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+# REPRESENTING INFERENTIAL UNCERTAINTY IN DEEP NEURAL NETWORKS THROUGH SAMPLING
+
+Patrick McClure & Nikolaus Kriegeskorte   
+MRC Cognition and Brain Sciences Unit   
+University of Cambridge   
+Cambridge, UK   
+{patrick.mcclure,nikolaus.kriegeskorte}@mrc-cbu.cam.ac.uk
+
+# ABSTRACT
+
+As deep neural networks (DNNs) are applied to increasingly challenging problems, they will need to be able to represent their own uncertainty. Modelling uncertainty is one of the key features of Bayesian methods. Bayesian DNNs that use dropout-based variational distributions and scale to complex tasks have recently been proposed. We evaluate Bayesian DNNs trained with Bernoulli or Gaussian multiplicative masking of either the units (dropout) or the weights (dropconnect). We compare these Bayesian DNNs ability to represent their uncertainty about their outputs through sampling during inference. We tested the calibration of these Bayesian fully connected and convolutional DNNs on two visual inference tasks (MNIST and CIFAR-10). By adding different levels of Gaussian noise to the test images in z-score space, we assessed how these DNNs represented their uncertainty about regions of input space not covered by the training set. These Bayesian DNNs represented their own uncertainty more accurately than traditional DNNs with a softmax output. We find that sampling of weights, whether Gaussian or Bernoulli, led to more accurate representation of uncertainty compared to sampling of units. However, sampling units using either Gaussian or Bernoulli dropout led to increased convolutional neural network (CNN) classification accuracy. Based on these findings we use both Bernoulli dropout and Gaussian dropconnect concurrently, which approximates the use of a spike-andslab variational distribution. We find that networks with spike-and-slab sampling combine the advantages of the other methods: they classify with high accuracy and robustly represent the uncertainty of their classifications for all tested architectures.
+
+# 1 INTRODUCTION
+
+Deep neural networks (DNNs), particularly convolutional neural networks (CNN), have recently been used to solve complex perceptual and decision tasks (Krizhevsky et al., 2012; Mnih et al., 2015; Silver et al., 2016). However, these networks fail to model the uncertainty of their predictions or actions. Although many networks deterministically map an input to a probabilistic prediction, they do not model the uncertainty of that mapping. In contrast, Bayesian neural networks (NNs) attempt to learn a distribution over their parameters thereby offering uncertainty estimates for their outputs (MacKay, 1992; Neal, 2012). However, these methods do not scale well due to the difficulty in computing the posterior of a network’s parameters.
+
+One type of method for sampling from the posteriors of these networks is Hamiltonian Monte Carlo (HMC) (Neal, 2012). These techniques use the gradient information calculated using backpropagation to perform Markov chain Monte Carlo (MCMC) sampling by randomly walking through parameter space. proposed stochastic gadient Langevien dynamcis (SGLD)
+
+Approximate methods, in particular variational inference, have been used to make Bayesian NNs more tractable (Hinton & Van Camp, 1993; Barber & Bishop, 1998; Graves, 2011; Blundell et al., 2015). Due in large part to the fact that these methods substantially increase the number of parameters in a network, they have not been applied to large DNNs, such as CNNs. Gal & Ghahramani (2016) and Kingma et al. (2015) bypassed this issue by developing Bayesian CNNs using dropout (Srivastava et al., 2014). Dropout is a widely used regularization technique where units are dropped out of a network with a probability $p$ during training and the output of all unit are multiplied by $p$ during inference. A similar technique is dropconnect (Wan et al., 2013), which drops network connections instead of units. Gal & Ghahramani (2015) detailed how dropping units was equivalent to sampling weights from a Bernoulli-based variational distribution and that in order to make a DNN with dropout Bayesian, sampling should be used during both training and inference. Monte-Carlo (MC) sampling at inference allows a DNN to efficiently model a distribution over its outputs. One limitation of the Bayesian dropout method is that it does not model the uncertatiniy of each network parameter. The uncertainty of a DNN can then be calculated using this probability distribution. In addition to Bernoulli and Gaussian distributions, there has also been work done using spike-anslab distributions (Louizos, 2015), a combination of the two, which has been shown to increase the quality of linear regression (Ishwaran & Rao, 2005). Interestingly, dropout and dropconnect can be seen as approximations to spike-and-slab distributions for units and weights, respectively (Louizos, 2015; Gal, 2016; Li et al., 2016). Dropout- and dropconnect-based variational DNNs are dependent on the dropout probability, which is often used as a hyperparameter. However, work has been done on automatically learning the dropout probability during training for dropconnect (Louizos, 2015) using spike-and-slab distributions and Gaussian dropout (Kingma et al., 2015).
+
+In this paper, we investigate how using MC sampling to model uncertainty affects a network’s probabilistic predictions. Specifically, we test if using MC sampling improves the calibration of the probabilistic predictions made by Bayesian DNNs with softmax output layers. We used variational distributions based on dropout and dropconnect with either Bernoulli or Gaussian sampling during both training and inference. Additonally, we propose a formulation of a spike-and-slab variational distribution based on Bernoulli dropout and Gaussian dropconnect. We find that the spike-and-slab networks robustly represented their uncertainty like Bayesian dropconnect networks and have the increased CNN classification accuracy of Bayesian dropout networks. Each of these variational distributions scale extremely well and make the results of this work applicable to a large range of state-of-the-art DNNs.
+
+# 2 METHODS
+
+# 2.1 BAYESIAN NEURAL NETWORKS
+
+Artificial neural networks (NNs) can be trained using Bayesian learning by finding the maximum a posteriori (MAP) weights given the training data $( D _ { t r a i n } )$ and a prior over the weight matrix $W$ $( p ( W ) )$ :
+
+$$
+\operatorname* { m a x } _ { W } p ( W | D _ { t r a i n } ) = \operatorname* { m a x } _ { W } p ( D _ { t r a i n } | W ) p ( W )
+$$
+
+This is usually done by minimizing the mean squared error (MSE) or cross entropy error for either regression or classification, respectively, while using L2 regularization, which corresponds to a Gaussian prior over the weights. At inference, the probability of the test data $( D _ { t e s t } )$ is then calculated using only the maximum likelihood estimate (MLE) of the weights $( W ^ { * } )$ :
+
+$$
+p ( D _ { t e s t } | W ^ { * } )
+$$
+
+However, ideally the full posterior distribution over the weights would be learned instead of just the MLE:
+
+$$
+p ( W | D _ { t r a i n } ) = \frac { p ( D _ { t r a i n } | W ) p ( W ) } { p ( D _ { t r a i n } ) }
+$$
+
+This can be intractable due to both the difficulty in calculating $p ( D _ { t r a i n } )$ and in calculating the joint distribution of a large number of parameters. Instead, $p ( W | D _ { t r a i n } )$ can be approximated using a variational distribution $q ( W )$ . This distribution is constructed to allow for easy generation of samples. Using variational inference, $q ( W )$ is learned by minimizing:
+
+$$
+- \int \log \ p ( D _ { t r a i n } | W ) q ( W ) d W + K L ( q ( W ) | | p ( W ) )
+$$
+
+Monte-Carlo (MC) sampling can then be used to estimate the probability of test data using $q ( W )$
+
+$$
+p ( D _ { t e s t } ) \approx \frac { 1 } { n } \sum _ { i } ^ { n } p ( D _ { t e s t } | \hat { W } ^ { i } ) ~ w h e r e ~ \hat { W } ^ { i } \sim q ( W )
+$$
+
+![](images/ebf36505305400ebfea2455be798a5425ee463c578f90d65ea7251041a7987f6.jpg)  
+Figure 1: A visualization of the different variational distributions on a simple neural network.
+
+# 2.2 VARIATIONAL DISTRIBUTIONS
+
+The number and continuous nature of the parameters in DNNs makes sampling from the entire distribution of possible weight matrices computationally challenging. However, variational distributions can make sampling easier. In deep learning, the most common sampling method is dropout with Bernoulli variables. However, dropconnect, which independently samples a Bernoulli for each weight, and Gaussian weights have also been used. A visualization of the different methods is shown in Figure 1. All of these methods can be formulated as variational distributions where weights are sampled by element-wise multiplying the variational parameters $V$ , the $n \times n$ connection matrix with an element for each connection between the $n$ units in the network, by a mask $\hat { M }$ , which is sampled from some probability distribution. Mathematically, this can be written as:
+
+$$
+\hat { W } = V \circ \hat { M } w h e r e \hat { M } \sim p ( M )
+$$
+
+From this perspective, the difference between dropout and dropconnect, as well as Bernoulli and Gaussian methods, is simply the probability distribution used to generate the mask sample, $\hat { M }$ (Figure 2).
+
+![](images/5fd29ce6bf9a42bf5eb9412eccdb30721d4bd6bbbbf2e2cc5dc088871bfe0e65.jpg)  
+Figure 2: An illustration of sampling network weights using the different variational distributions.
+
+# 2.2.1 BERNOULLI DROPCONNECT & DROPOUT
+
+Bernoulli distributions are simple distributions which return 1 with probability $p$ and 0 with probability $( 1 - p )$ . In Bernoulli dropconnect, each element of the mask is sampled independently, so $\hat { m } _ { i , j } \sim B e r n o u l l i ( p )$ . This sets $\hat { w } _ { i , j }$ to $v _ { i , j }$ with probability $p$ and 0 with a probability $( 1 - p )$ .
+
+In dropout, however, the weights are not sampled independently. Instead, one Bernoulli variable is sampled for each row of the weight matrix, so $\hat { m } _ { i , * } \sim B e r n o u l l i ( p )$ .
+
+# 2.2.2 GAUSSIAN DROPCONNECT & DROPOUT
+
+In Gaussian dropconnect and dropout, the elements of the mask are sampled from normal distributions. This corresponds to sampling $\hat { w } _ { i , j }$ from a Gaussian distribution centered at variational parameter $v _ { i , j }$ . Srivastava et al. (2014) proposed using Gaussian distribution with a mean of 1 and a variance of $\sigma _ { d c } ^ { 2 } = ( 1 - p ) / p$ , which matches the mean and variance of dropout when training time scaling is used. In Gaussian dropconnect, each element of the mask is sampled independently, which results in $\hat { m } _ { i , j } \sim { \mathcal N } ( 1 , \sigma _ { d c } ^ { 2 } )$ . In Gaussian dropout, each element in a row has the same random variable, so $\hat { m } _ { i , * } \sim \mathcal N ( 1 , \sigma _ { d c } ^ { 2 } )$ .
+
+# 2.2.3 SPIKE-AND-SLAB DROPOUT
+
+A spike-and-slab distribution is the normalized linear combination of a ”spike” of probability mass at zero andprobability $p _ { s p i k e }$ b” consisting of a Gaussian distribution. This spor a random sample from a Gaussian distribution $\mathcal { N } ( \mu _ { s l a b } , \sigma _ { s l a b } ^ { 2 } )$ urns a 0 with. We propose and-slab variational distribution by optimizing a lower-bound of the objective function (See Appendix A). In this formulation, $m _ { i , j } \sim b _ { i , * } \mathcal { N } ( \bar { 1 } , \sigma _ { d c } ^ { 2 } )$ , where $b _ { i , * } \sim B e r n ( p _ { d o } )$ for each mask row and $\sigma _ { d c } ^ { 2 } = p _ { d c } / ( 1 - p _ { d c } )$ . As for Bernoulli dropout, each row of the mask $M$ , $m _ { i , * }$ , is multiplied by 0 with probability $\left( 1 - p _ { d o } \right)$ , otherwise each element in that row is multiplied by a value independently sampled from a Gaussian distribution as in Gaussian dropconnect. During non-sampling inference, spike-and-slab dropout uses the mean weight values and, per Bernoulli dropout, multiplies unit outputs by $p _ { d o }$ . This differs from the work done by Louizos (2015) and Gal (2016) in that they used additive Gaussian noise and learn separate means and variances for each weight. In contrast, we define the variance as a function of the learned weight mean $v _ { i , j }$ . Tying the variance of a weight to its magnitude makes it only beneficial to learn large weights if they are robust to variance (Wang & Manning, 2013). Although we treat $p _ { d o }$ and $p _ { d c }$ as a hyperparameters thereby reducing the space of variational distributions we optimize over, similar methods could potentially learn these during training (Louizos, 2015; Kingma et al., 2015; Gal, 2016).
+
+![](images/3af3171b184d75daf489e8d01e3d53b15332531ac388ee6b9c7a8d1df4a13daa.jpg)  
+Figure 3: Examples of MNIST images with added Gaussian noise with varying standard deviations.
+
+# 3 RESULTS
+
+In this paper, we investigate how using MC sampling affects a DNN’s ability to represent the uncertainty of it’s probabalistic predictions. To test this, we trained several networks differing only in whether no sampling was performed (baseline DNN and DNN with L2-regularization), sampling was only performed during training (dropout and dropconnect), or sampling was performed both during training and inference (MC dropout and MC dropconnect). We used the MNIST and CIFAR10 datasets to train networks that sampled from different variational distribution using the above methods.
+
+For these DNNs, we compared the test classification error, the uncertainty of the softmax output, and the calibration of the softmax output for each type of sampling and variational distribution. The test classification error shows how well the probability distribution learned by each DNN models the data. The uncertainty shows how the probability distribution learned by each DNN is distributed across classes. A low entropy means that the probability mass is primarily located at a few labels and a high entropy means that the probability mass is distributed across many labels. The calibration shows how well the probability distribution learned by the DNN models it’s own uncertainty. We evaluated how calibrated a prediction was by the following procedure: (1) We binned test set predictions by predicted probability. (2) We calculated the percentage of predictions in each predicted-probability bin that correctly predicted a target label. Perfect calibration means that targets predicted with probability $z$ are correct in $z$ times $100 \%$ of the cases. We therefore (3) calculated the mean squared calibration error (i.e. the mean across bins of the squared deviations between the bin-mean predicted probability and the proportion of correct predictions in that bin). We evaluated these three measures for the trained networks on the MNIST and CIFAR test set with noise sampled from Gaussian distributions with varying standard deviations (Figure 3). This tested how well modelled each network’s uncertainty was for the test sets and the regions of input space not seen in the training set. For dropout and dropconnect, $p$ was set to 0.5, which corresponds to the best value for regularizing a linear layer Baldi & Sadowski (2013). However in practice, different values for $p$ have been used Srivastava et al. (2014). We found that 0.5 was a robust choice for different networks, measures and sampling methods we used. The one exception were the dropconnect parameter used for spike-and-slab distributions where 0.5 made learning difficult due to the variance during training. Through validation, we found that using larger values spike-and-slab probabilities (0.75 for the fully connected and 0.9 for the convolutional) allowed the networks to fit to the training data better while still maintaining good generalization.
+
+![](images/f9bbb36a30f17a436418786ef4e6823d485564aa13cafeab003021f582ded0d4.jpg)  
+Figure 4: The MNIST test classification error, entropy, and calibration of the predictions of the fully connected networks: NN, $_ { \mathrm { N N + L } 2 }$ , Bernoulli DropConnect (BDC) with and without Monte-Carlo (MC) sampling, Gaussian DropConnect (GDC) with and without MC sampling, Bernoulli Dropout (BDO) with and without MC sampling, Gaussian Dropout with and without MC sampling, and spike-and-slab Dropout (SSD) with and without MC sampling.
+
+![](images/8ba652d0bc5e0638f545cbd8e7287f5b36c65535eb1d2efa902e56f3654579e4.jpg)  
+Figure 5: The calibration curves for the MNIST test set with and without Gaussian noise of the softmax outputs of the fully connected networks: NN, $_ { \mathrm { N N + L } 2 }$ , Bernoulli DropConnect (BDC) with and without Monte-Carlo (MC) sampling, Gaussian DropConnect (GDC) with and without MC sampling, Bernoulli Dropout (BDO) with and without MC sampling, Gaussian Dropout with and without MC sampling, and spike-and-slab Dropout (SSD) with and without MC sampling.
+
+Table 1: MNIST test error for the trained fully connected neural networks with and without MonteCarlo (MC) sampling using 100 samples.   
+
+<table><tr><td>Method</td><td>Mean Error(%)</td><td>Std. Dev.</td></tr><tr><td>NN</td><td>1.68</td><td>-</td></tr><tr><td>NN+L2</td><td>1.64</td><td></td></tr><tr><td>Bernoulli DropConnect</td><td>1.33</td><td></td></tr><tr><td>MC Bernoulli DropConnect</td><td>1.30</td><td>0.04</td></tr><tr><td>Gaussian DropConnect</td><td>1.24</td><td></td></tr><tr><td>MC Gaussian DropConnect</td><td>1.27</td><td>0.03</td></tr><tr><td>Bernoulli Dropout</td><td>1.45</td><td>-</td></tr><tr><td>MC Bernoulli Dropout</td><td>1.42</td><td>0.03</td></tr><tr><td>Gaussian Dropout</td><td>1.36</td><td></td></tr><tr><td>MC Gaussian Dropout</td><td>1.37</td><td>0.03</td></tr><tr><td>Spike-and-Slab Dropout</td><td>1.23</td><td></td></tr><tr><td>MC Spike-and-Slab Dropout</td><td>1.23</td><td>0.03</td></tr></table>
+
+# 3.1 MNIST
+
+We trained two groups of DNNs, one with a fully connected (FC) architecture and one with a convolutional architecture, on digit classification using the MNIST dataset (LeCun et al., 1998). This set contains 60,000 training images and 10,000 testing images. No data augmentation was used.
+
+# 3.1.1 FULLY CONNECTED NEURAL NETWORKS
+
+First, we trained DNNs with two FC hidden layers, each with 800 units and ReLU non-linearities. For the L2-regularized network, an L2-coefficient of 1e-5 was used for all weights. For the dropout methods, unit sampling was performed after each FC layer. For the dropconnect methods, every weight was sampled. The classification errors of the FC networks on the MNIST test set are shown in Table 1. Sampling during learning significantly increased accuracy in comparison to the baseline NNs, with the dropconnect-based networks being the most accurate. MC sampling at inference did not significantly increase accuracy. We found that Gaussian dropconnect and spike-and-slab dropout had the best accuracy.
+
+The classification error, uncertainty, and calibration of the learned probability distributions of each FC network for varying levels of noise are shown in Figure 4. While not improving accuracy, MC sampling led to networks that better represent their own uncertainty. As the noise in the test set was increased, the uncertainty of the networks with MC sampling highly increased, especially when compared to networks with no sampling at inference. This resulted in better calibrated FC networks for all levels of noise.
+
+The calibration curves show that sampling only during training, especially when using only dropout, led to overconfidence through placing too much probability mass on the most predicted label (Figure 5). In particular, sampling only during training resulted in under-confidence for low predicted probabilities and over-confidence for high predicted probabilities. By distributing probability mass over several labels, the DNNs that sampled at inference better represented the uncertainty of their predictions.
+
+![](images/4569abac1e0eef01c33168f64efac9c35bb9c2d269ce5812ad28a0a1d25984cc.jpg)  
+Figure 6: The MNIST test classification error, entropy, and calibration of the predictions of the convolutional networks: CNN, $\mathrm { C N N + L } 2$ , Bernoulli DropConnect (BDC) with and without MonteCarlo (MC) sampling, Gaussian DropConnect (GDC) with and without MC sampling, Bernoulli Dropout (BDO) with and without MC sampling, Gaussian Dropout with and without MC sampling, and spike-and-slab Dropout (SSD) with and without MC sampling.
+
+![](images/5115246f909d10143a32429f907d51728a3b3bbb2fd8f022558bd6238546771e.jpg)  
+Figure 7: The calibration curves for the MNIST test set with and without Gaussian noise of the softmax outputs of the convolutional networks: CNN, $\mathrm { C N N + L } 2$ , Bernoulli DropConnect (BDC) with and without Monte-Carlo (MC) sampling, Gaussian DropConnect (GDC) with and without MC sampling, Bernoulli Dropout (BDO) with and without MC sampling, Gaussian Dropout with and without MC sampling, and spike-and-slab Dropout (SSD) with and without MC sampling.
+
+# 3.1.2 CONVOLUTIONAL NEURAL NETWORKS
+
+We also trained CNNs on MNIST. Every network had two convolutional layers and a fully-connected layer (See Appendix B for details). For the L2-regularized network, an L2-coefficient of 1e-5 was used for all weights. For Bernoulli and Gaussian dropout, dropout was performed after each convolutional layer and after the FC layer. For Bernoulli and Gaussian dropconnect, every weight was sampled. The classification error of the CNNs on the MNIST test set is shown in Table 2. Sampling during training significantly increased the accuracy for the all of the networks, but especially for the Gaussian dropout network. However, unlike for the FC networks, the dropout-based methods were more accurate than the dropconnect-based methods. Unlike for the FC networks, spike-and-slab had accuracies more similar to Bernoulli dropout, which classified more accurately than Gaussian dropconnect. MC sampling during inference did not significantly increase the accuracy of the networks.
+
+Table 2: MNIST test error for the trained convolutional neural networks (CNNs) with and without Monte-Carlo (MC) sampling using 100 samples.   
+
+<table><tr><td>Method</td><td>Mean Error(%)</td><td>Error Std.Dev.</td></tr><tr><td>CNN</td><td>0.70</td><td>-</td></tr><tr><td>CNN+L2</td><td>0.70</td><td></td></tr><tr><td>Bernoulli DropConnect</td><td>0.59</td><td>-</td></tr><tr><td>MC Bernoulli DropConnect</td><td>0.59</td><td>0.02</td></tr><tr><td>Gaussian DropConnect</td><td>0.49</td><td></td></tr><tr><td>MC Gaussian DropConnect</td><td>0.49</td><td>0.01</td></tr><tr><td>Bernoulli Dropout</td><td>0.45</td><td>-</td></tr><tr><td>MC Bernoulli Dropout</td><td>0.46</td><td>0.01</td></tr><tr><td>Gaussian Dropout</td><td>0.38</td><td>-</td></tr><tr><td>MC Gaussian Dropout</td><td>0.37</td><td>0.01</td></tr><tr><td>Spike-and-Slab Dropout</td><td>0.43</td><td>1</td></tr><tr><td>MC Spike-and-Slab Dropout</td><td>0.44</td><td>0.01</td></tr></table>
+
+The classification error, uncertainty, and calibration of the learned probability distributions of each network for varying levels of noise are shown in Figure 6. As with the FC networks, MC sampling at inference greatly increased the CNNs’ ability to estimate their own uncertainty, particularly for inputs that are different from the training set. MC sampling led to increased entropy as inputs became more noisy, which resulted in better calibration. In particular, this was true of both the Bernoulli and Gaussian dropconnect networks, which very accurately represented their uncertainty even for highly noisy inputs. The spike-and-slab CNN had similar robust calibration. The calibration curves show that not using MC sampling at inference led networks that were under-confident when making low probability predictions and over-confident when making high probability predictions (Figure 7).
+
+# 3.2 CIFAR-10
+
+We trained large CNNs on natural image classification using the CIFAR-10 dataset, which contains 50,000 training images and 10,000 testing images (Krizhevsky & Hinton, 2009). The CNNs had 13 convolutional layer followed by a fully connected layer (See Appendix B for details). For L2- regularization, an L2-coefficient of 5e-4 was used for all weights. For the dropout networks, was used after each convolutional layer, but before the non-linearities. For the dropconnect networks, all weights were sampled. During training, random horizontal flipping was used. The classification error of the CNNs on the CIFAR-10 test set is shown in Table 3. For each variational distribution, MC sampling significantly increased test accuracy. Also, the that used dropout, including spike-andslab, had significantly higher accuracies than the networks that only used dropconnect.
+
+The classification error, uncertainty, and calibration of the learned probability distributions of each network for varying levels of noise are shown in Figure 8. One of the major differences between the CIFAR-10 and the MNIST results was that using the layer-wise expectation for dropout did not produce good models, regardless of what variational distribution was used. Instead, the standard test time dropout methods led to relatively inaccurate networks with very high output entropy even when no input noise was used. This agrees with the results reported by Gal & Ghahramani (2015)), who also found that using dropout at every layer can reduce accuracy if MC sampling is not used. However, these results differ from those of Srivastava et al. (2014). In our experience, deeper networks with higher regularization (e.g. Bernoulli dropout probabilities closer to 0.5) result in traditional dropout inference performing significantly worse than MC dropout. As for the MNIST networks, MC sampling at inference overall greatly increased the CIFAR-10 trained CNNs’ ability to estimate their own uncertainty when no or little noise was added to the test images.
+
+Table 3: CIFAR-10 test error for the trained convolutional neural networks (CNNs) with and without Monte-Carlo (MC) sampling using 100 samples.   
+
+<table><tr><td>Method</td><td>Mean Error(%)</td><td>Error Std. Dev.</td></tr><tr><td>CNN</td><td>19.63</td><td>1</td></tr><tr><td>CNN+L2</td><td>19.44</td><td>1</td></tr><tr><td>Bernoulli DropConnect</td><td>17.64</td><td>=</td></tr><tr><td>MC Bernoulli DropConnect</td><td>17.29</td><td>0.05</td></tr><tr><td>Gaussian DropConnect</td><td>16.00</td><td></td></tr><tr><td>MC Gaussian DropConnect</td><td>15.63</td><td>0.04</td></tr><tr><td>Bernoulli Dropout</td><td>37.47</td><td></td></tr><tr><td>MC Bernoulli Dropout</td><td>10.19</td><td>0.06</td></tr><tr><td>Gaussian Dropout</td><td>24.10</td><td></td></tr><tr><td>MC Gaussian Dropout</td><td>9.29</td><td>0.10</td></tr><tr><td>Spike-and-Slab Dropout</td><td>18.05</td><td></td></tr><tr><td>MCSpike-and-Slab Dropout</td><td>10.44</td><td>0.03</td></tr></table>
+
+![](images/e5b981d48a35b2905daa6461fdfd4d81a72b0d75025b054d06fb8e6d87bc14f1.jpg)  
+Figure 8: The CIFAR-10 test classification error, entropy, and calibration of the predictions of the convolutional neural networks: CNN, $\mathrm { C N N + L } 2$ , Bernoulli DropConnect (BDC) with and without Monte-Carlo (MC) sampling, Gaussian DropConnect (GDC) with and without MC sampling, Bernoulli Dropout (BDO) with and without MC sampling, Gaussian Dropout with and without MC sampling, and spike-and-slab Dropout (SSD) with and without MC sampling.
+
+![](images/fc26828ec759e0f563f0d693175073c62b1146eb1b437ec6973876d91418b1c8.jpg)  
+Figure 9: The calibration curves for the CIFAR-10 test set with and without Gaussian noise of the softmax outputs of the convolutional neural networks: CNN, $\mathrm { C N N + L } 2$ , Bernoulli DropConnect (BDC) with and without Monte-Carlo (MC) sampling, Gaussian DropConnect (GDC) with and without MC sampling, Bernoulli Dropout (BDO) with and without MC sampling, Gaussian Dropout with and without MC sampling, and spike-and-slab Dropout (SSD) with and without MC sampling.
+
+The classification accuracies and the ability to model uncertainty of the networks with dropconnect sampling were far more robust to noise than the networks with only dropout. However, the MC dropconnect networks are significantly less accurate than the MC dropout networks for the CIFAR
+
+10 test set when no noise was added. Networks that used traditional dropout inference instead of sampling were consistently uncertain, regardless of the noise. These networks have worse calibration than the MC dropout networks at low levels of noise but better calibration than the MC dropout networks at low levels of noise because they always had high uncertainty. For CIFAR-10, not using MC sampling resulted in networks that were generally over-confident when making predictions (Figure 9). However, this was not true for the non-sampling dropout networks when no input noise was used. In that case, the networks were highly under-confident.
+
+# 4 DISCUSSION
+
+In this paper, we investigated the ability of MC sampling to improve a DNN’s representation of its own uncertainty. We did this by training Bayesian DNNs with either multiplicative masking of the weights (dropconnect) or units (dropout) using Bernoulli, Gaussian, or spike-and-slab sampling. Based on the results, we draw the following main conclusions:
+
+1. Sampling during both learning and inference improved a network’s ability to represent its own uncertainty
+
+MC sampling at inference improved the calibration of a network’s predictions. Overall, this improvement was particularly large for inputs from outside the training set, which traditional models classified with high confidence despite not being trained on similar inputs.
+
+2. Sampling weights independently led to networks that best represented their own uncertainty
+
+For all the network architectures and datasets tested, using dropconnect sampling at training and inference resulted in the best calibrated networks overall. This was true regardless of whether dropconnect sampling led to the most accurate network. This is in contrast to CNNs with Gaussian dropout sampling, which were significantly the most accurate and also the worst calibrated of the networks with sampling both during training an inference
+
+3. Sampling weights independently led to the most accurate FC networks, but sampling units led to the most accurate CNNs
+
+For the FC networks, using dropconnect, particularly with Gaussian sampling, resulted in the most accurate networks. However, using dropout led to the most accurate CNNs. A potential cause of this is the large correlation in the information contained by adjacent elements in an image, which are often covered by the same convolutional kernel. This could mean that sampling the weights of a kernel does not provide as much regularization as the dropout methods.
+
+4. Sampling using both Bernoulli dropout and Gaussian dropconnect led to accurate and well calibrated networks
+
+Using spike-and-slab dropout, which combines Bernoulli dropout and Gaussian dropconnect, resulted in networks that performed well for all architectures. Spike-and-slab networks had accuracies similar to the Bernoulli dropout or Gaussian dropconnect depending on which performed better for a given architecture and task, Gaussian dropconnect for FC networks and Bernoulli dropout for CNNs. Spike-and-slab networks also were robustly well calibrated similar to all of the other dropconnect methods.
+
+These scalable methods for improving a network’s representation of its own uncertainty are widely applicable, since most DNNs already use dropout and getting uncertainty estimates only requires using MC sampling at inference. We plan to further investigate the use of different variational distributions. We also plan to evaluate the use of dropout and dropconnect sampling on large recurrent neural networks. Our results suggest that sampling at inference allows DNNs to efficiently represent their own uncertainty, an essential part of real-world perception and decision making.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Yarin Gal and Sergii Strelchuk for their helpful discussions regarding the manuscript. This research was funded by the Cambridge Commonwealth, European & International Trust, the UK Medical Research Council (Program MC-A060-5PR20), and a European Research Council Starting Grant (ERC-2010-StG 261352).
+
+# REFERENCES
+
+Pierre Baldi and Peter J Sadowski. Understanding dropout. In Advances in Neural Information Processing Systems, pp. 2814–2822, 2013.
+
+David Barber and Christopher M Bishop. Ensemble learning in bayesian neural networks. NATO ASI SERIES F COMPUTER AND SYSTEMS SCIENCES, 168:215–238, 1998.
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+Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. In Proceedings of The 32nd International Conference on Machine Learning, pp. 1613–1622, 2015.
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+Yarin Gal. Uncertainty in Deep Learning. PhD thesis, University of Cambridge, 2016.
+
+Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Insights and applications. In Deep Learning Workshop, ICML, 2015.
+
+Yarin Gal and Zoubin Ghahramani. Bayesian convolutional neural networks with Bernoulli approximate variational inference. In 4th International Conference on Learning Representations (ICLR) workshop track, 2016.
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+Alex Graves. Practical variational inference for neural networks. In Advances in Neural Information Processing Systems, pp. 2348–2356, 2011.
+
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+Diederik P Kingma, Tim Salimans, and Max Welling. Variational dropout and the local reparameterization trick. In Advances in Neural Information Processing Systems, pp. 2575–2583, 2015.
+
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images, 2009.
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+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
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+Christos Louizos. Smart regularization of deep architectures. Master’s thesis, University of Amsterdam, 2015.
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+David JC MacKay. A practical bayesian framework for backpropagation networks. Neural computation, 4(3):448–472, 1992.
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+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015.
+
+Radford M Neal. Bayesian learning for neural networks, volume 118. Springer Science & Business Media, 2012.
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+David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. Nature, 529(7587):484–489, 2016.
+
+Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1):1929–1958, 2014.
+
+Li Wan, Matthew Zeiler, Sixin Zhang, Yann L Cun, and Rob Fergus. Regularization of neural networks using dropconnect. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pp. 1058–1066, 2013.
+
+Sida Wang and Christopher Manning. Fast dropout training. In Proceedings of the 30th International Conference on Machine Learning (ICML), pp. 118–126, 2013.
+
+# A DERIVATION OF APPROXIMATE SPIKE-AND-SLAB DROPOUT
+
+For Bayesian inference:
+
+$$
+p ( D _ { t e s t } | D _ { t r a i n } ) = \int p ( D _ { t e s t } | W ) p ( W | D _ { t r a i n } ) d W
+$$
+
+Using variational inference:
+
+$$
+p ( D _ { t e s t } ) = \int p ( D _ { t e s t } | W ) q ( W ) d W
+$$
+
+Where the variational distribution $q ( W )$ is learned by maximizing the log-evidence lower bound:
+
+$$
+l o g ( p ( D _ { t r a i n } ) ) \ge \int \log ( p ( D _ { t r a i n } | W ) ) q ( W ) d W - K L ( q ( W ) | | p ( W ) )
+$$
+
+For spike-and-slab dropout, as when using Bernoulli dropout, $W = B \circ V$ where $b _ { i , * } \sim B e r n ( p _ { d o } )$ so if we assume independence between the random variables $B$ and $V$ :
+
+$$
+\begin{array} { c } { { l o g ( p ( D _ { t r a i n } ) ) \ge \displaystyle \sum _ { B } \int _ { V } \log ( p ( D _ { t r a i n } | B , V ) ) q ( B ) q ( V ) d V d B } } \\ { { - K L ( q ( B ) | | p ( B ) ) - K L ( q ( V ) | | p ( V ) ) } } \end{array}
+$$
+
+For a spike-distribution, bution, each element of . As in Gaussian dropco $V$ is iect, s. pled from a Gaussiancan be sampled using $\mathcal { N } ( \mu _ { v _ { i , j } } , \sigma _ { v _ { i , j } } ^ { 2 } )$ $\sigma _ { v _ { i , j } } ^ { 2 } = \alpha \mu _ { v _ { i , j } } ^ { 2 }$ $V$ the “reparameterization trick”:
+
+$$
+v _ { i , j } = N ( \mu _ { v _ { i , j } } , \alpha \mu _ { v _ { i , j } } ^ { 2 } ) = g ( \mu _ { v _ { i , j } } , \epsilon _ { i , j } ) = \mu _ { v _ { i , j } } + \sqrt { \alpha } \mu _ { v _ { i , j } } \epsilon _ { i , j }
+$$
+
+where $\epsilon \sim N ( 0 , 1 )$ , $\alpha = ( 1 - p _ { d c } ) / p _ { d c }$ , and $p _ { d c }$ is the dropconnect keep probability.
+
+This leads to:
+
+$$
+\begin{array} { r l r } & { } & { l o g ( p ( D _ { t r a i n } ) ) \ge \displaystyle \sum _ { B } \int _ { \epsilon } \log ( p ( D _ { t r a i n } | B , V ) q ( \epsilon ) q ( B ) d \epsilon d B } \\ & { } & { \quad - K L ( q ( B ) | | p ( B ) ) - K L ( q ( V ) | | p ( V ) ) } \end{array}
+$$
+
+This results in the following minimization objective function:
+
+$$
+\mathcal { L } _ { \mu _ { V } } : = - \sum _ { B } \int _ { \epsilon } \log ( p ( D _ { t r a i n } | B , V ) ) q ( \epsilon ) q ( B ) d \epsilon d B + K L ( q ( B ) | | p ( B ) ) + K L ( q ( V ) | | p ( V ) ) ,
+$$
+
+Using $B e r n ( p _ { d o } )$ as a prior for $B$ leads to a constant KLD of zero. Using a prior of $\mathcal { N } ( 0 , \sigma _ { p } ^ { 2 } )$ for each element of $V$ leads to the following:
+
+$$
+K L ( q ( v _ { i , j } ) | | p ( v _ { i , j } ) ) = \frac { ( \mu _ { v _ { i , j } } - 0 ) ^ { 2 } } { 2 \sigma _ { p } ^ { 2 } } + \log \frac { \sigma _ { p } } { \sigma _ { v _ { i , j } } } + \frac { \sigma _ { v _ { i , j } } ^ { 2 } } { 2 \sigma _ { p } ^ { 2 } } - \frac { 1 } { 2 }
+$$
+
+By using L2 regularization, we are optimizing a lower-bound of the KLD between $q ( V )$ and $\dot { \mathcal { N } } ( 0 , \lambda ^ { - 1 } )$ by only matching the first moment (i.e. the mean):
+
+$$
+\mathcal { L } _ { \mu \nu } \geq \widetilde { \mathcal { L } } _ { \mu \nu } : = - \sum _ { B } \int _ { \epsilon } \log ( p ( D _ { t r a i n } | B , V ) ) q ( \epsilon ) q ( B ) d \epsilon d B + \frac { \lambda } { 2 } \mu _ { V } \mu _ { V } ^ { \intercal }
+$$
+
+where $\mu _ { V }$ is a vector containing each $\mu _ { v _ { i , j } }$ and $\epsilon$ is a vector containing each $\epsilon _ { i , j }$ .
+
+Approximating using Monte Carlo integration for learning (Eq. A.10) and inference (Eq. A.11):
+
+$$
+\begin{array} { c l c r } { { \displaystyle \widetilde { \mathcal { L } } _ { \mu \nu } : \approx - \frac { 1 } { n } \sum _ { ( B , \epsilon ) } \log ( p ( D _ { t r a i n } | B , V ) ) + \frac { \lambda } { 2 } \mu _ { V } \mu _ { V } ^ { \intercal } } } \\ { { \displaystyle p ( D _ { t e s t } ) \approx \frac { 1 } { n } \sum _ { ( B , \epsilon ) } p ( D _ { t e s t } | B , V ) } } \end{array}
+$$
+
+where $b _ { i , * } \sim B e r n ( p _ { d o } )$ and $\epsilon \sim N ( 0 , 1 )$ .
+
+# B CONVOLUTIONAL NEURAL NETWORK ARCHITECTURES
+
+# B.1 MNIST
+
+Table B.1: The convolutional neural network (CNN) architecture used for MNIST.   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Kernel Size</td><td rowspan=1 colspan=1>#Features</td><td rowspan=1 colspan=1>Stride</td><td rowspan=1 colspan=1>Non-linearity</td></tr><tr><td rowspan=1 colspan=1>Conv-1</td><td rowspan=1 colspan=1>5x5</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>MaxPool-1</td><td rowspan=1 colspan=1>2x2</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>Max</td></tr><tr><td rowspan=1 colspan=1>Conv-2</td><td rowspan=1 colspan=1>5x5</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>MaxPool-2</td><td rowspan=1 colspan=1>2x2</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>Max</td></tr><tr><td rowspan=1 colspan=1>FC</td><td rowspan=1 colspan=1>1500</td><td rowspan=1 colspan=1>500</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Linear</td><td rowspan=1 colspan=1>500</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr></table>
+
+# B.2 CIFAR-10
+
+Table B.2: The convolutional neural network (CNN) architecture used for CIFAR-10.   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Kernel Size</td><td rowspan=1 colspan=1>#Features</td><td rowspan=1 colspan=1>Stride</td><td rowspan=1 colspan=1>Non-linearity</td></tr><tr><td rowspan=1 colspan=1>Conv-1</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Conv-2</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>MaxPool-1</td><td rowspan=1 colspan=1>2x2</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>Max</td></tr><tr><td rowspan=1 colspan=1>Conv-3</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Conv-4</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>MaxPool-2</td><td rowspan=1 colspan=1>2x2</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>Max</td></tr><tr><td rowspan=1 colspan=1>Conv-5</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Conv-6</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Conv-7</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>MaxPool-3</td><td rowspan=1 colspan=1>2x2</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>Max</td></tr><tr><td rowspan=1 colspan=1>Conv-8</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Conv-9</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Conv-10</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>MaxPool-4</td><td rowspan=1 colspan=1>2x2</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>Max</td></tr><tr><td rowspan=1 colspan=1>Conv-11</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Conv-12</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Conv-13</td><td rowspan=1 colspan=1>3x3</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>MaxPool-5</td><td rowspan=1 colspan=1>2x2</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>Max</td></tr><tr><td rowspan=1 colspan=1>FC</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Linear</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr></table>

parse/train/S1M6Z2Cctm/S1M6Z2Cctm.md

@@ -0,0 +1,354 @@
+# HARMONIC UNPAIRED IMAGE-TO-IMAGE TRANSLATION
+
+Tomas Pfister   
+Google Cloud AI   
+Sunnyvale, USA   
+tpfister@google.com   
+Rui Zhang   
+Google Cloud AI &   
+Chinese Academy of Sciences   
+Beijing, China   
+zhangrui@ict.ac.cn   
+Jia Li   
+Google Cloud AI   
+Sunnyvale, USA   
+lijiali@google.com
+
+# ABSTRACT
+
+The recent direction of unpaired image-to-image translation is on one hand very exciting as it alleviates the big burden in obtaining label-intensive pixel-to-pixel supervision, but it is on the other hand not fully satisfactory due to the presence of artifacts and degenerated transformations. In this paper, we take a manifold view of the problem by introducing a smoothness term over the sample graph to attain harmonic functions to enforce consistent mappings during the translation. We develop HarmonicGAN to learn bi-directional translations between the source and the target domains. With the help of similarity-consistency, the inherent selfconsistency property of samples can be maintained. Distance metrics defined on two types of features including histogram and CNN are exploited. Under an identical problem setting as CycleGAN, without additional manual inputs and only at a small training-time cost, HarmonicGAN demonstrates a significant qualitative and quantitative improvement over the state of the art, as well as improved interpretability. We show experimental results in a number of applications including medical imaging, object transfiguration, and semantic labeling. We outperform the competing methods in all tasks, and for a medical imaging task in particular our method turns CycleGAN from a failure to a success, halving the mean-squared error, and generating images that radiologists prefer over competing methods in $9 5 \%$ of cases.
+
+# 1 INTRODUCTION
+
+Image-to-image translation (Isola et al., 2017) aims to learn a mapping from a source domain to a target domain. As a significant and challenging task in computer vision, image-to-image translation benefits many vision and graphics tasks, such as realistic image synthesis (Isola et al., 2017; Zhu et al., 2017a), medical image generation (Zhang et al., 2018; Dar et al., 2018), and domain adaptation (Hoffman et al., 2018). Given a pair of training images with detailed pixel-to-pixel correspondences between the source and the target, image-to-image translation can be cast as a regression problem using e.g. Fully Convolutional Neural Networks (FCNs) (Long et al., 2015) by minimizing e.g. the per-pixel prediction loss. Recently, approaches using rich generative models based on Generative Adaptive Networks (GANs) (Goodfellow et al., 2014; Radford et al., 2016; Arjovsky et al., 2017) have achieved astonishing success. The main benefit of introducing GANs (Goodfellow et al., 2014) to image-to-image translation (Isola et al., 2017) is to attain additional image-level (often through patches) feedback about the overall quality of the translation, and information which is not directly accessible through the per-pixel regression objective.
+
+The method by Isola et al. (2017) is able to generate high-quality images, but it requires paired training data which is difficult to collect and often does not exist. To perform translation without paired data, circularity-based approaches (Zhu et al., 2017a; Kim et al., 2017; Yi et al., 2017) have been proposed to learn translations from a set to another set, using a circularity constraint to establish relationships between the source and target domains and forcing the result generated from a sample in the source domain to map back and generate the original sample. The original image-to-image translation problem (Isola et al., 2017) is supervised in pixel-level, whereas the unpaired image-to-image translation task (Zhu et al., 2017a) is considered unsupervised, with pixel-level supervision absent but with adversarial supervision at the image-level (in the target domain) present. By using a cycled regression for the pixel-level prediction (source target source) plus a term for the adversarial difference between the transferred images and the target images, CycleGAN is able to successfully, in many cases, train a translation model without paired source target supervision. However, lacking a mechanism to enforce regularity in the translation creates problems like in Fig. 1 (a) and Fig. 2, making undesirable changes to the image contents, superficially removing tumors (the first row) or creating tumors (the second row) at the wrong positions in the target domain. Fig. 1 (b) also shows some artifacts of CycleGAN on natural images when translating horses into zebras.
+
+![](images/9a4775015eff999787f42dc541df64f20f0afb26e08cd8a9aa3dabaab50f15af.jpg)  
+Figure 1: HarmonicGAN corrects major failures in multiple domains: (a) for medical images it corrects incorrectly removed (top) and added (bottom) tumors; and (b) for horse zebra transfiguration it does not incorrectly transform the background (top) and performs a complete translation (bottom).
+
+To combat the above issue, in this paper we look at the problem of unpaired image-to-image translation from a manifold learning perspective (Tenenbaum et al., 2000; Roweis & Saul, 2000). Intuitively, the problem can be alleviated by introducing a regularization term in the translation, encouraging similar contents (based on textures or semantics) in the same image to undergo similar translations/transformations. A common principle in manifold learning is to preserve local distances after the unfolding: forcing neighboring (similar) samples in the original space to be neighbors in the new space. The same principle has been applied to graph-based semisupervised learning (Zhu, 2006) where harmonic functions with graph Laplacians (Zhu et al., 2003; Belkin et al., 2006) are used to obtain regularized labels of unlabeled data points.
+
+![](images/96d5ce59b87862c975114c52e9602e0c2e08be96e69bec6f76b7d3b4d8d6f0e5.jpg)  
+Figure 2: For CycleGAN, the reconstructed image may perfectly match the source image under the circularity constraint while translated image dose not maintain the inherent property of the source image (e.g. a tumor), thus generating unexpected results (e.g. incorrectly removing a tumor).
+
+During the translation/transformation, some domain-specific attributes are changed, such as the colors, texture, and semantics of certain image regions. Although there is no supervised information for these changes, certain consistency during the transformation is desirable, meaning that for image contents similar in the source space should also be similar in the target space. Inspired by graphbased semi-supervised learning (Zhu et al., 2003; Zhu, 2006), we introduce smoothness terms to unpaired image-to-image translation (Zhu et al., 2017a) by providing a stronger regularization for the translation/transformation between the source and target domains, aiming to exploit the “manifold structure” of the source and target domains. For a pair of similar samples (two different locations in an image; one can think of them as two patches although the receptive fields of CNN are quite large), we add the smoothness term to minimize a weighted distance of the corresponding locations in the target image. Note that two spatially distant samples might be neighbors in the feature space. We name our algorithm HarmonicGAN as it behaves harmonically along with the circularity and adversarial constraints to learn a pair of dual translations between the source and target domains, as shown in Fig. 1. Distance metrics defined on two alternative features are adopted: (1) a low-level soft RGB histograms; and (2) CNN (VGG) features with pre-trained semantics.
+
+We conduct experiments in a number of applications, showing that in each of them our method outperforms existing methods quantitatively, qualitatively, and with user studies. For a medical imaging task (Cohen et al., 2018) that was recently calling attention to a major CycleGAN failure case (learning to accidentally add/remove tumors in an MRI image translation task), our proposed method provides a large improvement over CycleGAN, halving the mean-squared error, and generating images that radiologists prefer over competing methods in $9 5 \%$ of cases.
+
+# CONTRIBUTIONS
+
+1. We introduce smooth regularization over the graph for unpaired image-to-image translation to attain harmonic translations.   
+2. When building an end-to-end learning pipeline, we adopt two alternative types of feature measures to compute the weight matrix for the graph Laplacian, one based on a soft histogram (Wang et al., 2016) and another based on semantic CNN (VGG) features (Simonyan & Zisserman, 2015).   
+3. We show that this method results in significantly improved consistency for transformations. With experiments on multiple translation tasks, we demonstrate that HarmonicGAN outperforms the state-of-the-art.
+
+# 2 RELATED WORK
+
+As discussed in the introduction, the general image-to-image translation task in the deep learning era was pioneered by (Isola et al., 2017), but there are prior works such as image analogies (Hertzmann et al., 2001) that aim at a similar goal, along with other exemplar-based methods (Efros & Freeman, 2001; Criminisi et al., 2004; Barnes et al., 2009). After (Isola et al., 2017), a series of other works have also exploited pixel-level reconstruction constraints to build connections between the source and target domains (Zhang et al., 2017; Wang et al., 2018). The image-to-image translation framework (Isola et al., 2017) is very powerful but it requires a sufficient amount of training data with paired source to target images, which are often laborious to obtain in the general tasks such as labeling (Long et al., 2015), synthesis (Chen & Koltun, 2017), and style transfer (Huang & Belongie, 2017).
+
+Unpaired image-to-image translation frameworks (Zhu et al., 2017a;b; Liu et al., 2017; Shrivastava et al., 2017; Kim et al., 2017) such as CycleGAN remove the requirement of having detailed pixellevel supervision. In CycleGAN this is achieved by enforcing a bi-directional prediction from source to target and target back to source, with an adversarial penalty in the translated images in the target domain. Similar unsupervised circularity-based approaches (Kim et al., 2017; Yi et al., 2017) have also been developed. The CycleGAN family models (Zhu et al., 2017a;b) point to an exciting direction of unsupervised approaches but they also create artifacts in many applications. As shown in Fig. 2, one reason for this is that the circularity constraint in CycleGAN lacks the straightforward description of the target domain, so it may change the inherent properties of the original samples and generate unexpected results which are inconsistent at different image locations. These failures have been prominently explored in recent works, showing that CycleGAN (Zhu et al., 2017a) may add or remove tumors accidentally in cross-modal medical image synthesis (Cohen et al., 2018), and that in the task of natural image transfiguration, e.g. from a horse to zebra, regions in the background may also be translated into a zebra-like texture (Zhu et al., 2018) (see Fig. 1 (b)).
+
+Here we propose HarmonicGAN that introduces a smoothness term into the CycleGAN framework to enforce a regularized translation, enforcing similar image content in the source space to also be similar in the target space. We follow the general design principle in manifold learning (Tenenbaum et al., 2000; Roweis & Saul, 2000) and the development of harmonic functions in the graph-based semi-supervised learning literature (Zhu et al., 2003; Belkin et al., 2006; Zhu, 2006; Weston et al., 2012). There has been previous work, DistanceGAN (Benaim & Wolf, 2017), in which distance preservation was also implemented. However, DistanceGAN differs from HarmonicGAN in (1) motivation, (2) formulation, (3) implementation, and (4) performance. The primary motivation of DistanceGAN demonstrates an alternative loss term for the per-pixel difference in CycleGAN. In HarmonicGAN, we observe that the cycled per-pixel loss is effective and we aim to make the translation harmonic by introducing additional regularization. The smoothness term acts as a graph Laplacian imposed on all pairs of samples (using random samples in implementation). In the experimental results, we show that the artifacts in CycleGAN are still present in DistanceGAN, whereas HarmonicGAN provides a significant boost to the performance of CycleGAN.
+
+In addition, it is worth mentioning that the smoothness term proposed here is quite different from the binary term used in the Conditional Random Fields literature (Lafferty et al., 2001; Krahenb ¨ uhl &¨ Koltun, 2011), either fully supervised (Chen et al., 2018; Zheng et al., 2015) or weakly-supervised (Tang et al., 2018; Lin et al., 2016). The two differ in (1) output space (multi-class label vs. highdimensional features), (2) mathematical formulation (a joint conditional probably for the neighboring labels vs. a Laplacian function over the graph), (3) application domain (image labeling vs. image translation), (4) effectiveness (boundary smoothing vs. manifold structure preserving), and (5) the role in the overall algorithm (post-processing effect with relatively small improvement vs. large-area error correction).
+
+# 3 HARMONICGAN FOR UNPAIRED IMAGE-TO-IMAGE TRANSLATION
+
+Follothe tarwhere baain . T formulation in CycleGAN (Zhu et al.,, we consider unpaired training samples  goal of image-to-image translation is to for the where ir of du doma, and ings, $X$ $Y$ $\{ \mathbf { x } _ { k } \} _ { k = 1 } ^ { N }$ $\mathbf { x } _ { k } \in X$ $\{ \mathbf { y } _ { k } \} _ { k = 1 } ^ { N }$ $\mathbf { y } _ { k } \in Y$   
+forward mapping $G : X  Y$ and backward mapping $F : Y  X$ . Two discriminators $D _ { X }$ and   
+$D _ { Y }$ are adopted in (Zhu et al., 2017a) to distinguish between real images and generated images.   
+In particular, the discriminator $D _ { X }$ aims to distinguish real image $\{ { \bf x } \}$ from the generated image   
+$\{ \bar { F ( \mathbf { y } ) } \}$ ; similarly discriminator $D _ { Y }$ distinguishes $\{ \mathbf { y } \}$ from $\{ G ( \mathbf { x } ) \}$ .
+
+Therefore, the objective of adversarial constraint is applied in both source and target domains, expressed in (Zhu et al., 2017a) as:
+
+$$
+\mathcal { L } _ { \mathrm { G A N } } ( G , D _ { Y } , X , Y ) = \mathbb { E } _ { { \mathbf { y } } \in Y } [ \log D _ { Y } ( { \mathbf { y } } ) ] + \mathbb { E } _ { { \mathbf { x } } \in X } [ \log ( 1 - D _ { Y } ( G ( { \mathbf { x } } ) ) ) ] ,
+$$
+
+and
+
+$$
+{ \mathcal { L } } _ { \mathrm { G A N } } ( F , D _ { X } , X , Y ) = \mathbb { E } _ { \mathbf { x } \in X } [ \log D _ { X } ( \mathbf { x } ) ] + \mathbb { E } _ { \mathbf { y } \in Y } [ \log ( 1 - D _ { X } ( F ( \mathbf { y } ) ) ) ] .
+$$
+
+For notational simplicity, we denote the GAN loss as
+
+$$
+\mathcal { L } _ { \mathrm { G A N } } ( G , F ) = \arg \operatorname* { m a x } _ { D _ { Y } , D _ { X } } [ \mathcal { L } _ { \mathrm { G A N } } ( G , D _ { Y } , X , Y ) + \mathcal { L } _ { \mathrm { G A N } } ( F , D _ { X } , X , Y ) ] .
+$$
+
+Since the data in the two domains are unpaired, a circularity constraint is introduced in (Zhu et al., 2017a) to establish relationships between $X$ and $Y$ . The circularity constraint enforces that $G$ and $F$ are a pair of inverse mappings, and that the translated sample can be mapped back to the original sample. The circularity constraint contains consistencies in two aspects: the forward cycle $\mathbf { x } \to G ( X ) \to F ( G ( \mathbf { x } ) ) \sim \mathbf { x }$ and the backward cycle $\mathbf { y } \to F ( \mathbf { y } ) \to G ( F ( \mathbf { y } ) ) \sim \mathbf { y }$ . Thus, the circularity constraint is formulated as (Zhu et al., 2017a):
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { c y c } } ( G , F ) = \mathbb { E } _ { \mathbf { x } \in X } | | F ( G ( \mathbf { x } ) ) - \mathbf { x } | | _ { 1 } + \mathbb { E } _ { \mathbf { y } \in Y } | | G ( F ( \mathbf { y } ) ) - \mathbf { y } | | _ { 1 } . } \end{array}
+$$
+
+Here we rewrite the overall objective in (Zhu et al., 2017a) to minimize as:
+
+$$
+\mathcal { L } _ { \mathrm { C y c l e G A N } } ( G , F ) = \lambda _ { \mathrm { G A N } } \times \mathcal { L } _ { \mathrm { G A N } } ( G , F ) + \lambda _ { \mathrm { c y c } } \times \mathcal { L } _ { \mathrm { c y c } } ( G , F ) ,
+$$
+
+where the weights $\lambda _ { \mathrm { G A N } }$ and $\lambda _ { \mathrm { c y c } }$ control the importance of the corresponding objectives.
+
+# 3.1 SMOOTHNESS TERM OVER THE GRAPH
+
+The full objective of circularity-based approach contains adversarial constraints and a circularity constraint. The adversarial constraints ensure the generated samples are in the distribution of the source or target domain, but ignore the relationship between the input and output of the forward or backward translations. The circularity constraint establishes connections between the source and target domain by forcing the forward and backward translations to be the inverse of each other. However, CycleGAN has limitations: as shown in Fig. 2, the circular projection might perfectly match the input, and the translated image might look very well like a real one, but the translated image may not maintain the inherent property of the input and contain a large artifact that is not connected to the input.
+
+![](images/adfdd441402b193fda323c1a37669af4bbf6c0840412563d205f9709ae0962c3.jpg)  
+Figure 3: Architecture of HarmonicGAN, consisting of a pair of inverse generators $G$ , $F$ and two discriminators $D _ { X }$ , $D _ { Y }$ . The objective combines an adversarial constraint, circularity constraint and smoothness term.
+
+Here we propose a smoothness term to enforce a stronger correlation between the source and target domains that focuses on providing similarity-consistency between image patches during the translation. The smoothness term defines a graph Laplacian with the minimal value achieved as a harmonic function. We define the set consisting of individual image patches as the nodes of the graph $\mathcal { G }$ . $\vec { x } _ { i }$ is referred to as the feature vector of the $i$ -th image patch in $\mathbf { x } \in X$ . For the image set $X$ , we define the set that consists of individual samples (image patches) of source image set $X$ as $S = \{ \vec { x } ( i ) , i = 1 . . M \}$ where $M$ is the total number of the samples/patches. An affinity measure (similarity) computed on image patch $\vec { x } ( i )$ and image patch $\vec { x } ( j )$ , $w _ { i j } ( X )$ (a scalar), defines the edge on the graph $\mathcal { G }$ of $S$ . The smoothness term acts as a graph Laplacian imposed on all pairs of image patches. Therefore, we define a smoothness term over the graph as
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { S m o u t } } ( G , X , Y ) = \mathbb { E } _ { \mathbf { x } \in X } \left[ \sum _ { i , j } w _ { i j } ( X ) \times D i s t [ G ( \vec { x } ) ( i ) , G ( \vec { x } ) ( j ) ] + \sum _ { i , j } w _ { i j } ( G ( X ) ) \times D i s t [ F ( G ( \vec { x } ) ) ( i ) , F ( G ( \vec { x } ) ) ( j ) ] \right] , } \end{array}
+$$
+
+where $w _ { i j } ( X ) = \exp \{ - D i s t [ \vec { x } ( i ) , \vec { x } ( j ) ] / \sigma ^ { 2 } \}$ (Zhu et al., 2003) defines the affinity between two patches $\vec { x } ( i )$ and $\vec { x } ( j )$ based on their distances (e.g. measured on histogram or CNN features). $D i s t [ G ( \vec { y } ) ( i ) , G ( \vec { y } ) ( j ) ]$ defines the distance between two image patches after translation at the same locations. In implementation, we first normalize the features to the scale of [0,1] and then use the L1 distance of normalized features as the Dist function (for both histogram and CNN features). Similarly, we define a smoothness term for the backward part as
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { S m o u t } } ( F , Y , X ) = \mathbb { E } _ { \mathbf { y } \in Y } \Big [ \sum _ { i , j } w _ { i j } ( Y ) \times D i s t [ F ( \vec { y } ) ( i ) , F ( \vec { y } ) ( j ) ] + \sum _ { i , j } w _ { i j } ( F ( Y ) ) \times D i s t [ G ( F ( \vec { y } ) ) ( i ) , G ( F ( \vec { y } ) ) ( j ) ] \Big ] , } \end{array}
+$$
+
+The combined loss for the smoothness thus becomes
+
+$$
+{ \mathcal { L } } _ { \mathrm { S m o o t h } } ( G , F ) = { \mathcal { L } } _ { \mathrm { S m o o t h } } ( G , X , Y ) + { \mathcal { L } } _ { \mathrm { S m o o t h } } ( F , Y , X ) .
+$$
+
+# 3.2 OVERALL OBJECTIVE FUNCTION
+
+As shown in Fig. 3, HarmonicGAN consists of a pair of inverse generators $G$ , $F$ and two discriminators $D _ { X } , D _ { Y }$ , defined in Eqn. (1) and Eqn. (2). The full objective combines an adversarial constraint (see Eqn. (3)), a circularity constraint (see Eqn. (4)), and a smoothness term (see Eqn. (8)). The adversarial constraint forces the translated images to be plausible and indistinguishable from the real images; the circularity constraint ensures the cycle-consistency of translated images; and the smoothness term provides a stronger similarity-consistency between patches to maintain inherent properties of the images.
+
+![](images/46e6c62829e8f49b4103ba63cc0b44325c2be733a85eee92f4bb0d85c7dc7f29.jpg)  
+Figure 4: Visualization using t-SNE (Maaten & Hinton, 2008) to illustrate the effectiveness of the smoothness term in HarmonicGAN (best viewed in color). As shown in the top two figures (source and target respectively), the smoothness term acts as a graph Laplacian imposed on all pairs of image patches. Bottom-left: For two similar patches in the original sample, if one patch is translated to a tumor region while the other is not, the two patches will have a large distance in the target space, resulting in a translation that incorrectly adds a tumor into the original sample (result for CycleGAN shown). Bottom-right: For two similar patches in the original sample, if the translation maintains the non-tumor property of these two patches in the translated sample then the two patches will also be similar in the target space (result for HarmonicGAN shown).
+
+Combining Eqn. (5) and Eqn. (8), the overall objective for our proposed HarmonicGAN under the smoothness constraint becomes
+
+$$
+{ \mathcal { L } } _ { \mathrm { H a r m o n i c G A N } } ( G , F ) = { \mathcal { L } } _ { \mathrm { C y c l e G A N } } ( G , F ) + \lambda _ { \mathrm { S m o o t h } } \times { \mathcal { L } } _ { \mathrm { S m o o t h } } ( G , F ) .
+$$
+
+Similar to the graph-based semi-supervised learning definition (Zhu et al., 2003; Zhu, 2006), the solution to Eqn. (9) leads to a harmonic function. The optimization process during training obtains:
+
+$$
+G ^ { * } , F ^ { * } = \arg \operatorname* { m i n } _ { G , F } { \mathcal { L } } _ { \mathrm { H a r m o n i c G A N } } ( G , F ) .
+$$
+
+The effectiveness of the smoothness term of Eqn. (8) is evident. In Fig. 4, we show (using tSNE (Maaten & Hinton, 2008)) that the local neighborhood structure is being preserved by HarmonicGAN, whereas CycleGAN results in two similar patches being far apart after translation.
+
+# 3.3 FEATURE DESIGN
+
+In the smoothness constraint, the similarity of a pair of patches is measured on the features for each patch (sample point). All the patches in an image form a graph. Here we adopt two types of features: (1) a low-level soft histogram, and (2) pre-trained CNN (VGG) features that carry semantic information. Soft histogram features are lightweight and easy to implement but without much semantic information; VGG requires an additional CNN network but carries more semantics.
+
+# 3.3.1 SOFT RGB HISTOGRAM FEATURES
+
+We first design a weight matrix based on simple low-level RGB histograms. To make the end-to-end learning system work, it is crucial to make the computation of gradient in the histograms derivable.
+
+We adopt a soft histogram representation proposed in (Wang et al., 2016) but fix the means and the bin size. This histogram representation is differentiable and its gradient is back-propagateable. This soft histogram function contains a family of linear basis functions $\psi _ { b } , b = 1 , \ldots , B$ , where $B$ is the number of bins in the histogram. As $\vec { x } _ { i }$ represents the $i$ -th patch in image domain $X$ , for each pixel $j$ in $\vec { x } _ { i }$ , $\psi _ { b } ( \vec { x } _ { i } ( j ) )$ represents pixel $j$ voting for the b-th bin, expressed as:
+
+$$
+\begin{array} { r } { \psi _ { b } ( \vec { x } _ { i } ( j ) ) = \mathrm { m a x } \{ 0 , 1 - | \vec { x } _ { i } ( j ) - \mu _ { b } | \times w _ { b } \} , } \end{array}
+$$
+
+where $\mu _ { b }$ and $w _ { b }$ are the center and width of the b-th bin. The representation of $\vec { x } _ { i }$ in the RGB space is the linear combination of linear basis functions on all the pixels in $\vec { x } _ { i }$ , expressed as:
+
+$$
+\phi _ { h } ( X , i , b ) = \phi _ { h } ( \vec { x } _ { i } , b ) = \sum _ { j } \psi _ { b } ( \vec { x } _ { i } ( j ) ) ,
+$$
+
+where $\phi _ { h }$ is the RGB histogram feature, $b$ is the index of dimension of the RGB histogram representation, and $j$ represents any pixel in the patch $\vec { x } _ { i }$ . The RGB histogram representation $\phi _ { h } ( X , i )$ of $\vec { x } _ { i }$ is a $B$ -dimensional vector.
+
+# 3.3.2 SEMANTIC CNN FEATURES
+
+For some domains we instead use semantic features to acquire higher-level representations of patches. The semantic representations are extracted from a pre-trained Convolutional Neural Network (CNN). The CNN encodes semantically relevant features from training on a large-scale dataset. It extracts semantic information of local patches in the image through multiple pooling or stride operators. Each point in the feature maps of the CNN is a semantic descriptor of the corresponding image patch. Additionally, the semantic features learned from the CNN are differentiable and the CNN can be integrated into HarmonicGAN and be trained end-to-end. We instantiate the semantic feature $\phi _ { s }$ as a pre-trained CNN model e.g. VGGNet (Simonyan & Zisserman, 2014). In implementation, we select the layer 4 3 after ReLU from VGG-16 network for computing the semantic features.
+
+# 4 EXPERIMENTS
+
+We evaluate the proposed method on three different applications: medical imaging, semantic labeling, and object transfiguration. We compare against several unpaired image-to-image translation methods: CycleGAN (Zhu et al., 2017a), DiscoGAN (Kim et al., 2017), DistanceGAN (Benaim & Wolf, 2017), and UNIT (Liu et al., 2017). We also provide two user studies as well as qualitative results. The appendix provides additional results and analysis.
+
+# 4.1 DATASETS AND EVALUATION METRICS
+
+Medical imaging. This task evaluates cross-modal medical image synthesis, Flair $ \mathrm { T } 1$ . The models are trained on the BRATS dataset (Menze et al., 2015) which contains paired MRI data to allow quantitative evaluation. Similar to the previous work (Cohen et al., 2018), we use a training set of 1400 image slices $50 \%$ healthy and $50 \%$ tumors) and a test set of 300, and use their unpaired training scenario. We adopt the Mean Absolute Error (MAE) and the Mean Squared Error (MSE) between the generated images and the real images to evaluate the reconstruction errors, and further use the Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM) to evaluate the reconstruction quality of generated images.
+
+Semantic labeling. We also test our method on the labels photos task using the Cityscapes dataset (Cordts et al., 2016) under the unpaired setting as in the original CycleGAN paper. For quantitative evaluation, in line with previous work, for labels photos we adopt the “FCN score” (Isola et al., 2017), which evaluates how interpretable the generated photos are according to a semantic segmentation algorithm. For photos labels, we use the standard segmentation metrics, including per-pixel accuracy, per-class accuracy, and mean class Intersection-Over-Union (Class IOU).
+
+Object transfiguration. Finally, we test our method on the horse zebra task using the standard CycleGAN dataset (2401 training images, 260 test images). This task does not have a quantitative evaluation measure, so we instead provide a user study together with qualitative results.
+
+# 4.2 IMPLEMENTATION DETAILS
+
+We apply the proposed smoothness term on the framework of CycleGAN (Zhu et al., 2017a). Similar with CycleGAN, we adopt the architecture of (Johnson et al., 2016) as the generator and the PatchGAN (Isola et al., 2017) as the discriminator. The log likelihood objective in the original GAN is replaced with a least-squared loss (Mao et al., 2017) for more stable training. We resize the input images to the size of $2 5 6 \times 2 5 6$ . For the histogram feature, we equally split the RGB range of [0, 255] to 16 bins, each with a range of 16. Images are divided into non-overlapping patches of $8 ~ \times$ 8 and the histogram feature is computed on each patch. For the semantic feature, we adopt a VGG network pre-trained on ImageNet to obtain semantic features. We select the feature map of layer relu4 3 in VGG. The loss weights are set as $\lambda _ { \mathrm { G A N } } = \lambda _ { \mathrm { S m o o t h } } = 1$ , $\lambda _ { \mathrm { c y c } } = 1 0$ . Following CycleGAN, we adopt the Adam optimizer (Kingma & Ba, 2015) with a learning rate of 0.0002. The learning rate is fixed for the first 100 epochs and linearly decayed to zero over the next 100 epochs.
+
+# 4.3 QUANTITATIVE COMPARISON
+
+Medical imaging. Table 1 shows the reconstruction performance on medical image synthesis, Flair $ \mathrm { { T } 1 }$ . The proposed method yields a large improvement over CycleGAN, showing lower MAE and MSE reconstruction losses, and higher PSNR and SSIM reconstruction scores, highlighting the significance of the proposed smoothness regularization. HarmonicGAN based on histogram and VGG features shows similar performance; the reconstruction losses of histogram-based HarmonicGAN are slightly lower than the VGG-based one in Flair $ \mathrm { T } 1$ , while they are slightly higher in $\mathrm { T } 1 $ Flair, indicating that both low-level RGB values and high-level CNN features can represent the inherent property of medical images well and help to maintain the smoothness-consistency of samples.
+
+Table 1: Reconstruction evaluation of cross-modal medical image synthesis on the BRATS dataset.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Flair→T1</td><td colspan="4">T1→Flair</td></tr><tr><td>MAE↓</td><td>MSE↓</td><td>PSNR↑</td><td>SSIM↑</td><td>MAE↓</td><td>MSE↓</td><td>PSNR ↑</td><td>SSIM↑</td></tr><tr><td>CycleGAN</td><td>10.47</td><td>674.40</td><td>22.35</td><td>0.80</td><td>11.81</td><td>1026.19</td><td>18.73</td><td>0.74</td></tr><tr><td>DiscoGAN</td><td>10.63</td><td>641.35</td><td>20.06</td><td>0.79</td><td>10.66</td><td>839.15</td><td>19.14</td><td>0.69</td></tr><tr><td>DistanceGAN</td><td>14.93</td><td>1197.64</td><td>17.92</td><td>0.67</td><td>10.57</td><td>716.75</td><td>19.95</td><td>0.64</td></tr><tr><td>UNIT</td><td>9.48</td><td>439.33</td><td>22.24</td><td>0.76</td><td>6.69</td><td>261.26</td><td>25.11</td><td>0.76</td></tr><tr><td colspan="9">HarmonicGAN (ours)</td></tr><tr><td>Histogram</td><td>6.38</td><td>216.83</td><td>24.34</td><td>0.83</td><td>5.04</td><td>163.29</td><td>26.72</td><td>0.75</td></tr><tr><td>VGG</td><td>6.86</td><td>237.94</td><td>24.14</td><td>0.81</td><td>4.69</td><td>127.84</td><td>27.22</td><td>0.76</td></tr></table>
+
+Semantic labeling. We report semantic labeling results in Table 2. The proposed method using VGG features yields a $3 \%$ improvement in Pixel Accuracy in translation scores for photo label and also shows stable improvements in other metrics, clearly outperforming all competing methods. The performance using a histogram is slightly lower than CycleGAN; we hypothesize that the reason is that the objects in photos have a large intra-class variance and inter-class similarity in appearance, e.g. cars have different colors, while vegetation and terrain have similar colors, thus the regularization of the RGB histogram is not appropriate to extract the inherent property of photos.
+
+# 4.4 USER STUDIES
+
+Medical imaging. We randomly selected 100 images from BRATS test set. For each image, we showed one radiologist the real ground truth image, followed by images generated by CycleGAN, DistanceGAN and HarmonicGAN (different order for each image set to avoid bias). The radiologist was told to evaluate similarity by how likely they would lead to the same clinical diagnosis, and was asked to rate similarity of the generation methods on a Likert scale from 1 to 5 (1 is not similar at all, 5 is exactly same). Results are in shown in Table 3. In $9 5 \%$ of cases, the radiologist preferred images generated by our method over the competing methods, and the average Likert score was 4.00 compared to 1.68 for CycleGAN, confirming that our generated images are significantly better. This is significant as it confirms that we solve the issue presented in a recent paper (Cohen et al., 2018) showing that CycleGAN can learn to accidentally add/remove tumors in images.
+
+![](images/cda88029c448fdb9092c985dc74da17ec1c06eb7f1004111ce4db63c0d1275ff.jpg)  
+Figure 5: Qualitative comparison for BRATS, Cityscapes and horse zebra (see appendix for more images).
+
+Table 2: FCN scores of Photo Label translation on the Cityscapes dataset.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Label →Photo</td><td colspan="3">Photo →Label</td></tr><tr><td>Pixel Acc. ↑</td><td>Class Acc.↑</td><td>Class IoU↑</td><td>Pixel Acc. ↑</td><td>Class Acc. ↑</td><td>Class IoU↑</td></tr><tr><td>CycleGAN</td><td>52.7</td><td>15.2</td><td>11.0</td><td>57.2</td><td>21.0</td><td>15.7</td></tr><tr><td>DiscoGAN</td><td>45.0</td><td>11.1</td><td>7.0</td><td>45.2</td><td>10.9</td><td>6.3</td></tr><tr><td>DistanceGAN</td><td>48.5</td><td>10.9</td><td>7.3</td><td>20.5</td><td>8.2</td><td>3.4</td></tr><tr><td>UNIT</td><td>48.5</td><td>12.9</td><td>7.9</td><td>56.0</td><td>20.5</td><td>14.3</td></tr><tr><td colspan="7">HarmonicGAN (ours)</td></tr><tr><td>Histogram</td><td>52.2</td><td>14.8</td><td>10.9</td><td>56.6</td><td>20.9</td><td>15.7</td></tr><tr><td>VGG</td><td>55.9</td><td>17.6</td><td>13.3</td><td>59.8</td><td>22.1</td><td>17.2</td></tr></table>
+
+Table 3: User study on the BRATS dataset.   
+
+<table><tr><td>Metric</td><td>CycleGAN</td><td>DistanceGAN</td><td>HarmonicGAN</td></tr><tr><td>Prefer[%]↑</td><td>5</td><td>0</td><td>95</td></tr><tr><td>Mean Likert 个</td><td>1.68</td><td>1.62</td><td>4.00</td></tr><tr><td>Std Likert</td><td>0.99</td><td>0.95</td><td>0.88</td></tr></table>
+
+Table 4: User study on the horse to zebra dataset.   
+
+<table><tr><td>Metric</td><td>CycleGAN</td><td>DistanceGAN</td><td>HarmonicGAN</td></tr><tr><td>Prefer[%]↑</td><td>28</td><td>0</td><td>72</td></tr><tr><td>Mean Likert↑</td><td>3.16</td><td>1.08</td><td>3.60</td></tr><tr><td>Std Likert</td><td>0.81</td><td>0.23</td><td>0.78</td></tr></table>
+
+Object transfiguration. We evaluate our algorithm on horse zebra with a human perceptual study. We randomly selected 50 images from the horse2zebra test set and showed the input images and three generated images from CycleGAN, DistanceGAN and HarmonicGAN (with generated images in random order). 10 participants were asked to score the generated images on a Likert scale from 1 to 5 (as above). As shown in Table 4, the participants give the highest score to the proposed method (in $72 \%$ of cases), significantly more often than CycleGAN (in $28 \%$ of cases). Additionally, the average Likert score of our method was 3.60, outperforming 3.16 of CycleGAN and 1.08 of DistanceGAN, indicating that our method generates better results.
+
+# 4.5 QUALITATIVE COMPARISON
+
+Medical imaging. Fig. 5 shows the qualitative comparison of the proposed method (HarmonicGAN) and the baseline methods CycleGAN on Flair $ \mathrm { T } 1$ . It shows that CycleGAN may remove tumors in the original images and add tumors to other locations in the brain. In contrast, our method preserves the location of tumors, confirming that the harmonic regularization can maintain the inherent property of tumor/non-tumor regions and solves the tumor add/remove problem introduced in Cohen et al. (2018). More results and analysis are shown in Fig. 6 and Fig. 9.
+
+Object transfiguration. Fig. 5 shows a qualitative comparison of our method on the horse zebra task. We observe that we correct several problems in CycleGAN, including not changing the background and performing more complete transformations. More results and analysis are shown in Fig. 7 and Fig. 10.
+
+# 5 CONCLUSION
+
+We introduce a smoothness term over the sample graph to enforce smoothness-consistency between the source and target domains. We have shown that by introducing additional regularization to enforce consistent mappings during the image-to-image translation, the inherent self-consistency property of samples can be maintained. Through a set of quantitative, qualitative and user studies, we have demonstrated that this results in a significant improvement over the current state-of-the-art methods in a number of applications including medical imaging, object transfiguration, and semantic labeling. In a medical imaging task in particular our method provides a very significant improvement over CycleGAN.
+
+# REFERENCES
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+Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In ECCV, 2016.
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+Jonathan Long, Evan Shelhamer, and Trevor Darrell. Fully convolutional networks for semantic segmentation. In CVPR, 2015.
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+Zili Yi, Hao (Richard) Zhang, Ping Tan, and Minglun Gong. Dualgan: Unsupervised dual learning for image-to-image translation. In ICCV, 2017.
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+Han Zhang, Tao Xu, Hongsheng Li, Shaoting Zhang, Xiaolei Huang, Xiaogang Wang, and Dimitris Metaxas. StackGAN: Text to photo-realistic image synthesis with stacked generative adversarial networks. arXiv preprint, 2017.
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+Zizhao Zhang, Lin Yang, and Yefeng Zheng. Translating and segmenting multimodal medical volumes with cycle-and shapeconsistency generative adversarial network. In CVPR, 2018.
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+Shuai Zheng, Sadeep Jayasumana, Bernardino Romera-Paredes, Vibhav Vineet, Zhizhong Su, Dalong Du, Chang Huang, and Philip HS Torr. Conditional random fields as recurrent neural networks. In ICCV, 2015.
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+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In ICCV, 2017a.   
+Jun-Yan Zhu, Richard Zhang, Deepak Pathak, Trevor Darrell, Alexei A Efros, Oliver Wang, and Eli Shechtman. Toward multimodal image-to-image translation. In NIPS, 2017b.   
+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. CycleGAN failure cases. https: //github.com/junyanz/CycleGAN#failure-cases, 2018.   
+Xiaojin Zhu. Semi-supervised learning literature survey. Computer Science, University of Wisconsin-Madison, 2006.   
+Xiaojin Zhu, Zoubin Ghahramani, and John D Lafferty. Semi-supervised learning using gaussian fields and harmonic functions. In ICML, 2003.
+
+# 6 APPENDIX
+
+# 6.1 COMPARISON TO DISTANCEGAN
+
+The proposed HarmonicGAN looks a bit similar to DistanceGAN Benaim & Wolf (2017), but actually there is a large difference between them. DistanceGAN encourages the distance of samples to an absolute mean during translation. In contrast, HarmonicGAN enforces a smoothness term naturally under the graph Laplacian, making the motivations of DistanceGAN and HarmonicGAN quite different. Comparing the distance constraint in DistanceGAN and the smoothness constraint in HarmonicGAN, we can conclude the following main differences between them:
+
+(1) They show different motivations and formulations. The distance constraint aims to preserve the distance between samples in the mapping in a direct way, so it minimizes the expectation of differences between distances in two domains. The distance constraint in DistanceGAN is not doing a graph-based Laplacian to explicitly enforce smoothness. In contrast, the smoothness constraint is designed from a graph Laplacian to build the similarity-consistency between image patches. Thus, the smoothness constraint uses the affinity between two patches as weight to measure the similarityconsistency between two domains. The whole idea is based on manifold learning. The smoothness term defines a Laplacian $\Delta = D - W$ , where $W$ is our weight matrix and $D$ is a diagonal matrix with $\begin{array} { r } { D _ { i } = \sum _ { j } \bar { w _ { i j } } } \end{array}$ , thus, the smoothness term defines a graph Laplacian with the minimal value achieved as a harmonic function.
+
+(2) They are different in implementation. The smoothness constraint in HarmonicGAN is computed on image patches while the distance constraint in DistanceGAN is computed on whole image samples. Therefore, the smoothness constraint is fine-grained compared to the distance constraint. Moreover, the distances in DistanceGAN is directly computed from the samples in each domain. They scale the distances with the precomputed means and stds of two domains to reduce the effect of the gap between two domains. Differently, the smoothness constraint in HarmonicGAN is measured on the features (Histogram or CNN features) of each patch, which maps samples in two domains into the same feature space and removes the gap between two domains.
+
+(3) They show different results. Fig. 6 shows the qualitative results of CycleGAN, DistanceGAN and the proposed HarmonicGAN on the BRATS dataset. As shown in Fig. 6, the problem of randomly adding/removing tumors in the translation of CycleGAN is still present in the results of DistanceGAN, while HarmonicGAN can correct the location of tumors. Table 1 shows the quantitative results on the whole test set, which also yields the same conclusion. The results of DistanceGAN on four metrics are even worse than CycleGAN, while HarmonicGAN yields a large improvement over CycleGAN.
+
+![](images/ad84f316926cf2190e66e0fe76a39d6f9178aa038c1ff14a3714b39753746362.jpg)  
+Figure 6: Comparison of CycleGAN, DistanceGAN and the proposed HarmonicGAN on BRATS dataset.
+
+# 6.2 COMPARISON TO CRF
+
+There are some fundamental differences between the CRF literature and our work. They differ in output space, mathematical formulation, application domain, effectiveness, and the role in the overall algorithm. The similarity between CRF and HarmonicGAN lies the adoption of a regularization term: a binary term in the CRF case and a Laplacian term in HarmonicGAN.
+
+The smoothness term in HarmonicGAN is not about obtaining ‘smoother’ images/labels in the translated domain, as seen in the experiments; instead, HarmonicGAN is about preserving the overall integrity of the translation itself for the image manifold. This is the main reason for the large improvement of HarmonicGAN over CycleGAN.
+
+To further demonstrate the difference of HarmonicGAN and CRF, we perform an experiment applying the pairwise regularization of CRFs to the CycleGAN framework. For each pixel of the generated image, we compute the unary term and binary term with its 8 neighbors, and then minimize the objective function of CRF. The results are shown in Table 5. The pairwise regularization of CRF is unable to handle the problem of CycleGAN illustrated in Fig. 1. What’s worse, using the pairwise regularization may over-smooth the boundary of generated images, which results in extra artifacts. In contrast, HarmonicGAN aims at preserving similarity from the overall view of the image manifold, and can thus exploit similarity-consistency of the generated images, rather than over-smooth the boundary.
+
+Table 5: Reconstruction evaluation on the BRATS dataset with comparison to CRF.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Flair→T1</td><td colspan="2">T1→Flair</td></tr><tr><td>MAE↓</td><td>MSE↓</td><td>MAE↓</td><td>MSE↓</td></tr><tr><td>CycleGAN</td><td>10.47</td><td>674.40</td><td>11.81</td><td>1026.19</td></tr><tr><td>CycleGAN+CRF</td><td>11.24</td><td>839.47</td><td>12.25</td><td>1138.42</td></tr><tr><td>HarmonicGAN-Histogram (ours)</td><td>6.38</td><td>216.83</td><td>5.04</td><td>163.29</td></tr><tr><td>HarmonicGAN-VGG (ours)</td><td>6.86</td><td>237.94</td><td>4.69</td><td>127.84</td></tr></table>
+
+![](images/5b137aea5fc7a75338e7e48ee7a01aa6ade906e6ab5516996c2a80f1150f1294.jpg)  
+Figure 7: Comparison on horse zebra for the Putin photo. CycleGAN translates both the background and human to a zebra-like texture. In contrast, HarmonicGAN does better in background region and achieves an improvement in some regions of of the human (Putin’s face), but it still fails on the human body. We hypothesize this is because the semantic features used by HarmonicGAN have not been trained on humans without a shirt.
+
+![](images/7485832a797a6eafb17db676d5cac03dc090762c94b00b56dac27313411c675b.jpg)  
+Figure 8: For similar image patches in input images, visualizing the average distance of corresponding image patches in translated images for cross-modal medical image synthesis on BRATS dataset. The distance of image patches in the target space shows the inconsistently translated patches.
+
+![](images/f359d07ea9fc2cc3e5b41db9aabb779da52b2a3c3aaa790813faa2e73c63326f.jpg)  
+Figure 9: Comparison on BRATS dataset.
+
+![](images/e6e239af50aa223e27fe0949d0898570941a7391f0687a3221499d3966c693a2.jpg)  
+Figure 10: Comparison on horse zebra.
+
+![](images/5274b54835d86cc916ba2c37397785dd45038636c947fc17207f80c471479df6.jpg)  
+Figure 11: Additional results of the proposed HarmonicGAN. From top to bottom are: apple to orange, orange to apple, facade to label, label to facade, aerial to map, map to aerial, summer to winter, winter to summer.

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+        "type": "text",
+        "text": "HARMONIC UNPAIRED IMAGE-TO-IMAGE TRANSLATION ",
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+        "text": "Tomas Pfister   \nGoogle Cloud AI   \nSunnyvale, USA   \ntpfister@google.com   \nRui Zhang   \nGoogle Cloud AI &   \nChinese Academy of Sciences   \nBeijing, China   \nzhangrui@ict.ac.cn   \nJia Li   \nGoogle Cloud AI   \nSunnyvale, USA   \nlijiali@google.com ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The recent direction of unpaired image-to-image translation is on one hand very exciting as it alleviates the big burden in obtaining label-intensive pixel-to-pixel supervision, but it is on the other hand not fully satisfactory due to the presence of artifacts and degenerated transformations. In this paper, we take a manifold view of the problem by introducing a smoothness term over the sample graph to attain harmonic functions to enforce consistent mappings during the translation. We develop HarmonicGAN to learn bi-directional translations between the source and the target domains. With the help of similarity-consistency, the inherent selfconsistency property of samples can be maintained. Distance metrics defined on two types of features including histogram and CNN are exploited. Under an identical problem setting as CycleGAN, without additional manual inputs and only at a small training-time cost, HarmonicGAN demonstrates a significant qualitative and quantitative improvement over the state of the art, as well as improved interpretability. We show experimental results in a number of applications including medical imaging, object transfiguration, and semantic labeling. We outperform the competing methods in all tasks, and for a medical imaging task in particular our method turns CycleGAN from a failure to a success, halving the mean-squared error, and generating images that radiologists prefer over competing methods in $9 5 \\%$ of cases. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Image-to-image translation (Isola et al., 2017) aims to learn a mapping from a source domain to a target domain. As a significant and challenging task in computer vision, image-to-image translation benefits many vision and graphics tasks, such as realistic image synthesis (Isola et al., 2017; Zhu et al., 2017a), medical image generation (Zhang et al., 2018; Dar et al., 2018), and domain adaptation (Hoffman et al., 2018). Given a pair of training images with detailed pixel-to-pixel correspondences between the source and the target, image-to-image translation can be cast as a regression problem using e.g. Fully Convolutional Neural Networks (FCNs) (Long et al., 2015) by minimizing e.g. the per-pixel prediction loss. Recently, approaches using rich generative models based on Generative Adaptive Networks (GANs) (Goodfellow et al., 2014; Radford et al., 2016; Arjovsky et al., 2017) have achieved astonishing success. The main benefit of introducing GANs (Goodfellow et al., 2014) to image-to-image translation (Isola et al., 2017) is to attain additional image-level (often through patches) feedback about the overall quality of the translation, and information which is not directly accessible through the per-pixel regression objective. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The method by Isola et al. (2017) is able to generate high-quality images, but it requires paired training data which is difficult to collect and often does not exist. To perform translation without paired data, circularity-based approaches (Zhu et al., 2017a; Kim et al., 2017; Yi et al., 2017) have been proposed to learn translations from a set to another set, using a circularity constraint to establish relationships between the source and target domains and forcing the result generated from a sample in the source domain to map back and generate the original sample. The original image-to-image translation problem (Isola et al., 2017) is supervised in pixel-level, whereas the unpaired image-to-image translation task (Zhu et al., 2017a) is considered unsupervised, with pixel-level supervision absent but with adversarial supervision at the image-level (in the target domain) present. By using a cycled regression for the pixel-level prediction (source target source) plus a term for the adversarial difference between the transferred images and the target images, CycleGAN is able to successfully, in many cases, train a translation model without paired source target supervision. However, lacking a mechanism to enforce regularity in the translation creates problems like in Fig. 1 (a) and Fig. 2, making undesirable changes to the image contents, superficially removing tumors (the first row) or creating tumors (the second row) at the wrong positions in the target domain. Fig. 1 (b) also shows some artifacts of CycleGAN on natural images when translating horses into zebras. ",
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+        "image_caption": [
+            "Figure 1: HarmonicGAN corrects major failures in multiple domains: (a) for medical images it corrects incorrectly removed (top) and added (bottom) tumors; and (b) for horse zebra transfiguration it does not incorrectly transform the background (top) and performs a complete translation (bottom). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "To combat the above issue, in this paper we look at the problem of unpaired image-to-image translation from a manifold learning perspective (Tenenbaum et al., 2000; Roweis & Saul, 2000). Intuitively, the problem can be alleviated by introducing a regularization term in the translation, encouraging similar contents (based on textures or semantics) in the same image to undergo similar translations/transformations. A common principle in manifold learning is to preserve local distances after the unfolding: forcing neighboring (similar) samples in the original space to be neighbors in the new space. The same principle has been applied to graph-based semisupervised learning (Zhu, 2006) where harmonic functions with graph Laplacians (Zhu et al., 2003; Belkin et al., 2006) are used to obtain regularized labels of unlabeled data points. ",
+        "bbox": [
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+    {
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+        "image_caption": [
+            "Figure 2: For CycleGAN, the reconstructed image may perfectly match the source image under the circularity constraint while translated image dose not maintain the inherent property of the source image (e.g. a tumor), thus generating unexpected results (e.g. incorrectly removing a tumor). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "During the translation/transformation, some domain-specific attributes are changed, such as the colors, texture, and semantics of certain image regions. Although there is no supervised information for these changes, certain consistency during the transformation is desirable, meaning that for image contents similar in the source space should also be similar in the target space. Inspired by graphbased semi-supervised learning (Zhu et al., 2003; Zhu, 2006), we introduce smoothness terms to unpaired image-to-image translation (Zhu et al., 2017a) by providing a stronger regularization for the translation/transformation between the source and target domains, aiming to exploit the “manifold structure” of the source and target domains. For a pair of similar samples (two different locations in an image; one can think of them as two patches although the receptive fields of CNN are quite large), we add the smoothness term to minimize a weighted distance of the corresponding locations in the target image. Note that two spatially distant samples might be neighbors in the feature space. We name our algorithm HarmonicGAN as it behaves harmonically along with the circularity and adversarial constraints to learn a pair of dual translations between the source and target domains, as shown in Fig. 1. Distance metrics defined on two alternative features are adopted: (1) a low-level soft RGB histograms; and (2) CNN (VGG) features with pre-trained semantics. ",
+        "bbox": [
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+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments in a number of applications, showing that in each of them our method outperforms existing methods quantitatively, qualitatively, and with user studies. For a medical imaging task (Cohen et al., 2018) that was recently calling attention to a major CycleGAN failure case (learning to accidentally add/remove tumors in an MRI image translation task), our proposed method provides a large improvement over CycleGAN, halving the mean-squared error, and generating images that radiologists prefer over competing methods in $9 5 \\%$ of cases. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "CONTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
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+            281,
+            292,
+            295
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. We introduce smooth regularization over the graph for unpaired image-to-image translation to attain harmonic translations.   \n2. When building an end-to-end learning pipeline, we adopt two alternative types of feature measures to compute the weight matrix for the graph Laplacian, one based on a soft histogram (Wang et al., 2016) and another based on semantic CNN (VGG) features (Simonyan & Zisserman, 2015).   \n3. We show that this method results in significantly improved consistency for transformations. With experiments on multiple translation tasks, we demonstrate that HarmonicGAN outperforms the state-of-the-art. ",
+        "bbox": [
+            212,
+            301,
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+            439
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            344,
+            477
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As discussed in the introduction, the general image-to-image translation task in the deep learning era was pioneered by (Isola et al., 2017), but there are prior works such as image analogies (Hertzmann et al., 2001) that aim at a similar goal, along with other exemplar-based methods (Efros & Freeman, 2001; Criminisi et al., 2004; Barnes et al., 2009). After (Isola et al., 2017), a series of other works have also exploited pixel-level reconstruction constraints to build connections between the source and target domains (Zhang et al., 2017; Wang et al., 2018). The image-to-image translation framework (Isola et al., 2017) is very powerful but it requires a sufficient amount of training data with paired source to target images, which are often laborious to obtain in the general tasks such as labeling (Long et al., 2015), synthesis (Chen & Koltun, 2017), and style transfer (Huang & Belongie, 2017). ",
+        "bbox": [
+            174,
+            492,
+            825,
+            631
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unpaired image-to-image translation frameworks (Zhu et al., 2017a;b; Liu et al., 2017; Shrivastava et al., 2017; Kim et al., 2017) such as CycleGAN remove the requirement of having detailed pixellevel supervision. In CycleGAN this is achieved by enforcing a bi-directional prediction from source to target and target back to source, with an adversarial penalty in the translated images in the target domain. Similar unsupervised circularity-based approaches (Kim et al., 2017; Yi et al., 2017) have also been developed. The CycleGAN family models (Zhu et al., 2017a;b) point to an exciting direction of unsupervised approaches but they also create artifacts in many applications. As shown in Fig. 2, one reason for this is that the circularity constraint in CycleGAN lacks the straightforward description of the target domain, so it may change the inherent properties of the original samples and generate unexpected results which are inconsistent at different image locations. These failures have been prominently explored in recent works, showing that CycleGAN (Zhu et al., 2017a) may add or remove tumors accidentally in cross-modal medical image synthesis (Cohen et al., 2018), and that in the task of natural image transfiguration, e.g. from a horse to zebra, regions in the background may also be translated into a zebra-like texture (Zhu et al., 2018) (see Fig. 1 (b)). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here we propose HarmonicGAN that introduces a smoothness term into the CycleGAN framework to enforce a regularized translation, enforcing similar image content in the source space to also be similar in the target space. We follow the general design principle in manifold learning (Tenenbaum et al., 2000; Roweis & Saul, 2000) and the development of harmonic functions in the graph-based semi-supervised learning literature (Zhu et al., 2003; Belkin et al., 2006; Zhu, 2006; Weston et al., 2012). There has been previous work, DistanceGAN (Benaim & Wolf, 2017), in which distance preservation was also implemented. However, DistanceGAN differs from HarmonicGAN in (1) motivation, (2) formulation, (3) implementation, and (4) performance. The primary motivation of DistanceGAN demonstrates an alternative loss term for the per-pixel difference in CycleGAN. In HarmonicGAN, we observe that the cycled per-pixel loss is effective and we aim to make the translation harmonic by introducing additional regularization. The smoothness term acts as a graph Laplacian imposed on all pairs of samples (using random samples in implementation). In the experimental results, we show that the artifacts in CycleGAN are still present in DistanceGAN, whereas HarmonicGAN provides a significant boost to the performance of CycleGAN. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In addition, it is worth mentioning that the smoothness term proposed here is quite different from the binary term used in the Conditional Random Fields literature (Lafferty et al., 2001; Krahenb ¨ uhl &¨ Koltun, 2011), either fully supervised (Chen et al., 2018; Zheng et al., 2015) or weakly-supervised (Tang et al., 2018; Lin et al., 2016). The two differ in (1) output space (multi-class label vs. highdimensional features), (2) mathematical formulation (a joint conditional probably for the neighboring labels vs. a Laplacian function over the graph), (3) application domain (image labeling vs. image translation), (4) effectiveness (boundary smoothing vs. manifold structure preserving), and (5) the role in the overall algorithm (post-processing effect with relatively small improvement vs. large-area error correction). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 HARMONICGAN FOR UNPAIRED IMAGE-TO-IMAGE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Follothe tarwhere baain . T formulation in CycleGAN (Zhu et al.,, we consider unpaired training samples  goal of image-to-image translation is to for the where ir of du doma, and ings, $X$ $Y$ $\\{ \\mathbf { x } _ { k } \\} _ { k = 1 } ^ { N }$ $\\mathbf { x } _ { k } \\in X$ $\\{ \\mathbf { y } _ { k } \\} _ { k = 1 } ^ { N }$ $\\mathbf { y } _ { k } \\in Y$   \nforward mapping $G : X  Y$ and backward mapping $F : Y  X$ . Two discriminators $D _ { X }$ and   \n$D _ { Y }$ are adopted in (Zhu et al., 2017a) to distinguish between real images and generated images.   \nIn particular, the discriminator $D _ { X }$ aims to distinguish real image $\\{ { \\bf x } \\}$ from the generated image   \n$\\{ \\bar { F ( \\mathbf { y } ) } \\}$ ; similarly discriminator $D _ { Y }$ distinguishes $\\{ \\mathbf { y } \\}$ from $\\{ G ( \\mathbf { x } ) \\}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the objective of adversarial constraint is applied in both source and target domains, expressed in (Zhu et al., 2017a) as: ",
+        "bbox": [
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+            502,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9cfa318610eedad6997b6703858b0c3421febebcf720e8f9b0b1c4643a78d98.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { G A N } } ( G , D _ { Y } , X , Y ) = \\mathbb { E } _ { { \\mathbf { y } } \\in Y } [ \\log D _ { Y } ( { \\mathbf { y } } ) ] + \\mathbb { E } _ { { \\mathbf { x } } \\in X } [ \\log ( 1 - D _ { Y } ( G ( { \\mathbf { x } } ) ) ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            537,
+            743,
+            556
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and ",
+        "bbox": [
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+            563,
+            200,
+            575
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1f9bea9aa55f507949ee13c036aa543b5e855578615c432d24855513af8ae1d.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { G A N } } ( F , D _ { X } , X , Y ) = \\mathbb { E } _ { \\mathbf { x } \\in X } [ \\log D _ { X } ( \\mathbf { x } ) ] + \\mathbb { E } _ { \\mathbf { y } \\in Y } [ \\log ( 1 - D _ { X } ( F ( \\mathbf { y } ) ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            574,
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+            590
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For notational simplicity, we denote the GAN loss as ",
+        "bbox": [
+            173,
+            593,
+            522,
+            608
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7deb5ea7bdf6bcfe2d165f05ee452cca0c834846808d84c38b4c71cf5c1b9e9e.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { G A N } } ( G , F ) = \\arg \\operatorname* { m a x } _ { D _ { Y } , D _ { X } } [ \\mathcal { L } _ { \\mathrm { G A N } } ( G , D _ { Y } , X , Y ) + \\mathcal { L } _ { \\mathrm { G A N } } ( F , D _ { X } , X , Y ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            613,
+            740,
+            638
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since the data in the two domains are unpaired, a circularity constraint is introduced in (Zhu et al., 2017a) to establish relationships between $X$ and $Y$ . The circularity constraint enforces that $G$ and $F$ are a pair of inverse mappings, and that the translated sample can be mapped back to the original sample. The circularity constraint contains consistencies in two aspects: the forward cycle $\\mathbf { x } \\to G ( X ) \\to F ( G ( \\mathbf { x } ) ) \\sim \\mathbf { x }$ and the backward cycle $\\mathbf { y } \\to F ( \\mathbf { y } ) \\to G ( F ( \\mathbf { y } ) ) \\sim \\mathbf { y }$ . Thus, the circularity constraint is formulated as (Zhu et al., 2017a): ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ead1e00dc2f392580793739684a50d9d4c21315b5bd5fbaf22db0586601fd2fe.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { c y c } } ( G , F ) = \\mathbb { E } _ { \\mathbf { x } \\in X } | | F ( G ( \\mathbf { x } ) ) - \\mathbf { x } | | _ { 1 } + \\mathbb { E } _ { \\mathbf { y } \\in Y } | | G ( F ( \\mathbf { y } ) ) - \\mathbf { y } | | _ { 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            742,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here we rewrite the overall objective in (Zhu et al., 2017a) to minimize as: ",
+        "bbox": [
+            178,
+            765,
+            661,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64ddd8923562ae7345b5bbcfa1d192a7be344c7d029f725365e59b97825b1473.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C y c l e G A N } } ( G , F ) = \\lambda _ { \\mathrm { G A N } } \\times \\mathcal { L } _ { \\mathrm { G A N } } ( G , F ) + \\lambda _ { \\mathrm { c y c } } \\times \\mathcal { L } _ { \\mathrm { c y c } } ( G , F ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            786,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the weights $\\lambda _ { \\mathrm { G A N } }$ and $\\lambda _ { \\mathrm { c y c } }$ control the importance of the corresponding objectives. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 SMOOTHNESS TERM OVER THE GRAPH ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The full objective of circularity-based approach contains adversarial constraints and a circularity constraint. The adversarial constraints ensure the generated samples are in the distribution of the source or target domain, but ignore the relationship between the input and output of the forward or backward translations. The circularity constraint establishes connections between the source and target domain by forcing the forward and backward translations to be the inverse of each other. However, CycleGAN has limitations: as shown in Fig. 2, the circular projection might perfectly match the input, and the translated image might look very well like a real one, but the translated image may not maintain the inherent property of the input and contain a large artifact that is not connected to the input. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/adfdd441402b193fda323c1a37669af4bbf6c0840412563d205f9709ae0962c3.jpg",
+        "image_caption": [
+            "Figure 3: Architecture of HarmonicGAN, consisting of a pair of inverse generators $G$ , $F$ and two discriminators $D _ { X }$ , $D _ { Y }$ . The objective combines an adversarial constraint, circularity constraint and smoothness term. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here we propose a smoothness term to enforce a stronger correlation between the source and target domains that focuses on providing similarity-consistency between image patches during the translation. The smoothness term defines a graph Laplacian with the minimal value achieved as a harmonic function. We define the set consisting of individual image patches as the nodes of the graph $\\mathcal { G }$ . $\\vec { x } _ { i }$ is referred to as the feature vector of the $i$ -th image patch in $\\mathbf { x } \\in X$ . For the image set $X$ , we define the set that consists of individual samples (image patches) of source image set $X$ as $S = \\{ \\vec { x } ( i ) , i = 1 . . M \\}$ where $M$ is the total number of the samples/patches. An affinity measure (similarity) computed on image patch $\\vec { x } ( i )$ and image patch $\\vec { x } ( j )$ , $w _ { i j } ( X )$ (a scalar), defines the edge on the graph $\\mathcal { G }$ of $S$ . The smoothness term acts as a graph Laplacian imposed on all pairs of image patches. Therefore, we define a smoothness term over the graph as ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ba878ad513a5983dbe228fa4e31d6f4c1b59093965e7ffa2cd44289531d76f33.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { S m o u t } } ( G , X , Y ) = \\mathbb { E } _ { \\mathbf { x } \\in X } \\left[ \\sum _ { i , j } w _ { i j } ( X ) \\times D i s t [ G ( \\vec { x } ) ( i ) , G ( \\vec { x } ) ( j ) ] + \\sum _ { i , j } w _ { i j } ( G ( X ) ) \\times D i s t [ F ( G ( \\vec { x } ) ) ( i ) , F ( G ( \\vec { x } ) ) ( j ) ] \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            627,
+            813,
+            646
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { i j } ( X ) = \\exp \\{ - D i s t [ \\vec { x } ( i ) , \\vec { x } ( j ) ] / \\sigma ^ { 2 } \\}$ (Zhu et al., 2003) defines the affinity between two patches $\\vec { x } ( i )$ and $\\vec { x } ( j )$ based on their distances (e.g. measured on histogram or CNN features). $D i s t [ G ( \\vec { y } ) ( i ) , G ( \\vec { y } ) ( j ) ]$ defines the distance between two image patches after translation at the same locations. In implementation, we first normalize the features to the scale of [0,1] and then use the L1 distance of normalized features as the Dist function (for both histogram and CNN features). Similarly, we define a smoothness term for the backward part as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/99b31867fcc9047c659676520a8e28673a72c8b9cf2f8ec1766ce463f62c9ff2.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { S m o u t } } ( F , Y , X ) = \\mathbb { E } _ { \\mathbf { y } \\in Y } \\Big [ \\sum _ { i , j } w _ { i j } ( Y ) \\times D i s t [ F ( \\vec { y } ) ( i ) , F ( \\vec { y } ) ( j ) ] + \\sum _ { i , j } w _ { i j } ( F ( Y ) ) \\times D i s t [ G ( F ( \\vec { y } ) ) ( i ) , G ( F ( \\vec { y } ) ) ( j ) ] \\Big ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            746,
+            810,
+            765
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The combined loss for the smoothness thus becomes ",
+        "bbox": [
+            176,
+            775,
+            519,
+            790
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab73dc43fa2b5cf505a1f097475e66bb2731e523eb02d845db25c3997660d5c0.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { S m o o t h } } ( G , F ) = { \\mathcal { L } } _ { \\mathrm { S m o o t h } } ( G , X , Y ) + { \\mathcal { L } } _ { \\mathrm { S m o o t h } } ( F , Y , X ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 OVERALL OBJECTIVE FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            828,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As shown in Fig. 3, HarmonicGAN consists of a pair of inverse generators $G$ , $F$ and two discriminators $D _ { X } , D _ { Y }$ , defined in Eqn. (1) and Eqn. (2). The full objective combines an adversarial constraint (see Eqn. (3)), a circularity constraint (see Eqn. (4)), and a smoothness term (see Eqn. (8)). The adversarial constraint forces the translated images to be plausible and indistinguishable from the real images; the circularity constraint ensures the cycle-consistency of translated images; and the smoothness term provides a stronger similarity-consistency between patches to maintain inherent properties of the images. ",
+        "bbox": [
+            174,
+            853,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/46e6c62829e8f49b4103ba63cc0b44325c2be733a85eee92f4bb0d85c7dc7f29.jpg",
+        "image_caption": [
+            "Figure 4: Visualization using t-SNE (Maaten & Hinton, 2008) to illustrate the effectiveness of the smoothness term in HarmonicGAN (best viewed in color). As shown in the top two figures (source and target respectively), the smoothness term acts as a graph Laplacian imposed on all pairs of image patches. Bottom-left: For two similar patches in the original sample, if one patch is translated to a tumor region while the other is not, the two patches will have a large distance in the target space, resulting in a translation that incorrectly adds a tumor into the original sample (result for CycleGAN shown). Bottom-right: For two similar patches in the original sample, if the translation maintains the non-tumor property of these two patches in the translated sample then the two patches will also be similar in the target space (result for HarmonicGAN shown). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            117,
+            816,
+            410
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            556,
+            823,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Combining Eqn. (5) and Eqn. (8), the overall objective for our proposed HarmonicGAN under the smoothness constraint becomes ",
+        "bbox": [
+            173,
+            590,
+            823,
+            619
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b5da6232b865e1088756617fd3f284e5cdd7b2285682ecde2274c15e496514e.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { H a r m o n i c G A N } } ( G , F ) = { \\mathcal { L } } _ { \\mathrm { C y c l e G A N } } ( G , F ) + \\lambda _ { \\mathrm { S m o o t h } } \\times { \\mathcal { L } } _ { \\mathrm { S m o o t h } } ( G , F ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            623,
+            722,
+            641
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similar to the graph-based semi-supervised learning definition (Zhu et al., 2003; Zhu, 2006), the solution to Eqn. (9) leads to a harmonic function. The optimization process during training obtains: ",
+        "bbox": [
+            173,
+            643,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8b567d2400ff2a9fb2452c7802822bcaa3adf114f8e82c6ab80fdf39c5d499a8.jpg",
+        "text": "$$\nG ^ { * } , F ^ { * } = \\arg \\operatorname* { m i n } _ { G , F } { \\mathcal { L } } _ { \\mathrm { H a r m o n i c G A N } } ( G , F ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            674,
+            629,
+            699
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The effectiveness of the smoothness term of Eqn. (8) is evident. In Fig. 4, we show (using tSNE (Maaten & Hinton, 2008)) that the local neighborhood structure is being preserved by HarmonicGAN, whereas CycleGAN results in two similar patches being far apart after translation. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 FEATURE DESIGN ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            761,
+            336,
+            775
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the smoothness constraint, the similarity of a pair of patches is measured on the features for each patch (sample point). All the patches in an image form a graph. Here we adopt two types of features: (1) a low-level soft histogram, and (2) pre-trained CNN (VGG) features that carry semantic information. Soft histogram features are lightweight and easy to implement but without much semantic information; VGG requires an additional CNN network but carries more semantics. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 SOFT RGB HISTOGRAM FEATURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            871,
+            468,
+            886
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We first design a weight matrix based on simple low-level RGB histograms. To make the end-to-end learning system work, it is crucial to make the computation of gradient in the histograms derivable. ",
+        "bbox": [
+            174,
+            895,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We adopt a soft histogram representation proposed in (Wang et al., 2016) but fix the means and the bin size. This histogram representation is differentiable and its gradient is back-propagateable. This soft histogram function contains a family of linear basis functions $\\psi _ { b } , b = 1 , \\ldots , B$ , where $B$ is the number of bins in the histogram. As $\\vec { x } _ { i }$ represents the $i$ -th patch in image domain $X$ , for each pixel $j$ in $\\vec { x } _ { i }$ , $\\psi _ { b } ( \\vec { x } _ { i } ( j ) )$ represents pixel $j$ voting for the b-th bin, expressed as: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fbb0e3b07a8b59159918aa5bc058abe3a30ed64c98fb7f3a6d0ed7b235f5dd73.jpg",
+        "text": "$$\n\\begin{array} { r } { \\psi _ { b } ( \\vec { x } _ { i } ( j ) ) = \\mathrm { m a x } \\{ 0 , 1 - | \\vec { x } _ { i } ( j ) - \\mu _ { b } | \\times w _ { b } \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            650,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\mu _ { b }$ and $w _ { b }$ are the center and width of the b-th bin. The representation of $\\vec { x } _ { i }$ in the RGB space is the linear combination of linear basis functions on all the pixels in $\\vec { x } _ { i }$ , expressed as: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1bf002b98bdfbc1cb3fff9b95edf03827ffb9cfb54d97691312befaf4ff1fb96.jpg",
+        "text": "$$\n\\phi _ { h } ( X , i , b ) = \\phi _ { h } ( \\vec { x } _ { i } , b ) = \\sum _ { j } \\psi _ { b } ( \\vec { x } _ { i } ( j ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            233,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\phi _ { h }$ is the RGB histogram feature, $b$ is the index of dimension of the RGB histogram representation, and $j$ represents any pixel in the patch $\\vec { x } _ { i }$ . The RGB histogram representation $\\phi _ { h } ( X , i )$ of $\\vec { x } _ { i }$ is a $B$ -dimensional vector. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3.2 SEMANTIC CNN FEATURES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For some domains we instead use semantic features to acquire higher-level representations of patches. The semantic representations are extracted from a pre-trained Convolutional Neural Network (CNN). The CNN encodes semantically relevant features from training on a large-scale dataset. It extracts semantic information of local patches in the image through multiple pooling or stride operators. Each point in the feature maps of the CNN is a semantic descriptor of the corresponding image patch. Additionally, the semantic features learned from the CNN are differentiable and the CNN can be integrated into HarmonicGAN and be trained end-to-end. We instantiate the semantic feature $\\phi _ { s }$ as a pre-trained CNN model e.g. VGGNet (Simonyan & Zisserman, 2014). In implementation, we select the layer 4 3 after ReLU from VGG-16 network for computing the semantic features. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate the proposed method on three different applications: medical imaging, semantic labeling, and object transfiguration. We compare against several unpaired image-to-image translation methods: CycleGAN (Zhu et al., 2017a), DiscoGAN (Kim et al., 2017), DistanceGAN (Benaim & Wolf, 2017), and UNIT (Liu et al., 2017). We also provide two user studies as well as qualitative results. The appendix provides additional results and analysis. ",
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+    {
+        "type": "text",
+        "text": "4.1 DATASETS AND EVALUATION METRICS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Medical imaging. This task evaluates cross-modal medical image synthesis, Flair $ \\mathrm { T } 1$ . The models are trained on the BRATS dataset (Menze et al., 2015) which contains paired MRI data to allow quantitative evaluation. Similar to the previous work (Cohen et al., 2018), we use a training set of 1400 image slices $50 \\%$ healthy and $50 \\%$ tumors) and a test set of 300, and use their unpaired training scenario. We adopt the Mean Absolute Error (MAE) and the Mean Squared Error (MSE) between the generated images and the real images to evaluate the reconstruction errors, and further use the Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM) to evaluate the reconstruction quality of generated images. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Semantic labeling. We also test our method on the labels photos task using the Cityscapes dataset (Cordts et al., 2016) under the unpaired setting as in the original CycleGAN paper. For quantitative evaluation, in line with previous work, for labels photos we adopt the “FCN score” (Isola et al., 2017), which evaluates how interpretable the generated photos are according to a semantic segmentation algorithm. For photos labels, we use the standard segmentation metrics, including per-pixel accuracy, per-class accuracy, and mean class Intersection-Over-Union (Class IOU). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Object transfiguration. Finally, we test our method on the horse zebra task using the standard CycleGAN dataset (2401 training images, 260 test images). This task does not have a quantitative evaluation measure, so we instead provide a user study together with qualitative results. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We apply the proposed smoothness term on the framework of CycleGAN (Zhu et al., 2017a). Similar with CycleGAN, we adopt the architecture of (Johnson et al., 2016) as the generator and the PatchGAN (Isola et al., 2017) as the discriminator. The log likelihood objective in the original GAN is replaced with a least-squared loss (Mao et al., 2017) for more stable training. We resize the input images to the size of $2 5 6 \\times 2 5 6$ . For the histogram feature, we equally split the RGB range of [0, 255] to 16 bins, each with a range of 16. Images are divided into non-overlapping patches of $8 ~ \\times$ 8 and the histogram feature is computed on each patch. For the semantic feature, we adopt a VGG network pre-trained on ImageNet to obtain semantic features. We select the feature map of layer relu4 3 in VGG. The loss weights are set as $\\lambda _ { \\mathrm { G A N } } = \\lambda _ { \\mathrm { S m o o t h } } = 1$ , $\\lambda _ { \\mathrm { c y c } } = 1 0$ . Following CycleGAN, we adopt the Adam optimizer (Kingma & Ba, 2015) with a learning rate of 0.0002. The learning rate is fixed for the first 100 epochs and linearly decayed to zero over the next 100 epochs. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 QUANTITATIVE COMPARISON ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Medical imaging. Table 1 shows the reconstruction performance on medical image synthesis, Flair $ \\mathrm { { T } 1 }$ . The proposed method yields a large improvement over CycleGAN, showing lower MAE and MSE reconstruction losses, and higher PSNR and SSIM reconstruction scores, highlighting the significance of the proposed smoothness regularization. HarmonicGAN based on histogram and VGG features shows similar performance; the reconstruction losses of histogram-based HarmonicGAN are slightly lower than the VGG-based one in Flair $ \\mathrm { T } 1$ , while they are slightly higher in $\\mathrm { T } 1 $ Flair, indicating that both low-level RGB values and high-level CNN features can represent the inherent property of medical images well and help to maintain the smoothness-consistency of samples. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/902d727e8c406ff74795529af1eeb9bacc2d6699581834d7b5448be3f9459b6e.jpg",
+        "table_caption": [
+            "Table 1: Reconstruction evaluation of cross-modal medical image synthesis on the BRATS dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">Flair→T1</td><td colspan=\"4\">T1→Flair</td></tr><tr><td>MAE↓</td><td>MSE↓</td><td>PSNR↑</td><td>SSIM↑</td><td>MAE↓</td><td>MSE↓</td><td>PSNR ↑</td><td>SSIM↑</td></tr><tr><td>CycleGAN</td><td>10.47</td><td>674.40</td><td>22.35</td><td>0.80</td><td>11.81</td><td>1026.19</td><td>18.73</td><td>0.74</td></tr><tr><td>DiscoGAN</td><td>10.63</td><td>641.35</td><td>20.06</td><td>0.79</td><td>10.66</td><td>839.15</td><td>19.14</td><td>0.69</td></tr><tr><td>DistanceGAN</td><td>14.93</td><td>1197.64</td><td>17.92</td><td>0.67</td><td>10.57</td><td>716.75</td><td>19.95</td><td>0.64</td></tr><tr><td>UNIT</td><td>9.48</td><td>439.33</td><td>22.24</td><td>0.76</td><td>6.69</td><td>261.26</td><td>25.11</td><td>0.76</td></tr><tr><td colspan=\"9\">HarmonicGAN (ours)</td></tr><tr><td>Histogram</td><td>6.38</td><td>216.83</td><td>24.34</td><td>0.83</td><td>5.04</td><td>163.29</td><td>26.72</td><td>0.75</td></tr><tr><td>VGG</td><td>6.86</td><td>237.94</td><td>24.14</td><td>0.81</td><td>4.69</td><td>127.84</td><td>27.22</td><td>0.76</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Semantic labeling. We report semantic labeling results in Table 2. The proposed method using VGG features yields a $3 \\%$ improvement in Pixel Accuracy in translation scores for photo label and also shows stable improvements in other metrics, clearly outperforming all competing methods. The performance using a histogram is slightly lower than CycleGAN; we hypothesize that the reason is that the objects in photos have a large intra-class variance and inter-class similarity in appearance, e.g. cars have different colors, while vegetation and terrain have similar colors, thus the regularization of the RGB histogram is not appropriate to extract the inherent property of photos. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 USER STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Medical imaging. We randomly selected 100 images from BRATS test set. For each image, we showed one radiologist the real ground truth image, followed by images generated by CycleGAN, DistanceGAN and HarmonicGAN (different order for each image set to avoid bias). The radiologist was told to evaluate similarity by how likely they would lead to the same clinical diagnosis, and was asked to rate similarity of the generation methods on a Likert scale from 1 to 5 (1 is not similar at all, 5 is exactly same). Results are in shown in Table 3. In $9 5 \\%$ of cases, the radiologist preferred images generated by our method over the competing methods, and the average Likert score was 4.00 compared to 1.68 for CycleGAN, confirming that our generated images are significantly better. This is significant as it confirms that we solve the issue presented in a recent paper (Cohen et al., 2018) showing that CycleGAN can learn to accidentally add/remove tumors in images. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/cda88029c448fdb9092c985dc74da17ec1c06eb7f1004111ce4db63c0d1275ff.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative comparison for BRATS, Cityscapes and horse zebra (see appendix for more images). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/7745dd45cf4a0fd17f390dc133a2362ae259bee6e2a18ea762f21785fa6e30af.jpg",
+        "table_caption": [
+            "Table 2: FCN scores of Photo Label translation on the Cityscapes dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">Label →Photo</td><td colspan=\"3\">Photo →Label</td></tr><tr><td>Pixel Acc. ↑</td><td>Class Acc.↑</td><td>Class IoU↑</td><td>Pixel Acc. ↑</td><td>Class Acc. ↑</td><td>Class IoU↑</td></tr><tr><td>CycleGAN</td><td>52.7</td><td>15.2</td><td>11.0</td><td>57.2</td><td>21.0</td><td>15.7</td></tr><tr><td>DiscoGAN</td><td>45.0</td><td>11.1</td><td>7.0</td><td>45.2</td><td>10.9</td><td>6.3</td></tr><tr><td>DistanceGAN</td><td>48.5</td><td>10.9</td><td>7.3</td><td>20.5</td><td>8.2</td><td>3.4</td></tr><tr><td>UNIT</td><td>48.5</td><td>12.9</td><td>7.9</td><td>56.0</td><td>20.5</td><td>14.3</td></tr><tr><td colspan=\"7\">HarmonicGAN (ours)</td></tr><tr><td>Histogram</td><td>52.2</td><td>14.8</td><td>10.9</td><td>56.6</td><td>20.9</td><td>15.7</td></tr><tr><td>VGG</td><td>55.9</td><td>17.6</td><td>13.3</td><td>59.8</td><td>22.1</td><td>17.2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/760c877a93dfb32acd27cbffe1c345c5a5c2ffc81477e5da4ffd3dcd54caa6d4.jpg",
+        "table_caption": [
+            "Table 3: User study on the BRATS dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Metric</td><td>CycleGAN</td><td>DistanceGAN</td><td>HarmonicGAN</td></tr><tr><td>Prefer[%]↑</td><td>5</td><td>0</td><td>95</td></tr><tr><td>Mean Likert 个</td><td>1.68</td><td>1.62</td><td>4.00</td></tr><tr><td>Std Likert</td><td>0.99</td><td>0.95</td><td>0.88</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/69378398a689a6641adc7780ef5757d4604846b9fce8830e8e556fdfa1dd6a8b.jpg",
+        "table_caption": [
+            "Table 4: User study on the horse to zebra dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Metric</td><td>CycleGAN</td><td>DistanceGAN</td><td>HarmonicGAN</td></tr><tr><td>Prefer[%]↑</td><td>28</td><td>0</td><td>72</td></tr><tr><td>Mean Likert↑</td><td>3.16</td><td>1.08</td><td>3.60</td></tr><tr><td>Std Likert</td><td>0.81</td><td>0.23</td><td>0.78</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Object transfiguration. We evaluate our algorithm on horse zebra with a human perceptual study. We randomly selected 50 images from the horse2zebra test set and showed the input images and three generated images from CycleGAN, DistanceGAN and HarmonicGAN (with generated images in random order). 10 participants were asked to score the generated images on a Likert scale from 1 to 5 (as above). As shown in Table 4, the participants give the highest score to the proposed method (in $72 \\%$ of cases), significantly more often than CycleGAN (in $28 \\%$ of cases). Additionally, the average Likert score of our method was 3.60, outperforming 3.16 of CycleGAN and 1.08 of DistanceGAN, indicating that our method generates better results. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.5 QUALITATIVE COMPARISON ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            513,
+            408,
+            527
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Medical imaging. Fig. 5 shows the qualitative comparison of the proposed method (HarmonicGAN) and the baseline methods CycleGAN on Flair $ \\mathrm { T } 1$ . It shows that CycleGAN may remove tumors in the original images and add tumors to other locations in the brain. In contrast, our method preserves the location of tumors, confirming that the harmonic regularization can maintain the inherent property of tumor/non-tumor regions and solves the tumor add/remove problem introduced in Cohen et al. (2018). More results and analysis are shown in Fig. 6 and Fig. 9. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Object transfiguration. Fig. 5 shows a qualitative comparison of our method on the horse zebra task. We observe that we correct several problems in CycleGAN, including not changing the background and performing more complete transformations. More results and analysis are shown in Fig. 7 and Fig. 10. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            714,
+            318,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We introduce a smoothness term over the sample graph to enforce smoothness-consistency between the source and target domains. We have shown that by introducing additional regularization to enforce consistent mappings during the image-to-image translation, the inherent self-consistency property of samples can be maintained. Through a set of quantitative, qualitative and user studies, we have demonstrated that this results in a significant improvement over the current state-of-the-art methods in a number of applications including medical imaging, object transfiguration, and semantic labeling. In a medical imaging task in particular our method provides a very significant improvement over CycleGAN. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "6 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "6.1 COMPARISON TO DISTANCEGAN ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            136,
+            444,
+            150
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The proposed HarmonicGAN looks a bit similar to DistanceGAN Benaim & Wolf (2017), but actually there is a large difference between them. DistanceGAN encourages the distance of samples to an absolute mean during translation. In contrast, HarmonicGAN enforces a smoothness term naturally under the graph Laplacian, making the motivations of DistanceGAN and HarmonicGAN quite different. Comparing the distance constraint in DistanceGAN and the smoothness constraint in HarmonicGAN, we can conclude the following main differences between them: ",
+        "bbox": [
+            174,
+            162,
+            825,
+            247
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(1) They show different motivations and formulations. The distance constraint aims to preserve the distance between samples in the mapping in a direct way, so it minimizes the expectation of differences between distances in two domains. The distance constraint in DistanceGAN is not doing a graph-based Laplacian to explicitly enforce smoothness. In contrast, the smoothness constraint is designed from a graph Laplacian to build the similarity-consistency between image patches. Thus, the smoothness constraint uses the affinity between two patches as weight to measure the similarityconsistency between two domains. The whole idea is based on manifold learning. The smoothness term defines a Laplacian $\\Delta = D - W$ , where $W$ is our weight matrix and $D$ is a diagonal matrix with $\\begin{array} { r } { D _ { i } = \\sum _ { j } \\bar { w _ { i j } } } \\end{array}$ , thus, the smoothness term defines a graph Laplacian with the minimal value achieved as a harmonic function. ",
+        "bbox": [
+            173,
+            255,
+            825,
+            395
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(2) They are different in implementation. The smoothness constraint in HarmonicGAN is computed on image patches while the distance constraint in DistanceGAN is computed on whole image samples. Therefore, the smoothness constraint is fine-grained compared to the distance constraint. Moreover, the distances in DistanceGAN is directly computed from the samples in each domain. They scale the distances with the precomputed means and stds of two domains to reduce the effect of the gap between two domains. Differently, the smoothness constraint in HarmonicGAN is measured on the features (Histogram or CNN features) of each patch, which maps samples in two domains into the same feature space and removes the gap between two domains. ",
+        "bbox": [
+            173,
+            401,
+            825,
+            513
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(3) They show different results. Fig. 6 shows the qualitative results of CycleGAN, DistanceGAN and the proposed HarmonicGAN on the BRATS dataset. As shown in Fig. 6, the problem of randomly adding/removing tumors in the translation of CycleGAN is still present in the results of DistanceGAN, while HarmonicGAN can correct the location of tumors. Table 1 shows the quantitative results on the whole test set, which also yields the same conclusion. The results of DistanceGAN on four metrics are even worse than CycleGAN, while HarmonicGAN yields a large improvement over CycleGAN. ",
+        "bbox": [
+            173,
+            520,
+            825,
+            617
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad84f316926cf2190e66e0fe76a39d6f9178aa038c1ff14a3714b39753746362.jpg",
+        "image_caption": [
+            "Figure 6: Comparison of CycleGAN, DistanceGAN and the proposed HarmonicGAN on BRATS dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            636,
+            818,
+            803
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "6.2 COMPARISON TO CRF ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            867,
+            369,
+            882
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "There are some fundamental differences between the CRF literature and our work. They differ in output space, mathematical formulation, application domain, effectiveness, and the role in the overall algorithm. The similarity between CRF and HarmonicGAN lies the adoption of a regularization term: a binary term in the CRF case and a Laplacian term in HarmonicGAN. ",
+        "bbox": [
+            176,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            821,
+            132
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The smoothness term in HarmonicGAN is not about obtaining ‘smoother’ images/labels in the translated domain, as seen in the experiments; instead, HarmonicGAN is about preserving the overall integrity of the translation itself for the image manifold. This is the main reason for the large improvement of HarmonicGAN over CycleGAN. ",
+        "bbox": [
+            174,
+            138,
+            823,
+            194
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To further demonstrate the difference of HarmonicGAN and CRF, we perform an experiment applying the pairwise regularization of CRFs to the CycleGAN framework. For each pixel of the generated image, we compute the unary term and binary term with its 8 neighbors, and then minimize the objective function of CRF. The results are shown in Table 5. The pairwise regularization of CRF is unable to handle the problem of CycleGAN illustrated in Fig. 1. What’s worse, using the pairwise regularization may over-smooth the boundary of generated images, which results in extra artifacts. In contrast, HarmonicGAN aims at preserving similarity from the overall view of the image manifold, and can thus exploit similarity-consistency of the generated images, rather than over-smooth the boundary. ",
+        "bbox": [
+            174,
+            202,
+            825,
+            327
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/adb204aa22d6b0f40ef025631b0641a58ef8355689435eb774f049625d649a21.jpg",
+        "table_caption": [
+            "Table 5: Reconstruction evaluation on the BRATS dataset with comparison to CRF. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Flair→T1</td><td colspan=\"2\">T1→Flair</td></tr><tr><td>MAE↓</td><td>MSE↓</td><td>MAE↓</td><td>MSE↓</td></tr><tr><td>CycleGAN</td><td>10.47</td><td>674.40</td><td>11.81</td><td>1026.19</td></tr><tr><td>CycleGAN+CRF</td><td>11.24</td><td>839.47</td><td>12.25</td><td>1138.42</td></tr><tr><td>HarmonicGAN-Histogram (ours)</td><td>6.38</td><td>216.83</td><td>5.04</td><td>163.29</td></tr><tr><td>HarmonicGAN-VGG (ours)</td><td>6.86</td><td>237.94</td><td>4.69</td><td>127.84</td></tr></table>",
+        "bbox": [
+            261,
+            366,
+            735,
+            462
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/5b137aea5fc7a75338e7e48ee7a01aa6ade906e6ab5516996c2a80f1150f1294.jpg",
+        "image_caption": [
+            "Figure 7: Comparison on horse zebra for the Putin photo. CycleGAN translates both the background and human to a zebra-like texture. In contrast, HarmonicGAN does better in background region and achieves an improvement in some regions of of the human (Putin’s face), but it still fails on the human body. We hypothesize this is because the semantic features used by HarmonicGAN have not been trained on humans without a shirt. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            520,
+            756,
+            731
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/7485832a797a6eafb17db676d5cac03dc090762c94b00b56dac27313411c675b.jpg",
+        "image_caption": [
+            "Figure 8: For similar image patches in input images, visualizing the average distance of corresponding image patches in translated images for cross-modal medical image synthesis on BRATS dataset. The distance of image patches in the target space shows the inconsistently translated patches. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            378,
+            816,
+            587
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/f359d07ea9fc2cc3e5b41db9aabb779da52b2a3c3aaa790813faa2e73c63326f.jpg",
+        "image_caption": [
+            "Figure 9: Comparison on BRATS dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            112,
+            818,
+            890
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/e6e239af50aa223e27fe0949d0898570941a7391f0687a3221499d3966c693a2.jpg",
+        "image_caption": [
+            "Figure 10: Comparison on horse zebra. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            131,
+            818,
+            887
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/5274b54835d86cc916ba2c37397785dd45038636c947fc17207f80c471479df6.jpg",
+        "image_caption": [
+            "Figure 11: Additional results of the proposed HarmonicGAN. From top to bottom are: apple to orange, orange to apple, facade to label, label to facade, aerial to map, map to aerial, summer to winter, winter to summer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            181,
+            815,
+            804
+        ],
+        "page_idx": 18
+    }
+]

parse/train/TiXl51SCNw8/TiXl51SCNw8.md

@@ -0,0 +1,278 @@
+# BSQ: EXPLORING BIT-LEVEL SPARSITY FOR MIXEDPRECISION NEURAL NETWORK QUANTIZATION
+
+Huanrui Yang, Lin Duan, Yiran Chen & Hai Li
+
+Department of Electrical and Computer Engineering   
+Duke University   
+Durham, NC 27708, USA   
+{huanrui.yang, lin.duan, yiran.chen, hai.li}@duke.edu
+
+# ABSTRACT
+
+Mixed-precision quantization can potentially achieve the optimal tradeoff between performance and compression rate of deep neural networks, and thus, have been widely investigated. However, it lacks a systematic method to determine the exact quantization scheme. Previous methods either examine only a small manuallydesigned search space or utilize a cumbersome neural architecture search to explore the vast search space. These approaches cannot lead to an optimal quantization scheme efficiently. This work proposes bit-level sparsity quantization (BSQ) to tackle the mixed-precision quantization from a new angle of inducing bit-level sparsity. We consider each bit of quantized weights as an independent trainable variable and introduce a differentiable bit-sparsity regularizer. BSQ can induce all-zero bits across a group of weight elements and realize the dynamic precision reduction, leading to a mixed-precision quantization scheme of the original model. Our method enables the exploration of the full mixed-precision space with a single gradient-based optimization process, with only one hyperparameter to tradeoff the performance and compression. BSQ achieves both higher accuracy and higher bit reduction on various model architectures on the CIFAR-10 and ImageNet datasets comparing to previous methods.
+
+# 1 INTRODUCTION
+
+Numerous deep neural network (DNN) models have been designed to tackle real-world problems and achieved beyond-human performance. DNN models commonly demand extremely high computation cost and large memory consumption, making the deployment and real-time processing on embedded and edge devices difficult (Han et al., 2015b; Wen et al., 2016). To address this challenge, model compression techniques, such as pruning (Han et al., 2015b; Wen et al., 2016; Yang et al., 2020), factorization (Jaderberg et al., 2014; Zhang et al., 2015) and fixed-point quantization (Zhou et al., 2016; Wu et al., 2019; Dong et al., 2019), have been extensively studied. Among them, fixed-point quantization works directly on the data representation by converting weight parameters originally in the 32-bit floating-point form to low-precision values in a fixed-point format. For a DNN model, its quantized version requires much less memory for weight storage. Moreover, it can better utilize fixed-point processing units in mobile and edge devices to run much faster and more efficiently.
+
+Typically, model compression techniques aim to reduce a DNN model size while maintaining its performance. The two optimization objectives in this tradeoff, however, have a contrary nature: the performance can be formulated as a differentiable loss function $\mathcal { L } ( W )$ w.r.t. the model’s weights $W$ ; yet the model size, typically measured by the number of non-zero parameters or operations, is a discrete function determined mainly by the model architecture. To co-optimize the performance and model size, some previous pruning and factorization methods relax the representation of model size as a differentiable regularization term $R ( W )$ . For example, group Lasso (Wen et al., 2016) and DeepHoyer (Yang et al., 2020) induce weight sparsity for pruning, and the attractive force regularizer (Wen et al., 2017) and nuclear norm $\mathrm { X u }$ et al., 2018) are utilized to induce low rank. The combined objective ${ \mathcal { L } } ( W ) + \alpha R ( W )$ can be directly minimized with a gradient-based optimizer for optimizing the performance and model size simultaneously. Here, the hyperparameter $\alpha$ controls the strength of the regularization and governs the performance-size tradeoff of the compressed model.
+
+Unlike for pruning and factorization, there lacks a well-defined differentiable regularization term that can effectively induce quantization schemes. Early works in quantization mitigate the tradeoff exploration complexity by applying the same precision to the entire model. This line of research focuses on improving the accuracy of ultra low-precision DNN models, e.g., quantizing all the weights to 3 or less bits (Zhou et al., 2016; Zhang et al., 2018), even to 1-bit (Rastegari et al., 2016). These models commonly incur significant accuracy loss, even after integrating emerging training techniques like straight-through estimator (Bengio et al., 2013; Zhou et al., 2016), dynamic range scaling (Polino et al., 2018) and non-linear trainable quantizers (Zhang et al., 2018). As different layers of a DNN model present different sensitivities with performance, a mixed-precision quantization scheme would be ideal for the performance-size tradeoff (Dong et al., 2019). There have also been accelerator designs to support the efficient inference of mixed-precision DNN models (Sharma et al., 2018). However, to achieve the optimal layer-wise precision configuration, it needs to exhaustively explore the aforementioned discrete search space, the size of which grows exponentially with the number of layers. Moreover, the dynamic change of each layer’s precision cannot be formulated into a differentiable objective, which hinders the efficiency of the design space exploration. Prior studies (Wu et al., 2019; Wang et al., 2019) utilize neural architecture search (NAS), which suffers from extremely high searching cost due to the large space of mixed-precision quantization scheme. Recently, Dong et al. (2019) propose to rank each layer based on the corresponding Hessian information and then determine the relative precision order of layers based on their ranking. The method, however, still requires to manually select the precision level for each layer.
+
+Here, we propose to revisit the fixed-point quantization process from a new angle of bit-level sparsity: decreasing the precision of a fixed-point number can be taken as forcing one or a few bits, most likely the least significant bit (LSB), to be zero; and reducing the precision of a layer is equivalent to zeroing out a specific bit of all the weight parameters of the layer. In other words, the precision reduction can be viewed as increasing the layer-wise bit-level sparsity. By considering the bits of fixed-point DNN parameters as continuous trainable variables during DNN training, we can utilize a sparsity-inducing regularizer to explore the bit-level sparsity with gradient-based optimization, dynamically reduce the layer precision and lead to a series of mixed-precision quantization schemes. More specific, we propose Bit-level Sparsity Quantization (BSQ) method with the following contributions:
+
+• We propose a gradient based training algorithm for bit-level quantized DNN models. The algorithm considers each bit of quantized weights as an independent trainable variable and enables the gradient-based optimization with straight-through estimator (STE). • We propose a bit-level group Lasso regularizer to dynamically reduce the weight precision of every layer and therefore induce mixed-precision quantization schemes. • BSQ uses only one hyperparameter, the strength of the regularizer, to trade-off the model performance and size, making the exploration more efficient.
+
+This work exclusively focuses on layer-wise mixed-precision quantization, which is the granularity considered in most previous works. However, the flexibility of BSQ enables it to explore mixedprecision quantization of any granularity with the same cost regardless of the search space size.
+
+# 2 RELATED WORKS ON DNN QUANTIZATION
+
+Quantization techniques convert floating-point weight parameters to low-precision fixed-point representations. Directly quantizing a pre-trained model inevitably introduces significant accuracy loss. So many of early research focus on how to finetune quantized models in low-precision configurations. As the quantized weights adopt discrete values, conventional gradient-based methods that are designed for continuous space cannot be directly used for training quantized models. To mitigate this problem, algorithms like DoReFa-Net utilize a straight-through estimator (STE) to approximate the quantized model training with trainable floating-point parameters (Zhou et al., 2016). As shown in Equation (1), a floating-point weight element $w$ is kept throughout the entire training process. Along the forward pass, the STE will quantize $w$ to $n$ -bit fixed-point representation $w _ { q }$ , which will be used to compute the model output and loss $\mathcal { L }$ . During the backward pass, the STE will directly pass the gradient w.r.t. $w _ { q }$ onto $w$ , which enables $w$ to be updated with the standard gradient-based optimizer.
+
+$$
+w _ { q } = \frac { 1 } { 2 ^ { n } - 1 } R o u n d [ ( 2 ^ { n } - 1 ) w ] ; ~ \mathbf { B a c k w a r d : } ~ \frac { \partial \mathcal { L } } { \partial w } = \frac { \partial \mathcal { L } } { \partial w _ { q } } .
+$$
+
+Early studies revealed that weights of different layers have different dynamic ranges. It is important to keep the dynamic range of each layer for maintaining the model performance, especially for quantized models. He et al. (2016b) and Polino et al. (2018) propose to explicitly keep track of the dynamic range of each layer by scaling all the weight elements in a layer to the range of [0,1] at every training step, before applying the quantization STE. Other techniques, such as learnable nonlinear quantifier function (Zhang et al., 2018) and incremental quantization (Zhou et al., 2017), are also useful in improving the performance of quantized models. However, it is still very difficult to quantize the entire DNN model to a unified ultra-low precision without incurring significant accuracy loss.
+
+Recent research shows that different layers in a DNN model contribute to the overall performance in varying extents. Therefore mixed-precision quantization scheme that assigns different precision to layers (Wu et al., 2019; Dong et al., 2019) presents a better accuracy-compression tradeoff. The challenge lies in how to determine the quantization scheme, i.e., the precision of each layer, as it needs to explore a large and discrete search space. Some works design quantization criteria based on concepts like “noise gain” (Sakr & Shanbhag, 2018; 2019) to constraint the relationship between each layer’s precision and thus largely reduce the search space, yet those criteria are often heuristic, preventing these methods to reach ultra-low precision and find the optimal tradeoff point between model size and accuracy. Other works utilize neural architecture search (NAS). For example, Wang et al. (2019) consider the precision assignment of each layer as an action and seek for the optimal design policy via reinforcement learning. Wu et al. (2019) combine all possible design choices into a “stochastic super net” and approximate the optimal scheme via sampling. However, the cost of NAS methods scales up quickly as the quantization search space grows exponentially with the number of layers. Common practices of constraining the search cost include limiting the precision choices or designing the quantization scheme in a coarse granularity. A recent line of research work attempts to rank layers based on their importance measured by the sensitivity or Hessian information. Higher precision will then be assigned to more important layers (Dong et al., 2019). The exact precision of each layer, however, needs to be manually selected. So these methods cannot adequately explore the whole search space for the optimal quantization scheme.
+
+# 3 THE BSQ METHOD
+
+BSQ aims to obtain an optimal mixed-precision quantization scheme through a single-pass training process of a quantized model. In this section, we first introduce how to convert a DNN model to the bit representation and propose a gradient-based algorithm for training the resulted bit-level model. A bit-level group Lasso regularizer is then proposed to induce precision reduction. In the end, we elaborate the overall training objective of BSQ and the dynamical precision adjustment procedure.
+
+# 3.1 TRAINING THE BIT REPRESENTATION OF DNN
+
+As illustrated in Figure 1(a), we convert a floating-point weight matrix $W$ of a pretrained network to its bit representation through a pipeline of scaling, quantization and binary conversion. Similar to the practice in (He et al., 2016b; Polino et al., 2018), we retain the dynamic range of $W$ by scaling all the elements to the range of [0, 1] before applying quantization. However, these prior works always scale the largest element to 1 to fully utilize all the quantized bins at every training step, which makes the dynamic precision reduction impossible. Instead, our method conducts the scaling only
+
+$$
+{ \begin{array} { c c } { { \underline { { { \mathrm { s c a l i n g } } } } } } & { { \leq \underline { { { \mathrm { s c a l i n g } } } } } } \\ { { \mathrm { 2 . 0 } } } \\ { { \mathrm { 1 . 2 } } } \\ { { \begin{array} { l } { { \mathrm { 8 . 0 } } } \\ { { \mathrm { 8 . 8 } } } \\ { { \mathrm { W } } } \end{array} = [ { \begin{array} { l } { 1 . 0 } \\ { 0 . 6 } \\ { 0 . 1 } \end{array} } ] } } & { { [ \begin{array} { l l l } { { \mathrm { 1 . ~ 1 } } } & { { \mathrm { 1 . } } } \\ { { \mathrm { 1 . } } } & { { \mathrm { 1 . } } } \\ { { \mathrm { 0 . } } } & { { 0 } } & { { 0 } } \\ { { \mathrm { - } } } & { { \mathrm { 8 . } } } \end{array} ] } } } \\ { { \mathrm { \begin{array} { l l l l } { { \mathrm { 0 . 0 } } } & { { \mathrm { 1 . 1 } } } & { { \mathrm { 1 . } } } \\ { { \mathrm { 0 . 0 } } } & { { \mathrm { 1 . } } } \end{array} } } & { { [ \begin{array} { l } { { \mathrm { 1 . } } } \\ { { \mathrm { 1 . } } } \\ { { \mathrm { - } } } \\ { { \mathrm { 0 . } } } & { { 0 } } \end{array} ] } } & { { [ \begin{array} { l l l } { { \mathrm { 2 . } } } & { { \mathrm { 1 . } } } & { { \mathrm { 0 . 8 } } } \\ { { \mathrm { - } } } & { { \mathrm { 1 . } } } & { { \mathrm { 0 . 8 } } } \end{array} ] - \begin{array} { l } { { \mathrm { 1 . } } } & { { \mathrm { 1 . } } } \\ { { \mathrm { - } } } & { { \mathrm { 5 7 . } } } \\ { { \mathrm { 1 . } } } & { { \mathrm { 1 . } } } \end{array} ] } } & { { \mathrm { 1 . } } } \\ { { \mathrm { ~ } } } & { { \mathrm { ~ } } } \\ { { \mathrm { ~ c a u a r i t z a t i o n } } } & { { \mathrm { 3 . } } } \\ { { \mathrm { ~ } } } & { { \mathrm { 0 u a r i t z a t i o n } } } \\ { { \mathrm { ( a ) } } } & { { \mathrm { ~ } } } \end{array} } & { { \mathrm { U p d a t e d a n d } } } &   \begin{array} { l l l }   \end{array}
+$$
+
+Figure 1: An example of DNN training under the bit representation with precision $n = 3$ . (a) Pipeline of converting from the floating-point weight $W$ to the bit representation; (b) Training the bit-level model weight with STE.
+
+once, which is right before the bit representation training. Formally, before converting $W$ to its bit representation, we first extract its dynamic range as $W = s \cdot W _ { s }$ , where $s = \operatorname* { m a x } | W |$ is the scaling factor and $W _ { s }$ is the scaled weight matrix. The absolute value of any element $w _ { s }$ in $W _ { s }$ is within the range of $[ 0 , 1 ]$ . Now we apply an $n$ -bit uniform quantization to the absolute value of $w _ { s }$ such as $w _ { q } = R o \bar { u } n d [ | w _ { s } | \times ( 2 ^ { n } - 1 ) ] / ( 2 ^ { n } - 1 )$ . Then $w _ { q }$ can be exactly represented by a $n$ -bit binary number as $\begin{array} { r } { w _ { q } = [ \sum _ { b = 0 } ^ { n - 1 } w _ { s } ^ { ( b ) } 2 ^ { b } ] / ( 2 ^ { n } - 1 ) . } \end{array}$ , where $w _ { s } ^ { ( b ) }$ denotes the $b ^ { t h }$ bit in the binary representation. Till this point, $W$ in the floating-point form is replaced with
+
+$$
+W \equiv s i g n ( W ) \odot s W _ { q } \equiv s i g n ( W ) \odot \frac { s } { 2 ^ { n } - 1 } \sum _ { b = 0 } ^ { n - 1 } W _ { s } ^ { ( b ) } 2 ^ { b } ,
+$$
+
+where $\odot$ denotes the element-wise Hadamard product. We consider the bit representation $W _ { s } ^ { ( b ) }$ where $b \in [ 0 , n - 1 ]$ and the scaling factor $s$ as independent trainable variables in the training process.
+
+Note that $W _ { s } ^ { ( b ) }$ is composed of binary values by definition and $s i g n ( W )$ is a discrete function. Neither of them can be directly trained with gradient descent. To mitigate the binary constraint of $W _ { s } ^ { ( b ) }$ , we adopt the STE proposed by Bengio et al. (2013) during the training process. As shown in Equation (1), STE enables a quantized model to be trained with continuous floating-point weights. Specifically, the STE for the bit representation training is defined as:
+
+$$
+: W _ { q } = \frac { 1 } { 2 ^ { n } - 1 } R o u n d \left[ \sum _ { b = 0 } ^ { n - 1 } W _ { s } ^ { ( b ) } 2 ^ { b } \right] ; ~ \mathbf { B a c k w a r d } \colon \frac { \partial \mathcal { L } } { \partial W _ { s } ^ { ( b ) } } = \frac { 2 ^ { b } } { 2 ^ { n } - 1 } \frac { \partial \mathcal { L } } { \partial W _ { q } } .
+$$
+
+STE relaxes the binary constraint and allows gradient updates for the elements in $W _ { s } ^ { ( b ) }$ . As illustrated in Figure 1(b), during the forward pass, $s \cdot W _ { q }$ will be used to reconstruct the model weight $W$ and compute the loss, which demonstrates the performance of the current model after quantization. The gradient w.r.t. $W _ { q }$ from the back-propagation will be passed through the rounding function and updated on the continuous values of $W _ { s } ^ { ( b ) }$ . The proposed bit representation can therefore be trained with any gradient-based optimizer.
+
+The proposed bit representation training only leads to minimal computational and run-time memory overhead comparing to the normal back propagation procedure. From the memory consumption perspective, the bit representation training treats each bit as separated floating-point trainable variables, so a $N$ -bit model in bit representation will have $N$ times more parameters and gradients to be stored comparing to that of the baseline training. Though for actual run-time memory consumption, the hidden feature between each layer consumes a significantly larger memory than weights and gradients. As the bit representation does not affect the hidden features, the increase in trainable variables does not lead to significant increase in run-time memory consumption. From the perspective of computation cost, note that the gradient w.r.t. each $W _ { s } ^ { ( b ) }$ can be computed as the gradient w.r.t. the corresponding $W _ { q }$ scaled by a power of 2. So under a $N$ -bit scheme there will only be $N$ additional scaling for each parameter comparing to the normal training. These additional computations are very cheap comparing to the floating-point operations involved in back propagation. So the proposed bit representation training only leads to minimal computational overhead comparing to a normal back propagation.
+
+We restrict the value of $W _ { s } ^ { ( b ) }$ within $[ 0 , 2 ]$ throughout the training, so that the corresponding $W _ { q }$ has the chance to increase or decrease its precision in the “precision adjustment” step, which will be discussed in Section 3.3. This is enforced by trimming $\bar { W } _ { s } ^ { ( b ) }$ to 0 or 2 if it exceeds the range after a training step.
+
+To enable the dynamic update of $s i g n ( W )$ during training, we separate the positive and negative elements in $W _ { s }$ as $W _ { s } = \left( W _ { p } - W _ { n } \right)$ before quantization. Here $W _ { p } = W _ { s } \odot \mathbb { 1 } ( W _ { s } \ge 0 )$ contains all the positive elements and $\hat { W _ { n } } = - W _ { s } \odot \mathbb { 1 } ( W _ { s } < 0 )$ includes the absolute value of all the negative weight elements. $W _ { p }$ and $W _ { n }$ will be respectively converted to $W _ { p } ^ { ( b ) }$ and $W _ { n } ^ { ( b ) }$ by following the process in Equation (2), so that $W _ { s } ^ { ( b ) } = W _ { p } ^ { ( b ) } - W _ { n } ^ { ( b ) }$ . Note that the replacement of $W _ { s } ^ { ( b ) }$ with W (b)p − $W _ { p } ^ { ( b ) } - W _ { n } ^ { ( b ) }$ does not introduce any non-differentiable function. Therefore all elements in $W _ { p } ^ { ( b ) }$ and W (b)n can take continuous values between $[ 0 , 2 ]$ and be trained with the bit representation STE in Equation (3). As such, the original weight matrix $W$ is converted into trainable variables $W _ { p } ^ { ( b ) }$ , $W _ { n } ^ { ( b ) }$ and $s$ throughout the BSQ training process.
+
+To induce the mixed-precision quantization scheme of a DNN model during training, we propose a bit-level group Lasso $( B _ { G L } )$ regularizer based on the group Lasso (Hastie et al., 2015) and apply it to W (b)p a nd W (b)n t hat are converted from a group of weights $W ^ { g }$ . The regularizer is defined as:
+
+$$
+B _ { G L } ( W ^ { g } ) = \sum _ { b = 0 } ^ { n - 1 } \left. \left[ W _ { p } ^ { ( b ) } ; W _ { n } ^ { ( b ) } \right] \right. _ { 2 } ,
+$$
+
+where W (b)p a nd $W _ { n } ^ { ( b ) }$ are bit representations converted from $W ^ { g }$ , and $[ \cdot ; \cdot ]$ denotes the concatenation of matrices. $B _ { G L }$ could make a certain bit $b$ of all elements in both $W _ { p } ^ { ( b ) }$ and $W _ { n } ^ { ( b ) }$ zero simultaneously. The bit can thus be safely removed for the precision reduction. Note that the granularity of the quantization scheme induced by $B _ { G L }$ is determined by how $W ^ { g }$ is grouped. Our experiments organize $W ^ { g }$ in a layer-wise fashion. So all elements in a layer have the same precision, which is a common setting in previous mixed-precision quantization work. $W ^ { g }$ can also be arranged as any group of weight elements, such as block-wise, filter-wise or even element-wise if needed. Accordingly, the formulation of the regularizer need to be revised to assist the exploration of the mixed-precision quantization at the given granularity. The cost for evaluating and optimizing the regularizer will remain the same under different granularity settings.
+
+# 3.3 OVERALL TRAINING PROCESS
+
+The overall training process starts with converting each layer of a pretrained floating-point model to the bit representation with a relatively high initial precision (e.g., 8-bit fixed-point). BSQ training is then preformed on the achieved bit representation with bit-level group Lasso integrated into the training objective. Re-quantization steps are conducted periodically to identify the bit-level sparsity induced by the regularizer and allow dynamic precision adjustment. As the mixed-precision quantization scheme is finalized, the achieved model is further finetuned for a higher accuracy.
+
+Objective of BSQ training. For higher memory efficiency it is desired to find a mixed-precision quantization scheme that minimizes the total number of bits in the model. Thus, in BSQ training we propose to penalize more on the layers with more bits by performing a memory consumption-aware reweighing to $B _ { G L }$ across layers. Specifically, the overall objective of training a $L$ -layer DNN model with BSQ is formulated as:
+
+$$
+\mathcal { L } = \mathcal { L } _ { C E } ( W _ { q } ^ { ( 1 : L ) } ) + \alpha \sum _ { l = 1 } ^ { L } \frac { \# P a r a ( W ^ { l } ) \times \# B i t ( W ^ { l } ) } { \# P a r a ( W ^ { ( 1 : L ) } ) } B _ { G L } ( W ^ { l } ) .
+$$
+
+Here $\mathcal { L } _ { C E } ( W _ { q } ^ { ( 1 : L ) } )$ is the original cross entropy loss evaluated with the quantized weight $W _ { q }$ acquired from the STE in Equation (3), $\alpha$ is a hyperparameter controlling the regularization strength, and $\# P a r a ( W ^ { l } )$ and $\# B i t ( W ^ { l } )$ respectively denote the parameter number and precision of layer $l$ The loss function in Equation (5) enables a layer-wise adjustment in the regularization strength by applying a stronger regularization on a layer with higher memory usage.
+
+Re-quantization and precision adjustment. As BSQ trains the bit representation of the model with floating-point variables, we perform re-quantization to convert W (b)p and $W _ { n } ^ { ( b ) }$ to exact binary values and identify the all-zero bits that can be removed for precision reduction. The requantization step reconstructs the quantized scaled weight $W _ { q } ^ { \prime }$ from $W _ { p } ^ { ( b ) }$ an d W (b)n as: $W _ { q } ^ { \prime } =$ Round $\begin{array} { r } { \left[ \sum _ { b = 0 } ^ { n - 1 } W _ { p } ^ { ( b ) } 2 ^ { b } - \sum _ { b = 0 } ^ { n - 1 } W _ { n } ^ { ( b ) } 2 ^ { b } \right] . } \end{array}$ . As we allow the values of $W _ { p } ^ { ( b ) }$ ) and W (b)n t o be within $[ 0 , 2 ]$ , the reconstructed $W _ { q } ^ { \prime }$ has a maximum absolute value of $2 ^ { n + 1 }$ . In this way, $W _ { q } ^ { \prime }$ is converted to a $( n + 1 )$ -bit binary number, where each bit is denoted by $W _ { q } ^ { ( b ) }$ . After the re-quantization, we will adjust the precision of each layer. Specifically, we first check $W _ { q } ^ { ( b ) }$ from the MSB down to the LSB and remove the bits with all zero elements until the first non-zero bit. The scaling factor $s$ of the layer remains unchanged during this process. A similar check will then be conducted from the LSB up to the MSB. $s$ needs to be doubled when a bit from the LSB side is removed, as all elements in $W _ { \underline { { q } } } ^ { \prime }$ are shifted right for one bit. Assume that the precision adjustment makes the precision of a layer change from $n$ to $n ^ { \prime }$ , the scaling factor will be updated as $s ^ { \prime } = s \frac { 2 ^ { n ^ { \prime } } - 1 } { 2 ^ { n } - 1 }$ . In this way, the bit representations of $W$ before and after the precision adjustment are equivalent, as indicated in Equation (6). The precision of a $n$ -bit layer may change between 0 and $( n + 1 )$ -bit after the precision adjustment.
+
+![](images/3a871dfc30f72199c5672520fcd242261320dd2a26a786f73cb674047c426e6f.jpg)  
+Figure 2: Quantization schemes achieved with or without layer-wise regularization reweighing. The compression rate and the accuracy after finetuning are listed in the legend.
+
+$$
+W \equiv \frac { s } { 2 ^ { n } - 1 } \sum _ { b = 0 } ^ { n - 1 } W _ { q } ^ { ( b ) } 2 ^ { b } \equiv \frac { s ^ { \prime } } { 2 ^ { n ^ { \prime } } - 1 } \sum _ { b = 0 } ^ { n ^ { \prime } - 1 } W _ { q } ^ { ( b ) } 2 ^ { b } .
+$$
+
+As formulated in Equation (5), the regularization strength assigned to each layer will change with the quantization scheme of the model. The re-quantization and precision adjustment step will be performed periodically during the training process, with an interval of several training epochs. After each precision adjustment, we separate the positive elements and negative elements in $\mathbf { \nabla } W _ { q } ^ { \prime }$ to form the new $W _ { p } ^ { ( b ) }$ and $W _ { n } ^ { ( b ) }$ , respectively. The training can then resume with the newly adjusted $W _ { p } ^ { ( b ) }$ and $W _ { n } ^ { ( b ) }$ and scaling factor $s ^ { \prime }$ . It is worth mentioning that $s W _ { q }$ from the forward pass STE remains unchanged before and after the re-quantization and precision adjustment, so the model performance and the gradient from the loss $\mathcal { L } _ { C E }$ will not be affected. The interval between re-quantizations needs to be carefully chosen: it shall promptly and properly adjust the regularization strength for stable convergence. The ablation study on re-quantization interval selection is presented in Appendix B.1.
+
+Activation quantization. Since BSQ modifies only the precision of weights but not affecting the precision of activations, we predetermine the activation precision and fix it throughout the BSQ training process. The activations are quantized in the same way as proposed by Polino et al. (2018). For training stability, we use ReLU-6 activation function for layers with 4-bit or above activations, and use PACT (Choi et al., 2018) for layers with a lower activation precision.
+
+Post-training finetuning. At the end of the BSQ training, we perform a final re-quantization and precision adjustment to get the final mixed-quantization scheme. The achieved model can be further finetuned under the obtained precision for improving the overall accuracy. As the quantization scheme is fixed, we adopt the quantization-aware training method proposed by Polino et al. (2018) for finetuning in our experiment.
+
+# 4 ABLATION STUDY
+
+We perform the ablation studies on key design choices of the BSQ algorithm. This section presents the effectiveness of layer-wise memory consumption-aware regularization reweighing and the model size-accuracy tradeoff under different regularization strengths. All experiments are conducted with ResNet-20 models (He et al., 2016a) with 4-bit activation on the CIFAR-10 dataset (Krizhevsky & Hinton, 2009). Detailed experiment setup and hyperparameter choices can be found in Appendix A.
+
+# 4.1 EFFECT OF LAYER-WISE REGULARIZATION REWEIGHING
+
+As stated in Equation (5), we propose to apply layer-wise memory consumption-aware reweighing on the $B _ { G L }$ regularizer to penalize more on larger layers during the BSQ training. Figure 2 compares the quantization scheme and the model performance achieved when performing the BSQ training with or without such a reweighing term. Here we set the regularization strength $\alpha$ to 5e-3 when training with the reweighing, and to 2e-3 when training without the reweighing to achieve comparable compression rates. All the other hyperparameters are kept the same. As shown in the figure, training without the reweighing term will lead to over-penalization on earlier layers with fewer parameters, while later layers with more parameters are not compressed enough. Therefore the achieved quantized model will have less accuracy even with a smaller compression rate comparing to the model achieved with layer-wise regularization reweighing. As we only show one pair of comparison here, the difference between BSQ training with or without the reweighing term is consistent when varying the regularization strength $\alpha$ . Additional results with other $\alpha$ values are shown in Appendix B.2.
+
+![](images/c858b8cca86de69750b1a11c62b608bc004ce82809981010525b06ef498c4427.jpg)  
+Figure 3: Layer-wise precision comparison of the quantization schemes achieved under different regularization strengths.
+
+Table 1: Accuracy-#Bits tradeoff under different regularization strengths. “FT” stands for finetuning. The last row is achieved by training with quantization schemes achieved by BSQ from scratch.   
+
+<table><tr><td>Strength α</td><td>3e-3</td><td>5e-3</td><td>7e-3</td><td>1e-2</td><td>2e-2</td></tr><tr><td>#Bits per Para /Comp(×）</td><td>3.02 / 10.60</td><td>2.25/14.24</td><td>1.66 / 19.24</td><td>1.37 / 23.44</td><td>0.87 / 36.63</td></tr><tr><td>BSQ acc before /after FT (%)</td><td>91.30 /92.60</td><td>90.98 /92.32</td><td>90.42 / 91.48</td><td>90.35 / 91.16</td><td>85.77 / 89.49</td></tr><tr><td>Train from scratch acc (%)</td><td>91.72</td><td>91.45</td><td>91.12</td><td>89.57</td><td>89.14</td></tr></table>
+
+# 4.2 ACCURACY-#BITS TRADEOFF UNDER DIFFERENT REGULARIZATION STRENGTHS
+
+We fix all the other hyperparameters while varying only the regularization strength $\alpha$ from 3e3 to 2e-2, to control the tradeoff between the model size and accuracy achieved by BSQ. The quantization schemes achieved by running BSQ with different $\alpha$ ’s are shown in Figure 3, and the detailed comparison on the compression rate comparing to the 32-bit floating point model (denoted as “Comp”) and the validation accuracy (denoted as “Acc”) is summarized in Table 1. As shown in Figure 3, the relative ranking of the precision assignment is mostly consistent under different $\alpha$ ’s, which is consistent with the previous observation that more important layers should be assigned with higher precision. This effect is further illustrated in Appendix B.3, where we compare the quantization scheme achieved by BSQ with the layer importance measured in HAWQ (Dong et al., 2019). Furthermore, as $\alpha$ increases, the overall bit reduction increases with the cost of a small performance loss. This tradeoff is also observed on models trained with 2-bit or 3-bit activation as we show their quantization schemes and performances in Appendix B.4. Note that some layers achieve 0-bit precision under large regularization strength, indicating all the weights become zero and the layer can be skipped. This is possible as the shortcut connection existing in the ResNet architecture enables the pass of information even if the weights are all zero in some layers. We also note that BSQ not only finds the desired mixed-precision quantization scheme, but also provides a model with higher performance under the same quantization scheme. As shown in Table 1, when training a model with the same quantization scheme as achieved by BSQ using the DoReFa-Net algorithm (Zhou et al., 2016) from scratch, the resulted accuracy is always lower than the BSQ model after finetuning.
+
+# 5 EXPERIMENTAL RESULTS
+
+In this section we compare BSQ with previous state-of-the-art methods. Here, ResNet-20 models are used for the comparison on the CIFAR-10 dataset, and ResNet-50 and Inception-V3 models (Szegedy et al., 2016) are utilized for the experiments on the ImageNet dataset (Russakovsky et al., 2015). The hyperparameters used for BSQ training and finetuning are listed in Appendix A. All the compression rates reported in Table 2 and Table 3 are compared to the 32-bit floating point model, and all the accuracy reported is the testing accuracy evaluated on models after finetuning.
+
+Table 2: Quantization results of ResNet-20 models on the CIFAR-10 dataset. BSQ is compared with DoReFa-Net (Zhou et al., 2016), PACT (Choi et al., 2018), LQ-Net (Zhang et al., 2018), DNAS (Wu et al., 2019) and HAWQ (Dong et al., 2019). “MP” denotes mixed-precision quantization.   
+
+<table><tr><td colspan="5">Benchmarks</td><td colspan="3">BSQ</td></tr><tr><td>Act. Prec.</td><td>Method</td><td>Weight Prec.</td><td>Comp(×)</td><td>Acc (%)</td><td>a</td><td>Comp(×)</td><td>Acc (%)</td></tr><tr><td>32-bit</td><td>Baseline</td><td>32</td><td>1.00</td><td>92.62</td><td rowspan="3">5e-3</td><td rowspan="3"></td><td>92.77</td></tr><tr><td></td><td>LQ-Nets</td><td>3</td><td>10.67</td><td>92.00</td><td>14.24 91.87</td></tr><tr><td></td><td>DNAS</td><td>MP</td><td>11.60</td><td>92.72</td><td>19.24</td></tr><tr><td></td><td>LQ-Nets</td><td>2</td><td>16.00</td><td>91.80</td><td rowspan="2">5e-3</td><td rowspan="2"></td><td></td></tr><tr><td>4-bit</td><td>HAWQ</td><td>MP</td><td>13.11</td><td>92.22</td><td>14.24</td></tr><tr><td>3-bit</td><td>LQ-Nets</td><td>3</td><td>10.67</td><td>91.60</td><td rowspan="2">2e-3 5e-3</td><td>11.04</td><td>92.16</td></tr><tr><td></td><td>PACT</td><td>3</td><td>10.67</td><td>91.10</td><td>16.37</td><td>91.72</td></tr><tr><td></td><td>DoReFa</td><td>3</td><td>10.67</td><td>89.90</td><td rowspan="2"></td><td rowspan="2"></td><td rowspan="2"></td></tr><tr><td>2-bit</td><td>LQ-Nets</td><td>2</td><td>16.00</td><td>90.20</td></tr><tr><td></td><td>PACT</td><td>2</td><td>16.00</td><td>89.70</td><td rowspan="2">5e-3</td><td rowspan="2">18.85</td><td rowspan="2">90.19</td></tr><tr><td></td><td>DoReFa</td><td>2</td><td>16.00</td><td>88.20</td></tr></table>
+
+Table 3: Quantization results of ResNet-50 and Inception-V3 models on the ImageNet dataset. BSQ is compared with DoReFa-Net (Zhou et al., 2016), PACT (Choi et al., 2018), LSQ (Esser et al., 2019), LQ-Net (Zhang et al., 2018), Deep Compression (DC) (Han et al., 2015a), Integer (Jacob et al., 2018), RVQ (Park et al., 2018), HAQ (Wang et al., 2019) and HAWQ (Dong et al., 2019).   
+
+<table><tr><td colspan="4">ResNet-50</td><td colspan="4">Inception-V3</td></tr><tr><td>Method</td><td>Prec.</td><td>Comp (x)</td><td>Top1 (%)</td><td>Method</td><td>Prec.</td><td>Comp(x)</td><td>Top1 (%)</td></tr><tr><td>Baseline</td><td>32</td><td>1.00</td><td>76.13</td><td>Baseline</td><td>32</td><td>1.00</td><td>77.21</td></tr><tr><td>DoReFa</td><td>3</td><td>10.67</td><td>69.90</td><td>Integer</td><td>8</td><td>4.00</td><td>75.40</td></tr><tr><td>PACT</td><td>3</td><td>10.67</td><td>75.30</td><td>Integer</td><td>7</td><td>4.57</td><td>75.00</td></tr><tr><td>LQ-Nets</td><td>3</td><td>10.67</td><td>74.20</td><td>RVQ</td><td>MP</td><td>10.67</td><td>74.14</td></tr><tr><td>DC</td><td>3</td><td>10.41</td><td>75.10</td><td>HAWQ</td><td>MP</td><td>12.04</td><td>75.52</td></tr><tr><td>HAQ</td><td>MP</td><td>10.57</td><td>75.30</td><td></td><td></td><td></td><td></td></tr><tr><td>LSQ</td><td>3</td><td>10.67</td><td>75.80</td><td></td><td></td><td></td><td></td></tr><tr><td>BSQ 5e-3</td><td>MP</td><td>11.90</td><td>75.29</td><td>BSQ 1e-2</td><td>MP</td><td>11.38</td><td>76.60</td></tr><tr><td>BSQ 7e-3</td><td>MP</td><td>13.90</td><td>75.16</td><td>BSQ2e-2</td><td>MP</td><td>12.89</td><td>75.90</td></tr></table>
+
+Table 2 reports the quantization results of ResNet-20 models on the CIFAR-10 dataset. Here we set the activation of the first convolutional layer and the final FC layer to 8 bits while all the other activations to 4, 3 or 2 bits respectively to match the settings of previous methods. The reported 32-bit activation model performance is achieved by finetuning the 4-bit activation model under full precision activation. The exact BSQ quantization schemes of the 4-bit activation models are listed in Figure 3, while those of the 2-bit and 3-bit activation models can be found in Appendix B.4. Comparing to previous mixed-precision quantization methods, the model obtained by BSQ with 4-bit activation and $\alpha = 5 \mathrm { e } { - 3 }$ has slightly higher accuracy as the model achieved by HAWQ (Dong et al., 2019), but a higher compression rate ( $1 4 . 2 4 \times$ vs. $1 3 . 1 1 \times$ ). The same model with 32-bit activation obtains $23 \%$ more compression rate with the same accuracy as the model found by DNAS (Wu et al., 2019), with a much less training cost as our method does not involve the costly neural architecture search. The advantage of BSQ is even larger comparing to single-precision quantization methods (Zhou et al., 2016; Choi et al., 2018; Zhang et al., 2018), as BSQ achieves both higher compression rate and higher accuracy comparing to all methods with the same activation precision.
+
+The results of BSQ and previous quantization methods on the ImageNet dataset are summarized in Table 3. The exact BSQ quantization schemes can be found in Appendix C. For ResNet models, the activation of the first and the final layer are set to 8 bits while all the other activations are set to 4 bits. For Inception-V3 models the activation of all the layers are set to 6 bits. On ResNet-50 models, BSQ with $\alpha = 5 \mathrm { e } { - 3 }$ achieves the same top-1 accuracy as PACT (Choi et al., 2018) and $0 . 5 \%$ less top-1 accuracy as the best available method LSQ (Esser et al., 2019) with a higher compression rate ( $1 1 . 9 0 \times$ vs. $1 0 . 6 7 \times )$ , showing competitive accuracy-compression tradeoff. BSQ can further increase the compression rate of ResNet-50 to $1 3 . 9 0 \times$ with $\alpha = 7 \mathrm { e } { - 3 }$ , with only $0 . 1 3 \%$ top-1 accuracy loss over the “5e-3” model. On Inception-V3 models, BSQ with $\alpha = 2 \mathrm { e } { - } 2$ achieves both higher accuracy $( 7 5 . 9 0 \%$ vs. $7 5 . 5 2 \%$ ) and higher compression rate ( $. 2 . 8 9 \times$ vs $1 2 . 0 4 \times $ ) comparing to the best previous method HAWQ (Dong et al., 2019). Adopting a smaller $\alpha = 1 0 ^ { - 2 }$ makes BSQ to achieve $0 . 7 \%$ accuracy improvement trading off ${ \sim } 1 0 \%$ less compression rate comparing to the $" 2 \mathrm { e } \mathrm { - } 2 "$ model.
+
+# 6 CONCLUSIONS
+
+In this work, we propose BSQ, which fully explores the accuracy-model size tradeoff of DNN’s mixed-precision quantization schemes with a differentiable training algorithm using DNN’s bit representation as trainable variables. A bit-level group Lasso regularizer with memory consumptionaware layer-wise reweighing is applied to induce bit-level sparsity, which leads to the dynamic adjustment of each layer’s precision and finally a mixed-precision quantization scheme through a single-pass gradient-based training process. This enables BSQ to dynamically produce a series of quantization schemes trading off accuracy and model size and provides models with higher accuracy comparing to training from scratch under the same quantization scheme. We apply BSQ in training ResNet-20 models on the CIFAR-10 dataset and training ResNet-50 and Inception-V3 models on the ImageNet dataset. In all the experiments, BSQ demonstrates the ability to reach both a better accuracy and a higher compression rate comparing to previous quantization methods. Our results prove that BSQ can successfully fill in the gap of inducing a mixed-precision quantization scheme with a differentiable regularizer, so as to effectively explore the tradeoff between accuracy and compression rate for finding DNN models with both higher accuracy and fewer bits.
+
+# ACKNOWLEDGMENTS
+
+This work is supported in part by NSF CCF-1910299 and NSF CNS-1822085.
+
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+
+# A HYPERPARAMETER CHOICES IN THE EXPERIMENTS
+
+# A.1 CIFAR-10 EXPERIMENTS
+
+We use ResNet-20 models on the CIFAR-10 dataset (Krizhevsky & Hinton, 2009) to do all of our ablation studies and evaluate the performance of BSQ. The CIFAR-10 dataset can be directly accessed through the dataset API provided in the “torchvision” python package. We do not change the splitting between the training and the test set. Standard preprocessing procedures, including random crop with a padding of 4, random horizontal flip and normalization, are used on the training set to train the model. The validation set is normalized with the same mean and variance as the training set. We implemented ResNet-20 models following the description in (He et al., 2016a), and pretrain the model for 350 epochs. The learning rate is set to 0.1 initially, and decayed by 0.1 at epoch 150, 250 and 325. The weights of all the layers except the batch normalization are then quantized to 8-bit before the BSQ training. The batch normalization layers are kept in the floating-point format throughout the training process. Similar to previous quantization works, we also apply the activation quantization during the training. For 4-bit or above activation precision we replace all the ReLU activation function in the model with the ReLU6 activation function. For lower activation precision we use the trainable PACT activation (Choi et al., 2018) with weight decay 0.0001. These changes will help achieving higher accuracy and better training stability when the activation is quantized as it eliminates extremely large activation values. As BSQ does not consider activation quantization as an objective, we fix the activation precision throughout the BSQ training and the finetuning process.
+
+We start the BSQ training with the 8-bit quantized pretrained model following the process described in Section 3.3. The BSQ training is done for 350 epochs, with the first 250 epochs using learning rate 0.1 and the rest using learning rate 0.01. Unless otherwise specified, the re-quantization and precision adjustment is done every 100 epochs, as well as after the BSQ training is finished to adjust and finalize the quantization scheme. Different regularization strengths $\alpha$ are tried to explore the tradeoff between accuracy and compression rate. The exact $\alpha$ used for each set of experiment is reported alongside the results in the main article. For comparing with previous methods, we further finetune the achieved mixed-precision model with the DoReFa-Net algorithm (Zhou et al., 2016) while fixing the quantization scheme. The finetuning is performed for 300 epochs with an initial learning rate 0.01 and the learning rate decay by 0.1 at epoch 150 and 250. The “train from scratch” accuracy reported in Table 1 is achieved by first quantizing a pretrained floating-point model to the mixed precision quantization scheme achieved by BSQ, then performing DoReFa-Net quantization aware training on the model. The training is done for 350 epochs, with an initial learning rate 0.1 and the learning rate decay by 0.1 at epoch 150, 250 and 325. All the training tasks are optimized with the SGD optimizer (Sutskever et al., 2013) with momentum 0.9 and weight decay 0.0001, and the batch size is set to 128. All the training processes are done on a single TITAN XP GPU.
+
+# A.2 IMAGENET EXPERIMENTS
+
+The ImageNet dataset is used to further compare BSQ with previous methods in Table 3. The ImageNet dataset is a large-scale color-image dataset containing 1.2 million images of 1,000 categories (Russakovsky et al., 2015), which has long been utilized as an important benchmark on image classification problems. In this paper, we use the “ILSVRC2012” version of the dataset, which can be found at http://www.image-net.org/challenges/LSVRC/ 2012/nonpub-downloads. We use all the data in the provided training set to train our model, and use the provided validation set to evaluate our model and report the testing accuracy. We follow the data reading and preprocessing pipeline suggested by the official PyTorch ImageNet example. For training images, we first perform the random sized crop on the training images with the desired input size, then apply random horizontal flipping and finally normalize them before feeding them into the network. We use an input size of $2 2 4 \times 2 2 4$ for experiments on the ResNet-50, and use an size of $2 9 9 \times 2 9 9$ for the Inception-V3 experiments. Validation images are resized then center cropped to the desired input size and normalized before used for testing. For both the ResNet-50 and the Inception-V3 model, the model architecture and the pretrained model provided in the “torchvision” package are directly utilized. The first convolutional layer of the ResNet-50 model and the first 5 conventional layers of the Inception-V3 model are quantized to 8-bit, while all the other layers are quantized to 6-bits before the BSQ training. Similar to the CIFAR-10 experiments, the batch normalization layers are kept as floating-point.
+
+![](images/7c23a1e6f4a5335b0d7011845a00bd4ebe35e9b6c39e8affdaa8770faa1d8019.jpg)  
+Figure 4: Range of testing accuracy and bit reduction rate achieved from 5 repeated runs with different random seeds. Solid line links the average performance, error bar marks the maximal and minimal performance achieved with each set of hyperparameters.
+
+We start the BSQ training with the quantized pretrained model. For both ResNet-50 and Inception-V3 models, the BSQ training is done for 90 epochs, with the first 30 epochs using learning rate 0.01 and the rest using learning rate 0.001. The re-quantization interval is set to 10 epochs for all the ImageNet experiments. The regularization strength $\alpha$ used is reported alongside the results in Table 3. The model after BSQ training is further finetuned with DoReFa-Net for 90 epochs, with the initial learning rate 0.001 and a learning rate decay by 0.1 after 30 epochs. All the models are optimized with the SGD optimizer with momentum 0.9 and weight decay 0.0001, and the batch size is set as 256 for all the experiments. Two TITAN RTX GPUs are used in parallel for the BSQ training and finetuning of both ResNet-50 and Inception-V3 models.
+
+# B ADDITIONAL ABLATION STUDY RESULTS
+
+# B.1 CHOICE OF RE-QUANTIZATION INTERVAL
+
+We propose the layer-wise regularization reweighing in Section 3.3 and show its importance in Section 4.1. This reweighing can be more effective if we adjust the precision of each layer regularly throughout the BSQ training routine. The precision adjustment is done through periodic re-quantization. From the one hand, a smaller re-quantization interval would help the precision to be adjusted in-time. From the other hand, it may cause the training unstable due to the frequent change in bit representation and regularizer values. So here we gradually increase the re-quantization interval to find the best choice that can reach high and stable performance. Figure 4 demonstrates the stability and performance under re-quantization intervals 20, 50, 100 and compare them with the performance achieved without re-quantization during the training. Each point in the figure corresponds to the averaged compression rate and accuracy after 5-time repeated BSQ training with a fixed regularization strength $\alpha$ but with different random seeds. The observation in the figure supports our analysis that as re-quantization is important to reach a better accuracy-# bits tradeoff, applying it too frequent will make the training unstable and hinders the overall performance. Comparing to not performing the re-quantization and applying it every 20 epochs, re-quantizing every 50 or 100 epochs yields similarly better tradeoff between accuracy and compression rate. Re-quantization interval 100 leads to a higher accuracy in a wider range of compression rate comparing to the Int 50 model, and the performance is more stable throughout the repeated trails. Therefore in all the other CIFAR-10 experiments we set the re-quantization interval to 100 epochs.
+
+# B.2 ADDITIONAL RESULTS ON REGULARIZATION REWEIGHING
+
+Figure 5 and Figure 6 compares the quantization scheme and the model performance achieved when performing the BSQ training with or without the memory consumption-aware reweighing of the bit-level group Lasso regularizer under additional choices of regularization strength $\alpha$ . The $\alpha$ used for each set of experiment are chosen so that comparable compression rates are achieved with or without reweighing. The $\alpha$ used are listed in the caption of the figures. All the other hyperparameters are kept the same. From both figures we can observe a consistent trend that training without the reweighing term will lead to less precision assigned to earlier layers with fewer parameters, while later layers with more parameters are not compressed enough. Therefore the achieved quantized model will have less accuracy and smaller compression rate comparing to the model achieved with layer-wise regularization reweighing. This observation is consistent with the results shown in Section 4.1. All these results show that the memory consumption-aware reweighing proposed in BSQ training is crucial for generating models with both higher compression rate and higher accuracy.
+
+![](images/ca507def97270cf74683b1b82132168b899c0420556d48f4b119fd31b604bde7.jpg)
+
+![](images/56909602b2724fe29c9ed0ba6115db9ff8191a175adbafa671c03b8ca807e335.jpg)  
+Figure 5: Quantization schemes achieved with or without layer-wise regularization reweighing. The compression rate and the accuracy after finetuning are listed in the legend. $\alpha { = } 6 \mathrm { e } { - } 3$ with reweighing and $\alpha = 3 \mathrm { e } { - 3 }$ without reweighing.   
+Figure 6: Quantization schemes achieved with or without layer-wise regularization reweighing. The compression rate and the accuracy after finetuning are listed in the legend. $\alpha { = } 0 . 0 1 5$ with reweighing and $\alpha = 5 \mathrm { e } { - 3 }$ without reweighing.
+
+# B.3 QUANTIZATION SCHEME COMPARISON WITH HAWQ
+
+As discussed in Section 4.2 and shown in Figure 3, for the same model architecture the relative ranking of the precision assignment by BSQ is mostly consistent under different $\alpha$ ’s. Here we compare quantization schemes achieved by BSQ with the “layer importance ranking” measured in HAWQ (Dong et al., 2019) to further analyze this consistency. HAWQ proposes to rank all the layers in the model with an importance score $S _ { i } = \lambda _ { i } / n _ { i }$ , where $\lambda _ { i }$ denotes the top eigenvalue of the Hessian matrix of layer $i$ , and $n _ { i }$ represents the number of parameters in layer $i$ . A layer with a higher $S _ { i }$ will be assigned with higher precision in the mixed-precision quantization scheme. The quantization schemes achieved by BSQ and HAWQ are compared in Figure 7, where the black dotted line shows the HAWQ scheme while the color solid lines demonstrate the schemes achieved by BSQ under different $\alpha$ . It can be observed that the relative ranking of BSQ’s precision is consistent with the ranking of precision in HAWQ, which to some extent shows that BSQ can dynamically identify the important layers during the training and assign higher precision to them. Note that HAWQ can only come up with the precision ranking of each layer, while the exact precision is designed manually. BSQ on the other hand is able to explicitly assign the precision to each layer during a single training process, and can dynamically tradeoff model size and accuracy by only changing $\alpha$ . Thus BSQ can easily find better tradeoff points with both higher accuracy and higher compression rate comparing to HAWQ and other quantization methods, as discussed in Section 5.
+
+![](images/e4ecfecb1ace1f3935f511426c2ea83faa0f91a22d11da0a5f9540db7a177207.jpg)  
+Figure 7: Layer-wise precision comparison between the quantization schemes achieved with BSQ and the scheme achieved with HAWQ (Dong et al., 2019) on the ResNet-20 model.
+
+![](images/7ff34bceb7ef000240edde61d361cb13047dafe758242e60c85269b7cb5e9d06.jpg)  
+Figure 8: Layer-wise precision comparison of the quantization schemes achieved under different regularization strengths with 2-bit activation.
+
+Table 4: Accuracy-#Bits tradeoff with 2-bit activation. “FT” stands for finetuning.   
+
+<table><tr><td>Strength α</td><td>1e-3</td><td>2e-3</td><td>3e-3</td><td>5e-3</td></tr><tr><td>#Bits per Para/Comp(×)</td><td>3.77 /8.48</td><td>2.86 /11.20</td><td>2.26 /14.13</td><td>1.70 / 18.85</td></tr><tr><td>BSQ acc before / after FT (%)</td><td>91.03 /91.21</td><td>90.19 /90.70</td><td>89.54 / 90.39</td><td>88.13 /90.19</td></tr></table>
+
+# B.4 MODELS ACHIEVED UNDER DIFFERENT ACTIVATION PRECISION
+
+As we discuss the BSQ quantization schemes and model performances under 4-bit activation in Figure 3 and Table 1, here we show the tradeoff between model size and accuracy under different regularization strength $\alpha$ with 2-bit activation in Figure 8 and Table 4, as well as those with 3-bit activation in Figure 9 and Table 5. In both cases we observe that the relative ranking of the precision assignment is mostly consistent under different $\alpha$ ’s. As $\alpha$ increases, less bits are assigned to each layer, leading to increasing overall bit reduction with the cost of a small performance loss. This tradeoff is consistent with our previous observations on the 4-bit activation models .
+
+# C DETAILED QUANTIZATION SCHEMES FOR IMAGENET EXPERIMENTS
+
+The quantization schemes of the reported ResNet-50 and Inception-V3 models can be found in Table 6 and Table 7 respectively.
+
+![](images/bf7e702ddcc67a5a47eac487a7c942f91624aa36e525cde8bf77847bb14374e2.jpg)  
+Figure 9: Layer-wise precision comparison of the quantization schemes achieved under different regularization strengths with 3-bit activation.
+
+Table 5: Accuracy-#Bits tradeoff with 3-bit activation. “FT” stands for finetuning.   
+
+<table><tr><td>Strength α</td><td>2e-3</td><td>5e-3</td><td>8e-3</td><td>1e-2</td></tr><tr><td>#Bits per Para /Comp(×)</td><td>2.90 / 11.04</td><td>1.95 /16.37</td><td>1.39 /23.04</td><td>1.28 / 25.06</td></tr><tr><td>BSQ acc before /after FT (%)</td><td>90.45 /92.16</td><td>90.44 / 91.72</td><td>89.01 /90.93</td><td>88.41 /90.51</td></tr></table>
+
+Table 6: Quantization schemes of ResNet-50 models on ImageNet dataset achieved by BSQ in Table 3. The scheme on the left is achieved with $\alpha = 5 \mathrm { e } { - 3 }$ and the one on the right is achieved with $\alpha = 7 \mathrm { e } { - 3 }$ . Except the first row for the leading convolutional layer and the last row for the FC layer, each row in the table reports the precision assigned to the 3 layers in a residual block, with layer 1-3 listed from left to right.   
+
+<table><tr><td></td><td>BSQ 5e-3</td><td></td><td>BSQ 7e-3</td><td></td></tr><tr><td>Conv 1</td><td>17</td><td></td><td>17</td><td></td></tr><tr><td>Block 1-0</td><td>7</td><td>6 6</td><td>7</td><td>6 6 6</td></tr><tr><td>Block 1-1</td><td>6</td><td>6 6 6</td><td>6</td><td>6 6</td></tr><tr><td>Block 1-2</td><td>6</td><td>6</td><td>6</td><td>5</td></tr><tr><td>Block 2-0</td><td>4 4</td><td>3 4</td><td>4</td><td>3</td></tr><tr><td>Block 2-1</td><td>4 4</td><td>4</td><td>4</td><td>3</td></tr><tr><td>Block 2-2</td><td>4</td><td>4</td><td>4</td><td>3</td></tr><tr><td>Block 2-3</td><td>4 3</td><td></td><td>43</td><td>3</td></tr><tr><td>Block 3-0</td><td>4 3</td><td>3</td><td>4</td><td>3</td></tr><tr><td>Block 3-1</td><td>33</td><td>3 ×</td><td>3</td><td>3</td></tr><tr><td>Block 3-2</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>Block 3-3</td><td>3</td><td></td><td>3</td><td>3</td></tr><tr><td></td><td>3</td><td>3</td><td></td><td>2 3 3</td></tr><tr><td>Block 3-4</td><td>3 3</td><td>3</td><td>3</td><td>3 3</td></tr><tr><td>Block 3-5</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>Block 4-0</td><td>3 2</td><td>2</td><td>3</td><td>2</td></tr><tr><td>Block 4-1</td><td>2</td><td>2 3</td><td>2</td><td>2</td></tr><tr><td>Block 4-2</td><td></td><td></td><td>2</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>2</td></tr><tr><td></td><td>13</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>2</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>3 3</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FC</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>2</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr></table>
+
+Table 7: Quantization schemes of Inception-V3 model on ImageNet dataset achieved by BSQ in Table 3. The scheme on the left is achieved with $\alpha = 1 { \mathrm e } { - } 2$ and the one on the right is achieved with $\alpha =$ 2e-2. Except for the first 5 convolutional layers and the final FC layer, each row in the table reports the precision assigned to the layers within the inception block. The order from left to right follows the parameter definition order provided in the torchvision package implementation (https://github. com/pytorch/vision/blob/master/torchvision/models/inception.py).   
+
+<table><tr><td></td><td></td><td colspan="10">BSQ 1e-2</td><td></td><td></td><td>BSQ 2e-2</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Conv 1a</td><td>8</td><td colspan="10"></td><td>8</td><td colspan="10"></td></tr><tr><td>Conv 2a</td><td>7</td><td colspan="10"></td><td colspan="10">7</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Conv 2b</td><td>6</td><td colspan="10"></td><td colspan="10">6</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Conv 3b</td><td>8</td><td colspan="10"></td><td colspan="10">8</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Conv 4a</td><td>5</td><td colspan="10"></td><td>4</td><td colspan="10"></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mixed 5b</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td></td><td>3</td><td>4</td><td></td><td></td><td></td><td></td><td>4</td><td>4</td><td></td><td>3</td><td>4</td><td>3</td><td></td><td>3</td><td></td><td>4</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mixed 5c</td><td>4</td><td>4</td><td>3</td><td>4</td><td>3</td><td>3</td><td></td><td>4</td><td></td><td></td><td></td><td></td><td>4</td><td>4</td><td>3</td><td></td><td>4 4</td><td>3 3</td><td></td><td>3</td><td></td><td>4</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mixed 5d</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td></td><td>3</td><td>4</td><td></td><td></td><td></td><td></td><td>4</td><td>4</td><td>3</td><td></td><td></td><td></td><td></td><td>3</td><td>4</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mixed 6a</td><td>2</td><td>4</td><td>4</td><td>3</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>2</td><td>4</td><td>3</td><td></td><td>3</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>3</td><td></td></tr><tr><td>Mixed 6b</td><td>4</td><td>3</td><td>3</td><td>3</td><td>3</td><td></td><td>3</td><td>3</td><td>3</td><td>3</td><td></td><td>3</td><td>3</td><td>3</td><td>3</td><td></td><td>3</td><td>3 3</td><td></td><td>3</td><td></td><td>3</td><td>3</td><td></td><td>3 2</td><td></td><td>3</td></tr><tr><td>Mixed 6c</td><td>4</td><td>4</td><td>3</td><td>3</td><td>4</td><td></td><td>3</td><td>3</td><td>3</td><td>3</td><td></td><td>3</td><td>3</td><td>3</td><td>3</td><td></td><td>3</td><td></td><td></td><td>3</td><td></td><td>3</td><td></td><td>2 3</td><td></td><td></td><td>3</td></tr><tr><td>Mixed 6d</td><td>5</td><td>3</td><td>3</td><td>3 3</td><td>4 3</td><td></td><td>3 3</td><td>3 3</td><td>3 3</td><td>3 3</td><td></td><td>3 4</td><td>5</td><td>3 3</td><td></td><td>3 2</td><td>3 2</td><td>3</td><td>4</td><td>3 3</td><td></td><td>3 3</td><td></td><td>3</td><td>2 3</td><td></td><td>3</td></tr><tr><td>Mixed 6e</td><td>5</td><td>4</td><td>3</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>4</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mixed 7a</td><td>3</td><td>3</td><td>4</td><td>3</td><td>3</td><td></td><td>2</td><td></td><td></td><td></td><td></td><td></td><td>3</td><td>3</td><td></td><td>3</td><td>3</td><td></td><td>3</td><td>2</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mixed 7b</td><td>2</td><td>3</td><td>3</td><td>3</td><td>2</td><td></td><td>2</td><td>3</td><td>2 3</td><td></td><td>3</td><td></td><td>2</td><td>2</td><td></td><td>3 3</td><td>3 3</td><td></td><td>2 2</td><td>1</td><td></td><td>2</td><td></td><td>2</td><td>2</td><td></td><td></td></tr><tr><td>Mixed 7c</td><td>2</td><td>2</td><td>3</td><td>3</td><td>3</td><td></td><td>2</td><td>3</td><td></td><td></td><td>2</td><td></td><td>2</td><td>2</td><td></td><td></td><td></td><td></td><td></td><td>2</td><td></td><td>3</td><td>3</td><td></td><td>2</td><td></td><td></td></tr><tr><td>FC</td><td>3</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>3</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>

parse/train/YFysbLCFdIe/YFysbLCFdIe.md

@@ -0,0 +1,263 @@
+# Neural Human Performer: Learning Generalizable Radiance Fields for Human Performance Rendering
+
+Youngjoong Kwon1, Dahun $\mathbf { K i m ^ { 2 } }$ , Duygu Ceylan3, Henry Fuchs1 1University of North Carolina at Chapel Hill. 2KAIST. 3Adobe Research. {youngjoong,fuchs}@cs.unc.edu {mcahny}@kaist.ac.kr {ceylan}@adobe.com
+
+# Abstract
+
+In this paper, we aim at synthesizing a free-viewpoint video of an arbitrary human performance using sparse multi-view cameras. Recently, several works have addressed this problem by learning person-specific neural radiance fields (NeRF) to capture the appearance of a particular human. In parallel, some work proposed to use pixel-aligned features to generalize radiance fields to arbitrary new scenes and objects. Adopting such generalization approaches to humans, however, is highly challenging due to the heavy occlusions and dynamic articulations of body parts. To tackle this, we propose Neural Human Performer, a novel approach that learns generalizable neural radiance fields based on a parametric human body model for robust performance capture. Specifically, we first introduce a temporal transformer that aggregates tracked visual features based on the skeletal body motion over time. Moreover, a multi-view transformer is proposed to perform cross-attention between the temporally-fused features and the pixel-aligned features at each time step to integrate observations on the fly from multiple views. Experiments on the ZJU-MoCap and AIST datasets show that our method significantly outperforms recent generalizable NeRF methods on unseen identities and poses. The video results and code are available at https://youngjoongunc.github.io/nhp.
+
+# 1 Introduction
+
+Free-viewpoint video of a human performer has a variety of applications in the area of telepresence, mixed reality, gaming and etc. Conventional free-viewpoint video systems require extremely expensive setups such as dense camera rigs [4, 7, 43] or accurate depth sensors [6, 13], to capture person-specific appearance information. In this paper, we aim at a scalable solution for free-viewpoint human performance rendering that generalizes across different human performers and requires only sparse camera views. However, representing and rendering arbitrary human performances is extremely challenging when the observations are highly sparse (up to three to four views) due to heavy self-occlusions and dynamic articulations of the body parts. In particular, an effective solution needs to coherently aggregate appearance information from sparse multi-view observations across time as the body undergoes a 3D motion. Furthermore, the solution needs to generalize to unseen motions and characters at test time.
+
+Recently, neural radiance fields (NeRF) [26, 11, 17, 30, 33, 35, 36, 47, 50, 52, 53] have shown photo-realistic novel view synthesis results in per-scene optimization settings. To avoid the expensive per-scene training and improve the practicality, generalizable NeRFs [36, 52, 47] have been proposed which use image-conditioned, pixel-aligned features and achieve feed-forward view synthesis from sparse input views [36, 52]. Direct application of these methods to complex and non-rigid human motion is not straightforward, however, and naive solutions suffer from significant artifacts as shown in Fig. 3. Some existing methods [37, 52] aggregate image features across multiple views by simple average pooling, which often leads to over-smoothed outputs when details observed from multiple views (e.g., front and side views) differ due to self occlusions of humans. Alternatively, several methods [23, 33] have proposed to learn person-specific global appearance features from multi-view observations. However, such methods are not able to generalize to new human performers.
+
+To address these challenges, we propose Neural Human Performer, a novel approach that learns generalizable radiance fields based on a parametric 3D body model for robust performance capture. In addition to exploiting a parametric body model as a geometric prior, the core of our method is a combination of temporal and multi-view transformers which help to effectively aggregate spatio-temporal observations to robustly compute the density and color of a query point. First, the temporal transformer aggregates trackable visual features based on the input skeletal body motion across time. The following multi-view transformer performs cross-attention between the temporallyaugmented skeletal features and the pixel-aligned features for each time step. The proposed modules collectively contribute to the adaptive aggregation of multi-time and multi-view information, resulting in significant improvements in synthesis results in different generalization settings of unseen motions and identities.
+
+We study the efficacy of Neural Human Performer on two multi-view human performance capture datasets, ZJU-MoCap [33] and AIST [16]. Experiments show that our method significantly outperforms recent generalizable radience field (NeRF) methods. Furthermore, we compare ours with identity-specific methods [33, 44, 49] that also utilize a 3D human body model prior. Surprisingly, our generalized method achieves better rendering quality than the person-specific dedicated methods when tested on novel poses demonstrating the effectiveness of our transformer-based generalizable representation.
+
+To summarize, our contributions are:
+
+• We present a new feed-forward method of synthesizing novel-view videos of arbitrary human performers from sparse camera views. We propose Neural Human Performer that learns generalizable neural radiance representations by leveraging a 3D body motion prior.   
+• We design a combination of temporal and multi-view transformers that can aggregate information on the fly over video frames across multiple views to render each frame of the novel-view video.   
+• We show significant improvements over recent generalizable NeRF methods on unseen identities and poses. Moreover, our generalization results can outperform even personspecific methods when tested on unseen poses.
+
+# 2 Related works
+
+Human performance capture. Novel view synthesis of human performance has a long history in computer vision and graphics. Traditional methods rely on complicated hardware such as dense camera rigs [4, 7, 43] or accurate depth sensors [6, 13]. To enable free-view video from sparse camera views, template-based methods [3, 5, 10, 42] exploit pre-scanned human models to track the motion of a person. However, their synthesis results are not photo-realistic and pre-scanned human models are not available in most cases. Recent methods [27, 37, 38, 54] learn 3D human geometry priors along with pixel aligned features to enable detailed 3D human reconstructions even from single images. However, these methods often suffer under complex human poses that are never seen during training and hence cannot be directly used for our purpose of human performance synthesis.
+
+Neural 3D representations. Recently, there has been great progress in learning neural networks to represent the shape and appearance of scenes. The 3D representations are learned from 2D images via differentiable rendering networks. Convolutional neural networks are used to predict volumetric representations via 3D voxel-grid features [40, 23, 29, 25, 14, 15], point clouds [1, 49], textured meshes [18, 21, 44] and multi-plane images [8, 55]. The learnt representations are projected by a 3D-to-2D operation to synthesize images. Some methods [14, 15] introduce consistency loss to improve the multi-view and temporal consistency. However, these methods often have difficulty in scaling to higher resolution due to memory restrictions.
+
+To eschew these problems, implicit function-based methods [20, 22, 28, 41] learn a multi-layer perceptron that directly translates a 3D positional feature into a pixel generator. The more recent NeRF [26] learns implicit fields of density and color with a volume rendering technique and achieves photo-realistic view synthesis. Among many following NeRF extensions, [30, 31, 35, 53, 17, 50] focus on dynamic scenes. While showing impressive results, it’s an extremely under-constrained problem to jointly learn NeRF and highly dynamic deformation fields. To regularize the training, Neural Body [33] combines NeRF with a deformable human body model (e.g., SMPL [24]). Despite the promising results, these general deformable NeRF [17, 53] and human-specific NeRF [11, 9, 33, 32] methods must be optimized for each new video separately, and generalize poorly on unseen scenarios. Generalizable NeRFs [36, 47, 52] try to avoid the expensive per-scene optimization by image-conditioning using pixel-aligned features. However, directly extending such methods to model complex and dynamic 3D humans is not straightforward when available observations are highly sparse. Unlike existing works, our method exploits temporal and multi-view information on-the-fly and achieves free-viewpoint human rendering in a feed-forward manner, also generalizing to new, unseen human identities and poses.
+
+![](images/d66549154edb235e1464e068e110b4ddedca832bc35c3eceb7e749af7272c231.jpg)  
+Figure 1: Given a sparse set of multi-view videos of a person, Neural Human Performer uses temporal and multi-view transformers to learn pixel aligned features on the surface of a deformable body template such as SMPL. Such features are utilized to predict the color and density value of query points to render novel views of possibly unseen poses of the person.
+
+# 3 Method
+
+# 3.1 Neural Human Performer
+
+Problem definition. In our setting, given a sparse set (e.g., 3 or 4) of multi-view cameras $c = 1 , \ldots , C$ , input videos of an arbitrary human performance $I _ { c , 1 : T } : = \{ I _ { c , 1 } , I _ { c , 2 } , . . . , I _ { c , T } \}$ are captured for each camera view $c$ defined by $\{ \mathbf { K } _ { c } , [ \mathbf { R } | \mathbf { t } ] _ { c } \}$ . We assume that the 3D human body model fit corresponding to each frame is given. Our goal is to synthesize a novel view video $\hat { I } _ { q , 1 : T }$ for a query viewpoint $q$ defined by $\{ \mathbf { K } _ { q } , \mathbf { \bar { [ R | t ] } } _ { q } \}$ .
+
+Overview. To compensate for the sparsity of available input views, we propose to exploit temporal information across video frames. In practice, we sample $M$ memory frames from the original input videos to augment each queried timestep $t$ . Our goal is to learn generalizable 3D representations of human performers from multi-time $( M )$ and multi-view $( C )$ observations.
+
+To this end, the Neural Human Performer is proposed with two main components as illustrated in Fig. 1. First, we construct the time-augmented skeletal features $\{ s \}$ . We exploit a human body model (SMPL [24]) to construct 3D skeletal features by projecting all the SMPL vertices onto each memory frame and picking up the pixel-aligned image features [36, 37, 52] at the projected 2D locations. The skeletal features are sampled from all memory frames to construct the skeletal feature bank. Inspired by Transformers[2, 46, 48], we propose a temporal Transformer that aggregates these memory features into the time-augmented skeletal features $\{ s ^ { \prime } \}$ .
+
+In the second stage, given a query 3D point $\mathbf { x }$ , skeletal features are sampled at $\mathbf { x }$ . In addition, pixel-aligned features $\{ p \}$ at each time $t$ are sampled by directly projecting $\mathbf { x }$ onto the input images $\{ I \} _ { t }$ . The multi-view Transformer is proposed to learn the correlation between the pixel-aligned features $\{ p \}$ and the time-augmented skeletal features $\{ s ^ { \prime } \}$ , and to adaptively fuse the multi-view information. The final fused feature of the query point $\mathbf { x }$ is fed into the radiance field module to predict its color and density.
+
+# 3.2 Construction of time-augmented skeletal features
+
+Unlike static scenes, video inputs of moving characters inherently contain more visual cues as the occluded regions in a frame may be visible in other (potentially distant) frames. To take advantage of this temporal information, we first build up the skeletal feature bank from memory frames by leveraging a parametric body model (see Fig. 1). Then, we propose a temporal Transformer module that aggregates such collected features.
+
+For each view $c$ and time $t$ , we first build frame-level skeletal features $s _ { c , t } \in \mathbb { R } ^ { L \times d }$ by sampling image features at the projection of the SMPL vertices in each image $I _ { c , t } \in \mathbb { R } ^ { H \times W \times 3 }$ . $L$ denotes the number of SMPL vertices and $d$ is the dimension of image features.
+
+After collecting all the skeletal features from all memory frames, we aggregate the time information in an attention-aware manner, instead of using simple average pooling. Specifically, for any $i ^ { t h }$ skeletal feature vertex $s _ { c , t } ^ { i } \in \mathbb { R } ^ { d }$ , the proposed temporal Transformer casts attention over all other features contained in the vertex’s memory bank we compute soft weights for all memory $\boldsymbol { s } _ { c , t _ { 1 } : t _ { M } } ^ { i } = \{ \boldsymbol { s } _ { c , t _ { 1 } } ^ { i } , \boldsymbol { s } _ { c , t _ { 2 } } ^ { i } , . . . , \boldsymbol { s } _ { c , t _ { M } } ^ { i } \} \in \mathbb { R } ^ { M \times d }$ . In particular,respect to the current timestep $t$ . Then, the values of the memory features are weighted summed as
+
+$$
+\begin{array} { l l } { { t _ { \perp } a t t _ { c , t } ^ { i } = \psi \big ( \frac { 1 } { \sqrt { d _ { 0 } } } q ( s _ { c , t } ^ { i } ) \cdot k ( s _ { c , t _ { 1 } : t _ { M } } ^ { i } ) ^ { T } \big ) , } } & { { t _ { - } a t t _ { c , t } ^ { i } \in \mathbb { R } ^ { 1 \times M } } } \\ { { s _ { c , t } ^ { \prime } { } ^ { i } = t _ { - } a t t _ { c , t } ^ { i } \cdot v \big ( s _ { c , t _ { 1 } : t _ { M } } ^ { i } \big ) + s _ { c , t } ^ { i } , } } & { { s _ { c , t } ^ { \prime } \in \mathbb { R } ^ { L \times d } \forall i } } \end{array}
+$$
+
+where $\psi$ represents the softmax operator along the second axis, $q ( \cdot ) , k ( \cdot )$ and $v ( \cdot )$ are learnable query, key and value embedding functions $\mathbb { R } ^ { d  d _ { 0 } }$ of the temporal Transformer.
+
+In other words, the representation $s _ { c , t } ^ { i }$ of each skeletal vertex at time $t$ is computed through a dynamically weighted combination of all its previous and next representations in the memory frames. This allows our network to incorporate helpful information and ignore irrelevant ones from other timesteps. In practice, the temporal Transformer operation in Eq. (1) is performed by a batch matrix multiplication for all skeletal vertices $L$ and all available viewpoints $C$ .
+
+# 3.3 Multi-view aggregation of skeletal and query features
+
+Given a query 3D point $\mathbf { x } \in \mathbb { R } ^ { 3 }$ , we retrieve the corresponding (time-augmented) skeletal feature $s _ { c , t } ^ { \prime } \in \mathring { \mathbb { R } } ^ { d }$ at the queried location via trilinear interpolation in the SMPL space with SparseConvNet [19], following [39, 34, 51, 33].
+
+In addition, we sample pixel-aligned image feature $p _ { c , t } ^ { \mathbf { x } }$ via direct $\mathbb { R } ^ { 3 \to 2 }$ projection of the query point $\mathbf { x }$ on $I _ { c , t }$ . It is important to note that the pixel-aligned feature $p _ { c , t } ^ { \mathbf { x } }$ is time-specific and represents the exact query location of $\mathbf { x }$ , while the skeletal feature $s _ { c , t } ^ { \prime } { } ^ { \mathbf { x } }$ is time-augmented (w.r.t $t$ ) and contains inherent geometric deviations in the SMPL vertices and the following trilinear interpolations. We propose to combine these two complementary features, which will be shown to be effective in Sec. 4.4.
+
+Given the two sets of multi-view features, skeletal s′1:C,tx $s _ { 1 : C , t } ^ { \prime } \mathbf { \Psi } ^ { \mathbf { x } } = \{ s _ { c , t } ^ { \prime } \mathbf { \Psi } ^ { \mathbf { x } } | c = 1 , . . . , C \} \in \mathbb { R } ^ { C \times d }$ and pixel-aligned $p _ { 1 : C , t } ^ { \bf x } = \{ p _ { c , t } ^ { \bf x } | c = 1 , . . . , C \} \in \mathbb { R } ^ { C \times d }$ , we propose a multi-view Transformer that performs cross-attention from skeletal to pixel-aligned features. Specifically, the values of pixelaligned features from all viewpoints is re-weighted based on how much compatible they are with each skeletal features. The non-local cross-attention mv_att is constructed as:
+
+$$
+\begin{array} { r l r } & { m v _ { - } a t t _ { t } ^ { \mathbf { x } } = \psi \bigl ( \frac { 1 } { \sqrt { d _ { 1 } } } k \bigl ( s _ { 1 : C , t } ^ { \prime } \bigr ) \cdot k \bigl ( p _ { 1 : C , t } ^ { \mathbf { x } } \bigr ) ^ { T } \bigr ) , \quad m v _ { - } a t t _ { t } ^ { \mathbf { x } } \in \mathbb { R } ^ { C \times C } } & \\ & { z _ { 1 : C , t } ^ { \mathbf { x } } = m v _ { - } a t t _ { t } ^ { \mathbf { x } } \cdot v \bigl ( p _ { 1 : C , t } ^ { \mathbf { x } } \bigr ) + v \bigl ( s _ { 1 : C , t } ^ { \prime } \bigr ) , \qquad z _ { 1 : C , t } ^ { \mathbf { x } } \in \mathbb { R } ^ { C \times d } , \quad z _ { c , t } ^ { \mathbf { x } } \in \mathbb { R } ^ { 1 \times d } } & \end{array}
+$$
+
+where $\psi$ represents the softmax operator along the second axis. Note that $k$ and $v$ are new layers independent from those in the temporal Transformer. The confident observations in each view will have large weights and be highlighted, and vice versa. Finally, we use the view-wise mean of $\begin{array} { r } { z _ { t } ^ { \mathbf { x } } = \frac { 1 } { C } \sum _ { c } z _ { c , t } ^ { \mathbf { x } } \in \mathbb { R } ^ { d } } \end{array}$ as our meta-time and meta-view representation of the query point $x$ .
+
+The final density $\sigma _ { t } ( \mathbf { x } )$ and color values $r g b _ { t } ( { \bf x } )$ at time $t$ are computed as:
+
+$$
+\sigma _ { t } ( \mathbf { x } ) = M L P _ { \sigma } ( z _ { t } ^ { \mathbf { x } } ) , \qquad r g b _ { t } ( \mathbf { X } ) = M L P _ { \mathbf { r g b } } ( \sum _ { c } { ( z _ { c , t } ^ { \mathbf { x } } ; \gamma _ { \mathbf { d } } ( \mathbf { d } ) ) } / C ) ,
+$$
+
+where $M L P _ { \sigma }$ and $M L P _ { \mathbf { r g b } }$ consist of four and two linear layers respectively, and $\gamma _ { \mathbf { d } } : \mathbb { R } ^ { 3  6 \times l }$ is a positional encoding of viewing direction $\mathbf { d } \in \mathbb { R } ^ { 3 }$ as in [26] with $2 \times l$ different basis functions.
+
+More details on the network architecture can be found in the supplementary material.
+
+# 3.4 Volume Rendering
+
+The predicted color of a pixel $p \in \mathbb { R } ^ { 2 }$ for a target viewpoint $q$ in the focal plane of the camera and center $\mathbf { r } _ { 0 } \in \mathbb { R } ^ { 3 }$ is obtained by marching rays into the scene using the camera-to-world projection matrix, P−1 = [Rq|tq]−1K−1q with the direction of the rays given by d = P−1p−r0∥P−1p−r0∥ .
+
+We then accumulate the radiance and opacity along the ray $\mathbf { r } ( z ) = \mathbf { r } _ { 0 } + z \mathbf { d }$ for $z \in [ z _ { \mathrm { n e a r } } , z _ { \mathrm { f a r } } ]$ as defined in NeRF [26] as follows:
+
+$$
+\mathbf { I } _ { q } ( p ) = \int _ { z _ { \mathrm { n e a r } } } ^ { z _ { \mathrm { f a r } } } \mathbf { T } ( z ) \sigma ( \mathbf { r } ( z ) ) \mathbf { c } ( \mathbf { r } ( z ) , \mathbf { d } ) d z , \quad w h e r e \quad \mathbf { T } ( z ) = \exp \left( - \int _ { z _ { \mathrm { n e a r } } } ^ { z } \sigma ( \mathbf { r } ( s ) ) d s \right)
+$$
+
+In practice, we uniformly sample a set of 64 points $z \sim [ z _ { n e a r } , z _ { f a r } ]$ . We set $\mathbf { X } = \mathbf { r } ( z )$ and use the quadrature rule to approximate the integral. We compute the 3D bounding box of the SMPL parameters at time $t$ and derive the bounds for ray sampling $z _ { n e a r } , z _ { f a r }$ .
+
+# 3.5 Loss Function
+
+Given the ground truth target images $\mathbf { I } _ { q , t }$ , we train both the radiance field and feature extraction networks using a simple photometric reconstruction loss $\mathcal { L } = \Vert \hat { \mathbf { I } } _ { q , t } - \mathbf { I } _ { q , t } \Vert _ { 2 }$ .
+
+# 4 Experiments
+
+We present novel view synthesis and 3d reconstruction results of human performances in different generalization scenarios. We compare our method against the current best view-synthesis methods from two classes: per-subject optimization methods that also use a human body model prior (Sec. 4.1) and generalizable NeRF methods (Sec. 4.2). We experiment on the ZJU-MoCap [33] and AIST datasets [45, 16]. For training and testing our model as well as the baselines, we remove the background using the foreground masks that are either provided by the dataset or pre-computed using an off-the-shelf method. Unless otherwise specified, we sample two memory frames $\{ t - \mathsf { 2 0 } , t + 2 0 \bar  \}$ at time $t$ (i.e., three timesteps in total) and take three canonical input views in all experiments. The details of the implementation, datasets, training process are provided in the supplementary material.
+
+# 4.1 Comparison with body model-based, per-subject optimization methods.
+
+Baselines. We perform comparisons with the state-of-the-art Neural Body (NB) [33] that combines the body model prior SMPL and NeRF in a per-subject optimization setting. Neural Textures (NT) [44] renders a coarse mesh with latent texture maps and uses a 2D CNN to render target images. We use the SMPL mesh as the coarse mesh input to NT in our experiments. NHR [49] extracts 3D features from input point clouds and renders them into 2D images. Since dense point clouds are difficult to obtain from sparse camera views, we use the SMPL vertices as input point clouds. These methods have reported that they can adapt to new poses of the same performer, hence we compare to them for the task of novel pose synthesis.
+
+Setup. We experiment with ZJU-MoCap dataset [33] which provides performance captures of 10 human subjects captured from 23 synchronized cameras, human body model parameters as well as the foreground mask corresponding to each frame. Each video contains complex motions such as kicking and Taichi and is between 1000 to 2000 frames long. We first split the dataset into two parts: source and target videos. In all comparisons, the first 300 frames of either source or target videos are used during training, and the remaining next frames (unseen poses) are held out for testing. Note that the baseline methods are always trained in a per-subject manner. To validate whether the training is reproducible, we experiment with 5 independent runs with random train/test splits and observe a variance of 0.15 PSNR, showing that the results are consistent. In each independent run, we used seven models for training and the other three for testing.
+
+<table><tr><td>Method</td><td>PSNR</td><td>SSIM</td></tr><tr><td colspan="2">Trained on target subjects</td><td></td></tr><tr><td>NB</td><td>22.88</td><td>0.880</td></tr><tr><td>NHR</td><td>22.03</td><td>0.875</td></tr><tr><td>NT</td><td>21.92</td><td>0.873</td></tr><tr><td>Trained on</td><td>source subjects</td><td></td></tr><tr><td>Ours</td><td>24.75</td><td>0.906</td></tr></table>
+
+b . Test results on unseen poses of target subjects
+
+Table 1: Comparison with other body model-based, per-subject optimization methods.   
+
+<table><tr><td>Method</td><td>PSNR</td><td>SSIM</td></tr><tr><td colspan="2">Trained on source subjects</td><td></td></tr><tr><td>NB NHR</td><td>23.79 22.31</td><td>0.887 0.871</td></tr><tr><td>NT</td><td>22.28</td><td>0.872</td></tr><tr><td colspan="3">Trained on source subjects</td></tr><tr><td>Ours</td><td>26.94</td><td>0.929</td></tr></table>
+
+a . Test results on unseen poses of source subjects
+
+<table><tr><td colspan="2">Method PSNR</td><td>SSIM</td></tr><tr><td colspan="3">Trained on source subjects</td></tr><tr><td>NB</td><td>28.51</td><td>0.947</td></tr><tr><td>NHR</td><td>23.95</td><td>0.897</td></tr><tr><td>NT</td><td>23.86</td><td>0.896</td></tr><tr><td colspan="3">Trained on source subjects</td></tr><tr><td>Ours</td><td>28.73</td><td>0.936</td></tr><tr><td>Ours per-subject</td><td>31.57</td><td>0.966</td></tr></table>
+
+c . Test results on seen poses of source subjects
+
+![](images/b0ead6611373a5fa611bac7276222a6f2cfa8e4096da2b85a72ed656de985623.jpg)  
+Figure 2: Pose generalization – comparison with per-scene optimization methods that use a human body prior. Results of NT(Neural textures) [44], NHR (Neural human rendering) [49], NB (Neural body) [33] and ours. Novel view synthesis on the ZJU-MoCap for unseen poses of source subjects. While ours is a single model trained on all source subjects; baselines are trained in a per-subject manner.
+
+Results. We present three different comparison settings to validate our method. We would like to point out that all the comparison settings place our method (‘ours’) in disadvantage. This is because our model is trained on all the source subjects at once (one network for all subjects), while the competing methods are per-subject trained on the subject to be tested (one network for one subject) - easier setting. First, we evaluate the pose generalization capability (Table. 1a and Fig. 2) where we train on the training splits of the source subjects and test on the testing splits, i.e., unseen poses, of the same source subjects. Our method significantly outperforms all the baselines and the state-of-the-art Neural Body [33] by ${ \bf + 3 . 1 5 }$ PSNR and $+ 0 . 0 4 2$ SSIM scores. We next evaluate the identity generalization capability (Table. 1b) by testing on the unseen poses of target subjects. While our method is trained on source subjects only, baselines are trained on target subjects. Note that this comparison is disadvantageous to us since unlike our method the baselines have seen the testing subjects as they must be trained separately for each subject. Surprisingly, however, our method still outperforms all the baselines by a healthy margin of $+ 1 . 8 7$ PSNR and $+ 0 . 0 2 6$ SSIM scores. These results indicate that our proposed architecture with the temporal and the multi-view transformers generalizes well across unseen identities and poses, and produces photo-realistic results. Finally, in Table. 1c, we evaluate our method and baselines on seen poses of seen subjects. Even when we train a single model of our method for all the source sobjects (‘ours’), ours achieves comparable results with the state-of-the-art per-subject based methods [33]. To make a more fair comparison, we also train our method in the same per-subject setting similar to the baselines. Table. 1c shows that in this case, our model (‘ours per-subject’) achieves a significant improvement over the best-performing baseline [33] by $+ 3$ PSNR and $+ 1 . 4 \%$ SSIM.
+
+![](images/c7a66ae91e16110cda6e3e003e86bae057370c4a950b769cf5a12046d3bf74dc.jpg)  
+Figure 3: Identity-and-pose generalization – comparison with generalizable NeRF methods. Results of Pixel-Nerf [52], PVA (Pixel-aligned volumetric avatar) [36] and ours. Novel view synthesis on the ZJU-MoCap dataset where all methods are trained on source subjects and tested on unseen poses of target subjects.
+
+<table><tr><td>Method</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Pixel-NeRF</td><td>18.06</td><td>0.7304</td></tr><tr><td>PVA</td><td>17.82</td><td>0.7211</td></tr><tr><td>Ours</td><td>19.03</td><td>0.8390</td></tr></table>
+
+b . Generalization results on AIST.
+
+Table 2: Comparison with generalizable NeRF methods.   
+
+<table><tr><td>Method</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Pixel-NeRF</td><td>23.17</td><td>0.8693</td></tr><tr><td>PVA</td><td>23.15</td><td>0.8663</td></tr><tr><td>Ours</td><td>24.75</td><td>0.9058</td></tr></table>
+
+a . Generalization results on ZJU-MoCap.
+
+# 4.2 Comparison with generalizable NeRF methods.
+
+Baselines. Among the recent generalizable NeRF methods [36, 52, 47], we compare with PixelNeRF [52] and PVA [36] which focus on very sparse (up to 3 or 4) input views. We reimplement [36] since the code is not available.
+
+Setup. In addition to the ZJU-MoCap dataset (see Sec. 4.1), we experiment on the larger AIST dataset [45, 16] to further evaluate the generalization abilities of different methods. AIST dataset provides dance videos of 30 human subjects captured from 9 cameras, together with the corresponding SMPL fits. We extract the foregroud mask of each image using an off-the-shelf human parser [12]. AIST dataset contains highly diverse motions, slow to fast, simple to complex. We split the dataset into 20 and 10 subjects for training and testing respectively, where the testing dataset contains unseen poses of unseen subjects.
+
+Novel view synthesis results. As shown in Table. 2, across all the datasets and all the metrics, our method consistently outperforms the baselines by healthy margins of $+ 1 . 6$ PSNR and $+ 0 . 0 3 7$ SSIM scores. Fig. 3 and Fig. 5 present the same tendency in visualizations. Pixel-NeRF and PVA aggregate multi-view observations via average pooling without explicitly considering the correlation between the views. In contrast, our temporal and multi-view transformers learn to model the correlation between input views and integrate different observations to help the NeRF module to produce higher quality results. Another advantage of our method is that the used body model prior provides a robust geometric cue to handle the self-occlusion of the human body.
+
+3D reconstruction results. We also evaluate the 3D reconstruction quality of generalizable NeRF methods and our method on the ZJU-MoCap (Fig. 4) and AIST datasets (Fig. 5) given three input views. The visualizations show that our 3D reconstructions align well with the input images, and is more accurate than even the per-subject method [33] (e.g., the shape of the upper cloth in Fig. 4).
+
+Overall, these results verify that since humans are complex and prone to occlusions, more sophisticated designs compared to simple image-conditioning are required to learn robust and accurate appearance representations.
+
+# 4.3 Cross-dataset generalization.
+
+We further study the generalization ability of our method across different datasets by training on one dataset and testing on another one as shown in Table. 4a. We would like to point out that the two datasets [33, 16] have significantly different statistics both in terms of color distribution (background, lighting) and distance of the camera to the subject, making the cross-dataset generalization task extremely challenging. Nevertheless, we found that only 8-minute fine-tuning on the target dataset can already outperform the baselines fully-trained on the target dataset. 16-minute fine-tuning performs on par with our model fully-trained on the target dataset.
+
+![](images/1cd66891c5362aeafa28deea6a66ea9144e0c29a9021293d7b68be10b88cb976.jpg)  
+Figure 4: 3D reconstruction on ZJU-MoCap. Tested on unseen poses of unseen subjects (except Neural Body which is a subject specific method). NB (Neural Body) [33], PVA (Pixel volumetric avatar) [36], Pixel-NeRF [52] and ours.
+
+![](images/f1d05313b8585928b673290f4d9d4351717d6170944eaa7a1a94db8088ca68a4.jpg)  
+Figure 5: Generalization results on AIST. Novel view synthesis and 3D reconstruction results on unseen poses of unseen subjects.
+
+# 4.4 Ablation studies
+
+Table. 3 shows the ablation study on the ZJU-MoCap dataset for unseen subjects and unseen poses, using three time-steps and three camera views as input. Note that all the variations without either temporal or multi-view transformer modules use simple average pooling instead, to fuse temporal or multi-view observations respectively.
+
+Complementariness of skeletal and pixel-aligned query features. ‘Sk’ uses only timeaugmented skeletal features (Sec. 3.2) without time-specific pixel-aligned features, while $\mathbf { \nabla } ^ { \circ } \mathbf { P } \mathbf { X } ^ { \prime }$ uses only the time-specific pixel-aligned features, on the contrary. Both ‘only’ models show the largest drops compared to our full model, and $\mathbf { \hat { S } k } + \mathbf { P x } ^ { \prime }$ model improves them by $+ 1 . 2$ PSNR and $+ 0 . 9$ PSNR respectively. This validates the complementariness between the skeletal and pixel-aligned features in that one is time-augmented but involves slight geometric deviations, while another is time-specific and represents exact query location, as discussed in Sec. 3.3.
+
+Impact of temporal and multi-view transformers. $\mathbf { \nabla } \cdot \mathbf { S } \mathbf { k } + \mathbf { P } \mathbf { X } ^ { \prime }$ uses no transformers so far, falling behind our full model by -1.3 PSNR score. Then $\mathbf { \hat { S } k } + \mathbf { P x } + \mathbf { T } ^ { \prime }$ adds the temporal transformer and improves $+ 0 . 7$ PSNR score, showing its effectiveness in aggregating information over video frames. $\mathbf { \dot { S } k } + \mathbf { P x } + \mathbf { M V } ^ { \prime }$ uses the multi-view transformer module and shows the largest gain of $+ 1 . 0$ PSNR, indicating the efficacy of learnt cross-attention between the skeletal features and pixel-aligned features, as well as the importance of learnt inter-view correlations. Our full model $\mathbf { \ddot { S } k } + \mathbf { P x } + \mathbf { T } +$ MV’ shows the best use of all the proposed components and achieves 24.75 PSNR and 0.9058 SSIM.
+
+Impact of number of camera views. Table. 4b shows the performance of our method when tested with different number of input views, fixing the number of timesteps to one. As expected, the performance degrades slightly with fewer input views (e.g., as few as one).
+
+<table><tr><td>Variant</td><td>Skeletal</td><td>Pixel-aligned</td><td>T-transformer MV-transformer</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Sk</td><td>√</td><td></td><td></td><td>22.31</td><td>0.8865</td></tr><tr><td>Px</td><td></td><td>√</td><td></td><td>22.58</td><td>0.8780</td></tr><tr><td>Sk + Px</td><td>√</td><td>√</td><td></td><td>23.47</td><td>0.8906</td></tr><tr><td>Sk + Px + T</td><td>√</td><td>√</td><td>1</td><td>24.21</td><td>0.9016</td></tr><tr><td>Sk + Px+ MV</td><td>√</td><td>√</td><td>√ √</td><td>24.44</td><td>0.9034</td></tr><tr><td>Sk+Px+T+MV</td><td>√</td><td>√</td><td>√</td><td>24.75</td><td>0.9058</td></tr></table>
+
+Table 3: Ablation study. Results on the ZJU-MoCap dataset for unseen subjects and unseen poses. Sk: skeletal features, $\mathrm { P x }$ : pixel-aligned features, T: temporal transformer, MV: multi-view transformer.
+
+<table><tr><td rowspan=1 colspan=1>Exp.protocol</td><td rowspan=1 colspan=1>Fine-tune</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td></tr><tr><td rowspan=2 colspan=1>Trained onAISTFine-tuned on ZJU-Mocap</td><td rowspan=1 colspan=1>8-min</td><td rowspan=1 colspan=1>24.25</td><td rowspan=1 colspan=1>0.8954</td></tr><tr><td rowspan=1 colspan=1>16-min</td><td rowspan=1 colspan=1>24.73</td><td rowspan=1 colspan=1>0.9023</td></tr><tr><td rowspan=2 colspan=1>Trained on ZJU-MocapFine-tuned on AIST</td><td rowspan=1 colspan=1>8-min</td><td rowspan=1 colspan=1>18.63</td><td rowspan=1 colspan=1>0.8242</td></tr><tr><td rowspan=1 colspan=1>16-min</td><td rowspan=1 colspan=1>18.83</td><td rowspan=1 colspan=1>0.8374</td></tr></table>
+
+a . Cross-dataset generalization.
+
+<table><tr><td># Timesteps</td><td>#Views</td><td>PSNR</td><td>SSIM</td></tr><tr><td rowspan="5">1</td><td>1</td><td>20.13</td><td>0.835</td></tr><tr><td>2</td><td>21.82</td><td>0.871</td></tr><tr><td>3</td><td>23.33</td><td>0.906</td></tr><tr><td>4</td><td>23.51</td><td>0.913</td></tr><tr><td></td><td></td><td></td></tr></table>
+
+b . Impact of number of camera views.   
+Table 4: Cross-dataset generalization (left) and impact of different number of camera views tested on the ZJU-Mocap dataset for unseen poses of unseen subjects (right).
+
+# 5 Limitations
+
+While our novel generalizable human performance rendering method outperforms recent per-subject and generalizable NERF methods, there are remaining challenges yet to be explored. 1) While we show that our method can generalize across datasets with finetuning, the generalization capability will be limited as the distribution of the datasets become significantly different. 2) The performance of our method will be affected as the SMPL parameter accuracy degrades. This is the reason for lower performance on the AIST dataset where the accuracy of the SMPL fits are low due to highly complex motion sequences. It is an interesting direction to jointly refine the SMPL parameters using differentiable rendering within our framework. 3) Our algorithm does not have an explicit assumption of static cameras. However, in practice, it might be hard to estimate the inputs to our method (SMPL fits, camera parameters, foreground masks) with moving cameras due to motion blur, changing background, lighting, and synchronization issues etc. We consider this as an orthogonal problem and expect that any advancements in unconstrained multi-view capture setups will help to generalize our method to in-the-wild settings.
+
+# 6 Societal impact
+
+We discuss the potential societal impact of our work. On a positive note, the human performance synthesis is the key component of realizing telepresence, which has become more important especially in this pandemic era. In the future, people physically apart can feel like they are in the same space and feel connected with a few inexpensive webcams and AR/VR headsets thanks to the development of the telepresence. Like any other technology, our method can also be used for other reasons by corporations, e.g., for identity recognition from a small number of surveillance cameras. We strongly hope that our research will be used in positive directions.
+
+# 7 Conclusion
+
+We present Neural Human Performer, a generalizable radiance field network based on a parametric body model that can synthesize free-viewpoint videos for arbitrary human performers from sparse camera views. Leveraging the trackable visual features from the input body motion prior, we propose a combination of a temporal and a multi-view Transformer that integrates multi-time and multi-view observations in a feed-forward manner. Our method can produce photo-realistic view synthesis of unseen poses and identities at test time. In various generalization settings on the ZJU-MoCap and AIST datasets, our method achieves state-of-the-art performance compared to other per-subject methods that utilize a human body prior as well as the generalizable NeRF methods.
+
+# Acknowledgments and Disclosure of Funding
+
+We thank Sida Peng of Zhejiang University, Hangzhou, China, for many very helpful discussions on a variety of implementation details of the Neural Body [33]. We thank Ruilong li and Alex Yu of UC Berkeley for many discussions on the $\mathrm { A I S T + + }$ dataset [16] and Pixel-NeRF [52] details. We thank Prof. Alex Berg of UNC for the generous offer of computational resources and Misha Shvets of UNC for a useful tutorial on it. This work was partially supported by National Science Foundation Award 1840131.
+
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+        "text": "Neural Human Performer: Learning Generalizable Radiance Fields for Human Performance Rendering ",
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+        "text": "Youngjoong Kwon1, Dahun $\\mathbf { K i m ^ { 2 } }$ , Duygu Ceylan3, Henry Fuchs1 1University of North Carolina at Chapel Hill. 2KAIST. 3Adobe Research. {youngjoong,fuchs}@cs.unc.edu {mcahny}@kaist.ac.kr {ceylan}@adobe.com ",
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+        "text": "Abstract ",
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+        "text": "In this paper, we aim at synthesizing a free-viewpoint video of an arbitrary human performance using sparse multi-view cameras. Recently, several works have addressed this problem by learning person-specific neural radiance fields (NeRF) to capture the appearance of a particular human. In parallel, some work proposed to use pixel-aligned features to generalize radiance fields to arbitrary new scenes and objects. Adopting such generalization approaches to humans, however, is highly challenging due to the heavy occlusions and dynamic articulations of body parts. To tackle this, we propose Neural Human Performer, a novel approach that learns generalizable neural radiance fields based on a parametric human body model for robust performance capture. Specifically, we first introduce a temporal transformer that aggregates tracked visual features based on the skeletal body motion over time. Moreover, a multi-view transformer is proposed to perform cross-attention between the temporally-fused features and the pixel-aligned features at each time step to integrate observations on the fly from multiple views. Experiments on the ZJU-MoCap and AIST datasets show that our method significantly outperforms recent generalizable NeRF methods on unseen identities and poses. The video results and code are available at https://youngjoongunc.github.io/nhp. ",
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+        "text": "1 Introduction ",
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+        "text": "Free-viewpoint video of a human performer has a variety of applications in the area of telepresence, mixed reality, gaming and etc. Conventional free-viewpoint video systems require extremely expensive setups such as dense camera rigs [4, 7, 43] or accurate depth sensors [6, 13], to capture person-specific appearance information. In this paper, we aim at a scalable solution for free-viewpoint human performance rendering that generalizes across different human performers and requires only sparse camera views. However, representing and rendering arbitrary human performances is extremely challenging when the observations are highly sparse (up to three to four views) due to heavy self-occlusions and dynamic articulations of the body parts. In particular, an effective solution needs to coherently aggregate appearance information from sparse multi-view observations across time as the body undergoes a 3D motion. Furthermore, the solution needs to generalize to unseen motions and characters at test time. ",
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+        "text": "Recently, neural radiance fields (NeRF) [26, 11, 17, 30, 33, 35, 36, 47, 50, 52, 53] have shown photo-realistic novel view synthesis results in per-scene optimization settings. To avoid the expensive per-scene training and improve the practicality, generalizable NeRFs [36, 52, 47] have been proposed which use image-conditioned, pixel-aligned features and achieve feed-forward view synthesis from sparse input views [36, 52]. Direct application of these methods to complex and non-rigid human motion is not straightforward, however, and naive solutions suffer from significant artifacts as shown in Fig. 3. Some existing methods [37, 52] aggregate image features across multiple views by simple average pooling, which often leads to over-smoothed outputs when details observed from multiple views (e.g., front and side views) differ due to self occlusions of humans. Alternatively, several methods [23, 33] have proposed to learn person-specific global appearance features from multi-view observations. However, such methods are not able to generalize to new human performers. ",
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+        "text": "To address these challenges, we propose Neural Human Performer, a novel approach that learns generalizable radiance fields based on a parametric 3D body model for robust performance capture. In addition to exploiting a parametric body model as a geometric prior, the core of our method is a combination of temporal and multi-view transformers which help to effectively aggregate spatio-temporal observations to robustly compute the density and color of a query point. First, the temporal transformer aggregates trackable visual features based on the input skeletal body motion across time. The following multi-view transformer performs cross-attention between the temporallyaugmented skeletal features and the pixel-aligned features for each time step. The proposed modules collectively contribute to the adaptive aggregation of multi-time and multi-view information, resulting in significant improvements in synthesis results in different generalization settings of unseen motions and identities. ",
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+        "text": "We study the efficacy of Neural Human Performer on two multi-view human performance capture datasets, ZJU-MoCap [33] and AIST [16]. Experiments show that our method significantly outperforms recent generalizable radience field (NeRF) methods. Furthermore, we compare ours with identity-specific methods [33, 44, 49] that also utilize a 3D human body model prior. Surprisingly, our generalized method achieves better rendering quality than the person-specific dedicated methods when tested on novel poses demonstrating the effectiveness of our transformer-based generalizable representation. ",
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+        "text": "To summarize, our contributions are: ",
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+        "text": "• We present a new feed-forward method of synthesizing novel-view videos of arbitrary human performers from sparse camera views. We propose Neural Human Performer that learns generalizable neural radiance representations by leveraging a 3D body motion prior.   \n• We design a combination of temporal and multi-view transformers that can aggregate information on the fly over video frames across multiple views to render each frame of the novel-view video.   \n• We show significant improvements over recent generalizable NeRF methods on unseen identities and poses. Moreover, our generalization results can outperform even personspecific methods when tested on unseen poses. ",
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+        "type": "text",
+        "text": "2 Related works ",
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+    {
+        "type": "text",
+        "text": "Human performance capture. Novel view synthesis of human performance has a long history in computer vision and graphics. Traditional methods rely on complicated hardware such as dense camera rigs [4, 7, 43] or accurate depth sensors [6, 13]. To enable free-view video from sparse camera views, template-based methods [3, 5, 10, 42] exploit pre-scanned human models to track the motion of a person. However, their synthesis results are not photo-realistic and pre-scanned human models are not available in most cases. Recent methods [27, 37, 38, 54] learn 3D human geometry priors along with pixel aligned features to enable detailed 3D human reconstructions even from single images. However, these methods often suffer under complex human poses that are never seen during training and hence cannot be directly used for our purpose of human performance synthesis. ",
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+    {
+        "type": "text",
+        "text": "Neural 3D representations. Recently, there has been great progress in learning neural networks to represent the shape and appearance of scenes. The 3D representations are learned from 2D images via differentiable rendering networks. Convolutional neural networks are used to predict volumetric representations via 3D voxel-grid features [40, 23, 29, 25, 14, 15], point clouds [1, 49], textured meshes [18, 21, 44] and multi-plane images [8, 55]. The learnt representations are projected by a 3D-to-2D operation to synthesize images. Some methods [14, 15] introduce consistency loss to improve the multi-view and temporal consistency. However, these methods often have difficulty in scaling to higher resolution due to memory restrictions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To eschew these problems, implicit function-based methods [20, 22, 28, 41] learn a multi-layer perceptron that directly translates a 3D positional feature into a pixel generator. The more recent NeRF [26] learns implicit fields of density and color with a volume rendering technique and achieves photo-realistic view synthesis. Among many following NeRF extensions, [30, 31, 35, 53, 17, 50] focus on dynamic scenes. While showing impressive results, it’s an extremely under-constrained problem to jointly learn NeRF and highly dynamic deformation fields. To regularize the training, Neural Body [33] combines NeRF with a deformable human body model (e.g., SMPL [24]). Despite the promising results, these general deformable NeRF [17, 53] and human-specific NeRF [11, 9, 33, 32] methods must be optimized for each new video separately, and generalize poorly on unseen scenarios. Generalizable NeRFs [36, 47, 52] try to avoid the expensive per-scene optimization by image-conditioning using pixel-aligned features. However, directly extending such methods to model complex and dynamic 3D humans is not straightforward when available observations are highly sparse. Unlike existing works, our method exploits temporal and multi-view information on-the-fly and achieves free-viewpoint human rendering in a feed-forward manner, also generalizing to new, unseen human identities and poses. ",
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+        "img_path": "images/d66549154edb235e1464e068e110b4ddedca832bc35c3eceb7e749af7272c231.jpg",
+        "image_caption": [
+            "Figure 1: Given a sparse set of multi-view videos of a person, Neural Human Performer uses temporal and multi-view transformers to learn pixel aligned features on the surface of a deformable body template such as SMPL. Such features are utilized to predict the color and density value of query points to render novel views of possibly unseen poses of the person. "
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+        "text": "3 Method ",
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+        "text": "3.1 Neural Human Performer ",
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+        "text": "Problem definition. In our setting, given a sparse set (e.g., 3 or 4) of multi-view cameras $c = 1 , \\ldots , C$ , input videos of an arbitrary human performance $I _ { c , 1 : T } : = \\{ I _ { c , 1 } , I _ { c , 2 } , . . . , I _ { c , T } \\}$ are captured for each camera view $c$ defined by $\\{ \\mathbf { K } _ { c } , [ \\mathbf { R } | \\mathbf { t } ] _ { c } \\}$ . We assume that the 3D human body model fit corresponding to each frame is given. Our goal is to synthesize a novel view video $\\hat { I } _ { q , 1 : T }$ for a query viewpoint $q$ defined by $\\{ \\mathbf { K } _ { q } , \\mathbf { \\bar { [ R | t ] } } _ { q } \\}$ . ",
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+        "type": "text",
+        "text": "Overview. To compensate for the sparsity of available input views, we propose to exploit temporal information across video frames. In practice, we sample $M$ memory frames from the original input videos to augment each queried timestep $t$ . Our goal is to learn generalizable 3D representations of human performers from multi-time $( M )$ and multi-view $( C )$ observations. ",
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+    {
+        "type": "text",
+        "text": "To this end, the Neural Human Performer is proposed with two main components as illustrated in Fig. 1. First, we construct the time-augmented skeletal features $\\{ s \\}$ . We exploit a human body model (SMPL [24]) to construct 3D skeletal features by projecting all the SMPL vertices onto each memory frame and picking up the pixel-aligned image features [36, 37, 52] at the projected 2D locations. The skeletal features are sampled from all memory frames to construct the skeletal feature bank. Inspired by Transformers[2, 46, 48], we propose a temporal Transformer that aggregates these memory features into the time-augmented skeletal features $\\{ s ^ { \\prime } \\}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the second stage, given a query 3D point $\\mathbf { x }$ , skeletal features are sampled at $\\mathbf { x }$ . In addition, pixel-aligned features $\\{ p \\}$ at each time $t$ are sampled by directly projecting $\\mathbf { x }$ onto the input images $\\{ I \\} _ { t }$ . The multi-view Transformer is proposed to learn the correlation between the pixel-aligned features $\\{ p \\}$ and the time-augmented skeletal features $\\{ s ^ { \\prime } \\}$ , and to adaptively fuse the multi-view information. The final fused feature of the query point $\\mathbf { x }$ is fed into the radiance field module to predict its color and density. ",
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+        "type": "text",
+        "text": "3.2 Construction of time-augmented skeletal features ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Unlike static scenes, video inputs of moving characters inherently contain more visual cues as the occluded regions in a frame may be visible in other (potentially distant) frames. To take advantage of this temporal information, we first build up the skeletal feature bank from memory frames by leveraging a parametric body model (see Fig. 1). Then, we propose a temporal Transformer module that aggregates such collected features. ",
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+        "text": "For each view $c$ and time $t$ , we first build frame-level skeletal features $s _ { c , t } \\in \\mathbb { R } ^ { L \\times d }$ by sampling image features at the projection of the SMPL vertices in each image $I _ { c , t } \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ . $L$ denotes the number of SMPL vertices and $d$ is the dimension of image features. ",
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+    },
+    {
+        "type": "text",
+        "text": "After collecting all the skeletal features from all memory frames, we aggregate the time information in an attention-aware manner, instead of using simple average pooling. Specifically, for any $i ^ { t h }$ skeletal feature vertex $s _ { c , t } ^ { i } \\in \\mathbb { R } ^ { d }$ , the proposed temporal Transformer casts attention over all other features contained in the vertex’s memory bank we compute soft weights for all memory $\\boldsymbol { s } _ { c , t _ { 1 } : t _ { M } } ^ { i } = \\{ \\boldsymbol { s } _ { c , t _ { 1 } } ^ { i } , \\boldsymbol { s } _ { c , t _ { 2 } } ^ { i } , . . . , \\boldsymbol { s } _ { c , t _ { M } } ^ { i } \\} \\in \\mathbb { R } ^ { M \\times d }$ . In particular,respect to the current timestep $t$ . Then, the values of the memory features are weighted summed as ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/fdf6a14df6a056b6ba54ffea93259b4aa51c1498676b711bdeb606ff3fa61b10.jpg",
+        "text": "$$\n\\begin{array} { l l } { { t _ { \\perp } a t t _ { c , t } ^ { i } = \\psi \\big ( \\frac { 1 } { \\sqrt { d _ { 0 } } } q ( s _ { c , t } ^ { i } ) \\cdot k ( s _ { c , t _ { 1 } : t _ { M } } ^ { i } ) ^ { T } \\big ) , } } & { { t _ { - } a t t _ { c , t } ^ { i } \\in \\mathbb { R } ^ { 1 \\times M } } } \\\\ { { s _ { c , t } ^ { \\prime } { } ^ { i } = t _ { - } a t t _ { c , t } ^ { i } \\cdot v \\big ( s _ { c , t _ { 1 } : t _ { M } } ^ { i } \\big ) + s _ { c , t } ^ { i } , } } & { { s _ { c , t } ^ { \\prime } \\in \\mathbb { R } ^ { L \\times d } \\forall i } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            378,
+            702,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\psi$ represents the softmax operator along the second axis, $q ( \\cdot ) , k ( \\cdot )$ and $v ( \\cdot )$ are learnable query, key and value embedding functions $\\mathbb { R } ^ { d  d _ { 0 } }$ of the temporal Transformer. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In other words, the representation $s _ { c , t } ^ { i }$ of each skeletal vertex at time $t$ is computed through a dynamically weighted combination of all its previous and next representations in the memory frames. This allows our network to incorporate helpful information and ignore irrelevant ones from other timesteps. In practice, the temporal Transformer operation in Eq. (1) is performed by a batch matrix multiplication for all skeletal vertices $L$ and all available viewpoints $C$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.3 Multi-view aggregation of skeletal and query features ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given a query 3D point $\\mathbf { x } \\in \\mathbb { R } ^ { 3 }$ , we retrieve the corresponding (time-augmented) skeletal feature $s _ { c , t } ^ { \\prime } \\in \\mathring { \\mathbb { R } } ^ { d }$ at the queried location via trilinear interpolation in the SMPL space with SparseConvNet [19], following [39, 34, 51, 33]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition, we sample pixel-aligned image feature $p _ { c , t } ^ { \\mathbf { x } }$ via direct $\\mathbb { R } ^ { 3 \\to 2 }$ projection of the query point $\\mathbf { x }$ on $I _ { c , t }$ . It is important to note that the pixel-aligned feature $p _ { c , t } ^ { \\mathbf { x } }$ is time-specific and represents the exact query location of $\\mathbf { x }$ , while the skeletal feature $s _ { c , t } ^ { \\prime } { } ^ { \\mathbf { x } }$ is time-augmented (w.r.t $t$ ) and contains inherent geometric deviations in the SMPL vertices and the following trilinear interpolations. We propose to combine these two complementary features, which will be shown to be effective in Sec. 4.4. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given the two sets of multi-view features, skeletal s′1:C,tx $s _ { 1 : C , t } ^ { \\prime } \\mathbf { \\Psi } ^ { \\mathbf { x } } = \\{ s _ { c , t } ^ { \\prime } \\mathbf { \\Psi } ^ { \\mathbf { x } } | c = 1 , . . . , C \\} \\in \\mathbb { R } ^ { C \\times d }$ and pixel-aligned $p _ { 1 : C , t } ^ { \\bf x } = \\{ p _ { c , t } ^ { \\bf x } | c = 1 , . . . , C \\} \\in \\mathbb { R } ^ { C \\times d }$ , we propose a multi-view Transformer that performs cross-attention from skeletal to pixel-aligned features. Specifically, the values of pixelaligned features from all viewpoints is re-weighted based on how much compatible they are with each skeletal features. The non-local cross-attention mv_att is constructed as: ",
+        "bbox": [
+            176,
+            710,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e820ddeb0af04143a7ff7bd08ebabdb85dc2177bee711328f48719b5f478645.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { m v _ { - } a t t _ { t } ^ { \\mathbf { x } } = \\psi \\bigl ( \\frac { 1 } { \\sqrt { d _ { 1 } } } k \\bigl ( s _ { 1 : C , t } ^ { \\prime } \\bigr ) \\cdot k \\bigl ( p _ { 1 : C , t } ^ { \\mathbf { x } } \\bigr ) ^ { T } \\bigr ) , \\quad m v _ { - } a t t _ { t } ^ { \\mathbf { x } } \\in \\mathbb { R } ^ { C \\times C } } & \\\\ & { z _ { 1 : C , t } ^ { \\mathbf { x } } = m v _ { - } a t t _ { t } ^ { \\mathbf { x } } \\cdot v \\bigl ( p _ { 1 : C , t } ^ { \\mathbf { x } } \\bigr ) + v \\bigl ( s _ { 1 : C , t } ^ { \\prime } \\bigr ) , \\qquad z _ { 1 : C , t } ^ { \\mathbf { x } } \\in \\mathbb { R } ^ { C \\times d } , \\quad z _ { c , t } ^ { \\mathbf { x } } \\in \\mathbb { R } ^ { 1 \\times d } } & \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            789,
+            767,
+            847
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\psi$ represents the softmax operator along the second axis. Note that $k$ and $v$ are new layers independent from those in the temporal Transformer. The confident observations in each view will have large weights and be highlighted, and vice versa. Finally, we use the view-wise mean of $\\begin{array} { r } { z _ { t } ^ { \\mathbf { x } } = \\frac { 1 } { C } \\sum _ { c } z _ { c , t } ^ { \\mathbf { x } } \\in \\mathbb { R } ^ { d } } \\end{array}$ as our meta-time and meta-view representation of the query point $x$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The final density $\\sigma _ { t } ( \\mathbf { x } )$ and color values $r g b _ { t } ( { \\bf x } )$ at time $t$ are computed as: ",
+        "bbox": [
+            173,
+            90,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/810bdde8e40066e657de495ae5e3994f8a277e6fc9c592fea6aa952dcf6da4a4.jpg",
+        "text": "$$\n\\sigma _ { t } ( \\mathbf { x } ) = M L P _ { \\sigma } ( z _ { t } ^ { \\mathbf { x } } ) , \\qquad r g b _ { t } ( \\mathbf { X } ) = M L P _ { \\mathbf { r g b } } ( \\sum _ { c } { ( z _ { c , t } ^ { \\mathbf { x } } ; \\gamma _ { \\mathbf { d } } ( \\mathbf { d } ) ) } / C ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            112,
+            732,
+            145
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $M L P _ { \\sigma }$ and $M L P _ { \\mathbf { r g b } }$ consist of four and two linear layers respectively, and $\\gamma _ { \\mathbf { d } } : \\mathbb { R } ^ { 3  6 \\times l }$ is a positional encoding of viewing direction $\\mathbf { d } \\in \\mathbb { R } ^ { 3 }$ as in [26] with $2 \\times l$ different basis functions. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "More details on the network architecture can be found in the supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 Volume Rendering ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The predicted color of a pixel $p \\in \\mathbb { R } ^ { 2 }$ for a target viewpoint $q$ in the focal plane of the camera and center $\\mathbf { r } _ { 0 } \\in \\mathbb { R } ^ { 3 }$ is obtained by marching rays into the scene using the camera-to-world projection matrix, P−1 = [Rq|tq]−1K−1q with the direction of the rays given by d = P−1p−r0∥P−1p−r0∥ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We then accumulate the radiance and opacity along the ray $\\mathbf { r } ( z ) = \\mathbf { r } _ { 0 } + z \\mathbf { d }$ for $z \\in [ z _ { \\mathrm { n e a r } } , z _ { \\mathrm { f a r } } ]$ as defined in NeRF [26] as follows: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea9181e5a0f4e07721d780469cf52ca73e804af6cfd37d213f0b4a2cbbb2c1db.jpg",
+        "text": "$$\n\\mathbf { I } _ { q } ( p ) = \\int _ { z _ { \\mathrm { n e a r } } } ^ { z _ { \\mathrm { f a r } } } \\mathbf { T } ( z ) \\sigma ( \\mathbf { r } ( z ) ) \\mathbf { c } ( \\mathbf { r } ( z ) , \\mathbf { d } ) d z , \\quad w h e r e \\quad \\mathbf { T } ( z ) = \\exp \\left( - \\int _ { z _ { \\mathrm { n e a r } } } ^ { z } \\sigma ( \\mathbf { r } ( s ) ) d s \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In practice, we uniformly sample a set of 64 points $z \\sim [ z _ { n e a r } , z _ { f a r } ]$ . We set $\\mathbf { X } = \\mathbf { r } ( z )$ and use the quadrature rule to approximate the integral. We compute the 3D bounding box of the SMPL parameters at time $t$ and derive the bounds for ray sampling $z _ { n e a r } , z _ { f a r }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.5 Loss Function ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given the ground truth target images $\\mathbf { I } _ { q , t }$ , we train both the radiance field and feature extraction networks using a simple photometric reconstruction loss $\\mathcal { L } = \\Vert \\hat { \\mathbf { I } } _ { q , t } - \\mathbf { I } _ { q , t } \\Vert _ { 2 }$ . ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We present novel view synthesis and 3d reconstruction results of human performances in different generalization scenarios. We compare our method against the current best view-synthesis methods from two classes: per-subject optimization methods that also use a human body model prior (Sec. 4.1) and generalizable NeRF methods (Sec. 4.2). We experiment on the ZJU-MoCap [33] and AIST datasets [45, 16]. For training and testing our model as well as the baselines, we remove the background using the foreground masks that are either provided by the dataset or pre-computed using an off-the-shelf method. Unless otherwise specified, we sample two memory frames $\\{ t - \\mathsf { 2 0 } , t + 2 0 \\bar  \\}$ at time $t$ (i.e., three timesteps in total) and take three canonical input views in all experiments. The details of the implementation, datasets, training process are provided in the supplementary material. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Comparison with body model-based, per-subject optimization methods. ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Baselines. We perform comparisons with the state-of-the-art Neural Body (NB) [33] that combines the body model prior SMPL and NeRF in a per-subject optimization setting. Neural Textures (NT) [44] renders a coarse mesh with latent texture maps and uses a 2D CNN to render target images. We use the SMPL mesh as the coarse mesh input to NT in our experiments. NHR [49] extracts 3D features from input point clouds and renders them into 2D images. Since dense point clouds are difficult to obtain from sparse camera views, we use the SMPL vertices as input point clouds. These methods have reported that they can adapt to new poses of the same performer, hence we compare to them for the task of novel pose synthesis. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Setup. We experiment with ZJU-MoCap dataset [33] which provides performance captures of 10 human subjects captured from 23 synchronized cameras, human body model parameters as well as the foreground mask corresponding to each frame. Each video contains complex motions such as kicking and Taichi and is between 1000 to 2000 frames long. We first split the dataset into two parts: source and target videos. In all comparisons, the first 300 frames of either source or target videos are used during training, and the remaining next frames (unseen poses) are held out for testing. Note that the baseline methods are always trained in a per-subject manner. To validate whether the training is reproducible, we experiment with 5 independent runs with random train/test splits and observe a variance of 0.15 PSNR, showing that the results are consistent. In each independent run, we used seven models for training and the other three for testing. ",
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+    {
+        "type": "table",
+        "img_path": "images/abfc2d50077ff08cbee1cc2f80333c8ffff1da4a15408f94efd57fdf00428597.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "b . Test results on unseen poses of target subjects "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>PSNR</td><td>SSIM</td></tr><tr><td colspan=\"2\">Trained on target subjects</td><td></td></tr><tr><td>NB</td><td>22.88</td><td>0.880</td></tr><tr><td>NHR</td><td>22.03</td><td>0.875</td></tr><tr><td>NT</td><td>21.92</td><td>0.873</td></tr><tr><td>Trained on</td><td>source subjects</td><td></td></tr><tr><td>Ours</td><td>24.75</td><td>0.906</td></tr></table>",
+        "bbox": [
+            388,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/fab390d68cb779aba2fe2a5ec5e41a3c2e42c86667fe5e05c7d8643288a70fdf.jpg",
+        "table_caption": [
+            "Table 1: Comparison with other body model-based, per-subject optimization methods. "
+        ],
+        "table_footnote": [
+            "a . Test results on unseen poses of source subjects "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>PSNR</td><td>SSIM</td></tr><tr><td colspan=\"2\">Trained on source subjects</td><td></td></tr><tr><td>NB NHR</td><td>23.79 22.31</td><td>0.887 0.871</td></tr><tr><td>NT</td><td>22.28</td><td>0.872</td></tr><tr><td colspan=\"3\">Trained on source subjects</td></tr><tr><td>Ours</td><td>26.94</td><td>0.929</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/e0d61156068d50ed1cb77fdf41725ecd043d635922061e3579ecfcfd65d59d27.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "c . Test results on seen poses of source subjects "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">Method PSNR</td><td>SSIM</td></tr><tr><td colspan=\"3\">Trained on source subjects</td></tr><tr><td>NB</td><td>28.51</td><td>0.947</td></tr><tr><td>NHR</td><td>23.95</td><td>0.897</td></tr><tr><td>NT</td><td>23.86</td><td>0.896</td></tr><tr><td colspan=\"3\">Trained on source subjects</td></tr><tr><td>Ours</td><td>28.73</td><td>0.936</td></tr><tr><td>Ours per-subject</td><td>31.57</td><td>0.966</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/b0ead6611373a5fa611bac7276222a6f2cfa8e4096da2b85a72ed656de985623.jpg",
+        "image_caption": [
+            "Figure 2: Pose generalization – comparison with per-scene optimization methods that use a human body prior. Results of NT(Neural textures) [44], NHR (Neural human rendering) [49], NB (Neural body) [33] and ours. Novel view synthesis on the ZJU-MoCap for unseen poses of source subjects. While ours is a single model trained on all source subjects; baselines are trained in a per-subject manner. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results. We present three different comparison settings to validate our method. We would like to point out that all the comparison settings place our method (‘ours’) in disadvantage. This is because our model is trained on all the source subjects at once (one network for all subjects), while the competing methods are per-subject trained on the subject to be tested (one network for one subject) - easier setting. First, we evaluate the pose generalization capability (Table. 1a and Fig. 2) where we train on the training splits of the source subjects and test on the testing splits, i.e., unseen poses, of the same source subjects. Our method significantly outperforms all the baselines and the state-of-the-art Neural Body [33] by ${ \\bf + 3 . 1 5 }$ PSNR and $+ 0 . 0 4 2$ SSIM scores. We next evaluate the identity generalization capability (Table. 1b) by testing on the unseen poses of target subjects. While our method is trained on source subjects only, baselines are trained on target subjects. Note that this comparison is disadvantageous to us since unlike our method the baselines have seen the testing subjects as they must be trained separately for each subject. Surprisingly, however, our method still outperforms all the baselines by a healthy margin of $+ 1 . 8 7$ PSNR and $+ 0 . 0 2 6$ SSIM scores. These results indicate that our proposed architecture with the temporal and the multi-view transformers generalizes well across unseen identities and poses, and produces photo-realistic results. Finally, in Table. 1c, we evaluate our method and baselines on seen poses of seen subjects. Even when we train a single model of our method for all the source sobjects (‘ours’), ours achieves comparable results with the state-of-the-art per-subject based methods [33]. To make a more fair comparison, we also train our method in the same per-subject setting similar to the baselines. Table. 1c shows that in this case, our model (‘ours per-subject’) achieves a significant improvement over the best-performing baseline [33] by $+ 3$ PSNR and $+ 1 . 4 \\%$ SSIM. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/c7a66ae91e16110cda6e3e003e86bae057370c4a950b769cf5a12046d3bf74dc.jpg",
+        "image_caption": [
+            "Figure 3: Identity-and-pose generalization – comparison with generalizable NeRF methods. Results of Pixel-Nerf [52], PVA (Pixel-aligned volumetric avatar) [36] and ours. Novel view synthesis on the ZJU-MoCap dataset where all methods are trained on source subjects and tested on unseen poses of target subjects. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            88,
+            825,
+            280
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/541d0cb284f767734060c617a8159fa3ad1ce400373431e5093579de25fc6ba6.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "b . Generalization results on AIST. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Pixel-NeRF</td><td>18.06</td><td>0.7304</td></tr><tr><td>PVA</td><td>17.82</td><td>0.7211</td></tr><tr><td>Ours</td><td>19.03</td><td>0.8390</td></tr></table>",
+        "bbox": [
+            521,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/fca13be3a2d59a76e834c82c219aa612c01fc6151691fc40057d1b1280e9fee5.jpg",
+        "table_caption": [
+            "Table 2: Comparison with generalizable NeRF methods. "
+        ],
+        "table_footnote": [
+            "a . Generalization results on ZJU-MoCap. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Pixel-NeRF</td><td>23.17</td><td>0.8693</td></tr><tr><td>PVA</td><td>23.15</td><td>0.8663</td></tr><tr><td>Ours</td><td>24.75</td><td>0.9058</td></tr></table>",
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+        "type": "text",
+        "text": "4.2 Comparison with generalizable NeRF methods. ",
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+        "type": "text",
+        "text": "Baselines. Among the recent generalizable NeRF methods [36, 52, 47], we compare with PixelNeRF [52] and PVA [36] which focus on very sparse (up to 3 or 4) input views. We reimplement [36] since the code is not available. ",
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+        "type": "text",
+        "text": "Setup. In addition to the ZJU-MoCap dataset (see Sec. 4.1), we experiment on the larger AIST dataset [45, 16] to further evaluate the generalization abilities of different methods. AIST dataset provides dance videos of 30 human subjects captured from 9 cameras, together with the corresponding SMPL fits. We extract the foregroud mask of each image using an off-the-shelf human parser [12]. AIST dataset contains highly diverse motions, slow to fast, simple to complex. We split the dataset into 20 and 10 subjects for training and testing respectively, where the testing dataset contains unseen poses of unseen subjects. ",
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+        "type": "text",
+        "text": "Novel view synthesis results. As shown in Table. 2, across all the datasets and all the metrics, our method consistently outperforms the baselines by healthy margins of $+ 1 . 6$ PSNR and $+ 0 . 0 3 7$ SSIM scores. Fig. 3 and Fig. 5 present the same tendency in visualizations. Pixel-NeRF and PVA aggregate multi-view observations via average pooling without explicitly considering the correlation between the views. In contrast, our temporal and multi-view transformers learn to model the correlation between input views and integrate different observations to help the NeRF module to produce higher quality results. Another advantage of our method is that the used body model prior provides a robust geometric cue to handle the self-occlusion of the human body. ",
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+        "type": "text",
+        "text": "3D reconstruction results. We also evaluate the 3D reconstruction quality of generalizable NeRF methods and our method on the ZJU-MoCap (Fig. 4) and AIST datasets (Fig. 5) given three input views. The visualizations show that our 3D reconstructions align well with the input images, and is more accurate than even the per-subject method [33] (e.g., the shape of the upper cloth in Fig. 4). ",
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+        "type": "text",
+        "text": "Overall, these results verify that since humans are complex and prone to occlusions, more sophisticated designs compared to simple image-conditioning are required to learn robust and accurate appearance representations. ",
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+        "text": "4.3 Cross-dataset generalization. ",
+        "text_level": 1,
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+        "text": "We further study the generalization ability of our method across different datasets by training on one dataset and testing on another one as shown in Table. 4a. We would like to point out that the two datasets [33, 16] have significantly different statistics both in terms of color distribution (background, lighting) and distance of the camera to the subject, making the cross-dataset generalization task extremely challenging. Nevertheless, we found that only 8-minute fine-tuning on the target dataset can already outperform the baselines fully-trained on the target dataset. 16-minute fine-tuning performs on par with our model fully-trained on the target dataset. ",
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+        "image_caption": [
+            "Figure 4: 3D reconstruction on ZJU-MoCap. Tested on unseen poses of unseen subjects (except Neural Body which is a subject specific method). NB (Neural Body) [33], PVA (Pixel volumetric avatar) [36], Pixel-NeRF [52] and ours. "
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+        "image_caption": [
+            "Figure 5: Generalization results on AIST. Novel view synthesis and 3D reconstruction results on unseen poses of unseen subjects. "
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+        "text": "4.4 Ablation studies ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Table. 3 shows the ablation study on the ZJU-MoCap dataset for unseen subjects and unseen poses, using three time-steps and three camera views as input. Note that all the variations without either temporal or multi-view transformer modules use simple average pooling instead, to fuse temporal or multi-view observations respectively. ",
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+        "type": "text",
+        "text": "Complementariness of skeletal and pixel-aligned query features. ‘Sk’ uses only timeaugmented skeletal features (Sec. 3.2) without time-specific pixel-aligned features, while $\\mathbf { \\nabla } ^ { \\circ } \\mathbf { P } \\mathbf { X } ^ { \\prime }$ uses only the time-specific pixel-aligned features, on the contrary. Both ‘only’ models show the largest drops compared to our full model, and $\\mathbf { \\hat { S } k } + \\mathbf { P x } ^ { \\prime }$ model improves them by $+ 1 . 2$ PSNR and $+ 0 . 9$ PSNR respectively. This validates the complementariness between the skeletal and pixel-aligned features in that one is time-augmented but involves slight geometric deviations, while another is time-specific and represents exact query location, as discussed in Sec. 3.3. ",
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+    {
+        "type": "text",
+        "text": "Impact of temporal and multi-view transformers. $\\mathbf { \\nabla } \\cdot \\mathbf { S } \\mathbf { k } + \\mathbf { P } \\mathbf { X } ^ { \\prime }$ uses no transformers so far, falling behind our full model by -1.3 PSNR score. Then $\\mathbf { \\hat { S } k } + \\mathbf { P x } + \\mathbf { T } ^ { \\prime }$ adds the temporal transformer and improves $+ 0 . 7$ PSNR score, showing its effectiveness in aggregating information over video frames. $\\mathbf { \\dot { S } k } + \\mathbf { P x } + \\mathbf { M V } ^ { \\prime }$ uses the multi-view transformer module and shows the largest gain of $+ 1 . 0$ PSNR, indicating the efficacy of learnt cross-attention between the skeletal features and pixel-aligned features, as well as the importance of learnt inter-view correlations. Our full model $\\mathbf { \\ddot { S } k } + \\mathbf { P x } + \\mathbf { T } +$ MV’ shows the best use of all the proposed components and achieves 24.75 PSNR and 0.9058 SSIM. ",
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+    {
+        "type": "text",
+        "text": "Impact of number of camera views. Table. 4b shows the performance of our method when tested with different number of input views, fixing the number of timesteps to one. As expected, the performance degrades slightly with fewer input views (e.g., as few as one). ",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Ablation study. Results on the ZJU-MoCap dataset for unseen subjects and unseen poses. Sk: skeletal features, $\\mathrm { P x }$ : pixel-aligned features, T: temporal transformer, MV: multi-view transformer. "
+        ],
+        "table_body": "<table><tr><td>Variant</td><td>Skeletal</td><td>Pixel-aligned</td><td>T-transformer MV-transformer</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Sk</td><td>√</td><td></td><td></td><td>22.31</td><td>0.8865</td></tr><tr><td>Px</td><td></td><td>√</td><td></td><td>22.58</td><td>0.8780</td></tr><tr><td>Sk + Px</td><td>√</td><td>√</td><td></td><td>23.47</td><td>0.8906</td></tr><tr><td>Sk + Px + T</td><td>√</td><td>√</td><td>1</td><td>24.21</td><td>0.9016</td></tr><tr><td>Sk + Px+ MV</td><td>√</td><td>√</td><td>√ √</td><td>24.44</td><td>0.9034</td></tr><tr><td>Sk+Px+T+MV</td><td>√</td><td>√</td><td>√</td><td>24.75</td><td>0.9058</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "a . Cross-dataset generalization. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Exp.protocol</td><td rowspan=1 colspan=1>Fine-tune</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td></tr><tr><td rowspan=2 colspan=1>Trained onAISTFine-tuned on ZJU-Mocap</td><td rowspan=1 colspan=1>8-min</td><td rowspan=1 colspan=1>24.25</td><td rowspan=1 colspan=1>0.8954</td></tr><tr><td rowspan=1 colspan=1>16-min</td><td rowspan=1 colspan=1>24.73</td><td rowspan=1 colspan=1>0.9023</td></tr><tr><td rowspan=2 colspan=1>Trained on ZJU-MocapFine-tuned on AIST</td><td rowspan=1 colspan=1>8-min</td><td rowspan=1 colspan=1>18.63</td><td rowspan=1 colspan=1>0.8242</td></tr><tr><td rowspan=1 colspan=1>16-min</td><td rowspan=1 colspan=1>18.83</td><td rowspan=1 colspan=1>0.8374</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "b . Impact of number of camera views. ",
+            "Table 4: Cross-dataset generalization (left) and impact of different number of camera views tested on the ZJU-Mocap dataset for unseen poses of unseen subjects (right). "
+        ],
+        "table_body": "<table><tr><td># Timesteps</td><td>#Views</td><td>PSNR</td><td>SSIM</td></tr><tr><td rowspan=\"5\">1</td><td>1</td><td>20.13</td><td>0.835</td></tr><tr><td>2</td><td>21.82</td><td>0.871</td></tr><tr><td>3</td><td>23.33</td><td>0.906</td></tr><tr><td>4</td><td>23.51</td><td>0.913</td></tr><tr><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 Limitations ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While our novel generalizable human performance rendering method outperforms recent per-subject and generalizable NERF methods, there are remaining challenges yet to be explored. 1) While we show that our method can generalize across datasets with finetuning, the generalization capability will be limited as the distribution of the datasets become significantly different. 2) The performance of our method will be affected as the SMPL parameter accuracy degrades. This is the reason for lower performance on the AIST dataset where the accuracy of the SMPL fits are low due to highly complex motion sequences. It is an interesting direction to jointly refine the SMPL parameters using differentiable rendering within our framework. 3) Our algorithm does not have an explicit assumption of static cameras. However, in practice, it might be hard to estimate the inputs to our method (SMPL fits, camera parameters, foreground masks) with moving cameras due to motion blur, changing background, lighting, and synchronization issues etc. We consider this as an orthogonal problem and expect that any advancements in unconstrained multi-view capture setups will help to generalize our method to in-the-wild settings. ",
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+    {
+        "type": "text",
+        "text": "6 Societal impact ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We discuss the potential societal impact of our work. On a positive note, the human performance synthesis is the key component of realizing telepresence, which has become more important especially in this pandemic era. In the future, people physically apart can feel like they are in the same space and feel connected with a few inexpensive webcams and AR/VR headsets thanks to the development of the telepresence. Like any other technology, our method can also be used for other reasons by corporations, e.g., for identity recognition from a small number of surveillance cameras. We strongly hope that our research will be used in positive directions. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We present Neural Human Performer, a generalizable radiance field network based on a parametric body model that can synthesize free-viewpoint videos for arbitrary human performers from sparse camera views. Leveraging the trackable visual features from the input body motion prior, we propose a combination of a temporal and a multi-view Transformer that integrates multi-time and multi-view observations in a feed-forward manner. Our method can produce photo-realistic view synthesis of unseen poses and identities at test time. In various generalization settings on the ZJU-MoCap and AIST datasets, our method achieves state-of-the-art performance compared to other per-subject methods that utilize a human body prior as well as the generalizable NeRF methods. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We thank Sida Peng of Zhejiang University, Hangzhou, China, for many very helpful discussions on a variety of implementation details of the Neural Body [33]. We thank Ruilong li and Alex Yu of UC Berkeley for many discussions on the $\\mathrm { A I S T + + }$ dataset [16] and Pixel-NeRF [52] details. We thank Prof. Alex Berg of UNC for the generous offer of computational resources and Misha Shvets of UNC for a useful tutorial on it. This work was partially supported by National Science Foundation Award 1840131. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4578–4587, 2021.   \n[53] Wentao Yuan, Zhaoyang Lv, Tanner Schmidt, and Steven Lovegrove. Star: Self-supervised tracking and reconstruction of rigid objects in motion with neural rendering. arXiv preprint arXiv:2101.01602, 2020.   \n[54] Zerong Zheng, Tao Yu, Yixuan Wei, Qionghai Dai, and Yebin Liu. Deephuman: 3d human reconstruction from a single image. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 7739–7749, 2019.   \n[55] Tinghui Zhou, Richard Tucker, John Flynn, Graham Fyffe, and Noah Snavely. Stereo magnification: Learning view synthesis using multiplane images. arXiv preprint arXiv:1805.09817, 2018. ",
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parse/train/hwoK62_GkiT/hwoK62_GkiT.md

@@ -0,0 +1,238 @@
+# Multi-Agent Reinforcement Learning for Active Voltage Control on Power Distribution Networks
+
+Jianhong Wang∗ Imperial College London jianhong.wang16@imperial.ac.uk
+
+Wangkun Xu∗ Imperial College London wangkun.xu18@imperial.ac.uk
+
+Yunjie $\mathbf { G u } ^ { \dagger }$ University of Bath yg934@bath.ac.uk
+
+Wenbin Song Shanghaitech University songwb@shanghaitech.edu.cn
+
+Tim C. Green‡ Imperial College London t.green@imperial.ac.uk
+
+# Abstract
+
+This paper presents a problem in power networks that creates an exciting and yet challenging real-world scenario for application of multi-agent reinforcement learning (MARL). The emerging trend of decarbonisation is placing excessive stress on power distribution networks. Active voltage control is seen as a promising solution to relieve power congestion and improve voltage quality without extra hardware investment, taking advantage of the controllable apparatuses in the network, such as roof-top photovoltaics (PVs) and static var compensators (SVCs). These controllable apparatuses appear in a vast number and are distributed in a wide geographic area, making MARL a natural candidate. This paper formulates the active voltage control problem in the framework of Dec-POMDP and establishes an open-source environment. It aims to bridge the gap between the power community and the MARL community and be a drive force towards real-world applications of MARL algorithms. Finally, we analyse the special characteristics of the active voltage control problems that cause challenges (e.g. interpretability) for state-of-the-art MARL approaches, and summarise the potential directions.
+
+# 1 Introduction
+
+Multi-agent reinforcement learning (MARL) has demonstrated impressive performances on games (e.g., Chess [1], StarCraft [2, 3], and etc.) and robotics [4]. There are now increasing interests and thrusts to apply MARL in real-world problems. Power network is a natural test field for MARL algorithms. There are many problems in power networks involving the collaborative or competitive actions of a vast number of agents, such as market bidding [5], voltage control, frequency control, and emergency handling [6]. One of the obstacles for applying MARL in power networks is the lack of transparency and guarantee, which is unacceptable considering the importance of reliable power supply for the whole society. Therefore, it is sensible to start with problems that are less sensitive to reliability but hard to be solved by conventional methods.
+
+Active voltage control in power distribution networks is such a candidate problem. The voltage control problem has been studied for years but only comes under spot light recently due to the increasing penetration of distributed resources, e.g. roof-top photovoltaics (PVs) [7]. The excessive active power injection may cause voltage fluctuations beyond the threshold of grid standards [8]. The voltage fluctuations can be relieved by exploiting the control flexibility of PV inverters themselves along with other controllable apparatuses, such as static var compensators (SVCs) and on load tap changers (OLTCs). An elaborate scheme is needed to coordinate these multiple apparatuses at scale to regulate voltage throughout the network with limited local information, which is called active voltage control [9–12]. The active voltage control problem has many interesting properties. (1) It is a combination of local and global problem, i.e., the voltage of each node is influenced by the powers (real and reactive) of all other nodes but the impact recedes with increasing distance between nodes. (2) It is a constrained optimisation problem where the constraint is the voltage threshold and the objective is the total power loss, but there is no explicit relationship between the constraint and the control action. (3) A distribution network has a radial topology involving a rich structure that can be taken as prior knowledge for control, but the node-branch parameters of the topology may be significantly uncertain. (4) Voltage control has a relatively large tolerance and less severe consequences if the control fails to meet standard requirements.
+
+There have been several attempts to apply MARL in active voltage control [6, 13–16]. Each work on a particular case showed promising performances of state-of-the-art MARL approaches with modifications adapting to the active voltage control problem. However, it is not clear if these methods scale well to a larger network, and the robustness against different scenarios, such as penetration levels and load profiles, are yet to be investigated. There is no commonly accepted benchmark to provide the basis for fair comparison of different solutions.
+
+To facilitate further research on this topic, and to bridge the gap between the power community and MARL community, we present a comprehensive test-bench and open-source environment for MARL based active voltage control. We formally define the active voltage control problem as a Dec-POMDP [17] that is widely acknowledged in the MARL community. We present an environment in Python and construct 3 scenarios with real public data that span from small scale (i.e. 6 agents) to large scale (i.e. 38 agents).
+
+The contributions of this paper are summarised as follows: (1) We formally define the active voltage control problem as a Dec-POMDP and develop an open-source environment.2 (2) We conduct large scale experimentation with 7 state-of-the-art MARL algorithms on different scenarios of the active voltage control problem. (3) We convert voltage constraints to barrier functions and observe the importance of designing an appropriate voltage barrier function from the experimental results. (4) By analysing the experimental results, we imply the possible challenges (e.g. interpretability) for MARL to solve the active voltage control problem and suggest potential directions for future works.
+
+# 2 Related Work
+
+Traditional Methods for Active Voltage Control. Voltage rising and fluctuation problem in distribution networks has been studied for 20 years [18]. The traditional voltage regulation devices such as OLTC and capacitor banks [10] are often installed at substations and therefore may not be effective in regulating voltages at the far end of the line [19]. The emergence of distributed generation, such as root-top PVs, introduces new approaches for voltage regulation by the active reactive power control of grid-connected inverters [11]. The state-of-the-art active voltage control strategies can be roughly classified into two categories: (1) reactive power dispatch based on optimal power flow (OPF) [20, 21]; and (2) droop control based on local voltage and power measurements [22, 23]. Specifically, centralised OPF [24–26] minimises the power loss while fulfilling voltage constraints (e.g. power flow equation defined in Eq.1); distributed OPF [27, 28] used distributed optimization techniques, such as alternating direction method of multipliers (ADMM), to replace the centralised solver. The primary limitation of OPF is the need of exact system model [29]. Besides, solving constrained optimisation problem is time-consuming, so it is difficult to react to the rapid change of load profile [30]. On the other hand, droop control only depends on its local measurements, but its performance relies on the manually-designed parameters and is often sub-optimal due to the lack of global information [30]. It is possible to enhance droop control by distributed algorithms, but extra communications are needed [31, 19]. In this paper we investigate the possibility of applying MARL techniques on the active voltage control problem. Compared with the previous works on traditional methods, (1) MARL is model-free, so no exact system model is needed; and (2) the response of MARL is fast so as to handle rapid changes of environments (e.g. the intermittency of renewable energy).
+
+Multi-Agent Reinforcement Learning for Active Voltage Control. In this section, we discuss the works that applied MARL to active voltage control problem in the power system community. [14, 16] applied MADDPG with the reactive power of inverters or static var compensators (SVCs) as control actions. [32] applied MADDPG with a manually designed voltage inner loop, so that agents set reference voltage instead of reactive power as their control actions. [13] applied MATD3 also with reactive power as control actions. [15] applied MASAC, where both reactive power and the curtailment of active power are used as control actions. In the above works, distribution networks are divided into regions, with each region controlled by a single agent [33]. It is not clear if these MARL approaches scales well for increasing number of agents. In particular, it is not clear if each single inverter in a distribution network can behave as an independent agent. In this work, we model the active voltage control problem as a Dec-POMDP [17], where each inverter is controlled by an agent. We propose Bowl-shape as a barrier function to represent voltage constraint as part of the reward. We build up an open-source environment for this specific problem that can be easily deployed with the existing MARL algorithms.
+
+Concurrent Works. L2RPN [34] is an environment mainly for the centralised control over power transmission networks with node-splitting and lines switch-off for topology management. Gym-ANM [35] is an environment for centralised power distribution network management. Compared with these 2 works, our environment mainly focuses on solving the active voltage control problem in power distribution networks, with the decentralised/distributed manner. Moreover, we provide more complicated scenarios with the real-world data. Finally, our environment can be easily and flexibly extended with more network topologies and data, thanks to the supports of PandaPower [36] and SimBench [37].
+
+# 3 Background
+
+# 3.1 Power Distribution Network
+
+![](images/747260f59aa5d8a2a2dcdd58c74e3c45331ddccbc08b3de7bcbc688b98e3b323.jpg)  
+Figure 1: Illustration on distribution network (block a-b-c) under PV penetration. The solid and dotted lines represent the power and information flows respectively. Block d is the detailed version of distribution network and block e is the circuit model of block d.
+
+An electric power distribution network is illustrated in Figure 1 stage a to c. The electricity is generated from power plant and transmitted through transmission lines. Muti-stage transformers are applied to reduce the voltage levels while the electricity is being delivered to the distribution network. The electricity is then consumed by residential and industrial clients. A typical PV unit consists of PV panels and voltage-source inverters which can be installed either on roof-top or in the solar farm. Conventionally, there exist management entities such as distributed system operator (DSO) monitoring and operating the PV resources through the local communication channels. With emergent PV penetration, distribution network gradually grows to be an active participant in power networks that can deliver power and service to its users and the main grid (see the bidirectional power flows in Figure 1 stage-d).
+
+System Model and Voltage Deviation. In this paper, we consider medium (10-24kV) and low (0.23-1kV) voltage distribution networks where PVs are highly penetrated. We model the distribution network in Figure 1 as a tree graph $\mathcal { G } = ( V , E )$ , where $V = \{ 0 , 1 , \ldots , N \}$ and $E = \{ 1 , 2 , \dots , N \}$ represent the set of nodes (buses) and edges (branches) respectively [24]. Bus 0 is considered as the connection to the main grid, balancing the active and reactive power in the distribution network. For each bus $i \in V$ , let $v _ { i }$ and $\theta _ { i }$ be the magnitude and phase angle of the complex voltage and $s _ { j } = p _ { i } + j q _ { i }$ be the complex power injection. Then the active and reactive power injection can be
+
+defined as follows:
+
+$$
+\begin{array} { r l r } { \quad } & { { } \displaystyle p _ { i } ^ { P V } - p _ { i } ^ { L } = v _ { i } ^ { 2 } \sum _ { j \in V _ { i } } g _ { i j } - v _ { i } \sum _ { j \in V _ { i } } v _ { j } \left( g _ { i j } \cos \theta _ { i j } + b _ { i j } \sin \theta _ { i j } \right) , } & { \forall i \in V \setminus \{ 0 \} } \\ { \displaystyle q _ { i } ^ { P V } - q _ { i } ^ { L } = - v _ { i } ^ { 2 } \sum _ { j \in V _ { i } } b _ { i j } + v _ { i } \sum _ { j \in V _ { i } } v _ { j } \left( g _ { i j } \sin \theta _ { i j } + b _ { i j } \cos \theta _ { i j } \right) , } & { \forall i \in V \setminus \{ 0 \} } \end{array}
+$$
+
+where $V _ { i } : = \{ j \mid ( i , j ) \in E \}$ is the index set of buses connected to bus $i$ . $g _ { i j }$ and $b _ { i j }$ are the conductance and susceptance on branch $( i , j ) . \theta _ { i j } = \theta _ { i } - \theta _ { j }$ is the phase difference between bus $i$ and $j . p _ { i } ^ { P V }$ and $q _ { i } ^ { P V }$ are the active power and reactive power of the PV on the bus $i$ (that are zeros if there is no PV on the bus $i$ ). $p _ { i } ^ { L }$ and $q _ { i } ^ { L }$ are the active power and reactive power of the loads on the bus $i$ (that are zeros if there is no loads on the bus $i$ ). Eq.1 can represent the power system dynamics which is essential for solving the power flow problem and active voltage control problem [38] (details in Appendix A.1-A.2). For safe and optimal operation, $5 \%$ voltage deviation is usually allowed, i.e., $v _ { 0 } = 1 . 0$ per unit $( p . u . )$ and $0 . 9 5 p . u . \leq v _ { i } \leq 1 . 0 5 p . u . , \forall i \in V \setminus \{ 0 \}$ . When the load is heavy during the nighttime, the end-user voltage could be smaller than $0 . 9 5 p . u$ . [39]. In contrast, to export its power, large penetration of $p _ { i } ^ { P V }$ leads to reverse current flow that would increase $v _ { i }$ out of the nominal range (Figure 1-d) [18, 40].
+
+Optimal Power Flow. In this paper, OPF is regarded as an optimization problem minimizing the total power loss subject to the power balance constraints defined in Eq.1, PV reactive power limits, and bus voltage limits [41]. As the centralized OPF has full access to the system topology, measurements, and PV resources, it provides the optimal active voltage control performance and can be used as a benchmark method (details in Appendix A.3). However, the performance of the OPF depends on the accuracy of the grid model and the optimisation is time-consuming which makes it difficult to be deployed online.
+
+Droop Control. To regulate local voltage deviation, the standard droop control defines a piece-wise linear relationship between PV reactive power generation and voltage deviation at a bus equipped with inverter-based PVs [30, 42] (details in Appendix A.4). It is a fully decentralised control and ignore both the total voltage divisions and the power loss.
+
+# 3.2 Dec-POMDP
+
+Multi-agent reinforcement learning (MARL) is an extension from the reinforcement learning problem, with multiple agents in the same environment. The problem of MARL for cooperation among agents is conventionally formulated as a Dec-POMDP [17]. It is usually formulated as a tuple such that $\langle \mathcal { T } , \mathcal { S } , \mathcal { A } , \mathcal { O } , \mathcal { T } , \dot { r } , \Omega , \rho , \gamma \rangle$ . $\mathcal { T }$ is an agent set; $s$ is a state set; $\mathcal { A } = \times _ { i \in \mathcal { T } } \mathcal { A } _ { i }$ is the joint action set, where $\mathbf { \mathcal { A } } _ { i }$ is each agent’s action set; $O = \times _ { i \in \tau } O _ { i }$ is the joint observation set, where $\mathcal { O } _ { i }$ is each agent’s observation set; $\mathcal { T } : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  [ 0 , 1 ]$ is a transition probability function that describes the dynamics of an environment; $r : S \times \mathcal { A }  \mathbb { R }$ is a global reward function that describes the award to the whole agents given their decisions; $\Omega : \mathcal { S } \times \mathcal { A } \times \mathcal { O }  [ 0 , 1 ]$ describes the perturbation of the observers (or sensors) for agents’ joint observations over the states after decisions; $\rho : { \mathcal { S } }  [ 0 , 1 ]$ is a probability function of initial states; and $\gamma \in ( 0 , 1 )$ is a discount factor. The objective of Dec-POMDP is finding an optimal joint policy $\pi = \times _ { i \in \boldsymbol { \mathbb { Z } } } \pi _ { i }$ that solves $\begin{array} { r } { \operatorname* { m a x } _ { \pi } \mathbb { E } _ { \pi } \big [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } \big ] } \end{array}$ .
+
+# 4 Distributed Active Voltage Control Problem
+
+# 4.1 Problem Formulation
+
+For the ease of operations, a large-scale power network is divided into multiple regions and there are several PVs installed into each region that is managed by the responsible distribution network owner. Each PV is with an inverter that generates reactive power to control the voltage around a stationary value denoted as $v _ { \mathrm { r e f } }$ . In our problem, we assume that the PV ownership (e.g. individual homeowners or enterprise) is independent and separated from the distribution network ownership (DNO) [33]. Specifically, a PV owner is responsible for operating a PV infrastructure. In other words, each PV is able to be controlled under a distributed manner so that it is natural to be considered as an agent. To enforce the safety of distribution power networks, all agents (i.e. PVs) within a region share the observation of this region.3 Since each agent can only observe partial information of the whole gird and maintaining the safety of the power network is a common goal among agents, it is reasonable to model the problem as a Dec-POMDP [17] that can be mathematically described as a 10-tuple such that $\langle \mathcal { T } , \mathcal { S } , \mathcal { A } , \mathcal { R } , \mathcal { O } , \mathcal { T } , r , \Omega , \rho , \gamma \rangle$ , where $\rho$ is the probability distribution for drawing the initial state and $\gamma$ is the discount factor.
+
+Agent Set. There is a set of agents controlling a set of PV inverters denoted as $\mathcal { T }$ . Each agent is located at some node in $\mathcal { G }$ (i.e. a graph representing the power network defined as before). We define a function $g : { \mathcal { T } }  V$ to indicate the node where an agent is located.
+
+Region Set. The whole power network is separated into $M$ regions, whose union is denoted as $\mathcal { R } \overset { \cdot } { = } \{ \mathcal { R } _ { k } \subset V | k < M , k \in \mathbb { N } \}$ , where $\textstyle \bigcup _ { \mathcal { R } _ { k } \in \mathcal { R } } \mathcal { R } _ { k } \subseteq V$ and $\mathcal { R } _ { k _ { 1 } } \cap \mathcal { R } _ { k _ { 2 } } = \emptyset$ if $k _ { 1 } \neq k _ { 2 }$ . We define a function $f : V \to \mathcal { R }$ that maps a node to the region where it is involved.
+
+State and Observation Set. The state set is defined as $\begin{array} { r } { S = \mathcal { L } \times \mathcal { P } \times \mathcal { Q } \times \mathcal { V } } \end{array}$ , where $\mathcal { L } = \{ ( \mathbf { p } ^ { L } , \mathbf { q } ^ { L } ) :$ $\mathbf { p } ^ { L } , \mathbf { q } ^ { L } \in ( 0 , \infty ) ^ { | V | } \}$ is a set of (active and reactive) powers of loads; $\mathcal { P } = \{ \mathbf { p } ^ { P V } : \mathbf { p } ^ { P V } \in \bar { ( 0 , \infty ) } ^ { | \mathcal { Z } | } \}$ is a set of active powers generated by PVs; $\mathcal { Q } = \{ \mathbf { q } ^ { P V } : \mathbf { q } ^ { P V } \in ( 0 , \infty ) ^ { | \mathcal { L } | } \}$ is a set of reactive powers generated by PV inverters at the preceding step; $\mathcal { V } = \left. ( \mathbf { v } , \theta ) : \mathbf { v } \in ( 0 , \infty ) ^ { | V | } , \theta \in [ - \pi , \pi ] ^ { | V | } \right.$ is a set of voltage wherein $\mathbf { v }$ is a vector of voltage magnitudes and $\theta$ is a vector of voltage phases measured in radius. $v _ { i } , p _ { i } ^ { L } , q _ { i } ^ { L } , p _ { i } ^ { P V }$ and $q _ { i } ^ { P V }$ are denoted as the components of the vectors $\mathbf { v } , \mathbf { p } ^ { L } , \mathbf { q } ^ { L } , \mathbf { p } ^ { P ^ { V } }$ and $\mathbf { q } ^ { P V }$ respectively. We define a function $h : \mathbb { P } ( V ) \to \mathbb { P } ( S )$ that maps a subset of $V$ to its correlated measures, where $\mathbb { P } ( \cdot )$ denotes the power set. The observation set is defined as $O = \times _ { i \in \tau } O _ { i }$ , where $\mathcal { O } _ { i } = ( h \circ f \circ g ) ( i )$ indicates the measures within the region where agent $i$ is located.
+
+Action Set. Each agent $i \in \mathcal { Z }$ is equipped with a continuous action set $\mathcal { A } _ { i } = \{ a _ { i } \ : \ - c \leq a _ { i } \leq$ $c , c > 0 \}$ . The continuous action represents the ratio of maximum reactive power it generates, i.e., the reactive power generated from the $k$ th PV inverter is $q _ { k } ^ { P V } = a _ { k } \ \sqrt { ( s _ { k } ^ { \mathrm { m a x } } ) ^ { 2 } - ( p _ { k } ^ { P V } ) ^ { 2 } }$ , where $s _ { k } ^ { \mathrm { m a x } }$ is the maximum apparent power of the $k$ th node that is dependent on the physical capacity of the PV inverter.4, 5 If $a _ { k } > 0$ , it means penetrating reactive powers to the distribution network. If $a _ { k } < 0$ , it means absorbing reactive powers from the distribution network. The value of $c$ is usually selected as per the loading capacity of a distribution network, which is for the safety of operations. The joint action set is denoted as $\mathcal { A } = \times _ { i \in \mathcal { T } } \mathcal { A } _ { i }$ .
+
+State Transition Probability Function. Since the state includes the last action and the change of loads is random (that theoretically can be modelled as any probabilistic distribution), we can naturally define the state transition probability function as $\mathcal { T } : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  [ 0 , 1 ]$ that follows Markov decision process. Specifically, $\begin{array} { r } { \mathcal { T } ( \mathbf { s } _ { t + 1 } \vert \mathbf { s } _ { t } , \mathbf { a } _ { t } ) = P r ( \mathbf { s } _ { t + 1 } \vert \delta ( \mathbf { s } _ { t } , \mathbf { a } _ { t } ) ) } \end{array}$ , where $\mathbf { a } _ { t } \in \mathcal A$ and $\mathbf { s } _ { t } , \mathbf { s } _ { t + 1 } \in S$ $\delta ( \mathbf { s } _ { t } , \mathbf { a } ) \mapsto \mathbf { s } _ { t + \tau }$ denotes the solution of the power flow, whereas $\begin{array} { r } { P r \big ( \mathbf { s } _ { t + 1 } \big | \mathbf { s } _ { t + \tau } \big ) } \end{array}$ describes the change of loads (i.e. highly correlated to the user behaviours). $\tau \ll \Delta t$ is an extremely short interval much less than the time interval between two controls (i.e. a time step) and $\Delta t = 1$ in this paper.
+
+Observation Probability Function. We now define the observation probability function. In the context of electric power network, it describes the measurement errors that may occur in sensors. Mathematically, we can define it as $\Omega : S \times A \times \mathcal { O } \to [ 0 , 1 ]$ . Specifically, $\Omega ( \mathbf { o } _ { t + 1 } | \mathbf { s } _ { t + 1 } , \mathbf { a } _ { t } ) = \mathbf { s } _ { t + 1 } +$ $\mathcal { N } ( \mathbf { 0 } , \Sigma )$ , where $\mathcal { N } ( \mathbf { 0 } , \Sigma )$ is an isotropic multi-variable Gaussian distribution and $\Sigma$ is dependent on the physical properties of sensors (e.g. smart meters).
+
+Reward Function. The reward function is defined as follows:
+
+$$
+r = - \frac { 1 } { | V | } \sum _ { i \in V } l _ { v } ( v _ { i } ) - \alpha \cdot l _ { q } ( \mathbf { q } ^ { P V } ) ,
+$$
+
+where $l _ { v } ( \cdot )$ is a voltage barrier function and $\begin{array} { r } { l _ { q } ( \mathbf q ^ { P V } ) = { \frac { 1 } { | \mathcal I | } } | | \mathbf q ^ { P V } | | _ { 1 } } \end{array}$ is the reactive power generation loss (i.e. a type of power loss approximation easy for computation). The objective is to control the voltage within a safety range around $v _ { \mathrm { r e f } }$ , while the reactive power generation is as less as possible, i.e., $l _ { q } ( \mathbf { q } ^ { P V } ) < \epsilon$ and $\epsilon > 0$ . Similar to the mathematical tricks used in $\beta$ -VAE [43], by KKT conditions we can transform a constrained reward to a unconstrained reward with a Lagrangian multiplier $\alpha \in ( 0 , 1 )$ shown in Eq.2. Since $l _ { v } ( \cdot )$ is not easy to define in practice (i.e., it affects $l _ { q } ( \mathbf { q } ^ { P V } ) )$ , we aim to study for a good choice in this paper.
+
+Objective Function. The objective function of this problem is $\begin{array} { r } { \operatorname* { m a x } _ { \pi } \mathbb { E } _ { \pi } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } ] } \end{array}$ , where ${ \boldsymbol { \pi } } = \times _ { i \in { \mathcal { I } } } { \boldsymbol { \pi } } _ { i }$ ; $\pi _ { i } : { \bar { \mathcal { O } } } _ { i } \times { \mathcal { A } } _ { i } \to [ 0 , 1 ]$ and $\bar { \mathcal { O } } _ { i } = ( \mathcal { O } _ { i } ^ { \tau } ) _ { \tau = 1 } ^ { h }$ is a history of observations with the length as $h$ . Literally, we need to find an optimal joint policy $\pi$ to maximize the discounted cumulative rewards.
+
+# 4.2 Voltage Barrier Function
+
+![](images/84ad91532ec1fef6df5ef513a69a50c5cee0b193f4244d8c10a84f32a74d0120.jpg)  
+Figure 2: This figure shows 3 voltage barrier functions, where L1-shape and L2-shape are 2 baselines while Bowl-shape is proposed in this paper.
+
+We define $v _ { \mathrm { r e f } } = 1 p . u$ . in this paper, and the voltage needs to be controlled within the safety range from $0 . 9 5 \ : p . u$ . to $1 . 0 5 p . u .$ , which sets the constraint of control. The voltage constraint is difficult to be handled in MARL, so we use a barrier function to represent the constraint. L1-shape (see Figure 2a) was most frequently used in the previous work [13–15], however, this may lead to wasteful reactive power generations since $\frac { | \Delta l _ { v } | } { \alpha | \Delta l _ { q } | } \gg 1$ within the safety range of voltage. Although L2-shape (see Figure 2b) may alleviate this problem, it may be slow to guide the policy outside the safety range. To address these problems, we propose a barrier function called Bowl-shape that combines the advantages of L1-shape and L2-shape. It gives a steep gradient outside the safety range, while it provides a slighter gradient as voltage tends to the $v _ { \mathrm { r e f } }$ that enables $\begin{array} { r } { \frac { | \Delta l _ { v } | } { \alpha | \Delta l _ { q } | }  0 } \end{array}$ as $v  v _ { \mathrm { r e f } }$ .
+
+# 5 Experiments
+
+# 5.1 Experimental Settings
+
+Power Network Topology. Two MV networks, IEEE 33-bus [44] and 141-bus [45] are modified as systems under test.6 To show the flexibility on network with multi-voltage levels, we construct a 110kV-20kV-0.4kV (high-medium-low voltage) 322-bus network using benchmark topology from SimBench [37]. For each network, a main branch is firstly determined and the control regions are partitioned by the shortest path between the terminal bus and the coupling point on the main branch. Each region consists of 1-4 PVs dependent on various regional sizes. The specific network descriptions and partitions are shown in Appendix D.1. To give a picture of the tasks, we demonstrate the 33-bus network in Figure 3.
+
+Data Descriptions. The load profile of each network is modified based on the real-time Portuguese electricity consumption accounting for 232 consumers of 3 years.7 To highlight the differences between residential and industrial users, we randomly perturb $\pm 5 \%$ on the default power factors defined in the case files and accordingly generate real-time reactive power consumption. The solar data is collected from Elia group,8 i.e. a Belgiums power network operator. The load and PV data are then interpolated with 3-min resolution that is consistent with the real-time control period in the grid. To distinguish among different solar radiation levels in various regions, the 3-year PV generations from 10 cites/regions are collected and PVs in the same control region possess the same generation profiles. We define the PV penetration rate $( P R )$ as the ratio between rated PV generation and rated load consumption. In this paper, we set $P R \in \{ 2 . 5 , 4 , 2 . 5 \}$ as the default $P R$ for different topologies. We oversize each PV inverter by $20 \%$ of its maximum active power generation to satisfy the IEEE grid code [42]. Besides, each PV inverter is considered to be able to generate reactive power in the STATCOM mode during night [47]. The median and $2 5 \% - 7 5 \%$ quantile shadings of PV generations, and the mean and minima-maxima shading of the loads are illustrated in Figure 4. The details can be found in Appendix D.2.
+
+![](images/0a7554b25b736f502735721952d4663acca780916517f382fc4d28d20a888a3d.jpg)  
+Figure 3: Illustration on 33-bus network. Each bus is indexed by a circle with a number. 4 control regions are partitioned by the smallest path from the terminal to the main branch (bus 1-6). We control the voltages on bus 2-33 whereas bus 0-1 represent the substation or main grid with the constant voltage and infinite active and reactive power capacity. G represents an external generator; small Ls represent loads; and the sun emoji represents the location where a PV is installed.
+
+![](images/8ab554fe97ec166a1b8ede7f2be29bc7bd10dad2fa98b25d2d220c2818a9ef56.jpg)  
+Figure 4: Active PV generations and load consumption.
+
+MARL Simulation Settings. We now describe the simulation settings standing by the view of MARL. In 33-bus network, there are 4 regions with 6 agents. In 141-bus network, there are 9 regions with 22 agents. In 322-bus network, there are 22 regions with 38 agents. The discount factor $\gamma$ is set to 0.99. $\alpha$ in Eq.2 is set to 0.1. To guarantee the safety of distribution networks, we manually set the range of actions for each scenario, with $[ - 0 . 8 , 0 . 8 ]$ for 33-bus network, $[ - 0 . 6 , 0 . 6 ]$ for 141-bus network, and $[ - 0 . 8 , 0 . 8 ]$ for 322-bus network. During training, we randomly sample the initial state for an episode and each episode lasts for 240 time steps (i.e. a half day). Every experiment is run with 5 random seeds and the test results during training are given by the median and the $2 5 \% - 7 5 \%$ quartile shading. Each test is conducted every 20 episodes with 10 randomly selected episodes for evaluation.
+
+Evaluation Metrics. In experiments, we use two metrics to evaluate the performance of algorithms.
+
+• Controllable rate $( C R )$ : It calculates the ratio of time steps where all buses’ voltages being under control during each episode.   
+• Power loss $( P L )$ : It calculates the average of the total power loss over all buses per time step during each episode.
+
+We aim to find algorithms and reward functions with high CR and low PL.
+
+MARL Algorithm Settings. We evaluate the performances of state-of-the-art MARL algorithms, i.e. IDDPG [48], MADDPG [49], COMA [50], IPPO [2], MAPPO [3], SQDDPG [48] and MATD3 [51] on this real-world problem with continuous actions. Since the original COMA can only work for discrete actions, we conduct some modifications so that it can work for continuous actions (see Appendix B). The details of settings are shown in Appendix C.
+
+# 5.2 Main Results
+
+![](images/82f718994cac468ac970519c3156001014894eac35d7944def6623bc169e5e27.jpg)
+
+Figure 5: Median CR and PL of algorithms with different voltage barrier functions. The sub-caption indicates metric-Barrier-scenario and BL is the contraction of Bowl.
+
+Algorithm Performances. We first show the main results of all algorithms on all scenarios in Figure 5. MADDPG and MATD3 generally perform well on all scenarios with different voltage barier functions. COMA performs well over CR on 33-bus networks and the performances fall on the large scale scenarios, but its PL is high. Similarly, SQDDPG performs generally well over CR on 33-bus and 141-bus networks, but it performs the worst on 322-bus networks and its PL on 141-bus networks is high. This reveals the limitations of COMA and SQDDPG on the scaling to many agents. Although MAPPO and IPPO performed well on games [2, 3], their performances on the real-world power network problems are poor. The most probable reason could be that the variations of dynamics and uncertainties in distribution networks are far faster and more complicated than games and their conservative policy updates cannot fast respond. IDDPG performs at the middle place in all scenarios, which may be due to non-stationary dynamics caused by multiple agents [49].
+
+![](images/decbf7717ac3061e320aba1ce5ba1455e7eaa19231f9ee6158898fc5dabee485.jpg)  
+Figure 6: Median performance of overall algorithms with different voltage barrier functions. The sub-caption indicates metric-scenario.
+
+Voltage Barrier Function Comparisons. To study the effects of different voltage barrier functions, we show the median performance of overall 7 MARL algorithms in Figure 6. It can be observed that Bowl-shape can preserve the high CR, while maintain the low PL on 33-bus and 141-bus networks. Although L1-shape can achieve the best CR on 33-bus and 141-bus networks, its PL on the 141-bus network is the highest. L2-shape performs the worst on 33-bus and 141-bus networks, but performs the best on the 322-bus network with the highest CR and the lowest PL. The reason could be that its slighter gradients is more suitable for the adaption among many agents. From the results, we can conclude that L1-shape, Bowl-shape and L2-shape are the best choices for the 33-bus network, the 141-bus network and the 322-bus network respectively.
+
+![](images/4c326a0c932a164750e09479bd378087c9d7e43c518b72bb677cfc0d79ad11cf.jpg)  
+Figure 7: Median performances of overall algorithms trained with various rewards consist of distinct voltage barrier functions shown in 3D surfaces. The sub-caption indicates metric-scenario.
+
+Diverse Algorithm Performances under Distinct Rewards. To clearly show the relationship between algorithms and reward functions, we also plot 3D surfaces over CR and PL w.r.t. algorithm types and reward types (consisting of distinct voltage barrier functions) in Figure 7. It is obvious that the performances of algorithms are highly correlated with the reward types. In other words, the same algorithm could perform diversely even trained by different reward functions with the same objective but different shapes.
+
+# 5.3 Comparison between MARL and Traditional Control Methods
+
+To compare MARL algorithms with the traditional control methods, we conduct a series of tests on various network topologies (i.e. 33-bus, 141-bus, and 322-bus networks). MADDPG trained by Bowl-shape is selected as the candidate for MARL. The traditional control methods that we select are OPF [24] and droop control [22]. For conciseness, we only demonstrate the voltages and powers on a typical bus with a PV installed (i.e. one of the most difficult buses to control) during a day (i.e. 480 consecutive time steps) in summer and winter respectively. The results for the 33-bus network is presented here and the results for other typologies are given in the Appendix E.2. From Figure 8, it can be seen that all methods control the voltage within the safety range in both summer and winter. Additionally, the power losses of MARL are lower than droop control but higher than OPF. This phenomenon is possibly due to the fact that droop control is a fully distributed algorithm which cannot explicitly reduce the power loss and OPF is a centralised algorithm with the known system model, while MARL lies between these 2 types of algorithms. This implies that MARL may outperform droop control for particular cases but it needs to be kept in mind that the droop gain used here may not be optimal. It is also worth noting that the control actions of MARL has a similar feature to droop control in the 33-bus case, and this feature is also observed in the 141-bus and 322-bus cases (see Figure 14-15 in the Appendix E.2 for details). However, this droop-like behaviour is missing in the winter case of 322-bus network, where the MARL control action is opposite to the desired direction, resulting in higher power losses and voltage deviation. This might be due to the over-generation as the MARL share the same policy for different seasons, so that it leads to the problem of relative overgeneralisation [52]. In summary, MARL seems able to learn the features of droop control but fails to outperform droop control significantly, and is even worse than droop control for certain cases.
+
+As a result, there is still significant headroom of improvement for MARL, in performance, robustness, and interpretability.
+
+![](images/a7a8081aaf36a2127c0195b301e307b7ad1a7bdc0ea53bdcdfcddef9a06de9b4.jpg)  
+Figure 8: Compare MARL with traditional control methods on bus 18 during a day for 33-bus network. 1st row: results for a summer day. 2nd row: results for a winter day. None and limit in (a) represent the voltage with no control and the safety voltage range respectively. P and Q in (b) indicate the PV active power and the reactive power by various methods.
+
+# 5.4 Discussion
+
+We now discuss the phenomenons that we observe from the experimental results.
+
+• It is obvious that the voltage barrier function may impact the performance of an algorithm (even with tiny changes for the common goal). This may be of general importance as many real-world problems may contain constraints that are not indirect in the objective function. An overall methodology is needed for designing barrier functions in state-constraint MARL.   
+• MARL may scale well for the number of agents and the complexity of networks, and only requires a very low control rate for active voltage control.   
+• The results above show the evidence that MARL may behave diversely from the traditional control methods. Some of the learnt behaviours may induce better performance, but some others deteriorate the performance. This shows the promising benefits of MARL in industrial applications but also highlights the drawbacks in interpretability and robustness.   
+• The combination of learning algorithms with domain knowledge is a potential roadmap towards interpretable MARL. For the active voltage control problem, the domain knowledge may present as network topology, inner control loops (say droop control), and load pattern. The exploitation of such domain knowledge reduces the dimensions of MARL exploration space and may offer a lower bound of performance as a guarantee. Encoding domain knowledge such as the network topology as a priori for model-based MARL is also a potential direction.
+
+# 6 Conclusion
+
+This paper investigates the potential of applying multi-agent reinforcement learning (MARL) to the active voltage control in power distribution networks. We firstly formulate this problem as a DecPOMDP and then study the behaviours of MARL with various voltage barrier functions (reflecting voltage constraints as barrier penalties). Moreover, we compare the behaviours of MARL with the traditional control methods (i.e. droop control and OPF), and observe that MARL is possible to generate inexplicable behaviours, so an interpretable and trustable algorithm is highly desired for industrial applications. Finally, the environment used in this work is open-sourced and easy to follow so that the machine learning community is able to contribute to this challenging real-world problem.
+
+# Acknowledgement
+
+This work is supported by the Engineering and Physical Sciences Research Council of UK (EPSRC) under awards EP/S000909/1.
+
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parse/train/hwoK62_GkiT/hwoK62_GkiT_content_list.json

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+    {
+        "type": "text",
+        "text": "Multi-Agent Reinforcement Learning for Active Voltage Control on Power Distribution Networks ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Jianhong Wang∗ Imperial College London jianhong.wang16@imperial.ac.uk ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Wangkun Xu∗ Imperial College London wangkun.xu18@imperial.ac.uk ",
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+    {
+        "type": "text",
+        "text": "Yunjie $\\mathbf { G u } ^ { \\dagger }$ University of Bath yg934@bath.ac.uk ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Wenbin Song Shanghaitech University songwb@shanghaitech.edu.cn ",
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+    {
+        "type": "text",
+        "text": "Tim C. Green‡ Imperial College London t.green@imperial.ac.uk ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This paper presents a problem in power networks that creates an exciting and yet challenging real-world scenario for application of multi-agent reinforcement learning (MARL). The emerging trend of decarbonisation is placing excessive stress on power distribution networks. Active voltage control is seen as a promising solution to relieve power congestion and improve voltage quality without extra hardware investment, taking advantage of the controllable apparatuses in the network, such as roof-top photovoltaics (PVs) and static var compensators (SVCs). These controllable apparatuses appear in a vast number and are distributed in a wide geographic area, making MARL a natural candidate. This paper formulates the active voltage control problem in the framework of Dec-POMDP and establishes an open-source environment. It aims to bridge the gap between the power community and the MARL community and be a drive force towards real-world applications of MARL algorithms. Finally, we analyse the special characteristics of the active voltage control problems that cause challenges (e.g. interpretability) for state-of-the-art MARL approaches, and summarise the potential directions. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multi-agent reinforcement learning (MARL) has demonstrated impressive performances on games (e.g., Chess [1], StarCraft [2, 3], and etc.) and robotics [4]. There are now increasing interests and thrusts to apply MARL in real-world problems. Power network is a natural test field for MARL algorithms. There are many problems in power networks involving the collaborative or competitive actions of a vast number of agents, such as market bidding [5], voltage control, frequency control, and emergency handling [6]. One of the obstacles for applying MARL in power networks is the lack of transparency and guarantee, which is unacceptable considering the importance of reliable power supply for the whole society. Therefore, it is sensible to start with problems that are less sensitive to reliability but hard to be solved by conventional methods. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Active voltage control in power distribution networks is such a candidate problem. The voltage control problem has been studied for years but only comes under spot light recently due to the increasing penetration of distributed resources, e.g. roof-top photovoltaics (PVs) [7]. The excessive active power injection may cause voltage fluctuations beyond the threshold of grid standards [8]. The voltage fluctuations can be relieved by exploiting the control flexibility of PV inverters themselves along with other controllable apparatuses, such as static var compensators (SVCs) and on load tap changers (OLTCs). An elaborate scheme is needed to coordinate these multiple apparatuses at scale to regulate voltage throughout the network with limited local information, which is called active voltage control [9–12]. The active voltage control problem has many interesting properties. (1) It is a combination of local and global problem, i.e., the voltage of each node is influenced by the powers (real and reactive) of all other nodes but the impact recedes with increasing distance between nodes. (2) It is a constrained optimisation problem where the constraint is the voltage threshold and the objective is the total power loss, but there is no explicit relationship between the constraint and the control action. (3) A distribution network has a radial topology involving a rich structure that can be taken as prior knowledge for control, but the node-branch parameters of the topology may be significantly uncertain. (4) Voltage control has a relatively large tolerance and less severe consequences if the control fails to meet standard requirements. ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "There have been several attempts to apply MARL in active voltage control [6, 13–16]. Each work on a particular case showed promising performances of state-of-the-art MARL approaches with modifications adapting to the active voltage control problem. However, it is not clear if these methods scale well to a larger network, and the robustness against different scenarios, such as penetration levels and load profiles, are yet to be investigated. There is no commonly accepted benchmark to provide the basis for fair comparison of different solutions. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To facilitate further research on this topic, and to bridge the gap between the power community and MARL community, we present a comprehensive test-bench and open-source environment for MARL based active voltage control. We formally define the active voltage control problem as a Dec-POMDP [17] that is widely acknowledged in the MARL community. We present an environment in Python and construct 3 scenarios with real public data that span from small scale (i.e. 6 agents) to large scale (i.e. 38 agents). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The contributions of this paper are summarised as follows: (1) We formally define the active voltage control problem as a Dec-POMDP and develop an open-source environment.2 (2) We conduct large scale experimentation with 7 state-of-the-art MARL algorithms on different scenarios of the active voltage control problem. (3) We convert voltage constraints to barrier functions and observe the importance of designing an appropriate voltage barrier function from the experimental results. (4) By analysing the experimental results, we imply the possible challenges (e.g. interpretability) for MARL to solve the active voltage control problem and suggest potential directions for future works. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            560,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Traditional Methods for Active Voltage Control. Voltage rising and fluctuation problem in distribution networks has been studied for 20 years [18]. The traditional voltage regulation devices such as OLTC and capacitor banks [10] are often installed at substations and therefore may not be effective in regulating voltages at the far end of the line [19]. The emergence of distributed generation, such as root-top PVs, introduces new approaches for voltage regulation by the active reactive power control of grid-connected inverters [11]. The state-of-the-art active voltage control strategies can be roughly classified into two categories: (1) reactive power dispatch based on optimal power flow (OPF) [20, 21]; and (2) droop control based on local voltage and power measurements [22, 23]. Specifically, centralised OPF [24–26] minimises the power loss while fulfilling voltage constraints (e.g. power flow equation defined in Eq.1); distributed OPF [27, 28] used distributed optimization techniques, such as alternating direction method of multipliers (ADMM), to replace the centralised solver. The primary limitation of OPF is the need of exact system model [29]. Besides, solving constrained optimisation problem is time-consuming, so it is difficult to react to the rapid change of load profile [30]. On the other hand, droop control only depends on its local measurements, but its performance relies on the manually-designed parameters and is often sub-optimal due to the lack of global information [30]. It is possible to enhance droop control by distributed algorithms, but extra communications are needed [31, 19]. In this paper we investigate the possibility of applying MARL techniques on the active voltage control problem. Compared with the previous works on traditional methods, (1) MARL is model-free, so no exact system model is needed; and (2) the response of MARL is fast so as to handle rapid changes of environments (e.g. the intermittency of renewable energy). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Multi-Agent Reinforcement Learning for Active Voltage Control. In this section, we discuss the works that applied MARL to active voltage control problem in the power system community. [14, 16] applied MADDPG with the reactive power of inverters or static var compensators (SVCs) as control actions. [32] applied MADDPG with a manually designed voltage inner loop, so that agents set reference voltage instead of reactive power as their control actions. [13] applied MATD3 also with reactive power as control actions. [15] applied MASAC, where both reactive power and the curtailment of active power are used as control actions. In the above works, distribution networks are divided into regions, with each region controlled by a single agent [33]. It is not clear if these MARL approaches scales well for increasing number of agents. In particular, it is not clear if each single inverter in a distribution network can behave as an independent agent. In this work, we model the active voltage control problem as a Dec-POMDP [17], where each inverter is controlled by an agent. We propose Bowl-shape as a barrier function to represent voltage constraint as part of the reward. We build up an open-source environment for this specific problem that can be easily deployed with the existing MARL algorithms. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Concurrent Works. L2RPN [34] is an environment mainly for the centralised control over power transmission networks with node-splitting and lines switch-off for topology management. Gym-ANM [35] is an environment for centralised power distribution network management. Compared with these 2 works, our environment mainly focuses on solving the active voltage control problem in power distribution networks, with the decentralised/distributed manner. Moreover, we provide more complicated scenarios with the real-world data. Finally, our environment can be easily and flexibly extended with more network topologies and data, thanks to the supports of PandaPower [36] and SimBench [37]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 Background ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Power Distribution Network ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/747260f59aa5d8a2a2dcdd58c74e3c45331ddccbc08b3de7bcbc688b98e3b323.jpg",
+        "image_caption": [
+            "Figure 1: Illustration on distribution network (block a-b-c) under PV penetration. The solid and dotted lines represent the power and information flows respectively. Block d is the detailed version of distribution network and block e is the circuit model of block d. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "An electric power distribution network is illustrated in Figure 1 stage a to c. The electricity is generated from power plant and transmitted through transmission lines. Muti-stage transformers are applied to reduce the voltage levels while the electricity is being delivered to the distribution network. The electricity is then consumed by residential and industrial clients. A typical PV unit consists of PV panels and voltage-source inverters which can be installed either on roof-top or in the solar farm. Conventionally, there exist management entities such as distributed system operator (DSO) monitoring and operating the PV resources through the local communication channels. With emergent PV penetration, distribution network gradually grows to be an active participant in power networks that can deliver power and service to its users and the main grid (see the bidirectional power flows in Figure 1 stage-d). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "System Model and Voltage Deviation. In this paper, we consider medium (10-24kV) and low (0.23-1kV) voltage distribution networks where PVs are highly penetrated. We model the distribution network in Figure 1 as a tree graph $\\mathcal { G } = ( V , E )$ , where $V = \\{ 0 , 1 , \\ldots , N \\}$ and $E = \\{ 1 , 2 , \\dots , N \\}$ represent the set of nodes (buses) and edges (branches) respectively [24]. Bus 0 is considered as the connection to the main grid, balancing the active and reactive power in the distribution network. For each bus $i \\in V$ , let $v _ { i }$ and $\\theta _ { i }$ be the magnitude and phase angle of the complex voltage and $s _ { j } = p _ { i } + j q _ { i }$ be the complex power injection. Then the active and reactive power injection can be ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "defined as follows: ",
+        "bbox": [
+            173,
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+            299,
+            104
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/44a7e295a8f3fcb1dad334b168e2c39a4358ec3c6fd7f0710fe78954de5c2aa5.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\quad } & { { } \\displaystyle p _ { i } ^ { P V } - p _ { i } ^ { L } = v _ { i } ^ { 2 } \\sum _ { j \\in V _ { i } } g _ { i j } - v _ { i } \\sum _ { j \\in V _ { i } } v _ { j } \\left( g _ { i j } \\cos \\theta _ { i j } + b _ { i j } \\sin \\theta _ { i j } \\right) , } & { \\forall i \\in V \\setminus \\{ 0 \\} } \\\\ { \\displaystyle q _ { i } ^ { P V } - q _ { i } ^ { L } = - v _ { i } ^ { 2 } \\sum _ { j \\in V _ { i } } b _ { i j } + v _ { i } \\sum _ { j \\in V _ { i } } v _ { j } \\left( g _ { i j } \\sin \\theta _ { i j } + b _ { i j } \\cos \\theta _ { i j } \\right) , } & { \\forall i \\in V \\setminus \\{ 0 \\} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            235,
+            113,
+            764,
+            188
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $V _ { i } : = \\{ j \\mid ( i , j ) \\in E \\}$ is the index set of buses connected to bus $i$ . $g _ { i j }$ and $b _ { i j }$ are the conductance and susceptance on branch $( i , j ) . \\theta _ { i j } = \\theta _ { i } - \\theta _ { j }$ is the phase difference between bus $i$ and $j . p _ { i } ^ { P V }$ and $q _ { i } ^ { P V }$ are the active power and reactive power of the PV on the bus $i$ (that are zeros if there is no PV on the bus $i$ ). $p _ { i } ^ { L }$ and $q _ { i } ^ { L }$ are the active power and reactive power of the loads on the bus $i$ (that are zeros if there is no loads on the bus $i$ ). Eq.1 can represent the power system dynamics which is essential for solving the power flow problem and active voltage control problem [38] (details in Appendix A.1-A.2). For safe and optimal operation, $5 \\%$ voltage deviation is usually allowed, i.e., $v _ { 0 } = 1 . 0$ per unit $( p . u . )$ and $0 . 9 5 p . u . \\leq v _ { i } \\leq 1 . 0 5 p . u . , \\forall i \\in V \\setminus \\{ 0 \\}$ . When the load is heavy during the nighttime, the end-user voltage could be smaller than $0 . 9 5 p . u$ . [39]. In contrast, to export its power, large penetration of $p _ { i } ^ { P V }$ leads to reverse current flow that would increase $v _ { i }$ out of the nominal range (Figure 1-d) [18, 40]. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Optimal Power Flow. In this paper, OPF is regarded as an optimization problem minimizing the total power loss subject to the power balance constraints defined in Eq.1, PV reactive power limits, and bus voltage limits [41]. As the centralized OPF has full access to the system topology, measurements, and PV resources, it provides the optimal active voltage control performance and can be used as a benchmark method (details in Appendix A.3). However, the performance of the OPF depends on the accuracy of the grid model and the optimisation is time-consuming which makes it difficult to be deployed online. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Droop Control. To regulate local voltage deviation, the standard droop control defines a piece-wise linear relationship between PV reactive power generation and voltage deviation at a bus equipped with inverter-based PVs [30, 42] (details in Appendix A.4). It is a fully decentralised control and ignore both the total voltage divisions and the power loss. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Dec-POMDP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Multi-agent reinforcement learning (MARL) is an extension from the reinforcement learning problem, with multiple agents in the same environment. The problem of MARL for cooperation among agents is conventionally formulated as a Dec-POMDP [17]. It is usually formulated as a tuple such that $\\langle \\mathcal { T } , \\mathcal { S } , \\mathcal { A } , \\mathcal { O } , \\mathcal { T } , \\dot { r } , \\Omega , \\rho , \\gamma \\rangle$ . $\\mathcal { T }$ is an agent set; $s$ is a state set; $\\mathcal { A } = \\times _ { i \\in \\mathcal { T } } \\mathcal { A } _ { i }$ is the joint action set, where $\\mathbf { \\mathcal { A } } _ { i }$ is each agent’s action set; $O = \\times _ { i \\in \\tau } O _ { i }$ is the joint observation set, where $\\mathcal { O } _ { i }$ is each agent’s observation set; $\\mathcal { T } : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S }  [ 0 , 1 ]$ is a transition probability function that describes the dynamics of an environment; $r : S \\times \\mathcal { A }  \\mathbb { R }$ is a global reward function that describes the award to the whole agents given their decisions; $\\Omega : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { O }  [ 0 , 1 ]$ describes the perturbation of the observers (or sensors) for agents’ joint observations over the states after decisions; $\\rho : { \\mathcal { S } }  [ 0 , 1 ]$ is a probability function of initial states; and $\\gamma \\in ( 0 , 1 )$ is a discount factor. The objective of Dec-POMDP is finding an optimal joint policy $\\pi = \\times _ { i \\in \\boldsymbol { \\mathbb { Z } } } \\pi _ { i }$ that solves $\\begin{array} { r } { \\operatorname* { m a x } _ { \\pi } \\mathbb { E } _ { \\pi } \\big [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t } \\big ] } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Distributed Active Voltage Control Problem ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 Problem Formulation ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the ease of operations, a large-scale power network is divided into multiple regions and there are several PVs installed into each region that is managed by the responsible distribution network owner. Each PV is with an inverter that generates reactive power to control the voltage around a stationary value denoted as $v _ { \\mathrm { r e f } }$ . In our problem, we assume that the PV ownership (e.g. individual homeowners or enterprise) is independent and separated from the distribution network ownership (DNO) [33]. Specifically, a PV owner is responsible for operating a PV infrastructure. In other words, each PV is able to be controlled under a distributed manner so that it is natural to be considered as an agent. To enforce the safety of distribution power networks, all agents (i.e. PVs) within a region share the observation of this region.3 Since each agent can only observe partial information of the whole gird and maintaining the safety of the power network is a common goal among agents, it is reasonable to model the problem as a Dec-POMDP [17] that can be mathematically described as a 10-tuple such that $\\langle \\mathcal { T } , \\mathcal { S } , \\mathcal { A } , \\mathcal { R } , \\mathcal { O } , \\mathcal { T } , r , \\Omega , \\rho , \\gamma \\rangle$ , where $\\rho$ is the probability distribution for drawing the initial state and $\\gamma$ is the discount factor. ",
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Agent Set. There is a set of agents controlling a set of PV inverters denoted as $\\mathcal { T }$ . Each agent is located at some node in $\\mathcal { G }$ (i.e. a graph representing the power network defined as before). We define a function $g : { \\mathcal { T } }  V$ to indicate the node where an agent is located. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Region Set. The whole power network is separated into $M$ regions, whose union is denoted as $\\mathcal { R } \\overset { \\cdot } { = } \\{ \\mathcal { R } _ { k } \\subset V | k < M , k \\in \\mathbb { N } \\}$ , where $\\textstyle \\bigcup _ { \\mathcal { R } _ { k } \\in \\mathcal { R } } \\mathcal { R } _ { k } \\subseteq V$ and $\\mathcal { R } _ { k _ { 1 } } \\cap \\mathcal { R } _ { k _ { 2 } } = \\emptyset$ if $k _ { 1 } \\neq k _ { 2 }$ . We define a function $f : V \\to \\mathcal { R }$ that maps a node to the region where it is involved. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "State and Observation Set. The state set is defined as $\\begin{array} { r } { S = \\mathcal { L } \\times \\mathcal { P } \\times \\mathcal { Q } \\times \\mathcal { V } } \\end{array}$ , where $\\mathcal { L } = \\{ ( \\mathbf { p } ^ { L } , \\mathbf { q } ^ { L } ) :$ $\\mathbf { p } ^ { L } , \\mathbf { q } ^ { L } \\in ( 0 , \\infty ) ^ { | V | } \\}$ is a set of (active and reactive) powers of loads; $\\mathcal { P } = \\{ \\mathbf { p } ^ { P V } : \\mathbf { p } ^ { P V } \\in \\bar { ( 0 , \\infty ) } ^ { | \\mathcal { Z } | } \\}$ is a set of active powers generated by PVs; $\\mathcal { Q } = \\{ \\mathbf { q } ^ { P V } : \\mathbf { q } ^ { P V } \\in ( 0 , \\infty ) ^ { | \\mathcal { L } | } \\}$ is a set of reactive powers generated by PV inverters at the preceding step; $\\mathcal { V } = \\left. ( \\mathbf { v } , \\theta ) : \\mathbf { v } \\in ( 0 , \\infty ) ^ { | V | } , \\theta \\in [ - \\pi , \\pi ] ^ { | V | } \\right.$ is a set of voltage wherein $\\mathbf { v }$ is a vector of voltage magnitudes and $\\theta$ is a vector of voltage phases measured in radius. $v _ { i } , p _ { i } ^ { L } , q _ { i } ^ { L } , p _ { i } ^ { P V }$ and $q _ { i } ^ { P V }$ are denoted as the components of the vectors $\\mathbf { v } , \\mathbf { p } ^ { L } , \\mathbf { q } ^ { L } , \\mathbf { p } ^ { P ^ { V } }$ and $\\mathbf { q } ^ { P V }$ respectively. We define a function $h : \\mathbb { P } ( V ) \\to \\mathbb { P } ( S )$ that maps a subset of $V$ to its correlated measures, where $\\mathbb { P } ( \\cdot )$ denotes the power set. The observation set is defined as $O = \\times _ { i \\in \\tau } O _ { i }$ , where $\\mathcal { O } _ { i } = ( h \\circ f \\circ g ) ( i )$ indicates the measures within the region where agent $i$ is located. ",
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+    },
+    {
+        "type": "text",
+        "text": "Action Set. Each agent $i \\in \\mathcal { Z }$ is equipped with a continuous action set $\\mathcal { A } _ { i } = \\{ a _ { i } \\ : \\ - c \\leq a _ { i } \\leq$ $c , c > 0 \\}$ . The continuous action represents the ratio of maximum reactive power it generates, i.e., the reactive power generated from the $k$ th PV inverter is $q _ { k } ^ { P V } = a _ { k } \\ \\sqrt { ( s _ { k } ^ { \\mathrm { m a x } } ) ^ { 2 } - ( p _ { k } ^ { P V } ) ^ { 2 } }$ , where $s _ { k } ^ { \\mathrm { m a x } }$ is the maximum apparent power of the $k$ th node that is dependent on the physical capacity of the PV inverter.4, 5 If $a _ { k } > 0$ , it means penetrating reactive powers to the distribution network. If $a _ { k } < 0$ , it means absorbing reactive powers from the distribution network. The value of $c$ is usually selected as per the loading capacity of a distribution network, which is for the safety of operations. The joint action set is denoted as $\\mathcal { A } = \\times _ { i \\in \\mathcal { T } } \\mathcal { A } _ { i }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "State Transition Probability Function. Since the state includes the last action and the change of loads is random (that theoretically can be modelled as any probabilistic distribution), we can naturally define the state transition probability function as $\\mathcal { T } : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S }  [ 0 , 1 ]$ that follows Markov decision process. Specifically, $\\begin{array} { r } { \\mathcal { T } ( \\mathbf { s } _ { t + 1 } \\vert \\mathbf { s } _ { t } , \\mathbf { a } _ { t } ) = P r ( \\mathbf { s } _ { t + 1 } \\vert \\delta ( \\mathbf { s } _ { t } , \\mathbf { a } _ { t } ) ) } \\end{array}$ , where $\\mathbf { a } _ { t } \\in \\mathcal A$ and $\\mathbf { s } _ { t } , \\mathbf { s } _ { t + 1 } \\in S$ $\\delta ( \\mathbf { s } _ { t } , \\mathbf { a } ) \\mapsto \\mathbf { s } _ { t + \\tau }$ denotes the solution of the power flow, whereas $\\begin{array} { r } { P r \\big ( \\mathbf { s } _ { t + 1 } \\big | \\mathbf { s } _ { t + \\tau } \\big ) } \\end{array}$ describes the change of loads (i.e. highly correlated to the user behaviours). $\\tau \\ll \\Delta t$ is an extremely short interval much less than the time interval between two controls (i.e. a time step) and $\\Delta t = 1$ in this paper. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Observation Probability Function. We now define the observation probability function. In the context of electric power network, it describes the measurement errors that may occur in sensors. Mathematically, we can define it as $\\Omega : S \\times A \\times \\mathcal { O } \\to [ 0 , 1 ]$ . Specifically, $\\Omega ( \\mathbf { o } _ { t + 1 } | \\mathbf { s } _ { t + 1 } , \\mathbf { a } _ { t } ) = \\mathbf { s } _ { t + 1 } +$ $\\mathcal { N } ( \\mathbf { 0 } , \\Sigma )$ , where $\\mathcal { N } ( \\mathbf { 0 } , \\Sigma )$ is an isotropic multi-variable Gaussian distribution and $\\Sigma$ is dependent on the physical properties of sensors (e.g. smart meters). ",
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+    {
+        "type": "text",
+        "text": "Reward Function. The reward function is defined as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9e615cf552352ca4dd3b9646638bb9b6bb59f30de56eab5d10c2f43655c9a3c.jpg",
+        "text": "$$\nr = - \\frac { 1 } { | V | } \\sum _ { i \\in V } l _ { v } ( v _ { i } ) - \\alpha \\cdot l _ { q } ( \\mathbf { q } ^ { P V } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $l _ { v } ( \\cdot )$ is a voltage barrier function and $\\begin{array} { r } { l _ { q } ( \\mathbf q ^ { P V } ) = { \\frac { 1 } { | \\mathcal I | } } | | \\mathbf q ^ { P V } | | _ { 1 } } \\end{array}$ is the reactive power generation loss (i.e. a type of power loss approximation easy for computation). The objective is to control the voltage within a safety range around $v _ { \\mathrm { r e f } }$ , while the reactive power generation is as less as possible, i.e., $l _ { q } ( \\mathbf { q } ^ { P V } ) < \\epsilon$ and $\\epsilon > 0$ . Similar to the mathematical tricks used in $\\beta$ -VAE [43], by KKT conditions we can transform a constrained reward to a unconstrained reward with a Lagrangian multiplier $\\alpha \\in ( 0 , 1 )$ shown in Eq.2. Since $l _ { v } ( \\cdot )$ is not easy to define in practice (i.e., it affects $l _ { q } ( \\mathbf { q } ^ { P V } ) )$ , we aim to study for a good choice in this paper. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Objective Function. The objective function of this problem is $\\begin{array} { r } { \\operatorname* { m a x } _ { \\pi } \\mathbb { E } _ { \\pi } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t } ] } \\end{array}$ , where ${ \\boldsymbol { \\pi } } = \\times _ { i \\in { \\mathcal { I } } } { \\boldsymbol { \\pi } } _ { i }$ ; $\\pi _ { i } : { \\bar { \\mathcal { O } } } _ { i } \\times { \\mathcal { A } } _ { i } \\to [ 0 , 1 ]$ and $\\bar { \\mathcal { O } } _ { i } = ( \\mathcal { O } _ { i } ^ { \\tau } ) _ { \\tau = 1 } ^ { h }$ is a history of observations with the length as $h$ . Literally, we need to find an optimal joint policy $\\pi$ to maximize the discounted cumulative rewards. ",
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+    {
+        "type": "text",
+        "text": "4.2 Voltage Barrier Function ",
+        "text_level": 1,
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+    {
+        "type": "image",
+        "img_path": "images/84ad91532ec1fef6df5ef513a69a50c5cee0b193f4244d8c10a84f32a74d0120.jpg",
+        "image_caption": [
+            "Figure 2: This figure shows 3 voltage barrier functions, where L1-shape and L2-shape are 2 baselines while Bowl-shape is proposed in this paper. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "We define $v _ { \\mathrm { r e f } } = 1 p . u$ . in this paper, and the voltage needs to be controlled within the safety range from $0 . 9 5 \\ : p . u$ . to $1 . 0 5 p . u .$ , which sets the constraint of control. The voltage constraint is difficult to be handled in MARL, so we use a barrier function to represent the constraint. L1-shape (see Figure 2a) was most frequently used in the previous work [13–15], however, this may lead to wasteful reactive power generations since $\\frac { | \\Delta l _ { v } | } { \\alpha | \\Delta l _ { q } | } \\gg 1$ within the safety range of voltage. Although L2-shape (see Figure 2b) may alleviate this problem, it may be slow to guide the policy outside the safety range. To address these problems, we propose a barrier function called Bowl-shape that combines the advantages of L1-shape and L2-shape. It gives a steep gradient outside the safety range, while it provides a slighter gradient as voltage tends to the $v _ { \\mathrm { r e f } }$ that enables $\\begin{array} { r } { \\frac { | \\Delta l _ { v } | } { \\alpha | \\Delta l _ { q } | }  0 } \\end{array}$ as $v  v _ { \\mathrm { r e f } }$ . ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "5.1 Experimental Settings ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Power Network Topology. Two MV networks, IEEE 33-bus [44] and 141-bus [45] are modified as systems under test.6 To show the flexibility on network with multi-voltage levels, we construct a 110kV-20kV-0.4kV (high-medium-low voltage) 322-bus network using benchmark topology from SimBench [37]. For each network, a main branch is firstly determined and the control regions are partitioned by the shortest path between the terminal bus and the coupling point on the main branch. Each region consists of 1-4 PVs dependent on various regional sizes. The specific network descriptions and partitions are shown in Appendix D.1. To give a picture of the tasks, we demonstrate the 33-bus network in Figure 3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Data Descriptions. The load profile of each network is modified based on the real-time Portuguese electricity consumption accounting for 232 consumers of 3 years.7 To highlight the differences between residential and industrial users, we randomly perturb $\\pm 5 \\%$ on the default power factors defined in the case files and accordingly generate real-time reactive power consumption. The solar data is collected from Elia group,8 i.e. a Belgiums power network operator. The load and PV data are then interpolated with 3-min resolution that is consistent with the real-time control period in the grid. To distinguish among different solar radiation levels in various regions, the 3-year PV generations from 10 cites/regions are collected and PVs in the same control region possess the same generation profiles. We define the PV penetration rate $( P R )$ as the ratio between rated PV generation and rated load consumption. In this paper, we set $P R \\in \\{ 2 . 5 , 4 , 2 . 5 \\}$ as the default $P R$ for different topologies. We oversize each PV inverter by $20 \\%$ of its maximum active power generation to satisfy the IEEE grid code [42]. Besides, each PV inverter is considered to be able to generate reactive power in the STATCOM mode during night [47]. The median and $2 5 \\% - 7 5 \\%$ quantile shadings of PV generations, and the mean and minima-maxima shading of the loads are illustrated in Figure 4. The details can be found in Appendix D.2. ",
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+    {
+        "type": "image",
+        "img_path": "images/0a7554b25b736f502735721952d4663acca780916517f382fc4d28d20a888a3d.jpg",
+        "image_caption": [
+            "Figure 3: Illustration on 33-bus network. Each bus is indexed by a circle with a number. 4 control regions are partitioned by the smallest path from the terminal to the main branch (bus 1-6). We control the voltages on bus 2-33 whereas bus 0-1 represent the substation or main grid with the constant voltage and infinite active and reactive power capacity. G represents an external generator; small Ls represent loads; and the sun emoji represents the location where a PV is installed. "
+        ],
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+        "text": "",
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+    {
+        "type": "image",
+        "img_path": "images/8ab554fe97ec166a1b8ede7f2be29bc7bd10dad2fa98b25d2d220c2818a9ef56.jpg",
+        "image_caption": [
+            "Figure 4: Active PV generations and load consumption. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "MARL Simulation Settings. We now describe the simulation settings standing by the view of MARL. In 33-bus network, there are 4 regions with 6 agents. In 141-bus network, there are 9 regions with 22 agents. In 322-bus network, there are 22 regions with 38 agents. The discount factor $\\gamma$ is set to 0.99. $\\alpha$ in Eq.2 is set to 0.1. To guarantee the safety of distribution networks, we manually set the range of actions for each scenario, with $[ - 0 . 8 , 0 . 8 ]$ for 33-bus network, $[ - 0 . 6 , 0 . 6 ]$ for 141-bus network, and $[ - 0 . 8 , 0 . 8 ]$ for 322-bus network. During training, we randomly sample the initial state for an episode and each episode lasts for 240 time steps (i.e. a half day). Every experiment is run with 5 random seeds and the test results during training are given by the median and the $2 5 \\% - 7 5 \\%$ quartile shading. Each test is conducted every 20 episodes with 10 randomly selected episodes for evaluation. ",
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+    {
+        "type": "text",
+        "text": "Evaluation Metrics. In experiments, we use two metrics to evaluate the performance of algorithms. ",
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+    {
+        "type": "text",
+        "text": "• Controllable rate $( C R )$ : It calculates the ratio of time steps where all buses’ voltages being under control during each episode.   \n• Power loss $( P L )$ : It calculates the average of the total power loss over all buses per time step during each episode. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We aim to find algorithms and reward functions with high CR and low PL. ",
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+        "type": "text",
+        "text": "MARL Algorithm Settings. We evaluate the performances of state-of-the-art MARL algorithms, i.e. IDDPG [48], MADDPG [49], COMA [50], IPPO [2], MAPPO [3], SQDDPG [48] and MATD3 [51] on this real-world problem with continuous actions. Since the original COMA can only work for discrete actions, we conduct some modifications so that it can work for continuous actions (see Appendix B). The details of settings are shown in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "5.2 Main Results ",
+        "text_level": 1,
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+    {
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+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "Figure 5: Median CR and PL of algorithms with different voltage barrier functions. The sub-caption indicates metric-Barrier-scenario and BL is the contraction of Bowl. ",
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+        "type": "text",
+        "text": "Algorithm Performances. We first show the main results of all algorithms on all scenarios in Figure 5. MADDPG and MATD3 generally perform well on all scenarios with different voltage barier functions. COMA performs well over CR on 33-bus networks and the performances fall on the large scale scenarios, but its PL is high. Similarly, SQDDPG performs generally well over CR on 33-bus and 141-bus networks, but it performs the worst on 322-bus networks and its PL on 141-bus networks is high. This reveals the limitations of COMA and SQDDPG on the scaling to many agents. Although MAPPO and IPPO performed well on games [2, 3], their performances on the real-world power network problems are poor. The most probable reason could be that the variations of dynamics and uncertainties in distribution networks are far faster and more complicated than games and their conservative policy updates cannot fast respond. IDDPG performs at the middle place in all scenarios, which may be due to non-stationary dynamics caused by multiple agents [49]. ",
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+    {
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+        "img_path": "images/decbf7717ac3061e320aba1ce5ba1455e7eaa19231f9ee6158898fc5dabee485.jpg",
+        "image_caption": [
+            "Figure 6: Median performance of overall algorithms with different voltage barrier functions. The sub-caption indicates metric-scenario. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Voltage Barrier Function Comparisons. To study the effects of different voltage barrier functions, we show the median performance of overall 7 MARL algorithms in Figure 6. It can be observed that Bowl-shape can preserve the high CR, while maintain the low PL on 33-bus and 141-bus networks. Although L1-shape can achieve the best CR on 33-bus and 141-bus networks, its PL on the 141-bus network is the highest. L2-shape performs the worst on 33-bus and 141-bus networks, but performs the best on the 322-bus network with the highest CR and the lowest PL. The reason could be that its slighter gradients is more suitable for the adaption among many agents. From the results, we can conclude that L1-shape, Bowl-shape and L2-shape are the best choices for the 33-bus network, the 141-bus network and the 322-bus network respectively. ",
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+    {
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+        "img_path": "images/4c326a0c932a164750e09479bd378087c9d7e43c518b72bb677cfc0d79ad11cf.jpg",
+        "image_caption": [
+            "Figure 7: Median performances of overall algorithms trained with various rewards consist of distinct voltage barrier functions shown in 3D surfaces. The sub-caption indicates metric-scenario. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Diverse Algorithm Performances under Distinct Rewards. To clearly show the relationship between algorithms and reward functions, we also plot 3D surfaces over CR and PL w.r.t. algorithm types and reward types (consisting of distinct voltage barrier functions) in Figure 7. It is obvious that the performances of algorithms are highly correlated with the reward types. In other words, the same algorithm could perform diversely even trained by different reward functions with the same objective but different shapes. ",
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+    {
+        "type": "text",
+        "text": "5.3 Comparison between MARL and Traditional Control Methods ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To compare MARL algorithms with the traditional control methods, we conduct a series of tests on various network topologies (i.e. 33-bus, 141-bus, and 322-bus networks). MADDPG trained by Bowl-shape is selected as the candidate for MARL. The traditional control methods that we select are OPF [24] and droop control [22]. For conciseness, we only demonstrate the voltages and powers on a typical bus with a PV installed (i.e. one of the most difficult buses to control) during a day (i.e. 480 consecutive time steps) in summer and winter respectively. The results for the 33-bus network is presented here and the results for other typologies are given in the Appendix E.2. From Figure 8, it can be seen that all methods control the voltage within the safety range in both summer and winter. Additionally, the power losses of MARL are lower than droop control but higher than OPF. This phenomenon is possibly due to the fact that droop control is a fully distributed algorithm which cannot explicitly reduce the power loss and OPF is a centralised algorithm with the known system model, while MARL lies between these 2 types of algorithms. This implies that MARL may outperform droop control for particular cases but it needs to be kept in mind that the droop gain used here may not be optimal. It is also worth noting that the control actions of MARL has a similar feature to droop control in the 33-bus case, and this feature is also observed in the 141-bus and 322-bus cases (see Figure 14-15 in the Appendix E.2 for details). However, this droop-like behaviour is missing in the winter case of 322-bus network, where the MARL control action is opposite to the desired direction, resulting in higher power losses and voltage deviation. This might be due to the over-generation as the MARL share the same policy for different seasons, so that it leads to the problem of relative overgeneralisation [52]. In summary, MARL seems able to learn the features of droop control but fails to outperform droop control significantly, and is even worse than droop control for certain cases. ",
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+        "type": "text",
+        "text": "As a result, there is still significant headroom of improvement for MARL, in performance, robustness, and interpretability. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/a7a8081aaf36a2127c0195b301e307b7ad1a7bdc0ea53bdcdfcddef9a06de9b4.jpg",
+        "image_caption": [
+            "Figure 8: Compare MARL with traditional control methods on bus 18 during a day for 33-bus network. 1st row: results for a summer day. 2nd row: results for a winter day. None and limit in (a) represent the voltage with no control and the safety voltage range respectively. P and Q in (b) indicate the PV active power and the reactive power by various methods. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5.4 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We now discuss the phenomenons that we observe from the experimental results. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• It is obvious that the voltage barrier function may impact the performance of an algorithm (even with tiny changes for the common goal). This may be of general importance as many real-world problems may contain constraints that are not indirect in the objective function. An overall methodology is needed for designing barrier functions in state-constraint MARL.   \n• MARL may scale well for the number of agents and the complexity of networks, and only requires a very low control rate for active voltage control.   \n• The results above show the evidence that MARL may behave diversely from the traditional control methods. Some of the learnt behaviours may induce better performance, but some others deteriorate the performance. This shows the promising benefits of MARL in industrial applications but also highlights the drawbacks in interpretability and robustness.   \n• The combination of learning algorithms with domain knowledge is a potential roadmap towards interpretable MARL. For the active voltage control problem, the domain knowledge may present as network topology, inner control loops (say droop control), and load pattern. The exploitation of such domain knowledge reduces the dimensions of MARL exploration space and may offer a lower bound of performance as a guarantee. Encoding domain knowledge such as the network topology as a priori for model-based MARL is also a potential direction. ",
+        "bbox": [
+            171,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper investigates the potential of applying multi-agent reinforcement learning (MARL) to the active voltage control in power distribution networks. We firstly formulate this problem as a DecPOMDP and then study the behaviours of MARL with various voltage barrier functions (reflecting voltage constraints as barrier penalties). Moreover, we compare the behaviours of MARL with the traditional control methods (i.e. droop control and OPF), and observe that MARL is possible to generate inexplicable behaviours, so an interpretable and trustable algorithm is highly desired for industrial applications. Finally, the environment used in this work is open-sourced and easy to follow so that the machine learning community is able to contribute to this challenging real-world problem. ",
+        "bbox": [
+            174,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            89,
+            330,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This work is supported by the Engineering and Physical Sciences Research Council of UK (EPSRC) under awards EP/S000909/1. ",
+        "bbox": [
+            176,
+            121,
+            823,
+            148
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
+            174,
+            169,
+            266,
+            184
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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parse/train/i7aMbliTkHs/i7aMbliTkHs.md

@@ -0,0 +1,352 @@
+# TAM: TEMPORAL ADAPTIVE MODULE FOR VIDEO RECOGNITION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Video data is with complex temporal dynamics due to various factors such as camera motion, speed variation, and different activities. To effectively capture this diverse motion pattern, this paper presents a new temporal adaptive module (TAM) to generate video-specific temporal kernels based on its own feature maps. TAM proposes a unique two-level adaptive modeling scheme by decoupling dynamic kernel into a location sensitive importance map and a location invariant aggregation weight. The importance map is learned in a local temporal window to capture short term information, while the aggregation weight is generated from a global view with a focus on long-term structure. TAM is a principled module and could be integrated into 2D CNNs to yield a powerful video architecture (TANet) with a very small extra computational cost. The extensive experiments on Kinetics400 and Something-Something datasets demonstrate that the TAM outperforms other temporal modeling methods consistently, and achieves the state-of-the-art performance under the similar complexity.
+
+# 1 INTRODUCTION
+
+Deep learning has brought great progress for various recognition tasks in image domain, such as image classification (Krizhevsky et al., 2012; He et al., 2016), object detection (Ren et al., 2017), and instance segmentation (He et al., 2017). The key to these successes is to devise flexible and efficient architectures that are capable of learning powerful visual representations from large-scale image datasets (Deng et al., 2009). However, deep learning research progress in video understanding is relatively more slowly, partially due to the high complexity of video data. The core technical problem in video understanding is to design an effective temporal module, that is expected to be able to capture complex temporal structure with high flexibility, while yet to be of low computational consumption for processing high dimensional video data efficiently.
+
+3D Convolutional Neural Networks (3D CNNs) (Ji et al., 2010; Tran et al., 2015) have turned out to be mainstream architectures for video modeling (Carreira & Zisserman, 2017; Feichtenhofer et al., 2019; Tran et al., 2018; Qiu et al., 2017). The 3D convolution is a natural extension over its 2D counterparts and provides a learnable operator for video recognition. However, this simple extension lacks specific consideration about the temporal properties in video data and might as well lead to high computational cost. Therefore, recent methods aim to improve 3D CNNs from two different aspects by combining a lightweight temporal module with 2D CNNs to improve efficiency (e.g., TSN (Wang et al., 2016), TSM (Lin et al., 2019)), or designing a dedicated temporal module to better capture temporal relation (e.g., Nonlocal Net (Wang et al., 2018b), ARTNet (Wang et al., 2018a), STM (Jiang et al., 2019)). However, how to devise a temporal module with high efficiency and strong flexibility still remains to be an unsolved problem in video recognition. Consequently, we aim at advancing the current video architectures along this direction.
+
+In this paper, we focus on devising a principled adaptive module to capture temporal information in a more flexible way. In general, we observe that video data is with extremely complex dynamics along the temporal dimension due to factors such as camera motion and various speed. Thus 3D convolutions (temporal convolutions) might lack enough representation power to describe motion diversity by simply employing a fixed number of video invariant kernels. To deal with such complex temporal variations in videos, we argue that adaptive temporal kernels for each video are effective and as well necessary to describe motion patterns. To this end, as shown in Figure 1, we present a two-level adaptive modeling scheme to decompose this video specific temporal kernel into a location sensitive importance map and a location invariant (also video adaptive) aggregation kernel. This unique design allows the location sensitive importance map to focus on enhancing discriminative temporal information from a local view, and enables the location invariant aggregation weights to capture temporal dependencies guided by a global view of the input video sequence.
+
+![](images/81893ddb34cd860ec9b42fe467a822d7657cdd27ca72f9f58fbc1b946d2ecd67.jpg)  
+Figure 1: Temporal module comparison: The standard temporal convolution shares weights among videos and may lack the flexibility to handle video variations due to the diversity of videos. The temporal attention learns position sensitive weights by assigning varied importance for different time without any temporal interaction, and may ignore the long-range temporal dependencies. Our proposed temporal adaptive module (TAM) presents a two-level adaptive scheme by learning the local importance weights for location adaptive enhancement and the global kernel weights for video adaptive aggregation. $\odot$ denotes attention operation, and $\otimes$ denotes convolution operation.
+
+Specifically, the design of temporal adaptive module (TAM) strictly follows two principles: high efficiency and strong flexibility. To ensure our TAM with a low computational cost, we first squeeze the feature map by employing a global spatial pooling, and then establish our TAM in a channelwise manner to keep the efficiency. Our TAM is composed of two branches: a local branch $( \mathcal { L } )$ and a global branch $( { \mathcal { G } } )$ . As shown in Fig. 2, TAM is implemented in an efficient way. The local branch employs temporal convolutions to produce the location sensitive importance maps to discriminate the local feature, while the global branch uses fully connected layers to produce the location invariant kernel for temporal aggregation. The importance map generated by a local temporal window focuses on short-term motion modeling and the aggregation kernel using a global view pays more attention to the long-term temporal information. Furthermore, our TAM could be flexibly plugged into the existing 2D CNNs to yield an efficient video recognition architecture, termed as TANet.
+
+We validate the proposed TANet on the task of action classification in video recognition. Particularly, we first study the performance of the TANet on the Kinetics-400 dataset. We demonstrate that our TAM is better at capturing temporal information than other several counterparts, such as temporal pooling, temporal convolution, TSM (Lin et al., 2019), and Non-local block (Wang et al., 2018b). Our TANet is able to yield a very competitive accuracy with the FLOPs similar to 2D CNNs. We further test our TANet on the motion dominated dataset of Something-Something, where the state-of-the-art performance is also achieved.
+
+# 2 RELATED WORKS
+
+Video understanding is a core topic in the field of computer vision. At early stage, a lot of traditional methods (Le et al., 2011; Klaser et al., 2008; Sadanand & Corso, 2012; Willems et al., 2008) have ¨ designed various hand-crafted features to encode the video data, but these methods are too inflexible when generalized to other video tasks. Recently, since the rapid development of video understanding has been much benefited from deep learning methods (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; He et al., 2016), especially in video recognition, a series of CNNs-based methods were proposed to learn spatiotemporal representation, and the differences with our method will be clarified later. Furthermore, our work also relates to dynamic convolution and attention in CNNs.
+
+CNNs-based Methods for Action Recognition. Since the deep learning method has been wildly used in the image tasks, there are many attempts (Karpathy et al., 2014; Simonyan & Zisserman, 2014; Wang et al., 2016; Zhou et al., 2018; He et al., 2019; Lin et al., 2019) based on 2D CNNs devoted to modeling the video clips. In particular, Wang et al. (2016) used the frames sparsely sampled from the whole video to learn the long-range information by aggregating scores after the last fully-connected layer. Lin et al. (2019) shifted the channels along the temporal dimension in an efficient way, which yields a good performance with 2D CNNs. By a simple extension from spatial domain to spatiotemporal domain, 3D convolution (Ji et al., 2010; Tran et al., 2015) was proposed to capture the motion information encoded in video clips. Due to the release of large-scale Kinetics dataset (Kay et al., 2017), 3D CNNs (Carreira & Zisserman, 2017) were wildly used in action recognition. Its variants (Qiu et al., 2017; Tran et al., 2018; Xie et al., 2018) decomposed the 3D convolution into a spatial 2D convolution and a temporal 1D convolution to learn the spatiotemporal features. And Feichtenhofer et al. (2019) designed a network with dual paths to learn the spatiotemporal features and achieved a promising accuracy in video understanding.
+
+The methods aforementioned all share a common insight that they are video invariant and ignore the inherent temporal diversities in videos. As opposed to these methods, we design a two-level adaptive modeling scheme by decomposing the video specific operation into a location sensitive excitation and a location invariant convolution with adaptive kernel for each video clip.
+
+Attention in Action Recognition. The local branch in TAM mostly relates to SENet (Hu et al., 2018). But the SENet learned modulation weights for each channel of feature maps. Several methods (Liu et al., 2019b; Diba et al., 2018) also resorted to the attention to learn more discriminative features in videos. Different from these methods, the local branch keeps the temporal information to learn the location sensitive importances. Wang et al. (2018b) designed a non-local block which can be seen as self-attention to capture long-range dependencies. Our TANet captures the long-range dependencies by simply stacking more TAM, and keep the efficiency of networks.
+
+Dynamic Convolutions. Jia et al. (2016) first proposed the dynamic filters on the tasks of video and stereo prediction, and designed a convolutional encoder-decoder as filter-generating network. Several works (Yang et al., 2019; Chen et al., 2020) in image tasks attempted to generate aggregation weights for a set of convolutional kernels, and then produce a dynamic kernel. Our motivation are different from these methods. We aim to use this temporal adaptive module to deal with temporal variations in videos. Specifically, we design an efficient form to implement this temporal dynamic kernel based on input feature maps, which is critical for understanding the video content.
+
+# 3 METHOD
+
+# 3.1 THE OVERVIEW OF TEMPORAL ADAPTIVE MODULE
+
+As we discussed in Sec.1, video data typically exhibit the complex temporal dynamics caused by many factors such as camera motion and speed variations. Therefore, we aim to tackle this issue by introducing a temporal adaptive module (TAM) with video specific kernels, unlike the sharing convolutional kernel in 3D CNNs. our TAM could be easily integrated into the existing 2D CNNs (e.g., ResNet) to yield a video network architecture, as shown in Figure 2. We will give an overview of TAM and then describe its technical details.
+
+Formally, let $X \in \mathbb { R } ^ { C \times T \times H \times W }$ denote the feature maps for a video clip, where $C$ represents the number of channels, and $T , H , W$ are its spatiotemporal dimensions. For efficiency, our TAM only focus on temporal modeling and the spatial pattern is expected to captured by 2D convolutions. Therefore, we first employ a global spatial average pooling to squeeze the feature map as follows:
+
+$$
+\hat { X } _ { c , t } = \phi ( X ) _ { c , t } = \frac { 1 } { H \times W } \sum _ { i , j } X _ { c , t , j , i } ,
+$$
+
+where $c , t , j , i$ is the index of different dimensions (in channel, time, height and width), and $\hat { X } \in$ $\mathbb { R } ^ { C \times T }$ aggregates the spatial information of $X$ . For simplicity, we here use $\phi$ to denote the function that aggregates the spatial information. Our proposed temporal adaptive module is established based on this squeezed 1D temporal signal for a high efficiency.
+
+![](images/48e87e930690c7939e12fae0a524031b74c47b124dee8230d3da7dda7cdd54bb.jpg)  
+Figure 2: The overall architecture of TANet: ResNet-Block vs. TA-Block. The whole workflow of temporal adaptive module (TAM) in the lower right shows how it works. The shape of tensor has noted after each step. $\oplus$ denotes element-wise addition, $\odot$ denotes element-wise multiplication, and $\otimes$ denotes convolution operator. The symbols appeared in figure will be explained in Sec. 3.1
+
+Our TAM is composed of two branches: a local branch $\mathcal { L }$ and a global branch $\mathcal { G }$ , which aims to learn a location sensitive importance map to enhance discriminative features and then produces the location invariant weights to adaptively aggregate temporal information in a convolutional manner. More specifically, the TAM is formulated as follows:
+
+$$
+Y = { \mathcal { G } } ( X ) \otimes ( { \mathcal { L } } ( X ) \odot X ) ,
+$$
+
+where $\otimes$ denotes convolutional operator and $\odot$ denotes element-wise multiplication. It is worth noting that these two branches focus on different aspects of temporal information, where the local branch tries to capture the short term information to attend important features by using a temporal convolution, while the global branch aims to incorporate long-range temporal structure to guide adaptive temporal aggregation with fully connected layers. Disentangling kernel learning procedures into local and global branches turns out to be an effective way in experiments. The two branches will be introduced in the following sections.
+
+# 3.2 LOCAL BRANCH IN TAM
+
+As discussed above, the local branch is location sensitive and aims to leverage short-term temporal dynamics to perform video specific operation. Given that the short-term information varies slowly along the temporal dimension, it is thus required to learn a location sensitive importance map to discriminate the local temporal semantics.
+
+As shown in Figure 2, we build the local branch with a sequence of temporal convolutional layers with ReLU non-linearity As the goal of local branch is to capture short term information, we set the kernel size $K$ as 3 to learn importance map solely based on a local temporal window. To control the model complexity, the first Conv1D followed by BN (Ioffe & Szegedy, 2015) reduces the number of channels from $C$ to $\frac { C } { \beta }$ . Then, the second Conv1D with a sigmoid activation yields the importance weights $V \in \mathbb { R } ^ { C \times T ^ { \stackrel { r } { } } }$ which are sensitive to temporal location. Finally, the temporal excitation is formulated as follows:
+
+$$
+Z = \operatorname { F } _ { \mathrm { r e s c a l e } } ( V ) \odot X = { \mathcal { L } } ( X ) \odot X ,
+$$
+
+where $\odot$ denotes the element-wise multiplication and $Z ~ \in ~ \mathbb { R } ^ { C \times T \times H \times W }$ . To match size of $X$ , $\mathrm { F } _ { \mathrm { r e s c a l e } } ( V )$ rescales the $V$ to $\hat { V } \in \mathbb { R } ^ { C \times T \times H \times W }$ by replicating in spatial dimension.
+
+# 3.3 GLOBAL BRANCH IN TAM
+
+The global branch, also termed as location invariant branch, focuses on generating an adaptive kernel based on long-term temporal information. It incorporates global context information and learns to produce the location invariant and also video adaptive convolution kernel for dynamic aggregation.
+
+Learning the Adaptive Kernels. We here opt to generate the dynamic kernel for each video clip and aggregate temporal information in a convolutional manner. To simply this procedure and as well as preserve high efficiency, The adaptive convolution will be applied in a channel-wise manner. In this sense, we expect our learned adaptive kernel only considers the temporal relations without taking channel correlation into account. Thus, our TAM would not change the number of channels of input feature maps, and the learned adaptive kernel convolves the input feature maps in a channel-wise manner. More formally, for the $c ^ { t h }$ channel, the adaptive kernel is learned as follows:
+
+$$
+\begin{array} { r } { \Theta _ { c } = \mathcal { G } ( X ) _ { c } = \operatorname { s o f t m a x } ( \mathcal { F } ( \mathbf { W _ { 2 } } , \delta ( \mathcal { F } ( \mathbf { W _ { 1 } } , \phi ( X ) _ { c } ) ) ) ) , } \end{array}
+$$
+
+where $\Theta _ { c } \in \mathcal { R } ^ { K }$ is generated adaptive kernel (aggregation weights) for $c ^ { t h }$ channel, $K$ is the adaptive kernel size, $\delta$ denotes the activation function $R e L U$ . The adaptive kernel is also learned based on the squeezed feature map $\hat { X } _ { c } \in \mathbb { R } ^ { T }$ without taking the spatial structure into account for modeling efficiency. But different with the local branch, we use fully connected $( f c )$ layers $\mathcal { F }$ to learn the adaptive kernel by leveraging long-term information. The learned adaptive kernel with the global receptive field, thus could aggregate temporal features guided by the global context. To increase the modeling capabilities of the global branch, we stack two $f c$ layers and the learned kernel is normalized with a softmax function to yield a positive aggregation weight. The learned aggregation weights $\boldsymbol { \Theta } = \{ \Theta _ { 1 } , \Theta _ { 2 } , . . . , \Theta _ { C } \}$ will be employed to perform video adaptive convolution.
+
+Temporal Adaptive Aggregation. Before introducing the adaptive aggregation, we can look back on how a vanilla temporal convolution aggregates the spatio-temporal visual information:
+
+$$
+Y = W \otimes X ,
+$$
+
+Where $W$ is the weights of convolution kernel and has no concern with input video samples in inference. We argue this fashion ignores the temporal dynamics in videos, and thus propose a video adaptive aggregation to model video clips:
+
+$$
+Y = { \mathcal { G } } ( X ) \otimes X ,
+$$
+
+where $\mathcal { G }$ can be seen as a kernel generator function. Kernel generated by $\mathcal { G }$ can perform adaptive convolution, but is still location invariant and shared cross temporal dimension. To address this issue, the local branch produces $Z$ with location sensitive importance map. The whole procedures can be expressed as follows:
+
+$$
+Y _ { c , t , j , i } = \mathcal { G } ( X ) \otimes Z = \Theta \otimes Z = \sum _ { k } \Theta _ { c , k } \cdot Z _ { c , t + k , j , i } ,
+$$
+
+where $\cdot$ denotes the scalar multiplication, $Y \in \mathbb { R } ^ { C \times T \times H \times W }$ is the output feature maps.
+
+In summary, TAM presents an adaptive module with a unique two-step aggregation scheme, where the location sensitive excitation and location invariant aggregation all derive from input features, but focus on capturing different structures (i.e., short-term and long-term temporal structure).
+
+# 3.4 EXEMPLAR: TANET
+
+We here intend to describe how to instantiate the TANet. Temporal adaptive module as a novel temporal modeling method can endow the existing 2D CNNs with a strong ability to model different temporal structures in video clips. In practice, TAM only causes limited computing overhead, but obviously improves the performance on different types of datasets.
+
+ResNets (He et al., 2016) are employed as backbones to verify the effectiveness of TAM. As illustrated in Fig. 2, the TAM is embedded into ResNet-Block after the first Conv2D, which easily turns the vanilla ResNet-Block into TA-Block. This fashion will not excessively alter the topology of networks and can reuse the weights of ResNet-Block. Supposing we sample T frames as an input clip, the scores of T frames after $f c$ will be aggregated by average pooling to yield the clip-level scores. No temporal downsampling is performed before $f c$ layer. The extensive experiments are conducted in Sec. 4 to demonstrate the flexibility and efficacy of TANet.
+
+# 3.4.1 DISCUSSIONS
+
+We have noticed that the structure of local branch is similar to the SENet (Hu et al., 2018) and STC (Diba et al., 2018). The first obvious difference is our local branch does not squeeze the temporal information. We thus use temporal 1D convolution as a basic layer, instead of using $f c$ layer. Two-layer design only seeks to make a trade-off between non-linear fitting capability and model complexity. Furthermore, the local branch provides the location sensitive information, and thus addresses the issue that the global branch is insensitive to temporal location.
+
+TSN (Wang et al., 2016), TSM (Lin et al., 2019), etc. only aggregate the temporal features with a fixed scheme, but our TAM can yield the video specific weights to adaptively aggregate the temporal features in different stages. In the extreme cases, our global branch in TAM can degenerate into TSN when dynamic kernel weights $\Theta$ is learned to equal to $[ 0 , 1 , 0 ]$ . From another perspective, if the kernel weights $\Theta$ is set to $[ 1 , 0 , 0 ]$ or $[ 0 , 0 , 1 ]$ , global branch can be turned into TSM. It seems that our TAM theoretically provides a more general and flexible form to model the video data. When it refers to 3D convolution (Ji et al., 2010), all input samples share the same convolution kernel without being aware of the temporal diversities in videos as well. In addition, our global branch essentially performs a video adaptive convolution whose filter has size $1 \times k \times 1 \times 1$ , while each filter in a normal 3D convolution has size $C \times k \times k \times k$ , where $C$ is the number of channels and $\mathbf { k }$ denotes the receptive field. Thus our method is more efficient than 3D CNNs. Unlike some current dynamic convolution (Chen et al., 2020; Yang et al., 2019), TAM is more flexible, and can directly generate the kernel weights to perform video adaptive convolution.
+
+# 4 EXPERIMENTS
+
+In this section, we elaborately study the effectiveness of TANet on several standard benchmarks.   
+The training recipe and inference protocol is described in Appendix (A.1).
+
+# 4.1 DATASETS
+
+Our experiments are conducted on three large scale datasets, namely, Kinetics-400 (Kay et al., 2017) and Something-Something (Sth-Sth) V1&V2 (Goyal et al., 2017). Kinetics-400 contains ${ \sim } 3 0 0 \mathrm { k }$ video clips with 400 human action categories. The videos in Kinetics-400, trimmed from raw YouTube videos, are around 10s. We here train models on training set ${ \sim } 2 4 0 \mathrm { k }$ video clips), and test models on validation set ( ${ \sim } 2 0 \mathrm { k }$ video clips). The Sth-Sth datasets focus on fine-grained action, which contains a series of pre-defined basic actions interacted with daily objects. The Sth-Sth V1 comprises ${ \sim } 8 6 \mathrm { k }$ video clips in training set and ${ \sim } 1 2 \mathrm { k }$ video clips in validation set. Sth-Sth V2 is an updated version of Sth-Sth V1, which contains ${ \sim } 1 6 9 \mathrm { k }$ video clips in training set and ${ \sim } 2 5 \mathrm { k }$ video clips in validation set. They both have 174 action categories.
+
+# 4.2 EXPLORATION STUDIES ON KINETICS-400
+
+The exploration studies are performed on Kinetics-400 to investigate different aspects of TANet. The ResNet architecture we used is the same with He et al. (2016). Our TANet replaces all ResNetBlocks with TA-Blocks by default.
+
+Table 1: Ablation studies on Kinetics-400. All models use ResNet50 as backbone.   
+(a) Studying on parameter choices of $\alpha$ and $\beta$ .   
+
+<table><tr><td>seting</td><td>Frames</td><td>Top-1</td><td>Top-5</td></tr><tr><td>α=1,β=4</td><td>8</td><td>75.63%</td><td>92.10%</td></tr><tr><td>α=2, β=4</td><td>8</td><td>76.09%</td><td>92.30%</td></tr><tr><td>α=4, β=4</td><td>8</td><td>75.72%</td><td>92.14%</td></tr><tr><td>a=2, β=2</td><td>8</td><td>75.91%</td><td>92.38%</td></tr><tr><td>Q=2, β=4</td><td>8</td><td>76.09%</td><td>92.30%</td></tr><tr><td>α=2,β=8</td><td>8</td><td>75.63%</td><td>92.20%</td></tr></table>
+
+(b) Trying larger temporal receptive fields of $\Theta$ .   
+
+<table><tr><td>Kernel</td><td>Frames</td><td>Top-1</td><td>Top-5</td></tr><tr><td>K=3</td><td>8</td><td>76.09%</td><td>92.30%</td></tr><tr><td>K=5</td><td>8</td><td>75.62%</td><td>92.14%</td></tr><tr><td>K=3</td><td>16</td><td>76.87%</td><td>92.88%</td></tr><tr><td>K=5</td><td>16</td><td>77.19%</td><td>93.17%</td></tr></table>
+
+Table 2: Study on the effectiveness of TAM. All models use ResNet50 as backbone and take 8 frames with sampling stride 8 as inputs. To be consistent with testing, the FLOPs are calculated with spatial size $2 5 6 \times 2 5 6$ . All methods share the same training setting and inference protocol.   
+
+<table><tr><td>Models</td><td>FLOPs (of single view)</td><td>Params</td><td>Top-1</td><td>Top-5</td></tr><tr><td>C2D</td><td>42.95G</td><td>24.33M</td><td>70.2%</td><td>88.9%</td></tr><tr><td>C2D-Pool</td><td>42.95G</td><td>24.33M</td><td>73.1%</td><td>90.6%</td></tr><tr><td>C2D-TConv</td><td>53.02G</td><td>28.10M</td><td>73.3%</td><td>90.7%</td></tr><tr><td>TSM (Lin et al., 2019)</td><td>42.95G</td><td>24.33M</td><td>74.1%</td><td>91.2%</td></tr><tr><td>TEINet (Liu et al.,2019b)</td><td>43.01G</td><td>25.11M</td><td>74.9%</td><td>91.8%</td></tr><tr><td>I3D3×1×1 (Wang et al.,2018b)</td><td>62.55G</td><td>32.99M</td><td>74.3%</td><td>91.6%</td></tr><tr><td>NL C2D (Wang et al.,2018b)</td><td>64.49G</td><td>31.69M</td><td>74.4%</td><td>91.5%</td></tr><tr><td>Global branch</td><td>43.00G</td><td>24.33M</td><td>74.9%</td><td>91.7%</td></tr><tr><td>Local branch</td><td>43.00G</td><td>25.59M</td><td>73.3%</td><td>90.7%</td></tr><tr><td>Global branch + SE (Hu et al.,2018)</td><td>43.02G</td><td>24.65M</td><td>75.4%</td><td>92.0%</td></tr><tr><td>TANet-R</td><td>43.02G</td><td>25.59M</td><td>76.0%</td><td>92.2%</td></tr><tr><td>TANet</td><td>43.02G</td><td>25.59M</td><td>76.1%</td><td>92.3%</td></tr></table>
+
+Parameter choices. We use different combinations of $\alpha$ and $\beta$ to figure out the optimal hyperparameters in TAM. The TANet is instantiated as in Fig. 2. TANet with $\alpha = 2$ and $\beta = 4$ achieves the highest performance shown in Table 1a, which will be applied in following experiments.
+
+Temporal receptive fields. We also try to increase the temporal receptive fields for learned kernel $\Theta$ in the global branch. From the Table 1b, it seems the larger $K$ is beneficial to the accuracy when TANet takes more sampled frames as inputs. On the other hand, it even degenerates the performance of TANet when sampling 8 frames. In our following experiments, the $K$ will be set to 3.
+
+The results in Table 1 have revealed that our TANet is insensitive to these Hyper-parameters, which can save a lot of time to find optimal settings in practice. We also provide other more exploration studies in Appendix (A.2), and show how these hyper-parameters impact the performance of TANet.
+
+# 4.3 COMPARISON WITH OTHER TEMPORAL MODULES
+
+As a principled temporal operator, we intend to describe its competitive counterparts and then make fair comparisons with TAM in Table 2. The optimal configurations studied in Sec. 4.2 will be employed in the following experiments. The default inference protocol samples 10 clips $\times 3$ crops to evaluate the performance of each model.
+
+2D ConvNet (C2D). We use ResNet50 as backbone to build 2D ConvNet. The 2D ConvNet focuses on learning the spatial clues, which operates on each frame independently without any temporal interaction before the last $f c$ layer.
+
+C2D-Pool. To probe into the impacts of temporal fusion, C2D-Pool utilizes the average pooling layer whose kernel size is $K \times 1 \times 1$ to perform temporal fusion without any temporal downsampling, which can be built by easily replacing all TAMs in network with pooling layers.
+
+C2D-TConv. We also replace each TAM with a standard temporal convolution, and this comparison is able to demonstrate the importance of adaptive modeling in temporal aggregation.
+
+Table 3: Extending to other backbones. All models share the same inference protocol, e.g., 10 clips $\times 3$ crops.   
+
+<table><tr><td rowspan="2">Models</td><td colspan="2">ShuffleNet V2</td><td colspan="2">MobileNet V2</td><td colspan="2">Inception V3</td><td colspan="2">ResNet-50</td><td colspan="2">ResNext-50</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td></tr><tr><td>w/o TAM</td><td>62.1%</td><td>84.3%</td><td>64.1%</td><td>85.6%</td><td>71.4%</td><td>89.8%</td><td>70.7%</td><td>88.9%</td><td>70.1%</td><td>88.8%</td></tr><tr><td>with TAM</td><td>67.3%</td><td>87.6%</td><td>71.6%</td><td>90.1%</td><td>75.6%</td><td>92.0%</td><td>76.1%</td><td>92.3%</td><td>76.4%</td><td>92.0%</td></tr><tr><td>△Acc.</td><td>+ 5.2%</td><td>+ 3.3%</td><td>+ 7.5%</td><td>+ 4.5%</td><td>+ 4.2%</td><td>+ 2.2%</td><td>+ 5.4%</td><td>+ 3.4%</td><td>+ 6.3%</td><td>+3.2%</td></tr></table>
+
+There are some competitive methods based on C2D, i.e., TSM (Lin et al., 2019) and TEINet (Liu et al., 2019b). The new methods focus on designing a lightweight temporal module and can be inserted into the 2D CNN for efficiently capturing temporal information.
+
+Inflated 3D ConvNet (I3D). I3D (Carreira & Zisserman, 2017) is most frequently used models in action recognition. In our implementation, we inflate the first $1 \times 1$ kernel in ResNet-Block to $3 \times 1 \times 1$ , which can provide more fair comparisons with our TANet. Following the (Wang et al., 2018b), we use $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ to denote this variant.
+
+The aforementioned methods share the same temporal modeling scheme with a fixed pooling or convolution operation. As shown in Table 2, our method yield a superior performance that higher than C2D by $5 . 9 \%$ accuracy, and even outperforms $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ $7 6 . 1 \%$ vs. $7 4 . 3 \%$ ), which exhibits the fixed schemes for modeling videos may be insufficient to learn the temporal clues. As the extra convolutions in C2D-TConv might destroy the ImageNet pretrained weights, C2D-TConv even achieves a degenerated performance compared with $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ . In addition, compared with other temporal counterparts in video recognition, TANet only brings a small portion of FLOPs and parameters.
+
+Non-local C2D (NL C2D). The non-local block was proposed to capture the long-range dependencies in videos. The preferable settings with 5 non-local blocks mentioned in (Wang et al., 2018b) are employed to compare with TANet. As seen in Table 2, TANet achieves higher accuracy than NL C2D $7 6 . 1 \%$ vs. $7 4 . 4 \%$ ). In addition, TANet is more efficient than NL C2D. TANet only has 43G FLOPs of single view and 25.6M parameters.
+
+To study the each part of temporal adaptive module, we separately validate the Global branch and Local branch. Furthermore, Global branch $\mathbf { \mu } + \mathbf { S E }$ uses global branch with SE module (Hu et al., 2018) to compare with TANet, which can prove the complementarity of local branch and global branch. TANet has achieved the highest accuracy among these models, which proves the efficacy of each part of TAM and as well as the strong complementarity between local branch and global branch. As opposed to Equ. 2, TANet-R combines the global and local branch in reverse order:
+
+$$
+Y = { \mathcal { L } } ( X ) \odot ( { \mathcal { G } } ( X ) \otimes X ) ,
+$$
+
+We found TANet is slightly better than TANet-R.
+
+Generalization to other Backbones. One critical issue that we need to figure out is the generalization of our proposed method extending to other backbone networks. To this end, we extend the TAM from ResNet to other well known classification backbones, like ShuffleNet V2 (Ma et al., 2018), MobileNet V2 (Sandler et al., 2018), Inception V3 (Szegedy et al., 2016) and ResNeXt50 (Xie et al., 2017). From the Table 3, we can observe that the backbone networks equipped with our TAM outperform their C2D baselines by a large margin, which strongly exhibits the powerful generalization as well as the huge potential of our temporal adaptive module.
+
+# 4.4 COMPARISONS WITH THE STATE OF THE ART
+
+Comparisons on Kinetics-400. Table 4 shows the state-of-the-art results on Kinetics-400. Our method, as an adaptive modeling scheme, has achieved competitive performance compared with other models. TANet-50 with 8-frame also outperforms SlowFast (Feichtenhofer et al., 2019) by $0 . 5 \%$ when using similar FLOPs per view. The 16-frame TANet only uses 4 clips and 3 crops for evaluation such that it provides higher inference efficiency and more fair comparisons with other models. It is worth noting that our 16-frame TANet-50 is still more accurate than 32-frame NL I3D by $2 . 2 \%$ . As ip-CSN (Tran et al., 2019) is pretrained on Sports-1M (Karpathy et al., 2014), it also achieves the promising accuracy with deeper backbone, i.e., ResNet152. Furthermore, TAM is compatible with existing video frameworks like SlowFast. Specifically, our TAM is more lightweight than $3 \times 1 \times 1$ convolution when taking the same number of frames as inputs, but yields a better performance. TAM thus can easily replace the $3 \times 1 \times 1$ convolution in SlowFast to achieve higher accuracy with lower computational costs. It seems that X3D has achieved a great success in video recognition. In other ways, X3D was searched by massive computing resource and can not be easily obtained in any situation. Although our method does not beat all state-of-the-art methods equipped with deeper networks, TAM as a lightweight operator can enjoy the benefits from more powerful backbones and video frameworks. In general, the proposed TANet makes a good practice on adaptively modeling the temporal relations in videos.
+
+Table 4: Comparisons with the state-of-the-art methods on Kinetics-400. As described in Feichtenhofer et al. (2019), the GFLOPs (of a single view) $\times$ the number of views (temporal clips with spatial crops) represents the model complexity. The GFLOPs are calculated with spatial size $2 5 6 \times 2 5 6$ . The models in gray rows represent it is unjust to directly compare with our method.   
+
+<table><tr><td>Methods</td><td>Backbones</td><td>Training Input</td><td>GFLOPs×views</td><td>Top-1</td><td>Top-5</td></tr><tr><td>TSN (Wang et al., 2016)</td><td>InceptionV3</td><td>3×224×224</td><td>3×250</td><td>72.5%</td><td>90.2%</td></tr><tr><td>ARTNet (Wang et al.,2018a)</td><td>ResNet18</td><td>16 ×112×112</td><td>23.5×250</td><td>70.7%</td><td>89.3%</td></tr><tr><td>S3D-G (Xie et al.,2018)</td><td>InceptionV1</td><td>64×224×224</td><td>71×30</td><td>74.7%</td><td>93.4%</td></tr><tr><td>I3D (Carreira &amp; Zisserman, 2017)</td><td>InceptionV1</td><td>64×224×224</td><td>108×N/A</td><td>72.1%</td><td>90.3%</td></tr><tr><td>R(2+1)D(Tran et al.,2018)</td><td>ResNet34</td><td>32×112×112</td><td>152×10</td><td>74.3%</td><td>91.4%</td></tr><tr><td>NL I3D (Wang et al., 2018b)</td><td>ResNet50</td><td>32×224×224</td><td>N/A</td><td>74.9%</td><td>91.6%</td></tr><tr><td>NL I3D (Wang et al.,2018b)</td><td>ResNet50</td><td>128×224×224</td><td>282×30</td><td>76.5%</td><td>92.6%</td></tr><tr><td>ip-CSN (Tran et al., 2019)</td><td>ResNet50</td><td>8×224×224</td><td>1.2×10</td><td>70.8%</td><td>-%</td></tr><tr><td>TSM (Lin et al., 2019)</td><td>ResNet50</td><td>16×224×224</td><td>65×30</td><td>74.7%</td><td>91.4%</td></tr><tr><td>TEINet (Liu et al.,2019b)</td><td>ResNet50</td><td>16×224×224</td><td>86×30</td><td>76.2%</td><td>92.5%</td></tr><tr><td>bLVNet-TAM-24×2</td><td>bLResNet50</td><td>48×224×224</td><td>93×9</td><td>73.5%</td><td>91.2%</td></tr><tr><td>SlowOnly (Feichtenhofer et al.,2019)</td><td>ResNet50</td><td>8×224×224</td><td>42×30</td><td>74.8%</td><td>91.6%</td></tr><tr><td>SlowFast (Feichtenhofer et al., 2019)</td><td>ResNet50</td><td>(4+32)×224×224</td><td>36×30</td><td>75.6%</td><td>92.1%</td></tr><tr><td>TANet-50</td><td>ResNet50</td><td>8×224×224</td><td>43×30</td><td>76.1%</td><td>92.3%</td></tr><tr><td>TANet-50</td><td>ResNet50</td><td>16×224×224</td><td>86×12</td><td>76.9%</td><td>92.9%</td></tr><tr><td>X3D-XL (Feichtenhofer,2020)</td><td></td><td>16×312x312</td><td>48×30</td><td>79.1%</td><td>93.9%</td></tr><tr><td>ip-CSN (Tran et al., 2019)</td><td>ResNet152</td><td>32 ×224×224</td><td>83×30</td><td>79.2%</td><td>93.8%</td></tr><tr><td>SlowFast+NL (Feichtenhofer et al., 2019)</td><td>ResNet101</td><td>(16+64)×224×224</td><td>234×30</td><td>79.8%</td><td>93.9%</td></tr></table>
+
+More Results and Analysis. The results on Sth-Sth V1 & V2 is presented in Appendix (A.3) in which TAM also achieves a competitive performance compared with other methods. To have more intuitive understandings of temporal adaptive module, we also visualize the learned kernels in Appendix (A.4), which are expected to provide more insights for TAM. As shown in Figure 4, the diversities in our learned kernels have shown that the complex dynamics are indeed existing in videos, and learning temporal clues in an adaptive scheme has proven to be effective yet reasonable.
+
+# 5 CONCLUSION
+
+In this paper, we have presented a novel temporal modeling operator, i.e., temporal adaptive module (TAM), to capture complex motion information in videos and built a powerful video architecture (TANet). Our TAM is able to yield a video-specific kernels with the combination of a local importance map and a global aggregation weight. The local and global branches designed in TAM are helpful to capture temporal structure by different views and contribute to making temporal modeling more effective and robust. As demonstrated on the Kinetics-400, the networks equipped with TAM are better than the existing temporal modules in action recognition, which confirms the efficacy of our TAM in video temporal modeling. TANet also achieved the state-of-the-art performance on the motion dominated datasets of Sth-Sth V1&V2.
+
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+![](images/8c4a966b4d8f2f6af7c0d801575795f955c749e29fef0957dd2d206fa27def15.jpg)  
+Figure 3: The four styles of TA-Block. The (b) is actually the model we used in the main text.
+
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+
+# A APPENDIX
+
+# A.1 IMPLEMENTATION DETAILS
+
+Training. In our experiments, we only train the models using 8 frames and 16 frames as inputs. On Kinetics-400, following the practice in Wang et al. (2018b), The frames are sampled from 64 consecutive frames in the video. On Sth-Sth V1&V2, we employ the uniform sampling strategy in TSN (Wang et al., 2016) to train TANet. We first resize the shorter side of frames to 256, and apply the multi-scale cropping and randomly horizontal flipping as data augmentation. The cropped frames are resized to $2 2 4 \times 2 2 4$ for training the networks. The batch size is set to 64. Our models are initialized by ImageNet pre-trained weights to reduce the training time. Specifically, on Kinetics400, the epoch for training is 100. The initial learning rate is set 0.01 and divided by 10 at 50, 75, 90 epoch. We use SGD with a momentum of 0.9 and a weight decay of 1e-4 to train TANet. On Sth-Sth V1&V2, We train models with 50 epochs. The learning rate starts at 0.01 and divided by 10 at 30, 40, 45 epoch. We use a momentum of 0.9 and a weight decay of 1e-3 to address the issue of overfitting.
+
+Testing. We apply different inference schemes to fairly compare with other state-of-the-art models. On kinetics-400, we resize the shorter to 256 and take 3 crops of $2 5 6 \times 2 5 6$ to cover the spatial dimensions. In the temporal dimension, we uniformly sample 10 clips for 8-frame models and 4 clips for 16-frame models. The final video-level prediction is yielded by averaging the scores of all spatio-temporal views. On Sth-Sth V1, we scale the shorter side of frames to 256 and use center crop of $2 2 4 \times 2 2 4$ for evaluation. On Sth-Sth V2, we employ similar evaluation protocols to Kinetics, but only uniformly sample 2 clips.
+
+# A.2 MORE EXPLORATION STUDIES ON KINETICS-400
+
+TAM in the different position. Table 5a tries to study the effects of TAM in different position. TANet-a, TANet-b, TANet-c, and TANet-d denotes the TAM is inserted before the first convolution, after the first convolution, after the second convolution, and after the last convolution in the block, respectively. These four styles are graphically presented in Fig. 3. The style-(b) in Fig. 2 actually is TANet-b, which has a slighter advantage than other styles as shown in Table 5a. The TANet-b will be abbreviated as TANet by default.
+
+Table 5: Ablation studies on Kinetics-400. All models use ResNet50 as backbone.   
+(a) Where to insert TAM into TA-Block.   
+
+<table><tr><td>model</td><td>Frames</td><td>Top-1</td><td>Top-5</td></tr><tr><td>TANet-a</td><td>8</td><td>75.95%</td><td>92.18%</td></tr><tr><td>TANet-b</td><td>8</td><td>76.09 %</td><td>92.30%</td></tr><tr><td>TANet-c</td><td>8</td><td>75.75%</td><td>92.13%</td></tr><tr><td>TANet-d</td><td>8</td><td>75.20%</td><td>91.78%</td></tr></table>
+
+(b) The number of TA-Blocks inserted into ResNet50.
+
+<table><tr><td>stages</td><td>Frames</td><td>Blocks</td><td>Top-1</td><td>Top-5</td></tr><tr><td>res5</td><td>8</td><td>3</td><td>74.12%</td><td>91.45%</td></tr><tr><td>res4-5</td><td>8</td><td>9</td><td>75.15%</td><td>92.04%</td></tr><tr><td>res3-5</td><td>8</td><td>13</td><td>75.90%</td><td>92.22%</td></tr><tr><td>res2-5</td><td>8</td><td>16</td><td>76.09%</td><td>92.30%</td></tr></table>
+
+The number of TA-Blocks. To make a trade-off between performance and efficiency, we gradually add more TA-Blocks into ResNet. As shown in Table 5b, we find that the performance is nearly saturated when adding more than 9 TA-Blocks into network. The $\mathrm { r e s } _ { 2 - 5 }$ achieves the highest performance and will be used in our experiments.
+
+# A.3 COMPARISONS ON STH-STH V1 & V2
+
+Table 6: Comparisons with the state-of-the-art methods on Sth-Sth V1. The models only taking RGB frames as inputs are listed in table. To be consistent with testing, we use spatial size $2 2 4 \times 2 2 4$ to compute the FLOPs.
+
+<table><tr><td rowspan=3 colspan=1>Methods</td><td rowspan=3 colspan=6>Backbones</td><td rowspan=3 colspan=4>Pre-train</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=2 colspan=1>Frames</td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>FLOPs</td><td rowspan=1 colspan=1>Top-1</td><td rowspan=1 colspan=1>Top-5</td></tr><tr><td rowspan=1 colspan=1>TSN-RGB (Wang et al.,2016)TRN-Multiscale (Zhou et al., 2018)S3D-G (Xie et al.,2018)ECO (Zolfaghari et al.,2018)ECOEnLite (Zolfaghari et al.,2018)</td><td rowspan=1 colspan=6>BNInceptionBNInceptionInceptionBNIncep+Res18BNIncep+Res18</td><td rowspan=1 colspan=4>ImgNetImgNetImgNetK400K400</td><td rowspan=1 colspan=1>8f8f64f16f92f</td><td rowspan=1 colspan=1>16G33G71.38G64G267G</td><td rowspan=1 colspan=1>19.5%34.4%48.2%41.6%46.4%</td><td rowspan=1 colspan=1>=-78.7%-■</td></tr><tr><td rowspan=6 colspan=1>TSN (Wang et al.,2016)I3D (Wang&amp; Gupta,2018)NL I3D(Wang&amp; Gupta,2018)NL I3D+GCN(Wang &amp; Gupta,2018)TSM (Lin et al., 2019)TSM (Lin et al., 2019)TSMEn (Lin et al.,2019)TAM (Fan et al.,2019)bLVNet-TAM (Fan et al.,2019)GST (Luo &amp; Yuille,2019)GST (Luo &amp; Yuille,2019)TEINet Liu et al. (2019b)TEINet Liu et al. (2019b)</td><td rowspan=6 colspan=6>ResNet50ResNet50ResNet50ResNet50+GCNResNet50ResNet50ResNet50ResNet50ResNet50ResNet50ResNet50ResNet50ResNet50</td><td rowspan=1 colspan=4>ImgNetImgNet+K400ImgNet+K400ImgNet+K400ImgNetImgNet</td><td rowspan=1 colspan=1>8f32f×232f×232f×28f16f</td><td rowspan=6 colspan=1>33G306G334G606G33G65G98G-48G30G59G33G66G</td><td rowspan=6 colspan=1>19.7%41.6%44.4%46.1%45.6%47.2%49.7%46.1%48.4%47.0%48.6%47.4%49.9%</td><td rowspan=4 colspan=1>46.6%72.2%76.0%76.8%74.2%77.1%78.5%-%78.8%76.1%77.9%</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=2 colspan=4>ImgNetImgNet</td><td rowspan=1 colspan=1>16f+8f</td></tr><tr><td rowspan=2 colspan=1>8f32f8f16f</td></tr><tr><td rowspan=1 colspan=2>50</td><td rowspan=1 colspan=4>Sth-Sth V2ImgNetImgNet</td><td rowspan=1 colspan=2>SthV2</td></tr><tr><td rowspan=2 colspan=3>ResNet50</td><td rowspan=2 colspan=1></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=4>ImgNetImgNet</td><td rowspan=1 colspan=1>8f16f</td><td rowspan=1 colspan=1>-%-%</td></tr><tr><td rowspan=2 colspan=1>TANetTANetTANetEn</td><td rowspan=1 colspan=3></td><td rowspan=2 colspan=4>ResNet50ResNet50ResNet50</td><td rowspan=1 colspan=3>ResNet50</td><td rowspan=1 colspan=1>ImgNetImgNet</td><td rowspan=1 colspan=1>8f16f</td><td rowspan=2 colspan=1>33G66G99G</td><td rowspan=2 colspan=1>46.5%47.6%50.6%</td></tr><tr><td></td><td></td><td></td><td rowspan=1 colspan=4>ImgNet</td><td rowspan=1 colspan=1>8f+16f</td><td></td></tr></table>
+
+Comparisons on Sth-Sth V1 & V2. As shown in Table 6, our method achieves state-of-the-art accuracy comparing with other models on Sth-Sth V1. For fair comparisons, the Table 6 only reports the results taking a single clip with a center crop as inputs. $\mathrm { T A N e t } _ { E n }$ is higher than $\mathrm { T S M } _ { E n }$ equipped with same backbone (Top-1: $5 0 . 6 \%$ vs. Top-1: $4 9 . 7 \%$ ). We also conduct the experiments on Sth-Sth V2. V2 has more video clips than V1, which can further unleash the full capabilities of TANet without suffering the overfitting. Following the common practice in Lin et al. (2019), TANets use 2 clips with 3 crops to evaluate the accuracy. As shown in Table 7, our models have achieved the state-of-art performance on Sth-Sth V2. As a result, the $\mathrm { T A N e t } _ { E n }$ yields a competitive accuracy (Top-1: $6 6 . 0 \%$ ) compared with current SOTA results. The results on Sth-Sth V1 & V2 have demonstrated that our method is also good at modeling the fine-grained and temporal-related video clips.
+
+![](images/428212b3502a21e051359a1ee726241c75caba20c68d421e895f8101f7d1a083.jpg)  
+Figure 4: The statistics of kernel weights training on Kinetics-400, which plots the distributions in different temporal offsets $( t \in \{ - 1 , \bar { 0 } , 1 \} )$ ). Each filled area in violinplot represents the entire data range, where has noted the minimum, the median and the maximum. The first four columns in the left figure are the distributions of learned kernels in TANet. In the fifth column, we also visualize the filters of $3 \times 1 \times 1$ kernel in $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ to compare with the TANet. The stage4 6b denotes the kernel comes from the 6th block in stage4.
+
+Table 7: Comparisons with the SOTA on Sth-Sth V2.   
+
+<table><tr><td>Methods</td><td>Backbones</td><td>Pre-train</td><td>frames×clips×crops</td><td>Top-1</td><td>Top-5</td></tr><tr><td>TRN (Zhou et al.,2018)</td><td>BNInception</td><td>ImgNet</td><td>8f×2x3</td><td>48.8%</td><td>77.6%</td></tr><tr><td>TSM (Lin et al., 2019)</td><td>ResNet50</td><td>ImgNet</td><td>8f×2x3</td><td>59.1%</td><td>85.6%</td></tr><tr><td>TSM (Lin et al., 2019)</td><td>ResNet50</td><td>ImgNet</td><td>16×2×3</td><td>63.4%</td><td>88.5%</td></tr><tr><td>TSMRGB+Flow (Lin et al.,2019)</td><td>ResNet50</td><td>ImgNet</td><td>(16+16)×2×3</td><td>66.0%</td><td>90.5%</td></tr><tr><td>CPNet (Liu et al.,2019a)</td><td>ResNet50</td><td>ImgNet</td><td>24f×16×16</td><td>57.7%</td><td>84.0%</td></tr><tr><td>GST(Luo &amp; Yuille,2019)</td><td>ResNet50</td><td>ImgNet</td><td>8fx1x1</td><td>61.6%</td><td>87.2%</td></tr><tr><td>GST(Luo &amp; Yuille,2019)</td><td>ResNet50</td><td>ImgNet</td><td>16fx1×1</td><td>62.6%</td><td>87.9%</td></tr><tr><td>bLVNet-TAM (Fan et al.,2019)</td><td>ResNet50</td><td>Sth-Sth V2</td><td>32fx1×1</td><td>61.7%</td><td>88.1%</td></tr><tr><td>TEINet Liu et al. (2019b)</td><td>ResNet50</td><td>ImgNet</td><td>8fx1x1</td><td>61.3%</td><td>-%</td></tr><tr><td>TEINet Liu et al. (2019b)</td><td>ResNet50</td><td>ImgNet</td><td>16fx1×1</td><td>62.1%</td><td>-%</td></tr><tr><td>TANet</td><td>ResNet50</td><td>ImgNet</td><td>8fx1×1</td><td>60.5%</td><td>86.2%</td></tr><tr><td>TANet</td><td>ResNet50</td><td>ImgNet</td><td>8f×2x3</td><td>62.7%</td><td>88.0%</td></tr><tr><td>TANet</td><td>ResNet50</td><td>ImgNet</td><td>16×1×1</td><td>62.5%</td><td>87.6%</td></tr><tr><td>TANet</td><td>ResNet50</td><td>ImgNet</td><td>16×2×3</td><td>64.6%</td><td>89.5%</td></tr><tr><td>TANetEn</td><td>ResNet50</td><td>ImgNet</td><td>(8f+16f)x2x3</td><td>66.0%</td><td>90.1%</td></tr></table>
+
+# A.4 VISUALIZATIONS OF LEARNED KERNEL
+
+To understand the behavior of TANet, we visualize the distribution of kernel $\Theta$ generated by global branch in the last block of stage4 and stage5. For clear contrast, the kernel weights in $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ at the same stages are also visualized to find more insights. As depicted in Fig. 4, we find that the learned kernel $\Theta$ has an evident character: the shapes and scales of distribution are more diverse than $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ . Since all video clips share the same kernels in $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ , it causes the kernel weights clusters together excessively. As opposed to temporal convolution, even modeling the same action in different videos, TAM can generate the kernel with slightly different distributions. Taking driving car as an example, the shapes of the distribution shown in Fig. 4 are similar to each other but the medians of distributions are not equal. For different actions like drinking beer and skydiving, the shapes and medians of distributions are greatly varied. Even the same action in different videos, TAM would learn a different distribution of kernel weights. Concerning that the motion patterns in different videos may share varied inherence, it is necessary to employ an adaptive scheme when modeling video sequences.
+
+To probe into the effects on learning kernels in the different stages, the visualized kernels are further chosen in stage4 6b and stage5 3b, respectively. The videos are randomly selected from Kinetics400 and Sth-Sth V2 to show the diversities in different video datasets. As depicted in Fig. 5 and Fig. 6, We can observe that the distributions of importance map $V$ in the local branch are smoother than the kernel $\Theta$ in the global branch, and the local branch pays different attention to each video when modeling the temporal relations. Futhermore, our learned kernels visualized in figures have exhibited the clear differences between two datasets (Kinetics-400 vs. Sth-Sth V2). This fact is in line with our prior knowledge that there is an obvious domain gap between two datasets. The Kinetics-400 mainly focuses on appearance and Sth-Sth V2 is a motion dominated dataset. However, this point can not be easily inferred from the kernels in $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ , because the overall distributions of kernels in $\mathrm { I } 3 \mathrm { D } _ { 3 \times 1 \times 1 }$ on two datasets show minor differences.
+
+We visualize the histogram of kernel weights in the global branch to intuitively show the insightful patterns between kernel weights and visual content. As observing the Fig. 7 and Fig. 8, we found the distribution of kernel is associated with the motion magnitude. The smaller magnitude of motion in video leads to a relatively lower weight at $t = 0$ , and the larger magnitude of motion or the actions severely related to scenes may cause the higher weight at $t = 0$ . It is worth noting that the weight at $\scriptstyle \mathrm { t = } 0$ usually plays a dominate role in learning the spatio-temporal representations. These findings are expected to provide more insights for designing the temporal module in video recognition.
+
+Generally, the diversities in our learned kernels have demonstrated that the diversities are indeed existing in videos, and it is reasonable to learn spatiotemporal representation in an adaptive scheme. These findings are again in line with our motivation claimed in the paper.
+
+![](images/f7327f70f28ac4e7f244b553af5cf2bc6565f7e7b004aaf96ce817c3639f485b.jpg)  
+Figure 5: The distribution of learned kernel $V$ and $\Theta$ in the stage4 6b.
+
+![](images/7ea765cdbb74fc052b4d9f2723a374dd17e9cdad7df4b5e1388a98264d902f21.jpg)  
+Figure 6: The distribution of learned kernel $V$ and $\Theta$ in the stage5 3b.
+
+![](images/f19f5004020475f5879e68840c1965553b1454f768d7d02bb0b7f5bdae506cd6.jpg)  
+Figure 7: The visualization for the videos which contain small magnitude of motion. The moving persons appeared in videos are usually far away from camera. Video clips are sampled from validation set of Kinetics-400, and kernel $\Theta$ is selected from stage4 6b.
+
+![](images/1900f633649d2c088be82f1a2e551811b71e8659283ff74dca85559312348f56.jpg)  
+Figure 8: The visualization for the videos with large magnitude of motion. The moving persons in videos are usually closed to camera. Video clips are sampled from validation set of Kinetics-400, and kernel $\Theta$ is selected from stage4 6b.