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+# Global Filter Networks for Image Classification
+
+Yongming Rao∗ Wenliang Zhao∗ Zheng Zhu Jiwen $\mathbf { L } \mathbf { u } ^ { \dagger }$ Jie Zhou
+
+Department of Automation, Tsinghua University State Key Lab of Intelligent Technologies and Systems Beijing National Research Center for Information Science and Technology
+
+# Abstract
+
+Recent advances in self-attention and pure multi-layer perceptrons (MLP) models for vision have shown great potential in achieving promising performance with fewer inductive biases. These models are generally based on learning interaction among spatial locations from raw data. The complexity of self-attention and MLP grows quadratically as the image size increases, which makes these models hard to scale up when high-resolution features are required. In this paper, we present the Global Filter Network (GFNet), a conceptually simple yet computationally efficient architecture, that learns long-term spatial dependencies in the frequency domain with log-linear complexity. Our architecture replaces the self-attention layer in vision transformers with three key operations: a 2D discrete Fourier transform, an element-wise multiplication between frequency-domain features and learnable global filters, and a 2D inverse Fourier transform. We exhibit favorable accuracy/complexity trade-offs of our models on both ImageNet and downstream tasks. Our results demonstrate that GFNet can be a very competitive alternative to transformer-style models and CNNs in efficiency, generalization ability and robustness. Code is available at https://github.com/raoyongming/GFNet.
+
+# 1 Introduction
+
+The transformer architecture, originally designed for the natural language processing (NLP) tasks [42], has shown promising performance on various vision problems recently [10, 40, 27, 49, 4, 47, 35, 5]. Different from convolutional neural networks (CNNs), vision transformer models use self-attention layers to capture long-term dependencies, which are able to learn more diverse interactions between spatial locations. The pure multi-layer perceptrons (MLP) models [38, 39] further simplify the vision transformers by replacing the self-attention layers with MLPs that are applied across spatial locations. Since fewer inductive biases are introduced, these two kinds of models have the potential to learn more generic and flexible interactions among spatial locations from raw data.
+
+One primary challenge of applying self-attention and pure MLP models to vision tasks is the considerable computational complexity that grows quadratically as the number of tokens increases. Therefore, typical vision transformer style models usually consider a relatively small resolution for the intermediate features (e.g. $1 4 \times 1 4$ tokens are extracted from the input images in both ViT [10] and MLP-Mixer [38]). This design may limit the applications of downstream dense prediction tasks like detection and segmentation. A possible solution is to replace the global self-attention with several local self-attention like Swin transformer [27]. Despite the effectiveness in practice, local self-attention brings quite a few hand-made choices (e.g., window size, padding strategy, etc.) and limits the receptive field of each layer.
+
+![](images/8156734ec27d2da454b6f04c895b494dad72fc5c974408d7d893bfd19e9f8e4c.jpg)  
+Figure 1: The overall architecture of the Global Filter Network. Our architecture is based on Vision Transformer (ViT) models with some minimal modifications. We replace the self-attention sub-layer with the proposed global filter layer, which consists of three key operations: a 2D discrete Fourier transform to convert the input spatial features to the frequency domain, an element-wise multiplication between frequency-domain features and the global filters, and a 2D inverse Fourier transform to map the features back to the spatial domain. The efficient fast Fourier transform (FFT) enables us to learn arbitrary interactions among spatial locations with log-linear complexity.
+
+In this paper, we present a new conceptually simple yet computationally efficient architecture called Global Filter Network (GFNet), which follows the trend of removing inductive biases from vision models while enjoying the log-linear complexity in computation. The basic idea behind our architecture is to learn the interactions among spatial locations in the frequency domain. Different from the self-attention mechanism in vision transformers and the fully connected layers in MLP models, the interactions among tokens are modeled as a set of learnable global filters that are applied to the spectrum of the input features. Since the global filters are able to cover all the frequencies, our model can capture both long-term and short-term interactions. The filters are directly learned from the raw data without introducing human priors. Our architecture is largely based on the vision transformers only with some minimal modifications. We replace the self-attention sub-layer in vision transformers with three key operations: a 2D discrete Fourier transform to convert the input spatial features to the frequency domain, an element-wise multiplication between frequency-domain features and the global filters, and a 2D inverse Fourier transform to map the features back to the spatial domain. Since the Fourier transform is used to mix the information of different tokens, the global filter is much more efficient compared to the self-attention and MLP thanks to the $\mathcal { O } ( L \log L )$ complexity of the fast Fourier transform algorithm (FFT) [7]. Benefiting from this, the proposed global filter layer is less sensitive to the token length $L$ and thus is compatible with larger feature maps and CNN-style hierarchical architectures without modifications. The overall architecture of GFNet is illustrated in Figure 1. We also compare our global filter with prevalent operations in deep vision models in Table 1.
+
+Our experiments on ImageNet verify the effectiveness of GFNet. With a similar architecture, our model outperform the recent vision transformer and MLP models including DeiT [40], ResMLP [39] and gMLP [26]. When using the hierarchical architecture, GFNet can further enlarge the gap. GFNet also works well on downstream transfer learning and semantic segmentation tasks. Our results demonstrate that GFNet can be a very competitive alternative to transformer-style models and CNNs in efficiency, generalization ability and robustness.
+
+# 2 Related works
+
+Vision transformers. Since Dosovitskiy et al. [10] introduce transformers to the image classification and achieve a competitive performance compared to CNNs, transformers begin to exhibit their potential in various vision tasks [3, 4, 49]. Recently, there are a large number of works which aim to improve the transformers [40, 41, 27, 44, 18, 11, 48]. These works either seek for better training strategies [40, 11] or design better architectures [27, 44, 48] or both [41, 11]. However, most of the architecture modification of the transformers [44, 18, 27, 48] introduces additional inductive biases similar to CNNs. In this work, we only focus on the standard transformer architecture [10, 40] and our goal is to replace the heavy self-attention layer $( \mathcal { O } ( L ^ { 2 } ) )$ to an more efficient operation which can still model the interactions among different spatial locations without introducing the inductive biases associated with CNNs.
+
+Table 1: Comparisons of the proposed Global Filter with prevalent operations in deep vision models. $H$ , $W$ and $D$ are the height, width and the number of channels of the feature maps. $k$ is the kernel size of the convolution operation. The proposed global filter is much more efficient than self-attention and spatial MLP.   
+
+<table><tr><td></td><td>Complexity (FLOPs)</td><td># Parameters</td></tr><tr><td>Depthwise Convolution</td><td>O(k² HWD)</td><td>k²D</td></tr><tr><td> Self-Attention</td><td>O(HWD² + H²W²D)</td><td>4D2</td></tr><tr><td> Spatial MLP</td><td>O(H²W² D)</td><td>H²W2</td></tr><tr><td>Global Filter</td><td>O(HWD[log2(HW)] + HWD)</td><td>HWD</td></tr></table>
+
+MLP-like models. More recently, there are several works that question the importance of selfattention in the vision transformers and propose to use MLP to replace the self-attention layer in the transformers [38, 39, 26]. The MLP-Mixer [38] employs MLPs to perform token mixing and channel mixing alternatively in each block. ResMLP [39] adopts a similar idea but substitutes the Layer Normalization with an Affine transformation for acceleration. The recently proposed gMLP [26] uses a spatial gating unit to re-weight tokens in the spatial dimension. However, all of the above models include MLPs to mix the tokens spatially, which brings two drawbacks: (1) like the self-attention in the transformers, the spatial MLP still requires computational complexity quadratic to the length of tokens. (2) unlike transformers, MLP models are hard to scale up to higher resolution since the weights of the spatial MLPs have fixed sizes. Our work follows this trend and successfully resolves the above issues in MLP-like models. The proposed GFNet enjoys log-linear complexity and can be easily scaled up to any resolution.
+
+Applications of Fourier transform in vision. Fourier transform has been an important tool in digital image processing for decades [32, 1]. With the breakthroughs of CNNs in vision [14, 13], there are a variety of works that start to incorporate Fourier transform in some deep learning method [24, 46, 9, 22, 6] for vision tasks. Some of these works employ discrete Fourier transform to convert the images to the frequency domain and leverage the frequency information to improve the performance in certain tasks [22, 46], while others utilize the convolution theorem to accelerate the CNNs via fast Fourier transform (FFT) [24, 9]. FFC [6] replaces the convolution in CNNs with an Local Fourier Unit and perform convolutions in the frequency domain. Very recent works also try to leverage Fourier transform to develop deep learning models to solve partial differential equations [25] and NLP tasks [23]. In this work, we propose to use learnable filters to interchange information globally among the tokens in the Fourier domain, inspired by the frequency filters in the digital image processing [32]. We also take advantage of some properties of FFT to reduce the computational costs and the number of parameters.
+
+# 3 Method
+
+# 3.1 Preliminaries: discrete Fourier transform
+
+We start by introducing the discrete Fourier transform (DFT), which plays an important role in the area of digital signal processing and is a crucial component in our GFNet. For clarity, We first consider the 1D DFT. Given a sequence of $N$ complex numbers $x [ n ] , 0 \leq n \leq N - 1$ , the 1D DFT converts the sequence into the frequency domain by:
+
+$$
+X [ k ] = \sum _ { n = 0 } ^ { N - 1 } x [ n ] e ^ { - j ( 2 \pi / N ) k n } : = \sum _ { n = 0 } ^ { N - 1 } x [ n ] W _ { N } ^ { k n }
+$$
+
+# Algorithm 1 Pseudocode of Global Filter Layer.
+
+X = rfft2(x, dim=(1, 2)) X_tilde $= \texttt { X * K }$ $\texttt { x } =$ irfft2(X_tilde, ${ \mathsf { d i m } } { = } ( 1 , \ 2 ) .$ )
+
+rfft2/irfft2: 2D FFT/IFFT for real signal
+
+where $j$ is the imaginary unit and $W _ { N } = e ^ { - j ( 2 \pi / N ) }$ . The formulation of DFT in Equation (3.1) can be derived from the Fourier transform for continuous signal by sampling in both the time domain and the frequency domain (see Appendix A for details). Since $X [ k ]$ repeats on intervals of length $N$ , it is suffice to take the value of $X [ k ]$ at $N$ consecutive points $k = 0 , 1 , \ldots , N - 1$ . Specifically, $X [ k ]$ represents to the spectrum of the sequence $x [ n ]$ at the frequency $\omega _ { k } = 2 \pi \boldsymbol { k } / N$ .
+
+It is also worth noting that DFT is a one-to-one transformation. Given the DFT $X [ k ]$ , we can recover the original signal $x [ n ]$ by the inverse DFT (IDFT):
+
+$$
+x [ n ] = { \frac { 1 } { N } } \sum _ { k = 0 } ^ { N - 1 } X [ k ] e ^ { j ( 2 \pi / N ) k n } .
+$$
+
+For real input $x [ n ]$ , it can be proved that (see Appendix A) its DFT is conjugate symmetric, i.e., $X [ N - k ] { \ ' } = X ^ { * } [ { \dot { k } } ]$ . The reverse is true as well: if we perform IDFT to $X [ k ]$ which is conjugate symmetric, a real discrete signal can be recovered. This property implies that the half of the DFT $\{ X [ k ] : 0 \le k \le \lceil N / 2 \rceil \}$ contains the full information about the frequency characteristics of $x [ n ]$ .
+
+DFT is widely used in modern signal processing algorithms for mainly two reasons: (1) the input and output of DFT are both discrete thus can be easily processed by computers; (2) there exist efficient algorithms for computing the DFT. The fast Fourier transform (FFT) algorithms take advantage of the symmetry and periodicity properties of $W _ { N } ^ { k n }$ and reduce the complexity to compute DFT from $\mathcal { O } ( N ^ { 2 } )$ to $\mathcal { O } ( N \log N )$ . The inverse DFT (3.2), which has a similar form to the DFT, can also be computed efficiently using the inverse fast Fourier transform (IFFT).
+
+The DFT described above can be extend to 2D signals. Given the 2D signal $X [ m , n ] , 0 \leq m \leq$ $M - 1 , 0 \leq n \leq N - 1$ , the 2D DFT of $x [ m , n ]$ is given by:
+
+$$
+X [ u , v ] = \sum _ { m = 0 } ^ { M - 1 } \sum _ { n = 0 } ^ { N - 1 } x [ m , n ] e ^ { - j 2 \pi \left( { \frac { u m } { M } } + { \frac { v n } { N } } \right) } .
+$$
+
+The 2D DFT can be viewed as performing 1D DFT on the two dimensions alternatively. Similar to 1D DFT, 2D DFT of real input $x [ m , n ]$ satisfied the conjugate symmetry property $X [ M - u , N - v ] =$ $X ^ { * } [ u , v ]$ . The FFT algorithms can also be applied to 2D DFT to improve computational efficiency.
+
+# 3.2 Global Filter Networks
+
+Overall architecture. Recent advances in vision transformers [10, 40] demonstrate that models based on self-attention can achieve competitive performance even without the inductive biases associated with the convolutions. Henceforth, there are several works [39, 38] that exploit approaches (e.g., MLPs) other than self-attention to mix the information among the tokens. The proposed Global Filter Networks (GFNet) follows this line of work and aims to replace the heavy self-attention layer $( \mathcal { O } ( N ^ { 2 } ) )$ ) with a simpler and more efficient one.
+
+The overall architecture of our model is depicted in Figure 1. Our model takes as an input $H \times W$ non-overlapping patches and projects the flattened patches into $L = H W$ tokens with dimension $D$ . The basic building block of GFNet consists of: 1) a global filter layer that can exchange spatial information efficiently $( { \mathcal { O } } ( L \log L ) )$ ; 2) a feedforward network (FFN) as in [10, 40]. The output tokens of the last block are fed into a global average pooling layer followed by a linear classifier.
+
+Global filter layer. We propose global filter layer as an alternative to the self-attention layer which can mix tokens representing different spatial locations. Given the tokens $\pmb { x } \in \mathbb { R } ^ { H \times W \times \tilde { D } }$ , we first perform 2D FFT (see Section 3.1) along the spatial dimensions to convert $_ { \textbf { \em x } }$ to the frequency domain:
+
+$$
+\pmb { X } = \mathcal { F } [ \pmb { x } ] \in \mathbb { C } ^ { H \times W \times D } ,
+$$
+
+where $\mathcal { F } [ \cdot ]$ denotes the 2D FFT. Note that $\boldsymbol { X }$ is a complex tensor and represents the spectrum of $_ { x }$ We can then modulate the spectrum by multiplying a learnable filter $\pmb { K } \in \mathbb { C } ^ { H \times W \times D }$ to the $\boldsymbol { X }$ :
+
+$$
+\tilde { \cal X } = { \cal K } \odot { \cal X } ,
+$$
+
+where $\odot$ is the element-wise multiplication (also known as the Hadamard product). The filter $\kappa$ is called the global filter since it has the same dimension with $\boldsymbol { X }$ , which can represent an arbitrary filter in the frequency domain. Finally, we adopt the inverse FFT to transform the modulated spectrum $\tilde { X }$ back to the spatial domain and update the tokens:
+
+$$
+\pmb { x }  \mathcal { F } ^ { - 1 } [ \tilde { \pmb { X } } ] .
+$$
+
+The formulation of the global filter layer is motivated by the frequency filters in the digital image processing [32], where the global filter $\kappa$ can be regarded as a set of learnable frequency filters for different hidden dimensions. It can be proved (see Appendix A) that the global filter layer is equivalent to a depthwise global circular convolution with the filter size $H \times W$ . Therefore, the global filter layer is different from the standard convolutional layer which adopts a relatively small filter size to enforce the inductive biases of the locality. We also find although the proposed global filter can also be interpreted as a spatial domain operation, the filters learned in our networks exhibit more clear patterns in the frequency domain than the spatial domain, which indicates our models tend to capture relation in the frequency domain instead of spatial domain (see Figure 4). Note that the global filter implemented in the frequency domain is also much more efficient compared to the spatial domain, which enjoys a complexity of $\mathcal { O } ( D L \log L )$ while the vanilla depthwise global circular convolution in the spatial domain has $\mathcal { O } ( D L ^ { 2 } )$ complexity. We will also show that the global filter layer is better than its local convolution counterparts in the experiments.
+
+It is also worth noting that in the implementation, we make use of the property of DFT to reduce the redundant computation. Since $_ { \textbf { \em x } }$ is a real tensor, its DFT $\boldsymbol { X }$ is conjugate symmetric, i.e. $X [ H -$ $u , W - v , : ] = \mathbf { \bar { X } } ^ { * } [ H , W , : ]$ . Therefore, we can take only the half of the values in the $\boldsymbol { X }$ but preserve the full information at the same time:
+
+$$
+\pmb { X } _ { r } = \pmb { X } [ : , 0 : \widehat { W } ] : = \mathcal { F } _ { r } [ \pmb { x } ] , \quad \widehat { W } = \lceil W / 2 \rceil ,
+$$
+
+where $\mathcal { F } _ { r }$ denotes the 2D FFT for real input. In this way, we can implement the global filter as $\boldsymbol { K _ { r } } \in \mathbb { C } ^ { H \times \widehat { W } \times D }$ , which can reduce half the parameters. This can also ensure $\mathcal { F } _ { r } ^ { - 1 } [ K _ { r } \odot X _ { r } ]$ is a real tensor, thus it can be added directly to the input $_ { \textbf { \em x } }$ . The global filter layer can be easily in modern deep learning frameworks (e.g., PyTorch [31]), as is shown in Algorithm 1. The FFT and IFFT are well supported by GPU and CPU thanks to the acceleration libraries like cuFFT and mkl-fft, which makes our models perform well on hardware.
+
+Relationship to other transformer-style models. The GFNet follows the line of research about the exploration of approaches to mix the tokens. Compared to existing architectures like vision transformers and pure MLP models, we exhibit that GFNet has several favorable properties: 1) GFNet is more efficient. The complexity of both the vision transformers [10, 40, 41] and the MLP models [38, 39] is $\mathcal { O } ( L ^ { 2 } )$ . Different from them, global filter layer only consists an FFT $( { \mathcal { O } } ( L \log L ) )$ , an element-wise multiplication $( \mathcal { O } ( L ) )$ and an IFFT $\scriptstyle ( { \mathcal { O } } ( L \log L ) )$ ), which means the total computational complexity is $\mathcal { O } ( \bar { L } \log L )$ . 2) Although pure MLP models are simpler compared to transformers, it is hard to fine-tune them on higher resolution (e.g., from $2 2 4 \times 2 2 4$ resolution to $3 8 4 \times 3 8 4$ resolution) since they can only process a fixed number of tokens. As opposed to pure MLP models, we will show that our GFNet can be easily scaled up to higher resolution. Our model is more flexible since both the FFT and the IFFT have no learnable parameters and can process sequences with arbitrary length. We can simply interpolate the global filter K to K0 ∈ CH0×W0×D for different inputs, where $H ^ { \prime } \times W ^ { \prime }$ is the target size. The interpolation is reasonable due to the property of DFT. Each element of the global filter $K [ u , v ]$ corresponds to the spectrum of the filter at $\omega _ { u } = 2 \pi u / H , \omega _ { v } = 2 \pi v / W$ and thus, the global filter $\kappa$ can be viewed as a sampling of a continuous spectrum $\pmb { K } ( \omega _ { u } , \omega _ { v } )$ , where $\omega _ { u } , \omega _ { v } \in [ 0 , 2 \pi ]$ . Hence, changing the resolution is equivalent to changing the sampling interval of $\pmb { K } ( \omega _ { u } , \omega _ { v } )$ . Therefore, we only need to perform interpolation to shift from one resolution to another.
+
+Table 2: Detailed configurations of different variants of GFNet. For hierarchical models, we provide the number of channels and blocks in 4 stages. The FLOPs are calculated with $2 2 4 \times 2 2 4$ input.   
+
+<table><tr><td>Model</td><td>#Blocks</td><td>#Channels</td><td>Params (M)</td><td>FLOPs (G)</td></tr><tr><td>GFNet-Ti</td><td>12</td><td>256</td><td>7</td><td>1.3</td></tr><tr><td>GFNet-XS</td><td>12</td><td>384</td><td>16</td><td>2.9</td></tr><tr><td>GFNet-S</td><td>19</td><td>384</td><td>25</td><td>4.5</td></tr><tr><td>GFNet-B</td><td>19</td><td>512</td><td>43</td><td>7.9</td></tr><tr><td>GFNet-H-Ti</td><td>[3,3,10, 3]</td><td>[64, 128,256, 512]</td><td>15</td><td>2.1</td></tr><tr><td>GFNet-H-S</td><td>[3,3,10, 3]</td><td>[96,192, 384, 768]</td><td>32</td><td>4.6</td></tr><tr><td>GFNet-H-B</td><td>[3,3, 27, 3]</td><td>[96,192, 384, 768]</td><td>54</td><td>8.6</td></tr></table>
+
+We also notice recently a concurrent work FNet [23] leverages Fourier transform to mix tokens. Our work is distinct from FNet in three aspects: (1) FNet performs FFT to the input and directly adds the real part of the spectrum to the input tokens, which blends the information from different domains (spatial/frequency) together. On the other hand, GFNet draws motivation from the frequency filters, which is more reasonable. (2) FNet only keeps the real part of the spectrum. Note that the spectrum of real input is conjugate symmetric, which means the real part is exactly symmetric and thus contains redundant information. Our GFNet, however, utilizes this property to simplify the computation. (3) FNet is designed for NLP tasks, while our GFNet focuses on vision tasks. In our experiments, we also implement the FNet and show that our model outperforms it.
+
+Architecture variants. Due to the limitation from the quadratic complexity in the self-attention, vision transformers [10, 40] are usually designed to process a relatively small feature map (e.g., $1 4 \times 1 4$ ). However, our GFNet, which enjoys log-linear complexity, avoids that problem. Since in our GFNet the computational costs do not grow such significantly when the feature map size increases, we can adopt a hierarchical architecture inspired by the success of CNNs [21, 14]. Generally speaking, we can start from a large feature map (e.g., $5 6 \times 5 6$ ) and gradually perform downsampling after a few blocks. In this paper, we mainly investigate two kinds of variants of GFNet: transformer-style models with a fixed number of tokens in each block and CNN-style hierarchical models with gradually downsampled tokens. For transformer-style models, we begin with a 12-layer model (GFNet-XS) with a similar architecture with DeiT-S and ResMLP-12. Then, we obtain 3 variants of the model (GFNet-Ti, GFNet-S and GFNet-B) by simply adjusting the depth and embedding dimension, which have similar computational costs with ResNet-18, 50 and 101 [14]. For hierarchical models, we also design three models (GFNet-H-Ti, GFNet-H-S and GFNet-H-B) that have these three levels of complexity following the design of PVT [43]. We use $4 \times 4$ patch embedding to form the input tokens and use a non-overlapping convolution layer to downsample tokens following [43, 27]. Unlike PVT [43] and Swin [27], we directly apply our building block on different stages without any modifications. The detailed architectures are summarized in Table 2.
+
+# 4 Experiments
+
+We conduct extensive experiments to verify the effectiveness of our GFNet. We present the main results on ImageNet [8] and compare them with various architectures. We also test our models on the downstream transfer learning datasets including CIFAR-10/100 [20], Stanford Cars [19] and Flowers-102 [30]. Lastly, we investigate the efficiency and robustness of the proposed models and provide visualization to have an intuitive understanding of our method.
+
+# 4.1 ImageNet Classification
+
+Setups. We conduct our main experiments on ImageNet [8], which is a widely used large-scale benchmark for image classification. ImageNet contains roughly 1.2M images from 1,000 categories. Following common practice [14, 40], we train our models on the training set of ImageNet and report the single-crop top-1 accuracy on 50,000 validation images. To fairly compare with previous works [40, 39], we follow the most training details for our models and do not add extra regularization methods like [18]. Different from [40], we does not use EMA model [33], RandomEarse [50] and repeated augmentation [17], which are important to train DeiT while sightly hurting the performance of our models. We set the gradient clipping norm to 1 for all of our models. During finetuning at the higher resolution, we use the hyper-parameters suggested by the implementation of [40] and train the model for 30 epochs. All of our models are trained on a single machine with 8 GPUs. More details can be found in Appendix B.
+
+Table 3: Comparisons with transformer-style architectures on ImageNet. We compare different transformer-style architectures for image classification including vision transformers [40], MLP-like models [39, 26] and our models that have comparable FLOPs and the number of parameters. We report the top-1 accuracy on the validation set of ImageNet as well as the number of parameters and FLOPs. All of our models are trained with $2 2 4 \times 2 2 4$ images. We use “ $\uparrow 3 8 4 "$ to represent models finetuned on $3 8 4 \times 3 8 4$ images for 30 epochs.   
+
+<table><tr><td>Model</td><td>Params (M)</td><td>FLOPs (G)</td><td>Resolution</td><td>Top-1 Acc. (%)</td><td>Top-5 Acc. (%)</td></tr><tr><td>DeiT-Ti40]</td><td>5</td><td>1.2</td><td>224</td><td>72.2</td><td>91.1</td></tr><tr><td>gMLP-Ti[26]</td><td>6</td><td>1.4</td><td>224</td><td>72.0</td><td>1</td></tr><tr><td>GFNet-Ti</td><td>7</td><td>1.3</td><td>224</td><td>74.6</td><td>92.2</td></tr><tr><td>ResMLP-12 [39]</td><td>15</td><td>3.0</td><td>224</td><td>76.6</td><td>1</td></tr><tr><td>GFNet-XS</td><td>16</td><td>2.9</td><td>224</td><td>78.6</td><td>94.2</td></tr><tr><td>DeiT-S40]</td><td>22</td><td>4.6</td><td>224</td><td>79.8</td><td>95.0</td></tr><tr><td>gMLP-S [26]</td><td>20</td><td>4.5</td><td>224</td><td>79.4</td><td>1</td></tr><tr><td>GFNet-S</td><td>25</td><td>4.5</td><td>224</td><td>80.0</td><td>94.9</td></tr><tr><td>ResMLP-36 [39]</td><td>45</td><td>8.9</td><td>224</td><td>79.7</td><td>1</td></tr><tr><td>GFNet-B</td><td>43</td><td>7.9</td><td>224</td><td>80.7</td><td>95.1</td></tr><tr><td>GFNet-XS1384</td><td>18</td><td>8.4</td><td>384</td><td>80.6</td><td>95.4</td></tr><tr><td>DeiT-B40]</td><td>86</td><td>17.5</td><td>224</td><td>81.8</td><td>95.6</td></tr><tr><td>gMLP-B [26]</td><td>73</td><td>15.8</td><td>224</td><td>81.6</td><td></td></tr><tr><td>GFNet-S↑384</td><td>28</td><td>13.2</td><td>384</td><td>81.7</td><td>95.8</td></tr><tr><td>GFNet-B↑384</td><td>47</td><td>23.3</td><td>384</td><td>82.1</td><td>95.8</td></tr></table>
+
+Table 4: Comparisons with hierarchical architectures on ImageNet. We compare different hierarchical architectures for image classification including convolutional neural networks [14, 34], hierarchical vision transformers [43, 27] and our hierarchical models that have comparable FLOPs and number of parameters. We report the top-1 accuracy on the validation set of ImageNet as well as the number of parameters and FLOPs. All models are trained and tested with $2 2 4 \times 2 2 4$ images.   
+
+<table><tr><td>Model</td><td>Params (M)</td><td>FLOPs (G)</td><td>Top-1 Acc. (%)</td><td>Top-5 Acc. (%)</td></tr><tr><td>ResNet-18 [14]</td><td>12</td><td>1.8</td><td>69.8</td><td>89.1</td></tr><tr><td>RegNetY-1.6GF [34]</td><td>11</td><td>1.6</td><td>78.0</td><td>1</td></tr><tr><td>PVT-Ti [26]</td><td>13</td><td>1.9</td><td>75.1</td><td>=</td></tr><tr><td>GFNet-H-Ti</td><td>15</td><td>2.1</td><td>80.1</td><td>95.1</td></tr><tr><td>ResNet-50 [40]</td><td>26</td><td>4.1</td><td>76.1</td><td>92.9</td></tr><tr><td>RegNetY-4.0GF [34]</td><td>21</td><td>4.0</td><td>80.0</td><td>-</td></tr><tr><td>PVT-S [26]</td><td>25</td><td>3.8</td><td>79.8</td><td>-</td></tr><tr><td>Swin-Ti [27]</td><td>29</td><td>4.5</td><td>81.3</td><td>1</td></tr><tr><td>GFNet-H-S</td><td>32</td><td>4.6</td><td>81.5</td><td>95.6</td></tr><tr><td>ResNet-101 [40]</td><td>45</td><td>7.9</td><td>77.4</td><td>93.5</td></tr><tr><td>RegNetY-8.0GF [34]</td><td>39</td><td>8.0</td><td>81.7</td><td>1</td></tr><tr><td>PVT-M[26]</td><td>44</td><td>6.7</td><td>81.2</td><td>=</td></tr><tr><td>Swin-S [27]</td><td>50</td><td>8.7</td><td>83.0</td><td></td></tr><tr><td>GFNet-H-B</td><td>54</td><td>8.6</td><td>82.9</td><td>96.2</td></tr></table>
+
+Comparisons with transformer-style architectures. The results are presented in Table 3. We compare our method with different transformer-style architectures for image classification including vision transformers (DeiT [40]) and MLP-like models (ResMLP [39] and $\mathrm { g M L P }$ [26]) that have similar complexity and number of parameters. We see that our method can clearly outperform recent MLP-like models such as ResMLP [39] and $\mathrm { g M L P }$ [26], and show similar performance to DeiT. Specifically, GFNet-XS outperforms ResMLP-12 by $2 . 0 \%$ while having slightly fewer FLOPs. GFNet-S also achieves better top-1 accuracy compared to gMLP-S and DeiT-S. Our tiny model is significantly better compared to both DeiT-Ti $( + 2 . 4 \% )$ and gMLP-Ti $( + 2 . 6 \% )$ with the similar level of complexity.
+
+Comparisons with hierarchical architectures. We compare different kinds of hierarchical models in Figure 4. ResNet [14] is the most widely used convolutional model while RegNet [34] is a family of carefully designed CNN models. We also compare with recent hierarchical vision transformers PVT [43] and Swin [27]. Benefiting from the log-linear complexity, GFNet-H models show significantly better performance than ResNet, RegNet and PVT and achieve similar performance with Swin while having a much simpler and more generic design.
+
+Fine-tuning at higher resolution. One prominent problem of MLP-like models is that the feature resolution is not adjustable. On the contrary, the proposed global filter is more flexible. We demonstrate the advantage of GFNet by finetuning the model trained at $2 2 4 \times 2 2 4$ resolution to higher resolution following the practice in vision transformers [40]. As shown in Table 3, our model can easily adapt to higher resolution with only 30 epoch finetuning and achieve better performance.
+
+Table 5: Results on transfer learning datasets. We report the top-1 accuracy on the four datasets as well as the number of parameters and FLOPs.   
+
+<table><tr><td>Model</td><td>FLOPs</td><td>Params</td><td>CIFAR-10</td><td>CIFAR-100</td><td>Flowers-102</td><td>Cars-196</td></tr><tr><td>ResNet50 [14]</td><td>4.1G</td><td>26M</td><td>=</td><td></td><td>96.2</td><td>90.0</td></tr><tr><td>EfficientNet-B7 [37]</td><td>37G</td><td>66M</td><td>98.9</td><td>91.7</td><td>98.8</td><td>94.7</td></tr><tr><td>ViT-B/16 [10]</td><td>55.4G</td><td>86M</td><td>98.1</td><td>87.1</td><td>89.5</td><td>-</td></tr><tr><td>ViT-L/16 [10]</td><td>190.7G</td><td>307M</td><td>97.9</td><td>86.4</td><td>89.7</td><td>=</td></tr><tr><td>Deit-B/16 [40]</td><td>17.5G</td><td>86M</td><td>99.1</td><td>90.8</td><td>98.4</td><td>92.1</td></tr><tr><td>ResMLP-12[39]</td><td>3.0G</td><td>15M</td><td>98.1</td><td>87.0</td><td>97.4</td><td>84.6</td></tr><tr><td>ResMLP-24 [39]</td><td>6.0G</td><td>30M</td><td>98.7</td><td>89.5</td><td>97.9</td><td>89.5</td></tr><tr><td>GFNet-XS</td><td>2.9G</td><td>16M</td><td>98.6</td><td>89.1</td><td>98.1</td><td>92.8</td></tr><tr><td>GFNet-H-B</td><td>8.6G</td><td>54M</td><td>99.0</td><td>90.3</td><td>98.8</td><td>93.2</td></tr></table>
+
+# 4.2 Transfer learning
+
+To test the generality of our architecture and the learned representation, we evaluate GFNet on a set of commonly used transfer learning benchmark datasets including CIFAR-10 [20], CIFAR-100 [20], Stanford Cars [19] and Flowers-102 [30]. We follow the setting of previous works [37, 10, 40, 39], where the model is initialized by the ImageNet pre-trained weights and finetuned on the new datasets. We evaluate the transfer learning performance of our basic model and best model. The results are presented in Table 5. The proposed models generally work well on downstream datasets. GFNet models outperform ResMLP models by a large margin and achieve very competitive performance with state-of-the-art EfficientNet-B7. Our models also show competitive performance compared to state-of-the-art CNNs and vision transformers.
+
+# 4.3 Analysis and visualization
+
+Efficiency of GFNet. We demonstrate the efficiency of our GFNet in Figure 2, where the models are compared in theoretical FLOPs, actual latency and peak memory usage on GPU. We test a single building block of each model (including one token mixing layer and one FFN) with respect to the different numbers of tokens and set the feature dimension and batch size to 384 and 32 respectively. The self-attention model quickly runs out of memory when feature resolution exceeds $5 6 ^ { 2 }$ , which is also the feature resolution of our hierarchical model. The advantage of the proposed architecture becomes larger as the resolution increases, which strongly shows the potential of our model in vision tasks requiring high-resolution feature maps.
+
+![](images/3211ca51af5b9d445d6134414166f71b40db289c91e3eccaacf4c9df0de46a48.jpg)  
+Figure 2: Comparisons among GFNet, ViT [10] and ResMLP [39] in (a) FLOPs (b) latency and (c) GPU memory with respect to the number of tokens (feature resolution). The dotted lines indicate the estimated values when the GPU memory has run out. The latency and GPU memory is measured using a single NVIDIA RTX 3090 GPU with batch size 32 and feature dimension 384.
+
+![](images/b03ce60b4d0897451f906dcc82cc18a30470840802fbe32925187887ba3bc25f.jpg)  
+Figure 3: ImageNet acc. vs model complexity.
+
+Table 6: Comparisons among the GFNet and other variants based on the transformer-like architecture on ImageNet. We show that GFNet outperforms the ResMLP [39], FNet [23] and models with local depthwise convolutions. We also report the number of parameters and theoretical complexity in FLOPs.   
+
+<table><tr><td>Model</td><td>Acc (%)</td><td>Param (M)</td><td>FLOPs (G)</td></tr><tr><td>DeiT-S [40]</td><td>79.8</td><td>22</td><td>4.6</td></tr><tr><td>Local Conv (3 × 3)</td><td>77.7</td><td>15</td><td>2.8</td></tr><tr><td>Local Conv (5 × 5)</td><td>78.1</td><td>15</td><td>2.9</td></tr><tr><td>Local Conv (7 × 7)</td><td>78.2</td><td>15</td><td>2.9</td></tr><tr><td>ResMLP [39]</td><td>76.6</td><td>15</td><td>3.0</td></tr><tr><td>FNet [23]</td><td>71.2</td><td>15</td><td>2.9</td></tr><tr><td>GFNet-XS</td><td>78.6</td><td>16</td><td>2.9</td></tr></table>
+
+Complexity/accuracy trade-offs. We show the computational complexity and accuracy trade-offs of various transformer-style architectures in Figure 3. It is clear that GFNet achieves the best trade-off among all kinds of models.
+
+Ablation study on the global filter. To more clearly show the effectiveness of the proposed global filters, we compare GFNet-XS with several baseline models that are equipped with different token mixing operations. The results are presented in Table 6. All models have a similar building block ( token mixing layer $^ +$ FFN ) and the same feature dimension of $D = 3 8 4$ . We also implement the recent FNet [23] for comparison, where a 1D FFT on feature dimension and a 2D FFT on spatial dimensions are used to mix tokens. As shown in Table 6, our method outperforms all baseline methods except DeiT-S that has $64 \%$ higher FLOPs.
+
+Robustness & generalization ability. Inspired by the [29], we further conduct experiments to evaluate the robustness and the generalization ability of the GFNet. For robustness, we consider ImageNet-A, ImageNet-C, FGSM and PGD. ImageNet-A [16] (IN-A) is a challenging dataset that contains natural adversarial examples. ImageNet-C [15] (IN-C) is used to validate the robustness of the model under various types of corruption. We use the mean corruption error (mCE, lower is better) on ImageNet-C as the evaluation metric. FGSM [12] and PGD [28] are two widely used algorithms that are targeted to evaluate the adversarial robustness of the model by single-step attack and multistep attack, respectively. For generalization ability, we adopt two variants of ImageNet validation set: ImageNet-V2 [36] (IN-V2) and ImageNet-Real [2] (IN-Real). ImageNet-V2 is a re-collected version of ImageNet validation set following the same data collection procedure of ImageNet, while ImageNet-Real contains the same images as ImageNet validation set but has reassessed labels. We compare GFNet-S with various baselines in Table 7 including CNNs, Transformers and MLP-like architectures and find the GFNet enjoys both favorable robustness and generalization ability.
+
+Table 7: Evaluation of robustness and generalization ability. We measure the robustness from different aspects, including the adversarial robustness by adopting adversarial attack algorithms including FGSM and PGD and the performance on corrupted/out-of-distribution datasets including ImageNet-A [16] (top-1 accuracy) and ImageNet-C [15] (mCE, lower is better). The generalization ability is evaluated on ImageNet-V2 [36] and ImageNet-Real [2].   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">FLOPs (G)</td><td rowspan="2">Params (M)</td><td colspan="2">ImageNet</td><td colspan="2">Generalization</td><td colspan="4">Robustness</td></tr><tr><td>Top-1个</td><td>Top-5个</td><td>IN-V2↑</td><td>IN-Real个</td><td>FGSM↑</td><td>PGD↑</td><td>IN-C↓</td><td>IN-A↑</td></tr><tr><td>ResNet-50[14]</td><td>4.1</td><td>26</td><td>76.1</td><td>92.9</td><td>67.4</td><td>85.8</td><td>12.2</td><td>0.9</td><td>76.7</td><td>0.0</td></tr><tr><td>ResNeXt50-32x4d [45]</td><td>4.3</td><td>25</td><td>79.8</td><td>94.6</td><td>68.2</td><td>85.2</td><td>34.7</td><td>13.5</td><td>64.7</td><td>10.7</td></tr><tr><td>DeiT-S[40]</td><td>4.6</td><td>22</td><td>79.8</td><td>95.0</td><td>68.4</td><td>85.6</td><td>40.7</td><td>16.7</td><td>54.6</td><td>18.9</td></tr><tr><td>ResMLP-12 [39]</td><td>3.0</td><td>15</td><td>76.6</td><td>93.2</td><td>64.4</td><td>83.3</td><td>23.9</td><td>8.5</td><td>66.0</td><td>7.1</td></tr><tr><td>GFNet-S</td><td>4.5</td><td>25</td><td>80.1</td><td>94.9</td><td>68.5</td><td>85.8</td><td>42.6</td><td>21.0</td><td>53.8</td><td>14.3</td></tr></table>
+
+![](images/8a458037fb6470589930cee1ac1eb171b69323f82c27f6c9821446afc75a0a70.jpg)  
+Figure 4: Visualization of the learned global filters in GFNet-XS. We visualize the original frequency domain global filters in (a) and show the corresponding spatial domain filters for the first 6 columns in (b). There are more clear patterns in the frequency domain than the spatial domain.
+
+Visualization. The core operation in GFNet is the element-wise multiplication between frequencydomain features and the global filter. Therefore, it is easy to visualize and interpret. We visualize the frequency domain filters as well as their corresponding spatial domain filters in Figure 4. The learned global filters have more clear patterns in the frequency domain, where different layers have different characteristics. Interestingly, the filters in the last layer particularly focus on the low-frequency component. The corresponding filters in the spatial domain are less interpretable for humans.
+
+# 5 Conclusion
+
+We have presented the Global Filter Network (GFNet), which is a conceptually simple yet computationally efficient architecture for image classification. Our model replaces the self-attention sub-layer in vision transformer with 2D FFT/IFFT and a set of learnable global filters in the frequency domain. Benefiting from the token mixing operation with log-linear complexity, our architecture is highly efficient. Our experimental results demonstrated that GFNet can be a very competitive alternative to vision transformers, MLP-like models and CNNs in accuracy/complexity trade-offs.
+
+# Acknowledgment
+
+This work was supported in part by the National Key Research and Development Program of China under Grant 2017YFA0700802, in part by the National Natural Science Foundation of China under Grant 62125603, Grant 61822603, Grant U1813218, Grant U1713214, in part by Beijing Academy of Artificial Intelligence (BAAI), and in part by a grant from the Institute for Guo Qiang, Tsinghua University.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 5.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] We develop a general framework for image classification in this paper. Our model is not for specific applications.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We follow the common practice used in our baseline methods, where no error bars are reported.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See our implementation details in the experiment part.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/K_Mnsw5VoOW/K_Mnsw5VoOW_content_list.json

@@ -0,0 +1,1127 @@
+[
+    {
+        "type": "text",
+        "text": "Global Filter Networks for Image Classification ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yongming Rao∗ Wenliang Zhao∗ Zheng Zhu Jiwen $\\mathbf { L } \\mathbf { u } ^ { \\dagger }$ Jie Zhou ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Automation, Tsinghua University State Key Lab of Intelligent Technologies and Systems Beijing National Research Center for Information Science and Technology ",
+        "bbox": [
+            253,
+            223,
+            745,
+            265
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            301,
+            535,
+            318
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent advances in self-attention and pure multi-layer perceptrons (MLP) models for vision have shown great potential in achieving promising performance with fewer inductive biases. These models are generally based on learning interaction among spatial locations from raw data. The complexity of self-attention and MLP grows quadratically as the image size increases, which makes these models hard to scale up when high-resolution features are required. In this paper, we present the Global Filter Network (GFNet), a conceptually simple yet computationally efficient architecture, that learns long-term spatial dependencies in the frequency domain with log-linear complexity. Our architecture replaces the self-attention layer in vision transformers with three key operations: a 2D discrete Fourier transform, an element-wise multiplication between frequency-domain features and learnable global filters, and a 2D inverse Fourier transform. We exhibit favorable accuracy/complexity trade-offs of our models on both ImageNet and downstream tasks. Our results demonstrate that GFNet can be a very competitive alternative to transformer-style models and CNNs in efficiency, generalization ability and robustness. Code is available at https://github.com/raoyongming/GFNet. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The transformer architecture, originally designed for the natural language processing (NLP) tasks [42], has shown promising performance on various vision problems recently [10, 40, 27, 49, 4, 47, 35, 5]. Different from convolutional neural networks (CNNs), vision transformer models use self-attention layers to capture long-term dependencies, which are able to learn more diverse interactions between spatial locations. The pure multi-layer perceptrons (MLP) models [38, 39] further simplify the vision transformers by replacing the self-attention layers with MLPs that are applied across spatial locations. Since fewer inductive biases are introduced, these two kinds of models have the potential to learn more generic and flexible interactions among spatial locations from raw data. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One primary challenge of applying self-attention and pure MLP models to vision tasks is the considerable computational complexity that grows quadratically as the number of tokens increases. Therefore, typical vision transformer style models usually consider a relatively small resolution for the intermediate features (e.g. $1 4 \\times 1 4$ tokens are extracted from the input images in both ViT [10] and MLP-Mixer [38]). This design may limit the applications of downstream dense prediction tasks like detection and segmentation. A possible solution is to replace the global self-attention with several local self-attention like Swin transformer [27]. Despite the effectiveness in practice, local self-attention brings quite a few hand-made choices (e.g., window size, padding strategy, etc.) and limits the receptive field of each layer. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/8156734ec27d2da454b6f04c895b494dad72fc5c974408d7d893bfd19e9f8e4c.jpg",
+        "image_caption": [
+            "Figure 1: The overall architecture of the Global Filter Network. Our architecture is based on Vision Transformer (ViT) models with some minimal modifications. We replace the self-attention sub-layer with the proposed global filter layer, which consists of three key operations: a 2D discrete Fourier transform to convert the input spatial features to the frequency domain, an element-wise multiplication between frequency-domain features and the global filters, and a 2D inverse Fourier transform to map the features back to the spatial domain. The efficient fast Fourier transform (FFT) enables us to learn arbitrary interactions among spatial locations with log-linear complexity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In this paper, we present a new conceptually simple yet computationally efficient architecture called Global Filter Network (GFNet), which follows the trend of removing inductive biases from vision models while enjoying the log-linear complexity in computation. The basic idea behind our architecture is to learn the interactions among spatial locations in the frequency domain. Different from the self-attention mechanism in vision transformers and the fully connected layers in MLP models, the interactions among tokens are modeled as a set of learnable global filters that are applied to the spectrum of the input features. Since the global filters are able to cover all the frequencies, our model can capture both long-term and short-term interactions. The filters are directly learned from the raw data without introducing human priors. Our architecture is largely based on the vision transformers only with some minimal modifications. We replace the self-attention sub-layer in vision transformers with three key operations: a 2D discrete Fourier transform to convert the input spatial features to the frequency domain, an element-wise multiplication between frequency-domain features and the global filters, and a 2D inverse Fourier transform to map the features back to the spatial domain. Since the Fourier transform is used to mix the information of different tokens, the global filter is much more efficient compared to the self-attention and MLP thanks to the $\\mathcal { O } ( L \\log L )$ complexity of the fast Fourier transform algorithm (FFT) [7]. Benefiting from this, the proposed global filter layer is less sensitive to the token length $L$ and thus is compatible with larger feature maps and CNN-style hierarchical architectures without modifications. The overall architecture of GFNet is illustrated in Figure 1. We also compare our global filter with prevalent operations in deep vision models in Table 1. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our experiments on ImageNet verify the effectiveness of GFNet. With a similar architecture, our model outperform the recent vision transformer and MLP models including DeiT [40], ResMLP [39] and gMLP [26]. When using the hierarchical architecture, GFNet can further enlarge the gap. GFNet also works well on downstream transfer learning and semantic segmentation tasks. Our results demonstrate that GFNet can be a very competitive alternative to transformer-style models and CNNs in efficiency, generalization ability and robustness. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related works ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Vision transformers. Since Dosovitskiy et al. [10] introduce transformers to the image classification and achieve a competitive performance compared to CNNs, transformers begin to exhibit their potential in various vision tasks [3, 4, 49]. Recently, there are a large number of works which aim to improve the transformers [40, 41, 27, 44, 18, 11, 48]. These works either seek for better training strategies [40, 11] or design better architectures [27, 44, 48] or both [41, 11]. However, most of the architecture modification of the transformers [44, 18, 27, 48] introduces additional inductive biases similar to CNNs. In this work, we only focus on the standard transformer architecture [10, 40] and our goal is to replace the heavy self-attention layer $( \\mathcal { O } ( L ^ { 2 } ) )$ to an more efficient operation which can still model the interactions among different spatial locations without introducing the inductive biases associated with CNNs. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/c07d9517c5ae039af4fd9058f6285ef10507398b0b9bf6471ac45adec4bfa97f.jpg",
+        "table_caption": [
+            "Table 1: Comparisons of the proposed Global Filter with prevalent operations in deep vision models. $H$ , $W$ and $D$ are the height, width and the number of channels of the feature maps. $k$ is the kernel size of the convolution operation. The proposed global filter is much more efficient than self-attention and spatial MLP. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Complexity (FLOPs)</td><td># Parameters</td></tr><tr><td>Depthwise Convolution</td><td>O(k² HWD)</td><td>k²D</td></tr><tr><td> Self-Attention</td><td>O(HWD² + H²W²D)</td><td>4D2</td></tr><tr><td> Spatial MLP</td><td>O(H²W² D)</td><td>H²W2</td></tr><tr><td>Global Filter</td><td>O(HWD[log2(HW)] + HWD)</td><td>HWD</td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "MLP-like models. More recently, there are several works that question the importance of selfattention in the vision transformers and propose to use MLP to replace the self-attention layer in the transformers [38, 39, 26]. The MLP-Mixer [38] employs MLPs to perform token mixing and channel mixing alternatively in each block. ResMLP [39] adopts a similar idea but substitutes the Layer Normalization with an Affine transformation for acceleration. The recently proposed gMLP [26] uses a spatial gating unit to re-weight tokens in the spatial dimension. However, all of the above models include MLPs to mix the tokens spatially, which brings two drawbacks: (1) like the self-attention in the transformers, the spatial MLP still requires computational complexity quadratic to the length of tokens. (2) unlike transformers, MLP models are hard to scale up to higher resolution since the weights of the spatial MLPs have fixed sizes. Our work follows this trend and successfully resolves the above issues in MLP-like models. The proposed GFNet enjoys log-linear complexity and can be easily scaled up to any resolution. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Applications of Fourier transform in vision. Fourier transform has been an important tool in digital image processing for decades [32, 1]. With the breakthroughs of CNNs in vision [14, 13], there are a variety of works that start to incorporate Fourier transform in some deep learning method [24, 46, 9, 22, 6] for vision tasks. Some of these works employ discrete Fourier transform to convert the images to the frequency domain and leverage the frequency information to improve the performance in certain tasks [22, 46], while others utilize the convolution theorem to accelerate the CNNs via fast Fourier transform (FFT) [24, 9]. FFC [6] replaces the convolution in CNNs with an Local Fourier Unit and perform convolutions in the frequency domain. Very recent works also try to leverage Fourier transform to develop deep learning models to solve partial differential equations [25] and NLP tasks [23]. In this work, we propose to use learnable filters to interchange information globally among the tokens in the Fourier domain, inspired by the frequency filters in the digital image processing [32]. We also take advantage of some properties of FFT to reduce the computational costs and the number of parameters. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Preliminaries: discrete Fourier transform ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We start by introducing the discrete Fourier transform (DFT), which plays an important role in the area of digital signal processing and is a crucial component in our GFNet. For clarity, We first consider the 1D DFT. Given a sequence of $N$ complex numbers $x [ n ] , 0 \\leq n \\leq N - 1$ , the 1D DFT converts the sequence into the frequency domain by: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/41b7091450ca28316e77089f5ae664eb20aee70f5409b8c5387a76e5ad70a9af.jpg",
+        "text": "$$\nX [ k ] = \\sum _ { n = 0 } ^ { N - 1 } x [ n ] e ^ { - j ( 2 \\pi / N ) k n } : = \\sum _ { n = 0 } ^ { N - 1 } x [ n ] W _ { N } ^ { k n }\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Pseudocode of Global Filter Layer. ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "X = rfft2(x, dim=(1, 2)) X_tilde $= \\texttt { X * K }$ $\\texttt { x } =$ irfft2(X_tilde, ${ \\mathsf { d i m } } { = } ( 1 , \\ 2 ) .$ ) ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "rfft2/irfft2: 2D FFT/IFFT for real signal ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $j$ is the imaginary unit and $W _ { N } = e ^ { - j ( 2 \\pi / N ) }$ . The formulation of DFT in Equation (3.1) can be derived from the Fourier transform for continuous signal by sampling in both the time domain and the frequency domain (see Appendix A for details). Since $X [ k ]$ repeats on intervals of length $N$ , it is suffice to take the value of $X [ k ]$ at $N$ consecutive points $k = 0 , 1 , \\ldots , N - 1$ . Specifically, $X [ k ]$ represents to the spectrum of the sequence $x [ n ]$ at the frequency $\\omega _ { k } = 2 \\pi \\boldsymbol { k } / N$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is also worth noting that DFT is a one-to-one transformation. Given the DFT $X [ k ]$ , we can recover the original signal $x [ n ]$ by the inverse DFT (IDFT): ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/0042ac3d1acd0d2e5b4e6704a57106027f6c328ceddf64d8ad9ca01b929f00f2.jpg",
+        "text": "$$\nx [ n ] = { \\frac { 1 } { N } } \\sum _ { k = 0 } ^ { N - 1 } X [ k ] e ^ { j ( 2 \\pi / N ) k n } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For real input $x [ n ]$ , it can be proved that (see Appendix A) its DFT is conjugate symmetric, i.e., $X [ N - k ] { \\ ' } = X ^ { * } [ { \\dot { k } } ]$ . The reverse is true as well: if we perform IDFT to $X [ k ]$ which is conjugate symmetric, a real discrete signal can be recovered. This property implies that the half of the DFT $\\{ X [ k ] : 0 \\le k \\le \\lceil N / 2 \\rceil \\}$ contains the full information about the frequency characteristics of $x [ n ]$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "DFT is widely used in modern signal processing algorithms for mainly two reasons: (1) the input and output of DFT are both discrete thus can be easily processed by computers; (2) there exist efficient algorithms for computing the DFT. The fast Fourier transform (FFT) algorithms take advantage of the symmetry and periodicity properties of $W _ { N } ^ { k n }$ and reduce the complexity to compute DFT from $\\mathcal { O } ( N ^ { 2 } )$ to $\\mathcal { O } ( N \\log N )$ . The inverse DFT (3.2), which has a similar form to the DFT, can also be computed efficiently using the inverse fast Fourier transform (IFFT). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The DFT described above can be extend to 2D signals. Given the 2D signal $X [ m , n ] , 0 \\leq m \\leq$ $M - 1 , 0 \\leq n \\leq N - 1$ , the 2D DFT of $x [ m , n ]$ is given by: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/8b1cedb56ceef897047bfd82a83add39e2ea9a379bc0d317ff0cbb789c3ebbe7.jpg",
+        "text": "$$\nX [ u , v ] = \\sum _ { m = 0 } ^ { M - 1 } \\sum _ { n = 0 } ^ { N - 1 } x [ m , n ] e ^ { - j 2 \\pi \\left( { \\frac { u m } { M } } + { \\frac { v n } { N } } \\right) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The 2D DFT can be viewed as performing 1D DFT on the two dimensions alternatively. Similar to 1D DFT, 2D DFT of real input $x [ m , n ]$ satisfied the conjugate symmetry property $X [ M - u , N - v ] =$ $X ^ { * } [ u , v ]$ . The FFT algorithms can also be applied to 2D DFT to improve computational efficiency. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 Global Filter Networks ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Overall architecture. Recent advances in vision transformers [10, 40] demonstrate that models based on self-attention can achieve competitive performance even without the inductive biases associated with the convolutions. Henceforth, there are several works [39, 38] that exploit approaches (e.g., MLPs) other than self-attention to mix the information among the tokens. The proposed Global Filter Networks (GFNet) follows this line of work and aims to replace the heavy self-attention layer $( \\mathcal { O } ( N ^ { 2 } ) )$ ) with a simpler and more efficient one. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The overall architecture of our model is depicted in Figure 1. Our model takes as an input $H \\times W$ non-overlapping patches and projects the flattened patches into $L = H W$ tokens with dimension $D$ . The basic building block of GFNet consists of: 1) a global filter layer that can exchange spatial information efficiently $( { \\mathcal { O } } ( L \\log L ) )$ ; 2) a feedforward network (FFN) as in [10, 40]. The output tokens of the last block are fed into a global average pooling layer followed by a linear classifier. ",
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+        "type": "text",
+        "text": "Global filter layer. We propose global filter layer as an alternative to the self-attention layer which can mix tokens representing different spatial locations. Given the tokens $\\pmb { x } \\in \\mathbb { R } ^ { H \\times W \\times \\tilde { D } }$ , we first perform 2D FFT (see Section 3.1) along the spatial dimensions to convert $_ { \\textbf { \\em x } }$ to the frequency domain: ",
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+        "type": "text",
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+        "bbox": [
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+        "text": "$$\n\\pmb { X } = \\mathcal { F } [ \\pmb { x } ] \\in \\mathbb { C } ^ { H \\times W \\times D } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { F } [ \\cdot ]$ denotes the 2D FFT. Note that $\\boldsymbol { X }$ is a complex tensor and represents the spectrum of $_ { x }$ We can then modulate the spectrum by multiplying a learnable filter $\\pmb { K } \\in \\mathbb { C } ^ { H \\times W \\times D }$ to the $\\boldsymbol { X }$ : ",
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+        "img_path": "images/cf14913398cfb1f7f6bede932b13f67d635146b6cbe33e158d75147caf33f8ed.jpg",
+        "text": "$$\n\\tilde { \\cal X } = { \\cal K } \\odot { \\cal X } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\odot$ is the element-wise multiplication (also known as the Hadamard product). The filter $\\kappa$ is called the global filter since it has the same dimension with $\\boldsymbol { X }$ , which can represent an arbitrary filter in the frequency domain. Finally, we adopt the inverse FFT to transform the modulated spectrum $\\tilde { X }$ back to the spatial domain and update the tokens: ",
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+        "type": "equation",
+        "img_path": "images/bc7c276000705b108b223168026cb9650f3bd1423943d3d806a128bfb199d268.jpg",
+        "text": "$$\n\\pmb { x }  \\mathcal { F } ^ { - 1 } [ \\tilde { \\pmb { X } } ] .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The formulation of the global filter layer is motivated by the frequency filters in the digital image processing [32], where the global filter $\\kappa$ can be regarded as a set of learnable frequency filters for different hidden dimensions. It can be proved (see Appendix A) that the global filter layer is equivalent to a depthwise global circular convolution with the filter size $H \\times W$ . Therefore, the global filter layer is different from the standard convolutional layer which adopts a relatively small filter size to enforce the inductive biases of the locality. We also find although the proposed global filter can also be interpreted as a spatial domain operation, the filters learned in our networks exhibit more clear patterns in the frequency domain than the spatial domain, which indicates our models tend to capture relation in the frequency domain instead of spatial domain (see Figure 4). Note that the global filter implemented in the frequency domain is also much more efficient compared to the spatial domain, which enjoys a complexity of $\\mathcal { O } ( D L \\log L )$ while the vanilla depthwise global circular convolution in the spatial domain has $\\mathcal { O } ( D L ^ { 2 } )$ complexity. We will also show that the global filter layer is better than its local convolution counterparts in the experiments. ",
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+        "type": "text",
+        "text": "It is also worth noting that in the implementation, we make use of the property of DFT to reduce the redundant computation. Since $_ { \\textbf { \\em x } }$ is a real tensor, its DFT $\\boldsymbol { X }$ is conjugate symmetric, i.e. $X [ H -$ $u , W - v , : ] = \\mathbf { \\bar { X } } ^ { * } [ H , W , : ]$ . Therefore, we can take only the half of the values in the $\\boldsymbol { X }$ but preserve the full information at the same time: ",
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+    {
+        "type": "equation",
+        "img_path": "images/77f7bad99cc49570dfa1086bde703c7bb2d647f44f92a5fbc19a8f2fc443c89d.jpg",
+        "text": "$$\n\\pmb { X } _ { r } = \\pmb { X } [ : , 0 : \\widehat { W } ] : = \\mathcal { F } _ { r } [ \\pmb { x } ] , \\quad \\widehat { W } = \\lceil W / 2 \\rceil ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { F } _ { r }$ denotes the 2D FFT for real input. In this way, we can implement the global filter as $\\boldsymbol { K _ { r } } \\in \\mathbb { C } ^ { H \\times \\widehat { W } \\times D }$ , which can reduce half the parameters. This can also ensure $\\mathcal { F } _ { r } ^ { - 1 } [ K _ { r } \\odot X _ { r } ]$ is a real tensor, thus it can be added directly to the input $_ { \\textbf { \\em x } }$ . The global filter layer can be easily in modern deep learning frameworks (e.g., PyTorch [31]), as is shown in Algorithm 1. The FFT and IFFT are well supported by GPU and CPU thanks to the acceleration libraries like cuFFT and mkl-fft, which makes our models perform well on hardware. ",
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+    {
+        "type": "text",
+        "text": "Relationship to other transformer-style models. The GFNet follows the line of research about the exploration of approaches to mix the tokens. Compared to existing architectures like vision transformers and pure MLP models, we exhibit that GFNet has several favorable properties: 1) GFNet is more efficient. The complexity of both the vision transformers [10, 40, 41] and the MLP models [38, 39] is $\\mathcal { O } ( L ^ { 2 } )$ . Different from them, global filter layer only consists an FFT $( { \\mathcal { O } } ( L \\log L ) )$ , an element-wise multiplication $( \\mathcal { O } ( L ) )$ and an IFFT $\\scriptstyle ( { \\mathcal { O } } ( L \\log L ) )$ ), which means the total computational complexity is $\\mathcal { O } ( \\bar { L } \\log L )$ . 2) Although pure MLP models are simpler compared to transformers, it is hard to fine-tune them on higher resolution (e.g., from $2 2 4 \\times 2 2 4$ resolution to $3 8 4 \\times 3 8 4$ resolution) since they can only process a fixed number of tokens. As opposed to pure MLP models, we will show that our GFNet can be easily scaled up to higher resolution. Our model is more flexible since both the FFT and the IFFT have no learnable parameters and can process sequences with arbitrary length. We can simply interpolate the global filter K to K0 ∈ CH0×W0×D for different inputs, where $H ^ { \\prime } \\times W ^ { \\prime }$ is the target size. The interpolation is reasonable due to the property of DFT. Each element of the global filter $K [ u , v ]$ corresponds to the spectrum of the filter at $\\omega _ { u } = 2 \\pi u / H , \\omega _ { v } = 2 \\pi v / W$ and thus, the global filter $\\kappa$ can be viewed as a sampling of a continuous spectrum $\\pmb { K } ( \\omega _ { u } , \\omega _ { v } )$ , where $\\omega _ { u } , \\omega _ { v } \\in [ 0 , 2 \\pi ]$ . Hence, changing the resolution is equivalent to changing the sampling interval of $\\pmb { K } ( \\omega _ { u } , \\omega _ { v } )$ . Therefore, we only need to perform interpolation to shift from one resolution to another. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/176f7f47b078dfb2a7f5e4d67280fbcad4f4fbd6319beaae99afc9316ccdf7e4.jpg",
+        "table_caption": [
+            "Table 2: Detailed configurations of different variants of GFNet. For hierarchical models, we provide the number of channels and blocks in 4 stages. The FLOPs are calculated with $2 2 4 \\times 2 2 4$ input. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>#Blocks</td><td>#Channels</td><td>Params (M)</td><td>FLOPs (G)</td></tr><tr><td>GFNet-Ti</td><td>12</td><td>256</td><td>7</td><td>1.3</td></tr><tr><td>GFNet-XS</td><td>12</td><td>384</td><td>16</td><td>2.9</td></tr><tr><td>GFNet-S</td><td>19</td><td>384</td><td>25</td><td>4.5</td></tr><tr><td>GFNet-B</td><td>19</td><td>512</td><td>43</td><td>7.9</td></tr><tr><td>GFNet-H-Ti</td><td>[3,3,10, 3]</td><td>[64, 128,256, 512]</td><td>15</td><td>2.1</td></tr><tr><td>GFNet-H-S</td><td>[3,3,10, 3]</td><td>[96,192, 384, 768]</td><td>32</td><td>4.6</td></tr><tr><td>GFNet-H-B</td><td>[3,3, 27, 3]</td><td>[96,192, 384, 768]</td><td>54</td><td>8.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "We also notice recently a concurrent work FNet [23] leverages Fourier transform to mix tokens. Our work is distinct from FNet in three aspects: (1) FNet performs FFT to the input and directly adds the real part of the spectrum to the input tokens, which blends the information from different domains (spatial/frequency) together. On the other hand, GFNet draws motivation from the frequency filters, which is more reasonable. (2) FNet only keeps the real part of the spectrum. Note that the spectrum of real input is conjugate symmetric, which means the real part is exactly symmetric and thus contains redundant information. Our GFNet, however, utilizes this property to simplify the computation. (3) FNet is designed for NLP tasks, while our GFNet focuses on vision tasks. In our experiments, we also implement the FNet and show that our model outperforms it. ",
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+    {
+        "type": "text",
+        "text": "Architecture variants. Due to the limitation from the quadratic complexity in the self-attention, vision transformers [10, 40] are usually designed to process a relatively small feature map (e.g., $1 4 \\times 1 4$ ). However, our GFNet, which enjoys log-linear complexity, avoids that problem. Since in our GFNet the computational costs do not grow such significantly when the feature map size increases, we can adopt a hierarchical architecture inspired by the success of CNNs [21, 14]. Generally speaking, we can start from a large feature map (e.g., $5 6 \\times 5 6$ ) and gradually perform downsampling after a few blocks. In this paper, we mainly investigate two kinds of variants of GFNet: transformer-style models with a fixed number of tokens in each block and CNN-style hierarchical models with gradually downsampled tokens. For transformer-style models, we begin with a 12-layer model (GFNet-XS) with a similar architecture with DeiT-S and ResMLP-12. Then, we obtain 3 variants of the model (GFNet-Ti, GFNet-S and GFNet-B) by simply adjusting the depth and embedding dimension, which have similar computational costs with ResNet-18, 50 and 101 [14]. For hierarchical models, we also design three models (GFNet-H-Ti, GFNet-H-S and GFNet-H-B) that have these three levels of complexity following the design of PVT [43]. We use $4 \\times 4$ patch embedding to form the input tokens and use a non-overlapping convolution layer to downsample tokens following [43, 27]. Unlike PVT [43] and Swin [27], we directly apply our building block on different stages without any modifications. The detailed architectures are summarized in Table 2. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We conduct extensive experiments to verify the effectiveness of our GFNet. We present the main results on ImageNet [8] and compare them with various architectures. We also test our models on the downstream transfer learning datasets including CIFAR-10/100 [20], Stanford Cars [19] and Flowers-102 [30]. Lastly, we investigate the efficiency and robustness of the proposed models and provide visualization to have an intuitive understanding of our method. ",
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+    {
+        "type": "text",
+        "text": "4.1 ImageNet Classification ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Setups. We conduct our main experiments on ImageNet [8], which is a widely used large-scale benchmark for image classification. ImageNet contains roughly 1.2M images from 1,000 categories. Following common practice [14, 40], we train our models on the training set of ImageNet and report the single-crop top-1 accuracy on 50,000 validation images. To fairly compare with previous works [40, 39], we follow the most training details for our models and do not add extra regularization methods like [18]. Different from [40], we does not use EMA model [33], RandomEarse [50] and repeated augmentation [17], which are important to train DeiT while sightly hurting the performance of our models. We set the gradient clipping norm to 1 for all of our models. During finetuning at the higher resolution, we use the hyper-parameters suggested by the implementation of [40] and train the model for 30 epochs. All of our models are trained on a single machine with 8 GPUs. More details can be found in Appendix B. ",
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+        "type": "table",
+        "img_path": "images/56675fb36202727989f953dd9c6386565f7e07bfd58adc0fbce22ef764ba2d70.jpg",
+        "table_caption": [
+            "Table 3: Comparisons with transformer-style architectures on ImageNet. We compare different transformer-style architectures for image classification including vision transformers [40], MLP-like models [39, 26] and our models that have comparable FLOPs and the number of parameters. We report the top-1 accuracy on the validation set of ImageNet as well as the number of parameters and FLOPs. All of our models are trained with $2 2 4 \\times 2 2 4$ images. We use “ $\\uparrow 3 8 4 \"$ to represent models finetuned on $3 8 4 \\times 3 8 4$ images for 30 epochs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Params (M)</td><td>FLOPs (G)</td><td>Resolution</td><td>Top-1 Acc. (%)</td><td>Top-5 Acc. (%)</td></tr><tr><td>DeiT-Ti40]</td><td>5</td><td>1.2</td><td>224</td><td>72.2</td><td>91.1</td></tr><tr><td>gMLP-Ti[26]</td><td>6</td><td>1.4</td><td>224</td><td>72.0</td><td>1</td></tr><tr><td>GFNet-Ti</td><td>7</td><td>1.3</td><td>224</td><td>74.6</td><td>92.2</td></tr><tr><td>ResMLP-12 [39]</td><td>15</td><td>3.0</td><td>224</td><td>76.6</td><td>1</td></tr><tr><td>GFNet-XS</td><td>16</td><td>2.9</td><td>224</td><td>78.6</td><td>94.2</td></tr><tr><td>DeiT-S40]</td><td>22</td><td>4.6</td><td>224</td><td>79.8</td><td>95.0</td></tr><tr><td>gMLP-S [26]</td><td>20</td><td>4.5</td><td>224</td><td>79.4</td><td>1</td></tr><tr><td>GFNet-S</td><td>25</td><td>4.5</td><td>224</td><td>80.0</td><td>94.9</td></tr><tr><td>ResMLP-36 [39]</td><td>45</td><td>8.9</td><td>224</td><td>79.7</td><td>1</td></tr><tr><td>GFNet-B</td><td>43</td><td>7.9</td><td>224</td><td>80.7</td><td>95.1</td></tr><tr><td>GFNet-XS1384</td><td>18</td><td>8.4</td><td>384</td><td>80.6</td><td>95.4</td></tr><tr><td>DeiT-B40]</td><td>86</td><td>17.5</td><td>224</td><td>81.8</td><td>95.6</td></tr><tr><td>gMLP-B [26]</td><td>73</td><td>15.8</td><td>224</td><td>81.6</td><td></td></tr><tr><td>GFNet-S↑384</td><td>28</td><td>13.2</td><td>384</td><td>81.7</td><td>95.8</td></tr><tr><td>GFNet-B↑384</td><td>47</td><td>23.3</td><td>384</td><td>82.1</td><td>95.8</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8c28d1a481c75f2feacca6f6d0a1b098ab30d44a7357af8b8b3e1b3d9d61402e.jpg",
+        "table_caption": [
+            "Table 4: Comparisons with hierarchical architectures on ImageNet. We compare different hierarchical architectures for image classification including convolutional neural networks [14, 34], hierarchical vision transformers [43, 27] and our hierarchical models that have comparable FLOPs and number of parameters. We report the top-1 accuracy on the validation set of ImageNet as well as the number of parameters and FLOPs. All models are trained and tested with $2 2 4 \\times 2 2 4$ images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Params (M)</td><td>FLOPs (G)</td><td>Top-1 Acc. (%)</td><td>Top-5 Acc. (%)</td></tr><tr><td>ResNet-18 [14]</td><td>12</td><td>1.8</td><td>69.8</td><td>89.1</td></tr><tr><td>RegNetY-1.6GF [34]</td><td>11</td><td>1.6</td><td>78.0</td><td>1</td></tr><tr><td>PVT-Ti [26]</td><td>13</td><td>1.9</td><td>75.1</td><td>=</td></tr><tr><td>GFNet-H-Ti</td><td>15</td><td>2.1</td><td>80.1</td><td>95.1</td></tr><tr><td>ResNet-50 [40]</td><td>26</td><td>4.1</td><td>76.1</td><td>92.9</td></tr><tr><td>RegNetY-4.0GF [34]</td><td>21</td><td>4.0</td><td>80.0</td><td>-</td></tr><tr><td>PVT-S [26]</td><td>25</td><td>3.8</td><td>79.8</td><td>-</td></tr><tr><td>Swin-Ti [27]</td><td>29</td><td>4.5</td><td>81.3</td><td>1</td></tr><tr><td>GFNet-H-S</td><td>32</td><td>4.6</td><td>81.5</td><td>95.6</td></tr><tr><td>ResNet-101 [40]</td><td>45</td><td>7.9</td><td>77.4</td><td>93.5</td></tr><tr><td>RegNetY-8.0GF [34]</td><td>39</td><td>8.0</td><td>81.7</td><td>1</td></tr><tr><td>PVT-M[26]</td><td>44</td><td>6.7</td><td>81.2</td><td>=</td></tr><tr><td>Swin-S [27]</td><td>50</td><td>8.7</td><td>83.0</td><td></td></tr><tr><td>GFNet-H-B</td><td>54</td><td>8.6</td><td>82.9</td><td>96.2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparisons with transformer-style architectures. The results are presented in Table 3. We compare our method with different transformer-style architectures for image classification including vision transformers (DeiT [40]) and MLP-like models (ResMLP [39] and $\\mathrm { g M L P }$ [26]) that have similar complexity and number of parameters. We see that our method can clearly outperform recent MLP-like models such as ResMLP [39] and $\\mathrm { g M L P }$ [26], and show similar performance to DeiT. Specifically, GFNet-XS outperforms ResMLP-12 by $2 . 0 \\%$ while having slightly fewer FLOPs. GFNet-S also achieves better top-1 accuracy compared to gMLP-S and DeiT-S. Our tiny model is significantly better compared to both DeiT-Ti $( + 2 . 4 \\% )$ and gMLP-Ti $( + 2 . 6 \\% )$ with the similar level of complexity. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparisons with hierarchical architectures. We compare different kinds of hierarchical models in Figure 4. ResNet [14] is the most widely used convolutional model while RegNet [34] is a family of carefully designed CNN models. We also compare with recent hierarchical vision transformers PVT [43] and Swin [27]. Benefiting from the log-linear complexity, GFNet-H models show significantly better performance than ResNet, RegNet and PVT and achieve similar performance with Swin while having a much simpler and more generic design. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Fine-tuning at higher resolution. One prominent problem of MLP-like models is that the feature resolution is not adjustable. On the contrary, the proposed global filter is more flexible. We demonstrate the advantage of GFNet by finetuning the model trained at $2 2 4 \\times 2 2 4$ resolution to higher resolution following the practice in vision transformers [40]. As shown in Table 3, our model can easily adapt to higher resolution with only 30 epoch finetuning and achieve better performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/673906fb2a44cd862b53e7d0f7e37a3a19b846090e67e675fba39244c9df1a9f.jpg",
+        "table_caption": [
+            "Table 5: Results on transfer learning datasets. We report the top-1 accuracy on the four datasets as well as the number of parameters and FLOPs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>FLOPs</td><td>Params</td><td>CIFAR-10</td><td>CIFAR-100</td><td>Flowers-102</td><td>Cars-196</td></tr><tr><td>ResNet50 [14]</td><td>4.1G</td><td>26M</td><td>=</td><td></td><td>96.2</td><td>90.0</td></tr><tr><td>EfficientNet-B7 [37]</td><td>37G</td><td>66M</td><td>98.9</td><td>91.7</td><td>98.8</td><td>94.7</td></tr><tr><td>ViT-B/16 [10]</td><td>55.4G</td><td>86M</td><td>98.1</td><td>87.1</td><td>89.5</td><td>-</td></tr><tr><td>ViT-L/16 [10]</td><td>190.7G</td><td>307M</td><td>97.9</td><td>86.4</td><td>89.7</td><td>=</td></tr><tr><td>Deit-B/16 [40]</td><td>17.5G</td><td>86M</td><td>99.1</td><td>90.8</td><td>98.4</td><td>92.1</td></tr><tr><td>ResMLP-12[39]</td><td>3.0G</td><td>15M</td><td>98.1</td><td>87.0</td><td>97.4</td><td>84.6</td></tr><tr><td>ResMLP-24 [39]</td><td>6.0G</td><td>30M</td><td>98.7</td><td>89.5</td><td>97.9</td><td>89.5</td></tr><tr><td>GFNet-XS</td><td>2.9G</td><td>16M</td><td>98.6</td><td>89.1</td><td>98.1</td><td>92.8</td></tr><tr><td>GFNet-H-B</td><td>8.6G</td><td>54M</td><td>99.0</td><td>90.3</td><td>98.8</td><td>93.2</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 Transfer learning ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To test the generality of our architecture and the learned representation, we evaluate GFNet on a set of commonly used transfer learning benchmark datasets including CIFAR-10 [20], CIFAR-100 [20], Stanford Cars [19] and Flowers-102 [30]. We follow the setting of previous works [37, 10, 40, 39], where the model is initialized by the ImageNet pre-trained weights and finetuned on the new datasets. We evaluate the transfer learning performance of our basic model and best model. The results are presented in Table 5. The proposed models generally work well on downstream datasets. GFNet models outperform ResMLP models by a large margin and achieve very competitive performance with state-of-the-art EfficientNet-B7. Our models also show competitive performance compared to state-of-the-art CNNs and vision transformers. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 Analysis and visualization ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Efficiency of GFNet. We demonstrate the efficiency of our GFNet in Figure 2, where the models are compared in theoretical FLOPs, actual latency and peak memory usage on GPU. We test a single building block of each model (including one token mixing layer and one FFN) with respect to the different numbers of tokens and set the feature dimension and batch size to 384 and 32 respectively. The self-attention model quickly runs out of memory when feature resolution exceeds $5 6 ^ { 2 }$ , which is also the feature resolution of our hierarchical model. The advantage of the proposed architecture becomes larger as the resolution increases, which strongly shows the potential of our model in vision tasks requiring high-resolution feature maps. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3211ca51af5b9d445d6134414166f71b40db289c91e3eccaacf4c9df0de46a48.jpg",
+        "image_caption": [
+            "Figure 2: Comparisons among GFNet, ViT [10] and ResMLP [39] in (a) FLOPs (b) latency and (c) GPU memory with respect to the number of tokens (feature resolution). The dotted lines indicate the estimated values when the GPU memory has run out. The latency and GPU memory is measured using a single NVIDIA RTX 3090 GPU with batch size 32 and feature dimension 384. "
+        ],
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+        "image_caption": [
+            "Figure 3: ImageNet acc. vs model complexity. "
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+            "Table 6: Comparisons among the GFNet and other variants based on the transformer-like architecture on ImageNet. We show that GFNet outperforms the ResMLP [39], FNet [23] and models with local depthwise convolutions. We also report the number of parameters and theoretical complexity in FLOPs. "
+        ],
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+        "table_body": "<table><tr><td>Model</td><td>Acc (%)</td><td>Param (M)</td><td>FLOPs (G)</td></tr><tr><td>DeiT-S [40]</td><td>79.8</td><td>22</td><td>4.6</td></tr><tr><td>Local Conv (3 × 3)</td><td>77.7</td><td>15</td><td>2.8</td></tr><tr><td>Local Conv (5 × 5)</td><td>78.1</td><td>15</td><td>2.9</td></tr><tr><td>Local Conv (7 × 7)</td><td>78.2</td><td>15</td><td>2.9</td></tr><tr><td>ResMLP [39]</td><td>76.6</td><td>15</td><td>3.0</td></tr><tr><td>FNet [23]</td><td>71.2</td><td>15</td><td>2.9</td></tr><tr><td>GFNet-XS</td><td>78.6</td><td>16</td><td>2.9</td></tr></table>",
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+        "text": "Complexity/accuracy trade-offs. We show the computational complexity and accuracy trade-offs of various transformer-style architectures in Figure 3. It is clear that GFNet achieves the best trade-off among all kinds of models. ",
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+        "text": "Ablation study on the global filter. To more clearly show the effectiveness of the proposed global filters, we compare GFNet-XS with several baseline models that are equipped with different token mixing operations. The results are presented in Table 6. All models have a similar building block ( token mixing layer $^ +$ FFN ) and the same feature dimension of $D = 3 8 4$ . We also implement the recent FNet [23] for comparison, where a 1D FFT on feature dimension and a 2D FFT on spatial dimensions are used to mix tokens. As shown in Table 6, our method outperforms all baseline methods except DeiT-S that has $64 \\%$ higher FLOPs. ",
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+        "text": "Robustness & generalization ability. Inspired by the [29], we further conduct experiments to evaluate the robustness and the generalization ability of the GFNet. For robustness, we consider ImageNet-A, ImageNet-C, FGSM and PGD. ImageNet-A [16] (IN-A) is a challenging dataset that contains natural adversarial examples. ImageNet-C [15] (IN-C) is used to validate the robustness of the model under various types of corruption. We use the mean corruption error (mCE, lower is better) on ImageNet-C as the evaluation metric. FGSM [12] and PGD [28] are two widely used algorithms that are targeted to evaluate the adversarial robustness of the model by single-step attack and multistep attack, respectively. For generalization ability, we adopt two variants of ImageNet validation set: ImageNet-V2 [36] (IN-V2) and ImageNet-Real [2] (IN-Real). ImageNet-V2 is a re-collected version of ImageNet validation set following the same data collection procedure of ImageNet, while ImageNet-Real contains the same images as ImageNet validation set but has reassessed labels. We compare GFNet-S with various baselines in Table 7 including CNNs, Transformers and MLP-like architectures and find the GFNet enjoys both favorable robustness and generalization ability. ",
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+        "table_caption": [
+            "Table 7: Evaluation of robustness and generalization ability. We measure the robustness from different aspects, including the adversarial robustness by adopting adversarial attack algorithms including FGSM and PGD and the performance on corrupted/out-of-distribution datasets including ImageNet-A [16] (top-1 accuracy) and ImageNet-C [15] (mCE, lower is better). The generalization ability is evaluated on ImageNet-V2 [36] and ImageNet-Real [2]. "
+        ],
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+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">FLOPs (G)</td><td rowspan=\"2\">Params (M)</td><td colspan=\"2\">ImageNet</td><td colspan=\"2\">Generalization</td><td colspan=\"4\">Robustness</td></tr><tr><td>Top-1个</td><td>Top-5个</td><td>IN-V2↑</td><td>IN-Real个</td><td>FGSM↑</td><td>PGD↑</td><td>IN-C↓</td><td>IN-A↑</td></tr><tr><td>ResNet-50[14]</td><td>4.1</td><td>26</td><td>76.1</td><td>92.9</td><td>67.4</td><td>85.8</td><td>12.2</td><td>0.9</td><td>76.7</td><td>0.0</td></tr><tr><td>ResNeXt50-32x4d [45]</td><td>4.3</td><td>25</td><td>79.8</td><td>94.6</td><td>68.2</td><td>85.2</td><td>34.7</td><td>13.5</td><td>64.7</td><td>10.7</td></tr><tr><td>DeiT-S[40]</td><td>4.6</td><td>22</td><td>79.8</td><td>95.0</td><td>68.4</td><td>85.6</td><td>40.7</td><td>16.7</td><td>54.6</td><td>18.9</td></tr><tr><td>ResMLP-12 [39]</td><td>3.0</td><td>15</td><td>76.6</td><td>93.2</td><td>64.4</td><td>83.3</td><td>23.9</td><td>8.5</td><td>66.0</td><td>7.1</td></tr><tr><td>GFNet-S</td><td>4.5</td><td>25</td><td>80.1</td><td>94.9</td><td>68.5</td><td>85.8</td><td>42.6</td><td>21.0</td><td>53.8</td><td>14.3</td></tr></table>",
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+            "Figure 4: Visualization of the learned global filters in GFNet-XS. We visualize the original frequency domain global filters in (a) and show the corresponding spatial domain filters for the first 6 columns in (b). There are more clear patterns in the frequency domain than the spatial domain. "
+        ],
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+        "text": "Visualization. The core operation in GFNet is the element-wise multiplication between frequencydomain features and the global filter. Therefore, it is easy to visualize and interpret. We visualize the frequency domain filters as well as their corresponding spatial domain filters in Figure 4. The learned global filters have more clear patterns in the frequency domain, where different layers have different characteristics. Interestingly, the filters in the last layer particularly focus on the low-frequency component. The corresponding filters in the spatial domain are less interpretable for humans. ",
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+        "text": "5 Conclusion ",
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+        "text": "We have presented the Global Filter Network (GFNet), which is a conceptually simple yet computationally efficient architecture for image classification. Our model replaces the self-attention sub-layer in vision transformer with 2D FFT/IFFT and a set of learnable global filters in the frequency domain. Benefiting from the token mixing operation with log-linear complexity, our architecture is highly efficient. Our experimental results demonstrated that GFNet can be a very competitive alternative to vision transformers, MLP-like models and CNNs in accuracy/complexity trade-offs. ",
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+        "type": "text",
+        "text": "Acknowledgment ",
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+        "text": "This work was supported in part by the National Key Research and Development Program of China under Grant 2017YFA0700802, in part by the National Natural Science Foundation of China under Grant 62125603, Grant 61822603, Grant U1813218, Grant U1713214, in part by Beijing Academy of Artificial Intelligence (BAAI), and in part by a grant from the Institute for Guo Qiang, Tsinghua University. ",
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parse/train/S1eALyrYDH/S1eALyrYDH.md

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+# RNA SECONDARY STRUCTURE PREDICTION BY LEARNING UNROLLED ALGORITHMS
+
+Xinshi Chen1∗, $\mathbf { Y } \mathbf { u } \mathbf { L i } ^ { 2 * }$ , Ramzan Umarov2, Xin Gao2,†, Le Song1,3,†
+
+1Georgia Tech 2KAUST 3Ant Financial   
+xinshi.chen@gatech.edu   
+{yu.li;ramzan.umarov;xin.gao}@kaust.edu.sa   
+lsong@cc.gatech.edu
+
+# ABSTRACT
+
+In this paper, we propose an end-to-end deep learning model, called E2Efold, for RNA secondary structure prediction which can effectively take into account the inherent constraints in the problem. The key idea of E2Efold is to directly predict the RNA base-pairing matrix, and use an unrolled algorithm for constrained programming as the template for deep architectures to enforce constraints. With comprehensive experiments on benchmark datasets, we demonstrate the superior performance of E2Efold: it predicts significantly better structures compared to previous SOTA (especially for pseudoknotted structures), while being as efficient as the fastest algorithms in terms of inference time.
+
+# 1 INTRODUCTION
+
+Ribonucleic acid (RNA) is a molecule playing essential roles in numerous cellular processes and regulating expression of genes (Crick, 1970). It consists of an ordered sequence of nucleotides, with each nucleotide containing one of four bases: Adenine $( A )$ , Guanine $( G )$ , Cytosine $( C )$ and Uracile $( U )$ . This sequence of bases can be represented as
+
+$$
+\pmb { x } : = ( x _ { 1 } , \ldots , x _ { L } ) \mathrm { w h e r e } x _ { i } \in \{ A , G , C , U \} ,
+$$
+
+![](images/0e00bdbe75a720d6e67a58f26ea8fa82beb73db52998581ffe4ba6a4a80bb744.jpg)  
+Figure 1: Graph and matrix representations of RNA secondary structure.
+
+which is known as the primary structure of RNA. The bases can bond with one another to form a set of base-pairs, which defines the secondary structure. A secondary structure can be represented by a binary matrix $A ^ { * }$ where $A _ { i j } ^ { * } = 1$ if the $i , j$ -th bases are paired (Fig 1). Discovering the secondary structure of RNA is important for understanding functions of RNA since the structure essentially affects the interaction and reaction between RNA and other cellular components. Although secondary structure can be determined by experimental assays (e.g. X-ray diffraction), it is slow, expensive and technically challenging. Therefore, computational prediction of RNA secondary structure becomes an important task in RNA research and is useful in many applications such as drug design (Iorns et al., 2007).
+
+Research on computational prediction of RNA secondary structure from knowledge of primary structure has been carried out for decades. Most existing methods assume the secondary structure is a result of energy minimization, i.e., $A ^ { * } = \arg \operatorname* { m i n } _ { A } E _ { \pmb { x } } ( A )$ . The energy function is either estimated by physics-based thermodynamic experiments (Lorenz et al., 2011; Bellaousov et al., 2013; Markham & Zuker, 2008) or learned from data (Do et al.,
+
+![](images/fc82549788b515e25c21370a4b35752d64cf69c6945d4df92d0456dc4c6cdd1c.jpg)  
+Figure 2: Nested and non-nested structures.
+
+2006). These approaches are faced with a common problem that the search space of all valid secondary structures is exponentially-large with respect to the length $L$ of the sequence. To make the minimization tractable, it is often assumed the base-pairing has a nested structure (Fig 2 left), and the energy function factorizes pairwisely. With this assumption, dynamic programming (DP) based algorithms can iteratively find the optimal structure for subsequences and thus consider an enormous number of structures in time $\mathcal { O } ( L ^ { 3 } )$ .
+
+Although DP-based algorithms have dominated RNA structure prediction, it is notable that they restrict the search space to nested structures, which excludes some valid yet biologically important RNA secondary structures that contain ‘pseudoknots’, i.e., elements with at least two non-nested base-pairs (Fig 2 right). Pseudoknots make up roughly $1 . 4 \%$ of base-pairs (Mathews & Turner, 2006), and are overrepresented in functionally important regions (Hajdin et al., 2013; Staple & Butcher, 2005). Furthermore, pseudoknots are present in around $40 \%$ of the RNAs. They also assist folding into 3D structures (Fechter et al., 2001) and thus should not be ignored. To predict RNA structures with pseudoknots, energy-based methods need to run more computationally intensive algorithms to decode the structures.
+
+In summary, in the presence of more complex structured output (i.e., pseudoknots), it is challenging for energy-based approaches to simultaneously take into account the complex constraints while being efficient. In this paper, we adopt a different viewpoint by assuming that the secondary structure is the output of a feed-forward function, i.e., $A ^ { * } = { \mathcal { F } } _ { \theta } ( { \pmb x } )$ , and propose to learn $\theta$ from data in an end-to-end fashion. It avoids the second minimization step needed in energy function based approach, and does not require the output structure to be nested. Furthermore, the feed-forward model can be fitted by directly optimizing the loss that one is interested in.
+
+Despite the above advantages of using a feed-forward model, the architecture design is challenging. To be more concrete, in the RNA case, $\mathcal { F } _ { \theta }$ is difficult to design for the following reasons:
+
+(i) RNA secondary structure needs to obey certain hard constraints (see details in Section 3), which means certain kinds of pairings cannot occur at all (Steeg, 1993). Ideally, the output of $\mathcal { F } _ { \theta }$ needs to satisfy these constraints.   
+(ii) The number of RNA data points is limited, so we cannot expect that a naive fully connected network can learn the predictive information and constraints directly from data. Thus, inductive biases need to be encoded into the network architecture.   
+(iii) One may take a two-step approach, where a post-processing step can be carried out to enforce the constraints when $\mathcal { F } _ { \theta }$ predicts an invalid structure. However, in this design, the deep network trained in the first stage is unaware of the post-processing stage, making less effective use of the potential prior knowledge encoded in the constraints.
+
+In this paper, we present an end-to-end deep learning solution which integrates the two stages. The first part of the architecture is a transformer-based deep model called Deep Score Network which represents sequence information useful for structure prediction. The second part is a multilayer network called Post-Processing Network which gradually enforces the constraints and restrict the output space. It is designed based on an unrolled algorithm for solving a constrained optimiza-tion. These two networks are coupled together and learned jointly in an end-to-end fashion. Therefore, we call our model E2Efold.
+
+![](images/d75aa594315789a9ef399948aa5e2e96e64214beaf481b195befe2cc283de5a5.jpg)  
+Figure 3: Output space of E2Efold.
+
+By using an unrolled algorithm as the inductive bias to design Post-Processing Network, the output space of E2Efold is constrained (illustrated in Fig 3), which makes it easier to learn a good model in the case of limited data and also reduces the overfitting issue. Yet, the constraints encoded in E2Efold are flexible enough such that pseudoknots are not excluded. In summary, E2Efold strikes a nice balance between model biases for learning and expressiveness for valid RNA structures.
+
+We conduct extensive experiments to compare E2Efold with state-of-the-art (SOTA) methods on several RNA benchmark datasets, showing superior performance of E2Efold including:
+
+• being able to predict valid RNA secondary structures including pseudoknots;   
+• running as efficient as the fastest algorithm in terms of inference time;   
+• producing structures that are visually close to the true structure;   
+• better than previous SOTA in terms of F1 score, precision and recall.
+
+Although in this paper we focus on RNA secondary structure prediction, which presents an important and concrete problem where E2Efold leads to significant improvements, our method is generic and can be applied to other problems where constraints need to be enforced or prior knowledge is provided. We imagine that our design idea of learning unrolled algorithm to enforce constraints can also be transferred to problems such as protein folding and natural language understanding problems (e.g., building correspondence structure between different parts in a document).
+
+# 2 RELATED WORK
+
+Classical RNA folding methods identify candidate structures for an RNA sequence energy minimization through DP and rely on thousands of experimentally-measured thermodynamic parameters. A few widely used methods such as RNAstructure (Bellaousov et al., 2013), Vienna RNAfold (Lorenz et al., 2011) and UNAFold (Markham & Zuker, 2008) adpoted this approach. These methods typically scale as $\mathcal { O } ( L ^ { 3 } )$ in time and $\mathcal { O } ( L ^ { 2 } )$ in storage (Mathews, 2006), making them slow for long sequences. A recent advance called LinearFold (Huang et al., 2019) achieved linear run time ${ \mathcal { O } } ( { \bar { L } } )$ by applying beam search, but it can not handle pseudoknots in RNA structures. The prediction of lowest free energy structures with pseudoknots is NP-complete (Lyngsø & Pedersen, 2000), so pseudoknots are not considered in most algorithms. Heuristic algorithms such as HotKnots (Andronescu et al., 2010) and Probknots (Bellaousov & Mathews, 2010) have been made to predict structures with pseudoknots, but the predictive accuracy and efficiency still need to be improved.
+
+Learning-based RNA folding methods such as ContraFold (Do et al., 2006) and ContextFold (Zakov et al., 2011) have been proposed for energy parameters estimation due to the increasing availability of known RNA structures, resulting in higher prediction accuracies, but these methods still rely on the above DP-based algorithms for energy minimization. A recent deep learning model, CDPfold (Zhang et al., 2019), applied convolutional neural networks to predict base-pairings, but it adopts the dot-bracket representation for RNA secondary structure, which can not represent pseudoknotted structures. Moreover, it requires a DP-based post-processing step whose computational complexity is prohibitive for sequences longer than a few hundreds.
+
+Learning with differentiable algorithms is a useful idea that inspires a series of works (Hershey et al., 2014; Belanger et al., 2017; Ingraham et al., 2018; Chen et al., 2018; Shrivastava et al., 2019), which shared similar idea of using differentiable unrolled algorithms as a building block in neural architectures. Some models are also applied to structured prediction problems (Hershey et al., 2014; Pillutla et al., 2018; Ingraham et al., 2018), but they did not consider the challenging RNA secondary structure problem or discuss how to properly incorporating constraints into the architecture. OptNet (Amos & Kolter, 2017) integrates constraints by differentiating KKT conditions, but it has cubic complexity in the number of variables and constraints, which is prohibitive for the RNA case.
+
+Dependency parsing in NLP is a different but related problem to RNA folding. It predicts the dependency between the words in a sentence. Similar to nested/non-nested structures, the corresponding terms in NLP are projective/non-projective parsing, where most works focus on the former and DP-based inference algorithms are commonly used (McDonald et al., 2005). Deep learning models (Dozat & Manning, 2016; Kiperwasser & Goldberg, 2016) are proposed to proposed to score the dependency between words, which has a similar flavor to the Deep Score Network in our work.
+
+# 3 RNA SECONDARY STRUCTURE PREDICTION PROBLEM
+
+In the RNA secondary structure prediction problem, the input is the ordered sequence of bases $\pmb { x } = ( x _ { 1 } , \dots , x _ { L } )$ and the output is the RNA secondary structure represented by a matrix $A ^ { * } \in$ $\{ 0 , 1 \} ^ { L \times L }$ . Hard constraints on the forming of an RNA secondary structure dictate that certain kinds of pairings cannot occur at all (Steeg, 1993). Formally, these constraints are:
+
+<table><tr><td>i）（</td><td>Only three types of nucleotides combinations, B := {AU, UA}U {GC,CG} U {GU,UG},can form base-pairs.</td><td>Vi,j,if xixj B, then Aij = 0.</td></tr><tr><td>(ii） No sharp loops are allowed.</td><td></td><td>∀|i-jl&lt;4,Aij = 0. ,∑=1Aij ≤1.</td></tr><tr><td></td><td>(iii) There is no overlap of pairs,i.e.,it is a matching.</td><td></td></tr></table>
+
+(i) and (ii) prevent pairing of certain base-pairs based on their types and relative locations. Incorporating these two constraints can help the model exclude lots of illegal pairs. (iii) is a global constraint among the entries of $A ^ { * }$ .
+
+The space of all valid secondary structures contains all symmetric matrices $A \in \{ 0 , 1 \} ^ { L \times L }$ that satisfy the above three constraints. This space is much smaller than the space of all binary matrices $\{ 0 , 1 \} ^ { L \times L }$ . Therefore, if we could incorporate these constraints in our deep model, the reduced output space could help us train a better predictive model with less training data. We do this by using an unrolled algorithm as the inductive bias to design deep architecture.
+
+# 4 E2EFOLD: DEEP LEARNING MODEL BASED ON UNROLLED ALGORITHM
+
+In the literature on feed-forward networks for structured prediction, most models are designed using traditional deep learning architectures. However, for RNA secondary structure prediction, directly using these architectures does not work well due to the limited amount of RNA data points and the hard constraints on forming an RNA secondary structure. These challenges motivate the design of our E2Efold deep model, which combines a Deep Score Network with a Post-Processing Network based on an unrolled algorithm for solving a constrained optimization problem.
+
+# 4.1 DEEP SCORE NETWORK
+
+The first part of E2Efold is a Deep Score Network $U _ { \theta } ( { \pmb x } )$ whose output is an $L \times L$ symmetric matrix. Each entry of this matrix, i.e., $U _ { \theta } ( { \pmb x } ) _ { i j }$ , indicates the score of nucleotides $x _ { i }$ and $x _ { j }$ being paired. The $_ { \textbf { \em x } }$ input to the network here is the $L \times 4$ dimensional one-hot embedding. The specific architecture of $U _ { \theta }$ is shown in $\operatorname { F i g } 4$ . It mainly consists of
+
+• a position embedding matrix $_ { P }$ which distinguishes $\{ x _ { i } \} _ { i = 1 } ^ { L }$ by their exact and relative positions: $P _ { i } = \mathrm { M L P } \big ( \psi _ { 1 } ( i ) , \ldots , \psi _ { \ell } ( i ) , \psi _ { \ell + 1 } ( i / L ) , \ldots , \psi _ { n } ( i / L ) \big )$ , where $\{ \psi _ { j } \}$ is a set of $n$ feature maps such as $\sin ( \cdot ) , \mathrm { p o l y } ( \cdot )$ , sigmoid $( \cdot )$ , etc, and $\mathrm { \mathbf { M L P } ( \cdot ) }$ denotes multi-layer perceptions. Such position embedding idea has been used in natural language modeling such as BERT (Devlin et al., 2018), but we adapted for RNA sequence representation;   
+• a stack of Transformer Encoders (Vaswani et al., 2017) which encode the sequence information and the global dependency between nucleotides;   
+• a 2D Convolution layers (Wang et al., 2017) for outputting the pairwise scores.
+
+With the representation power of neural networks, the hope is that we can learn an informative $U _ { \theta }$ such that higher scoring entries in $U _ { \theta } ( { \pmb x } )$ correspond well to actual paired bases in RNA structure. Once the score matrix $U _ { \theta } ( { \pmb x } )$ is computed, a naive approach to use it is to choose an offset term $s \in \mathbb { R }$ (e.g., $s = 0$ ) and let $A _ { i j } ~ = ~ 1$ if $U _ { \theta } ( { \pmb x } ) _ { i j } ~ > ~ s$ . However, such entry-wise independent predictions of $A _ { i j }$ may result in a matrix $A$ that violates the constraints for a valid RNA secondary structure. Therefore, a postprocessing step is needed to make sure the predicted $A$ is valid. This step could be carried out separately after $U _ { \theta }$ is learned. But such decoupling of base-pair scoring and post-processing for constraints may lead to sub-optimal results, where the errors in these two stages can not be considered together and tuned together. Instead, we will introduce a Post-Processing Network which can be trained end-to-end together with $U _ { \theta }$ to enforce the constraints.
+
+# 4.2 POST-PROCESSING NETWORK
+
+![](images/71b2fbddddce5637757e1188d6114da618e5931297c705e2f0e7b6b9caddd433.jpg)  
+Figure 4: Architecture of Deep Score Network.
+
+The second part of E2Efold is a Post-Processing Network $\mathrm { P P } _ { \phi }$ which is an unrolled and parameterized algorithm for solving a constrained optimization prob
+
+lem. We first present how we formulate the post-processing step as a constrained optimization problem and the algorithm for solving it. After that, we show how we use the algorithm as a template to design deep architecture $\mathrm { P P } _ { \phi }$ .
+
+# 4.2.1 POST-PROCESSING WITH CONSTRAINED OPTIMIZATION
+
+Formulation of constrained optimization. Given the scores predicted by $U _ { \theta } ( { \pmb x } )$ , we define the total score $\begin{array} { r } { \frac 1 2 \sum _ { i , j } ( U _ { \theta } ( \pmb { x } ) _ { i j } - s ) \hat { \textmd { A } } _ { i j } } \end{array}$ as the objective to maximize, where $s$ is an offset term. Clearly, without structure constraints, the optimal solution is to take $A _ { i j } = 1$ when $U _ { \theta } ( { \pmb x } ) _ { i j } > s$ . Intuitively, the objective measures the covariation between the entries in the scoring matrix and the $A$ matrix. With constraints, the exact maximization becomes intractable. To make it tractable, we consider a convex relaxation of this discrete optimization to a continuous one by allowing $A _ { i j } ~ \in ~ [ 0 , 1 ]$ . Consequently, the solution space that we consider to optimize over is $\begin{array} { r l } { A ( \pmb { x } ) } & { { } : = } \end{array}$ $\{ A \in [ 0 , 1 ] ^ { L \times L } \mid A$ is symmetric and satisfies constraints (i)-(iii) in Section $3 \}$ .
+
+To further simplify the search space, we define a nonlinear transformation $\tau$ on $\mathbb { R } ^ { L \times L }$ as $\mathcal { T } ( \hat { A } ) : =$ $\begin{array} { r } { \frac { 1 } { 2 } \left( \hat { A } \circ \hat { A } + ( \hat { A } \circ \hat { A } ) ^ { \top } \right) \circ M ( { \pmb x } ) } \end{array}$ , where $\circ$ denotes element-wise multiplication. Matrix $M$ is defined as $M ( { \pmb x } ) _ { i j } : = 1$ if $x _ { i } x _ { j } \in B$ and also $| i - j | \geq 4$ , and $M ( { \pmb x } ) _ { i j } : = 0$ otherwise. From this definition we can see that $M ( { \pmb x } )$ encodes both constraint (i) and (ii). With transformation $\tau$ , the resulting matrix is non-negative, symmetric, and satisfies constraint (i) and (ii). Hence, by defining $A : = { \mathcal { T } } ( { \hat { A } } )$ , the solution space is simplified as $\mathcal { A } ( \pmb { x } ) = \{ A = \mathcal { T } ( \hat { A } ) \ | \ \hat { A } \in \mathbb { R } ^ { L \times L } , A \mathbf { 1 } \leq \mathbf { 1 } \}$ .
+
+Finally, we introduce a $\ell _ { 1 }$ penalty term $\begin{array} { r } { \| \hat { A } \| _ { 1 } : = \sum _ { i , j } | \hat { A } _ { i j } | } \end{array}$ to make $A$ sparse and formulate the post-processing step as: $\langle \cdot , \cdot \rangle$ denotes matrix inner product, i.e., sum of entry-wise multiplication)
+
+$$
+\begin{array} { r } { \operatorname* { m a x } _ { \hat { A } \in \mathbb { R } ^ { L \times L } } \frac { 1 } { 2 } \left. U _ { \theta } ( \pmb { x } ) - s , A : = \mathcal { T } ( \hat { A } ) \right. - \rho \| \hat { A } \| _ { 1 } \quad \mathrm { s . t . } ~ A \mathbf { 1 } \leq \mathbf { 1 } } \end{array}
+$$
+
+The advantages of this formulation are that the variables ${ \hat { A } } _ { i j }$ are free variables in $\mathbb { R }$ and there are only $L$ inequality constraints $A \mathbf { 1 } \leq \mathbf { 1 }$ . This system of linear inequalities can be replaced by a set of nonlinear equalities $\operatorname { r e l u } ( A \mathbf { 1 } - \mathbf { 1 } ) = \mathbf { 0 }$ so that the constrained problem can be easily transformed into an unconstrained problem by introducing a Lagrange multiplier $\lambda \in \mathbb { R } _ { + } ^ { L }$ :
+
+$$
+\operatorname* { m i n } _ { \lambda \geq 0 } \operatorname* { m a x } _ { \hat { A } \in \mathbb { R } ^ { L \times L } } \underbrace { \frac { 1 } { 2 } \langle U _ { \theta } ( \pmb { x } ) - s , A \rangle - \langle \pmb { \lambda } , \mathrm { r e l u } ( A \mathbf { 1 } - \mathbf { 1 } ) \rangle } _ { f } - \rho \| \hat { A } \| _ { 1 } .
+$$
+
+Algorithm for solving it. We use a primal-dual method for solving Eq. 2 (derived in Appendix B). In each iteration, $\hat { A }$ and $\boldsymbol { \lambda }$ are updated alternatively by:
+
+$$
+\dot { A } _ { t + 1 } \gets \hat { A } _ { t } + \alpha \cdot \gamma _ { \alpha } ^ { t } \cdot \hat { A } _ { t } \circ M ( { \pmb x } ) \circ \Big ( \partial f / \partial A _ { t } + ( \partial f / \partial A _ { t } ) ^ { \top } \Big ) ,
+$$
+
+where $\alpha$ , $\beta$ are step sizes and $\gamma _ { \alpha } , \gamma _ { \beta }$ are decaying coefficients. When it converges at $T$ , an approximate solution $R o u n d \big ( A _ { T } = \mathcal { T } ( \hat { A } _ { T } ) \big )$ is obtained. With this algorithm operated on the learned $U _ { \theta } ( { \pmb x } )$ , even if this step is disconnected to the training phase of $U _ { \theta } ( { \pmb x } )$ , the final prediction works much better than many other existing methods (as reported in Section 6). Next, we introduce how to couple this post-processing step with the training of $U _ { \theta } ( { \pmb x } )$ to further improve the performance.
+
+# 4.2.2 POST-PROCESSING NETWORK VIA AN UNROLLED ALGORITHM
+
+We design a Post-Processing Network, denoted by $\mathrm { P P } _ { \phi }$ , based on the above algorithm. After it is defined, we can connect it with the deep score network $U _ { \theta }$ and train them jointly in an end-to-end fashion, so that the training phase of $U _ { \theta } ( { \pmb x } )$ is aware of the post-processing step.
+
+# Algorithm 1: Post-Processing Network $\mathrm { P P } _ { \phi } ( U , M )$
+
+# Algorithm 2: Neural Cell PPcellφ
+
+The specific computation graph of $\mathrm { P P } _ { \phi }$ is given in Algorithm 1, whose main component is a recurrent cell which we call ${ \mathrm { P P c e l l } } _ { \phi }$ . The computation graph is almost the same as the iterative update from Eq. 3 to Eq. 6, except for several modifications:
+
+• (learnable hyperparameters) The hyperparameters including step sizes $\alpha , \beta$ , decaying rate $\gamma _ { \alpha } , \gamma _ { \beta }$ , sparsity coefficient $\rho$ and the offset term $s$ are treated as learnable parameters in $\phi$ , so that there is no need to tune the hyperparameters by hand but automatically learn them from data instead.   
+• (fixed # iterations) Instead of running the iterative updates until convergence, ${ \mathrm { P P c e l l } } _ { \phi }$ is applied recursively for $T$ iterations where $T$ is a manually fixed number. This is why in Fig 3 the output space of E2Efold is slightly larger than the true solution space. (smoothed sign function) Resulted from the gradient of $\mathrm { r e l u } ( \cdot )$ , the update step in Eq. 4 contains a $\mathrm { s i g n } ( \cdot )$ function. However, to push gradient through $\mathrm { P P } _ { \phi }$ , we require a differentiable update step. Therefore, we use a smoothed sign function defined as softsign $( c ) : = 1 / ( 1 + \exp ( - k c ) )$ , where $k$ is a temperature.   
+• (clip $\hat { A }$ ) An additional step, $\hat { A } \gets \operatorname* { m i n } ( \hat { A } , 1 )$ , is included to make the output $A _ { t }$ at each iteration stay in the range $[ 0 , 1 ] ^ { L \times L }$ . This is useful for computing the loss over intermediate results $\{ A _ { t } \} _ { t = 1 } ^ { T }$ , for which we will explain more in Section 5.
+
+With these modifications, the Post-Processing Network $\mathrm { P P } _ { \phi }$ is a tuning-free and differentiable unrolled algorithm with meaningful intermediate outputs. Combining it with the deep score network, the final deep model is
+
+$$
+\mathbf { E } 2 \mathbf { E } \mathbf { f o l d : } \quad \{ A _ { t } \} _ { t = 1 } ^ { T } = \widetilde { \mathbf { P } \mathbf { P } _ { \phi } ( \mathbf { \mathscr { \textbf { \textsf { \textbf { U } } } } } \mathbf { U } _ { \theta } ( \mathbf { x } ) , \quad , M ( \mathbf { x } ) ) } .
+$$
+
+# 5 END-TO-END TRAINING ALGORITHM
+
+Given a dataset $\mathcal { D }$ containing examples of input-output pairs $( x , A ^ { * } )$ , the training procedure of E2Efold is similar to standard gradient-based supervised learning. However, for RNA secondary structure prediction problems, commonly used metrics for evaluating predictive performances are F1 score, precision and recall, which are non-differentiable.
+
+Differentiable F1 Loss. To directly optimize these metrics, we mimic true positive (TP), false positive (FP), true negative (TN) and false negative (FN) by defining continuous functions on $[ 0 , 1 ] ^ { \hat { L } \times L }$
+
+$$
+\mathrm { T P } = \langle A , A ^ { * } \rangle , \mathrm { F P } = \langle A , 1 - A ^ { * } \rangle , \mathrm { F N } = \langle 1 - A , A ^ { * } \rangle , \mathrm { T N } = \langle 1 - A , 1 - A ^ { * } \rangle .
+$$
+
+Since $\mathrm { F 1 } = 2 \mathrm { T P } / ( 2 \mathrm { T P } + \mathrm { F P } + \mathrm { F N } )$ , we define a loss function to mimic the negative of F1 score as:
+
+$$
+\begin{array} { r } { \textstyle \mathcal { L } _ { - \mathrm { F l } } ( A , A ^ { * } ) : = - 2 \langle A , A ^ { * } \rangle / \left( 2 \langle A , A ^ { * } \rangle + \langle A , 1 - A ^ { * } \rangle + \langle 1 - A , A ^ { * } \rangle \right) . } \end{array}
+$$
+
+Assuming that $\textstyle \sum _ { i j } A _ { i j } ^ { * } \neq 0$ , this loss is well-defined and differentiable on $[ 0 , 1 ] ^ { L \times L }$ . Precision and recall losses can be defined in a similar way, but we optimize F1 score in this paper.
+
+It is notable that this F1 loss takes advantages over other differentiable losses including $\ell _ { 2 }$ and cross-entropy losses, because there are much more negative samples (i.e. $A _ { i j } = 0$ ) than positive samples (i.e. $A _ { i j } = 1$ ). A hand-tuned weight is needed to balance them while using $\ell _ { 2 }$ or crossentropy losses, but F1 loss handles this issue automatically, which can be useful for a number of problems (Wang et al., 2016; Li et al., 2017).
+
+Overall Loss Function. As noted earlier, E2Efold outputs a matrix $A _ { t } \in [ 0 , 1 ] ^ { L \times L }$ in each iteration. This allows us to add auxiliary losses to regularize the intermediate results, guiding it to learn parameters which can generate a smooth solution trajectory. More specifically, we use an objective that depends on the entire trajectory of optimization:
+
+$$
+\operatorname* { m i n } _ { \theta , \phi } \frac { 1 } { | \mathscr { D } | } \sum _ { ( x , A ^ { * } ) \in \mathcal { D } } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \gamma ^ { T - t } \mathcal { L } _ { - \mathrm { F } 1 } ( A _ { t } , A ^ { * } ) ,
+$$
+
+where $\{ A _ { t } \} _ { t = 1 } ^ { T } = \mathrm { P P } _ { \phi } ( U _ { \theta } ( \pmb { x } ) , M ( \pmb { x } ) )$ and $\gamma \leq 1$ is a discounting factor. Empirically, we find it very useful to pre-train $U _ { \theta }$ using logistic regression loss. Also, it is helpful to add this additional loss to Eq. 9 as a regularization.
+
+# 6 EXPERIMENTS
+
+We compare E2Efold with the SOTA and also the most commonly used methods in the RNA secondary structure prediction field on two benchmark datasets. It is revealed from the experimental results that E2Efold achieves $2 9 . 7 \%$ improvement in terms of F1 score on RNAstralign dataset and it infers the RNA secondary structure as fast as the most efficient algorithm (LinearFold) among existing ones. An ablation study is also conducted to show the necessity of pushing gradient through the post-processing step. The codes for reproducing the experimental results are released.1
+
+Dataset. We use two benchmark datasets: (i) ArchiveII (Sloma & Mathews, 2016), containing 3975 RNA structures from 10 RNA types, is a widely used benchmark dataset for classical RNA folding methods. (ii) RNAStralign (Tan et al., 2017), composed of 37149 structures from 8 RNA types, is one of the most comprehensive collections of RNA structures in the market. After removing redundant sequences and structures, 30451 structures remain. See Table 1 for statistics about these two datasets.
+
+Experiments On RNAStralign. We divide RNAStralign dataset into training, testing and validation sets by stratified sampling (see details in Table 7 and Fig 6), so that each set contains all RNA types. We compare the performance of E2Efold to six methods including CDPfold, LinearFold, Mfold, RNAstructure (ProbKnot), RNAfold and CONTRAfold. Both E2Efold and CDPfold are learned from the same training/validation sets. For other methods, we directly use the provided packages or web-servers to generate predicted structures. We evaluate the F1 score, Precision and Recall for each sequence in the test set. Averaged values are reported in Table 2. As suggested by Mathews (2019), for a base pair $( i , j )$ , the following predictions are also considered as correct: $( i + 1 , j )$ , $( i - 1 , j )$ , $( i , j + 1 )$ , $( i , j - 1 )$ , so we also reported the metrics when one-position shift is allowed.
+
+Table 1: Dataset Statistics   
+
+<table><tr><td rowspan="2">Type</td><td colspan="2">ArchiveII</td><td colspan="2">RNAStralign</td></tr><tr><td>length</td><td>#samples</td><td>length #samples</td><td></td></tr><tr><td>All</td><td>28~2968</td><td>3975</td><td>30~1851</td><td>30451 11620</td></tr><tr><td>16SrRNA 5SrRNA</td><td>73~1995 102~135</td><td>110 1283</td><td>54~1851 104~132</td><td>9385</td></tr><tr><td>tRNA</td><td>54~93</td><td>557</td><td>59~95</td><td>6443</td></tr><tr><td>grp1</td><td>210~736</td><td>98</td><td>163~615</td><td>1502</td></tr><tr><td>SRP</td><td>28~533</td><td>928</td><td>30~553</td><td>468</td></tr><tr><td>tmRNA</td><td>102~437</td><td>462</td><td>102~437</td><td>572</td></tr><tr><td>RNaseP</td><td>120~486</td><td>454</td><td></td><td>434</td></tr><tr><td></td><td></td><td></td><td>189~486</td><td></td></tr><tr><td></td><td>telomerase 382~559</td><td>37</td><td>382~559</td><td>37</td></tr><tr><td>23SrRNA 242~2968</td><td></td><td>35</td><td></td><td>=</td></tr><tr><td>grp2</td><td>619~780</td><td>11</td><td>-</td><td>=</td></tr></table>
+
+Table 2: Results on RNAStralign test set. “(S)” indicates the results when one-position shift is allowed.   
+
+<table><tr><td>Method</td><td>Prec</td><td>Rec</td><td>F1</td><td>Prec(S) Rec(S) F1(S)</td></tr><tr><td>E2Efold</td><td>0.866</td><td>0.788</td><td>0.821</td><td>0.880 0.798 0.833</td></tr><tr><td>CDPfold</td><td>0.633</td><td>0.597</td><td>0.614</td><td>0.720 0.677 0.697</td></tr><tr><td>LinearFold</td><td>0.620</td><td>0.606</td><td>0.609</td><td>0.635 0.622 0.624</td></tr><tr><td>Mfold</td><td>0.450</td><td>0.398</td><td>0.420</td><td>0.463 0.409 0.433</td></tr><tr><td>RNAstructure</td><td>0.537</td><td>0.568</td><td>0.550</td><td>0.559 0.592 0.573</td></tr><tr><td>RNAfold</td><td>0.516 0.568</td><td></td><td>0.540</td><td>0.533 0.587 0.558</td></tr><tr><td>CONTRAfold</td><td>0.608</td><td>：0.663</td><td>0.633</td><td>0.624 0.681 0.650</td></tr></table>
+
+![](images/b14d3f7527033b23a84d3ceb16be643583f1daf1a700c533bf5e1f968ec644a7.jpg)  
+Figure 5: Distribution of F1 score.
+
+As shown in Table 2, traditional methods can achieve a F1 score ranging from 0.433 to 0.624, which is consistent with the performance reported with their original papers. The two learning-based methods, CONTRAfold and CDPfold, can outperform classical methods with reasonable margin on some criteria. E2Efold, on the other hand, significantly outperforms all previous methods across all criteria, with at least $20 \%$ improvement. Notice that, for almost all the other methods, the recall is usually higher than precision, while for E2Efold, the precision is higher than recall. That can be the result of incorporating constraints during neural network training. Fig 5 shows the distributions of F1 scores for each method. It suggests that E2Efold has consistently good performance.
+
+To estimate the performance of E2Efold on long sequences, we also compute the F1 scores weighted by the length of sequences, such that the results are more dominated by longer sequences. Detailed results are given in Appendix D.3.
+
+Table 3: Performance comparison on ArchiveII   
+
+<table><tr><td>Method</td><td>Prec</td><td>Rec</td><td>F1</td><td>Prec(S) Rec(S)</td><td>F1(S)</td></tr><tr><td>E2Efold</td><td>0.734</td><td>0.66</td><td>0.686</td><td>0.758</td><td>0.676 0.704</td></tr><tr><td>CDPfold</td><td>0.557</td><td>0.535</td><td>0.545</td><td>0.612 0.585</td><td>0.597</td></tr><tr><td>LinearFold</td><td>0.641</td><td>0.617</td><td>0.621</td><td>0.668 0.644</td><td>0.647</td></tr><tr><td>Mfold</td><td>0.428</td><td>0.383</td><td>0.401</td><td>0.450 0.403</td><td>0.421</td></tr><tr><td>RNAstructure</td><td>0.563</td><td>0.615</td><td>0.585</td><td>0.590 0.645</td><td>0.613</td></tr><tr><td>RNAfold</td><td>0.565</td><td>0.627</td><td>0.592</td><td>0.586</td><td>0.652 0.615</td></tr><tr><td>CONTRAfold</td><td>0.607</td><td>0.679</td><td>0.638</td><td>0.629</td><td>0.705 0.662</td></tr></table>
+
+Table 4: Inference time on RNAStralign   
+
+<table><tr><td>Method</td><td>total run time time per seq</td></tr><tr><td>E2Efold (Pytorch)</td><td>19m (GPU) 0.40s</td></tr><tr><td>CDPfold (Pytorch)</td><td>440m*32 threads 300.107s</td></tr><tr><td>LinearFold (C)</td><td>20m 0.43s</td></tr><tr><td>Mfold (C)</td><td>360m 7.65s</td></tr><tr><td>RNAstructure (C) 3 days</td><td>142.02s</td></tr><tr><td>RNAfold (C)</td><td>26m 0.55s</td></tr><tr><td>CONTRAfold (C)</td><td>1 day 30.58s</td></tr></table>
+
+Test On ArchiveII Without Re-training. To mimic the real world scenario where the users want to predict newly discovered RNA’s structures which may have a distribution different from the training dataset, we directly test the model learned from RNAStralign training set on the ArchiveII dataset, without re-training the model. To make the comparison fair, we exclude sequences that are overlapped with the RNAStralign dataset. We then test the model on sequences in ArchiveII that have overlapping RNA types (5SrRNA, 16SrRNA, etc) with the RNAStralign dataset. Results are shown in Table 3. It is understandable that the performances of classical methods which are not learningbased are consistent with that on RNAStralign. The performance of E2Efold, though is not as good as that on RNAStralign, is still better than all the other methods across different evaluation criteria. In addition, since the original ArchiveII dataset contains domain sequences (subsequences), we remove the domains and report the results in Appendix D.4, which are similar to results in Table 3.
+
+Inference Time Comparison. We record the running time of all algorithms for predicting RNA secondary structures on the RNAStralign test set, which is summarized in Table 4. LinearFold is the most efficient among baselines because it uses beam pruning heuristic to accelerate DP. CDPfold, which achieves higher F1 score than other baselines, however, is extremely slow due to its DP post-processing step. Since we use a gradient-based algorithm which is simple to design the PostProcessing Network, E2Efold is fast. On GPU, E2Efold has similar inference time as LinearFold.
+
+Pseudoknot Prediction. Even though E2Efold does not exclude pseudoknots, it is not sure whether it actually generates pseudoknotted structures. Therefore, we pick all sequences containing pseudoknots and compute the averaged F1 score only on this set. Besides, we count the number of pseudoknotted sequences that are predicted as pseudoknotted and report this count as true positive (TP). Similarly we report TN, F and FN in Table 5 along with the F1 score. Most tools exclude pseudoknots while RNAstructure i the most famous one that can predict pseudoknots, so we choose it for comparison.
+
+Table 5: Evaluation of pseudoknot prediction   
+
+<table><tr><td>Method</td><td>SetF1 TP FP TN FN</td></tr><tr><td>E2Efold</td><td>0.710 ）1312 242 1271 0</td></tr><tr><td></td><td></td></tr><tr><td>RNAstructure 0.472</td><td>1248 307 983 286</td></tr></table>
+
+![](images/4cebc007e1c12649c00cf0ca0f6cb398b35db6546ff4d4c58080f56563f16127.jpg)
+
+Visualization. We visualize predicted structures of three RNA sequences in the main text. More examples are provided in appendix (Fig 8 to 14). In these figures, purple lines indicate edges of pseudoknotted elements. Although CDPfold has higher F1 score than other basetrue structurelines, its predictions are visually far from the groundtruth. Instead, RNAstructure and CONTRAfold produce comparatively more reasonable visualizations among all baselines, so we compare with them. These two methods can capture a rough sketch of the structure, but not good enough. For most cases, E2Efold produces structures most similar to the ground-truths. Moreover, it works surprisingly well for some RNA sequences that are long and very difficult to predict.
+
+![](images/165f36aef60bb7d08957e5852e615201582bc1bc0aae49760f1c937ecea2e78f.jpg)
+
+Ablation Study. To exam whether integrating the two stages by pushing gradient through the postprocess is necessary for performance of E2Efold, we conduct an ablation study (Table 6). We test the performance when the post-processing step is disconnected with the training of Deep Score Network $U _ { \theta }$ . We apply the post-processing step (i.e., for solving augmented Lagrangian) after $U _ { \theta }$ is learned (thus the notation $ { ^ { \circ } U _ { \theta } } +  { \operatorname { P P } } ^ { \beta }$ in Table 6). Although $\mathbf { \cdots } U _ { \theta } + \mathbf { P } \mathbf { P } ^ { \boldsymbol { \cdots } }$ performs decently well, with constraints incorporated into training, E2Efold still has significant advantages over it.
+
+Table 6: Ablation study (RNAStralign test set)   
+
+<table><tr><td>Method Prec</td><td>Rec</td><td>F1 Prec(S)Rec(S)F1(S)</td></tr><tr><td>E2Efold 0.866 0.788 U+PP 0.755 0.712</td><td>0.821 0.880 0.721 0.782</td><td>0.798 0.833 0.737 0.752</td></tr></table>
+
+Discussion. To better estimate the performance of E2Efold on different RNA types, we include the per-family F1 scores in Appendix D.5. E2Efold performs significantly better than other methods in 16S rRNA, tRNA, 5S RNA, tmRNA, and telomerase. These results are from a single model. In the future, we can view it as multi-task learning and further improve the performance by learning multiple models for different RNA families and learning an additional classifier to predict which model to use for the input sequence.
+
+# 7 CONCLUSION
+
+We propose a novel DL model, E2Efold, for RNA secondary structure prediction, which incorporates hard constraints in its architecture design. Comprehensive experiments are conducted to show the superior performance of E2Efold, no matter on quantitative criteria, running time, or visualization. Further studies need to be conducted to deal with the RNA types with less samples. Finally, we believe the idea of unrolling constrained programming and pushing gradient through post-processing can be generic and useful for other constrained structured prediction problems.
+
+# ACKNOWLEDGEMENT
+
+We would like to thank anonymous reviewers for providing constructive feedbacks. This work is supported in part by NSF grants CDS&E-1900017 D3SC, CCF-1836936 FMitF, IIS-1841351, CAREER IIS-1350983 to L.S. and grants from King Abdullah University of Science and Technology, under award numbers BAS/1/1624-01, FCC/1/1976-18-01, FCC/1/1976-23-01, FCC/1/1976-25-01, FCC/1/1976-26-01, REI/1/0018-01-01, and URF/1/4098-01-01.
+
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+# A MORE DISCUSSION ON RELATED WORKS
+
+Here we explain the difference between our approach and other works on unrolling optimization problems.
+
+First, our view of incorporating constraints to reduce output space and to reduce sample complexity is novel. Previous works (Hershey et al., 2014; Belanger et al., 2017; Ingraham et al., 2018) did not discuss these aspects. The most related work which also integrates constraints is OptNet (Amos & Kolter, 2017), but its very expensive and can not scale to the RNA problem. Therefore, our proposed approach is a simple and effective one.
+
+Second, compared to (Chen et al., 2018; Shrivastava et al., 2019), our approach has a different purpose of using the algorithm. Their goal is to learn a better algorithm, so they commonly make their architecture more flexible than the original algorithm for the room of improvement. However, we aim at enforcing constraints. To ensure that constraints are nicely incorporated, we keep the original structure of the algorithm and only make the hyperparameters learnable.
+
+Finally, although all works consider end-to-end training, none of them can directly optimize the F1 score. We proposed a differentiable loss function to mimic the F1 score/precision/recall, which is effective and also very useful when negative samples are much fewer than positive samples (or the inverse).
+
+# B DERIVATION OF THE PROXIMAL GRADIENT STEP
+
+The maximization step in Eq. 1 can be written as the following minimization:
+
+$$
+\operatorname* { m i n } _ { \hat { A } \in \mathbb { R } ^ { L \times L } } ~ \underbrace { - \frac { 1 } { 2 } \langle U _ { \theta } ( { \boldsymbol x } ) - s , A \rangle + \langle \lambda , \operatorname { r e l u } ( A \mathbf { 1 } - \mathbf { 1 } ) \rangle } _ { - f ( \hat { A } ) } + \rho \| \hat { A } \| _ { 1 } .
+$$
+
+Consider the quadratic approximation of $- f ( { \hat { A } } )$ centered at $\hat { A } _ { t }$
+
+$$
+\begin{array} { r l } & { - \tilde { f } _ { \alpha } ( \hat { A } ) : = - f ( \hat { A } _ { t } ) + \langle - \frac { \partial f } { \partial \hat { A } _ { t } } , \hat { A } - \hat { A } _ { t } \rangle + \displaystyle \frac { 1 } { 2 \alpha } \| \hat { A } - \hat { A } _ { t } \| _ { F } ^ { 2 } } \\ & { \quad \quad \quad \quad = - f ( \hat { A } _ { t } ) + \displaystyle \frac { 1 } { 2 \alpha } \Big \| \hat { A } - \left( \hat { A } _ { t } + \alpha \frac { \partial f } { \partial \hat { A } _ { t } } \right) \Big \| _ { F } ^ { 2 } , } \end{array}
+$$
+
+and rewrite the optimization in Eq. 10 as
+
+$$
+\begin{array} { r l } & { \underset { \hat { A } \in \mathbb R ^ { L \times L } } { \mathrm { m i n } } ~ - ~ { f ( \hat { A } _ { t } ) } + \frac { 1 } { 2 \alpha } \Big \| \hat { A } - \dot { A } _ { t + 1 } \Big \| _ { F } ^ { 2 } + \rho \| \hat { A } \| _ { 1 } } \\ & { \underset { \hat { A } \in \mathbb R ^ { L \times L } } { \equiv } \frac { 1 } { 2 \alpha } \Big \| \hat { A } - \dot { A } _ { t + 1 } \Big \| _ { F } ^ { 2 } + \rho \| \hat { A } \| _ { 1 } , } \end{array}
+$$
+
+where
+
+$$
+\dot { A } _ { t + 1 } : = \hat { A } _ { t } + \alpha \frac { \partial f } { \partial \hat { A } _ { t } } .
+$$
+
+Next, we define proximal mapping as a function depending on $\alpha$ as follows:
+
+$$
+\begin{array} { r l } & { \displaystyle p r o x _ { \alpha } ( \dot { A } _ { t + 1 } ) = \underset { \dot { A } \in \mathbb R ^ { L \times L } } { \arg \operatorname* { m i n } } \ \frac { 1 } { 2 \alpha } \Bigl \| \hat { A } - \dot { A } _ { t + 1 } \Bigr \| _ { F } ^ { 2 } + \rho \| \hat { A } \| _ { 1 } } \\ & { \quad \quad \quad = \underset { \dot { A } \in \mathbb R ^ { L \times L } } { \arg \operatorname* { m i n } } \ \frac { 1 } { 2 } \Bigl \| \hat { A } - \dot { A } _ { t + 1 } \Bigr \| _ { F } ^ { 2 } + \alpha \rho \| \hat { A } \| _ { 1 } } \\ & { \quad \quad \quad = \mathrm { s i g n } ( \dot { A } _ { t + 1 } ) \operatorname* { m a x } ( | \dot { A } _ { t + 1 } | - \alpha \rho , 0 ) } \\ & { \quad \quad \quad = \mathrm { s i g n } ( \dot { A } _ { t + 1 } ) \mathrm { r e l u } ( | \dot { A } _ { t + 1 } | - \alpha \rho ) . } \end{array}
+$$
+
+Since we always use $\hat { A } \circ \hat { A }$ instead of $\hat { A }$ in our problem, we can take the absolute value $| p r o x _ { \alpha } ( \dot { A } _ { t + 1 } ) | = \mathrm { r e l u } ( | \dot { A } _ { t + 1 } | - \alpha \rho )$ without loss of generality. Therefore, the proximal gradient
+
+step is
+
+$$
+\begin{array} { r l } & { \dot { A } _ { t + 1 }  \hat { A } _ { t } + \alpha \cfrac { \partial f } { \partial \hat { A } _ { t } } \quad \mathrm { ( c o r r e s p o n d t o E q . 3 ) } } \\ & { \dot { A } _ { t + 1 }  \mathrm { r e l u } ( \vert \dot { A } _ { t + 1 } \vert - \alpha \rho ) \quad \mathrm { ( c o r r e s p o n d t o E q . 5 ) } . } \end{array}
+$$
+
+More specifically, in the main text, we write $\frac { \partial f } { \partial \hat { A } _ { t } }$ as
+
+$$
+\begin{array} { r l } & { \frac { \partial f } { \partial \hat { A } _ { t } } = \frac { 1 } { 2 } \left( \frac { \partial f } { \partial A _ { t } } + \frac { \partial f } { \partial A _ { t } } ^ { \top } \right) \circ \frac { \partial A _ { t } } { \partial \hat { A } _ { t } } } \\ & { \qquad = \left( \frac { 1 } { 2 } \frac { \partial A _ { t } } { \partial \hat { A } _ { t } } \right) \circ \left( \frac { \partial f } { \partial A _ { t } } + \frac { \partial f } { \partial A _ { t } } ^ { \top } \right) } \\ & { \qquad = \left( \frac { 1 } { 2 ^ { 2 } } \circ M \circ ( 2 \hat { A } _ { t } + 2 \hat { A } _ { t } ^ { \top } ) \right) \circ \left( \frac { \partial f } { \partial A _ { t } } + \frac { \partial f } { \partial A _ { t } } ^ { \top } \right) } \\ & { \qquad = \left( \frac { 1 } { 2 ^ { 2 } } \circ M \circ ( 2 \hat { A } _ { t } + 2 \hat { A } _ { t } ^ { \top } ) \right) \circ \left( \frac { \partial f } { \partial A _ { t } } + \frac { \partial f } { \partial A _ { t } } ^ { \top } \right) } \\ & { \qquad = M \circ \hat { A } _ { t } \circ \left( \frac { \partial f } { \partial A _ { t } } + \frac { \partial f } { \partial A _ { t } } ^ { \top } \right) . } \end{array}
+$$
+
+The last equation holds since $\hat { A } _ { t }$ will remain symmetric in our algorithm if the initial $\hat { A } _ { 0 }$ is symmetric. Moreover, in the main text, $\alpha$ is replaced by $\boldsymbol { \alpha } \cdot \boldsymbol { \gamma } _ { \alpha } ^ { t }$ .
+
+# C IMPLEMENTATION AND TRAINING DETAILS
+
+We used Pytorch to implement the whole package of E2Efold.
+
+Deep Score Network. In the deep score network, we used a hyper-parameter, $d$ , which was set as 10 in the final model, to control the model capacity. In the transformer encoder layers, we set the number of heads as 2, the dimension of the feed-forward network as 2048, the dropout rate as 0.1. As for the position encoding, we used 58 base functions to form the position feature map, which goes through a 3-layer fully-connected neural network (the number of hidden neurons is $5 * d$ ) to generate the final position embedding, whose dimension is $L$ by $d$ . In the final output layer, the pairwise concatenation is carried out in the following way: Let $\boldsymbol { X } \in \mathbb { R } ^ { L \times 3 d }$ be the input to the final output layers in Figure 4 (which is the concatenation of the sequence embedding and position embedding). The pairwise concatenation results in a tensor $Y \in \mathbb { R } ^ { L \times L \times 6 d }$ defined as
+
+$$
+Y ( i , j , : ) = [ X ( i , : ) , X ( j , : ) ] ,
+$$
+
+where $Y ( i , j , : ) \in \mathbb { R } ^ { 6 d }$ , $X ( i , : ) \in \mathbb { R } ^ { 3 d }$ , and $X ( j , : ) \in \mathbb { R } ^ { 3 d }$ .
+
+In the 2D convolution layers, the the channel of the feature map gradually change from $6 { * d }$ to $d$ , and finally to 1. We set the kernel size as 1 to translate the feature map into the final score matrix. Each 2D convolution layer is followed by a batch normalization layer. We used ReLU as the activation function within the whole score network.
+
+Post-Processing Network. In the PP network, we initialized $w$ as 1, $s$ as $\log ( 9 )$ , $\alpha$ as 0.01, $\beta$ as 0.1, $\gamma _ { \alpha }$ as 0.99, $\gamma _ { \beta }$ as 0.99, and $\rho$ as 1. We set $T$ as 20.
+
+Training details. During training, we first pre-trained a deep score network and then fine-tuned the score network and the PP network together. To pre-train the score network, we used binary crossentropy loss and Adam optimizer. Since, in the contact map, most entries are 0, we used weighted loss and set the positive sample weight as 300. The batch size was set to fully use the GPU memory, which was 20 for the Titan Xp card. We pre-train the score network for 100 epochs. As for the fine-tuning, we used binary cross-entropy loss for the score network and F1 loss for the PP network and summed up these two losses as the final loss. The user can also choose to only use the F1 loss or use another coefficient to weight the loss estimated on the score network $U _ { \theta }$ . Due to the limitation of the GPU memory, we set the batch size as 8. However, we updated the model’s parameters every 30 steps to stabilize the training process. We fine-tuned the whole model for 20 epochs. Also, since the data for different RNA families are imbalanced, we up-sampled the data in the small RNA families based on their size. For the training of the score network $U _ { \theta }$ in the ablation study, it is exactly the same as the training of the above mentioned process. Except that during the fine-tune process, there is the unrolled number of iterations is set to be 0.
+
+# D MORE EXPERIMENTAL DETAILS
+
+# D.1 DATASET STATISTICS
+
+![](images/445430595c93431ed574baae67b7fb1b4a4a1b242415f39b377948a0a1acb6cb.jpg)  
+Figure 6: The RNAStralign length distribution.
+
+Table 7: RNAStralign dataset splits statistics   
+
+<table><tr><td>RNA type</td><td>All</td><td>Training</td><td>Validation</td><td>Testing</td></tr><tr><td>16SrRNA</td><td>11620</td><td>9325</td><td>1145</td><td>1150</td></tr><tr><td>5SrRNA</td><td>9385</td><td>7687</td><td>819</td><td>879</td></tr><tr><td>tRNA</td><td>6443</td><td>5412</td><td>527</td><td>504</td></tr><tr><td>grp1</td><td>1502</td><td>1243</td><td>123</td><td>136</td></tr><tr><td>SRP</td><td>468</td><td>379</td><td>36</td><td>53</td></tr><tr><td>tmRNA</td><td>572</td><td>461</td><td>50</td><td>61</td></tr><tr><td>RNaseP</td><td>434</td><td>360</td><td>37</td><td>37</td></tr><tr><td>telomerase</td><td>37</td><td>28</td><td>4</td><td>5</td></tr><tr><td>RNAStralign</td><td>30451</td><td>24895</td><td>2702</td><td>2854</td></tr></table>
+
+# D.2 TWO-SAMPLE HYPOTHESIS TESTING
+
+To better understand the data distribution in different datasets, we provide statistical hypothesis test results in this section.
+
+We can assume that
+
+(i) Samples in RNAStralign training set are i.i.d. from the distribution $\mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t r a i n } } )$
+
+(ii) Samples in RNAStralign testing set are i.i.d. from the distribution $\mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t e s t } } )$ ;
+
+(iii) Samples in ArchiveII dataset are i.i.d. from the distribution $\mathcal { P } ( \mathrm { A r c I I } )$ .
+
+To compare the differences among these data distributions, we can test the following hypothesis:
+
+(a) $\mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t r a i n } } ) = \mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t e s t } } )$ (b) $\mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t r a i n } } ) = \mathcal { P } ( \mathrm { A r c h i v e I I } )$
+
+The approach that we adopted is the permutation test on the unbiased empirical Maximum Mean Discrepancy (MMD) estimator:
+
+$$
+ { \mathbf { M M D } } _ { u } ( X , Y ) : = \Big ( \sum _ { i = 1 } ^ { N } \sum _ { j \neq i } ^ { N } k ( x _ { i } , x _ { j } ) + \sum _ { i = 1 } ^ { M } \sum _ { j \neq i } ^ { M } k ( y _ { i } , y _ { j } ) - \frac { 2 } { m n } \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { M } k ( x _ { i } , y _ { j } ) \Big ) ^ { \frac { 1 } { 2 } } ,
+$$
+
+where $X = \{ x _ { i } \} _ { i = 1 } ^ { N }$ contains $N$ i.i.d. samples from a distribution $\mathcal { P } _ { 1 }$ , $Y = \{ y _ { i } \} _ { i = 1 } ^ { M }$ contains $M$ i.i.d. samples from a distribution $\mathcal { P } _ { 2 }$ , and $k ( \cdot , \cdot )$ is a string kernel.
+
+Since we conduct stratified sampling to split the training and testing dataset, when we perform permutation test, we use stratified re-sampling as well (for both Hypothese (a) and (b)). The result of the permutation test (permuted 1000 times) is reported in Figure 7.
+
+![](images/258a691b364aebf724acc6efbb31e6b2a2c9d18b3d3b699f058590ea74eac833.jpg)  
+Figure 7: Left: Distribution of $\mathbf { M M D } _ { u }$ under Hypothesis $\mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t r a i n } } ) = \mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t e s t } } )$ . Right: Distribution of $\mathbf { M M D } _ { u }$ under Hypothesis $\mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t r a i n } } ) = \mathcal { P } ($ (ArchiveII).
+
+The result shows
+
+(a) Hypothesis $\mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t r a i n } } ) = \mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t e s t } } )$ can be accepted with significance level 0.1.   
+(b) Hypothesis $\mathcal { P } ( \mathrm { R N A S t r } _ { \mathrm { t r a i n } } ) = \mathcal { P } ($ (ArchiveII) is rejected since the p-value is 0.
+
+Therefore, the data distribution in ArchiveII is very different from the RNAStralign training set. A good performance on ArchiveII shows a significant generalization power of E2Efold.
+
+D.3 PERFORMANCE ON LONG SEQUENCES: WEIGHTED F1 SCORE
+
+For long sequences, E2Efold still performs better than other methods. We compute F1 scores weighted by the length of sequences (Table 8), such that the results are more dominated by longer sequences.
+
+Table 8: RNAStralign: F1 after a weighted average by sequence length.   
+
+<table><tr><td>Method</td><td>E2Efold</td><td>CDPfold</td><td>LinearFold</td><td>Mfold</td><td>RNAstructure</td><td>RNAfold</td><td>CONTRAfold</td></tr><tr><td>non-weighted</td><td>0.821</td><td>0.614</td><td>0.609</td><td>0.420</td><td>0.550</td><td>0.540</td><td>0.633</td></tr><tr><td>weighted</td><td>0.720</td><td>0.691</td><td>0.509</td><td>0.366</td><td>0.471</td><td>0.444</td><td>0.542</td></tr><tr><td>change</td><td>-12.3%</td><td>+12.5%</td><td>-16.4%</td><td>-12.8%</td><td>-14.3%</td><td>-17.7%</td><td>-14.3%</td></tr></table>
+
+The third row reports how much F1 score drops after reweighting.
+
+# D.4 ARCHIVEII RESULTS AFTER DOMAIN SEQUENCES ARE REMOVED
+
+Since domain sequence (subsequences) in ArchiveII are explicitly labeled, we filter them out in ArchiveII and recompute the F1 scores (Table 9).
+
+The results do not change too much before or after filtering out subsequences.
+
+Table 9: ArchiveII: F1 after subsequences are filtered out.   
+
+<table><tr><td>Method</td><td>E2Efold</td><td>CDPfold</td><td>LinearFold</td><td>Mfold</td><td>RNAstructure</td><td>RNAfold</td><td>CONTRAfold</td></tr><tr><td>original</td><td>0.704</td><td>0.597</td><td>0.647</td><td>0.421</td><td>0.613</td><td>0.615</td><td>0.662</td></tr><tr><td>filtered</td><td>0.723</td><td>0.605</td><td>0.645</td><td>0.419</td><td>0.611</td><td>0.615</td><td>0.659</td></tr></table>
+
+# D.5 PER-FAMILY PERFORMANCES
+
+To balance the performance among different families, during the training phase we conducted weighted sampling of the data based on their family size. With weighted sampling, the overall F1 score (S) is 0.83, which is the same as when we did equal-weighted sampling. The per-family results are shown in Table 10.
+
+Table 10: RNAStralign: per-family performances   
+
+<table><tr><td rowspan="2"></td><td colspan="2">16S rRNA</td><td colspan="2">tRNA</td><td colspan="2">5S RNA</td><td colspan="2">SRP</td></tr><tr><td>F1</td><td>F1(S)</td><td>F1</td><td>F1(S)</td><td>F1</td><td>F1(S)</td><td>F1</td><td>F1(S)</td></tr><tr><td>E2Efold</td><td>0.783</td><td>0.795</td><td>0.917</td><td>0.939</td><td>0.906</td><td>0.936</td><td>0.550</td><td>0.614</td></tr><tr><td>LinearFold</td><td>0.493</td><td>0.504</td><td>0.734</td><td>0.739</td><td>0.713</td><td>0.738</td><td>0.618</td><td>0.648</td></tr><tr><td>Mfold</td><td>0.362</td><td>0.373</td><td>0.662</td><td>0.675</td><td>0.356</td><td>0.367</td><td>0.350</td><td>0.378</td></tr><tr><td>RNAstructure</td><td>0.464</td><td>0.485</td><td>0.709</td><td>0.736</td><td>0.578</td><td>0.597</td><td>0.579</td><td>0.617</td></tr><tr><td>RNAfold</td><td>0.430</td><td>0.449</td><td>0.695</td><td>0.706</td><td>0.592</td><td>0.612</td><td>0.617</td><td>0.651</td></tr><tr><td>CONTRAfold</td><td>0.529</td><td>0.546</td><td>0.758</td><td>0.765</td><td>0.717</td><td>0.740</td><td>0.563</td><td>0.596</td></tr><tr><td></td><td colspan="2">tmRNA</td><td>Group Iintron</td><td></td><td colspan="2">RNaseP</td><td colspan="2">telomerase</td></tr><tr><td></td><td>F1</td><td>F1(S)</td><td>F1</td><td>F1(S)</td><td>F1</td><td>F1(S)</td><td>F1</td><td>F1(S)</td></tr><tr><td>E2Efold</td><td>0.588</td><td>0.653</td><td>0.387</td><td>0.428</td><td>0.565</td><td>0.604</td><td>0.954</td><td>0.961</td></tr><tr><td>LinearFold</td><td>0.393</td><td>0.412</td><td>0.565</td><td>0.579</td><td>0.567</td><td>0.578</td><td>0.515</td><td>0.531</td></tr><tr><td>Mfold</td><td>0.290</td><td>0.308</td><td>0.483</td><td>0.498</td><td>0.562</td><td>0.579</td><td>0.403</td><td>0.531</td></tr><tr><td>RNAstructure</td><td>0.400</td><td>0.423</td><td>0.566</td><td>0.599</td><td>0.589</td><td>0.616</td><td>0.512</td><td>0.545</td></tr><tr><td>RNAfold</td><td>0.411</td><td>0.430</td><td>0.589</td><td>0.599</td><td>0.544</td><td>0.563</td><td>0.471</td><td>0.496</td></tr><tr><td>CONTRAfold</td><td>0.463</td><td>0.482</td><td>0.603</td><td>0.620</td><td>0.645</td><td>0.662</td><td>0.529</td><td>0.548</td></tr></table>
+
+![](images/eea787fd6595d57642aa73a5d26dd6760fa1ce67428aaa82823346cca926448d.jpg)  
+Figure 8: Visualization of 5S rRNA, B01865.
+
+![](images/dca67c5aa7b1fe2e2ddfd6f3fb357068b9ef5edac9bd3be6936e2712de78fa56.jpg)  
+Figure 9: Visualization of 16S rRNA, DQ170870.
+
+![](images/d92a110db26faf6c2820b004a4742fdd20dee835bdb1fbaf65601704f5fec892.jpg)  
+Figure 10: Visualization of Group I intron, IC3, Kaf.c.trnL.
+
+![](images/f3b94cba807c5deca9ab3a7bb9d7cdb4f17e8e4b7eecda89e30b5f9ad24f8d29.jpg)  
+Figure 11: Visualization of RNaseP, A.salinestris-184.
+
+![](images/8362cee7b2575102890b0c08813cebbdafa6788ba5953d0a793280d9a78024ea.jpg)  
+Figure 12: Visualization of SRP, Homo.sapi. BU56690.
+
+![](images/db29663aeb283cc29ddc3b917ee2ced586aec34c55bf86425e8565a955f28ac3.jpg)  
+Figure 13: Visualization of tmRNA, uncu.bact. AF389956.
+
+![](images/8233219a50f9fba107301ff937117c9b05ed93159f7a3e87a6126e47b0952075.jpg)  
+Figure 14: Visualization of tRNA, tdbD00012019.

parse/train/S1l-C0NtwS/S1l-C0NtwS.md

@@ -0,0 +1,333 @@
+# CROSS-LINGUAL ALIGNMENT VS JOINT TRAINING:A COMPARATIVE STUDY AND A SIMPLE UNIFIEDFRAMEWORK
+
+Zirui Wang∗, Jiateng Xie∗, Ruochen Xu, Yiming Yang, Graham Neubig, Jaime Carbonell Language Technologies Institute, Carnegie Mellon University, Pittsburgh, PA 15213, USA {ziruiw, jiatengx, ruochenx, yiming, gneubig, jgc}@cs.cmu.edu
+
+# ABSTRACT
+
+Learning multilingual representations of text has proven a successful method for many cross-lingual transfer learning tasks. There are two main paradigms for learning such representations: (1) alignment, which maps different independently trained monolingual representations into a shared space, and (2) joint training, which directly learns unified multilingual representations using monolingual and cross-lingual objectives jointly. In this paper, we first conduct direct comparisons of representations learned using both of these methods across diverse crosslingual tasks. Our empirical results reveal a set of pros and cons for both methods, and show that the relative performance of alignment versus joint training is taskdependent. Stemming from this analysis, we propose a simple and novel framework that combines these two previously mutually-exclusive approaches. Extensive experiments demonstrate that our proposed framework alleviates limitations of both approaches, and outperforms existing methods on the MUSE bilingual lexicon induction (BLI) benchmark. We further show that this framework can generalize to contextualized representations such as Multilingual BERT, and produces state-of-the-art results on the CoNLL cross-lingual NER benchmark.1
+
+# 1 INTRODUCTION
+
+Continuous word representations (Mikolov et al., 2013a; Pennington et al., 2014; Bojanowski et al., 2017) have become ubiquitous across a wide range of NLP tasks. In particular, methods for crosslingual word embeddings (CLWE) have proven a powerful tool for cross-lingual transfer for downstream tasks, such as text classification (Klementiev et al., 2012), dependency parsing (Ahmad et al., 2019), named entity recognition (NER) (Xie et al., 2018; Chen et al., 2019), natural language inference (Conneau et al., 2018b), language modeling (Adams et al., 2017), and machine translation (MT) (Zou et al., 2013; Lample et al., 2018a; Artetxe et al., 2018b; Lample et al., 2018b). The goal of these CLWE methods is to learn embeddings in a shared vector space for two or more languages. There are two main paradigms for learning CLWE: cross-lingual alignment and joint training.
+
+The most successful approach has been the cross-lingual embedding alignment method (Mikolov et al., 2013b), which relies on the assumption that monolingually-trained continuous word embedding spaces share similar structure across different languages. The underlying idea is to first independently train embeddings in different languages using monolingual corpora alone, and then learn a mapping to align them to a shared vector space. Such a mapping can be trained in a supervised fashion using parallel resources such as bilingual lexicons (Xing et al., 2015; Smith et al., 2017; Joulin et al., 2018b; Jawanpuria et al., 2019), or even in an unsupervised2 manner based on distribution matching (Zhang et al., 2017a; Conneau et al., 2018a; Artetxe et al., 2018a; Zhou et al., 2019). Recently, it has been shown that alignment methods can also be effectively applied to contextualized word representations (Schuster et al., 2019; Aldarmaki & Diab, 2019).
+
+Another successful line of research for CLWE considers joint training methods, which optimize a monolingual objective predicting the context of a word in a monolingual corpus along with either a hard or soft cross-lingual constraint. Similar to alignment methods, some early works rely on bilingual dictionaries (Ammar et al., 2016; Duong et al., 2016) or parallel corpora (Luong et al., 2015; Gouws et al., 2015) for direct supervision. More recently, a seemingly naive unsupervised joint training approach has received growing attention due to its simplicity and effectiveness. In particular, Lample et al. (2018b) reports that simply training embeddings on concatenated monolingual corpora of two related languages using a shared vocabulary without any cross-lingual resources is able to produce higher accuracy than the more sophisticated alignment methods on unsupervised MT tasks. Besides, for contextualized representations, unsupervised multilingual language model pretraining using a shared vocabulary has produced state-of-the-art results on multiple benchmarks3 (Devlin et al., 2019; Artetxe & Schwenk, 2019; Lample & Conneau, 2019).
+
+Despite a large amount of research on both alignment and joint training, previous work has neither performed a systematic comparison between the two, analyzed their pros and cons, nor elucidated when we may prefer one method over the other. Particularly, it’s natural to ask: (1) Does the phenomenon reported in Lample et al. (2018b) extend to other cross-lingual tasks? (2) Can we employ alignment methods to further improve unsupervised joint training? (3) If so, how would such a framework compare to supervised joint training methods that exploit equivalent resources, i.e., bilingual dictionaries? (4) And lastly, can this framework generalize to contextualized representations?
+
+In this work, we attempt to address these questions. Specifically, we first evaluate and compare alignment versus joint training methods across three diverse tasks: BLI, cross-lingual NER, and unsupervised MT. We seek to characterize the conditions under which one approach outperforms the other, and glean insight on the reasons behind these differences. Based on our analysis, we further propose a simple, novel, and highly generic framework that uses unsupervised joint training as initialization and alignment as refinement to combine both paradigms. Our experiments demonstrate that our framework improves over both alignment and joint training baselines, and outperforms existing methods on the MUSE BLI benchmark. Moreover, we show that our framework can generalize to contextualized representations such as Multilingual BERT, producing state-of-the-art results on the CoNLL cross-lingual NER benchmark. To the best of our knowledge, this is the first framework that combines previously mutually-exclusive alignment and joint training methods.
+
+# 2 BACKGROUND: CROSS-LINGUAL REPRESENTATIONS
+
+Notation. We assume we have two different languages $\{ L _ { 1 } , L _ { 2 } \}$ and access to their corresponding training corpora. We use $V _ { L _ { i } } = { \{ w _ { L _ { i } } ^ { j } \} } _ { j = 1 } ^ { n _ { L _ { i } } }$ to denote the vocabulary set of the ith language where each $w _ { L _ { i } } ^ { j }$ represents a unique token, such as a word or subword. The goal is to learn a set of embeddings $E = \{ \pmb { x } ^ { j } \} _ { j = 1 } ^ { m }$ , with $\pmb { x } ^ { j } \in \mathbb { R } ^ { d }$ , in a shared vector space, where each token $w _ { L _ { i } } ^ { j }$ is mapped to a vector in $E$ . Ideally, these vectorial representations should have similar values for tokens with similar meanings or syntactic properties, so they can better facilitate cross-lingual transfer.
+
+# 2.1 ALIGNMENT METHODS
+
+Given the notation, alignment methods consist of the following steps:
+
+Step 1: Train an embedding set $E _ { 0 } = E _ { L _ { 1 } } \cup E _ { L _ { 2 } }$ , where each subset $E _ { L _ { i } } = \{ x _ { L _ { i } } ^ { j } \} _ { j = 1 } ^ { n _ { L _ { i } } }$ is trained independently using the ith language corpus and contains an embedding $x _ { L _ { i } } ^ { j }$ for each token $w _ { L _ { i } } ^ { j }$ .
+
+Step 2: Obtain a seed dictionary $D = \{ ( w _ { L _ { 1 } } ^ { i } , w _ { L _ { 2 } } ^ { j } ) \} _ { k = 1 } ^ { K }$ , either provided or learnt unsupervised.
+
+Step 3: Learn a projection matrix $W \in \mathbb { R } ^ { d \times d }$ based on $D$ , resulting in a final embedding set $\ c E _ { A } \overset { - } { = } ( W \cdot E _ { L _ { 1 } } ) \overset { - } { \cup } \bar { E } _ { L _ { 2 } }$ in a shared vector space.
+
+To find the optimal projection matrix $W$ , Mikolov et al. (2013b) proposed to solve the following optimization problem:
+
+$$
+\operatorname* { m i n } _ { W \in \mathbb { R } ^ { d \times d } } \| W X _ { L _ { 1 } } - X _ { L _ { 2 } } \| _ { F }
+$$
+
+where $X _ { L _ { 1 } }$ and $X _ { L _ { 2 } }$ are matrices of size $d \times K$ containing embeddings of the words in $D$ . Xing et al. (2015) later showed further improvement could be achieved by restricting $W$ to an orthogonal matrix, which turns the Eq.(1) into the Procrustes problem with the following closed form solution:
+
+$$
+\begin{array} { r } { W ^ { * } = U V ^ { T } , \quad \quad } \\ { \mathrm { w i t h ~ } U \Sigma V ^ { T } = \mathrm { S V D } ( X _ { L _ { 2 } } X _ { L _ { 1 } } ^ { T } ) } \end{array}
+$$
+
+where $W ^ { * }$ denotes the optimal solution and SVD(·) stands for the singular value decomposition.
+
+As surveyed in Section 5, different methods (Smith et al., 2017; Conneau et al., 2018a; Joulin et al., 2018b; Artetxe et al., 2018a) differ in the way how they obtain the dictionary $D$ and how they solve for $W$ in step 3. However, most of them still involve solving the Eq.(2) as a crucial step.
+
+# 2.2 JOINT TRAINING METHODS
+
+Joint training methods in general have the following objective:
+
+$$
+\mathscr { L } _ { J } = \mathscr { L } _ { 1 } + \mathscr { L } _ { 2 } + \mathscr { R } \left( L _ { 1 } , L _ { 2 } \right)
+$$
+
+where $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ are monolingual objectives and $\mathcal { R } ( L _ { 1 } , L _ { 2 } )$ is a cross-lingual regularization term. For example, Klementiev et al. (2012) use language modeling objectives for $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ . The term $\mathcal { R } ( L _ { 1 } , L _ { 2 } )$ encourages alignment of representations of words that are translations. Training an embedding set $E _ { J } = E _ { L _ { 1 } } \cup E _ { L _ { 2 } }$ is usually done by directly optimizing $\mathcal { L } _ { J }$ .
+
+While supervised joint training requires access to parallel resources, recent studies (Lample et al., 2018b; Devlin et al., 2019; Artetxe & Schwenk, 2019; Lample & Conneau, 2019) have suggested that unsupervised joint training without such resources is also effective. Specifically, they show that the cross-lingual regularization term $\mathcal { R } ( L _ { 1 } , L _ { 2 } )$ does not require direct cross-lingual supervision to achieve highly competitive results. This is because the shared words between $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ can serve implicitly as anchors by sharing their embeddings to ensure that representations of different languages lie in a shared space. Using our notation, the unsupervised joint training approach takes the following steps:
+
+Step 1: Construct a joint vocabulary $V _ { J } = V _ { L _ { 1 } } \cup V _ { L _ { 2 } }$ that is shared across two languages.
+
+Step 2: Concatenate the two training corpora and learn an embedding set $E _ { J }$ corresponding to $V _ { J }$
+
+The joint vocabulary is composed of three disjoint sets: $V _ { J } ^ { 1 } , V _ { J } ^ { 2 } , V _ { J } ^ { s }$ , where $V _ { J } ^ { s } = V _ { L _ { 1 } } \cap V _ { L _ { 2 } }$ is the shared vocabulary set and $V _ { J } ^ { i }$ is the set of tokens that appear in the $i$ th language only. Note that a key difference of existing supervised joint training methods is that embeddings corresponding to $V _ { J } ^ { s }$ are not shared between $E _ { L _ { 1 } }$ and $E _ { L _ { 2 } }$ , meaning that they are disjoint, as in alignment methods.
+
+# 2.3 DISCUSSION
+
+While alignment methods have had great success, there are still some critical downsides, among which we stress the following points:
+
+1. While recent studies in unsupervised joint training have suggested the potential benefits of word sharing, alignment methods rely on two disjoint sets of embeddings. Along with some possible loss of information due to no sharing, one consequence is that finetuning the aligned embeddings on downstream tasks may be sub-optimal due to the lack of crosslingual constraints at the finetuning stage, whereas shared words can fulfill this role in jointly trained models.   
+2. A key assumption of alignment methods is the isomorphism of monolingual embedding spaces. However, some recent papers have challenged this assumption, showing that it does not hold for many language pairs (Søgaard et al., 2018; Patra et al., 2019). Also notably, Ormazabal et al. (2019) suggests that this limitation results from the fact that the two sets of monolingual embeddings are independently trained.
+
+On the other hand, the unsupervised joint training method is much simpler and doesn’t share these disadvantages with the alignment methods, but there are also some key limitations:
+
+![](images/b6891e82933db0590976c1a8dfda71666b5167846eb8b62f827eff934a7e3443.jpg)  
+Figure 1: PCA visualization of English and Spanish embeddings learnt by unsupervised joint training as in Lample et al. (2018b). As shown by plots (a) and (b), most words are shared in the initial embedding space but not well-aligned, hence the oversharing problem. Plots (b) and (c) shows that the vocabulary reallocation step effectively mitigates oversharing while the alignment refinement step further improves the poorly aligned embeddings by projecting them into a close neighborhood.
+
+1. It assumes that all shared words across two languages serve implicitly as anchors and thus need not be aligned to other words. Nonetheless, this assumption is not always true, leading to misalignment. For example, the English word “the” will most likely also appear in the training corpus of Spanish, but preferably it should be paired with Spanish words such as “el” and “la” instead of itself. We refer to this problem as oversharing.   
+2. It does not utilize any explicit form of seed dictionary as in alignment methods, resulting in potentially less accurate alignments, especially for words that are not shared.
+
+Lastly, while the supervised joint training approach does not have the same issues of unsupervised joint training, it shares limitation 1 of the alignment methods.
+
+We empirically compare both joint training and alignment approaches in Section 4 and shed light on some of these pros and cons for both paradigms (See Section 4.3.1).
+
+# 3 PROPOSED FRAMEWORK
+
+Motivated by the pros and cons of both paradigms, we propose a unified framework that first uses unsupervised joint training as a coarse initialization and then applies alignment methods for refinement, as demonstrated in Figure 1. Specifically, we first build a single set of embeddings with a shared vocabulary through unsupervised joint training, so as to alleviate the limitations of alignment methods. Next, we use a vocabulary reallocation technique to mitigate oversharing, before finally resorting back to alignment methods to further improve the embeddings’ quality. Lastly, we show that this framework can generalize to contextualized representations.
+
+# 3.1 UNIFYING ALIGNMENT WITH JOINT TRAINING
+
+Our proposed framework mainly involves three components and we discuss each of them as follows.
+
+Joint Initialization. We use unsupervised joint training (Lample et al., 2018b) to train the initial CLWE. As described in Section 2.2, we first obtain a joint vocabulary $V _ { J }$ and train its corresponding set of embeddings $E _ { J }$ on the concatenated corpora of two languages. This allows us to obtain a single set of embeddings that maximizes sharing across two languages. To train embeddings, we used fastText4 (Bojanowski et al., 2017) in all our experiments for both word and subword tokens.
+
+Vocabulary Reallocation. As discussed in Section 2.3, a key issue of unsupervised joint training is oversharing, which prohibits further refinement as shown in Figure 1. To alleviate this drawback, we attempt to “unshare” some of the overshared words, so their embeddings can be better aligned in the next step. Particularly, we perform a vocabulary reallocation step such that words appearing mostly exclusively in the ith language are reallocated from the shared vocabulary $V _ { J } ^ { s }$ to $V _ { J } ^ { i }$ , whereas words that appear similarly frequent in both languages stay still in $V _ { J } ^ { s }$ . Formally, for each token $w$ in the shared vocabulary $V _ { J } ^ { s }$ , we use the ratio of counts within each language to determine whether it belongs to the shared vocabulary:
+
+$$
+r = \frac { T _ { L _ { 2 } } } { T _ { L _ { 1 } } } \cdot \frac { C _ { L _ { 1 } } ( w ) } { C _ { L _ { 2 } } ( w ) } ,
+$$
+
+where $C _ { L _ { i } } ( w )$ is the count of $w$ in the training corpus of the $i$ th language and $\begin{array} { r } { T _ { L _ { i } } = \sum _ { w } C _ { L _ { i } } ( w ) } \end{array}$ is the total number of tokens. The token $w$ is allocated to the shared vocabulary if
+
+$$
+\frac { 1 - \gamma } { \gamma } \leq r \leq \frac { \gamma } { 1 - \gamma } ,
+$$
+
+where $\gamma$ is a hyper-parameter. Otherwise, we put $w$ into either $V _ { J } ^ { 1 }$ or $V _ { J } ^ { 2 }$ , where it appears mostly frequent. The above process generates three new disjoint vocabulary sets $V _ { J } ^ { 1 ^ { \prime } } , V _ { J } ^ { 2 ^ { \prime } } , V _ { J } ^ { s ^ { \prime } }$ and their corresponding embeddings $E _ { J } ^ { 1 ^ { \prime } } , E _ { J } ^ { 2 ^ { \prime } } , E _ { J } ^ { s ^ { \prime } }$ that are used thereafter. Note that, $V _ { J } ^ { ' } = V _ { J }$ and $E _ { J } ^ { ' } = E _ { J }$ .
+
+Alignment Refinement. The unsupervised joint training method does not explicitly utilize any dictionary or form of alignment. Thus, the resulting embedding set is coarse and ill-aligned in the shared vector space, as demonstrated in Figure 1. As a final refinement step, we utilize any off-the-shelf alignment method to refine alignments across the non-sharing embedding sets, i.e. mapping $E _ { J } ^ { 1 ^ { \prime } }$ to $\mathrm { ~ \bar { \it E } _ { \it J } ^ { 2 ^ { \prime } } ~ }$ and leaving $E _ { J } ^ { s ^ { \prime } }$ untouched. This step could be conducted by either supervised or unsupervised alignment method and we compare both in our experiments.
+
+# 3.2 EXTENSION TO CONTEXTUALIZED REPRESENTATIONS
+
+As our framework is highly generic and applicable to any alignment and unsupervised joint training methods, it can naturally generalize to contextualized word representations by aligning the fixed outputs of a multilingual encoder such as multilingual BERT (M-BERT) (Devlin et al., 2019). While our vocab reallocation technique is no longer necessary as contextualized representations are dependent on context and thus dynamic, we can still apply alignment refinement on extracted contextualized features for further improvement. For instance, as proposed by Aldarmaki & Diab (2019), one method to perform alignment on contextualized representations is to first use word alignment pairs extracted from parallel corpora as a dictionary, learn an alignment matrix $W$ based on it, and apply $W$ back to the extracted representations. To obtain $W$ , we can solve Eq.( 1) as described in Section 2.1, where the embedding matrices $X _ { L _ { 1 } }$ and $X _ { L _ { 2 } }$ now contain contextualized representations of aligned word pairs. Note that this method is applicable to fixed representations but not finetuning.
+
+# 4 EXPERIMENTS
+
+We evaluate the proposed approach and compare with alignment and joint training methods on three NLP benchmarks. This evaluation aims to: (1) systematically compare alignment vs. joint training paradigms and reveal their pros and cons discussed in Section 2.3, (2) show that the proposed framework can effectively alleviate limitations of both alignment and joint training, and (3) demonstrate the effectiveness of the proposed framework in both non-contextualized and contextualized settings.
+
+# 4.1 EVALUATION TASKS
+
+Bilingual Lexicon Induction (BLI) This task has been the de facto evaluation task for CLWE methods. It considers the problem of retrieving the target language translations of source langauge words. We use bilingual dictionaries complied by Conneau et al. (2018a) and test on six diverse language pairs, including Chinese and Russian, which use a different writing script than English. Each test set consists of 1500 queries and we report precision at $^ { l }$ scores $( \mathrm { P } @ 1 )$ , following standard evaluation practices (Conneau et al., 2018a; Glavas et al., 2019).
+
+Name Entity Recognition (NER) We also evaluate our proposed framework on cross-lingual NER, a sequence labeling task, where we assign a label to each token in a sequence. We evaluate both non-contextualized and contextualized word representations on the CoNLL 2002 and 2003 benchmarks (Tjong Kim Sang, 2002; Tjong Kim Sang & De Meulder, 2003), which contain 4 European languages. To measure the quality of CLWE, we perform zero-shot cross-lingual classification, where we train a model on English and directly apply it to each of the other 3 languages.
+
+Table 1: Precision $@ 1$ for the BLI task on the MUSE dataset6. Within each category, unsupervised methods are listed at the top while supervised methods are at the bottom. The best result for unsupervised methods is underlined while bold signifies the overall best. “IN” refers to iterative normalization proposed in Zhang et al. (2019), “AR” refers to alignment refinement and “VR” refers to vocabulary reallocation.   
+
+<table><tr><td></td><td>en-es</td><td>es-en</td><td>en-fr</td><td>fr-en</td><td>en-de</td><td>de-en</td><td>en-it</td><td>it-en</td><td>en-ru</td><td>ru-en</td><td>en-zh</td><td>zh-en</td><td>avg</td></tr><tr><td colspan="10">Alignment Methods</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(1)MUSE(Conneau et al.,2018a)</td><td>81.7</td><td>83.3</td><td>82.3</td><td>82.1</td><td>74.0</td><td>72.0</td><td>77.7</td><td>78.2</td><td>44.0</td><td>59.1</td><td>32.5</td><td>31.4</td><td></td><td>66.5</td></tr><tr><td>(2) VECMAP(Artetxe etal.,218a)</td><td>82.3</td><td>84.7</td><td>82.3</td><td>83.6</td><td>75.1</td><td>74.3</td><td>-</td><td>-</td><td>49.2</td><td></td><td>65.6</td><td>0.0</td><td>0.0</td><td>1</td></tr><tr><td>(3)DeMa-BWE(Zhou et al.,2019)</td><td>82.8</td><td>84.9</td><td>83.1</td><td>83.5</td><td>77.2</td><td>74.4</td><td>-</td><td>-</td><td></td><td>49.2</td><td>65.7</td><td>42.5</td><td>37.9</td><td>-</td></tr><tr><td>(4) Procrustes (Smith et al.,2017)</td><td>81.4</td><td>82.9</td><td>81.1</td><td>82.4</td><td>73.5</td><td>72.4</td><td>77.5</td><td>77.9</td><td></td><td>51.7</td><td>63.7</td><td>42.7</td><td>36.7</td><td>68.7</td></tr><tr><td>(5) GeoMM (Jawanpuria et al., 2019)</td><td>81.4</td><td>85.5</td><td>82.1</td><td>84.1</td><td>74.7</td><td>76.7</td><td>77.9</td><td>80.9</td><td></td><td>51.3</td><td>67.6</td><td>49.1</td><td>45.3</td><td>71.4</td></tr><tr><td>(6) RCSLS (Joulin et al.,2018b) (7) RCSLS + IN (Zhang et al., 2019)</td><td>84.1</td><td>86.3</td><td>83.3 83.9</td><td>84.1</td><td>79.1</td><td>76.3</td><td>78.5 79.1</td><td>79.8 -</td><td>57.9</td><td></td><td>67.2</td><td>45.9</td><td>46.4</td><td>72.4</td></tr><tr><td></td><td>83.9</td><td>-</td><td></td><td>-</td><td>78.1</td><td></td><td></td><td></td><td></td><td>57.9</td><td>1</td><td>48.6</td><td>-</td><td>1</td></tr><tr><td colspan="10">Joint Traing Methods</td><td></td><td></td><td>17.9</td><td></td><td></td></tr><tr><td>(8) Unsupervised Joint</td><td>33.4</td><td>36.6</td><td>42.2</td><td>47.4</td><td>39.5</td><td>41.4</td><td>36.8</td><td>38.8</td><td>4.0</td><td>3.5</td><td></td><td>10.2</td><td></td><td>29.3</td></tr><tr><td>(9) Supervised Joint (Duong et al.,2016)</td><td>79.7</td><td>79.8</td><td>78.1</td><td>76.7</td><td>67.5</td><td>68.9</td><td>74.4</td><td>74.1</td><td>41.8</td><td></td><td>51.8</td><td>46.7</td><td></td><td>65.2</td></tr><tr><td>(10) Joint - Replace</td><td>48.2</td><td>47.7</td><td>49.4</td><td>52.1</td><td>46.5</td><td>46.9</td><td>43.8</td><td>45.8</td><td></td><td>20.3</td><td>36.6</td><td>32.7</td><td>43.3 34.1</td><td>42.0</td></tr><tr><td colspan="10">Joint Align Framework</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(11) Joint_Align (w/o AR)</td><td>55.9</td><td>62.8</td><td>61.8</td><td>67.0</td><td>49.1</td><td>54.6</td><td>50.2</td><td>51.4</td><td>8.7</td><td></td><td>8.2</td><td>19.4</td><td></td><td>42.3</td></tr><tr><td>(12) Joint_Align+MUSE</td><td>81.4</td><td>84.2</td><td>82.8</td><td>83.6</td><td>74.2</td><td>72.2</td><td>77.5</td><td>81.5</td><td>45.0</td><td></td><td>58.3</td><td>36.1</td><td>18.2 35.3</td><td>67.7</td></tr><tr><td>(13)Joint_Align+RCSLS(w/o VR)</td><td>34.2</td><td>37.0</td><td>41.2</td><td>46.8</td><td>34.0</td><td>35.6</td><td>35.3</td><td>35.1</td><td></td><td>7.7</td><td>5.2</td><td>20.2</td><td>15.7</td><td>29.0</td></tr><tr><td>(14)Joint_Align+GeoMM</td><td>82.6</td><td>85.7</td><td>82.5</td><td>84.2</td><td>75.5</td><td>77.2</td><td>78.2</td><td>81.4</td><td></td><td>52.4</td><td>67.7</td><td>50.4</td><td>46.5</td><td>72.0</td></tr><tr><td>(15) Joint_Align + RCSLS</td><td>84.7</td><td>87.9</td><td>83.5</td><td>85.6</td><td>79.6</td><td>78.0</td><td>80.6</td><td>84.0</td><td></td><td>59.8</td><td>67.8</td><td>54.3</td><td>48.7</td><td>74.5</td></tr></table>
+
+Unsupervised Machine Translation (UMT) Lastly, we test our approach using the unsupervised MT task, on which the initialization of CLWE plays a crucial role (Lample et al., 2018b). Note that our purpose here is to directly compare with similar studies in Lample et al. (2018b), and thus we follow their settings and consider two language pairs, English-French and English-German, and evaluate on the widely used WMT’14 en-fr and WMT’16 en-de benchmarks.
+
+# 4.2 EXPERIMENTAL SETUP
+
+For the BLI task, we compare our framework to recent state-of-the-art methods. We obtain numbers from the corresponding papers or Zhou et al. (2019), and use the official tools for MUSE (Conneau et al., 2018a), GeoMM (Jawanpuria et al., 2019) and RCSLS (Joulin et al., 2018b) to obtain missing results. We consider the method of Duong et al. (2016) for supervised joint training based on bilingual dictionaries, which is comparable to supervised alignment methods in terms of resources used. For unsupervised joint training, we train uncased joint fastText word vectors of dimension 300 on concatenated Wikipedia corpora of each language pair with default parameters. The hyperparameter $\gamma$ is selected from $\{ 0 . 7 , \bar { 0 } . 8 , 0 . 9 , 0 . 9 5 \}$ on validation sets. For the alignment refinement step in our proposed framework, we use RCSLS and GeoMM to compare with supervised methods, and MUSE for unsupervised methods. In addition, we include an additional baseline of joint training, denoted as Joint - Replace, which is identical to unsupervised joint training except that it utilizes a seed dictionary to randomly replace words with their translations in the training corpus. Following standard practices, we consider the top $2 0 0 \mathrm { k }$ most frequent words and use the crossdomain similarity local scaling (CSLS) (Conneau et al., 2018a) as the retrieval criteria. Note that a concurrent work (Artetxe et al., 2019) proposed a new retrieval method based on MT systems and produced state-of-the-art results. Although their method is applicable to our framework, it has high computational cost and is out of the scope of this work.
+
+For the NER task: (1) For non-contextualized representations, we train embeddings the same way as in the BLI task and use a vanilla Bi-LSTM-CRF model (Lample et al., 2016). For all alignment steps, we apply the supervised Procrustes method using dictionaries from the MUSE library for simplicity. (2) For contextualized representations, we use M-BERT, an unsupervised joint training model, as our base model and apply our proposed framework on it by first aligning its extracted features and then feeding them to a task-specific model (M-BERT Feature $^ +$ Align). Specifically, we use the sum of the last $4 \mathbf { M }$ -BERT layers’ outputs as the extracted features. To obtain the alignment matrices, one for each layer, we use 30k parallel sentences from the Europarl corpus for each language pair and follow the procedure of Section 3.2. We feed the extracted features as inputs to a task-specific model with 2 Bi-LSTM layers and a CRF layer (see appendix A). We compare our framework to both finetuning (M-BERT Finetune), which has been studied by previous papers, and feature extraction (M-BERT Feature). Lastly, we also compare against XLM, a supervised joint training model.
+
+Table 2: Precision $@ 1$ for the BLI task on the MUSE dataset with test pairs of same surface form removed. The best result for unsupervised methods is underlined while bold signifies the overall best.   
+
+<table><tr><td></td><td>en-es</td><td>es-en</td><td>en-fr</td><td>fr-en</td><td>en-de</td><td>de-en</td><td>en-it</td><td>it-en</td><td>en-ru</td><td>ru-en</td><td>en-zh</td><td>zh-en</td><td>avg</td></tr><tr><td colspan="10"></td><td></td><td></td><td></td><td></td></tr><tr><td>(1) MUSE(Conneau et al.,2018a)</td><td>77.1</td><td>82.5</td><td>76.4</td><td>Unsupervised 78.0</td><td>674</td><td>67.8 10.5</td><td>72.5 78</td><td>77.5</td><td>42.7</td><td>50.2</td><td>28.7</td><td>29.1</td><td>62.5</td></tr><tr><td>(2) Unsupervised Joint</td><td>3.7</td><td>10.2</td><td>5.1</td><td>10.7</td><td>8.5</td><td></td><td></td><td>8.1</td><td>0.4</td><td>2.7</td><td>2.5</td><td>6.4</td><td>6.4</td></tr><tr><td>(3) Joint_Align + MUSE</td><td>77.5</td><td>83.0</td><td>770</td><td>79.5</td><td>66.7</td><td>68.0</td><td>70.9</td><td>78.0</td><td>43.5</td><td>55.1</td><td>32.3</td><td>32.7</td><td>63.7</td></tr><tr><td></td><td></td><td></td><td></td><td>Supervised</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(4) RCSLS (Joulin et al.,2018b) 78.0</td><td></td><td>83.9</td><td>76.0</td><td>78.6</td><td>68.2</td><td>68.4</td><td>71.8</td><td>78.2</td><td>50.7</td><td>56.9</td><td>51.0</td><td>41.7</td><td>67.0</td></tr><tr><td>(5) Supervised Joint (Duong et al.,2016)</td><td>76.8</td><td>80.8</td><td>73.4</td><td>76.1</td><td>60.1</td><td>61.7</td><td>69.7</td><td>76.2</td><td>41.0</td><td>51.8</td><td>52.3</td><td>43.3</td><td>63.6</td></tr><tr><td>(6) Joint_Align + RCSLS</td><td>82.1</td><td>84.6</td><td>78.1</td><td>80.4</td><td>68.4</td><td>70.4</td><td>73.7</td><td>79.0</td><td>59.0</td><td>66.8</td><td>51.4</td><td>45.7</td><td>70.0</td></tr></table>
+
+For the UMT task, we use the exact same data, architecture and parameters released by Lample et al.   
+(2018b)7. We simply use different embeddings trained with the same data as inputs to the model.
+
+# 4.3 RESULTS AND ANALYSIS
+
+# 4.3.1 ALIGNMENT VS. JOINT TRAINING
+
+We compare alignment methods with joint training on all three downstream tasks. As shown in Table 1 and Table 3, we find alignment methods significantly outperform the joint training approach by a large margin in all language pairs for both BLI and NER. However, the unsupervised joint training method is superior than its alignment counterpart on the unsupervised MT task as demonstrated in 2(c). While these results demonstrate that their relative performance is task-dependent, we conduct further analysis to reveal three limitations as discussed in Sec 2.3.
+
+First, their poor performance on BLI and NER tasks shows that unsupervised joint training fails to generate high-quality alignments due to the lack of a fine-grained seed dictionary as discussed in its limitation 2. To evaluate accuracy on words that are not shared, we further remove test pairs of the same surface form (e.g. (hate, hate) as a test pair for en-de) of the BLI task and report their results in Table 2. We find unsupervised joint training (row 2) to achieve extremely low scores which shows that emebddings of non-shared parts are poorly aligned, consistent with the PCA visualization shown in Figure 1.
+
+Moreover, we delve into the relative performance of the two paradigms on the MT task by plotting their test BLEU scores of the first 20 epochs in Figure 2(a) and 2(b). We observe that the alignment method actually obtains higher BLEU scores in the first few epochs, but gets surpassed by joint training in later epochs. This shows the importance of parameter sharing as discussed in limitation 1 of alignment methods: shared words can be used as a cross-lingual constraint for unsupervised joint training during fine-tuning but this constraint cannot easily be used in alignment methods. The lack of sharing is also a limitation for the supervised joint training method, which performs poorly on the MT task even with supervision as shown in Figure 2(c).
+
+Lastly, we demonstrate that oversharing can be sub-optimal for unsupervised joint training as discussed in its limitation 2. Specifically, we conduct ablation studies for our framework in Table 1. Applying alignment refinement on unsupervised joint training without any vocabulary reallocation does not improve its performance (row 13). On the other hand, simple vocabulary reallocation alone boosts the performance by quite a margin (row 11). This shows some words are shared erroneously across languages in unsupervised joint training, thereby hindering its performance. In addition, while utilizing a seed dictionary (row 10) improves the performance of unsupervised joint train
+
+Table 3: F1 score for the crosslingual NER task. “Adv” refers to adversarial training. ‡ denotes results that are not directly comparable due to different resources and architectures used. ∗ denotes supervised XLM model trained with MLM and TLM objectives. Its Dutch (nl) result is blank because the model is not pretrained on it. Bold signifies state-of-the-art results. We report the average of 5 runs.
+
+<table><tr><td></td><td>es</td><td>nl</td><td>de</td><td>avg</td></tr><tr><td colspan="5">Non-contextualized</td></tr><tr><td>Unsupervised Joint</td><td>50.28</td><td>42.77</td><td>21.49</td><td>38.18</td></tr><tr><td>Supervised Joint (Duong et al.,2016)</td><td>63.16</td><td>63.60</td><td>36.24</td><td>54.33</td></tr><tr><td>Joint- Replace</td><td>65.28</td><td>68.44</td><td>51.59</td><td>61.77</td></tr><tr><td>Align</td><td>69.00</td><td>71.33</td><td>52.17</td><td>64.17</td></tr><tr><td>Joint_Align</td><td>70.46</td><td>72.10</td><td>56.47</td><td>66.34</td></tr><tr><td>Xie et al. (2018)</td><td>71.67</td><td>70.90</td><td>57.43</td><td>66.67</td></tr><tr><td>Chen et al. (2019)‡</td><td>73.50</td><td>72.40</td><td>56.00</td><td>67.30</td></tr><tr><td colspan="5">Contextualized</td></tr><tr><td>XLMFinetune (Lample&amp; Conneau,2019)*</td><td>63.18</td><td>-</td><td>67.55</td><td>-</td></tr><tr><td>M-BERT Finetune (Pires et al., 2019)</td><td>73.59</td><td>77.36</td><td>69.74</td><td>73.56</td></tr><tr><td>M-BERT Finetune (Wu&amp; Dredze,2019)</td><td>74.96</td><td>77.57</td><td>69.56</td><td>74.03</td></tr><tr><td>M-BERT Finetune (Keung et al., 2019)</td><td>75.00</td><td>77.50</td><td>68.60</td><td>73.70</td></tr><tr><td>M-BERT Finetune + Adv (Keung et al.,2019)</td><td>74.30</td><td>77.60</td><td>71.90</td><td>74.60</td></tr><tr><td>M-BERT Feature</td><td>74.23</td><td>78.65</td><td>67.63</td><td>73.50</td></tr><tr><td>M-BERT Feature + Align</td><td>75.77</td><td>79.03</td><td>70.54</td><td>75.11</td></tr></table>
+
+ing, it still suffers from the oversharing problem and performs worse compared to supervised joint training (row 9).
+
+# 4.3.2 EVALUATION OF PROPOSED FRAMEWORK
+
+As shown in Table 1, Table 3, and Figure 2, our proposed framework substantially improves over the alignment and joint training baselines on all three tasks. In particular, it outperforms existing methods on all language pairs for the BLI task (using the CSLS as retrieval metric) and achieves state-of-the-art results on 2 out of 3 language pairs for the NER task. Besides, we show that it alleviates limitations of alignment and joint training methods shown in the previous section.
+
+First, the proposed framework largely improves the poor alignment of unsupervised joint training, especially for non-sharing parts. As shown in Table 1, the proposed Joint Align framework achieves comparable results to prior methods in the unsupervised case (row 12) and it outperforms previous state-of-the-art methods in the supervised setting (row 15). Specifically, our proposed framework can generate well-aligned embeddings after alignment refinement is applied to the initially ill-aligned embeddings, as demonstrated in Figure 1. This is further verified by results in Table 2, where our proposed framework largely improves accuracy on words not shared between two languages over the unsupervised joint training baseline (row 3 and 6 vs row 2).
+
+Besides, our ablation study in Table 1 further shows the effectiveness of the proposed vocabulary reallocation technique, which alleviates the issue of oversharing. Particularly, we observe no improvement compared to unsupervised joint training baseline (row 8) when an alignment refinement step is used without vocabulary reallocation (row 13), while a vocabulary reallocation step alone significantly improves the performance (row 11). This is consistent with Figure 1 and shows that the oversharing is a bottleneck for applying alignment methods to joint training. It also suggests detecting what to share is crucial to achieve better cross-lingual transfer.
+
+Lastly, while supervised joint training shares the limitation 1 of alignment methods and performs poorly when finetuned, our proposed framework exploits the same idea of vocabulary sharing used in unsupervised joint training. In the MT tasks, our framework obtains a maximum gain of 2.97 BLEU over baselines we ran and consistently performs better than results reported in Lample et al. (2018b). In addition, Figure 2 shows that Joint Align not only converges faster in earlier training epochs but also consistently outperforms the two baselines thereafter. These empirical findings demonstrate the effectiveness of our proposed methods in the non-contextualized case.
+
+# 4.3.3 CONTEXTUALIZED WORD REPRESENTATIONS
+
+As can be seen in Table 3, when using our framework (M-BERT Feature $^ +$ Align), we achieve state-of-the-art results on cross-lingual NER on 2 out of 3 languages and the overall average. This shows that our framework can effectively generalize to contextualized representations. Specifically, our framework improves over both the M-BERT feature extraction and finetuning baselines on all three language pairs. However, when compared to non-contextualized results, the gain of using alignment refinement on top of unsupervised joint training is much smaller. This suggests that, as the contextualized unsupervised joint training model performs very well already even without any supervision, it is harder to achieve large improvements. While alignment refinement relies on word alignment, a process that is noisy itself, a better alignment approach may be warranted. Lastly, the reason why a supervised joint training model, XLM, performs worse than its unsupervised counterpart, M-BERT, is likely that XLM uses an uncased vocabulary, where casing information is important for NER tasks.
+
+![](images/0ed39ba542e2114a70a133ccf587b90c94874585c425f7eae793846915ce0375.jpg)  
+Figure 2: (a)(b): Results on MT of Align, Joint and our framework for the first 20 training epochs. Results after 20 epochs have similar patterns. (c): BLEU scores for the MT task. Results evaluated on the WMT’14 English-French and WMT’16 German-English. All training settings are the same for each language pair except the embedding initialization. Note that we are not trying to outperform state-of-the-art methods (Song et al., 2019) but rather to observe improvements afforded by embedding initialization. †Results reported by Lample et al. (2018b). Our results are obtained using the official code released by the author. ‡Duong et al. (2016) is a supervised method that we include for analysis purpose only and is not directly comparable to other results in this table.
+
+# 5 RELATED WORK
+
+Word embeddings (Mikolov et al., 2013a; Ruder et al., 2019) are a key ingredient to achieving success in monolingual NLP tasks. However, directly using word embeddings independently trained for each language may cause negative transfer (Wang et al., 2019) in cross-lingual transfer tasks. In order to capture the cross-lingual mapping, a rich body of existing work relying on cross-lingual supervision, including bilingual dictionaries (Mikolov et al., 2013a; Faruqui & Dyer, 2014; Artetxe et al., 2016; Xing et al., 2015; Duong et al., 2016; Gouws & Søgaard, 2015; Joulin et al., 2018a), sentence-aligned corpora (Kocisk ˇ y et al., 2014; Hermann & Blunsom, 2014; Gouws et al., 2015) \` and document-aligned corpora (Vulic & Moens, 2016; Søgaard et al., 2015). ´
+
+Besides, unsupervised alignment methods aim to eliminate the requirement for cross-lingual supervision. Early work of Cao et al. (2016) matches the mean and the standard deviation of two embedding spaces after alignment. Barone (2016); Zhang et al. (2017a;b); Conneau et al. (2018a) adapted a generative adversarial network (GAN) (Goodfellow et al., 2014) to make the distributions of two word embedding spaces indistinguishable. Follow-up works improve upon GAN-based training for better stability and robustness by introducing Sinkhorn distance (Xu et al., 2018), by stochastic self-training (Artetxe et al., 2018a), or by introducing latent variables (Dou et al., 2018).
+
+While alignment methods utilize embeddings trained independently on different languages, joint training methods train word embeddings at the same time. Klementiev et al. (2012) train a bilingual dictionary-based regularization term jointly with monolingual language model objectives while Kocisk ˇ y et al. (2014) defines the cross-lingual regularization with the parallel corpus. Another \` branch of methods (Xiao & Guo, 2014; Gouws & Søgaard, 2015; Ammar et al., 2016; Duong et al., 2016) build a pseudo-bilingual corpus by randomly replacing words in monolingual corpus with their translations and use monolingual word embedding algorithms to induce bilingual representations. The unsupervised joint method by Lample & Conneau (2019) simply exploit words that share the same surface form as bilingual “supervision” and directly train a shared set of embedding with joint vocabulary. Recently, unsupervised joint training of contextualized word embeddings through the form of multilingual language model pretraining using shared subword vocabularies has produced state-of-the-art results on various benchmarks (Devlin et al., 2019; Artetxe & Schwenk, 2019; Lample & Conneau, 2019; Pires et al., 2019; Wu & Dredze, 2019).
+
+A concurrent work by Ormazabal et al. (2019) also compares alignment and joint method in the bilingual lexicon induction task. Different from their setup which only tests on supervised settings, we conduct analysis across various tasks and experiment with both supervised and unsupervised conditions. While Ormazabal et al. (2019) suggests that the combination of the alignment and joint model could potentially advance the state-of-art of both worlds, we propose such a framework and empirically verify its effectiveness on various tasks and settings.
+
+# 6 CONCLUSION
+
+In this paper, we systematically compare the alignment and joint training methods for CLWE. We point out that the nature of each category of methods leads to certain strengths and limitations. The empirical experiments on extensive benchmark datasets and various NLP tasks verified our analysis. To further improve the state-of-art of CLWE, we propose a simple hybrid framework which combines the strength from both worlds and achieves significantly better performance in the BLI, MT and NER tasks. Our work opens a promising new direction that combines two previously exclusive lines of research. For future work, an interesting direction is to find a more effective word sharing strategy.
+
+Acknowledgments: This research was sponsored by Defense Advanced Research Projects Agency Information Innovation Office (I2O) under the Low Resource Languages for Emergent Incidents (LORELEI) program, issued by DARPA/I2O under Contract No. HR0011-15-C0114. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. government. The U.S. government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation here on.
+
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+
+# APPENDIX
+
+# A NER EXPERIMENT DETAILS
+
+Here we include some additional details for the NER experiments with contextualized representations:
+
+• Alignment As described in Section 3.2, we apply word alignment methods, such as fastalign (Dyer et al., 2013), on parallel data to extract word-aligned pairs for learning the alignment matrix. As M-BERT is based on subword tokens, we use the average of the representations of all subword tokens that correspond to a word as the representation for that word. For instance, assume that an English word “Resumption” is aligned to a German word “Wiederaufnahme”, and they are tokenized by M-BERT as “Res”, “##sumption”, and “Wie”, “##dera”, “##uf”, “##nahme”, respectively. Then the representation for “Resumption” is the average of the representations of subword tokens “Res” and “##sumption”, and the same goes for “Wiederaufnahme”. • Hyperparameters For the task-specific NER model, we use a 2-layer Bi-LSTM with a hidden size of 768 followed by a CRF layer. We apply a dropout rate of 0.5 on the input and the output of the Bi-LSTM, and use Adam with default parameters and a learning rate of 0.0001 for optimization. We train the model for 40 epochs with a batch size of 10, and evaluate the model per 150 steps. For prediction, we feed the outputs of the Bi-LSTM that correspond to the first subword tokens of each word to the CRF model. This is identical to finetuning BERT on the NER task, except that in our case the outputs that correspond to the first subword token are fed into a CRF, rather than a linear layer as done in BERT.
+
+Table 4: Precision $@ 1$ for the BLI task on the MUSE dataset using test set produced by vocabulary reallocation. Within each category, unsupervised methods are listed at the top while supervised methods are at the bottom. Bold signifies the overall best results. “AR” refers to alignment refinement and “VR” refers to vocabulary reallocation.   
+
+<table><tr><td></td><td>en-es</td><td>es-en</td><td>en-fr</td><td>fr-en</td><td>en-de</td><td>de-en</td><td>en-it</td><td>it-en</td><td>en-ru</td><td>ru-en</td><td>en-zh</td><td>zh-en</td><td>avg</td></tr><tr><td colspan="10">Alignment Methods</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(1) MUSE(Conneau et al.,2018a)</td><td>82.1</td><td>83.5</td><td>82.6</td><td>83.1</td><td>74.1</td><td>71.8</td><td>77.4</td><td>79.8</td><td>44.1</td><td>59.1</td><td>34.1</td><td>31.6</td><td>66.9</td></tr><tr><td>(4) Procrustes (Smith et al.,2017)</td><td>81.6</td><td>82.0</td><td>80.5</td><td>81.5</td><td>74.2</td><td>73.3</td><td>77.0</td><td>77.8</td><td>50.8</td><td>63.5</td><td>44.2</td><td>37.0</td><td>68.6</td></tr><tr><td>(5) GeoMM (Jawanpuria et al., 2019)</td><td>82.1</td><td>86.9</td><td>82.3</td><td>85.1</td><td>75.3</td><td>77.2</td><td>78.6</td><td>82.2</td><td>51.7</td><td>67.8</td><td>50.5</td><td>45.6</td><td>72.1</td></tr><tr><td>(6) RCSLS (Joulin et al.,2018b)</td><td>82.8</td><td>84.3</td><td>82.4</td><td>83.3</td><td>78.6</td><td>75.6</td><td>78.3</td><td>81.0</td><td>57.7</td><td>66.8</td><td>48.3</td><td>45.6</td><td>72.1</td></tr><tr><td colspan="10">Joint Traing Methods</td><td></td><td></td><td></td><td></td></tr><tr><td>(7) Unsupervised Joint</td><td>33.2</td><td>36.3</td><td>41.5</td><td>46.8</td><td>39.1</td><td>40.7</td><td>35.8</td><td>38.1</td><td>4.1</td><td>3.7</td><td>8.2</td><td>5.7</td><td>27.8</td></tr><tr><td>(8) Supervised Joint (Duong et al., 2016)</td><td>80.1</td><td>80.5</td><td>78.6</td><td>77.1</td><td>67.2</td><td>68.3</td><td>74.6</td><td>74.5</td><td>41.7</td><td>51.8</td><td>47.2</td><td>44.0</td><td>65.5</td></tr><tr><td colspan="10">Joint Align Framework</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(9)Joint_Align (w/o AR)</td><td>56.8</td><td>63.2</td><td>62.2</td><td>67.2</td><td>49.2</td><td>55.1</td><td>50.6</td><td>51.9</td><td>8.7</td><td>8.2</td><td>19.5</td><td>18.4</td><td>42.6</td></tr><tr><td>(10) Joint_Align +MUSE</td><td>82.4</td><td>85.0</td><td>83.5</td><td>84.7</td><td>74.6</td><td>72.9</td><td>78.2</td><td>82.6</td><td>46.1</td><td>58.7</td><td>39.9</td><td>36.2</td><td>68.7</td></tr><tr><td>(11) Joint_Align +RCSLS(w/o VR) (12) Joint_Align +GeoMM</td><td>34.3</td><td>36.8</td><td>41.0</td><td>47.0</td><td>34.3</td><td>35.9</td><td>35.5</td><td>35.2</td><td>7.6</td><td>5.2</td><td>21.3 54.8</td><td>16.1 48.0</td><td>29.2 73.3</td></tr><tr><td></td><td>83.9</td><td>86.2</td><td>83.1</td><td>85.2</td><td>76.1</td><td>77.9</td><td>79.0</td><td>82.8</td><td>53.7</td><td>68.3</td><td></td><td></td><td></td></tr><tr><td>(13) Joint_Align + RCSLS</td><td>87.1</td><td>88.5</td><td>84.2</td><td>86.6</td><td>80.1</td><td>78.7</td><td>81.3</td><td>85.2</td><td>61.3</td><td>68.3</td><td>59.6</td><td>50.7</td><td>76.0</td></tr></table>
+
+# B BLI TEST PAIRS
+
+The official evaluation script8 of MUSE only includes test pairs whose source words and target words both appear in their corresponding vocabularies, leaving out those that are OOV on either side. Since our proposed vocabulary reallocation step modifies both the source and target vocabularies, the script may exclude some test pairs when evaluating our model. For example, the word “age” from the test pair (age, age) for en-fr could be allocated as an en (not shared) word, so it is OOV on the fr side and the pair would thus be left out by the script. As a result, the total number of test pairs would be smaller, thereby changing the denominator when we calculate accuracy. To ensure fair comparison, we include these OOV pairs so the total number of test pairs stays the same. Specifically, if the source word of a test pair is OOV, we retrieve itself. Otherwise, we count it as incorrect.
+
+In addition, we further investigate which pairs are left out and found that the MUSE benchmark contains some noisy test data. Specifically, we find that the majority of these pairs are in the same surface form, such as (sit, sit), but many target words are not actual translations of the source words. For example, we found test pairs such as {(age, age), (century, century)} for en- $f r$ and {(mickey, mickey), (uncredited, uncredited) $\}$ for en-zh. Clearly, these are English words and should not be considered as appropriate translations for French or Chinese. In Table 1, we mark these pairs as incorrect for our framework to ensure fair comparison, while in fact our framework correctly allocates these words to English. To reveal the full picture, we also conduct BLI experiments without test pairs that got left out due to vocabulary reallocation. The results are shown in Table 4. We observe that our proposed framework obtains a gain of 3.9 accuracy on average over the RCSLS baseline.

parse/train/S1l-C0NtwS/S1l-C0NtwS_content_list.json

@@ -0,0 +1,1799 @@
+[
+    {
+        "type": "text",
+        "text": "CROSS-LINGUAL ALIGNMENT VS JOINT TRAINING:A COMPARATIVE STUDY AND A SIMPLE UNIFIEDFRAMEWORK",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zirui Wang∗, Jiateng Xie∗, Ruochen Xu, Yiming Yang, Graham Neubig, Jaime Carbonell Language Technologies Institute, Carnegie Mellon University, Pittsburgh, PA 15213, USA {ziruiw, jiatengx, ruochenx, yiming, gneubig, jgc}@cs.cmu.edu ",
+        "bbox": [
+            183,
+            195,
+            790,
+            238
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            273,
+            544,
+            289
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Learning multilingual representations of text has proven a successful method for many cross-lingual transfer learning tasks. There are two main paradigms for learning such representations: (1) alignment, which maps different independently trained monolingual representations into a shared space, and (2) joint training, which directly learns unified multilingual representations using monolingual and cross-lingual objectives jointly. In this paper, we first conduct direct comparisons of representations learned using both of these methods across diverse crosslingual tasks. Our empirical results reveal a set of pros and cons for both methods, and show that the relative performance of alignment versus joint training is taskdependent. Stemming from this analysis, we propose a simple and novel framework that combines these two previously mutually-exclusive approaches. Extensive experiments demonstrate that our proposed framework alleviates limitations of both approaches, and outperforms existing methods on the MUSE bilingual lexicon induction (BLI) benchmark. We further show that this framework can generalize to contextualized representations such as Multilingual BERT, and produces state-of-the-art results on the CoNLL cross-lingual NER benchmark.1 ",
+        "bbox": [
+            233,
+            308,
+            764,
+            526
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            556,
+            334,
+            571
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Continuous word representations (Mikolov et al., 2013a; Pennington et al., 2014; Bojanowski et al., 2017) have become ubiquitous across a wide range of NLP tasks. In particular, methods for crosslingual word embeddings (CLWE) have proven a powerful tool for cross-lingual transfer for downstream tasks, such as text classification (Klementiev et al., 2012), dependency parsing (Ahmad et al., 2019), named entity recognition (NER) (Xie et al., 2018; Chen et al., 2019), natural language inference (Conneau et al., 2018b), language modeling (Adams et al., 2017), and machine translation (MT) (Zou et al., 2013; Lample et al., 2018a; Artetxe et al., 2018b; Lample et al., 2018b). The goal of these CLWE methods is to learn embeddings in a shared vector space for two or more languages. There are two main paradigms for learning CLWE: cross-lingual alignment and joint training. ",
+        "bbox": [
+            174,
+            588,
+            825,
+            713
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The most successful approach has been the cross-lingual embedding alignment method (Mikolov et al., 2013b), which relies on the assumption that monolingually-trained continuous word embedding spaces share similar structure across different languages. The underlying idea is to first independently train embeddings in different languages using monolingual corpora alone, and then learn a mapping to align them to a shared vector space. Such a mapping can be trained in a supervised fashion using parallel resources such as bilingual lexicons (Xing et al., 2015; Smith et al., 2017; Joulin et al., 2018b; Jawanpuria et al., 2019), or even in an unsupervised2 manner based on distribution matching (Zhang et al., 2017a; Conneau et al., 2018a; Artetxe et al., 2018a; Zhou et al., 2019). Recently, it has been shown that alignment methods can also be effectively applied to contextualized word representations (Schuster et al., 2019; Aldarmaki & Diab, 2019). ",
+        "bbox": [
+            174,
+            720,
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+            859
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Another successful line of research for CLWE considers joint training methods, which optimize a monolingual objective predicting the context of a word in a monolingual corpus along with either a hard or soft cross-lingual constraint. Similar to alignment methods, some early works rely on bilingual dictionaries (Ammar et al., 2016; Duong et al., 2016) or parallel corpora (Luong et al., 2015; Gouws et al., 2015) for direct supervision. More recently, a seemingly naive unsupervised joint training approach has received growing attention due to its simplicity and effectiveness. In particular, Lample et al. (2018b) reports that simply training embeddings on concatenated monolingual corpora of two related languages using a shared vocabulary without any cross-lingual resources is able to produce higher accuracy than the more sophisticated alignment methods on unsupervised MT tasks. Besides, for contextualized representations, unsupervised multilingual language model pretraining using a shared vocabulary has produced state-of-the-art results on multiple benchmarks3 (Devlin et al., 2019; Artetxe & Schwenk, 2019; Lample & Conneau, 2019). ",
+        "bbox": [
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+            103,
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+            270
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Despite a large amount of research on both alignment and joint training, previous work has neither performed a systematic comparison between the two, analyzed their pros and cons, nor elucidated when we may prefer one method over the other. Particularly, it’s natural to ask: (1) Does the phenomenon reported in Lample et al. (2018b) extend to other cross-lingual tasks? (2) Can we employ alignment methods to further improve unsupervised joint training? (3) If so, how would such a framework compare to supervised joint training methods that exploit equivalent resources, i.e., bilingual dictionaries? (4) And lastly, can this framework generalize to contextualized representations? ",
+        "bbox": [
+            174,
+            277,
+            825,
+            375
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we attempt to address these questions. Specifically, we first evaluate and compare alignment versus joint training methods across three diverse tasks: BLI, cross-lingual NER, and unsupervised MT. We seek to characterize the conditions under which one approach outperforms the other, and glean insight on the reasons behind these differences. Based on our analysis, we further propose a simple, novel, and highly generic framework that uses unsupervised joint training as initialization and alignment as refinement to combine both paradigms. Our experiments demonstrate that our framework improves over both alignment and joint training baselines, and outperforms existing methods on the MUSE BLI benchmark. Moreover, we show that our framework can generalize to contextualized representations such as Multilingual BERT, producing state-of-the-art results on the CoNLL cross-lingual NER benchmark. To the best of our knowledge, this is the first framework that combines previously mutually-exclusive alignment and joint training methods. ",
+        "bbox": [
+            174,
+            382,
+            825,
+            535
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND: CROSS-LINGUAL REPRESENTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            558,
+            642,
+            574
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Notation. We assume we have two different languages $\\{ L _ { 1 } , L _ { 2 } \\}$ and access to their corresponding training corpora. We use $V _ { L _ { i } } = { \\{ w _ { L _ { i } } ^ { j } \\} } _ { j = 1 } ^ { n _ { L _ { i } } }$ to denote the vocabulary set of the ith language where each $w _ { L _ { i } } ^ { j }$ represents a unique token, such as a word or subword. The goal is to learn a set of embeddings $E = \\{ \\pmb { x } ^ { j } \\} _ { j = 1 } ^ { m }$ , with $\\pmb { x } ^ { j } \\in \\mathbb { R } ^ { d }$ , in a shared vector space, where each token $w _ { L _ { i } } ^ { j }$ is mapped to a vector in $E$ . Ideally, these vectorial representations should have similar values for tokens with similar meanings or syntactic properties, so they can better facilitate cross-lingual transfer. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 ALIGNMENT METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            707,
+            379,
+            722
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given the notation, alignment methods consist of the following steps: ",
+        "bbox": [
+            174,
+            734,
+            629,
+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Step 1: Train an embedding set $E _ { 0 } = E _ { L _ { 1 } } \\cup E _ { L _ { 2 } }$ , where each subset $E _ { L _ { i } } = \\{ x _ { L _ { i } } ^ { j } \\} _ { j = 1 } ^ { n _ { L _ { i } } }$ is trained independently using the ith language corpus and contains an embedding $x _ { L _ { i } } ^ { j }$ for each token $w _ { L _ { i } } ^ { j }$ . ",
+        "bbox": [
+            173,
+            751,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Step 2: Obtain a seed dictionary $D = \\{ ( w _ { L _ { 1 } } ^ { i } , w _ { L _ { 2 } } ^ { j } ) \\} _ { k = 1 } ^ { K }$ , either provided or learnt unsupervised. ",
+        "bbox": [
+            173,
+            785,
+            807,
+            803
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Step 3: Learn a projection matrix $W \\in \\mathbb { R } ^ { d \\times d }$ based on $D$ , resulting in a final embedding set $\\ c E _ { A } \\overset { - } { = } ( W \\cdot E _ { L _ { 1 } } ) \\overset { - } { \\cup } \\bar { E } _ { L _ { 2 } }$ in a shared vector space. ",
+        "bbox": [
+            173,
+            804,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To find the optimal projection matrix $W$ , Mikolov et al. (2013b) proposed to solve the following optimization problem: ",
+        "bbox": [
+            173,
+            838,
+            825,
+            867
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffbe63811888b837f9c0c1aae793bd0d28976553157f2f8c8a7aa6c37721a077.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { W \\in \\mathbb { R } ^ { d \\times d } } \\| W X _ { L _ { 1 } } - X _ { L _ { 2 } } \\| _ { F }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            869,
+            591,
+            893
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $X _ { L _ { 1 } }$ and $X _ { L _ { 2 } }$ are matrices of size $d \\times K$ containing embeddings of the words in $D$ . Xing et al. (2015) later showed further improvement could be achieved by restricting $W$ to an orthogonal matrix, which turns the Eq.(1) into the Procrustes problem with the following closed form solution: ",
+        "bbox": [
+            174,
+            103,
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+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca812aafd61f2255bae50ed38b3918ed33543ff042c58c5839a5ea4257a98678.jpg",
+        "text": "$$\n\\begin{array} { r } { W ^ { * } = U V ^ { T } , \\quad \\quad } \\\\ { \\mathrm { w i t h ~ } U \\Sigma V ^ { T } = \\mathrm { S V D } ( X _ { L _ { 2 } } X _ { L _ { 1 } } ^ { T } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            152,
+            606,
+            196
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $W ^ { * }$ denotes the optimal solution and SVD(·) stands for the singular value decomposition. ",
+        "bbox": [
+            171,
+            202,
+            803,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As surveyed in Section 5, different methods (Smith et al., 2017; Conneau et al., 2018a; Joulin et al., 2018b; Artetxe et al., 2018a) differ in the way how they obtain the dictionary $D$ and how they solve for $W$ in step 3. However, most of them still involve solving the Eq.(2) as a crucial step. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 JOINT TRAINING METHODS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            285,
+            406,
+            299
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Joint training methods in general have the following objective: ",
+        "bbox": [
+            174,
+            311,
+            583,
+            327
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/970aeca484b0c4e7d257b76f445daf272b7af4988a454a45a7c684e3339d8086.jpg",
+        "text": "$$\n\\mathscr { L } _ { J } = \\mathscr { L } _ { 1 } + \\mathscr { L } _ { 2 } + \\mathscr { R } \\left( L _ { 1 } , L _ { 2 } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            335,
+            596,
+            353
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { 1 }$ and $\\mathcal { L } _ { 2 }$ are monolingual objectives and $\\mathcal { R } ( L _ { 1 } , L _ { 2 } )$ is a cross-lingual regularization term. For example, Klementiev et al. (2012) use language modeling objectives for $\\mathcal { L } _ { 1 }$ and $\\mathcal { L } _ { 2 }$ . The term $\\mathcal { R } ( L _ { 1 } , L _ { 2 } )$ encourages alignment of representations of words that are translations. Training an embedding set $E _ { J } = E _ { L _ { 1 } } \\cup E _ { L _ { 2 } }$ is usually done by directly optimizing $\\mathcal { L } _ { J }$ . ",
+        "bbox": [
+            174,
+            368,
+            825,
+            425
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While supervised joint training requires access to parallel resources, recent studies (Lample et al., 2018b; Devlin et al., 2019; Artetxe & Schwenk, 2019; Lample & Conneau, 2019) have suggested that unsupervised joint training without such resources is also effective. Specifically, they show that the cross-lingual regularization term $\\mathcal { R } ( L _ { 1 } , L _ { 2 } )$ does not require direct cross-lingual supervision to achieve highly competitive results. This is because the shared words between $\\mathcal { L } _ { 1 }$ and $\\mathcal { L } _ { 2 }$ can serve implicitly as anchors by sharing their embeddings to ensure that representations of different languages lie in a shared space. Using our notation, the unsupervised joint training approach takes the following steps: ",
+        "bbox": [
+            173,
+            431,
+            825,
+            542
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Step 1: Construct a joint vocabulary $V _ { J } = V _ { L _ { 1 } } \\cup V _ { L _ { 2 } }$ that is shared across two languages. ",
+        "bbox": [
+            169,
+            542,
+            802,
+            559
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Step 2: Concatenate the two training corpora and learn an embedding set $E _ { J }$ corresponding to $V _ { J }$ ",
+        "bbox": [
+            174,
+            558,
+            820,
+            571
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The joint vocabulary is composed of three disjoint sets: $V _ { J } ^ { 1 } , V _ { J } ^ { 2 } , V _ { J } ^ { s }$ , where $V _ { J } ^ { s } = V _ { L _ { 1 } } \\cap V _ { L _ { 2 } }$ is the shared vocabulary set and $V _ { J } ^ { i }$ is the set of tokens that appear in the $i$ th language only. Note that a key difference of existing supervised joint training methods is that embeddings corresponding to $V _ { J } ^ { s }$ are not shared between $E _ { L _ { 1 } }$ and $E _ { L _ { 2 } }$ , meaning that they are disjoint, as in alignment methods. ",
+        "bbox": [
+            174,
+            577,
+            825,
+            636
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            654,
+            302,
+            669
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While alignment methods have had great success, there are still some critical downsides, among which we stress the following points: ",
+        "bbox": [
+            178,
+            681,
+            823,
+            709
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. While recent studies in unsupervised joint training have suggested the potential benefits of word sharing, alignment methods rely on two disjoint sets of embeddings. Along with some possible loss of information due to no sharing, one consequence is that finetuning the aligned embeddings on downstream tasks may be sub-optimal due to the lack of crosslingual constraints at the finetuning stage, whereas shared words can fulfill this role in jointly trained models.   \n2. A key assumption of alignment methods is the isomorphism of monolingual embedding spaces. However, some recent papers have challenged this assumption, showing that it does not hold for many language pairs (Søgaard et al., 2018; Patra et al., 2019). Also notably, Ormazabal et al. (2019) suggests that this limitation results from the fact that the two sets of monolingual embeddings are independently trained. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, the unsupervised joint training method is much simpler and doesn’t share these disadvantages with the alignment methods, but there are also some key limitations: ",
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+    {
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+        "img_path": "images/b6891e82933db0590976c1a8dfda71666b5167846eb8b62f827eff934a7e3443.jpg",
+        "image_caption": [
+            "Figure 1: PCA visualization of English and Spanish embeddings learnt by unsupervised joint training as in Lample et al. (2018b). As shown by plots (a) and (b), most words are shared in the initial embedding space but not well-aligned, hence the oversharing problem. Plots (b) and (c) shows that the vocabulary reallocation step effectively mitigates oversharing while the alignment refinement step further improves the poorly aligned embeddings by projecting them into a close neighborhood. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "1. It assumes that all shared words across two languages serve implicitly as anchors and thus need not be aligned to other words. Nonetheless, this assumption is not always true, leading to misalignment. For example, the English word “the” will most likely also appear in the training corpus of Spanish, but preferably it should be paired with Spanish words such as “el” and “la” instead of itself. We refer to this problem as oversharing.   \n2. It does not utilize any explicit form of seed dictionary as in alignment methods, resulting in potentially less accurate alignments, especially for words that are not shared. ",
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+    {
+        "type": "text",
+        "text": "Lastly, while the supervised joint training approach does not have the same issues of unsupervised joint training, it shares limitation 1 of the alignment methods. ",
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+        "type": "text",
+        "text": "We empirically compare both joint training and alignment approaches in Section 4 and shed light on some of these pros and cons for both paradigms (See Section 4.3.1). ",
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+    {
+        "type": "text",
+        "text": "3 PROPOSED FRAMEWORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Motivated by the pros and cons of both paradigms, we propose a unified framework that first uses unsupervised joint training as a coarse initialization and then applies alignment methods for refinement, as demonstrated in Figure 1. Specifically, we first build a single set of embeddings with a shared vocabulary through unsupervised joint training, so as to alleviate the limitations of alignment methods. Next, we use a vocabulary reallocation technique to mitigate oversharing, before finally resorting back to alignment methods to further improve the embeddings’ quality. Lastly, we show that this framework can generalize to contextualized representations. ",
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+    {
+        "type": "text",
+        "text": "3.1 UNIFYING ALIGNMENT WITH JOINT TRAINING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our proposed framework mainly involves three components and we discuss each of them as follows. ",
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+        "type": "text",
+        "text": "Joint Initialization. We use unsupervised joint training (Lample et al., 2018b) to train the initial CLWE. As described in Section 2.2, we first obtain a joint vocabulary $V _ { J }$ and train its corresponding set of embeddings $E _ { J }$ on the concatenated corpora of two languages. This allows us to obtain a single set of embeddings that maximizes sharing across two languages. To train embeddings, we used fastText4 (Bojanowski et al., 2017) in all our experiments for both word and subword tokens. ",
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+    {
+        "type": "text",
+        "text": "Vocabulary Reallocation. As discussed in Section 2.3, a key issue of unsupervised joint training is oversharing, which prohibits further refinement as shown in Figure 1. To alleviate this drawback, we attempt to “unshare” some of the overshared words, so their embeddings can be better aligned in the next step. Particularly, we perform a vocabulary reallocation step such that words appearing mostly exclusively in the ith language are reallocated from the shared vocabulary $V _ { J } ^ { s }$ to $V _ { J } ^ { i }$ , whereas words that appear similarly frequent in both languages stay still in $V _ { J } ^ { s }$ . Formally, for each token $w$ in the shared vocabulary $V _ { J } ^ { s }$ , we use the ratio of counts within each language to determine whether it belongs to the shared vocabulary: ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/cc657fc1a1e48241706f9ac3f54bb5cb5e9bc159c978bba9bc47207b1bbd7e48.jpg",
+        "text": "$$\nr = \\frac { T _ { L _ { 2 } } } { T _ { L _ { 1 } } } \\cdot \\frac { C _ { L _ { 1 } } ( w ) } { C _ { L _ { 2 } } ( w ) } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $C _ { L _ { i } } ( w )$ is the count of $w$ in the training corpus of the $i$ th language and $\\begin{array} { r } { T _ { L _ { i } } = \\sum _ { w } C _ { L _ { i } } ( w ) } \\end{array}$ is the total number of tokens. The token $w$ is allocated to the shared vocabulary if ",
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+    {
+        "type": "equation",
+        "img_path": "images/071a169edb6ab333291b8e050f369a0734fc982d34d955ffb981c1fb712dfd34.jpg",
+        "text": "$$\n\\frac { 1 - \\gamma } { \\gamma } \\leq r \\leq \\frac { \\gamma } { 1 - \\gamma } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\gamma$ is a hyper-parameter. Otherwise, we put $w$ into either $V _ { J } ^ { 1 }$ or $V _ { J } ^ { 2 }$ , where it appears mostly frequent. The above process generates three new disjoint vocabulary sets $V _ { J } ^ { 1 ^ { \\prime } } , V _ { J } ^ { 2 ^ { \\prime } } , V _ { J } ^ { s ^ { \\prime } }$ and their corresponding embeddings $E _ { J } ^ { 1 ^ { \\prime } } , E _ { J } ^ { 2 ^ { \\prime } } , E _ { J } ^ { s ^ { \\prime } }$ that are used thereafter. Note that, $V _ { J } ^ { ' } = V _ { J }$ and $E _ { J } ^ { ' } = E _ { J }$ . ",
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+    {
+        "type": "text",
+        "text": "Alignment Refinement. The unsupervised joint training method does not explicitly utilize any dictionary or form of alignment. Thus, the resulting embedding set is coarse and ill-aligned in the shared vector space, as demonstrated in Figure 1. As a final refinement step, we utilize any off-the-shelf alignment method to refine alignments across the non-sharing embedding sets, i.e. mapping $E _ { J } ^ { 1 ^ { \\prime } }$ to $\\mathrm { ~ \\bar { \\it E } _ { \\it J } ^ { 2 ^ { \\prime } } ~ }$ and leaving $E _ { J } ^ { s ^ { \\prime } }$ untouched. This step could be conducted by either supervised or unsupervised alignment method and we compare both in our experiments. ",
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+    {
+        "type": "text",
+        "text": "3.2 EXTENSION TO CONTEXTUALIZED REPRESENTATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As our framework is highly generic and applicable to any alignment and unsupervised joint training methods, it can naturally generalize to contextualized word representations by aligning the fixed outputs of a multilingual encoder such as multilingual BERT (M-BERT) (Devlin et al., 2019). While our vocab reallocation technique is no longer necessary as contextualized representations are dependent on context and thus dynamic, we can still apply alignment refinement on extracted contextualized features for further improvement. For instance, as proposed by Aldarmaki & Diab (2019), one method to perform alignment on contextualized representations is to first use word alignment pairs extracted from parallel corpora as a dictionary, learn an alignment matrix $W$ based on it, and apply $W$ back to the extracted representations. To obtain $W$ , we can solve Eq.( 1) as described in Section 2.1, where the embedding matrices $X _ { L _ { 1 } }$ and $X _ { L _ { 2 } }$ now contain contextualized representations of aligned word pairs. Note that this method is applicable to fixed representations but not finetuning. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate the proposed approach and compare with alignment and joint training methods on three NLP benchmarks. This evaluation aims to: (1) systematically compare alignment vs. joint training paradigms and reveal their pros and cons discussed in Section 2.3, (2) show that the proposed framework can effectively alleviate limitations of both alignment and joint training, and (3) demonstrate the effectiveness of the proposed framework in both non-contextualized and contextualized settings. ",
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+    {
+        "type": "text",
+        "text": "4.1 EVALUATION TASKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Bilingual Lexicon Induction (BLI) This task has been the de facto evaluation task for CLWE methods. It considers the problem of retrieving the target language translations of source langauge words. We use bilingual dictionaries complied by Conneau et al. (2018a) and test on six diverse language pairs, including Chinese and Russian, which use a different writing script than English. Each test set consists of 1500 queries and we report precision at $^ { l }$ scores $( \\mathrm { P } @ 1 )$ , following standard evaluation practices (Conneau et al., 2018a; Glavas et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "Name Entity Recognition (NER) We also evaluate our proposed framework on cross-lingual NER, a sequence labeling task, where we assign a label to each token in a sequence. We evaluate both non-contextualized and contextualized word representations on the CoNLL 2002 and 2003 benchmarks (Tjong Kim Sang, 2002; Tjong Kim Sang & De Meulder, 2003), which contain 4 European languages. To measure the quality of CLWE, we perform zero-shot cross-lingual classification, where we train a model on English and directly apply it to each of the other 3 languages. ",
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+    {
+        "type": "table",
+        "img_path": "images/1752a672c3357cb205d3fb46002e5f7c1a18b8c9f08ff63f381a4b032e1b2b6f.jpg",
+        "table_caption": [
+            "Table 1: Precision $@ 1$ for the BLI task on the MUSE dataset6. Within each category, unsupervised methods are listed at the top while supervised methods are at the bottom. The best result for unsupervised methods is underlined while bold signifies the overall best. “IN” refers to iterative normalization proposed in Zhang et al. (2019), “AR” refers to alignment refinement and “VR” refers to vocabulary reallocation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>en-es</td><td>es-en</td><td>en-fr</td><td>fr-en</td><td>en-de</td><td>de-en</td><td>en-it</td><td>it-en</td><td>en-ru</td><td>ru-en</td><td>en-zh</td><td>zh-en</td><td>avg</td></tr><tr><td colspan=\"10\">Alignment Methods</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(1)MUSE(Conneau et al.,2018a)</td><td>81.7</td><td>83.3</td><td>82.3</td><td>82.1</td><td>74.0</td><td>72.0</td><td>77.7</td><td>78.2</td><td>44.0</td><td>59.1</td><td>32.5</td><td>31.4</td><td></td><td>66.5</td></tr><tr><td>(2) VECMAP(Artetxe etal.,218a)</td><td>82.3</td><td>84.7</td><td>82.3</td><td>83.6</td><td>75.1</td><td>74.3</td><td>-</td><td>-</td><td>49.2</td><td></td><td>65.6</td><td>0.0</td><td>0.0</td><td>1</td></tr><tr><td>(3)DeMa-BWE(Zhou et al.,2019)</td><td>82.8</td><td>84.9</td><td>83.1</td><td>83.5</td><td>77.2</td><td>74.4</td><td>-</td><td>-</td><td></td><td>49.2</td><td>65.7</td><td>42.5</td><td>37.9</td><td>-</td></tr><tr><td>(4) Procrustes (Smith et al.,2017)</td><td>81.4</td><td>82.9</td><td>81.1</td><td>82.4</td><td>73.5</td><td>72.4</td><td>77.5</td><td>77.9</td><td></td><td>51.7</td><td>63.7</td><td>42.7</td><td>36.7</td><td>68.7</td></tr><tr><td>(5) GeoMM (Jawanpuria et al., 2019)</td><td>81.4</td><td>85.5</td><td>82.1</td><td>84.1</td><td>74.7</td><td>76.7</td><td>77.9</td><td>80.9</td><td></td><td>51.3</td><td>67.6</td><td>49.1</td><td>45.3</td><td>71.4</td></tr><tr><td>(6) RCSLS (Joulin et al.,2018b) (7) RCSLS + IN (Zhang et al., 2019)</td><td>84.1</td><td>86.3</td><td>83.3 83.9</td><td>84.1</td><td>79.1</td><td>76.3</td><td>78.5 79.1</td><td>79.8 -</td><td>57.9</td><td></td><td>67.2</td><td>45.9</td><td>46.4</td><td>72.4</td></tr><tr><td></td><td>83.9</td><td>-</td><td></td><td>-</td><td>78.1</td><td></td><td></td><td></td><td></td><td>57.9</td><td>1</td><td>48.6</td><td>-</td><td>1</td></tr><tr><td colspan=\"10\">Joint Traing Methods</td><td></td><td></td><td>17.9</td><td></td><td></td></tr><tr><td>(8) Unsupervised Joint</td><td>33.4</td><td>36.6</td><td>42.2</td><td>47.4</td><td>39.5</td><td>41.4</td><td>36.8</td><td>38.8</td><td>4.0</td><td>3.5</td><td></td><td>10.2</td><td></td><td>29.3</td></tr><tr><td>(9) Supervised Joint (Duong et al.,2016)</td><td>79.7</td><td>79.8</td><td>78.1</td><td>76.7</td><td>67.5</td><td>68.9</td><td>74.4</td><td>74.1</td><td>41.8</td><td></td><td>51.8</td><td>46.7</td><td></td><td>65.2</td></tr><tr><td>(10) Joint - Replace</td><td>48.2</td><td>47.7</td><td>49.4</td><td>52.1</td><td>46.5</td><td>46.9</td><td>43.8</td><td>45.8</td><td></td><td>20.3</td><td>36.6</td><td>32.7</td><td>43.3 34.1</td><td>42.0</td></tr><tr><td colspan=\"10\">Joint Align Framework</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(11) Joint_Align (w/o AR)</td><td>55.9</td><td>62.8</td><td>61.8</td><td>67.0</td><td>49.1</td><td>54.6</td><td>50.2</td><td>51.4</td><td>8.7</td><td></td><td>8.2</td><td>19.4</td><td></td><td>42.3</td></tr><tr><td>(12) Joint_Align+MUSE</td><td>81.4</td><td>84.2</td><td>82.8</td><td>83.6</td><td>74.2</td><td>72.2</td><td>77.5</td><td>81.5</td><td>45.0</td><td></td><td>58.3</td><td>36.1</td><td>18.2 35.3</td><td>67.7</td></tr><tr><td>(13)Joint_Align+RCSLS(w/o VR)</td><td>34.2</td><td>37.0</td><td>41.2</td><td>46.8</td><td>34.0</td><td>35.6</td><td>35.3</td><td>35.1</td><td></td><td>7.7</td><td>5.2</td><td>20.2</td><td>15.7</td><td>29.0</td></tr><tr><td>(14)Joint_Align+GeoMM</td><td>82.6</td><td>85.7</td><td>82.5</td><td>84.2</td><td>75.5</td><td>77.2</td><td>78.2</td><td>81.4</td><td></td><td>52.4</td><td>67.7</td><td>50.4</td><td>46.5</td><td>72.0</td></tr><tr><td>(15) Joint_Align + RCSLS</td><td>84.7</td><td>87.9</td><td>83.5</td><td>85.6</td><td>79.6</td><td>78.0</td><td>80.6</td><td>84.0</td><td></td><td>59.8</td><td>67.8</td><td>54.3</td><td>48.7</td><td>74.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Unsupervised Machine Translation (UMT) Lastly, we test our approach using the unsupervised MT task, on which the initialization of CLWE plays a crucial role (Lample et al., 2018b). Note that our purpose here is to directly compare with similar studies in Lample et al. (2018b), and thus we follow their settings and consider two language pairs, English-French and English-German, and evaluate on the widely used WMT’14 en-fr and WMT’16 en-de benchmarks. ",
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+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    {
+        "type": "text",
+        "text": "For the BLI task, we compare our framework to recent state-of-the-art methods. We obtain numbers from the corresponding papers or Zhou et al. (2019), and use the official tools for MUSE (Conneau et al., 2018a), GeoMM (Jawanpuria et al., 2019) and RCSLS (Joulin et al., 2018b) to obtain missing results. We consider the method of Duong et al. (2016) for supervised joint training based on bilingual dictionaries, which is comparable to supervised alignment methods in terms of resources used. For unsupervised joint training, we train uncased joint fastText word vectors of dimension 300 on concatenated Wikipedia corpora of each language pair with default parameters. The hyperparameter $\\gamma$ is selected from $\\{ 0 . 7 , \\bar { 0 } . 8 , 0 . 9 , 0 . 9 5 \\}$ on validation sets. For the alignment refinement step in our proposed framework, we use RCSLS and GeoMM to compare with supervised methods, and MUSE for unsupervised methods. In addition, we include an additional baseline of joint training, denoted as Joint - Replace, which is identical to unsupervised joint training except that it utilizes a seed dictionary to randomly replace words with their translations in the training corpus. Following standard practices, we consider the top $2 0 0 \\mathrm { k }$ most frequent words and use the crossdomain similarity local scaling (CSLS) (Conneau et al., 2018a) as the retrieval criteria. Note that a concurrent work (Artetxe et al., 2019) proposed a new retrieval method based on MT systems and produced state-of-the-art results. Although their method is applicable to our framework, it has high computational cost and is out of the scope of this work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the NER task: (1) For non-contextualized representations, we train embeddings the same way as in the BLI task and use a vanilla Bi-LSTM-CRF model (Lample et al., 2016). For all alignment steps, we apply the supervised Procrustes method using dictionaries from the MUSE library for simplicity. (2) For contextualized representations, we use M-BERT, an unsupervised joint training model, as our base model and apply our proposed framework on it by first aligning its extracted features and then feeding them to a task-specific model (M-BERT Feature $^ +$ Align). Specifically, we use the sum of the last $4 \\mathbf { M }$ -BERT layers’ outputs as the extracted features. To obtain the alignment matrices, one for each layer, we use 30k parallel sentences from the Europarl corpus for each language pair and follow the procedure of Section 3.2. We feed the extracted features as inputs to a task-specific model with 2 Bi-LSTM layers and a CRF layer (see appendix A). We compare our framework to both finetuning (M-BERT Finetune), which has been studied by previous papers, and feature extraction (M-BERT Feature). Lastly, we also compare against XLM, a supervised joint training model. ",
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+    {
+        "type": "table",
+        "img_path": "images/54e6d93bac850917e03d9c2562c6e038e9c336a473c79c066ff31f2f6acb0e03.jpg",
+        "table_caption": [
+            "Table 2: Precision $@ 1$ for the BLI task on the MUSE dataset with test pairs of same surface form removed. The best result for unsupervised methods is underlined while bold signifies the overall best. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>en-es</td><td>es-en</td><td>en-fr</td><td>fr-en</td><td>en-de</td><td>de-en</td><td>en-it</td><td>it-en</td><td>en-ru</td><td>ru-en</td><td>en-zh</td><td>zh-en</td><td>avg</td></tr><tr><td colspan=\"10\"></td><td></td><td></td><td></td><td></td></tr><tr><td>(1) MUSE(Conneau et al.,2018a)</td><td>77.1</td><td>82.5</td><td>76.4</td><td>Unsupervised 78.0</td><td>674</td><td>67.8 10.5</td><td>72.5 78</td><td>77.5</td><td>42.7</td><td>50.2</td><td>28.7</td><td>29.1</td><td>62.5</td></tr><tr><td>(2) Unsupervised Joint</td><td>3.7</td><td>10.2</td><td>5.1</td><td>10.7</td><td>8.5</td><td></td><td></td><td>8.1</td><td>0.4</td><td>2.7</td><td>2.5</td><td>6.4</td><td>6.4</td></tr><tr><td>(3) Joint_Align + MUSE</td><td>77.5</td><td>83.0</td><td>770</td><td>79.5</td><td>66.7</td><td>68.0</td><td>70.9</td><td>78.0</td><td>43.5</td><td>55.1</td><td>32.3</td><td>32.7</td><td>63.7</td></tr><tr><td></td><td></td><td></td><td></td><td>Supervised</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(4) RCSLS (Joulin et al.,2018b) 78.0</td><td></td><td>83.9</td><td>76.0</td><td>78.6</td><td>68.2</td><td>68.4</td><td>71.8</td><td>78.2</td><td>50.7</td><td>56.9</td><td>51.0</td><td>41.7</td><td>67.0</td></tr><tr><td>(5) Supervised Joint (Duong et al.,2016)</td><td>76.8</td><td>80.8</td><td>73.4</td><td>76.1</td><td>60.1</td><td>61.7</td><td>69.7</td><td>76.2</td><td>41.0</td><td>51.8</td><td>52.3</td><td>43.3</td><td>63.6</td></tr><tr><td>(6) Joint_Align + RCSLS</td><td>82.1</td><td>84.6</td><td>78.1</td><td>80.4</td><td>68.4</td><td>70.4</td><td>73.7</td><td>79.0</td><td>59.0</td><td>66.8</td><td>51.4</td><td>45.7</td><td>70.0</td></tr></table>",
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+        "text": "For the UMT task, we use the exact same data, architecture and parameters released by Lample et al.   \n(2018b)7. We simply use different embeddings trained with the same data as inputs to the model. ",
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+        "text": "4.3 RESULTS AND ANALYSIS ",
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+        "text": "4.3.1 ALIGNMENT VS. JOINT TRAINING ",
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+        "text": "We compare alignment methods with joint training on all three downstream tasks. As shown in Table 1 and Table 3, we find alignment methods significantly outperform the joint training approach by a large margin in all language pairs for both BLI and NER. However, the unsupervised joint training method is superior than its alignment counterpart on the unsupervised MT task as demonstrated in 2(c). While these results demonstrate that their relative performance is task-dependent, we conduct further analysis to reveal three limitations as discussed in Sec 2.3. ",
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+        "text": "First, their poor performance on BLI and NER tasks shows that unsupervised joint training fails to generate high-quality alignments due to the lack of a fine-grained seed dictionary as discussed in its limitation 2. To evaluate accuracy on words that are not shared, we further remove test pairs of the same surface form (e.g. (hate, hate) as a test pair for en-de) of the BLI task and report their results in Table 2. We find unsupervised joint training (row 2) to achieve extremely low scores which shows that emebddings of non-shared parts are poorly aligned, consistent with the PCA visualization shown in Figure 1. ",
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+        "type": "text",
+        "text": "Moreover, we delve into the relative performance of the two paradigms on the MT task by plotting their test BLEU scores of the first 20 epochs in Figure 2(a) and 2(b). We observe that the alignment method actually obtains higher BLEU scores in the first few epochs, but gets surpassed by joint training in later epochs. This shows the importance of parameter sharing as discussed in limitation 1 of alignment methods: shared words can be used as a cross-lingual constraint for unsupervised joint training during fine-tuning but this constraint cannot easily be used in alignment methods. The lack of sharing is also a limitation for the supervised joint training method, which performs poorly on the MT task even with supervision as shown in Figure 2(c). ",
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+        "type": "text",
+        "text": "Lastly, we demonstrate that oversharing can be sub-optimal for unsupervised joint training as discussed in its limitation 2. Specifically, we conduct ablation studies for our framework in Table 1. Applying alignment refinement on unsupervised joint training without any vocabulary reallocation does not improve its performance (row 13). On the other hand, simple vocabulary reallocation alone boosts the performance by quite a margin (row 11). This shows some words are shared erroneously across languages in unsupervised joint training, thereby hindering its performance. In addition, while utilizing a seed dictionary (row 10) improves the performance of unsupervised joint train",
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+        "type": "text",
+        "text": "Table 3: F1 score for the crosslingual NER task. “Adv” refers to adversarial training. ‡ denotes results that are not directly comparable due to different resources and architectures used. ∗ denotes supervised XLM model trained with MLM and TLM objectives. Its Dutch (nl) result is blank because the model is not pretrained on it. Bold signifies state-of-the-art results. We report the average of 5 runs. ",
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+        "type": "table",
+        "img_path": "images/07d2d5e3bec392d13ccecfb057c3a2ba6b8f8317b5457426c196b8b1e664af5d.jpg",
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+        "table_body": "<table><tr><td></td><td>es</td><td>nl</td><td>de</td><td>avg</td></tr><tr><td colspan=\"5\">Non-contextualized</td></tr><tr><td>Unsupervised Joint</td><td>50.28</td><td>42.77</td><td>21.49</td><td>38.18</td></tr><tr><td>Supervised Joint (Duong et al.,2016)</td><td>63.16</td><td>63.60</td><td>36.24</td><td>54.33</td></tr><tr><td>Joint- Replace</td><td>65.28</td><td>68.44</td><td>51.59</td><td>61.77</td></tr><tr><td>Align</td><td>69.00</td><td>71.33</td><td>52.17</td><td>64.17</td></tr><tr><td>Joint_Align</td><td>70.46</td><td>72.10</td><td>56.47</td><td>66.34</td></tr><tr><td>Xie et al. (2018)</td><td>71.67</td><td>70.90</td><td>57.43</td><td>66.67</td></tr><tr><td>Chen et al. (2019)‡</td><td>73.50</td><td>72.40</td><td>56.00</td><td>67.30</td></tr><tr><td colspan=\"5\">Contextualized</td></tr><tr><td>XLMFinetune (Lample&amp; Conneau,2019)*</td><td>63.18</td><td>-</td><td>67.55</td><td>-</td></tr><tr><td>M-BERT Finetune (Pires et al., 2019)</td><td>73.59</td><td>77.36</td><td>69.74</td><td>73.56</td></tr><tr><td>M-BERT Finetune (Wu&amp; Dredze,2019)</td><td>74.96</td><td>77.57</td><td>69.56</td><td>74.03</td></tr><tr><td>M-BERT Finetune (Keung et al., 2019)</td><td>75.00</td><td>77.50</td><td>68.60</td><td>73.70</td></tr><tr><td>M-BERT Finetune + Adv (Keung et al.,2019)</td><td>74.30</td><td>77.60</td><td>71.90</td><td>74.60</td></tr><tr><td>M-BERT Feature</td><td>74.23</td><td>78.65</td><td>67.63</td><td>73.50</td></tr><tr><td>M-BERT Feature + Align</td><td>75.77</td><td>79.03</td><td>70.54</td><td>75.11</td></tr></table>",
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+        "type": "text",
+        "text": "ing, it still suffers from the oversharing problem and performs worse compared to supervised joint training (row 9). ",
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+        "type": "text",
+        "text": "4.3.2 EVALUATION OF PROPOSED FRAMEWORK ",
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+        "type": "text",
+        "text": "As shown in Table 1, Table 3, and Figure 2, our proposed framework substantially improves over the alignment and joint training baselines on all three tasks. In particular, it outperforms existing methods on all language pairs for the BLI task (using the CSLS as retrieval metric) and achieves state-of-the-art results on 2 out of 3 language pairs for the NER task. Besides, we show that it alleviates limitations of alignment and joint training methods shown in the previous section. ",
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+        "type": "text",
+        "text": "First, the proposed framework largely improves the poor alignment of unsupervised joint training, especially for non-sharing parts. As shown in Table 1, the proposed Joint Align framework achieves comparable results to prior methods in the unsupervised case (row 12) and it outperforms previous state-of-the-art methods in the supervised setting (row 15). Specifically, our proposed framework can generate well-aligned embeddings after alignment refinement is applied to the initially ill-aligned embeddings, as demonstrated in Figure 1. This is further verified by results in Table 2, where our proposed framework largely improves accuracy on words not shared between two languages over the unsupervised joint training baseline (row 3 and 6 vs row 2). ",
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+        "text": "Besides, our ablation study in Table 1 further shows the effectiveness of the proposed vocabulary reallocation technique, which alleviates the issue of oversharing. Particularly, we observe no improvement compared to unsupervised joint training baseline (row 8) when an alignment refinement step is used without vocabulary reallocation (row 13), while a vocabulary reallocation step alone significantly improves the performance (row 11). This is consistent with Figure 1 and shows that the oversharing is a bottleneck for applying alignment methods to joint training. It also suggests detecting what to share is crucial to achieve better cross-lingual transfer. ",
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+        "type": "text",
+        "text": "Lastly, while supervised joint training shares the limitation 1 of alignment methods and performs poorly when finetuned, our proposed framework exploits the same idea of vocabulary sharing used in unsupervised joint training. In the MT tasks, our framework obtains a maximum gain of 2.97 BLEU over baselines we ran and consistently performs better than results reported in Lample et al. (2018b). In addition, Figure 2 shows that Joint Align not only converges faster in earlier training epochs but also consistently outperforms the two baselines thereafter. These empirical findings demonstrate the effectiveness of our proposed methods in the non-contextualized case. ",
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+    {
+        "type": "text",
+        "text": "4.3.3 CONTEXTUALIZED WORD REPRESENTATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As can be seen in Table 3, when using our framework (M-BERT Feature $^ +$ Align), we achieve state-of-the-art results on cross-lingual NER on 2 out of 3 languages and the overall average. This shows that our framework can effectively generalize to contextualized representations. Specifically, our framework improves over both the M-BERT feature extraction and finetuning baselines on all three language pairs. However, when compared to non-contextualized results, the gain of using alignment refinement on top of unsupervised joint training is much smaller. This suggests that, as the contextualized unsupervised joint training model performs very well already even without any supervision, it is harder to achieve large improvements. While alignment refinement relies on word alignment, a process that is noisy itself, a better alignment approach may be warranted. Lastly, the reason why a supervised joint training model, XLM, performs worse than its unsupervised counterpart, M-BERT, is likely that XLM uses an uncased vocabulary, where casing information is important for NER tasks. ",
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+        "image_caption": [
+            "Figure 2: (a)(b): Results on MT of Align, Joint and our framework for the first 20 training epochs. Results after 20 epochs have similar patterns. (c): BLEU scores for the MT task. Results evaluated on the WMT’14 English-French and WMT’16 German-English. All training settings are the same for each language pair except the embedding initialization. Note that we are not trying to outperform state-of-the-art methods (Song et al., 2019) but rather to observe improvements afforded by embedding initialization. †Results reported by Lample et al. (2018b). Our results are obtained using the official code released by the author. ‡Duong et al. (2016) is a supervised method that we include for analysis purpose only and is not directly comparable to other results in this table. "
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+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Word embeddings (Mikolov et al., 2013a; Ruder et al., 2019) are a key ingredient to achieving success in monolingual NLP tasks. However, directly using word embeddings independently trained for each language may cause negative transfer (Wang et al., 2019) in cross-lingual transfer tasks. In order to capture the cross-lingual mapping, a rich body of existing work relying on cross-lingual supervision, including bilingual dictionaries (Mikolov et al., 2013a; Faruqui & Dyer, 2014; Artetxe et al., 2016; Xing et al., 2015; Duong et al., 2016; Gouws & Søgaard, 2015; Joulin et al., 2018a), sentence-aligned corpora (Kocisk ˇ y et al., 2014; Hermann & Blunsom, 2014; Gouws et al., 2015) \\` and document-aligned corpora (Vulic & Moens, 2016; Søgaard et al., 2015). ´ ",
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+    {
+        "type": "text",
+        "text": "Besides, unsupervised alignment methods aim to eliminate the requirement for cross-lingual supervision. Early work of Cao et al. (2016) matches the mean and the standard deviation of two embedding spaces after alignment. Barone (2016); Zhang et al. (2017a;b); Conneau et al. (2018a) adapted a generative adversarial network (GAN) (Goodfellow et al., 2014) to make the distributions of two word embedding spaces indistinguishable. Follow-up works improve upon GAN-based training for better stability and robustness by introducing Sinkhorn distance (Xu et al., 2018), by stochastic self-training (Artetxe et al., 2018a), or by introducing latent variables (Dou et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "While alignment methods utilize embeddings trained independently on different languages, joint training methods train word embeddings at the same time. Klementiev et al. (2012) train a bilingual dictionary-based regularization term jointly with monolingual language model objectives while Kocisk ˇ y et al. (2014) defines the cross-lingual regularization with the parallel corpus. Another \\` branch of methods (Xiao & Guo, 2014; Gouws & Søgaard, 2015; Ammar et al., 2016; Duong et al., 2016) build a pseudo-bilingual corpus by randomly replacing words in monolingual corpus with their translations and use monolingual word embedding algorithms to induce bilingual representations. The unsupervised joint method by Lample & Conneau (2019) simply exploit words that share the same surface form as bilingual “supervision” and directly train a shared set of embedding with joint vocabulary. Recently, unsupervised joint training of contextualized word embeddings through the form of multilingual language model pretraining using shared subword vocabularies has produced state-of-the-art results on various benchmarks (Devlin et al., 2019; Artetxe & Schwenk, 2019; Lample & Conneau, 2019; Pires et al., 2019; Wu & Dredze, 2019). ",
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+    {
+        "type": "text",
+        "text": "A concurrent work by Ormazabal et al. (2019) also compares alignment and joint method in the bilingual lexicon induction task. Different from their setup which only tests on supervised settings, we conduct analysis across various tasks and experiment with both supervised and unsupervised conditions. While Ormazabal et al. (2019) suggests that the combination of the alignment and joint model could potentially advance the state-of-art of both worlds, we propose such a framework and empirically verify its effectiveness on various tasks and settings. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we systematically compare the alignment and joint training methods for CLWE. We point out that the nature of each category of methods leads to certain strengths and limitations. The empirical experiments on extensive benchmark datasets and various NLP tasks verified our analysis. To further improve the state-of-art of CLWE, we propose a simple hybrid framework which combines the strength from both worlds and achieves significantly better performance in the BLI, MT and NER tasks. Our work opens a promising new direction that combines two previously exclusive lines of research. For future work, an interesting direction is to find a more effective word sharing strategy. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments: This research was sponsored by Defense Advanced Research Projects Agency Information Innovation Office (I2O) under the Low Resource Languages for Emergent Incidents (LORELEI) program, issued by DARPA/I2O under Contract No. HR0011-15-C0114. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. government. The U.S. government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation here on. ",
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+    {
+        "type": "text",
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+        "text": "Will Y. Zou, Richard Socher, Daniel Cer, and Christopher D. Manning. Bilingual word embeddings for phrase-based machine translation. In Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing, pp. 1393–1398, Seattle, Washington, USA, October 2013. Association for Computational Linguistics. URL https://www.aclweb.org/ anthology/D13-1141. ",
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A NER EXPERIMENT DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            691,
+            446,
+            708
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Here we include some additional details for the NER experiments with contextualized representations: ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "• Alignment As described in Section 3.2, we apply word alignment methods, such as fastalign (Dyer et al., 2013), on parallel data to extract word-aligned pairs for learning the alignment matrix. As M-BERT is based on subword tokens, we use the average of the representations of all subword tokens that correspond to a word as the representation for that word. For instance, assume that an English word “Resumption” is aligned to a German word “Wiederaufnahme”, and they are tokenized by M-BERT as “Res”, “##sumption”, and “Wie”, “##dera”, “##uf”, “##nahme”, respectively. Then the representation for “Resumption” is the average of the representations of subword tokens “Res” and “##sumption”, and the same goes for “Wiederaufnahme”. • Hyperparameters For the task-specific NER model, we use a 2-layer Bi-LSTM with a hidden size of 768 followed by a CRF layer. We apply a dropout rate of 0.5 on the input and the output of the Bi-LSTM, and use Adam with default parameters and a learning rate of 0.0001 for optimization. We train the model for 40 epochs with a batch size of 10, and evaluate the model per 150 steps. For prediction, we feed the outputs of the Bi-LSTM that correspond to the first subword tokens of each word to the CRF model. This is identical to finetuning BERT on the NER task, except that in our case the outputs that correspond to the first subword token are fed into a CRF, rather than a linear layer as done in BERT. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/8302085309ea76b84f005a5ce136a497525d56b4b505aecb9350797ea4c4acce.jpg",
+        "table_caption": [
+            "Table 4: Precision $@ 1$ for the BLI task on the MUSE dataset using test set produced by vocabulary reallocation. Within each category, unsupervised methods are listed at the top while supervised methods are at the bottom. Bold signifies the overall best results. “AR” refers to alignment refinement and “VR” refers to vocabulary reallocation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>en-es</td><td>es-en</td><td>en-fr</td><td>fr-en</td><td>en-de</td><td>de-en</td><td>en-it</td><td>it-en</td><td>en-ru</td><td>ru-en</td><td>en-zh</td><td>zh-en</td><td>avg</td></tr><tr><td colspan=\"10\">Alignment Methods</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(1) MUSE(Conneau et al.,2018a)</td><td>82.1</td><td>83.5</td><td>82.6</td><td>83.1</td><td>74.1</td><td>71.8</td><td>77.4</td><td>79.8</td><td>44.1</td><td>59.1</td><td>34.1</td><td>31.6</td><td>66.9</td></tr><tr><td>(4) Procrustes (Smith et al.,2017)</td><td>81.6</td><td>82.0</td><td>80.5</td><td>81.5</td><td>74.2</td><td>73.3</td><td>77.0</td><td>77.8</td><td>50.8</td><td>63.5</td><td>44.2</td><td>37.0</td><td>68.6</td></tr><tr><td>(5) GeoMM (Jawanpuria et al., 2019)</td><td>82.1</td><td>86.9</td><td>82.3</td><td>85.1</td><td>75.3</td><td>77.2</td><td>78.6</td><td>82.2</td><td>51.7</td><td>67.8</td><td>50.5</td><td>45.6</td><td>72.1</td></tr><tr><td>(6) RCSLS (Joulin et al.,2018b)</td><td>82.8</td><td>84.3</td><td>82.4</td><td>83.3</td><td>78.6</td><td>75.6</td><td>78.3</td><td>81.0</td><td>57.7</td><td>66.8</td><td>48.3</td><td>45.6</td><td>72.1</td></tr><tr><td colspan=\"10\">Joint Traing Methods</td><td></td><td></td><td></td><td></td></tr><tr><td>(7) Unsupervised Joint</td><td>33.2</td><td>36.3</td><td>41.5</td><td>46.8</td><td>39.1</td><td>40.7</td><td>35.8</td><td>38.1</td><td>4.1</td><td>3.7</td><td>8.2</td><td>5.7</td><td>27.8</td></tr><tr><td>(8) Supervised Joint (Duong et al., 2016)</td><td>80.1</td><td>80.5</td><td>78.6</td><td>77.1</td><td>67.2</td><td>68.3</td><td>74.6</td><td>74.5</td><td>41.7</td><td>51.8</td><td>47.2</td><td>44.0</td><td>65.5</td></tr><tr><td colspan=\"10\">Joint Align Framework</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(9)Joint_Align (w/o AR)</td><td>56.8</td><td>63.2</td><td>62.2</td><td>67.2</td><td>49.2</td><td>55.1</td><td>50.6</td><td>51.9</td><td>8.7</td><td>8.2</td><td>19.5</td><td>18.4</td><td>42.6</td></tr><tr><td>(10) Joint_Align +MUSE</td><td>82.4</td><td>85.0</td><td>83.5</td><td>84.7</td><td>74.6</td><td>72.9</td><td>78.2</td><td>82.6</td><td>46.1</td><td>58.7</td><td>39.9</td><td>36.2</td><td>68.7</td></tr><tr><td>(11) Joint_Align +RCSLS(w/o VR) (12) Joint_Align +GeoMM</td><td>34.3</td><td>36.8</td><td>41.0</td><td>47.0</td><td>34.3</td><td>35.9</td><td>35.5</td><td>35.2</td><td>7.6</td><td>5.2</td><td>21.3 54.8</td><td>16.1 48.0</td><td>29.2 73.3</td></tr><tr><td></td><td>83.9</td><td>86.2</td><td>83.1</td><td>85.2</td><td>76.1</td><td>77.9</td><td>79.0</td><td>82.8</td><td>53.7</td><td>68.3</td><td></td><td></td><td></td></tr><tr><td>(13) Joint_Align + RCSLS</td><td>87.1</td><td>88.5</td><td>84.2</td><td>86.6</td><td>80.1</td><td>78.7</td><td>81.3</td><td>85.2</td><td>61.3</td><td>68.3</td><td>59.6</td><td>50.7</td><td>76.0</td></tr></table>",
+        "bbox": [
+            173,
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            232,
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+    },
+    {
+        "type": "text",
+        "text": "B BLI TEST PAIRS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            438,
+            346,
+            454
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The official evaluation script8 of MUSE only includes test pairs whose source words and target words both appear in their corresponding vocabularies, leaving out those that are OOV on either side. Since our proposed vocabulary reallocation step modifies both the source and target vocabularies, the script may exclude some test pairs when evaluating our model. For example, the word “age” from the test pair (age, age) for en-fr could be allocated as an en (not shared) word, so it is OOV on the fr side and the pair would thus be left out by the script. As a result, the total number of test pairs would be smaller, thereby changing the denominator when we calculate accuracy. To ensure fair comparison, we include these OOV pairs so the total number of test pairs stays the same. Specifically, if the source word of a test pair is OOV, we retrieve itself. Otherwise, we count it as incorrect. ",
+        "bbox": [
+            174,
+            468,
+            825,
+            594
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In addition, we further investigate which pairs are left out and found that the MUSE benchmark contains some noisy test data. Specifically, we find that the majority of these pairs are in the same surface form, such as (sit, sit), but many target words are not actual translations of the source words. For example, we found test pairs such as {(age, age), (century, century)} for en- $f r$ and {(mickey, mickey), (uncredited, uncredited) $\\}$ for en-zh. Clearly, these are English words and should not be considered as appropriate translations for French or Chinese. In Table 1, we mark these pairs as incorrect for our framework to ensure fair comparison, while in fact our framework correctly allocates these words to English. To reveal the full picture, we also conduct BLI experiments without test pairs that got left out due to vocabulary reallocation. The results are shown in Table 4. We observe that our proposed framework obtains a gain of 3.9 accuracy on average over the RCSLS baseline. ",
+        "bbox": [
+            173,
+            602,
+            825,
+            741
+        ],
+        "page_idx": 14
+    }
+]

parse/train/rkePU0VYDr/rkePU0VYDr.md

@@ -0,0 +1,565 @@
+# A PERTURBATION ANALYSIS OF INPUT TRANSFORMATIONS FOR ADVERSARIAL ATTACKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Many defenses for Convolutional Neural Networks are based on a simple observation that the adversarial examples are not robust and small perturbations to the attacking input often recover the desired prediction. Intuitively, the adversarial examples occupy a very small cone in the decision space which is surrounded by a large area corresponding to the correct class. While the intuition is simple, a detailed understanding of this phenomenon is missing from the research literature.
+
+We identify a family of defense techniques that are based on the instability assumption. The defenses include deterministic lossy compression algorithms and randomized perturbations to the input that all lead to similar gains in robustness. We present a comprehensive experimental analysis of when and why perturbation defenses work and potential mechanisms that could explain their effectiveness (or ineffectiveness) in different settings.
+
+# 1 INTRODUCTION
+
+The attacks on Convolutional Neural Networks, such as Carlini & Wanger (Carlini & Wagner, 2017) or PGD (Madry et al., 2017), generate strategically placed modifications that can be easily dominated by different types of perturbations resulting in correct predictions (Dziugaite et al., 2016; Roth et al., 2019). This suggests that the standard adversarial examples are not robust. Many defense techniques explicitly leverage this property and can be retrospectively interpreted as perturbations of the input images. However, a detailed understanding of this phenomenon is lacking from the research literature including: (1) what types of perturbations work and what is their underlying mechanism, (2) whether all attacks exhibit this property, and (3) possible counter-measures attackers can employ to defeat perturbation defenses.
+
+We can interpret a large number of recent defenses as a type of input perturbations, for example, feature squeezing (Xu et al., 2017), frequency or JPEG compression (Dziugaite et al., 2016), randomized smoothing (Cohen et al., 2019), and perturbation of network structure or the inputs randomly (JafarniaJahromi et al., 2018; Zhang & Liang, 2019; Guo et al., 2017). The defense techniques exhibit very similar gains in robustness. To show it, we start with a simple model where every example is passed through a lossy channel (stochastic or deterministic) prior to model inference. This channel induces a small perturbation to the input. We optimize the perturbation to be small enough as not to affect the prediction accuracy on clean examples but large enough to dominate any adversarial attack. We find that this trade-off is surprisingly consistent across very different families of input perturbations, where the relationship between channel distortion (the $L _ { 2 }$ distance between channel input and output) and robustness is very similar.
+
+Why are some state-of-the-art attacks are sensitive to perturbation-based defenses? We find that many attacks execute an optimization procedure that finds an adversarial image that is very close to the original image in terms of of $L _ { 1 }$ , $L _ { 2 }$ , or $L _ { \infty }$ norm. The resultant optimum, i.e., the adversarial image, tends to exhibit a higher level of instability than natural examples, which we demonstrate from the perspective of a first-order and second-order analysis. By instability we mean that small perturbations of the example can affect the prediction confidences.
+
+The unification of perturbation-based defense also gives us some insight into how an attacker might avoid them.
+
+Our experiments suggest that all the perturbation based defenses are vulnerable to the same types of attack strategies. We argue that the optimization procedure in the attacker should find the smallest distance from the original image that closes the recovery window. In fact, we can devise a generic attacker that attacks a particularly strong lossy channel, based on the additive Laplace noise, and adaptive attacks designed on this channel are often successful against other defenses. This result implies that for many input perturbation defenses the attacker need not be fully adaptive, i.e., they do not need to know exactly what kind of transformation is used to defend the network.
+
+# 2 RELATED WORK
+
+Much of the community’s current understanding of adversarial sensitivity in neural networks is based on the seminal work by Szegedy et al. (2014). Multiple contemporaneous works also studied different aspects of this problem, postulating linearity and over-parametrization as possible explanations (Goodfellow et al., 2014; Biggio et al., 2013). Since the beginning of this line of work, the connection between compression and adversarial robustness has been recognized. The main defense strategies include: the idea of defensive network distillation1 (Papernot et al., 2015), quantizing inputs using feature squeezing (Xu et al., 2017), the thermometer encoding as another form of quantization (Buckman et al., 2018), JPEG compression harnessed by (Dziugaite et al., 2016; Guo et al., 2017; Das et al., 2017; 2018; Aydemir et al., 2018; Liu et al., 2019). Other line of research leveraged connection between randomization and adversarial robustness: Pixel Deflection (Prakash et al., 2018), random resizing and padding of the input image (Xie et al., 2017), and total variance minimization (Guo et al., 2017). In our work we unify the methods based on compression and randomization that are applied to the input images.
+
+While all of the aforementioned defenses were later broken (Carlini & Wagner, 2017; Athalye et al., 2018), it is important to understand why these approaches afforded any form of robustness. The community actually lacks consensus on this point: Szegedy et al. (2014) suggest that neural networks have blind spots, Xu et al. (2017) suggest that quantization makes the adversarial search space smaller, Buckman et al. (2018) suggest the linearity is the main culprit and quantized inputs break up linearity.
+
+Zhang & Liang (2019) inject random Gaussian noise into an image and then discretize it. This method fits into the noisy channel framework with different definitions of $C ( x )$ . They show improved performance of this combined model in the non-adaptive setting. Our experiments find that simply injecting Gaussian noise (or even Uniform or Laplace noise) is an equally effective defense method. We also show that the discretization is not essential to good performance if the level of noise is appropriately tuned. The imprecise channel defense in a neural network is also related to the idea of gradient masking or gradient obfuscation, i.e., a hard to differentiate layer (Papernot et al., 2017). In this work, the backward pass computation is perturbed to make it difficult for a gradient-based attack to synthesize an adversarial image (while the forward pass is kept the same). We implement both non-adaptive attacks and adaptive that can observe the channel and take an approximate gradient through it. We further focus our study on the families of white-box attacks proposed by Carlini & Wagner (2017), and their adaptive variants.
+
+Noise injection can be much more powerful than regularization or a dataset augmentation method. The dropout algorithm can be seen as applying noise to the hidden units. The dropout randomization (Feinman et al., 2017) was used to create a defense that was not completely broken and required a high distortion added to the adversarial examples (Carlini & Wagner, 2017). Many new defenses propose randomization through noise injection without considering the adversarial training (Zhang & Liang, 2019; Cohen et al., 2019). The work on injection of noise into inputs and each of the layers of neural networks by Liu et al. (2018) is a strong heuristic that led to defenses with theoretical guarantees. The random smoothing provides a certified robustness up to a certain threshold of input distortion (and is not designed to be robust beyond the threshold) by utilizing inequalities from the differential privacy literature (Lecuyer et al., 2018). Cohen et al. (2019) improve the theoretical bounds of methods that randomly smooth the input examples. Recent work focuses on combination of randomized smoothing with adversarial training and achieves state of the art in terms of the provable robustness (Salman et al., 2019).
+
+# 3 LOSSY CHANNEL MODEL
+
+We consider convolutional neural networks that take $w \times h$ (width times height) RGB digital images as input, giving an example space of $\mathcal { X } \in ( 2 5 5 ) ^ { w \times h \times 3 }$ , where $( z )$ denotes the integer numbers from 0 to $z$ . We consider a discrete label space of $k$ classes represented as a confidence value $\mathcal { V } \in [ 0 , 1 ] ^ { k }$ . Neural networks are parametrized functions (by a weight vector $\theta$ ) between the example and label spaces $f ( x ; \theta ) : \mathcal { X } \mapsto \mathcal { Y }$ .
+
+An adversarial input $x _ { a d v }$ is a perturbation of a correctly predicted example $x$ that is incorrectly predicted by $f$ .
+
+$$
+f ( x ) \neq f ( x _ { a d v } )
+$$
+
+The distortion is the $\ell _ { 2 }$ error between the original example and the adversarial one:
+
+$$
+\delta _ { a d v } = \lVert \boldsymbol { x } - \boldsymbol { x } _ { a d v } \rVert _ { 2 }
+$$
+
+# 3.1 MODEL
+
+Approximating $f ( \cdot )$ with a less precise version $\bar { f } ( \cdot )$ can counter-intuitively make it more robust (Dziugaite et al., 2016):
+
+$$
+f ( x ) = { \bar { f } } ( x _ { a d v } )
+$$
+
+Intuitively, a lossy version of $f$ introduces noise into a prediction which dominates the strategic perturbations found by an adversarial attack procedure. It turns out that we can characterize a number of popular defense methodologies with this basic framework.
+
+Let $x$ be an example and $f$ be a trained neural network. Precise evaluation means running $f ( x )$ and observing the predicted label. Imprecise evaluation involves first transforming $x$ through a deterministic or stochastic noise process $C ( x ) = C [ x ^ { \prime } \mid x ]$ , and then evaluating the neural network
+
+$$
+y = f ( x ^ { \prime } ) \quad x ^ { \prime } \sim C ( x )
+$$
+
+We can think of $C ( x )$ as a noisy channel (as in signal processing). The distortion of a $C ( x )$ is the expected $\ell _ { 2 }$ reconstruction error:
+
+$$
+\delta _ { c } = \mathbf { E } [ \| C ( x ) - x \| _ { 2 } ] ,
+$$
+
+which is a measure of how much information is lost passing the example through a channel.
+
+This paper shows that there is a subtle trade-off between $\delta _ { c }$ and $\delta _ { a d v }$ . In particular, we can find $\delta _ { c }$ such that $\delta _ { c } > > \delta _ { a d v }$ and $f ( x ) = f ( C ( x _ { a d v } ) )$ . We show that compression and randomization based techniques exhibit this property.
+
+EXAMPLE OF DETERMINISTIC CHANNEL
+
+When $C ( x )$ is deterministic it can be thought of as a lossy compression technique. Essentially, we run the following operation on each input example:
+
+$$
+x ^ { \prime } = \mathtt { c o m p r e s s } ( x )
+$$
+
+One form of compression for CNNs is color-depth compression. Most common image classification neural network architectures convert the integer valued inputs into floating point numbers. We abstract this process with the norm function that for each pixel $n \in ( 2 5 5 )$ maps it to a real number $v \in [ 0 , 1 ]$ by normalizing the value and the corresponding denorm function that retrieves the original integer value (where $\lfloor \rceil$ denotes the nearest integer function) 2:
+
+$$
+\operatorname { n o r m } ( n ) : = { \frac { n } { 2 5 5 } } \qquad \operatorname { d e n o r m } ( v ) : = \left\lfloor 2 5 5 * v \right\rceil
+$$
+
+This process is normally reversible $v = \mathtt { n o r m } ( \mathtt { d e n o r m } ( v ) )$ , but we can artificially make this process lossy. Consider a parametrized $C ( \cdot )$ version of the color-depth compression function:
+
+$$
+C ( v , b ) : = \frac { 1 } { 2 ^ { b } - 1 } \cdot \lfloor ( 2 ^ { b } - 1 ) * v \rceil
+$$
+
+By decreasing $b$ by $\Delta b$ we reduce the fidelity of representing $v$ by a factor of $2 ^ { \Delta b }$ (for the $b$ bits of precision).
+
+# EXAMPLE OF STOCHASTIC PERTURBATION
+
+The channel model is particularly interesting when $C ( x )$ is stochastic. Randomization has also been noted to play a big role in strong defenses in prior work (Madry et al., 2017; Zhang & Liang, 2019; Cohen et al., 2019). For example, we could add independent random noise to each input pixel:
+
+$$
+x ^ { \prime } = x + \epsilon
+$$
+
+We consider two schemes, Gaussian $\epsilon \sim N ( 0 , \sigma )$ and additive Uniform noise $\epsilon \sim U ( - B , B )$ which add independent noise to each pixel.
+
+One of the advantages of randomization is that an adversary cannot anticipate how the particular channel $C$ will transform an input before prediction. However, there is another subtle advantage to randomization. Randomized approaches can partially recover their loss in accuracy due to imprecision by averaging over multiple instances of the perturbation. In classification problems, we can take the most frequent label seen after $T$ perturbation trials:
+
+$$
+\bar { f } ( x ) = \arg \operatorname* { m a x } _ { 1 \ldots k } \sum _ { i } ^ { T } f ( x + \epsilon )
+$$
+
+# 3.2 PERTURBATION ANALYSIS
+
+While the intuition is that the channel’s perturbations dominate strategically placed distortions in an adversarial example, the underlying mathematical mechanism of why recovery is possible is not clear. We start with the hypothesis that synthesized adversarial examples have unstable predictions–meaning that small perturbations to the input space can change confidence values drastically. How do we quantify instability?
+
+Let $f ( x )$ be a function that maps an image to a single class confidence value (i.e., a scalar output). We want to understand how $f ( x )$ changes if $x$ is perturbed by $\epsilon$ . We can apply a Taylor expansion of $f$ around the given example $x$ :
+
+$$
+f ( x + \epsilon ) \approx f ( x ) + \epsilon ^ { T } \nabla _ { x } f ( x ) + \frac { 1 } { 2 } \epsilon ^ { T } \nabla _ { x } ^ { 2 } f ( x ) \epsilon + . . .
+$$
+
+where $\nabla _ { x } f ( x )$ denotes the gradient of the function $f$ with respect to $x$ and $\nabla _ { x } ^ { 2 } f ( x )$ denotes the Hessian of the function $f$ with respect to $x$ . The magnitude of the change in confidence is governed by the Taylor series terms in factorially decreasing importance. $\lVert \epsilon \rVert _ { 2 }$ is exactly the distortion measure $\delta _ { c }$ described at the beginning of Section 3.1. Thus, the expression is bounded in terms of the operator norm, or the maximal change in norm that could be induced, of each of the terms (see Appendix A):
+
+$$
+\epsilon ^ { T } \nabla _ { x } f ( x ) + \frac { 1 } { 2 } \epsilon ^ { T } \nabla _ { x } ^ { 2 } f ( x ) \boldsymbol { \epsilon } + \ldots \leq \delta _ { c } M _ { 1 } ( x ) + \frac { 1 } { 2 } \delta _ { c } ^ { 2 } M _ { 2 } ( x ) + \ldots
+$$
+
+As $\nabla _ { x } f ( x )$ is a vector, this is simply the familiar $\ell _ { 2 }$ norm, and for the second order term this is the maximal eigenvalue:
+
+$$
+\begin{array} { r l } { M _ { 1 } ( x ) = \| \nabla _ { x } f ( x ) \| _ { 2 } } & { { } M _ { 2 } ( x ) = \lambda _ { m a x } ( \nabla _ { x } ^ { 2 } f ( x ) ) } \end{array}
+$$
+
+When $M _ { 1 }$ and $M _ { 2 }$ are larger this means there is a greater propensity to change the prediction for small perturbations. We will show experimentally that for certain types of attacks the $M _ { 1 }$ and $M _ { 2 }$ values around adversarial examples exhibit signs of instability compared to those around natural examples–suggesting a mathematical mechanism of why recovery is possible.
+
+# 4 EXPERIMENTS
+
+Our experiments evaluate the efficacy of imprecision based defenses in a number of different adversarial problem settings.
+
+# 4.1 EXPERIMENTAL SETUP
+
+We run our experiments using ResNet-18 on CIFAR-10 and ResNet-50 on ImageNet dataset using P-100 GPUs (16GB memory). We explore a number of different attacks that are implemented in the foolbox library (Rauber et al., 2017). In each experiment we measure the test accuracy $( \% )$ , the confidence of predictions, and distances between the original images and either their adversarial counterparts or the recovered images after applying one of the defenses. We present our results for non-targeted attacks; if the adversary is successful it induces any misclassification. We experiment with many gradient-based attacks provided in the foolbox library that explore different optimization algorithms and distance measures, for instance:
+
+• LBFGS minimizes the distance between the input image and the adversarial example as well as the cross-entropy between the predictions for the adversarial and the input image; introduced by Szegedy et al. (2014) and further extended in Tabacof & Valle (2015). Carlini-Wagner $L _ { 2 }$ (C&W $L _ { 2 }$ ) is a generalization of the LBFGS attack that is devised after exhaustive search over possible space of: norms, loss functions, box optimization procedures, etc. (Carlini & Wagner, 2017). BIM $L _ { 1 }$ is a modified version of the Basic Iterative Method that minimizes the $L _ { 1 }$ distance (Kurakin et al., 2016).   
+• FGSM adds the sign of the gradient to the image, gradually increasing the magnitude until the image is misclassified Goodfellow et al. (2014).   
+• PGD $L _ { \infty }$ the Projected Gradient Descent Attack that is an iterative version of the FGSM attack; we use the version that minimizes the $L _ { \infty }$ distance (Madry et al., 2017).
+
+We extend the Carlini-Wagner $L _ { 2 }$ attack in its adaptive version so that the gradients are not obfuscated. We approximate the gradients for the backward pass on the compression layers as an identity function, similarly to (He et al., 2017; Athalye et al., 2018). More details on the setup can be found in the Appendix, Section C.13
+
+# 1. PERTURBATION DEFENSES GIVE SIMILAR GAINS IN ROBUSTNESS
+
+In the first experiment, we select 1000 random images from ImageNet and CIFAR-10 datasets. For each of these images we generate an adversarial attack with popular white-box gradient-based methods. Then, we run the adversarial images through different channels: Frequency Compression (FC), Color Depth reduction (CD), Uniform noise (Unif), Gaussian Noise (Gauss), SVD-based compression (SVD), Identity (Iden). The identity channel just passes through the adversarial image with no modification. We measure the accuracy of $f ( C ( x ) )$ , which indicates the ability of the imprecise channels to recover the original label. We present the results in Figure 1 and also in the Supplement in Figure 7 (for all images in the test CIFAR-10 and dev ImageNet sets) and in Table 3 (for different channel parameters and five attacks).
+
+When there is an identity channel, the adversarial attack is always successful. However, each of the imprecise channels is able to recover a substantial portion of original labels from the adversarial examples. It is important to note that we are evaluating these attacks in the setting, where any mis-classification is considered a success and the adversary is not aware of the defense.
+
+Importantly, all the channels can be tuned to recover very similar maximum accuracy after attacks. For example, all the channels can achieve about $85 \%$ accuracy for CIFAR-10 after nonadaptive PGD or C&W attacks. This suggests that any form of imprecision with the right error magnitude is effective at defending against these types of attacks. We observe that the attacks that incur higher distortions such as FGSM or LBFGS decrease the accuracy more than the iterative attacks such as C&W $L _ { 2 }$ or PGD $L _ { \infty }$ . This is because the iterative attacks find the adversarial images that are closer to the original images in terms of the corresponding distance measure that they optimize for in the input space. The key is to ensure that the error introduced by the imprecise channels is big enough to dominate the adversarial perturbations but small enough to generate valid predictions. Figure 1 illustrates this relationship (see also a detailed analysis for a single image presented in Figure 9 in the Appendix). For five different imprecise channels, we plot the channel distortion against the accuracy for CIFAR-10 and ImageNet datasets. We analyze FC, CD, Unif, Gauss, and SVD channels. The curves are qualitatively similar in the low noise regime, but they show more differences when higher distortions are incurred. The lowest accuracy across the datasets for high distortions is observed for content-preserving FC and SVD compression channels. The Gaussian and Uniform channels have very similar trends. They outperform other channels on ImageNet for
+
+![](images/bb5e58d8eef172f8146e4a6e0c34537f72c5bb9fc3622966cc863cb65fae9cfb.jpg)  
+Figure 1: We plot the channel distortion against test accuracy $( \% )$ . The distortion of the imprecise channels has to be large enough to recover the correct label but not so large that it degrades model performance. The base test accuracy is about $9 3 . 5 \%$ for CIFAR-10 and $8 3 . 5 \%$ for ImageNet on 1000 randomly chosen images (results for the full test CIFAR-10 set and the full dev ImageNet set can be found in the Appendix in Figure 7). The experiment is run for the C&W $L _ { 2 }$ attack with 100 iterations and the PGD attack with 40 iterations.
+
+Table 1: Transferability of the adversarial images created against a given noisy channel denoted as A (adaptive attack specified in the first column) to the defense protected with a noisy channel denoted as $D$ (the defense with a noisy channel specified in the first row). Each result represents a recovery $( \% )$ of the adversarial examples (generated for $A$ ) to correct labels after applying the defense $( D )$ . We use $30 \%$ FC compression, $50 \%$ SVD compression, 4 bit values in CD, 0.03 noise level for Gauss and Laplace, and 0.04 noise level for the Uniform channel. We use 2000 images from the CIFAR-10 test set and 100 attack iterations with 5 binary steps to find the $c$ value (with initial $c$ value set to 0.01) for the adaptive C&W $L _ { 2 }$ attack. The baseline test accuracy is $9 3 . 5 6 \%$ . The test accuracy of the noisy channels on clean images is given in the first row denoted: Empty (an empty attack).
+
+<table><tr><td>D</td><td rowspan="3">FC</td><td rowspan="3">CD</td><td rowspan="3">SVD</td><td rowspan="3">Gauss</td><td rowspan="3">Uniform</td><td rowspan="3">Laplace</td></tr><tr><td>A</td><td></td></tr><tr><td>Empty</td><td>93.12</td></tr><tr><td></td><td>93.32 0.20</td><td>93.01 80.75</td><td>83.05</td><td>92.53 81.15</td><td>91.6 79.65</td><td>91.35 78.70</td></tr><tr><td>FC CD</td><td>3.85</td><td>0.70</td><td>43.60</td><td>47.30</td><td>60.45</td><td>62.35</td></tr><tr><td>SVD</td><td>1.99</td><td>47.96</td><td>0.77</td><td>46.52</td><td>62.87</td><td>65.75</td></tr><tr><td>Gauss</td><td>4.45</td><td>48.70</td><td>44.80</td><td>51.50</td><td>61.75</td><td>60.15</td></tr><tr><td>Uniform</td><td>3.45</td><td>30.30</td><td>30.60</td><td>30.15</td><td>48.05</td><td>51.55</td></tr><tr><td>Laplace</td><td>3.05</td><td>23.35</td><td>24.60</td><td>23.80</td><td>39.15</td><td>46.70</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+large distortions but are less performant on low-resolution CIFAR-10 images, where the CD channel achieves higher accuracy.
+
+# 2. ATTACKS ARE TRANSFERABLE
+
+Many input transformation defenses are broken. If the attacker has full knowledge of the defense, it is possible to construct an attack that is impervious to the defense. This is called the adaptive setting. Since the underlying mechanisms of input transformations are similar, we find that an attacker does not need to be fully adaptive. The attacker can assume a particular strong defense and that same adversarial input often transfers to other defenses. We narrow the attacker to a single adaptive step (for details see Section C.8 in the Appendix). Even in this weak adaptive setting, the deterministic channels are fully broken but the randomized channels retain relatively high accuracy above $2 3 . 8 \%$ . We show in the Appendix in Figure 14 that the randomized defenses can also be broken when the adversary is given an unlimited number of adaptive steps.
+
+Laplace attacked images transfer the best to other defenses. They decrease the accuracy of the defense models by at least $4 4 . 3 \%$ (for Laplace itself). Table 1 shows that FC attacked images do not transfer well to other defenses; the maximum drop in accuracy of the model protected by other defenses is $1 2 . 2 6 \%$ . Most adversarial images (against a given defense) transfer very well to the FC defense, i.e. an adversarial image against any defense (e.g. CD, SVD, Gauss, Uniform, or Laplace) is also adversarial against the FC defense. The adversarial images generated against the Uniform defense show better transfer to other defenses in comparison to the adversarial images generated against the Gaussian defense. This is because the higher noise level is applied in the Uniform defense. We observe analogous trends for the ImageNet dataset and present the results in the supplement (Tables 6 and 7).
+
+# 3. RECOVERABLE RANGE SHRINKS WITH HIGHER ATTACK DISTORTION
+
+We use an example from the ImageNet dataset, the ResNet-50 model, and set the stochastic channel to the Gaussian noise. We start from an adversarial example generated with the Carlini & Wagner (non-adaptive) $L _ { 2 }$ attack and for consecutive subplots (in the left to right and top to bottom sequence), we increase the attack strength and incur higher distortion of the adversarial image from the original image. For a single plot, we increase the $L _ { 2 }$ distance of the output from the channel to the adversarial example by increasing the Gaussian noise (controlled by parameter $\sigma$ ). For $L _ { 2 }$ distances incurred by different noise levels, we execute 100 predictions. In Figure 9 in the Appendix, we use the frequency count and report how many times the model predicts the original, adversarial or other class. The plot shows what range of distances from the adversarial image reveal the correct class. For the adversarial examples that are very close to the original image (e.g. adversarial distance of 0.006 for the top-left figure), the window of recovery (which indicates which strengths of the random noise can recover the correct label) is relatively wide, however, as we increase the distance of the adversarial image from the original image, the window shrinks and finally we are not enable to recover the correct label.
+
+For attacks that incur more input distortion or the stronger white-box attacks, we can resort to other statistics as proposed in Roth et al. (2019). They show that adversarial examples are much closer to the unperturbed sample than to any other neighbor and use the fact that the probability of the correct class increases faster than the probability of the highest other class when adding noise with a small to intermediate magnitude to the adversarial example.
+
+# 4. COMPARISON WITH OTHER METHODS
+
+![](images/4c037a052e0f6f386da2f5bcdbf8e8f6ca08ce02e7983db97720223c6ccb376b.jpg)  
+Figure 2: Non-adaptive attack and the recoverable ranges in terms of the c parameter that controls the strength of the C&W $L _ { 2 }$ attack. We systematically change the c parameter and keep the parameters for the channels unchanged. We use the VGG-16 network and all images from the CIFAR-10 test set. The test accuracy of the model without any noise layers and on the clean data is $8 5 . 2 3 \%$ .   
+Figure 3: The top eigenvalues of the Hessians with respect to (w.r.t.) the input 1024 images from the CIFAR-10 dataset trained on the ResNet-18 architecture. We plot the histogram that shows counts of magnitudes for the eigenvalues. Yao et al. (2018) show analysis of the Hessian w.r.t. parameters and we extend it to analyze Hessian w.r.t. inputs.
+
+In Figure 2 we present the accuracy of different channels as the strength of the C&W attack is systematically increased. The strength is controlled by the $c$ tradeoff-parameter that is used to set the relative importance of distance and confidence.
+
+We present a related approach which is the RSE (Random Self-Ensemble) network with 0.2 noise level in the first layer and 0.1 noise level in the remaining layers (as recommended in Liu et al. (2018)). This defense does better for lower distortion levels (c value below 0.1) than other noisy channels, but then its accuracy deterioration is faster for higher distortion levels. The Laplace channel gives the highest accuracy for high values of the $c$ parameter (above 1.0). The CD, FC, SVD, Gauss, and
+
+Uniform channels show similar trends. We also include a very simple channel that reduces brightness of an image by subtracting an arbitrary value from each pixel. The comparison between very complex approaches and a simple input transformation is informative–as they largely follow the same trends.
+
+![](images/4e47be33d1fb121c148da1abb7638a32ff8352766a081d4ac4a171f32b70bd13.jpg)  
+Figure 4: Comparison of magnitude of gradients for original and adversarial images.
+
+First-order analysis. The key question is why adversarial examples are more sensitive to perturbations than natural inputs, when there is evidence that from an input perspective they are statistically indistinguishable. Our experiments suggest that this sensitivity arises from the optimization process that generates adversarial inputs. Based on the operator-norm analysis presented in Section 3.2, we plot in Figure 4 the $L _ { 2 }$ norm of gradients w.r.t. the original $x _ { o r g }$ and adversarial $x _ { a d v }$ images for original (correct) $c _ { o r g }$ and adversarial classes $c _ { a d v }$ . The work by Simon-Gabriel et al. (2019) also evaluates the norm of gradients of the network output with respect to its inputs. They show that the adversarial examples are primarily caused by large gradients of the classifier as captured via the induced loss. At first-order approximation in $\epsilon$ , an $\epsilon$ -sized $L _ { 2 }$ untargeted adversarial attack increases the loss $\mathcal { L }$ at point $x$ by $\epsilon | | \partial _ { x } \mathcal { L } ( x , c _ { o r g } ) | | _ { 2 }$ . Analogously, at first-order approximation in $\epsilon$ , an $\epsilon$ -sized $L _ { 2 }$ targeted adversarial attack decreases the loss $\mathcal { L }$ at point $x$ by $\epsilon | | \partial _ { x } \mathcal { L } ( \bar { x } , c _ { a d v } ) | | _ { 2 }$ . In our experiments, the recovered original images have lower norms of the gradients for both original and adversarial classes than the not recovered images. Additionally, the gradient w.r.t. original image for the original class is much smaller than the gradient w.r.t. adversarial image for the adversarial class. We also observe that the magnitudes of gradients change smoothly as we systematically add more Gaussian noise to the original or adversarial images (Figures 15, 16). For the adversarial image, the gradient w.r.t. adversarial image and for adversarial class decreases as we increase the attack strength and the confidence of adversarial predictions.
+
+Second-order analysis. Our results in Figure 3 show that the adversarial inputs lead to noticeably higher Hessian spectrum than the original inputs. This suggests that the model predictions for the adversarial inputs are less stable than for the original images. Thus, perturbations of the adversarial images with some form of noise can easily change the classification outcome while the prediction for the original images are much more robust and do not lead to such unstable predictions.
+
+# 5 CONCLUSION, LIMITATIONS, AND FUTURE WORK
+
+The non-adaptive attacks are not robust since small changes to the adversarial input often recover the original label. This is an obvious corollary to the very existence of adversarial examples that by definition are relatively close to correctly predicted examples in the input space. Random perturbations of the input can dominate the strategically placed perturbations synthesized by an attack. In fact, the results are consistent across both deterministic and stochastic channels that degrade the fidelity of the input example. From the perspective of the attacker, the recovery window can be closed to make the perturbation based recovery techniques ineffective.
+
+# REFERENCES
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+
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+
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+Mathias Lecuyer, Vaggelis Atlidakis, Roxana Geambasu, Daniel Hsu, and Suman Jana. Certified Robustness to Adversarial Examples with Differential Privacy. arXiv e-prints, art. arXiv:1802.03471, Feb 2018.
+
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+
+Zihao Liu, Qi Liu, Tao Liu, Nuo Xu, Xue Lin, Yanzhi Wang, and Wujie Wen. Feature distillation: Dnnoriented jpeg compression against adversarial examples. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
+
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXiv:1706.06083, 2017.
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+Nicolas Papernot, Patrick McDaniel, Xi Wu, Somesh Jha, and Ananthram Swami. Distillation as a defense to adversarial perturbations against deep neural networks. arXiv preprint arXiv:1511.04508, 2015.
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+
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+
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+
+Yuchen Zhang and Percy Liang. Defending against whitebox adversarial attacks via randomized discretization. AISTATS, 2019.
+
+# A PERTURBATION ANALYSIS: ADDENDUM
+
+$$
+\epsilon ^ { T } \nabla _ { x } f ( x ) + \frac { 1 } { 2 } \epsilon ^ { T } \nabla _ { x } ^ { 2 } f ( x ) \boldsymbol { \epsilon } + \ldots \leq \delta _ { c } M _ { 1 } ( x ) + \frac { 1 } { 2 } \delta _ { c } ^ { 2 } M _ { 2 } ( x ) + \ldots
+$$
+
+$$
+\begin{array} { r l } { M _ { 1 } ( x ) = \| \nabla _ { x } f ( x ) \| _ { 2 } } & { { } M _ { 2 } ( x ) = \lambda _ { m a x } ( \nabla _ { x } ^ { 2 } f ( x ) ) } \end{array}
+$$
+
+$$
+\epsilon ^ { T } \nabla _ { x } f ( x ) \leq \delta _ { c } M _ { 1 } ( x )
+$$
+
+From the Cauchy-Schwarz inequality: $\epsilon ^ { T } \nabla _ { x } f ( x ) \leq | | \epsilon | | _ { 2 } M _ { 1 } ( x )$
+
+$$
+| | \epsilon | | _ { 2 } M _ { 1 } ( x ) = \delta _ { a d v } M _ { 1 } ( x ) \leq \delta _ { c } M _ { 1 } ( x ) ( \mathrm { s i n c e } \delta _ { a d v } < < \delta _ { c } )
+$$
+
+$$
+\nabla _ { x } ^ { 2 } f ( x ) \epsilon \leq \delta _ { c } ^ { 2 } M _ { 2 } ( x )
+$$
+
+From the definition of maximum eigenvalue : $: \lambda _ { m a x } \geq \frac { \epsilon ^ { T } \nabla _ { x } ^ { 2 } f ( x ) \epsilon } { \epsilon ^ { T } \epsilon }$
+
+$$
+\epsilon ^ { T } \nabla _ { x } ^ { 2 } f ( x ) \epsilon \leq | | \epsilon | | _ { 2 } ^ { 2 } \lambda _ { m a x } = \delta _ { a d v } ^ { 2 } \lambda _ { m a x } \leq \delta _ { c } ^ { 2 } \lambda _ { m a x }
+$$
+
+# B COMPRESSION TECHNIQUES
+
+# B.1 FFT-BASED COMPRESSION
+
+We apply compression in the frequency domain to reduce the precision of the input images. Let $x$ be an input image, which has corresponding Fourier representation that re-indexes each tensor in the frequency domain:
+
+$$
+F [ \omega ] = F ( x [ \mathbf { n } ] )
+$$
+
+This Fourier representation can be efficiently computed with an FFT. The mapping is invertible $x = F ^ { - 1 } ( F ( x ) )$ . Let $M _ { f } [ \omega ]$ be a discrete indicator function defined as follows:
+
+$$
+M _ { f } [ \omega ] = \left\{ 1 , \omega \leq f \right.
+$$
+
+$M _ { f } [ \omega ]$ is a mask that limits the $F [ \omega ]$ to a certain band of frequencies. $f$ represents how much of the frequency domain is considered. The band-limited spectrum is defined as, $F [ \omega ] \cdot M _ { f } [ \omega ]$ , and the band-limited filter is defined as:
+
+$$
+x ^ { \prime } = F ^ { - 1 } ( F [ \omega ] \cdot M _ { f } [ \omega ] )
+$$
+
+# B.2 SVD-BASED COMPRESSION
+
+Analogously to the FFT-based method, we decompose an image with SVD transformation and reconstruct its compressed version with dominant singular values. The basis used in SVD are adaptive and determined by an image, as opposed to pre-selected basis used in FFT. This can result in higher quality for the same compression rate in case of SVD, however it is more computationally intensive than FFT-based compression.
+
+# C ADDITIONAL EXPERIMENTS FOR WHITE-BOX ATTACKS
+
+# C.1 ACCURACY OF PERTURBATION DEFENSES ON CLEAN DATA
+
+One pitfall of the imprecise channel defense is that it introduces errors whether or not there are any adversarial examples. The errors act as an upper-bound for the best possible test accuracy we can get under adversarial perturbations. For frequency compression, color depth compression, and uniform noise injection, we compare the test accuracy for different levels of imprecision. Table 2 shows the results for all test images from CIFAR-10 on the ResNet-18 architecture, for three of the imprecise channels, and for different noise settings.
+
+Table 2: On CIFAR-10 with ResNet-18, we measure the max test accuracy for imprecise channels without any adversarial perturbation. This signifies the amount of accuracy we sacrifice with respect to the baseline test accuracy of the model (without any perturbations of the images): $9 3 . 5 6 \%$ .   
+
+<table><tr><td>FC (%)</td><td>Acc. (%)</td><td>CD (bits)</td><td>Acc. (%)</td><td>Uniform (ε)</td><td>Acc. (%)</td></tr><tr><td>1</td><td>93.5</td><td>8</td><td>93.4</td><td>0.009</td><td>93.52</td></tr><tr><td>10</td><td>93.42</td><td>6</td><td>93.3</td><td>0.03</td><td>92.59</td></tr><tr><td>50</td><td>91.6</td><td>4</td><td>91.9</td><td>0.07</td><td>85.2</td></tr><tr><td>75</td><td>79.53</td><td>2</td><td>87.4</td><td>0.1</td><td>70.67</td></tr></table>
+
+![](images/f492cd590dcbc09ae5e3e0dee865e3eab14ab579208c64e71299848351ee3cc1.jpg)  
+Figure 5: The test accuracy after passing clean images through six different noisy channels, where the added noise is controlled by the compression rate and epsilon parameters. We use full CIFAR-10 test set for ResNet-18, and full ImageNet validation set for ResNet-50.
+
+The test accuracy of the models can be increased by training with compression, e.g., by using FFT based convolutions with $50 \%$ compression in the frequency domain increases the accuracy to $9 2 . 3 2 \%$ .
+
+We present the results for six different noisy channels; three of them are compression based: FC, CD, SVD, and other three add different type of noise: Gauss, Uniform, and Laplace. For each of the compression based channels, we increase the compression rate systematically from 0 to about $90 \%$ (in case of the CD channel, the compression rate is computed based on how many bits are used per value). For the noise based channels, we increase the strength of the noise by controlling the epsilon parameter $\epsilon$ (in case of the Gaussian noise, it corresponds to the sigma parameter $\sigma$ ). The full result is presented in Figure 5.
+
+![](images/56ebad6a48346aeddaf478f5dee4dc0a60096e2f497767194e97ed6e55a05a39.jpg)  
+Figure 6: The test accuracy after four different attacks and the recovery via the uniform noise stochastic channel with different parameters . For the Carlini-Wagner L2 attack, the best performing parameter $\epsilon = 0 . 0 3$ . We run the experiment for ResNet-50 on ImageNet (1000 samples).
+
+![](images/1b55dd579fa6888d04682ee25057b8d956b49e0de0b2ec0b9c43f1df2d73b674.jpg)  
+Figure 7: We plot the channel distortion against accuracy $( \% )$ , analogously to Figure 1. The experiment is run for the PGD attack with 40 iterations and C&W $L _ { 2 }$ attack with 100 iterations on all images from the test CIFAR-10 and the dev ImageNet datasets. The adversary is not aware of the defense. The test accuracy on the full clean data is $9 3 . 5 6 \%$ and $7 6 . 1 3 \%$ for CIFAR-10 and ImageNet, respectively. Interestingly enough, we observe that the C&W attack is on the sub-pixel level for ImageNet and rounding to the nearest 8 bit integers in the CD channel has high recovery rate, while for 7 bit integer the accuracy drops slightly due to imprecision.
+
+Figure 6 presents the test accuracy $( \% )$ after four different attacks (3 different norms) and the recovery via the uniform noise stochastic channel in more detail.
+
+# C.2 CHANNEL DISTORTION VS ACCURACY
+
+We show channel distortion vs accuracy for the C&W $L _ { 2 }$ attack on the full CIFAR-10 test set and full ImageNet dev set in Figure 7.
+
+We further show the performance of the defenses across different attacks and 1000 images in Table 3.
+
+Table 3: Given an adversarial input, we pass the input through a channel before prediction. We evaluate the accuracy $( \% )$ of the classifier over 1000 images, the best possible accuracy is listed in the Baseline column. For the channels, we report in parentheses: compression used $( \% )$ , number of bits per value, and the strength of the attack (). We use the attacks described in Section: 4.1   
+
+<table><tr><td>Attack</td><td>Data set</td><td>Baseline</td><td>FC (%)</td><td>CD (bits)</td><td>Uniform (ε)</td><td>Gauss (e)</td><td>Iden</td></tr><tr><td>BIM L1</td><td>CIFAR10</td><td>93.5</td><td>86.2 (20)</td><td>85.1(4)</td><td>82.8 (0.03)</td><td>82.1 (0.03)</td><td>0</td></tr><tr><td>LBFGS</td><td>CIFAR10</td><td>93.5</td><td>82.8 (50)</td><td>80.2 (4)</td><td>79.2 (0.04)</td><td>79.3 (0.05)</td><td>0</td></tr><tr><td>C&amp;WL2</td><td>CIFAR10</td><td>93.5</td><td>85.2 (20)</td><td>84.4 (4)</td><td>84.3 (0.01)</td><td>84.8 (0.02)</td><td>0</td></tr><tr><td>FGSM</td><td>CIFAR10</td><td>93.5</td><td>79.0 (50)</td><td>49.2 (4)</td><td>49.4 (0.03)</td><td>49.9 (0.03)</td><td>0</td></tr><tr><td>PGDL</td><td>CIFAR10</td><td>93.5</td><td>88.6 (10)</td><td>84.3 (5)</td><td>85.7 (0.01)</td><td>84.9 (0.02)</td><td>0</td></tr><tr><td>BIM L1</td><td>ImageNet</td><td>83.5</td><td>81.5 (10)</td><td>82.0 (4)</td><td>81.2 (0.009)</td><td>81.2 (0.009)</td><td>0</td></tr><tr><td>LBFGS</td><td>ImageNet</td><td>83.5</td><td>71.7 (70)</td><td>77.6 (4)</td><td>76.4 (0.07)</td><td>76.5 (0.07)</td><td>0</td></tr><tr><td>C&amp;WL2</td><td>ImageNet</td><td>83.5</td><td>78.7 (50)</td><td>80.9 (4)</td><td>81.4 (0.03)</td><td>80.4 (0.03)</td><td>0</td></tr><tr><td>FGSM</td><td>ImageNet</td><td>83.5</td><td>73.8 (50)</td><td>76.0 (4)</td><td>75.4 (0.03)</td><td>75.4 (0.02)</td><td>0</td></tr><tr><td>PGD Lo</td><td>ImageNet</td><td>83.5</td><td>82.1 (5)</td><td>82.9 (4)</td><td>82.0 (0.007)</td><td>80.9 (0.01)</td><td>0</td></tr></table>
+
+# C.3 CHANNEL ACCURACY ON CLEAN AND ADVERSARIAL EXAMPLES
+
+We compare the accuracy of the perturbation channels on clean and adversarial examples in Figure 8. We plot the results for the whole test set from CIFAR-10 and the whole validation set from ImageNet. We use two deterministic channels (FFT compression denoted by FC and SVD compression) and two noisy channels (Gaussian and Uniform noise). We use C&W and PGD attacks. The FC Clean label denotes that we pass clean images through the channel that applies FFT compression. The SVD C&W label denotes that we pass adversarial images found with the C&W attack through the channel that applies SVD compression. We tune the channels for a given dataset across the attacks, and present the results in Table 4. The accuracy of the channels on clean data gives us the upper bound for the accuracy on the adversarial examples. Thus, we choose the channel parameter based on the highest accuracy on the adversarial images.
+
+Table 4: Channel tuning. The best parameters for the perturbation channels when tuned on the PGD and C&W attacks.   
+
+<table><tr><td></td><td>Dataset</td><td>CIFAR-10</td><td>ImageNet</td></tr><tr><td>Channel</td><td></td><td></td><td></td></tr><tr><td>FC (%)</td><td></td><td>20</td><td>60</td></tr><tr><td>SVD (%)</td><td></td><td>40</td><td>70</td></tr><tr><td>Gauss (∈)</td><td></td><td>0.015 0.025</td><td>0.04 0.04</td></tr><tr><td>Uniform (ε)</td><td></td><td></td><td></td></tr></table>
+
+# C.4 NEIGHBORHOOD OF ADVERSARIAL EXAMPLES
+
+Please, see Figure 9 and description in Section 4.1.
+
+C.5 DISTRIBUTIONS OF DELTAS BETWEEN INPUTS & OUTPUTS FOR CHANNELS
+
+We plot the distribution of deltas for six imprecise channels in Figure 10. We compute the deltas by subtracting an original image from the perturbed adversarial image and plot the histograms of differences. We use an image from the ImageNet dataset. For all the examples, the correct labels were recovered. We use the C&W attack with 1000 iterations and the initial value $c = 0 . 0 1$ .
+
+The CD channel resembles the Uniform distribution. The FFT and SVD compression methods belong to double-sided exponential distributions, thus they are more related to the Laplace distribution than to the Gaussian distribution.
+
+![](images/5c2c3b2a541977774333345321a0fdeeba231ce2f302e5c3d101e42f446ad321.jpg)  
+Figure 8: Comparison of accuracy of the perturbation channels on clean and adversarial images. We pass either clean or adversarial images through the channels and measure their accuracy. The accuracy on clean inputs gives us an upper bound for the accuracy on the adversarial examples. The channels can be tuned based on their accuracy after different attacks.
+
+![](images/04119d4bca258e69e3c5e66a9200613333550417ec9e0400ffcc867532fa9fa4.jpg)  
+CarliniWagnerL2Attack (adv. dist.: 0.006)
+
+![](images/2a8805ccb7c9f49a44e3d2f8b40a5698decc1baaafcaadcadf3402b4a9f5af59.jpg)  
+CarliniWagnerL2Attack (adv. dist.: 0.033)
+
+![](images/0fe15dd459f608d665440ea18c5c31dbc186e61ba8f8a571943577d96e77d0e3.jpg)  
+CarliniWagnerL2Attack (adv. dist.: 0.297)   
+Figure 9: Frequency of model predictions for original, adversarial, and other classes as we increase the distance from an adversarial example using Gaussian noise.
+
+![](images/6033857c901b8c493d94d29a3d0f7ddaa3c0d1e8c65ae24092a3cd613d5f2736.jpg)  
+CarliniWagnerL2Attack (adv. dist.: 0.016)
+
+![](images/4af1f73746aa28a4291cf60bdb845112225b3725e39e4b819fe6b2123e6c78e7.jpg)  
+CarliniWagnerL2Attack (adv. dist.: 0.104)
+
+![](images/7f1b8ae323bf704e8a26c1f519be5626d63ab913910c4af791bd49448daad32d.jpg)  
+CarliniWagnerL2Attack (adv. dist.: 1.590)
+
+![](images/c8d979c65725110143a0ce79a9769f9f261bc7c53f8675f90496ae039750e1be.jpg)  
+Figure 10: Distribution of deltas for imprecise channels.
+
+# C.6 VISUALIZATIONS OF ATTACKS AND IMPRECISE CHANNELS
+
+Figure 11 presents a sample image from ImageNet for the Carlini-Wagner L2 attack.
+
+Figure 12 shows the effect of the FC (frequency-based imprecise channel) in both spatial and Fourier domains.
+
+# C.7 MULTIPLE TRIALS FOR STOCHASTIC CHANNELS
+
+In addition to the robustness to non-adaptive attacks, another compelling reason to use a stochastic channel (like Uniform noise) as a defense is that it can be run repeatedly in a number of random trials. We show that doing so improves the efficacy of the defense. We randomly choose 1000 images from the CIFAR-10 test set. For each of these images, we generate an adversarial attack. We then pass each image through the same stochastic channel multiple times. We take the most frequent prediction. Figure 13 illustrates the results.
+
+Only 16 trials are needed to get a relatively strong defense. We argue that this result is significant. Randomized defenses are difficult to attack. The attacker cannot anticipate which particular perturbation to the model will happen. The downside is a potential of erratic predictions. We show that a relatively small number of trials can greatly reduce this noise. Furthermore, the expense of running multiple trials of a randomized defense is small relative to the expense of synthesizing an attack in the first place.
+
+# C.8 WHITE-BOX ADAPTIVE ATTACK
+
+We consider the problem setting when the adversary knows the defense method (i.e., has full knowledge of $C ( x )$ ). We use the strategy described in He et al. (2017) to construct attacks for each case. Not surprisingly, deterministic channels (CD, FC, and SVD) are easy for an adversary to fool when they are known. However, such attacks incur higher distortion (distance to the original image)
+
+when compared to attacks against unprotected networks. Intuitively, when the channel is deterministic and known, the adversary can account for the error introduced by the channel. Table 5 illustrates the results.
+
+Table 5: The distortion and accuracy for the adaptive setting where the adversary knows the defense method. We report top-1 class, use 100 image samples, run 1000 iterations of $\mathrm { C } \& \mathbf { W }$ attack, $\epsilon = 0 . 0 4$ , apply a single random noise injection.   
+
+<table><tr><td>CIFAR-10</td><td>L2 Distortion</td><td>Acc. (%)</td></tr><tr><td>Iden</td><td>0.17</td><td>0</td></tr><tr><td>CD (b=5)</td><td>0.5</td><td>0</td></tr><tr><td>FC (c=30)</td><td>0.74</td><td>0</td></tr></table>
+
+<table><tr><td>ImageNet</td><td>L2 Distortion</td><td>Acc. (%)</td></tr><tr><td>Iden</td><td>0.28</td><td>0</td></tr><tr><td>CD (b=5)</td><td>3.6</td><td>0</td></tr><tr><td>FC (c=30)</td><td>3.64</td><td>0</td></tr></table>
+
+The stochastic channel is harder to attack with a gradient-based adaptive methods. Unlike for deterministic compression, the adversary cannot anticipate the particular error pattern. We build an adaptive attack against the additive uniform noise channel. Our strategy is to send an output from the adversarial algorithm through the stochastic channel and the network at least as many times as set in the defense. We mark the attack as successful if the most frequent output label is different from the ground truth. The more passes through the noisy channel we optimize for, the stronger the attack. Furthermore, we run many iterations of the attack to decrease the $L _ { 2 }$ distortion. An attack that always evades the noise injection defense is more difficult to generate because of randomization. The other randomized approach was introduced in dropout (Feinman et al., 2017). The attacks against randomized defenses require optimization of complex loss functions, incur higher distortion, and the attacks are not fully successful (Carlini & Wagner, 2017).
+
+In the Figure 14, we present the result of running attacks and defenses on CIFAR-10 data with single and many iterations. The defense with many trials can be drawn to $0 \%$ accuracy, however, the defense not fully optimized by the adversary (single noise injection) can result in about $40 \%$ or higher accuracy.
+
+# C.9 HYBRID APPROACHES
+
+A natural thought is whether these defenses can be made more effective by combining them. This is an approach that has been applied, for example, in random discretization (Zhang & Liang, 2019). Our experiments contrast with the previous work and suggest that there is little benefit to hybrid approaches. We present an illustrative example for pair-wise combinations of Frequency Compression (FC), Color Depth (CD) compression, and Uniform (Unif) noise. Other combinations are possible but for brevity, we exclude them from the manuscript; we found no to very small benefit to combining for properly tuned channels. We show the recovery rate, i.e. a fraction of original labels recovered. Table C.9 presents results for the L2 Carlini-Wagner attack on the MNIST, the whole test set of CIFAR-10, and the whole development set of ImageNet.
+
+<table><tr><td></td><td>CD (bits)</td><td>FC (%)</td><td>Unif ()</td><td>CD+FC</td><td>FC+CD</td><td>CD+Unif</td><td>FC+Unif</td></tr><tr><td>MNIST</td><td>100</td><td>95</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td></tr><tr><td>CIFAR-10</td><td>84.4(4)</td><td>85.2(20)</td><td>83.4(0.03)</td><td>86.0</td><td>83.6</td><td>85.18</td><td>86.45</td></tr><tr><td>ImageNet</td><td>80.9(4)</td><td>78.7(50)</td><td>80.3(0.03)</td><td>78.6</td><td>77.5</td><td>73.15</td><td>76.13</td></tr></table>
+
+We believe these results suggest that the noisy channels do not exploit anything inherent to the images. Simply the addition of noise (through reconstruction error) is the mechanism for robustness. Composing two different schemes usually increases this noise.
+
+# C.10 TRANSFERABILITY OF THE ADVERSARIAL IMAGES
+
+We present more results on the transferability of adversarial examples between different channels in Table 1 and Table 7).
+
+Table 6: Transferability of the adversarial images that extend results from Table 1 for ImageNet dataset. We use $30 \%$ FC compression, $50 \%$ SVD compression, 4 bit values in CD, 0.03 noise level for Laplace, and 0.04 noise level for the Gauss and Uniform channels. We use 3000 images from the ImageNet-10 validation set and 100 attack iterations.   
+
+<table><tr><td>D A</td><td>FC</td><td>CD</td><td>SVD</td><td>Gauss</td><td>Uniform</td><td>Laplace</td></tr><tr><td>FC</td><td>0.10</td><td>75.50</td><td>75.83</td><td>77.04</td><td>77.49</td><td>76.29</td></tr><tr><td>CD</td><td>0.17</td><td>1.16</td><td>6.77</td><td>62.60</td><td>62.04</td><td>65.46</td></tr><tr><td>SVD</td><td>12.02</td><td>72.33</td><td>0.46</td><td>72.79</td><td>72.52</td><td>73.09</td></tr><tr><td>Gauss</td><td>0.57</td><td>26.67</td><td>6.67</td><td>58.68</td><td>58.62</td><td>64.95</td></tr><tr><td>Uniform</td><td>0.50</td><td>26.71</td><td>6.99</td><td>58.48</td><td>59.06</td><td>64.59</td></tr><tr><td>Laplace</td><td>0.33</td><td>18.59</td><td>4.16</td><td>29.76</td><td>29.84</td><td>50.00</td></tr></table>
+
+Table 7: Transferability of the adversarial images. The results are presented similarly to Figure 1 but for different parameters. We use $50 \%$ FC compression, $50 \%$ SVD compression, 4 bit values in CD, 0.03 noise level for Laplace, and 0.04 noise level for the Gauss and Uniform channels. We use 3000 images from the CIFAR-10 validation set and 1000 attack iterations.   
+
+<table><tr><td>D A</td><td>FC</td><td>CD</td><td>SVD</td><td>Gauss</td><td>Uniform</td><td>Laplace</td></tr><tr><td>FC</td><td>0.19</td><td>79.00</td><td>83.73</td><td>79.19</td><td>79.38</td><td>76.70</td></tr><tr><td>CD</td><td>6.74</td><td>0.93</td><td>47.57</td><td>62.98</td><td>63.04</td><td>65.22</td></tr><tr><td>SVD</td><td>78.89</td><td>47.04</td><td>0.50</td><td>59.85</td><td>63.42</td><td>67.94</td></tr><tr><td>Gauss</td><td>4.77</td><td>38.52</td><td>36.95</td><td>51.36</td><td>51.27</td><td>52.61</td></tr><tr><td>Uniform</td><td>4.65</td><td>38.14</td><td>36.35</td><td>50.14</td><td>51.08</td><td>53.21</td></tr><tr><td>Laplace</td><td>46.75</td><td>22.22</td><td>22.68</td><td>34.03</td><td>33.93</td><td>46.39</td></tr></table>
+
+# C.11 GRADIENT-BASED ANALYSIS
+
+We run the experiment on the ImageNet dataset. We analyze only the clean images that were classified correctly.
+
+We present how the gradient of the loss w.r.t. the input image changes for the correct class as we add the Gaussian noise to the original image in Figure 15. The norm of the gradient smoothly increases.
+
+In Figure 16, we start from an adversarial image found with the default C&W attack from the foolbox library. Then, we systematically add Gaussian noise to the adversarial image and collect data on the norm of gradients for the original and adversarial classes. The norm of the gradients for the adversarial class increases while the norm of the gradients for the original class decreases. We cross the decision boundary to the correct class very early and recover the correct labels for images. Then, as we add more and more Gaussian noise, the predictions of the classifier become random and the norm of the gradients converge to a single value.
+
+In Figure 17, we plot the gradients also for a random class. We observe that for an untargeted attack, the gradients for the original and adversarial classes are larger than for the other classes. The targeted attack decreases the loss for the target class and the gradients for the adversarial classes are lower when compared with gradients from the untargeted attacks, so fewer images can be recovered in the former case. The targeted attack causes a smaller increase of the norms of gradients for the original class than the untargted attack. However, it is still higher than for a random class.
+
+# C.12 HESSIAN-BASED ANALYSIS
+
+We present the Hessian spectrum in Figure 18 for top 20 eigenvalues and in Figure 19 the distribution of top eigenvalues of Hessians on ImageNet. For the adversarial images, the eigenvalues are clearly higher in both cases, which indicates higher instability and proclivity to prediction changes.
+
+# C.13 DETAILS ON THE EXPERIMENTAL SETUP
+
+We use the foolbox library (Rauber et al., 2017) in our experiments. We borrowed the nomenclature used for the attacks from the library. For example, the name for the attack initially proposed in Szegedy et al. (2014) and extended in Tabacof & Valle (2015) is LBFGS. In most of our experiments, we also use the default foolbox parameters for the attacks. For example, for PGD the initial limit on the perturbation size epsilon is set to 0.3, step size to 0.01, default number of iterations is 40. For Carlini & Wagner, we set maximum number of iterations to 1000, learning rate to 0.005, initial value of the constant $c$ to 0.01. Note that for the Carlini & Wagner attack presented in 2, we use the $c$ parameter as described in (Carlini & Wagner, 2017) and the code from Liu et al. (2018). For the LBFGS attack, we use the epsilon parameter set to 0.00001 and up to 150 iterations. For the FGSM attack, different epsilons starting from 50 and up to 1000 are tried until the adversarial image is found. For the BIM $L _ { 1 }$ attack, we set epsilon to 0.3, step size to 0.05, and number of iterations to 10.
+
+# D ADDITIONAL EXPERIMENTS FOR BLACK-BOX ATTACKS
+
+As a black box attack, we define an attack that does not need the knowledge about the gradient or the model.
+
+# D.1 DECISION-BASED ATTACKS
+
+The attacks require neither gradients nor probabilities. They operate directly on the images.
+
+D.1.1 ROBUSTNESS TO UNIFORM AND GAUSSIAN NOISE
+
+We evaluate the robustness of band-limited CNNs. Specifically, models trained with more compression discard part of the noise by removing the high frequency Fourier coefficients (FC channel). In Figure 20, we show the test accuracy for input images perturbed with different levels of uniform and Gaussian noise, which is controlled systematically by the sigma parameter, fed into models trained with different compression levels (i.e., $0 \%$ , $50 \%$ , or $8 5 \%$ ) and methods (i.e., band-limited vs. RPA-based3). Our results demonstrate that models trained with higher compression are more robust to the inserted noise. Interestingly, band-limited CNNs also outperform the RPA-based method and under-fitted models (e.g., via early stopping), which do not exhibit the robustness to noise.
+
+Input test images are perturbed with uniform or Gaussian noise, where the sigma parameter is changed from 0 to 1 or 0 to 2, respectively. The more band-limited model, the more robust it is to the introduced noise.
+
+# D.1.2 CONTRAST REDUCTION ATTACK
+
+This black-box attack gradually distorts all the pixels:
+
+$$
+\begin{array} { c } { { \mathrm { t a r g e t } = \displaystyle \frac { { \operatorname* { m a x } } + { \operatorname* { m i n } } } { 2 } } } \\ { { \mathrm { p e r t u r b e d } = ( 1 - \epsilon ) * \displaystyle \operatorname* { i m a g e } + \epsilon * \mathrm { t a r g e t } } } \end{array}
+$$
+
+where min and max values are computed across all pixels of images in the dataset.
+
+We can defend the attack with CD (Color Depth reduction) until certain value of epsilon, but then every pixel is perturbed in a smooth way so there are no high-frequency coefficients increased in the FFT domain of the image. The contrast reduction attack becomes a low-frequency based attack when considered in the frequency domain. Another way to defend the attack is to run a high-pass filter in the frequency domain instead of the low-pass filter.
+
+We run the experiments for different models with CD and two band-limited models (the model with full spectra and no compression as well as model with $85 \%$ of compression - with FC layers). The CD does defend the attack to some extent and the fewer pixels per channel (the stronger the CD in a model), the more robust the model is against the contrast reduction attack.
+
+Test accuracy as a function of the contrast reduction attack for ResNet-18 on CIFAR-10 (after 350 epochs) is plotted in Figure 20. We control the strength of the attack with parameter epsilon that is changed systematically from 0.0 to 1.0. We use the whole test set for CIFAR-10. R denotes the number of values used per channel (e.g., $\scriptstyle \mathrm { R } = 3 2$ means that we use 32 values instead of standard 256).
+
+# D.1.3 MULTIPLE PIXELS ATTACK
+
+The foolbox library supports a single pixel attack, where a given pixel is set to white or black4. A certain number of pixels (e.g., 1000) is chosen and each of them is checked separately if it can lead to the misclassification of the image. The natural extension is to increase the number of pixels to be perturbed, in case where the single pixel attack does not succeed. We present results for the multiple pixel attack in Figure 20.
+
+# D.2 SPATIAL-BASED ATTACKS
+
+Spatial attacks apply adversarial rotations and translations that can be easily added to the data augmentation during training. However, these attacks are defended neither by removing the high frequency coefficients nor by quantization (CD) . We separately apply rotation by changing its angle from 0 to 20 degrees and do the translations within a horizontal and vertical limit of shifted pixels (Figure 20).
+
+# D.3 SCORE-BASED ATTACKS
+
+The score based attack require access to the model predictions and its probabilities (the inputs to the softmax) or the logits to estimate the gradients.
+
+# D.3.1 LOCAL SEARCH ATTACK
+
+The local search attack estimates the sensitivity of individual pixels by applying extreme perturbations and observing the effect on the probability of the correct class. Next, it perturbs the pixels to which the model is most sensitive. The procedure is repeated until the image is misclassified, searching for additional critical pixels in the neighborhood of previously found ones. We run the experiments for the attack on 100 test images from CIFAR-10, since the attack is relatively slow (Figure 20).
+
+Original label: Loafer confidence: 0.9561 L2 distance: 0.0
+
+Adv. label: pencil box, pencil case confidence: 0.5466 L2 distance: 0.4032
+
+![](images/d34db35d404a5323bbb4cd8216bc3db0f5370b69ddb0620ba11df3281a00b86d.jpg)
+
+![](images/6d404921c9efb892ea18a93639ebe6b7a37bea845e71cd1eae587371bc5d8ac5.jpg)
+
+CD (32) label: Loafer confidence: 0.7438 L2 distance: 7.2698
+
+FC (30) label: Loafer confidence: 0.8995 L2 distance: 12.6384
+
+![](images/1f26494dce7b28a8394280f97e9703c72a955a9bf19e1a35c36727ea1399d1a4.jpg)
+
+![](images/820be13f928019415e2054d6210a10143fb22156b1fab923f2b61293d2a20930.jpg)
+
+Gauss (0.03) label: Loafer confidence: 0.9642 L2 distance: 16.0992
+
+Uniform (0.03) label: Loafer confidence: 0.966 L2 distance: 16.088
+
+![](images/9ebb03963700d20de1bab1e07cc00d9cd65d750d0302575adc1e56156582eba3.jpg)
+
+![](images/b636788c23e7efa7396829c0eba7bc76bc52e0f704ec2b0d898f0d619c61cd75.jpg)  
+Figure 11: We plot a sample image from the ImageNet dataset in its original state, after adversarial (white-box, non-adaptive) attack, and then after recovery via imprecise channels: CD (color depth reduction with 32 bits), FC $30 \%$ compression in the frequency domain), Gaussian, and uniform noise $\epsilon = 0 . 0 3$ ). The order is from left to right, and from top to bottom.
+
+![](images/9e8c8e25900037f4fbe0053fdb576c5640199c1e357b17b6500e9ffac9213a06.jpg)
+
+![](images/13674108198ea547075801d175465732098a8f7bfbbcb5351fef691a6d80ac6a.jpg)  
+Figure 12: The original image from the ImageNet dataset, its adversarial example, and the state of the image after recovery from the attack via the $30 \%$ compressed Fourier Channel (FC). The heat maps of magnitudes of Fourier coefficients are presented in a logarithmic scale (dB) with linear interpolation and the max value is colored with white while the min value is colored with black. The black part of the (bottom-right) image represents the removed high-frequency coefficients. The Fourier-ed representation is plotted for a single (0-th) channel. The lowest frequency coefficients are placed in the corners of the FFT maps (with the DC component in the top-left corner).   
+Figure 13: For the CIFAR-10 dataset, we run multiple trials of the uniform noise channel and take the most frequent prediction. We further test multiple noise levels. The multiple trials improve overall accuracy for different noise levels significantly. After 128 trials for the best setting we are within of the overall model accuracy (of about $9 3 . 5 \%$ ).   
+Figure 14: For the CIFAR-10 dataset, we run multiple trials of the uniform noise channel and take the most frequent prediction in the defense (many noise iterations). We also run just a single noise injection and return the predicated label. The attacker runs the same number of many uniform trials as the defender. The experiment is run on 100 images, with 100 C&W $L _ { 2 }$ attack iterations.
+
+![](images/944b48bfc25b61660e74b7bb9e6768ab2486655aa3a4c929434343b7718b4bd2.jpg)  
+Figure 15: The changes in the $L _ { 2 }$ norm of the gradient of the loss w.r.t. the input image $x$ for the correct class $c _ { o r g }$ as we add Gaussian noise to the original image. The experiment is run on 1000 images from the ImageNet dataset.
+
+![](images/6dcd19bf0024ced3e98b5925735c001e499efda32e76e3504257e8f9d6bd0d42.jpg)  
+Figure 16: The changes in the $L _ { 2 }$ norm of the gradient for the correct class $c _ { o r g }$ and the adversarial class $c _ { a d v }$ as we add Gaussian noise to the adversarial image generated with C&W $L _ { 2 }$ attack (we label such an image as a gauss image $x$ ). The experiment is run on 1000 images from the CIFAR-10 dataset.
+
+![](images/26cb102d4af86d61263b618b9fb46876c8bc0df7b2f5027b78058bd20ce64ece.jpg)  
+Figure 17: The changes in the $L _ { 2 }$ norm of the gradient of the loss for the correct class $c _ { o r g }$ , the adversarial class $c _ { a d v }$ , and a random class $c _ { r a n }$ as we add Gaussian noise to the adversarial image generated with C&W $L _ { 2 }$ attack (we label such an image as a gauss image $x$ ). The experiment is run on 1000 images from the CIFAR-10 dataset.
+
+![](images/d8519241cf549224ca88ab7cd46d7f91e70a2671c63c2bd0a7dc0c00c4f6ccb6.jpg)  
+Figure 18: The spectrum of the Hessian with respect to the original and adversarial inputs. We use the symmetric log scale on the y axis and plot the min and max ranges of the eigenvalues for 100 images from the ImageNet dataset trained the ResNet-50 architecture.
+
+![](images/f58df95b50c9cf723f1880fb58ff82ee4cc1bf2af276de3b17f10247a0c1a57f.jpg)  
+Figure 19: The top eigenvalues of the Hessians on ImageNet using ResNet-50.
+
+![](images/1525e7a55e6f102257dc513f15a8a1b8c8d4d1266f3a45974976d6f7e7a40780.jpg)  
+Figure 20: Test accuracy as a function of the strenghts of the attacks for ResNet-18 on CIFAR-10.