n32 applying meta-learning with the right inductive biases $\\textcircled { 1 3 } ; \\textcircled { 1 4 }$ and designing appropriate benchmarks   \n33 for meta-learning. Our evidence suggests this is not the case.   \n34 In this work, we show that when the task diversity – a novel measure of variability across tasks – is   \n35 low, then MAML (Model Agnostic Meta-Learning) $\\textcircled{1 1 5 }$ learned solutions have the same accuracy   \n36 as transfer learning (i.e., a supervised learned model with a fine-tuned final linear layer). We want   \n37 to emphasize the importance of doing such an analysis fairly: with the same architecture, same   \n38 optimizer and all models trained to convergence. This empirical equivalence remained true even as   \n39 the model size changed – thus further suggesting this equivalence is more a property of the data than   \n40 of the model. Therefore, we suggest taking a problem-centric approach to meta-learning and suggest   \n41 applying Marr’s level of analysis $\\textcircled { 1 6 } ; \\textcircled { 1 7 } \\textcircled { }$ to few-shot learning – to identify the family of problems   \n42 suitable for meta-learning. Marr emphasized the importance of understanding the computational   \n43 problem being solved and not only analyzing the algorithms or hardware that attempts to solve   \n44 them. An example given by Marr is marveling at the rich structure of bird feathers without also   \n45 understanding the problem they solve is flight. Similarly, there has been analysis of MAML solutions   \n46 and transfer learning without putting the problem such solutions should solve into perspective $( \\overline { { 1 8 } } ; \\overline { { 1 9 } } )$ .   \n47 Therefore, in this work, we hope to clarify some of these results by partially placing the current   \n48 state of affairs in meta-learning from a problem-centric view. In addition, the novelty of our analysis   \n49 compared to previous work is that we make analysis intrinsic of the data as a first class citizen. ",
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+    },
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+    {
+        "type": "text",
+        "text": "50 Our contributions summarized as follows: ",
+        "bbox": [
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+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We propose a novel metric that quantifies the intrinsic diversity of the data of a few-shot learning benchmark. We call it the diversity coefficient. It enables analysis of meta-learning algorithms through a problem-centric framework. It also goes beyond counting the number of classes or number of data points or counting the number concatenated data sets – and instead quantifies the expected diversity/variability of tasks in a few-shot learning benchmark. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. We analyze the two most prominent few-shot learning benchmarks – MiniImagenet and Cifar-fs – and show that their diversity is low. These results are robust across different ways to measure the diversity coefficient, suggesting that our approach is robust. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. With this context, we partially clarify the surprising results from $\\textcircled { 1 1 9 }$ by comparing their transfer learning method against models trained with MAML $\\textcircled{1 1 5 }$ . In particular, when making a fair comparison, transfer learning method with a fixed feature extractor fails to outperform MAML. We define a fair comparison when the two methods are compared using the same architecture (backbone), same optimizer and all models trained to convergence. We also show that their final layer makes similar predictions according to neural network distance techniques like distance based Singular Value Canonical Correlation Analysis (SVCCA), Projection Weighted (PWCCA), Linear Centered Kernel Analysis (LINCKA) and Orthogonal Procrustes Distance (OPD). This equivalence holds even as the model size increases. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "4. Interestingly, we also find that even in the regime where task diversity is low (in MiniImagenet and Cifar-fs), the features extracted by supervised learning and MAML are different – implying that the mechanism by which they function is different despite the similarity of their final predictions. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "5. As an actionable conclusion, we provide a metric that can be used to analyze the intrinsic diversity of the data in a few-shot learning benchmarks and therefore build more thoughtful environments to drive research in meta-learning. In addition, our evidence suggests the following test to predict the empirical equivalence of MAML and transfer learning: if the task diversity is low, then transfer learned solutions might fail to outperform meta-learned solutions. This test is easy to run because our diversity coefficient can be done using the Task2Vec method $\\textcircled { 1 2 0 }$ using pre-trained neural network. We also found that random networks were consistent with the results of pre-trained networks on Imagenet. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "81 We hope that this line of work inspires a problem-centric first approach to meta-learning – which   \n82 appears to be especially sensitive to the properties of the problem in question. Therefore, we hope   \n83 future work takes a more thoughtful and quantitative approach to benchmark creation – instead of   \n84 focusing only on making huge data sets. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Background ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide a summary of the background needed to understand our main results. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "87 Model-Agnostic Meta-Learning (MAML): The MAML algorithm $\\textcircled{1 1 5 }$ attempts to meta-learn   \n88 an initialization of parameters for a neural network so that it is primed for fast gradient descent ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "adaptation. It consists of two main optimization loops: 1) an outer loop used to prime the parameters for fast adaptation, and 2) an inner loop that does the fast adaptation. During meta-testing, only the inner loop is used to adapt the representation learned by the outer loop. ",
+        "bbox": [
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+            92,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "92 Transfer Learning with Union Supervised Learning (USL): Previous work $\\textcircled { 1 9 }$ shows that   \n93 an initialization trained with supervised learning, on a union of all tasks, can outperform many   \n94 sophisticated methods in meta-learning. In particular, their method consists of two stages: 1) first   \n95 they use a union of all the labels in the few-shot learning benchmark during meta-training and train   \n96 with standard supervised learning (SL), then 2) during the meta-testing, they use an inference method   \n97 common in transfer learning: extract a fixed feature from the neural network and fully fine-tune the   \n98 final classification layer (i.e., the head). Note that our experiments only consider when the final layer   \n99 is regularized Logistic Regression trained with LBGFS.   \n100 Distances for Deep Neural Network Feature Analysis: To compute the distance between neural   \n101 networks we use the distance versions of Singular Value Canonical Correlation Analysis (SVCCA)   \n102 $\\textcircled { 2 1 }$ , Projection Weighted Canonical Correlation (PWCCA) $\\textcircled { 1 2 2 }$ , Linear Centered Kernel Analysis   \n103 (LINCKA) $\\textcircled { 1 2 3 }$ and Orthogonal Procrustes Distance (OPD) $\\textcircled { 1 2 4 }$ . These distances are in the interval   \n104 $[ 0 , 1 ]$ and are not necessarily a formal distance metric but are guaranteed to be zero when their   \n105 inputs are equal and nonzero otherwise. This is true because SVCCA, PWCCA, LINCKA are based   \n106 on similarity metrics and OPD is already a distance. Note that we use the formula $d ( X , Y ) =$   \n107 $1 - s i m ( X , Y )$ for our distance metrics where sim is one either SVCCA, PWCCA, LINCKA   \n108 similarity metric and $X , Y$ are matrices of activations (called layer matrices). The distance between   \n109 two models is computed by choosing a layer and then comparing the features/activations after   \n110 adaptation for that layer given a batch of tasks represented as a support and query set. A more   \n111 thorough overview of these metrics for the analysis of internal representations for convolutional   \n112 neural networks (CNNS) can be found in the appendix, section G.   \n113 Task2Vec Embeddings for Distances between Tasks: The diversity coefficient we propose is   \n114 the expectation of distance between tasks (explain in more detail in section $\\textcircled { 3 }$ ). Therefore, it is   \n115 essential to define the distance between different pairs of tasks. We choose the cosine distance   \n116 between Task2Vec (vectorial) embeddings as in $\\textcircled { 1 2 0 }$ . Therefore, we provide a summary of the   \n117 Task2Vec method to compute task embeddings. The vectorial representation of tasks provided by   \n118 Task2Vec $\\textcircled { 1 2 0 }$ is the vector of diagonal entries of the Fisher Information Matrix (FIM) given a fix   \n119 neural network as a feature extractor – also called a probe network – after fine-tuning the final   \n120 classification layer to the task. The authors explain this is a good vectorial representation of tasks   \n121 because 1. It approximately indicates the most informative weights for solving the current task   \n122 (up to a second order approximation) 2. For rich probe networks like CNNs, the diagonal is more   \n123 computationally tractable. We choose Task2Vec because the original authors provide extensive   \n124 evidence that their embeddings correlate with semantic and taxonomic relations between different   \n125 visual classes – making it a convincing embedding for tasks $\\textcircled { 1 2 0 }$ . The Task2Vec embedding of task $\\tau$   \n126 is the diagonal of the following matrix: ",
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+    },
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+    {
+        "type": "equation",
+        "img_path": "images/d565f89a75a741130d2b2ac1b006b7f47c44fe88a365ef18a6d744d24ae7ef05.jpg",
+        "text": "$$\n\\hat { F } _ { D _ { \\tau } , f _ { w } } = \\hat { F } ( D _ { \\tau } , f _ { w } ) = \\mathbb { E } _ { \\boldsymbol { x } , \\boldsymbol { y } \\sim \\hat { p } ( \\boldsymbol { x } | \\tau ) p ( \\boldsymbol { y } | \\boldsymbol { x } , f _ { w } ) } [ \\nabla _ { w } \\log p ( \\boldsymbol { y } \\mid \\boldsymbol { x } , f _ { w } ) \\nabla _ { w } p ( \\boldsymbol { y } \\mid \\boldsymbol { x } , f _ { w } ) ^ { \\top } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "127 where $f _ { w }$ is the neural networks used as a feature extractor with architecture $f$ and weights $w$ ,   \n128 $\\hat { p } ( x \\mid \\tau )$ is the empirical distribution defined by the training data $D _ { \\tau } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { \\bar { n } }$ for task $\\tau$ , and   \n129 $p ( y \\mid x , f _ { w } )$ is a deep neural network trained to approximate the (empirical) posterior $\\hat { p } ( y \\mid x , \\tau )$ .   \n130 We’d like to emphasize that the there is a dependence on target label since Task2Vec fixes the   \n131 feature extractor (using $f _ { w , }$ ) and then fits the final layer (or “head\") to approximate the task posterior   \n132 distribution $\\hat { p } ( y \\mid x , \\tau )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "133 3 Definition of the Diversity Coefficient ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            768,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "134 The diversity coefficient aims to measure the intrinsic diversity (or variability) of tasks in a few-shot   \n135 learning benchmark. At a high level, the diversity coefficient is the expected distance between a   \n136 pair of different tasks given a fixed probe network. In this work, we choose the distance to be the   \n137 cosine distance between vectorial representations (i.e. embeddings) of tasks according to Task2Vec   \n138 as described in section $2 .$ Using a fixed probe networks is essential because: 1. Using a fixed probe   \n139 network means that the distances between different tasks are comparable, as discussed in the original   \n140 Task2Vec $\\textcircled{20 }$ and 2. Since we are computing the distance between different tasks, we need to make   \n141 sure the difference comes from intrinsic properties of the data and not from a different source, e.g. if   \n142 one uses different models then this might confound the source of variability in our metric. We define   \n143 the diversity coefficient of a few-shot learning benchmark $B$ as follows: ",
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+    },
+    {
+        "type": "text",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/19a7b4f3cf0df083d6e9b534a9eecfeacb5289beb80c9c30008a9a375688c452.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { d i } v ( B ) = \\mathbb { E } _ { \\tau _ { 1 } \\sim \\hat { p } ( \\tau | B ) , \\tau _ { 2 } \\sim \\hat { p } ( \\tau | B ) } \\mathbb { E } _ { D _ { 1 } \\sim \\hat { p } ( x _ { 1 } , y _ { 1 } | \\tau _ { 1 } ) , D _ { 2 } \\sim \\hat { p } ( x _ { 2 } , y _ { 2 } | \\tau _ { 2 } ) } \\left[ d ( \\hat { F } _ { D _ { 1 } , f _ { w } } , \\hat { F } _ { D _ { 2 } , f _ { w } } ) \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "144 where $f _ { w }$ is the neural networks used as a feature extractor with architecture $f$ and weights $w$ ,   \n145 $\\hat { p } ( x \\mid \\tau )$ is the empirical distribution defined by the training data $D _ { \\tau } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ for task $\\tau$   \n146 $\\tau _ { 1 } , \\tau _ { 2 }$ are tasks sampled from the empirical distribution of tasks $\\hat { p } ( \\tau \\mid B )$ for the current benchmark   \n147 $B$ (i.e. a batch of tasks with their data sets $\\mathcal { D } = ( \\tau _ { i } , D _ { \\tau _ { i } } ) _ { i = 1 } ^ { N } )$ , a task $\\tau _ { i }$ is the probability distribution   \n148 $p ( x , y \\mid \\tau )$ of the data, is a distance metric (for us cosine), $f _ { w }$ is the neural networks used as   \n149 a feature extractor with architecture $f$ and weights $w$ , and $\\hat { p } ( x \\mid \\tau )$ is the empirical distribution   \n150 defined by the training data $D _ { \\tau } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ for task $\\tau$ . We’d also like to recall the reader that the   \n151 definition of a task in this setting is of a n-way, $\\mathbf { k }$ -shot few-shot learning task. Therefore, each task has   \n152 n classes sampled with $\\mathrm { k }$ examples used for the adaptation. We’d like to emphasize that the adaptation   \n153 here is only to fine-tune the final layer according to the Task2Vec method for the correct computation   \n154 of the FIM. Therefore, in this setting we combine the support and query set as the split is not relevant   \n155 for the computation of the task embedding using Task2Vec. Note that the above formulation can be   \n156 easily adapted to any distance function between tasks, and is not necessarily specific to using the   \n157 FIM or cosine distance. For example, given the true distributions for tasks one can use real distances   \n158 between probability distributions e.g. Hellinger distance. In addition, it is obvious one can use a   \n159 distance function besides the cosine distance – but choose it in accordance to the original work of   \n160 Task2Vec $\\textcircled { 1 2 0 }$ . ",
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            163,
+            409,
+            312,
+            426
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "62 This section explains the experiments backing up our main results outlined in our list of contributions.   \n63 Experimental details are provided in the supplementary section $\\bigstar$ and the learning curves displaying   \n64 the convergence for a fair comparison are in supplementary section B. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "165 4.1 The Diversity Coefficient of MiniImagenet and Cifar-fs ",
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+        "text": "66 To put our analysis into a problem-centric framework, we first analyze the problem they are trying   \n7 to solve through the diversity coefficient. Recall that the diversity coefficient aims to quantify the   \n8 intrinsic variation of tasks in a few-shot learning benchmark. We show that the diversity coefficient   \n69 of the popular MiniImagenet and Cifar-fs benchmarks are low with good confidence intervals using   \n70 four different probe networks in table 1. ",
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+        "table_body": "<table><tr><td>Probe Network</td><td>Diversity on MI</td><td>Diversity on Cifar-fs</td></tr><tr><td>Resnet18 (pt)</td><td>0.117 ± 2.098e-5</td><td>0.100 ± 2.18e-5</td></tr><tr><td>Resnet18 (rand)</td><td>0.0955 ± 1.29e-5</td><td>0.103 ± 1.05e-5</td></tr><tr><td>Resnet34 (pt)</td><td>0.0999 ± 1.95e-5</td><td>0.0847 ± 3.06e-5</td></tr><tr><td>Resnet34 (rand)</td><td>0.0620 ± 8.12e-6</td><td>0.0643 ± 9.64e-6</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: The diversity coefficient of MiniImagenet (MI) and Cifar-fs is low. The diversity coefficient was computed using the cosine distance between different standard n-way, $\\mathbf { k }$ -shot classification tasks from the few-shot learning benchmark using the Task2Vec method described in section $3 .$ We used $\\mathrm { n } { = } 5$ (number of classes) and ${ \\bf k } = 2 0$ (number of examples per class) since we can use the whole task data to compute the diversity coefficient (no splitting of support and query set required). We used Resnet18 and Resnet34 networks as probe networks – both pre-trained on ImageNet (indicated as “pt\" on table) and randomly initialized (indicated as “rand\" on table). We observe that both type of networks and weights give similar diversity results. All confidence intervals were at $9 5 \\%$ . To compute results, we used 500 few-shot learning tasks and only compare pairs of different tasks. This results in $( 5 0 0 ^ { 2 } - 5 0 0 ) / 2 = 1 2 4 ,$ 750 pair-wise distances used to compute the diversity coefficient. ",
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+        "type": "text",
+        "text": "171 4.2 Low Diversity Correlates with Equivalence of MAML and Transfer Learning ",
+        "text_level": 1,
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+        "text": "172 Now that we have placed ourselves in a problem-centric framework and shown the diversity coefficient   \n173 of the popular MiniImagenet and Cifar-fs benchmarks are low – we proceed to show the failure of   \n174 transfer learning (with USL) to outperform MAML. Crucially, the analysis was done using a fair   \n175 comparison: using the same model architecture, optimizer, and training all models to convergence   \n176 – details in section A. We used the five-layer CNN used in $\\textcircled { 1 5 } \\textcircled { 2 5 }$ and Resnet12 as in $\\textcircled { 1 9 }$ . We   \n177 provide evidence that in the setting of low diversity: ",
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+        "text": "1. The accuracy of an adapted MAML meta-learner vs. an adapted USL pre-trained model are similar and statistically significant, except for one result where transfer learning with USL is worse. This is shown in table 2 and 1.   \n2. The distance for the classification layer decreases sharply according to four distance-based metrics – SVCCA, PWCCA, LINCKA, and OPD – as shown in figure $2 .$ This implies the predictions of the two are similar. ",
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+        "text": "184 For the first point, we emphasize that tables 1 and table 2 taken together support our central hypothesis:   \n185 that models trained with meta-learning are not inferior to transfer learning models (using USL) when   \n186 the diversity coefficient is low. Careful inspection reveals that the methods have the same meta-test   \n187 accuracy with intersecting confidence intervals – making the results statistically significant across   \n188 few-shot benchmarks and architectures. The one exception is the third set of bar plots, where transfer   \n189 learning with USL is in fact worse.   \n190 For the second point, refer to figure $\\nsubseteq$ and observe that as the depth of the network increases, the   \n191 distance between the activation layers of a model trained with MAML vs USL increases until it   \n192 reaches the final classification layer – where all four metrics display a noticeable dip. In particular,   \n193 PWCCA considers the two prediction layers identical (approximately zero distance). This final point   \n194 is particularly interesting because PWCCA is weighted according to the CCA weights that stabilize   \n195 with the final predictions of the network. This means that the PWCCA distance value is reflective of   \n196 what the networked actually learned and gives a more reliable distance metric (for details, refer to the   \n197 appendix section G.5). This is important because this supports our main hypothesis: that at prediction   \n198 time there is an equivalence between transfer learning and MAML when the diversity coefficient is   \n199 low. ",
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+        "img_path": "images/a9b91dba46f4e1bb66a582521775272ad42f050a687fee3846c532e28aaf2892.jpg",
+        "image_caption": [
+            "Figure 1: MAML trained models and union supervised trained (USL) models have statistically equivalent meta-test accuracy for MiniImagenet and Cifar-fs with Resnet12 and five layer CNNs. This holds for both the Resnet12 architecture used in $\\textcircled { 1 9 }$ and the 5 layer CNN (indicated as “5CNN\") in $\\textcircled { 1 2 5 }$ . Results used a (meta) batch-size of 100 tasks and $9 5 \\%$ confidence intervals. All MAML models were trained with 5 inner steps during meta-training. “MAML5\" and “MAML10\" in the bar plot indicates the adaptation method used at test time i.e. we used 5 inner steps and 10 inner steps at test time. MiniImagenet is abbreviated as “MI\" in the figure. "
+        ],
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+        "type": "text",
+        "text": "00 4.3 Is the Equivalence of MAML and Transfer Learning related to Model Size or Low 01 Diversity? ",
+        "text_level": 1,
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+        "text": "An alternative hypothesis to explain the equivalence of transfer learning (with USL) and MAML could be due to the capabilities of large neural networks to be better meta-learners in general. Inspired by the impressive ability of large language models to be few-shot (or even zero-shot) learners $\\textcircled { 1 4 } \\textcircled { 2 7 } \\textcircled { 2 8 } \\textcircled { 3 } \\textcircled { - }$ we hypothesized that perhaps the meta-learning capabilities of deep learning models is a function of the model size. If this were true, then we expected to see the difference in meta-test accuracy ",
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+        "table_caption": [],
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+        "table_body": "<table><tr><td>Meta-train Initialization </td><td>Adaptation at Inference</td><td>Meta-test Accuracy</td></tr><tr><td>Random</td><td>no adaptation</td><td>19.3 ± 0.80</td></tr><tr><td>MAML0</td><td>no adaptation</td><td>20.0 ± 0.00</td></tr><tr><td>USL</td><td>no adaptation</td><td>15.0 ± 0.26</td></tr><tr><td>Random</td><td>MAML5 adaptation</td><td>34.2 ± 1.16</td></tr><tr><td>MAML5</td><td>MAML5 adaptation</td><td>62.4 ± 1.64</td></tr><tr><td>USL</td><td>MAML5 adaptation</td><td>25.1 ± 0.98</td></tr><tr><td>Random</td><td>MAML10 adaptation</td><td>34.1 ± 1.23</td></tr><tr><td>MAML5</td><td>MAML10 adaptation</td><td>62.3 ± 1.50</td></tr><tr><td>USL</td><td>MAML10 adaptation</td><td>25.1 ± 0.97</td></tr><tr><td>Random</td><td>Adapt Head only (with LR)</td><td>40.2 ± 1.30</td></tr><tr><td>MAML5</td><td>Adapt Head only (with LR)</td><td>59.7 ± 1.37</td></tr><tr><td>USL</td><td>Adapt Head only (with LR)</td><td>60.1 ± 1.37</td></tr></table>",
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+        "text": "Table 2: MAML trained representations and supervised trained representation have statistically equivalent meta-test accuracy on MiniImagenet – which has low diversity. The transfer model’s adaptation is labeled as “Adapted Head only (with LR)\" – which stands for “Logistic Regression (LR)\" used in $\\textcircled{1 1 9 }$ . More precisely, we used Logistic Regression (LR) with LBFGS with the default value for the l2 regularization parameter given by Python’s Sklearn. Note that an increase in inner steps from 5 to 10 with the MAML5 trained model does not provide an additional meta-test accuracy boost, consistent with previous work $\\textcircled { 1 2 6 }$ . Note that the fact that the MAML5 representation matches the USL representation when both use the same adaptation method is not surprising – given that: 1) previous work has shown that the distance between the body of an adapted MAML model is minimal compared to the unadapted MAML (which we reproduce in $\\boxed { 5 }$ in the green line) and 2) the fact that a MAML5 adaptation is only 5 steps of MAML while LR fully converges the prediction layer. We want to highlight that only the MAML5 model achieved the maximum meta-test performance of 0.6 with the MAML5 adaptation – suggesting that the USL and MAML5 meta-learning algorithms might learn different representations. For USL to have a fair comparison during meta-test time when using the MAML adaptation, we provide the MAML final layer learned initialization parameters to the USL model (but any is fine due to convexity when using a fixed feature extractor). This is needed since during meta-training USL is trained with a union of all the labels (64) – so it does not even have the right output size of 5 for few-shot prediction. Meta-testing was done in the standard 5-way, 5-shot regime. ",
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+    {
+        "type": "text",
+        "text": "207 of MAML and USL to be larger for smaller models and the difference to decrease as the model   \n208 size increased. Once the two models were, of the same size but large enough, we hypothesized that   \n209 the meta-test accuracy would be the same. We tested this to rule out that our observations were a   \n210 consequence of the model size. The results were negative and surprisingly the equivalence between   \n211 MAML and USL seems to hold even as the model increased – strengthening our hypothesis that the   \n212 low task diversity might be a bigger factor explaining our observations. We show this in figure $^ { 3 , }$   \n213 and we want to draw attention to the fact this statistical equivalence holds even when using only four   \n214 filters – the case where we expected the biggest difference. ",
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+        "type": "text",
+        "text": "15 4.4 MAML learns a different base model compared to Union Supervised Learned models – 16 even in the presence of low task diversity ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The first four layers of figure 2 shows how large the distance is of a MAML representation compared to a SL representation. In particular, it is much larger than the distance value in the range $[ 0 , 0 . 1 ]$ from previous work that compared MAML vs. adapted MAML $\\textcircled { 1 8 }$ . We reproduced that and indeed MAML vs. adapted MAML has a small difference (smaller for us) – supporting our observations that a MAML vs. a USL learned representations are different at the feature extractor layer even when the diversity is low. Results are statistically significant. ",
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+        "image_caption": [
+            "Figure 2: The classification layer of transfer learning and a MAML5 model decrease in distance – implying similar predictions. More precisely, an initialization trained with 5 inner steps (MAML5) has an increasingly similar head (classifier) after adaptation with MAML5 compared to the classifier layer of the union supervise learned (USL) model that has been adapted only at the final layer. In particular, the USL model has been adapted with Logistic Regression (LR) with LBFGS with the default value for the l2 regularization parameter given by Python’s Sklearn (as in $\\textcircled { 1 1 9 }$ ). We showed this trend with four different distance metrics SVCCA, PWCCA, LICKA, and OPD referenced in section $\\bigstar$ Observe that according to PWCCA the distance between the predictions is zero. This is true because the distance of classification layer (indicated as “head\" in the figure) is zero. The architecture used here is a five layer CNN as in $\\textcircled { 1 5 } , \\textcircled { 2 5 } )$ with their same setup. The benchmark used for this analysis is MiniImagenet. "
+        ],
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+    {
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+        "image_caption": [
+            "Figure 3: The meta-test accuracy of MAML and transfer learning using USL is similar in a statistically significant way – regardless of the model size. In this experiment, we used the MiniImagenet benchmark, the five layer CNN used in $\\textcircled { 1 5 } \\textcircled { 2 5 }$ , and only increased the filter size using sizes 4, 8, 16, and 32. We made sure the comparison was fair by using the same architecture, optimizer, and trained all models to convergence. During meta-training, the MAML model was trained using 5 inner steps. The legends indicating MAMl5 and MAML10 refer to the number of inner steps used at test time. We used a (meta) batch size of 100 tasks. "
+        ],
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+    {
+        "type": "text",
+        "text": "In this section, we show the closeness of MAML and transfer learning (with USL) for synthetic experiments for low and high diversity regimes in Figure $\\textcircled { 4 }$ In the low regime, the two methods are equivalent in a statistically significant way – which supports the main claims of our paper. As the diversity increases, however, the difference between USL and MAML increases (in favor of USL). This will be explored further in future work. ",
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+        "type": "text",
+        "text": "The task is the usual n-way, $\\mathbf { k }$ -shot tasks, but the data comes from a Gaussian and the meta-learners are tasked with classifying from which Gaussian the data points came from in a few-shot learning manner. Benchmarks are created by sampling a Gaussian distribution with means moving away from the origin as the benchmark changes. Therefore, the Gaussian benchmark with the highest diversity coefficient has Gaussians that are the furthest from the origin. We computed the diversity coefficient using a proper distance between distributions using the Hellinger distance eluded in section $3$ instead of the FIM distance. We can do this because we know the ground truth distribution in our synthetic experiments, and Gaussians have a closed form Hellinger distance. Details on the n-way Gaussian benchmark and diversity coefficient using the Hellinger distance can be found in supplementary section E and F. ",
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+        "img_path": "images/699c005e17187b208d460b65ed9d829c88d2ab9c6d3faa2d76636d5608875fd0.jpg",
+        "image_caption": [
+            "Figure 4: The meta-test accuracy of MAML and transfer learning using USL is similar in a statistically equivalent way in the low diversity regime in the 5-way, 10-shot Gaussian Benchmarks. MAML models were trained with 5 inner steps. MAML5 and MAML10 indicate the adaptation procedure at test time. Results used a (meta) batch-size of 500 tasks and $9 5 \\%$ confidence intervals. As the diversity of the benchmark increases, the Gaussian tasks are sampled further away from the origin. Note, as the diversity increases, the difference between USL and MAML increases (in favor of USL). "
+        ],
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+    {
+        "type": "text",
+        "text": "240 5 Related Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "241   \n242   \n243   \n244   \n245   \n246   \n247   \n248   \n249   \n250   \n251   \n252   \n253   \n254 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our work proposes a problem-centric framework for the analysis of meta-learning algorithms inspired from previous puzzling results $\\textcircled { 1 9 }$ . We propose to use a pair-wise distance between tasks and analyze how this metric might correlate with meta-learning. The closest line of work for this is the long line of work by $\\textcircled { 1 2 0 }$ where they suggest methods to analyze the complexity of a task, propose unsymmetrical distance metrics for data sets, reachability of tasks with SGD, ways to embed entire data sets and more $( | 2 0 ; | 2 9 ; | 3 0 ; | 3 1 | )$ . We believe this line of work to be very fruitful and hope that more people adopt tools like the ones they suggest and we propose in this paper before researching or deploying meta-learning algorithms. We hope this helps meta-learning methods succeed in practice – since cognitive science suggests meta-learning is a powerful method humans use to learn $\\textcircled{3 2 }$ . In the future, we hope to compare $\\textcircled { 1 2 0 }$ ’s distance metrics between tasks with ours to provide a further unified understanding of meta-learning and transfer learning. A contrast between their work and ours is that we focus our analysis from a meta-learning perspective applied to few-shot learning – while their focus is understanding transfer learning methods between data sets. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The use of a distance metric in our definition of the diversity coefficient is inspired by the analysis 255 done by $\\textcircled { 1 8 }$ . They showed that MAML functions mainly via feature re-use than by rapid learning i.e., 256 that a model trained with MAML changes very little after the MAML adaptation. The main difference ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "257 of their work with our is: 1) that we compare MAML trained models against union supervised learned   \n258 models (USL) instead of only comparing MAML against adapted MAML, and 2) that we explicitly   \n259 analyzed properties of the data sets. In addition, we use a large set of distance metrics for our analysis   \n260 including: SVCCA, PWCCA, LINCKA and OPD as proposed by (21; 22; 23; 24). ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Our work is most influenced by previous work suggesting modern meta-learning requires rethinking $\\textcircled{1 1 9 }$ The main difference of our work with theirs is that we analyzed the internal representation of the meta-learning algorithms and contextualize these with quantifiable metrics of the problem being solved. Unlike their work, we focused on a fair comparison between meta-learning methods by ensuring the same neural network backbone was used. Another difference is that they gained further accuracy gains by using distillation – a method we did not analyze and leave for future work. ",
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+    {
+        "type": "text",
+        "text": "267 A related line of work $\\textcircled { 1 3 3 } \\textcircled { 2 6 }$ first showed that there exist synthetic data sets that are capable of   \n268 exhibiting higher degrees of adaptation as compared to the original work by $\\textcircled { 1 1 8 }$ . The difference is   \n269 that they did not compare MAML models against transfer learning methods like we did here. Instead,   \n270 they focused on comparing adapted MAML models vs. unadapted MAML models.   \n271 Another related line of work is the predictability of adversarial transferability and transfer learning.   \n272 They show this both theoretically and with extensive experiments $\\textcircled{3 4 }$ . The main difference between   \n273 their work and ours is that they focus their analysis mainly on transfer learning, while we concentrated   \n274 on meta-learning for few-shot learning. In addition, we did not consider adversarial transferability –   \n275 while that was a central piece of their analysis. Further, related work is outlined in the supplementary   \n276 section I. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "77 6 Discussion and Future Work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "In this work, we presented a problem-centric framework when comparing transfer learning methods with meta-learning algorithms – using USL and MAML as the canonical representatives of transfer and meta-learning methods respectively. We showed the diversity coefficient of the popular MiniImagenet and Cifar-fs benchmark is low and that under a fair comparison – MAML is very similar to transfer learning with USL. This was also true even when decreasing the model size – removing the alternative hypothesis that the equivalence of MAML and transfer learning with USL held due to large models. Instead, this suggests strengthens our hypothesis that the diversity of the data might be the driving factor. The equivalence of MAML and USL also replicated in our synthetic experiments. Therefore, we challenge the suggestions from previous work $\\textcircled { 1 1 9 }$ that only a good embedding can beat more effective than sophisticated meta-learning – especially in the low diversity regime. In addition, our synthetic experiments show a promising scenario where we can systematically differentiate meta-learning algorithms from transfer learning algorithms – which supports our actionable suggestion to use the diversity coefficient to effectively study meta-learning and transfer learning algorithms. We hope to study this in more depth in the future with real and synthetic data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also have theoretical results from a statistical decision perspective in the supplementary section ?? that inspired this work and suggest that when the distance between tasks is zero – then the predictions of transfer learning, meta-learning and even a fixed model with no adaptation are all equivalent (with the l2 loss). The results are theoretically limited because we can only reason when the diversity is exactly zero, but regardless provided an interesting perspective to study and inspire empirical work. ",
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+    {
+        "type": "text",
+        "text": "We hope this work inspires the community in meta-learning and machine learning to construct benchmarks from a problem-centric perspective – that go beyond large scale data sets – using have quantitative metrics. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "[1] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet Classification with Deep Convolutional Neural Networks,”   \n[2] K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2016-December, pp. 770–778, dec 2015.   \n[3] J. Devlin, M. W. Chang, K. Lee, and K. Toutanova, “BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding,” NAACL HLT 2019 - 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies - Proceedings of the Conference, vol. 1, pp. 4171–4186, oct 2018.   \n[4] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, S. Agarwal, A. Herbert-Voss, G. Krueger, T. Henighan, R. Child, A. Ramesh, D. M. Ziegler, J. Wu, C. Winter, C. Hesse, M. Chen, E. Sigler, M. Litwin, S. Gray, B. Chess, J. Clark, C. Berner, S. Mccandlish, A. Radford, I. Sutskever, and D. A. Openai, “Language Models are Few-Shot Learners,” tech. rep., 2020.   \n[5] D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 2016 529:7587, vol. 529, pp. 484–489, jan 2016.   \n[6] V. Mnih, K. Kavukcuoglu, D. Silver, A. Graves, I. Antonoglou, D. Wierstra, and M. Riedmiller, “Playing Atari with Deep Reinforcement Learning,”   \n[7] W. Ye, S. Liu, T. Kurutach, P. Abbeel, Y. Gao, T. University, U. C. Berkeley, S. Qi, and Z. Institute, “Mastering Atari Games with Limited Data,” oct 2021.   \n[8] Y. Tian, Y. Wang, D. Krishnan, J. B. Tenenbaum, and P. Isola, “Rethinking Few-Shot Image Classification: a Good Embedding Is All You Need?,” 2020.   \n[9] W.-Y. Chen, Y.-C. Liu, Z. Kira, Y.-C. F. Wang, and J.-B. Huang, “A Closer Look at Few-shot Classification,” 7th International Conference on Learning Representations, ICLR 2019, 2019.   \n[10] Y. Chen, X. Wang, Z. Liu, H. Xu, and T. Darrell, “A New Meta-Baseline for Few-Shot Learning,” tech. rep.   \n[11] G. S. Dhillon, P. Chaudhari, A. Ravichandran, and S. Soatto, “A Baseline for Few-Shot Image Classification,” 2019.   \n[12] S. Huang and D. Tao, “All you need is a good representation: A multi-level and classifier-centric representation for few-shot learning,” 2019.   \n[13] T. M. Mitchell, “The Need for Biases in Learning Generalizations by The Need for Biases in Learning Generalizations,” 1980.   \n[14] S. B.-D. Shai Shalev-Shwartz, “Understanding Machine Learning: From Theory to Algorithms,” Cambridge University Press, 2014.   \n[15] C. Finn, P. Abbeel, and S. Levine, “Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks,” 2017.   \n[16] J. B. Hamrick Deepmind and S. M. Deepmind, “LEVELS OF ANALYSIS FOR MACHINE LEARNING,” 2020.   \n[17] D. Marr, “Vision: A Computational Investigation into the Human Representation and Processing of Visual Information,” Phenomenology and the Cognitive Sciences, vol. 8, no. 4, p. 397, 1982.   \n[18] A. Raghu, M. Raghu, S. Bengio, and G. Brain, “Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAML,” tech. rep., 2020.   \n[19] Y. Tian, Y. Wang, D. Krishnan, J. B. Tenenbaum, and P. Isola, “Rethinking Few-Shot Image Classification: a Good Embedding Is All You Need?,” 2020.   \n[20] A. Achille UCLA, M. Lam AWS, R. Tewari AWS, A. Ravichandran AWS, S. Maji UMass, S. Soatto UCLA, and P. Perona Caltech, “TASK2VEC: Task Embedding for Meta-Learning Charless Fowlkes UCI and AWS,” tech. rep., 2019.   \n[21] M. Raghu, J. Gilmer, J. Yosinski, and J. Sohl-Dickstein, “SVCCA: Singular Vector Canonical Correlation Analysis for Deep Learning Dynamics and Interpretability,” tech. rep., 2017. ",
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+        "text": "353 [22] A. S. Morcos, Deepmind, M. Raghu, S. Bengio, and G. Brain, “Insights on representational   \n354 similarity in neural networks with canonical correlation,” tech. rep., 2018.   \n355 [23] S. Kornblith, M. Norouzi, H. Lee, and G. Hinton, “Similarity of Neural Network Representations   \n356 Revisited,” tech. rep., may 2019.   \n357 [24] F. Ding, J.-S. Denain, and J. Steinhardt, “Grounding Representation Similarity with Statistical   \n358 Testing,” 2021.   \n359 [25] S. Ravi and H. Larochelle, “Optimization as a model for few-shot learning,” tech. rep., 2017.   \n360 [26] B. Miranda, “An empirical study of the properties of meta-learning - presentation,” Illinois   \n361 Digital Environment for Access to Learning and Scholarship (IDEALS), dec 2020.   \n362 [27] R. Bommasani, D. A. Hudson, E. Adeli, R. Altman, S. Arora, S. von Arx, M. S. Bernstein,   \n363 J. Bohg, A. Bosselut, E. Brunskill, E. Brynjolfsson, S. Buch, D. Card, R. Castellon, N. Chatterji,   \n364 A. Chen, K. Creel, J. Q. Davis, D. Demszky, C. Donahue, M. Doumbouya, E. Durmus, S. Ermon,   \n365 J. Etchemendy, K. Ethayarajh, L. Fei-Fei, C. Finn, T. Gale, L. Gillespie, K. Goel, N. Goodman,   \n366 S. Grossman, N. Guha, T. Hashimoto, P. Henderson, J. Hewitt, D. E. Ho, J. Hong, K. Hsu,   \n367 J. Huang, T. Icard, S. Jain, D. Jurafsky, P. Kalluri, S. Karamcheti, G. Keeling, F. Khani,   \n368 O. Khattab, P. W. Kohd, M. Krass, R. Krishna, R. Kuditipudi, A. Kumar, F. Ladhak, M. Lee,   \n369 T. Lee, J. Leskovec, I. Levent, X. L. Li, X. Li, T. Ma, A. Malik, C. D. Manning, S. Mirchandani,   \n370 E. Mitchell, Z. Munyikwa, S. Nair, A. Narayan, D. Narayanan, B. Newman, A. Nie, J. C.   \n371 Niebles, H. Nilforoshan, J. Nyarko, G. Ogut, L. Orr, I. Papadimitriou, J. S. Park, C. Piech,   \n372 E. Portelance, C. Potts, A. Raghunathan, R. Reich, H. Ren, F. Rong, Y. Roohani, C. Ruiz,   \n373 J. Ryan, C. Ré, D. Sadigh, S. Sagawa, K. Santhanam, A. Shih, K. Srinivasan, A. Tamkin,   \n374 R. Taori, A. W. Thomas, F. Tramèr, R. E. Wang, and W. Wang, “On the Opportunities and Risks   \n375 of Foundation Models,” aug 2021.   \n376 [28] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell,   \n377 P. Mishkin, J. Clark, G. Krueger, and I. Sutskever, “Learning Transferable Visual Models From   \n378 Natural Language Supervision,” feb 2021.   \n379 [29] “The Dynamic Distance Between Learning Tasks: \\* From Kolmogorov Complexity to Transfer   \n380 Learning via Quantum Physics and the Information Bottleneck of the Weights of Deep Networks,”   \n381 2018.   \n382 [30] A. Achille, G. B. Mbeng, and S. Soatto, “Dynamics and Reachability of Learning Tasks,” 2019.   \n383 [31] A. Achille, G. Paolini, G. Mbeng, and S. Soatto, “The Information Complexity of Learning   \n384 Tasks, their Structure and their Distance,” 2020.   \n385 [32] B. M. Lake, T. D. Ullman, J. B. Tenenbaum, and S. J. Gershman, “Building Machines That   \n386 Learn and Think Like People,” Behavioral and Brain Sciences, vol. 40, 2016.   \n387 [33] B. Miranda, “An empirical study of the properties of meta-learning - presentation,” Illinois   \n388 Digital Environment for Access to Learning and Scholarship (IDEALS), 2020.   \n389 [34] K. Liang, J. Y. Zhang, B. Wang, Z. Yang, O. Koyejo, and B. Li, “Uncovering the Connections   \n390 Between Adversarial Transferability and Knowledge Transferability,” 2021. ",
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+        "text": "The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example: ",
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+        "text": "• Did you include the license to the code and datasets? [Yes] See Section ??. • Did you include the license to the code and datasets? [No] Code and data will be released if accepted. ",
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+        "text": "400 Please do not modify the questions and only use the provided macros for your answers. Note that the   \n401 Checklist section does not count towards the page limit. In your paper, please delete this instructions   \n402 block and only keep the Checklist section heading above along with the questions/answers below. ",
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