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+# Learning Generative Models with Visual Attention
+
+Yichuan Tang, Nitish Srivastava, Ruslan Salakhutdinov
+
+Department of Computer Science University of Toronto Toronto, Ontario, Canada {tang,nitish,rsalakhu}@cs.toronto.edu
+
+# Abstract
+
+Attention has long been proposed by psychologists to be important for efficiently dealing with the massive amounts of sensory stimulus in the neocortex. Inspired by the attention models in visual neuroscience and the need for object-centered data for generative models, we propose a deep-learning based generative framework using attention. The attentional mechanism propagates signals from the region of interest in a scene to an aligned canonical representation for generative modeling. By ignoring scene background clutter, the generative model can concentrate its resources on the object of interest. A convolutional neural net is employed to provide good initializations during posterior inference which uses Hamiltonian Monte Carlo. Upon learning images of faces, our model can robustly attend to the face region of novel test subjects. More importantly, our model can learn generative models of new faces from a novel dataset of large images where the face locations are not known.1
+
+# 1 Introduction
+
+Building rich generative models that are capable of extracting useful, high-level latent representations from high-dimensional sensory input lies at the core of solving many AI-related tasks, including object recognition, speech perception and language understanding. These models capture underlying structure in data by defining flexible probability distributions over high-dimensional data as part of a complex, partially observed system. Some of the successful generative models that are able to discover meaningful high-level latent representations include the Boltzmann Machine family of models: Restricted Boltzmann Machines, Deep Belief Nets [1], and Deep Boltzmann Machines [2]. Mixture models, such as Mixtures of Factor Analyzers [3] and Mixtures of Gaussians, have also been used for modeling natural image patches [4]. More recently, denoising auto-encoders have been proposed as a way to model the transition operator that has the same invariant distribution as the data generating distribution [5].
+
+Generative models have an advantage over discriminative models when part of the images are occluded or missing. Occlusions are very common in realistic settings and have been largely ignored in recent literature on deep learning. In addition, prior knowledge can be easily incorporated in generative models in the forms of structured latent variables, such as lighting and deformable parts. However, the enormous amount of content in high-resolution images makes generative learning difficult [6, 7]. Therefore, generative models have found most success in learning to model small patches of natural images and objects: Zoran and Weiss [4] learned a mixture of Gaussians model over $8 { \times } 8$ image patches; Salakhutdinov and Hinton [2] used $6 4 { \times } 6 4$ centered and uncluttered stereo images of toy objects on a clear background; Tang et al. [8] used $2 4 \times 2 4$ images of centered and cropped faces. The fact that these models require curated training data limits their applicability on using the (virtually) unlimited unlabeled data.
+
+In this paper, we propose a framework to infer the region of interest in a big image for generative modeling. This will allow us to learn a generative model of faces on a very large dataset of (unlabeled) images containing faces. Our framework is able to dynamically route the relevant information to the generative model and can ignore the background clutter. The need to dynamically and selectively route information is also present in the biological brain. Plethora of evidence points to the presence of attention in the visual cortex [9, 10]. Recently, in visual neuroscience, attention has been shown to exist not only in extrastriate areas, but also all the way down to V1 [11].
+
+Attention as a form of routing was originally proposed by Anderson and Van Essen [12] and then extended by Olshausen et al. [13]. Dynamic routing has been hypothesized as providing a way for achieving shift and size invariance in the visual cortex [14, 15]. Tsotsos et al. [16] proposed a model combining search and attention called the Selective Tuning model. Larochelle and Hinton [17] proposed a way of using third-order Boltzmann Machines to combine information gathered from many foveal glimpses. Their model chooses where to look next to find locations that are most informative of the object class. Reichert et al. [18] proposed a hierarchical model to show that certain aspects of covert object-based attention can be modeled by Deep Boltzmann Machines. Several other related models attempt to learn where to look for objects [19, 20] and for video based tracking [21]. Inspired by Olshausen et al. [13], we use 2D similarity transformations to implement the scaling, rotation, and shift operation required for routing. Our main motivation is to enable the learning of generative models in big images where the location of the object of interest is unknown a-priori.
+
+# 2 Gaussian Restricted Boltzmann Machines
+
+Before we describe our model, we briefly review the Gaussian Restricted Boltzmann Machine (GRBM) [22], as it will serve as the building block for our attention-based model. GRBMs are a type of Markov Random Field model that has a bipartite structure with real-valued visible variables $\mathbf { v } \in \mathbb { R } ^ { D }$ connected to binary stochastic hidden variables $\mathbf { h } \in \{ 0 , 1 \} ^ { H }$ . The energy of the joint configuration $\{ \mathbf { v } , \mathbf { h } \}$ of the Gaussian RBM is defined as follows:
+
+$$
+\begin{array} { r c l } { { \displaystyle E _ { G R B M } ( { \bf v } , { \bf h } ; \Theta ) } } & { { = } } & { { \displaystyle \frac { 1 } { 2 } \sum _ { i } \frac { ( v _ { i } - b _ { i } ) ^ { 2 } } { \sigma _ { i } ^ { 2 } } - \sum _ { j } c _ { j } h _ { j } - \sum _ { i j } W _ { i j } v _ { i } h _ { j } , } } \end{array}
+$$
+
+where $\boldsymbol { \Theta } = \{ \mathbf { W } , \mathbf { b } , \mathbf { c } , \pmb { \sigma } \}$ are the model parameters. The marginal distribution over the visible vector $\mathbf { v }$ is $\begin{array} { r } { P ( \mathbf { v } ; \Theta ) = \frac { 1 } { \mathcal { Z } ( \Theta ) } \sum _ { \mathbf { h } } \exp \left( - E ( \mathbf { v } , \mathbf { h } ; \Theta ) \right) } \end{array}$ and the corresponding conditional distributions take the following form:
+
+$$
+\begin{array} { r c l } { p ( h _ { j } = 1 | \mathbf { v } ) } & { = } & { \displaystyle 1 / \big ( 1 + \exp ( - \sum _ { i } W _ { i j } v _ { i } - c _ { j } ) \big ) , } \\ { p ( v _ { i } | \mathbf { h } ) } & { = } & { \displaystyle \mathcal { N } ( v _ { i } ; \mu _ { i } , \sigma _ { i } ^ { 2 } ) , ~ \mathrm { w h e r e } ~ \mu _ { i } = b _ { i } + \sigma _ { i } ^ { 2 } \sum _ { j } W _ { i j } h _ { j } . } \end{array}
+$$
+
+Observe that conditioned on the states of the hidden variables (Eq. 3), each visible unit is modeled by a Gaussian distribution, whose mean is shifted by the weighted combination of the hidden unit activations. Unlike directed models, an RBM’s conditional distribution over hidden nodes is factorial and can be easily computed.
+
+We can also add a binary RBM on top of the learned GRBM by treating the inferred h as the “visible” layer together with a second hidden layer $\mathbf { h } ^ { 2 }$ . This results in a 2-layer Gaussian Deep Belief Network (GDBN) [1] that is a more powerful model of $\mathbf { v }$ .
+
+Specifically, in a GDBN model, $p ( \mathbf { h } ^ { 1 } , \mathbf { h } ^ { 2 } )$ is modeled by the energy function of the 2nd-layer RBM, while $p ( \mathbf { v } ^ { \tilde { 1 } } \vert \mathbf { h } ^ { 1 } )$ is given by Eq. 3. Efficient inference can be performed using the greedy approach of [1] by treating each DBN layer as a separate RBM model. GDBNs have been applied to various tasks, including image classification, video action and speech recognition [6, 23, 24, 25].
+
+# 3 The Model
+
+Let $\mathcal { T }$ be a high resolution image of a scene, e.g. a $2 5 6 \times 2 5 6$ image. We want to use attention to propagate regions of interest from $\mathcal { T }$ up to a canonical representation. For example, in order to learn a model of faces, the canonical representation could be a $2 4 \times 2 4$ aligned and cropped frontal face image. Let $\mathbf { v } \in \mathbb { R } ^ { D }$ represent this low resolution canonical image. In this work, we focus on a Deep Belief Network2 to model $\mathbf { v }$ .
+
+![](images/15c14e094912a2e46700a68681a73b0dc66dbfeb37706dcbae5bd7b4a96b302d.jpg)  
+Figure 1: Left: The Shifter Circuit, a well-known neuroscience model for visual attention [13]; Right: The proposed model uses 2D similarity transformations from geometry and a Gaussian DBN to model canonical face images. Associative memory corresponds to the DBN, object-centered frame correspond to the visible layer and the attentional mechanism is modeled by 2D similarity transformations.
+
+This is illustrated in the diagrams of Fig. 1. The left panel displays the model of Olshausen et.al. [13], whereas the right panel shows a graphical diagram of our proposed generative model with an attentional mechanism. Here, ${ \bf h } ^ { 1 }$ and $\mathbf { h } ^ { 2 }$ represent the latent hidden variables of the DBN model, and $\triangle x , \triangle y , \triangle \theta , \triangle s$ (position, rotation, and scale) are the parameters of the 2D similarity transformation.
+
+The 2D similarity transformation is used to rotate, scale, and translate the canonical image v onto the canvas that we denote by $\mathcal { T }$ . Let $\mathbf { p } = [ x y ] ^ { \mathsf { T } }$ be a pixel coordinate (e.g. $[ 0 , 0 ]$ or $[ 0 , 1 ] \cdot$ ) of the canonical image $\mathbf { v }$ . Let $\{ \mathbf { p } \}$ be the set of all coordinates of $\mathbf { v }$ . For example, if $\mathbf { v }$ is $2 4 \times 2 4$ , then $\{ \mathbf { p } \}$ ranges from $[ 0 , 0 ]$ to [23, 23]. Let the “gaze” variables $\mathbf { u } \in \mathbb { R } ^ { 4 } \equiv [ \widehat { \triangle } x , \triangle y , \triangle \theta , \triangle s ]$ be the parameter of the Similarity transformation. In order to simplify derivations and to make transformations be linear w.r.t. the transformation parameters, we can equivalently redefine $\mathbf { u } = [ a , \ b , \ \triangle x , \ \triangle y ]$ , where $a = s \sin ( \theta ) - 1$ and $b = s \cos ( \theta )$ (see [26] for details). We further define a function $\mathsf { w } : = \mathsf { w } ( \mathbf { p } , \mathbf { u } ) \to \mathbf { p } ^ { \prime }$ as the transformation function to warp points $\mathbf { p }$ to $\mathbf { p } ^ { \prime }$ :
+
+$$
+\mathbf { p } ^ { \prime } \triangleq \left[ \begin{array} { l } { x ^ { \prime } } \\ { y ^ { \prime } } \end{array} \right] = \left[ \begin{array} { c c } { 1 + a } & { - b } \\ { b } & { 1 + a } \end{array} \right] \left[ \begin{array} { l } { x } \\ { y } \end{array} \right] + \left[ \begin{array} { l } { \bigtriangleup x } \\ { \bigtriangleup y } \end{array} \right] .
+$$
+
+We use the notation $\mathcal { T } ( \{ \mathbf { p } \} )$ to denote the bilinear interpolation of $\mathcal { T }$ at coordinates $\{ \mathbf { p } \}$ with antialiasing. Let $\mathbf { x } ( \mathbf { u } )$ be the extracted low-resolution image at warped locations $\mathbf { p } ^ { \prime }$ :
+
+$$
+\mathbf { x } ( \mathbf { u } ) \triangleq { \mathcal { T } } ( \mathbf { w } ( \{ \mathbf { p } \} , \mathbf { u } ) ) .
+$$
+
+Intuitively, $\mathbf { x } ( \mathbf { u } )$ is a patch extracted from $\mathcal { T }$ according to the shift, rotation and scale parameters of $\mathbf { u }$ , as shown in Fig. 1, right panel. It is this patch of data that we seek to model generatively. Note that the dimensionality of $\bar { \bf x } ( { \bf u } )$ is equal to the cardinality of $\{ \mathbf { p } \}$ , where $\{ \mathbf { p } \}$ denotes the set of pixel coordinates of the canonical image $\mathbf { v }$ . Unlike standard generative learning tasks, the data $\mathbf { x } ( \mathbf { u } )$ is not static but changes with the latent variables $\mathbf { u }$ . Given $\mathbf { v }$ and $\mathbf { u }$ , we model the top-down generative process over $^ 3 \mathrm { \bf ~ x }$ with a Gaussian distribution having a diagonal covariance matrix $\sigma ^ { 2 } \boldsymbol { \mathrm { I } }$ :
+
+$$
+p ( \mathbf { x } | \mathbf { v } , \mathbf { u } , \mathcal { T } ) \propto \exp \bigg ( - \frac { 1 } { 2 } \sum _ { i } \frac { ( x _ { i } ( \mathbf { u } ) - v _ { i } ) ^ { 2 } } { \sigma _ { i } ^ { 2 } } \bigg ) .
+$$
+
+The fact that we do not seek to model the rest of the regions/pixels of $\mathcal { T }$ is by design. By using 2D similarity transformation to mimic attention, we can discard the complex background of the scene and let the generative model focus on the object of interest. The proposed generative model takes the following form:
+
+$$
+\begin{array} { r } { p ( \mathbf { x } , \mathbf { v } , \mathbf { u } | \mathcal { T } ) = p ( \mathbf { x } | \mathbf { v } , \mathbf { u } , \mathcal { T } ) p ( \mathbf { v } ) p ( \mathbf { u } ) , } \end{array}
+$$
+
+where for $p ( \mathbf { u } )$ we use a flat prior that is constant for all $\mathbf { u }$ , and $p ( \mathbf { v } )$ is defined by a 2-layer Gaussian Deep Belief Network. The conditional $p ( \mathbf { x } | \mathbf { v } , \mathbf { u } , \mathcal { T } )$ is given by a Gaussian distribution as in Eq. 6. To simplify the inference procedure, $p ( \mathbf { x } | \mathbf { v } , \mathbf { u } , \mathcal { T } )$ and the GDBN model of $\mathbf { v }$ , $p ( \mathbf { v } )$ , will share the same noise parameters $\sigma _ { i }$ .
+
+# 4 Inference
+
+While the generative equations in the last section are straightforward and intuitive, inference in these models is typically intractable due to the complicated energy landscape of the posterior. During inference, we wish to compute the distribution over the gaze variables u and canonical object v given the big image $\mathcal { T }$ . Unlike in standard RBMs and DBNs, there are no simplifying factorial assumptions about the conditional distribution of the latent variable u. Having a 2D similarity transformation is reminiscent of third-order Boltzmann machines with $\mathbf { u }$ performing top-down multiplicative gating of the connections between $\mathbf { v }$ and $\mathcal { T }$ . It is well known that inference in these higher-order models is rather complicated.
+
+One way to perform inference in our model is to resort to Gibbs sampling by computing the set of alternating conditional posteriors: The conditional distribution over the canonical image v takes the following form:
+
+$$
+p ( \mathbf { v } | \mathbf { u } , \mathbf { h } ^ { 1 } , \mathcal { T } ) = \mathcal { N } \Big ( \frac { \mu + \mathbf { x } ( \mathbf { u } ) } { 2 } ; \pmb { \sigma } ^ { 2 } \Big ) ,
+$$
+
+where $\mu _ { i } = b _ { i } + \sigma _ { i } ^ { 2 } \sum _ { j } W _ { i j } h _ { j } ^ { 1 }$ is the top-down influence of the DBN. Note that if we know the gaze variable $\mathbf { u }$ and the first layer of hidden variables ${ \bf h } ^ { 1 }$ , then $\mathbf { v }$ is simply defined by a Gaussian distribution, where the mean is given by the average of the top-down influence and bottom-up information from $\mathbf { x }$ . The conditional distributions over ${ \bf h } ^ { 1 }$ and $\bar { \mathbf { h } ^ { 2 } }$ given $\mathbf { v }$ are given by the standard DBN inference equations [1]. The conditional posterior over the gaze variables $\mathbf { u }$ is given by:
+
+$$
+\begin{array} { c } { { \displaystyle p ( { \bf u } | { \bf x } , { \bf v } ) = \frac { p ( { \bf x } | { \bf u } , { \bf v } ) p ( { \bf u } ) } { p ( { \bf x } | { \bf v } ) } , } } \\ { { \displaystyle \log p ( { \bf u } | { \bf x } , { \bf v } ) \propto \log p ( { \bf x } | { \bf u } , { \bf v } ) + \log p ( { \bf u } ) = \frac { 1 } { 2 } \sum _ { i } \frac { ( x _ { i } ( { \bf u } ) - v _ { i } ) ^ { 2 } } { \sigma _ { i } ^ { 2 } } + c o n s t . } } \end{array}
+$$
+
+Using Bayes’ rule, the unnormalized log probability of $p ( \mathbf { u } | \mathbf { x } , \mathbf { v } )$ is defined in Eq. 9. We stress that this equation is atypical in that the random variable of interest $\mathbf { u }$ actually affects the conditioning variable $\mathbf { x }$ (see Eq. 5) We can explore the gaze variables using Hamiltonian Monte Carlo (HMC) algorithm [27, 28]. Intuitively, conditioned on the canonical object $\mathbf { v }$ that our model has in “mind”, HMC searches over the entire image $\mathcal { T }$ to find a region $\mathbf { x }$ with a good match to $\mathbf { v }$ .
+
+If the goal is only to find the MAP estimate of $p ( \mathbf { u } | \mathbf { x } , \mathbf { v } )$ , then we may want to use second-order methods for optimizing $\mathbf { u }$ . This would be equivalent to the Lucas-Kanade framework in computer vision, developed for image alignment [29]. However, HMC has the advantage of being a proper MCMC sampler that satisfies detailed balance and fits nicely with our probabilistic framework.
+
+The HMC algorithm first specifies the Hamiltonian over the position variables $\mathbf { u }$ and auxiliary momentum variables r: $\mathcal { H } ( \mathbf { u } , \mathbf { r } ) ~ = ~ U ( \mathbf { u } ) + K ( \mathbf { r } )$ , where the potential function is defined by $\begin{array} { r } { U ( \mathbf { u } ) = \frac { 1 } { 2 } \sum _ { i } \frac { ( x _ { i } ( \mathbf { u } ) - v _ { i } ) ^ { 2 } } { \sigma _ { i } ^ { 2 } } } \end{array}$ and the kinetic energy function is given by $\begin{array} { r } { K ( { \bf r } ) = \frac { 1 } { 2 } \sum _ { i } r _ { i } ^ { 2 } } \end{array}$ . The dyinamics of the system is defined by:
+
+$$
+\begin{array} { r l } & { \mathrm { c u t ~ \mathbf { 1 } s ~ u c t u n c u ~ \mathbf { y } . } } \\ & { ~ \displaystyle \frac { \partial \mathbf { u } } { \partial t } = \mathbf { r } , \qquad \frac { \partial \mathbf { r } } { \partial t } = - \frac { \partial \mathbf { \mathcal { H } } } { \partial \mathbf { u } } } \\ & { \displaystyle \frac { \partial \mathcal { H } } { \partial \mathbf { u } } = \frac { ( \mathbf { x } ( \mathbf { u } ) - \mathbf { v } ) } { \sigma ^ { 2 } } \frac { \partial \mathbf { x } ( \mathbf { u } ) } { \partial \mathbf { u } } , } \\ & { \displaystyle \frac { \partial \mathbf { x } } { \partial \mathbf { u } } = \frac { \partial \mathbf { x } } { \partial \mathbf { w } ( \{ \mathbf { p } \} , \mathbf { u } ) } \frac { \partial \mathbf { w } ( \{ \mathbf { p } \} , \mathbf { u } ) } { \partial \mathbf { u } } = \sum _ { i } \frac { \partial x _ { i } } { \partial \mathbf { w } ( \mathbf { p } _ { i } , \mathbf { u } ) } \frac { \partial \mathbf { w } ( \mathbf { p } _ { i } , \mathbf { u } ) } { \partial \mathbf { u } } . } \end{array}
+$$
+
+Observe that Eq. 12 decomposes into sums over single coordinate positions $\mathbf { p } _ { i } = [ x y ] ^ { \mathsf { T } }$ . Let us denote $\mathbf { p ^ { \prime } } _ { i } = \mathsf { w } ( \mathbf { p } _ { i } , \mathbf { u } )$ to be the coordinate $\mathbf { p } _ { i }$ warped by $\mathbf { u }$ . For the first term on the RHS of Eq. 12,
+
+$$
+{ \frac { \partial x _ { i } } { \partial \mathbf { w } ( \mathbf { p } _ { i } , \mathbf { u } ) } } = \nabla I ( \mathbf { p ^ { \prime } } _ { i } ) , \quad ( { \mathrm { d i m e n s i o n ~ } } 1 \ \mathbf { b y } \ 2 \ )
+$$
+
+where $\nabla I ( \mathbf { p } ^ { \prime } _ { i } )$ denotes the sampling of the gradient images of $I$ at the warped location $\mathbf { p } _ { i }$ . For the second term on the RHS of Eq. 12, we note that we can re-write Eq. 4 as:
+
+$$
+{ \left[ \begin{array} { l } { x ^ { \prime } } \\ { y ^ { \prime } } \end{array} \right] } = { \left[ \begin{array} { l l l l } { x } & { - y } & { 1 } & { 0 } \\ { y } & { x } & { 0 } & { 1 } \end{array} \right] } { \left[ \begin{array} { l } { a } \\ { b } \\ { \bigtriangleup x } \\ { \bigtriangleup y } \end{array} \right] } + { \left[ \begin{array} { l } { x } \\ { y } \end{array} \right] } ,
+$$
+
+giving us
+
+$$
+\frac { \partial \mathbf { w } ( \mathbf { p } _ { i } , \mathbf { u } ) } { \partial \mathbf { u } } = \left[ \begin{array} { c c c c } { x } & { - y } & { 1 } & { 0 } \\ { y } & { x } & { 0 } & { 1 } \end{array} \right] .
+$$
+
+HMC simulates the discretized system by performing leap-frog updates of $\mathbf { u }$ and $\mathbf { r }$ using Eq. 10. Additional hyperparameters that need to be specified include the step size $\epsilon$ , number of leap-frog steps, and the mass of the variables (see [28] for details).
+
+# 4.1 Approximate Inference
+
+HMC essentially performs gradient descent with momentum, therefore it is prone to getting stuck at local optimums. This is especially a problem for our task of finding the best transformation parameters. While the posterior over u should be unimodal near the optimum, many local minima exist away from the global optimum. For example, in Fig. 2(a), the big image $\mathcal { T }$ is enclosed by the blue box, and the canonical image $\mathbf { v }$ is enclosed by the green box. The current setting of u aligns together the wrong eyes. However, it is hard to move the green box to the left due to the local optima created by the dark intensities of the eye. Resampling the momentum variable every iteration in HMC does not help significantly because we are modeling real-valued images using a Gaussian distribution as the residual, leading to quadratic costs in the difference between $\mathbf { x } ( \mathbf { u } )$ and v (see Eq. 9). This makes the energy barriers between modes extremely high.
+
+To alleviate this problem we need to find good initializations of u. We use a Convolutional Network (ConvNet) to perform efficient approximate inference, resulting in good initial guesses. Specifically, given v, $\mathbf { u }$ and $\mathcal { T }$ , we predict the change in u that will lead to the maximum $\log p ( { \bf u } | { \bf x } , { \bf v } )$ . In other words, instead of using the gradient field for updating u, we learn a ConvNet to output a better vector field in the space of u. We used a fairly standard ConvNet architecture and the standard stochastic gradient descent learning procedure.
+
+![](images/6ad22dcaefcc12e375c444941ddbe9cc9830b1ea2964d1d1916af8d7ef825e97.jpg)  
+Figure 2: (a) HMC can easily get stuck at local optima. (b) Importance of modeling $p ( \mathbf { u } | \mathbf { v } , \mathcal { T } )$ . Best in color.
+
+We note that standard feedforward face detectors seek to model $p ( \mathbf { u } | \mathcal { T } )$ , while completely ignoring the canonical face v. In contrast, here we take $\mathbf { v }$ into account as well. The ConvNet is used to initialize $\mathbf { u }$ for the HMC algorithm. This is important in a proper generative model because conditioning on $\mathbf { v }$ is appealing when multiple faces are present in the scene. Fig. 2(b) is a hypothesized Euclidean space of $\mathbf { v }$ , where the black manifold represents canonical faces and the blue manifold represents cropped faces $\mathbf { x } ( \mathbf { u } )$ . The blue manifold has a low intrinsic dimensionality of 4, spanned by u. At A and B, the blue comes close to black manifold. This means that there are at least two modes in the posterior over u. By conditioning on $\mathbf { v }$ , we can narrow the posterior to a single mode, depending on whom we want to focus our attention. We demonstrate this exact capability in Sec. 6.3.
+
+Fig. 3 demonstrates the iterative process of how approximate inference works in our model. Specifically, based on u, the ConvNet takes a window patch around $\mathbf { x } ( \mathbf { u } )$ $( 7 2 \times 7 2 )$ and v $( 2 4 \times 2 4 )$ as input, and predicts the output $[ \triangle x , \triangle y , \triangle \theta , \triangle s ]$ . In step 2, $\mathbf { u }$ is updated accordingly, followed by step 3 of alternating Gibbs updates of $\mathbf { v }$ and $\mathbf { h }$ , as discussed in Sec. 4. The process is repeated. For the details of the ConvNet see the supplementary materials.
+
+# 5 Learning
+
+While inference in our framework localizes objects of interest and is akin to object detection, it is not the main objective. Our motivation is not to compete with state-of-the-art object detectors but rather propose a probabilistic generative framework capable of generative modeling of objects which are at unknown locations in big images. This is because labels are expensive to obtain and are often not available for images in an unconstrained environment.
+
+To learn generatively without labels we propose a simple Monte Carlo based ExpectationMaximization algorithm. This algorithm is an unbiased estimator of the maximum likelihood objective. During the E-step, we use the Gibbs sampling algorithm developed in Sec. 4 to draw samples from the posterior over the latent gaze variables u, the canonical variables $\mathbf { v }$ , and the hidden variables ${ \bf h } ^ { 1 }$ , $\mathbf { h } ^ { 2 }$ of a Gaussian DBN model. During the M-step, we can update the weights of the Gaussian DBN by using the posterior samples as its training data. In addition, we can update the parameters of the ConvNet that performs approximate inference. Due to the fact that the first E-step requires a good inference algorithm, we need to pretrain the ConvNet using labeled gaze data as part of a bootstrap process. Obtaining training data for this initial phase is not a problem as we can jitter/rotate/scale to create data. In Sec. 6.2, we demonstrate the ability to learn a good generative model of face images from the CMU Multi-PIE dataset.
+
+![](images/6095db78056547c5d915d4c552d83090d0ac619c5b47f372e7b406624d98f5dd.jpg)  
+Figure 3: Inference process: u in step 1 is randomly initialized. The average v and the extracted $\mathbf { x } ( \mathbf { u } )$ form the input to a ConvNet for approximate inference, giving a new u. The new u is used to sample $p ( \mathbf { v } \vert \mathcal { T } , \mathbf { u } , \mathbf { h } )$ . In step 3, one step of Gibbs sampling of the GDBN is performed. Step 4 repeats the approximate inference using the updated $\mathbf { v }$ and $\mathbf { x } ( \mathbf { u } )$ .
+
+![](images/8fd52db5f77c15bb4c59ff1692945a2c3a68b56579a21097e3ec07f0064a03ee.jpg)  
+Figure 4: Example of an inference step. v is $2 4 \times 2 4$ , $\mathbf { x }$ is $7 2 \times 7 2$ . Approximate inference quickly finds a good initialization for u, while HMC provides further adjustments. Intermediate inference steps on the right are subsampled from 10 actual iterations.
+
+# 6 Experiments
+
+We used two face datasets in our experiments. The first dataset is a frontal face dataset, called the Caltech Faces from 1999, collected by Markus Weber. In this dataset, there are 450 faces of 27 unique individuals under different lighting conditions, expressions, and backgrounds. We downsampled the images from their native 896 by 692 by a factor of 2. The dataset also contains manually labeled eyes and mouth coordinates, which will serve as the gaze labels. We also used the CMU Multi-PIE dataset [30], which contains 337 subjects, captured under 15 viewpoints and 19 illumination conditions in four recording sessions for a total of more than 750,000 images. We demonstrate our model’s ability to perform approximate inference, to learn without labels, and to perform identity-based attention given an image with two people.
+
+# 6.1 Approximate inference
+
+We first investigate the critical inference algorithm of $p ( \mathbf { u } | \mathbf { v } , \mathcal { T } )$ on the Caltech Faces dataset. We run 4 steps of approximate inference detailed in Sec. 4.1 and diagrammed in Fig. 3, followed by three iterations of 20 leap-frog steps of HMC. Since we do not initially know the correct v, we initialize v to be the average face across all subjects.
+
+Fig. 4 shows the image of v and $\mathbf { x }$ during inference for a test subject. The initial gaze box is colored yellow on the left. Subsequent gaze updates progress from yellow to blue. Once ConvNet-based approximate inference gives a good initialization, starting from step 5, five iterations of 20 leap-frog steps of HMC are used to sample from the the posterior.
+
+Fig. 5 shows the quantitative results of Intersection over Union (IOU) of the ground truth face box and the inferred face box. The results show that inference is very robust to initialization and requires only a few steps of approximate inference to converge. HMC clearly improves model performance, resulting in an IOU increase of about $5 \%$ for localization. This is impressive given that none of the test subjects were part of the training and the background is different from backgrounds in the training set.
+
+![](images/fab1f63a903168eedabf95bfa8d89a9fe20840c9d3986d6067f2cb221a82714b.jpg)  
+Figure 5: (a) Accuracy as a function of gaze initialization (pixel offset). Blue curve is the percentage success of at least $50 \%$ IOU. Red curve is the average IOU. (b) Accuracy as a function of the number of approximate inference steps when initializing 50 pixels away. (c) Accuracy improvements of HMC as a function of gaze initializations.
+
+![](images/8c371c74eaa860b9c3b45a5f9bd20153bb10d00a8c26359f0f5e9ed91d001c04.jpg)  
+Figure 6: Left: Samples from a 2-layer DBN trained on Caltech. Right: samples from an updated DBN after training on CMU Multi-PIE without labels. Samples highlighted in green are similar to faces from CMU.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Ourmethod</td><td rowspan=1 colspan=1>OpenCV</td><td rowspan=1 colspan=1>NCC</td><td rowspan=1 colspan=1>template</td></tr><tr><td rowspan=1 colspan=1>IOU&gt; 0.5</td><td rowspan=1 colspan=1>97%</td><td rowspan=1 colspan=1>97%</td><td rowspan=1 colspan=1>93%</td><td rowspan=1 colspan=1>78%</td></tr><tr><td rowspan=1 colspan=1>#evaluations</td><td rowspan=1 colspan=1>O(c)</td><td rowspan=1 colspan=1>O(whs)</td><td rowspan=1 colspan=1>O(whs)</td><td rowspan=1 colspan=1>O(whs)</td></tr></table>
+
+Table 1: Face localization accuracy. $w$ : image width; $h$ : image height; $s$ : image scales; $c { \mathrm { : } }$ : number of inference steps used.
+
+We also compared our inference algorithm to the template matching in the task of face detection. We took the first 5 subjects as test subjects and the rest as training. We can localize with $97 \%$ accuracy $\mathrm { T O U } > 0 . 5$ ) using our inference algorithm4. In comparison, a near state-of-the-art face detection system from OpenCV 2.4.9 obtains the same $9 7 \%$ ac
+
+curacy. It uses Haar Cascades, which is a form of AdaBoost5. Normalized Cross Correlation [31] obtained $93 \%$ accuracy, while Euclidean distance template matching achieved an accuracy of only $78 \%$ . However, note that our algorithm looks at a constant number of windows while the other baselines are all based on scanning windows.
+
+# 6.2 Generative learning without labels
+
+Table 2: Variational lower-bound estimates on the log-density of the Gaussian DBNs (higher is better).   
+
+<table><tr><td rowspan=1 colspan=1>nats</td><td rowspan=1 colspan=1>No CMU training</td><td rowspan=1 colspan=1>CMUw/olabels</td><td rowspan=1 colspan=1>CMUw/labels</td></tr><tr><td rowspan=1 colspan=1>Caltech Train</td><td rowspan=1 colspan=1>617±0.4</td><td rowspan=1 colspan=1>627±0.5</td><td rowspan=1 colspan=1>569±0.6</td></tr><tr><td rowspan=1 colspan=1>Caltech Valid</td><td rowspan=1 colspan=1>512±1.1</td><td rowspan=1 colspan=1>503±1.8</td><td rowspan=1 colspan=1>494±1.7</td></tr><tr><td rowspan=1 colspan=1>CMUTrain</td><td rowspan=1 colspan=1>96±0.8</td><td rowspan=1 colspan=1>499±0.1</td><td rowspan=1 colspan=1>594±0.5</td></tr><tr><td rowspan=1 colspan=1>CMUValid</td><td rowspan=1 colspan=1>85±0.5</td><td rowspan=1 colspan=1>387±0.3</td><td rowspan=1 colspan=1>503±0.7</td></tr><tr><td rowspan=1 colspan=1>log2</td><td rowspan=1 colspan=1>454.6</td><td rowspan=1 colspan=1>687.8</td><td rowspan=1 colspan=1>694.2</td></tr></table>
+
+The main advantage of our model is that it can learn on large images of faces without localization label information (no manual cropping required). To demonstrate, we use both the Caltech and the CMU faces dataset. For the CMU faces, a subset of 2526 frontal faces with
+
+ground truth labels are used. We split the Caltech dataset into a training and a validation set. For the CMU faces, we first took $10 \%$ of the images as training cases for the ConvNet for approximate inference. This is needed due to the completely different backgrounds of the Caltech and CMU datasets. The remaining $90 \%$ of the CMU faces are split into a training and validation set. We first trained a GDBN with $\bar { 1 0 2 4 } \mathbf { h } ^ { 1 }$ and $2 5 6 \mathbf { h } ^ { 2 }$ hidden units on the Caltech training set. We also trained a ConvNet for approximate inference using the Caltech training set and $10 \%$ of the CMU training images.
+
+![](images/560194f3ef509e3587acead12e7f26ead1c64290fdf8d9bdc8610d5942576721.jpg)  
+Figure 7: Left: Conditioned on different v will result in a different $\triangle \mathbf { u }$ . Note that the initial $\mathbf { u }$ is exactly the same for two trials. Right: Additional examples. The only difference between the top and bottom panels is the conditioned v. Best viewed in color.
+
+Table 2 shows the estimates of the variational lower-bounds on the average log-density (higher is better) that the GDBN models assign to the ground-truth cropped face images from the training/test sets under different scenarios. In the left column, the model is only trained on Caltech faces. Thus it gives very low probabilities to the CMU faces. Indeed, GDBNs achieve a variational lower-bound of only 85 nats per test image. In the middle column, we use our approximate inference to estimate the location of the CMU training faces and further trained the GDBN on the newly localized faces. This gives a dramatic increase of the model performance on the CMU Validation $\mathrm { s e t } ^ { 6 }$ , achieving a lowerbound of 387 nats per test image. The right column gives the best possible results if we can train with the CMU manual localization labels. In this case, GDBNs achieve a lower-bound of 503 nats. We used Annealed Importance Sampling (AIS) to estimate the partition function for the top-layer RBM. Details on estimating the variational lower bound are in the supplementary materials.
+
+Fig. 6(a) further shows samples drawn from the Caltech trained DBN, whereas Fig. 6(b) shows samples after training with the CMU dataset using estimated u. Observe that samples in Fig. 6(b) show a more diverse set of faces. We trained GDBNs using a greedy, layer-wise algorithm of [1]. For the top layer we use Fast Persistent Contrastive Divergence [32], which substantially improved generative performance of GDBNs (see supplementary material for more details).
+
+# 6.3 Inference with ambiguity
+
+Our attentional mechanism can also be useful when multiple objects/faces are present in the scene. Indeed, the posterior $p ( \mathbf { u } | \mathbf { x } , \mathbf { v } )$ is conditioned on v, which means that where to attend is a function of the canonical object $\mathbf { v }$ the model has in “mind” (see Fig. 2(b)). To explore this, we first synthetically generate a dataset by concatenating together two faces from the Caltech dataset. We then train approximate inference ConvNet as in Sec. 4.1 and test on the held-out subjects. Indeed, as predicted, Fig. 7 shows that depending on which canonical image is conditioned, the same exact gaze initialization leads to two very different gaze shifts. Note that this phenomenon is observed across different scales and location of the initial gaze. For example, in Fig. 7, right-bottom panel, the initialized yellow box is mostly on the female’s face to the left, but because the conditioned canonical face $\mathbf { v }$ is that of the right male, attention is shifted to the right.
+
+# 7 Conclusion
+
+In this paper we have proposed a probabilistic graphical model framework for learning generative models using attention. Experiments on face modeling have shown that ConvNet based approximate inference combined with HMC sampling is sufficient to explore the complicated posterior distribution. More importantly, we can generatively learn objects of interest from novel big images. Future work will include experimenting with faces as well as other objects in a large scene. Currently the ConvNet approximate inference is trained in a supervised manner, but reinforcement learning could also be used instead.
+
+# Acknowledgements
+
+The authors gratefully acknowledge the support and generosity from Samsung, Google, and ONR grant N00014-14-1-0232.
+
+# References
+
+tation, 18(7):1527–1554, 2006.   
+[2] R. Salakhutdinov and G. Hinton. Deep Boltzmann machines. In AISTATS, 2009.   
+[3] Geoffrey E. Hinton, Peter Dayan, and Michael Revow. Modeling the manifolds of images of handwritten digits. IEEE Transactions on Neural Networks, 8(1):65–74, 1997.   
+[4] Daniel Zoran and Yair Weiss. From learning models of natural image patches to whole image restoration. In ICCV. IEEE, 2011.   
+[5] Yoshua Bengio, Li Yao, Guillaume Alain, and Pascal Vincent. Generalized denoising auto-encoders as generative models. In Advances in Neural Information Processing Systems 26, 2013.   
+[6] H. Lee, R. Grosse, R. Ranganath, and A. Y. Ng. Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations. In ICML, pages 609–616, 2009. [7] Marc’Aurelio Ranzato, Joshua Susskind, Volodymyr Mnih, and Geoffrey Hinton. On Deep Generative Models with Applications to Recognition. In CVPR, 2011. [8] Yichuan Tang, Ruslan Salakhutdinov, and Geoffrey E. Hinton. Deep mixtures of factor analysers. In ICML. icml.cc / Omnipress, 2012.   
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+[10] E. A. Buffalo, P. Fries, R. Landman, H. Liang, and R. Desimone. A backward progression of attentional effects in the ventral stream. PNAS, 107(1):361–365, Jan. 2010.   
+[11] N Kanwisher and E Wojciulik. Visual attention: Insights from brain imaging. Nature Reviews Neuroscience, 1:91–100, 2000.   
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+[14] Laurenz Wiskott. How does our visual system achieve shift and size invariance?, 2004.   
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+[17] Hugo Larochelle and Geoffrey E. Hinton. Learning to combine foveal glimpses with a third-order boltzmann machine. In NIPS, pages 1243–1251. Curran Associates, Inc., 2010.   
+[18] D. P. Reichert, P. Seriès, and A. J. Storkey. A hierarchical generative model of recurrent object-based attention in the visual cortex. In ICANN (1), volume 6791, pages 18–25. Springer, 2011.   
+[19] B. Alexe, N. Heess, Y. W. Teh, and V. Ferrari. Searching for objects driven by context. In NIPS 2012, December 2012.   
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+[21] M. Denil, L. Bazzani, H. Larochelle, and N. de Freitas. Learning where to attend with deep architectures for image tracking. Neural Computation, 28:2151–2184, 2012.   
+[22] G. E. Hinton and R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313:504–507, 2006.   
+[23] A. Krizhevsky. Learning multiple layers of features from tiny images. Master’s thesis, University of Toronto, Toronto, Ontario, Canada, 2009.   
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+[27] S. Duane, A. D. Kennedy, B. J Pendleton, and D. Roweth. Hybrid Monte Carlo. Physics Letters B, 195(2):216–222, 1987.   
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+
+![](images/be71cbc5181417920d82329590b5f9235fd959aff334cd817ca0e56060d77e56.jpg)  
+Figure 8: A visual diagram of the convolutional net used for approximate inference.
+
+# APPENDIX
+
+# Convolutional Neural Network
+
+The training of ConvNet for approximate inference is standard and did not involve any special ’tricks’. We used SGD with minibatch size of 128 samples. We used a standard ConvNet architecture with convolution C layers followed by max-pooling S layers. The ConvNet takes as input $\mathbf { x }$ and $\mathbf { v }$ to predict change in u such that to maximize $\log p ( { \bf u } | { \bf x } , \bar { \bf v } )$ . In order to better predict change of u, $\mathbf { x }$ as well as a bigger border around $\mathbf { x }$ are used as the input to the ConvNet. Therefore, $\mathbf { x }$ has resolution $7 2 \times 7 2$ and $\mathbf { v }$ has resolution of $2 4 \times 2 4$ .
+
+Two handle two different inputs with different resolutions, two different “streams" are used in this ConvNet architecture. One stream will process $\mathbf { x }$ and another one for $\mathbf { v }$ . These two streams will be combined multiplicatively after subsampling the $\mathbf { x }$ stream by a factor of 3. The rest of the ConvNet is same as the standard classification ConvNets, except that we use mean squared error as our cost function. See Figure 8 for a visual diagram of what the convolutional neural network architecture used.
+
+Table 3: Model architectures of the convolutional neural network used during approximate inference.   
+
+<table><tr><td>layer</td><td>type</td><td>latent variables</td><td>filter size</td><td># weights</td></tr><tr><td>0</td><td>input x</td><td>maps:3 72x72</td><td></td><td>=</td></tr><tr><td>1</td><td>input v</td><td>maps:3 24x24</td><td>-</td><td>=</td></tr><tr><td>2</td><td>Conv of layer 0</td><td>maps:16 66x66</td><td>7x7</td><td>2352</td></tr><tr><td>3</td><td>Pooling</td><td>maps:16 22x22</td><td>3x3</td><td>-</td></tr><tr><td>4</td><td>Conv of layer 1</td><td>maps:16 22x22</td><td>5x5</td><td>1200</td></tr><tr><td>5</td><td>Combine layers 3,4</td><td>maps:16 22x22</td><td>-</td><td>=</td></tr><tr><td>6</td><td>Fully connected</td><td>1024</td><td></td><td>7.9M</td></tr><tr><td>7</td><td>Fully connected</td><td>4</td><td>=</td><td>4K</td></tr></table>
+
+Table 3 details the exact model architecture used. In layer 5, the two streams have the same number of hidden maps and hidden topography. We combine these two multiplicatively by multiplying their activation elementwise. This creates a third-order flavor and is more powerful for the task of determining where to shift attention to next.
+
+# Gaussian Deep Belief Network
+
+The training of the Gaussian Deep Belief Network is performed in a standard greedy layerwise fashion. The first layer Gaussian Restricted Boltzmann Machine is trained with the Contrastive Divergence algorithm where the standard deviation of each visible unit is learned as well. After training the first layer, we use Eq. 3 to obtain first hidden layer binary probabilities. We then train a 2nd binary-binary Restricted Boltzmann Machine using the fast persistent contrastive divergence learning algorithm. This greedy training leads us to the Gaussian Deep Belief Network. No finetuning of the entire network is performed.
+
+# Quantitative evaluation for Gaussian Deep Belief Network
+
+For quantitative evaluation, we approximate the standard variational lower bound on the log likelihood of the Gaussian Deep Belief Network. The model is a directed model:
+
+$$
+p ( { \mathbf { v } } , \mathbf { h } ^ { 1 } , \mathbf { h } ^ { 2 } ) = p ( { \mathbf { v } } | \mathbf { h } ^ { 1 } ) p ( { \mathbf { h } } ^ { 1 } , \mathbf { h } ^ { 2 } )
+$$
+
+For any approximating posterior distribution $q ( \mathbf { h } ^ { 1 } | \mathbf { v } )$ , the GDBN’s log-likelihood has this lower variational
+
+$$
+\log \sum _ { \mathbf { h } ^ { 1 } } p ( \mathbf { v } , \mathbf { h } ^ { 1 } ) \geq \sum _ { \mathbf { h } ^ { 1 } } q ( \mathbf { h } ^ { 1 } | \mathbf { v } ) [ \log p ( \mathbf { v } | \mathbf { h } ^ { 1 } ) + \log p ^ { * } ( \mathbf { h } ^ { 1 } ) ] - \log Z + \mathcal { H } ( q ( \mathbf { h } ^ { 1 } | \mathbf { v } ) )
+$$
+
+The entropy $\mathcal { H } ( q ( \mathbf { h } ^ { 1 } | \mathbf { v } ) )$ can be calculated since we made the factorial assumption on $q ( \mathbf { h } ^ { 1 } | \mathbf { v } )$ .
+
+$$
+\begin{array} { l } { { \displaystyle \log p ( { \bf v } | { \bf h } ^ { 1 } ) = - \sum \log \sigma _ { i } - \frac { D } { 2 } \log 2 \pi - \frac { 1 } { 2 } \sum _ { i } ^ { D } \frac { ( x - \mu ) ^ { 2 } } { \sigma _ { i } ^ { 2 } } } } \\ { { \displaystyle \log p ^ { * } ( { \bf h } ^ { 1 } ) = { \bf b } ^ { \mathsf { T } } { \bf h } ^ { 1 } + \log \sum _ { j } \exp \{ { \bf h } ^ { 1 } { \bf W } _ { j } + c _ { j } \} } } \end{array}
+$$
+
+In order to calculate the expectation of the approximating posteriors, we use Monte Carlo sampling.
+
+$$
+\begin{array} { l } { { \displaystyle \sum _ { \mathbf { h } ^ { 1 } } q ( \mathbf { h } ^ { 1 } | \mathbf { v } ) \log p ^ { * } ( \mathbf { v } , \mathbf { h } ^ { 1 } ) \approx \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \log p ^ { * } ( \mathbf { v } , \mathbf { h } ^ { 1 ( m ) } ) } \ ~ } \\ { { \displaystyle = - \sum _ { i = 1 } ^ { D } \log \sigma _ { i } - \frac { D } { 2 } \log 2 \pi - \frac { 1 } { 2 } \sum _ { i } ^ { D } \frac { ( x - \mu ) ^ { 2 } } { \sigma _ { i } ^ { 2 } } } \ ~ } \\ { { \displaystyle ~ + \mathbf { b } ^ { \mathsf { T } } \mathbf { h } ^ { 1 } + \log \sum _ { j } \exp \{ \mathbf { h } ^ { 1 } \mathbf { W } _ { j } + c _ { j } \} } } \end{array}
+$$
+
+where $\mathbf { h } ^ { 1 ( m ) }$ is the $m$ -th sample from the posterior $q ( \mathbf { h } ^ { 1 } | \mathbf { v } )$
+
+In order to calculate the partition function of the top-most layer of the GDBN, we use Annealed Importance Sampling (AIS). We used 100 chains with 50,000 intermediate distributions to estimate the partition function of the binary-binary RBM which forms the top layer of the GDBN. Even though AIS is an unbiased estimator of the partition function, it is prone to under-estimating it due to bad mixing of the chains. This causes the log probability to be over-estimated. Therefore the variational lower bounds reported in our paper are not strictly guaranteed to be lower bounds and are subject to errors. However, we believe that the margin of error is unlikely to be high enough to affect our conclusions.
+
+# Additional Results
+
+We present some more examples of inference process of our framework.
+
+Below, we show some success cases and a failure case for inference on the CMU Multi-PIE dataset. The initial gaze variables of $\mathbf { u }$ are highlighted in yellow and later iterations are highlighted with color gradually changing to blue.
+
+![](images/22135e80c1fa41d282fe74bc86f2b896a1c7808c289b962886426cad59ce6f39.jpg)  
+Figure 9: Example of an approximate inference steps. v is $2 4 { \times } 2 4$ , $\mathbf { x }$ is $7 2 { \times } 7 2$ . Approximate inference quickly finds a good initialization for u, while HMC makes small adjustments.
+
+![](images/0a8956de8d0141024c2d7ab80524190d6ce85a39211d2a5f295dcca994434be5.jpg)  
+Figure 10: E-step for learning on CMU Multi-PIE. (a),(b),(c) are successful. (d) is a failure case.

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+        "type": "text",
+        "text": "Learning Generative Models with Visual Attention ",
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+        "text": "Yichuan Tang, Nitish Srivastava, Ruslan Salakhutdinov ",
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+        "type": "text",
+        "text": "Department of Computer Science University of Toronto Toronto, Ontario, Canada {tang,nitish,rsalakhu}@cs.toronto.edu ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "Attention has long been proposed by psychologists to be important for efficiently dealing with the massive amounts of sensory stimulus in the neocortex. Inspired by the attention models in visual neuroscience and the need for object-centered data for generative models, we propose a deep-learning based generative framework using attention. The attentional mechanism propagates signals from the region of interest in a scene to an aligned canonical representation for generative modeling. By ignoring scene background clutter, the generative model can concentrate its resources on the object of interest. A convolutional neural net is employed to provide good initializations during posterior inference which uses Hamiltonian Monte Carlo. Upon learning images of faces, our model can robustly attend to the face region of novel test subjects. More importantly, our model can learn generative models of new faces from a novel dataset of large images where the face locations are not known.1 ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Building rich generative models that are capable of extracting useful, high-level latent representations from high-dimensional sensory input lies at the core of solving many AI-related tasks, including object recognition, speech perception and language understanding. These models capture underlying structure in data by defining flexible probability distributions over high-dimensional data as part of a complex, partially observed system. Some of the successful generative models that are able to discover meaningful high-level latent representations include the Boltzmann Machine family of models: Restricted Boltzmann Machines, Deep Belief Nets [1], and Deep Boltzmann Machines [2]. Mixture models, such as Mixtures of Factor Analyzers [3] and Mixtures of Gaussians, have also been used for modeling natural image patches [4]. More recently, denoising auto-encoders have been proposed as a way to model the transition operator that has the same invariant distribution as the data generating distribution [5]. ",
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+    {
+        "type": "text",
+        "text": "Generative models have an advantage over discriminative models when part of the images are occluded or missing. Occlusions are very common in realistic settings and have been largely ignored in recent literature on deep learning. In addition, prior knowledge can be easily incorporated in generative models in the forms of structured latent variables, such as lighting and deformable parts. However, the enormous amount of content in high-resolution images makes generative learning difficult [6, 7]. Therefore, generative models have found most success in learning to model small patches of natural images and objects: Zoran and Weiss [4] learned a mixture of Gaussians model over $8 { \\times } 8$ image patches; Salakhutdinov and Hinton [2] used $6 4 { \\times } 6 4$ centered and uncluttered stereo images of toy objects on a clear background; Tang et al. [8] used $2 4 \\times 2 4$ images of centered and cropped faces. The fact that these models require curated training data limits their applicability on using the (virtually) unlimited unlabeled data. ",
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+    {
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+        "text": "In this paper, we propose a framework to infer the region of interest in a big image for generative modeling. This will allow us to learn a generative model of faces on a very large dataset of (unlabeled) images containing faces. Our framework is able to dynamically route the relevant information to the generative model and can ignore the background clutter. The need to dynamically and selectively route information is also present in the biological brain. Plethora of evidence points to the presence of attention in the visual cortex [9, 10]. Recently, in visual neuroscience, attention has been shown to exist not only in extrastriate areas, but also all the way down to V1 [11]. ",
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+        "text": "Attention as a form of routing was originally proposed by Anderson and Van Essen [12] and then extended by Olshausen et al. [13]. Dynamic routing has been hypothesized as providing a way for achieving shift and size invariance in the visual cortex [14, 15]. Tsotsos et al. [16] proposed a model combining search and attention called the Selective Tuning model. Larochelle and Hinton [17] proposed a way of using third-order Boltzmann Machines to combine information gathered from many foveal glimpses. Their model chooses where to look next to find locations that are most informative of the object class. Reichert et al. [18] proposed a hierarchical model to show that certain aspects of covert object-based attention can be modeled by Deep Boltzmann Machines. Several other related models attempt to learn where to look for objects [19, 20] and for video based tracking [21]. Inspired by Olshausen et al. [13], we use 2D similarity transformations to implement the scaling, rotation, and shift operation required for routing. Our main motivation is to enable the learning of generative models in big images where the location of the object of interest is unknown a-priori. ",
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+    {
+        "type": "text",
+        "text": "2 Gaussian Restricted Boltzmann Machines ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Before we describe our model, we briefly review the Gaussian Restricted Boltzmann Machine (GRBM) [22], as it will serve as the building block for our attention-based model. GRBMs are a type of Markov Random Field model that has a bipartite structure with real-valued visible variables $\\mathbf { v } \\in \\mathbb { R } ^ { D }$ connected to binary stochastic hidden variables $\\mathbf { h } \\in \\{ 0 , 1 \\} ^ { H }$ . The energy of the joint configuration $\\{ \\mathbf { v } , \\mathbf { h } \\}$ of the Gaussian RBM is defined as follows: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/da45d2c2e029934ba3cf5b7015c1a0110d30a83ebf0a19c1a092fa9bdfcd952d.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { \\displaystyle E _ { G R B M } ( { \\bf v } , { \\bf h } ; \\Theta ) } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\sum _ { i } \\frac { ( v _ { i } - b _ { i } ) ^ { 2 } } { \\sigma _ { i } ^ { 2 } } - \\sum _ { j } c _ { j } h _ { j } - \\sum _ { i j } W _ { i j } v _ { i } h _ { j } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            450,
+            728,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { \\Theta } = \\{ \\mathbf { W } , \\mathbf { b } , \\mathbf { c } , \\pmb { \\sigma } \\}$ are the model parameters. The marginal distribution over the visible vector $\\mathbf { v }$ is $\\begin{array} { r } { P ( \\mathbf { v } ; \\Theta ) = \\frac { 1 } { \\mathcal { Z } ( \\Theta ) } \\sum _ { \\mathbf { h } } \\exp \\left( - E ( \\mathbf { v } , \\mathbf { h } ; \\Theta ) \\right) } \\end{array}$ and the corresponding conditional distributions take the following form: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bd1c5675d4dbd91d5c93a23641cf2898b254c2ed3c0ad4dc3299583b09aa8c5a.jpg",
+        "text": "$$\n\\begin{array} { r c l } { p ( h _ { j } = 1 | \\mathbf { v } ) } & { = } & { \\displaystyle 1 / \\big ( 1 + \\exp ( - \\sum _ { i } W _ { i j } v _ { i } - c _ { j } ) \\big ) , } \\\\ { p ( v _ { i } | \\mathbf { h } ) } & { = } & { \\displaystyle \\mathcal { N } ( v _ { i } ; \\mu _ { i } , \\sigma _ { i } ^ { 2 } ) , ~ \\mathrm { w h e r e } ~ \\mu _ { i } = b _ { i } + \\sigma _ { i } ^ { 2 } \\sum _ { j } W _ { i j } h _ { j } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Observe that conditioned on the states of the hidden variables (Eq. 3), each visible unit is modeled by a Gaussian distribution, whose mean is shifted by the weighted combination of the hidden unit activations. Unlike directed models, an RBM’s conditional distribution over hidden nodes is factorial and can be easily computed. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We can also add a binary RBM on top of the learned GRBM by treating the inferred h as the “visible” layer together with a second hidden layer $\\mathbf { h } ^ { 2 }$ . This results in a 2-layer Gaussian Deep Belief Network (GDBN) [1] that is a more powerful model of $\\mathbf { v }$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Specifically, in a GDBN model, $p ( \\mathbf { h } ^ { 1 } , \\mathbf { h } ^ { 2 } )$ is modeled by the energy function of the 2nd-layer RBM, while $p ( \\mathbf { v } ^ { \\tilde { 1 } } \\vert \\mathbf { h } ^ { 1 } )$ is given by Eq. 3. Efficient inference can be performed using the greedy approach of [1] by treating each DBN layer as a separate RBM model. GDBNs have been applied to various tasks, including image classification, video action and speech recognition [6, 23, 24, 25]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 The Model ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathcal { T }$ be a high resolution image of a scene, e.g. a $2 5 6 \\times 2 5 6$ image. We want to use attention to propagate regions of interest from $\\mathcal { T }$ up to a canonical representation. For example, in order to learn a model of faces, the canonical representation could be a $2 4 \\times 2 4$ aligned and cropped frontal face image. Let $\\mathbf { v } \\in \\mathbb { R } ^ { D }$ represent this low resolution canonical image. In this work, we focus on a Deep Belief Network2 to model $\\mathbf { v }$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/15c14e094912a2e46700a68681a73b0dc66dbfeb37706dcbae5bd7b4a96b302d.jpg",
+        "image_caption": [
+            "Figure 1: Left: The Shifter Circuit, a well-known neuroscience model for visual attention [13]; Right: The proposed model uses 2D similarity transformations from geometry and a Gaussian DBN to model canonical face images. Associative memory corresponds to the DBN, object-centered frame correspond to the visible layer and the attentional mechanism is modeled by 2D similarity transformations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This is illustrated in the diagrams of Fig. 1. The left panel displays the model of Olshausen et.al. [13], whereas the right panel shows a graphical diagram of our proposed generative model with an attentional mechanism. Here, ${ \\bf h } ^ { 1 }$ and $\\mathbf { h } ^ { 2 }$ represent the latent hidden variables of the DBN model, and $\\triangle x , \\triangle y , \\triangle \\theta , \\triangle s$ (position, rotation, and scale) are the parameters of the 2D similarity transformation. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The 2D similarity transformation is used to rotate, scale, and translate the canonical image v onto the canvas that we denote by $\\mathcal { T }$ . Let $\\mathbf { p } = [ x y ] ^ { \\mathsf { T } }$ be a pixel coordinate (e.g. $[ 0 , 0 ]$ or $[ 0 , 1 ] \\cdot$ ) of the canonical image $\\mathbf { v }$ . Let $\\{ \\mathbf { p } \\}$ be the set of all coordinates of $\\mathbf { v }$ . For example, if $\\mathbf { v }$ is $2 4 \\times 2 4$ , then $\\{ \\mathbf { p } \\}$ ranges from $[ 0 , 0 ]$ to [23, 23]. Let the “gaze” variables $\\mathbf { u } \\in \\mathbb { R } ^ { 4 } \\equiv [ \\widehat { \\triangle } x , \\triangle y , \\triangle \\theta , \\triangle s ]$ be the parameter of the Similarity transformation. In order to simplify derivations and to make transformations be linear w.r.t. the transformation parameters, we can equivalently redefine $\\mathbf { u } = [ a , \\ b , \\ \\triangle x , \\ \\triangle y ]$ , where $a = s \\sin ( \\theta ) - 1$ and $b = s \\cos ( \\theta )$ (see [26] for details). We further define a function $\\mathsf { w } : = \\mathsf { w } ( \\mathbf { p } , \\mathbf { u } ) \\to \\mathbf { p } ^ { \\prime }$ as the transformation function to warp points $\\mathbf { p }$ to $\\mathbf { p } ^ { \\prime }$ : ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9fd2b443201706e3aeabbcab732f1a71dd6ce8fc20f03f8eb8d5aec48a870cd5.jpg",
+        "text": "$$\n\\mathbf { p } ^ { \\prime } \\triangleq \\left[ \\begin{array} { l } { x ^ { \\prime } } \\\\ { y ^ { \\prime } } \\end{array} \\right] = \\left[ \\begin{array} { c c } { 1 + a } & { - b } \\\\ { b } & { 1 + a } \\end{array} \\right] \\left[ \\begin{array} { l } { x } \\\\ { y } \\end{array} \\right] + \\left[ \\begin{array} { l } { \\bigtriangleup x } \\\\ { \\bigtriangleup y } \\end{array} \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We use the notation $\\mathcal { T } ( \\{ \\mathbf { p } \\} )$ to denote the bilinear interpolation of $\\mathcal { T }$ at coordinates $\\{ \\mathbf { p } \\}$ with antialiasing. Let $\\mathbf { x } ( \\mathbf { u } )$ be the extracted low-resolution image at warped locations $\\mathbf { p } ^ { \\prime }$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4a2fa34db3872c355979f29f309accc178bef9ef43c022f859641f7f95da755d.jpg",
+        "text": "$$\n\\mathbf { x } ( \\mathbf { u } ) \\triangleq { \\mathcal { T } } ( \\mathbf { w } ( \\{ \\mathbf { p } \\} , \\mathbf { u } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            622,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, $\\mathbf { x } ( \\mathbf { u } )$ is a patch extracted from $\\mathcal { T }$ according to the shift, rotation and scale parameters of $\\mathbf { u }$ , as shown in Fig. 1, right panel. It is this patch of data that we seek to model generatively. Note that the dimensionality of $\\bar { \\bf x } ( { \\bf u } )$ is equal to the cardinality of $\\{ \\mathbf { p } \\}$ , where $\\{ \\mathbf { p } \\}$ denotes the set of pixel coordinates of the canonical image $\\mathbf { v }$ . Unlike standard generative learning tasks, the data $\\mathbf { x } ( \\mathbf { u } )$ is not static but changes with the latent variables $\\mathbf { u }$ . Given $\\mathbf { v }$ and $\\mathbf { u }$ , we model the top-down generative process over $^ 3 \\mathrm { \\bf ~ x }$ with a Gaussian distribution having a diagonal covariance matrix $\\sigma ^ { 2 } \\boldsymbol { \\mathrm { I } }$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46612f055882ea34f782d41615a0d56fb3eccf2ac8deee4f82b21165b07f45aa.jpg",
+        "text": "$$\np ( \\mathbf { x } | \\mathbf { v } , \\mathbf { u } , \\mathcal { T } ) \\propto \\exp \\bigg ( - \\frac { 1 } { 2 } \\sum _ { i } \\frac { ( x _ { i } ( \\mathbf { u } ) - v _ { i } ) ^ { 2 } } { \\sigma _ { i } ^ { 2 } } \\bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The fact that we do not seek to model the rest of the regions/pixels of $\\mathcal { T }$ is by design. By using 2D similarity transformation to mimic attention, we can discard the complex background of the scene and let the generative model focus on the object of interest. The proposed generative model takes the following form: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2c867ae0b46e3176b11eb861ad7f2b1d331afc9b9ca7425b37cd55d9f0bc9b23.jpg",
+        "text": "$$\n\\begin{array} { r } { p ( \\mathbf { x } , \\mathbf { v } , \\mathbf { u } | \\mathcal { T } ) = p ( \\mathbf { x } | \\mathbf { v } , \\mathbf { u } , \\mathcal { T } ) p ( \\mathbf { v } ) p ( \\mathbf { u } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where for $p ( \\mathbf { u } )$ we use a flat prior that is constant for all $\\mathbf { u }$ , and $p ( \\mathbf { v } )$ is defined by a 2-layer Gaussian Deep Belief Network. The conditional $p ( \\mathbf { x } | \\mathbf { v } , \\mathbf { u } , \\mathcal { T } )$ is given by a Gaussian distribution as in Eq. 6. To simplify the inference procedure, $p ( \\mathbf { x } | \\mathbf { v } , \\mathbf { u } , \\mathcal { T } )$ and the GDBN model of $\\mathbf { v }$ , $p ( \\mathbf { v } )$ , will share the same noise parameters $\\sigma _ { i }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 Inference ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While the generative equations in the last section are straightforward and intuitive, inference in these models is typically intractable due to the complicated energy landscape of the posterior. During inference, we wish to compute the distribution over the gaze variables u and canonical object v given the big image $\\mathcal { T }$ . Unlike in standard RBMs and DBNs, there are no simplifying factorial assumptions about the conditional distribution of the latent variable u. Having a 2D similarity transformation is reminiscent of third-order Boltzmann machines with $\\mathbf { u }$ performing top-down multiplicative gating of the connections between $\\mathbf { v }$ and $\\mathcal { T }$ . It is well known that inference in these higher-order models is rather complicated. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "One way to perform inference in our model is to resort to Gibbs sampling by computing the set of alternating conditional posteriors: The conditional distribution over the canonical image v takes the following form: ",
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/596d133b520e7000e4f69b91b7374ca9ad2296d8c3991bf86dc539b2568603a4.jpg",
+        "text": "$$\np ( \\mathbf { v } | \\mathbf { u } , \\mathbf { h } ^ { 1 } , \\mathcal { T } ) = \\mathcal { N } \\Big ( \\frac { \\mu + \\mathbf { x } ( \\mathbf { u } ) } { 2 } ; \\pmb { \\sigma } ^ { 2 } \\Big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            280,
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+            310
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mu _ { i } = b _ { i } + \\sigma _ { i } ^ { 2 } \\sum _ { j } W _ { i j } h _ { j } ^ { 1 }$ is the top-down influence of the DBN. Note that if we know the gaze variable $\\mathbf { u }$ and the first layer of hidden variables ${ \\bf h } ^ { 1 }$ , then $\\mathbf { v }$ is simply defined by a Gaussian distribution, where the mean is given by the average of the top-down influence and bottom-up information from $\\mathbf { x }$ . The conditional distributions over ${ \\bf h } ^ { 1 }$ and $\\bar { \\mathbf { h } ^ { 2 } }$ given $\\mathbf { v }$ are given by the standard DBN inference equations [1]. The conditional posterior over the gaze variables $\\mathbf { u }$ is given by: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/10ab4e8b12eb7629cdacb8ea570d10e261caa2f1198c044c8467ebc17b859202.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\displaystyle p ( { \\bf u } | { \\bf x } , { \\bf v } ) = \\frac { p ( { \\bf x } | { \\bf u } , { \\bf v } ) p ( { \\bf u } ) } { p ( { \\bf x } | { \\bf v } ) } , } } \\\\ { { \\displaystyle \\log p ( { \\bf u } | { \\bf x } , { \\bf v } ) \\propto \\log p ( { \\bf x } | { \\bf u } , { \\bf v } ) + \\log p ( { \\bf u } ) = \\frac { 1 } { 2 } \\sum _ { i } \\frac { ( x _ { i } ( { \\bf u } ) - v _ { i } ) ^ { 2 } } { \\sigma _ { i } ^ { 2 } } + c o n s t . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            383,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Using Bayes’ rule, the unnormalized log probability of $p ( \\mathbf { u } | \\mathbf { x } , \\mathbf { v } )$ is defined in Eq. 9. We stress that this equation is atypical in that the random variable of interest $\\mathbf { u }$ actually affects the conditioning variable $\\mathbf { x }$ (see Eq. 5) We can explore the gaze variables using Hamiltonian Monte Carlo (HMC) algorithm [27, 28]. Intuitively, conditioned on the canonical object $\\mathbf { v }$ that our model has in “mind”, HMC searches over the entire image $\\mathcal { T }$ to find a region $\\mathbf { x }$ with a good match to $\\mathbf { v }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "If the goal is only to find the MAP estimate of $p ( \\mathbf { u } | \\mathbf { x } , \\mathbf { v } )$ , then we may want to use second-order methods for optimizing $\\mathbf { u }$ . This would be equivalent to the Lucas-Kanade framework in computer vision, developed for image alignment [29]. However, HMC has the advantage of being a proper MCMC sampler that satisfies detailed balance and fits nicely with our probabilistic framework. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The HMC algorithm first specifies the Hamiltonian over the position variables $\\mathbf { u }$ and auxiliary momentum variables r: $\\mathcal { H } ( \\mathbf { u } , \\mathbf { r } ) ~ = ~ U ( \\mathbf { u } ) + K ( \\mathbf { r } )$ , where the potential function is defined by $\\begin{array} { r } { U ( \\mathbf { u } ) = \\frac { 1 } { 2 } \\sum _ { i } \\frac { ( x _ { i } ( \\mathbf { u } ) - v _ { i } ) ^ { 2 } } { \\sigma _ { i } ^ { 2 } } } \\end{array}$ and the kinetic energy function is given by $\\begin{array} { r } { K ( { \\bf r } ) = \\frac { 1 } { 2 } \\sum _ { i } r _ { i } ^ { 2 } } \\end{array}$ . The dyinamics of the system is defined by: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb427a972661b4717ad49f2a261e3842d6a8e3cfa0424e787e8935ae8611c8a9.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { c u t ~ \\mathbf { 1 } s ~ u c t u n c u ~ \\mathbf { y } . } } \\\\ & { ~ \\displaystyle \\frac { \\partial \\mathbf { u } } { \\partial t } = \\mathbf { r } , \\qquad \\frac { \\partial \\mathbf { r } } { \\partial t } = - \\frac { \\partial \\mathbf { \\mathcal { H } } } { \\partial \\mathbf { u } } } \\\\ & { \\displaystyle \\frac { \\partial \\mathcal { H } } { \\partial \\mathbf { u } } = \\frac { ( \\mathbf { x } ( \\mathbf { u } ) - \\mathbf { v } ) } { \\sigma ^ { 2 } } \\frac { \\partial \\mathbf { x } ( \\mathbf { u } ) } { \\partial \\mathbf { u } } , } \\\\ & { \\displaystyle \\frac { \\partial \\mathbf { x } } { \\partial \\mathbf { u } } = \\frac { \\partial \\mathbf { x } } { \\partial \\mathbf { w } ( \\{ \\mathbf { p } \\} , \\mathbf { u } ) } \\frac { \\partial \\mathbf { w } ( \\{ \\mathbf { p } \\} , \\mathbf { u } ) } { \\partial \\mathbf { u } } = \\sum _ { i } \\frac { \\partial x _ { i } } { \\partial \\mathbf { w } ( \\mathbf { p } _ { i } , \\mathbf { u } ) } \\frac { \\partial \\mathbf { w } ( \\mathbf { p } _ { i } , \\mathbf { u } ) } { \\partial \\mathbf { u } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            652,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Observe that Eq. 12 decomposes into sums over single coordinate positions $\\mathbf { p } _ { i } = [ x y ] ^ { \\mathsf { T } }$ . Let us denote $\\mathbf { p ^ { \\prime } } _ { i } = \\mathsf { w } ( \\mathbf { p } _ { i } , \\mathbf { u } )$ to be the coordinate $\\mathbf { p } _ { i }$ warped by $\\mathbf { u }$ . For the first term on the RHS of Eq. 12, ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f11769fd40e486871c314cd288e9434127d7e2904aa26ac1ccf1862670a66af.jpg",
+        "text": "$$\n{ \\frac { \\partial x _ { i } } { \\partial \\mathbf { w } ( \\mathbf { p } _ { i } , \\mathbf { u } ) } } = \\nabla I ( \\mathbf { p ^ { \\prime } } _ { i } ) , \\quad ( { \\mathrm { d i m e n s i o n ~ } } 1 \\ \\mathbf { b y } \\ 2 \\ )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            808,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\nabla I ( \\mathbf { p } ^ { \\prime } _ { i } )$ denotes the sampling of the gradient images of $I$ at the warped location $\\mathbf { p } _ { i }$ . For the second term on the RHS of Eq. 12, we note that we can re-write Eq. 4 as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be77de7b25cc41357fdf823a2df37233f4e01e819b0667761d4c57db2e4a5b05.jpg",
+        "text": "$$\n{ \\left[ \\begin{array} { l } { x ^ { \\prime } } \\\\ { y ^ { \\prime } } \\end{array} \\right] } = { \\left[ \\begin{array} { l l l l } { x } & { - y } & { 1 } & { 0 } \\\\ { y } & { x } & { 0 } & { 1 } \\end{array} \\right] } { \\left[ \\begin{array} { l } { a } \\\\ { b } \\\\ { \\bigtriangleup x } \\\\ { \\bigtriangleup y } \\end{array} \\right] } + { \\left[ \\begin{array} { l } { x } \\\\ { y } \\end{array} \\right] } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "giving us ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/3ad23d83c563c7f5b49df8461b1909d0648fc356dac38690c2c4dad3757ff34d.jpg",
+        "text": "$$\n\\frac { \\partial \\mathbf { w } ( \\mathbf { p } _ { i } , \\mathbf { u } ) } { \\partial \\mathbf { u } } = \\left[ \\begin{array} { c c c c } { x } & { - y } & { 1 } & { 0 } \\\\ { y } & { x } & { 0 } & { 1 } \\end{array} \\right] .\n$$",
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+        "type": "text",
+        "text": "HMC simulates the discretized system by performing leap-frog updates of $\\mathbf { u }$ and $\\mathbf { r }$ using Eq. 10. Additional hyperparameters that need to be specified include the step size $\\epsilon$ , number of leap-frog steps, and the mass of the variables (see [28] for details). ",
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+    {
+        "type": "text",
+        "text": "4.1 Approximate Inference ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "HMC essentially performs gradient descent with momentum, therefore it is prone to getting stuck at local optimums. This is especially a problem for our task of finding the best transformation parameters. While the posterior over u should be unimodal near the optimum, many local minima exist away from the global optimum. For example, in Fig. 2(a), the big image $\\mathcal { T }$ is enclosed by the blue box, and the canonical image $\\mathbf { v }$ is enclosed by the green box. The current setting of u aligns together the wrong eyes. However, it is hard to move the green box to the left due to the local optima created by the dark intensities of the eye. Resampling the momentum variable every iteration in HMC does not help significantly because we are modeling real-valued images using a Gaussian distribution as the residual, leading to quadratic costs in the difference between $\\mathbf { x } ( \\mathbf { u } )$ and v (see Eq. 9). This makes the energy barriers between modes extremely high. ",
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+    {
+        "type": "text",
+        "text": "To alleviate this problem we need to find good initializations of u. We use a Convolutional Network (ConvNet) to perform efficient approximate inference, resulting in good initial guesses. Specifically, given v, $\\mathbf { u }$ and $\\mathcal { T }$ , we predict the change in u that will lead to the maximum $\\log p ( { \\bf u } | { \\bf x } , { \\bf v } )$ . In other words, instead of using the gradient field for updating u, we learn a ConvNet to output a better vector field in the space of u. We used a fairly standard ConvNet architecture and the standard stochastic gradient descent learning procedure. ",
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+        "img_path": "images/6ad22dcaefcc12e375c444941ddbe9cc9830b1ea2964d1d1916af8d7ef825e97.jpg",
+        "image_caption": [
+            "Figure 2: (a) HMC can easily get stuck at local optima. (b) Importance of modeling $p ( \\mathbf { u } | \\mathbf { v } , \\mathcal { T } )$ . Best in color. "
+        ],
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+        "type": "text",
+        "text": "We note that standard feedforward face detectors seek to model $p ( \\mathbf { u } | \\mathcal { T } )$ , while completely ignoring the canonical face v. In contrast, here we take $\\mathbf { v }$ into account as well. The ConvNet is used to initialize $\\mathbf { u }$ for the HMC algorithm. This is important in a proper generative model because conditioning on $\\mathbf { v }$ is appealing when multiple faces are present in the scene. Fig. 2(b) is a hypothesized Euclidean space of $\\mathbf { v }$ , where the black manifold represents canonical faces and the blue manifold represents cropped faces $\\mathbf { x } ( \\mathbf { u } )$ . The blue manifold has a low intrinsic dimensionality of 4, spanned by u. At A and B, the blue comes close to black manifold. This means that there are at least two modes in the posterior over u. By conditioning on $\\mathbf { v }$ , we can narrow the posterior to a single mode, depending on whom we want to focus our attention. We demonstrate this exact capability in Sec. 6.3. ",
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+    {
+        "type": "text",
+        "text": "Fig. 3 demonstrates the iterative process of how approximate inference works in our model. Specifically, based on u, the ConvNet takes a window patch around $\\mathbf { x } ( \\mathbf { u } )$ $( 7 2 \\times 7 2 )$ and v $( 2 4 \\times 2 4 )$ as input, and predicts the output $[ \\triangle x , \\triangle y , \\triangle \\theta , \\triangle s ]$ . In step 2, $\\mathbf { u }$ is updated accordingly, followed by step 3 of alternating Gibbs updates of $\\mathbf { v }$ and $\\mathbf { h }$ , as discussed in Sec. 4. The process is repeated. For the details of the ConvNet see the supplementary materials. ",
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+        "type": "text",
+        "text": "5 Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "While inference in our framework localizes objects of interest and is akin to object detection, it is not the main objective. Our motivation is not to compete with state-of-the-art object detectors but rather propose a probabilistic generative framework capable of generative modeling of objects which are at unknown locations in big images. This is because labels are expensive to obtain and are often not available for images in an unconstrained environment. ",
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+        "type": "text",
+        "text": "To learn generatively without labels we propose a simple Monte Carlo based ExpectationMaximization algorithm. This algorithm is an unbiased estimator of the maximum likelihood objective. During the E-step, we use the Gibbs sampling algorithm developed in Sec. 4 to draw samples from the posterior over the latent gaze variables u, the canonical variables $\\mathbf { v }$ , and the hidden variables ${ \\bf h } ^ { 1 }$ , $\\mathbf { h } ^ { 2 }$ of a Gaussian DBN model. During the M-step, we can update the weights of the Gaussian DBN by using the posterior samples as its training data. In addition, we can update the parameters of the ConvNet that performs approximate inference. Due to the fact that the first E-step requires a good inference algorithm, we need to pretrain the ConvNet using labeled gaze data as part of a bootstrap process. Obtaining training data for this initial phase is not a problem as we can jitter/rotate/scale to create data. In Sec. 6.2, we demonstrate the ability to learn a good generative model of face images from the CMU Multi-PIE dataset. ",
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+        "img_path": "images/6095db78056547c5d915d4c552d83090d0ac619c5b47f372e7b406624d98f5dd.jpg",
+        "image_caption": [
+            "Figure 3: Inference process: u in step 1 is randomly initialized. The average v and the extracted $\\mathbf { x } ( \\mathbf { u } )$ form the input to a ConvNet for approximate inference, giving a new u. The new u is used to sample $p ( \\mathbf { v } \\vert \\mathcal { T } , \\mathbf { u } , \\mathbf { h } )$ . In step 3, one step of Gibbs sampling of the GDBN is performed. Step 4 repeats the approximate inference using the updated $\\mathbf { v }$ and $\\mathbf { x } ( \\mathbf { u } )$ . "
+        ],
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+        "image_caption": [
+            "Figure 4: Example of an inference step. v is $2 4 \\times 2 4$ , $\\mathbf { x }$ is $7 2 \\times 7 2$ . Approximate inference quickly finds a good initialization for u, while HMC provides further adjustments. Intermediate inference steps on the right are subsampled from 10 actual iterations. "
+        ],
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+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We used two face datasets in our experiments. The first dataset is a frontal face dataset, called the Caltech Faces from 1999, collected by Markus Weber. In this dataset, there are 450 faces of 27 unique individuals under different lighting conditions, expressions, and backgrounds. We downsampled the images from their native 896 by 692 by a factor of 2. The dataset also contains manually labeled eyes and mouth coordinates, which will serve as the gaze labels. We also used the CMU Multi-PIE dataset [30], which contains 337 subjects, captured under 15 viewpoints and 19 illumination conditions in four recording sessions for a total of more than 750,000 images. We demonstrate our model’s ability to perform approximate inference, to learn without labels, and to perform identity-based attention given an image with two people. ",
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+        "type": "text",
+        "text": "6.1 Approximate inference ",
+        "text_level": 1,
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+        "text": "We first investigate the critical inference algorithm of $p ( \\mathbf { u } | \\mathbf { v } , \\mathcal { T } )$ on the Caltech Faces dataset. We run 4 steps of approximate inference detailed in Sec. 4.1 and diagrammed in Fig. 3, followed by three iterations of 20 leap-frog steps of HMC. Since we do not initially know the correct v, we initialize v to be the average face across all subjects. ",
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+    {
+        "type": "text",
+        "text": "Fig. 4 shows the image of v and $\\mathbf { x }$ during inference for a test subject. The initial gaze box is colored yellow on the left. Subsequent gaze updates progress from yellow to blue. Once ConvNet-based approximate inference gives a good initialization, starting from step 5, five iterations of 20 leap-frog steps of HMC are used to sample from the the posterior. ",
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+    {
+        "type": "text",
+        "text": "Fig. 5 shows the quantitative results of Intersection over Union (IOU) of the ground truth face box and the inferred face box. The results show that inference is very robust to initialization and requires only a few steps of approximate inference to converge. HMC clearly improves model performance, resulting in an IOU increase of about $5 \\%$ for localization. This is impressive given that none of the test subjects were part of the training and the background is different from backgrounds in the training set. ",
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+        "img_path": "images/fab1f63a903168eedabf95bfa8d89a9fe20840c9d3986d6067f2cb221a82714b.jpg",
+        "image_caption": [
+            "Figure 5: (a) Accuracy as a function of gaze initialization (pixel offset). Blue curve is the percentage success of at least $50 \\%$ IOU. Red curve is the average IOU. (b) Accuracy as a function of the number of approximate inference steps when initializing 50 pixels away. (c) Accuracy improvements of HMC as a function of gaze initializations. "
+        ],
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+        "image_caption": [
+            "Figure 6: Left: Samples from a 2-layer DBN trained on Caltech. Right: samples from an updated DBN after training on CMU Multi-PIE without labels. Samples highlighted in green are similar to faces from CMU. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/5b12f4c5be3d84ef2b4630cb9c3a91fced62fdc8b864bd380a65c55b72884a61.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Face localization accuracy. $w$ : image width; $h$ : image height; $s$ : image scales; $c { \\mathrm { : } }$ : number of inference steps used. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Ourmethod</td><td rowspan=1 colspan=1>OpenCV</td><td rowspan=1 colspan=1>NCC</td><td rowspan=1 colspan=1>template</td></tr><tr><td rowspan=1 colspan=1>IOU&gt; 0.5</td><td rowspan=1 colspan=1>97%</td><td rowspan=1 colspan=1>97%</td><td rowspan=1 colspan=1>93%</td><td rowspan=1 colspan=1>78%</td></tr><tr><td rowspan=1 colspan=1>#evaluations</td><td rowspan=1 colspan=1>O(c)</td><td rowspan=1 colspan=1>O(whs)</td><td rowspan=1 colspan=1>O(whs)</td><td rowspan=1 colspan=1>O(whs)</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also compared our inference algorithm to the template matching in the task of face detection. We took the first 5 subjects as test subjects and the rest as training. We can localize with $97 \\%$ accuracy $\\mathrm { T O U } > 0 . 5$ ) using our inference algorithm4. In comparison, a near state-of-the-art face detection system from OpenCV 2.4.9 obtains the same $9 7 \\%$ ac",
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+    {
+        "type": "text",
+        "text": "curacy. It uses Haar Cascades, which is a form of AdaBoost5. Normalized Cross Correlation [31] obtained $93 \\%$ accuracy, while Euclidean distance template matching achieved an accuracy of only $78 \\%$ . However, note that our algorithm looks at a constant number of windows while the other baselines are all based on scanning windows. ",
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+    {
+        "type": "text",
+        "text": "6.2 Generative learning without labels ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/f04ef1c51b6bf2c2cf7aad56265e89b6d51f87bf5311c8bf3e2ee27ce9613e37.jpg",
+        "table_caption": [
+            "Table 2: Variational lower-bound estimates on the log-density of the Gaussian DBNs (higher is better). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>nats</td><td rowspan=1 colspan=1>No CMU training</td><td rowspan=1 colspan=1>CMUw/olabels</td><td rowspan=1 colspan=1>CMUw/labels</td></tr><tr><td rowspan=1 colspan=1>Caltech Train</td><td rowspan=1 colspan=1>617±0.4</td><td rowspan=1 colspan=1>627±0.5</td><td rowspan=1 colspan=1>569±0.6</td></tr><tr><td rowspan=1 colspan=1>Caltech Valid</td><td rowspan=1 colspan=1>512±1.1</td><td rowspan=1 colspan=1>503±1.8</td><td rowspan=1 colspan=1>494±1.7</td></tr><tr><td rowspan=1 colspan=1>CMUTrain</td><td rowspan=1 colspan=1>96±0.8</td><td rowspan=1 colspan=1>499±0.1</td><td rowspan=1 colspan=1>594±0.5</td></tr><tr><td rowspan=1 colspan=1>CMUValid</td><td rowspan=1 colspan=1>85±0.5</td><td rowspan=1 colspan=1>387±0.3</td><td rowspan=1 colspan=1>503±0.7</td></tr><tr><td rowspan=1 colspan=1>log2</td><td rowspan=1 colspan=1>454.6</td><td rowspan=1 colspan=1>687.8</td><td rowspan=1 colspan=1>694.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "The main advantage of our model is that it can learn on large images of faces without localization label information (no manual cropping required). To demonstrate, we use both the Caltech and the CMU faces dataset. For the CMU faces, a subset of 2526 frontal faces with ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "ground truth labels are used. We split the Caltech dataset into a training and a validation set. For the CMU faces, we first took $10 \\%$ of the images as training cases for the ConvNet for approximate inference. This is needed due to the completely different backgrounds of the Caltech and CMU datasets. The remaining $90 \\%$ of the CMU faces are split into a training and validation set. We first trained a GDBN with $\\bar { 1 0 2 4 } \\mathbf { h } ^ { 1 }$ and $2 5 6 \\mathbf { h } ^ { 2 }$ hidden units on the Caltech training set. We also trained a ConvNet for approximate inference using the Caltech training set and $10 \\%$ of the CMU training images. ",
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+    {
+        "type": "image",
+        "img_path": "images/560194f3ef509e3587acead12e7f26ead1c64290fdf8d9bdc8610d5942576721.jpg",
+        "image_caption": [
+            "Figure 7: Left: Conditioned on different v will result in a different $\\triangle \\mathbf { u }$ . Note that the initial $\\mathbf { u }$ is exactly the same for two trials. Right: Additional examples. The only difference between the top and bottom panels is the conditioned v. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows the estimates of the variational lower-bounds on the average log-density (higher is better) that the GDBN models assign to the ground-truth cropped face images from the training/test sets under different scenarios. In the left column, the model is only trained on Caltech faces. Thus it gives very low probabilities to the CMU faces. Indeed, GDBNs achieve a variational lower-bound of only 85 nats per test image. In the middle column, we use our approximate inference to estimate the location of the CMU training faces and further trained the GDBN on the newly localized faces. This gives a dramatic increase of the model performance on the CMU Validation $\\mathrm { s e t } ^ { 6 }$ , achieving a lowerbound of 387 nats per test image. The right column gives the best possible results if we can train with the CMU manual localization labels. In this case, GDBNs achieve a lower-bound of 503 nats. We used Annealed Importance Sampling (AIS) to estimate the partition function for the top-layer RBM. Details on estimating the variational lower bound are in the supplementary materials. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Fig. 6(a) further shows samples drawn from the Caltech trained DBN, whereas Fig. 6(b) shows samples after training with the CMU dataset using estimated u. Observe that samples in Fig. 6(b) show a more diverse set of faces. We trained GDBNs using a greedy, layer-wise algorithm of [1]. For the top layer we use Fast Persistent Contrastive Divergence [32], which substantially improved generative performance of GDBNs (see supplementary material for more details). ",
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+    {
+        "type": "text",
+        "text": "6.3 Inference with ambiguity ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our attentional mechanism can also be useful when multiple objects/faces are present in the scene. Indeed, the posterior $p ( \\mathbf { u } | \\mathbf { x } , \\mathbf { v } )$ is conditioned on v, which means that where to attend is a function of the canonical object $\\mathbf { v }$ the model has in “mind” (see Fig. 2(b)). To explore this, we first synthetically generate a dataset by concatenating together two faces from the Caltech dataset. We then train approximate inference ConvNet as in Sec. 4.1 and test on the held-out subjects. Indeed, as predicted, Fig. 7 shows that depending on which canonical image is conditioned, the same exact gaze initialization leads to two very different gaze shifts. Note that this phenomenon is observed across different scales and location of the initial gaze. For example, in Fig. 7, right-bottom panel, the initialized yellow box is mostly on the female’s face to the left, but because the conditioned canonical face $\\mathbf { v }$ is that of the right male, attention is shifted to the right. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper we have proposed a probabilistic graphical model framework for learning generative models using attention. Experiments on face modeling have shown that ConvNet based approximate inference combined with HMC sampling is sufficient to explore the complicated posterior distribution. More importantly, we can generatively learn objects of interest from novel big images. Future work will include experimenting with faces as well as other objects in a large scene. Currently the ConvNet approximate inference is trained in a supervised manner, but reinforcement learning could also be used instead. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The authors gratefully acknowledge the support and generosity from Samsung, Google, and ONR grant N00014-14-1-0232. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            104,
+            251,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "tation, 18(7):1527–1554, 2006.   \n[2] R. Salakhutdinov and G. Hinton. Deep Boltzmann machines. In AISTATS, 2009.   \n[3] Geoffrey E. Hinton, Peter Dayan, and Michael Revow. Modeling the manifolds of images of handwritten digits. IEEE Transactions on Neural Networks, 8(1):65–74, 1997.   \n[4] Daniel Zoran and Yair Weiss. From learning models of natural image patches to whole image restoration. In ICCV. IEEE, 2011.   \n[5] Yoshua Bengio, Li Yao, Guillaume Alain, and Pascal Vincent. Generalized denoising auto-encoders as generative models. In Advances in Neural Information Processing Systems 26, 2013.   \n[6] H. Lee, R. Grosse, R. Ranganath, and A. Y. Ng. Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations. In ICML, pages 609–616, 2009. [7] Marc’Aurelio Ranzato, Joshua Susskind, Volodymyr Mnih, and Geoffrey Hinton. On Deep Generative Models with Applications to Recognition. In CVPR, 2011. [8] Yichuan Tang, Ruslan Salakhutdinov, and Geoffrey E. Hinton. Deep mixtures of factor analysers. In ICML. icml.cc / Omnipress, 2012.   \n[9] M. I. Posner and C. D. Gilbert. Attention and primary visual cortex. Proc. of the National Academy of Sciences, 96(6), March 1999.   \n[10] E. A. Buffalo, P. Fries, R. Landman, H. Liang, and R. Desimone. A backward progression of attentional effects in the ventral stream. PNAS, 107(1):361–365, Jan. 2010.   \n[11] N Kanwisher and E Wojciulik. Visual attention: Insights from brain imaging. Nature Reviews Neuroscience, 1:91–100, 2000.   \n[12] C. H. Anderson and D. C. Van Essen. Shifter circuits: A computational strategy for dynamic aspects of visual processing. National Academy of Sciences, 84:6297–6301, 1987.   \n[13] B. A. Olshausen, C. H. Anderson, and D. C. Van Essen. A neurobiological model of visual attention and invariant pattern recognition based on dynamic routing of information. The Journal of neuroscience : the official journal of the Society for Neuroscience, 13(11):4700–4719, 1993.   \n[14] Laurenz Wiskott. How does our visual system achieve shift and size invariance?, 2004.   \n[15] S. Chikkerur, T. Serre, C. Tan, and T. Poggio. What and where: a Bayesian inference theory of attention. Vision Research, 50(22):2233–2247, October 2010.   \n[16] J. K. Tsotsos, S. M. Culhane, W. Y. K. Wai, Y. H. Lai, N. Davis, and F. Nuflo. Modeling visual-attention via selective tuning. Artificial Intelligence, 78(1-2):507–545, October 1995.   \n[17] Hugo Larochelle and Geoffrey E. Hinton. Learning to combine foveal glimpses with a third-order boltzmann machine. In NIPS, pages 1243–1251. Curran Associates, Inc., 2010.   \n[18] D. P. Reichert, P. Seriès, and A. J. Storkey. A hierarchical generative model of recurrent object-based attention in the visual cortex. In ICANN (1), volume 6791, pages 18–25. Springer, 2011.   \n[19] B. Alexe, N. Heess, Y. W. Teh, and V. Ferrari. Searching for objects driven by context. In NIPS 2012, December 2012.   \n[20] Marc’Aurelio Ranzato. On learning where to look. arXiv, arXiv:1405.5488, 2014.   \n[21] M. Denil, L. Bazzani, H. Larochelle, and N. de Freitas. Learning where to attend with deep architectures for image tracking. Neural Computation, 28:2151–2184, 2012.   \n[22] G. E. Hinton and R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313:504–507, 2006.   \n[23] A. Krizhevsky. Learning multiple layers of features from tiny images. Master’s thesis, University of Toronto, Toronto, Ontario, Canada, 2009.   \n[24] Graham W. Taylor, Rob Fergus, Yann LeCun, and Christoph Bregler. Convolutional learning of spatiotemporal features. In ECCV 2010. Springer, 2010.   \n[25] A. Mohamed, G. Dahl, and G. Hinton. Acoustic modeling using deep belief networks. IEEE Transactions on Audio, Speech, and Language Processing, 2011.   \n[26] Richard Szeliski. Computer Vision - Algorithms and Applications. Texts in Computer Science. Springer, 2011.   \n[27] S. Duane, A. D. Kennedy, B. J Pendleton, and D. Roweth. Hybrid Monte Carlo. Physics Letters B, 195(2):216–222, 1987.   \n[28] R. M. Neal. MCMC using Hamiltonian dynamics. in Handbook of Markov Chain Monte Carlo (eds S. Brooks, A. Gelman, G. Jones, XL Meng). Chapman and Hall/CRC Press, 2010.   \n[29] Simon Baker and Iain Matthews. Lucas-kanade 20 years on: A unifying framework. International Journal of Computer Vision, 56:221–255, 2002.   \n[30] Ralph Gross, Iain Matthews, Jeffrey F. Cohn, Takeo Kanade, and Simon Baker. Multi-pie. Image Vision Comput., 28(5):807–813, 2010.   \n[31] J. P. Lewis. Fast normalized cross-correlation, 1995.   \n[32] T. Tieleman and G. E. Hinton. Using fast weights to improve persistent contrastive divergence. In Proceedings of the 26th Annual International Conference on Machine Learning, ICML 2009, volume 382, page 130. ACM, 2009. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/be71cbc5181417920d82329590b5f9235fd959aff334cd817ca0e56060d77e56.jpg",
+        "image_caption": [
+            "Figure 8: A visual diagram of the convolutional net used for approximate inference. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            331,
+            99,
+            661,
+            422
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            472,
+            276,
+            488
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Convolutional Neural Network ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            508,
+            390,
+            523
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The training of ConvNet for approximate inference is standard and did not involve any special ’tricks’. We used SGD with minibatch size of 128 samples. We used a standard ConvNet architecture with convolution C layers followed by max-pooling S layers. The ConvNet takes as input $\\mathbf { x }$ and $\\mathbf { v }$ to predict change in u such that to maximize $\\log p ( { \\bf u } | { \\bf x } , \\bar { \\bf v } )$ . In order to better predict change of u, $\\mathbf { x }$ as well as a bigger border around $\\mathbf { x }$ are used as the input to the ConvNet. Therefore, $\\mathbf { x }$ has resolution $7 2 \\times 7 2$ and $\\mathbf { v }$ has resolution of $2 4 \\times 2 4$ . ",
+        "bbox": [
+            174,
+            536,
+            825,
+            599
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Two handle two different inputs with different resolutions, two different “streams\" are used in this ConvNet architecture. One stream will process $\\mathbf { x }$ and another one for $\\mathbf { v }$ . These two streams will be combined multiplicatively after subsampling the $\\mathbf { x }$ stream by a factor of 3. The rest of the ConvNet is same as the standard classification ConvNets, except that we use mean squared error as our cost function. See Figure 8 for a visual diagram of what the convolutional neural network architecture used. ",
+        "bbox": [
+            174,
+            606,
+            825,
+            669
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/f5811a694918a2006ebd699fd2cf5b2fb12b59544d1d4805529cbab7d3e08c87.jpg",
+        "table_caption": [
+            "Table 3: Model architectures of the convolutional neural network used during approximate inference. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>layer</td><td>type</td><td>latent variables</td><td>filter size</td><td># weights</td></tr><tr><td>0</td><td>input x</td><td>maps:3 72x72</td><td></td><td>=</td></tr><tr><td>1</td><td>input v</td><td>maps:3 24x24</td><td>-</td><td>=</td></tr><tr><td>2</td><td>Conv of layer 0</td><td>maps:16 66x66</td><td>7x7</td><td>2352</td></tr><tr><td>3</td><td>Pooling</td><td>maps:16 22x22</td><td>3x3</td><td>-</td></tr><tr><td>4</td><td>Conv of layer 1</td><td>maps:16 22x22</td><td>5x5</td><td>1200</td></tr><tr><td>5</td><td>Combine layers 3,4</td><td>maps:16 22x22</td><td>-</td><td>=</td></tr><tr><td>6</td><td>Fully connected</td><td>1024</td><td></td><td>7.9M</td></tr><tr><td>7</td><td>Fully connected</td><td>4</td><td>=</td><td>4K</td></tr></table>",
+        "bbox": [
+            274,
+            691,
+            720,
+            819
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 3 details the exact model architecture used. In layer 5, the two streams have the same number of hidden maps and hidden topography. We combine these two multiplicatively by multiplying their activation elementwise. This creates a third-order flavor and is more powerful for the task of determining where to shift attention to next. ",
+        "bbox": [
+            174,
+            872,
+            823,
+            922
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gaussian Deep Belief Network ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            387,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The training of the Gaussian Deep Belief Network is performed in a standard greedy layerwise fashion. The first layer Gaussian Restricted Boltzmann Machine is trained with the Contrastive Divergence algorithm where the standard deviation of each visible unit is learned as well. After training the first layer, we use Eq. 3 to obtain first hidden layer binary probabilities. We then train a 2nd binary-binary Restricted Boltzmann Machine using the fast persistent contrastive divergence learning algorithm. This greedy training leads us to the Gaussian Deep Belief Network. No finetuning of the entire network is performed. ",
+        "bbox": [
+            174,
+            128,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Quantitative evaluation for Gaussian Deep Belief Network ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            218,
+            580,
+            233
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For quantitative evaluation, we approximate the standard variational lower bound on the log likelihood of the Gaussian Deep Belief Network. The model is a directed model: ",
+        "bbox": [
+            173,
+            241,
+            823,
+            267
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2347eaa180410b3be5330c72cdc6da20d58b14223e5d9f52be46e7f60157c497.jpg",
+        "text": "$$\np ( { \\mathbf { v } } , \\mathbf { h } ^ { 1 } , \\mathbf { h } ^ { 2 } ) = p ( { \\mathbf { v } } | \\mathbf { h } ^ { 1 } ) p ( { \\mathbf { h } } ^ { 1 } , \\mathbf { h } ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            273,
+            602,
+            291
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For any approximating posterior distribution $q ( \\mathbf { h } ^ { 1 } | \\mathbf { v } )$ , the GDBN’s log-likelihood has this lower variational ",
+        "bbox": [
+            176,
+            305,
+            825,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f9027ef85bd84555c1b549947877b93ccb9a47e17e27122624b7c8343e50b0c2.jpg",
+        "text": "$$\n\\log \\sum _ { \\mathbf { h } ^ { 1 } } p ( \\mathbf { v } , \\mathbf { h } ^ { 1 } ) \\geq \\sum _ { \\mathbf { h } ^ { 1 } } q ( \\mathbf { h } ^ { 1 } | \\mathbf { v } ) [ \\log p ( \\mathbf { v } | \\mathbf { h } ^ { 1 } ) + \\log p ^ { * } ( \\mathbf { h } ^ { 1 } ) ] - \\log Z + \\mathcal { H } ( q ( \\mathbf { h } ^ { 1 } | \\mathbf { v } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            327,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The entropy $\\mathcal { H } ( q ( \\mathbf { h } ^ { 1 } | \\mathbf { v } ) )$ can be calculated since we made the factorial assumption on $q ( \\mathbf { h } ^ { 1 } | \\mathbf { v } )$ . ",
+        "bbox": [
+            169,
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+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3dc633ce36b448b5fdaa3eccfec6f59279ae6182cbcd466e03ebc4b9555a554f.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\log p ( { \\bf v } | { \\bf h } ^ { 1 } ) = - \\sum \\log \\sigma _ { i } - \\frac { D } { 2 } \\log 2 \\pi - \\frac { 1 } { 2 } \\sum _ { i } ^ { D } \\frac { ( x - \\mu ) ^ { 2 } } { \\sigma _ { i } ^ { 2 } } } } \\\\ { { \\displaystyle \\log p ^ { * } ( { \\bf h } ^ { 1 } ) = { \\bf b } ^ { \\mathsf { T } } { \\bf h } ^ { 1 } + \\log \\sum _ { j } \\exp \\{ { \\bf h } ^ { 1 } { \\bf W } _ { j } + c _ { j } \\} } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In order to calculate the expectation of the approximating posteriors, we use Monte Carlo sampling. ",
+        "bbox": [
+            169,
+            468,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/582af4bfdf227ef139ab5b9a53879ee82e9767ff1e8d843d95dcd974d8644f77.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\sum _ { \\mathbf { h } ^ { 1 } } q ( \\mathbf { h } ^ { 1 } | \\mathbf { v } ) \\log p ^ { * } ( \\mathbf { v } , \\mathbf { h } ^ { 1 } ) \\approx \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\log p ^ { * } ( \\mathbf { v } , \\mathbf { h } ^ { 1 ( m ) } ) } \\ ~ } \\\\ { { \\displaystyle = - \\sum _ { i = 1 } ^ { D } \\log \\sigma _ { i } - \\frac { D } { 2 } \\log 2 \\pi - \\frac { 1 } { 2 } \\sum _ { i } ^ { D } \\frac { ( x - \\mu ) ^ { 2 } } { \\sigma _ { i } ^ { 2 } } } \\ ~ } \\\\ { { \\displaystyle ~ + \\mathbf { b } ^ { \\mathsf { T } } \\mathbf { h } ^ { 1 } + \\log \\sum _ { j } \\exp \\{ \\mathbf { h } ^ { 1 } \\mathbf { W } _ { j } + c _ { j } \\} } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            488,
+            722,
+            603
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { h } ^ { 1 ( m ) }$ is the $m$ -th sample from the posterior $q ( \\mathbf { h } ^ { 1 } | \\mathbf { v } )$ ",
+        "bbox": [
+            174,
+            609,
+            531,
+            625
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In order to calculate the partition function of the top-most layer of the GDBN, we use Annealed Importance Sampling (AIS). We used 100 chains with 50,000 intermediate distributions to estimate the partition function of the binary-binary RBM which forms the top layer of the GDBN. Even though AIS is an unbiased estimator of the partition function, it is prone to under-estimating it due to bad mixing of the chains. This causes the log probability to be over-estimated. Therefore the variational lower bounds reported in our paper are not strictly guaranteed to be lower bounds and are subject to errors. However, we believe that the margin of error is unlikely to be high enough to affect our conclusions. ",
+        "bbox": [
+            174,
+            630,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Additional Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            733,
+            305,
+            748
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We present some more examples of inference process of our framework. ",
+        "bbox": [
+            174,
+            758,
+            599,
+            772
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Below, we show some success cases and a failure case for inference on the CMU Multi-PIE dataset. The initial gaze variables of $\\mathbf { u }$ are highlighted in yellow and later iterations are highlighted with color gradually changing to blue. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/22135e80c1fa41d282fe74bc86f2b896a1c7808c289b962886426cad59ce6f39.jpg",
+        "image_caption": [
+            "Figure 9: Example of an approximate inference steps. v is $2 4 { \\times } 2 4$ , $\\mathbf { x }$ is $7 2 { \\times } 7 2$ . Approximate inference quickly finds a good initialization for u, while HMC makes small adjustments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            199,
+            789,
+            448
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a8956de8d0141024c2d7ab80524190d6ce85a39211d2a5f295dcca994434be5.jpg",
+        "image_caption": [
+            "Figure 10: E-step for learning on CMU Multi-PIE. (a),(b),(c) are successful. (d) is a failure case. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
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+        ],
+        "page_idx": 11
+    }
+]

parse/train/BM64dm9HvN/BM64dm9HvN.md

@@ -0,0 +1,395 @@
+# Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+![](images/4aea6773a272b2fa49bcf2c895f5c0d84686c92df2e836446e6be0f770b0a17c.jpg)  
+Figure 1: Our proposal.
+
+# Abstract
+
+1 The training of neural networks is usually monitored with a validation (holdout)   
+2 set to estimate the generalization of the model. This is done instead of measuring   
+3 intrinsic properties of the model to determine whether it is learning appropriately.   
+4 In this work, we suggest studying the training of neural networks with Algebraic   
+5 Topology, specifically Persistent Homology (PH). Using simplicial complex repre  
+6 sentations of neural networks, we study the PH diagram distance evolution on the   
+7 neural network learning process with different architectures and several datasets.   
+8 Results show that the PH diagram distance between consecutive neural network   
+9 states correlates with the validation accuracy, implying that the generalization error   
+10 of a neural network could be intrinsically estimated without any holdout set.
+
+# 11 1 Introduction
+
+12 Generalization is what makes a machine learning model useful; the uncertainty of its behaviour with   
+13 unseen data is what makes it potentially dangerous. Thus, understanding the generalization error of a   
+14 model can be considered one of the holy grails of the entire machine learning field.   
+15 Machine learning practitioners typically monitor some metrics of the model to estimate its generaliza  
+16 tion error and stop the training even before the numerical convergence to prevent the overfitting of   
+17 the model. Usually, the error measure or the metric relevant to the task is computed for a holdout   
+18 set, the validation set. Since these data have not been directly used for updating the parameters, it   
+19 is assumed that the performance of the model on the validation set can be used as a proxy of the
+
+generalization error, provided it is representative of the data that will be used in inference. One can, though, potentially overfit to this holdout set if is repeatedly used for guiding a hyperparameter search.
+
+23 Instead of relying on an external set, though, the question of whether it could be possible to estimate   
+24 the generalization error with some intrinsic property of the model is highly relevant, and it has been   
+25 barely explored in the literature. On the other hand, Algebraic Topology has recently been gaining   
+26 momentum as a mathematical tool for studying graphs, machine learning algorithms, and data.   
+27 In this work, we have the goal of, once having characterized neural networks as weighted, acyclic   
+28 graphs, represented as Algebraic Topology objects (following previous works), computing distances   
+29 between consecutive neural network states. More specifically, we can calculate the Persistent   
+30 Homology (PH) diagram distances between a give state (i.e., when having a specific weights during   
+31 the training process) and the next one (i.e., after having updated the weights in a training step) (see   
+32 Figure 1. We observe that during the training procedure of neural networks we can measure this   
+33 distance in each learning step, and show that there exists a high correlation with the corresponding   
+34 validation accuracy of the model. We do so in a diverse set of deep learning benchmarks and model   
+35 hyperparameters. This shines light on the question of whether the generalization error could be   
+36 estimated from intrinsic properties of the model, and opens the path towards a better theoretical   
+37 understanding of the dynamics of the training of neural networks.
+
+38 In summary, our contributions are as follows:
+
+• Based on principles of Algebraic Topology, we propose measuring the distances (Silhouette and Heat) between the PH persistence diagrams obtained from a given state of a neural network during the training procedure and the one in the immediately previous weights update.   
+• We empirically show that the evolution of these measures during training correlate with the accuracy in the validation set. We do so in diverse benchmarks (MNIST, CIFAR10, CIFAR100, Reuters text classification), and models (MLPs in MNIST and Reuters, MLPs and CNNs in CIFAR100 and CIFAR100).   
+• We thus provide empirical proof of the fact that valuable information related to the learning process of neural networks can be obtained from PH distances between persistence diagrams (homological convergence). In particular, we show that homological convergence is related to learning process and the generalization properties of neural networks.   
+• In practice, we provide a new tool for monitoring the training of neural networks, and open the path to estimating their generalization error without a validation set.
+
+53 The remainder of this article is as follows. In Section 2 we describe the theoretical background of our   
+54 proposal in terms of Algebraic Topology, while in Section 3 we go through the related work. Then, in   
+55 Section 4 we formalize our method. Finally, in sections 6 and 7 we present and discuss our empirical   
+56 results, respectively.
+
+# 57 2 Background
+
+58 In this section we introduce the mathematical foundations of this paper. A detailed mathematical   
+59 description is included in the Supplementary Material.   
+60 A simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional   
+61 counterparts, named simplex $( K )$ . In particular, a simplicial complex must comply with two properties:   
+62 1. Every face of a simplex is also in the simplicial complex (of lower dimension). 2. The non-empty   
+63 intersection of any two simplices contained on a simplicial complex is a face of both. 0,1,2,3-simplex   
+64 and non simplex examples are shown in Figure 2.   
+65 We can associate to an undirected graph, $G = \left( V , E \right)$ , a simplicial complex where all the vertices   
+66 of $\mathbf { G }$ are the 0-simplex of the simplicial complex and the complete subgraphs with i vertices, in $G$   
+67 corresponds to a $( i - 1 )$ -simplex. This type of construction is usually called a complex clique on the   
+68 graph G, and is denoted by $C l ( G )$ . Figure 3 shows a graph clique complex Cl(G) example.   
+69 The boundary function is defined as a map, from   
+70 an $i$ -simplex to an $( i - 1 )$ -simplex, as the sum   
+71 of its $( i - 1 )$ -dimensional faces. A boundary   
+72 function sample is shown in Figure 4.   
+73 In algebraic topology, a $k$ -chain is a combination   
+74 of $k$ -simplices (sometimes symbolized as a lin  
+75 ear combination of simplices that compose the   
+76 chain). The boundary of a $k$ -chain is a $\left( k - 1 \right)$ -   
+77 chain. It is the linear and signed combination of   
+78 chain element boundary simplices. The space of   
+79 $i$ -chains is denoted by $C _ { i } ( K )$ .
+
+![](images/de4681b948ab23cf4e4885d2d41c848915506c2dff51424de01cfbc429c8ad93.jpg)  
+Figure 2: Simplex and non-simplex examples.
+
+![](images/5f7df6fe10b6faf7e31994260d5040989cac5974f6905f8df0a71dd4966c3c07.jpg)  
+Figure 4: Boundary function sample.
+
+![](images/c3b577eefcf443015365ad03338552041223c006305dfa5938eb382f2ab82c4f.jpg)  
+Figure 3: Graph clique complex Cl(G) example.
+
+80 There are two special cases of chains that will be useful to define homology:
+
+• Closed chain or $i$ -cycle: $i .$ -chain with empty boundary. An $i$ -chain $c$ is an $i$ -cycle if and only if $\partial _ { i } c = 0$ , i.e. $c \in \dot { k e r } ( \partial _ { i } )$ . This subspace of $C _ { i } ( K )$ is denoted as $\mathbb { Z } _ { i } ( K )$ . • Exact chain or $i$ -boundary: An $i \cdot$ -chain $c$ is an $i \cdot$ -boundary if there exists an $( i + 1 )$ -chain $d$ such that $c = \partial _ { i + 1 } ( d )$ , i.e. $c \in i m ( \partial i + 1 )$ . This subspace of $C _ { i } ( K )$ , the set of all such i-boundaries forms, is denoted by $\mathbb { B } _ { i } ( K )$ .
+
+86 Now, if we think in the $i$ -cycles that do not bound an $( i + 1 )$ -simplicial complex, this is the definition   
+87 $i$ -th homology of the simplicial complex $K$ . The precise definition is the quotient space of $\mathbb { B } _ { i } ( K )$ a   
+88 subspace of $\mathbb { Z } _ { i } ( K )$ (see Supplementary Material). The number of non equivalent $i \cdot$ -cycles (Figure 5)   
+89 is the dimension of the homology group $H _ { i } ( K )$ , also named Betti numbers.
+
+We can create a nested family of simplicial complexes, $K _ { \varepsilon }$ , where at each step $t$ , $K _ { \varepsilon _ { t } }$ is embedded in the simplicial complex $K _ { \varepsilon _ { t + 1 } }$ . We call this set a simplicial complex filtration.
+
+![](images/c673fa13230b91d7c3a60512e7dd55d64bd3ab3d1e9d2c21d54d4c5d49323ef4.jpg)  
+Figure 5: The two blue dashed cycles are homologically equivalent, the pink isn’t.
+
+For each filtration simplicial complex, we can calculate the homology groups. Then, we can look at the birth, that is, when a homology class appears, and death, the time when the homology class disappears. The PH treats the birth and the death of these homological features in $K _ { \varepsilon }$ for different ε values. The lifespan of each homological feature can be represented as an interval $( b \bar { i } r t h , d e a t h )$ , of the homological class. Given a filtration, this collection of intervals is named a Persistence Diagram (PD) [5].
+
+105 It is possible to compare two PDs using specific distances (Wasserstein and Bottleneck). To efficiently   
+106 perform this operation, due to the size of these diagrams, it is sometimes necessary to simplify them   
+107 by means of a discretization process (such as Weighted Silhouette and Heat vectorizations).
+
+# 108 3 Related Work
+
+109 Algebraic Topology and Machine Learning The use of Algebraic Topology in the fields of   
+110 data science and machine learning has been gaining momentum in recent years (see Carlsson [5]).   
+111 Specifically in the case of neural networks, some works have applied topology for improving the   
+112 training procedure of the models [15, 8], or pruning the model afterwards [30]. Other works have   
+113 focused on analyzing the capacity of neural networks [14, 26, 17] or the complexity of input data   
+114 [17]. Furthermore, recent works have provided topological analysis of the decision boundaries of   
+115 classifiers based on PH and Betti numbers [24, 22].   
+116 Graph and topological representations of neural networks Gebhart et al. [12] suggest a method   
+117 for computing the PH over the graphical activation neural networks, while Watanabe and Yamana   
+118 [29] propose representing neural networks via simplicial complexes based on Taylor decomposition,   
+119 from which one can compute the PH. Chowdhury et al. [7] show that directed homology can be used   
+120 to represent MLPs. Anonymous [2] concurrently show neural networks, when represented as directed,   
+121 acyclic graphs, can be associated to an Algebraic Topology object. By computing the PH diagram,   
+122 one can effectively characterize neural networks, and even compute distances between two given   
+123 neural networks, which can be used to measure their similarity. This is unlike other works [11, 13]   
+124 approximating neural networks representations with regard to the input space.   
+125 Estimating the generalization and studying the learning process We are, though, specifically   
+126 interested in the use of PH for analyzing the learning process, especially with the goal of estimating   
+127 generalization. In this regard, the literature is perhaps more limited. Jiang et al. [16] work on   
+128 understanding what drives generalization in deep networks from a Bayesian of view. Neyshabur et al.   
+129 [23] study the generalization gap prediction from the training data and network parameters using a   
+130 margin distribution, which are the distances of training points to the decision boundary. In Li et al.   
+131 [21], authors propose an alternative to cross-validation for model selection based on training once on   
+132 the whole train set, without any data split, deriving a validation set with data augmentation.   
+133 Corneanu et al. [10] try to estimate the performance gap between training and testing using PH   
+134 measures. They claim. However, one can observe some caveats. The first one is that their regression   
+135 fitted to predict the test error has a considerably high error, making it not usable in practice. The   
+136 second caveat is that for fitting the regression one needs at least part of the sequestered testing set.   
+137 In this work, motivated by the interest of having a better understanding of whether it would be   
+138 possible to estimate the generalization of neural networks without a holdout set, we suggest using the   
+139 topological characterization and distances concurrently proposed in Anonymous [2] but, crucially,   
+140 measured between consecutive weight updates. We will show that the evolution of this distance   
+141 is similar to the one of the validation accuracy. Unlike Li et al. [21], we do not use any data at   
+142 all. Unlike [10], we do not build a statistical or machine learning model (linear regression) for   
+143 predicting the testing error. Instead, we propose a new measure, and we empirically show that it   
+144 highly correlates with the validation accuracy. Note that in this work we do not work with any input   
+145 data and activations, but with the parameters of the neural network themselves.
+
+# 146 4 Approach
+
+Representation For representing neural networks as graphs, we follow the approach proposed concurrently in Anonymous [2]. We associate to the neural network, at each learning state (defined by its weights), a weighted directed graph that is analyzed as an abstract simplicial complex. It is important to note that abstract simplicial complex are used in opposition to geometric simplicial complex.
+
+152 For every training state, neural network connections are considered as directed and weighted edges   
+153 between neurons, represented by graph nodes. Biases are considered as new edges that join to isolate   
+154 vertices. In this representation, activation functions are lost. Bias information could also have been   
+155 ignored because, as we will see, it is not very informative in terms of homology, but we decided to   
+56 preserve it.   
+157 Negative edge weights are represented with reverse edges with the same weight absolute value. We   
+158 discard the use of weight absolute value as neural networks are not invariant under weight sign   
+159 transformations. This representation is consistent with the fact that every neuron can be replaced by a   
+160 neuron from which two edges with opposite weights emerge and converge again on another neuron   
+161 with opposite weights. From an homological point of view, this would be represented as a closed   
+162 cycle. Weights are normalized following the Equation 1. $\zeta$ is an smoothing parameter that we set to   
+163 1e-6. This smoothing parameter is necessary as we want to avoid normalized weights of edges to be   
+164 0 (in our representation 0 implies a lack of connection):
+
+$$
+m a x ( 1 - \frac { | w | } { m a x ( | m a x ( W ) | , | m i n ( W ) | ) } , \zeta )
+$$
+
+165 Algebraic Topology object For each weighted directed graph associated with the state of a neural   
+166 network, we link a directed flag complex to it. The topological properties of this directed flag complex   
+167 are studied using homology groups $H _ { n }$ . We calculate the homology groups up to degree 3 $\left( H _ { 0 } – H _ { 3 } \right)$ .   
+168 For each state, we use a family of simplicial complexes, $K _ { \varepsilon }$ , for a range of values of $\varepsilon \in \mathbb { R }$ . The   
+169 simplicial complex at step $\varepsilon _ { t }$ is embedded in the complex at $\varepsilon _ { t + 1 }$ , for $\varepsilon _ { t } \leq \varepsilon _ { t + 1 }$ , i.e. $K _ { \varepsilon } \subseteq K _ { \varepsilon _ { t + 1 } }$ . $\varepsilon$   
+170 is used as a filter that establish the minimum weight of the graph representation edges included on   
+171 the simplicial complex. This collection of contained simplicial complex (associated to a directed   
+172 weighted graph), called filtration, $K _ { \varepsilon _ { m i n } } \subseteq \ldots \subseteq K _ { \varepsilon _ { t } } \subseteq K _ { \varepsilon _ { t + 1 } } \subseteq \ldots \subseteq K _ { \varepsilon _ { m a x } }$ , where $t \in [ 0 , 1 ]$ and $\varepsilon _ { m i n } = 0$   
+173 $\varepsilon _ { m a x } = 1$ (remember that edge weights are normalized).   
+174 The sequence of homology groups is calculated by varying the $\varepsilon$ parameter to obtain the persistence   
+175 homology diagram. In our case, persistent homology calculations are performed on $\mathbb { Z } _ { 2 }$ . In other   
+176 words, once the corresponding filter has been applied to the weight of the edges, all connected edges   
+177 are considered equally.   
+178 Distances between persistence diagrams of consecutive states In this paper, we are interested in   
+179 comparing PDs between different simplicial complex associated to each training state of the neural   
+180 network. There are two distances traditionally used to compare PDs, Bottleneck distance (the length   
+181 of the longest edge) and Wasserstein distance (using the sum of all edges lengths, instead of the   
+182 maximum). Their stability with respect to perturbations on PDs has been object of different studies   
+183 [6, 9].   
+184 In order to make computations feasible and obviate noisy intervals, we filter the PDs by limiting   
+185 the minimum PD interval size. We do so by setting a minimum threshold $\eta = 0 . 0 1$ . Intervals with   
+186 a lifespan under this value are not considered (spurious homological features). Additionally, for   
+187 computing distances, we need to remove infinity values. As we are only interested in the deaths until   
+188 the maximum weight value, we replace all the infinity values by 1.0.   
+189 In our case, our neural networks have millions of persistence intervals per Persistence Diagram,   
+190 while Wasserstein distance calculations are computationally hard for large PDs. In order to make   
+191 calculations computationally feasible, we will use a vectorized version of PDs, also called PD   
+192 discretization. This vectorized version summaries have been proposed and used on recent literature   
+193 [1, 3, 4, 19, 25]. For persistence diagram distance calculation, we use weighted Silhouette and Heat   
+194 vectorizations, using the Giotto-TDA library [27].
+
+# 95 5 Experiments
+
+Data We validate our method in several heterogeneous (vision, natural language), well-known datasets, namely 1. MNIST [20], 2. CIFAR-10, 3. CIFAR-100 [18], and 4. the Reuters dataset [28] (multi-class and multi-label document classification dataset).
+
+Models We experiment with two neural architectures,1. MLPs and 2. CNNs. In the latter case, we use the convolutional layers as a pre-trained model with frozen weights, and we learn an MLP on top of it. The reason we do so is that our method is based in a representation that, at least in the basic form, does not allow capturing information from convolutional layers. Thus, we need a single (exact
+
+203 same weights) feature extractor, to abstract away distances related to the CNN layers and focus on   
+204 the MLP.
+
+Conducted experiments We define the base MLP architecture as {Input, Linear(512), Dropout(0.2), Linear(512), Dropout(0.2), Output}. In the case of CNNs, the pre-trained model is defined as 3 convolutional blocks with kernel size 3 (starting with 32 channels), interleaved with max pooling (its linear layers are thrown away after the pre-training). On top of the pre-trained CNN, we also define the same base MLP architecture. Then, for each dataset and model (MLP and CNN), we experiment with varying (while keeping the rest fixed to the base architecture)
+
+1. Layer size (number of units per layer): 4, 16, 32, 128, 256.   
+2. Number of labels (the other classes are removed): 2, 4, 6, 8, 10.   
+3. Learning rate: 1e-e05, 0.0001, 0.001, 0.01, 0.1   
+4. Dropout: 0.0, 0.2, 0.4, 0.5, 0.8.   
+5. Input order: 5 random input orders. As a control experiment, for each analyzed problem we run the same configuration with 5 different input orders. If the measured distances are, indeed, related with the learning process of neural networks, these variations should not have any noticeable effect.
+
+We run each configuration 5 times with different random seeds (and, thus, weight initializations1) to see if the results are consistent across runs. All models are trained with the RMSProp optimizer with 221 a batch size of 256.
+
+Distances and validation accuracy computation Note that homological distances are obtained at the end of each batch, while validation metrics are only computed on each epoch. The methodology we follow to analyze the learning process on each different problem can be summarized with the following steps:
+
+1. In each training step (i.e., for each batch) we extract the weights from the MLP current state and use them to build an abstract simplicial complex from the associated weighted directed graph.   
+2. We calculate the homological persistence diagram of the simplicial complex.   
+3. We then calculate the distance between consecutive persistence diagrams (we will call this sequence homological convergence). We use two different distances, namely, Heat and Silhouette.   
+4. We compare the homological convergence with the evolution of the validation results on neural network learning process.
+
+Hardware All experiments were executed in a machine with 2 NVIDIA V100 of 32GB, 2 Intel(R) Xeon(R) Platinum 8176 CPU $\textcircled { a } 2 . 1 0 \mathrm { G H z }$ , and of 1.5TB RAM, for a total of around 7 days. We note that our method is considerably demanding in terms of both compute and memory.
+
+The code and outputs are fully available in the Supplementary Material under MIT License.
+
+# 6 Results
+
+In this section, we highlight the main results, omitting the ones with Silhouette (since the obtained results were clearer with Heat). See the Supplementary Material for the full results (plots and correlations), including the ones with Silhouette distance.
+
+We study the relation between the evolution of the PH diagram distances with the one of the validation score with the cumulative values of the distance between homologous persistence diagrams because this value seems much more stable. The information of the distance between the persistence diagrams has been normalized to visualize clearly the type of evolution of each curve on the same scale. Some of the non-normalized plots can be found in the Supplementary Material. Figure 6 shows the cumulative and non-cumulative homology the MNIST experiment with layer size.
+
+![](images/33bb98bbb4dab01096e1139cbe1b0f0ee157590f81a64030c97e410898a0f9ed.jpg)  
+Figure 6: Heat distance and validation accuracy curves on the MNIST experiment with layer size. Normalized.
+
+249 For each experiment (e.g., layer size in MNIST), we plot both the evolution of the PH diagram   
+250 distance and the validation score (accuracy). The plotted values are the corresponding means of   
+251 the 5 repetitions with different seeds. In addition, we compute the Pearson correlation for these   
+252 values. Plots show on the x-axis each training step (for each batch) of the evolution in the training   
+253 state of the neural network. On the y-axis, two scales are shown that apply to the distance curves   
+254 between accumulated persistence diagrams (solid lines), scale on the right side, and the neural   
+255 network validation (dotted lines), numerical scale on the left side. For each sub-experiment (for   
+256 example, different values of layer size) a different color was used.   
+257 The general result is that the evolution of the homological convergence of the MLPs seems to be   
+258 very similar to the one of the validation score. This is generally consistent across experiments (see   
+259 the Supplementary Material). Table 1 shows the mean (and standard deviations) of the Pearson   
+260 correlations for all datasets. All means are above 0.8, implying that there is strong correlation.   
+261 Intuitively, this is also observed in the plots, although once the distances are normalized it is not as   
+262 clear to visualize. Interestingly, we find that the very few exceptions in which the correlation is low   
+263 corresponds to extreme values (very small number of neurons per layer, very high learning rate, very   
+264 high dropout), in which the neural network doesn’t end up learning properly.   
+265 In the case of CNNs, the correlations are lower (although still almost always above 0.8 in experiments   
+266 such as the one of increasing the number of layers). Recall that in the case of CNN we froze a   
+267 single convolutional feature extractor, since our method only supports MLPs. We believe these lower   
+268 correlations can be explained because an important part of the learning process happened in the   
+269 convolutional layers (in the pre-training), which we do not capture.   
+70 Another finding is that the method obtains consistent results across runs, meaning that it is capturing   
+71 information related to important properties of the networks themselves instead of random artifacts.   
+272 When varying the studied hyperparameters, we observe that the curves for each configuration are   
+273 indeed, different. Remarkably, in the control experiments, this is not the case; results show that the   
+274 homological convergence during the learning of the same problem with the same model but with   
+275 different input order is very similar. The alteration of the order of the input doesn’t have any effect in   
+276 the homological convergence. The results of two of these experiments are shown in Figure 7.   
+277 In addition, we observe that when the neural network learns the given problem, homological conver  
+278 gence occurs. For example, when the layer size is modified, the capacity of the neural network to   
+279 learn the problem changes (Figure 6). When it can’t learn the problem, because the network does not   
+280 have sufficient capacity (the layer size is too small, 4 units), the homology does not seem to converge.
+
+Regarding the learning rate, the results are coherent with the intuition that it is a fundamental parameter that controls how much to change the model in response to the estimated error during the learning process. A too small learning rate may result in a long training process that could be stalled, while a too large value may fall in a fast suboptimal solution or an unstable training process. Using homological convergence we find similar behaviour, as can be seen in Figure 9.
+
+![](images/3a2b67b2fc2e6de4e2fc17d2bc3005ca409730639ea045f722c4f0b56c840741.jpg)  
+Figure 7: Learning evolution on input order experiments (control experiments). Normalized.
+
+<table><tr><td colspan="2">Heat distance</td><td colspan="3">Silhouete distance</td></tr><tr><td>Dataset</td><td>Means mean</td><td>Deviations mean</td><td>Means mean</td><td>Deviations mean</td></tr><tr><td>MNIST</td><td>0.8910</td><td>0.0424</td><td>0.8910</td><td>0.0424</td></tr><tr><td>Reuters</td><td>0.6220</td><td>0.0700</td><td>0.6220</td><td>0.0700</td></tr><tr><td>CIFAR-10MLP</td><td>0.8233</td><td>0.0649</td><td>0.8233</td><td>0.0649</td></tr><tr><td>CIFAR-10 CNN</td><td>0.4241</td><td>0.1915</td><td>0.4241</td><td>0.1915</td></tr><tr><td>CIFAR-100MLP</td><td>0.8420</td><td>0.0566</td><td>0.8420</td><td>0.0566</td></tr><tr><td>CIFAR-100 CNN</td><td>0.6130</td><td>0.0800</td><td>0.6130</td><td>0.0800</td></tr></table>
+
+Table 1: Correlation of validation values with topological difference cumulative. Correlation is computed with 20 points.
+
+286 Finally, we note that even if the two convergences (validation and homological convergence) are   
+287 correlated, they are not the same process. This is especially visible in the case of the learning rate   
+288 experiments. For instance, in Figure 9, homological convergence is reached before the stabilization of   
+289 the validation accuracy. Presumably, they are not capturing the exact same information; specifically,   
+290 we believe that the difference is due to the fact that the validation accuracy depends on the specifics   
+291 of the data sampled in the validation subset, while the homological convergence is independent of the   
+292 validation data.
+
+# 293 7 Discussion
+
+We posed the of question whether homological convergence (in terms of distances between PH diagrams in consecutive neural network states) is related to the learning process of neural networks. We have seen that, indeed, it is the case, with strong empirical results backing our claim.
+
+297 This finding has a remarkable implication. If the homological convergence evolution mirrors the   
+298 validation accuracy curve, one could ignore the validation set to monitor the training. This opens the   
+299 path towards estimating the generalization of neural networks without the need of any holdout set.   
+300 Researchers have wondered for a long time whether generalization could be predicted from intrinsic   
+301 properties of the model or training data alone (i.e., without a holdout set), and in fact other works   
+302 have claimed to do so. Although we do not provide any predictive model, we show that our proposed   
+303 measures strongly correlate with the validation accuracy. In addition, we do so by not using any data   
+304 at all; we just look at the neural network itself.   
+305 Our contribution aims pushing towards having a better understanding of the learning process of neural   
+306 networks, not targeting any specific direct application. However, we note that it can be effectively   
+307 used for monitoring the training of neural networks in terms of convergence expected generalization,   
+308 as we have extensively shown in the experiments. Apart from the cases without access to a validation   
+309 set, this is relevant because depending on a validation set has the risk of overfitting to it. Having an   
+310 intrinsic, well-principled measure should be more robust to random noise in a specific data sample.   
+311 The main limitation of our method is its computational scalability. As we said in Section 5, our   
+312 method took more than 7 days of compute in a HPC machine, even if we restricted the experiments   
+313 to small datasets and parameter count. However, we note that our approach computes the exact   
+314 persistence diagram distances, that is, we do not simplify the graph representation of the neural   
+315 networks (we keep every single neuron and connections) and we do not approximate any computation.   
+316 This leaves room for finding efficient approximations, opening a new research line. In addition, this   
+317 lack of scalability has prevented us from validating our method on bigger models and datasets.   
+318 Finally, we note that instead of computing correlations, serving as a basic quantitative study, it would   
+319 be interesting to perform a time-series analysis to gain more insights on how the two curves vary   
+320 together. Moreover, it would have been interesting to investigate how to build a predictive model of   
+321 the validation accuracy from the PH distances, but it is was of the scope of this work.
+
+![](images/3a0752f9154791e10f8a71b4bf154cc55ae94ffbedae5fd456b3fb3b03c928cc.jpg)  
+Figure 8: Learning evolution when dropout parameter is changed. Normalized.
+
+![](images/761557a44d7a2b8b9d5c87e20e8faa7ff1bf7f68fc75b1642ef5128ae6015564.jpg)  
+Figure 9: Learning evolution when modifying the learning rate parameter. Not normalized.
+
+# 22 8 Conclusions & Future Work
+
+In this work, we have provided an empirical proof of the fact that homological convergence is related to the learning process and generalization properties of neural networks. Furthermore, we have shown that it can be used to monitor the training of a neural network (and potentially estimating its generalization) without a validation set. As future work, we suggest generalizing our representation to other neural architectures and scaling up the experiments to larger models and datasets, for which finding efficient approximations of our method will be crucial.
+
+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 the Discussion Section.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (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] Both code and outputs.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Check the Experiments Section and the code.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We include means, standard deviations and raw outputs.   
+(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] Check Experiments Section.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] In the case of the datasets. We do not use any other additional asset.   
+(b) Did you mention the license of the assets? [No]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Code, results and pictures we have made for explanations.   
+(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] References [1] H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta, and L. Ziegelmeier. Persistence images: A stable vector representation of persistent homology. J. Mach. Learn. Res., 18:8:1–8:35, 2017. [2] A. Anonymous. Characterizing and measuring the similarity of neural networks with persistent homology, 2021. [3] E. Berry, Y.-C. Chen, J. Cisewski-Kehe, and B. T. Fasy. Functional summaries of persistence diagrams. Journal of Applied and Computational Topology, 4:211–262, 2020. [4] P. Bubenik. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res., 16:77–102, 2015. [5] G. Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46:255–308, 2009. [6] F. Chazal, V. D. Silva, and S. Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173:193–214, 2012. [7] S. Chowdhury, T. Gebhart, S. Huntsman, and M. Yutin. Path homologies of deep feedforward networks. 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pages 1077–1082, 2019. [8] J. Clough, I. Öksüz, N. Byrne, V. Zimmer, J. A. Schnabel, and A. P. King. A topological loss function for deep-learning based image segmentation using persistent homology. IEEE transactions on pattern analysis and machine intelligence, PP, 2020. [9] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Proceedings of the twenty-first annual symposium on Computational geometry, 2005.   
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parse/train/BM64dm9HvN/BM64dm9HvN_content_list.json

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+    {
+        "type": "text",
+        "text": "Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "image_caption": [
+            "Figure 1: Our proposal. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "1 The training of neural networks is usually monitored with a validation (holdout)   \n2 set to estimate the generalization of the model. This is done instead of measuring   \n3 intrinsic properties of the model to determine whether it is learning appropriately.   \n4 In this work, we suggest studying the training of neural networks with Algebraic   \n5 Topology, specifically Persistent Homology (PH). Using simplicial complex repre  \n6 sentations of neural networks, we study the PH diagram distance evolution on the   \n7 neural network learning process with different architectures and several datasets.   \n8 Results show that the PH diagram distance between consecutive neural network   \n9 states correlates with the validation accuracy, implying that the generalization error   \n10 of a neural network could be intrinsically estimated without any holdout set. ",
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+        "text": "11 1 Introduction ",
+        "text_level": 1,
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+        "text": "12 Generalization is what makes a machine learning model useful; the uncertainty of its behaviour with   \n13 unseen data is what makes it potentially dangerous. Thus, understanding the generalization error of a   \n14 model can be considered one of the holy grails of the entire machine learning field.   \n15 Machine learning practitioners typically monitor some metrics of the model to estimate its generaliza  \n16 tion error and stop the training even before the numerical convergence to prevent the overfitting of   \n17 the model. Usually, the error measure or the metric relevant to the task is computed for a holdout   \n18 set, the validation set. Since these data have not been directly used for updating the parameters, it   \n19 is assumed that the performance of the model on the validation set can be used as a proxy of the ",
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+    {
+        "type": "text",
+        "text": "generalization error, provided it is representative of the data that will be used in inference. One can, though, potentially overfit to this holdout set if is repeatedly used for guiding a hyperparameter search. ",
+        "bbox": [
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+        "text": "23 Instead of relying on an external set, though, the question of whether it could be possible to estimate   \n24 the generalization error with some intrinsic property of the model is highly relevant, and it has been   \n25 barely explored in the literature. On the other hand, Algebraic Topology has recently been gaining   \n26 momentum as a mathematical tool for studying graphs, machine learning algorithms, and data.   \n27 In this work, we have the goal of, once having characterized neural networks as weighted, acyclic   \n28 graphs, represented as Algebraic Topology objects (following previous works), computing distances   \n29 between consecutive neural network states. More specifically, we can calculate the Persistent   \n30 Homology (PH) diagram distances between a give state (i.e., when having a specific weights during   \n31 the training process) and the next one (i.e., after having updated the weights in a training step) (see   \n32 Figure 1. We observe that during the training procedure of neural networks we can measure this   \n33 distance in each learning step, and show that there exists a high correlation with the corresponding   \n34 validation accuracy of the model. We do so in a diverse set of deep learning benchmarks and model   \n35 hyperparameters. This shines light on the question of whether the generalization error could be   \n36 estimated from intrinsic properties of the model, and opens the path towards a better theoretical   \n37 understanding of the dynamics of the training of neural networks. ",
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+    {
+        "type": "text",
+        "text": "38 In summary, our contributions are as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Based on principles of Algebraic Topology, we propose measuring the distances (Silhouette and Heat) between the PH persistence diagrams obtained from a given state of a neural network during the training procedure and the one in the immediately previous weights update.   \n• We empirically show that the evolution of these measures during training correlate with the accuracy in the validation set. We do so in diverse benchmarks (MNIST, CIFAR10, CIFAR100, Reuters text classification), and models (MLPs in MNIST and Reuters, MLPs and CNNs in CIFAR100 and CIFAR100).   \n• We thus provide empirical proof of the fact that valuable information related to the learning process of neural networks can be obtained from PH distances between persistence diagrams (homological convergence). In particular, we show that homological convergence is related to learning process and the generalization properties of neural networks.   \n• In practice, we provide a new tool for monitoring the training of neural networks, and open the path to estimating their generalization error without a validation set. ",
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+        "text": "53 The remainder of this article is as follows. In Section 2 we describe the theoretical background of our   \n54 proposal in terms of Algebraic Topology, while in Section 3 we go through the related work. Then, in   \n55 Section 4 we formalize our method. Finally, in sections 6 and 7 we present and discuss our empirical   \n56 results, respectively. ",
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+    {
+        "type": "text",
+        "text": "57 2 Background ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "58 In this section we introduce the mathematical foundations of this paper. A detailed mathematical   \n59 description is included in the Supplementary Material.   \n60 A simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional   \n61 counterparts, named simplex $( K )$ . In particular, a simplicial complex must comply with two properties:   \n62 1. Every face of a simplex is also in the simplicial complex (of lower dimension). 2. The non-empty   \n63 intersection of any two simplices contained on a simplicial complex is a face of both. 0,1,2,3-simplex   \n64 and non simplex examples are shown in Figure 2.   \n65 We can associate to an undirected graph, $G = \\left( V , E \\right)$ , a simplicial complex where all the vertices   \n66 of $\\mathbf { G }$ are the 0-simplex of the simplicial complex and the complete subgraphs with i vertices, in $G$   \n67 corresponds to a $( i - 1 )$ -simplex. This type of construction is usually called a complex clique on the   \n68 graph G, and is denoted by $C l ( G )$ . Figure 3 shows a graph clique complex Cl(G) example.   \n69 The boundary function is defined as a map, from   \n70 an $i$ -simplex to an $( i - 1 )$ -simplex, as the sum   \n71 of its $( i - 1 )$ -dimensional faces. A boundary   \n72 function sample is shown in Figure 4.   \n73 In algebraic topology, a $k$ -chain is a combination   \n74 of $k$ -simplices (sometimes symbolized as a lin  \n75 ear combination of simplices that compose the   \n76 chain). The boundary of a $k$ -chain is a $\\left( k - 1 \\right)$ -   \n77 chain. It is the linear and signed combination of   \n78 chain element boundary simplices. The space of   \n79 $i$ -chains is denoted by $C _ { i } ( K )$ . ",
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+        "image_caption": [
+            "Figure 2: Simplex and non-simplex examples. "
+        ],
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+        "image_caption": [
+            "Figure 4: Boundary function sample. "
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+    {
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+        "img_path": "images/c3b577eefcf443015365ad03338552041223c006305dfa5938eb382f2ab82c4f.jpg",
+        "image_caption": [
+            "Figure 3: Graph clique complex Cl(G) example. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "80 There are two special cases of chains that will be useful to define homology: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• Closed chain or $i$ -cycle: $i .$ -chain with empty boundary. An $i$ -chain $c$ is an $i$ -cycle if and only if $\\partial _ { i } c = 0$ , i.e. $c \\in \\dot { k e r } ( \\partial _ { i } )$ . This subspace of $C _ { i } ( K )$ is denoted as $\\mathbb { Z } _ { i } ( K )$ . • Exact chain or $i$ -boundary: An $i \\cdot$ -chain $c$ is an $i \\cdot$ -boundary if there exists an $( i + 1 )$ -chain $d$ such that $c = \\partial _ { i + 1 } ( d )$ , i.e. $c \\in i m ( \\partial i + 1 )$ . This subspace of $C _ { i } ( K )$ , the set of all such i-boundaries forms, is denoted by $\\mathbb { B } _ { i } ( K )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "86 Now, if we think in the $i$ -cycles that do not bound an $( i + 1 )$ -simplicial complex, this is the definition   \n87 $i$ -th homology of the simplicial complex $K$ . The precise definition is the quotient space of $\\mathbb { B } _ { i } ( K )$ a   \n88 subspace of $\\mathbb { Z } _ { i } ( K )$ (see Supplementary Material). The number of non equivalent $i \\cdot$ -cycles (Figure 5)   \n89 is the dimension of the homology group $H _ { i } ( K )$ , also named Betti numbers. ",
+        "bbox": [
+            148,
+            635,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We can create a nested family of simplicial complexes, $K _ { \\varepsilon }$ , where at each step $t$ , $K _ { \\varepsilon _ { t } }$ is embedded in the simplicial complex $K _ { \\varepsilon _ { t + 1 } }$ . We call this set a simplicial complex filtration. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/c673fa13230b91d7c3a60512e7dd55d64bd3ab3d1e9d2c21d54d4c5d49323ef4.jpg",
+        "image_caption": [
+            "Figure 5: The two blue dashed cycles are homologically equivalent, the pink isn’t. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            272,
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+    },
+    {
+        "type": "text",
+        "text": "For each filtration simplicial complex, we can calculate the homology groups. Then, we can look at the birth, that is, when a homology class appears, and death, the time when the homology class disappears. The PH treats the birth and the death of these homological features in $K _ { \\varepsilon }$ for different ε values. The lifespan of each homological feature can be represented as an interval $( b \\bar { i } r t h , d e a t h )$ , of the homological class. Given a filtration, this collection of intervals is named a Persistence Diagram (PD) [5]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "105 It is possible to compare two PDs using specific distances (Wasserstein and Bottleneck). To efficiently   \n106 perform this operation, due to the size of these diagrams, it is sometimes necessary to simplify them   \n107 by means of a discretization process (such as Weighted Silhouette and Heat vectorizations). ",
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+    {
+        "type": "text",
+        "text": "108 3 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+            154,
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+    },
+    {
+        "type": "text",
+        "text": "109 Algebraic Topology and Machine Learning The use of Algebraic Topology in the fields of   \n110 data science and machine learning has been gaining momentum in recent years (see Carlsson [5]).   \n111 Specifically in the case of neural networks, some works have applied topology for improving the   \n112 training procedure of the models [15, 8], or pruning the model afterwards [30]. Other works have   \n113 focused on analyzing the capacity of neural networks [14, 26, 17] or the complexity of input data   \n114 [17]. Furthermore, recent works have provided topological analysis of the decision boundaries of   \n115 classifiers based on PH and Betti numbers [24, 22].   \n116 Graph and topological representations of neural networks Gebhart et al. [12] suggest a method   \n117 for computing the PH over the graphical activation neural networks, while Watanabe and Yamana   \n118 [29] propose representing neural networks via simplicial complexes based on Taylor decomposition,   \n119 from which one can compute the PH. Chowdhury et al. [7] show that directed homology can be used   \n120 to represent MLPs. Anonymous [2] concurrently show neural networks, when represented as directed,   \n121 acyclic graphs, can be associated to an Algebraic Topology object. By computing the PH diagram,   \n122 one can effectively characterize neural networks, and even compute distances between two given   \n123 neural networks, which can be used to measure their similarity. This is unlike other works [11, 13]   \n124 approximating neural networks representations with regard to the input space.   \n125 Estimating the generalization and studying the learning process We are, though, specifically   \n126 interested in the use of PH for analyzing the learning process, especially with the goal of estimating   \n127 generalization. In this regard, the literature is perhaps more limited. Jiang et al. [16] work on   \n128 understanding what drives generalization in deep networks from a Bayesian of view. Neyshabur et al.   \n129 [23] study the generalization gap prediction from the training data and network parameters using a   \n130 margin distribution, which are the distances of training points to the decision boundary. In Li et al.   \n131 [21], authors propose an alternative to cross-validation for model selection based on training once on   \n132 the whole train set, without any data split, deriving a validation set with data augmentation.   \n133 Corneanu et al. [10] try to estimate the performance gap between training and testing using PH   \n134 measures. They claim. However, one can observe some caveats. The first one is that their regression   \n135 fitted to predict the test error has a considerably high error, making it not usable in practice. The   \n136 second caveat is that for fitting the regression one needs at least part of the sequestered testing set.   \n137 In this work, motivated by the interest of having a better understanding of whether it would be   \n138 possible to estimate the generalization of neural networks without a holdout set, we suggest using the   \n139 topological characterization and distances concurrently proposed in Anonymous [2] but, crucially,   \n140 measured between consecutive weight updates. We will show that the evolution of this distance   \n141 is similar to the one of the validation accuracy. Unlike Li et al. [21], we do not use any data at   \n142 all. Unlike [10], we do not build a statistical or machine learning model (linear regression) for   \n143 predicting the testing error. Instead, we propose a new measure, and we empirically show that it   \n144 highly correlates with the validation accuracy. Note that in this work we do not work with any input   \n145 data and activations, but with the parameters of the neural network themselves. ",
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+        "type": "text",
+        "text": "146 4 Approach ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Representation For representing neural networks as graphs, we follow the approach proposed concurrently in Anonymous [2]. We associate to the neural network, at each learning state (defined by its weights), a weighted directed graph that is analyzed as an abstract simplicial complex. It is important to note that abstract simplicial complex are used in opposition to geometric simplicial complex. ",
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+        "text": "152 For every training state, neural network connections are considered as directed and weighted edges   \n153 between neurons, represented by graph nodes. Biases are considered as new edges that join to isolate   \n154 vertices. In this representation, activation functions are lost. Bias information could also have been   \n155 ignored because, as we will see, it is not very informative in terms of homology, but we decided to   \n56 preserve it.   \n157 Negative edge weights are represented with reverse edges with the same weight absolute value. We   \n158 discard the use of weight absolute value as neural networks are not invariant under weight sign   \n159 transformations. This representation is consistent with the fact that every neuron can be replaced by a   \n160 neuron from which two edges with opposite weights emerge and converge again on another neuron   \n161 with opposite weights. From an homological point of view, this would be represented as a closed   \n162 cycle. Weights are normalized following the Equation 1. $\\zeta$ is an smoothing parameter that we set to   \n163 1e-6. This smoothing parameter is necessary as we want to avoid normalized weights of edges to be   \n164 0 (in our representation 0 implies a lack of connection): ",
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+        "text": "165 Algebraic Topology object For each weighted directed graph associated with the state of a neural   \n166 network, we link a directed flag complex to it. The topological properties of this directed flag complex   \n167 are studied using homology groups $H _ { n }$ . We calculate the homology groups up to degree 3 $\\left( H _ { 0 } – H _ { 3 } \\right)$ .   \n168 For each state, we use a family of simplicial complexes, $K _ { \\varepsilon }$ , for a range of values of $\\varepsilon \\in \\mathbb { R }$ . The   \n169 simplicial complex at step $\\varepsilon _ { t }$ is embedded in the complex at $\\varepsilon _ { t + 1 }$ , for $\\varepsilon _ { t } \\leq \\varepsilon _ { t + 1 }$ , i.e. $K _ { \\varepsilon } \\subseteq K _ { \\varepsilon _ { t + 1 } }$ . $\\varepsilon$   \n170 is used as a filter that establish the minimum weight of the graph representation edges included on   \n171 the simplicial complex. This collection of contained simplicial complex (associated to a directed   \n172 weighted graph), called filtration, $K _ { \\varepsilon _ { m i n } } \\subseteq \\ldots \\subseteq K _ { \\varepsilon _ { t } } \\subseteq K _ { \\varepsilon _ { t + 1 } } \\subseteq \\ldots \\subseteq K _ { \\varepsilon _ { m a x } }$ , where $t \\in [ 0 , 1 ]$ and $\\varepsilon _ { m i n } = 0$   \n173 $\\varepsilon _ { m a x } = 1$ (remember that edge weights are normalized).   \n174 The sequence of homology groups is calculated by varying the $\\varepsilon$ parameter to obtain the persistence   \n175 homology diagram. In our case, persistent homology calculations are performed on $\\mathbb { Z } _ { 2 }$ . In other   \n176 words, once the corresponding filter has been applied to the weight of the edges, all connected edges   \n177 are considered equally.   \n178 Distances between persistence diagrams of consecutive states In this paper, we are interested in   \n179 comparing PDs between different simplicial complex associated to each training state of the neural   \n180 network. There are two distances traditionally used to compare PDs, Bottleneck distance (the length   \n181 of the longest edge) and Wasserstein distance (using the sum of all edges lengths, instead of the   \n182 maximum). Their stability with respect to perturbations on PDs has been object of different studies   \n183 [6, 9].   \n184 In order to make computations feasible and obviate noisy intervals, we filter the PDs by limiting   \n185 the minimum PD interval size. We do so by setting a minimum threshold $\\eta = 0 . 0 1$ . Intervals with   \n186 a lifespan under this value are not considered (spurious homological features). Additionally, for   \n187 computing distances, we need to remove infinity values. As we are only interested in the deaths until   \n188 the maximum weight value, we replace all the infinity values by 1.0.   \n189 In our case, our neural networks have millions of persistence intervals per Persistence Diagram,   \n190 while Wasserstein distance calculations are computationally hard for large PDs. In order to make   \n191 calculations computationally feasible, we will use a vectorized version of PDs, also called PD   \n192 discretization. This vectorized version summaries have been proposed and used on recent literature   \n193 [1, 3, 4, 19, 25]. For persistence diagram distance calculation, we use weighted Silhouette and Heat   \n194 vectorizations, using the Giotto-TDA library [27]. ",
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+        "type": "text",
+        "text": "95 5 Experiments ",
+        "text_level": 1,
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+        "text": "Data We validate our method in several heterogeneous (vision, natural language), well-known datasets, namely 1. MNIST [20], 2. CIFAR-10, 3. CIFAR-100 [18], and 4. the Reuters dataset [28] (multi-class and multi-label document classification dataset). ",
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+        "text": "Models We experiment with two neural architectures,1. MLPs and 2. CNNs. In the latter case, we use the convolutional layers as a pre-trained model with frozen weights, and we learn an MLP on top of it. The reason we do so is that our method is based in a representation that, at least in the basic form, does not allow capturing information from convolutional layers. Thus, we need a single (exact ",
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+        "type": "text",
+        "text": "203 same weights) feature extractor, to abstract away distances related to the CNN layers and focus on   \n204 the MLP. ",
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+        "type": "text",
+        "text": "Conducted experiments We define the base MLP architecture as {Input, Linear(512), Dropout(0.2), Linear(512), Dropout(0.2), Output}. In the case of CNNs, the pre-trained model is defined as 3 convolutional blocks with kernel size 3 (starting with 32 channels), interleaved with max pooling (its linear layers are thrown away after the pre-training). On top of the pre-trained CNN, we also define the same base MLP architecture. Then, for each dataset and model (MLP and CNN), we experiment with varying (while keeping the rest fixed to the base architecture) ",
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+        "text": "1. Layer size (number of units per layer): 4, 16, 32, 128, 256.   \n2. Number of labels (the other classes are removed): 2, 4, 6, 8, 10.   \n3. Learning rate: 1e-e05, 0.0001, 0.001, 0.01, 0.1   \n4. Dropout: 0.0, 0.2, 0.4, 0.5, 0.8.   \n5. Input order: 5 random input orders. As a control experiment, for each analyzed problem we run the same configuration with 5 different input orders. If the measured distances are, indeed, related with the learning process of neural networks, these variations should not have any noticeable effect. ",
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+        "type": "text",
+        "text": "We run each configuration 5 times with different random seeds (and, thus, weight initializations1) to see if the results are consistent across runs. All models are trained with the RMSProp optimizer with 221 a batch size of 256. ",
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+        "text": "Distances and validation accuracy computation Note that homological distances are obtained at the end of each batch, while validation metrics are only computed on each epoch. The methodology we follow to analyze the learning process on each different problem can be summarized with the following steps: ",
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+        "type": "text",
+        "text": "1. In each training step (i.e., for each batch) we extract the weights from the MLP current state and use them to build an abstract simplicial complex from the associated weighted directed graph.   \n2. We calculate the homological persistence diagram of the simplicial complex.   \n3. We then calculate the distance between consecutive persistence diagrams (we will call this sequence homological convergence). We use two different distances, namely, Heat and Silhouette.   \n4. We compare the homological convergence with the evolution of the validation results on neural network learning process. ",
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+        "text": "Hardware All experiments were executed in a machine with 2 NVIDIA V100 of 32GB, 2 Intel(R) Xeon(R) Platinum 8176 CPU $\\textcircled { a } 2 . 1 0 \\mathrm { G H z }$ , and of 1.5TB RAM, for a total of around 7 days. We note that our method is considerably demanding in terms of both compute and memory. ",
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+        "type": "text",
+        "text": "The code and outputs are fully available in the Supplementary Material under MIT License. ",
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+        "type": "text",
+        "text": "6 Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we highlight the main results, omitting the ones with Silhouette (since the obtained results were clearer with Heat). See the Supplementary Material for the full results (plots and correlations), including the ones with Silhouette distance. ",
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+        "type": "text",
+        "text": "We study the relation between the evolution of the PH diagram distances with the one of the validation score with the cumulative values of the distance between homologous persistence diagrams because this value seems much more stable. The information of the distance between the persistence diagrams has been normalized to visualize clearly the type of evolution of each curve on the same scale. Some of the non-normalized plots can be found in the Supplementary Material. Figure 6 shows the cumulative and non-cumulative homology the MNIST experiment with layer size. ",
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+        "type": "image",
+        "img_path": "images/33bb98bbb4dab01096e1139cbe1b0f0ee157590f81a64030c97e410898a0f9ed.jpg",
+        "image_caption": [
+            "Figure 6: Heat distance and validation accuracy curves on the MNIST experiment with layer size. Normalized. "
+        ],
+        "image_footnote": [],
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+        "text": "249 For each experiment (e.g., layer size in MNIST), we plot both the evolution of the PH diagram   \n250 distance and the validation score (accuracy). The plotted values are the corresponding means of   \n251 the 5 repetitions with different seeds. In addition, we compute the Pearson correlation for these   \n252 values. Plots show on the x-axis each training step (for each batch) of the evolution in the training   \n253 state of the neural network. On the y-axis, two scales are shown that apply to the distance curves   \n254 between accumulated persistence diagrams (solid lines), scale on the right side, and the neural   \n255 network validation (dotted lines), numerical scale on the left side. For each sub-experiment (for   \n256 example, different values of layer size) a different color was used.   \n257 The general result is that the evolution of the homological convergence of the MLPs seems to be   \n258 very similar to the one of the validation score. This is generally consistent across experiments (see   \n259 the Supplementary Material). Table 1 shows the mean (and standard deviations) of the Pearson   \n260 correlations for all datasets. All means are above 0.8, implying that there is strong correlation.   \n261 Intuitively, this is also observed in the plots, although once the distances are normalized it is not as   \n262 clear to visualize. Interestingly, we find that the very few exceptions in which the correlation is low   \n263 corresponds to extreme values (very small number of neurons per layer, very high learning rate, very   \n264 high dropout), in which the neural network doesn’t end up learning properly.   \n265 In the case of CNNs, the correlations are lower (although still almost always above 0.8 in experiments   \n266 such as the one of increasing the number of layers). Recall that in the case of CNN we froze a   \n267 single convolutional feature extractor, since our method only supports MLPs. We believe these lower   \n268 correlations can be explained because an important part of the learning process happened in the   \n269 convolutional layers (in the pre-training), which we do not capture.   \n70 Another finding is that the method obtains consistent results across runs, meaning that it is capturing   \n71 information related to important properties of the networks themselves instead of random artifacts.   \n272 When varying the studied hyperparameters, we observe that the curves for each configuration are   \n273 indeed, different. Remarkably, in the control experiments, this is not the case; results show that the   \n274 homological convergence during the learning of the same problem with the same model but with   \n275 different input order is very similar. The alteration of the order of the input doesn’t have any effect in   \n276 the homological convergence. The results of two of these experiments are shown in Figure 7.   \n277 In addition, we observe that when the neural network learns the given problem, homological conver  \n278 gence occurs. For example, when the layer size is modified, the capacity of the neural network to   \n279 learn the problem changes (Figure 6). When it can’t learn the problem, because the network does not   \n280 have sufficient capacity (the layer size is too small, 4 units), the homology does not seem to converge. ",
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+        "type": "text",
+        "text": "Regarding the learning rate, the results are coherent with the intuition that it is a fundamental parameter that controls how much to change the model in response to the estimated error during the learning process. A too small learning rate may result in a long training process that could be stalled, while a too large value may fall in a fast suboptimal solution or an unstable training process. Using homological convergence we find similar behaviour, as can be seen in Figure 9. ",
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+        "image_caption": [
+            "Figure 7: Learning evolution on input order experiments (control experiments). Normalized. "
+        ],
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+        "table_footnote": [
+            "Table 1: Correlation of validation values with topological difference cumulative. Correlation is computed with 20 points. "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">Heat distance</td><td colspan=\"3\">Silhouete distance</td></tr><tr><td>Dataset</td><td>Means mean</td><td>Deviations mean</td><td>Means mean</td><td>Deviations mean</td></tr><tr><td>MNIST</td><td>0.8910</td><td>0.0424</td><td>0.8910</td><td>0.0424</td></tr><tr><td>Reuters</td><td>0.6220</td><td>0.0700</td><td>0.6220</td><td>0.0700</td></tr><tr><td>CIFAR-10MLP</td><td>0.8233</td><td>0.0649</td><td>0.8233</td><td>0.0649</td></tr><tr><td>CIFAR-10 CNN</td><td>0.4241</td><td>0.1915</td><td>0.4241</td><td>0.1915</td></tr><tr><td>CIFAR-100MLP</td><td>0.8420</td><td>0.0566</td><td>0.8420</td><td>0.0566</td></tr><tr><td>CIFAR-100 CNN</td><td>0.6130</td><td>0.0800</td><td>0.6130</td><td>0.0800</td></tr></table>",
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+        "type": "text",
+        "text": "286 Finally, we note that even if the two convergences (validation and homological convergence) are   \n287 correlated, they are not the same process. This is especially visible in the case of the learning rate   \n288 experiments. For instance, in Figure 9, homological convergence is reached before the stabilization of   \n289 the validation accuracy. Presumably, they are not capturing the exact same information; specifically,   \n290 we believe that the difference is due to the fact that the validation accuracy depends on the specifics   \n291 of the data sampled in the validation subset, while the homological convergence is independent of the   \n292 validation data. ",
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+    {
+        "type": "text",
+        "text": "293 7 Discussion ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We posed the of question whether homological convergence (in terms of distances between PH diagrams in consecutive neural network states) is related to the learning process of neural networks. We have seen that, indeed, it is the case, with strong empirical results backing our claim. ",
+        "bbox": [
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+        "type": "text",
+        "text": "297 This finding has a remarkable implication. If the homological convergence evolution mirrors the   \n298 validation accuracy curve, one could ignore the validation set to monitor the training. This opens the   \n299 path towards estimating the generalization of neural networks without the need of any holdout set.   \n300 Researchers have wondered for a long time whether generalization could be predicted from intrinsic   \n301 properties of the model or training data alone (i.e., without a holdout set), and in fact other works   \n302 have claimed to do so. Although we do not provide any predictive model, we show that our proposed   \n303 measures strongly correlate with the validation accuracy. In addition, we do so by not using any data   \n304 at all; we just look at the neural network itself.   \n305 Our contribution aims pushing towards having a better understanding of the learning process of neural   \n306 networks, not targeting any specific direct application. However, we note that it can be effectively   \n307 used for monitoring the training of neural networks in terms of convergence expected generalization,   \n308 as we have extensively shown in the experiments. Apart from the cases without access to a validation   \n309 set, this is relevant because depending on a validation set has the risk of overfitting to it. Having an   \n310 intrinsic, well-principled measure should be more robust to random noise in a specific data sample.   \n311 The main limitation of our method is its computational scalability. As we said in Section 5, our   \n312 method took more than 7 days of compute in a HPC machine, even if we restricted the experiments   \n313 to small datasets and parameter count. However, we note that our approach computes the exact   \n314 persistence diagram distances, that is, we do not simplify the graph representation of the neural   \n315 networks (we keep every single neuron and connections) and we do not approximate any computation.   \n316 This leaves room for finding efficient approximations, opening a new research line. In addition, this   \n317 lack of scalability has prevented us from validating our method on bigger models and datasets.   \n318 Finally, we note that instead of computing correlations, serving as a basic quantitative study, it would   \n319 be interesting to perform a time-series analysis to gain more insights on how the two curves vary   \n320 together. Moreover, it would have been interesting to investigate how to build a predictive model of   \n321 the validation accuracy from the PH distances, but it is was of the scope of this work. ",
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+        "image_caption": [
+            "Figure 8: Learning evolution when dropout parameter is changed. Normalized. "
+        ],
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+            "Figure 9: Learning evolution when modifying the learning rate parameter. Not normalized. "
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+    {
+        "type": "text",
+        "text": "22 8 Conclusions & Future Work ",
+        "text_level": 1,
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+        "text": "In this work, we have provided an empirical proof of the fact that homological convergence is related to the learning process and generalization properties of neural networks. Furthermore, we have shown that it can be used to monitor the training of a neural network (and potentially estimating its generalization) without a validation set. As future work, we suggest generalizing our representation to other neural architectures and scaling up the experiments to larger models and datasets, for which finding efficient approximations of our method will be crucial. ",
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+    {
+        "type": "text",
+        "text": "1. For all authors... ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See the Discussion Section.   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
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+    {
+        "type": "text",
+        "text": "(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] ",
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+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
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+    },
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+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] References [1] H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta, and L. Ziegelmeier. Persistence images: A stable vector representation of persistent homology. J. Mach. Learn. Res., 18:8:1–8:35, 2017. [2] A. Anonymous. Characterizing and measuring the similarity of neural networks with persistent homology, 2021. [3] E. Berry, Y.-C. Chen, J. Cisewski-Kehe, and B. T. Fasy. Functional summaries of persistence diagrams. Journal of Applied and Computational Topology, 4:211–262, 2020. [4] P. Bubenik. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res., 16:77–102, 2015. [5] G. Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46:255–308, 2009. [6] F. Chazal, V. D. Silva, and S. Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173:193–214, 2012. [7] S. Chowdhury, T. Gebhart, S. Huntsman, and M. Yutin. Path homologies of deep feedforward networks. 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pages 1077–1082, 2019. [8] J. Clough, I. Öksüz, N. Byrne, V. Zimmer, J. A. Schnabel, and A. P. King. A topological loss function for deep-learning based image segmentation using persistent homology. IEEE transactions on pattern analysis and machine intelligence, PP, 2020. [9] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Proceedings of the twenty-first annual symposium on Computational geometry, 2005.   \n390 [10] C. Corneanu, M. Madadi, S. Escalera, and A. Martínez. Computing the testing error without a testing set. 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 2674–2682, 2020. [11] C. A. Corneanu, M. Madadi, S. Escalera, and A. M. Martinez. What does it mean to learn in deep networks? and, how does one detect adversarial attacks? In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 4752–4761, 2019. doi: 10.1109/ CVPR.2019.00489. [12] T. Gebhart, P. Schrater, and A. Hylton. Characterizing the shape of activation space in deep neural networks. 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pages 1537–1542, 2019. [13] W. H. Guss and R. Salakhutdinov. On characterizing the capacity of neural networks using algebraic topology. CoRR, abs/1802.04443, 2018. URL http://arxiv.org/abs/1802. 04443. [14] W. H. Guss and R. Salakhutdinov. On characterizing the capacity of neural networks using algebraic topology. ArXiv, abs/1802.04443, 2018. [15] C. Hofer, F. Graf, M. Niethammer, and R. Kwitt. Topologically densified distributions. ArXiv, abs/2002.04805, 2020.   \n407 [16] Y. Jiang, D. Krishnan, H. Mobahi, and S. Bengio. Predicting the generalization gap in deep networks with margin distributions. ArXiv, abs/1810.00113, 2019. [17] E. Konuk and K. Smith. An empirical study of the relation between network architecture and complexity. 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), pages 4597–4599, 2019. [18] A. Krizhevsky. Learning multiple layers of features from tiny images. 2009. [19] P. Lawson, A. Sholl, J. Brown, B. T. Fasy, and C. Wenk. Persistent homology for the quantitative evaluation of architectural features in prostate cancer histology. Scientific Reports, 9, 2019. [20] Y. LeCun and C. Cortes. MNIST handwritten digit database. 2010. URL http://yann.lecun. com/exdb/mnist/. [21] W. Li, C. Geng, and S. Chen. Leave zero out: Towards a no-cross-validation approach for model selection. CoRR, abs/2012.13309, 2020. URL https://arxiv.org/abs/2012.13309. ",
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+]

parse/train/ByljMaNKwB/ByljMaNKwB.md

@@ -0,0 +1,397 @@
+# DOMAIN AGGREGATION NETWORKS FORMULTI-SOURCE DOMAIN ADAPTATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In many real-world applications, we want to exploit multiple source datasets of similar tasks to learn a model for a different but related target dataset – e.g., recognizing characters of a new font using a set of different fonts. While most recent research has considered ad-hoc combination rules to address this problem, we extend previous work on domain discrepancy minimization to develop a finite-sample generalization bound, and accordingly propose a theoretically justified optimization procedure. The algorithm we develop, Domain AggRegation Network (DARN), is able to effectively adjust the weight of each source domain during training to ensure relevant domains are given more importance for adaptation. We evaluate the proposed method on real-world sentiment analysis, digit recognition and object recognition datasets and show that DARN can significantly outperform the state-of-the-art alternatives.
+
+# 1 INTRODUCTION
+
+Many machine learning algorithms assume the learned predictor will be tested on the data from the same distribution as the training data. This assumption, although reasonable, is not necessarily true for many real-world applications. For example, patients from one hospital may have a different distribution of gender, height and weight from another hospital. Consequently, a diagnostic model constructed at one location may not be directly applicable to another location without proper adjustment. The situation becomes even more challenging when we want to use data from multiple source domains to build a model for a target domain, as this requires deciding, e.g., how to rank the source domains and how to effectively aggregate these domains when training complex models like deep neural networks. Including irrelevant or worse, adversarial, data from certain source domains can severely reduce the performance on the target domain, leading to undesired consequences.
+
+To address this problem, researchers have been exploring methods of transfer learning (Pan & Yang, 2009) or domain adaptation (Mansour et al., $2 0 0 9 \mathrm { a }$ ; Cortes et al., 2019), where a model is trained based on labelled source data and unlabelled target data. Most existing works have focused on single-source to single-target (“one-to-one”) domain adaptation, using different assumptions such as covariate shift (Shimodaira, 2000; Gretton et al., 2009; Sugiyama & Kawanabe, 2012) or concept drift (Jiang & Zhai, 2007; Gama et al., 2014). When dealing with multiple source domains, one may attempt to directly use these approaches by combining all source data into a large joint dataset and then apply one-to-one adaptation. This na¨ıve aggregation method will often fail as not all source domains are equally important when transferring to a specific target domain. There are some works on multi-source to single-target adaptation. Although many of them are theoretically motivated with cross-domain generalization bounds, they either use ad-hoc aggregation rules when developing actual algorithms (Zhao et al., 2018; Li et al., 2018) or lack finite-sample analysis (Mansour et al., 2009b;c; Hoffman et al., 2018a). This leaves a gap between the theory for multi-source adaptation and theoretically sound algorithm for domain aggregation.
+
+This research has three contributions: First, we extend prior work on one-to-one adaptation using discrepancy (Cortes et al., 2019) to develop a finite-sample cross-domain generalization bound for multi-source adaptation. We show that in order to improve performance on the specific target domain, there is a trade-off between utilizing all source domains to increase effective sample size, versus removing source domains that are underperforming or not similar to the target domain. Second, motivated by our theory and domain adversarial method (Ganin & Lempitsky, 2015; Ganin et al., 2016), we propose Domain AggRegation Network (DARN), that can effectively aggregate multiple source domains dynamically during the course of training. Unlike previous works, our aggregation scheme (Eq. (6)), which itself is of independent interest in some other contexts, is a direct optimization of our generalization upper bound (Eq. (3)) without resorting to heuristics. Third, our experiments on sentiment analysis, digit recognition and object recognition show that DARN can significantly outperform state-of-the-art methods.
+
+Section 2 introduces necessary background on one-to-one adaptation based on discrepancy. Then Section 3 elaborates our theoretical analysis and the corresponding algorithm deployment. Section 4 discusses about related approaches in more detail, highlighting the key differences to the approach developed here. Section 5 empirically compares the performance of the proposed method to other alternatives. Finally, Section 6 concludes this work.
+
+# 2 BACKGROUND ON CROSS-DOMAIN GENERALIZATION
+
+This section provides necessary background from previous work on one-to-one domain adaptation (Cortes et al., 2019). Let $\mathcal { X }$ be the input space, $\mathcal { V } \subseteq \mathbb { R }$ be output space and $\mathcal { H } \subseteq \{ h : \mathcal { X } \mapsto \mathsf { \bar { y } } \}$ be a hypothesis set. A loss function $L : \mathcal { V } \times \mathcal { V } \mapsto \mathbb { R } ^ { + }$ is $\mu$ -admissible1 if
+
+$$
+\forall y , y ^ { \prime } , y ^ { \prime \prime } \in \mathcal { V } \qquad | L ( y ^ { \prime } , y ) - L ( y ^ { \prime \prime } , y ) | \leq \mu | y ^ { \prime } - y ^ { \prime \prime } | .
+$$
+
+The discrepancy (Mansour et al., 2009a) between two distributions $P , Q$ over $\mathcal { X }$ is defined as
+
+$$
+\mathrm { d i s c } ( P , Q ) = \operatorname* { m a x } _ { h , h ^ { \prime } \in \mathcal { H } } | \mathcal { L } _ { P } ( h , h ^ { \prime } ) - \mathcal { L } _ { Q } ( h , h ^ { \prime } ) | \quad \mathrm { w h e r e } \quad \mathcal { L } _ { P } ( h , h ^ { \prime } ) = \mathbb { E } _ { x \sim P } [ L ( h ( x ) , h ^ { \prime } ( x ) ) ] .
+$$
+
+This quantity can be computed or approximated empirically given samples from both distributions. For classification problems with 0-1 loss, it reduces to the well-known $d _ { \mathcal { A } }$ -distance (Kifer et al., 2004; Blitzer et al., 2008; Ben-David et al., 2010) and can be approximated using a domain-classifier loss w.r.t. $\mathcal { H }$ (Zhao et al., 2018; Ben-David et al., 2007, Sec.4), while for regression problems with $L _ { 2 }$ loss, it reduces to a maximum eigenvalue (Cortes & Mohri, 2011, Sec.5); see Section 3.2 for more details. For two domains $( P , f _ { P } ) , ( Q , f _ { Q } )$ where $f _ { P } , f _ { Q } : \mathcal { X } \mapsto \mathcal { Y }$ are the corresponding labeling functions, we have the following cross-domain generalization bound:
+
+Theorem 1 (Proposition 5 & 8, Cortes et al. (2019)) Let $\Re _ { m } ( \mathcal { H } )$ be the Rademacher complexity of $\mathcal { H }$ given sample size m, $\mathcal { H } _ { Q } = \{ x \mapsto L ( h ( x ) , f _ { Q } ( x ) ) : h \in \mathcal { H } \}$ be the set of functions mapping $x$ to its loss w.r.t. $f _ { Q }$ and $\mathcal { H }$ ,
+
+$$
+\eta _ { \mathcal H } = \mu \times \operatorname* { m i n } _ { h \in \mathcal H } \left( \operatorname* { m a x } _ { x \in \mathrm { s u p p } ( \widehat P ) } | f _ { P } ( x ) - h ( x ) | + \operatorname* { m a x } _ { x \in \mathrm { s u p p } ( \widehat Q ) } | f _ { Q } ( x ) - h ( x ) | \right) ,
+$$
+
+be a constant measuring how well $\mathcal { H }$ can fit the true models where $\operatorname { s u p p } ( \widehat { P } )$ means the support of the empirical distribution $\widehat { P }$ (using the $\mu$ from Eq. (1)), and $\begin{array} { r } { M _ { Q } = \operatorname* { s u p } _ { x \in \mathcal { X } , h \in \mathcal { H } } L ( \mathbf { \Psi } h ( x ) , \mathbf { \Psi } f _ { Q } ( x ) \mathbf { \Psi } ) } \end{array}$ be the upper bound on loss for $Q$ . Given $\widehat { Q }$ with m points sampled iid from $Q$ labelled according to $f _ { Q }$ , for $\delta \in ( 0 , 1 ) , \forall h \in \mathcal { H }$ , w.p. at least $1 - \delta$ ,
+
+$$
+\mathcal { L } _ { P } ( h , f _ { P } ) \ : \le \ : \mathcal { L } _ { \widehat { Q } } ( h , f _ { Q } ) + \mathrm { d i s c } ( P , Q ) + 2 \Re _ { m } ( \mathcal { H } _ { Q } ) + \eta _ { \mathcal { H } } + M _ { Q } \sqrt { \frac { \log ( 1 / \delta ) } { 2 m } } .
+$$
+
+This theorem provides a way to generalize across domains when we have sample $\widehat { Q }$ labelled according to $f _ { Q }$ and an unlabelled sample $\widehat { P }$ . The first term is the usual loss function for the sample $\widehat { Q }$ , while the second term $\mathrm { d i s c } ( P , Q )$ can be estimated based on the unlabelled data ${ \widehat { Q } } , { \widehat { P } }$ . $\eta _ { \mathcal { H } }$ measures how well the model family $\mathcal { H }$ can fit the example from the datasets, and it is not controllable once $\mathcal { H }$ is given. The final term, as a function of sample size $m$ , determines the convergence speed.
+
+# 3 DOMAIN AGGREGATION NETWORKS
+
+# 3.1 THEORY
+
+Suppose we are given $k$ source domains $\{ ( S _ { i } , f _ { S _ { i } } ) : i \in [ k ] \stackrel { \mathrm { d e f } } { = } \{ 1 , 2 , \ldots , k \} \}$ and a target domain $( T , f _ { T } )$ where $S _ { i } , T$ are distributions over $\mathcal { X }$ and $f _ { S _ { i } } , f _ { T } : \mathcal { X } \mapsto \mathcal { Y }$ are their respective labelling functions. For simplicity, assume that each sample $\widehat { S } _ { i }$ has $m$ points, drawn iid from $S _ { i }$ and labelled according to $f _ { S _ { i } }$ . We are also given $m$ unlabelled points $\widehat { T }$ drawn iid from $T$ . We want to leverage all source domains’ information to learn a model $h \in \mathcal H$ minimizing $\mathcal { L } _ { T } ( h , f _ { T } )$ .
+
+One na¨ıve approach could be to combine all the source domains into a large joint dataset and conduct one-to-one adaptation to the target domain using Theorem 1. However, including data from irrelevant or even adversarial domains is likely to jeopardize the performance on the target domain, which is sometimes referred to as negative transfer (Pan & Yang, 2009). Moreover, as certain source domains may be more similar or relevant to the target domain than the others, it makes more sense to adjust their importance according to their utilities. We propose to find domain weight $\alpha _ { i } \geq 0$ such that combine $\textstyle \sum _ { i = 1 } ^ { k } \alpha _ { i } = 1$ to achieve this. Our main theorem below sheds some light on how we shouldmains (the proof is provided in Appendix A):
+
+Theorem 2 Given $k$ source domains datasets $\{ ( x _ { j } ^ { ( i ) } , y _ { j } ^ { ( i ) } ) : i \in [ k ] , j \in [ m ] \}$ with m iid examples each where $\widehat { S } _ { i } = \{ x _ { j } ^ { ( i ) } \}$ and $y _ { j } ^ { ( i ) } = f _ { S _ { i } } ( x _ { j } ^ { ( i ) } )$ , for any $\begin{array} { r } { \pmb { \alpha } \in \Delta = \{ \pmb { \alpha } : \alpha _ { i } \geq 0 , \sum _ { i } \alpha _ { i } = 1 \} , \delta \in } \end{array}$ $( 0 , 1 )$ , and $\forall h \in { \mathcal { H } }$ , w.p. at least $1 - \delta$
+
+$$
+\mathcal { L } _ { T } ( h , f _ { T } ) \leq \sum _ { i } \alpha _ { i } \left( \mathcal { L } _ { \widehat { S } _ { i } } ( h , f _ { S _ { i } } ) + \mathrm { { d i s c } } ( T , S _ { i } ) + 2 \mathfrak { R } _ { m } ( \mathcal { H } _ { S _ { i } } ) + \eta _ { \mathcal { H } , i } \right) + \| \alpha \| _ { 2 } M _ { S } \sqrt { \frac { \log ( 1 / \delta ) } { 2 m } } ,
+$$
+
+where $\mathcal { H } _ { S _ { i } } ~ = ~ \{ x ~ \mapsto ~ L \big ( h ( x ) , f _ { S _ { i } } ( x ) \big ) ~ : ~ h ~ \in ~ \mathcal { H } \}$ is the set of functions mapping $x$ to the corresponding loss, $\eta _ { \mathcal { H } , i }$ is a constant similar to Eq. (2) with $\widehat { Q } \ = \ \widehat { S } _ { i }$ , $\widehat { P } = \widehat { T }$ and $\quad M _ { S } \ =$ $\begin{array} { r } { \operatorname* { s u p } _ { i \in [ k ] , x \in \mathcal { X } , h \in \mathcal { H } } L ( h ( x ) , f _ { S _ { i } } ( x ) ) } \end{array}$ is the upper bound on loss on the source domains.
+
+There are several observations. (1) In the last term of the bound, $m / \lVert { \boldsymbol { \alpha } } \rVert _ { 2 } ^ { 2 }$ serves as the effective sample size. If $_ { \pmb { \alpha } }$ is uniform (i.e., $[ 1 / k , \ldots , 1 / k ] ^ { \top } )$ , then the effective sample size is $k m$ ; if $_ \alpha$ is onehot, then we effectively only have $m$ points from exactly one domain. (2) $\Re _ { m } ( \mathcal { H } _ { S _ { i } } )$ determines how expressive the model family $\mathcal { H }$ is w.r.t. the source data $S _ { i }$ . It can be estimated from samples (Bartlett & Mendelson, 2002, Theorem 11), but the computation is non-trivial for a model family like deep neural networks. The $\eta _ { \mathcal { H } , i }$ is uncontrollable once the hypothesis class $\mathcal { H }$ (e.g., neural network architecture) is given. Using a richer $\mathcal { H }$ is not always beneficial. Richer $\mathcal { H }$ can reduce the $\eta _ { \mathcal { H } , i }$ and also help us find a better function $h$ with smaller source losses $\mathcal { L } _ { \widehat { S } _ { i } }$ , but it will increase the $\Re _ { m } ( \mathcal { H } _ { S _ { i } } )$ . As removing these two terms does not seem to be empirically significant, we ignore them below for simplicity. (3) Let $g _ { h , i } \ { \stackrel { \mathrm { d e f } } { = } } \ { \mathcal { L } } _ { { \widehat { S } } _ { i } } ( h , f _ { S _ { i } } ) + { \mathrm { d i s c } } ( T , S _ { i } )$ . Small $g _ { h , i }$ indicates that we can achieve small loss on domain $S _ { i }$ , and it is similar to the target domain (i.e., small $\mathrm { d i s c } ( T , S _ { i } )$ , estimated from $\widehat { T } , \widehat { S } _ { i } )$ . We may want to emphasize on $S _ { i }$ by setting $\alpha _ { i }$ close to 1, but this will reduce the effective sample size. Therefore, we have to trade-off between the terms in this bound by choosing a proper $_ { \pmb { \alpha } }$ . (4) When $S _ { i }$ and $T$ are only partially overlapped, it might be difficult to find a suitable $\alpha _ { i }$ . In such cases, we can artificially split the given source domains into smaller datasets (e.g., by using clustering) then apply our method on the finer scale. This strategy requires that the learning algorithm has low computational complexities w.r.t. $k$ , the number of source domains. As we will see later in Section 3.3, this is indeed the case for our algorithm.
+
+Before we proceed to develop an algorithm based on the theorem, let us compare Eq. (3) to existing finite-sample bounds. The bound of Blitzer et al. (2008, Theorem 3) is informative when we have access to a small set of labelled target examples. In such cases, we can improve our bound by using this small labelled target set as an additional source domain $S _ { k + 1 }$ . We can also perform better model selection using such labelled set. The bound of Zhao et al. (2018, Theorem 2) is based on the $d _ { \mathcal { A } }$ -distance, which is a special case of disc in our bound. Moreover, our bound (Eq. (3)) use samplebased Rademacher complexity, which is generally tighter than other complexity measures such as VC-dimension (Bartlett & Mendelson, 2002; Koltchinskii et al., 2002; Bousquet et al., 2003).
+
+# 3.2 ALGORITHM
+
+In this section, we illustrate how to develop a practical algorithm based on Theorem 2. Ignoring the constants, we would like to minimize the upper bound of Eq. (3):
+
+$$
+\operatorname* { m i n } _ { h \in \mathcal { H } } \operatorname* { m i n } _ { \alpha \in \Delta } \quad U _ { h } ( \alpha ) = \langle \mathbf { g } _ { h } , \pmb { \alpha } \rangle + \tau \| \pmb { \alpha } \| _ { 2 }
+$$
+
+where $\mathbf { g } _ { h } = [ g _ { h , 1 } , \ldots , g _ { h , k } ] ^ { \top }$ and $\tau > 0$ is a hyper-parameter. If we can solve the inner minimization exactly given $h$ , then we can treat the optimal $\alpha ^ { * } ( h )$ as a function of $h$ and solve the outer minimization over $h$ effectively. In the following, we show how to achieve this.
+
+Given $\mathbf { g } _ { h }$ , the inner minimization can be reformulated as a second-order cone programming problem, but it has no closed-form solution due to the $\tau \| \alpha \| _ { 2 }$ term. Consider the following problem $\mathbf { \delta } ( \mathbf { z } = - \mathbf { g } _ { h } / \tau$ recovers the inner minimization):
+
+$$
+\begin{array} { r l } { \underset { { \pmb { \alpha } } \in \Delta } { \operatorname* { m i n } } } & { { } - \langle { \bf z } , { \pmb { \alpha } } \rangle + \| { \pmb { \alpha } } \| _ { 2 } } \end{array}
+$$
+
+The Lagrangian for its dual problem is
+
+$$
+\Lambda ( \alpha , \lambda , \nu ) = - \mathbf { z } ^ { \top } { \boldsymbol { \alpha } } + \| { \boldsymbol { \alpha } } \| _ { 2 } - \lambda ^ { \top } { \boldsymbol { \alpha } } + \nu ( \mathbf { 1 } ^ { \top } { \boldsymbol { \alpha } } - 1 ) \qquad \nu \in \mathbb { R } , \lambda \succeq 0
+$$
+
+Taking the derivative w.r.t. $_ \alpha$ and setting it to zero gives
+
+$$
+\frac { \partial \Lambda } { \partial \alpha } = - \mathbf { z } + \alpha / \| \alpha \| _ { 2 } - \lambda + \nu \mathbf { 1 } = 0 \quad \Longrightarrow \quad \alpha ^ { * } / \| \alpha ^ { * } \| _ { 2 } = \mathbf { z } - \nu \mathbf { 1 } + \lambda .
+$$
+
+Notice that $\alpha \neq \mathbf { 0 }$ so we have the constraint $\| \mathbf { z } - \nu \mathbf { 1 } + \lambda \| _ { 2 } = 1$ . Using this $\alpha ^ { * }$ in $\Lambda$ gives the following dual problem
+
+$$
+\operatorname* { m a x } _ { \nu , \lambda } \quad - \nu \qquad \mathrm { s . t . } \quad \| \mathbf { z } - \nu \mathbf { 1 } + \lambda \| _ { 2 } = 1 \quad \mathrm { a n d } \quad \lambda \succeq 0
+$$
+
+We would like to decrease $\nu$ as much as possible, and the best we can attain is the $\nu ^ { * }$ satisfying $\| [ \mathbf { z } \ - \ \nu ^ { * } \mathbf { 1 } ] _ { + } \| _ { 2 } \ = \ 1$ where $[ \mathbf { v } ] _ { + } ~ =$ $\operatorname* { m a x } ( \mathbf { 0 } , \mathbf { v } )$ . In this case, the optimal $\lambda ^ { * }$ can be attained as $\lambda _ { i } ^ { * } = 0$ if $z _ { i } - \nu ^ { * } > 0$ , otherwise $\lambda _ { i } ^ { * } = \nu ^ { * } - z _ { i }$ ; see Fig. 1. Although we do not have a closed-form expression for the optimal $\nu ^ { * }$ , we can use binary search to find it, starting from the interval $[ z _ { \mathrm { m i n } } - 1 , z _ { \mathrm { m a x } } ]$ where $z _ { \mathrm { m i n } } , z _ { \mathrm { m a x } }$ are the minimum and maximum of $\mathbf { z }$ respectively. Then we can recover the primal solution as
+
+$$
+\begin{array} { r } { \pmb { \alpha } ^ { * } = [ \mathbf { z } - \nu ^ { * } \mathbf { 1 } ] _ { + } / \| [ \mathbf { z } - \nu ^ { * } \mathbf { 1 } ] _ { + } \| _ { 1 } . } \end{array}
+$$
+
+![](images/eb5e4ad1811c0695772e0d2a205f02d764d0722497db315ad81de5d4dccfeb41.jpg)  
+Figure 1: Optimal solution to Eq. (5) (best viewed in color). $\lambda ^ { * }$ compensates what is below the $\nu ^ { * }$ , while $\nu ^ { * }$ is chosen such that the vector of the green bars has $L _ { 2 }$ norm of 1.
+
+Eq. (6) gives rise to a new way to project any vector $\mathbf { z }$ to the probability simplex, which is of independent interest and may be used in some other contexts.2 It resembles the standard projection onto the simplex based on squared Euclidean distance (Duchi et al., 2008, Eq.(3)). One subtle but crucial difference is that our Eq. (5) uses $\lVert \alpha \rVert _ { 2 }$ instead of $\| \alpha \| _ { 2 } ^ { 2 }$ . Recall that $\mathbf { z } = - \mathbf { g } _ { h } / \tau$ . Here $\tau$ can be interpreted as a temperature parameter. On one hand, if $\tau \gg 0$ , all $\mathbf { z }$ will have similar values and thus the optimal $\nu ^ { * }$ will be close to $z _ { \mathrm { m a x } }$ and $\alpha ^ { * }$ will be close to uniform. On the other hand, as $\tau  0$ , $z _ { \mathrm { m a x } }$ will stand out from the rest $z _ { i }$ and eventually $\nu ^ { * } = z _ { \mathrm { m a x } } - 1$ . This means the $g _ { h , i }$ corresponding to the $z _ { \mathrm { m a x } }$ is small enough so we focus solely on this domain and ignore all other domains (even though this will reduce effective sample size as discussed in Section 3.1).
+
+From a different perspective, if we define $F _ { \Delta } ^ { * } ( \alpha ) = \| \alpha \| _ { 2 }$ , then Eq. (5) equals $- F _ { \Delta } ( \mathbf { z } )$ , where $F _ { \Delta } , F _ { \Delta } ^ { * }$ are convex conjugates of each other, where we use $\Delta$ to emphasize their dependency on the simplex domain. Then our objective Eq. (4) can be expressed as
+
+$$
+\begin{array} { r l } { \underset { h \in \mathcal { H } } { \operatorname* { m i n } } } & { { } - F _ { \Delta } ( \mathbf { z } ) = - F _ { \Delta } ( - \mathbf { g } _ { h } / \tau ) } \end{array}
+$$
+
+![](images/ea8fec7c9b0f37a7286ba263febae4dffeab23bed712ddedb59faba1bf06933c.jpg)  
+Figure 2: Model architecture with two source domains (best viewed in color). Mini-batches of $x _ { S _ { i } }$ and $x _ { T }$ are fed to the network. $x _ { S _ { i } }$ will go through the classification/regression path on the upper $h _ { y }$ box, while all $x$ will go through to the corresponding discrepancies (in the lower $h _ { d }$ box). The gradients from the discrepancies will be reverted during backpropagation.
+
+We can optimize our objective and train a neural network $h$ using gradient-based optimizer. The optimal $\alpha ^ { * } ( h )$ is merely a function of $h$ and we can backprop through it. To facilitate the gradient computation and show that we can efficiently backprop through the projection of Eq. (6), we derive the Jacobian $J = \partial \pmb { \alpha } / \partial \mathbf { z }$ in Appendix B. This ensures effective end-to-end training.
+
+Now we elaborate more on how to compute $g _ { h , i }$ . The $\mathcal { L } _ { \widehat { S } _ { i } } ( h , f _ { S _ { i } } )$ is the task loss. The $\mathrm { d i s c } ( T , S _ { i } )$ b depends on whether the task is classification or regression. For classification, disc coincides with the $d _ { \mathcal { A } }$ -distance, so we use the domain classification loss (Zhao et al., 2018; Ben-David et al., 2007):
+
+$$
+\operatorname { d i s c } ( \widehat { T } , \widehat { S } _ { i } ) = 2 \left( 1 - \operatorname * { m i n } _ { h _ { d } } \widehat { \epsilon } _ { T , S _ { i } } ( h _ { d } ) \right) \qquad \widehat { \epsilon } _ { T , S _ { i } } ( h _ { d } ) = \frac { 1 } { 2 m } \sum _ { i = 1 } ^ { 2 m } \left| h _ { d } ( x ) - \delta _ { x \in \widehat { T } } \right|
+$$
+
+where $\widehat { \epsilon } _ { T , S _ { i } } ( h _ { d } )$ is the sample domain classification loss of a domain classifier $h _ { d } : \mathcal { X } \mapsto \{ 0 , 1 \}$ . bThis minimization over $h _ { d }$ will become maximization once we move it outside of the disc due to the minus sign. Then our objective consists of $\operatorname* { m i n } _ { h }$ and $\mathrm { m a x } _ { h _ { d } }$ , which resembles adversarial training (Goodfellow et al., 2014): learning a task classifier $h$ to minimize loss and a domain classifier $h _ { d }$ to maximize domain confusion. More specifically, if we decompose the neural network $h$ into a feature extractor $h _ { \mathrm { f e a } }$ and a label predictor $h _ { y }$ (i.e., $h ( x ) = h _ { y } ( h _ { \mathrm { f e a } } ( x ) ) )$ , we can learn a domainclassifier $h _ { d }$ on top of $h _ { \mathrm { f e a } }$ to classify $h _ { \mathrm { f e a } } ( x )$ between $S _ { i }$ and $T$ as a binary classification problem, where the domain itself is the label (see Fig. 2). To achieve this, we use the logistic loss to approximate $\widehat { \epsilon } _ { T , S _ { i } } ( h _ { d } )$ and apply the gradient reversal layer (Ganin & Lempitsky, 2015; Ganin et al., 2016) bwhen optimizing disc through backpropagation.
+
+If we are solving regression problems with eigenvalue (in magnitude) of the difference $L _ { 2 }$ loss, then  two matri $\mathrm { d i s c } ( \widehat { T } , \widehat { S } _ { i } ) = \| M _ { T } - M _ { S _ { i } } \| _ { 2 }$ gest and $\begin{array} { r } { M _ { T } = \frac { 1 } { m } \sum _ { j } h _ { \mathrm { f e a } } ( x _ { j } ^ { ( T ) } ) h _ { \mathrm { f e a } } ^ { \top } ( x _ { j } ^ { ( T ) } ) } \end{array}$ $\begin{array} { r } { M _ { S _ { i } } = \frac { 1 } { m } \sum _ { j } h _ { \mathrm { f e a } } ( x _ { j } ^ { ( i ) } ) h _ { \mathrm { f e a } } ^ { \top } ( x _ { j } ^ { ( i ) } ) } \end{array}$ (Mansour et al., $2 0 0 9 \mathrm { a }$ ; Cortes & Mohri, 2011, Sec.5), which can be conveniently approximated using mini-batches and a few steps of power iteration.
+
+# 3.3 COMPLEXITY
+
+Here we analyze the time and space complexities of the algorithm in each iteration. Similar to MDAN (Zhao et al., 2018), in each gradient step, we need to compute the task loss $\mathcal { L } _ { \widehat { S } _ { i } } ( h , f _ { S _ { i } } )$ and the $\mathrm { d i s c } ( T , S _ { i } )$ (or $d _ { \mathcal { A } }$ -distance) using mini-batches from each source domain $i \in [ k ]$ b. The question is whether one can maintain the $O ( k )$ complexity given that we need to compute the weights using Eq. (6) and backprop through it. For the forward computation of $\alpha ^ { * }$ , in order to compute the threshold $\nu ^ { * }$ to the $\epsilon > 0$ relative precision, the binary search will cost ${ \cal O } ( k \log ( 1 / \epsilon ) )$ . As for the backward pass of gradient computation, according to our calculation in Appendix B, the Jacobian $J = \partial \pmb { \alpha } / \partial \mathbf { z }$ has a concise form, meaning that it is possible to compute the matrix-vector product $J \mathbf { v }$ for a given vector $\mathbf { v }$ in $O ( k )$ time and space. Therefore, our space complexity is the same as MDAN but our time complexity is slightly slower by a factor of $\log ( 1 / \epsilon )$ . In comparison, the time complexity for MDMN (Li et al., 2018) is $O ( k ^ { 2 } )$ because it requires computing the pairwise weights within the $k$ source domains. When we have a lot of source domains, MDMN will be noticeably slower than MDAN and our DARN.
+
+# 4 RELATED WORK
+
+The idea of utilizing data from the source domain $( S , f _ { S } )$ to train a model for a different but related target domain $( T , f _ { T } )$ has been explored extensively for the last decade using different assumptions (Pan & Yang, 2009; Zhang et al., 2015). For instance, the covariate shift scenario (Shimodaira, 2000; Gretton et al., 2009; Sugiyama & Kawanabe, 2012; Wen et al., 2014) assumes $S \ne T$ but $f _ { S } = f _ { T }$ , while concept drift (Jiang & Zhai, 2007; Gama et al., 2014) assumes $S = T$ but $f _ { S } \neq f _ { T }$ . More specifically, Theorem 1 (Cortes et al., 2019) shows that both covariate shift (as measured by the discrepancy disc) and model misspecification (as controlled by $\eta _ { \mathcal { H } }$ ) contribute to the adaptation performance (Wen et al., 2014).
+
+Finding a domain-invariant feature space by minimizing a distance measure is common practice in domain adaptation, especially for training neural networks. Tzeng et al. (2017) provided a comprehensive framework that subsumes several prior efforts on learning shared representations across domains (Tzeng et al., 2015; Ganin et al., 2016). DARN uses adversarial domain classifier and the gradient reversal trick from Ganin et al. (2016). Instead of proposing a new loss for each pair of the source and target domains, our main contribution is the aggregation technique of computing the mixing coefficients $_ { \pmb { \alpha } }$ , which is derived from theoretical guarantees. When dealing with multiple source domains, our aggregation method can certainly be applied to other forms of discrepancies such as MMD (Gretton et al., 2012; Long et al., 2015; 2016), and other model architectures such as Domain Separation Network (Bousmalis et al., 2016), cycle-consistent model (Hoffman et al., 2018b), class-dependent adversarial domain classifier (Pei et al., 2018) and Known Unknown Discrimination (Schoenauer-Sebag et al., 2019).
+
+Our work focuses on multi-source to single-target adaptation, which has been investigated in the literature. Sun et al. (2011) developed a generalization bound but resorted to heuristic algorithms to adjust distribution shifts. Zhao et al. (2018) proposed a certain ad-hoc scheme for the combination coefficients $_ { \pmb { \alpha } }$ , which, unlike ours, are not theoretically justified. Multiple Domain Matching Network (MDMN) (Li et al., 2018) computes domain similarities not only between the source and target domains but also within the source domain themselves based on Wasserstein-like measure. Calculating such pairwise weights can be computationally demanding when we have a lot of source domains. Their bound requires additional smooth assumptions on the labelling functions $f _ { S _ { i } } , f _ { T }$ , and is not a finite-sample bound, as opposed to ours. As for the actual algorithm, they also use adhoc coefficients $_ { \pmb { \alpha } }$ without theoretical justification. Mansour et al. (2009b;c) consider multi-source adaptation where $\begin{array} { r } { T = \sum _ { i } \beta _ { i } S _ { i } } \end{array}$ is a convex mixture of source distributions with some weights $\beta _ { i }$ . Our analysis does not require this assumption. Hoffman et al. (2018a) provides similar guarantees with different assumptions, but unlike ours, their bounds are not finite-sample bounds.
+
+# 5 EXPERIMENTS
+
+In this section, we compare DARN with several other baselines and state-of-the-art methods for the popular tasks: sentiment analysis, digit recognition and object recognition. The following methods are compared. (1) The SRC (for source) method uses only labelled source data to train the model. It merges all available source examples to form a large dataset to perform training without adaptation. (2) TAR (for target) is another baseline that uses only $m$ labelled target data instances. It serves as upper bound of the best we can achieve if we had access to the true label of the target data. (3) Domain Adversarial Neural Network (DANN) (Ganin et al., 2016) is similar to our method in that we both use adversarial training objectives. It is not obvious how to adapt DANN to the multisource setting. Here we follow previous protocol (Zhao et al., 2018) and merge all source data to form a large joint source dataset of $k m$ instances for DANN to perform adaptation. (4) Moment Matching for Multi-Source Domain Adaptation (M3SDA) (Peng et al., 2019) is a recent state-ofthe-art method that combines moment matching and maximizing classifier discrepancy (Saito et al., 2018). We use their public code with a few necessary adjustments (change classification head based on the number of classes etc.) (5) Multisource Domain Adversarial Network (MDAN) (Zhao et al.,
+
+2018) resembles our method in that we both dynamically assign each source domain an importance weight during training. However, unlike ours, their weights are not theoretically justified. We use the soft version of MDAN since they reported that it performs better than the hard version, and we use their code. (6) Multiple Domain Matching Network (MDMN) (Li et al., 2018) computes weights not only between source and target domains but also within source domain themselves. We use their code of computing weights in our implementation. All the methods are applied to the same neural network structure to ensure fair comparison. Our PyTorch implementation will be available online after acceptance.
+
+# 5.1 SENTIMENT ANALYSIS
+
+Setup. We use the Amazon review dataset (Blitzer et al., 2007; Chen et al., 2012) that consists of positive and negative product reviews from four domains (Books, DVD, Electronics and Kitchen). Each of them is used in turn as the target domain and the other three are used as source domains. Their sample sizes are 6465, 5586, 7681, 7945 respectively. We follow the common protocol (Chen et al., 2012; Zhao et al., 2018) of using the top-5000 frequent unigrams/bigrams of all reviews as bag-of-words features and train a fully connected model (MLP) with [1000, 500, 100] hidden units for classifying positive versus negative reviews. The dropout drop rate is 0.7 for the input and hidden layers. In each run, we randomly sample 2000 reviews from each domain as labelled source or unlabelled target training examples, while the remaining instances are used as test examples for evaluation. The hyper-parameters are chosen based on cross-validation. The model is trained for 50 epochs and the mini-batch size is 20 per domain. The optimizer is Adadelta with a learning rate of 1.0. The soft version of MDAN has an additional parameter $\gamma = 1 / \tau$ which is the inverse of our temperature $\tau$ . The chosen parameters are $\gamma = 1 0 . 0$ for MDAN and $\gamma = 0 . 9$ for our DARN, which are selected from a wide range of candidate values.
+
+Results and Analysis. Table 1 summarizes the classification accuracies. The last column is the average accuracy of four domains, and the standard errors are calculated based on 20 runs. (1) Most of the time, DANN with joint source data performs worse than SRC which has no adaptation. This may be because it does not adjust for the each source domain, and as a result, it fails to ignore irrelevant data to avoid negative transfer. (2) Some domains are harder to adapt to than the others. For example, the accuracies of SRC and TAR on the Electronics domain are very close to each other, indicating that this requires little to no adaptation. Yet, our DARN is the closest to the TAR performance here. The Books domain is more challenging as the improvement over the SRC method is small, even though there exists a large gap between SRC and TAR. (3) DARN is always within the best performing methods and significantly outperforms others in the Electronics domain. Note that MDMN additionally computes similarities within source domains in each iteration, which can be computationally expensive $\operatorname { \mathcal { O } } ( k ^ { 2 } )$ per iteration) if the number of source domains is large. Instead, our method focuses on the discrepancy between source and target domains ( $O ( k )$ per iteration) and can achieve similar performance on this problem.
+
+Table 1: Classification accuracy $( \% )$ of the target sentiment datasets. Mean and standard error over 20 runs. The best method(s) (excluding TAR) based on one-sided Wilcoxon signed-rank test at the $5 \%$ significance level is(are) shown in bold for each domain.   
+
+<table><tr><td>Method</td><td>Books</td><td>DVD</td><td>Electronics</td><td>Kitchen</td><td>Avg.</td></tr><tr><td>SRC</td><td>79.15 ± 0.39</td><td>80.38 ± 0.30</td><td>85.48 ± 0.10</td><td>85.46 ± 0.34</td><td>82.62 ± 0.20</td></tr><tr><td>DANN</td><td>79.13 ± 0.29</td><td>80.60 ± 0.29</td><td>85.27 ± 0.14</td><td>85.56 ± 0.28</td><td>82.64 ± 0.14</td></tr><tr><td>M3SDA</td><td>79.42 ± 0.17</td><td>80.82 ± 0.35</td><td>85.52 ± 0.19</td><td>86.45 ± 0.43</td><td>83.05 ± 0.14</td></tr><tr><td>MDAN</td><td>79.99 ±0.20</td><td>81.66 ±0.19</td><td>84.76 ± 0.17</td><td>86.82 ± 0.13</td><td>83.31 ±0.08</td></tr><tr><td>MDMN</td><td>80.13 ±0.20</td><td>81.58 ±0.21</td><td>85.61 ±0.13</td><td>87.13 ± 0.11</td><td>83.61 ±0.07</td></tr><tr><td>DARN</td><td>79.93 ± 0.19</td><td>81.57 ± 0.16</td><td>85.75 ± 0.16</td><td>87.15 ± 0.14</td><td>83.60 ± 0.08</td></tr><tr><td>TAR</td><td>84.10 ± 0.13</td><td>83.68 ± 0.12</td><td>86.11 ±0.32</td><td>88.72 ±0.14</td><td>85.65 ± 0.09</td></tr></table>
+
+# 5.2 DIGIT RECOGNITION
+
+Setup. Following the setting from previous work (Ganin et al., 2016; Zhao et al., 2018), we use the four digit recognition datasets in this experiment (MNIST, MNIST-M, SVHN and Synth). MNIST is a well-known gray-scale images for digit recognition, and MINST-M (Ganin & Lempitsky, 2015) is a variant where the black and white pixels are masked with color patches. Street View House Number (SVHN) (Netzer et al., 2011) is a standard digit dataset taken from house numbers in Google Street View images. Synthetic Digits (Synth) (Ganin & Lempitsky, 2015) is a synthetic dataset that mimic SVHN using various transformations. One of the four datasets is chosen as unlabelled target domain in turn and the other three are used as labelled source domains.
+
+MNIST images are resized to $3 2 \times 3 2$ and represented as 3-channel color images in order to match the shape of the other three datasets. Each domain has its own given training and test sets when downloaded. Their respective training sample sizes are 60000, 59001, 73257, 479400, and the respective test sample sizes are 10000, 9001, 26032, 9553. In each run, 20000 images are randomly sampled from each domain’s training set as actual labelled source or unlabelled target training examples, and 9000 images are randomly sampled from each domain’s test set as actual test examples for evaluation. The model structure is shown in Appendix D. There is no dropout and the hyperparameters are chosen based on cross-validation. It is trained for 50 epochs and the mini-batch size is 128 per domain. The optimizer is Adadelta with a learning rate of 1.0. Validation selected $\gamma = 0 . 5$ for MDAN and $\gamma = 0 . 1$ for DARN.
+
+Results and Analysis. Table 2 shows the classification accuracy of each target dataset over 20 runs. (1) All methods except DANN can consistently outperform SRC. Again, without proper adjustment for each source domain, DANN with joint source data can perform worse than SRC. This suggests the importance of ignoring irrelevant data to avoid negative transfer. (2) DARN significantly outperforms MDAN and MDMN across all four domains, especially on the MNIST-M and SVHN domains. Notice that even though MDAN and MDMN have generalization guarantees, they both resort to ad-hoc aggregation rules to combine the source domains during training. Instead, our aggregation (Eq. (6)) is a direct optimization of the upper bound (Theorem 2) thus is theoretically justified and empirically superior for this problem.
+
+Table 2: Classification accuracy $( \% )$ of the target digit datasets. Mean and standard error over 20 runs. The best method (excluding TAR) based on one-sided Wilcoxon signed-rank test at the $5 \%$ significance level is shown in bold for each domain.   
+
+<table><tr><td>Method</td><td>MNIST</td><td>MNIST-M</td><td>SVHN</td><td>Synth</td><td>Avg.</td></tr><tr><td>SRC</td><td>96.78 ±0.08</td><td>60.80 ± 0.21</td><td>68.99 ± 0.69</td><td>84.09 ± 0.27</td><td>77.66 ± 0.14</td></tr><tr><td>DANN</td><td>96.41 ± 0.13</td><td>60.10 ± 0.27</td><td>70.19 ± 1.30</td><td>83.83 ± 0.25</td><td>77.63 ± 0.35</td></tr><tr><td>M3SDA</td><td>96.95 ± 0.06</td><td>65.03 ± 0.80</td><td>71.66 ± 1.16</td><td>80.12 ± 0.56</td><td>78.44 ± 0.36</td></tr><tr><td>MDAN</td><td>97.10 ± 0.10</td><td>64.09 ± 0.31</td><td>77.72 ± 0.60</td><td>85.52 ± 0.19</td><td>81.11 ± 0.21</td></tr><tr><td>MDMN</td><td>97.15 ± 0.09</td><td>64.34 ± 0.27</td><td>76.43 ± 0.48</td><td>85.80 ± 0.21</td><td>80.93 ± 0.16</td></tr><tr><td>DARN</td><td>98.09 ± 0.03</td><td>67.06 ± 0.14</td><td>81.58 ± 0.14</td><td>86.79 ± 0.09</td><td>83.38 ± 0.06</td></tr><tr><td>TAR</td><td>99.02 ± 0.02</td><td>94.66 ± 0.10</td><td>87.40 ± 0.17</td><td>96.90 ± 0.09</td><td>94.49 ± 0.07</td></tr></table>
+
+# 5.3 OBJECT RECOGNITION: OFFICE-HOME
+
+To showcase the applicability of our method to more complicated real-world tasks, we use the challenging Office-Home dataset (Venkateswara et al., 2017). It contains images of 65 everyday objects such as spoon, sink, mug and pen from four different domains: Art, Clipart, Product and Real-World. One of the four datasets is chosen as unlabelled target domain in turn and the other three are used as labelled source domains.
+
+The respective sample sizes are 2427, 4365, 4439, 4357. In each run, 2000 images are randomly sampled from each domain as labelled source or unlabelled target training examples, and the rest images are used as test images for evaluation. We use the ResNet50 He et al. (2016) pretrained from the ImageNet in PyTorch as the base network for feature learning and put an MLP with [1000, 500, 100, 65] units on top for classification. It is trained for 50 epochs and the mini-batch size is 32 per domain. The optimizer is Adadelta with a learning rate of 1.0. MDAN uses $\gamma = 1 . 0$ while DARN uses $\gamma = 0 . 5$ .
+
+Table 3 shows the classification accuracy of each target dataset over 20 runs. Most existing works in the literature focused on single-source to single-target adaptation on this problem (e.g., see (Long et al., 2018)). Compared to them, using multi-source methods can significantly boost performance, revealing the importance of using multiple source domains when possible. Even though this is a significantly more challenging problem with more classes and much fewer images compared to the digits datasets, our DARN achieves state-of-the-art performance in this setting, excelling existing methods by a noticeable margin.
+
+Table 3: Classification accuracy $( \% )$ of the Office-Home datasets. Mean and standard error over 20 runs. The best method (excluding TAR) based on one-sided Wilcoxon signed-rank test at the $5 \%$ significance level is shown in bold for each domain.   
+
+<table><tr><td>Method</td><td>Art</td><td>Clipart</td><td>Product</td><td>Real-World</td><td>Avg.</td></tr><tr><td>SRC</td><td>58.02 ± 0.47</td><td>57.29 ± 0.30</td><td>74.26 ± 0.22</td><td>77.98 ± 0.25</td><td>66.89 ± 0.16</td></tr><tr><td>DANN</td><td>57.39 ± 0.69</td><td>57.35 ± 0.35</td><td>73.78 ± 0.27</td><td>78.12 ± 0.21</td><td>66.66 ± 0.19</td></tr><tr><td>M3SDA</td><td>64.05 ± 0.61</td><td>62.79 ± 0.37</td><td>76.21 ± 0.30</td><td>78.63 ± 0.22</td><td>70.42 ± 0.18</td></tr><tr><td>MDAN</td><td>68.14 ± 0.58</td><td>67.04 ± 0.21</td><td>81.03 ± 0.22</td><td>82.79 ± 0.15</td><td>74.75 ± 0.18</td></tr><tr><td>MDMN</td><td>68.67 ± 0.55</td><td>67.75 ± 0.20</td><td>81.37 ± 0.18</td><td>83.32 ± 0.14</td><td>75.28 ± 0.15</td></tr><tr><td>DARN</td><td>70.00 ± 0.38</td><td>68.42 ± 0.14</td><td>82.75 ± 0.21</td><td>83.88 ± 0.16</td><td>76.26 ± 0.13</td></tr><tr><td>TAR</td><td>71.19 ± 0.38</td><td>79.16 ± 0.16</td><td>90.66 ± 0.15</td><td>85.60 ± 0.14</td><td>81.65 ± 0.12</td></tr></table>
+
+# 5.4 VISUALIZING DOMAIN IMPORTANCE
+
+To demonstrate how DARN can effectively aggregate multiple source domains, we visualize the source domain weights (i.e., $_ \alpha$ in DARN) for the Amazon dataset. We also compared to the weights produced by MDMN, using the original authors’ code.
+
+![](images/62e04dadbe5bfe9a4580e1a269ed124bd7fed8455dec07278e12a04c6d3cc6e9.jpg)
+
+Fig. 3 and Fig. 4 compare the evolution of source domain weights during training. In each subfigure, every row corresponds to the weights of the source domains when learning for one target domain. The white stripe indicates there is no target domain weight and darker stripe means less weight. They are evaluated at the end of each epoch over 50 epochs. To avoid noisy values due to small mini-batch size, the values are exponential moving averages with a decay rate of 0.95. (1) For DARN, as Electronics and Kitchen are more related to each other than Books and DVD, their respective weights remain higher during training. This is reasonable since we have overlapping products (e.g., blenders) in both domains. The $_ { \pmb { \alpha } }$ is changing dynamically during training, showing the flexibility of our method to adjust domain importance when needed. (2) In comparison, the domain weights produced by MDMN are not very stable. We take a closer look at the MDMN weights and notice that they change drastically, especially towards the end of the training. It can produce alternating one-hot vectors $_ { \pmb { \alpha } }$ , changing from one domain to a different domain and ignoring the rest. This instability makes their domain weights hard to interpret.
+
+![](images/6b4d4cdea0b80f67aaa590012c1c3312a0bb22cb56659b0246dcc7dc99b53f8c.jpg)  
+Figure 3: Domain weights of DARN for the Amazon data.   
+Figure 4: Domain weights of MDMN for the Amazon data.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+Our work uses the discrepancy (Mansour et al., $2 0 0 9 \mathrm { a }$ ; Cortes et al., 2019) to derive a finite-sample generalization bound for multi-source to single-target adaptation. Our theorem shows that, in order to achieve the best possible generalization upper bound for a target domain, we need to trade-off between including all source domains to increase effective sample size and removing source domains that are underperforming or not similar to the target domain. Based on this observation, we derive an algorithm, Domain AggRegation Network (DARN), that can dynamically adjust the weight of each source domain during end-to-end training. Experiments on sentiment analysis, digits recognition and object recognition show that DARN outperforms state-of-the-art alternatives. Recent analysis (Zhao et al., 2019; Johansson et al., 2019) show that solely focusing on learning domain invariant features can be problematic when the marginal label distributions on $\mathcal { V }$ between source and target domains are significantly different. Thus it makes sense to take $\eta _ { \mathcal { H } }$ into consideration when a small amount of labelled data is available for the target domain, which we will explore in the future.
+
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+
+# A PROOF OF THEOREM 2
+
+Theorem 2 Given $k$ source domains datasets $\{ ( x _ { j } ^ { ( i ) } , y _ { j } ^ { ( i ) } ) : i \in [ k ] , j \in [ m ] \}$ with m iid examples each where $\widehat { S } _ { i } = \{ x _ { j } ^ { ( i ) } \}$ and $y _ { j } ^ { ( i ) } = f _ { S _ { i } } ( x _ { j } ^ { ( i ) } )$ , for any $\begin{array} { r } { \pmb { \alpha } \in \Delta = \{ \pmb { \alpha } : \alpha _ { i } \geq 0 , \sum _ { i } \alpha _ { i } = 1 \} , \delta \in } \end{array}$ $( 0 , 1 )$ , and $\forall h \in { \mathcal { H } }$ , w.p. at least $1 - \delta$
+
+$$
+\mathcal { L } _ { T } ( h , f _ { T } ) \leq \sum _ { i } \alpha _ { i } \left( \mathcal { L } _ { \widehat { S } _ { i } } ( h , f _ { S _ { i } } ) + \mathrm { { d i s c } } ( T , S _ { i } ) + 2 \mathfrak { R } _ { m } ( \mathcal { H } _ { S _ { i } } ) + \eta _ { \mathcal { H } , i } \right) + \| \alpha \| _ { 2 } M _ { S } \sqrt { \frac { \log ( 1 / \delta ) } { 2 m } } ,
+$$
+
+where $\mathcal { H } _ { S _ { i } } ~ = ~ \{ x ~ \mapsto ~ L \big ( h ( x ) , f _ { S _ { i } } ( x ) \big ) ~ : ~ h ~ \in ~ \mathcal { H } \}$ is the set of functions mapping $x$ to the corresponding loss, $\eta _ { \mathcal { H } , i }$ is a constant similar to Eq. (2) with ${ \widehat { Q } } \ = \ { \widehat { S } } _ { i } , \ { \widehat { P } } \ = \ { \widehat { T } }$ and $\quad M _ { S } \ =$ $\begin{array} { r } { \operatorname* { s u p } _ { i \in [ k ] , x \in \mathcal { X } , h \in \mathcal { H } } L ( h ( x ) , f _ { S _ { i } } ( x ) ) } \end{array}$ is the upper bound on loss on the source domains.
+
+Proof: The proof is similar to Cortes et al. (2019); Zhao et al. (2018). Given $\alpha \in \Delta$ , the mixture $\widehat { S } = \textstyle \sum _ { i } { \alpha _ { i } } \widehat { S } _ { i }$ can be considered as the joint source data with $k m$ points, where a point $x ^ { ( i ) }$ from $\widehat { S } _ { i }$ has weight $\alpha _ { i } / m$ . Define $\begin{array} { r } { \Phi = \operatorname* { s u p } _ { h \in \mathcal { H } } \mathcal { L } _ { T } ( h , f _ { T } ) - \sum _ { i } \alpha _ { i } \mathcal { L } _ { \widehat { S } _ { i } } ( h , f _ { S _ { i } } ) } \end{array}$ . Changing a point $\boldsymbol { x } ^ { ( i ) }$ from $\widehat { S } _ { i }$ $\Phi$ at most . As a re $\frac { M _ { S } \alpha _ { i } } { m }$ . or McDiarmid’s, w.p. at least ality, we have , the followin $\mathrm { P r } ( \Phi - \mathbb { E } [ \Phi ] > \epsilon ) \leq$ $\exp \left( - \frac { 2 \epsilon ^ { 2 } m } { M _ { S } ^ { 2 } \| \alpha \| _ { 2 } ^ { 2 } } \right)$ $\delta \in ( 0 , 1 )$ $1 - \delta$ $h \in \mathcal H$
+
+$$
+\mathcal { L } _ { T } ( h , f _ { T } ) \leq \sum _ { i } \alpha _ { i } \mathcal { L } _ { \widehat { S } _ { i } } ( h , f _ { S _ { i } } ) + \mathbb { E } [ \Phi ] + \| \alpha \| _ { 2 } M _ { S } \sqrt { \frac { \log ( 1 / \delta ) } { 2 m } } .
+$$
+
+Now we bound $\mathbb { E } [ \Phi ]$ . Let $\mathcal { H } _ { S _ { i } } = \{ x \mapsto L ( h ( x ) , f _ { S _ { i } } ( x ) ) : h \in \mathcal { H } \} .$ .
+
+$$
+\begin{array} { r l } & { \displaystyle \{ \Phi \} = \mathbb { E } _ { \hat { S } } [ \operatorname* { s u p } _ { \hat { S } } \int _ { x ( \hat { R } , f _ { S } ) } \sum _ { t = 1 } \alpha _ { i } C _ { \hat { S } _ { i } } ( h , f _ { S _ { i } } ) ] } \\ & { \quad \le \mathbb { E } _ { \hat { S } } [ \operatorname* { s u p } _ { \hat { S } } \displaystyle \sum _ { i = 0 } \alpha _ { i } C _ { \hat { S } _ { i } } ( h , f _ { S _ { i } } ) - \sum _ { t } \alpha _ { i } C _ { \hat { S } _ { i } } ( h , f _ { S _ { i } } ) ] + \operatorname* { s u p } _ { \hat { S } \in \mathcal { R } } \Bigg ( \mathcal { L } _ { \mathcal { X } } ( h , f _ { T } ) - \sum _ { t } \alpha _ { i } C _ { \hat { S } _ { i } } ( h , f _ { S _ { i } } ) \Bigg ) } \\ & { \quad \le \mathbb { E } _ { \hat { S } } [ \operatorname* { s u p } _ { \hat { S } } ( \mathcal { L } _ { S _ { i } } ( h , f _ { S _ { i } } ) - \sum _ { t _ { S _ { i } } ( h , f _ { S _ { i } } ) } ) + \sum _ { t } \alpha _ { i } \operatorname* { s u p } _ { \hat { S } \in \mathcal { R } } ( \mathcal { L } _ { T } ( h , f _ { T } ) - \mathcal { L } _ { S _ { i } } ( h , f _ { S _ { i } } ) )  } \\ & { \quad  = \sum _ { t } \alpha _ { i } \mathbb { E } _ { \hat { S } } [ \operatorname* { s u p } _ { \hat { S } } ( \mathcal { L } _ { S _ { i } } ( h , f _ { S _ { i } } ) - \mathcal { L } _ { \hat { S } _ { i } } ( h , f _ { S _ { i } } ) ) ] + \sum _ { t } \alpha _ { i \in \mathcal { R } } ( \mathcal { L } _ { T } ( h , f _ { T } ) - \mathcal { L } _ { S _ { i } } ( h , f _ { S _ { i } } ) )  } \\ &  \quad \le 2 \sum _ { t } \alpha _ { i } \mathbb { E } _ { \hat { S } } [ \operatorname* { s u p } _ { \hat { S } } ( \mathcal { L } _ { S _ { i } } ( h , f _ { S _ { i } } ) - \mathcal { L } _ { \hat { S } _ { i } } ( h , f _  S \end{array}
+$$
+
+where first and second inequalities are using the subadditivity of sup, followed by the equality using the independence between the domains $\left. \widehat { S } _ { i } \right.$ , the second last inequality is due to the standard “ghost sample” argument in terms of the Rademacher complexity and the last inequality is due to Cortes et al. (2019, Proposition 8) for each individual $S _ { i }$ . ■
+
+# B JACOBIAN
+
+Here we calculate the Jacobian $J _ { i j } = \partial \alpha _ { i } / \partial z _ { j }$ for Eq. (6) in the main text:
+
+$$
+\begin{array} { r } { \pmb { \alpha } ^ { * } = [ \mathbf { z } - \nu ^ { * } \mathbf { 1 } ] _ { + } / \| [ \mathbf { z } - \nu ^ { * } \mathbf { 1 } ] _ { + } \| _ { 1 } . } \end{array}
+$$
+
+In the following, we write $\pmb { \alpha } = \pmb { \alpha } ^ { * } , \pmb { \nu } = \pmb { \nu } ^ { * }$ to simplify notations. Let $S = \{ i : z _ { i } - \nu > 0 \}$ be the support of the probability vector $_ { \pmb { \alpha } }$ . $J _ { i j } = 0$ if $i \not \in S$ or $j \not \in S$ since $\alpha _ { i } = 0$ in the former case while $z _ { j }$ does not contribute to the $_ { \pmb { \alpha } }$ in the latter case. Now consider the case $i , j \in S$ . Let $\begin{array} { r } { K = \| [ \mathbf { z } - \nu \mathbf { 1 } ] _ { + } \| _ { 1 } = \sum _ { j \in S } ( z _ { j } - \nu ) } \end{array}$ . Then $\alpha _ { i } = ( z _ { i } - \nu ) \cdot { \frac { 1 } { K } }$ and
+
+$$
+\frac { \partial \alpha _ { i } } { \partial z _ { j } } = \left( \delta _ { i = j } - \frac { \partial \nu } { \partial z _ { j } } \right) \cdot \frac { 1 } { K } - \frac { 1 } { K ^ { 2 } } \cdot \frac { \partial K } { \partial z _ { j } } \cdot \left( z _ { i } - \nu \right) = \frac { 1 } { K } \left( \delta _ { i = j } - \frac { \partial \nu } { \partial z _ { j } } - \frac { \partial K } { \partial z _ { j } } \cdot \alpha _ { i } \right) ,
+$$
+
+where $\delta _ { i = j }$ is the indicator or delta function. Now we compute $\textstyle { \frac { \partial \nu } { \partial z _ { j } } }$ and $\frac { \partial K } { \partial z _ { j } }$ . By the definition of $\nu$ , we know that
+
+$$
+\begin{array} { r l } & { ~ \displaystyle \sum _ { j \in S } ( z _ { j } - \nu ) ^ { 2 } = | S | \nu ^ { 2 } - 2 \nu \sum _ { j \in S } z _ { j } + \sum _ { j \in S } z _ { j } ^ { 2 } = 1 } \\ { \implies } & { \nu = \frac { \sum _ { j \in S } z _ { j } } { | S | } - \frac { \sqrt { A } } { | S | } \quad \mathrm { w h e r e } \quad A = \left( \displaystyle \sum _ { j \in S } z _ { j } \right) ^ { 2 } - | S | \left( \displaystyle \sum _ { j \in S } z _ { j } ^ { 2 } - 1 \right) } \\ { \implies } & { \displaystyle \frac { \partial \nu } { \partial z _ { j } } = \frac { 1 } { | S | } - \frac { B _ { j } } { | S | } \quad \mathrm { w h e r e } \quad B _ { j } = \displaystyle \sum _ { j \prime \in S } z _ { j \prime } - | S | z _ { j } } \end{array}
+$$
+
+The first right-arrow is due to the quadratic formula and realizing that $\textstyle \sum _ { j \in S } z _ { j } / | S |$ is the mean of the supported $z _ { j }$ so $\nu$ must be smaller than it (i.e., we take $-$ in the $\pm$ of the quadratic formula, otherwise some of the $z _ { j }$ will not be in the support anymore). And
+
+$$
+\frac { \partial K } { \partial z _ { j } } = 1 - | \boldsymbol { S } | \cdot \frac { \partial \nu } { \partial z _ { j } } = \frac { B _ { j } } { \sqrt { A } } .
+$$
+
+Plugging Eq. (8) and Eq. (9) in Eq. (7) gives
+
+$$
+\frac { \partial \alpha _ { i } } { \partial z _ { j } } = \frac { 1 } { K } \left[ \delta _ { i = j } - \frac { 1 } { | S | } + \frac { B _ { j } } { \sqrt { A } } \cdot \left( \frac { 1 } { | S | } - \alpha _ { i } \right) \right]
+$$
+
+Note that
+
+$$
+\frac { B _ { j } } { K } = \frac { \sum _ { j ^ { \prime } \in S } z _ { j ^ { \prime } } - | S | z _ { j } } { \sum _ { j ^ { \prime } \in S } ( z _ { j ^ { \prime } } - \nu ) } = \frac { \sum _ { j ^ { \prime } \in S } ( z _ { j ^ { \prime } } - \nu ) + | S | ( \nu - z _ { j } ) } { \sum _ { j ^ { \prime } \in S } ( z _ { j ^ { \prime } } - \nu ) } = 1 - | S | \alpha _ { j } .
+$$
+
+Then
+
+$$
+J _ { i j } = \frac { \partial \alpha _ { i } } { \partial z _ { j } } = \frac { 1 } { K } \left( \delta _ { i = j } - \frac { 1 } { | S | } \right) + \frac { | S | } { \sqrt { A } } \left( \frac { 1 } { | S | } - \alpha _ { i } \right) \left( \frac { 1 } { | S | } - \alpha _ { j } \right) .
+$$
+
+In matrix form,
+
+$$
+J = { \frac { 1 } { K } } \left( \operatorname { D i a g } ( \mathbf { s } ) - { \frac { \mathbf { s s } ^ { \top } } { | S | } } \right) + { \frac { | S | } { \sqrt { A } } } \left( { \frac { \mathbf { s } } { | S | } } - \alpha \circ \mathbf { s } \right) \left( { \frac { \mathbf { s } } { | S | } } - \alpha \circ \mathbf { s } \right) ^ { \top } ,
+$$
+
+where $\mathbf { s } = [ s _ { 1 } , \ldots , s _ { k } ] ^ { \top }$ is a vector indicating the support $s _ { i } = \delta _ { i \in S }$ and $\circ$ is element-wise multiplication. More often, we need to compute its multiplication with a vector $\mathbf { v }$
+
+$$
+J \mathbf { v } = { \frac { \mathbf { s } } { K } } \circ \left( \mathbf { v } - { \frac { \mathbf { s } ^ { \mathsf { T } } \mathbf { v } } { | S | } } \right) + { \frac { | S | } { \sqrt { A } } } \left( { \frac { \mathbf { s } } { | S | } } - \alpha \circ \mathbf { s } \right) \left( { \frac { \mathbf { s } } { | S | } } - \alpha \circ \mathbf { s } \right) ^ { \mathsf { T } } \mathbf { v } .
+$$
+
+Note that all quantities except $A$ have been computed during the forward pass of calculating Eq. (6).   
+$A$ can be computed in $O ( | S | )$ time so the overall computation is still $O ( k )$ since $| S | \le k$ .
+
+![](images/f9cd470cc85623a387ae86e2cf3053ca48e01e4934132d87851623a860fab801.jpg)  
+Figure 5: Regression experiment (best viewed in color).
+
+# C REGRESSION ON SYNTHETIC DATA
+
+Due to the lack of publicly available multi-source regression datasets, we demonstrate our method on a synthetic regression problem here.
+
+Setup. We use eight source domains, where the ith $\begin{array} { r } { x _ { i } \sim \dot { \mathcal { N } } ( \frac { \pi } { 4 } i - \frac { 7 \pi } { 8 } , 0 . 2 ^ { 2 } ) } \end{array}$ and the output is $y = \sin ( x ) + \dot { \epsilon }$ $( i = \{ 0 , 1 , \ldots , 7 \} ,$ where $\epsilon \sim \mathcal { N } ( 0 , 0 . 0 5 ^ { 2 } )$ ) domain data is generated by is random noise. These eight domains evenly cover the sin function on $[ - \pi , \pi ]$ (see Fig. 5a). Next we construct four target domains, where the $j$ th $( j = \{ 0 , 1 , 2 , 3 \} )$ ) domain is generated by $\begin{array} { r } { x _ { j } \sim \mathcal { N } ( \frac { \pi } { 2 } j - \frac { 3 \pi } { 4 } , 0 . 4 ^ { 2 } ) } \end{array}$ . Similar to the source domains, these four target domains evenly cover $[ - \pi , \pi ]$ (see Fig. 5b). Each source/target domain has 100 data points.
+
+We use labelled source data and unlabelled target data for learning. We take on one target domain at a time, and learn a linear model from all eight source domains with MSE loss. The goal is to see whether our method can focus on the relevant source domains and learn a linear model that can perform well on the target domain.
+
+Results and Analysis. First, Fig. 5b shows the learned models. The learned linear models can fit the target data very well. This shows that our DARN can learn a meaningful model for a specific target domain, using only labelled source data and unlabelled target data. Second, Fig. 5c shows the source domain weights (the $\alpha$ ) for each target domain after training. The weight colors correspond to the colors in Fig. 5a. It is noticeable that DARN can focus well on the respective relevant source domains for each target domain and ignore the rest. Note that these values are automatically learned during the training of the model.
+
+![](images/cfd5ab8a3618f8d52a8447297bb3a26d1fe8751451a9ce0f4a567982578e57a4.jpg)  
+D MODEL ARCHITECTURE FOR DIGIT RECOGNITION   
+Figure 6: Model architecture for the digit recognition.

parse/train/DEsIX_D_vR/DEsIX_D_vR.md

@@ -0,0 +1,323 @@
+# Neural Bellman-Ford Networks: A General Graph Neural Network Framework for Link Prediction
+
+Zhaocheng $\mathbf { Z } \mathbf { h } \mathbf { u } ^ { 1 , 2 }$ , Zuobai Zhang1,2, Louis-Pascal Xhonneux1,2, Jian Tang1,3,4
+
+Mila - Québec AI Institute1, Université de Montréal2 HEC Montréal3, CIFAR AI Chair4 {zhaocheng.zhu, zuobai.zhang, louis-pascal.xhonneux}@mila.quebec jian.tang@hec.ca
+
+# Abstract
+
+Link prediction is a very fundamental task on graphs. Inspired by traditional path-based methods, in this paper we propose a general and flexible representation learning framework based on paths for link prediction. Specifically, we define the representation of a pair of nodes as the generalized sum of all path representations between the nodes, with each path representation as the generalized product of the edge representations in the path. Motivated by the Bellman-Ford algorithm for solving the shortest path problem, we show that the proposed path formulation can be efficiently solved by the generalized Bellman-Ford algorithm. To further improve the capacity of the path formulation, we propose the Neural Bellman-Ford Network (NBFNet), a general graph neural network framework that solves the path formulation with learned operators in the generalized Bellman-Ford algorithm. The NBFNet parameterizes the generalized Bellman-Ford algorithm with 3 neural components, namely INDICATOR, MESSAGE and AGGREGATE functions, which corresponds to the boundary condition, multiplication operator, and summation operator respectively1. The NBFNet covers many traditional path-based methods, and can be applied to both homogeneous graphs and multi-relational graphs (e.g., knowledge graphs) in both transductive and inductive settings. Experiments on both homogeneous graphs and knowledge graphs show that the proposed NBFNet outperforms existing methods by a large margin in both transductive and inductive settings, achieving new state-of-the-art results2.
+
+# 1 Introduction
+
+Predicting the interactions between nodes (a.k.a. link prediction) is a fundamental task in the field of graph machine learning. Given the ubiquitous existence of graphs, such a task has many applications, such as recommender system [34], knowledge graph completion [41] and drug repurposing [27].
+
+Traditional methods of link prediction usually define different heuristic metrics over the paths between a pair of nodes. For example, Katz index [30] is defined as a weighted count of paths between two nodes. Personalized PageRank [42] measures the similarity of two nodes as the random walk probability from one to the other. Graph distance [37] uses the length of the shortest path between two nodes to predict their association. These methods can be directly applied to new graphs, i.e., inductive setting, enjoy good interpretability and scale up to large graphs. However, they are designed based on handcrafted metrics and may not be optimal for link prediction on real-world graphs.
+
+To address these limitations, some link prediction methods adopt graph neural networks (GNNs) [32, 48, 59] to automatically extract important features from local neighborhoods for link prediction. Thanks to the high expressiveness of GNNs, these methods have shown state-of-the-art performance. However, these methods can only be applied to predict new links on the training graph, i.e. transductive setting, and lack interpretability. While some recent methods [73, 55] extract features from local subgraphs with GNNs and support inductive setting, the scalability of these methods is compromised.
+
+Therefore, we wonder if there exists an approach that enjoys the advantages of both traditional path-based methods and recent approaches based on graph neural networks, i.e., generalization in the inductive setting, interpretability, high model capacity and scalability.
+
+In this paper, we propose such a solution. Inspired by traditional path-based methods, our goal is to develop a general and flexible representation learning framework for link prediction based on the paths between two nodes. Specifically, we define the representation of a pair of nodes as the generalized sum of all the path representations between them, where each path representation is defined as the generalized product of the edge representations in the path. Many link prediction methods, such as Katz index [30], personalized PageRank [42], graph distance [37], as well as graph theory algorithms like widest path [4] and most reliable path [4], are special instances of this path formulation with different summation and multiplication operators. Motivated by the polynomial-time algorithm for the shortest path problem [5], we show that such a formulation can be efficiently solved via the generalized Bellman-Ford algorithm [4] under mild conditions and scale up to large graphs.
+
+The operators in the generalized Bellman-Ford algorithm—summation and multiplication—are handcrafted, which have limited flexibility. Therefore, we further propose the Neural Bellman-Ford Networks (NBFNet), a graph neural network framework that solves the above path formulation with learned operators in the generalized Bellman-Ford algorithm. Specifically, NBFNet parameterizes the generalized Bellman-Ford algorithm with three neural components, namely INDICATOR, MESSAGE and AGGREGATE functions. The INDICATOR function initializes a representation on each node, which is taken as the boundary condition of the generalized Bellman-Ford algorithm. The MESSAGE and the AGGREGATE functions learn the multiplication and summation operators respectively.
+
+We show that the MESSAGE function can be defined according to the relational operators in knowledge graph embeddings [6, 68, 58, 31, 52], e.g., as a translation in Euclidean space induced by the relational operators of TransE [6]. The AGGREGATE function can be defined as learnable set aggregation functions [71, 65, 9]. With such parameterization, NBFNet can generalize to the inductive setting, meanwhile achieve one of the lowest time complexity among inductive GNN methods. A comparison of NBFNet and other GNN frameworks for link prediction is showed in Table 1. With other instantiations of MESSAGE and AGGREGATE functions, our framework can also recover some existing works on learning logic rules [69, 46] for link prediction on knowledge graphs (Table 2).
+
+Our NBFNet framework can be applied to several link prediction variants, covering not only singlerelational graphs (e.g., homogeneous graphs) but also multi-relational graphs (e.g., knowledge graphs). We empirically evaluate the proposed NBFNet for link prediction on homogeneous graphs and knowledge graphs in both transductive and inductive settings. Experimental results show that the proposed NBFNet outperforms existing state-of-the-art methods by a large margin in all settings, with an average relative performance gain of $18 \%$ on knowledge graph completion $( \mathrm { H I T S } @ 1 )$ ) and $22 \%$ on inductive relation prediction $( \mathrm { H I T S } @ 1 0 )$ . We also show that the proposed NBFNet is indeed interpretable by visualizing the top-k relevant paths for link prediction on knowledge graphs.
+
+Table 1: Comparison of GNN frameworks for link prediction. The time complexity refers to the amortized time for predicting a single edge or triplet. $| \nu |$ is the number of nodes, $\lvert \mathcal { E } \rvert$ is the number of edges, and $d$ is the dimension of representations. The wall time is measured on FB15k-237 test set with 40 CPU cores and 4 GPUs. We estimate the wall time of GraIL based on a downsampled test set.   
+
+<table><tr><td>Method</td><td>Inductive3</td><td>Interpretable</td><td>Learned Representation</td><td>Time Complexity</td><td>Wall Time</td></tr><tr><td>VGAE [32] / RGCN [48]</td><td></td><td></td><td>√</td><td>0(d)</td><td>18 secs</td></tr><tr><td>NeuralLP [69] / DRUM[46]</td><td>√</td><td>√</td><td></td><td>(+R）</td><td>2.1 mins</td></tr><tr><td>SEAL [73]] GraIL [55]</td><td>√</td><td></td><td>√</td><td>0(1ε|d²)</td><td>~1 month</td></tr><tr><td>NBFNet</td><td>√</td><td>√</td><td>√</td><td>（+）</td><td>4.0 mins</td></tr></table>
+
+# 2 Related Work
+
+Existing work on link prediction can be generally classified into 3 main paradigms: path-based methods, embedding methods, and graph neural networks.
+
+Path-based Methods. Early methods on homogeneous graphs compute the similarity between two nodes based on the weighted count of paths (Katz index [30]), random walk probability (personalized PageRank [42]) or the length of the shortest path (graph distance [37]). SimRank [28] uses advanced metrics such as the expected meeting distance on homogeneous graphs, which is extended by PathSim [51] to heterogeneous graphs. On knowledge graphs, Path Ranking [35, 15] directly uses relational paths as symbolic features for prediction. Rule mining methods, such as NeuralLP [69] and DRUM [46], learn probabilistic logical rules to weight different paths. Path representation methods, such as Path-RNN [40] and its successors [11, 62], encode each path with recurrent neural networks (RNNs), and aggregate paths for prediction. However, these methods need to traverse an exponential number of paths and are limited to very short paths, e.g., $\leq 3$ edges. To scale up path-based methods, All-Paths [57] proposes to efficiently aggregate all paths with dynamic programming. However, All-Paths is restricted to bilinear models and has limited model capacity. Another stream of works [64, 10, 22] learns an agent to collect useful paths for link prediction. While these methods can produce interpretable paths, they suffer from extremely sparse rewards and require careful engineering of the reward function [38] or the search strategy [50]. Some other works [8, 44] adopt variational inference to learn a path finder and a path reasoner for link prediction.
+
+Embedding Methods. Embedding methods learn a distributed representation for each node and edge by preserving the edge structure of the graph. Representative methods include DeepWalk [43] and LINE [53] on homogeneous graphs, and TransE [6], DistMult [68] and RotatE [52] on knowledge graphs. Later works improve embedding methods with new score functions [58, 13, 31, 52, 54, 76] that capture common semantic patterns of the relations, or search the score function in a general design space [75]. Embedding methods achieve promising results on link prediction, and can be scaled to very large graphs using multiple GPUs [78]. However, embedding methods do not explicitly encode local subgraphs between node pairs and cannot be applied to the inductive setting.
+
+Graph Neural Networks. Graph neural networks (GNNs) [47, 33, 60, 65] are a family of representation learning models that encode topological structures of graphs. For link prediction, the prevalent frameworks [32, 48, 12, 59] adopt an auto-encoder formulation, which uses GNNs to encode node representations, and decodes edges as a function over node pairs. Such frameworks are potentially inductive if the dataset provides node features, but are transductive only when node features are unavailable. Another stream of frameworks, such as SEAL [73] and GraIL [55], explicitly encodes the subgraph around each node pair for link prediction. While these frameworks are proved to be more powerful than the auto-encoder formulation [74] and can solve the inductive setting, they require to materialize a subgraph for each link, which is not scalable to large graphs. By contrast, our NBFNet explicitly captures the paths between two nodes for link prediction, meanwhile achieves a relatively low time complexity (Table 1). ID-GNN [70] formalizes link prediction as a conditional node classification task, and augments GNNs with the identity of the source node. While the architecture of NBFNet shares some spirits with ID-GNN, our model is motivated by the generalized Bellman-Ford algorithm and has theoretical connections with traditional path-based methods. There are also some works trying to scale up GNNs for link prediction by dynamically pruning the set of nodes in message passing [66, 20]. These methods are complementary to NBFNet, and may be incorporated into our method to further improve scalability.
+
+# 3 Methodology
+
+In this section, we first define a path formulation for link prediction. Our path formulation generalizes several traditional methods, and can be efficiently solved by the generalized Bellman-Ford algorithm. Then we propose Neural Bellman-Ford Networks to learn the path formulation with neural functions.
+
+# 3.1 Path Formulation for Link Prediction
+
+We consider the link prediction problem on both knowledge graphs and homogeneous graphs. A knowledge graph is denoted by $\mathcal { G } = ( \nu , \mathcal { E } , \mathcal { R } )$ , where $\nu$ and $\mathcal { E }$ represent the set of entities (nodes) and relations (edges) respectively, and $\mathcal { R }$ is the set of relation types. We use $\mathcal { N } ( u )$ to denote the set of nodes connected to $u$ , and $\mathcal { E } ( u )$ to denote the set of edges ending with node $u$ . A homogeneous graph $\mathcal { G } = ( \nu , \mathcal { E } )$ can be viewed as a special case of knowledge graphs, with only one relation type for all edges. Throughout this paper, we use bold terms, ${ \pmb w } _ { q } ( e )$ or $h _ { q } ( u , v )$ , to denote vector representations, and italic terms, $w _ { e }$ or $w _ { u v }$ , to denote scalars like the weight of edge $( u , v )$ in homogeneous graphs or triplet $( u , r , v )$ in knowledge graphs. Without loss of generality, we derive our method based on knowledge graphs, while our method can also be applied to homogeneous graphs.
+
+Path Formulation. Link prediction is aimed at predicting the existence of a query relation $q$ between a head entity $u$ and a tail entity $v$ . From a representation learning perspective, this requires to learn a pair representation $h _ { q } ( u , v )$ , which captures the local subgraph structure between $u$ and $v$ w.r.t. the query relation $q$ . In traditional methods, such a local structure is encoded by counting different types of random walks from $u$ to $v$ [35, 15]. Inspired by this construction, we formulate the pair representation as a generalized sum of path representations between $u$ and $v$ with a commutative summation operator $\oplus$ . Each path representation $h _ { q } ( P )$ is defined as a generalized product of the edge representations in the path with the multiplication operator $\otimes$ .
+
+$$
+h _ { q } ( u , v ) = h _ { q } ( P _ { 1 } ) \oplus h _ { q } ( P _ { 2 } ) \oplus \ldots \oplus h _ { q } ( P _ { | \mathcal { P } _ { u v } | } ) | _ { P _ { i } \in \mathcal { P } _ { u v } } \stackrel { \Delta } { = } \bigoplus _ { P \in \mathcal { P } _ { u v } } h _ { q } ( P )
+$$
+
+$$
+h _ { q } ( P = ( e _ { 1 } , e _ { 2 } , . . . , e _ { | P | } ) ) = w _ { q } ( e _ { 1 } ) \otimes w _ { q } ( e _ { 2 } ) \otimes . . . \otimes w _ { q } ( e _ { | P | } ) \triangleq \bigotimes _ { i = 1 } ^ { | P | } w _ { q } ( e _ { i } )
+$$
+
+where $\mathcal { P } _ { u v }$ denotes the set of paths from $u$ to $v$ and ${ \pmb w } _ { q } ( e _ { i } )$ is the representation of edge $e _ { i }$ . Note the multiplication operator $\otimes$ is not required to be commutative (e.g., matrix multiplication), therefore we define $\otimes$ to compute the product following the exact order. Intuitively, the path formulation can be interpreted as a depth-first-search (DFS) algorithm, where one searches all possible paths from $u$ to $v$ , computes their representations (Equation 2) and aggregates the results (Equation 1). Such a formulation is capable of modeling several traditional link prediction methods, as well as graph theory algorithms. Formally, Theorem 1-5 state the corresponding path formulations for 3 link prediction methods and 2 graph theory algorithms respectively. See Appendix A for proofs.
+
+Theorem 1 Katz index is a path formulation with $\begin{array} { r } { \oplus = + , \otimes = \times a n d { \pmb w } _ { q } ( e ) = \beta { \pmb w } _ { e } . } \end{array}$
+
+Theorem 2 Personalized PageRank is a path formulation with $\oplus = + , \otimes = \times a n d \pmb { w } _ { q } ( e ) =$ $\textstyle \alpha w _ { u v } / \sum _ { v ^ { \prime } \in { \mathcal { N } } ( u ) } w _ { u v ^ { \prime } }$ .
+
+Theorem 3 Graph distance is a path formulation with $\oplus = \operatorname* { m i n }$ , $\otimes = +$ and ${ \pmb w } _ { q } ( e ) = { \pmb w } _ { e }$ .
+
+Theorem 4 Widest path is a path formulation with $\Phi = { \mathrm { m a x } }$ , $\otimes =$ min and ${ \pmb w } _ { q } ( e ) = { \pmb w } _ { e }$
+
+Theorem 5 Most reliable path is a path formulation with $\oplus = \operatorname* { m a x }$ , $\otimes = \times$ and ${ \pmb w } _ { q } ( e ) = { \pmb w } _ { e }$
+
+Generalized Bellman-Ford Algorithm. While the above formulation is able to model important heuristics for link prediction, it is computationally expensive since the number of paths grows exponentially with the path length. Previous works [40, 11, 62] that directly computes the exponential number of paths can only afford a maximal path length of 3. A more scalable solution is to use the generalized Bellman-Ford algorithm [4]. Specifically, assuming the operators $\langle \oplus , \otimes \rangle$ satisfy a semiring system [21] with summation identity ${ \widehat { ( 0 ) } } _ { q }$ and multiplication identity $\textcircled{1} _ { q }$ , we have the following algorithm.
+
+$$
+\begin{array} { r l } & { \pmb { h } _ { q } ^ { ( 0 ) } ( u , v )  \mathbb { 1 } _ { q } ( u = v ) } \\ & { \pmb { h } _ { q } ^ { ( t ) } ( u , v )  ( \displaystyle \bigoplus _ { ( \boldsymbol { x } , \boldsymbol { r } , v ) \in \mathcal { E } ( v ) } \pmb { h } _ { q } ^ { ( t - 1 ) } ( u , \boldsymbol { x } ) \otimes \pmb { w } _ { q } ( \boldsymbol { x } , \boldsymbol { r } , v ) ) \oplus \pmb { h } _ { q } ^ { ( 0 ) } ( u , v ) } \end{array}
+$$
+
+where $\mathbb { 1 } _ { q } ( u = v )$ is the indicator function that outputs $\textcircled{1} _ { q }$ if $u = v$ and ${ \widehat { ( 0 ) } } _ { q }$ otherwise. ${ \pmb w } _ { q } ( x , r , v )$ is the representation for edge $\boldsymbol { e } = ( x , r , v )$ and $r$ is the relation type of the edge. Equation 3 is known as the boundary condition, while Equation 4 is known as the Bellman-Ford iteration. The high-level idea of the generalized Bellman-Ford algorithm is to compute the pair representation $\bar { h _ { q } ( u , v ) }$ for a given entity $u$ , a given query relation $q$ and all $v \in \nu$ in parallel, and reduce the total computation by the distributive property of multiplication over summation. Since $u$ and $q$ are fixed in the generalized Bellman-Ford algorithm, we may abbreviate $h _ { q } ^ { ( t ) } ( u , v )$ as $h _ { v } ^ { ( t ) }$ when the context is clear. When $\Phi = m i n$ and $\otimes = +$ , it recovers the original Bellman-Ford algorithm for the shortest path problem [5]. See Appendix B for preliminaries and the proof of the above algorithm.
+
+Theorem 6 Katz index, personalized PageRank, graph distance, widest path and most reliable path can be solved via the generalized Bellman-Ford algorithm.   
+Table 2: Comparison of operators in NBFNet and other methods from the view of path formulation.   
+
+<table><tr><td>Class</td><td>Method</td><td>MESSAGE wq(ei)wq(ej)</td><td>AGGREGATE hq(Pi)hq(Pj)</td><td>INDICATOR ，①q</td><td>Edge Representation wq(e)</td></tr><tr><td>Traditional Link</td><td>Katz Index [30]</td><td>wq(ei)xw（ej)</td><td>h(P）+hq(P）</td><td>0,1</td><td>βwe</td></tr><tr><td>Prediction</td><td>Personalized PageRank [42] Graph Distance [37]</td><td>wq(ei）xwq(ej) wq(ei)+wq(ej)</td><td>hq(Pi)+hq(Pj) min(hq(Pi),hq(Pj))</td><td>0,1 +8,0</td><td>awuu/∑&#x27;EN(u) Wuv&#x27; we</td></tr><tr><td>Graph Theory</td><td>Widest Path [4]</td><td>min(wq(ei),wq(ej))</td><td>max(hq(Pi),hq(Pj))</td><td>18,+8</td><td>we</td></tr><tr><td>Algorithms</td><td>Most Reliable Path [4]</td><td>wq(ei)xwq(ej)</td><td>max(hq(Pi),hq(P))</td><td>0,1</td><td>We</td></tr><tr><td>Logic Rules</td><td>NeuralLP [69] / DRUM[46]</td><td>wq(ei) xwq(ej)</td><td>hq(Pi) +hq(Pj)</td><td>0,1</td><td>Weights learned by LSTM [23]</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>Relational operators of</td><td></td><td></td><td></td></tr><tr><td></td><td>NBFNet</td><td>knowledge graph embeddings [6,68,52]</td><td>Learned set aggregators [9]</td><td>Learned indicator functions</td><td>Learned relation embeddings</td></tr></table>
+
+# 3.2 Neural Bellman-Ford Networks
+
+While the generalized Bellman-Ford algorithm can solve many classical methods (Theorem 6), these methods instantiate the path formulation with handcrafted operators (Table 2), and may not be optimal for link prediction. To improve the capacity of path formulation, we propose a general framework, Neural Bellman-Ford Networks (NBFNet), to learn the operators in the pair representations.
+
+Neural Parameterization. We relax the semiring assumption and parameterize the generalized Bellman-Ford algorithm (Equation 3 and 4) with 3 neural functions, namely INDICATOR, MESSAGE and AGGREGATE functions. The INDICATOR function replaces the indicator function $\mathbb { 1 } _ { q } ( u \ = \ \bar { v } )$ . The MESSAGE function replaces the binary multiplication operator $\otimes$ . The AGGREGATE function is a permutation invariant function over sets that replaces the n-ary summation operator $\oplus$ . Note that one may alternatively define AGGREGATE as the commutative binary operator $\oplus$ and apply it to a sequence of messages. However, this will make the parameterization more complicated.
+
+<table><tr><td>Algorithm1NeuralBellman-FordNetworks</td></tr><tr><td>Input: source node u, query relation q, #layers T Output: pair representations hq(u,v) for all v ∈V 1:for v ∈Vdo</td></tr><tr><td>Boundary condition 2: 3: end for</td></tr><tr><td>fort←1toTdo Bellman-Ford iteration forv ∈V do</td></tr><tr><td>Ms←{h} &gt;Message augmentation</td></tr><tr><td>for (x,r,v) ∈ε(v) do (t)</td></tr><tr><td></td></tr><tr><td>9:</td></tr><tr><td>10: end for</td></tr><tr><td></td></tr><tr><td>h（） 11: ← AGGREGATE(t)(M(t))</td></tr><tr><td>12: end for</td></tr><tr><td></td></tr><tr><td>13: end for 14: return h(T) as hq(u,v) for all v ∈V</td></tr></table>
+
+Now consider the generalized Bellman-Ford algorithm for a given entity $u$ and relation $q$ . In this context, we abbreviate $h _ { q } ^ { ( t ) } ( u , v )$ as ${ h } _ { v } ^ { ( t ) }$ , i.e., a representation on entity $v$ in the $t { \cdot }$ -th iteration. It should be stressed that $h _ { v } ^ { ( t ) }$ is still a pair representation, rather than a node representation. By substituting the neural functions into Equation 3 and 4, we get our Neural Bellman-Ford Networks.
+
+$$
+\begin{array} { r l } & { h _ { v } ^ { ( 0 ) } \gets \mathrm { I N D I C A T O R } ( u , v , q ) } \\ & { h _ { v } ^ { ( t ) } \gets \mathrm { A G G R E G A T E } \left( \left\{ \mathbf { M E S S A G E } \left( h _ { x } ^ { ( t - 1 ) } , w _ { q } ( x , r , v ) \right) \Big | ( x , r , v ) \in \mathcal { E } ( v ) \right\} \cup \left\{ h _ { v } ^ { ( 0 ) } \right\} \right) } \end{array}
+$$
+
+NBFNet can be interpreted as a novel GNN framework for learning pair representations. Compared to common GNN frameworks [32, 48] that compute the pair representation as two independent node representations $ { \boldsymbol { h } } _ { q } ( u )$ and $ { \boldsymbol { h } } _ { q } ( v )$ , NBFNet initializes a representation on the source node $u$ , and readouts the pair representation on the target node $v$ . Intuitively, our framework can be viewed as a source-specific message passing process, where every node learns a representation conditioned on the source node. The pseudo code of NBFNet is outlined in Algorithm 1.
+
+Design Space. Now we discuss some principled designs for MESSAGE, AGGREGATE and INDICATOR functions by drawing insights from traditional methods. Note the potential design space for NBFNet is way larger than what is presented here, as one can always borrow MESSAGE and AGGREGATE from the arsenal of message-passing GNNs [19, 16, 60, 65].
+
+For the MESSAGE function, traditional methods instantiate it as natural summation, natural multiplication or min over scalars. Therefore, we may use the vectorized version of summation or multiplication. Intuitively, summation of $h _ { x } ^ { ( t - 1 ) }$ and ${ \pmb w } _ { q } ( x , r , v )$ can be interpreted as a translation of $h _ { x } ^ { ( t - 1 ) }$ by ${ \pmb w } _ { q } ( x , r , v )$ in the pair representation space, while multiplication corresponds to scaling. Such transformations correspond to the relational operators [18, 45] in knowledge graph embeddings [6, 68, 58, 31, 52]. For example, translation and scaling are the relational operators used in TransE [6] and DistMult [68] respectively. We also consider the rotation operator in RotatE [52].
+
+The AGGREGATE function is instantiated as natural summation, max or min in traditional methods, which are reminiscent of set aggregation functions [71, 65, 9] used in GNNs. Therefore, we specify the AGGREGATE function to be sum, mean, or max, followed by a linear transformation and a non-linear activation. We also consider the principal neighborhood aggregation (PNA) proposed in a recent work [9], which jointly learns the types and scales of the aggregation function.
+
+The INDICATOR function is aimed at providing a non-trivial representation for the source node $u$ as the boundary condition. Therefore, we learn a query embedding $\pmb q$ for $\textcircled{1} _ { q }$ and define INDICATOR function as $\mathbb { 1 } ( u = v ) \ast q$ . Note it is also possible to additionally learn an embedding for ${ \widehat { ( 0 ) } } _ { q }$ . However, we find a single query embedding works better in practice.
+
+The edge representations are instantiated as transition probabilities or length in traditional methods. We notice that an edge may have different contribution in answering different query relations. Therefore, we parameterize the edge representations as a linear function over the query relation, i.e., ${ \pmb w } _ { q } ( x , r , v ) = { \pmb W } _ { r } { \pmb q } + { \pmb b } _ { r }$ . For homogeneous graphs or knowledge graphs with very few relations, we simplify the parameterization to ${ \pmb w } _ { q } ( x , r , v ) = { \pmb b } _ { r }$ to prevent overfitting. Note that one may also parameterize ${ \pmb w } _ { q } ( x , r , v )$ with learnable entity embeddings $_ { \textbf { \em x } }$ and $\textbf {  { v } }$ , but such a parameterization cannot solve the inductive setting. Similar to NeuralLP [69] & DRUM [46], we use different edge representations for different iterations, which is able to distinguish noncommutative edges in paths, e.g., father’s mother v.s. mother’s father.
+
+Link Prediction. We now show how to apply the learned pair representations $h _ { q } ( u , v )$ to the link prediction problem. We predict the conditional likelihood of the tail entity $v$ as $p ( v | u , q ) =$ $\sigma ( f ( \bar { h } _ { q } ( u , v ) ) )$ , where $\sigma ( \cdot )$ is the sigmoid function and $f ( \cdot )$ is a feed-forward neural network. The conditional likelihood of the head entity $u$ can be predicted by $p ( u | v , q ^ { - 1 } ) = \sigma ( f ( h _ { q ^ { - 1 } } ( v , u ) ) )$ with the same model. Following previous works [6, 52], we minimize the negative log-likelihood of positive and negative triplets (Equation 7). The negative samples are generated according to Partial Completeness Assumption (PCA) [14], which corrupts one of the entities in a positive triplet to create a negative sample. For undirected graphs, we symmetrize the representations and define $p _ { q } ( u , v ) = \sigma ( \bar { f } ( h _ { q } ( u , v ) + h _ { q } ( v , u ) )$ ). Equation 8 shows the loss for homogeneous graphs.
+
+$$
+\begin{array} { l l } { \mathcal { L } _ { K G } = - \log p ( u , q , v ) - \displaystyle \sum _ { i = 1 } ^ { n } \frac { 1 } { n } \log ( 1 - p ( u _ { i } ^ { \prime } , q , v _ { i } ^ { \prime } ) ) } \\ { \mathcal { L } _ { h o m o } = - \log p ( u , v ) - \displaystyle \sum _ { i = 1 } ^ { n } \frac { 1 } { n } \log ( 1 - p ( u _ { i } ^ { \prime } , v _ { i } ^ { \prime } ) ) , } \end{array}
+$$
+
+where $n$ is the number of negative samples per positive sample and $( u _ { i } ^ { \prime } , q , v _ { i } ^ { \prime } )$ and $( u _ { i } ^ { \prime } , v _ { i } ^ { \prime } )$ are the $i$ -th negative samples for knowledge graphs and homogeneous graphs, respectively.
+
+Time Complexity. One advantage of NBFNet is that it has a relatively low time complexity during inference4. Consider a scenario where a model is required to infer the conditional likelihood of all possible triplets $p ( v | u , q )$ . We group triplets with the same condition $u , q$ together, where each group contains $| \nu |$ triplets. For each group, we only need to execute Algorithm 1 once to get their predictions. Since a small constant number of iterations $T$ is enough for NBFNet to converge (Table 6b), Algorithm 1 has a time complexity of $O ( | \mathcal { E } | d + | \mathcal { V } | d ^ { 2 } )$ , where $d$ is the dimension of representations. Therefore, the amortized time complexity for a single triplet is $O \left( \frac { | \varepsilon | d } { | \nu | } + d ^ { 2 } \right)$ . For a detailed derivation of time complexity of other GNN frameworks, please refer to Appendix C.
+
+# 4 Experiment
+
+# 4.1 Experiment Setup
+
+We evaluate NBFNet in three settings, knowledge graph completion, homogeneous graph link prediction and inductive relation prediction. The former two are transductive settings, while the last is an inductive setting. For knowledge graphs, we use FB15k-237 [56] and WN18RR [13]. We use the standard transductive splits [56, 13] and inductive splits [55] of these datasets. For homogeneous graphs, we use Cora, Citeseer and PubMed [49]. Following previous works [32, 12], we split the edges into train/valid/test with a ratio of 85:5:10. Statistics of datasets can be found in Appendix E. Additional experiments of NBFNet on OGB [25] datasets can be found in Appendix G.
+
+Implementation Details. Our implementation generally follows the open source codebases of knowledge graph completion5 and homogeneous graph link prediction6. For knowledge graphs, we follow [69, 46] and augment each triplet $\langle u , q , \nu \rangle$ with a flipped triplet $\langle \nu , q ^ { - 1 } , u \rangle$ . For homogeneous graphs, we follow [33, 32] and augment each node $u$ with a self loop $\langle u , u \rangle$ . We instantiate NBFNet with 6 layers, each with 32 hidden units. The feed-forward network $f ( \cdot )$ is set to a 2-layer MLP with 64 hidden units. ReLU is used as the activation function for all hidden layers. We drop out edges that directly connect query node pairs during training to encourage the model to capture longer paths and prevent overfitting. Our model is trained on 4 Tesla V100 GPUs for 20 epochs. We select the models based on their performance on the validation set. See Appendix F for more details.
+
+Evaluation. We follow the filtered ranking protocol [6] for knowledge graph completion. For a test triplet $\langle u , q , \nu \rangle$ , we rank it against all negative triplets $\left. u , q , \nu ^ { \prime } \right.$ or $\langle u ^ { \prime } , q , \nu \rangle$ that do not appear in the knowledge graph. We report mean rank (MR), mean reciprocal rank (MRR) and HITS at N $( \mathrm { H } @ \mathrm { N } )$ for knowledge graph completion. For inductive relation prediction, we follow [55] and draw 50 negative triplets for each positive triplet and use the above filtered ranking. We report $\mathrm { H I T S } @ 1 0$ for inductive relation prediction. For homogeneous graph link prediction, we follow [32] and compare the positive edges against the same number of negative edges. We report area under the receiver operating characteristic curve (AUROC) and average precision (AP) for homogeneous graphs.
+
+Baselines. We compare NBFNet against path-based methods, embedding methods, and GNNs. These include 11 baselines for knowledge graph completion, 10 baselines for homogeneous graph link prediction and 4 baselines for inductive relation prediction. Note the inductive setting only includes path-based methods and GNNs, since existing embedding methods cannot handle this setting.
+
+# 4.2 Main Results
+
+Table 3 summarizes the results on knowledge graph completion. NBFNet significantly outperforms existing methods on all metrics and both datasets. NBFNet achieves an average relative gain of $21 \%$ in $\mathrm { H I T S } @ 1$ compared to the best path-based method, DRUM [46], on two datasets. Since DRUM is a special instance of NBFNet with natural summation and multiplication operators, this indicates the importance of learning MESSAGE and AGGREGATE functions in NBFNet. NBFNet also outperforms the best embedding method, LowFER [1], with an average relative performance gain of $18 \%$ in $\mathrm { H I T S } @ 1$ on two datasets. Meanwhile, NBFNet requires much less parameters than embedding methods. NBFNet only uses 3M parameters on FB15k-237, while TransE needs 30M parameters. See Appendix D for details on the number of parameters.
+
+Table 4 shows the results on homogeneous graph link prediction. NBFNet gets the best results on Cora and PubMed, meanwhile achieves competitive results on CiteSeer. Note CiteSeer is extremely sparse (Appendix E), which makes it hard to learn good representations with NBFNet. One thing to note here is that unlike other GNN methods, NBFNet does not use the node features provided by the datasets but is still able to outperform most other methods. We leave how to effectively combine node features and structural representations for link prediction as our future work.
+
+Table 3: Knowledge graph completion results. Results of NeuraLP and DRUM are taken from [46]. Results of RotatE, HAKE and LowFER are taken from their original papers [52, 76, 1]. Results of the other embedding methods are taken from [52]. Since GraIL has scalability issues in this setting, we evaluate it with 50 and 100 negative triplets for FB15k-237 and WN18RR respectively and report MR based on an unbiased estimation.   
+
+<table><tr><td rowspan="2">Class</td><td rowspan="2">Method</td><td colspan="5">FB15k-237</td><td colspan="5">WN18RR</td></tr><tr><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td></tr><tr><td rowspan="3">Path-based</td><td>Path Ranking [35]</td><td>3521</td><td>0.174</td><td>0.119</td><td>0.186</td><td>0.285</td><td>22438</td><td>0.324</td><td>0.276</td><td>0.360</td><td>0.406</td></tr><tr><td>NeuralLP [69]</td><td>-</td><td>0.240</td><td>=</td><td>-</td><td>0.362</td><td>-</td><td>0.435</td><td>0.371</td><td>0.434</td><td>0.566</td></tr><tr><td>DRUM[46]</td><td>1</td><td>0.343</td><td>0.255</td><td>0.378</td><td>0.516</td><td>-</td><td>0.486</td><td>0.425</td><td>0.513</td><td>0.586</td></tr><tr><td rowspan="6">Embeddings</td><td>TransE [6]</td><td>357</td><td>0.294</td><td>-</td><td>1</td><td>0.465</td><td>3384</td><td>0.226</td><td>-</td><td>-</td><td>0.501</td></tr><tr><td>DistMult [68]</td><td>254</td><td>0.241</td><td>0.155</td><td>0.263</td><td>0.419</td><td>5110</td><td>0.43</td><td>0.39</td><td>0.44</td><td>0.49</td></tr><tr><td>ComplEx [58]</td><td>339</td><td>0.247</td><td>0.158</td><td>0.275</td><td>0.428</td><td>5261</td><td>0.44</td><td>0.41</td><td>0.46</td><td>0.51</td></tr><tr><td>RotatE [52]</td><td>177</td><td>0.338</td><td>0.241</td><td>0.375</td><td>0.553</td><td>3340</td><td>0.476</td><td>0.428</td><td>0.492</td><td>0.571</td></tr><tr><td>HAKE [76]</td><td>-</td><td>0.346</td><td>0.250</td><td>0.381</td><td>0.542</td><td>-</td><td>0.497</td><td>0.452</td><td>0.516</td><td>0.582</td></tr><tr><td>LowFER[1]</td><td>-</td><td>0.359</td><td>0.266</td><td>0.396</td><td>0.544</td><td>-</td><td>0.465</td><td>0.434</td><td>0.479</td><td>0.526</td></tr><tr><td rowspan="3">GNNs</td><td>RGCN[48]</td><td>221</td><td>0.273</td><td>0.182</td><td>0.303</td><td>0.456</td><td>2719</td><td>0.402</td><td>0.345</td><td>0.437</td><td>0.494</td></tr><tr><td>GraIL [55]</td><td>2053</td><td>-</td><td>-</td><td>-</td><td>-</td><td>2539</td><td>-</td><td>=</td><td>-</td><td></td></tr><tr><td>NBFNet</td><td>114</td><td>0.415</td><td>0.321</td><td>0.454</td><td>0.599</td><td>636</td><td>0.551</td><td>0.497</td><td>0.573</td><td>0.666</td></tr></table>
+
+Table 4: Homogeneous graph link prediction results. Results of VGAE and S-VGAE are taken from their original papers [32, 12].   
+
+<table><tr><td rowspan="2">Class</td><td rowspan="2">Method</td><td colspan="2">Cora</td><td colspan="2">Citeseer</td><td colspan="2">PubMed</td></tr><tr><td>AUROC</td><td>AP</td><td>AUROC</td><td>AP</td><td>AUROC</td><td>AP</td></tr><tr><td rowspan="3">Path-based</td><td>Katz Index [30]</td><td>0.834</td><td>0.889</td><td>0.768</td><td>0.810</td><td>0.757</td><td>0.856</td></tr><tr><td>Personalized PageRank [42]</td><td>0.845</td><td>0.899</td><td>0.762</td><td>0.814</td><td>0.763</td><td>0.860</td></tr><tr><td>SimRank [28]</td><td>0.838</td><td>0.888</td><td>0.755</td><td>0.805</td><td>0.743</td><td>0.829</td></tr><tr><td rowspan="3">Embeddings</td><td>DeepWalk [43]</td><td>0.831</td><td>0.850</td><td>0.805</td><td>0.836</td><td>0.844</td><td>0.841</td></tr><tr><td>LINE [53]</td><td>0.844</td><td>0.876</td><td>0.791</td><td>0.826</td><td>0.849</td><td>0.888</td></tr><tr><td>node2vec [17]</td><td>0.872</td><td>0.879</td><td>0.838</td><td>0.868</td><td>0.891</td><td>0.914</td></tr><tr><td rowspan="5">GNNs</td><td>VGAE [32]</td><td>0.914</td><td>0.926</td><td>0.908</td><td>0.920</td><td>0.944</td><td>0.947</td></tr><tr><td>S-VGAE [12]</td><td>0.941</td><td>0.941</td><td>0.947</td><td>0.952</td><td>0.960</td><td>0.960</td></tr><tr><td>SEAL[73]</td><td>0.933</td><td>0.942</td><td>0.905</td><td>0.924</td><td>0.978</td><td>0.979</td></tr><tr><td>TLC-GNN [67]</td><td>0.934</td><td>0.931</td><td>0.909</td><td>0.916</td><td>0.970</td><td>0.968</td></tr><tr><td>NBFNet</td><td>0.956</td><td>0.962</td><td>0.923</td><td>0.936</td><td>0.983</td><td>0.982</td></tr></table>
+
+Table 5: Inductive relation prediction results $( \mathrm { H I T S } @ 1 0 )$ . V1-v4 corresponds to the 4 standard versions of inductive splits. Results of compared methods are taken from [55].   
+
+<table><tr><td rowspan="2">Class</td><td rowspan="2">Method</td><td colspan="4">FB15k-237</td><td colspan="4">WN18RR</td></tr><tr><td>v1</td><td>v2</td><td>v3</td><td>v4</td><td>v1</td><td>v2</td><td>v3</td><td>v4</td></tr><tr><td rowspan="3">Path-based</td><td>NeuralLP[16]</td><td>0.529</td><td>0.589</td><td>0.529</td><td>0.559</td><td>0.744</td><td>0.689</td><td>0.462</td><td>0.671</td></tr><tr><td>DRUM [46]</td><td>0.529</td><td>0.587</td><td>0.529</td><td>0.559</td><td>0.744</td><td>0.689</td><td>0.462</td><td>0.671</td></tr><tr><td>RuleN [39]</td><td>0.498</td><td>0.778</td><td>0.877</td><td>0.856</td><td>0.809</td><td>0.782</td><td>0.534</td><td>0.716</td></tr><tr><td rowspan="2">GNNs</td><td>GraIL [55]</td><td>0.642</td><td>0.818</td><td>0.828</td><td>0.893</td><td>0.825</td><td>0.787</td><td>0.584</td><td>0.734</td></tr><tr><td>NBFNet</td><td>0.834</td><td>0.949</td><td>0.951</td><td>0.960</td><td>0.948</td><td>0.905</td><td>0.893</td><td>0.890</td></tr></table>
+
+Table 5 summarizes the results on inductive relation prediction. On all inductive splits of two datasets, NBFNet achieves the best result. NBFNet outperforms the previous best method, GraIL [55], with an average relative performance gain of $22 \%$ in $\mathrm { H I T S } @ 1 0$ . Note that GraIL explicitly encodes the local subgraph surrounding each node pair and has a high time complexity (Appendix C). Usually, GraIL can at most encode a 2-hop subgraph, while our NBFNet can efficiently explore longer paths.
+
+# 4.3 Ablation Study
+
+MESSAGE & AGGREGATE Functions. Table 6a shows the results of different MESSAGE and AGGREGATE functions. Generally, NBFNet benefits from advanced embedding methods (DistMult,
+
+RotatE $>$ TransE) and aggregation functions ( $\mathrm { P N A } >$ sum, mean, max). Among simple AGGREGATE functions (sum, mean, max), combinations of MESSAGE and AGGREGATE functions (TransE & max, DistMult & sum) that satisfy the semiring assumption7 of the generalized Bellman-Ford algorithm, achieve locally optimal performance. PNA significantly improves over simple counterparts, which highlights the importance of learning more powerful AGGREGATE functions.
+
+Number of GNN Layers. Table 6b compares the results of NBFNet with different number of layers. Although it has been reported that GNNs with deep layers often result in significant performance drop [36, 77], we observe NBFNet does not have this issue. The performance increases monotonically with more layers, hitting a saturation after 6 layers. We conjecture the reason is that longer paths have negligible contribution, and paths not longer than 6 are enough for link prediction.
+
+Performance by Relation Category. We break down the performance of NBFNet by the categories of query relations: one-to-one, one-to-many, many-to-one and many-to-many8. Table 6c shows the prediction results for each category. It is observed that NBFNet not only improves on easy one-to-one cases, but also on hard cases where there are multiple true answers for the query.
+
+Table 6: Ablation studies of NBFNet on FB15k-237. Due to space constraints, we only report MRR here. For full results on all metrics, please refer to Appendix H.
+
+(a) Different MESSAGE and AGGREGATE functions.   
+
+<table><tr><td rowspan="2">MESSAGE</td><td colspan="4">AGGREGATE</td></tr><tr><td>Sum</td><td>Mean</td><td>Max</td><td>PNA [9]</td></tr><tr><td>TransE [6]</td><td>0.297</td><td>0.310</td><td>0.377</td><td>0.383</td></tr><tr><td>DistMult [69]</td><td>0.388</td><td>0.384</td><td>0.374</td><td>0.415</td></tr><tr><td>RotatE [52]</td><td>0.392</td><td>0.376</td><td>0.385</td><td>0.414</td></tr></table>
+
+(b) Different number of layers.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">#Layers (T)</td></tr><tr><td>2</td><td>4</td><td>6</td><td>8</td></tr><tr><td>NBFNet</td><td>0.345</td><td>0.409</td><td>0.415</td><td>0.416</td></tr></table>
+
+(c) Performance w.r.t. relation category. The two scores are the rankings over heads and tails respectively.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Relation Category</td></tr><tr><td>1-to-1</td><td>1-to-N</td><td>N-to-1</td><td>N-to-N</td></tr><tr><td>TransE [6]</td><td>0.498/0.488</td><td>0.455/0.071</td><td>0.079/0.744</td><td>0.224/0.330</td></tr><tr><td>RotatE [51]</td><td>0.487/0.484</td><td>0.467/0.070</td><td>0.081/0.747</td><td>0.234/0.338</td></tr><tr><td>NBFNet</td><td>0.578/0.600</td><td>0.499/0.122</td><td>0.165/0.790</td><td>0.348/0.456</td></tr></table>
+
+# 4.4 Path Interpretations of Predictions
+
+One advantage of NBFNet is that we can interpret its predictions through paths, which may be important for users to understand and debug the model. Intuitively, the interpretations should contain paths that contribute most to the prediction $p ( u , q , v )$ . Following local interpretation methods [3, 72], we approximate the local landscape of NBFNet with a linear model over the set of all paths, i.e., 1st-order Taylor polynomial. We define the importance of a path as its weight in the linear model, which can be computed by the partial derivative of the prediction w.r.t. the path. Formally, the top- $\mathbf { \nabla } \cdot \mathbf { k }$ path interpretations for $p ( u , q , v )$ are defined as
+
+$$
+P _ { 1 } , P _ { 2 } , . . . , P _ { k } = \underset { P \in \mathcal { P } _ { u v } } { \mathrm { t o p - k } } \frac { \partial p ( u , q , v ) } { \partial P }
+$$
+
+Note this formulation generalizes the definition of logical rules [69, 46] to non-linear models. While directly computing the importance of all paths is intractable, we approximate them with edge importance. Specifically, the importance of each path is approximated by the sum of the importance of edges in that path, where edge importance is obtained via auto differentiation. Then the top- $\mathbf { \nabla } \cdot \mathbf { k }$ path interpretations are equivalent to the top- $\mathbf { \nabla \cdot k }$ longest paths on the edge importance graph, which can be solved by a Bellman-Ford-style beam search. Better approximation is left as a future work.
+
+Table 7 visualizes path interpretations from FB15k-237 test set. While users may have different insights towards the visualization, here is our understanding. 1) In the first example, NBFNet learns soft logical entailment, such as impersonate $^ { - 1 } \wedge$ nationality $\Longrightarrow$ nationality and ethnicity $^ { - 1 } \wedge$ distribution $\Longrightarrow$ nationality. 2) In second example, NBFNet performs analogical reasoning by leveraging the fact that Florence is similar to Rome. 3) In the last example, NBFNet extracts longer paths, since there is no obvious connection between Pearl Harbor (film) and Japanese language.
+
+Table 7: Path interpretations of predictions on FB15k-237 test set. For each query triplet, we visualize the top-2 path interpretations and their weights. Inverse relations are denoted with a superscript $^ { - 1 }$ .   
+
+<table><tr><td>Query</td><td>(u,q,v): (O. Hardy, nationality, U.S.)</td></tr><tr><td>0.243</td><td>(O.Hardy, impersonate-1,R.Little)^ (R.Little, nationality, U.S.)</td></tr><tr><td>0.224</td><td>(O.Hardy,ethnicity-1,Scottish American)^ (Scottish American,distribution,U.S.)</td></tr><tr><td>Query</td><td>{u,q, v): (Florence, vacationer, D.C. Henrie)</td></tr><tr><td>0.251</td><td>(Florence,contain-1,Italy)^ (Italy,capital,Rome)^ (Rome,vacationer, D.C.Henrie)</td></tr><tr><td>0.183</td><td>(Florence,place live-1,G.F.Handel)(G.F.Handel,place live,Rome) (Rome,vacationer,D.C.Henrie)</td></tr><tr><td>Query 0.211</td><td>(u,q,v): (Pearl Harbor (film), language,Japanese)</td></tr><tr><td></td><td>(Pearl Harbor(film),film actor, C.-H.Tagawa)^ (C.-H.Tagawa,nationality,Japan) (Japan,country of origin, Yu-Gi-Oh!)^ (Yu-Gi-Oh!,language,Japanese)</td></tr><tr><td>0.208</td><td>(Pearl Harbor (film),film actor,C.-H.Tagawa) ^ (C.-H.Tagawa,nationality,Japan) (Japan,official language,Japanese)</td></tr></table>
+
+# 5 Discussion and Conclusion
+
+Limitations. There are a few limitations for NBFNet. First, the assumption of the generalized Bellman-Ford algorithm requires the operators $\langle \oplus , \otimes \rangle$ to satisfy a semiring. Due to the non-linear activation functions in neural networks, this assumption does not hold for NBFNet, and we do not have a theoretical guarantee on the loss incurred by this relaxation. Second, NBFNet is only verified on simple edge prediction, while there are other link prediction variants, e.g., complex logical queries with conjunctions (∧) and disjunctions (∨) [18, 45]. In the future, we would like to how NBFNet approximates the path formulation, as well as apply NBFNet to other link prediction settings.
+
+Social Impacts. Link prediction has a wide range of beneficial applications, including recommender systems, knowledge graph completion and drug repurposing. However, there are also some potentially negative impacts. First, NBFNet may encode the bias present in the training data, which leads to stereotyped predictions when the prediction is applied to a user on a social or e-commerce platform. Second, some harmful network activities could be augmented by powerful link prediction models, e.g., spamming, phishing, and social engineering. We expect future studies will mitigate these issues.
+
+Conclusion. We present a representation learning framework based on paths for link prediction. Our path formulation generalizes several traditional methods, and can be efficiently solved via the generalized Bellman-Ford algorithm. To improve the capacity of the path formulation, we propose NBFNet, which parameterizes the generalized Bellman-Ford algorithm with learned INDICATOR, MESSAGE, AGGREGATE functions. Experiments on knowledge graphs and homogeneous graphs show that NBFNet outperforms a wide range of methods in both transductive and inductive settings.
+
+# Acknowledgements
+
+We would like to thank Komal Teru for discussion on inductive relation prediction, Guyue Huang for discussion on fused message passing implementation, and Yao Lu for assistance on large-scale GPU training. We thank Meng Qu, Chence Shi and Minghao Xu for providing feedback on our manuscript.
+
+This project is supported by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program, collaboration grants between Microsoft Research and Mila, Samsung Electronics Co., Ltd., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund and a NRC Collaborative R&D Project (AI4D-CORE-06). This project was also partially funded by IVADO Fundamental Research Project grant PRF-2019-3583139727. The computation resource of this project is supported by Calcul Québec9 and Compute Canada10.
+
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+        "type": "text",
+        "text": "Neural Bellman-Ford Networks: A General Graph Neural Network Framework for Link Prediction ",
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+        "text": "Zhaocheng $\\mathbf { Z } \\mathbf { h } \\mathbf { u } ^ { 1 , 2 }$ , Zuobai Zhang1,2, Louis-Pascal Xhonneux1,2, Jian Tang1,3,4 ",
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+        "text": "Mila - Québec AI Institute1, Université de Montréal2 HEC Montréal3, CIFAR AI Chair4 {zhaocheng.zhu, zuobai.zhang, louis-pascal.xhonneux}@mila.quebec jian.tang@hec.ca ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "Link prediction is a very fundamental task on graphs. Inspired by traditional path-based methods, in this paper we propose a general and flexible representation learning framework based on paths for link prediction. Specifically, we define the representation of a pair of nodes as the generalized sum of all path representations between the nodes, with each path representation as the generalized product of the edge representations in the path. Motivated by the Bellman-Ford algorithm for solving the shortest path problem, we show that the proposed path formulation can be efficiently solved by the generalized Bellman-Ford algorithm. To further improve the capacity of the path formulation, we propose the Neural Bellman-Ford Network (NBFNet), a general graph neural network framework that solves the path formulation with learned operators in the generalized Bellman-Ford algorithm. The NBFNet parameterizes the generalized Bellman-Ford algorithm with 3 neural components, namely INDICATOR, MESSAGE and AGGREGATE functions, which corresponds to the boundary condition, multiplication operator, and summation operator respectively1. The NBFNet covers many traditional path-based methods, and can be applied to both homogeneous graphs and multi-relational graphs (e.g., knowledge graphs) in both transductive and inductive settings. Experiments on both homogeneous graphs and knowledge graphs show that the proposed NBFNet outperforms existing methods by a large margin in both transductive and inductive settings, achieving new state-of-the-art results2. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Predicting the interactions between nodes (a.k.a. link prediction) is a fundamental task in the field of graph machine learning. Given the ubiquitous existence of graphs, such a task has many applications, such as recommender system [34], knowledge graph completion [41] and drug repurposing [27]. ",
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+    {
+        "type": "text",
+        "text": "Traditional methods of link prediction usually define different heuristic metrics over the paths between a pair of nodes. For example, Katz index [30] is defined as a weighted count of paths between two nodes. Personalized PageRank [42] measures the similarity of two nodes as the random walk probability from one to the other. Graph distance [37] uses the length of the shortest path between two nodes to predict their association. These methods can be directly applied to new graphs, i.e., inductive setting, enjoy good interpretability and scale up to large graphs. However, they are designed based on handcrafted metrics and may not be optimal for link prediction on real-world graphs. ",
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+        "type": "text",
+        "text": "To address these limitations, some link prediction methods adopt graph neural networks (GNNs) [32, 48, 59] to automatically extract important features from local neighborhoods for link prediction. Thanks to the high expressiveness of GNNs, these methods have shown state-of-the-art performance. However, these methods can only be applied to predict new links on the training graph, i.e. transductive setting, and lack interpretability. While some recent methods [73, 55] extract features from local subgraphs with GNNs and support inductive setting, the scalability of these methods is compromised. ",
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+        "text": "Therefore, we wonder if there exists an approach that enjoys the advantages of both traditional path-based methods and recent approaches based on graph neural networks, i.e., generalization in the inductive setting, interpretability, high model capacity and scalability. ",
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+    {
+        "type": "text",
+        "text": "In this paper, we propose such a solution. Inspired by traditional path-based methods, our goal is to develop a general and flexible representation learning framework for link prediction based on the paths between two nodes. Specifically, we define the representation of a pair of nodes as the generalized sum of all the path representations between them, where each path representation is defined as the generalized product of the edge representations in the path. Many link prediction methods, such as Katz index [30], personalized PageRank [42], graph distance [37], as well as graph theory algorithms like widest path [4] and most reliable path [4], are special instances of this path formulation with different summation and multiplication operators. Motivated by the polynomial-time algorithm for the shortest path problem [5], we show that such a formulation can be efficiently solved via the generalized Bellman-Ford algorithm [4] under mild conditions and scale up to large graphs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The operators in the generalized Bellman-Ford algorithm—summation and multiplication—are handcrafted, which have limited flexibility. Therefore, we further propose the Neural Bellman-Ford Networks (NBFNet), a graph neural network framework that solves the above path formulation with learned operators in the generalized Bellman-Ford algorithm. Specifically, NBFNet parameterizes the generalized Bellman-Ford algorithm with three neural components, namely INDICATOR, MESSAGE and AGGREGATE functions. The INDICATOR function initializes a representation on each node, which is taken as the boundary condition of the generalized Bellman-Ford algorithm. The MESSAGE and the AGGREGATE functions learn the multiplication and summation operators respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show that the MESSAGE function can be defined according to the relational operators in knowledge graph embeddings [6, 68, 58, 31, 52], e.g., as a translation in Euclidean space induced by the relational operators of TransE [6]. The AGGREGATE function can be defined as learnable set aggregation functions [71, 65, 9]. With such parameterization, NBFNet can generalize to the inductive setting, meanwhile achieve one of the lowest time complexity among inductive GNN methods. A comparison of NBFNet and other GNN frameworks for link prediction is showed in Table 1. With other instantiations of MESSAGE and AGGREGATE functions, our framework can also recover some existing works on learning logic rules [69, 46] for link prediction on knowledge graphs (Table 2). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our NBFNet framework can be applied to several link prediction variants, covering not only singlerelational graphs (e.g., homogeneous graphs) but also multi-relational graphs (e.g., knowledge graphs). We empirically evaluate the proposed NBFNet for link prediction on homogeneous graphs and knowledge graphs in both transductive and inductive settings. Experimental results show that the proposed NBFNet outperforms existing state-of-the-art methods by a large margin in all settings, with an average relative performance gain of $18 \\%$ on knowledge graph completion $( \\mathrm { H I T S } @ 1 )$ ) and $22 \\%$ on inductive relation prediction $( \\mathrm { H I T S } @ 1 0 )$ . We also show that the proposed NBFNet is indeed interpretable by visualizing the top-k relevant paths for link prediction on knowledge graphs. ",
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+    {
+        "type": "table",
+        "img_path": "images/199d1f89d4b4d3af4ca500f3c333356f512bbc81691d9a9d6172cebf5f1e9fe5.jpg",
+        "table_caption": [
+            "Table 1: Comparison of GNN frameworks for link prediction. The time complexity refers to the amortized time for predicting a single edge or triplet. $| \\nu |$ is the number of nodes, $\\lvert \\mathcal { E } \\rvert$ is the number of edges, and $d$ is the dimension of representations. The wall time is measured on FB15k-237 test set with 40 CPU cores and 4 GPUs. We estimate the wall time of GraIL based on a downsampled test set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Inductive3</td><td>Interpretable</td><td>Learned Representation</td><td>Time Complexity</td><td>Wall Time</td></tr><tr><td>VGAE [32] / RGCN [48]</td><td></td><td></td><td>√</td><td>0(d)</td><td>18 secs</td></tr><tr><td>NeuralLP [69] / DRUM[46]</td><td>√</td><td>√</td><td></td><td>(+R）</td><td>2.1 mins</td></tr><tr><td>SEAL [73]] GraIL [55]</td><td>√</td><td></td><td>√</td><td>0(1ε|d²)</td><td>~1 month</td></tr><tr><td>NBFNet</td><td>√</td><td>√</td><td>√</td><td>（+）</td><td>4.0 mins</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Existing work on link prediction can be generally classified into 3 main paradigms: path-based methods, embedding methods, and graph neural networks. ",
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+    {
+        "type": "text",
+        "text": "Path-based Methods. Early methods on homogeneous graphs compute the similarity between two nodes based on the weighted count of paths (Katz index [30]), random walk probability (personalized PageRank [42]) or the length of the shortest path (graph distance [37]). SimRank [28] uses advanced metrics such as the expected meeting distance on homogeneous graphs, which is extended by PathSim [51] to heterogeneous graphs. On knowledge graphs, Path Ranking [35, 15] directly uses relational paths as symbolic features for prediction. Rule mining methods, such as NeuralLP [69] and DRUM [46], learn probabilistic logical rules to weight different paths. Path representation methods, such as Path-RNN [40] and its successors [11, 62], encode each path with recurrent neural networks (RNNs), and aggregate paths for prediction. However, these methods need to traverse an exponential number of paths and are limited to very short paths, e.g., $\\leq 3$ edges. To scale up path-based methods, All-Paths [57] proposes to efficiently aggregate all paths with dynamic programming. However, All-Paths is restricted to bilinear models and has limited model capacity. Another stream of works [64, 10, 22] learns an agent to collect useful paths for link prediction. While these methods can produce interpretable paths, they suffer from extremely sparse rewards and require careful engineering of the reward function [38] or the search strategy [50]. Some other works [8, 44] adopt variational inference to learn a path finder and a path reasoner for link prediction. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Embedding Methods. Embedding methods learn a distributed representation for each node and edge by preserving the edge structure of the graph. Representative methods include DeepWalk [43] and LINE [53] on homogeneous graphs, and TransE [6], DistMult [68] and RotatE [52] on knowledge graphs. Later works improve embedding methods with new score functions [58, 13, 31, 52, 54, 76] that capture common semantic patterns of the relations, or search the score function in a general design space [75]. Embedding methods achieve promising results on link prediction, and can be scaled to very large graphs using multiple GPUs [78]. However, embedding methods do not explicitly encode local subgraphs between node pairs and cannot be applied to the inductive setting. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Graph Neural Networks. Graph neural networks (GNNs) [47, 33, 60, 65] are a family of representation learning models that encode topological structures of graphs. For link prediction, the prevalent frameworks [32, 48, 12, 59] adopt an auto-encoder formulation, which uses GNNs to encode node representations, and decodes edges as a function over node pairs. Such frameworks are potentially inductive if the dataset provides node features, but are transductive only when node features are unavailable. Another stream of frameworks, such as SEAL [73] and GraIL [55], explicitly encodes the subgraph around each node pair for link prediction. While these frameworks are proved to be more powerful than the auto-encoder formulation [74] and can solve the inductive setting, they require to materialize a subgraph for each link, which is not scalable to large graphs. By contrast, our NBFNet explicitly captures the paths between two nodes for link prediction, meanwhile achieves a relatively low time complexity (Table 1). ID-GNN [70] formalizes link prediction as a conditional node classification task, and augments GNNs with the identity of the source node. While the architecture of NBFNet shares some spirits with ID-GNN, our model is motivated by the generalized Bellman-Ford algorithm and has theoretical connections with traditional path-based methods. There are also some works trying to scale up GNNs for link prediction by dynamically pruning the set of nodes in message passing [66, 20]. These methods are complementary to NBFNet, and may be incorporated into our method to further improve scalability. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first define a path formulation for link prediction. Our path formulation generalizes several traditional methods, and can be efficiently solved by the generalized Bellman-Ford algorithm. Then we propose Neural Bellman-Ford Networks to learn the path formulation with neural functions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Path Formulation for Link Prediction ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We consider the link prediction problem on both knowledge graphs and homogeneous graphs. A knowledge graph is denoted by $\\mathcal { G } = ( \\nu , \\mathcal { E } , \\mathcal { R } )$ , where $\\nu$ and $\\mathcal { E }$ represent the set of entities (nodes) and relations (edges) respectively, and $\\mathcal { R }$ is the set of relation types. We use $\\mathcal { N } ( u )$ to denote the set of nodes connected to $u$ , and $\\mathcal { E } ( u )$ to denote the set of edges ending with node $u$ . A homogeneous graph $\\mathcal { G } = ( \\nu , \\mathcal { E } )$ can be viewed as a special case of knowledge graphs, with only one relation type for all edges. Throughout this paper, we use bold terms, ${ \\pmb w } _ { q } ( e )$ or $h _ { q } ( u , v )$ , to denote vector representations, and italic terms, $w _ { e }$ or $w _ { u v }$ , to denote scalars like the weight of edge $( u , v )$ in homogeneous graphs or triplet $( u , r , v )$ in knowledge graphs. Without loss of generality, we derive our method based on knowledge graphs, while our method can also be applied to homogeneous graphs. ",
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+        "text": "Path Formulation. Link prediction is aimed at predicting the existence of a query relation $q$ between a head entity $u$ and a tail entity $v$ . From a representation learning perspective, this requires to learn a pair representation $h _ { q } ( u , v )$ , which captures the local subgraph structure between $u$ and $v$ w.r.t. the query relation $q$ . In traditional methods, such a local structure is encoded by counting different types of random walks from $u$ to $v$ [35, 15]. Inspired by this construction, we formulate the pair representation as a generalized sum of path representations between $u$ and $v$ with a commutative summation operator $\\oplus$ . Each path representation $h _ { q } ( P )$ is defined as a generalized product of the edge representations in the path with the multiplication operator $\\otimes$ . ",
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+        "img_path": "images/657ed00271ab2f6b0577ba1b04a458b9e7821d4a8bc81f41ca40d8b52b15725f.jpg",
+        "text": "$$\nh _ { q } ( u , v ) = h _ { q } ( P _ { 1 } ) \\oplus h _ { q } ( P _ { 2 } ) \\oplus \\ldots \\oplus h _ { q } ( P _ { | \\mathcal { P } _ { u v } | } ) | _ { P _ { i } \\in \\mathcal { P } _ { u v } } \\stackrel { \\Delta } { = } \\bigoplus _ { P \\in \\mathcal { P } _ { u v } } h _ { q } ( P )\n$$",
+        "text_format": "latex",
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+        "img_path": "images/37cf667f340c208d1a1279b359c29c90e4623efb111bbc405c6e889433d2424a.jpg",
+        "text": "$$\nh _ { q } ( P = ( e _ { 1 } , e _ { 2 } , . . . , e _ { | P | } ) ) = w _ { q } ( e _ { 1 } ) \\otimes w _ { q } ( e _ { 2 } ) \\otimes . . . \\otimes w _ { q } ( e _ { | P | } ) \\triangleq \\bigotimes _ { i = 1 } ^ { | P | } w _ { q } ( e _ { i } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { P } _ { u v }$ denotes the set of paths from $u$ to $v$ and ${ \\pmb w } _ { q } ( e _ { i } )$ is the representation of edge $e _ { i }$ . Note the multiplication operator $\\otimes$ is not required to be commutative (e.g., matrix multiplication), therefore we define $\\otimes$ to compute the product following the exact order. Intuitively, the path formulation can be interpreted as a depth-first-search (DFS) algorithm, where one searches all possible paths from $u$ to $v$ , computes their representations (Equation 2) and aggregates the results (Equation 1). Such a formulation is capable of modeling several traditional link prediction methods, as well as graph theory algorithms. Formally, Theorem 1-5 state the corresponding path formulations for 3 link prediction methods and 2 graph theory algorithms respectively. See Appendix A for proofs. ",
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+        "type": "text",
+        "text": "Theorem 1 Katz index is a path formulation with $\\begin{array} { r } { \\oplus = + , \\otimes = \\times a n d { \\pmb w } _ { q } ( e ) = \\beta { \\pmb w } _ { e } . } \\end{array}$ ",
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+        "text": "Theorem 2 Personalized PageRank is a path formulation with $\\oplus = + , \\otimes = \\times a n d \\pmb { w } _ { q } ( e ) =$ $\\textstyle \\alpha w _ { u v } / \\sum _ { v ^ { \\prime } \\in { \\mathcal { N } } ( u ) } w _ { u v ^ { \\prime } }$ . ",
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+        "type": "text",
+        "text": "Theorem 3 Graph distance is a path formulation with $\\oplus = \\operatorname* { m i n }$ , $\\otimes = +$ and ${ \\pmb w } _ { q } ( e ) = { \\pmb w } _ { e }$ . ",
+        "bbox": [
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+        "text": "Theorem 4 Widest path is a path formulation with $\\Phi = { \\mathrm { m a x } }$ , $\\otimes =$ min and ${ \\pmb w } _ { q } ( e ) = { \\pmb w } _ { e }$ ",
+        "bbox": [
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+        "text": "Theorem 5 Most reliable path is a path formulation with $\\oplus = \\operatorname* { m a x }$ , $\\otimes = \\times$ and ${ \\pmb w } _ { q } ( e ) = { \\pmb w } _ { e }$ ",
+        "bbox": [
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+        "type": "text",
+        "text": "Generalized Bellman-Ford Algorithm. While the above formulation is able to model important heuristics for link prediction, it is computationally expensive since the number of paths grows exponentially with the path length. Previous works [40, 11, 62] that directly computes the exponential number of paths can only afford a maximal path length of 3. A more scalable solution is to use the generalized Bellman-Ford algorithm [4]. Specifically, assuming the operators $\\langle \\oplus , \\otimes \\rangle$ satisfy a semiring system [21] with summation identity ${ \\widehat { ( 0 ) } } _ { q }$ and multiplication identity $\\textcircled{1} _ { q }$ , we have the following algorithm. ",
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+        "type": "equation",
+        "img_path": "images/b05e2fb527ef208a07491befd957c543e5c263c360d2dda7c4806361c8f07b2c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\pmb { h } _ { q } ^ { ( 0 ) } ( u , v )  \\mathbb { 1 } _ { q } ( u = v ) } \\\\ & { \\pmb { h } _ { q } ^ { ( t ) } ( u , v )  ( \\displaystyle \\bigoplus _ { ( \\boldsymbol { x } , \\boldsymbol { r } , v ) \\in \\mathcal { E } ( v ) } \\pmb { h } _ { q } ^ { ( t - 1 ) } ( u , \\boldsymbol { x } ) \\otimes \\pmb { w } _ { q } ( \\boldsymbol { x } , \\boldsymbol { r } , v ) ) \\oplus \\pmb { h } _ { q } ^ { ( 0 ) } ( u , v ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbb { 1 } _ { q } ( u = v )$ is the indicator function that outputs $\\textcircled{1} _ { q }$ if $u = v$ and ${ \\widehat { ( 0 ) } } _ { q }$ otherwise. ${ \\pmb w } _ { q } ( x , r , v )$ is the representation for edge $\\boldsymbol { e } = ( x , r , v )$ and $r$ is the relation type of the edge. Equation 3 is known as the boundary condition, while Equation 4 is known as the Bellman-Ford iteration. The high-level idea of the generalized Bellman-Ford algorithm is to compute the pair representation $\\bar { h _ { q } ( u , v ) }$ for a given entity $u$ , a given query relation $q$ and all $v \\in \\nu$ in parallel, and reduce the total computation by the distributive property of multiplication over summation. Since $u$ and $q$ are fixed in the generalized Bellman-Ford algorithm, we may abbreviate $h _ { q } ^ { ( t ) } ( u , v )$ as $h _ { v } ^ { ( t ) }$ when the context is clear. When $\\Phi = m i n$ and $\\otimes = +$ , it recovers the original Bellman-Ford algorithm for the shortest path problem [5]. See Appendix B for preliminaries and the proof of the above algorithm. ",
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+    {
+        "type": "table",
+        "img_path": "images/371cba06b43e2851758c7e5321835ee7149a0515af6e80d8119924ff0a0bd1e7.jpg",
+        "table_caption": [
+            "Theorem 6 Katz index, personalized PageRank, graph distance, widest path and most reliable path can be solved via the generalized Bellman-Ford algorithm. ",
+            "Table 2: Comparison of operators in NBFNet and other methods from the view of path formulation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Class</td><td>Method</td><td>MESSAGE wq(ei)wq(ej)</td><td>AGGREGATE hq(Pi)hq(Pj)</td><td>INDICATOR ，①q</td><td>Edge Representation wq(e)</td></tr><tr><td>Traditional Link</td><td>Katz Index [30]</td><td>wq(ei)xw（ej)</td><td>h(P）+hq(P）</td><td>0,1</td><td>βwe</td></tr><tr><td>Prediction</td><td>Personalized PageRank [42] Graph Distance [37]</td><td>wq(ei）xwq(ej) wq(ei)+wq(ej)</td><td>hq(Pi)+hq(Pj) min(hq(Pi),hq(Pj))</td><td>0,1 +8,0</td><td>awuu/∑&#x27;EN(u) Wuv&#x27; we</td></tr><tr><td>Graph Theory</td><td>Widest Path [4]</td><td>min(wq(ei),wq(ej))</td><td>max(hq(Pi),hq(Pj))</td><td>18,+8</td><td>we</td></tr><tr><td>Algorithms</td><td>Most Reliable Path [4]</td><td>wq(ei)xwq(ej)</td><td>max(hq(Pi),hq(P))</td><td>0,1</td><td>We</td></tr><tr><td>Logic Rules</td><td>NeuralLP [69] / DRUM[46]</td><td>wq(ei) xwq(ej)</td><td>hq(Pi) +hq(Pj)</td><td>0,1</td><td>Weights learned by LSTM [23]</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>Relational operators of</td><td></td><td></td><td></td></tr><tr><td></td><td>NBFNet</td><td>knowledge graph embeddings [6,68,52]</td><td>Learned set aggregators [9]</td><td>Learned indicator functions</td><td>Learned relation embeddings</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.2 Neural Bellman-Ford Networks ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "While the generalized Bellman-Ford algorithm can solve many classical methods (Theorem 6), these methods instantiate the path formulation with handcrafted operators (Table 2), and may not be optimal for link prediction. To improve the capacity of path formulation, we propose a general framework, Neural Bellman-Ford Networks (NBFNet), to learn the operators in the pair representations. ",
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+        "type": "text",
+        "text": "Neural Parameterization. We relax the semiring assumption and parameterize the generalized Bellman-Ford algorithm (Equation 3 and 4) with 3 neural functions, namely INDICATOR, MESSAGE and AGGREGATE functions. The INDICATOR function replaces the indicator function $\\mathbb { 1 } _ { q } ( u \\ = \\ \\bar { v } )$ . The MESSAGE function replaces the binary multiplication operator $\\otimes$ . The AGGREGATE function is a permutation invariant function over sets that replaces the n-ary summation operator $\\oplus$ . Note that one may alternatively define AGGREGATE as the commutative binary operator $\\oplus$ and apply it to a sequence of messages. However, this will make the parameterization more complicated. ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm1NeuralBellman-FordNetworks</td></tr><tr><td>Input: source node u, query relation q, #layers T Output: pair representations hq(u,v) for all v ∈V 1:for v ∈Vdo</td></tr><tr><td>Boundary condition 2: 3: end for</td></tr><tr><td>fort←1toTdo Bellman-Ford iteration forv ∈V do</td></tr><tr><td>Ms←{h} &gt;Message augmentation</td></tr><tr><td>for (x,r,v) ∈ε(v) do (t)</td></tr><tr><td></td></tr><tr><td>9:</td></tr><tr><td>10: end for</td></tr><tr><td></td></tr><tr><td>h（） 11: ← AGGREGATE(t)(M(t))</td></tr><tr><td>12: end for</td></tr><tr><td></td></tr><tr><td>13: end for 14: return h(T) as hq(u,v) for all v ∈V</td></tr></table>",
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+        "type": "text",
+        "text": "Now consider the generalized Bellman-Ford algorithm for a given entity $u$ and relation $q$ . In this context, we abbreviate $h _ { q } ^ { ( t ) } ( u , v )$ as ${ h } _ { v } ^ { ( t ) }$ , i.e., a representation on entity $v$ in the $t { \\cdot }$ -th iteration. It should be stressed that $h _ { v } ^ { ( t ) }$ is still a pair representation, rather than a node representation. By substituting the neural functions into Equation 3 and 4, we get our Neural Bellman-Ford Networks. ",
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+        "img_path": "images/148290ac44554d14efe6b5491cbe354779098c1bacc7a73b7d1501cba0379354.jpg",
+        "text": "$$\n\\begin{array} { r l } & { h _ { v } ^ { ( 0 ) } \\gets \\mathrm { I N D I C A T O R } ( u , v , q ) } \\\\ & { h _ { v } ^ { ( t ) } \\gets \\mathrm { A G G R E G A T E } \\left( \\left\\{ \\mathbf { M E S S A G E } \\left( h _ { x } ^ { ( t - 1 ) } , w _ { q } ( x , r , v ) \\right) \\Big | ( x , r , v ) \\in \\mathcal { E } ( v ) \\right\\} \\cup \\left\\{ h _ { v } ^ { ( 0 ) } \\right\\} \\right) } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "NBFNet can be interpreted as a novel GNN framework for learning pair representations. Compared to common GNN frameworks [32, 48] that compute the pair representation as two independent node representations $ { \\boldsymbol { h } } _ { q } ( u )$ and $ { \\boldsymbol { h } } _ { q } ( v )$ , NBFNet initializes a representation on the source node $u$ , and readouts the pair representation on the target node $v$ . Intuitively, our framework can be viewed as a source-specific message passing process, where every node learns a representation conditioned on the source node. The pseudo code of NBFNet is outlined in Algorithm 1. ",
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+        "type": "text",
+        "text": "Design Space. Now we discuss some principled designs for MESSAGE, AGGREGATE and INDICATOR functions by drawing insights from traditional methods. Note the potential design space for NBFNet is way larger than what is presented here, as one can always borrow MESSAGE and AGGREGATE from the arsenal of message-passing GNNs [19, 16, 60, 65]. ",
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+    {
+        "type": "text",
+        "text": "For the MESSAGE function, traditional methods instantiate it as natural summation, natural multiplication or min over scalars. Therefore, we may use the vectorized version of summation or multiplication. Intuitively, summation of $h _ { x } ^ { ( t - 1 ) }$ and ${ \\pmb w } _ { q } ( x , r , v )$ can be interpreted as a translation of $h _ { x } ^ { ( t - 1 ) }$ by ${ \\pmb w } _ { q } ( x , r , v )$ in the pair representation space, while multiplication corresponds to scaling. Such transformations correspond to the relational operators [18, 45] in knowledge graph embeddings [6, 68, 58, 31, 52]. For example, translation and scaling are the relational operators used in TransE [6] and DistMult [68] respectively. We also consider the rotation operator in RotatE [52]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The AGGREGATE function is instantiated as natural summation, max or min in traditional methods, which are reminiscent of set aggregation functions [71, 65, 9] used in GNNs. Therefore, we specify the AGGREGATE function to be sum, mean, or max, followed by a linear transformation and a non-linear activation. We also consider the principal neighborhood aggregation (PNA) proposed in a recent work [9], which jointly learns the types and scales of the aggregation function. ",
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+    {
+        "type": "text",
+        "text": "The INDICATOR function is aimed at providing a non-trivial representation for the source node $u$ as the boundary condition. Therefore, we learn a query embedding $\\pmb q$ for $\\textcircled{1} _ { q }$ and define INDICATOR function as $\\mathbb { 1 } ( u = v ) \\ast q$ . Note it is also possible to additionally learn an embedding for ${ \\widehat { ( 0 ) } } _ { q }$ . However, we find a single query embedding works better in practice. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The edge representations are instantiated as transition probabilities or length in traditional methods. We notice that an edge may have different contribution in answering different query relations. Therefore, we parameterize the edge representations as a linear function over the query relation, i.e., ${ \\pmb w } _ { q } ( x , r , v ) = { \\pmb W } _ { r } { \\pmb q } + { \\pmb b } _ { r }$ . For homogeneous graphs or knowledge graphs with very few relations, we simplify the parameterization to ${ \\pmb w } _ { q } ( x , r , v ) = { \\pmb b } _ { r }$ to prevent overfitting. Note that one may also parameterize ${ \\pmb w } _ { q } ( x , r , v )$ with learnable entity embeddings $_ { \\textbf { \\em x } }$ and $\\textbf {  { v } }$ , but such a parameterization cannot solve the inductive setting. Similar to NeuralLP [69] & DRUM [46], we use different edge representations for different iterations, which is able to distinguish noncommutative edges in paths, e.g., father’s mother v.s. mother’s father. ",
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+        "text": "Link Prediction. We now show how to apply the learned pair representations $h _ { q } ( u , v )$ to the link prediction problem. We predict the conditional likelihood of the tail entity $v$ as $p ( v | u , q ) =$ $\\sigma ( f ( \\bar { h } _ { q } ( u , v ) ) )$ , where $\\sigma ( \\cdot )$ is the sigmoid function and $f ( \\cdot )$ is a feed-forward neural network. The conditional likelihood of the head entity $u$ can be predicted by $p ( u | v , q ^ { - 1 } ) = \\sigma ( f ( h _ { q ^ { - 1 } } ( v , u ) ) )$ with the same model. Following previous works [6, 52], we minimize the negative log-likelihood of positive and negative triplets (Equation 7). The negative samples are generated according to Partial Completeness Assumption (PCA) [14], which corrupts one of the entities in a positive triplet to create a negative sample. For undirected graphs, we symmetrize the representations and define $p _ { q } ( u , v ) = \\sigma ( \\bar { f } ( h _ { q } ( u , v ) + h _ { q } ( v , u ) )$ ). Equation 8 shows the loss for homogeneous graphs. ",
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+        "text": "$$\n\\begin{array} { l l } { \\mathcal { L } _ { K G } = - \\log p ( u , q , v ) - \\displaystyle \\sum _ { i = 1 } ^ { n } \\frac { 1 } { n } \\log ( 1 - p ( u _ { i } ^ { \\prime } , q , v _ { i } ^ { \\prime } ) ) } \\\\ { \\mathcal { L } _ { h o m o } = - \\log p ( u , v ) - \\displaystyle \\sum _ { i = 1 } ^ { n } \\frac { 1 } { n } \\log ( 1 - p ( u _ { i } ^ { \\prime } , v _ { i } ^ { \\prime } ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $n$ is the number of negative samples per positive sample and $( u _ { i } ^ { \\prime } , q , v _ { i } ^ { \\prime } )$ and $( u _ { i } ^ { \\prime } , v _ { i } ^ { \\prime } )$ are the $i$ -th negative samples for knowledge graphs and homogeneous graphs, respectively. ",
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+        "type": "text",
+        "text": "Time Complexity. One advantage of NBFNet is that it has a relatively low time complexity during inference4. Consider a scenario where a model is required to infer the conditional likelihood of all possible triplets $p ( v | u , q )$ . We group triplets with the same condition $u , q$ together, where each group contains $| \\nu |$ triplets. For each group, we only need to execute Algorithm 1 once to get their predictions. Since a small constant number of iterations $T$ is enough for NBFNet to converge (Table 6b), Algorithm 1 has a time complexity of $O ( | \\mathcal { E } | d + | \\mathcal { V } | d ^ { 2 } )$ , where $d$ is the dimension of representations. Therefore, the amortized time complexity for a single triplet is $O \\left( \\frac { | \\varepsilon | d } { | \\nu | } + d ^ { 2 } \\right)$ . For a detailed derivation of time complexity of other GNN frameworks, please refer to Appendix C. ",
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+    {
+        "type": "text",
+        "text": "4 Experiment ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 Experiment Setup ",
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+        "type": "text",
+        "text": "We evaluate NBFNet in three settings, knowledge graph completion, homogeneous graph link prediction and inductive relation prediction. The former two are transductive settings, while the last is an inductive setting. For knowledge graphs, we use FB15k-237 [56] and WN18RR [13]. We use the standard transductive splits [56, 13] and inductive splits [55] of these datasets. For homogeneous graphs, we use Cora, Citeseer and PubMed [49]. Following previous works [32, 12], we split the edges into train/valid/test with a ratio of 85:5:10. Statistics of datasets can be found in Appendix E. Additional experiments of NBFNet on OGB [25] datasets can be found in Appendix G. ",
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+        "type": "text",
+        "text": "Implementation Details. Our implementation generally follows the open source codebases of knowledge graph completion5 and homogeneous graph link prediction6. For knowledge graphs, we follow [69, 46] and augment each triplet $\\langle u , q , \\nu \\rangle$ with a flipped triplet $\\langle \\nu , q ^ { - 1 } , u \\rangle$ . For homogeneous graphs, we follow [33, 32] and augment each node $u$ with a self loop $\\langle u , u \\rangle$ . We instantiate NBFNet with 6 layers, each with 32 hidden units. The feed-forward network $f ( \\cdot )$ is set to a 2-layer MLP with 64 hidden units. ReLU is used as the activation function for all hidden layers. We drop out edges that directly connect query node pairs during training to encourage the model to capture longer paths and prevent overfitting. Our model is trained on 4 Tesla V100 GPUs for 20 epochs. We select the models based on their performance on the validation set. See Appendix F for more details. ",
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+        "type": "text",
+        "text": "Evaluation. We follow the filtered ranking protocol [6] for knowledge graph completion. For a test triplet $\\langle u , q , \\nu \\rangle$ , we rank it against all negative triplets $\\left. u , q , \\nu ^ { \\prime } \\right.$ or $\\langle u ^ { \\prime } , q , \\nu \\rangle$ that do not appear in the knowledge graph. We report mean rank (MR), mean reciprocal rank (MRR) and HITS at N $( \\mathrm { H } @ \\mathrm { N } )$ for knowledge graph completion. For inductive relation prediction, we follow [55] and draw 50 negative triplets for each positive triplet and use the above filtered ranking. We report $\\mathrm { H I T S } @ 1 0$ for inductive relation prediction. For homogeneous graph link prediction, we follow [32] and compare the positive edges against the same number of negative edges. We report area under the receiver operating characteristic curve (AUROC) and average precision (AP) for homogeneous graphs. ",
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+        "text": "Baselines. We compare NBFNet against path-based methods, embedding methods, and GNNs. These include 11 baselines for knowledge graph completion, 10 baselines for homogeneous graph link prediction and 4 baselines for inductive relation prediction. Note the inductive setting only includes path-based methods and GNNs, since existing embedding methods cannot handle this setting. ",
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+    {
+        "type": "text",
+        "text": "4.2 Main Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 3 summarizes the results on knowledge graph completion. NBFNet significantly outperforms existing methods on all metrics and both datasets. NBFNet achieves an average relative gain of $21 \\%$ in $\\mathrm { H I T S } @ 1$ compared to the best path-based method, DRUM [46], on two datasets. Since DRUM is a special instance of NBFNet with natural summation and multiplication operators, this indicates the importance of learning MESSAGE and AGGREGATE functions in NBFNet. NBFNet also outperforms the best embedding method, LowFER [1], with an average relative performance gain of $18 \\%$ in $\\mathrm { H I T S } @ 1$ on two datasets. Meanwhile, NBFNet requires much less parameters than embedding methods. NBFNet only uses 3M parameters on FB15k-237, while TransE needs 30M parameters. See Appendix D for details on the number of parameters. ",
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+    {
+        "type": "text",
+        "text": "Table 4 shows the results on homogeneous graph link prediction. NBFNet gets the best results on Cora and PubMed, meanwhile achieves competitive results on CiteSeer. Note CiteSeer is extremely sparse (Appendix E), which makes it hard to learn good representations with NBFNet. One thing to note here is that unlike other GNN methods, NBFNet does not use the node features provided by the datasets but is still able to outperform most other methods. We leave how to effectively combine node features and structural representations for link prediction as our future work. ",
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+    {
+        "type": "table",
+        "img_path": "images/773b6a307ec47455b73a800597528ebc0d542fdcee80734c130a0040fe467a2a.jpg",
+        "table_caption": [
+            "Table 3: Knowledge graph completion results. Results of NeuraLP and DRUM are taken from [46]. Results of RotatE, HAKE and LowFER are taken from their original papers [52, 76, 1]. Results of the other embedding methods are taken from [52]. Since GraIL has scalability issues in this setting, we evaluate it with 50 and 100 negative triplets for FB15k-237 and WN18RR respectively and report MR based on an unbiased estimation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Class</td><td rowspan=\"2\">Method</td><td colspan=\"5\">FB15k-237</td><td colspan=\"5\">WN18RR</td></tr><tr><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td></tr><tr><td rowspan=\"3\">Path-based</td><td>Path Ranking [35]</td><td>3521</td><td>0.174</td><td>0.119</td><td>0.186</td><td>0.285</td><td>22438</td><td>0.324</td><td>0.276</td><td>0.360</td><td>0.406</td></tr><tr><td>NeuralLP [69]</td><td>-</td><td>0.240</td><td>=</td><td>-</td><td>0.362</td><td>-</td><td>0.435</td><td>0.371</td><td>0.434</td><td>0.566</td></tr><tr><td>DRUM[46]</td><td>1</td><td>0.343</td><td>0.255</td><td>0.378</td><td>0.516</td><td>-</td><td>0.486</td><td>0.425</td><td>0.513</td><td>0.586</td></tr><tr><td rowspan=\"6\">Embeddings</td><td>TransE [6]</td><td>357</td><td>0.294</td><td>-</td><td>1</td><td>0.465</td><td>3384</td><td>0.226</td><td>-</td><td>-</td><td>0.501</td></tr><tr><td>DistMult [68]</td><td>254</td><td>0.241</td><td>0.155</td><td>0.263</td><td>0.419</td><td>5110</td><td>0.43</td><td>0.39</td><td>0.44</td><td>0.49</td></tr><tr><td>ComplEx [58]</td><td>339</td><td>0.247</td><td>0.158</td><td>0.275</td><td>0.428</td><td>5261</td><td>0.44</td><td>0.41</td><td>0.46</td><td>0.51</td></tr><tr><td>RotatE [52]</td><td>177</td><td>0.338</td><td>0.241</td><td>0.375</td><td>0.553</td><td>3340</td><td>0.476</td><td>0.428</td><td>0.492</td><td>0.571</td></tr><tr><td>HAKE [76]</td><td>-</td><td>0.346</td><td>0.250</td><td>0.381</td><td>0.542</td><td>-</td><td>0.497</td><td>0.452</td><td>0.516</td><td>0.582</td></tr><tr><td>LowFER[1]</td><td>-</td><td>0.359</td><td>0.266</td><td>0.396</td><td>0.544</td><td>-</td><td>0.465</td><td>0.434</td><td>0.479</td><td>0.526</td></tr><tr><td rowspan=\"3\">GNNs</td><td>RGCN[48]</td><td>221</td><td>0.273</td><td>0.182</td><td>0.303</td><td>0.456</td><td>2719</td><td>0.402</td><td>0.345</td><td>0.437</td><td>0.494</td></tr><tr><td>GraIL [55]</td><td>2053</td><td>-</td><td>-</td><td>-</td><td>-</td><td>2539</td><td>-</td><td>=</td><td>-</td><td></td></tr><tr><td>NBFNet</td><td>114</td><td>0.415</td><td>0.321</td><td>0.454</td><td>0.599</td><td>636</td><td>0.551</td><td>0.497</td><td>0.573</td><td>0.666</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8934dd03ec8b5fb45a8734e04434c79b12a79498877aa11d903446bf334f4819.jpg",
+        "table_caption": [
+            "Table 4: Homogeneous graph link prediction results. Results of VGAE and S-VGAE are taken from their original papers [32, 12]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Class</td><td rowspan=\"2\">Method</td><td colspan=\"2\">Cora</td><td colspan=\"2\">Citeseer</td><td colspan=\"2\">PubMed</td></tr><tr><td>AUROC</td><td>AP</td><td>AUROC</td><td>AP</td><td>AUROC</td><td>AP</td></tr><tr><td rowspan=\"3\">Path-based</td><td>Katz Index [30]</td><td>0.834</td><td>0.889</td><td>0.768</td><td>0.810</td><td>0.757</td><td>0.856</td></tr><tr><td>Personalized PageRank [42]</td><td>0.845</td><td>0.899</td><td>0.762</td><td>0.814</td><td>0.763</td><td>0.860</td></tr><tr><td>SimRank [28]</td><td>0.838</td><td>0.888</td><td>0.755</td><td>0.805</td><td>0.743</td><td>0.829</td></tr><tr><td rowspan=\"3\">Embeddings</td><td>DeepWalk [43]</td><td>0.831</td><td>0.850</td><td>0.805</td><td>0.836</td><td>0.844</td><td>0.841</td></tr><tr><td>LINE [53]</td><td>0.844</td><td>0.876</td><td>0.791</td><td>0.826</td><td>0.849</td><td>0.888</td></tr><tr><td>node2vec [17]</td><td>0.872</td><td>0.879</td><td>0.838</td><td>0.868</td><td>0.891</td><td>0.914</td></tr><tr><td rowspan=\"5\">GNNs</td><td>VGAE [32]</td><td>0.914</td><td>0.926</td><td>0.908</td><td>0.920</td><td>0.944</td><td>0.947</td></tr><tr><td>S-VGAE [12]</td><td>0.941</td><td>0.941</td><td>0.947</td><td>0.952</td><td>0.960</td><td>0.960</td></tr><tr><td>SEAL[73]</td><td>0.933</td><td>0.942</td><td>0.905</td><td>0.924</td><td>0.978</td><td>0.979</td></tr><tr><td>TLC-GNN [67]</td><td>0.934</td><td>0.931</td><td>0.909</td><td>0.916</td><td>0.970</td><td>0.968</td></tr><tr><td>NBFNet</td><td>0.956</td><td>0.962</td><td>0.923</td><td>0.936</td><td>0.983</td><td>0.982</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ce6bd705e791734037c6e30d61cf34974213f13fdc9b05ac74c101f3b8035df8.jpg",
+        "table_caption": [
+            "Table 5: Inductive relation prediction results $( \\mathrm { H I T S } @ 1 0 )$ . V1-v4 corresponds to the 4 standard versions of inductive splits. Results of compared methods are taken from [55]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Class</td><td rowspan=\"2\">Method</td><td colspan=\"4\">FB15k-237</td><td colspan=\"4\">WN18RR</td></tr><tr><td>v1</td><td>v2</td><td>v3</td><td>v4</td><td>v1</td><td>v2</td><td>v3</td><td>v4</td></tr><tr><td rowspan=\"3\">Path-based</td><td>NeuralLP[16]</td><td>0.529</td><td>0.589</td><td>0.529</td><td>0.559</td><td>0.744</td><td>0.689</td><td>0.462</td><td>0.671</td></tr><tr><td>DRUM [46]</td><td>0.529</td><td>0.587</td><td>0.529</td><td>0.559</td><td>0.744</td><td>0.689</td><td>0.462</td><td>0.671</td></tr><tr><td>RuleN [39]</td><td>0.498</td><td>0.778</td><td>0.877</td><td>0.856</td><td>0.809</td><td>0.782</td><td>0.534</td><td>0.716</td></tr><tr><td rowspan=\"2\">GNNs</td><td>GraIL [55]</td><td>0.642</td><td>0.818</td><td>0.828</td><td>0.893</td><td>0.825</td><td>0.787</td><td>0.584</td><td>0.734</td></tr><tr><td>NBFNet</td><td>0.834</td><td>0.949</td><td>0.951</td><td>0.960</td><td>0.948</td><td>0.905</td><td>0.893</td><td>0.890</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Table 5 summarizes the results on inductive relation prediction. On all inductive splits of two datasets, NBFNet achieves the best result. NBFNet outperforms the previous best method, GraIL [55], with an average relative performance gain of $22 \\%$ in $\\mathrm { H I T S } @ 1 0$ . Note that GraIL explicitly encodes the local subgraph surrounding each node pair and has a high time complexity (Appendix C). Usually, GraIL can at most encode a 2-hop subgraph, while our NBFNet can efficiently explore longer paths. ",
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+    {
+        "type": "text",
+        "text": "4.3 Ablation Study ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "MESSAGE & AGGREGATE Functions. Table 6a shows the results of different MESSAGE and AGGREGATE functions. Generally, NBFNet benefits from advanced embedding methods (DistMult, ",
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+    {
+        "type": "text",
+        "text": "RotatE $>$ TransE) and aggregation functions ( $\\mathrm { P N A } >$ sum, mean, max). Among simple AGGREGATE functions (sum, mean, max), combinations of MESSAGE and AGGREGATE functions (TransE & max, DistMult & sum) that satisfy the semiring assumption7 of the generalized Bellman-Ford algorithm, achieve locally optimal performance. PNA significantly improves over simple counterparts, which highlights the importance of learning more powerful AGGREGATE functions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Number of GNN Layers. Table 6b compares the results of NBFNet with different number of layers. Although it has been reported that GNNs with deep layers often result in significant performance drop [36, 77], we observe NBFNet does not have this issue. The performance increases monotonically with more layers, hitting a saturation after 6 layers. We conjecture the reason is that longer paths have negligible contribution, and paths not longer than 6 are enough for link prediction. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Performance by Relation Category. We break down the performance of NBFNet by the categories of query relations: one-to-one, one-to-many, many-to-one and many-to-many8. Table 6c shows the prediction results for each category. It is observed that NBFNet not only improves on easy one-to-one cases, but also on hard cases where there are multiple true answers for the query. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Ablation studies of NBFNet on FB15k-237. Due to space constraints, we only report MRR here. For full results on all metrics, please refer to Appendix H. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b028315c7e25366a7a3ef970ecb098f1e8aa2f34d872cf0a2eb700702e91c8ec.jpg",
+        "table_caption": [
+            "(a) Different MESSAGE and AGGREGATE functions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">MESSAGE</td><td colspan=\"4\">AGGREGATE</td></tr><tr><td>Sum</td><td>Mean</td><td>Max</td><td>PNA [9]</td></tr><tr><td>TransE [6]</td><td>0.297</td><td>0.310</td><td>0.377</td><td>0.383</td></tr><tr><td>DistMult [69]</td><td>0.388</td><td>0.384</td><td>0.374</td><td>0.415</td></tr><tr><td>RotatE [52]</td><td>0.392</td><td>0.376</td><td>0.385</td><td>0.414</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/4dc159585f3199eeeb6d175e6c43727355e9190a14f03fd728f4d21b75872509.jpg",
+        "table_caption": [
+            "(b) Different number of layers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">#Layers (T)</td></tr><tr><td>2</td><td>4</td><td>6</td><td>8</td></tr><tr><td>NBFNet</td><td>0.345</td><td>0.409</td><td>0.415</td><td>0.416</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/70dca02c48d1b5925c9cdeb1a51bf9b64f3794b9ddfb1ed4b23226adfb2622fe.jpg",
+        "table_caption": [
+            "(c) Performance w.r.t. relation category. The two scores are the rankings over heads and tails respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">Relation Category</td></tr><tr><td>1-to-1</td><td>1-to-N</td><td>N-to-1</td><td>N-to-N</td></tr><tr><td>TransE [6]</td><td>0.498/0.488</td><td>0.455/0.071</td><td>0.079/0.744</td><td>0.224/0.330</td></tr><tr><td>RotatE [51]</td><td>0.487/0.484</td><td>0.467/0.070</td><td>0.081/0.747</td><td>0.234/0.338</td></tr><tr><td>NBFNet</td><td>0.578/0.600</td><td>0.499/0.122</td><td>0.165/0.790</td><td>0.348/0.456</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.4 Path Interpretations of Predictions ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "One advantage of NBFNet is that we can interpret its predictions through paths, which may be important for users to understand and debug the model. Intuitively, the interpretations should contain paths that contribute most to the prediction $p ( u , q , v )$ . Following local interpretation methods [3, 72], we approximate the local landscape of NBFNet with a linear model over the set of all paths, i.e., 1st-order Taylor polynomial. We define the importance of a path as its weight in the linear model, which can be computed by the partial derivative of the prediction w.r.t. the path. Formally, the top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ path interpretations for $p ( u , q , v )$ are defined as ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b64629d52f65171a139a1d561ab3dc8527c90df4a70db311444c6f206f679009.jpg",
+        "text": "$$\nP _ { 1 } , P _ { 2 } , . . . , P _ { k } = \\underset { P \\in \\mathcal { P } _ { u v } } { \\mathrm { t o p - k } } \\frac { \\partial p ( u , q , v ) } { \\partial P }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note this formulation generalizes the definition of logical rules [69, 46] to non-linear models. While directly computing the importance of all paths is intractable, we approximate them with edge importance. Specifically, the importance of each path is approximated by the sum of the importance of edges in that path, where edge importance is obtained via auto differentiation. Then the top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ path interpretations are equivalent to the top- $\\mathbf { \\nabla \\cdot k }$ longest paths on the edge importance graph, which can be solved by a Bellman-Ford-style beam search. Better approximation is left as a future work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 7 visualizes path interpretations from FB15k-237 test set. While users may have different insights towards the visualization, here is our understanding. 1) In the first example, NBFNet learns soft logical entailment, such as impersonate $^ { - 1 } \\wedge$ nationality $\\Longrightarrow$ nationality and ethnicity $^ { - 1 } \\wedge$ distribution $\\Longrightarrow$ nationality. 2) In second example, NBFNet performs analogical reasoning by leveraging the fact that Florence is similar to Rome. 3) In the last example, NBFNet extracts longer paths, since there is no obvious connection between Pearl Harbor (film) and Japanese language. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "table",
+        "img_path": "images/2c4b0d78502ef77074c73cde4860d7ec0128631aa99c80ee6bcb8df41fc76b34.jpg",
+        "table_caption": [
+            "Table 7: Path interpretations of predictions on FB15k-237 test set. For each query triplet, we visualize the top-2 path interpretations and their weights. Inverse relations are denoted with a superscript $^ { - 1 }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Query</td><td>(u,q,v): (O. Hardy, nationality, U.S.)</td></tr><tr><td>0.243</td><td>(O.Hardy, impersonate-1,R.Little)^ (R.Little, nationality, U.S.)</td></tr><tr><td>0.224</td><td>(O.Hardy,ethnicity-1,Scottish American)^ (Scottish American,distribution,U.S.)</td></tr><tr><td>Query</td><td>{u,q, v): (Florence, vacationer, D.C. Henrie)</td></tr><tr><td>0.251</td><td>(Florence,contain-1,Italy)^ (Italy,capital,Rome)^ (Rome,vacationer, D.C.Henrie)</td></tr><tr><td>0.183</td><td>(Florence,place live-1,G.F.Handel)(G.F.Handel,place live,Rome) (Rome,vacationer,D.C.Henrie)</td></tr><tr><td>Query 0.211</td><td>(u,q,v): (Pearl Harbor (film), language,Japanese)</td></tr><tr><td></td><td>(Pearl Harbor(film),film actor, C.-H.Tagawa)^ (C.-H.Tagawa,nationality,Japan) (Japan,country of origin, Yu-Gi-Oh!)^ (Yu-Gi-Oh!,language,Japanese)</td></tr><tr><td>0.208</td><td>(Pearl Harbor (film),film actor,C.-H.Tagawa) ^ (C.-H.Tagawa,nationality,Japan) (Japan,official language,Japanese)</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "5 Discussion and Conclusion ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Limitations. There are a few limitations for NBFNet. First, the assumption of the generalized Bellman-Ford algorithm requires the operators $\\langle \\oplus , \\otimes \\rangle$ to satisfy a semiring. Due to the non-linear activation functions in neural networks, this assumption does not hold for NBFNet, and we do not have a theoretical guarantee on the loss incurred by this relaxation. Second, NBFNet is only verified on simple edge prediction, while there are other link prediction variants, e.g., complex logical queries with conjunctions (∧) and disjunctions (∨) [18, 45]. In the future, we would like to how NBFNet approximates the path formulation, as well as apply NBFNet to other link prediction settings. ",
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+    },
+    {
+        "type": "text",
+        "text": "Social Impacts. Link prediction has a wide range of beneficial applications, including recommender systems, knowledge graph completion and drug repurposing. However, there are also some potentially negative impacts. First, NBFNet may encode the bias present in the training data, which leads to stereotyped predictions when the prediction is applied to a user on a social or e-commerce platform. Second, some harmful network activities could be augmented by powerful link prediction models, e.g., spamming, phishing, and social engineering. We expect future studies will mitigate these issues. ",
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+    },
+    {
+        "type": "text",
+        "text": "Conclusion. We present a representation learning framework based on paths for link prediction. Our path formulation generalizes several traditional methods, and can be efficiently solved via the generalized Bellman-Ford algorithm. To improve the capacity of the path formulation, we propose NBFNet, which parameterizes the generalized Bellman-Ford algorithm with learned INDICATOR, MESSAGE, AGGREGATE functions. Experiments on knowledge graphs and homogeneous graphs show that NBFNet outperforms a wide range of methods in both transductive and inductive settings. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Komal Teru for discussion on inductive relation prediction, Guyue Huang for discussion on fused message passing implementation, and Yao Lu for assistance on large-scale GPU training. We thank Meng Qu, Chence Shi and Minghao Xu for providing feedback on our manuscript. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This project is supported by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program, collaboration grants between Microsoft Research and Mila, Samsung Electronics Co., Ltd., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund and a NRC Collaborative R&D Project (AI4D-CORE-06). This project was also partially funded by IVADO Fundamental Research Project grant PRF-2019-3583139727. The computation resource of this project is supported by Calcul Québec9 and Compute Canada10. ",
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+# Unsupervised Object-Level Representation Learning from Scene Images
+
+Jiahao Xie1 Xiaohang Zhan2 Ziwei Liu1 Yew Soon Ong1,3 Chen Change Loy1
+
+1Nanyang Technological University 2The Chinese University of Hong Kong 3A\*STAR, Singapore {jiahao003, ziwei.liu, asysong, ccloy}@ntu.edu.sg xiaohangzhan@outlook.com
+
+# Abstract
+
+Contrastive self-supervised learning has largely narrowed the gap to supervised pre-training on ImageNet. However, its success highly relies on the object-centric priors of ImageNet, i.e., different augmented views of the same image correspond to the same object. Such a heavily curated constraint becomes immediately infeasible when pre-trained on more complex scene images with many objects. To overcome this limitation, we introduce Object-level Representation Learning (ORL), a new self-supervised learning framework towards scene images. Our key insight is to leverage image-level self-supervised pre-training as the prior to discover object-level semantic correspondence, thus realizing object-level representation learning from scene images. Extensive experiments on COCO show that ORL significantly improves the performance of self-supervised learning on scene images, even surpassing supervised ImageNet pre-training on several downstream tasks. Furthermore, ORL improves the downstream performance when more unlabeled scene images are available, demonstrating its great potential of harnessing unlabeled data in the wild. We hope our approach can motivate future research on more general-purpose unsupervised representation learning from scene data.1
+
+# 1 Introduction
+
+Unsupervised visual representation learning aims at obtaining transferable features with abundant unlabeled data. Recent self-supervised learning (SSL) methods based on contrastive learning [60, 22, 37, 5, 19, 4, 7] have largely narrowed the gap and even surpassed the supervised counterpart on a number of downstream tasks [30, 49, 15, 47, 35, 23]. These methods build upon the instance discrimination task that maximizes the agreement between different data-augmented views of the same image. Despite their success, current SSL methods are primarily pre-trained on the unlabeled ImageNet [8] dataset that contains iconic images with single object as shown in Figure 1(a). The underlying object-centric constraint of ImageNet makes it hard to be applied in real world scenarios where more complex scene images with multiple objects are available. Meanwhile, naïvely adopting the off-the-shelf contrastive learning methods on scene images introduces inconsistent learning signals since random crops of the same image may correspond to different objects as shown in Figure 1(b). Indeed, it has been shown that current contrastive learning methods tend to struggle on more complex scene datasets [19, 50, 34, 58] like COCO [33] or Places365 [72]. Therefore, it is imperative to design an effective object-level representation learning paradigm as illustrated in Figure 1(c) to harness massive unlabeled scene images in the wild.
+
+![](images/a919e3ea35f294be21ec244aa7ce625c2450aca7f8f498519807ea147f589e62.jpg)  
+  
+Figure 1: (a) Current image-level contrastive learning methods heavily rely on the object-centric bias of ImageNet, i.e., different crops correspond to the same object. Prior works use either the different views of the same image [60, 22, 37, 5, 19, 7] (i.e., intra-image) or similar images [74, 63, 1, 11] (i.e., inter-image) to form positive pairs. (b) Directly adopting image-level contrastive learning methods on scene images can cause inconsistent learning signals since different crops may correspond to different objects. (c) Object-level contrastive learning can overcome the limitation in (b) by enforcing object-level consistency. (d) We find that image-level contrastive learning encodes priors for region correspondence discovery across images, and high-response regions are usually objects or object parts (we show one discovered object-instance pair per image pair for clarity), which is useful for object-level representation learning.
+
+In this work, we are interested in going beyond ImageNet to obtain better representations on noniconic images. Apparently, it is challenging to learn representations from scene-level images since they are entangled with many concepts including structures, objects, backgrounds and relationships. It remains an open question how to take advantages of spatial information of multiple objects naturally residing in the scene images when no object annotations are available, let alone further deriving object-level correspondence to construct positive object-instance pairs.
+
+To tackle these challenges, we introduce a novel object-level unsupervised representation learning framework tailored for scene images. Our framework is based on a key insight of the current contrastive learning methods: they can implicitly group different images with similar visual concepts together even though they are explicitly optimized to group different views of the same image. This phenomenon reveals that image-level contrastive learning has already induced a latent space with rich visual concepts. Though the latent space usually entangles other scene concepts like structures, backgrounds and relationships, it will be useful for object discovery if appropriately deployed. Through computing the similarity of sampled regions between $k$ -nearest-neighbor (KNN) images, we conclude two observations: 1) image-level contrastive learning encodes priors for region correspondence discovery across images; 2) high-response regions are usually objects or object parts.
+
+Based on the observation above, we propose a multi-stage framework for unsupervised object-level representation learning. Specifically, we first extract potential object-based regions in scene images using the unsupervised region proposal algorithms (e.g., selective search [56]). We then propose a region correspondence generation scheme to leverage the off-the-shelf image-level contrastive learning pre-trained model to discover corresponding object-instance pairs for the proposed regions in the embedding space. Finally, we use the obtained object-instance pairs to construct positive sample pairs for object-level representation learning. Figure 1(d) shows several cross-image object-instance pairs discovered by our framework on COCO dataset using the latent prior of BYOL [19], the state-of-the-art image-level contrastive learning method. The discovered inter-corresponding pairs substantially provide diverse intra-class variances at the object-instance level to aid object-level representation learning.
+
+Overall, our main contributions are summarized as follows:
+
+1) We observe that existing image-level contrastive learning methods have priors to discover objectlevel correspondence across images. We leverage this prior for the first time for unsupervised cross-image object-level correspondence discovery.
+
+2) With the obtained correspondence, we introduce a novel multi-stage self-supervised learning pipeline, termed as ORL, for object-level representation learning from scene images, going beyond object-centric ImageNet.
+
+3) We contribute the first study for object-level SSL. ORL substantially outperforms image-level contrastive learning approaches pre-trained on COCO dataset $\mathord { \sim } 1 1 8 \mathrm { k }$ images with labels discarded), setting a new state of the art on this challenging dataset that contains diverse scenes in the wild. The COCO pre-trained ORL even surpasses supervised ImageNet pre-training on several considered downstream tasks. When SSL is conducted on a larger ${ } ^ { 6 6 } \mathrm { C O C O + } { } ^ { , }$ dataset (COCO train2017 set plus COCO unlabeled2017 set, ${ \sim } 2 4 1 \mathrm { k }$ images in total), ORL further improves the performance, demonstrating its potential to benefit from more unlabeled scene data.
+
+# 2 Related work
+
+Self-supervised learning. Self-supervised learning builds unsupervised representations by exploiting the internal priors or structures of data in the form of a pretext task. A wide range of pretext tasks have been proposed in the past few years. Examples include patch context prediction [10], jigsaw puzzles [39], inpainting [43], colorization [31, 70], cross-channel prediction [71], visual primitive counting [40], and rotation prediction [14]. Although good representations emerge with these pretext tasks, they are prone to lose generality due to their hand-crafted nature.
+
+Recently, contrastive learning [20] that performs instance discrimination [60, 22, 37, 5, 19, 4, 7] has shown great potential in this field, largely narrowing the gap to fully supervised learning. The core idea of contrastive learning is to gather positive pairs and separate negative pairs in the embedding space. A positive pair is usually formed with two transformed views of the same image while the negative pairs are formed with different images. Typically, contrastive learning methods require a large number of negative samples to avoid mode collapse. These samples can be maintained within a mini-batch [42, 27, 66, 26, 2, 5], a memory bank [60, 53, 74, 37] or a queue [22, 6]. BYOL [19] and SwAV [4] further remove the necessity of involving negative pairs. BYOL directly predicts the features of one view from another view, while SwAV predicts the cluster assignments between multiple views of the same image. Despite their improved performance, the existing image-level contrastive learning methods are largely confined to the underlying object-centric bias of ImageNet.
+
+More recently, a group of works that perform pixel-level [45, 58, 64, 50, 34, 25] or region-level [48, 65, 61, 62, 9] representation learning have emerged. Our work is more related to region-level representation learning but substantially different from this line of research in the following aspects: 1) they still largely pre-train on object-centric ImageNet while we pre-train on non-iconic scene images, 2) they align pre-training specifically for dense prediction downstream tasks while we target at more general-purpose representation learning that improves performance in both dense prediction and classification tasks, 3) their randomly cropped local regions do not contain the explicit object notion as ours, and 4) they only rely on intra-image transformations (e.g., random cropping) to construct corresponding positive pairs from the same image while we leverage the discovered high-level semantic correspondence to construct positive pairs across images.
+
+There are also a few prior attempts [18, 3, 16] for self-supervised learning on non-curated scene images. As opposed to our work, most of them consider larger models and datasets to explore the limit of current self-supervised learning methods without further considering the object-level information residing in scene images.
+
+Visual correspondence. Visual correspondence aims at finding pairwise pixels or regions across images that result from the same scene [67], which can be regarded as similarity learning of visual descriptors among matched points or patches. While early efforts learn dense correspondence with labeled data [21, 68, 29, 55, 41], some recent works learn the similarity between the parts or landmarks of the data in an unsupervised manner [52, 51]. Our work substantially differs from this line of research from original intention. Previous works aim at accurately detecting all correspondence given two images, whereas our work focuses on retrieving high-quality correspondence to improve representation learning.
+
+![](images/575ab9e1793d07665088f0a09cc6dc9ce09301fcb4142d1dae60de314b455d89.jpg)  
+Figure 2: Overview of our three-stage pipeline. In Stage 1, we pre-train an image-level contrastive learning model, e.g., BYOL. In Stage 2, we first use the pre-trained model to retrieve KNNs for each image in the embedding space to obtain image-level visually similar pairs. We then use unsupervised region proposal algorithms (e.g., selective search) to generate rough RoIs for each image pair. Afterwards, we reuse the pre-trained model to retrieve the top-ranked RoI pairs, i.e., correspondence. We find these pairs of RoIs are almost objects or object parts. In Stage 3, with the corresponding RoI pairs discovered across images, we finally perform object-level contrastive learning using the same architecture as Stage 1.
+
+# 3 Methodology
+
+We propose a new multi-stage self-supervised learning framework, i.e., ORL, for object-level representation learning from scene images. ORL extends the existing image-level contrastive learning framework to object level by leveraging priors from image-level instance discrimination. The overall pipeline of ORL is illustrated in Figure 2. It contains three stages: image-level pre-training, correspondence discovery, and object-level pre-training. We detail each stage as follows.
+
+# 3.1 ORL pipeline
+
+Preliminary: Contrastive learning. Our pipeline contains several contrastive learning modules in Stage 1 and 3. Without loss of generality, we consider BYOL [19] as our basic contrastive learning module. BYOL uses two neural networks: the online network $f _ { \boldsymbol { \theta } } ( \boldsymbol { x } )$ and the target network $g _ { \xi } ( x )$ The target network provides the regression target to train the online network while its weights $\xi$ are updated by an exponential moving average of the online parameters $\theta$ with a decay rate $\tau \in [ 0 , 1 ]$ following BYOL. Given two input images $x _ { 1 }$ and $x _ { 2 }$ , the loss function is defined as:
+
+$$
+\mathcal { L } \left( \boldsymbol { x } _ { 1 } , \boldsymbol { x } _ { 2 } \right) \triangleq \left. \boldsymbol { f } _ { \boldsymbol { \theta } } \left( \boldsymbol { x } _ { 1 } \right) - \boldsymbol { g } _ { \boldsymbol { \xi } } \left( \boldsymbol { x } _ { 2 } \right) \right. _ { 2 } ^ { 2 } ,
+$$
+
+We name it an “intra-” version of BYOL if $x _ { 1 }$ and $x _ { 2 }$ are two augmented views from the same image, otherwise an “inter-” one.
+
+Stage 1: Image-level pre-training. The foremost stage is to obtain an unsupervised pre-trained model from image-level tasks. As shown in Figure 2 Stage 1, given two augmented views $v$ and $v ^ { \prime }$ from the same input image $x$ , we pre-train the network following the loss function ${ \mathcal { L } } _ { \mathrm { i m a g e } } = { \mathcal { L } } \left( v , v ^ { \prime } \right)$ constituting a standard image-level BYOL pre-training. This stage can be freely replaced with other image-level contrastive learning methods. We adopt BYOL here for its simplicity and effectiveness.
+
+Stage 2: Correspondence discovery. We employ the pre-trained image-level contrastive learning model in Stage 1 to mine object-level correspondence for the whole dataset. As shown in Figure 2 Stage 2, the overall discovery process comprises three steps.
+
+(i) Image-level nearest-neighbor retrieval. Specifically, for each query image $x$ in the training set $\mathcal { D }$ , we first retrieve its top $K$ nearest neighbors $\mathcal { N } _ { k }$ , $k = 1 , . . . , K$ , by cosine distance in the embedding space using the features learned from the first stage to form image-level pairs that contain similar visual context.
+
+(ii) Region-of-interest (RoI) generation. To generate object-based RoIs, we apply unsupervised region proposal algorithms, e.g., selective search [56], for each image in the pair. Considering the redundancy of generated proposals (each image can have thousands of proposals), we filter certain number of them with some pre-defined thresholds2 including the minimal scale, the range of aspect ratio, and the maximal intersection-over-union (IoU) among the filtered boxes. After the filtering operation, we select the top 100 proposals ranked with objectiveness as the candidate RoI set for subsequent RoI pair retrieval. To extract features with the equally-sized input that is compatible with the backbone, we crop and resize each RoI to $2 2 4 \times 2 2 4$ . Note that even top RoIs ranked with objectiveness are still very noisy, containing a large proportion of non-object regions.
+
+(iii) Top-ranked RoI pair retrieval. For each query RoI from $x$ , we compute its cosine similarity in the embedding space with all RoIs from its nearest-neighbor image $\mathcal { N } _ { k }$ using the features learned from Stage 1 again. Within the calculated cosine similarity matrix $\bar { \mathbf { M } } _ { k } \in \mathbb { R } ^ { 1 0 0 \bar { \times } 1 0 0 }$ , we retrieve top-ranked $N$ RoI pairs to construct the set of object-level corresponding pairs $\{ B _ { k } ^ { n } \}$ , where $n = \{ 1 , . . . , N \}$ These high-response corresponding regions are almost objects or object parts. Finally, we save the nearest-neighbor image id and bounding box coordinate information of each corresponding pair.
+
+Stage 3: Object-level pre-training. With the corresponding inter-image RoI (inter-RoI) pairs obtained in Stage 2, we perform object-level representation learning following the BYOL framework as shown in Figure 2 Stage 3. Specifically, given an input image $x$ , we first randomly select one nearest-neighbor image $\mathcal { N } _ { k }$ to obtain the corresponding set of inter-RoI pairs $\{ B _ { k } ^ { n } \}$ . We then randomly select one inter-RoI pair $B _ { k } ^ { n }$ as a positive pair. With the bounding box coordinate stored in $B _ { k } ^ { n }$ , we crop the corresponding inter-RoIs from $x$ and $\mathcal { N } _ { k }$ , respectively, and resize each patch to $9 \ddot { 6 } \times 9 6$ , constituting two patches $p _ { 1 }$ and $p _ { 2 }$ . We feed the two patches to the online network and target network separately to compute the loss $\mathcal { L } _ { \mathrm { i n t e r - R o I } } = \mathcal { L } \left( p _ { 1 } , p _ { 2 } \right)$ .
+
+To make full use of discovered objects, we introduce the intra-RoI contrastive learning via augmenting object patches. Specifically, we randomly select one filtered bounding box from $x$ obtained in Stage 2, and spatially jitter the box around its original location with the following operations3: (i) a random box center shifting within $50 \%$ of its width and height, (ii) a random area scaling between $50 \%$ and $200 \%$ of the original box, and (iii) a random aspect ratio between $1 / 2$ and $2 / 1$ . Similarly, we crop the two intra-RoIs $p$ and $p ^ { \prime }$ , and resize each patch to $9 6 \times 9 6$ for forward propagation to compute the loss $\mathcal { L } _ { \mathrm { \scriptsize { i n t r a - R o I } } } = \mathcal { L } \left( p , p ^ { \prime } \right)$ . The diverse spatial jittering of the bounding box encourages the network to preserve common object information and disregard the background, thus further improving the localization ability.
+
+We keep the two original global views in BYOL as well since they preserve the global image-level information compared with the local patches. The final loss for our ORL can thus be formulated as:
+
+$$
+\mathcal { L } _ { \mathrm { O R L } } = \lambda _ { 1 } \mathcal { L } _ { \mathrm { i m a g e } } + \lambda _ { 2 } \mathcal { L } _ { \mathrm { i n t r a - R o I } } + \lambda _ { 3 } \mathcal { L } _ { \mathrm { i n t e r - R o I } } ,
+$$
+
+where $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$ are the loss weights to balance each term. We set all loss weights to 1 by default. Following BYOL, we also compute the symmetric loss $\widetilde { \mathcal { L } } _ { \mathrm { { o R L } } }$ by separately feeding $v ^ { \prime } , p ^ { \prime } , p _ { 2 }$ to the online network and $v , p , p _ { 1 }$ to the target network.
+
+# 3.2 Implementation details
+
+Dataset. We pre-train our models on the COCO train2017 set that contains ${ \sim } 1 1 8 \mathrm { k }$ images without using labels. Compared with the heavily curated object-centric ImageNet dataset, COCO contains more natural and diverse scenes in the wild, which is closer to real-world scenarios. We also perform self-supervised learning on a larger ${ } ^ { \cdot \cdot } \mathrm { C O C O } { + } ^ { \prime \cdot }$ dataset (COCO train2017 set plus COCO unlabeled2017 set) to verify whether our method can benefit from more unlabeled scene data.
+
+Image augmentations. The global image augmentation setting is the same as BYOL [19]: a $2 2 4 \times 2 2 4$ -pixel random resized crop with a random horizontal flip, followed by a random color distortion, random grayscale conversion, random Gaussian blur and solarization. For the local patch augmentation, we directly crop the corresponding intra-RoI and inter-RoI on the input images, and resize each cropped patch to $9 6 \times 9 6$ to take place of the random resized cropping. The subsequent augmentations exactly follow the global ones.
+
+Network architecture. We adopt ResNet-50 [24] as the default backbone. We use the same MLP projector and predictor as in BYOL: a linear layer with output size 4096 followed by batch normalization (BN) [28], rectified linear units (ReLU) [38], and a final linear layer with output dimension 256. We share the backbone and projector weights among the global and two local branches while the weights of predictor are not shared.
+
+Optimization. For pre-training in Stage 1 and Stage 3, we use the same training hyper-parameters. Specifically, we use the SGD optimizer with a weight decay of 0.0001 and a momentum of 0.9. We adopt the cosine learning rate decay schedule [36] with a base learning rate of 0.2, linearly scaled[17] with the batch size $l r = 0 . 2 \times \mathrm { ~ F ~ }$ atchSize/256). The batch size is set to 512 by default, which is friendly to typical 8-GPU implementations. To keep the training iterations comparable with the ImageNet supervised pre-training, we train our models for 800 epochs with a warm-up period of 4 epochs. The exponential moving average parameter $\tau$ starts from 0.99 and is increased to 1 during training, following [19]. For correspondence generation in Stage 2, we retrieve top $K = 1 0$ nearest neighbors for each image and select top-ranked $N = 1 0 \%$ RoI pairs for each image-level nearest-neighbor pair.
+
+# 4 Experiments
+
+# 4.1 Transferring to downstream tasks
+
+We evaluate the quality of learned representations by transferring them to multiple downstream tasks. Following common protocol [18, 37], we use two evaluation setups: (i) the pre-trained network is frozen as a feature extractor, and (ii) the network parameters are fully fine-tuned as weight initialization. We provide more experimental details in the supplementary material.
+
+Table 1: Image classification with linear models. All unsupervised methods are based on 800-epoch pretraining on $\mathrm { C O C O ( + ) }$ with ResNet-50. We report mAP on the VOC07 dataset and top-1 center-crop accuracy on all other datasets. Numbers for all other methods are reproduced by us.   
+
+<table><tr><td>Method</td><td>Pre-train data</td><td>VOC07 mAP</td><td>ImageNet Top-1</td><td>Places205 Top-1</td><td>iNat. Top-1</td></tr><tr><td>Random [18]</td><td>=</td><td>9.6</td><td>13.7</td><td>16.6</td><td>4.8</td></tr><tr><td>Supervised [37]</td><td>ImageNet</td><td>87.5</td><td>75.9</td><td>51.5</td><td>45.4</td></tr><tr><td>SimCLR [5]</td><td>COCO</td><td>78.1</td><td>50.9</td><td>48.0</td><td>22.7</td></tr><tr><td>MoCo v2 [6]</td><td>COCO</td><td>82.2</td><td>55.1</td><td>48.8</td><td>27.8</td></tr><tr><td>BYOL[19]</td><td>COCO</td><td>84.5</td><td>57.8</td><td>50.5</td><td>29.5</td></tr><tr><td>ORL (ours)</td><td>COCO</td><td>86.7</td><td>59.0</td><td>52.7</td><td>31.8</td></tr><tr><td>BYOL [19]</td><td>COCO+</td><td>87.0</td><td>59.6</td><td>52.7</td><td>30.9</td></tr><tr><td>ORL (ours)</td><td>COCO+</td><td>88.6</td><td>60.7</td><td>54.1</td><td>32.0</td></tr></table>
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Pre-train data</td><td colspan="7">VOC07 low-shot (mAP)</td></tr><tr><td>1</td><td>2</td><td>4</td><td>8</td><td>16</td><td>32</td><td>64</td></tr><tr><td>Random</td><td>=</td><td>9.2</td><td>9.4</td><td>11.1</td><td>12.3</td><td>14.3</td><td>17.4</td><td>21.3</td><td>23.8</td></tr><tr><td>Supervied</td><td>ImageNet</td><td>53.0</td><td>63.6</td><td>73.7</td><td>78.8</td><td>81.8</td><td>83.8</td><td>85.2</td><td>86.0</td></tr><tr><td>SimCLR [5] MoCo v2 [6]</td><td>CoCo</td><td>33.3</td><td>43.5</td><td>52.5</td><td>61.1</td><td>66.7</td><td>70.5</td><td>73.7</td><td>75.0</td></tr><tr><td>BYOL [19]</td><td>COCO</td><td>39.5</td><td>49.3</td><td>60.4</td><td>69.3</td><td>74.1</td><td>76.8</td><td>79.1</td><td>80.1</td></tr><tr><td>ORL (ours)</td><td>COCO COCO</td><td>39.4</td><td>50.9</td><td>62.2</td><td>71.7</td><td>76.6</td><td>79.2</td><td>81.3</td><td>82.2</td></tr><tr><td></td><td></td><td>39.6</td><td>51.2</td><td>63.4</td><td>72.6</td><td>78.2</td><td>81.3</td><td>83.6</td><td>84.7</td></tr><tr><td>BYOL [19] ORL (ours)</td><td>COCO+</td><td>41.1</td><td>54.3</td><td>66.6</td><td>75.2</td><td>80.1</td><td>82.6</td><td>84.6</td><td>85.4</td></tr><tr><td></td><td>COCO+</td><td>42.1</td><td>54.9</td><td>67.4</td><td>75.7</td><td>81.3</td><td>83.7</td><td>85.8</td><td>86.7</td></tr></table>
+
+Table 2: Low-shot image classification on VOC07 using linear SVMs trained on the fixed representations. All unsupervised methods are pre-trained on $\mathrm { C O C O ( + ) }$ for 800 epochs with ResNet-50. We report mAP for each case across five runs.
+
+Image classification with linear models. Following [18, 37], we assess the quality of features by training linear classifiers on top of the fixed representations extracted from different depths of the network for four datasets: VOC07 [12], ImageNet [8], Places205 [73], and iNaturalist18 [57]. These datasets involve diverse classification tasks ranging from object classification, scene recognition to fine-grained recognition. For VOC07, we train linear SVMs using LIBLINEAR package [13]
+
+Table 3: Semi-supervised learning on ImageNet. All unsupervised methods are pre-tained on $\mathrm { C O C O ( + ) }$ for 800 epochs with ResNet-50. We fine-tune all models with $1 \%$ and $10 \%$ ImageNet labels, and report both top-1 and top-5 center-crop accuracy on the ImageNet validation set.   
+
+<table><tr><td>Method</td><td>Pre-train</td><td colspan="2">1% labels</td><td colspan="2">10% labels</td></tr><tr><td></td><td>data</td><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Random</td><td></td><td>1.6</td><td>5.0</td><td>21.8</td><td>44.2</td></tr><tr><td>Supervised [69]</td><td>ImageNet</td><td>25.4</td><td>48.4</td><td>56.4</td><td>80.4</td></tr><tr><td>SimCLR [5]</td><td>CoCo</td><td>23.4</td><td>46.4</td><td>52.2</td><td>77.4</td></tr><tr><td>MoCo v2 [6]</td><td>COCO</td><td>28.2</td><td>54.7</td><td>57.1</td><td>81.7</td></tr><tr><td>BYOL[19]</td><td>COCO</td><td>28.4</td><td>55.9</td><td>58.4</td><td>82.7</td></tr><tr><td>ORL (ours)</td><td>CoCo</td><td>31.0</td><td>58.9</td><td>60.5</td><td>84.2</td></tr><tr><td>BYOL[19]</td><td>COCO+</td><td>28.3</td><td>56.0</td><td>59.4</td><td>83.6</td></tr><tr><td>ORL (ours)</td><td>COCO+</td><td>31.8</td><td>60.1</td><td>60.9</td><td>84.4</td></tr></table>
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Pre-train data</td><td colspan="3">COCO detection</td><td colspan="3">COCO instance seg.</td></tr><tr><td>Ap66</td><td>AP</td><td>AP</td><td>Apmk</td><td>APk</td><td>AP</td></tr><tr><td>Random [54]</td><td></td><td>32.8</td><td>50.9</td><td>35.3</td><td>29.9</td><td>47.9</td><td>32.0</td></tr><tr><td>Supervised [54]</td><td>ImageNet</td><td>39.7</td><td>59.5</td><td>43.3</td><td>35.9</td><td>56.6</td><td>38.6</td></tr><tr><td>SimCLR [5]</td><td>COCO</td><td>37.0</td><td>56.8</td><td>40.3</td><td>33.7</td><td>53.8</td><td>36.1</td></tr><tr><td>MoCo v2 [6]</td><td>COCO</td><td>38.5</td><td>58.1</td><td>42.1</td><td>34.8</td><td>55.3</td><td>37.3</td></tr><tr><td>Self-EMD [34]</td><td>COCo</td><td>39.3</td><td>60.1</td><td>42.8</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DenseCL [58]</td><td>COCo</td><td>39.6</td><td>59.3</td><td>43.3</td><td>35.7</td><td>56.5</td><td>38.4</td></tr><tr><td>BYOL[19]</td><td>COCO</td><td>39.5</td><td>59.3</td><td>43.2</td><td>35.6</td><td>56.5</td><td>38.2</td></tr><tr><td>ORL (ours)</td><td>COCO</td><td>40.3</td><td>60.2</td><td>44.4</td><td>36.3</td><td>57.3</td><td>38.9</td></tr><tr><td>BYOL[19]</td><td>COCO+</td><td>40.0</td><td>60.1</td><td>44.0</td><td>36.2</td><td>57.1</td><td>39.0</td></tr><tr><td>ORL (ours)</td><td>COCO+</td><td>40.6</td><td>60.8</td><td>44.5</td><td>36.7</td><td>57.9</td><td>39.3</td></tr></table>
+
+Table 4: Object detection and instance segmentation fine-tuned on COCO. All unsupervised methods are based on 800-epoch pre-training on $\mathrm { C O C O ( + ) }$ . We use Mask R-CNN R50-FPN ( $1 \times$ schedule), and report bounding-box AP $( \mathrm { A } \mathbf { \hat { P } } ^ { b b } )$ and mask AP $( \mathbf { A } \mathbf { P } ^ { m k } )$ . Numbers for MoCo v2 are adopted from [58].
+
+following the setup in [18, 37]. We train on trainval split of VOC07 and evaluate mAP on test split. For ImageNet, Places205 and iNaturalist18, we follow [71, 18, 37] and train a 1000-way, 205-way and 8142-way linear classifier, respectively. We train on train split of each dataset, and report top-1 center-crop accuracy on the respective val split. Table 1 reports the results for the best-performing layer of each method. ORL substantially outperforms the BYOL baseline on all four datasets. We also observe that the COCO pre-trained ORL surpasses the supervised ImageNet pre-trained counterpart on Places205 by $1 . 2 \%$ in top-1 accuracy. This is the first time that a selfsupervised learner outperforms the ImageNet pre-training using only ${ \sim } 1 / 1 0$ images compared with ImageNet. When pre-trained on a larger $\mathrm { C O C O + }$ dataset, ORL again outperforms BYOL. Note that apart from Places205 $2 . 6 \%$ gains), ORL also surpasses the supervised ImageNet counterpart on VOC07 by $1 . 1 \%$ mAP, using merely ${ \sim } 1 / 5$ images.
+
+Low-shot image classification. We perform low-shot image classification with few training examples per class on VOC07 dataset following the same setup in [18]. We vary the number of labeled examples per category used to train linear SVMs on train split of VOC07 and report the average mAP across five independent samples for each low-shot case evaluated on test split. Table 2 provides the results. ORL shows consistent performance improvement over BYOL for each low-shot value, with larger gains achieved as the number of labeled examples per class is increasing. ORL also gradually bridges the gap to the supervised ImageNet pre-training under this scenario. We observe consistent performance boost when pre-training on the $\mathrm { C O C O + }$ dataset. Note that the $\mathrm { C O C O + }$ pre-trained ORL again outperforms the supervised ImageNet pre-training when the low-shot samples are 64 and 96.
+
+Semi-supervised learning. We perform semi-supervised learning on ImageNet following the protocol of previous studies [60, 26, 37, 5, 19]. Specifically, we first randomly select $1 \%$ and $10 \%$ labeled data from ImageNet train split. We then fine-tune our models on these two training subsets and report both top-1 and top-5 accuracy on the official val split of ImageNet in Table 3. Again, ORL outperforms BYOL as well as the supervised ImageNet counterpart by large margins.
+
+Object detection and segmentation. We train a Mask R-CNN model [23] with R50-FPN backbone [32] implemented in Detectron2 [59]. We fine-tune all layers end-to-end on COCO train2017 split with the standard $1 \times$ schedule and evaluate on COCO val2017 split. We follow the same setup in [54], with batch normalization layers synchronized across GPUs [44]. As shown in Table 4, ORL yields $0 . 8 \%$ AP and $0 . 7 \%$ AP improvements over BYOL for object detection and instance segmentation, respectively. The improvements are consistent over all evaluation metrics. When pre-trained on the $\mathrm { C O C O + }$ dataset, ORL again outperforms BYOL. It should be well noted that ORL even outperforms the most recent Self-EMD and DenseCL that are specifically designed for dense prediction downstream tasks. More importantly, either COCO or $\mathrm { C O C O + }$ pre-trained ORL can surpass the supervised ImageNet pre-training on all metrics. This further demonstrates the superiority of learning unsupervised representations at the object level.
+
+Table 5: Ablations for ORL. (a) Effect of intra-RoI and inter-RoI losses. (b) Effect of NNs and RoI pairs. (c) Comparison with multi-crop BYOL. (d) Comparison with ground truth bounding boxes. (e) Pre-trainig schedules. We report mAP of linear SVMs on VOC07 classification benchmark.   
+
+<table><tr><td colspan="5">(a)</td><td colspan="9">(b)</td></tr><tr><td colspan="2">pre-train</td><td>intra-RoI</td><td>inter-RoI</td><td>VOC07</td><td></td><td colspan="4"># of NNs</td><td colspan="4"># of RoI pairs</td></tr><tr><td colspan="2">BYOL</td><td></td><td></td><td>84.5 85.7</td><td></td><td></td><td>1</td><td>10</td><td></td><td>20</td><td>5%</td><td>10%</td><td>20%</td></tr><tr><td colspan="3">ORL</td><td>√ √</td><td>85.9</td><td></td><td>VOC07</td><td>84.7</td><td></td><td>86.7</td><td>87.0</td><td>86.1</td><td>86.7</td><td>86.4</td></tr><tr><td colspan="3"></td><td></td><td>86.7</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">(C）</td><td>(d)</td><td></td><td></td><td></td><td></td><td>(e)</td><td></td><td></td><td></td></tr><tr><td colspan="2">pre-train</td><td>input views</td><td>VOC07</td><td></td><td>boxes</td><td>VOC07</td><td>pre-train</td><td></td><td>100</td><td>200</td><td>400</td><td>800</td><td>1600</td></tr><tr><td colspan="2">BYOL</td><td>2 × 224+4×96</td><td>84.0</td><td></td><td>GT</td><td>85.4</td><td>BYOL</td><td>77.1</td><td></td><td>81.8</td><td>83.7</td><td>84.5</td><td>84.9</td></tr><tr><td colspan="2">ORL</td><td>2 × 224+4×96</td><td>86.7</td><td></td><td>SS</td><td>86.7</td><td>ORL</td><td></td><td>83.5</td><td>85.2</td><td>86.3</td><td>86.7</td><td>87.1</td></tr></table>
+
+# 4.2 Ablation study
+
+In this subsection, we conduct extensive ablation experiments to examine the effect of each component that contributes to ORL. We pre-train our models on COCO and observe the downstream performance of all ablations on VOC07 SVM classification benchmark as introduced in Section 4.1.
+
+Effect of intra-RoI and inter-RoI losses. Table 5a ablates the effect of our introduced $\mathcal { L } _ { \mathrm { i n t r a - R o I } }$ and $\mathcal { L } _ { \mathrm { i n t e r - R o I } }$ losses in Equation 2. Adding either $\mathcal { L } _ { \mathrm { i n t r a - R o I } }$ or $\mathcal { L } _ { \mathrm { i n t e r - R o I } }$ can improve the performance, with the best results obtained by adding both terms.
+
+Effect of nearest neighbors and RoI pairs. Table 5b ablates the effect of the number of nearest neighbors $K$ and RoI pairs $N$ used for generating inter-RoI pairs in Stage 2 of ORL. We set $N = 1 0 \%$ when ablating $K$ , and set $K = 1 0$ when ablating $N$ . We observe that retrieving more nearest neighbors leads to better performance since more nearest neighbors provide more diverse image-level pairs to the subsequent generation of inter-RoI pairs. Although setting $K = 2 0$ produces a slightly better performance, we choose $K = 1 0$ by default as a trade-off considering the tendency of the saturated performance. Our method is more robust to the number of retrieved top-ranked RoI pairs after image-level nearest-neighbor retrieval, with $N = 1 0 \%$ performing slightly better.
+
+Comparison with multi-crop BYOL. Prior work [4] has indicated that cropping multiple views of the same image can improve the performance of self-supervised learning methods pre-trained on ImageNet. To investigate whether our gains are due to more accurate object-instance comparison or simply more number of mixed views, we randomly crop four additional smaller views for BYOL to ensure the number and size of the input patches are equal to ORL (i.e., $2 \times 2 2 4 + 4 \times 9 6 )$ . As shown in Table 5c, different from the observation on ImageNet, simply adding more low-resolution crops tends to hurt the performance since it will further intensify the inconsistent noise on scene images. In contrast, ORL substantially outperforms this multi-crop variant, validating that the gains are truly due to our object-level representation learning mechanism.
+
+Comparison with ground truth bounding boxes. In Stage 2, ORL requires an unsupervised region proposal algorithm to extract approximate object-based regions, which is inaccurate to some extent. We further investigate whether the performance can be improved when more accurate object regions are available, i.e., with bounding box annotations. To this end, we replace our selective-search generated object proposals with ground truth bounding boxes provided from COCO train2017 set, while keeping all other procedures unchanged. As shown in Table 5d, adopting ground truth (GT) bounding boxes performs inferior to selective search (SS). This is mainly due to that although the ground truth bounding boxes can provide more accurate object location, their numbers are too scarce compared with a large amount of region proposals generated by selective search. The more diverse region proposals can potentially induce more unknown object or object-part discovery beyond the manually annotated objects. In this case, the diversity can make up for the inaccuracy.
+
+![](images/d5a97181c8fbf71ada6b1e82ebc6e9e6d7847945aa7ef08265f4a8ce01c87937.jpg)  
+Figure 3: Top-ranked region correspondence discovered by ORL in Stage 2. We show a pair of discovered object-instance per image pair for clarity. More discovered correspondence pairs are provided in the supplementary material.
+
+![](images/c987186734ba1c985d90dfb47d4ca86f92feb602cb9747213e1d53d835617045.jpg)  
+Figure 4: Attention maps generated by BYOL and ORL. ORL can activate more object regions and produce more accurate object boundary in the heatmap than BYOL. We provide more attention maps in the supplementary material. Best viewed with zoom in.
+
+Pre-training schedules. Table 5e shows the results with different pre-training schedules, from 100 epochs to 1600 epochs. The performance of both ORL and BYOL improves when pre-trained for longer epochs, while ORL consistently outperforms BYOL by at least $2 . 2 \%$ mAP. Note that our 200-epoch ORL has already surpassed the 1600-epoch BYOL $8 5 . 2 \%$ mAP vs. $8 4 . 9 \%$ mAP), demonstrating that the performance efficiency of ORL is at least $8 \times$ than BYOL below 1600 epochs.
+
+# 4.3 Visualization
+
+Correspondence pairs. Figure 3 visualizes some top-ranked region correspondence discovered in Stage 2 of ORL. We observe that each generated inter-RoI pair largely correspond to the regions with similar visual concepts (i.e., objects or object parts) across images. In contrast to typical contrastive learning methods that perform aggressive intra-image augmentations to simulate intra-class variances, our discovered inter-RoI pairs can substantially provide more natural and diverse intra-class variances at the object-instance level.
+
+Attention maps. Figure 4 visualizes the attention maps generated by BYOL and ORL. We observe that both BYOL and ORL can produce relatively high-quality attention maps that focus on the foreground objects. This reflects from the side that current image-level contrastive learning methods have already induced a latent space with rich visual concepts. Nevertheless, ORL can activate more object regions and produce more accurate object boundary than BYOL in the generated attention maps. It is mainly due to introducing object-level similarity learning into ORL, which can minimize the inconsistent noise caused by image-level contrastive learning. In contrast, BYOL only uses the whole image to extract features, thus activating the most discriminative region.
+
+# 5 Conclusion
+
+In this work, we have presented a new self-supervised learning framework, ORL, for object-level representation learning from scene images. We leverage the latent prior of image-level self-supervised pre-training for discovering object-based region correspondence across images. The generated objectinstance correspondence enables us to perform pairwise contrastive learning at the object level. ORL significantly improves the performance of self-supervised learning from scene images in a variety of downstream tasks. We expect that our method can be applied to larger-scale unlabeled data in the wild to fully realize its potential, and hope that our study can attract the community’s attention to more general-purpose unsupervised representation learning from scene images.
+
+# Limitations
+
+In this paper, we mainly perform pre-training experiments with ResNet-50 on COCO dataset, and further scale them up on $\mathrm { C O C O + }$ dataset. However, the promise of self-supervised learning is to harness massive unlabeled data by scaling up to ever-larger datasets. Some prior works [18, 3, 16] have attempted to leverage larger models and datasets to explore the limit of current self-supervised learning methods. For instance, a recent representative work SEER [16] performs billion-scale selfsupervised pre-training on internet images using the RegNet architectures [46] with 700M parameters over 512 GPUs. Training at scale requires huge computational resources that are inaccessible to many researchers, which is not the core of our paper. We wish to highlight that our general-purpose ORL has yielded better performance than concurrent works [58, 34] that are tailored for dense prediction downstream tasks when pre-trained on COCO (Table 4), even surpassing the supervised ImageNet pre-training on several downstream tasks (Table 1-4). We expect that scaling ORL with larger architectures and datasets can further unleash its potential. Besides, ORL may not handle well on images with cluttered backgrounds since they will deviate the generated proposals to focus on these background regions. A possible remedy is to use some heuristic algorithms like saliency estimation to avoid the background regions. Another limitation is that ORL is a multi-stage framework. We expect an end-to-end framework to further improve the efficiency. We leave these explorations to future work.
+
+# Broader impact
+
+We present a more effective approach for learning unsupervised visual representations. Compared to supervised learning, it can liberate humans from expensive annotations as well as take advantages of rapidly growing real-world data. Like other learning algorithms, self-supervised learning should be applied with cautions when deployed in the real-world scenario. First, it is susceptible to biased learning if the algorithm is given with biased data. The exposure to unlabeled data may amplify such biases. Thus, debiasing measures have to be taken. Second, it remains non-trivial to dissect what is learned by self-supervised models. Similar concerns about the calibration, robustness, and interpretability of supervised models are equally applicable to the unsupervised counterpart. Our work is limited to the improvement of self-supervised learning within our scope. However, we acknowledge the importance of providing more transparent explanations for classification decisions, as well as the credibility of each prediction. Finally, our method still relies on the traditional regime of centralized learning. Privacy can be compromised if the method is applied on an unsaved platform. Federated learning can be a solution. How to scale self-supervised learning to the regime of decentralized learning will be an interesting research question to answer.
+
+# Acknowledgements
+
+This study is supported under the RIE2020 Industry Alignment Fund – Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s). The project is also supported by Singapore MOE AcRF Tier 2 (T2EP20120-0005), the Data Science and Artificial Intelligence Research Center at NTU.
+
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parse/train/X2K8KVEaAXG/X2K8KVEaAXG_content_list.json

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+    {
+        "type": "text",
+        "text": "Unsupervised Object-Level Representation Learning from Scene Images ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Jiahao Xie1 Xiaohang Zhan2 Ziwei Liu1 Yew Soon Ong1,3 Chen Change Loy1 ",
+        "bbox": [
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+        "text": "1Nanyang Technological University 2The Chinese University of Hong Kong 3A\\*STAR, Singapore {jiahao003, ziwei.liu, asysong, ccloy}@ntu.edu.sg xiaohangzhan@outlook.com ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Contrastive self-supervised learning has largely narrowed the gap to supervised pre-training on ImageNet. However, its success highly relies on the object-centric priors of ImageNet, i.e., different augmented views of the same image correspond to the same object. Such a heavily curated constraint becomes immediately infeasible when pre-trained on more complex scene images with many objects. To overcome this limitation, we introduce Object-level Representation Learning (ORL), a new self-supervised learning framework towards scene images. Our key insight is to leverage image-level self-supervised pre-training as the prior to discover object-level semantic correspondence, thus realizing object-level representation learning from scene images. Extensive experiments on COCO show that ORL significantly improves the performance of self-supervised learning on scene images, even surpassing supervised ImageNet pre-training on several downstream tasks. Furthermore, ORL improves the downstream performance when more unlabeled scene images are available, demonstrating its great potential of harnessing unlabeled data in the wild. We hope our approach can motivate future research on more general-purpose unsupervised representation learning from scene data.1 ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
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+        "type": "text",
+        "text": "Unsupervised visual representation learning aims at obtaining transferable features with abundant unlabeled data. Recent self-supervised learning (SSL) methods based on contrastive learning [60, 22, 37, 5, 19, 4, 7] have largely narrowed the gap and even surpassed the supervised counterpart on a number of downstream tasks [30, 49, 15, 47, 35, 23]. These methods build upon the instance discrimination task that maximizes the agreement between different data-augmented views of the same image. Despite their success, current SSL methods are primarily pre-trained on the unlabeled ImageNet [8] dataset that contains iconic images with single object as shown in Figure 1(a). The underlying object-centric constraint of ImageNet makes it hard to be applied in real world scenarios where more complex scene images with multiple objects are available. Meanwhile, naïvely adopting the off-the-shelf contrastive learning methods on scene images introduces inconsistent learning signals since random crops of the same image may correspond to different objects as shown in Figure 1(b). Indeed, it has been shown that current contrastive learning methods tend to struggle on more complex scene datasets [19, 50, 34, 58] like COCO [33] or Places365 [72]. Therefore, it is imperative to design an effective object-level representation learning paradigm as illustrated in Figure 1(c) to harness massive unlabeled scene images in the wild. ",
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+    {
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+        "img_path": "images/a919e3ea35f294be21ec244aa7ce625c2450aca7f8f498519807ea147f589e62.jpg",
+        "image_caption": [
+            "",
+            "Figure 1: (a) Current image-level contrastive learning methods heavily rely on the object-centric bias of ImageNet, i.e., different crops correspond to the same object. Prior works use either the different views of the same image [60, 22, 37, 5, 19, 7] (i.e., intra-image) or similar images [74, 63, 1, 11] (i.e., inter-image) to form positive pairs. (b) Directly adopting image-level contrastive learning methods on scene images can cause inconsistent learning signals since different crops may correspond to different objects. (c) Object-level contrastive learning can overcome the limitation in (b) by enforcing object-level consistency. (d) We find that image-level contrastive learning encodes priors for region correspondence discovery across images, and high-response regions are usually objects or object parts (we show one discovered object-instance pair per image pair for clarity), which is useful for object-level representation learning. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In this work, we are interested in going beyond ImageNet to obtain better representations on noniconic images. Apparently, it is challenging to learn representations from scene-level images since they are entangled with many concepts including structures, objects, backgrounds and relationships. It remains an open question how to take advantages of spatial information of multiple objects naturally residing in the scene images when no object annotations are available, let alone further deriving object-level correspondence to construct positive object-instance pairs. ",
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+    },
+    {
+        "type": "text",
+        "text": "To tackle these challenges, we introduce a novel object-level unsupervised representation learning framework tailored for scene images. Our framework is based on a key insight of the current contrastive learning methods: they can implicitly group different images with similar visual concepts together even though they are explicitly optimized to group different views of the same image. This phenomenon reveals that image-level contrastive learning has already induced a latent space with rich visual concepts. Though the latent space usually entangles other scene concepts like structures, backgrounds and relationships, it will be useful for object discovery if appropriately deployed. Through computing the similarity of sampled regions between $k$ -nearest-neighbor (KNN) images, we conclude two observations: 1) image-level contrastive learning encodes priors for region correspondence discovery across images; 2) high-response regions are usually objects or object parts. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Based on the observation above, we propose a multi-stage framework for unsupervised object-level representation learning. Specifically, we first extract potential object-based regions in scene images using the unsupervised region proposal algorithms (e.g., selective search [56]). We then propose a region correspondence generation scheme to leverage the off-the-shelf image-level contrastive learning pre-trained model to discover corresponding object-instance pairs for the proposed regions in the embedding space. Finally, we use the obtained object-instance pairs to construct positive sample pairs for object-level representation learning. Figure 1(d) shows several cross-image object-instance pairs discovered by our framework on COCO dataset using the latent prior of BYOL [19], the state-of-the-art image-level contrastive learning method. The discovered inter-corresponding pairs substantially provide diverse intra-class variances at the object-instance level to aid object-level representation learning. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Overall, our main contributions are summarized as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1) We observe that existing image-level contrastive learning methods have priors to discover objectlevel correspondence across images. We leverage this prior for the first time for unsupervised cross-image object-level correspondence discovery. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2) With the obtained correspondence, we introduce a novel multi-stage self-supervised learning pipeline, termed as ORL, for object-level representation learning from scene images, going beyond object-centric ImageNet. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3) We contribute the first study for object-level SSL. ORL substantially outperforms image-level contrastive learning approaches pre-trained on COCO dataset $\\mathord { \\sim } 1 1 8 \\mathrm { k }$ images with labels discarded), setting a new state of the art on this challenging dataset that contains diverse scenes in the wild. The COCO pre-trained ORL even surpasses supervised ImageNet pre-training on several considered downstream tasks. When SSL is conducted on a larger ${ } ^ { 6 6 } \\mathrm { C O C O + } { } ^ { , }$ dataset (COCO train2017 set plus COCO unlabeled2017 set, ${ \\sim } 2 4 1 \\mathrm { k }$ images in total), ORL further improves the performance, demonstrating its potential to benefit from more unlabeled scene data. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised learning. Self-supervised learning builds unsupervised representations by exploiting the internal priors or structures of data in the form of a pretext task. A wide range of pretext tasks have been proposed in the past few years. Examples include patch context prediction [10], jigsaw puzzles [39], inpainting [43], colorization [31, 70], cross-channel prediction [71], visual primitive counting [40], and rotation prediction [14]. Although good representations emerge with these pretext tasks, they are prone to lose generality due to their hand-crafted nature. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, contrastive learning [20] that performs instance discrimination [60, 22, 37, 5, 19, 4, 7] has shown great potential in this field, largely narrowing the gap to fully supervised learning. The core idea of contrastive learning is to gather positive pairs and separate negative pairs in the embedding space. A positive pair is usually formed with two transformed views of the same image while the negative pairs are formed with different images. Typically, contrastive learning methods require a large number of negative samples to avoid mode collapse. These samples can be maintained within a mini-batch [42, 27, 66, 26, 2, 5], a memory bank [60, 53, 74, 37] or a queue [22, 6]. BYOL [19] and SwAV [4] further remove the necessity of involving negative pairs. BYOL directly predicts the features of one view from another view, while SwAV predicts the cluster assignments between multiple views of the same image. Despite their improved performance, the existing image-level contrastive learning methods are largely confined to the underlying object-centric bias of ImageNet. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "More recently, a group of works that perform pixel-level [45, 58, 64, 50, 34, 25] or region-level [48, 65, 61, 62, 9] representation learning have emerged. Our work is more related to region-level representation learning but substantially different from this line of research in the following aspects: 1) they still largely pre-train on object-centric ImageNet while we pre-train on non-iconic scene images, 2) they align pre-training specifically for dense prediction downstream tasks while we target at more general-purpose representation learning that improves performance in both dense prediction and classification tasks, 3) their randomly cropped local regions do not contain the explicit object notion as ours, and 4) they only rely on intra-image transformations (e.g., random cropping) to construct corresponding positive pairs from the same image while we leverage the discovered high-level semantic correspondence to construct positive pairs across images. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are also a few prior attempts [18, 3, 16] for self-supervised learning on non-curated scene images. As opposed to our work, most of them consider larger models and datasets to explore the limit of current self-supervised learning methods without further considering the object-level information residing in scene images. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Visual correspondence. Visual correspondence aims at finding pairwise pixels or regions across images that result from the same scene [67], which can be regarded as similarity learning of visual descriptors among matched points or patches. While early efforts learn dense correspondence with labeled data [21, 68, 29, 55, 41], some recent works learn the similarity between the parts or landmarks of the data in an unsupervised manner [52, 51]. Our work substantially differs from this line of research from original intention. Previous works aim at accurately detecting all correspondence given two images, whereas our work focuses on retrieving high-quality correspondence to improve representation learning. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/575ab9e1793d07665088f0a09cc6dc9ce09301fcb4142d1dae60de314b455d89.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our three-stage pipeline. In Stage 1, we pre-train an image-level contrastive learning model, e.g., BYOL. In Stage 2, we first use the pre-trained model to retrieve KNNs for each image in the embedding space to obtain image-level visually similar pairs. We then use unsupervised region proposal algorithms (e.g., selective search) to generate rough RoIs for each image pair. Afterwards, we reuse the pre-trained model to retrieve the top-ranked RoI pairs, i.e., correspondence. We find these pairs of RoIs are almost objects or object parts. In Stage 3, with the corresponding RoI pairs discovered across images, we finally perform object-level contrastive learning using the same architecture as Stage 1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            426,
+            313,
+            444
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We propose a new multi-stage self-supervised learning framework, i.e., ORL, for object-level representation learning from scene images. ORL extends the existing image-level contrastive learning framework to object level by leveraging priors from image-level instance discrimination. The overall pipeline of ORL is illustrated in Figure 2. It contains three stages: image-level pre-training, correspondence discovery, and object-level pre-training. We detail each stage as follows. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 ORL pipeline ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            549,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Preliminary: Contrastive learning. Our pipeline contains several contrastive learning modules in Stage 1 and 3. Without loss of generality, we consider BYOL [19] as our basic contrastive learning module. BYOL uses two neural networks: the online network $f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } )$ and the target network $g _ { \\xi } ( x )$ The target network provides the regression target to train the online network while its weights $\\xi$ are updated by an exponential moving average of the online parameters $\\theta$ with a decay rate $\\tau \\in [ 0 , 1 ]$ following BYOL. Given two input images $x _ { 1 }$ and $x _ { 2 }$ , the loss function is defined as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e7be6fedccbb421f42702baa8175d74ad7093ae0f57078396f906d7963239c18.jpg",
+        "text": "$$\n\\mathcal { L } \\left( \\boldsymbol { x } _ { 1 } , \\boldsymbol { x } _ { 2 } \\right) \\triangleq \\left. \\boldsymbol { f } _ { \\boldsymbol { \\theta } } \\left( \\boldsymbol { x } _ { 1 } \\right) - \\boldsymbol { g } _ { \\boldsymbol { \\xi } } \\left( \\boldsymbol { x } _ { 2 } \\right) \\right. _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We name it an “intra-” version of BYOL if $x _ { 1 }$ and $x _ { 2 }$ are two augmented views from the same image, otherwise an “inter-” one. ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Stage 1: Image-level pre-training. The foremost stage is to obtain an unsupervised pre-trained model from image-level tasks. As shown in Figure 2 Stage 1, given two augmented views $v$ and $v ^ { \\prime }$ from the same input image $x$ , we pre-train the network following the loss function ${ \\mathcal { L } } _ { \\mathrm { i m a g e } } = { \\mathcal { L } } \\left( v , v ^ { \\prime } \\right)$ constituting a standard image-level BYOL pre-training. This stage can be freely replaced with other image-level contrastive learning methods. We adopt BYOL here for its simplicity and effectiveness. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Stage 2: Correspondence discovery. We employ the pre-trained image-level contrastive learning model in Stage 1 to mine object-level correspondence for the whole dataset. As shown in Figure 2 Stage 2, the overall discovery process comprises three steps. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(i) Image-level nearest-neighbor retrieval. Specifically, for each query image $x$ in the training set $\\mathcal { D }$ , we first retrieve its top $K$ nearest neighbors $\\mathcal { N } _ { k }$ , $k = 1 , . . . , K$ , by cosine distance in the embedding space using the features learned from the first stage to form image-level pairs that contain similar visual context. ",
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+    {
+        "type": "text",
+        "text": "(ii) Region-of-interest (RoI) generation. To generate object-based RoIs, we apply unsupervised region proposal algorithms, e.g., selective search [56], for each image in the pair. Considering the redundancy of generated proposals (each image can have thousands of proposals), we filter certain number of them with some pre-defined thresholds2 including the minimal scale, the range of aspect ratio, and the maximal intersection-over-union (IoU) among the filtered boxes. After the filtering operation, we select the top 100 proposals ranked with objectiveness as the candidate RoI set for subsequent RoI pair retrieval. To extract features with the equally-sized input that is compatible with the backbone, we crop and resize each RoI to $2 2 4 \\times 2 2 4$ . Note that even top RoIs ranked with objectiveness are still very noisy, containing a large proportion of non-object regions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(iii) Top-ranked RoI pair retrieval. For each query RoI from $x$ , we compute its cosine similarity in the embedding space with all RoIs from its nearest-neighbor image $\\mathcal { N } _ { k }$ using the features learned from Stage 1 again. Within the calculated cosine similarity matrix $\\bar { \\mathbf { M } } _ { k } \\in \\mathbb { R } ^ { 1 0 0 \\bar { \\times } 1 0 0 }$ , we retrieve top-ranked $N$ RoI pairs to construct the set of object-level corresponding pairs $\\{ B _ { k } ^ { n } \\}$ , where $n = \\{ 1 , . . . , N \\}$ These high-response corresponding regions are almost objects or object parts. Finally, we save the nearest-neighbor image id and bounding box coordinate information of each corresponding pair. ",
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+    {
+        "type": "text",
+        "text": "Stage 3: Object-level pre-training. With the corresponding inter-image RoI (inter-RoI) pairs obtained in Stage 2, we perform object-level representation learning following the BYOL framework as shown in Figure 2 Stage 3. Specifically, given an input image $x$ , we first randomly select one nearest-neighbor image $\\mathcal { N } _ { k }$ to obtain the corresponding set of inter-RoI pairs $\\{ B _ { k } ^ { n } \\}$ . We then randomly select one inter-RoI pair $B _ { k } ^ { n }$ as a positive pair. With the bounding box coordinate stored in $B _ { k } ^ { n }$ , we crop the corresponding inter-RoIs from $x$ and $\\mathcal { N } _ { k }$ , respectively, and resize each patch to $9 \\ddot { 6 } \\times 9 6$ , constituting two patches $p _ { 1 }$ and $p _ { 2 }$ . We feed the two patches to the online network and target network separately to compute the loss $\\mathcal { L } _ { \\mathrm { i n t e r - R o I } } = \\mathcal { L } \\left( p _ { 1 } , p _ { 2 } \\right)$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To make full use of discovered objects, we introduce the intra-RoI contrastive learning via augmenting object patches. Specifically, we randomly select one filtered bounding box from $x$ obtained in Stage 2, and spatially jitter the box around its original location with the following operations3: (i) a random box center shifting within $50 \\%$ of its width and height, (ii) a random area scaling between $50 \\%$ and $200 \\%$ of the original box, and (iii) a random aspect ratio between $1 / 2$ and $2 / 1$ . Similarly, we crop the two intra-RoIs $p$ and $p ^ { \\prime }$ , and resize each patch to $9 6 \\times 9 6$ for forward propagation to compute the loss $\\mathcal { L } _ { \\mathrm { \\scriptsize { i n t r a - R o I } } } = \\mathcal { L } \\left( p , p ^ { \\prime } \\right)$ . The diverse spatial jittering of the bounding box encourages the network to preserve common object information and disregard the background, thus further improving the localization ability. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We keep the two original global views in BYOL as well since they preserve the global image-level information compared with the local patches. The final loss for our ORL can thus be formulated as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/3e91f34a0a8de980d270b553ab1800cb5a35d4de4a0244fa030e71fc1055e48a.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { O R L } } = \\lambda _ { 1 } \\mathcal { L } _ { \\mathrm { i m a g e } } + \\lambda _ { 2 } \\mathcal { L } _ { \\mathrm { i n t r a - R o I } } + \\lambda _ { 3 } \\mathcal { L } _ { \\mathrm { i n t e r - R o I } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda _ { 1 } , \\lambda _ { 2 } , \\lambda _ { 3 }$ are the loss weights to balance each term. We set all loss weights to 1 by default. Following BYOL, we also compute the symmetric loss $\\widetilde { \\mathcal { L } } _ { \\mathrm { { o R L } } }$ by separately feeding $v ^ { \\prime } , p ^ { \\prime } , p _ { 2 }$ to the online network and $v , p , p _ { 1 }$ to the target network. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.2 Implementation details ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Dataset. We pre-train our models on the COCO train2017 set that contains ${ \\sim } 1 1 8 \\mathrm { k }$ images without using labels. Compared with the heavily curated object-centric ImageNet dataset, COCO contains more natural and diverse scenes in the wild, which is closer to real-world scenarios. We also perform self-supervised learning on a larger ${ } ^ { \\cdot \\cdot } \\mathrm { C O C O } { + } ^ { \\prime \\cdot }$ dataset (COCO train2017 set plus COCO unlabeled2017 set) to verify whether our method can benefit from more unlabeled scene data. ",
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+    {
+        "type": "text",
+        "text": "Image augmentations. The global image augmentation setting is the same as BYOL [19]: a $2 2 4 \\times 2 2 4$ -pixel random resized crop with a random horizontal flip, followed by a random color distortion, random grayscale conversion, random Gaussian blur and solarization. For the local patch augmentation, we directly crop the corresponding intra-RoI and inter-RoI on the input images, and resize each cropped patch to $9 6 \\times 9 6$ to take place of the random resized cropping. The subsequent augmentations exactly follow the global ones. ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Network architecture. We adopt ResNet-50 [24] as the default backbone. We use the same MLP projector and predictor as in BYOL: a linear layer with output size 4096 followed by batch normalization (BN) [28], rectified linear units (ReLU) [38], and a final linear layer with output dimension 256. We share the backbone and projector weights among the global and two local branches while the weights of predictor are not shared. ",
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+    {
+        "type": "text",
+        "text": "Optimization. For pre-training in Stage 1 and Stage 3, we use the same training hyper-parameters. Specifically, we use the SGD optimizer with a weight decay of 0.0001 and a momentum of 0.9. We adopt the cosine learning rate decay schedule [36] with a base learning rate of 0.2, linearly scaled[17] with the batch size $l r = 0 . 2 \\times \\mathrm { ~ F ~ }$ atchSize/256). The batch size is set to 512 by default, which is friendly to typical 8-GPU implementations. To keep the training iterations comparable with the ImageNet supervised pre-training, we train our models for 800 epochs with a warm-up period of 4 epochs. The exponential moving average parameter $\\tau$ starts from 0.99 and is increased to 1 during training, following [19]. For correspondence generation in Stage 2, we retrieve top $K = 1 0$ nearest neighbors for each image and select top-ranked $N = 1 0 \\%$ RoI pairs for each image-level nearest-neighbor pair. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Transferring to downstream tasks ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate the quality of learned representations by transferring them to multiple downstream tasks. Following common protocol [18, 37], we use two evaluation setups: (i) the pre-trained network is frozen as a feature extractor, and (ii) the network parameters are fully fine-tuned as weight initialization. We provide more experimental details in the supplementary material. ",
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+    {
+        "type": "table",
+        "img_path": "images/99b0c9fdc50fc7ab2febd925a8844d152818cdb870885606acd9df8bd5e23ca2.jpg",
+        "table_caption": [
+            "Table 1: Image classification with linear models. All unsupervised methods are based on 800-epoch pretraining on $\\mathrm { C O C O ( + ) }$ with ResNet-50. We report mAP on the VOC07 dataset and top-1 center-crop accuracy on all other datasets. Numbers for all other methods are reproduced by us. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Pre-train data</td><td>VOC07 mAP</td><td>ImageNet Top-1</td><td>Places205 Top-1</td><td>iNat. Top-1</td></tr><tr><td>Random [18]</td><td>=</td><td>9.6</td><td>13.7</td><td>16.6</td><td>4.8</td></tr><tr><td>Supervised [37]</td><td>ImageNet</td><td>87.5</td><td>75.9</td><td>51.5</td><td>45.4</td></tr><tr><td>SimCLR [5]</td><td>COCO</td><td>78.1</td><td>50.9</td><td>48.0</td><td>22.7</td></tr><tr><td>MoCo v2 [6]</td><td>COCO</td><td>82.2</td><td>55.1</td><td>48.8</td><td>27.8</td></tr><tr><td>BYOL[19]</td><td>COCO</td><td>84.5</td><td>57.8</td><td>50.5</td><td>29.5</td></tr><tr><td>ORL (ours)</td><td>COCO</td><td>86.7</td><td>59.0</td><td>52.7</td><td>31.8</td></tr><tr><td>BYOL [19]</td><td>COCO+</td><td>87.0</td><td>59.6</td><td>52.7</td><td>30.9</td></tr><tr><td>ORL (ours)</td><td>COCO+</td><td>88.6</td><td>60.7</td><td>54.1</td><td>32.0</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4e04116bf986630ffe896e51a52276ca0887002b0c3d912e58c6ad73ce0f2eba.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Pre-train data</td><td colspan=\"7\">VOC07 low-shot (mAP)</td></tr><tr><td>1</td><td>2</td><td>4</td><td>8</td><td>16</td><td>32</td><td>64</td></tr><tr><td>Random</td><td>=</td><td>9.2</td><td>9.4</td><td>11.1</td><td>12.3</td><td>14.3</td><td>17.4</td><td>21.3</td><td>23.8</td></tr><tr><td>Supervied</td><td>ImageNet</td><td>53.0</td><td>63.6</td><td>73.7</td><td>78.8</td><td>81.8</td><td>83.8</td><td>85.2</td><td>86.0</td></tr><tr><td>SimCLR [5] MoCo v2 [6]</td><td>CoCo</td><td>33.3</td><td>43.5</td><td>52.5</td><td>61.1</td><td>66.7</td><td>70.5</td><td>73.7</td><td>75.0</td></tr><tr><td>BYOL [19]</td><td>COCO</td><td>39.5</td><td>49.3</td><td>60.4</td><td>69.3</td><td>74.1</td><td>76.8</td><td>79.1</td><td>80.1</td></tr><tr><td>ORL (ours)</td><td>COCO COCO</td><td>39.4</td><td>50.9</td><td>62.2</td><td>71.7</td><td>76.6</td><td>79.2</td><td>81.3</td><td>82.2</td></tr><tr><td></td><td></td><td>39.6</td><td>51.2</td><td>63.4</td><td>72.6</td><td>78.2</td><td>81.3</td><td>83.6</td><td>84.7</td></tr><tr><td>BYOL [19] ORL (ours)</td><td>COCO+</td><td>41.1</td><td>54.3</td><td>66.6</td><td>75.2</td><td>80.1</td><td>82.6</td><td>84.6</td><td>85.4</td></tr><tr><td></td><td>COCO+</td><td>42.1</td><td>54.9</td><td>67.4</td><td>75.7</td><td>81.3</td><td>83.7</td><td>85.8</td><td>86.7</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Low-shot image classification on VOC07 using linear SVMs trained on the fixed representations. All unsupervised methods are pre-trained on $\\mathrm { C O C O ( + ) }$ for 800 epochs with ResNet-50. We report mAP for each case across five runs. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Image classification with linear models. Following [18, 37], we assess the quality of features by training linear classifiers on top of the fixed representations extracted from different depths of the network for four datasets: VOC07 [12], ImageNet [8], Places205 [73], and iNaturalist18 [57]. These datasets involve diverse classification tasks ranging from object classification, scene recognition to fine-grained recognition. For VOC07, we train linear SVMs using LIBLINEAR package [13] ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f243b084645bffaa487cd0b8325012d99433e9be1ef49016b7cbf0b3e1dffb88.jpg",
+        "table_caption": [
+            "Table 3: Semi-supervised learning on ImageNet. All unsupervised methods are pre-tained on $\\mathrm { C O C O ( + ) }$ for 800 epochs with ResNet-50. We fine-tune all models with $1 \\%$ and $10 \\%$ ImageNet labels, and report both top-1 and top-5 center-crop accuracy on the ImageNet validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Pre-train</td><td colspan=\"2\">1% labels</td><td colspan=\"2\">10% labels</td></tr><tr><td></td><td>data</td><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Random</td><td></td><td>1.6</td><td>5.0</td><td>21.8</td><td>44.2</td></tr><tr><td>Supervised [69]</td><td>ImageNet</td><td>25.4</td><td>48.4</td><td>56.4</td><td>80.4</td></tr><tr><td>SimCLR [5]</td><td>CoCo</td><td>23.4</td><td>46.4</td><td>52.2</td><td>77.4</td></tr><tr><td>MoCo v2 [6]</td><td>COCO</td><td>28.2</td><td>54.7</td><td>57.1</td><td>81.7</td></tr><tr><td>BYOL[19]</td><td>COCO</td><td>28.4</td><td>55.9</td><td>58.4</td><td>82.7</td></tr><tr><td>ORL (ours)</td><td>CoCo</td><td>31.0</td><td>58.9</td><td>60.5</td><td>84.2</td></tr><tr><td>BYOL[19]</td><td>COCO+</td><td>28.3</td><td>56.0</td><td>59.4</td><td>83.6</td></tr><tr><td>ORL (ours)</td><td>COCO+</td><td>31.8</td><td>60.1</td><td>60.9</td><td>84.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/aea583fe5bce155691e70a6e5bf8d32a20db1cef2bce29118a16159308bbf598.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Pre-train data</td><td colspan=\"3\">COCO detection</td><td colspan=\"3\">COCO instance seg.</td></tr><tr><td>Ap66</td><td>AP</td><td>AP</td><td>Apmk</td><td>APk</td><td>AP</td></tr><tr><td>Random [54]</td><td></td><td>32.8</td><td>50.9</td><td>35.3</td><td>29.9</td><td>47.9</td><td>32.0</td></tr><tr><td>Supervised [54]</td><td>ImageNet</td><td>39.7</td><td>59.5</td><td>43.3</td><td>35.9</td><td>56.6</td><td>38.6</td></tr><tr><td>SimCLR [5]</td><td>COCO</td><td>37.0</td><td>56.8</td><td>40.3</td><td>33.7</td><td>53.8</td><td>36.1</td></tr><tr><td>MoCo v2 [6]</td><td>COCO</td><td>38.5</td><td>58.1</td><td>42.1</td><td>34.8</td><td>55.3</td><td>37.3</td></tr><tr><td>Self-EMD [34]</td><td>COCo</td><td>39.3</td><td>60.1</td><td>42.8</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DenseCL [58]</td><td>COCo</td><td>39.6</td><td>59.3</td><td>43.3</td><td>35.7</td><td>56.5</td><td>38.4</td></tr><tr><td>BYOL[19]</td><td>COCO</td><td>39.5</td><td>59.3</td><td>43.2</td><td>35.6</td><td>56.5</td><td>38.2</td></tr><tr><td>ORL (ours)</td><td>COCO</td><td>40.3</td><td>60.2</td><td>44.4</td><td>36.3</td><td>57.3</td><td>38.9</td></tr><tr><td>BYOL[19]</td><td>COCO+</td><td>40.0</td><td>60.1</td><td>44.0</td><td>36.2</td><td>57.1</td><td>39.0</td></tr><tr><td>ORL (ours)</td><td>COCO+</td><td>40.6</td><td>60.8</td><td>44.5</td><td>36.7</td><td>57.9</td><td>39.3</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Object detection and instance segmentation fine-tuned on COCO. All unsupervised methods are based on 800-epoch pre-training on $\\mathrm { C O C O ( + ) }$ . We use Mask R-CNN R50-FPN ( $1 \\times$ schedule), and report bounding-box AP $( \\mathrm { A } \\mathbf { \\hat { P } } ^ { b b } )$ and mask AP $( \\mathbf { A } \\mathbf { P } ^ { m k } )$ . Numbers for MoCo v2 are adopted from [58]. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "following the setup in [18, 37]. We train on trainval split of VOC07 and evaluate mAP on test split. For ImageNet, Places205 and iNaturalist18, we follow [71, 18, 37] and train a 1000-way, 205-way and 8142-way linear classifier, respectively. We train on train split of each dataset, and report top-1 center-crop accuracy on the respective val split. Table 1 reports the results for the best-performing layer of each method. ORL substantially outperforms the BYOL baseline on all four datasets. We also observe that the COCO pre-trained ORL surpasses the supervised ImageNet pre-trained counterpart on Places205 by $1 . 2 \\%$ in top-1 accuracy. This is the first time that a selfsupervised learner outperforms the ImageNet pre-training using only ${ \\sim } 1 / 1 0$ images compared with ImageNet. When pre-trained on a larger $\\mathrm { C O C O + }$ dataset, ORL again outperforms BYOL. Note that apart from Places205 $2 . 6 \\%$ gains), ORL also surpasses the supervised ImageNet counterpart on VOC07 by $1 . 1 \\%$ mAP, using merely ${ \\sim } 1 / 5$ images. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Low-shot image classification. We perform low-shot image classification with few training examples per class on VOC07 dataset following the same setup in [18]. We vary the number of labeled examples per category used to train linear SVMs on train split of VOC07 and report the average mAP across five independent samples for each low-shot case evaluated on test split. Table 2 provides the results. ORL shows consistent performance improvement over BYOL for each low-shot value, with larger gains achieved as the number of labeled examples per class is increasing. ORL also gradually bridges the gap to the supervised ImageNet pre-training under this scenario. We observe consistent performance boost when pre-training on the $\\mathrm { C O C O + }$ dataset. Note that the $\\mathrm { C O C O + }$ pre-trained ORL again outperforms the supervised ImageNet pre-training when the low-shot samples are 64 and 96. ",
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+    {
+        "type": "text",
+        "text": "Semi-supervised learning. We perform semi-supervised learning on ImageNet following the protocol of previous studies [60, 26, 37, 5, 19]. Specifically, we first randomly select $1 \\%$ and $10 \\%$ labeled data from ImageNet train split. We then fine-tune our models on these two training subsets and report both top-1 and top-5 accuracy on the official val split of ImageNet in Table 3. Again, ORL outperforms BYOL as well as the supervised ImageNet counterpart by large margins. ",
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+    {
+        "type": "text",
+        "text": "Object detection and segmentation. We train a Mask R-CNN model [23] with R50-FPN backbone [32] implemented in Detectron2 [59]. We fine-tune all layers end-to-end on COCO train2017 split with the standard $1 \\times$ schedule and evaluate on COCO val2017 split. We follow the same setup in [54], with batch normalization layers synchronized across GPUs [44]. As shown in Table 4, ORL yields $0 . 8 \\%$ AP and $0 . 7 \\%$ AP improvements over BYOL for object detection and instance segmentation, respectively. The improvements are consistent over all evaluation metrics. When pre-trained on the $\\mathrm { C O C O + }$ dataset, ORL again outperforms BYOL. It should be well noted that ORL even outperforms the most recent Self-EMD and DenseCL that are specifically designed for dense prediction downstream tasks. More importantly, either COCO or $\\mathrm { C O C O + }$ pre-trained ORL can surpass the supervised ImageNet pre-training on all metrics. This further demonstrates the superiority of learning unsupervised representations at the object level. ",
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+            "Table 5: Ablations for ORL. (a) Effect of intra-RoI and inter-RoI losses. (b) Effect of NNs and RoI pairs. (c) Comparison with multi-crop BYOL. (d) Comparison with ground truth bounding boxes. (e) Pre-trainig schedules. We report mAP of linear SVMs on VOC07 classification benchmark. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"5\">(a)</td><td colspan=\"9\">(b)</td></tr><tr><td colspan=\"2\">pre-train</td><td>intra-RoI</td><td>inter-RoI</td><td>VOC07</td><td></td><td colspan=\"4\"># of NNs</td><td colspan=\"4\"># of RoI pairs</td></tr><tr><td colspan=\"2\">BYOL</td><td></td><td></td><td>84.5 85.7</td><td></td><td></td><td>1</td><td>10</td><td></td><td>20</td><td>5%</td><td>10%</td><td>20%</td></tr><tr><td colspan=\"3\">ORL</td><td>√ √</td><td>85.9</td><td></td><td>VOC07</td><td>84.7</td><td></td><td>86.7</td><td>87.0</td><td>86.1</td><td>86.7</td><td>86.4</td></tr><tr><td colspan=\"3\"></td><td></td><td>86.7</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"5\">(C）</td><td>(d)</td><td></td><td></td><td></td><td></td><td>(e)</td><td></td><td></td><td></td></tr><tr><td colspan=\"2\">pre-train</td><td>input views</td><td>VOC07</td><td></td><td>boxes</td><td>VOC07</td><td>pre-train</td><td></td><td>100</td><td>200</td><td>400</td><td>800</td><td>1600</td></tr><tr><td colspan=\"2\">BYOL</td><td>2 × 224+4×96</td><td>84.0</td><td></td><td>GT</td><td>85.4</td><td>BYOL</td><td>77.1</td><td></td><td>81.8</td><td>83.7</td><td>84.5</td><td>84.9</td></tr><tr><td colspan=\"2\">ORL</td><td>2 × 224+4×96</td><td>86.7</td><td></td><td>SS</td><td>86.7</td><td>ORL</td><td></td><td>83.5</td><td>85.2</td><td>86.3</td><td>86.7</td><td>87.1</td></tr></table>",
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+        "text": "4.2 Ablation study ",
+        "text_level": 1,
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+        "text": "In this subsection, we conduct extensive ablation experiments to examine the effect of each component that contributes to ORL. We pre-train our models on COCO and observe the downstream performance of all ablations on VOC07 SVM classification benchmark as introduced in Section 4.1. ",
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+        "text": "Effect of intra-RoI and inter-RoI losses. Table 5a ablates the effect of our introduced $\\mathcal { L } _ { \\mathrm { i n t r a - R o I } }$ and $\\mathcal { L } _ { \\mathrm { i n t e r - R o I } }$ losses in Equation 2. Adding either $\\mathcal { L } _ { \\mathrm { i n t r a - R o I } }$ or $\\mathcal { L } _ { \\mathrm { i n t e r - R o I } }$ can improve the performance, with the best results obtained by adding both terms. ",
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+        "type": "text",
+        "text": "Effect of nearest neighbors and RoI pairs. Table 5b ablates the effect of the number of nearest neighbors $K$ and RoI pairs $N$ used for generating inter-RoI pairs in Stage 2 of ORL. We set $N = 1 0 \\%$ when ablating $K$ , and set $K = 1 0$ when ablating $N$ . We observe that retrieving more nearest neighbors leads to better performance since more nearest neighbors provide more diverse image-level pairs to the subsequent generation of inter-RoI pairs. Although setting $K = 2 0$ produces a slightly better performance, we choose $K = 1 0$ by default as a trade-off considering the tendency of the saturated performance. Our method is more robust to the number of retrieved top-ranked RoI pairs after image-level nearest-neighbor retrieval, with $N = 1 0 \\%$ performing slightly better. ",
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+        "text": "Comparison with multi-crop BYOL. Prior work [4] has indicated that cropping multiple views of the same image can improve the performance of self-supervised learning methods pre-trained on ImageNet. To investigate whether our gains are due to more accurate object-instance comparison or simply more number of mixed views, we randomly crop four additional smaller views for BYOL to ensure the number and size of the input patches are equal to ORL (i.e., $2 \\times 2 2 4 + 4 \\times 9 6 )$ . As shown in Table 5c, different from the observation on ImageNet, simply adding more low-resolution crops tends to hurt the performance since it will further intensify the inconsistent noise on scene images. In contrast, ORL substantially outperforms this multi-crop variant, validating that the gains are truly due to our object-level representation learning mechanism. ",
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+        "type": "text",
+        "text": "Comparison with ground truth bounding boxes. In Stage 2, ORL requires an unsupervised region proposal algorithm to extract approximate object-based regions, which is inaccurate to some extent. We further investigate whether the performance can be improved when more accurate object regions are available, i.e., with bounding box annotations. To this end, we replace our selective-search generated object proposals with ground truth bounding boxes provided from COCO train2017 set, while keeping all other procedures unchanged. As shown in Table 5d, adopting ground truth (GT) bounding boxes performs inferior to selective search (SS). This is mainly due to that although the ground truth bounding boxes can provide more accurate object location, their numbers are too scarce compared with a large amount of region proposals generated by selective search. The more diverse region proposals can potentially induce more unknown object or object-part discovery beyond the manually annotated objects. In this case, the diversity can make up for the inaccuracy. ",
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+        "img_path": "images/d5a97181c8fbf71ada6b1e82ebc6e9e6d7847945aa7ef08265f4a8ce01c87937.jpg",
+        "image_caption": [
+            "Figure 3: Top-ranked region correspondence discovered by ORL in Stage 2. We show a pair of discovered object-instance per image pair for clarity. More discovered correspondence pairs are provided in the supplementary material. "
+        ],
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+        "image_caption": [
+            "Figure 4: Attention maps generated by BYOL and ORL. ORL can activate more object regions and produce more accurate object boundary in the heatmap than BYOL. We provide more attention maps in the supplementary material. Best viewed with zoom in. "
+        ],
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+    {
+        "type": "text",
+        "text": "Pre-training schedules. Table 5e shows the results with different pre-training schedules, from 100 epochs to 1600 epochs. The performance of both ORL and BYOL improves when pre-trained for longer epochs, while ORL consistently outperforms BYOL by at least $2 . 2 \\%$ mAP. Note that our 200-epoch ORL has already surpassed the 1600-epoch BYOL $8 5 . 2 \\%$ mAP vs. $8 4 . 9 \\%$ mAP), demonstrating that the performance efficiency of ORL is at least $8 \\times$ than BYOL below 1600 epochs. ",
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+    {
+        "type": "text",
+        "text": "4.3 Visualization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Correspondence pairs. Figure 3 visualizes some top-ranked region correspondence discovered in Stage 2 of ORL. We observe that each generated inter-RoI pair largely correspond to the regions with similar visual concepts (i.e., objects or object parts) across images. In contrast to typical contrastive learning methods that perform aggressive intra-image augmentations to simulate intra-class variances, our discovered inter-RoI pairs can substantially provide more natural and diverse intra-class variances at the object-instance level. ",
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+    {
+        "type": "text",
+        "text": "Attention maps. Figure 4 visualizes the attention maps generated by BYOL and ORL. We observe that both BYOL and ORL can produce relatively high-quality attention maps that focus on the foreground objects. This reflects from the side that current image-level contrastive learning methods have already induced a latent space with rich visual concepts. Nevertheless, ORL can activate more object regions and produce more accurate object boundary than BYOL in the generated attention maps. It is mainly due to introducing object-level similarity learning into ORL, which can minimize the inconsistent noise caused by image-level contrastive learning. In contrast, BYOL only uses the whole image to extract features, thus activating the most discriminative region. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work, we have presented a new self-supervised learning framework, ORL, for object-level representation learning from scene images. We leverage the latent prior of image-level self-supervised pre-training for discovering object-based region correspondence across images. The generated objectinstance correspondence enables us to perform pairwise contrastive learning at the object level. ORL significantly improves the performance of self-supervised learning from scene images in a variety of downstream tasks. We expect that our method can be applied to larger-scale unlabeled data in the wild to fully realize its potential, and hope that our study can attract the community’s attention to more general-purpose unsupervised representation learning from scene images. ",
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+    {
+        "type": "text",
+        "text": "Limitations ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we mainly perform pre-training experiments with ResNet-50 on COCO dataset, and further scale them up on $\\mathrm { C O C O + }$ dataset. However, the promise of self-supervised learning is to harness massive unlabeled data by scaling up to ever-larger datasets. Some prior works [18, 3, 16] have attempted to leverage larger models and datasets to explore the limit of current self-supervised learning methods. For instance, a recent representative work SEER [16] performs billion-scale selfsupervised pre-training on internet images using the RegNet architectures [46] with 700M parameters over 512 GPUs. Training at scale requires huge computational resources that are inaccessible to many researchers, which is not the core of our paper. We wish to highlight that our general-purpose ORL has yielded better performance than concurrent works [58, 34] that are tailored for dense prediction downstream tasks when pre-trained on COCO (Table 4), even surpassing the supervised ImageNet pre-training on several downstream tasks (Table 1-4). We expect that scaling ORL with larger architectures and datasets can further unleash its potential. Besides, ORL may not handle well on images with cluttered backgrounds since they will deviate the generated proposals to focus on these background regions. A possible remedy is to use some heuristic algorithms like saliency estimation to avoid the background regions. Another limitation is that ORL is a multi-stage framework. We expect an end-to-end framework to further improve the efficiency. We leave these explorations to future work. ",
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+    {
+        "type": "text",
+        "text": "Broader impact ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We present a more effective approach for learning unsupervised visual representations. Compared to supervised learning, it can liberate humans from expensive annotations as well as take advantages of rapidly growing real-world data. Like other learning algorithms, self-supervised learning should be applied with cautions when deployed in the real-world scenario. First, it is susceptible to biased learning if the algorithm is given with biased data. The exposure to unlabeled data may amplify such biases. Thus, debiasing measures have to be taken. Second, it remains non-trivial to dissect what is learned by self-supervised models. Similar concerns about the calibration, robustness, and interpretability of supervised models are equally applicable to the unsupervised counterpart. Our work is limited to the improvement of self-supervised learning within our scope. However, we acknowledge the importance of providing more transparent explanations for classification decisions, as well as the credibility of each prediction. Finally, our method still relies on the traditional regime of centralized learning. Privacy can be compromised if the method is applied on an unsaved platform. Federated learning can be a solution. How to scale self-supervised learning to the regime of decentralized learning will be an interesting research question to answer. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This study is supported under the RIE2020 Industry Alignment Fund – Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s). The project is also supported by Singapore MOE AcRF Tier 2 (T2EP20120-0005), the Data Science and Artificial Intelligence Research Center at NTU. ",
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+    {
+        "type": "text",
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