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+# A UNIFIED THEORY OF EARLY VISUAL REPRESENTATIONS FROM RETINA TO CORTEX THROUGH ANATOMICALLY CONSTRAINED DEEP CNNS
+
+Jack Lindsey∗ †, Samuel A. Ocko∗, Surya Ganguli1, Stephane Deny† Department of Applied Physics, Stanford and 1Google Brain, Mountain View, CA
+
+# ABSTRACT
+
+The vertebrate visual system is hierarchically organized to process visual information in successive stages. Neural representations vary drastically across the first stages of visual processing: at the output of the retina, ganglion cell receptive fields (RFs) exhibit a clear antagonistic center-surround structure, whereas in the primary visual cortex (V1), typical RFs are sharply tuned to a precise orientation. There is currently no unified theory explaining these differences in representations across layers. Here, using a deep convolutional neural network trained on image recognition as a model of the visual system, we show that such differences in representation can emerge as a direct consequence of different neural resource constraints on the retinal and cortical networks, and for the first time we find a single model from which both geometries spontaneously emerge at the appropriate stages of visual processing. The key constraint is a reduced number of neurons at the retinal output, consistent with the anatomy of the optic nerve as a stringent bottleneck. Second, we find that, for simple downstream cortical networks, visual representations at the retinal output emerge as nonlinear and lossy feature detectors, whereas they emerge as linear and faithful encoders of the visual scene for more complex cortical networks. This result predicts that the retinas of small vertebrates (e.g. salamander, frog) should perform sophisticated nonlinear computations, extracting features directly relevant to behavior, whereas retinas of large animals such as primates should mostly encode the visual scene linearly and respond to a much broader range of stimuli. These predictions could reconcile the two seemingly incompatible views of the retina as either performing feature extraction or efficient coding of natural scenes, by suggesting that all vertebrates lie on a spectrum between these two objectives, depending on the degree of neural resources allocated to their visual system.
+
+# 1 INTRODUCTION
+
+Why did natural selection shape our visual representations to be the way they are? Traditionally, the properties of the early visual system have been explained with theories of efficient coding, which are based on the premise that the neural representations are optimal at preserving information about the visual scene, under a set of metabolic constraints such as total firing rate or total number of synapses. These theories can successfully account for the antagonistic center-surround structure of receptive fields (RFs) found in the retina (Atick & Redlich, 1990; 1992; Vincent & Baddeley, 2003; Karklin & Simoncelli, 2011; Doi et al., 2012), as well as for the oriented structure of RFs found in the primary visual cortex V1 (Olshausen & Field, 1996; 1997; Bell & Sejnowski, 1997).
+
+However, a number of properties of the early visual system remain unexplained. First, it is unclear why RF geometries would be so different in the retina and V1. A study (Vincent et al., 2005) has proposed that both representations are optimal at preserving visual information under different metabolic constraints: a constraint on total number of synapses for the retina, and one on total firing rate in V1. However, it is unclear why the two systems would be optimized for these two different objectives. Second, there is a great diversity of ganglion cell types at the output the retina (Gollisch & Meister, 2010), with each cell type tiling the entire visual field and performing a specific computation. Interestingly, some of these types perform a highly nonlinear computation, extracting specific, behaviorally-relevant cues from the visual scene (e.g. direction-selective cells, objectmotion-selective cells), whereas other types are better approximated by a quasi-linear model, and respond to a broad range of stimuli (e.g. midget cells in the primate (Roska & Meister, 2014) and quasi-linear pixel-encoders in the mouse (Johnson et al., 2018)). Intriguingly, although quasi-linear and more nonlinear types exist in species of all sizes (e.g. primate parasol cells are nonlinear (Crook et al., 2008)), the proportion of cells performing a rather linear encoding versus a nonlinear feature detection seems to vary across species. For example, the most common ganglion cell type in the primate retina is fairly well approximated by a quasi-linear pixel-encoder (midget cells, $50 \%$ of all cells and ${ > } 9 5 \%$ in the central retina (Roska & Meister, 2014; Dacey, 2004)), whereas the most common cell type in mouse acts as a specific feature detector, thought to serve as an alarm system for overhead predators (W3 cells, $13 \%$ of all ganglion cells (Zhang et al., 2012)). Again, theories of efficient coding have not been able to account for this diversity of computations found across cell types and across species.
+
+The limitations of current efficient coding theories might reside in the simplistic assumption that the objective is to simply relay indiscriminately all visual information to the next stages of processing. Indeed, the ultimate goal of the visual system is to extract meaningful features from the visual scene in order to produce an adequate behavioral response, not necessarily to faithfully encode it. A recent line of work has proposed using the information bottleneck framework as a way to move beyond the simplistic objective of information preservation towards more realistic objectives (Chalk et al., 2016; 2018). Another study has shown that by changing the objective from efficiently encoding the present to efficiently encoding the future (predictive coding), one could better account for the spatio-temporal RFs of V1 cells (Singer et al., 2018). Although promising, these approaches were limited to the study of a single layer of neurons, and they did not answer the aforementioned questions about cross-layer or cross-species differences. On the other hand, deep convolutional networks have proven to be accurate models of the visual system, whether they are trained directly on reproducing neural activity (McIntosh et al., 2016; Cadena et al., 2017), or on a behaviorally relevant task (Yamins et al., 2014; Eberhardt et al., 2016; Cadena et al., 2017), but they have not yet been used to study the visual system through the lens of efficient coding theories.
+
+In this study, we trained deep convolutional neural networks on image recognition (CIFAR-10, Krizhevsky (2009)) and varied their architectures to explore the sets of constraints that could have shaped vertebrates’ early visual representations through natural selection. We modeled the visual system with a series of two convolutional networks, one corresponding to the retina and one downstream network corresponding to the ventral visual system in the brain. By varying the architecture of these networks, we first found that a reduction in the number of neurons at the retinal output – corresponding to a realistic physical constraint on the number of fibers in the optic nerve – accounted simultaneously for the emergence of center-surround RFs in our model of the retina, and for the emergence of oriented receptive fields in the primary visual relay of the brain. Second, we found that the degree of neural resources allocated to visual cortices in our model drastically reshaped retinal representations. Given a deep visual cortex, the retinal processing emerged as quasi-linear and retained substantial information about the visual scene. In contrast, for a shallow cortex, the retinal processing emerged as nonlinear and more information-lossy, but was better at extracting features relevant to the object classification task. These observations make testable predictions on the qualitative differences that should be found in retinal representations across species, and could reconcile the seemingly incompatible theories of retinal processing as either performing efficient encoding or feature detection.
+
+# 2 FRAMEWORK: A DEEP CONVOLUTIONAL NEURAL NETWORK MODEL OF THE VISUAL SYSTEM
+
+The retinal architecture is strongly conserved across species (Masland, 2001), and consists of three layers of feed-forward convolutional neurons (photoreceptors, bipolar cells, ganglion cells) and two layers of inhibitory interneurons (horizontal, amacrine cells). However, we chose to model the retina as a convolutional neural network (LeCun et al., 2015) with only two layers (fig. 1A). Indeed the retinal response of many species to complex stimuli has been modeled successfully with only one or two-layer models (Deny et al., 2017; Maheswaranathan et al., 2018; Gollisch & Meister, 2010), with some rare exceptions of models requiring more layers (McIntosh et al., 2016). We refer to this network as the retina-net. In our simulations, we varied the number of neurons in the second layer of the retina-net, which is the output of the retina, corresponding to the physical bottleneck of the optic nerve conveying all the visual information to the brain (fig. 1B).
+
+We modeled the ventral visual system – the system associated with object recognition in the brain (Hubel, 1995) – as a convolutional neural network taking its inputs from the retina-net (fig. 1A). We varied the neural resources allocated to the ventral visual system network (VVS-net) by changing the number of layers it is composed of (fig. 1B).
+
+We trained the neural network composed of the retina-net and VVS-net end-to-end on an object classification task (CIFAR-10, fig. 1A-B-C). Even though the visual system does much more than just classify objects in natural images, this objective is already much more complex and biologically realistic than the one used in previous studies of efficient coding, namely preserving all information about the visual scene. Moreover, we are encouraged by the fact that previous studies using this objective have found a good agreement between neural activity in artificial and biological visual networks (Yamins et al., 2014; Cadena et al., 2017).
+
+More specifically, we trained a convolutional neural network on a grayscale version of the standard CIFAR-10 dataset for image classification. The retina-net consisted of two convolutional layers with 32 channels and $N _ { B N }$ channels respectively, and with ReLU nonlinearities at each layer. The VVSnet consisted of a varying number $D _ { V V S }$ of convolutional layers with 32 channels followed by two fully connected layers (the first one with 1024 neurons and the second one with 10 neurons mapping to the 10 object categories), with ReLU nonlinearities at each layer and a softmax nonlinearity at the last layer. The full system encompassing the retina-net and VVS-net thus had $3 2  N _ { B N } $ $3 2  3 2  . . .$ channels respectively, where we varied the retinal bottleneck width, $N _ { B N }$ , as well as the number $D _ { V V S }$ of convolutional brain layers (not counting the fully connected layers). In each convolutional layer, we used $9 \mathrm { x } 9$ convolutional filters with a stride of 1 at each step. The large filter size was chosen to give the network flexibility in determining the optimal filter arrangement. We trained our network with the RMSProp optimizer for 20 epochs on the training set with batches of size 32. All optimizations were performed using Keras and TensorFlow. For all results presented, we tested statistical significance by training 10 identical networks with different random initializations of weights and biases taken from a Glorot-uniform distribution (Glorot & Bengio, 2010).
+
+![](images/4168d1511b9e76c27f3ca49da2c4be02beddc63a1cff2326d8a2bf973b6e99a8.jpg)  
+Figure 1: Illustration of the framework we used to model early visual representations. A: We trained convolutional neural networks on an image recognition task (CIFAR-10). The networks were composed of two parts, a retina-net and a ventral-visual-system-net (VVS-net), which receives input from the retina-net. B: We varied the number of layers in the VVS-net (white boxes) and the number of channels at the output of the retina-net (blue box). C: Key results: (1) A bottleneck at the output of the retina yielded center-surround retinal RFs. (2) A shallow VVS-net yielded more nonlinear retinal responses (linearity is schematized by the red arrow), which better disentangled image classes (represented as bent manifolds). D: Test-set accuracy of all model architectures on CIFAR-10, averaged over ten networks with random initial weights for each architecture. Performance increases with VVS-net depth and retinal channel, indicating that both factors are meaningful constraints on the network in the regime tested.
+
+After training, we determined the linear approximation of RFs of each convolutional channel of the network in each layer. This was achieved by computing the gradient of the activation of that channel with respect to a blank image. This gradient map gives a first-order approximation of the image pattern that maximally activates the cells in the channel of interest. In the limit of small noise variance, this computation is mathematically equivalent to measuring the cell’s spike-triggered average in response to a perturbative white-noise stimulus (Koelling & Nykamp, 2008; Schwartz et al., 2006), a commonly used method for determining receptive fields in experimental biology (Chichilnisky, 2001). This equivalence allowed us to compare directly the geometries of RFs experimentally measured in biological networks with the ones found in our models.
+
+The test accuracy of our neural network model of the visual system at the recognition task increased both with the number of channels in the retinal bottleneck, and with the number of layers in the VVS-net (fig. 1D), confirming that we were in a regime where the restrictions on neural resources in the VVS-net and at the output of the retina were critical to the ability of the network to perform the task.
+
+# 3 A UNIFIED MODEL FOR CENTER-SURROUND RFS IN THE RETINA AND ORIENTED RFS IN V1
+
+Here we investigate the effects of a dimensionality bottleneck at the retinal output on early visual representations in our model of the visual system.
+
+# 3.1 A DIMENSIONALITY BOTTLENECK AT THE RETINAL OUTPUT YIELDS THE EXPECTED REPRESENTATIONS IN RETINA AND V1
+
+When reducing the number of neurons at the output of the retina we found that RFs with antagonistic center and surround emerged. For $N _ { B N } = 3 2$ , our control setting with no bottleneck at the retinal output, we observed mostly oriented receptive fields in the second layer of the network (fig. 2A). For $N _ { B N } = 4 , 2$ , and 1, we observed center-surround receptive fields in the second layer of the network and mostly oriented receptive fields in the third layer, which is the first layer of the ventral visual system in our model (fig. 2B). We quantified these results in App. A. The RF geometries did not depend qualitatively on the VVS-net depth $D _ { V V S }$ (results shown for $D _ { V V S } = 2 $ ), except for the shallowest VVS-net tested ${ \cal D } _ { V V S } = 0$ , no convolutional layer, and thus no dimensionality expansion), for which the shape of emergent retinal RFs were variable across trials and difficult to interpret. These results are in good agreement with the organization of the biological visual system, where retinal RFs are center-surround and most downstream RFs in primary visual cortex (V1) are sharply oriented (Hubel, 1995), suggesting that the dimensionality bottleneck at the output of the retina is sufficient to explain these differences in representations. It is worth noting that for both conditions (bottleneck and no bottleneck), the RFs of downstream layers in the VVS-net after the first layer exhibited complex shapes that were neither clearly oriented, nor circular, and the RFs in the first layer of the retina did not appear to have any well-defined structure (data not shown).
+
+We then tested in our model the hypothesis of Hubel and Wiesel concerning how center-surround cells are pooled to give rise to oriented RFs in V1 (Hubel, 1995). We found that orientation-selective neurons in the VVS-net typically draw primarily from center-surround neurons in the retina-net that are aligned with the direction of the edge, with positive or negative weights corresponding to whether the polarity (light-selective / dark-selective) of the two neurons are consistent or inconsistent (fig. 2C, and App. A for a quantification). These qualitative results are in good agreement with Hubel and Wiesel’s hypothesis. Of course, this hypothesis remains to be tested in the real brain, since there is no evidence that the micro-circuitry of the brain matches that of our simulation.
+
+In the visual system of mammals, the main relay of visual information taking its input from the retina is the LGN (thalamus), which has center-surround RFs and a similar total number of neurons as the retinal output (Hubel, 1995). We created a network reflecting this architecture by having two lowdimensionality layers in a row instead of just one (fig. 2C). After training, we found center-surround RFs in the two layers with a bottleneck (retinal output and LGN), and oriented RFs in the next layer, corresponding to the primary visual cortex (V1). These results suggest that center-surround representations remain advantageous as long as the dimensionality of the representation remains low, and hence dimensionality expansion seems to be the crucial factor explaining the qualitative change of RFs found between LGN and V1.
+
+It is an interesting question to ask whether neurons in our model of the VVS are more similar to simple or complex cells (Hubel, 1995). To test this, we performed a one-step gradient ascent on the neural activity of VVS neurons with respect to the image, starting from several random initial images (App. B). If the neurons were acting as simple cells (i.e. are approximately linear in the stimulus), we would expect all optimized stimuli to converge to the same preferred stimulus. On the other hand, if the cells were complex (i.e. OR function between several preferred stimuli), we would expect the emergent preferred stimuli to depend on the exact initialization. Interestingly, we found that most neurons in the first layer of the VVS-net behaved as simple cells, whereas most neurons in the second layer of the VVS-net behaved as complex cells. Note that in biology, both simple and complex cells are found in V1. These results expose the fact that anatomical regions of visual cortex involve multiple nonlinearities and hence may map onto more than one layer of our simple model. Indeed, V1 itself is a multilayered cortical column, with LGN inputs coming in to layer 4, and layer 4 projecting to layers 2 and 3 (Hubel, 1995). Simple cells are predominantly found in layer 4 and complex cells are predominantly found in layers 2 and 3. These observations bolster the interpretation that biological V1 may correspond to multiple layers in our model.
+
+Local divisive normalization (i.e. local gain control) is an ubiquitous source of nonlinearity in the visual system (Geisler & Albrecht, 1992; Heeger, 1992; Deny et al., 2017). We thus tested the robustness of our main result to a more realistic model of the visual system with local normalization, by adding it at every layer of the network (App. C). We found that receptive fields still emerged as center-surround in the retina-net, and as oriented in our model of V1. We note that the local normalization slightly degraded the performance of the network on the task for all parameter settings we tried.
+
+![](images/98cc31250542547c9f81e82515536c39418d45eb4123699aabb6be149055e5be.jpg)  
+Figure 2: Effects of a bottleneck constraint on receptive fields (RFs). All results are shown for $D _ { V V S } = 2$ . A: Examples of RFs of cells at selected layers (layers 2 and 3) of a control network with no bottleneck. No center-surround RFs appear. B: Center-surround RFs emerge at the output of the retina-net (layer 2) and oriented RFs emerge in the first layer of the VVS-net when we impose a bottleneck constraint at the output of the retina $N _ { B N } = 1$ ) C: Top: Hubel and Wiesel’s hypothesis on oriented cell formation in V1 (Hubel, 1995). Bottom: A representative example of an orientationselective neuron (bottom RF) drawing from center-surround channels (top RFs) in the previous layer with weight matrices (center) according to their polarity. Light / dark-selective regions of a receptive field, and positive / negative weights, are represented with red / blue, respectively. D: Examples of RFs in a network with an extra bottleneck layer corresponding to mammalian LGN. Center-surround RFs appear at both the retinal output and LGN layer. E: Examples of ON and OFF center-surround RFs in the untied network $( N _ { B N } = 4 )$ ). F: t-SNE clustering of the retinal neurons of the untied network (see text). Two distinct cell type clusters form corresponding to ON and OFF center-surround receptive fields.
+
+3.2 EMERGENCE OF ON AND OFF POPULATIONS OF CENTER-SURROUND CELLS IN THERETINA
+
+We then verified that the emergence of center-surround RFs in the retina-net is a consequence of reducing the number of neurons at the retinal output, not of reducing the number of channels, our model’s equivalent of biological retinal cell types. In the retina, there exist 20-30 types of ganglion cells (Roska & Meister, 2014), each with a different and stereotyped receptive field, density, polarity (i.e. ON or OFF), and nonlinearities. Cells of each type tile the entire visual field like a convolutional channel in our model, so there is a direct analogy between channels in our model and ganglion cell types in the retina. In order to test whether the emergence of center-surround RFs depends on the number of types that we allow, or just on the number of neurons that we allow at the output of the retina (i.e. dimensionality bottleneck), we employed locally connected layers – equivalent to convolutional layers, but without parameter-tying between artificial neurons within a channel at different spatial locations. In this manner, we can limit the number of neurons at the retinal output without imposing a constraint on the number of cell types. Such a network contains too many parameters to be trained from scratch by gradient descent; to work around this, we trained the model stage-wise by first training our convolutional control network $N _ { B N } = 3 2$ with parameter tying) and then we trained a three-layers untied network (with bottleneck dimension $N _ { B N } = 4$ in the second layer) to reproduce the edge-like activations of the second layer of the control network. Even in the untied retina-net, in which each neuron is effectively its own channel, we found that centersurround RFs emerged (fig. 2E), indicating that center-surround RFs are the network’s preferred strategy for passing information through a dimensionality bottleneck even when no constraint on the number of cell types is imposed. We then found that the cells cluster in two distinct populations. To demonstrate this, we measured their activations in response to 10000 natural images, computed the first 20 principal components of this 10000-dimensional space, and ran t-SNE to visualize the clustering of neuron types. We found that two distinct clusters emerged, that corresponded visually to ON and OFF center-surround RFs (fig. 2F). We thus observe in our model the emergence of one of the most prominent axes of dichotomy of biological ganglion cell types, namely the classification of cells in ON and OFF populations with RFs of opposite polarity.
+
+# 4 RETINAL REPRESENTATIONS ARE A FUNCTION OF THE NEURAL RESOURCES ALLOCATED TO THE VENTRAL VISUAL STREAM
+
+To what extent are retinal representations in our model shaped by the degree of neural resources allocated to downstream processing? To investigate this question, we studied the effects of varying the degree of neural resources in the VVS-net, on emergent visual representations in the retina-net.
+
+# 4.1 THE RETINA BECOMES MORE LINEAR AS BRAIN COMPLEXITY INCREASES
+
+As we increased the number of layers in the VVS-net, the retinal computation became more linear (fig. 3A), as measured by the ability of the raw image to linearly map onto the neural representation at the retinal output (see methods, and App. F for a visualization of retinal representation as VVS-net depth increases). This observation is consistent with the current state of knowledge of the differences found in retinal representations across vertebrate species with different brain sizes. The linearization of the retinal response with increased brain complexity was true for different values of bottleneck $N _ { B N }$ . However, when we did not use any bottleneck ( $N _ { B N } = 3 2$ ), the trend became non-monotonic, with a peak in linearity of the response when the VVS-net had 1 conv layer (data not shown). Another interesting phenomenon to note is that linearity of the retinal response decreased as we increased the number of channels in the bottleneck, at any fixed brain depth (fig. 3A).
+
+The two main sources of nonlinearity in the retina are thought to be the inner retinal rectifications (bipolar and amacrine cells, corresponding to the first rectified layer in our model) and the ganglion cell rectification (corresponding to the second rectified layer in our model). As we decreased VVSnet depth, we observed that the retinal response became more nonlinear. Is this increase in response nonlinearity due to the first or second stage of nonlinearity in our retina-net? To test this, we plotted the real response against the response predicted by a purely linear model for the most shallow and for the deepest VVS-nets tested (fig. 3B). If the linear prediction were inaccurate because of the first stage of nonlinear processing in the retina-net, we would expect the points on the scatter plot to be scattered around the unit line. If the prediction error were due to the second-stage of nonlinearity, we would expect the linear approximation to make incorrect negative predictions for inactive neurons. In practice, we found that the prediction error of the linear model was partly explained by both stages of nonlinearity in the retina-net model, predicting that both inner retinal nonlinear processing and ganglion cell rectifications should be more pronounced in animals with fewer neural resources in their visual cortices.
+
+# 4.2 THE RETINAL REPRESENTATION IS THE RESULT OF A TRADE-OFF BETWEEN INFORMATION TRANSMISSION AND FEATURE EXTRACTION
+
+Why would retinal representations be more linear when the subsequent ventral visual stream has more resources? One hypothesis is that with a restricted number of neurons, the retina must trade-off between the two incentives of (1) compressing visual information in order to transmit it to down-1 Channel 1 Channel stream layers and (2) extracting nonlinear features from the scene to start disentangling the manifolds4 ChannelsRaw Pixels 4 ChannelsRaw Pixels corresponding to different classes of objects (Chung et al., 2018a;b). According to this hypothesis,VVS-net depth VVS-net depth VVS-net depth when the VVS is shallow, the priority of the retina should be to work toward extracting relevantB CE F  D E features. When the VVS is deep, the priority of the retina should be to transmit as much visual information as possible for downstream processing. We validated this hypothesis in two ways in our1 Channel model.
+
+![](images/5ff93b6301757e10e32a70529e050720018a2e3569dd5a57f35d7dd68daa863f.jpg)  
+Figure 3: Emergent retinal representations are function of the depth of downstream visual cortices. All error bars represent the $9 5 \%$ confidence interval about the mean (all simulations were repeated over 10 networks trained from random initial conditions). Three stars indicate t-test significance $\scriptstyle ( \mathbf { p } < 0 . 0 0 1$ ). A: Linearity of the retinal response increases with the number of layers in the VVSnet. Note that it also decreases with the number of cells at the retinal output (different lines). B: Responses of example retina-net output cell to natural images, vs. best linear fit prediction from 1 Channel \*\*\*raw image, for most (top) and least (bottom) deep VVS-nets. Nonlinearity arises from two sources: 2 Channels Raw Pixelsrectification within the retina-net (corresponds to the spread of the bulk of the point cloud) and rectification at the retina-net output (corresponds to inactive neurons being incorrectly predicted to have negative activations). C: Quality of image reconstruction from the retinal representation as a function of VVS-net depth. The retinal representation retains more information about the raw image for deep VVS-nets. D: Linear separability of classes of objects at the retinal output, as a function of VVS-net depth. Dashed line indicates separability of classes of images from the raw image pixels. Classes are less separable at the retinal output for deeper VVS-nets. E: Performance on CIFAR-10 for a two-layer densely connected network taking its input from the retina-net or from a raw image. Class information is more accessible from retinal representation. F: Class separability at all layers of network for a deep VVS-net $D _ { V V S } = 4 $ ) with and without bottleneck $\boldsymbol { N } _ { B N } = 1$ and $N _ { B N } = 3 2$ ). Retinal representation of bottleneck network has low separability. However, the first layer of the VVS-net has high separability (see text).
+
+First we showed that the retinal representation retained more information about the image as VVSnet complexity increased (fig. 3C). To estimate information retention, we trained a linear decoder (see methods) from the output of the retina to reconstruct the image and we measured the reconstruction error. The reconstruction error provided a lower bound on the information that the retina retained about the stimulus (note that more information for reconstruction might be accessible by a nonlinear decoder). This result corroborated our hypothesis that, as the VVS-net becomes more complex, the retinal representation gets better at retaining visual information for further processing by the VVS-net.
+
+Second, we found that different classes of objects of CIFAR-10 (e.g. trucks, frogs) were more linearly separable from the retina-net representation when the VVS-net was shallow than when it was deep (fig. 3D). To measure linear separability of manifolds, we trained a linear SVM decoder to separate all pairs of classes and evaluated the performance of the SVM classifier on held-out images (see methods). Moreover, we showed that a VVS-net consisting of two fully connected layers only (no convolutional layers) equipped and trained end-to-end with a retina with a tight bottleneck $N _ { B N } = 1$ (dimensionality of retinal output matches dimensionality of the input image) performed better at image recognition than the same VVS-net trained without a retina-net, taking raw images as input (fig. 3E). Both these results corroborate our hypothesis that retinas followed by a simple cortex performs meaningful feature extraction, whereas retinas followed by more complex visual cortices prioritize non-lossy encoding, postponing feature extraction to downstream layers that are better equipped to do it.
+
+Next, we show that within a single network, each retinal channel is trading-off between (1) linearly transmitting visual information to the brain, and (2) extracting relevant features for the object classification task. For 10 instantiations of a network with a retinal bottleneck containing 4 channels, we represented the linearity of each of these 4 channels against the linear separability of object categories obtained from each of these representations. We found, across all networks, a systematic negative correlation between linearity and linear separability across all 4 channels (App. D). Again, this result strongly suggests that extracting features and transmitting visual information are indeed two competing goals shaping representations in our model of the retina.
+
+In the case of the deepest VVS-nets tested, the retinal processing was quasi-linear for the tightest bottleneck (var.expl. $= 0 . 9$ , $N _ { B N } = 1$ , fig. 3A). One might take this result to suggest that the retinanet in such models does little more than copy image information. However the very first layer of the VVS-net after the retina disentangled classes (as measured by linear separability) almost as well as the second layer of a VVS-net without a retina (fig. 3F), suggesting that the retinal representation, while only moderately linearly separable itself, is especially transformable into a representation with a high linear separability. This result suggests that even when the retina-net is quasi-linear, it can still participate in extracting relevant features for downstream processing by the brain. The increased separability allowed by the retinal pre-processing for this deep VVS-net could be due to (1) the linear processing or (2) the slightly nonlinear part of the retinal processing (3) a combination of both linear and nonlinear processing. To distinguish between these hypotheses, we replaced the true retinal processing by its best linear approximation, retrained the VVS-net on the output of this linearized retina, and tested whether separability was as high as with the true retinal processing (App. E). We found that the first layer trained on the output of the linearized retinal representation was indeed much more separable than the first layer of the control network (trained directly on natural images) at separating classes of objects, suggesting that the linear operation done by the retina does indeed play a crucial role in making the representation especially separable for subsequent layers.
+
+# 5 METHODS
+
+To estimate the linearity of the response of retinal neurons, we fit a linear model to predict the neural response from the image on 8,000 images. In order to prevent overfitting, we regularized the linear weights with an L2 penalty and optimized the weights using ridge regression. The value of the penalty term was chosen by 10-fold cross-validation on the training set. We then measured the Pearson correlation between the linearized responses and original model responses on a testing set of 2,000 images.
+
+To estimate the information about the input image retained by the retinal output representation, we fit a linear model to reconstruct the image from the (fixed) outputs of the trained retina-net of interest. All numerical figures given are variance-explained results on the held-out test set.
+
+To estimate the linear separability of classes of objects from the neural representation, we trained an SVM classifier between all pairs of classes on half of the testing set of CIFAR-10 (1,000 images that were not used to train the network), and we tested the performance of the SVM classifier on 1,000 held-out images from the testing set, as measured by the percentage of images classified correctly. We then averaged the performance of the SVM across all pairs of classes to obtain the linear separability score.
+
+# 6 DISCUSSION
+
+A unified theoretical account for the structural differences between the receptive field shapes of retinal neurons and V1 neurons has until now been beyond the reach of efficient coding theories. Karklin & Simoncelli (2011) found that efficient encoding of images with added noise and a cost on firing rate produce center-surround RFs, whereas the same task without noise produces edge detectors. However, this observation (as they note) does not explain the discrepancy between retinal and cortical representations. Vincent et al. (2005) propose a different set of constraints for the retina and V1, in which the retina optimizes for a metabolic constraint on total number of synapses, whereas V1 optimizes for a constraint on total firing rate. It is not clear why each of these constraints would predominate in each respective system. Here we show that these two representations can emerge from the requirement to perform a biologically relevant task (extracting object identity from an image) with a bottleneck constraint on the dimensionality of the retinal output. Interestingly, this constraint differs from the ones used previously to account for center-surround RFs (number of synapses or total firing rate). It is worth noting that we unsuccessfully tried to reproduce the result of Karklin & Simoncelli (2011) in our network, by adding noise to the image and applying an L1 regularization to the retina-net activations. In our framework (different than the one of Karklin & Simoncelli (2011) in many ways), the receptive fields of the retina-net without bottleneck remained oriented across the full range of orders of magnitude of noise and L1 regularization that permitted successful task performance.
+
+There is a long-standing debate on whether the role of the retina is to extract relevant features from the environment (Lettvin et al., 1959; Gollisch & Meister, 2010; Roska & Meister, 2014), or to efficiently encode all visual information indistinctly (Barlow, 1961; Atick & Redlich, 1990; 1992). In this work, we show that our model of the visual system, trained on the same task and with the same input statistics, can exhibit different retinal representations depending on the degree of neural resources allocated to downstream processing by the ventral visual stream. These results suggest the hypothesis that, despite its conserved structure across evolution, the retina could prioritize different computations in different species. In species with fewer brain resources devoted to visual processing, the retina should nonlinearly extract relevant features from the environment for object recognition, and in species with a more complex ventral visual stream, the retina should prioritize a linear and efficient transmission of visual information for further processing by the brain. Although all species contain a mix of quasi-linear and nonlinear cell types, the proportion of quasi-linear cells seems to vary across species. In the mouse, the most numerous cell type is a two-stage nonlinear feature detector, thought to detect overhead predators (Zhang et al., 2012). In contrast, the most common ganglion cell type in the primate retina is fairly well approximated by a linear filter (midget cells, $50 \%$ of all cells and ${ > } 9 5 \%$ in the central retina (Roska & Meister, 2014; Dacey, 2004)). Note however that two-stage nonlinear models are also present in larger species, such as cat Y-type cells and primate parasol cells (Crook et al., 2008), making it difficult to make definitive statements about inter-species differences in retinal coding. To gain a better understanding of these differences, it would be useful to collect a dataset consisting of recordings of complete populations of ganglion cells of different species in response to a common bank of natural scenes.
+
+A related question is the role of the parcellation of visual information in many ganglion cell types at the retinal output. A recent theory of efficient coding has shown that properties of midget and parasol cells in the primate retina can emerge from the objective of faithfully encoding natural movies with a cost on the total firing rate traversing the optic nerve (Ocko et al., 2018). On the other hand, many cell types seem exquisitely sensitive to behaviorally relevant features, such as potential prey or predators (Gollisch & Meister, 2010). For example, some cell types in the frog are tuned to detect moving flies or looming predators (Lettvin et al., 1959). It is an intriguing possibility that different cell types could subserve different functions within a single species, namely efficient coding of natural scenes for some types and extraction of behaviorally-relevant features for others. In this study we allowed only a limited number of cell types (i.e. convolutional channels) at the retinal output (1 to 4), in order to have a dimensionality expansion between the retinal representation and the representation in the ventral visual stream (32 channels), an important condition to see the retinal center-surround representation emerge. By using larger networks with more channels in the retina-net and the VVS-net, we could study the emergence of a greater diversity of neuron types in our retina-net and compare their properties to real retinal cell types. It would also be interesting to extend our model to natural movies. Indeed, most feature detectors identified to date seem to process some form of image motion: wide-field, local or differential (Roska & Meister, 2014). Adding a temporal dimension to the model would be necessary to study their emergence.
+
+In conclusion, by studying emergent representations learned by a deep network trained on a biologically relevant task, we found that striking differences in retinal and cortical representations of visual information could be a consequence of the anatomical constraint of transmitting visual information through a low-dimensional communication channel, the optic nerve. Moreover, our computational explorations suggest that the rich diversity of retinal representations found across species could have adaptively co-evolved with the varying sophistication of subsequent processing performed by the ventral visual stream. These insights illustrate how deep neural networks, whose creation was once inspired by the visual system, can now be used to shed light on the constraints and objectives that have driven the evolution of our visual system.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Lane McIntosh, Niru Maheswaranathan, Aran Nayebi, SueYeon Chung, Vardan Papyan, Nora Brackbill, E.J. Chichilnisky for useful discussions and Stephen Baccus for his comments that greatly improved the manuscript. S.G. thanks the Burroughs-Wellcome, McKnight, James S. McDonnell and Simons foundations for support.
+
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+
+APPENDIX
+
+# A QUANTIFICATION OF RECEPTIVE FIELD ISOTROPY IN RETINA AND V1
+
+![](images/f9a61d69ffda3d2cbfb4ae39b653f0ce63f4698172fdda19fe3fb4da33cec1bc.jpg)  
+Figure 4: A: Left: Schematic re-illustrating the architecture of a vanilla (no bottleneck) network and showing examples oriented RFs in its second layer. Center: Visualization of average RF isotropy for cells in the second layer of a vanilla convolutional network $( N _ { B N } = 1$ , $D _ { V V S } = 2$ ). Orange error bars indicate $9 5 \%$ confidence intervals. Right: Visualization of RF isotropy for ten example RFs from the same network architecture. B: Left: Schematic re-illustrating the architecture of the retina-net $+ \mathrm { \Delta V V S }$ -net model $( N _ { B N } = 1 , D _ { V V S } = 2 )$ and showing example center-surround RFs at the retina-net output and oriented RFs in the following layer (V1). Center and right: Same RF isotropy visualizations as in part A. C: Left: re-illustration of V1 RFs pooling in oriented fashion from center-surround retinal RFs $( N _ { B N } = 1 , D _ { V V S } = 2 )$ . Right: Same isotropy visualizations as in panel A carried on the weight matrix from retina to V1.
+
+The following analysis corroborates our qualitative observation that a dimensionality bottleneck in the retina-net yields center-surround retinal receptive fields and oriented, edge-detecting receptive fields in the first layer of the VVS-net (V1). For a given receptive field, we quantified its orientedness as follows: we displayed rectangular bar stimuli of all possible combinations of width, orientations and spatial translations that fit in the input image window. Among all these combinations, we selected the bar stimulus width, orientation, and translation that yielded the strongest response from the RF. Bars with the same width as the best stimuli were presented at all orientations and translations, and for each orientation, we select the strongest response it produced (across all translations). In this manner we obtained a measure of the strength of a receptive field’s preference for all orientations.
+
+We measured the strength of each RF preference (maximum strength of response) for its preferred orientation and for the orthogonal orientation, and computed the ratio of these strengths. Completely isotropic filters would be expected to give a ratio of 1, while oriented filters should give higher ratios. Note however that some deviation from 1 may indicate noise in the filter rather than true orientedness. For each network layer, we averaged this ratio across filters (for convolutional layers with multiple layers) and trials (re-training of the same neural network architecture with different random initializations). We found that the average ratios were $1 . 5 6 ( \pm 0 . 2 2 )$ for the retinal output, $3 . 0 5 ( \pm 0 . 3 0 ) $ for the first VVS-net layer, and $2 . 5 7 ( \pm 0 . 2 7 )$ for the second VVS-net layer, where error margins given are $9 5 \%$ confidence intervals. To help assess whether retinal RFs were more isotropic than expected by chance, we compared them to receptive fields composed of random Gaussian noise as a baseline. These give an average ratio (as computed above) of $\bar { 1 . 9 7 } ( \pm 0 . 0 8 )$ , significantly higher than that for retinal RFs. Furthermore, the standard deviation of RF preference across orientations was significantly lower for the retinal RFs $( 0 . 1 1 8 \pm 0 . 0 3 6 )$ than for random RFs $( 0 . 1 7 7 \pm 0 . 0 0 7 )$ , also indicating that retinal RFs were more isotropic than expected by chance.
+
+We also plot the average RF preference for different orientations at each layer to more comprehensively assess the isotropy of RFs at each network layer. To aggregate results across multiple trials and filters, we rotated the coordinates of each receptive field such that its preferred orientation was vertical, and averaged our results across filters and trials. (See Figure 4).
+
+The results confirm our qualitative observations that (1) RFs in the second layer of a vanilla network $( N _ { B N } = 3 2 )$ ) are highly oriented (Figure 4A) (2) RFs in the second layer (retina output) of a bottleneck network $N _ { B N } = 1 \textgreater$ ) are much more isotropic, consistent with center-surround RFs (Figure 4B top), and (3) RFs in the layer immediately following the retina-net in the bottleneck network are oriented (Figure 4B bottom).
+
+We also quantitatively corroborate our observation that oriented receptive fields in the V1 layer pool input from oriented arrays of center-surround filters in the retina-net output layer. We apply our method of isotropy quantification described above to the weight matrix for each input-output filter combination in the V1 convolutional layer. We find that this weight matrix itself exhibits orientedness across filters and trials, confirming our observation (Figure 4C).
+
+# B SIMPLE AND COMPLEX CELLS
+
+![](images/d179773cc49e10c02fab446959e80cc8f6ae44ade2c76b6fcb6f3e906229eb9b.jpg)  
+Figure 5: A: Visualizations of retina-net output RFs for an example network $( N _ { B N } = 1 , D _ { V V S } = 2 )$ using different random initialization, as described in the text. B: Same as A, for the first layer of the VVS-net, and showing 5 of the layer’s 32 channels on the x axis. C: Same as B, for the second layer of the VVS-net. In contrast to the first layer, the emergent preferred stimuli are always different across different initializations, indicative of a complex-cell like behavior.
+
+To investigate whether neurons in our model’s early layers more closely resembled simple or complex cells, we performed the following analysis. As before, we obtained local linear approximations of receptive fields by computing the gradient in input space with respect to the response of a given neuron. Rather than beginning with a blank input, we ran multiple trials with different randomly initialized inputs. A purely linear cell would give the same result no matter the initialization; a somewhat nonlinear but still “simple” cell is expected to give similar results across initializations. A “complex” cell is expected to give different RF visualizations for different random inputs, reflecting multiple peaks in its response as a function of input. In Figure 5 we show examples of receptive fields at different layers of our retina-net $+ \mathrm { \Delta V V S }$ -net model (with $N _ { B N } = 1 , D _ { V V S } = 2 )$ for different random intializations of the image (uniform random in [0, 1]). The retina-net output and first VVS-net layer exhibit “simple” behavior, but the second VVS-net layer exhibits observably “complex” behavior. To quantify this effect, we measure the average (across filters within each layer and re-trainings of the same network architecture) standard deviation of computed RFs (normalized to the range [0, 1]) for each network layer. We found that the average standard deviations were $7 . 9 ( \pm 1 . 1 ) \times 1 0 ^ { - 3 }$ , $1 5 . 4 ( \pm 0 . 8 ) \times 1 0 ^ { - 3 }$ , and $3 5 . 9 ( \pm 0 . 8 ) \times 1 0 ^ { - 3 }$ for the retina-net output, first VVS-net layer, and second VVS-net layer, respectively, where the margins of error given are $9 5 \%$ confidence intervals. These results corroborate the observation of significantly more complex behavior in the second VVS-net layer, mirroring the biological phenomenon in which complex cells pool from simple cells in $\mathrm { V } 1$ .
+
+# C EFFECTS OF LOCAL RESPONSE NORMALIZATION ON EARLY VISUAL malization FigurREPRESENTATIONS
+
+![](images/679c7d4d1bf4021059a7f1a2d21611f59969d6766478b59e5a97dcfba385ad75.jpg)  
+Figure 6: Example RFs from the bottleneck network $( N _ { B N } = 1 , D _ { V V S } = 2$ without (A) and with (B) local response normalization (i.e. local gain control).
+
+We tested the robustness of our first main finding – that a bottlenecked retina-net $+ \mathrm { \nabla { V V S } }$ -net model yields center-surround receptive fields in the retina and oriented receptive felds in $\mathrm { { V } 1 - }$ to the use of biologically realistic local response normalization at every layer of the network. In particular, we normalized the output $x$ of each channel (row $r$ , column $c$ ) of each layer as follows (during training and testing):
+
+$$
+x _ { r , c } \gets \frac { x _ { r , c } } { \Big ( k + \alpha \sum _ { r ^ { \prime } \in [ r - \frac { n } { 2 } , r + \frac { n } { 2 } ] , c ^ { \prime } \in [ c - \frac { n } { 2 } , c + \frac { n } { 2 } ] } x _ { r ^ { \prime } , c ^ { \prime } } \Big ) ^ { \beta } }
+$$
+
+where the subscripts of $x$ indicate the spatial location (row/column), and $k , \alpha _ { \mathrm { { ; } } }$ , ad $\beta$ are constants. We used $k = 2$ , $\beta = 0 . 5$ and $\beta = 0 . 7 5$ , and $\alpha = 5 \times 1 0 ^ { - 4 }$ and $\alpha = 5 . 0$ . All parameter settings tested yielded RFs with the same qualitative properties as in the model without normalization. Figure 6 shows example RFs from the no-normalzation model next to example RFs from the normalization model with $k = 2 , \beta = 0 . 5 , \alpha = 5 . 0$ .
+
+# D RETINAL CELL TYPES TRADE OFF BETWEEN LINEAR INFORMATION TRANSMISSION AND NONLINEAR FEATURE EXTRACTION
+
+![](images/975e7596308fe16a0a7c943657318acd6b90291b6140f066a27a5a13c5f647da.jpg)  
+Fig XX Linearity vs. separability of retina-net output channels (as in XX), Figure 7: Linearity vs. Class separability for each retina-net output channels (i.e. bottleneck layer). network has a VVS-Net depth of 4 and a bottleneck size of 4. VVS-Net depth is equal to 4. Each network has a bottleneck size of 4 channels (i.e. $N _ { B N } { = } 4 )$ . DisDistributions are plotted across 8 network instances; each point represents a single channel, colored according to its network. Here, we tributions are plotted across 10 network instances; each point represents a single channel, colored can see that the tradeoff between efficient coding and feature extraction also happens within the retina-nets of individual networks.according to its network. The negative slope suggests that there is trade-off between linearly transmitting visual information for downstream processing (i.e. efficient coding) and extracting useful features for the object recognition task.
+
+# E LINEARIZED RETINA ALSO INCREASES SEPARABILITY IN SUBSEQUENT LAYERS
+
+![](images/be127bef6a36f1cc24b24501c50f4d931f9eb8892dfaa764893c38ef230465df.jpg)  
+representation of bottleneck network has low separability. However, the Figure 8: Class separability at all layers of network for a deep VVS-net $( D _ { V V S } = 4 )$ ) with and without bottleneck $\boldsymbol { N } _ { B N } = 1$ t layer oarability and $N _ { B N } = 3 2$ h separability. We additionally plot the leneck (NBN = 1) network (see test) as a ). Retinal representation of bottleneck network has low function of layer. That the jump in linear separability between layers 2,3 survives linearization suggests that the main effect of retinal processing separability. However, the first layer of the VVS-net has high separability. We additionally plot the in this network is whseparability of the linearized bottleneck $N _ { B N } = 1$ on-linear processing.) network (see test) as a function of layer. That the jump in linear separability between layers 2,3 survives linearization suggests that the main effect of retinal processing in this network is whitening (see Fig. 9) rather than nonlinear processing.
+
+In the case of the deepest VVS-nets tested, the retinal processing was quasi-linear for the tightest bottleneck (var.expl. $= 0 . 9$ , $N _ { B N } = 1$ , fig. 3A). However the very first layer of the VVS-net after the retina disentangled classes (as measured by linear separability) almost as well as the second layer of a VVS-net without retina (fig. 3F), suggesting that the retinal representation, while only moderately linearly separable itself, is especially transformable into a representation with a high linear separability. To determine to what degree this increased separability was due to (1) the linear processing or (2) the slightly nonlinear part of the retinal processing, we performed an ablation experiment to eliminate factor (2). We first replaced the true retinal processing by its best approximation by a onelayer linear convolution (of sufficient filter width to correspond to two convolutional layers with 9 by 9 filters). After this linearization process, we retrained the VVS-net using the linearized retinal representation as input, keeping the linearized retina weights frozen. We found that the first layer trained on the output of the linearized retinal representation was indeed much better than the first layer of the control network (trained directly on natural images) at separating classes of objects (Fig.
+
+8), suggesting that the linear operation done by the retina does indeed play a crucial role in making the representation especially separable for subsequent layers. Visualization of retinal processing in App. F suggest that whitening is an important part of this linear processing.
+
+# F RETINAL REPRESENTATION VISUALIZATION AS A FUNCTION OF VVS-NET DEPTH FOR BOTTLENECK $N _ { B N } = 1$
+
+![](images/e5dd316498caac5948f6a2c039010178896c096525cb6e3199a8661488b1d0b8.jpg)  
+examples (x axis) as a function of VVS-Net depth (y axis)Figure 9: Visualization of the output of the retina-net (one-channel-bottleneck, i.e. $N _ { B N } = 1$ ) for different images from the testing set (x-axis) as a function of VVS-net depth (y-axis). Each pixel intensity of the retinal image is proportional to the activation of the corresponding neuron of the retina, where light shades indicate high activities and dark shades low activities. While retinas for every VVS-net depth appear to whiten the input, we can see that the retinal image is more and more processed and less and less recognizable as VVS-net depth decreases.

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+        "text": "The vertebrate visual system is hierarchically organized to process visual information in successive stages. Neural representations vary drastically across the first stages of visual processing: at the output of the retina, ganglion cell receptive fields (RFs) exhibit a clear antagonistic center-surround structure, whereas in the primary visual cortex (V1), typical RFs are sharply tuned to a precise orientation. There is currently no unified theory explaining these differences in representations across layers. Here, using a deep convolutional neural network trained on image recognition as a model of the visual system, we show that such differences in representation can emerge as a direct consequence of different neural resource constraints on the retinal and cortical networks, and for the first time we find a single model from which both geometries spontaneously emerge at the appropriate stages of visual processing. The key constraint is a reduced number of neurons at the retinal output, consistent with the anatomy of the optic nerve as a stringent bottleneck. Second, we find that, for simple downstream cortical networks, visual representations at the retinal output emerge as nonlinear and lossy feature detectors, whereas they emerge as linear and faithful encoders of the visual scene for more complex cortical networks. This result predicts that the retinas of small vertebrates (e.g. salamander, frog) should perform sophisticated nonlinear computations, extracting features directly relevant to behavior, whereas retinas of large animals such as primates should mostly encode the visual scene linearly and respond to a much broader range of stimuli. These predictions could reconcile the two seemingly incompatible views of the retina as either performing feature extraction or efficient coding of natural scenes, by suggesting that all vertebrates lie on a spectrum between these two objectives, depending on the degree of neural resources allocated to their visual system. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Why did natural selection shape our visual representations to be the way they are? Traditionally, the properties of the early visual system have been explained with theories of efficient coding, which are based on the premise that the neural representations are optimal at preserving information about the visual scene, under a set of metabolic constraints such as total firing rate or total number of synapses. These theories can successfully account for the antagonistic center-surround structure of receptive fields (RFs) found in the retina (Atick & Redlich, 1990; 1992; Vincent & Baddeley, 2003; Karklin & Simoncelli, 2011; Doi et al., 2012), as well as for the oriented structure of RFs found in the primary visual cortex V1 (Olshausen & Field, 1996; 1997; Bell & Sejnowski, 1997). ",
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+        "text": "However, a number of properties of the early visual system remain unexplained. First, it is unclear why RF geometries would be so different in the retina and V1. A study (Vincent et al., 2005) has proposed that both representations are optimal at preserving visual information under different metabolic constraints: a constraint on total number of synapses for the retina, and one on total firing rate in V1. However, it is unclear why the two systems would be optimized for these two different objectives. Second, there is a great diversity of ganglion cell types at the output the retina (Gollisch & Meister, 2010), with each cell type tiling the entire visual field and performing a specific computation. Interestingly, some of these types perform a highly nonlinear computation, extracting specific, behaviorally-relevant cues from the visual scene (e.g. direction-selective cells, objectmotion-selective cells), whereas other types are better approximated by a quasi-linear model, and respond to a broad range of stimuli (e.g. midget cells in the primate (Roska & Meister, 2014) and quasi-linear pixel-encoders in the mouse (Johnson et al., 2018)). Intriguingly, although quasi-linear and more nonlinear types exist in species of all sizes (e.g. primate parasol cells are nonlinear (Crook et al., 2008)), the proportion of cells performing a rather linear encoding versus a nonlinear feature detection seems to vary across species. For example, the most common ganglion cell type in the primate retina is fairly well approximated by a quasi-linear pixel-encoder (midget cells, $50 \\%$ of all cells and ${ > } 9 5 \\%$ in the central retina (Roska & Meister, 2014; Dacey, 2004)), whereas the most common cell type in mouse acts as a specific feature detector, thought to serve as an alarm system for overhead predators (W3 cells, $13 \\%$ of all ganglion cells (Zhang et al., 2012)). Again, theories of efficient coding have not been able to account for this diversity of computations found across cell types and across species. ",
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+        "type": "text",
+        "text": "The limitations of current efficient coding theories might reside in the simplistic assumption that the objective is to simply relay indiscriminately all visual information to the next stages of processing. Indeed, the ultimate goal of the visual system is to extract meaningful features from the visual scene in order to produce an adequate behavioral response, not necessarily to faithfully encode it. A recent line of work has proposed using the information bottleneck framework as a way to move beyond the simplistic objective of information preservation towards more realistic objectives (Chalk et al., 2016; 2018). Another study has shown that by changing the objective from efficiently encoding the present to efficiently encoding the future (predictive coding), one could better account for the spatio-temporal RFs of V1 cells (Singer et al., 2018). Although promising, these approaches were limited to the study of a single layer of neurons, and they did not answer the aforementioned questions about cross-layer or cross-species differences. On the other hand, deep convolutional networks have proven to be accurate models of the visual system, whether they are trained directly on reproducing neural activity (McIntosh et al., 2016; Cadena et al., 2017), or on a behaviorally relevant task (Yamins et al., 2014; Eberhardt et al., 2016; Cadena et al., 2017), but they have not yet been used to study the visual system through the lens of efficient coding theories. ",
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+        "text": "In this study, we trained deep convolutional neural networks on image recognition (CIFAR-10, Krizhevsky (2009)) and varied their architectures to explore the sets of constraints that could have shaped vertebrates’ early visual representations through natural selection. We modeled the visual system with a series of two convolutional networks, one corresponding to the retina and one downstream network corresponding to the ventral visual system in the brain. By varying the architecture of these networks, we first found that a reduction in the number of neurons at the retinal output – corresponding to a realistic physical constraint on the number of fibers in the optic nerve – accounted simultaneously for the emergence of center-surround RFs in our model of the retina, and for the emergence of oriented receptive fields in the primary visual relay of the brain. Second, we found that the degree of neural resources allocated to visual cortices in our model drastically reshaped retinal representations. Given a deep visual cortex, the retinal processing emerged as quasi-linear and retained substantial information about the visual scene. In contrast, for a shallow cortex, the retinal processing emerged as nonlinear and more information-lossy, but was better at extracting features relevant to the object classification task. These observations make testable predictions on the qualitative differences that should be found in retinal representations across species, and could reconcile the seemingly incompatible theories of retinal processing as either performing efficient encoding or feature detection. ",
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+        "type": "text",
+        "text": "2 FRAMEWORK: A DEEP CONVOLUTIONAL NEURAL NETWORK MODEL OF THE VISUAL SYSTEM ",
+        "text_level": 1,
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+        "text": "The retinal architecture is strongly conserved across species (Masland, 2001), and consists of three layers of feed-forward convolutional neurons (photoreceptors, bipolar cells, ganglion cells) and two layers of inhibitory interneurons (horizontal, amacrine cells). However, we chose to model the retina as a convolutional neural network (LeCun et al., 2015) with only two layers (fig. 1A). Indeed the retinal response of many species to complex stimuli has been modeled successfully with only one or two-layer models (Deny et al., 2017; Maheswaranathan et al., 2018; Gollisch & Meister, 2010), with some rare exceptions of models requiring more layers (McIntosh et al., 2016). We refer to this network as the retina-net. In our simulations, we varied the number of neurons in the second layer of the retina-net, which is the output of the retina, corresponding to the physical bottleneck of the optic nerve conveying all the visual information to the brain (fig. 1B). ",
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+        "text": "We modeled the ventral visual system – the system associated with object recognition in the brain (Hubel, 1995) – as a convolutional neural network taking its inputs from the retina-net (fig. 1A). We varied the neural resources allocated to the ventral visual system network (VVS-net) by changing the number of layers it is composed of (fig. 1B). ",
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+        "text": "We trained the neural network composed of the retina-net and VVS-net end-to-end on an object classification task (CIFAR-10, fig. 1A-B-C). Even though the visual system does much more than just classify objects in natural images, this objective is already much more complex and biologically realistic than the one used in previous studies of efficient coding, namely preserving all information about the visual scene. Moreover, we are encouraged by the fact that previous studies using this objective have found a good agreement between neural activity in artificial and biological visual networks (Yamins et al., 2014; Cadena et al., 2017). ",
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+        "text": "More specifically, we trained a convolutional neural network on a grayscale version of the standard CIFAR-10 dataset for image classification. The retina-net consisted of two convolutional layers with 32 channels and $N _ { B N }$ channels respectively, and with ReLU nonlinearities at each layer. The VVSnet consisted of a varying number $D _ { V V S }$ of convolutional layers with 32 channels followed by two fully connected layers (the first one with 1024 neurons and the second one with 10 neurons mapping to the 10 object categories), with ReLU nonlinearities at each layer and a softmax nonlinearity at the last layer. The full system encompassing the retina-net and VVS-net thus had $3 2  N _ { B N } $ $3 2  3 2  . . .$ channels respectively, where we varied the retinal bottleneck width, $N _ { B N }$ , as well as the number $D _ { V V S }$ of convolutional brain layers (not counting the fully connected layers). In each convolutional layer, we used $9 \\mathrm { x } 9$ convolutional filters with a stride of 1 at each step. The large filter size was chosen to give the network flexibility in determining the optimal filter arrangement. We trained our network with the RMSProp optimizer for 20 epochs on the training set with batches of size 32. All optimizations were performed using Keras and TensorFlow. For all results presented, we tested statistical significance by training 10 identical networks with different random initializations of weights and biases taken from a Glorot-uniform distribution (Glorot & Bengio, 2010). ",
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+    {
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+        "image_caption": [
+            "Figure 1: Illustration of the framework we used to model early visual representations. A: We trained convolutional neural networks on an image recognition task (CIFAR-10). The networks were composed of two parts, a retina-net and a ventral-visual-system-net (VVS-net), which receives input from the retina-net. B: We varied the number of layers in the VVS-net (white boxes) and the number of channels at the output of the retina-net (blue box). C: Key results: (1) A bottleneck at the output of the retina yielded center-surround retinal RFs. (2) A shallow VVS-net yielded more nonlinear retinal responses (linearity is schematized by the red arrow), which better disentangled image classes (represented as bent manifolds). D: Test-set accuracy of all model architectures on CIFAR-10, averaged over ten networks with random initial weights for each architecture. Performance increases with VVS-net depth and retinal channel, indicating that both factors are meaningful constraints on the network in the regime tested. "
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+        "text": "After training, we determined the linear approximation of RFs of each convolutional channel of the network in each layer. This was achieved by computing the gradient of the activation of that channel with respect to a blank image. This gradient map gives a first-order approximation of the image pattern that maximally activates the cells in the channel of interest. In the limit of small noise variance, this computation is mathematically equivalent to measuring the cell’s spike-triggered average in response to a perturbative white-noise stimulus (Koelling & Nykamp, 2008; Schwartz et al., 2006), a commonly used method for determining receptive fields in experimental biology (Chichilnisky, 2001). This equivalence allowed us to compare directly the geometries of RFs experimentally measured in biological networks with the ones found in our models. ",
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+        "text": "The test accuracy of our neural network model of the visual system at the recognition task increased both with the number of channels in the retinal bottleneck, and with the number of layers in the VVS-net (fig. 1D), confirming that we were in a regime where the restrictions on neural resources in the VVS-net and at the output of the retina were critical to the ability of the network to perform the task. ",
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+        "type": "text",
+        "text": "3 A UNIFIED MODEL FOR CENTER-SURROUND RFS IN THE RETINA AND ORIENTED RFS IN V1 ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Here we investigate the effects of a dimensionality bottleneck at the retinal output on early visual representations in our model of the visual system. ",
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+        "text": "3.1 A DIMENSIONALITY BOTTLENECK AT THE RETINAL OUTPUT YIELDS THE EXPECTED REPRESENTATIONS IN RETINA AND V1 ",
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+        "text": "When reducing the number of neurons at the output of the retina we found that RFs with antagonistic center and surround emerged. For $N _ { B N } = 3 2$ , our control setting with no bottleneck at the retinal output, we observed mostly oriented receptive fields in the second layer of the network (fig. 2A). For $N _ { B N } = 4 , 2$ , and 1, we observed center-surround receptive fields in the second layer of the network and mostly oriented receptive fields in the third layer, which is the first layer of the ventral visual system in our model (fig. 2B). We quantified these results in App. A. The RF geometries did not depend qualitatively on the VVS-net depth $D _ { V V S }$ (results shown for $D _ { V V S } = 2 $ ), except for the shallowest VVS-net tested ${ \\cal D } _ { V V S } = 0$ , no convolutional layer, and thus no dimensionality expansion), for which the shape of emergent retinal RFs were variable across trials and difficult to interpret. These results are in good agreement with the organization of the biological visual system, where retinal RFs are center-surround and most downstream RFs in primary visual cortex (V1) are sharply oriented (Hubel, 1995), suggesting that the dimensionality bottleneck at the output of the retina is sufficient to explain these differences in representations. It is worth noting that for both conditions (bottleneck and no bottleneck), the RFs of downstream layers in the VVS-net after the first layer exhibited complex shapes that were neither clearly oriented, nor circular, and the RFs in the first layer of the retina did not appear to have any well-defined structure (data not shown). ",
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+        "text": "We then tested in our model the hypothesis of Hubel and Wiesel concerning how center-surround cells are pooled to give rise to oriented RFs in V1 (Hubel, 1995). We found that orientation-selective neurons in the VVS-net typically draw primarily from center-surround neurons in the retina-net that are aligned with the direction of the edge, with positive or negative weights corresponding to whether the polarity (light-selective / dark-selective) of the two neurons are consistent or inconsistent (fig. 2C, and App. A for a quantification). These qualitative results are in good agreement with Hubel and Wiesel’s hypothesis. Of course, this hypothesis remains to be tested in the real brain, since there is no evidence that the micro-circuitry of the brain matches that of our simulation. ",
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+        "text": "In the visual system of mammals, the main relay of visual information taking its input from the retina is the LGN (thalamus), which has center-surround RFs and a similar total number of neurons as the retinal output (Hubel, 1995). We created a network reflecting this architecture by having two lowdimensionality layers in a row instead of just one (fig. 2C). After training, we found center-surround RFs in the two layers with a bottleneck (retinal output and LGN), and oriented RFs in the next layer, corresponding to the primary visual cortex (V1). These results suggest that center-surround representations remain advantageous as long as the dimensionality of the representation remains low, and hence dimensionality expansion seems to be the crucial factor explaining the qualitative change of RFs found between LGN and V1. ",
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+        "text": "It is an interesting question to ask whether neurons in our model of the VVS are more similar to simple or complex cells (Hubel, 1995). To test this, we performed a one-step gradient ascent on the neural activity of VVS neurons with respect to the image, starting from several random initial images (App. B). If the neurons were acting as simple cells (i.e. are approximately linear in the stimulus), we would expect all optimized stimuli to converge to the same preferred stimulus. On the other hand, if the cells were complex (i.e. OR function between several preferred stimuli), we would expect the emergent preferred stimuli to depend on the exact initialization. Interestingly, we found that most neurons in the first layer of the VVS-net behaved as simple cells, whereas most neurons in the second layer of the VVS-net behaved as complex cells. Note that in biology, both simple and complex cells are found in V1. These results expose the fact that anatomical regions of visual cortex involve multiple nonlinearities and hence may map onto more than one layer of our simple model. Indeed, V1 itself is a multilayered cortical column, with LGN inputs coming in to layer 4, and layer 4 projecting to layers 2 and 3 (Hubel, 1995). Simple cells are predominantly found in layer 4 and complex cells are predominantly found in layers 2 and 3. These observations bolster the interpretation that biological V1 may correspond to multiple layers in our model. ",
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+        "text": "Local divisive normalization (i.e. local gain control) is an ubiquitous source of nonlinearity in the visual system (Geisler & Albrecht, 1992; Heeger, 1992; Deny et al., 2017). We thus tested the robustness of our main result to a more realistic model of the visual system with local normalization, by adding it at every layer of the network (App. C). We found that receptive fields still emerged as center-surround in the retina-net, and as oriented in our model of V1. We note that the local normalization slightly degraded the performance of the network on the task for all parameter settings we tried. ",
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+        "img_path": "images/98cc31250542547c9f81e82515536c39418d45eb4123699aabb6be149055e5be.jpg",
+        "image_caption": [
+            "Figure 2: Effects of a bottleneck constraint on receptive fields (RFs). All results are shown for $D _ { V V S } = 2$ . A: Examples of RFs of cells at selected layers (layers 2 and 3) of a control network with no bottleneck. No center-surround RFs appear. B: Center-surround RFs emerge at the output of the retina-net (layer 2) and oriented RFs emerge in the first layer of the VVS-net when we impose a bottleneck constraint at the output of the retina $N _ { B N } = 1$ ) C: Top: Hubel and Wiesel’s hypothesis on oriented cell formation in V1 (Hubel, 1995). Bottom: A representative example of an orientationselective neuron (bottom RF) drawing from center-surround channels (top RFs) in the previous layer with weight matrices (center) according to their polarity. Light / dark-selective regions of a receptive field, and positive / negative weights, are represented with red / blue, respectively. D: Examples of RFs in a network with an extra bottleneck layer corresponding to mammalian LGN. Center-surround RFs appear at both the retinal output and LGN layer. E: Examples of ON and OFF center-surround RFs in the untied network $( N _ { B N } = 4 )$ ). F: t-SNE clustering of the retinal neurons of the untied network (see text). Two distinct cell type clusters form corresponding to ON and OFF center-surround receptive fields. "
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+        "type": "text",
+        "text": "3.2 EMERGENCE OF ON AND OFF POPULATIONS OF CENTER-SURROUND CELLS IN THERETINA",
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+        "text": "We then verified that the emergence of center-surround RFs in the retina-net is a consequence of reducing the number of neurons at the retinal output, not of reducing the number of channels, our model’s equivalent of biological retinal cell types. In the retina, there exist 20-30 types of ganglion cells (Roska & Meister, 2014), each with a different and stereotyped receptive field, density, polarity (i.e. ON or OFF), and nonlinearities. Cells of each type tile the entire visual field like a convolutional channel in our model, so there is a direct analogy between channels in our model and ganglion cell types in the retina. In order to test whether the emergence of center-surround RFs depends on the number of types that we allow, or just on the number of neurons that we allow at the output of the retina (i.e. dimensionality bottleneck), we employed locally connected layers – equivalent to convolutional layers, but without parameter-tying between artificial neurons within a channel at different spatial locations. In this manner, we can limit the number of neurons at the retinal output without imposing a constraint on the number of cell types. Such a network contains too many parameters to be trained from scratch by gradient descent; to work around this, we trained the model stage-wise by first training our convolutional control network $N _ { B N } = 3 2$ with parameter tying) and then we trained a three-layers untied network (with bottleneck dimension $N _ { B N } = 4$ in the second layer) to reproduce the edge-like activations of the second layer of the control network. Even in the untied retina-net, in which each neuron is effectively its own channel, we found that centersurround RFs emerged (fig. 2E), indicating that center-surround RFs are the network’s preferred strategy for passing information through a dimensionality bottleneck even when no constraint on the number of cell types is imposed. We then found that the cells cluster in two distinct populations. To demonstrate this, we measured their activations in response to 10000 natural images, computed the first 20 principal components of this 10000-dimensional space, and ran t-SNE to visualize the clustering of neuron types. We found that two distinct clusters emerged, that corresponded visually to ON and OFF center-surround RFs (fig. 2F). We thus observe in our model the emergence of one of the most prominent axes of dichotomy of biological ganglion cell types, namely the classification of cells in ON and OFF populations with RFs of opposite polarity. ",
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+        "type": "text",
+        "text": "4 RETINAL REPRESENTATIONS ARE A FUNCTION OF THE NEURAL RESOURCES ALLOCATED TO THE VENTRAL VISUAL STREAM ",
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+        "text": "To what extent are retinal representations in our model shaped by the degree of neural resources allocated to downstream processing? To investigate this question, we studied the effects of varying the degree of neural resources in the VVS-net, on emergent visual representations in the retina-net. ",
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+        "text": "4.1 THE RETINA BECOMES MORE LINEAR AS BRAIN COMPLEXITY INCREASES ",
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+        "text": "As we increased the number of layers in the VVS-net, the retinal computation became more linear (fig. 3A), as measured by the ability of the raw image to linearly map onto the neural representation at the retinal output (see methods, and App. F for a visualization of retinal representation as VVS-net depth increases). This observation is consistent with the current state of knowledge of the differences found in retinal representations across vertebrate species with different brain sizes. The linearization of the retinal response with increased brain complexity was true for different values of bottleneck $N _ { B N }$ . However, when we did not use any bottleneck ( $N _ { B N } = 3 2$ ), the trend became non-monotonic, with a peak in linearity of the response when the VVS-net had 1 conv layer (data not shown). Another interesting phenomenon to note is that linearity of the retinal response decreased as we increased the number of channels in the bottleneck, at any fixed brain depth (fig. 3A). ",
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+        "text": "The two main sources of nonlinearity in the retina are thought to be the inner retinal rectifications (bipolar and amacrine cells, corresponding to the first rectified layer in our model) and the ganglion cell rectification (corresponding to the second rectified layer in our model). As we decreased VVSnet depth, we observed that the retinal response became more nonlinear. Is this increase in response nonlinearity due to the first or second stage of nonlinearity in our retina-net? To test this, we plotted the real response against the response predicted by a purely linear model for the most shallow and for the deepest VVS-nets tested (fig. 3B). If the linear prediction were inaccurate because of the first stage of nonlinear processing in the retina-net, we would expect the points on the scatter plot to be scattered around the unit line. If the prediction error were due to the second-stage of nonlinearity, we would expect the linear approximation to make incorrect negative predictions for inactive neurons. In practice, we found that the prediction error of the linear model was partly explained by both stages of nonlinearity in the retina-net model, predicting that both inner retinal nonlinear processing and ganglion cell rectifications should be more pronounced in animals with fewer neural resources in their visual cortices. ",
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+        "type": "text",
+        "text": "4.2 THE RETINAL REPRESENTATION IS THE RESULT OF A TRADE-OFF BETWEEN INFORMATION TRANSMISSION AND FEATURE EXTRACTION ",
+        "text_level": 1,
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+        "text": "Why would retinal representations be more linear when the subsequent ventral visual stream has more resources? One hypothesis is that with a restricted number of neurons, the retina must trade-off between the two incentives of (1) compressing visual information in order to transmit it to down-1 Channel 1 Channel stream layers and (2) extracting nonlinear features from the scene to start disentangling the manifolds4 ChannelsRaw Pixels 4 ChannelsRaw Pixels corresponding to different classes of objects (Chung et al., 2018a;b). According to this hypothesis,VVS-net depth VVS-net depth VVS-net depth when the VVS is shallow, the priority of the retina should be to work toward extracting relevantB CE F  D E features. When the VVS is deep, the priority of the retina should be to transmit as much visual information as possible for downstream processing. We validated this hypothesis in two ways in our1 Channel model. ",
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+        "img_path": "images/5ff93b6301757e10e32a70529e050720018a2e3569dd5a57f35d7dd68daa863f.jpg",
+        "image_caption": [
+            "Figure 3: Emergent retinal representations are function of the depth of downstream visual cortices. All error bars represent the $9 5 \\%$ confidence interval about the mean (all simulations were repeated over 10 networks trained from random initial conditions). Three stars indicate t-test significance $\\scriptstyle ( \\mathbf { p } < 0 . 0 0 1$ ). A: Linearity of the retinal response increases with the number of layers in the VVSnet. Note that it also decreases with the number of cells at the retinal output (different lines). B: Responses of example retina-net output cell to natural images, vs. best linear fit prediction from 1 Channel \\*\\*\\*raw image, for most (top) and least (bottom) deep VVS-nets. Nonlinearity arises from two sources: 2 Channels Raw Pixelsrectification within the retina-net (corresponds to the spread of the bulk of the point cloud) and rectification at the retina-net output (corresponds to inactive neurons being incorrectly predicted to have negative activations). C: Quality of image reconstruction from the retinal representation as a function of VVS-net depth. The retinal representation retains more information about the raw image for deep VVS-nets. D: Linear separability of classes of objects at the retinal output, as a function of VVS-net depth. Dashed line indicates separability of classes of images from the raw image pixels. Classes are less separable at the retinal output for deeper VVS-nets. E: Performance on CIFAR-10 for a two-layer densely connected network taking its input from the retina-net or from a raw image. Class information is more accessible from retinal representation. F: Class separability at all layers of network for a deep VVS-net $D _ { V V S } = 4 $ ) with and without bottleneck $\\boldsymbol { N } _ { B N } = 1$ and $N _ { B N } = 3 2$ ). Retinal representation of bottleneck network has low separability. However, the first layer of the VVS-net has high separability (see text). "
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+        "type": "text",
+        "text": "First we showed that the retinal representation retained more information about the image as VVSnet complexity increased (fig. 3C). To estimate information retention, we trained a linear decoder (see methods) from the output of the retina to reconstruct the image and we measured the reconstruction error. The reconstruction error provided a lower bound on the information that the retina retained about the stimulus (note that more information for reconstruction might be accessible by a nonlinear decoder). This result corroborated our hypothesis that, as the VVS-net becomes more complex, the retinal representation gets better at retaining visual information for further processing by the VVS-net. ",
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+        "text": "Second, we found that different classes of objects of CIFAR-10 (e.g. trucks, frogs) were more linearly separable from the retina-net representation when the VVS-net was shallow than when it was deep (fig. 3D). To measure linear separability of manifolds, we trained a linear SVM decoder to separate all pairs of classes and evaluated the performance of the SVM classifier on held-out images (see methods). Moreover, we showed that a VVS-net consisting of two fully connected layers only (no convolutional layers) equipped and trained end-to-end with a retina with a tight bottleneck $N _ { B N } = 1$ (dimensionality of retinal output matches dimensionality of the input image) performed better at image recognition than the same VVS-net trained without a retina-net, taking raw images as input (fig. 3E). Both these results corroborate our hypothesis that retinas followed by a simple cortex performs meaningful feature extraction, whereas retinas followed by more complex visual cortices prioritize non-lossy encoding, postponing feature extraction to downstream layers that are better equipped to do it. ",
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+        "type": "text",
+        "text": "Next, we show that within a single network, each retinal channel is trading-off between (1) linearly transmitting visual information to the brain, and (2) extracting relevant features for the object classification task. For 10 instantiations of a network with a retinal bottleneck containing 4 channels, we represented the linearity of each of these 4 channels against the linear separability of object categories obtained from each of these representations. We found, across all networks, a systematic negative correlation between linearity and linear separability across all 4 channels (App. D). Again, this result strongly suggests that extracting features and transmitting visual information are indeed two competing goals shaping representations in our model of the retina. ",
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+        "text": "In the case of the deepest VVS-nets tested, the retinal processing was quasi-linear for the tightest bottleneck (var.expl. $= 0 . 9$ , $N _ { B N } = 1$ , fig. 3A). One might take this result to suggest that the retinanet in such models does little more than copy image information. However the very first layer of the VVS-net after the retina disentangled classes (as measured by linear separability) almost as well as the second layer of a VVS-net without a retina (fig. 3F), suggesting that the retinal representation, while only moderately linearly separable itself, is especially transformable into a representation with a high linear separability. This result suggests that even when the retina-net is quasi-linear, it can still participate in extracting relevant features for downstream processing by the brain. The increased separability allowed by the retinal pre-processing for this deep VVS-net could be due to (1) the linear processing or (2) the slightly nonlinear part of the retinal processing (3) a combination of both linear and nonlinear processing. To distinguish between these hypotheses, we replaced the true retinal processing by its best linear approximation, retrained the VVS-net on the output of this linearized retina, and tested whether separability was as high as with the true retinal processing (App. E). We found that the first layer trained on the output of the linearized retinal representation was indeed much more separable than the first layer of the control network (trained directly on natural images) at separating classes of objects, suggesting that the linear operation done by the retina does indeed play a crucial role in making the representation especially separable for subsequent layers. ",
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+        "type": "text",
+        "text": "5 METHODS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To estimate the linearity of the response of retinal neurons, we fit a linear model to predict the neural response from the image on 8,000 images. In order to prevent overfitting, we regularized the linear weights with an L2 penalty and optimized the weights using ridge regression. The value of the penalty term was chosen by 10-fold cross-validation on the training set. We then measured the Pearson correlation between the linearized responses and original model responses on a testing set of 2,000 images. ",
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+    {
+        "type": "text",
+        "text": "To estimate the information about the input image retained by the retinal output representation, we fit a linear model to reconstruct the image from the (fixed) outputs of the trained retina-net of interest. All numerical figures given are variance-explained results on the held-out test set. ",
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+        "type": "text",
+        "text": "To estimate the linear separability of classes of objects from the neural representation, we trained an SVM classifier between all pairs of classes on half of the testing set of CIFAR-10 (1,000 images that were not used to train the network), and we tested the performance of the SVM classifier on 1,000 held-out images from the testing set, as measured by the percentage of images classified correctly. We then averaged the performance of the SVM across all pairs of classes to obtain the linear separability score. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A unified theoretical account for the structural differences between the receptive field shapes of retinal neurons and V1 neurons has until now been beyond the reach of efficient coding theories. Karklin & Simoncelli (2011) found that efficient encoding of images with added noise and a cost on firing rate produce center-surround RFs, whereas the same task without noise produces edge detectors. However, this observation (as they note) does not explain the discrepancy between retinal and cortical representations. Vincent et al. (2005) propose a different set of constraints for the retina and V1, in which the retina optimizes for a metabolic constraint on total number of synapses, whereas V1 optimizes for a constraint on total firing rate. It is not clear why each of these constraints would predominate in each respective system. Here we show that these two representations can emerge from the requirement to perform a biologically relevant task (extracting object identity from an image) with a bottleneck constraint on the dimensionality of the retinal output. Interestingly, this constraint differs from the ones used previously to account for center-surround RFs (number of synapses or total firing rate). It is worth noting that we unsuccessfully tried to reproduce the result of Karklin & Simoncelli (2011) in our network, by adding noise to the image and applying an L1 regularization to the retina-net activations. In our framework (different than the one of Karklin & Simoncelli (2011) in many ways), the receptive fields of the retina-net without bottleneck remained oriented across the full range of orders of magnitude of noise and L1 regularization that permitted successful task performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There is a long-standing debate on whether the role of the retina is to extract relevant features from the environment (Lettvin et al., 1959; Gollisch & Meister, 2010; Roska & Meister, 2014), or to efficiently encode all visual information indistinctly (Barlow, 1961; Atick & Redlich, 1990; 1992). In this work, we show that our model of the visual system, trained on the same task and with the same input statistics, can exhibit different retinal representations depending on the degree of neural resources allocated to downstream processing by the ventral visual stream. These results suggest the hypothesis that, despite its conserved structure across evolution, the retina could prioritize different computations in different species. In species with fewer brain resources devoted to visual processing, the retina should nonlinearly extract relevant features from the environment for object recognition, and in species with a more complex ventral visual stream, the retina should prioritize a linear and efficient transmission of visual information for further processing by the brain. Although all species contain a mix of quasi-linear and nonlinear cell types, the proportion of quasi-linear cells seems to vary across species. In the mouse, the most numerous cell type is a two-stage nonlinear feature detector, thought to detect overhead predators (Zhang et al., 2012). In contrast, the most common ganglion cell type in the primate retina is fairly well approximated by a linear filter (midget cells, $50 \\%$ of all cells and ${ > } 9 5 \\%$ in the central retina (Roska & Meister, 2014; Dacey, 2004)). Note however that two-stage nonlinear models are also present in larger species, such as cat Y-type cells and primate parasol cells (Crook et al., 2008), making it difficult to make definitive statements about inter-species differences in retinal coding. To gain a better understanding of these differences, it would be useful to collect a dataset consisting of recordings of complete populations of ganglion cells of different species in response to a common bank of natural scenes. ",
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+    {
+        "type": "text",
+        "text": "A related question is the role of the parcellation of visual information in many ganglion cell types at the retinal output. A recent theory of efficient coding has shown that properties of midget and parasol cells in the primate retina can emerge from the objective of faithfully encoding natural movies with a cost on the total firing rate traversing the optic nerve (Ocko et al., 2018). On the other hand, many cell types seem exquisitely sensitive to behaviorally relevant features, such as potential prey or predators (Gollisch & Meister, 2010). For example, some cell types in the frog are tuned to detect moving flies or looming predators (Lettvin et al., 1959). It is an intriguing possibility that different cell types could subserve different functions within a single species, namely efficient coding of natural scenes for some types and extraction of behaviorally-relevant features for others. In this study we allowed only a limited number of cell types (i.e. convolutional channels) at the retinal output (1 to 4), in order to have a dimensionality expansion between the retinal representation and the representation in the ventral visual stream (32 channels), an important condition to see the retinal center-surround representation emerge. By using larger networks with more channels in the retina-net and the VVS-net, we could study the emergence of a greater diversity of neuron types in our retina-net and compare their properties to real retinal cell types. It would also be interesting to extend our model to natural movies. Indeed, most feature detectors identified to date seem to process some form of image motion: wide-field, local or differential (Roska & Meister, 2014). Adding a temporal dimension to the model would be necessary to study their emergence. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In conclusion, by studying emergent representations learned by a deep network trained on a biologically relevant task, we found that striking differences in retinal and cortical representations of visual information could be a consequence of the anatomical constraint of transmitting visual information through a low-dimensional communication channel, the optic nerve. Moreover, our computational explorations suggest that the rich diversity of retinal representations found across species could have adaptively co-evolved with the varying sophistication of subsequent processing performed by the ventral visual stream. These insights illustrate how deep neural networks, whose creation was once inspired by the visual system, can now be used to shed light on the constraints and objectives that have driven the evolution of our visual system. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Lane McIntosh, Niru Maheswaranathan, Aran Nayebi, SueYeon Chung, Vardan Papyan, Nora Brackbill, E.J. Chichilnisky for useful discussions and Stephen Baccus for his comments that greatly improved the manuscript. S.G. thanks the Burroughs-Wellcome, McKnight, James S. McDonnell and Simons foundations for support. ",
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+        "bbox": [
+            174,
+            746,
+            825,
+            803
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "bbox": [
+            176,
+            103,
+            263,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A QUANTIFICATION OF RECEPTIVE FIELD ISOTROPY IN RETINA AND V1 ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            138,
+            784,
+            154
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/f9a61d69ffda3d2cbfb4ae39b653f0ce63f4698172fdda19fe3fb4da33cec1bc.jpg",
+        "image_caption": [
+            "Figure 4: A: Left: Schematic re-illustrating the architecture of a vanilla (no bottleneck) network and showing examples oriented RFs in its second layer. Center: Visualization of average RF isotropy for cells in the second layer of a vanilla convolutional network $( N _ { B N } = 1$ , $D _ { V V S } = 2$ ). Orange error bars indicate $9 5 \\%$ confidence intervals. Right: Visualization of RF isotropy for ten example RFs from the same network architecture. B: Left: Schematic re-illustrating the architecture of the retina-net $+ \\mathrm { \\Delta V V S }$ -net model $( N _ { B N } = 1 , D _ { V V S } = 2 )$ and showing example center-surround RFs at the retina-net output and oriented RFs in the following layer (V1). Center and right: Same RF isotropy visualizations as in part A. C: Left: re-illustration of V1 RFs pooling in oriented fashion from center-surround retinal RFs $( N _ { B N } = 1 , D _ { V V S } = 2 )$ . Right: Same isotropy visualizations as in panel A carried on the weight matrix from retina to V1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            180,
+            825,
+            425
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The following analysis corroborates our qualitative observation that a dimensionality bottleneck in the retina-net yields center-surround retinal receptive fields and oriented, edge-detecting receptive fields in the first layer of the VVS-net (V1). For a given receptive field, we quantified its orientedness as follows: we displayed rectangular bar stimuli of all possible combinations of width, orientations and spatial translations that fit in the input image window. Among all these combinations, we selected the bar stimulus width, orientation, and translation that yielded the strongest response from the RF. Bars with the same width as the best stimuli were presented at all orientations and translations, and for each orientation, we select the strongest response it produced (across all translations). In this manner we obtained a measure of the strength of a receptive field’s preference for all orientations. ",
+        "bbox": [
+            173,
+            597,
+            825,
+            723
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We measured the strength of each RF preference (maximum strength of response) for its preferred orientation and for the orthogonal orientation, and computed the ratio of these strengths. Completely isotropic filters would be expected to give a ratio of 1, while oriented filters should give higher ratios. Note however that some deviation from 1 may indicate noise in the filter rather than true orientedness. For each network layer, we averaged this ratio across filters (for convolutional layers with multiple layers) and trials (re-training of the same neural network architecture with different random initializations). We found that the average ratios were $1 . 5 6 ( \\pm 0 . 2 2 )$ for the retinal output, $3 . 0 5 ( \\pm 0 . 3 0 ) $ for the first VVS-net layer, and $2 . 5 7 ( \\pm 0 . 2 7 )$ for the second VVS-net layer, where error margins given are $9 5 \\%$ confidence intervals. To help assess whether retinal RFs were more isotropic than expected by chance, we compared them to receptive fields composed of random Gaussian noise as a baseline. These give an average ratio (as computed above) of $\\bar { 1 . 9 7 } ( \\pm 0 . 0 8 )$ , significantly higher than that for retinal RFs. Furthermore, the standard deviation of RF preference across orientations was significantly lower for the retinal RFs $( 0 . 1 1 8 \\pm 0 . 0 3 6 )$ than for random RFs $( 0 . 1 7 7 \\pm 0 . 0 0 7 )$ , also indicating that retinal RFs were more isotropic than expected by chance. ",
+        "bbox": [
+            174,
+            729,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We also plot the average RF preference for different orientations at each layer to more comprehensively assess the isotropy of RFs at each network layer. To aggregate results across multiple trials and filters, we rotated the coordinates of each receptive field such that its preferred orientation was vertical, and averaged our results across filters and trials. (See Figure 4). ",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The results confirm our qualitative observations that (1) RFs in the second layer of a vanilla network $( N _ { B N } = 3 2 )$ ) are highly oriented (Figure 4A) (2) RFs in the second layer (retina output) of a bottleneck network $N _ { B N } = 1 \\textgreater$ ) are much more isotropic, consistent with center-surround RFs (Figure 4B top), and (3) RFs in the layer immediately following the retina-net in the bottleneck network are oriented (Figure 4B bottom). ",
+        "bbox": [
+            174,
+            166,
+            823,
+            236
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We also quantitatively corroborate our observation that oriented receptive fields in the V1 layer pool input from oriented arrays of center-surround filters in the retina-net output layer. We apply our method of isotropy quantification described above to the weight matrix for each input-output filter combination in the V1 convolutional layer. We find that this weight matrix itself exhibits orientedness across filters and trials, confirming our observation (Figure 4C). ",
+        "bbox": [
+            173,
+            243,
+            825,
+            313
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B SIMPLE AND COMPLEX CELLS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            333,
+            459,
+            348
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/d179773cc49e10c02fab446959e80cc8f6ae44ade2c76b6fcb6f3e906229eb9b.jpg",
+        "image_caption": [
+            "Figure 5: A: Visualizations of retina-net output RFs for an example network $( N _ { B N } = 1 , D _ { V V S } = 2 )$ using different random initialization, as described in the text. B: Same as A, for the first layer of the VVS-net, and showing 5 of the layer’s 32 channels on the x axis. C: Same as B, for the second layer of the VVS-net. In contrast to the first layer, the emergent preferred stimuli are always different across different initializations, indicative of a complex-cell like behavior. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            363,
+            821,
+            628
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To investigate whether neurons in our model’s early layers more closely resembled simple or complex cells, we performed the following analysis. As before, we obtained local linear approximations of receptive fields by computing the gradient in input space with respect to the response of a given neuron. Rather than beginning with a blank input, we ran multiple trials with different randomly initialized inputs. A purely linear cell would give the same result no matter the initialization; a somewhat nonlinear but still “simple” cell is expected to give similar results across initializations. A “complex” cell is expected to give different RF visualizations for different random inputs, reflecting multiple peaks in its response as a function of input. In Figure 5 we show examples of receptive fields at different layers of our retina-net $+ \\mathrm { \\Delta V V S }$ -net model (with $N _ { B N } = 1 , D _ { V V S } = 2 )$ for different random intializations of the image (uniform random in [0, 1]). The retina-net output and first VVS-net layer exhibit “simple” behavior, but the second VVS-net layer exhibits observably “complex” behavior. To quantify this effect, we measure the average (across filters within each layer and re-trainings of the same network architecture) standard deviation of computed RFs (normalized to the range [0, 1]) for each network layer. We found that the average standard deviations were $7 . 9 ( \\pm 1 . 1 ) \\times 1 0 ^ { - 3 }$ , $1 5 . 4 ( \\pm 0 . 8 ) \\times 1 0 ^ { - 3 }$ , and $3 5 . 9 ( \\pm 0 . 8 ) \\times 1 0 ^ { - 3 }$ for the retina-net output, first VVS-net layer, and second VVS-net layer, respectively, where the margins of error given are $9 5 \\%$ confidence intervals. These results corroborate the observation of significantly more complex behavior in the second VVS-net layer, mirroring the biological phenomenon in which complex cells pool from simple cells in $\\mathrm { V } 1$ . ",
+        "bbox": [
+            174,
+            729,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            102,
+            825,
+            174
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C EFFECTS OF LOCAL RESPONSE NORMALIZATION ON EARLY VISUAL malization FigurREPRESENTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            194,
+            764,
+            227
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/679c7d4d1bf4021059a7f1a2d21611f59969d6766478b59e5a97dcfba385ad75.jpg",
+        "image_caption": [
+            "Figure 6: Example RFs from the bottleneck network $( N _ { B N } = 1 , D _ { V V S } = 2$ without (A) and with (B) local response normalization (i.e. local gain control). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            341,
+            247,
+            651,
+            402
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We tested the robustness of our first main finding – that a bottlenecked retina-net $+ \\mathrm { \\nabla { V V S } }$ -net model yields center-surround receptive fields in the retina and oriented receptive felds in $\\mathrm { { V } 1 - }$ to the use of biologically realistic local response normalization at every layer of the network. In particular, we normalized the output $x$ of each channel (row $r$ , column $c$ ) of each layer as follows (during training and testing): ",
+        "bbox": [
+            173,
+            465,
+            826,
+            535
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c8ea92573edc53e5e3ab8ff39498e0d3fb07bc293726c6a62d2761728d6c0761.jpg",
+        "text": "$$\nx _ { r , c } \\gets \\frac { x _ { r , c } } { \\Big ( k + \\alpha \\sum _ { r ^ { \\prime } \\in [ r - \\frac { n } { 2 } , r + \\frac { n } { 2 } ] , c ^ { \\prime } \\in [ c - \\frac { n } { 2 } , c + \\frac { n } { 2 } ] } x _ { r ^ { \\prime } , c ^ { \\prime } } \\Big ) ^ { \\beta } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            532,
+            678,
+            577
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where the subscripts of $x$ indicate the spatial location (row/column), and $k , \\alpha _ { \\mathrm { { ; } } }$ , ad $\\beta$ are constants. We used $k = 2$ , $\\beta = 0 . 5$ and $\\beta = 0 . 7 5$ , and $\\alpha = 5 \\times 1 0 ^ { - 4 }$ and $\\alpha = 5 . 0$ . All parameter settings tested yielded RFs with the same qualitative properties as in the model without normalization. Figure 6 shows example RFs from the no-normalzation model next to example RFs from the normalization model with $k = 2 , \\beta = 0 . 5 , \\alpha = 5 . 0$ . ",
+        "bbox": [
+            174,
+            579,
+            825,
+            648
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D RETINAL CELL TYPES TRADE OFF BETWEEN LINEAR INFORMATION TRANSMISSION AND NONLINEAR FEATURE EXTRACTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            767,
+            136
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/975e7596308fe16a0a7c943657318acd6b90291b6140f066a27a5a13c5f647da.jpg",
+        "image_caption": [
+            "Fig XX Linearity vs. separability of retina-net output channels (as in XX), Figure 7: Linearity vs. Class separability for each retina-net output channels (i.e. bottleneck layer). network has a VVS-Net depth of 4 and a bottleneck size of 4. VVS-Net depth is equal to 4. Each network has a bottleneck size of 4 channels (i.e. $N _ { B N } { = } 4 )$ . DisDistributions are plotted across 8 network instances; each point represents a single channel, colored according to its network. Here, we tributions are plotted across 10 network instances; each point represents a single channel, colored can see that the tradeoff between efficient coding and feature extraction also happens within the retina-nets of individual networks.according to its network. The negative slope suggests that there is trade-off between linearly transmitting visual information for downstream processing (i.e. efficient coding) and extracting useful features for the object recognition task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            303,
+            154,
+            692,
+            306
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "E LINEARIZED RETINA ALSO INCREASES SEPARABILITY IN SUBSEQUENT LAYERS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            436,
+            794,
+            469
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/be127bef6a36f1cc24b24501c50f4d931f9eb8892dfaa764893c38ef230465df.jpg",
+        "image_caption": [
+            "representation of bottleneck network has low separability. However, the Figure 8: Class separability at all layers of network for a deep VVS-net $( D _ { V V S } = 4 )$ ) with and without bottleneck $\\boldsymbol { N } _ { B N } = 1$ t layer oarability and $N _ { B N } = 3 2$ h separability. We additionally plot the leneck (NBN = 1) network (see test) as a ). Retinal representation of bottleneck network has low function of layer. That the jump in linear separability between layers 2,3 survives linearization suggests that the main effect of retinal processing separability. However, the first layer of the VVS-net has high separability. We additionally plot the in this network is whseparability of the linearized bottleneck $N _ { B N } = 1$ on-linear processing.) network (see test) as a function of layer. That the jump in linear separability between layers 2,3 survives linearization suggests that the main effect of retinal processing in this network is whitening (see Fig. 9) rather than nonlinear processing. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            341,
+            493,
+            656,
+            627
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In the case of the deepest VVS-nets tested, the retinal processing was quasi-linear for the tightest bottleneck (var.expl. $= 0 . 9$ , $N _ { B N } = 1$ , fig. 3A). However the very first layer of the VVS-net after the retina disentangled classes (as measured by linear separability) almost as well as the second layer of a VVS-net without retina (fig. 3F), suggesting that the retinal representation, while only moderately linearly separable itself, is especially transformable into a representation with a high linear separability. To determine to what degree this increased separability was due to (1) the linear processing or (2) the slightly nonlinear part of the retinal processing, we performed an ablation experiment to eliminate factor (2). We first replaced the true retinal processing by its best approximation by a onelayer linear convolution (of sufficient filter width to correspond to two convolutional layers with 9 by 9 filters). After this linearization process, we retrained the VVS-net using the linearized retinal representation as input, keeping the linearized retina weights frozen. We found that the first layer trained on the output of the linearized retinal representation was indeed much better than the first layer of the control network (trained directly on natural images) at separating classes of objects (Fig. ",
+        "bbox": [
+            173,
+            742,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "8), suggesting that the linear operation done by the retina does indeed play a crucial role in making the representation especially separable for subsequent layers. Visualization of retinal processing in App. F suggest that whitening is an important part of this linear processing. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "F RETINAL REPRESENTATION VISUALIZATION AS A FUNCTION OF VVS-NET DEPTH FOR BOTTLENECK $N _ { B N } = 1$ ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            167,
+            820,
+            200
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/e5dd316498caac5948f6a2c039010178896c096525cb6e3199a8661488b1d0b8.jpg",
+        "image_caption": [
+            "examples (x axis) as a function of VVS-Net depth (y axis)Figure 9: Visualization of the output of the retina-net (one-channel-bottleneck, i.e. $N _ { B N } = 1$ ) for different images from the testing set (x-axis) as a function of VVS-net depth (y-axis). Each pixel intensity of the retinal image is proportional to the activation of the corresponding neuron of the retina, where light shades indicate high activities and dark shades low activities. While retinas for every VVS-net depth appear to whiten the input, we can see that the retinal image is more and more processed and less and less recognizable as VVS-net depth decreases. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            239,
+            818,
+            351
+        ],
+        "page_idx": 16
+    }
+]

parse/train/Syzn9i05Ym/Syzn9i05Ym.md

@@ -0,0 +1,564 @@
+# LEARNING NEURAL RANDOM FIELDS WITH INCLUSIVE AUXILIARY GENERATORS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Neural random fields (NRFs), which are defined by using neural networks to implement potential functions in undirected models, provide an interesting family of model spaces for machine learning. In this paper we develop a new approach to learning NRFs with inclusive-divergence minimized auxiliary generator - the inclusive-NRF approach, for continuous data (e.g. images), with solid theoretical examination on exploiting gradient information in model sampling. We show that inclusive-NRFs can be flexibly used in unsupervised/supervised image generation and semi-supervised classification, and empirically to the best of our knowledge, represent the best-performed random fields in these tasks. Particularly, inclusiveNRFs achieve state-of-the-art sample generation quality on CIFAR-10 in both unsupervised and supervised settings. Semi-supervised inclusive-NRFs show strong classification results on par with state-of-the-art generative model based semi-supervised learning methods, and simultaneously achieve superior generation, on the widely benchmarked datasets - MNIST, SVHN and CIFAR-10.
+
+# 1 INTRODUCTION
+
+One of the core research problems in machine learning is learning with probabilistic models, which can be broadly classified into two classes - directed and undirected1 (Koller & Friedman, 2009). Significant progress has been made recently on learning with deep generative models (DGMs), which generally refer to models with multiple layers of stochastic or deterministic variables. There have emerged a bundle of deep directed generative models, such as variational AutoEncoders (VAEs) (Kingma & Welling, 2014), generative adversarial networks (GANs) (Goodfellow et al., 2014) and so on. In contrast, undirected generative models (also known as random fields (Koller & Friedman, 2009), energy-based models (LeCun et al., 2006)) received less attention with slow progress. This is presumably because fitting undirected models is more challenging than fitting directed models. In general, calculating the log-likelihood and its gradient is analytically intractable, because it involves the expectation with respect to (w.r.t.) the model distribution.
+
+In this paper, we aims to advance the learning of neural random fields (RFs) which use neural networks with multiple (deterministic) layers to define the potential function2 $u _ { \theta } ( x )$ over observation $x$ with parameter $\theta$ . The probability distribution $p _ { \theta } ( x ) \propto \exp ( u _ { \theta } ( x ) )$ is then defined by normalizing the exponentiated potential function. This type of RFs has been studied several times in different contexts, once called deep energy models (DEMs) (Ngiam et al., 2012; Kim & Bengio, 2016), descriptive models (Xie et al., 2016), generative ConvNet (Dai et al., 2014), neural random field language models (Wang & Ou, 2017). For convenience, we refer to such models as neural random fields (NRFs) in general.
+
+An important method of maximum likelihood (ML) learning of random fields is called stochastic maximum likelihood (SML) (Younes, 1989), which approximates the model expectations by Monte Carlo sampling for calculating the gradient. A recent progress in learning NRFs as studied in Kim & Bengio (2016); Xie et al. (2016); Wang & Ou (2017); Kuleshov & Ermon (2017) is to pair the target random field $p _ { \theta }$ with an auxiliary directed generative model (often called generator) $q _ { \phi } ( x )$ parameterized by $\phi$ , which approximates sampling from the target random field. Learning is performed by maximizing the log-likelihood of training data under $p _ { \theta }$ or some bound of the log-likelihood, and simultaneously minimizing some divergence between the target random field $p _ { \theta }$ and the auxiliary generator $q _ { \phi }$ . Different learning algorithms differ in the objective functions used in the joint training of $p _ { \theta }$ and $q _ { \phi }$ , and thus have different computational and statistical properties (partly illustrated in Figure 1). For example, minimizing the exclusive-divergence $K L [ q _ { \phi } | | p _ { \theta } ] \triangleq$ $\bar { \int { q _ { \phi } \log { \left( { q _ { \phi } } / { p _ { \theta } } \right) } } }$ w.r.t. $\phi$ , as employed in Kim & Bengio (2016), involves the intractable entropy term and tends to enforce the generator to seek modes, yielding missing modes. There are also other factors, e.g. modeling discrete or continuous data, different choices of the target RF and the generator, which lead to different algorithms. We leave detailed comparison and connection of our approach with existing studies to section 3 (related work).
+
+In this paper, we propose to use inclusive-divergence minimized auxiliary generators (section 2.1). And particularly for continuous data (e.g. images), we propose to use SGLD (stochastic gradient Langevin dynamics (Welling & Teh, 2011)) and SGHMC (stochastic gradient Hamiltonian Monte Carlo (Chen et al., 2014)) to exploit gradient information in model sampling with solid theoretical examination (section 2.2). The new approach, abbreviated as the inclusive-NRF approach, offers some advantages over previous methods. First, minimizing the inclusive-divergence $K L [ p _ { \theta } | | q _ { \phi } ] \triangleq$ $\int p _ { \theta } \log { ( p _ { \theta } / \bar { q } _ { \phi } ) }$ w.r.t. $\phi$ avoids the annoying entropy term and tends to drive the generator to cover modes of the target density $p _ { \theta }$ . The SGLD/SGHMC sampling further pushes the samples towards the modes of $p _ { \theta }$ . Presumably, this helps to produce Markov chains that mix fast between modes and facilitate model learning. Second, the new approach enables us to flexibly use NRFs in unsupervised/supervised image generation and semi-supervised classification (section 2.3), and empirically to the best of our knowledge, represents the best-performed random fields in these tasks.
+
+The main contributions of this paper can be summarized as follows:
+
+• We develop the inclusive-NRF approach, which learns NRFs with inclusive auxiliary generators and particularly for continuous data, exploits gradient information in model sampling with solid theoretical examination. • Inclusive-NRFs achieve state-of-the-art sample generation quality, measured by both Inception Score (IS) and Frechet Inception Distance (FID). On CIFAR-10, we obtain unsupervised IS 8.28 (FID 20.9) and supervised IS 9.06 (FID 18.1), both using unconditional generation. • Semi-supervised inclusive-NRFs show strong classification results on par with state-ofthe-art DGM-based semi-supervised learning (SSL) methods, and simultaneously achieve superior generation, on the widely benchmarked datasets - MNIST, SVHN and CIFAR-10.
+
+# 2 THE INCLUSIVE-NRF APPROACH
+
+Consider a random field for modeling observation $x$ with parameter $\theta$ :
+
+$$
+p _ { \theta } ( x ) = { \frac { 1 } { Z ( \theta ) } } \exp \left[ u _ { \theta } ( x ) \right]
+$$
+
+where $\begin{array} { r } { Z ( \theta ) = \int \exp ( u _ { \theta } ( x ) ) d x } \end{array}$ is the normalizing constant, $u _ { \theta } ( x )$ is the potential function3 which assigns a scalar value to each configuration of random variable $x$ . The general idea of neural random fields (NRFs) is to implement $u _ { \theta } ( \bar { x } ) : \mathbb { R } ^ { d _ { x } }  \mathbb { R }$ , by a neural network, taking the multi-dimensional $x \in \mathbb { R } ^ { d _ { x } }$ as input and outputting the scalar $u _ { \theta } ( x ) \in \mathbb { R }$ . In this manner, we can take advantage of the representation power of neural networks for RF modeling. It is usually intractable to maximize the data log-likelihood $l o g p \hat { \theta } ( \tilde { x } )$ for observed $\tilde { x }$ , since the gradient involves expectation w.r.t. the model distribution, as shown below:
+
+$$
+\nabla _ { \boldsymbol { \theta } } \log { p _ { \boldsymbol { \theta } } ( \boldsymbol { \tilde { x } } ) } = \nabla _ { \boldsymbol { \theta } } u _ { \boldsymbol { \theta } } ( \boldsymbol { \tilde { x } } ) - E _ { p _ { \boldsymbol { \theta } } ( \boldsymbol { x } ) } \left[ \nabla _ { \boldsymbol { \theta } } u _ { \boldsymbol { \theta } } ( \boldsymbol { x } ) \right]
+$$
+
+# 2.1 INTRODUCING INCLUSIVE-DIVERGENCE MINIMIZED AUXILIARY GENERATORS
+
+In this paper, we further develop NRF learning with auxiliary generators. We are mainly concerned with modeling fixed-dimensional continuous observations $\bar { \boldsymbol { x } _ { \mathrm { ~ \in ~ } } } \mathbb { R } ^ { d _ { x } }$ (e.g. images), and choose a
+
+<table><tr><td>Algorithm1Learning NRFs with inclusive auxiliary generators</td><td></td></tr><tr><td>repeat Sampling: Draw a minibatch M= {(xi,x𝑖,h𝑖),i=1,..: |M|} from p(x)pe(x)q(h|x) (see</td><td rowspan="3"></td></tr><tr><td>Algorithm 2);</td></tr><tr><td>Updating: 1</td></tr><tr><td>Update θ by ascending: 1∑(a,x,h)~M[Vθuθ(x)- Vθuθ(x); M</td><td rowspan="3"></td></tr><tr><td>Update Φ by ascending: ∑(x,x,h)~M V logq(x,h);</td></tr><tr><td>until convergence</td></tr></table>
+
+directed generative model, $q _ { \phi } ( x , h ) \triangleq q ( h ) q _ { \phi } ( x | h )$ , for the auxiliary generator, which specifically is defined as follows4:
+
+$$
+\begin{array} { l } { h \sim \mathcal { N } ( 0 , I _ { h } ) , } \\ { x = g _ { \phi } ( h ) + \epsilon , \epsilon \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I _ { \epsilon } ) , } \end{array}
+$$
+
+where $g _ { \phi } ( h ) : \mathbb { R } ^ { d _ { h } }  \mathbb { R } ^ { d _ { x } }$ is implemented as a neural network with parameter $\phi$ , which maps the latent code $h$ to the observation space. $I _ { h }$ and $I _ { \epsilon }$ denote the identity matrices, with dimensionality implied by $h$ and $\epsilon$ respectively. Drawing samples from the generator $q _ { \phi } ( x , h )$ is simple as it is just ancestral sampling from a 2-variable directed graphical model.
+
+Suppose that data $\mathcal { D } = \{ \tilde { x } _ { 1 } , \cdots , \tilde { x } _ { n } \}$ , consisting of $n$ observations, are drawn from the true but unknown data distribution $p _ { 0 } ( \cdot )$ . 1n Pnk=1 δ(˜x − x˜k) denotes the empirical data distribution. Then we formulate the maximum likelihood learning of $p _ { \theta } ( x )$ with the inclusive-divergence minimized generator $q _ { \phi } ( x )$ as optimizing5
+
+$$
+\{ \begin{array} { l l } { \underset { \theta } { \operatorname* { m i n } } K L [ \tilde { p } ( \tilde { x } ) | | p _ { \theta } ( \tilde { x } ) ] } \\ { \underset { \phi } { \operatorname* { m i n } } K L [ p _ { \theta } ( x ) | | q _ { \phi } ( x ) ] } \end{array} 
+$$
+
+The first line of Eq. (4) is equivalent to maximum likelihood training of the target RF $p _ { \theta }$ under the empirical data $\tilde { p }$ , which requires sampling from $p _ { \theta }$ . Simultaneously, the second line optimizes the generator $q _ { \phi }$ to be close to $p _ { \theta }$ so that $q _ { \phi }$ becomes a good proposal for sampling from $p _ { \theta }$ . It can be easily seen that the gradients w.r.t. $\theta$ and $\phi$ (to be ascended) are defined as follows:
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \nabla _ { \theta } = E _ { \tilde { p } ( \tilde { x } ) } \left[ \nabla _ { \theta } \log p _ { \theta } ( \tilde { x } ) \right] = E _ { \tilde { p } ( \tilde { x } ) } \left[ \nabla _ { \theta } u _ { \theta } ( \tilde { x } ) \right] - E _ { p _ { \theta } ( x ) } \left[ \nabla _ { \theta } u _ { \theta } ( x ) \right] , } \\ { \nabla _ { \phi } = E _ { p _ { \theta } ( x ) } \left[ \nabla _ { \phi } \log q _ { \phi } ( x ) \right] = E _ { p _ { \theta } ( x ) q _ { \phi } ( h \vert x ) } \left[ \nabla _ { \phi } \log q _ { \phi } ( x , h ) \right] . } \end{array} \right. } \end{array}
+$$
+
+Both lines of Eq. (5) hold, as proved in Proposition 1 in the Supplement. In practice, we calculate noisy gradient estimators, and apply minibatch based stochastic gradient descent (SGD) to solve the optimization problem Eq. (4), as shown in Algorithm 1.
+
+# 2.2 APPLYING SGLD/SGHMC FOR MODEL SAMPLING
+
+In Algorithm 1, we need to draw samples from $p _ { \theta } ( x ) q _ { \phi } ( h | x )$ given current $\theta$ and $\phi$ . For continuous observations, SGLD (stochastic gradient Langevin dynamics) (Welling & Teh, 2011) and SGHMC (Stochastic Gradient Hamiltonian Monte Carlo) (Chen et al., 2014) sampling provide mechanisms for exploiting (stochastic) gradients of the target density $p _ { \theta } ( x ) q _ { \phi } ( h | x )$ , enabling efficient exploration of the state space. We take the theoretical results about SGLD from Teh et al. (2016) and SGHMC from Chen et al. (2014), which are briefly summarized in Theorem 1 in the Supplement, and apply them in the sampling step in Algorithm 1. Denoting the target density as $p ( z ; \lambda )$ with given $\lambda$ , Theorem 1 shows that SGLD/SGHMC, by utilizing $\begin{array} { r } { \frac { \partial } { \partial z } \log p ( z ; \lambda ) } \end{array}$ , yields a non-homogeneous Markov chain $\{ z ^ { ( l ) } , l \ge 1 \}$ , which converges to the equilibrium distribution $p ( z ; \lambda )$ .
+
+By letting $z \triangleq ( x , h ) , p ( z ; \lambda ) \triangleq p _ { \boldsymbol { \theta } } ( x ) q _ { \boldsymbol { \phi } } ( h | x ) , \lambda \triangleq ( \boldsymbol { \theta } , \boldsymbol { \phi } ) ^ { T }$ in Theorem 1, we can perform the sampling step in Algorithm 1 by running $| { \mathcal { M } } |$ parallel chains, each chain being executed as shown
+
+# Algorithm 2 Sampling from $p _ { \theta } ( x ) q _ { \phi } ( h | x )$
+
+1. Do ancestral sampling by the generator, namely first drawing $h ^ { \prime } \sim p ( h ^ { \prime } )$ , and then drawing $x ^ { \prime } \sim q _ { \phi } ( x ^ { \prime } | h ^ { \prime } )$ ;   
+2. Starting from $\left( x ^ { \prime } , h ^ { \prime } \right) = z ^ { ( 0 ) }$ , run finite steps of SGLD/SGHMC $( l = 1 , \cdots , L )$ to obtain $( x , h ) = z ^ { ( L ) }$ , which we call sample revision, according to Eq. (11)/(12).   
+Return $( x , h )$ .
+
+in Algorithm 2. In sample revision, the calculation of the gradient w.r.t. $h$ , $\begin{array} { r } { \frac { \partial } { \partial h } \log p ( z ; \lambda ) = } \end{array}$ $\begin{array} { r } { \frac { \partial } { \partial h } \log q _ { \phi } ( h | x ) = \frac { \partial } { \partial h } \log q _ { \phi } ( h , x ) } \end{array}$ , is straightforward. For the gradient w.r.t. $x$ ∂h , we have
+
+$$
+\frac { \partial } { \partial x } \log p ( z ; \lambda ) = \frac { \partial } { \partial x } \log p _ { \theta } ( x ) + \frac { \partial } { \partial x } \log q _ { \phi } ( h , x ) - \frac { \partial } { \partial x } \log q _ { \phi } ( x ) \approx \frac { \partial } { \partial x } \log p _ { \theta } ( x ) .
+$$
+
+The reason is that $\begin{array} { r } { \frac { \partial } { \partial x } \log q _ { \phi } ( x ) } \end{array}$ can be approximated by an unbiased estimate, as proved in Proposition 2 in the Supplement:
+
+$$
+\frac { \partial } { \partial x } \log q _ { \phi } ( x ) \approx \frac { \partial } { \partial x } \log q _ { \phi } ( h , x ) .
+$$
+
+Therefore, we can use can apply Theorem 1 i $\textstyle { \frac { \partial } { \partial x } } \log p _ { \theta } ( x ) ^ { 6 }$ as an unbiased estimate of the gradienent with tractable gradients w.r.t. both $\begin{array} { r } { \frac { \partial } { \partial x } \log p ( z ; \lambda ) } \end{array}$ , and we $x$ $h$
+
+Remarks. Intuitively, the generator gives a proposal $( x ^ { \prime } , h ^ { \prime } )$ , and then the system follows the gradients of $p _ { \theta } ( x )$ and $q _ { \phi } ( h , x )$ (w.r.t. $x$ and $h$ respectively) to revise $( x ^ { \prime } , h ^ { \prime } )$ to $( x , h )$ . The gradient terms pull samples moving to low energy region of the random field and adjust the latent code of the generator, while the noise term brings randomness. In this manner, we obtain Markov chain samples from $p _ { \theta } ( x ) q _ { \phi } ( h | x )$ . Note that finite steps in sample revision will produce biased estimates of the gradients $\nabla _ { \theta }$ and $\nabla _ { \phi }$ in Eq. (5). We did not find this to pose problems to the SGD optimization in practice, as similarly found in Bornschein $\&$ Bengio (2015) and Kuleshov & Ermon (2017), which work with biased gradient estimators.
+
+# 2.3 SEMI-SUPERVISED LEARNING WITH INCLUSIVE NRFS
+
+In the following, we apply our inclusive-NRF approach in the SSL setting to show its flexibility. Note that different models are needed in unsupervised and semi-supervised learning, because SSL needs to additionally consider labels apart from observations.
+
+Model definition. In semi-supervised tasks, we consider the following RF for joint modeling of observation $x \in \mathbb { R } ^ { d _ { x } }$ and class label $y \in \{ 1 , \cdots , K \}$ :
+
+$$
+p _ { \theta } ( x , y ) = \frac { 1 } { Z ( \theta ) } \exp \left[ u _ { \theta } ( x , y ) \right]
+$$
+
+which is different from Eq.1 for unsupervised learning without labels. To implement the potential function $u _ { \boldsymbol { \theta } } ( \boldsymbol { x } , \boldsymbol { y } )$ , we consider a neural network $\Phi _ { \theta } ( x ) \backslash \mathbb { R } ^ { d _ { x } }  \mathbb { R } ^ { K }$ , with $x$ as the input and the output size being equal to the number of class labels, $K$ . Then we define $u _ { \theta } ( x , y ) = o n e \bar { h } o t ( y ) ^ { T } \Phi _ { \theta } ( x )$ , where onehot $( y )$ represents the one-hot encoding vector for the label $y$ . In this manner, the conditional density $p _ { \theta } ( y | x )$ is the classifier, defined as follows:
+
+$$
+p _ { \theta } ( y | x ) = \frac { p _ { \theta } ( x , y ) } { p _ { \theta } ( x ) } = \frac { \exp \left[ u _ { \theta } ( x , y ) \right] } { \sum _ { y } \exp \left[ u _ { \theta } ( x , y ) \right] }
+$$
+
+which acts like multi-class logistic regression using $K$ logits calculated from $x$ by the neural network $\Phi _ { \theta } ( x )$ . And we do not need to calculate $Z ( \theta )$ for classification. The auxiliary generator is implemented the same as in Eq. 3, i.e. an unconditional generator.
+
+With the definition the joint density in Eq. 7, it can be shown that, with abuse of notation, the marginal density $\begin{array} { r } { p _ { \theta } ( x ) = \frac { 1 } { Z ( \theta ) } \exp \left[ u _ { \theta } ( x ) \right] } \end{array}$ where $\begin{array} { r } { u _ { \theta } ( x ) \triangleq l o g \sum _ { y } \exp { [ u _ { \theta } ( x , y ) ] } } \end{array}$ .
+
+Model learning. Suppose that among the data $\mathcal { D } = \{ \tilde { x } _ { 1 } , \cdot \cdot \cdot , \tilde { x } _ { n } \}$ , only a small subset of the observations, for example the first $m$ observations, have class labels, $m \ll n$ . Denote these labeled data as $\mathcal { L } = \{ ( \tilde { x } _ { 1 } , \tilde { y } _ { 1 } ) , \cdot \cdot \cdot , ( \tilde { x } _ { m } , \tilde { y } _ { m } ) \}$ . Then we can formulate the semi-supervised learning as jointly optimizing
+
+$$
+\left\{ \begin{array} { l l } { \displaystyle { \operatorname* { m i n } _ { \theta } K L \left[ \tilde { p } ( \tilde { x } ) | | p _ { \theta } ( \tilde { x } ) \right] - \alpha _ { d } \sum _ { ( \tilde { x } , \tilde { y } ) \sim \mathcal { L } } l o g p _ { \theta } ( \tilde { y } | \tilde { x } ) } } \\ { \displaystyle { \operatorname* { m i n } _ { \phi } K L \left[ p _ { \theta } ( x ) | | q _ { \phi } ( x ) \right] } } \end{array} \right.
+$$
+
+which are defined by hybrids of generative and discriminative criteria, similar to Zhu (2006); Larochelle et al. (2012); Kingma et al. (2014). The hyper-parameter $\alpha _ { d }$ controls the relative weight between generative and discriminative criteria. Similar to deriving Eq. (5), it can be easily seen that the gradients w.r.t. $\theta$ and $\phi$ (to be ascended) are defined as follows:
+
+$$
+\left\{ \begin{array} { l l } { \nabla _ { \theta } ^ { \mathrm { s e m i } } = E _ { \tilde { p } ( \tilde { x } ) } \left[ \nabla _ { \theta } l o g p _ { \theta } ( \tilde { x } ) \right] + \alpha _ { d } \displaystyle \sum _ { ( \tilde { x } , \tilde { y } ) \sim \mathcal { L } } \nabla _ { \theta } l o g p _ { \theta } ( \tilde { y } | \tilde { x } ) } \\ { \quad \quad \quad } \\ { \quad \quad \quad = E _ { \tilde { p } ( \tilde { x } ) } \left[ \nabla _ { \theta } u _ { \theta } ( \tilde { x } ) \right] - E _ { p _ { \theta } ( x ) } \left[ \nabla _ { \theta } u _ { \theta } ( x ) \right] + \alpha _ { d } \displaystyle \sum _ { ( \tilde { x } , \tilde { y } ) \sim \mathcal { L } } \nabla _ { \theta } l o g p _ { \theta } ( \tilde { y } | \tilde { x } ) } \\ { \quad \quad \quad } \\ { \nabla _ { \phi } ^ { \mathrm { s e m i } } = E _ { p _ { \theta } ( x ) } \left[ \nabla _ { \phi } l o g q _ { \phi } ( x ) \right] = E _ { p _ { \theta } ( x ) q _ { \phi } ( h | x ) } \left[ \nabla _ { \phi } l o g q _ { \phi } ( x , h ) \right] } \end{array} \right.
+$$
+
+In practice, we calculate noisy gradient estimators, and apply minibatch based stochastic gradient descent (SGD) to solve the optimization problem Eq. (9), as shown in Algorithm 3 in the Supplement. Apart from the basic losses as shown in Eq. (9), there are some regularization losses that are found to be helpful to guide SSL learning and are presented in the Supplement. To conclude, we show that the inclusive-NRF can be easily applied to SSL. To the best of our knowledge, there are no priori studies in applying random fields to SSL. The semi-supervised inclusive-NRF model defined above is novel itself for SSL.
+
+# 3 RELATED WORK
+
+Comparison and connection of our inclusive-NRF approach with existing studies are provided in the following from three perspectives.
+
+Learning NRFs with auxiliary generators. These studies are most relevant to this work, which aims to learn NRFs. The classic method for learning RFs is the SML method (Younes, 1989), which works with the single target model $p _ { \theta }$ . Compared to learning traditional RFs which mainly use linear potential functions, learning NRFs which use NN based nonlinear potential functions, is more challenging. A recent progress in learning NRFs as studied in $\mathrm { K i m } \ \&$ Bengio (2016); Xie et al. (2016); Wang & Ou (2017); Kuleshov & Ermon (2017) is to jointly train the target random field $p _ { \theta } ( x )$ and an auxiliary generator $q _ { \phi } ( x )$ . Different studies differ in the objective functions used in the joint training, and thus have different computational and statistical properties.
+
+• It is shown in Proposition 3 in the Supplement that learning in Kim & Bengio (2016) minimizes the exclusive-divergence $K \bar { L } [ q _ { \phi } | | p _ { \theta } ]$ w.r.t. $\phi$ , which involves the intractable entropy term and tends to enforce the generator to seek modes, yielding missing modes. We refer to this approach as exclusive-NRF. Learning in Wang & Ou (2017) and in this paper minimizes the inclusive-divergence $K L [ p _ { \theta } | | \bar { q } _ { \phi } ]$ w.r.t. $\phi$ . But noticeably, this paper presents our innovation in development of NRFs for continuous data, which is fundamentally different from Wang & Ou (2017) for discrete data. The target NRF model, the generator and the sampler are all different. Wang & Ou (2017) mainly studies random field language models, using LSTM generators (autoregressive with no latent variables) and employing Metropolis independence sampler (MIS) - applicable for discrete data (natural sentences). In this paper, we mainly develop random field models for continuous data (images), using latent-variable generators and utilizing SGLD/SGHMC (with solid theoretical examination) to exploit gradient information in the continuous space.   
+• In Xie et al. (2016), motivated by interweaving maximum likelihood training of the random field $p _ { \theta }$ and the latent-variable generator $q _ { \phi }$ , a joint training method is introduced.
+
+Operationally, in learning $\theta$ and $\phi$ , this method also uses Langevin sampling to generate samples. Two Langevin sampling steps are intuitively interleaved according to $\begin{array} { r } { \frac { \partial } { \partial x } \operatorname* { l o g } p _ { \theta } ( x ) \ } \end{array}$ and $\begin{array} { r } { \frac { \partial } { \partial h } \log q _ { \phi } ( h , x ) } \end{array}$ separately. This is different from our sampling step, which moves $( h , x )$ jointly, as theoretically justified in section 2.2. Let $r ( h , x )$ denote the distribution obtained by running the interleaved Langevin transitions starting from $( h , x ) \sim q _ { \phi } ( h , x )$ . Interpretation presented in Xie et al. (2016) relates their method to the following joint optimization problem:
+
+$$
+\left\{ \begin{array} { l l } { \underset { \theta } { \operatorname* { m i n } } \left\{ K L \left[ \tilde { p } ( \tilde { x } ) | | p _ { \theta } ( \tilde { x } ) \right] - K L \left[ r ( h , x ) | | p _ { \theta } ( x ) \right] \right\} } \\ { \underset { \phi } { \operatorname* { m i n } } K L \left[ r ( h , x ) | | q _ { \phi } ( h , x ) \right] } \end{array} \right.
+$$
+
+which is also different from ours as shown in Eq. (4). Thus, learning in Xie et al. (2016) does not aim to minimize the inclusive-divergence $K L [ p _ { \theta } | | q _ { \phi } ]$ w.r.t. $\phi$ .
+
+• Learning in Kuleshov & Ermon (2017) minimizes the $\chi ^ { 2 }$ -divergence $\chi ^ { 2 } [ q _ { \phi } | | p _ { \theta } ] \triangleq$ $\int { \frac { \left( p _ { \theta } - q _ { \phi } \right) ^ { 2 } } { q _ { \phi } } }$ w.r.t. $\phi$ , which also tends to drive the generator to cover modes. But this approach is severely limited by the high variance of the gradient estimator w.r.t. $\phi$ , and is only tested on the simpler MNIST and Omniglot.
+
+Additionally, different NRF studies also differ in models used in the joint training. For example, the target NRF used in this work is different from those in previous studies Kim & Bengio (2016); Wang & Ou (2017); Xie et al. (2016). The differences are: Kim & Bengio (2016) includes linear and squared terms in $u _ { \theta } ( x )$ , Wang & Ou (2017) defines over sequences, and Xie et al. (2016) defines in the form of exponential tilting of a reference distribution (Gaussian white noise). There exist different choices for the generator, such as GAN models in Kim & Bengio (2016), LSTMs in Wang & Ou (2017), or latent-variable models in both Xie et al. (2016) and this work. All are easy to do sampling.
+
+Moreover, all the previous NRF studies examine unsupervised learning, and none shows application or extension of their methods or models for semi-supervised learning.
+
+Monte Carlo sampling. One step in our inclusive-NRF approach is to apply SGLD/SGHMC to draw samples from the target density $p _ { \theta }$ , starting from the proposal sample from the generator. Theoretically, improvements in NRF sampling methods could be potentially integrated into NRF learning algorithms. For example, it is recently studied in Levy et al. (2018) to learn MCMC transition kernels, also parameterized by neural networks, to improve the HMC sampling from the given target distribution. Integration into learning NRFs is interesting but outside the scope of this paper.
+
+Comparison and connection with GANs. On the one hand, there are some efforts that aim to address the inability of GANs to provide sensible energy estimates for samples. The energy-based GANs (Zhao et al., 2017) proposes to view the discriminator as an energy function by designing an auto-encoder discriminator. The recent work in Dai et al. (2017a) connects Zhao et al. (2017) and Kim & Bengio (2016), and show another two approximations for the entropy term. However, it is known that as the generator converges to the true data distribution, the GAN discriminator converges to a degenerate uniform solution. This basically afflicts the GAN discriminator to provide density information, though there are some modifications. In contrast, our inclusive-NRFs, unlike GANs, naturally provide (unnormalized) density estimate. Moreover, none of the above energy-related GAN studies examine their methods or models for SSL, except in EBGAN which performs moderately.
+
+On the other hand, there are interesting connections between inclusive-NRFs and GANs, as elaborated in section 11 in the Supplement. When interpreting the potential function $u _ { \theta } ( x )$ as the critic in Wasserstein GANs, inclusive-NRFs seem to be similar to Wasserstein GANs. A difference is that in optimizing $\theta$ in inclusive-NRFs, the generated samples are further revised by taking finite-stepgradient of $u _ { \theta } ( x )$ w.r.t. $x$ . However, the critic in Wasserstein GANs can hardly be interpreted as an unnormalized log-density. Thus strictly speaking, inclusive-NRFs are not GAN-like.
+
+# 4 EXPERIMENTS
+
+We conduct a series of experiments to evaluate the performances of our approach (inclusive-NRFs) and various existing methods on synthetic and real-world datasets for both unsupervised and semisupervised learning tasks, with both visual and numerical evaluation. We refer to the Supplement for experimental details and additional results.
+
+![](images/5ecbedd308440dd389e2ff57d70a9f93f36c595291451de38c7ace0aeebe7f1b.jpg)  
+Figure 1: Comparison of different methods over GMM synthetic data. Stochastic generations from GAN with logD trick, WGAN-GP, Exclusive-NRF, Inclusive-NRF generation (i.e. sampling from the auxiliary generator) and Inclusive-NRF revision (i.e. after performing sample revision over samples from the auxiliary generator), are shown in (b)-(f) respectively. Inclusive-NRF generation and inclusive-NRF revision are two manners to generate samples, given a trained NRF. For both manners, the NRF model is trained with the sample revision step. Each generation contains 1,000 samples. The learned potentials $u _ { \theta } ( x )$ from exclusive and inclusive NRFs are shown in (g) and (h) respectively, where the red dots indicate the mean of each Gaussian component. Inclusive NRFs are clearly superior in learning data density and sample generation.
+
+# 4.1 GMM SYNTHETIC EXPERIMENT
+
+The synthetic data consist of 1,600 training examples generated from a 2D Gaussian mixture model (GMM) with 32 equally-weighted, low-variance $\mathit { \check { \sigma } } = 0 . 1$ ) Gaussian components, uniformly laid out on four concentric circles as in Figure 1(a). The data distribution exhibits many modes separated by large low-probability regions, which makes it suitable to examine how well different learning methods can deal with multiple modes. For comparison, we experiment with GAN with logD trick (Goodfellow et al., 2014) and WGAN-GP (Gulrajani et al., 2017) for directed generative model, exclusive-NRF (Kim & Bengio, 2016) and inclusive-NRF for undirected generative model.
+
+Figure 1 visually shows the generated samples from the trained models using different methods. Table 1 reports the “covered modes” and “realistic ratio” as numerical measures of how the multi-modal data are fitted, similarly as in Dumoulin et al. (2017). The main observations are as follows. (1) GAN suffers from mode missing, generating realistic but not diverse samples. WGAN-GP increases “covered modes” but decreases “realistic ratio”. Inclusive-NRF performs much better than both GAN and WGAN-GP in sample generation. (2) Inclusive-NRF outperforms exclusive-NRF in both sample generation and density estimation. (3) After revision, samples from inclusive-NRF become more like real samples, achieving the best in both “covered modes" and “realistic ratio” metrics.
+
+# 4.2 IMAGE GENERATION ON CIFAR-10
+
+We examine both unsupervised and supervised learning over the widely used real-world dataset CIFAR-10 Krizhevsky (2009) for image generation. To evaluate generation quality quantitatively, we use inception score (IS) Salimans et al. (2016), and Frechet inception distance (FID) Heusel et al. (2017). Table 2 reports the inception score and FID for state of the art methods, for both unsupervised and supervised settings. The supervised learning of inclusive-NRF is conducted as a special case of semi-supervised learning over all labeled images $m = n$ ), which uses unconditional generation. We use ResNet in this experiment, see section 12.2 in the Supplement for experimental details.
+
+Table 1: Numerical evaluations over the GMM (32 components) synthetic data. The “covered modes” metric is defined as the number of covered modes by a set of generated samples. The “realistic ratio” metric is defined as the proportion of generated samples which are close to a mode. The measurement details are presented in section 12.1 in the Supplement. Mean and SD are from 10 independent runs.   
+
+<table><tr><td>Methods</td><td>covered modes</td><td>realistic ratio</td></tr><tr><td>GAN with logD trick</td><td>22.25 ± 1.54</td><td>0.90 ± 0.01</td></tr><tr><td>WGAN-GP (Gulrajani et al.,2017)</td><td>27.81 ± 1.40</td><td>0.74±0.04</td></tr><tr><td>Exclusive-NRF(Kim&amp; Bengio,2016)</td><td>28.14±0.68</td><td>0.73 ± 0.03</td></tr><tr><td>Inclusive-NRF generation</td><td>29.52± 0.54</td><td>0.84± 0.01</td></tr><tr><td>Inclusive-NRF revision</td><td>30.75 ± 0.43</td><td>0.97 ± 0.01</td></tr></table>
+
+Table 2: Inception score (IS) and FID on CIFAR-10 for unsupervised and supervised learning.   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">Unsupervised</td><td colspan="2">Supervised</td></tr><tr><td>IS</td><td>FID</td><td>IS</td><td>FID</td></tr><tr><td>DCGAN (Radford et al., 2015)</td><td>6.16 ± 0.07</td><td></td><td>6.58</td><td></td></tr><tr><td>Improved-GAN (Salimans et al., 2016)</td><td></td><td></td><td>8.09 ±0.07</td><td></td></tr><tr><td>WGAN-GP (Gulrajani et al., 2017)</td><td>7.86 ± 0.07</td><td></td><td>8.42 ±0.10</td><td></td></tr><tr><td>SGAN (Huang et al., 2017)</td><td></td><td></td><td>8.59 ± 0.12</td><td></td></tr><tr><td>DFM(Warde-Farley &amp; Bengio,2017)</td><td>7.72 ± 0.13</td><td></td><td></td><td></td></tr><tr><td>CT-GAN (Wei et al., 2018)</td><td>8.12 ±0.12</td><td></td><td>8.81 ± 0.13</td><td></td></tr><tr><td>Fisher-GAN (Mroueh &amp; Sercu,2017)</td><td>7.90 ± 0.05</td><td></td><td>8.16 ± 0.12</td><td></td></tr><tr><td>BWGAN (Adler &amp; Lunz, 2018)</td><td>8.26 ± 0.07</td><td></td><td></td><td></td></tr><tr><td>SNGAN (Miyato et al., 2018)</td><td>8.22 ± 0.05</td><td>21.7 ± 0.21</td><td></td><td></td></tr><tr><td>Inclusive-NRF generation</td><td>8.28 ± 0.09</td><td>20.9 ±0.25</td><td>9.06 ± 0.10</td><td>18.1 ± 0.23</td></tr></table>
+
+From the comparison results in Table 2, it can be seen that the proposed inclusive-NRF model achieves the best inception score over CIFAR-10, to the best of our knowledge, in both unsupervised and supervised settings. Some generated samples are shown in Figure 5(c)(d) for unsupervised and supervised settings respectively. We also show in the Supplement the capability of inclusive-NRFs in latent space interpolation (section 14) and conditional generation (section 15).
+
+# 4.3 SEMI-SUPERVISED LEARNING RESULTS
+
+For semi-supervised learning, we consider the three widely used benchmark datasets, namely MNIST (LeCun et al., 1998), SVHN (Netzer et al., 2011), and CIFAR-10 (Krizhevsky, 2009). As in previous work, we randomly sample 100, 1,000, and 4,000 labeled samples from MNIST, SVHN, and CIFAR10 respectively during training, and use the standard data split for testing. See section 12.3 in the Supplement for experimental details. We also provide a SSL toy experiment in section 13 in the Supplement to help understanding how semi-supervised inclusive-NRF works.
+
+It can be seen from Table 3 that semi-supervised inclusive-NRFs produce strong classification results on par with state-of-art DGM-based SSL methods. See Figure 5(a)(b) in the Supplement for generated samples. Bad-GANs achieve better classification results, but as indicated by the low inception score, their generation is much worse than semi-NRF-IAGs. In fact, among DGM-based SSL methods, inclusive-NRFs achieve the best performance in sample generation. This is in contrast to the conflict of good classification and good generation, as observed in GAN-based SSL (Salimans et al., 2016; Dai et al., 2017b). It is analyzed in Dai et al. (2017b) that good GAN-based SSL requires a bad generator7. This is embarrassing and in fact obviates the original idea of generative SSL - successful generative training, which indicates good generation, provides regularization for finding good classifiers (Zhu, 2006; Larochelle et al., 2012). In this sense, Bad-GANs could hardly be classified as a generative SSL method.
+
+Table 3: Comparison with state-of-the-art methods on three benchmark datasets. “CIFAR-10 IS” means the inception score for samples generated by SSL models trained on CIFAR-10. “†” is obtained by running the released code accompanied by the corresponding papers. “-” means the results are not reported in the original work and without released code. “/” means not applicable, e.g. the models cannot generate samples stochastically. “ $\ddag ^ { \prime \prime }$ uses image data augmentation which significantly helps classification performance. The upper/lower blocks show generative/discriminative SSL methods respectively.   
+
+<table><tr><td>Methods</td><td>error (%) MNIST</td><td>error (%) SVHN</td><td>error (%) CIFAR-10</td><td>IS CIFAR-10</td></tr><tr><td>CatGAN (Springenberg,2016)</td><td>1.91 ± 0.10</td><td></td><td>19.58 ± 0.46</td><td>3.57± 0.13†</td></tr><tr><td>SDGM (Maaloe et al., 2016)</td><td>1.32 ± 0.07</td><td>16.61 ± 0.24</td><td></td><td>1</td></tr><tr><td>Ladder network (Rasmus et al.,2015)</td><td>1.06 ± 0.37</td><td></td><td>20.40 ± 0.47</td><td>/</td></tr><tr><td>ADGM (Maaloe et al., 2016)</td><td>0.96 ± 0.02</td><td>22.86</td><td>1</td><td>-</td></tr><tr><td>Improved-GAN (Salimans et al.,2016)</td><td>0.93 ± 0.07</td><td>8.11 ±1.3</td><td>18.63 ± 2.32</td><td>3.87± 0.03</td></tr><tr><td>EBGAN (Zhao et al.,2017)</td><td>1.04 ± 0.12</td><td>-</td><td>=</td><td>-</td></tr><tr><td>ALI (Dumoulin et al., 2017)</td><td></td><td>7.42 ± 0.65</td><td>17.99 ± 1.62</td><td></td></tr><tr><td>Triple-GAN (Li et al., 2017)</td><td>0.91 ± 0.58</td><td>5.77 ± 0.17</td><td>16.99 ± 0.36</td><td>5.08 ±0.09</td></tr><tr><td>Triangle-GAN (Gan et al., 2017)</td><td></td><td>-</td><td>16.80 ±0.42</td><td>=</td></tr><tr><td>BadGAN (Dai et al., 2017b)</td><td>0.80 ± 0.10</td><td>4.25 ± 0.03</td><td>14.41 ± 0.30</td><td>3.46 ± 0.11†</td></tr><tr><td>Sobolev-GAN (Mroueh et al., 2018)</td><td>=</td><td></td><td>15.77 ± 0.19</td><td></td></tr><tr><td>Semi-supervised inclusive-NRF</td><td>0.97 ± 0.10</td><td>5.84 ±0.15</td><td>15.12 ± 0.36</td><td>7.72 ± 0.09</td></tr><tr><td colspan="5">Results below this line cannot be directly compared to those above.</td></tr><tr><td>VAT small (Miyato et al., 2017)</td><td>1.36</td><td>6.83</td><td>14.87</td><td>/</td></tr><tr><td>II model‡ (Laine &amp; Aila,2017)</td><td>-</td><td>4.82 ± 0.17</td><td>12.36 ± 0.31</td><td></td></tr><tr><td>Temporal Ensembling‡ (Laine &amp; Aila,2017)</td><td></td><td>4.42 ± 0.16</td><td>12.16 ± 0.31</td><td>/</td></tr><tr><td>Mean Teacher‡ (Tarvainen&amp; Valpola,2017)</td><td></td><td>3.95 ±0.19</td><td>12.31 ±0.28</td><td>/</td></tr><tr><td>VAT+EntMin‡ (Miyato et al.,2017)</td><td></td><td>3.86</td><td>10.55</td><td>/</td></tr><tr><td>CT-GAN‡ (Wei et al., 2018)</td><td>0.89 ± 0.13</td><td>-</td><td>9.98 ± 0.21</td><td>/</td></tr></table>
+
+Finally, note that some discriminative SSL methods, as listed in the lower block in Table 3 also produce superior performances, by utilizing data augmentation and consistency regularization. However, these methods are unable to generate (realistic) samples. It can be seen that discriminative SSL methods utilize different regularization from generative SSL methods and cannot be directly compared to generative SSL methods. Their combination, as an interesting future work, could yield further performance improvement.
+
+# 4.4 ABLATION STUDY
+
+We report the results of ablation study of our inclusive-NRF method on CIFAR-10 in Table 4. In this experiment, we use the standard CNN (Miyato et al., 2018) for unsupervised learning and the same networks as those used in Table 3 for semi-supervised learning. See section 12.4 in the Supplement for experimental details. We analyze the effects of different settings in model training, such as using SGLD or SGHMC and the revision step $L = 1 / 5 / 1 0$ used. For each training setting, we also compare the two manners to generate samples - whether applying sample revision or not in inference (generating samples) given a trained NRF, as previously illustrated in Figure 1 over synthetic GMM data. The main observations are as follows.
+
+First, given a trained NRF, after revision (i.e. following the gradient of the RF’s potential $u _ { \theta } ( x )$ w.r.t. $x$ ), the quality (IS) of samples is always improved, as shown by the consistent IS improvement from the second column (generation) to the third (revision). This is in accordance with the results in the GMM synthetic experiments. Moreover, noting that in revision, it is the the estimated density $p _ { \theta }$ that guides the samples towards low energy region of the random field. This demonstrates one benefit of random field modeling, which, unlike GANs, can learn density estimate about the data manifold.
+
+Second, a row-wise reading of Table 4 reveals that with more revision steps and using SGHMC in training, the SSL classification performance is improved. Utilizing SGHMC in inclusive-NRFs to exploit gradient information with momentum yields better performance than simple SGLD as used in Xie et al. (2016). It is also found that more revision steps in model training do not significantly improve unsupervised IS. So we can use $L = 1$ in unsupervised learning for generation, which can reduce the computational cost.
+
+Table 4: Ablation study of our inclusive-NRF method on CIFAR-10, regarding the effects of using SGLD or SGHMC in training and of applying sample revision in inference (generating samples). Mean and SD are from 5 independent runs for each training setting. In each training setting, for unsupervised learning, two manners to generate samples given a trained NRF are compared, as previously illustrated in Figure 1 over synthetic GMM data. We examine generated samples (i.e. directly from the generator) and revised samples (i.e. after sample revision) respectively, in term of inception scores (IS). For semi-supervised learning, we examine the classification error rates.   
+
+<table><tr><td rowspan="2">Training Setting</td><td colspan="2">Unsupervised</td><td rowspan="2">Semi-supervised error (%)</td></tr><tr><td>Generation IS</td><td>Revision IS</td></tr><tr><td>SGLD L =1</td><td>7.47 ± 0.15</td><td>7.53 ± 0.13</td><td>17.08 ± 0.39</td></tr><tr><td>SGLD L = 5</td><td>7.44± 0.16</td><td>7.49 ± 0.12</td><td>16.15 ± 0.44</td></tr><tr><td>SGLD L = 10</td><td>7.43 ± 0.18</td><td>7.50 ± 0.13</td><td>15.60 ± 0.31</td></tr><tr><td>SGHMC L = 10</td><td>7.46 ± 0.12</td><td>7.57 ± 0.10</td><td>15.12 ± 0.36</td></tr></table>
+
+# 5 DISCUSSION AND CONCLUSION
+
+In this paper we develop the inclusive-NRF approach, which learns NRFs with inclusive auxiliary generators and particularly for continuous data, exploits gradient information in model sampling with solid theoretical examination. Extensive empirical evaluations show that inclusive-NRFs obtain state-of-the-art sample generation quality and achieve strong semi-supervised learning results on par with state-of-the-art DGMs. The superior performances presumably are attributed to the two distinctive features in inclusive-NRFs - introducing the inclusive-divergence minimized auxiliary generator and utilizing sample revision by SGLD/SGHMC. Intuitively, the revised samples from the RF will guide the training of the generator, and subsequently the generator will propose samples for the RF to sense the data manifold. This forms positive interactions between the random field and the generator, which enables successful joint training of both models.
+
+The new approach enables us to flexibly use NRFs in unsupervised, supervised and semi-supervised settings and successfully train them in a black-box manner. Interesting future work will consider inclusive-NRFs in more challenging tasks, e.g. unsupervised and semi-supervised learning with sequential data (e.g. speech, language, video, etc.).
+
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+# Supplement for “Learning Neural Random Fields with Inclusive Auxiliary Generators”
+
+# 6 PROOF OF PROPOSITION 1
+
+Proposition 1. Both lines of Eq.(5) for gradient calculations hold.
+
+Proof. The first line of Eq.(5) can be obtained by directly taking derivative of $K L [ \tilde { p } ( \tilde { x } ) | | p _ { \theta } ( \tilde { x } ) ]$ w.r.t. $\theta$ , as shown below,
+
+$$
+\frac { \partial } { \partial \theta } K L \left[ \tilde { p } ( \tilde { x } ) | | p _ { \theta } ( \tilde { x } ) \right] = \frac { \partial } { \partial \theta } \int \tilde { p } ( \tilde { x } ) \log \frac { \tilde { p } ( \tilde { x } ) } { p _ { \theta } ( \tilde { x } ) } d \tilde { x } = - \int \tilde { p } ( \tilde { x } ) \frac { \partial } { \partial \theta } \log p _ { \theta } ( \tilde { x } ) d \tilde { x } ,
+$$
+
+and then applying the basic formula of Eq. (2).
+
+For the second line, by direct calculation, we first have
+
+$$
+\begin{array} { l } { E _ { q _ { \phi } ( h | x ) } \left[ \nabla _ { \phi } \log q _ { \phi } ( h | x ) \right] = \displaystyle \int q _ { \phi } ( h | x ) q _ { \phi } ( h | x ) ^ { - 1 } \nabla _ { \phi } q _ { \phi } ( h | x ) d h } \\ { \displaystyle \qquad = \int \nabla _ { \phi } q _ { \phi } ( h | x ) d h = \nabla _ { \phi } \int q _ { \phi } ( h | x ) d h = \nabla _ { \phi } 1 = 0 . } \end{array}
+$$
+
+Then combining $\begin{array} { r } { \frac { \partial } { \partial \phi } K L \left[ p _ { \theta } ( x ) | | q _ { \phi } ( x ) \right] = - E _ { p _ { \theta } ( x ) } \left[ \nabla _ { \phi } \log q _ { \phi } ( x ) \right] } \end{array}$ and
+
+$$
+\begin{array} { r l } & { \nabla _ { \phi } \log q _ { \phi } ( x ) = E _ { q _ { \phi } ( h \vert x ) } \left[ \nabla _ { \phi } \log q _ { \phi } ( x ) \right] = E _ { q _ { \phi } ( h \vert x ) } \left[ \nabla _ { \phi } \log q _ { \phi } ( x , h ) - \nabla _ { \phi } \log q _ { \phi } ( h \vert x ) \right] } \\ & { \qquad = E _ { q _ { \phi } ( h \vert x ) } \left[ \nabla _ { \phi } \log q _ { \phi } ( x , h ) \right] . } \end{array}
+$$
+
+will give the second line of Eq.(5).
+
+# 7 SGLD/SGHMC
+
+Theorem 1. Denote the target density as $p ( z ; \lambda )$ with given $\lambda .$ . Assume that one can compute a noisy, unbiased estimate which contains al $\Delta ( z , \xi ; \lambda )$ to the gradient mness involved $\begin{array} { r } { \frac { \partial } { \partial z } \log p ( z ; \lambda ) } \end{array}$ , where ng the e $\xi$ is an auxiliaryimate, namely $E \left[ \Delta ( z , \xi ; \lambda ) \right] =$ $\begin{array} { r } { \frac { \partial } { \partial z } \log p ( z ; \lambda ) } \end{array}$ . Assume the stability Assumptions $^ { 4 }$ in Teh et al. (2016) holds.
+
+For a sequence of asymptotically vanishing time-steps $\{ \delta \boldsymbol { { l } } , \boldsymbol { { l } } \ge 1 \}$ (satisfying $\textstyle \sum _ { l = 1 } ^ { \infty } \delta _ { l } = \infty$ and $\textstyle \sum _ { l = 1 } ^ { \infty } \delta _ { l } ^ { 2 } < \infty )$ , an i.i.d. sequence $\eta ^ { ( l ) }$ , and an independent and i.i.d. sequence $\xi _ { l }$ of auxiliary random variables, $l \geq 1$ , the SGLD iterates as follows, starting from $z ^ { ( 0 ) }$ :
+
+$$
+z ^ { ( l ) } = z ^ { ( l - 1 ) } + \frac { \delta _ { l } } { 2 } \Delta ( z ^ { ( l - 1 ) } , \xi _ { l } ; \lambda ) + \sqrt { \delta _ { l } } \eta ^ { ( l ) } , \eta ^ { ( l ) } \sim \mathcal { N } ( 0 , I ) , l = 1 , \cdots
+$$
+
+Starting from $z ^ { ( 0 ) }$ and $v ^ { ( 0 ) } = 0$ , the SGHMC iterates as follows:
+
+$$
+\left\{ \begin{array} { l l } { \displaystyle v ^ { ( l ) } = \beta v ^ { ( l - 1 ) } + \frac { \delta _ { l } } { 2 } \Delta ( z ^ { ( l - 1 ) } , \xi _ { l } ; \lambda ) + \sqrt { \delta _ { l } } \eta ^ { ( l ) } , \eta ^ { ( l ) } \sim \mathcal { N } ( 0 , I ) } \\ { z ^ { ( l ) } = z ^ { ( l - 1 ) } + v ^ { ( l ) } , l = 1 , \cdots } \end{array} \right.
+$$
+
+Then in both cases, the non-homogeneous Markov chain $\{ z ^ { ( l ) } , l \ge 1 \}$ converges to the equilibrium distribution $p ( z ; \lambda )$ .
+
+# 8 PROOF OF PROPOSITION 2
+
+Proposition 2. Let $z \triangleq ( x , h ) , p ( z ; \lambda ) \triangleq p _ { \theta } ( x ) q _ { \phi } ( h | x ) , \lambda \triangleq ( \theta , \phi ) ^ { T }$ in Theorem 1. The initial value $( x ^ { ( 0 ) } , h ^ { ( 0 ) } )$ is obtained from ancestral sampling by the generator. The SGLD/SGHMC as shown in Eq. $( l I ) / ( l 2 )$ iteratively generates $( x ^ { ( m ) } , h ^ { ( m ) } )$ , $m = 1 , \cdots$ . Then, $\begin{array} { r } { \frac { \partial } { \partial x ^ { ( m ) } } \log q _ { \phi } ( h ^ { ( m ) } , x ^ { ( m ) } ) } \end{array}$ is an unbiased estimate of the gradient $\frac { \partial } { \partial x ^ { ( m ) } } \log q _ { \phi } \big ( x ^ { ( m ) } \big )$ , $m = 0 , 1 , \cdots$ .
+
+Proof. Note that Langevin dynamics and Hamiltonian dynamics are reversible Neal (2011). Thus the SGLD/SGHMC transitions Eq.(11)/(12) satisfy the detailed balance condition:
+
+π $\tau ( h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } ) K ( h ^ { ( m ) } , x ^ { ( m ) } | h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } ) = \pi ( h ^ { ( m ) } , x ^ { ( m ) } ) K ( h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } | h ^ { ( m ) } , x ^ { ( m ) } ) ,$ where $\pi ( \cdot )$ denotes the target density, and $K ( \cdot | \cdot )$ denotes the transition kernel. Also note that $\begin{array} { r } { E _ { h \sim q _ { \phi } ( h | x ) } \left[ \frac { \partial } { \partial x } \log q _ { \phi } ( h , x ) \right] = \frac { \partial } { \partial x } \log q _ { \phi } ( x ) } \end{array}$ . Thus if we show that $h ^ { ( m ) }$ is indeed drawn from $q _ { \phi } ( h ^ { ( m ) } | x ^ { ( m ) } )$ during sample revision, $m = 0 , 1 , \cdots$ , then the unbiasedness will hold.
+
+Denote by $\pi ^ { ( m ) } ( h ^ { ( m ) } , x ^ { ( m ) } )$ the state-occupation density at step $m$ . Then we need to show that $\pi ^ { ( m ) } ( h ^ { ( m ) } | x ^ { ( m ) } )$ actually follows $\pi ( h ^ { ( m ) } | x ^ { ( m ) } )$ , i.e. $q _ { \phi } ( h ^ { ( m ) } | x ^ { ( m ) } )$ , $m = 0 , 1 , \cdots$ .
+
+First, it is obvious that this holds for $( h ^ { ( 0 ) } , x ^ { ( 0 ) } )$ . Then, we proceed by mathematical induction. Suppose $\pi ^ { ( m - 1 ) } ( h ^ { ( m - 1 ) } | x ^ { ( m - 1 ) } ) = \pi \big ( h ^ { ( m - 1 ) } | x ^ { ( m - 1 ) } \big )$ . Then we have
+
+$$
+\begin{array} { r l } & { \quad \pi ^ { ( m - 1 ) } ( h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } ) K ( h ^ { ( m ) } , x ^ { ( m ) } | h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } ) } \\ & { = \pi ^ { ( m - 1 ) } ( h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } ) \frac { \pi ( h ^ { ( m ) } , x ^ { ( m ) } ) } { \pi \left( h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } \right) } K ( h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } | h ^ { ( m ) } , x ^ { ( m ) } ) . } \end{array}
+$$
+
+Integrating out $( h ^ { ( m - 1 ) } , x ^ { ( m - 1 ) } )$ from both sides of Eq.13, we obtain
+
+$$
+\begin{array} { l } { \pi ^ { ( m ) } ( h ^ { ( m ) } , x ^ { ( m ) } ) = \pi ( h ^ { ( m ) } , x ^ { ( m ) } ) \displaystyle \sum _ { x ^ { ( m - 1 ) } } \frac { \pi ^ { ( m - 1 ) } ( x ^ { ( m - 1 ) } ) } { \pi ( x ^ { ( m - 1 ) } ) } K ( x ^ { ( m - 1 ) } | h ^ { ( m ) } , x ^ { ( m ) } ) } \\ { = \pi ( h ^ { ( m ) } , x ^ { ( m ) } ) \displaystyle \sum _ { x ^ { ( m - 1 ) } } \frac { \pi ^ { ( m - 1 ) } ( x ^ { ( m - 1 ) } ) } { \pi ( x ^ { ( m - 1 ) } ) } K ( x ^ { ( m - 1 ) } | x ^ { ( m ) } ) } \end{array}
+$$
+
+where the second equality, i.e. $K ( x | h ^ { \prime } , x ^ { \prime } ) = K ( x | x ^ { \prime } )$ , holds because in the SGLD/SGHMC transitions Eq.(11)/(12), generating next step $x$ only depends on current $x ^ { \prime }$ and is independent of current $h ^ { \prime }$ . Then we have
+
+$$
+\pi ^ { ( m ) } ( h ^ { ( m ) } | x ^ { ( m ) } ) = \pi ( h ^ { ( m ) } | x ^ { ( m ) } ) \frac { \pi ( x ^ { ( m ) } ) } { \pi ^ { ( m ) } ( x ^ { ( m ) } ) } \sum _ { x ^ { ( m - 1 ) } } \frac { \pi ^ { ( m - 1 ) } ( x ^ { ( m - 1 ) } ) } { \pi ( x ^ { ( m - 1 ) } ) } K ( x ^ { ( m - 1 ) } | x ^ { ( m ) } )
+$$
+
+where the second equality holds because we have
+
+$$
+\begin{array} { c } { { \displaystyle \frac { \pi ( x ^ { ( m ) } ) } { \pi ^ { ( m ) } ( x ^ { ( m ) } ) } \sum _ { x ^ { ( m - 1 ) } } \frac { \pi ^ { ( m - 1 ) } ( x ^ { ( m - 1 ) } ) } { \pi ( x ^ { ( m - 1 ) } ) } K ( x ^ { ( m - 1 ) } | x ^ { ( m ) } ) } } \\ { { = \displaystyle \frac { 1 } { \pi ^ { ( m ) } ( x ^ { ( m ) } ) } \sum _ { x ^ { ( m - 1 ) } } \pi ^ { ( m - 1 ) } ( x ^ { ( m - 1 ) } ) \frac { K ( x ^ { ( m - 1 ) } | x ^ { ( m ) } ) \pi ( x ^ { ( m ) } ) } { \pi ( x ^ { ( m - 1 ) } ) } } } \\ { { = 1 } } \end{array}
+$$
+
+Thereby, we show $h ^ { ( m ) } \sim \pi ( h ^ { ( m ) } | x ^ { ( m ) } )$ , i.e. $q _ { \phi } ( h ^ { ( m ) } | x ^ { ( m ) } )$ . This concludes the inductive step.
+
+# 9 SEMI-SUPERVISED LEARNING WITH INCLUSIVE-NRFS
+
+Apart from the basic losses, as shown in Eq.10, in applying inclusive-NRFs in SSL, there are some regularization losses that are helpful to guide the semi-supervised learning.
+
+Algorithm 3 Semi-supervised learning of inclusive-NRFs   
+
+<table><tr><td>repeat Sampling:</td></tr><tr><td>Draw a unsupervised minibatch U ~ p(x)pe(x)qp(h|x) and a supervised minibatch S ~ L; Updating:</td></tr><tr><td>Update θby ascending: ∑(x,x,h)~u[Vθuθ(x)-Vθue(x)]+ad∑(x,g)~s [Vθlogpe(i|x)]</td></tr><tr><td>-∑(x,x,h)~u [acVθH(pθ(y|z))+apVθ[uθ(x)]²];</td></tr><tr><td></td></tr><tr><td>Update by ascending:</td></tr><tr><td>∑(x,x,h)~u Vlogq(x,h); until convergence</td></tr></table>
+
+Confidence loss. Similar to Springenberg (2016); Li et al. (2017), we add the minimization of the conditional entropy of $p _ { \theta } ( y | \tilde { x } )$ averaged over training data to the loss w.r.t. $\theta$ (i.e. the first line in Eq.9) as follows:
+
+$$
+L _ { c } ( \theta ) = E _ { \tilde { p } ( \tilde { x } ) } \left[ H ( p _ { \theta } ( y | \tilde { x } ) ) \right] = - E _ { \tilde { p } ( \tilde { x } ) } \left[ \sum _ { y } p _ { \theta } ( y | \tilde { x } ) \log p _ { \theta } ( y | \tilde { x } ) \right]
+$$
+
+In this manner, we encourage the classifier $p _ { \theta } ( y | x )$ derived from the RF to make classifications confidently. In practice, we use stochastic gradients of $L _ { c } ( \theta )$ over minibatches in optimizing $\theta$ , as shown in Algorithm 3.
+
+Potential control loss. For random fields, the data log-likelihood $l o g p \theta ( \tilde { x } )$ is determined relatively by the potential value $u _ { \theta } ( \tilde { x } )$ . To avoid the potential values not to increase unreasonably, we could control the squared potential values, by minimizing:
+
+$$
+L _ { p } ( \theta ) = E _ { \tilde { p } ( \tilde { x } ) } \left[ u _ { \theta } ( \tilde { x } ) \right] ^ { 2 }
+$$
+
+In this manner, the potential values would be attracted to zeros. In practice, we use stochastic gradients of $L _ { p } ( \theta )$ over minibatches in optimizing $\theta$ , as shown in Algorithm 3.
+
+# 10 PROOF OF PROPOSITION 3
+
+Proposition 3. For the $R F$ as defined in Eq. 1, we have the following evidence upper bound:
+
+$$
+\begin{array} { r l } & { l o g p _ { \theta } ( \tilde { x } ) = \mathcal { U } ( \tilde { x } ; \theta , \phi ) - K L ( q _ { \phi } ( x ) | | p _ { \theta } ( x ) ) \leq \mathcal { U } ( \tilde { x } ; \theta , \phi ) , } \\ & { \mathcal { U } ( \tilde { x } ; \theta , \phi ) \triangleq u _ { \theta } ( \tilde { x } ) - \left( E _ { q _ { \phi } ( x ) } \left[ u _ { \theta } ( x ) \right] + H \left[ q _ { \phi } ( x ) \right] \right) . } \end{array}
+$$
+
+Proof. Note that $\begin{array} { r } { l o g p _ { \theta } ( \tilde { x } ) = u _ { \theta } ( \tilde { x } ) - l o g Z ( \theta ) } \end{array}$ . And we have the following lower bound on $Z ( \theta )$ $\begin{array} { r } { l o g Z ( \theta ) = l o g \int \exp ( u _ { \theta } ( x ) ) d x = l o g \int q _ { \phi } ( x ) \frac { \exp ( u _ { \theta } ( x ) ) } { q _ { \phi } ( x ) } d x \geq \int q _ { \phi } ( x ) l o g \frac { \exp ( u _ { \theta } ( x ) ) } { q _ { \phi } ( x ) } d x . } \end{array}$ g exp(uθ(x)) dx. This can be also seen from:
+
+$$
+\begin{array} { l } { { \displaystyle \int q _ { \phi } ( x ) u _ { \theta } ( x ) d x = \int q _ { \phi } ( x ) l o g p _ { \theta } ( x ) d x + l o g Z ( \theta ) } } \\ { { \displaystyle = - K L ( q _ { \phi } ( x ) | | p _ { \theta } ( x ) ) + l o g Z ( \theta ) + \int q _ { \phi } ( x ) l o g q _ { \phi } ( x ) d x . } } \end{array}
+$$
+
+Furthermore, it can be seen that learning in Kim & Bengio (2016) amounts to optimizing the following evidence upper bound:
+
+$$
+\operatorname* { m a x } _ { \theta } \operatorname* { m i n } _ { \phi } \mathcal { U } ( \tilde { x } ; \theta , \phi ) ,
+$$
+
+which is unfortunately not revealed in this manner in Kim & Bengio (2016).
+
+# 11 CONNECTION BETWEEN INCLUSIVE-NRFS AND GANS
+
+Note that for the generator as defined in Eq. 3, we have the following joint density
+
+$$
+l o g q _ { \phi } ( x , h ) = - \frac { 1 } { 2 \sigma ^ { 2 } } | | x - g _ { \phi } ( h ) | | ^ { 2 } + c o n s t a n t .
+$$
+
+The generator parameter $\phi$ is updated according to Eq. 5, which is rewritten as follows:
+
+$$
+{ { E } _ { p _ { \theta } ( x ) q _ { \phi } ( h | x ) } } \left[ \nabla _ { \phi } l o g q _ { \phi } ( x , h ) \right] = 0
+$$
+
+Specifically, we draw $( h ^ { \prime } , x ^ { \prime } ) \sim q _ { \phi }$ and then perform one-step SGLD to obtain $( h , x )$ . To simply the analysis of the connection, suppose $h \approx h ^ { \prime }$ , $\bar { x } ^ { \prime } \approx g _ { \phi } ( h ^ { \prime } ) \approx \bar { g } _ { \phi } ( h )$ . Then we have
+
+$$
+\begin{array} { l }  \displaystyle  \begin{array} { l } { \displaystyle x = x ^ { \prime } + \frac { \delta _ { 1 } } { 2 } [ \frac { \partial } { \partial x } l o g p _ { \theta } ( x ) ]  _ { x = x ^ { \prime } } + \sqrt { \delta _ { 1 } } \eta ^ { ( 1 ) } , \eta ^ { ( 1 ) } \sim \mathcal { N } ( 0 , I ) } \\ { \displaystyle x - g _ { \phi } ( h ) \approx \frac { \delta _ { 1 } } { 2 } [ \frac { \partial } { \partial x } l o g p _ { \theta } ( x ) ]  _ { x = g _ { \phi } ( h ) } = \frac { \delta _ { 1 } } { 2 } [ \frac { \partial } { \partial x } u _ { \theta } ( x ) ] \bigg \vert _ { x = g _ { \phi } ( h ) } } \end{array} \end{array}
+$$
+
+The gradient in the updating step in Algorithm 1 becomes:
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { \phi } l o g q _ { \phi } ( x , h ) = \frac { 1 } { \sigma ^ { 2 } } \left[ \frac { \partial } { \partial \phi } g _ { \phi } ( h ) \right] \left[ x - g _ { \phi } ( h ) \right] } \\ { \displaystyle \approx \frac { 1 } { \sigma ^ { 2 } } \left[ \frac { \partial } { \partial \phi } g _ { \phi } ( h ) \right] \left. \frac { \delta _ { 1 } } { 2 } \left[ \frac { \partial } { \partial x } u _ { \theta } ( x ) \right] \right. _ { x = g _ { \phi } ( h ) } } \\ { \displaystyle = \frac { 1 } { \sigma ^ { 2 } } \frac { \delta _ { 1 } } { 2 } \left[ \frac { \partial } { \partial \phi } u _ { \theta } ( g _ { \phi } ( h ) ) \right] } \end{array}
+$$
+
+where $\begin{array} { r l r } {  { \frac { \partial } { \partial \phi } g _ { \phi } ( h ) } } \end{array}$ is a matrix of size $d i m ( \phi ) \times d i m ( x )$ . Therefore, the inclusive-NRF Algorithm 1 can be viewed to perform the following steps:
+
+1. Draw an empirical example $\tilde { x } \sim p _ { 0 }$ .   
+2. Draw $h \sim p ( h )$ , $x ^ { \prime } = g _ { \phi } ( h )$ , and generate $x$ by one-step-gradient according to Eq. 14.   
+3. Update $\theta$ by ascending: $\nabla _ { \theta } u _ { \theta } ( \tilde { x } ) - \nabla _ { \theta } u _ { \theta } ( x )$ .   
+4. Update $\phi$ by descending: $\begin{array} { r l r } { \mathrm { - } \frac { \partial } { \partial \phi } u _ { \theta } ( g _ { \phi } ( h ) ) } \end{array}$ .
+
+Now suppose that we interpret the potential function $u _ { \theta } ( x )$ as the discriminator in GANs (or the critic in Wasserstein GANs), which assign high scalar scores to empirical samples $\tilde { x } \sim p _ { 0 }$ and low scalar scores to generated samples $x$ . Then, the inclusive-NRF training could be viewed as playing a two-player minimax game:
+
+$$
+\operatorname* { m i n } _ { \phi } \operatorname* { m a x } _ { \theta } E _ { \tilde { x } \sim p _ { 0 } } \left[ u _ { \theta } ( \tilde { x } ) \right] - E _ { h \sim p ( h ) } \left[ u _ { \theta } ( g _ { \phi } ( h ) ) \right] ,
+$$
+
+except that in optimizing $\theta$ , the generated sample are further revised by taking one-step-gradient of $u _ { \theta } ( x )$ w.r.t. $x$ (as shown in the above Step 2). The discriminator $u _ { \theta }$ is trained to discriminate between empirical samples and generated samples, while the generator $q _ { \phi }$ is trained to fool the discriminator by assigning higher scores to generated samples. From the above analysis, we find some interesting connections between inclusive-NRFs and existing studies in GANs.
+
+• The optimization shown in Eq. 15 is in fact the same as that in Wasserstein GANs (Theorem 3 in Arjovsky et al. (2017)), except that in Wasserstein GANs, the critic $u _ { \theta } ( x )$ is constrained to be 1-Lipschitz continuous. So hopefully we can improve the inclusive-NRF training by constraining the discriminator $u _ { \theta } ( x )$ to be 1-Lipschitz continuous, e.g. by utilizing the recently developed technique of spectral normalization of weight matrices in the discriminator as in Miyato et al. (2018).
+
+• To optimize $\theta$ , the generated sample is obtained by taking one-step-gradient of $u _ { \theta } ( x )$ w.r.t. $x$ . The tiny perturbation guided by the gradient to increase the score for the generated sample in fact creates an adversarial example. A similar idea is presented in Liu & Hsieh (2018) that when feeding real samples to the discriminator, 5 steps of PGD (Projected Gradient Descent) attack is taken to decrease the score to create adversarial samples. It is shown in Liu & Hsieh (2018) that training the discriminator with adversarial examples significantly improves the GAN traning. Hopefully in training the discriminator in inclusive-NRFs, the adversarial attack could be increasing scores for generated samples, or decreasing scores for real samples, or a mixed one. • The above analysis assume the use of one-step SGLD. It can be seen that running finite steps of SGLD in sample revision in fact create adversarial samples to fool the discriminator.
+
+# 12 DETAILS OF EXPERIMENTS
+
+# 12.1 GMM SYNTHETIC EXPERIMENT
+
+In the GMM experiment, we use the following procedure to estimate the metrics “covered modes” and “realistic ratio” for each trained model.
+
+1. Stochastically generate 100 samples.   
+2. A mode is defined to be covered (not missed) if there exist generated samples located closely   
+to the mode (with squared distance $< 0 . 0 2 $ ), and those samples are said to be realistic.   
+3. Count how many modes are covered and calculate the proportion of realistic samples.   
+4. Repeat the above steps 100 times and perform averaging.
+
+For each method, we independently train 10 models and calculate the mean and standard deviation (SD) across the 10 independent runs.
+
+The network architectures and hyperparameters are the same for all methods, as listed in Table 5. We use SGLD Welling & Teh (2011) for inclusive-NRFs on this synthetic dataset, with empirical revision hyperparameters $\delta _ { l } = 0 . 0 1$ .
+
+# 12.2 IMAGE GENERATION ON CIFAR-10
+
+Network architectures. For convenience, we refer to the two neural networks in implementing the potential $u _ { \theta }$ and the generator $q _ { \phi }$ in NRFs as the potential network and the generator network, respectively. For comparison of different methods, we use the same network architectures as in Table 4 in (Miyato et al., 2018) (ResNet using spectral normalization) for unsupervised learning of NRFs. For supervised learning, we use the semi-supervised inclusive-NRF Algorithm 3 over all labeled images. The difference in network architectures used for semi-supervised and unsupervised learning of inclusive-NRFs is that for SSL, the output layer of the potential network contains $K = 1 0$ scalar units, while a single scalar output unit is used for unsupervised learning.
+
+Hyperparameters. We use Adam optimizer with the hyperparameter $( \beta _ { 1 } ~ = ~ 0 , \beta _ { 2 } ~ = ~ 0 . 9$ and $\alpha = 0 . 0 0 0 3$ for random fields, $\alpha = 0 . 0 0 0 1$ for generators). For sample revision for inclusive-NRFs, we empirically choose SGLD with $L = 1$ $\langle \delta _ { l } = 0 . 0 0 3 )$ . More revision steps do not significantly improve unsupervised IS, as discussed in section 4.4 Note that we use the potential control loss in both unsupervised $\begin{array} { r } { \mathbf { \Phi } ( \alpha _ { p } = 0 . 1 ) } \end{array}$ ) and supervised $( \alpha _ { d } = 1 , \alpha _ { p } = 0 . 1 )$ settings, which is found beneficial for stable training.
+
+Evaluation. Figure 5(c)(d) show the generated samples from inclusive-NRFs for unsupervised and supervised settings respectively. We compute inception score (IS) and Frechet inception distance (FID) in the same way as in Miyato et al. (2018). We trained 10 models with different random seeds, and then generate 5000 images 10 times and compute the average inception score and the standard deviation. We compute FID between the true distribution and the generated distribution empirically over 10000 (test set) and 5000 samples.
+
+# 12.3 SEMI-SUPERVISED EXPERIMENT ON MNIST, SVHN AND CIFAR-10
+
+The network architectures (taken from the released code from Salimans et al. (2016) and widely used in Li et al. (2017); Dai et al. (2017b)) and hyperparameters for semi-supervised inclusiveNRFs on MNIST, SVHN and CIFAR-10 are listed in Table 6, Table 7 and Table 8 respectively. We use SGHMC for semi-supervised inclusive-NRFs for all three datasets, with empirical revision hyperparameters $( \beta = 0 . 5 , \bar { \delta _ { l } } = 0 . 0 0 3 )$ for MNIST and CIFAR-10, and $( \beta = 0 . 5 \bar { , } \delta _ { l } = 0 . 0 1 )$ for SVHN. The confidence loss is employed for semi-supervised inclusive-NRFs on MNIST and SVHN, and the potential control loss is employed on CIFAR-10.
+
+Figure 5(a)(b) show the generated samples from semi-supervised inclusive-NRFs trained over SVHN and CIFAR-10 respectively.
+
+# 12.4 ABLATION STUDY OF INCLUSIVE-NRFS ON CIFAR-10
+
+For unsupervised learning, we use the same networks as in Table 3 in Miyato et al. (2018) (standard CNN using spectral normalization). We use Adam optimizer with the hyperparameter $( \alpha = 0 . 0 0 0 2 , \beta _ { 1 } = 0 , \beta _ { 2 } = 0 . 9 )$ . For semi-supervised learning, the experimental setting is the same as in section 12.3 including the networks, number of labels, etc. For different revision steps, we use $\langle \delta _ { l } = 0 . 0 0 3 )$ for SGLD, and $( \beta = 0 . 5 , \delta _ { l } = 0 . 0 0 3 )$ for SGHMC. The potential control loss is employed in both unsupervised $\mathrm { \Delta } \alpha _ { p } = 0 . 1 )$ and semi-supervised $( \alpha _ { d } = 1 0 0 , \alpha _ { p } = 0 . 1 )$ learning.
+
+# 13 SSL TOY EXPERIMENT
+
+In Figure 2, we present the performance of semi-supervised inclusive-NRFs for SSL on a synthetic dataset, which emphasizes that inclusive-NRFs can provide (unnormalized) density estimates for $p _ { \theta } ( x )$ , $p _ { \theta } ( x , y = 1 )$ and $p _ { \theta } ( x , y = 2 )$ . In contrast, the use of GANs as general purpose probabilistic generative models has been limited by the difficulty in using them to provide density estimates or even unnormalized potential values for sample evaluation.
+
+The dataset is a 2D GMM with 16 Gaussian components, uniformly laid out on two concentric circles. The two circles represent two different classes, each class with 4 labeled data and 400 unlabeled data. The network architectures are the same as in Table 5, except that the neural network which implement the potential function $u _ { \boldsymbol { \theta } } ( \boldsymbol { x } , \boldsymbol { y } )$ for SSL now has two units in the output.
+
+![](images/1b15657e5f0222b34e4c135f5c523693a6b4ddd0194058b54335f5349276f8f0.jpg)  
+Figure 2: SSL toy experiment based on semi-supervised inclusive-NRFs. Each class has 4 labeled points, red dots for class 1 and blue for class 2. The learned potentials for $u _ { \theta } ( x )$ , $u _ { \theta } ( x , y = 1 )$ and $\overset { \cdot } { u } _ { \theta } ( x , y = 2 )$ are shown in (b)(c)(d) respectively.
+
+Figure 3 shows that the auxiliary generator smoothly outputs transitional samples as the latent code $h$ moves linearly in the latent space. The interpolated generation demonstrates that the model has indeed learned an abstract representation of the data.
+
+![](images/ec865b1908d6c1bac2116dc2ae82f1f353f69c500ab085e5d1549c7405bd788e.jpg)  
+Figure 3: Latent space interpolation with inclusive-NRFs on MNIST. The leftmost and rightmost columns are from stochastic generations $x _ { 1 }$ with latent code $h _ { 1 }$ and $x _ { 2 }$ with $h _ { 2 }$ . The columns in between correspond to the generations from the latent codes interpolated linearly from $h _ { 1 }$ to $h _ { 2 }$ .
+
+![](images/db3281866e1afab3ca5fc8ac7678f1db3fe0e8379adbf6b9074f4fe51f954daa.jpg)  
+Figure 4: Conditional generated samples from semi-supervised inclusive-NRFs trained on MNIST. Due to sample revision, the background pixels are not purely black.
+
+# 15 CLASS-CONDITIONAL GENERATION
+
+Figure 4 shows class-conditional generation results on MNIST with semi-supervised inclusive-NRFs. Notice that the generator does not explicitly include class labels, thus it is unable to perform classconditional generation directly. However, the random field has modeling of $p _ { \theta } ( x , y )$ , based on which we can perform class-conditional generation as follows:
+
+1. Generate a sample $x$ unconditionally, by ancestral sampling with the generator.   
+2. Predict the label $y$ for the sample $x$ by the random field;   
+3. Starting from $x$ , running SGLD/SGHMC revision with $p _ { \theta } ( x | y )$ as the target density by fixing $y$ . The resulting samples could be viewed as conditional generations, according to Theorem 1.
+
+![](images/c9ae503ee430f2a8db7bbe263985d614ab71b0461c2245c8d7db2c1f350c9c34.jpg)  
+Figure 5: Generated samples from semi-supervised inclusive-NRFs (i.e. trained for SSL) on SVHN and CIFAR-10 are shown in (a) and (b) respectively. Generated samples from unsupervised and supervised training of inclusive-NRFs on CIFAR-10 are shown in (c) and (d) respectively.
+
+Table 5: Network architectures and hyperparameters for the 2D GMM data.   
+
+<table><tr><td>Random Field</td><td>Generator</td></tr><tr><td>Input 2-dim data</td><td>Noise h (2-dim)</td></tr><tr><td>MLP100 units,Leaky ReLU</td><td>MLP 50 units,ReLU</td></tr><tr><td>MLP100 units,Leaky ReLU</td><td>MLP 50 units,ReLU</td></tr><tr><td>MLP1 unit, Linear</td><td>MLP 2 units,Linear</td></tr><tr><td>Batch size</td><td>100</td></tr><tr><td>Number of iterations</td><td>160,000</td></tr><tr><td>Leaky ReLU slope</td><td>0.2</td></tr><tr><td>Learning rate</td><td>0.001</td></tr><tr><td>Optimizer</td><td>Adam (β1 = 0.5,β2 =0.9)</td></tr><tr><td>Sample revision steps</td><td>L=10</td></tr></table>
+
+Table 6: Network architectures and hyperparameters for semi-supervised inclusive-NRFs on MNIST   
+
+<table><tr><td>Random Field</td><td>Generator</td></tr><tr><td>Input 28 × 28 Gray Image MLP 100O units,Leaky ReLU, Weight norm</td><td>Noise h (100-dim) MLP 50O units,Sotfplus,Batch norm</td></tr><tr><td>MLP 500 units,Leaky ReLU, Weight norm MLP 250 units,Leaky ReLU, Weight norm MLP 250 units,Leaky ReLU, Weight norm MLP 250 units,Leaky ReLU,Weight norm</td><td>MLP 50O units, Sotfplus,Batch norm MLP 784 units, Sigmoid</td></tr><tr><td>MLP 10 units,Linear,Weight norm Batch size</td><td>100</td></tr><tr><td>Number of epochs</td><td>200</td></tr><tr><td>Leaky ReLU slope</td><td>0.2</td></tr><tr><td>Learning rate</td><td>0.001</td></tr><tr><td>Optimizer</td><td>Adam (β1 = 0.0,β2 = 0.9)</td></tr><tr><td>Sample revision steps</td><td>L= 20</td></tr><tr><td>α in SSL</td><td>αd = 10,αc = 10,αp = 0</td></tr></table>
+
+Table 7: Network architectures and hyperparameters for semi-supervised inclusive-NRFs on SVHN   
+
+<table><tr><td>RandomField</td><td>Generator</td></tr><tr><td>Input 32 × 32 Colored Image</td><td>Noise h (100-dim)</td></tr><tr><td>3 × 3 conv. 64,Leaky ReLU, Weight norm 3 × 3 conv. 64, Leaky ReLU, Weight norm 3 × 3 conv. 64, Leaky ReLU, Weight norm</td><td>MLP 8192 units,ReLU, Batch norm Reshape 512 × 4×4 5 × 5 deconv. 256,ReLU, Stride=2</td></tr><tr><td>stride=2, dropout2d=0.5 3 × 3 conv. 128,Leaky ReLU, Weight norm 3 × 3 conv.128,Leaky ReLU,Weight norm 3 × 3 conv. 128,Leaky ReLU, Weight norm stride=2, dropout2d=0.5 3 × 3 conv. 128,Leaky ReLU,Weight norm 1 × 1 conv. 128,Leaky ReLU,Weight norm</td><td>5 × 5 deconv. 128,ReLU, Stride=2 5 × 5 deconv. 3, Tanh, Stride=2</td></tr><tr><td>1 × 1 conv. 128,Leaky ReLU, Weight norm MLP 10 units,Linear,Weight norm</td><td></td></tr><tr><td></td><td></td></tr><tr><td>Batch size Number of epochs</td><td>100</td></tr><tr><td>Leaky ReLU slope</td><td>400</td></tr><tr><td>Learning rate Optimizer</td><td>0.2 0.001</td></tr></table>
+
+Table 8: Network architectures and hyperparameters for semi-supervised inclusive-NRFs on CIFAR10   
+
+<table><tr><td>Random Field</td><td>Generator</td></tr><tr><td>Input 32 × 32 Colored Image</td><td>Noise h (100-dim)</td></tr><tr><td>3 × 3 conv. 128,Leaky ReLU, Weight norm</td><td>MLP 8192 units,ReLU, batch norm</td></tr><tr><td>3 × 3 conv. 128,Leaky ReLU, Weight norm</td><td>Reshape 512 × 4× 4</td></tr><tr><td>3 × 3 conv. 128,Leaky ReLU, Weight norm stride=2, dropout2d=0.5</td><td>5 × 5 deconv. 256, ReLU, Stride=2 5 × 5 deconv. 128 ReLU, stride=2</td></tr><tr><td>3 × 3 conv. 256,Leaky ReLU, Weight norm 3 × 3 conv. 256,Leaky ReLU, Weight norm 3 × 3 conv. 256,Leaky ReLU, Weight norm</td><td>5 × 5 deconv. 3, Tanh, Stride=2</td></tr><tr><td>stride=2, dropout2d=0.5 3 × 3 conv. 512,Leaky ReLU, Weight norm 1 × 1 conv. 256,Leaky ReLU, Weight norm</td><td></td></tr><tr><td>1 × 1 conv. 128,Leaky ReLU, Weight norm</td><td></td></tr><tr><td>MLP 10 units,Linear, Weight norm</td><td></td></tr><tr><td>Batch size</td><td>100</td></tr><tr><td>Number of epochs</td><td>600</td></tr><tr><td>Leaky ReLU slope</td><td>0.2</td></tr><tr><td>Learning rate</td><td></td></tr><tr><td></td><td>0.001</td></tr><tr><td>Optimizer</td><td>Adam (β1 = 0.0,β2 = 0.9)</td></tr><tr><td>Sample revision steps α in SSL</td><td>L=10 αd = 100,αc = 0,αp = 0.1</td></tr></table>

parse/train/kHromd7SNA/kHromd7SNA.md

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+# MOTION REPRESENTATIONS FOR ARTICULATED ANIMATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose novel motion representations for animating articulated objects consisting of distinct parts. In a completely unsupervised manner, our method identifies meaningful object parts, tracks them in a driving video, and infers their motions by considering their principal axes. In contrast to the previous keypoint-based works, our method extracts meaningful and consistent regions, describing locations, shape, and pose. The regions correspond to semantically relevant and distinct object parts, that are more easily detected in frames of the driving video. To force decoupling of foreground from background, we model non-object related global motion with a homography. Our model1 can animate a variety of objects, surpassing previous methods by a large margin on existing benchmarks. We present a challenging new benchmark with high-resolution videos and show that the improvement is particularly pronounced when articulated objects are considered.
+
+# 1 INTRODUCTION
+
+Animation—bringing static objects to life—has broad applications across education and entertainment. Animated characters and objects increase the creativity and appeal of content, improve the clarity of material through storytelling, and enhance user experiences. Imagine the Mona Lisa describing the manner in which she was painted, or Michelangelo’s David detailing the method with which he was sculpted, or an influential historical figure shedding light on key events of the past (Fig. 1); how much more engaging this would be.
+
+Until very recently, animation techniques necessary for achieving such results required a trained professional, specialized hardware, software, and a great deal of effort. Quality results generally still do, but vision and graphics communities have attempted to address some of these limitations by training data-driven methods (Wang
+
+![](images/f30413fadc3b455edf46dd95ca7f4a717bde631a83181d84719e4a045ad6cfb8.jpg)  
+Figure 1: An animation produced by our method.
+
+et al., 2018a; Chan et al., 2019; Ren et al., 2020; Geng et al., 2019; Gafni et al., 2019) on object classes for which prior knowledge of object shape and pose can be learned. This, however, requires ground truth pose and shape data to be available during training.
+
+Recent works have sought to avoid the need for ground truth data through unsupervised motion transfer (Wiles et al., 2018; Siarohin et al., 2019a;b). Significant progress on the several key challenges have been made, including training using image reconstruction as a loss, and disentangling motion from appearance. This has created the potential to animate a broader range of object categories, without any domain knowledge or labelled data, requiring only videos of objects in motion during training. However, two key problems remain open. The first is how to represent the parts of an articulated or non-rigid moving object, including their shapes and poses. The second is given the object parts, how to animate them using the sequence of motions in a driving video.
+
+Initial attempts involved extracting unsupervised keypoints (Lorenz et al., 2019; Kim et al., 2019) in end-to-end frameworks (Wiles et al., 2018; Siarohin et al., 2019b;a), then warping a feature embedding of a source image to align its keypoints with those of a driving video. Follow on work (Siarohin et al., 2019a) additionally modelled the motion around each keypoint with local, affine transformations, and introduced a generation module that both composites warped source image regions and inpaints occluded regions, to render the final image. This enabled a variety of creative applications2, for example needing only one source face image to generate a near photo-realistic animation, driven by a video of a different face.
+
+However, the resulting unsupervised keypoints are detected on the boundary of the objects. While points on edges are easier to identify, tracking such keypoints between frames is problematic, as any point on the boundary is a valid candidate, making it hard to establish correspondences between frames. A further problem is that the unsupervised keypoints do not correspond to semantically meaningful object parts, and represent location and direction, but not shape. Due to this limitation, animating articulated objects, such as bodies, remains challenging. Furthermore, these methods assume static backgrounds, i.e., no camera motion, leading to leakage of background motion information into one or several of the detected keypoints. Despite significant breakthroughs, these remaining deficiencies limit the scope of the core innovation to more trivial object categories and motions, and lower quality outputs, especially when objects are articulated.
+
+This work introduces two contributions critical to addressing these challenges. First, we redefine the underlying motion representation. Instead of using keypoints, we switch to regions that allow first-order motion to be measured, rather than regressed. This enables improved convergence, more stable, robust object and motion representations, and also empirically captures the shape of the underpinning object parts, leading to better motion segmentation. Fig. 3 contains several examples of region vs. keypoint-based motion representation.
+
+Secondly, we explicitly model background or camera motion between training frames by predicting the parameters of a global homography explaining non-object related motions. This enables the model to focus solely on the foreground object, making the identified points more stable, and further improves convergence.
+
+These contributions unlock significant gains in capability for unsupervised motion transfer methods, resulting in much improved animation of articulated objects in particular. Furthermore, the framework scales better in the number of unsupervised regions, resulting in more detailed motion. Our method outperforms previous unsupervised animation methods on a variety of datasets, including talking faces, taichi videos and animated pixel art. We additionally present a new dataset, TED talk speakers, to create a more challenging benchmark for the task of animating articulated objects.
+
+# 2 RELATED WORK
+
+Image animation methods can be separated into supervised, which require knowledge about the animated object during training, and unsupervised, which do not. Such knowledge typically includes landmarks (Cao et al., 2014; Zakharov et al., 2019; Qian et al., 2019; Ha et al., 2020), semantic segmentations (Nirkin et al., 2019), and parametric 3D models (Geng et al., 2019; Thies et al., 2016; Deng et al., 2020; Nagano et al., 2018; Liu et al., 2019). As a result, supervised methods are limited to a small number of object categories for which a lot of labelled data is available, such as faces and human bodies. Early face reenactment work (Thies et al., 2016) fitted a 3D morphable model to an image, animating and rendering it back using graphical techniques. Further works used neural networks to get higher quality rendering (Kim et al., 2018; Wang et al., 2018b), sometimes requiring multiple images per identity (Geng et al., 2019; Pumarola et al., 2018). A body of works treats animation as an image-to-image (Siarohin et al., 2018) or a video-to-video (Wang et al., 2018a; Chan et al., 2019; Ren et al., 2020) translation problem. Apart from some exceptions (Wang et al., 2019), these works further constrain the problem to animating a single instance of an object, such as a single face (Kim et al., 2018; Bansal et al., 2018) or a single human body (Chan et al., 2019; Ren et al., 2020; Wang et al., 2018a), requiring retraining (Bansal et al., 2018; Chan et al., 2019; Ren et al., 2020) or fine-tuning (Zakharov et al., 2019) for each new instance. Despite promising results, generalizing these methods beyond a limited range of object categories remains challenging. Additionally, they tend to transfer not only the motion but also the identity of the driving object, making the shape of the animated face or a body similar or identical to the driving face or body (Kim et al., 2018; Zakharov et al., 2019).
+
+![](images/5b23edf1ef336c348613954cba5bdb75e548528d675bde52781924f7396669c0.jpg)  
+Figure 2: Overview of our model. The region predictor returns heatmaps for each part in the source and the driving images. We then compute principal axes of each heatmap, to transform each region from the source to the driving frame through a whitened reference frame. Region and background transformations are combined by the pixel-wise flow prediction network. The target image is generated by warping the source image in a feature space using the pixel-wise flow, and inpainting newly introduced regions, as indicated by the confidence map.
+
+Unsupervised methods address some of these limitations. They do not require any labelled data regarding the shape or landmarks of the animated object. Video-generation-based animation methods predict future frames of a video, given the first frame and an animation class label, such as "make a happy face", "do jumping jack", or "play golf" (Tulyakov et al., 2018; Saito et al., 2017; Clark et al., 2019). A further group of works re-target animation from a driving video to a source frame. X2Face (Wiles et al., 2018) builds a canonical representation of an input face, and generates a warp field conditioned on the driving video. Monkey-Net (Siarohin et al., 2019b) learns a set of unsupervised keypoints to generate animations. Follow-up work substantially improves the quality of animation by considering a first order motion model (FOMM) (Siarohin et al., 2019a) for each keypoint, represented by regressing a local, affine transformation. Both of these works apply to a wider range of objects including faces, bodies, robots, and pixel art animations. Empirically, these methods extract keypoints on the boundary of the animated objects. Articulated objects such as human bodies are therefore challenging, as internal motion, for example, an arm moving across the body, is not well modeled, producing unconvincing animations.
+
+This work presents an unsupervised method. We argue that the limitations of previous such methods in animating articulated objects is due to an inability of their internal representations to capture complete object parts, their shape and pose. X2Face (Wiles et al., 2018) assumes an object can be represented with a single RGB texture, while other methods find keypoints on edges (Siarohin et al., 2019b;a). Our new region and background motion representations address these shortcomings.
+
+# 3 METHOD
+
+Our unsupervised animation framework consists of a system design, and methods for training this system using two different frames, source S, and driving D, from the same video.
+
+# 3.1 SYSTEM DESIGN
+
+FOMM (Siarohin et al., 2019a), the current state-of-the-art method in unsupervised animation learning, consists of two main parts: motion estimation and image generation. The contributions of our work lie in novel motion representations within the first part of this framework. Our system, outlined in Fig. 2, therefore follows the FOMM design as closely as possible, in order to demonstrate the impact due specifically to our contributions.
+
+![](images/b434e5f758996b8746e73620d93b6fcb9f93375ab8e8dd904e07b5ad278a1f48.jpg)  
+Figure 3: Comparison of motion/part representations. Regression-based keypoint representations do not provide consistent detection between frames (marked with red). Additionally, background motion leaks into one or several detected keypoints. Our PCA-based regions (with and without background motion) correctly identify meaningful parts, are consistent between frames, and use additional regions more effectively.
+
+# 3.1.1 REGIONS AND COARSE MOTION
+
+Regions FOMM learns to detect $K$ distinct object regions, where $K$ is a user-defined parameter. An encoder-decoder region predictor network takes an image as input, and outputs $K$ heatmaps, $\mathbf { M } ^ { 1 } , . . , \mathbf { M } ^ { K }$ . The final network layer is a softmax operation, s.t. $\mathbf { M } ^ { k } \in [ 0 , 1 ] ^ { H \times W }$ , where $H$ and $W$ are the height and width of the image respectively, and $\begin{array} { r } { \sum _ { z \in \mathcal { Z } } m _ { z } ^ { k } = 1 } \end{array}$ , where $z$ is a pixel location $\mathbf { \dot { x } }$ , y coordinates) in the image, the set of all pixel locations being $\mathcal { Z }$ , and $m _ { z } ^ { k }$ is the $k$ -th heatmap weight at pixel $z$ . We use the same region representation and encoder here. Nevertheless, the encoded regions differ significantly (see Fig. 3), ours mapping to meaningful object parts such as the limbs of an articulated body, due to our novel foreground motion representation, described below.
+
+Estimating foreground region motion FOMM estimates a first-order transformation from an image $\mathbf { X }$ to a reference frame $\mathbf { R }$ , for each region separately. The region heatmap encodes translation by its mean position, while other affine parameters are regressed per pixel and then pooled per region according to the heatmap weights. Here we change the way this transformation is represented: all motion is measured directly from the heatmap. Translation is given by the mean position, as before, while in-plane rotation and scaling in $\mathbf { X } ^ { - }$ and y-directions are computed via a principal component analysis (PCA) of the heatmap. Shear is not captured, therefore our transform isn’t fully affine, with only five degrees of freedom instead of six. Nevertheless, it captures sufficient motion, shear being a less significant component of the affine transform for this task. The transformation of a region from the reference frame to the image is computed as follows:
+
+$$
+{ \begin{array} { r l } & { \qquad \mu ^ { k } = \displaystyle \sum _ { z \in { \mathcal { Z } } } m _ { z } ^ { k } z , } \\ & { \qquad U ^ { k } S ^ { k } { V ^ { k } } ^ { \mathsf { T } } = \displaystyle \sum _ { z \in { \mathcal { Z } } } m _ { z } ^ { k } ( z - \mu ^ { k } ) ( z - \mu ^ { k } ) ^ { \mathsf { T } } , ~ \mathrm { ( v i a S V D ) } , } \\ & { \qquad A _ { \mathbf { X }  \mathbf { R } } = \displaystyle [ U ^ { k } S ^ { k ^ { \frac { 1 } { 2 } } } , \mu ^ { k } ] . } \end{array} }
+$$
+
+The singular value decomposition (SVD) approach to computing PCA (Wall et al., 2003) is used here. We refer to FOMM and our estimation approaches as regression-based and PCA-based, respectively. The reference frame in both is used only as an abstract, intermediate coordinate frame between the source and driving image coordinate frames. However, here (in contrast to FOMM) it is not in fact abstract, corresponding to the coordinate frame where the heatmap is whitened (i.e. has zero mean and identity covariance); see Fig. 2. Driving to source image motion is then
+
+$$
+\begin{array} { r } { A _ { \mathbf { S }  \mathbf { D } } ^ { k } = A _ { \mathbf { S }  \mathbf { R } } ^ { k } [ A _ { \mathbf { D }  \mathbf { R } } ^ { k } ] ^ { - 1 } . } \end{array}
+$$
+
+Estimating background motion FOMM has no background motion model. We observe that with significant background motion between frames, e.g. due to camera motion, predicted regions can therefore include the moving background, reducing test-time accuracy. To resolve this, we additionally regress a background homography transformation, $\mathbf { H }$ , using an encoder network that takes as input the source and driving images, concatenated along the channel dimension, and outputs eight real values, $h _ { 1 } , . . , h _ { 8 }$ , such that $\bar { { \bf H } } = \left[ [ h _ { 1 } , h _ { 2 } , h _ { 3 } ] ^ { \top } \left[ h _ { 4 } , h _ { 5 } , h _ { 6 } \right] ^ { \top } [ h _ { 7 } , h _ { 8 } , 1 ] ^ { \top } \right]$ . With background motion well modeled, we show that the network is able to separate background and object motion in a completely unsupervised manner.
+
+# 3.1.2 IMAGE GENERATION
+
+Given these coarse motions, FOMM then renders the target image in two stages: a pixel-wise flow generator converts coarse motions to dense optical flow, then a composition network warps the source image according to the flow, and also inpaints missing regions. We follow this architecture, and summarize these two modules here, but refer the reader to Siarohin et al. (2019a) for the full details.
+
+Pixel-wise flow generation Coarse motions are combined via a weighted sum, to compute a dense, per pixel motion, or flow. The per pixel weights, as well as a confidence map, are computed via an encoder-decoder network. The input is a $H \times W \times ( 4 K + 3 )$ tensor, with four channels per region, three for the source image warped according to the region’s motion model, and one for a heatmap of the region, which is a gaussian approximation to ${ \bf { M } } ^ { \breve { k } }$ , in order to avoid leakage of driving image appearance through the heatmap. Here we add a further three input channels (compared to FOMM) for the source image warped according to the background motion model.
+
+Warping and inpainting The source image is passed through an encoder network. The resulting feature map is warped and masked according to the pixel-wise flow and confidence map from the previous module, respectively. A decoder then renders the final image, inpainting missing parts. In contrast to FOMM, but similar to Monkey-Net (Siarohin et al., 2019b), here skip connections are used between the encoder and decoder. The skip connection feature maps are also warped and masked.
+
+# 3.2 TRAINING
+
+The proposed model is trained end-to-end using a reconstruction loss in the feature space of the pretrained VGG-19 network (Johnson et al., 2016; Wang et al., 2017). Following Siarohin et al. (2019a); Wang et al. (2003), we adopt a multi-resolution version of the reconstruction loss:
+
+$$
+\mathcal { L } _ { \mathrm { r e c } } ( \hat { \mathbf { D } } , \mathbf { D } ) = \sum _ { l } \sum _ { i } \left| \mathrm { V } _ { i } \big ( \mathrm { F } _ { l } \odot \hat { \mathbf { D } } \big ) - \mathrm { V } _ { i } \big ( \mathrm { F } _ { l } \odot \mathbf { D } \big ) \right| ,
+$$
+
+where $\hat { \bf D }$ is the generated image, $\mathrm { V } _ { i }$ is the $i ^ { \mathrm { { t h } } }$ -layer of the VGG-19 pretrained network, $\mathrm { F } _ { l }$ is a downsampling operator. Similarly to Wang et al. (2017); Siarohin et al. (2019a) we used conv1_2, conv2_2, conv3_2, conv4_2, conv5_2 layers and downsampled the images to 1, 0.5, 0.25, 0.125 of the original edge size. In total we have 20 reconstruction terms. To improve detection of unsupervised regions we follow the unsupervised keypoint detection literature (Jakab et al., 2018; Zhang et al., 2018) and adopt the equivariance loss, denoted as $\mathcal { L } _ { \mathrm { e q } }$ . We use a thin-plate spline implementation provided in FOMM (Siarohin et al., 2019a). The final loss is a sum of the two loss terms, $\mathcal { L } = \mathcal { L } _ { \mathrm { r e c } } + \mathcal { L } _ { \mathrm { e q } }$ .
+
+![](images/97dc64f529863911ea0d5ff8c5401b09ec59c12b8c5759765a48499e1054109f.jpg)  
+Figure 4: Qualitative comparisons. We show representative examples of articulated animation using our method and FOMM (Siarohin et al., 2019a), on two datasets of articulated objects: TED-talks (left) and TaiChiHD (right). Zoom in for greater detail.
+
+# 4 EVALUATION
+
+We now discuss the datasets, metrics and experiments used to evaluate the proposed method. Later we compare with prior work, as well as ablate our contributions.
+
+# 4.1 TOY MOTION REPRESENTATION EXPERIMENT
+
+To demonstrate the benefit of the proposed PCAbased motion representation, we devise an experiment on rotated rectangles (see Appendix E): the task is to predict the rotation angle of a rectangle in an image. To fully isolate our contribution, we consider a supervised task, where three different architectures learn to predict angles under the $L _ { 1 }$ loss. The first, a Naive architecture, directly regresses the angle using an encoder-like architecture. The second is Regression-based, as in to FOMM (Siarohin et al., 2019a). The third uses our PCA-based approach (see Appendix E). Test results are presented in Fig. 5, against training set size. The Naive baseline struggles to produce meaningful results for any size of training set, while Regression-based performance improves with more data. However, the PCA-based significantly improves accuracy over the Regression-based one, being over an order of magnitude better with a large number of samples. This shows that it is significantly easier for the network to infer geometric parameters of the image, such as angle, using our proposed PCA-based representation.
+
+![](images/977dd74fcd71d52ec274f24c38abdbbe09bcd9b20a9d9ebb057ddc69537a1d61.jpg)  
+Figure 5: Mean test-time absolute rotation error, as a function of training set size.
+
+# 4.2 BENCHMARKS
+
+We evaluate our method on several benchmark datasets for animating human faces and bodies. Each dataset has separate training and test videos. The datasets are as follows:
+
+• VoxCeleb (Nagrani et al., 2017) consists of interview videos of different celebrities. We extract square, face regions and downscale them to $2 5 6 \times 2 5 6$ , following FOMM (Siarohin et al., 2019a). The number of frames per video ranging from 64 to 1024. • TaiChiHD (Siarohin et al., 2019a) consists of cropped videos of full human bodies performing Tai Chi actions. We evaluate on two resolutions of the dataset: $2 5 6 \times 2 5 6$ (from FOMM (Siarohin et al., 2019a)), and a new, $5 1 2 \times 5 1 2$ subset, removing videos lacking sufficient resolution to support that size. • TED-talks is a new dataset, collected for this paper in order to demonstrate the generalization properties of our model. We cropped the upper part of the human body from the videos, downscaling to $3 8 4 \times 3 8 4$ . The number of frames per video ranges from 64 to 1024.
+
+Table 2: Video reconstruction: comparison with the state of the art on four different datasets. For all methods we use $K = 1 0$ regions. (Best result in bold.)   
+
+<table><tr><td></td><td colspan="3">TaiChiHD (256)</td><td colspan="3">TaiChiHD (512)</td><td colspan="3">TED-talks</td><td colspan="3">VoxCeleb</td></tr><tr><td></td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>L1</td><td>AKD</td><td>AED</td></tr><tr><td>FOMM</td><td>0.056</td><td>(6.53,0.033)</td><td>0.172</td><td>0.075</td><td>(17.12,0.66)</td><td>0.203</td><td>0.033</td><td>(7.07,0.014)</td><td>0.163</td><td>0.041</td><td>1.27</td><td>0.134</td></tr><tr><td>Ours</td><td>0.048</td><td>(5.45,0.028)</td><td>0.152</td><td>0.064</td><td>(14.00,0.44)</td><td>0.171</td><td>0.026</td><td>(4.02,0.007)</td><td>0.119</td><td>0.040</td><td>1.28</td><td>0.133</td></tr></table>
+
+Further datasets are used in the supplementary material.
+
+Since video animation is a relatively new problem, there are not currently many effective ways of evaluating it. For quantitative metrics, prior works (Siarohin et al., 2019b;a) use video reconstruction accuracy as a proxy for image animation quality. We adopt the same metrics here:
+
+• $\mathcal { L } _ { 1 }$ error measures the difference between reconstructed video and ground-truth video pixel values using the $\mathcal { L } _ { 1 }$ metric. • Average keypoint distance (AKD) and missing keypoint rate (MKR) evaluate the difference between poses of reconstructed and ground truth video. Landmarks are extracted from both videos using public, body (Cao et al., 2017) (for TaiChiHD and TED-talks) and face (Bulat & Tzimiropoulos, 2017) (for VoxCeleb) detectors. AKD is then the average distance between corresponding landmarks, while MKR is the proportion of landmarks present in the ground-truth that are missing in the reconstructed video. • Average Euclidean distance (AED) evaluates how well identity is preserved in reconstructed video. Public re-identification networks for bodies (Hermans et al., 2017) (for TaiChiHD and TED-talks) and faces (Amos et al., 2016) extract identity from reconstructed and ground truth frame pairs, then we compute the average $\mathcal { L } _ { 2 }$ norm of their difference across all pairs.
+
+# 4.3 COMPARISON WITH THE STATE OF THE ART
+
+We compare our method with the current state of the art for unsupervised animation, FOMM (Siarohin et al., 2019a), across all datasets, on both reconstruction (the training task) and animation (the testtime task). We used an extended training schedule compared to Siarohin et al. (2019a), with $50 \%$ more iterations. To compare fairly with FOMM (Siarohin et al., 2019a), we also re-trained it with the same training schedule.
+
+Reconstruction quality Quantitative reconstruction results are reported in Tab. 2. We first show that our method reaches state-of-the-art results on a dataset with non-articulated objects such as faces. Indeed, when compared with FOMM (Siarohin et al., 2019a) on VoxCeleb our method shows on-par results. The situation changes, however, when articulated objects are considered, such as human bodies in TaiChiHD and TED-talks datasets, on which our improved motion representations boost all the metrics. The advantage over the state of the art holds at different resolutions, for TaiChiHD (256), TaiChiHD (512) and TED-talks, as well as for different numbers of selected regions (discussed later).
+
+Animation quality Fig. 3 & 4 show selected and representative animations respectively, using our method and FOMM (Siarohin et al., 2019a), on articulated bodies, both using absolute motion. The results show clear improvements, in most cases, in animation quality, especially of limbs.
+
+Animation quality was evaluated quantitatively through a user preference study similar to that of Siarohin et al. (2019a). AMT users were presented with the source image, driving video, and the output from our method and FOMM (Siarohin et al., 2019a), and asked which of the two videos they preferred. 50 such videos were evaluated, by 50 users each, for a total of 2500 preferences per study. The results, shown in Tab. 4, further support the reconstruction scores in Tab. 2. When the animated object is not articulated (VoxCeleb), the method delivers results comparable to the previous work. When bodies are animated (TaiChiHD & TED-talks), FOMM (Siarohin et al., 2019a) fails to correctly detect and animate the articulated body parts such as hands. Our method renders them in the driving pose even for extreme cases, leading to a high preference in favor of it.
+
+Finally, we applied animation from a TED-talks video to a photograph of Winston Churchill, shown in Fig. 1, demonstrating animation of out of domain data.
+
+Table 3: Ablation study on TaiChiHD (256) dataset with $K = 1 0$ . (Best result in bold.)   
+
+<table><tr><td></td><td>L1</td><td>(AKD,MKR)</td><td>AED</td></tr><tr><td>No pca or bg model</td><td>0.060</td><td>(6.14, 0.033)</td><td>0.163</td></tr><tr><td>No pca</td><td>0.049</td><td>(6.04,0.034)</td><td>0.163</td></tr><tr><td>No bg model</td><td>0.059</td><td>(5.47, 0.027)</td><td>0.164</td></tr><tr><td>Full method</td><td>0.048</td><td>(5.45, 0.028)</td><td>0.152</td></tr></table>
+
+Table 4: User study: the proportion $( \% )$ of users that prefer our method over FOMM (Siarohin et al., 2019a).   
+
+<table><tr><td>Dataset</td><td>User preference (%)</td></tr><tr><td>VoxCeleb</td><td>52.2%</td></tr><tr><td>TaiChiHD (256)</td><td>83.0%</td></tr><tr><td>TED-talks</td><td>91.0%</td></tr></table>
+
+# 4.4 ABLATIONS
+
+In order to understand how much benefit each of our contributions bring, we ran a number of ablation experiments, detailed in Tab. 3.
+
+PCA-based vs. regression-based representations First we compare the PCA-based motion model with the previous, regression-based one (Siarohin et al., 2019a). From the qualitative, heatmap depictions in Fig. 3, we observe that the regression-based method localizes one edge of each corresponding part, while our method predicts regions that roughly correspond to the segmentation of the object into its constituent, articulated parts. This meaningful segmentation arises completely unsupervised.
+
+From Tab. 1 we note that adding the PCA-based representation alone (second row) had marginal impact on the $\mathcal { L } _ { 1 }$ score (dominated by the much larger background region), but it had a much larger impact on other metrics, which are more sensitive to object-part-related errors on articulated objects. This is corroborated by Tab. 3.
+
+We intuit that PCA-based estimation both captures regions and improves performance because it is much easier for the convolutional network to assign pixels of an object part to the corresponding heatmap than to directly regress motion parameters to an abstract reference frame. This is borne out by our toy experiment (sec. 4.1). In order to estimate the heatmap it need only learn all appearances of the corresponding object part, whereas regression-based networks must learn the joint space of all appearances of a part in all possible geometric configurations (e.g. rotated, scaled etc.).
+
+One of the most important hyper-parameters of our model is the number of regions, $K$ . The qualitative and quantitative ablations of this parameter are shown in Fig. 3 and Tab. 1 respectively. We can observe that, while the regression-based representation fails when the number of keypoints grows to 20, our PCA-based representation scales well with the number of regions.
+
+Modeling background motion Tab. 3 shows that methods with background motion modeling have much lower $\mathcal { L } _ { 1 }$ error. Since background constitutes a large portion of the image, and $\mathcal { L } _ { 1 }$ treats all pixels equally, this is to be expected. AED was also impacted, suggesting that the identity representation captures some background appearance. However, since AKD & MKR metrics evaluate object pose only, they are not improved by background modelling.
+
+# 5 CONCLUSION
+
+We have argued that previous unsupervised animation frameworks’ poor results on articulated objects are due to their representations. We propose a new, PCA-based, region motion representation, which we believe both makes it easier for the network to learn region motion, and encourages it to learn semantically meaningful object parts. In addition, we propose a background motion estimation module to decouple foreground and background motion. Qualitative and quantitative results across a range of datasets and tasks demonstrate several key benefits: improved region distribution and stability, improved reconstruction accuracy and user perceived quality, and an ability to scale to more regions. We also introduce a new, more challenging dataset, TED-talks, for benchmarking future improvements on this task.
+
+While we show some results on out of domain data (Fig. 1), generalization remains a significant challenge to making this method broadly practical in articulated animation of inanimate objects.
+
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+Albert Pumarola, Antonio Agudo, Aleix M Martinez, Alberto Sanfeliu, and Francesc Moreno-Noguer. Ganimation: Anatomically-aware facial animation from a single image. In Proceedings of the European Conference on Computer Vision, 2018.
+
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+
+# A TED-TALKS DATASET CREATION
+
+To create the TED-talks dataset, we downloaded 3,035 YouTube videos, shared under the “CC BY – $\mathrm { N C } - \mathrm { N D } 4 . 0$ International” license,3 using the "TED talks" query. From these initial candidates, we selected the videos where the upper part of the person is visible for at least 64 frames, and the height of the person bounding box was at least 384 pixels. After that, we manually filtered out static videos and videos in which a person is doing something other than presenting. We ended up with 411 videos, and split these videos in 369 training and 42 testing videos. We then split each video into chunks from a consistent camera angle (i.e. with no cuts to another camera), and for which the presenter didn’t move too far from their starting position in the chunk. We cropped the a square region around the presenter, such that they had a consistent scale, and downscaled this region to $3 8 4 \times 3 8 4$ pixels. Chunks that lacked sufficient resolution to be downscaled, or had a length shorter than 64 frames, were removed. Both the distance moved and the region cropping were achieved using a bounding box estimator for humans (Wu et al., 2019). Overall, we obtained 1,177 training video chunks and 145 test videos chunks.
+
+# B IMPLEMENTATION DETAILS
+
+For a fair comparison, in order to highlight our contributions, we mostly follow the architecture design of FOMM (Siarohin et al., 2019a). Similar to FOMM, our region predictor, background motion predictor and pixel-wise flow predictor operate on a quarter of the original resolution, e.g. $6 4 \times 6 4$ for $2 5 6 \times 2 5 6$ images, $9 6 \times 9 6$ for $3 8 4 \times 3 8 4$ and $1 2 8 \times 1 2 8$ for $5 1 2 \times 5 1 2$ . We use the U-Net (Ronneberger et al., 2015) architecture with five "convolution - batch norm - ReLU - pooling" blocks in the encoder and five "upsample - convolution - batch norm - ReLU" blocks in the decoder for both the region predictor and the pixel-wise flow predictor. For the background motion predictor, we use only the five block encoder part. Similarly to FOMM (Siarohin et al., 2019a), we use the Johnson architecture (Johnson et al., 2016) for image generation, with two down-sampling blocks, six residual-blocks, and two up-sampling blocks. However, we add skip connections that are warped and weighted by the confidence map. Our method is trained using Adam (Kingma & Ba, 2014) optimizer with learning rate $2 e - 4$ and batch size 48, 20, 12 for $2 5 6 \times 2 5 6$ , $3 8 4 \times 3 8 4$ and $5 1 2 \times 5 1 2$ resolutions respectively. During the training process, the networks observe 3M source-driving pairs, each pair selected at random from a random video chunk, and we drop the learning rate by a factor of 10 after 1.8M and 2.7M pairs. We use 4 Nvidia P100 GPUs for training.
+
+# C MGIF DATASET
+
+We run additional experiments on the MGif (Siarohin et al., 2019b) dataset to further demonstrate the superiority of PCA-based representations over regression-based ones. The dataset contains a set of animations of articulated, 2D, cartoon animals. The qualitative results are presented in the supplementary video. We can observe that our PCA-based representation successfully tracks all legs, while the regression-based representation often misses some of the legs, which leads to worse reconstruction quality. This observation is further confirmed by quantitative evaluation; the $\mathcal { L } _ { 1 }$ error for FOMM (Siarohin et al., 2019a) is 0.0223, while for our method it is 0.0206.
+
+# D COMPARISON WITH OTHER METHODS
+
+The main paper has focused on comparing our method to FOMM (Siarohin et al., 2019a), as it is both most similar to our work, and the current state-of-the-art. We show quantitative results using prior works (Wiles et al., 2018; Siarohin et al., 2019b) in Table 5. These are significantly inferior to both FOMM and our method.
+
+Table 5: Video reconstruction comparison. (Best result in bold.)   
+
+<table><tr><td rowspan="2"></td><td colspan="3">TaiChiHD (256)</td><td colspan="3">VoxCeleb</td></tr><tr><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>L1</td><td>AKD</td><td>AED</td></tr><tr><td>X2Face</td><td>0.080</td><td>(17.65, 0.109)</td><td>0.27</td><td>0.078</td><td>7.69</td><td>0.405</td></tr><tr><td>Monkey-Net</td><td>0.077</td><td>(10.80, 0.059)</td><td>0.228</td><td>0.049</td><td>1.89</td><td>0.199</td></tr><tr><td>FOMM</td><td>0.056</td><td>(6.53, 0.033)</td><td>0.172</td><td>0.041</td><td>1.27</td><td>0.134</td></tr><tr><td>Ours</td><td>0.048</td><td>(5.45, 0.028)</td><td>0.152</td><td>0.040</td><td>1.28</td><td>0.133</td></tr></table>
+
+# E TOY EXPERIMENT DETAILS
+
+The rotated rectangles dataset consists of images of rectangles randomly rotated from $0 ^ { \circ }$ to $9 0 °$ , along with labels that indicate the angle of rotation. The rectangles have different, random colors. Visual samples are shown in Fig. 6.
+
+![](images/77aea8e7c834470bac00866386bc4f7458ce91a18ec5c52d79bbcaf32aa67e40.jpg)  
+Figure 6: Examples of synthetic rectangle dataset.
+
+We tested three different networks: Naive, Regression-based and PCA-based. The Naive network directly predicts an angle from an image using an encoder and a fully-connected layer. Regressionbased is similar to FOMM (Siarohin et al., 2019a); the angle is regressed per pixel an using hourglass network, and pooled according to heatmap weights predicted using the same hourglass network. PCAbased is our method described in Sec. 3; we predict the heatmap using an hourglass network, PCA is performed according to eq. equation 2, and the angle is computed from matrix $U$ as arctan $\left( U _ { 1 0 } / U _ { 0 0 } \right)$ .
+
+Each of the networks was trained, on subsets of the dataset of varying sizes, to minimize the $\mathcal { L } _ { 1 }$ loss between predicted and ground truth rotation angle. All models were trained for 100 epochs, with batch size 8. We used the Adam optimizer, with a learning rate of $1 0 ^ { - 4 }$ . We varied the size of the training set from 32 to 1024. Results, on a separate, fixed test set of size 128, were then computed, shown in Fig. 5.

parse/train/kzPtpIpF8o/kzPtpIpF8o.md

@@ -0,0 +1,248 @@
+# XCiT: Cross-Covariance Image Transformers
+
+Alaaeldin El-Nouby1,2 Hugo Touvron1,3 Mathilde Caron1,2 Piotr Bojanowski1
+
+Matthijs Douze1 Armand Joulin1 Ivan Laptev2 Natalia Neverova1
+
+Gabriel Synnaeve1 Jakob Verbeek1 Hervé Jégou1
+
+1Facebook AI 2Inria 3Sorbonne University
+
+# Abstract
+
+Following tremendous success in natural language processing, transformers have recently shown much promise for computer vision. The self-attention operation underlying transformers yields global interactions between all tokens, i.e. words or image patches, and enables flexible modelling of image data beyond the local interactions of convolutions. This flexibility, however, comes with a quadratic complexity in time and memory, hindering application to long sequences and highresolution images. We propose a “transposed” version of self-attention that operates across feature channels rather than tokens, where the interactions are based on the cross-covariance matrix between keys and queries. The resulting cross-covariance attention (XCA) has linear complexity in the number of tokens, and allows efficient processing of high-resolution images. Our cross-covariance image transformer (XCiT) – built upon XCA – combines the accuracy of conventional transformers with the scalability of convolutional architectures. We validate the effectiveness and generality of XCiT by reporting excellent results on multiple vision benchmarks, including (self-supervised) image classification on ImageNet-1k, object detection and instance segmentation on COCO, and semantic segmentation on ADE20k.
+
+# 1 Introduction
+
+Transformers architectures $\left[ \left[ 6 8 \right] \right]$ have provided quantitative and qualitative breakthroughs in speech and natural language processing (NLP). After a few attempts to incorporate wide-range self-attention in vision architectures $[ 7 1 , 8 2 ]$ , Dosovitskiy et al. [21] established transformers as a viable architecture for learning visual representations, reporting competitive results for image classification while relying on large-scale pre-training. Touvron et al. $\textcircled { 6 4 } \textcircled { 1 6 }$ have shown on par or better accuracy/throughput compared to strong convolutional baselines such as EfficientNets $\lVert 5 8 \rVert$ when training transformers on ImageNet-1k using extensive data augmentation and improved training schemes. Promising results have been obtained for other vision tasks, including image retrieval $\pmb { \mathbb { Z } } 2 \mathbb { I }$ , object detection and semantic segmentation $\boxed { \boxed { 4 4 } } \boxed { 7 0 } \boxed { 8 1 } \boxed { 8 3 } \boxed { }$ , as well as video understanding [2, 7, 23].
+
+One major drawback of transformers is the time and memory complexity of the core self-attention operation, that increases quadratically with the number of input tokens, or similarly number of patches in computer vision. For $w \times h$ images, this translates to a complexity of $\mathcal { O } ( w ^ { 2 } \bar { h } ^ { 2 } )$ , which is prohibitive for most tasks involving high-resolution images, such as object detection and segmentation. Various strategies have been proposed to alleviate this complexity, for instance using approximate forms of self-attention $\textcircled { 1 4 4 } , \textcircled { 8 1 }$ , or pyramidal architectures which progressively downsample the feature maps $ { \mathbb { I } } ^ {  { \mathbb { Z } } 0 \| }$ . However, none of the existing solutions are fully satisfactory, as they either trade complexity for accuracy, or their complexity remains excessive for processing very large images.
+
+We replace the self-attention, as originally introduced by Vaswani et al. $\lVert \rVert \bigotimes \rVert$ , with a “transposed” attention that we denote as “cross-covariance attention” (XCA). Cross-covariance attention substitutes the explicit full pairwise interaction between tokens by self-attention among features, where the attention map is derived from the cross-covariance matrix computed over the key and query projections of the token features. Importantly, XCA has a linear complexity in the number of patches. To construct our Cross-Covariance Image Transformers (XCiT), we combine XCA with local patch interaction modules that rely on efficient depth-wise convolutions and point-wise feedforward networks commonly used in transformers, see Figure $\mathbb { L }$ XCA can be regarded as a form of a dynamic $1 \times 1$ convolution, which multiplies all tokens with the same data-dependent weight matrix. We find that the performance of our XCA layer can be further improved by applying it on blocks of channels, rather than directly mixing all channels together. This “block-diagonal” shape of XCA further reduces the computational complexity with a factor linear in the number of blocks.
+
+![](images/9dcfff922e69d5962cb57fbbc31fa8195fc5262914194a68fc36a933012c1196.jpg)  
+Figure 1: Our XCiT layer consists of three main blocks, each preceded by LayerNorm and followed by a residual connection: (i) the core cross-covariance attention (XCA) operation, (ii) the local patch interaction (LPI) module, and (iii) a feed-forward network (FFN). By transposing the query-key interaction, the computational complexity of XCA is linear in the number of data elements $N$ , rather than quadratic as in conventional self-attention.
+
+Given its linear complexity in the number of tokens, XCiT can efficiently process images with more than thousand pixels in each dimension. Notably, our experiments show that XCiT does not compromise the accuracy and achieves similar results to DeiT $\mathbb { \lVert \rVert }$ and CaiT $ { \mathbb { I } }  { \mathbb { I } }$ in comparable settings. Moreover, for dense prediction tasks such as object detection and image segmentation, our models outperform popular ResNet $\left[ \left[ 2 8 \right] \right]$ backbones as well as the recent transformer-based models [44, 70, 81]. Finally, we also successfully apply XCiT to the self-supervised feature learning using DINO [12], and demonstrate improved performance compared to a DeiT-based backbone $\pmb { \| 6 4 \| }$ .
+
+Overall, we summarize our contributions as follows:
+
+• We introduce cross-covariance attention (XCA), which provides a “transposed” alternative to conventional self-attention, attending over channels instead of tokens. Its complexity is linear in the number of tokens, allowing for efficient processing of high-resolution images, see Figure 2.   
+• XCA attends to a fixed number of channels, irrespective of the number of tokens. As a result, our models are significantly more robust to changes in image resolution at test time, and are therefore more amenable to process variable-size images.   
+• For image classification, we demonstrate that our models are on par with state-of-the-art vision transformers for multiple model sizes using a simple columnar architecture, i.e., in which we keep the resolution constant across layers. In particular, our XCiT-L24 model achieves $8 6 . 0 \%$ top-1 accuracy on ImageNet, outperforming its CaiT-M24 [67] and NFNet-F2 $\mathbb { \ m }$ counterparts with comparable numbers of parameters.   
+• For dense prediction tasks with high-resolution images, our models outperform ResNet and multiple transformer-based backbones. On the COCO benchmark, we achieve a strong performance of $4 8 . 5 \%$ and $4 3 . 7 \%$ mAP for object detection and instance segmentation respectively. Moreover, we report $4 8 . 4 \%$ mIoU for semantic segmentation on the ADE20k benchmark, outperforming the state-of-the-art Swin Transformer $\pm \ddagger { 4 } \rVert$ backbones across all comparable model sizes.   
+• Finally, our XCiT model is highly effective in self-supervised learning setups, achieving $8 0 . 9 \%$ top-1 accuracy on ImageNet-1k using DINO [12].
+
+# 2 Related work
+
+Deep vision transformers. Training deep vision transformers can be challenging due to instabilities and optimization issues. Touvron et al. $\dot { \left[ 6 7 \right] }$ successfully train models with up to 48 layers using LayerScale, which weighs contributions of residual blocks across layers and improves optimization. Additionally, the authors introduce class attention layers which decouple the learning of patch features and the feature aggregation stage for classification.
+
+Spatial structure in vision transformers. Yuan et al. $\pmb { \mathbb { Z } } 9 \|$ propose applying a soft split for patch projection with overlapping patches which is applied repeatedly across model layers, reducing the number of patches progressively. Han et al. $\mathbb { \left| \overline { { 2 7 } } \right| }$ introduce a transformer module for intra-patch structure, exploiting pixel-level information and integrating with an inter-patch transformer to attain higher representation power. d’Ascoli et al. $\mathbb { \ m }$ consider the initialization of self-attention blocks as a convolutional operator, and demonstrate that such initialization improves the performance of vision transformers in low-data regimes. Graham et al. $\pmb { \mathbb { D } } \pmb { \ 6 } \|$ introduce LeViT, which adopts a multistage architecture with progressively reduced feature resolution similar to popular convolutional architectures, allowing for models with high inference speed while retaining a strong performance. Moreover, the authors adopt a convolution-based module for extracting patch descriptors. Yuan et al. $\left[ \left[ 7 8 \right] \right]$ improve both the performance and the convergence speed of vision transformers by replacing the linear patch projection with convolutional layers and max-pooling, as well as modifying the feed-forward networks in each transformer layer to incorporate depth-wise convolutions.
+
+Efficient attention. Numerous methods for efficient self-attention have been proposed in the literature to address the quadratic complexity of self-attention in the number of input tokens. These include restricting the span of the self-attention to local windows [48, 50], strided patterns $\pmb { \mathbb { I } }$ , axial patterns $\textcircled { \lvert 3 0 \rvert }$ , or an adaptive computation across layers $ { \mathbb { I } }$ . Other methods provide an approximation of the self-attention matrix which can be achieved by a projection across the token dimension $\mathbb { \lVert \rVert }$ , or through a factorization of the softmax-attention kernel [15, 37, 56, 77], which avoids explicit computation of the attention matrix. While conceptually different, our XCA performs similar computations without being sensitive to the choice of the kernel. Similarly, Lee-Thorp et al. [41] achieve faster training by substituting self-attention with unparametrized Fourier Transform. Other efficient attention methods rely on local attention and adding a small number of global tokens, thus allowing interaction among all tokens only by hopping through the global tokens $\boxed { 1 } \boxed { 5 } \boxed { 3 4 } \boxed { 8 0 }$ . Similarly, Goyal et al. $\boldsymbol { \| 2 5 \| }$ use a global workspace though which items interact, albeit one that is shared across layers.
+
+Transformers for high-resolution images. Several works adopt visual transformers to highresolution image tasks beyond image classification, such as object detection and image segmentation. Wang et al. $\tilde { \left. 7 0 \right. }$ design a model with a pyramidal architecture and address complexity by gradually reducing the spatial resolution of keys and values. Similarly, for video recognition Fan et al. [23] utilize pooling to reduce the resolution across the spatial and temporal dimensions to allow for an efficient computation of the attention matrix. Zhang et al. $\textcircled { 8 1 }$ adopt global tokens and local attention to reduce the model complexity, while Liu et al. $\checkmark$ provide an efficient method for local attention with shifted windows. In addition, Zheng et al. $[ [ 8 3 ] ]$ and Ranftl et al. $[ [ 5 4 ]$ study problems like semantic segmentation and monocular depth estimation with the quadratic self-attention operation.
+
+Data-dependent layers. Our XCiT layer can be regarded as a “dynamic” $1 \times 1$ convolution, which multiplies all token features with the same data-dependent weight matrix, derived from the key and query cross-covariance matrix. In the context of convolutional networks, Dynamic Filter Networks [9] explore a related idea, using a filter generating subnetwork to produce convolutional filters based on features in previous layers. Squeeze-and-Excitation networks $\left[ \left[ 3 2 \right] \right]$ use data dependent $1 \times 1$ convolutions in convolutional architectures. Spatially average-pooled features are fed to a 2-layer MLP which produces per channel scaling parameters. Closer in spirit to our work, Lambda layers propose a way to ensure global interaction in ResNet models $\bar { \mathbb { H } }$ . Their “content-based lambda function” is computing a similar term as our cross-covariance attention, but differing in how the softmax and $\ell _ { 2 }$ normalizations are applied. Moreover, Lambda layers also include specific positionbased lambda functions, and LambdaNetworks are based on ResNets while XCiT follows the ViT architecture. Recently data-independent analogues of self-attention have also been found to be an effective alternative to convolutional and self-attention layers for vision tasks $\textcircled { 1 2 0 } , \textcircled { 4 6 } , \textcircled { 6 2 } , \textcircled { 6 6 } $ . These methods treat entries in the attention map as learnable parameters, rather than deriving the attention map dynamically from queries and keys, but their complexity remains quadratic in the number of tokens. Zhao et al. [82] consider alternative attention forms in computer vision.
+
+# 3 Method
+
+In this section, we first recall the self-attention mechanism, and the connection between the Gram and covariance matrices, which motivated our work. We then propose our cross-covariance attention operation (XCA) – which operates along the feature dimension instead of token dimension in conventional transformers – and combine it with local patch interaction and feedforward layers to construct our Cross-Covariance Image Transformer (XCiT). See Figure $\bigtriangledown$ for an overview.
+
+# 3.1 Background
+
+Token self-attention. Self-attention, as introduced by Vaswani et al. $\lVert \overline { { 6 8 } } \rVert$ , operates on an input matrix $X \in \mathbb { R } ^ { N \times d }$ , where $N$ is the number of tokens, each of dimensionality $d$ . The input $X$ is linearly projected to queries, keys and values, using the weight matrices $\dot { W _ { q } } \in \mathbb { R } ^ { d \times d _ { q } }$ , $W _ { k } \in$ $\mathbb { R } ^ { d \times d _ { k } }$ and $W _ { v } \in \mathbb { R } ^ { d \times d _ { v } }$ , such that $Q { = } X W _ { q }$ , $K { = } X W _ { k }$ and $V { = } X W _ { v }$ , where $d _ { q } = d _ { k }$ . Keys and values are used to compute an attention map $\mathcal { A } ( K , Q ) = \operatorname { S o f t m a x } ( Q K ^ { \top } / \sqrt { d _ { k } } )$ , and the output of the self-attention operation is defined as the weighted sum of $N$ token features in $V$ with the weights corresponding to the attention map: Attention $( Q , K , V ) = \mathcal { A } ( K , Q ) V$ . The computational complexity of self-attention scales quadratically in $N$ , due to pairwise interactions between all $N$ elements.
+
+Relationship between Gram and covariance matrices. To motivate our cross-covariance attention operation, we recall the relation between Gram and covariance matrices. The unnormalised $d \times d$ covariance matrix is obtained as $C { = } X ^ { \top } X$ . The $N \times N$ Gram matrix contains all pairwise innerproducts: $G { = } X X ^ { \top }$ . The non-zero part of the eigenspectrum of the Gram and covariance matrix are equivalent, and the eigenvectors of $C$ and $G$ can be computed in terms of each other. If $V$ are the eigenvectors of $G$ , then the eigenvectors of $C$ are given by $U { = } X V$ . To minimise the computational cost, the eigendecomposition of either the Gram or covariance matrix can be obtained in terms of the decomposition of the other, depending on which of the two matrices is the smallest.1
+
+We draw upon this strong connection between the Gram and covariance matrices to consider whether it is possible to avoid the quadratic cost to compute the $N \times N$ attention matrix, which is computed from the analogue of the $N \times N$ Gram matrix $\mathsf { \bar { Q } } K ^ { \top } { = } X W _ { q } W _ { k } ^ { \top } X ^ { \top }$ . Below we consider how we can use the $d _ { k } \times d _ { q }$ cross-covariance matrix, $K ^ { \top } Q { = } W _ { k } ^ { \top } X ^ { \top } X W _ { q }$ , which can be computed in linear time in the number of elements $N$ , to define an attention mechanism.
+
+# 3.2 Cross-covariance attention
+
+We propose a cross-covariance based self-attention function that operates along the feature dimension, rather than along the token dimension as in token self-attention. Using the definitions of queries, keys and values from above, the cross-covariance attention function is defined as:
+
+$$
+\operatorname { X C - A t t e n t i o n } ( Q , K , V ) = V A \operatorname { x c } ( K , Q ) , \qquad A \operatorname { x c } ( K , Q ) = \operatorname { S o f t m a x } \left( \hat { K } ^ { \top } \hat { Q } / \tau \right) ,
+$$
+
+where each output token embedding dimension is a convex combination of the $d _ { v }$ features of its corresponding token embedding in $V$ . The attention weights $\mathcal { A }$ are computed based on the crosscovariance matrix.
+
+$\ell _ { 2 }$ -Normalization and temperature scaling. In addition to building our attention operation on the cross-covariance matrix, we make a second modification compared to token self-attention. We restrict the magnitude of the query and key matrices by $\ell _ { 2 }$ -normalising them, such that each column of length $N$ of the normalised matrices $\hat { Q }$ and $\hat { K }$ has unit norm, and every element in $d { \times } d$ cross-covariance matrix $\hat { K } ^ { \top } \hat { Q }$ is in the range $[ - 1 , 1 ]$ . We observed that controlling the norm strongly enhances the stability of training, especially when trained with a variable numbers of tokens. However, restricting the norm reduces the representational power of the operation by removing a degree of freedom. Therefore, we introduce a learnable temperature parameter $\tau$ which scales the inner products before the Softmax, allowing for sharper or more uniform distribution of attention weights.
+
+![](images/301e7b358ccf93cc7973974e2bad457d5f17a3e86617dc31e723f62a68c6c260.jpg)  
+Figure 2: Inference memory usage of vision transformer variants. Our XCiT models scale linearly in the number of tokens, which makes it possible to scale to much larger image sizes, even in comparison to approaches employing approximate self-attention or a pyramidal design. All measurements are performed with a batch size of 64 on a single V100-32GB GPU.
+
+![](images/7a6dca3126c84125944af039122617444e1690e36b6fe95f4c1ffa293a03e1ef.jpg)  
+Figure 3: Performance when changing the resolution at test-time for models with a similar number of parameters. All networks were trained at resolution 224, w/o distillation. XCiT is more tolerant to changes of resolution than the Gram-based DeiT and benefit more from the “FixRes” effect $\mathbb { \lVert \rVert }$ when inference is performed at a larger resolution than at train-time.
+
+Block-diagonal cross-covariance attention. Instead of allowing all features to interact among each other, we divide them into a $h$ groups, or “heads”, in a similar fashion as multi-head token self-attention. We apply the cross-covariance attention separately per head where for each head, we learn separate weight matrices to project $X$ to queries, keys and values, and collect the corresponding weight matrices in the tensors $\dot { W _ { q } } \in \mathbb { R } ^ { h \times d \times d _ { q } }$ , $W _ { k } \in \mathbb { R } ^ { \mathbf { \bar { h } } \times d \times d _ { k } }$ and $\dot { W } _ { v } \in \mathbb { R } ^ { h \times d \times d _ { v } }$ , where we set $d _ { k } \bar { = } d _ { q }  { = } d _ { v }  { = } d / h$ . Restricting the attention within heads has two advantages: (i) the complexity of aggregating the values with the attention weights is reduced by a factor $h$ ; (ii) more importantly, we empirically observe that the block-diagonal version is easier to optimize, and typically leads to improved results. This observation is in line with observations made for Group Normalization $\mathbb { [   ]  }$ which normalizes groups of channels separately based on their statistics, and achieves favorable results for computer vision tasks compared to Layer Normalization $\pmb { \Vert 3 \Vert }$ , which combines all channels in a single group. Figure $\boxed { 4 }$ shows that each head learns to focus on semantically coherent parts of the image, while being flexible to change what type of features it attends to based on the image content.
+
+Complexity analysis. The usual token self-attention with $h$ heads has a time complexity of $\mathcal { O } ( N ^ { 2 } d )$ and memory complexity of $\mathcal { O } ( h N ^ { 2 } { + } N d )$ . Due to the quadratic complexity, it is problematic to scale token self-attention to images with a large number of tokens. Our cross-covariance attention overcomes this drawback as its computational cost of $\mathcal { O } ( N d ^ { 2 } / h )$ scales linearly with the number of tokens, as does the memory complexity of $\mathcal { O } ( d ^ { 2 } / h + N \dot { d } )$ . Therefore, our model scales much better to cases where the number of tokens $N$ is large, and the feature dimension $d$ is relatively small, as is typically the case, in particularly when splitting the features into $h$ heads.
+
+# 3.3 Cross-covariance image transformers
+
+To construct our cross-covariance image transformers (XCiT), we adopt a columnar architecture which maintains the same spatial resolution across layers, similarly to $\boxed { 1 2 1 } \boxed { 6 4 } \boxed { 6 7 }$ . We combine our cross-covariance attention (XCA) block with the following additional modules, each one being preceded by a LayerNorm $\pmb { \mathbb { B } } \|$ . See Figure $\perp$ for an overview. Since in this section we specifically design the model for computer vision tasks, tokens correspond to image patches in this context.
+
+Table 1: XCiT models. Design choices include model depth, patch embeddings dimensionality $d$ , and the number of heads $h$ used in XCA. By default our models are trained and tested at resolution 224 with patch sizes of $1 6 \times 1 6$ . We also train with distillation using a convolutional teacher (denoted $\Upsilon$ ) as proposed by Touvron et al. [64]. Finally, we report performance of our strongest models obtained with $8 \times 8$ patch size, fine-tuned $( \uparrow )$ and tested at resolution $3 8 4 \times 3 8 4$ (column $\textcircled { \alpha } 3 8 4 / 8 $ ), using distillation with a teacher that was also fine-tuned $@ 3 8 4$ .   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Depth</td><td rowspan="2">d</td><td rowspan="2">#heads</td><td rowspan="2">#params</td><td colspan="2">GFLOPs</td><td colspan="3">ImageNet-1k-val top-1 acc. (%)</td></tr><tr><td>@224/16</td><td>@384/8</td><td>@224/16 @224/16r</td><td></td><td>@384/8r ↑</td></tr><tr><td>XCiT-N12</td><td>12</td><td>128</td><td>4</td><td>3M</td><td>0.5</td><td>6.4</td><td>69.9</td><td>72.2</td><td>77.8</td></tr><tr><td>XCiT-T12</td><td>12</td><td>192</td><td>4</td><td>7M</td><td>1.2</td><td>14.3</td><td>77.1</td><td>78.6</td><td>82.4</td></tr><tr><td>XCiT-T24</td><td>24</td><td>192</td><td>4</td><td>12M</td><td>2.3</td><td>27.3</td><td>79.4</td><td>80.4</td><td>83.7</td></tr><tr><td>XCiT-S12</td><td>12</td><td>384</td><td>8</td><td>26M</td><td>4.8</td><td>55.6</td><td>82.0</td><td>83.3</td><td>85.1</td></tr><tr><td>XCiT-S24</td><td>24</td><td>384</td><td>8</td><td>48M</td><td>9.1</td><td>106.0</td><td>82.6</td><td>83.9</td><td>85.6</td></tr><tr><td>XCiT-M24</td><td>24</td><td>512</td><td>8</td><td>84M</td><td>16.2</td><td>188.0</td><td>82.7</td><td>84.3</td><td>85.8</td></tr><tr><td>XCiT-L24</td><td>24</td><td>768</td><td>16</td><td>189M</td><td>36.1</td><td>417.9</td><td>82.9</td><td>84.9</td><td>86.0</td></tr></table>
+
+Local patch interaction. In the XCA block communication between patches is only implicit through the shared statistics. To enable explicit communication across patches we add a simple Local Patch Interaction (LPI) block after each XCA block. LPI consists of two depth-wise $3 { \times } 3$ convolutional layers with Batch Normalization and GELU non-linearity in between. Due to its depth-wise structure, the LPI block has a negligible overhead in terms of parameters, as well as a very limited overhead in terms of throughput and memory usage during inference.
+
+Feed-forward network. As is common in transformer models, we add a point-wise feedforward network (FFN), which has a single hidden layer with $4 d$ hidden units. While interaction between features is confined within groups in the XCA block, and no feature interaction takes place in the LPI block, the FFN allows for interaction across all features.
+
+Global aggregation with class attention. When training our models for image classification, we utilize the class attention layers as proposed by Touvron et al. $ { \mathbb { I } }  { \mathbb { K } }  { \ b { 7 } }  { \mathbb { I } }$ . These layers aggregate the patch embeddings of the last XCiT layer through writing to a CLS token by one-way attention between the CLS tokens and the patch embeddings. The class attention is also applied per head, i.e. feature group.
+
+Handling images of varying resolution. In contrast to the attention map involved in token selfattention, in our case the covariance blocks are of fixed size independent of the input image resolution. The softmax always operates over the same number of elements, which may explain why our models behave better when dealing with images of varying resolutions (see Figure $\textcircled{3}$ . In XCiT we include additive sinusoidal positional encoding $\lVert \rVert$ with the input tokens. We generate them in 64 dimensions from the 2d patch coordinates and then linearly project to the transformer working dimension $d$ . This choice is orthogonal to the use of learned positional encoding, as in ViT [21]. However, it is more flexible since there is no need to interpolate or fine-tune the network when changing the image size.
+
+Model configurations. In Table $^ 1$ we list different variants of our model which we use in our experiments, with different choices for model width and depth. For the patch encoding layer, unless mentioned otherwise, we adopt the alternative used by Graham et al. $\pmb { \mathbb { D } } \pmb { 6 } \|$ with convolutional patch projection layers. We also experimented with a linear patch projection as described in $\pmb { \mathbb { D } } \mathbf { 1 } \mathbf { h }$ , see our ablation in Table 4. Our default patch size is $1 6 \times 1 6$ , as in other vision transformer models including ViT $\scriptstyle { \left[ \left[ 2 1 \right] \right] }$ , DeiT [64] and CaiT $\pmb { \mathbb { E 7 } }$ . We also experiment with smaller $8 \times 8$ patches, which has been observed to improve performance $\mathbb { [ [  2 ]  }$ . Note that this is efficient with XCiT as its complexity scales linearly which the number of patches, while ViT, DeiT and CaiT scale quadratically.
+
+# 4 Experimental evaluation
+
+In this section we demonstrate the effectiveness and versatility of XCiT on multiple computer vision benchmarks, and present ablations providing insight on the importance of its different components. In the supplementary material we provide additional analysis, including the impact on performance of image resolution in Section ${ \bf A . l } ^ { \dot { } }$ and of multiple approximate attention baselines in Section A.2.
+
+Table 2: ImageNet classification. Number of parameters, FLOPs, image resolution, and top-1 accuracy on ImageNet-1k and ImageNet-V2. Training strategies vary across models, transformer-based models and the reported RegNet mostly follow recipes from DeiT [64].   
+
+<table><tr><td>Model</td><td>#params</td><td>FLOPs Res.</td><td></td><td>ImNet</td><td>V2</td></tr><tr><td>EfficientNet-B5 RA [17]</td><td>30M</td><td>9.9B</td><td>456</td><td>83.7</td><td></td></tr><tr><td>RegNetY-4GFI531</td><td>21M</td><td>4.0B</td><td>224</td><td>80.0</td><td>72.4</td></tr><tr><td>DeiT-SY 64]</td><td>22M</td><td>4.6B</td><td>224</td><td>81.2</td><td>68.5</td></tr><tr><td>Swin-T [44]</td><td>29M</td><td>4.5B</td><td>224</td><td>81.3</td><td></td></tr><tr><td>CaiT-XS24Y ↑ 67]</td><td>26M</td><td>19.3B</td><td>384</td><td>84.1</td><td>74.1</td></tr><tr><td>XCiT-S12/16M</td><td>26M</td><td>4.8B</td><td>224</td><td>83.3</td><td>72.5</td></tr><tr><td>XCiT-S12/16Y↑</td><td>26M</td><td>14.3B</td><td>384</td><td>84.7</td><td>74.1</td></tr><tr><td>XCiT-S12/8Y↑</td><td>26M</td><td>55.6B</td><td>384</td><td>85.1</td><td>74.8</td></tr><tr><td>EfficientNet-B7RA [17]</td><td>66M</td><td>37.0B</td><td>600</td><td>84.7</td><td></td></tr><tr><td>NFNet-F0 [10]</td><td>72M</td><td>12.4B</td><td>256</td><td>83.6</td><td>72.6</td></tr><tr><td>RegNetY-8GF 园</td><td>39M</td><td>8.0B</td><td>224</td><td>81.7</td><td>72.4</td></tr><tr><td>TNT-B 四</td><td>66M</td><td>14.1B</td><td>224</td><td>82.8</td><td>1</td></tr><tr><td>国 Swin-S</td><td>50M</td><td>8.7B</td><td>224</td><td>83.0</td><td></td></tr><tr><td>CaiT-S24Y ↑ 67]</td><td>47M</td><td>32.2B</td><td>384</td><td>85.1</td><td>75.4</td></tr><tr><td>XCiT-S24/16M</td><td>48M</td><td>9.1B</td><td>224</td><td>83.9</td><td>73.3</td></tr><tr><td>XCiT-S24/16Y↑</td><td>48M</td><td>26.9B</td><td>384</td><td>85.1</td><td>74.6</td></tr><tr><td>XCiT-S24/8Y ↑</td><td>48M</td><td>105.9B</td><td>384</td><td>85.6</td><td>75.7</td></tr><tr><td>Fix-EfficientNet-B8</td><td>园 87M</td><td>89.5B</td><td>800</td><td>85.7</td><td>75.9</td></tr><tr><td>RegNetY-16GF[53]</td><td>84M</td><td>16.0B</td><td>224</td><td>82.9</td><td>72.4</td></tr><tr><td>Swin-B↑ 四</td><td>88M</td><td>47.0B</td><td>384</td><td>84.2</td><td></td></tr><tr><td>DeiT-BY↑[64</td><td>87M</td><td>55.5B</td><td>384</td><td>85.2</td><td>75.2</td></tr><tr><td>CaiT-S48Y ↑[67]</td><td>89M</td><td>63.8B</td><td>384</td><td>85.3</td><td>76.2</td></tr><tr><td>XCiT-M24/16T</td><td>84M</td><td>16.2B</td><td>224</td><td>84.3</td><td>73.6</td></tr><tr><td>XCiT-M24/16Y↑</td><td>84M</td><td>47.7B</td><td>384</td><td>85.4</td><td>75.1</td></tr><tr><td>XCiT-M24/8Y↑</td><td>84M</td><td>187.9B</td><td>384</td><td>85.8</td><td>76.1</td></tr><tr><td>NFNet-F2[ 目</td><td>194M</td><td>62.6B</td><td>352</td><td>85.1</td><td>74.3</td></tr><tr><td>NFNet-F3 目</td><td>255M</td><td>114.8B</td><td>416</td><td>85.7</td><td>75.2</td></tr><tr><td>CaiT-M24Y ↑[67</td><td>186M</td><td>116.1B</td><td>384</td><td>85.8</td><td>76.1</td></tr><tr><td>XCiT-L24/16T</td><td>189M</td><td>36.1B</td><td>224</td><td>84.9</td><td>74.6</td></tr><tr><td>XCiT-L24/16Y ↑</td><td>189M</td><td>106.0B</td><td>3384</td><td>85.8</td><td>75.8</td></tr><tr><td>XCiT-L24/8Y↑</td><td>189M</td><td>417.8B</td><td>384</td><td>86.0</td><td>76.6</td></tr></table>
+
+![](images/d87cd7c275cbbc0e18ed54c4731e8c41009d08d1f159fc54964749277f2df726.jpg)  
+Figure 4: Visualization of the attention map between the CLS token and individual patches in the class-attention stage. For each column, each row represents the attention map w.r.t. one head, corresponding to the image in the first row. Each head appears sensitive to semantically coherent regions. Heads are sensitive to similar features within the same or across images (e.g. people or bird faces). They are trigger by different concepts when such features are missing (e.g., cockpit for race cars).
+
+# 4.1 Image classification
+
+We use ImageNet-1k $\mathbb { \lVert 1 9 \rVert }$ to train and evaluate our models for image classification. It consists of 1.28M training images and $5 0 \mathrm { k }$ validation images, labeled across 1,000 semantic categories. Our training setup follows the DeiT recipe $\pmb { \mathbb { \lVert 6 4 \rVert } }$ . We train our model for 400 epochs with the AdamW optimizer $| \bar { \mathbf { \nabla } } 4 \bar { 5 } | |$ using a cosine learning rate decay. In order to enhance the training of larger models, we utilize LayerScale $ { \mathbb { I } }  { \mathbb { I } }$ and adjust the stochastic depth $\mathbb { \lVert 3 3 \rVert }$ for each of our models accordingly (see the supplementary material for details). Following $ { \mathbb { I } }  { \mathbb { K } } 7  { \mathbb { I } }$ , images are cropped with crop ratio of 1.0 for evaluation. In addition to the ImageNet-1k validation set, we report results for ImageNet-V2 [55] which has a distinct test set. Our implementation is based on the Timm library $\lVert \ b { 7 2 } \rVert$ .
+
+Results on ImageNet. We present a family of seven models in Table $\lfloor 1 \rfloor$ with different operating points in terms of parameters and FLOPs. We observe that the performance of the XCiT models benefits from increased capacity both in depth and width. Additionally, consistent with $\mathbb { B 4 } \mathbb { 6 7 } \mathbb { 1 }$ we find that using hard distillation with a convolutional teacher improves the performance. Because of its linear complexity in the number of tokens, it is feasible to train XCiT at $3 8 4 \times 3 8 4$ resolution with small $8 \times 8$ patches, i.e. 2304 tokens, which provides a strong boost in performance across all configurations.
+
+We compare to the state-of-the-art convolutional and transformer-based architectures [10, 44, 53, 58, $\textcircled { 6 7 }$ in Table $2 .$ By varying the input image resolution and/or patch size, our models provide competitive or superior performance across model sizes and FLOP budgets. First, the models operating on $2 2 4 \times 2 2 4$ and $1 6 \times 1 6$ (e.g. XCiT-S12/16) enjoy high accuracy at relatively few FLOPs compared to their counterparts with comparable parameter count and FLOPs. Second, our models with $1 6 \times 1 6$ and $3 8 4 \times 3 8 4$ resolution images (e.g. XCiT-S12/16") yield an improved accuracy at the expense of higher FLOPs, and provide superior or on-par performance compared to state-of-the-art models with comparable computational requirements. Finally, the linear complexity of XCiT allows us to scale to process $3 8 4 \times 3 8 4$ images with $8 \times 8$ patch sizes (e.g. XCiT-S12/8"), achieving the highest accuracy across the board, albeit at a relatively high FLOPs count.
+
+Table 3: Self-supervised learning. Top-1 acc. on ImageNet-1k. We report with a crop-ratio 0.875 for consistency with DINO. For the last row it is set to 1.0 (improves from $8 0 . 7 \%$ to $8 0 . 9 \%$ ). All models are trained for 300 epochs.   
+
+<table><tr><td>SSL Method</td><td>Model</td><td>#params</td><td>FLOPs</td><td>Linear</td><td>k-NN</td></tr><tr><td>MoBY 回</td><td>Swin-T 国</td><td>29M</td><td>4.5B</td><td>75.0</td><td>1</td></tr><tr><td>DINO 回</td><td>ResNet-50[28]</td><td>23M</td><td>4.1B</td><td>74.5</td><td>65.6</td></tr><tr><td>DINO 圆</td><td>ViT-S/16_21</td><td>22M</td><td>4.6B</td><td>76.1</td><td>72.8</td></tr><tr><td>DINO 圆</td><td>ViT-S/8 21</td><td>22M</td><td>22.4B</td><td>79.2</td><td>77.2</td></tr><tr><td>DINO 圆</td><td>XCiT-S12/16</td><td>26M</td><td>4.9B</td><td>77.8</td><td>76.0</td></tr><tr><td>DINO 国</td><td>XCiT-S12/8</td><td>26M</td><td>18.9B</td><td>79.2</td><td>77.1</td></tr><tr><td>DINO 圆</td><td>ViT-B/1621</td><td>87M</td><td>17.5B</td><td>78.2</td><td>76.1</td></tr><tr><td>DINO 园</td><td>ViT-B/8 日</td><td>87M</td><td>78.2B</td><td>80.1</td><td>77.4</td></tr><tr><td>DINO 园</td><td>XCiT-M24/16</td><td>84M</td><td>16.2B</td><td>78.8</td><td>76.4</td></tr><tr><td>DINO 圆</td><td>XCiT-M24/8</td><td>84M</td><td>64.0B</td><td>80.3</td><td>77.9</td></tr><tr><td>DINO 园</td><td>XCiT-M24/8↑384</td><td>84M</td><td>188.0B</td><td>80.9</td><td>78.3</td></tr></table>
+
+Table 4: Ablations of various architectural design choices on the task of ImageNet-1k classification using the XCiT-S12 model. Our baseline model uses the convolutional projection adopted from LeVit.   
+
+<table><tr><td>Model</td><td>Ablation</td><td>ImNet top-1 acc.</td></tr><tr><td>XCiT-S12/16</td><td rowspan="2">Baseline</td><td>82.0</td></tr><tr><td>XCiT-S12/8</td><td>83.4</td></tr><tr><td>XCiT-S12/16</td><td rowspan="2">Linear patch proj.</td><td>81.1</td></tr><tr><td>XCiT-S12/8</td><td>83.1</td></tr><tr><td rowspan="2">XCiT-S12/16</td><td>w/o LPI layer</td><td>80.8</td></tr><tr><td>w/o XCA layer</td><td>75.9</td></tr><tr><td rowspan="2">XCiT-S12/16</td><td>w/o l2-normal.</td><td>failed</td></tr><tr><td>w/o learned temp. T</td><td>81.8</td></tr></table>
+
+Class attention visualization. In Figure $\sharp$ we show the class attention map obtained in the feature aggregation stage. Each head focuses on different semantically coherent regions in the image (e.g. faces or umbrellas). Furthermore, heads tend to focus on similar patterns across images (e.g. bird head or human face), but adapts by focusing on other salient regions when such patterns are absent.
+
+Robustness to resolution changes. In Figure $3$ we report the accuracy of XCiT-S12, DeiT-S and ResNet-50 trained on $2 2 4 \times 2 2 4$ images and evaluated at different image resolutions. While DeiT outperforms ResNet-50 when train and test resolutions are similar, it suffers from a larger drop in performance as the image resolution deviates farther from the training resolution. XCiT displays a substantially increased accuracy when train and test resolutions are similar, while also being robust to resolution changes, in particular for the model with $8 \times 8$ patches.
+
+Self-supervised learning. We train XCiT in a self-supervised manner using DINO $\mathbb { \lVert \rVert }$ on ImageNet-1k. In Table $\bar { 3 }$ we report performance using the linear and $\mathbf { k }$ -NN protocols as in $\mathbb { \lVert \rVert }$ . Across model sizes XCiT obtains excellent accuracy with both protocols, substantially improving DINO with ResNet-50 or ViT architectures, as well as over those reported for Swin-Transformer trained with MoBY $\left[ \left[ 7 6 \right] \right]$ . Comparing the larger models to ViT, we also observed improved performance for XCiT achieving a strong $8 0 . 3 \%$ accuracy. For fair comparison, all reported models have been trained for 300 epochs. Further improved performance of small models is reported by Caron et al. $\mathbb { \lVert \rVert }$ when training for 800 epochs, which we expect to carryover to XCiT based on the results presented here.
+
+Analysis and ablations. In Table $\sharp$ we provide ablation experiments to analyse the impact of different design choices for our XCiT-S12 model. First, we observe the positive effect of using the convolutional patch projection as compared to using linear patch projection, for both $8 \times 8$ and $1 6 \times 1 6$ patches. Second, while removing the LPI layer reduces the accuracy by only $1 . 2 \%$ (from 82.0 to 80.8), removing the XCA layer results in a large drop of $6 . 1 \%$ , underlining the effectiveness of XCA. We noticed that the inclusion of two convolutional components – convolutional patch projection and LPI – not only brings improvements in accuracy, but also accelerates training. Third, although we were able to ensure proper convergence without $\ell _ { 2 }$ -normalization of queries and keys by tweaking the hyper-parameters, we found that it provides stability across model size (depth and width) and other hyper-parameters. Finally, while the learnable softmax temperature parameter is not critical, removing it drops accuracy by $0 . 2 \%$ . Additional ablations are provided in the supplementary material.
+
+Table 5: COCO object detection and instance segmentation performance on the mini-val set. All backbones are pre-trained on ImageNet-1k, use Mask R-CNN model [29] and are trained with the same 3x schedule.   
+
+<table><tr><td>Backbone</td><td>#params</td><td>AP</td><td>AP</td><td>AP5</td><td>Apm</td><td>AP</td><td>AP</td></tr><tr><td>ResNet18 国</td><td>31.2M</td><td>36.9</td><td>57.1</td><td>40.0</td><td>33.6</td><td>53.9</td><td>35.7</td></tr><tr><td>PVT-Tiny 凯</td><td>32.9M</td><td>39.8</td><td>62.2</td><td>43.0</td><td>37.4</td><td>59.3</td><td>39.9</td></tr><tr><td>ViL-Tiny I81</td><td>26.9M</td><td>41.2</td><td>64.0</td><td>44.7</td><td>37.9</td><td>59.8</td><td>40.6</td></tr><tr><td>XCiT-T12/16</td><td>26.1M</td><td>42.7</td><td>64.3</td><td>46.4</td><td>38.5</td><td>61.2</td><td>41.1</td></tr><tr><td>XCiT-T12/8</td><td>25.8M</td><td>44.5</td><td>66.4</td><td>48.8</td><td>40.3</td><td>63.5</td><td>43.2</td></tr><tr><td>ResNet50 I28]</td><td>44.2M</td><td>41.0</td><td>61.7</td><td>44.9</td><td>37.1</td><td>58.4</td><td>40.1</td></tr><tr><td>PVT-Small[70]</td><td>44.1M</td><td>43.0</td><td>65.3</td><td>46.9</td><td>39.9</td><td>62.5</td><td>42.8</td></tr><tr><td>ViL-Small81</td><td>45.0M</td><td>43.4</td><td>64.9</td><td>47.0</td><td>39.6</td><td>62.1</td><td>42.4</td></tr><tr><td>Swin-T [44</td><td>47.8M</td><td>46.0</td><td>68.1</td><td>50.3</td><td>41.6</td><td>65.1</td><td>44.9</td></tr><tr><td>XCiT-S12/16</td><td>44.3M</td><td>45.3</td><td>67.0</td><td>49.5</td><td>40.8</td><td>64.0</td><td>43.8</td></tr><tr><td>XCiT-S12/8</td><td>43.1M</td><td>47.0</td><td>68.9</td><td>51.7</td><td>42.3</td><td>66.0</td><td>45.4</td></tr><tr><td>ResNet101 28</td><td>63.2M</td><td>42.8</td><td>63.2</td><td>47.1</td><td>38.5</td><td>60.1</td><td>41.3</td></tr><tr><td>ResNeXt101-32</td><td>62.8M</td><td>44.0</td><td>64.4</td><td>48.0</td><td>39.2</td><td>61.4</td><td>41.9</td></tr><tr><td>PVT-Medium 目</td><td>63.9M</td><td>44.2</td><td>66.0</td><td>48.2</td><td>40.5</td><td>63.1</td><td>43.5</td></tr><tr><td>ViL-Medium 图</td><td>60.1M</td><td>44.6</td><td>66.3</td><td>48.5</td><td>40.7</td><td>63.8</td><td>43.7</td></tr><tr><td>Swin-S44</td><td>69.1M</td><td>48.5</td><td>70.2</td><td>53.5</td><td>43.3</td><td>67.3</td><td>46.6</td></tr><tr><td>XCiT-S24/16</td><td>65.8M</td><td>46.5</td><td>68.0</td><td>50.9</td><td>41.8</td><td>65.2</td><td>45.0</td></tr><tr><td>XCiT-S24/8</td><td>64.5M</td><td>48.1</td><td>69.5</td><td>53.0</td><td>43.0</td><td>66.5</td><td>46.1</td></tr><tr><td>ResNeXt101-6475</td><td>101.9M</td><td>44.4</td><td>64.9</td><td>48.8</td><td>39.7</td><td>61.9</td><td>42.6</td></tr><tr><td>PVT-Large 目</td><td>81.0M</td><td>44.5</td><td>66.0</td><td>48.3</td><td>40.7</td><td>63.4</td><td>43.7</td></tr><tr><td>ViL-Large 日</td><td>76.1M</td><td>45.7</td><td>67.2</td><td>49.9</td><td>41.3</td><td>64.4</td><td>44.5</td></tr><tr><td>XCiT-M24/16</td><td>101.1M</td><td>46.7</td><td>68.2</td><td>51.1</td><td>42.0</td><td>65.6</td><td>44.9</td></tr><tr><td>XCiT-M24/8</td><td>98.9M</td><td>48.5</td><td>70.3</td><td>53.4</td><td>43.7</td><td>67.5</td><td>46.9</td></tr></table>
+
+Table 6: ADE20k semantic segmentation performance using Semantic FPN [38] and UperNet [74] (in comparable settings). We do not include comparisons with other state-of-the-art models that are pre-trained on larger datasets [44, 54, 83].   
+
+<table><tr><td rowspan=2 colspan=1>Backbone</td><td rowspan=1 colspan=2>Semantic FPN</td><td rowspan=1 colspan=2>UperNet</td></tr><tr><td rowspan=1 colspan=1>#params</td><td rowspan=1 colspan=1>mIoU</td><td rowspan=1 colspan=1>#params</td><td rowspan=1 colspan=1>mIoU</td></tr><tr><td rowspan=1 colspan=1>ResNet1828PVT-Tiny70</td><td rowspan=1 colspan=1>15.5M17.0M</td><td rowspan=1 colspan=1>32.935.7M</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>--</td></tr><tr><td rowspan=1 colspan=1>XCiT-T12/16XCiT-T12/8</td><td rowspan=1 colspan=1>8.4M8.4M</td><td rowspan=1 colspan=1>38.139.9</td><td rowspan=1 colspan=1>33.7M33.7</td><td rowspan=1 colspan=1>41.543.5</td></tr><tr><td rowspan=1 colspan=1>ResNet5028PVT-Small[70Swin-T44</td><td rowspan=1 colspan=1>28.5M28.2M</td><td rowspan=1 colspan=1>36.739.8-</td><td rowspan=1 colspan=1>66.5M-59.9M</td><td rowspan=1 colspan=1>42.0-44.5</td></tr><tr><td rowspan=1 colspan=1>XCiT-S12/16XCiT-S12/8</td><td rowspan=1 colspan=1>30.4M30.4M</td><td rowspan=1 colspan=1>43.944.2</td><td rowspan=1 colspan=1>52.4M52.3M</td><td rowspan=1 colspan=1>45.946.6</td></tr><tr><td rowspan=2 colspan=1>ResNet10128ResNeXt101-3275PVT-Medium 70Swin-S44</td><td rowspan=2 colspan=1>47.5M47.1M48.0M=</td><td rowspan=1 colspan=1>38.8</td><td rowspan=1 colspan=1>85.5M</td><td rowspan=1 colspan=1>43.8</td></tr><tr><td rowspan=1 colspan=1>39.741.6-</td><td rowspan=1 colspan=1>--81.0M</td><td rowspan=1 colspan=1>--47.6</td></tr><tr><td rowspan=1 colspan=1>XCiT-S24/16XCiT-S24/8</td><td rowspan=1 colspan=1>51.8M51.8M</td><td rowspan=1 colspan=1>44.647.1</td><td rowspan=1 colspan=1>73.8M73.8M</td><td rowspan=1 colspan=1>46.948.1</td></tr><tr><td rowspan=2 colspan=1>ResNeXt101-6475PVT-Large[701Swin-B国</td><td rowspan=2 colspan=1>86.4M65.1M=</td><td rowspan=1 colspan=1>40.242.1</td><td rowspan=2 colspan=1>-=121.0M</td><td rowspan=2 colspan=1>--48.1</td></tr><tr><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=2 colspan=1>XCiT-M24/16XCiT-M24/8</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>90.8M</td><td rowspan=1 colspan=1>46.9</td><td rowspan=1 colspan=1>108.9M</td><td rowspan=1 colspan=1>48.4</td></tr></table>
+
+# 4.2 Object detection and instance segmentation
+
+Our XCiT models can efficiently process high-resolution images (see Figure 2). Additionally, XCiT has a better adaptability to varying image resolutions compared to ViT models (see Figure $3 )$ . These two properties make XCiT a good fit for dense prediction tasks including detection and segmentation.
+
+We evalutate XCiT for object detection and instance segmentation using the COCO benchmark [42] which consists of $1 1 8 \mathrm { k }$ training and $5 \mathrm { k }$ validation images including bounding boxes and mask labels for 80 categories. We integrate XCiT as backbone in the Mask R-CNN $\mathbb { \left[ \left[ 2 9 \right] \right. }$ detector with FPN [43]. Since the XCiT architecture is inherently columnar, we make it FPN-compatible by extracting features from different layers, e.g., layers 4, 6, 8, and 12 for XCiT-S12. All features have a constant stride of 8 or 16 based on the patch size, and the feature resolutions are adjusted to have strides of 4, 8, 16, and 32, similar to ResNet-FPN backbones, where the downsampling is achieved by max pooling and the upsampling is obtained using a single transposed convolution layer (see the supplementary material for details). The model is trained for 36 epochs (3x schedule) using the AdamW optimizer with learning rate of $1 0 ^ { - 4 }$ , 0.05 weight decay and 16 batch size. We adopt the multiscale training and augmentation strategy of DETR $\bar { \mathbb { W } }$ . Our implementation is based on the mmdetection library $\overline { { \mathbb { B } 3 } } \Vert$ .
+
+Results on COCO. In Table $\boxed { 5 }$ we report object detection and instance segmentation results of four variants of XCiT using $1 6 \times 1 6$ and $8 { \times } 8$ patches. We compare to ResNets $\bar { \mathbb { B } } \bar { \mathbb { B } }$ and concurrent efficient vision transformers [44, 70, 81]. All models are trained using the 3x schedule after ImageNet-1k pretraining. Note that other results with higher absolute numbers have been achieved when pre-training on larger datasets $[ \textcircled { 4 4 } ]$ or with longer schedules $\mathbb { H }$ , and are therefore not directly comparable to the reported results. First, across all model sizes XCiT outperforms the convolutional ResNet [28] and ResNeXt $\mathbb { \left. \overline { { \boldsymbol { \mathscr { Q } } \boldsymbol { 5 } } } \right. }$ by a large margin with either patch size. Second, we observe a similar increase in accuracy compared to PVT $\bar { \mathbb { I D } }$ and ViL $\mathbb { \left[ 8 1 \right] }$ backbones. Finally, XCiT provides a competitive performance with Swin $[ \overline { { 1 4 4 } } ] \cdot \big \rrangle ^ { 2 }$ For relatively small models, XCiT-S12/8 outperforms its Swin-T counterpart with a decent margin. On the other hand, Swin-S provides slightly stronger results compared to XCiT-S24/8. Utilizing smaller $8 \times 8$ patches leads to a consistent gain across all models.
+
+# 4.3 Semantic segmentation
+
+We further show transferability of our models with semantic segmentation experiments on the ADE20k dataset $\textcircled { 1 8 4 } \textcircled { 1 }$ , which consists of $2 0 \mathrm { k }$ training and 5k validation images with labels over 150 semantic categories. We integrate our backbones in two segmentation methods: Semantic FPN [38] and UperNet $\bar { \textregistered }$ . We train for 80k and 160k iterations for Semantic FPN and UperNet respectively. Following $\textcircled { | 4 4 | }$ , the models are trained using batch size 16 and an AdamW optimizer with learning rate of $6 \times 1 0 ^ { - 5 }$ and 0.01 weight decay. We apply the same method of extracting FPN features as explained in Section $\mathbb { H } . 2 \big \downarrow$ We report the performance using the standard single scale protocol (without multi-scale and flipping). Our implementation is based on the mmsegmentation library $\mathbb { \lVert 1 6 \rVert }$ .
+
+Results on ADE20k. We present the semantic segmentation performance using XCiT backbones in Table $6 .$ First, for Semantic FPN $\mathbb { B }$ , XCiT provides a superior performance compared to ResNet, ResNeXt and PVT backbones using either option of patch size. Second, compared to Swin Transformers using the same UperNet decoder $\pmb { \Vert 7 4 \Vert }$ , XCiT with $8 \times 8$ patches consistently achieves a higher mIoU for different models. XCiT with $1 6 \times 1 6$ patches provides a strong performance especially for smaller models where XCiT-S12/16 outperforms Swin-T.
+
+# 5 Conclusion
+
+Contributions. We present an alternative to token self-attention which operates on the feature dimension, eliminating the need for expensive computation of quadratic attention maps. We build our XCiT models with the cross-covariance attention as its core component and demonstrate the effectiveness and generality of our models on various computer vision tasks. In particular, it exhibits a strong image classification performance on par with state-of-the-art transformer models while similarly robust to changing image resolutions as convnets. XCiT is effective as a backbone for dense prediction tasks, providing excellent performance on object detection, instance and semantic segmentation. Finally, we showed that XCiT can be a strong backbone for self-supervised learning, matching the state-of-the-art results with less compute. XCiT is a generic architecture that can readily be deployed in other research domains where self-attention has shown success.
+
+Limitations. Our models enable training with smaller patches and on higher-resolution images, which leads to clear performance gains. However, for tasks like image classification this gain comes at a cost of relatively high number of FLOPs. In order to address this issue, other components, like FFN, could also be re-examined. Another point is that XCiT models seem to overfit more than their CaiT counterparts, see Table 2. They are more similar to some convnets in that respect.
+
+# References
+
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+        "text": "XCiT: Cross-Covariance Image Transformers ",
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+        "text": "Alaaeldin El-Nouby1,2 Hugo Touvron1,3 Mathilde Caron1,2 Piotr Bojanowski1 ",
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+        "text": "Gabriel Synnaeve1 Jakob Verbeek1 Hervé Jégou1 ",
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+        "text": "1Facebook AI 2Inria 3Sorbonne University ",
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+        "text": "Abstract ",
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+        "text": "Following tremendous success in natural language processing, transformers have recently shown much promise for computer vision. The self-attention operation underlying transformers yields global interactions between all tokens, i.e. words or image patches, and enables flexible modelling of image data beyond the local interactions of convolutions. This flexibility, however, comes with a quadratic complexity in time and memory, hindering application to long sequences and highresolution images. We propose a “transposed” version of self-attention that operates across feature channels rather than tokens, where the interactions are based on the cross-covariance matrix between keys and queries. The resulting cross-covariance attention (XCA) has linear complexity in the number of tokens, and allows efficient processing of high-resolution images. Our cross-covariance image transformer (XCiT) – built upon XCA – combines the accuracy of conventional transformers with the scalability of convolutional architectures. We validate the effectiveness and generality of XCiT by reporting excellent results on multiple vision benchmarks, including (self-supervised) image classification on ImageNet-1k, object detection and instance segmentation on COCO, and semantic segmentation on ADE20k. ",
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+        "text": "1 Introduction ",
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+        "text": "Transformers architectures $\\left[ \\left[ 6 8 \\right] \\right]$ have provided quantitative and qualitative breakthroughs in speech and natural language processing (NLP). After a few attempts to incorporate wide-range self-attention in vision architectures $[ 7 1 , 8 2 ]$ , Dosovitskiy et al. [21] established transformers as a viable architecture for learning visual representations, reporting competitive results for image classification while relying on large-scale pre-training. Touvron et al. $\\textcircled { 6 4 } \\textcircled { 1 6 }$ have shown on par or better accuracy/throughput compared to strong convolutional baselines such as EfficientNets $\\lVert 5 8 \\rVert$ when training transformers on ImageNet-1k using extensive data augmentation and improved training schemes. Promising results have been obtained for other vision tasks, including image retrieval $\\pmb { \\mathbb { Z } } 2 \\mathbb { I }$ , object detection and semantic segmentation $\\boxed { \\boxed { 4 4 } } \\boxed { 7 0 } \\boxed { 8 1 } \\boxed { 8 3 } \\boxed { }$ , as well as video understanding [2, 7, 23]. ",
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+        "text": "One major drawback of transformers is the time and memory complexity of the core self-attention operation, that increases quadratically with the number of input tokens, or similarly number of patches in computer vision. For $w \\times h$ images, this translates to a complexity of $\\mathcal { O } ( w ^ { 2 } \\bar { h } ^ { 2 } )$ , which is prohibitive for most tasks involving high-resolution images, such as object detection and segmentation. Various strategies have been proposed to alleviate this complexity, for instance using approximate forms of self-attention $\\textcircled { 1 4 4 } , \\textcircled { 8 1 }$ , or pyramidal architectures which progressively downsample the feature maps $ { \\mathbb { I } } ^ {  { \\mathbb { Z } } 0 \\| }$ . However, none of the existing solutions are fully satisfactory, as they either trade complexity for accuracy, or their complexity remains excessive for processing very large images. ",
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+        "text": "We replace the self-attention, as originally introduced by Vaswani et al. $\\lVert \\rVert \\bigotimes \\rVert$ , with a “transposed” attention that we denote as “cross-covariance attention” (XCA). Cross-covariance attention substitutes the explicit full pairwise interaction between tokens by self-attention among features, where the attention map is derived from the cross-covariance matrix computed over the key and query projections of the token features. Importantly, XCA has a linear complexity in the number of patches. To construct our Cross-Covariance Image Transformers (XCiT), we combine XCA with local patch interaction modules that rely on efficient depth-wise convolutions and point-wise feedforward networks commonly used in transformers, see Figure $\\mathbb { L }$ XCA can be regarded as a form of a dynamic $1 \\times 1$ convolution, which multiplies all tokens with the same data-dependent weight matrix. We find that the performance of our XCA layer can be further improved by applying it on blocks of channels, rather than directly mixing all channels together. This “block-diagonal” shape of XCA further reduces the computational complexity with a factor linear in the number of blocks. ",
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+        "image_caption": [
+            "Figure 1: Our XCiT layer consists of three main blocks, each preceded by LayerNorm and followed by a residual connection: (i) the core cross-covariance attention (XCA) operation, (ii) the local patch interaction (LPI) module, and (iii) a feed-forward network (FFN). By transposing the query-key interaction, the computational complexity of XCA is linear in the number of data elements $N$ , rather than quadratic as in conventional self-attention. "
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+    {
+        "type": "text",
+        "text": "Given its linear complexity in the number of tokens, XCiT can efficiently process images with more than thousand pixels in each dimension. Notably, our experiments show that XCiT does not compromise the accuracy and achieves similar results to DeiT $\\mathbb { \\lVert \\rVert }$ and CaiT $ { \\mathbb { I } }  { \\mathbb { I } }$ in comparable settings. Moreover, for dense prediction tasks such as object detection and image segmentation, our models outperform popular ResNet $\\left[ \\left[ 2 8 \\right] \\right]$ backbones as well as the recent transformer-based models [44, 70, 81]. Finally, we also successfully apply XCiT to the self-supervised feature learning using DINO [12], and demonstrate improved performance compared to a DeiT-based backbone $\\pmb { \\| 6 4 \\| }$ . ",
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+    {
+        "type": "text",
+        "text": "Overall, we summarize our contributions as follows: ",
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+        "text": "• We introduce cross-covariance attention (XCA), which provides a “transposed” alternative to conventional self-attention, attending over channels instead of tokens. Its complexity is linear in the number of tokens, allowing for efficient processing of high-resolution images, see Figure 2.   \n• XCA attends to a fixed number of channels, irrespective of the number of tokens. As a result, our models are significantly more robust to changes in image resolution at test time, and are therefore more amenable to process variable-size images.   \n• For image classification, we demonstrate that our models are on par with state-of-the-art vision transformers for multiple model sizes using a simple columnar architecture, i.e., in which we keep the resolution constant across layers. In particular, our XCiT-L24 model achieves $8 6 . 0 \\%$ top-1 accuracy on ImageNet, outperforming its CaiT-M24 [67] and NFNet-F2 $\\mathbb { \\ m }$ counterparts with comparable numbers of parameters.   \n• For dense prediction tasks with high-resolution images, our models outperform ResNet and multiple transformer-based backbones. On the COCO benchmark, we achieve a strong performance of $4 8 . 5 \\%$ and $4 3 . 7 \\%$ mAP for object detection and instance segmentation respectively. Moreover, we report $4 8 . 4 \\%$ mIoU for semantic segmentation on the ADE20k benchmark, outperforming the state-of-the-art Swin Transformer $\\pm \\ddagger { 4 } \\rVert$ backbones across all comparable model sizes.   \n• Finally, our XCiT model is highly effective in self-supervised learning setups, achieving $8 0 . 9 \\%$ top-1 accuracy on ImageNet-1k using DINO [12]. ",
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+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Deep vision transformers. Training deep vision transformers can be challenging due to instabilities and optimization issues. Touvron et al. $\\dot { \\left[ 6 7 \\right] }$ successfully train models with up to 48 layers using LayerScale, which weighs contributions of residual blocks across layers and improves optimization. Additionally, the authors introduce class attention layers which decouple the learning of patch features and the feature aggregation stage for classification. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Spatial structure in vision transformers. Yuan et al. $\\pmb { \\mathbb { Z } } 9 \\|$ propose applying a soft split for patch projection with overlapping patches which is applied repeatedly across model layers, reducing the number of patches progressively. Han et al. $\\mathbb { \\left| \\overline { { 2 7 } } \\right| }$ introduce a transformer module for intra-patch structure, exploiting pixel-level information and integrating with an inter-patch transformer to attain higher representation power. d’Ascoli et al. $\\mathbb { \\ m }$ consider the initialization of self-attention blocks as a convolutional operator, and demonstrate that such initialization improves the performance of vision transformers in low-data regimes. Graham et al. $\\pmb { \\mathbb { D } } \\pmb { \\ 6 } \\|$ introduce LeViT, which adopts a multistage architecture with progressively reduced feature resolution similar to popular convolutional architectures, allowing for models with high inference speed while retaining a strong performance. Moreover, the authors adopt a convolution-based module for extracting patch descriptors. Yuan et al. $\\left[ \\left[ 7 8 \\right] \\right]$ improve both the performance and the convergence speed of vision transformers by replacing the linear patch projection with convolutional layers and max-pooling, as well as modifying the feed-forward networks in each transformer layer to incorporate depth-wise convolutions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Efficient attention. Numerous methods for efficient self-attention have been proposed in the literature to address the quadratic complexity of self-attention in the number of input tokens. These include restricting the span of the self-attention to local windows [48, 50], strided patterns $\\pmb { \\mathbb { I } }$ , axial patterns $\\textcircled { \\lvert 3 0 \\rvert }$ , or an adaptive computation across layers $ { \\mathbb { I } }$ . Other methods provide an approximation of the self-attention matrix which can be achieved by a projection across the token dimension $\\mathbb { \\lVert \\rVert }$ , or through a factorization of the softmax-attention kernel [15, 37, 56, 77], which avoids explicit computation of the attention matrix. While conceptually different, our XCA performs similar computations without being sensitive to the choice of the kernel. Similarly, Lee-Thorp et al. [41] achieve faster training by substituting self-attention with unparametrized Fourier Transform. Other efficient attention methods rely on local attention and adding a small number of global tokens, thus allowing interaction among all tokens only by hopping through the global tokens $\\boxed { 1 } \\boxed { 5 } \\boxed { 3 4 } \\boxed { 8 0 }$ . Similarly, Goyal et al. $\\boldsymbol { \\| 2 5 \\| }$ use a global workspace though which items interact, albeit one that is shared across layers. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Transformers for high-resolution images. Several works adopt visual transformers to highresolution image tasks beyond image classification, such as object detection and image segmentation. Wang et al. $\\tilde { \\left. 7 0 \\right. }$ design a model with a pyramidal architecture and address complexity by gradually reducing the spatial resolution of keys and values. Similarly, for video recognition Fan et al. [23] utilize pooling to reduce the resolution across the spatial and temporal dimensions to allow for an efficient computation of the attention matrix. Zhang et al. $\\textcircled { 8 1 }$ adopt global tokens and local attention to reduce the model complexity, while Liu et al. $\\checkmark$ provide an efficient method for local attention with shifted windows. In addition, Zheng et al. $[ [ 8 3 ] ]$ and Ranftl et al. $[ [ 5 4 ]$ study problems like semantic segmentation and monocular depth estimation with the quadratic self-attention operation. ",
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+    {
+        "type": "text",
+        "text": "Data-dependent layers. Our XCiT layer can be regarded as a “dynamic” $1 \\times 1$ convolution, which multiplies all token features with the same data-dependent weight matrix, derived from the key and query cross-covariance matrix. In the context of convolutional networks, Dynamic Filter Networks [9] explore a related idea, using a filter generating subnetwork to produce convolutional filters based on features in previous layers. Squeeze-and-Excitation networks $\\left[ \\left[ 3 2 \\right] \\right]$ use data dependent $1 \\times 1$ convolutions in convolutional architectures. Spatially average-pooled features are fed to a 2-layer MLP which produces per channel scaling parameters. Closer in spirit to our work, Lambda layers propose a way to ensure global interaction in ResNet models $\\bar { \\mathbb { H } }$ . Their “content-based lambda function” is computing a similar term as our cross-covariance attention, but differing in how the softmax and $\\ell _ { 2 }$ normalizations are applied. Moreover, Lambda layers also include specific positionbased lambda functions, and LambdaNetworks are based on ResNets while XCiT follows the ViT architecture. Recently data-independent analogues of self-attention have also been found to be an effective alternative to convolutional and self-attention layers for vision tasks $\\textcircled { 1 2 0 } , \\textcircled { 4 6 } , \\textcircled { 6 2 } , \\textcircled { 6 6 } $ . These methods treat entries in the attention map as learnable parameters, rather than deriving the attention map dynamically from queries and keys, but their complexity remains quadratic in the number of tokens. Zhao et al. [82] consider alternative attention forms in computer vision. ",
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+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
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+        "text": "In this section, we first recall the self-attention mechanism, and the connection between the Gram and covariance matrices, which motivated our work. We then propose our cross-covariance attention operation (XCA) – which operates along the feature dimension instead of token dimension in conventional transformers – and combine it with local patch interaction and feedforward layers to construct our Cross-Covariance Image Transformer (XCiT). See Figure $\\bigtriangledown$ for an overview. ",
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+        "type": "text",
+        "text": "3.1 Background ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Token self-attention. Self-attention, as introduced by Vaswani et al. $\\lVert \\overline { { 6 8 } } \\rVert$ , operates on an input matrix $X \\in \\mathbb { R } ^ { N \\times d }$ , where $N$ is the number of tokens, each of dimensionality $d$ . The input $X$ is linearly projected to queries, keys and values, using the weight matrices $\\dot { W _ { q } } \\in \\mathbb { R } ^ { d \\times d _ { q } }$ , $W _ { k } \\in$ $\\mathbb { R } ^ { d \\times d _ { k } }$ and $W _ { v } \\in \\mathbb { R } ^ { d \\times d _ { v } }$ , such that $Q { = } X W _ { q }$ , $K { = } X W _ { k }$ and $V { = } X W _ { v }$ , where $d _ { q } = d _ { k }$ . Keys and values are used to compute an attention map $\\mathcal { A } ( K , Q ) = \\operatorname { S o f t m a x } ( Q K ^ { \\top } / \\sqrt { d _ { k } } )$ , and the output of the self-attention operation is defined as the weighted sum of $N$ token features in $V$ with the weights corresponding to the attention map: Attention $( Q , K , V ) = \\mathcal { A } ( K , Q ) V$ . The computational complexity of self-attention scales quadratically in $N$ , due to pairwise interactions between all $N$ elements. ",
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+        "type": "text",
+        "text": "Relationship between Gram and covariance matrices. To motivate our cross-covariance attention operation, we recall the relation between Gram and covariance matrices. The unnormalised $d \\times d$ covariance matrix is obtained as $C { = } X ^ { \\top } X$ . The $N \\times N$ Gram matrix contains all pairwise innerproducts: $G { = } X X ^ { \\top }$ . The non-zero part of the eigenspectrum of the Gram and covariance matrix are equivalent, and the eigenvectors of $C$ and $G$ can be computed in terms of each other. If $V$ are the eigenvectors of $G$ , then the eigenvectors of $C$ are given by $U { = } X V$ . To minimise the computational cost, the eigendecomposition of either the Gram or covariance matrix can be obtained in terms of the decomposition of the other, depending on which of the two matrices is the smallest.1 ",
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+        "type": "text",
+        "text": "We draw upon this strong connection between the Gram and covariance matrices to consider whether it is possible to avoid the quadratic cost to compute the $N \\times N$ attention matrix, which is computed from the analogue of the $N \\times N$ Gram matrix $\\mathsf { \\bar { Q } } K ^ { \\top } { = } X W _ { q } W _ { k } ^ { \\top } X ^ { \\top }$ . Below we consider how we can use the $d _ { k } \\times d _ { q }$ cross-covariance matrix, $K ^ { \\top } Q { = } W _ { k } ^ { \\top } X ^ { \\top } X W _ { q }$ , which can be computed in linear time in the number of elements $N$ , to define an attention mechanism. ",
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+        "type": "text",
+        "text": "3.2 Cross-covariance attention ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We propose a cross-covariance based self-attention function that operates along the feature dimension, rather than along the token dimension as in token self-attention. Using the definitions of queries, keys and values from above, the cross-covariance attention function is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/c782364387393975aaaceb61444e9a54aea92c55d937d4043f91c3097489166e.jpg",
+        "text": "$$\n\\operatorname { X C - A t t e n t i o n } ( Q , K , V ) = V A \\operatorname { x c } ( K , Q ) , \\qquad A \\operatorname { x c } ( K , Q ) = \\operatorname { S o f t m a x } \\left( \\hat { K } ^ { \\top } \\hat { Q } / \\tau \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where each output token embedding dimension is a convex combination of the $d _ { v }$ features of its corresponding token embedding in $V$ . The attention weights $\\mathcal { A }$ are computed based on the crosscovariance matrix. ",
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+        "text": "$\\ell _ { 2 }$ -Normalization and temperature scaling. In addition to building our attention operation on the cross-covariance matrix, we make a second modification compared to token self-attention. We restrict the magnitude of the query and key matrices by $\\ell _ { 2 }$ -normalising them, such that each column of length $N$ of the normalised matrices $\\hat { Q }$ and $\\hat { K }$ has unit norm, and every element in $d { \\times } d$ cross-covariance matrix $\\hat { K } ^ { \\top } \\hat { Q }$ is in the range $[ - 1 , 1 ]$ . We observed that controlling the norm strongly enhances the stability of training, especially when trained with a variable numbers of tokens. However, restricting the norm reduces the representational power of the operation by removing a degree of freedom. Therefore, we introduce a learnable temperature parameter $\\tau$ which scales the inner products before the Softmax, allowing for sharper or more uniform distribution of attention weights. ",
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+    {
+        "type": "image",
+        "img_path": "images/301e7b358ccf93cc7973974e2bad457d5f17a3e86617dc31e723f62a68c6c260.jpg",
+        "image_caption": [
+            "Figure 2: Inference memory usage of vision transformer variants. Our XCiT models scale linearly in the number of tokens, which makes it possible to scale to much larger image sizes, even in comparison to approaches employing approximate self-attention or a pyramidal design. All measurements are performed with a batch size of 64 on a single V100-32GB GPU. "
+        ],
+        "image_footnote": [],
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+    {
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+        "image_caption": [
+            "Figure 3: Performance when changing the resolution at test-time for models with a similar number of parameters. All networks were trained at resolution 224, w/o distillation. XCiT is more tolerant to changes of resolution than the Gram-based DeiT and benefit more from the “FixRes” effect $\\mathbb { \\lVert \\rVert }$ when inference is performed at a larger resolution than at train-time. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Block-diagonal cross-covariance attention. Instead of allowing all features to interact among each other, we divide them into a $h$ groups, or “heads”, in a similar fashion as multi-head token self-attention. We apply the cross-covariance attention separately per head where for each head, we learn separate weight matrices to project $X$ to queries, keys and values, and collect the corresponding weight matrices in the tensors $\\dot { W _ { q } } \\in \\mathbb { R } ^ { h \\times d \\times d _ { q } }$ , $W _ { k } \\in \\mathbb { R } ^ { \\mathbf { \\bar { h } } \\times d \\times d _ { k } }$ and $\\dot { W } _ { v } \\in \\mathbb { R } ^ { h \\times d \\times d _ { v } }$ , where we set $d _ { k } \\bar { = } d _ { q }  { = } d _ { v }  { = } d / h$ . Restricting the attention within heads has two advantages: (i) the complexity of aggregating the values with the attention weights is reduced by a factor $h$ ; (ii) more importantly, we empirically observe that the block-diagonal version is easier to optimize, and typically leads to improved results. This observation is in line with observations made for Group Normalization $\\mathbb { [   ]  }$ which normalizes groups of channels separately based on their statistics, and achieves favorable results for computer vision tasks compared to Layer Normalization $\\pmb { \\Vert 3 \\Vert }$ , which combines all channels in a single group. Figure $\\boxed { 4 }$ shows that each head learns to focus on semantically coherent parts of the image, while being flexible to change what type of features it attends to based on the image content. ",
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+    {
+        "type": "text",
+        "text": "Complexity analysis. The usual token self-attention with $h$ heads has a time complexity of $\\mathcal { O } ( N ^ { 2 } d )$ and memory complexity of $\\mathcal { O } ( h N ^ { 2 } { + } N d )$ . Due to the quadratic complexity, it is problematic to scale token self-attention to images with a large number of tokens. Our cross-covariance attention overcomes this drawback as its computational cost of $\\mathcal { O } ( N d ^ { 2 } / h )$ scales linearly with the number of tokens, as does the memory complexity of $\\mathcal { O } ( d ^ { 2 } / h + N \\dot { d } )$ . Therefore, our model scales much better to cases where the number of tokens $N$ is large, and the feature dimension $d$ is relatively small, as is typically the case, in particularly when splitting the features into $h$ heads. ",
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+        "type": "text",
+        "text": "3.3 Cross-covariance image transformers ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To construct our cross-covariance image transformers (XCiT), we adopt a columnar architecture which maintains the same spatial resolution across layers, similarly to $\\boxed { 1 2 1 } \\boxed { 6 4 } \\boxed { 6 7 }$ . We combine our cross-covariance attention (XCA) block with the following additional modules, each one being preceded by a LayerNorm $\\pmb { \\mathbb { B } } \\|$ . See Figure $\\perp$ for an overview. Since in this section we specifically design the model for computer vision tasks, tokens correspond to image patches in this context. ",
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+    {
+        "type": "table",
+        "img_path": "images/1dc7bc735004c7a44ab04cdc863cc2cd94c905ffd60ceef05cdc783162fad18e.jpg",
+        "table_caption": [
+            "Table 1: XCiT models. Design choices include model depth, patch embeddings dimensionality $d$ , and the number of heads $h$ used in XCA. By default our models are trained and tested at resolution 224 with patch sizes of $1 6 \\times 1 6$ . We also train with distillation using a convolutional teacher (denoted $\\Upsilon$ ) as proposed by Touvron et al. [64]. Finally, we report performance of our strongest models obtained with $8 \\times 8$ patch size, fine-tuned $( \\uparrow )$ and tested at resolution $3 8 4 \\times 3 8 4$ (column $\\textcircled { \\alpha } 3 8 4 / 8 $ ), using distillation with a teacher that was also fine-tuned $@ 3 8 4$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Depth</td><td rowspan=\"2\">d</td><td rowspan=\"2\">#heads</td><td rowspan=\"2\">#params</td><td colspan=\"2\">GFLOPs</td><td colspan=\"3\">ImageNet-1k-val top-1 acc. (%)</td></tr><tr><td>@224/16</td><td>@384/8</td><td>@224/16 @224/16r</td><td></td><td>@384/8r ↑</td></tr><tr><td>XCiT-N12</td><td>12</td><td>128</td><td>4</td><td>3M</td><td>0.5</td><td>6.4</td><td>69.9</td><td>72.2</td><td>77.8</td></tr><tr><td>XCiT-T12</td><td>12</td><td>192</td><td>4</td><td>7M</td><td>1.2</td><td>14.3</td><td>77.1</td><td>78.6</td><td>82.4</td></tr><tr><td>XCiT-T24</td><td>24</td><td>192</td><td>4</td><td>12M</td><td>2.3</td><td>27.3</td><td>79.4</td><td>80.4</td><td>83.7</td></tr><tr><td>XCiT-S12</td><td>12</td><td>384</td><td>8</td><td>26M</td><td>4.8</td><td>55.6</td><td>82.0</td><td>83.3</td><td>85.1</td></tr><tr><td>XCiT-S24</td><td>24</td><td>384</td><td>8</td><td>48M</td><td>9.1</td><td>106.0</td><td>82.6</td><td>83.9</td><td>85.6</td></tr><tr><td>XCiT-M24</td><td>24</td><td>512</td><td>8</td><td>84M</td><td>16.2</td><td>188.0</td><td>82.7</td><td>84.3</td><td>85.8</td></tr><tr><td>XCiT-L24</td><td>24</td><td>768</td><td>16</td><td>189M</td><td>36.1</td><td>417.9</td><td>82.9</td><td>84.9</td><td>86.0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Local patch interaction. In the XCA block communication between patches is only implicit through the shared statistics. To enable explicit communication across patches we add a simple Local Patch Interaction (LPI) block after each XCA block. LPI consists of two depth-wise $3 { \\times } 3$ convolutional layers with Batch Normalization and GELU non-linearity in between. Due to its depth-wise structure, the LPI block has a negligible overhead in terms of parameters, as well as a very limited overhead in terms of throughput and memory usage during inference. ",
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+    {
+        "type": "text",
+        "text": "Feed-forward network. As is common in transformer models, we add a point-wise feedforward network (FFN), which has a single hidden layer with $4 d$ hidden units. While interaction between features is confined within groups in the XCA block, and no feature interaction takes place in the LPI block, the FFN allows for interaction across all features. ",
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+        "type": "text",
+        "text": "Global aggregation with class attention. When training our models for image classification, we utilize the class attention layers as proposed by Touvron et al. $ { \\mathbb { I } }  { \\mathbb { K } }  { \\ b { 7 } }  { \\mathbb { I } }$ . These layers aggregate the patch embeddings of the last XCiT layer through writing to a CLS token by one-way attention between the CLS tokens and the patch embeddings. The class attention is also applied per head, i.e. feature group. ",
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+    {
+        "type": "text",
+        "text": "Handling images of varying resolution. In contrast to the attention map involved in token selfattention, in our case the covariance blocks are of fixed size independent of the input image resolution. The softmax always operates over the same number of elements, which may explain why our models behave better when dealing with images of varying resolutions (see Figure $\\textcircled{3}$ . In XCiT we include additive sinusoidal positional encoding $\\lVert \\rVert$ with the input tokens. We generate them in 64 dimensions from the 2d patch coordinates and then linearly project to the transformer working dimension $d$ . This choice is orthogonal to the use of learned positional encoding, as in ViT [21]. However, it is more flexible since there is no need to interpolate or fine-tune the network when changing the image size. ",
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+    {
+        "type": "text",
+        "text": "Model configurations. In Table $^ 1$ we list different variants of our model which we use in our experiments, with different choices for model width and depth. For the patch encoding layer, unless mentioned otherwise, we adopt the alternative used by Graham et al. $\\pmb { \\mathbb { D } } \\pmb { 6 } \\|$ with convolutional patch projection layers. We also experimented with a linear patch projection as described in $\\pmb { \\mathbb { D } } \\mathbf { 1 } \\mathbf { h }$ , see our ablation in Table 4. Our default patch size is $1 6 \\times 1 6$ , as in other vision transformer models including ViT $\\scriptstyle { \\left[ \\left[ 2 1 \\right] \\right] }$ , DeiT [64] and CaiT $\\pmb { \\mathbb { E 7 } }$ . We also experiment with smaller $8 \\times 8$ patches, which has been observed to improve performance $\\mathbb { [ [  2 ]  }$ . Note that this is efficient with XCiT as its complexity scales linearly which the number of patches, while ViT, DeiT and CaiT scale quadratically. ",
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+        "type": "text",
+        "text": "4 Experimental evaluation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we demonstrate the effectiveness and versatility of XCiT on multiple computer vision benchmarks, and present ablations providing insight on the importance of its different components. In the supplementary material we provide additional analysis, including the impact on performance of image resolution in Section ${ \\bf A . l } ^ { \\dot { } }$ and of multiple approximate attention baselines in Section A.2. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 2: ImageNet classification. Number of parameters, FLOPs, image resolution, and top-1 accuracy on ImageNet-1k and ImageNet-V2. Training strategies vary across models, transformer-based models and the reported RegNet mostly follow recipes from DeiT [64]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>#params</td><td>FLOPs Res.</td><td></td><td>ImNet</td><td>V2</td></tr><tr><td>EfficientNet-B5 RA [17]</td><td>30M</td><td>9.9B</td><td>456</td><td>83.7</td><td></td></tr><tr><td>RegNetY-4GFI531</td><td>21M</td><td>4.0B</td><td>224</td><td>80.0</td><td>72.4</td></tr><tr><td>DeiT-SY 64]</td><td>22M</td><td>4.6B</td><td>224</td><td>81.2</td><td>68.5</td></tr><tr><td>Swin-T [44]</td><td>29M</td><td>4.5B</td><td>224</td><td>81.3</td><td></td></tr><tr><td>CaiT-XS24Y ↑ 67]</td><td>26M</td><td>19.3B</td><td>384</td><td>84.1</td><td>74.1</td></tr><tr><td>XCiT-S12/16M</td><td>26M</td><td>4.8B</td><td>224</td><td>83.3</td><td>72.5</td></tr><tr><td>XCiT-S12/16Y↑</td><td>26M</td><td>14.3B</td><td>384</td><td>84.7</td><td>74.1</td></tr><tr><td>XCiT-S12/8Y↑</td><td>26M</td><td>55.6B</td><td>384</td><td>85.1</td><td>74.8</td></tr><tr><td>EfficientNet-B7RA [17]</td><td>66M</td><td>37.0B</td><td>600</td><td>84.7</td><td></td></tr><tr><td>NFNet-F0 [10]</td><td>72M</td><td>12.4B</td><td>256</td><td>83.6</td><td>72.6</td></tr><tr><td>RegNetY-8GF 园</td><td>39M</td><td>8.0B</td><td>224</td><td>81.7</td><td>72.4</td></tr><tr><td>TNT-B 四</td><td>66M</td><td>14.1B</td><td>224</td><td>82.8</td><td>1</td></tr><tr><td>国 Swin-S</td><td>50M</td><td>8.7B</td><td>224</td><td>83.0</td><td></td></tr><tr><td>CaiT-S24Y ↑ 67]</td><td>47M</td><td>32.2B</td><td>384</td><td>85.1</td><td>75.4</td></tr><tr><td>XCiT-S24/16M</td><td>48M</td><td>9.1B</td><td>224</td><td>83.9</td><td>73.3</td></tr><tr><td>XCiT-S24/16Y↑</td><td>48M</td><td>26.9B</td><td>384</td><td>85.1</td><td>74.6</td></tr><tr><td>XCiT-S24/8Y ↑</td><td>48M</td><td>105.9B</td><td>384</td><td>85.6</td><td>75.7</td></tr><tr><td>Fix-EfficientNet-B8</td><td>园 87M</td><td>89.5B</td><td>800</td><td>85.7</td><td>75.9</td></tr><tr><td>RegNetY-16GF[53]</td><td>84M</td><td>16.0B</td><td>224</td><td>82.9</td><td>72.4</td></tr><tr><td>Swin-B↑ 四</td><td>88M</td><td>47.0B</td><td>384</td><td>84.2</td><td></td></tr><tr><td>DeiT-BY↑[64</td><td>87M</td><td>55.5B</td><td>384</td><td>85.2</td><td>75.2</td></tr><tr><td>CaiT-S48Y ↑[67]</td><td>89M</td><td>63.8B</td><td>384</td><td>85.3</td><td>76.2</td></tr><tr><td>XCiT-M24/16T</td><td>84M</td><td>16.2B</td><td>224</td><td>84.3</td><td>73.6</td></tr><tr><td>XCiT-M24/16Y↑</td><td>84M</td><td>47.7B</td><td>384</td><td>85.4</td><td>75.1</td></tr><tr><td>XCiT-M24/8Y↑</td><td>84M</td><td>187.9B</td><td>384</td><td>85.8</td><td>76.1</td></tr><tr><td>NFNet-F2[ 目</td><td>194M</td><td>62.6B</td><td>352</td><td>85.1</td><td>74.3</td></tr><tr><td>NFNet-F3 目</td><td>255M</td><td>114.8B</td><td>416</td><td>85.7</td><td>75.2</td></tr><tr><td>CaiT-M24Y ↑[67</td><td>186M</td><td>116.1B</td><td>384</td><td>85.8</td><td>76.1</td></tr><tr><td>XCiT-L24/16T</td><td>189M</td><td>36.1B</td><td>224</td><td>84.9</td><td>74.6</td></tr><tr><td>XCiT-L24/16Y ↑</td><td>189M</td><td>106.0B</td><td>3384</td><td>85.8</td><td>75.8</td></tr><tr><td>XCiT-L24/8Y↑</td><td>189M</td><td>417.8B</td><td>384</td><td>86.0</td><td>76.6</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/d87cd7c275cbbc0e18ed54c4731e8c41009d08d1f159fc54964749277f2df726.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of the attention map between the CLS token and individual patches in the class-attention stage. For each column, each row represents the attention map w.r.t. one head, corresponding to the image in the first row. Each head appears sensitive to semantically coherent regions. Heads are sensitive to similar features within the same or across images (e.g. people or bird faces). They are trigger by different concepts when such features are missing (e.g., cockpit for race cars). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.1 Image classification ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We use ImageNet-1k $\\mathbb { \\lVert 1 9 \\rVert }$ to train and evaluate our models for image classification. It consists of 1.28M training images and $5 0 \\mathrm { k }$ validation images, labeled across 1,000 semantic categories. Our training setup follows the DeiT recipe $\\pmb { \\mathbb { \\lVert 6 4 \\rVert } }$ . We train our model for 400 epochs with the AdamW optimizer $| \\bar { \\mathbf { \\nabla } } 4 \\bar { 5 } | |$ using a cosine learning rate decay. In order to enhance the training of larger models, we utilize LayerScale $ { \\mathbb { I } }  { \\mathbb { I } }$ and adjust the stochastic depth $\\mathbb { \\lVert 3 3 \\rVert }$ for each of our models accordingly (see the supplementary material for details). Following $ { \\mathbb { I } }  { \\mathbb { K } } 7  { \\mathbb { I } }$ , images are cropped with crop ratio of 1.0 for evaluation. In addition to the ImageNet-1k validation set, we report results for ImageNet-V2 [55] which has a distinct test set. Our implementation is based on the Timm library $\\lVert \\ b { 7 2 } \\rVert$ . ",
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+    {
+        "type": "text",
+        "text": "Results on ImageNet. We present a family of seven models in Table $\\lfloor 1 \\rfloor$ with different operating points in terms of parameters and FLOPs. We observe that the performance of the XCiT models benefits from increased capacity both in depth and width. Additionally, consistent with $\\mathbb { B 4 } \\mathbb { 6 7 } \\mathbb { 1 }$ we find that using hard distillation with a convolutional teacher improves the performance. Because of its linear complexity in the number of tokens, it is feasible to train XCiT at $3 8 4 \\times 3 8 4$ resolution with small $8 \\times 8$ patches, i.e. 2304 tokens, which provides a strong boost in performance across all configurations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare to the state-of-the-art convolutional and transformer-based architectures [10, 44, 53, 58, $\\textcircled { 6 7 }$ in Table $2 .$ By varying the input image resolution and/or patch size, our models provide competitive or superior performance across model sizes and FLOP budgets. First, the models operating on $2 2 4 \\times 2 2 4$ and $1 6 \\times 1 6$ (e.g. XCiT-S12/16) enjoy high accuracy at relatively few FLOPs compared to their counterparts with comparable parameter count and FLOPs. Second, our models with $1 6 \\times 1 6$ and $3 8 4 \\times 3 8 4$ resolution images (e.g. XCiT-S12/16\") yield an improved accuracy at the expense of higher FLOPs, and provide superior or on-par performance compared to state-of-the-art models with comparable computational requirements. Finally, the linear complexity of XCiT allows us to scale to process $3 8 4 \\times 3 8 4$ images with $8 \\times 8$ patch sizes (e.g. XCiT-S12/8\"), achieving the highest accuracy across the board, albeit at a relatively high FLOPs count. ",
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+    {
+        "type": "table",
+        "img_path": "images/0cd706ed91865f761ffe03807f984655f712b992252942160ae51b7d4f8c1326.jpg",
+        "table_caption": [
+            "Table 3: Self-supervised learning. Top-1 acc. on ImageNet-1k. We report with a crop-ratio 0.875 for consistency with DINO. For the last row it is set to 1.0 (improves from $8 0 . 7 \\%$ to $8 0 . 9 \\%$ ). All models are trained for 300 epochs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>SSL Method</td><td>Model</td><td>#params</td><td>FLOPs</td><td>Linear</td><td>k-NN</td></tr><tr><td>MoBY 回</td><td>Swin-T 国</td><td>29M</td><td>4.5B</td><td>75.0</td><td>1</td></tr><tr><td>DINO 回</td><td>ResNet-50[28]</td><td>23M</td><td>4.1B</td><td>74.5</td><td>65.6</td></tr><tr><td>DINO 圆</td><td>ViT-S/16_21</td><td>22M</td><td>4.6B</td><td>76.1</td><td>72.8</td></tr><tr><td>DINO 圆</td><td>ViT-S/8 21</td><td>22M</td><td>22.4B</td><td>79.2</td><td>77.2</td></tr><tr><td>DINO 圆</td><td>XCiT-S12/16</td><td>26M</td><td>4.9B</td><td>77.8</td><td>76.0</td></tr><tr><td>DINO 国</td><td>XCiT-S12/8</td><td>26M</td><td>18.9B</td><td>79.2</td><td>77.1</td></tr><tr><td>DINO 圆</td><td>ViT-B/1621</td><td>87M</td><td>17.5B</td><td>78.2</td><td>76.1</td></tr><tr><td>DINO 园</td><td>ViT-B/8 日</td><td>87M</td><td>78.2B</td><td>80.1</td><td>77.4</td></tr><tr><td>DINO 园</td><td>XCiT-M24/16</td><td>84M</td><td>16.2B</td><td>78.8</td><td>76.4</td></tr><tr><td>DINO 圆</td><td>XCiT-M24/8</td><td>84M</td><td>64.0B</td><td>80.3</td><td>77.9</td></tr><tr><td>DINO 园</td><td>XCiT-M24/8↑384</td><td>84M</td><td>188.0B</td><td>80.9</td><td>78.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/41b6af8b59f66ad278cdcd83a3335febd98b78f503b2369982284d23ae743683.jpg",
+        "table_caption": [
+            "Table 4: Ablations of various architectural design choices on the task of ImageNet-1k classification using the XCiT-S12 model. Our baseline model uses the convolutional projection adopted from LeVit. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Ablation</td><td>ImNet top-1 acc.</td></tr><tr><td>XCiT-S12/16</td><td rowspan=\"2\">Baseline</td><td>82.0</td></tr><tr><td>XCiT-S12/8</td><td>83.4</td></tr><tr><td>XCiT-S12/16</td><td rowspan=\"2\">Linear patch proj.</td><td>81.1</td></tr><tr><td>XCiT-S12/8</td><td>83.1</td></tr><tr><td rowspan=\"2\">XCiT-S12/16</td><td>w/o LPI layer</td><td>80.8</td></tr><tr><td>w/o XCA layer</td><td>75.9</td></tr><tr><td rowspan=\"2\">XCiT-S12/16</td><td>w/o l2-normal.</td><td>failed</td></tr><tr><td>w/o learned temp. T</td><td>81.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Class attention visualization. In Figure $\\sharp$ we show the class attention map obtained in the feature aggregation stage. Each head focuses on different semantically coherent regions in the image (e.g. faces or umbrellas). Furthermore, heads tend to focus on similar patterns across images (e.g. bird head or human face), but adapts by focusing on other salient regions when such patterns are absent. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Robustness to resolution changes. In Figure $3$ we report the accuracy of XCiT-S12, DeiT-S and ResNet-50 trained on $2 2 4 \\times 2 2 4$ images and evaluated at different image resolutions. While DeiT outperforms ResNet-50 when train and test resolutions are similar, it suffers from a larger drop in performance as the image resolution deviates farther from the training resolution. XCiT displays a substantially increased accuracy when train and test resolutions are similar, while also being robust to resolution changes, in particular for the model with $8 \\times 8$ patches. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised learning. We train XCiT in a self-supervised manner using DINO $\\mathbb { \\lVert \\rVert }$ on ImageNet-1k. In Table $\\bar { 3 }$ we report performance using the linear and $\\mathbf { k }$ -NN protocols as in $\\mathbb { \\lVert \\rVert }$ . Across model sizes XCiT obtains excellent accuracy with both protocols, substantially improving DINO with ResNet-50 or ViT architectures, as well as over those reported for Swin-Transformer trained with MoBY $\\left[ \\left[ 7 6 \\right] \\right]$ . Comparing the larger models to ViT, we also observed improved performance for XCiT achieving a strong $8 0 . 3 \\%$ accuracy. For fair comparison, all reported models have been trained for 300 epochs. Further improved performance of small models is reported by Caron et al. $\\mathbb { \\lVert \\rVert }$ when training for 800 epochs, which we expect to carryover to XCiT based on the results presented here. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Analysis and ablations. In Table $\\sharp$ we provide ablation experiments to analyse the impact of different design choices for our XCiT-S12 model. First, we observe the positive effect of using the convolutional patch projection as compared to using linear patch projection, for both $8 \\times 8$ and $1 6 \\times 1 6$ patches. Second, while removing the LPI layer reduces the accuracy by only $1 . 2 \\%$ (from 82.0 to 80.8), removing the XCA layer results in a large drop of $6 . 1 \\%$ , underlining the effectiveness of XCA. We noticed that the inclusion of two convolutional components – convolutional patch projection and LPI – not only brings improvements in accuracy, but also accelerates training. Third, although we were able to ensure proper convergence without $\\ell _ { 2 }$ -normalization of queries and keys by tweaking the hyper-parameters, we found that it provides stability across model size (depth and width) and other hyper-parameters. Finally, while the learnable softmax temperature parameter is not critical, removing it drops accuracy by $0 . 2 \\%$ . Additional ablations are provided in the supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/57df260c30fed994ba9210c1bd037d5a095d94598b1a084873da3d7645b71f27.jpg",
+        "table_caption": [
+            "Table 5: COCO object detection and instance segmentation performance on the mini-val set. All backbones are pre-trained on ImageNet-1k, use Mask R-CNN model [29] and are trained with the same 3x schedule. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Backbone</td><td>#params</td><td>AP</td><td>AP</td><td>AP5</td><td>Apm</td><td>AP</td><td>AP</td></tr><tr><td>ResNet18 国</td><td>31.2M</td><td>36.9</td><td>57.1</td><td>40.0</td><td>33.6</td><td>53.9</td><td>35.7</td></tr><tr><td>PVT-Tiny 凯</td><td>32.9M</td><td>39.8</td><td>62.2</td><td>43.0</td><td>37.4</td><td>59.3</td><td>39.9</td></tr><tr><td>ViL-Tiny I81</td><td>26.9M</td><td>41.2</td><td>64.0</td><td>44.7</td><td>37.9</td><td>59.8</td><td>40.6</td></tr><tr><td>XCiT-T12/16</td><td>26.1M</td><td>42.7</td><td>64.3</td><td>46.4</td><td>38.5</td><td>61.2</td><td>41.1</td></tr><tr><td>XCiT-T12/8</td><td>25.8M</td><td>44.5</td><td>66.4</td><td>48.8</td><td>40.3</td><td>63.5</td><td>43.2</td></tr><tr><td>ResNet50 I28]</td><td>44.2M</td><td>41.0</td><td>61.7</td><td>44.9</td><td>37.1</td><td>58.4</td><td>40.1</td></tr><tr><td>PVT-Small[70]</td><td>44.1M</td><td>43.0</td><td>65.3</td><td>46.9</td><td>39.9</td><td>62.5</td><td>42.8</td></tr><tr><td>ViL-Small81</td><td>45.0M</td><td>43.4</td><td>64.9</td><td>47.0</td><td>39.6</td><td>62.1</td><td>42.4</td></tr><tr><td>Swin-T [44</td><td>47.8M</td><td>46.0</td><td>68.1</td><td>50.3</td><td>41.6</td><td>65.1</td><td>44.9</td></tr><tr><td>XCiT-S12/16</td><td>44.3M</td><td>45.3</td><td>67.0</td><td>49.5</td><td>40.8</td><td>64.0</td><td>43.8</td></tr><tr><td>XCiT-S12/8</td><td>43.1M</td><td>47.0</td><td>68.9</td><td>51.7</td><td>42.3</td><td>66.0</td><td>45.4</td></tr><tr><td>ResNet101 28</td><td>63.2M</td><td>42.8</td><td>63.2</td><td>47.1</td><td>38.5</td><td>60.1</td><td>41.3</td></tr><tr><td>ResNeXt101-32</td><td>62.8M</td><td>44.0</td><td>64.4</td><td>48.0</td><td>39.2</td><td>61.4</td><td>41.9</td></tr><tr><td>PVT-Medium 目</td><td>63.9M</td><td>44.2</td><td>66.0</td><td>48.2</td><td>40.5</td><td>63.1</td><td>43.5</td></tr><tr><td>ViL-Medium 图</td><td>60.1M</td><td>44.6</td><td>66.3</td><td>48.5</td><td>40.7</td><td>63.8</td><td>43.7</td></tr><tr><td>Swin-S44</td><td>69.1M</td><td>48.5</td><td>70.2</td><td>53.5</td><td>43.3</td><td>67.3</td><td>46.6</td></tr><tr><td>XCiT-S24/16</td><td>65.8M</td><td>46.5</td><td>68.0</td><td>50.9</td><td>41.8</td><td>65.2</td><td>45.0</td></tr><tr><td>XCiT-S24/8</td><td>64.5M</td><td>48.1</td><td>69.5</td><td>53.0</td><td>43.0</td><td>66.5</td><td>46.1</td></tr><tr><td>ResNeXt101-6475</td><td>101.9M</td><td>44.4</td><td>64.9</td><td>48.8</td><td>39.7</td><td>61.9</td><td>42.6</td></tr><tr><td>PVT-Large 目</td><td>81.0M</td><td>44.5</td><td>66.0</td><td>48.3</td><td>40.7</td><td>63.4</td><td>43.7</td></tr><tr><td>ViL-Large 日</td><td>76.1M</td><td>45.7</td><td>67.2</td><td>49.9</td><td>41.3</td><td>64.4</td><td>44.5</td></tr><tr><td>XCiT-M24/16</td><td>101.1M</td><td>46.7</td><td>68.2</td><td>51.1</td><td>42.0</td><td>65.6</td><td>44.9</td></tr><tr><td>XCiT-M24/8</td><td>98.9M</td><td>48.5</td><td>70.3</td><td>53.4</td><td>43.7</td><td>67.5</td><td>46.9</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/db3eb872f723a17ba91bfd5d84d0634fe013f403837f65f9b6c315677d32393e.jpg",
+        "table_caption": [
+            "Table 6: ADE20k semantic segmentation performance using Semantic FPN [38] and UperNet [74] (in comparable settings). We do not include comparisons with other state-of-the-art models that are pre-trained on larger datasets [44, 54, 83]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Backbone</td><td rowspan=1 colspan=2>Semantic FPN</td><td rowspan=1 colspan=2>UperNet</td></tr><tr><td rowspan=1 colspan=1>#params</td><td rowspan=1 colspan=1>mIoU</td><td rowspan=1 colspan=1>#params</td><td rowspan=1 colspan=1>mIoU</td></tr><tr><td rowspan=1 colspan=1>ResNet1828PVT-Tiny70</td><td rowspan=1 colspan=1>15.5M17.0M</td><td rowspan=1 colspan=1>32.935.7M</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>--</td></tr><tr><td rowspan=1 colspan=1>XCiT-T12/16XCiT-T12/8</td><td rowspan=1 colspan=1>8.4M8.4M</td><td rowspan=1 colspan=1>38.139.9</td><td rowspan=1 colspan=1>33.7M33.7</td><td rowspan=1 colspan=1>41.543.5</td></tr><tr><td rowspan=1 colspan=1>ResNet5028PVT-Small[70Swin-T44</td><td rowspan=1 colspan=1>28.5M28.2M</td><td rowspan=1 colspan=1>36.739.8-</td><td rowspan=1 colspan=1>66.5M-59.9M</td><td rowspan=1 colspan=1>42.0-44.5</td></tr><tr><td rowspan=1 colspan=1>XCiT-S12/16XCiT-S12/8</td><td rowspan=1 colspan=1>30.4M30.4M</td><td rowspan=1 colspan=1>43.944.2</td><td rowspan=1 colspan=1>52.4M52.3M</td><td rowspan=1 colspan=1>45.946.6</td></tr><tr><td rowspan=2 colspan=1>ResNet10128ResNeXt101-3275PVT-Medium 70Swin-S44</td><td rowspan=2 colspan=1>47.5M47.1M48.0M=</td><td rowspan=1 colspan=1>38.8</td><td rowspan=1 colspan=1>85.5M</td><td rowspan=1 colspan=1>43.8</td></tr><tr><td rowspan=1 colspan=1>39.741.6-</td><td rowspan=1 colspan=1>--81.0M</td><td rowspan=1 colspan=1>--47.6</td></tr><tr><td rowspan=1 colspan=1>XCiT-S24/16XCiT-S24/8</td><td rowspan=1 colspan=1>51.8M51.8M</td><td rowspan=1 colspan=1>44.647.1</td><td rowspan=1 colspan=1>73.8M73.8M</td><td rowspan=1 colspan=1>46.948.1</td></tr><tr><td rowspan=2 colspan=1>ResNeXt101-6475PVT-Large[701Swin-B国</td><td rowspan=2 colspan=1>86.4M65.1M=</td><td rowspan=1 colspan=1>40.242.1</td><td rowspan=2 colspan=1>-=121.0M</td><td rowspan=2 colspan=1>--48.1</td></tr><tr><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=2 colspan=1>XCiT-M24/16XCiT-M24/8</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>90.8M</td><td rowspan=1 colspan=1>46.9</td><td rowspan=1 colspan=1>108.9M</td><td rowspan=1 colspan=1>48.4</td></tr></table>",
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+        "text": "4.2 Object detection and instance segmentation ",
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+        "text": "Our XCiT models can efficiently process high-resolution images (see Figure 2). Additionally, XCiT has a better adaptability to varying image resolutions compared to ViT models (see Figure $3 )$ . These two properties make XCiT a good fit for dense prediction tasks including detection and segmentation. ",
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+        "text": "We evalutate XCiT for object detection and instance segmentation using the COCO benchmark [42] which consists of $1 1 8 \\mathrm { k }$ training and $5 \\mathrm { k }$ validation images including bounding boxes and mask labels for 80 categories. We integrate XCiT as backbone in the Mask R-CNN $\\mathbb { \\left[ \\left[ 2 9 \\right] \\right. }$ detector with FPN [43]. Since the XCiT architecture is inherently columnar, we make it FPN-compatible by extracting features from different layers, e.g., layers 4, 6, 8, and 12 for XCiT-S12. All features have a constant stride of 8 or 16 based on the patch size, and the feature resolutions are adjusted to have strides of 4, 8, 16, and 32, similar to ResNet-FPN backbones, where the downsampling is achieved by max pooling and the upsampling is obtained using a single transposed convolution layer (see the supplementary material for details). The model is trained for 36 epochs (3x schedule) using the AdamW optimizer with learning rate of $1 0 ^ { - 4 }$ , 0.05 weight decay and 16 batch size. We adopt the multiscale training and augmentation strategy of DETR $\\bar { \\mathbb { W } }$ . Our implementation is based on the mmdetection library $\\overline { { \\mathbb { B } 3 } } \\Vert$ . ",
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+        "text": "Results on COCO. In Table $\\boxed { 5 }$ we report object detection and instance segmentation results of four variants of XCiT using $1 6 \\times 1 6$ and $8 { \\times } 8$ patches. We compare to ResNets $\\bar { \\mathbb { B } } \\bar { \\mathbb { B } }$ and concurrent efficient vision transformers [44, 70, 81]. All models are trained using the 3x schedule after ImageNet-1k pretraining. Note that other results with higher absolute numbers have been achieved when pre-training on larger datasets $[ \\textcircled { 4 4 } ]$ or with longer schedules $\\mathbb { H }$ , and are therefore not directly comparable to the reported results. First, across all model sizes XCiT outperforms the convolutional ResNet [28] and ResNeXt $\\mathbb { \\left. \\overline { { \\boldsymbol { \\mathscr { Q } } \\boldsymbol { 5 } } } \\right. }$ by a large margin with either patch size. Second, we observe a similar increase in accuracy compared to PVT $\\bar { \\mathbb { I D } }$ and ViL $\\mathbb { \\left[ 8 1 \\right] }$ backbones. Finally, XCiT provides a competitive performance with Swin $[ \\overline { { 1 4 4 } } ] \\cdot \\big \\rrangle ^ { 2 }$ For relatively small models, XCiT-S12/8 outperforms its Swin-T counterpart with a decent margin. On the other hand, Swin-S provides slightly stronger results compared to XCiT-S24/8. Utilizing smaller $8 \\times 8$ patches leads to a consistent gain across all models. ",
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+        "text": "4.3 Semantic segmentation ",
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+        "text": "We further show transferability of our models with semantic segmentation experiments on the ADE20k dataset $\\textcircled { 1 8 4 } \\textcircled { 1 }$ , which consists of $2 0 \\mathrm { k }$ training and 5k validation images with labels over 150 semantic categories. We integrate our backbones in two segmentation methods: Semantic FPN [38] and UperNet $\\bar { \\textregistered }$ . We train for 80k and 160k iterations for Semantic FPN and UperNet respectively. Following $\\textcircled { | 4 4 | }$ , the models are trained using batch size 16 and an AdamW optimizer with learning rate of $6 \\times 1 0 ^ { - 5 }$ and 0.01 weight decay. We apply the same method of extracting FPN features as explained in Section $\\mathbb { H } . 2 \\big \\downarrow$ We report the performance using the standard single scale protocol (without multi-scale and flipping). Our implementation is based on the mmsegmentation library $\\mathbb { \\lVert 1 6 \\rVert }$ . ",
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+        "type": "text",
+        "text": "Results on ADE20k. We present the semantic segmentation performance using XCiT backbones in Table $6 .$ First, for Semantic FPN $\\mathbb { B }$ , XCiT provides a superior performance compared to ResNet, ResNeXt and PVT backbones using either option of patch size. Second, compared to Swin Transformers using the same UperNet decoder $\\pmb { \\Vert 7 4 \\Vert }$ , XCiT with $8 \\times 8$ patches consistently achieves a higher mIoU for different models. XCiT with $1 6 \\times 1 6$ patches provides a strong performance especially for smaller models where XCiT-S12/16 outperforms Swin-T. ",
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+        "type": "text",
+        "text": "5 Conclusion ",
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+        "text": "Contributions. We present an alternative to token self-attention which operates on the feature dimension, eliminating the need for expensive computation of quadratic attention maps. We build our XCiT models with the cross-covariance attention as its core component and demonstrate the effectiveness and generality of our models on various computer vision tasks. In particular, it exhibits a strong image classification performance on par with state-of-the-art transformer models while similarly robust to changing image resolutions as convnets. XCiT is effective as a backbone for dense prediction tasks, providing excellent performance on object detection, instance and semantic segmentation. Finally, we showed that XCiT can be a strong backbone for self-supervised learning, matching the state-of-the-art results with less compute. XCiT is a generic architecture that can readily be deployed in other research domains where self-attention has shown success. ",
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+        "type": "text",
+        "text": "Limitations. Our models enable training with smaller patches and on higher-resolution images, which leads to clear performance gains. However, for tasks like image classification this gain comes at a cost of relatively high number of FLOPs. In order to address this issue, other components, like FFN, could also be re-examined. Another point is that XCiT models seem to overfit more than their CaiT counterparts, see Table 2. They are more similar to some convnets in that respect. ",
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+    {
+        "type": "text",
+        "text": "References ",
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Multiscale vision longformer: A new vision transformer for high-resolution image encoding. arXiv preprint arXiv:2103.15358, 2021.   \n[82] Hengshuang Zhao, Jiaya Jia, and Vladlen Koltun. Exploring self-attention for image recognition. In Computer Vision and Pattern Recognition, 2020.   \n[83] Sixiao Zheng, Jiachen Lu, Hengshuang Zhao, Xiatian Zhu, Zekun Luo, Yabiao Wang, Yanwei Fu, Jianfeng Feng, Tao Xiang, Philip HS Torr, et al. Rethinking semantic segmentation from a sequence-to-sequence perspective with transformers. arXiv preprint arXiv:2012.15840, 2020.   \n[84] Bolei Zhou, Hang Zhao, Xavier Puig, Sanja Fidler, Adela Barriuso, and Antonio Torralba. Scene parsing through ade20k dataset. In Computer Vision and Pattern Recognition, 2017. ",
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parse/train/zGsRcuoR5-0/zGsRcuoR5-0.md

@@ -0,0 +1,432 @@
+# Sample Selection with Uncertainty of Losses for Learning with Noisy Labels
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 In learning with noisy labels, the sample selection approach is very popular, which   
+2 regards small-loss data as correctly labeled during training. However, losses are   
+3 generated on-the-fly based on the model being trained with noisy labels, and thus   
+4 large-loss data are likely but not certainly to be incorrect. There are actually   
+5 two possibilities of a large-loss data point: (a) it is mislabeled, and then its loss   
+6 decreases slower than other data, since deep neural networks “learn patterns first”;   
+7 (b) it belongs to an underrepresented group of data and has not been selected yet. In   
+8 this paper, we incorporate the uncertainty of losses by adopting interval estimation   
+9 instead of point estimation of losses, where lower bounds of the confidence intervals   
+10 of losses derived from distribution-free concentration inequalities, but not losses   
+11 themselves, are used for sample selection. In this way, we also give large-loss but   
+12 less selected data a try; then, we can better distinguish between the cases (a) and   
+13 (b) by seeing if the losses effectively decrease with the uncertainty after the try. As   
+14 a result, we can better explore underrepresented data that are correctly labeled but   
+15 seem to be mislabeled at first glance. Experiments demonstrate that the proposed   
+16 method is superior to baselines and robust to a broad range of label noise types.
+
+# 17 1 Introduction
+
+18 Learning with noisy labels is one of the most challenging problems in weakly-supervised learning,   
+19 since noisy labels are ubiquitous in the real world [36, 65, 40, 1, 61]. For instance, both crowdsourcing   
+20 and web crawling yield large numbers of noisy labels everyday [12]. Noisy labels can severely impair   
+21 the performance of deep neural networks with strong memorization capacities [67, 69, 42, 30].   
+22 To reduce the influence of noisy labels, a lot of approaches have been recently proposed [38, 29, 31,   
+23 68, 71, 55, 56, 46, 33, 25, 34, 47, 60, 49, 19, 17, 14]. They can be generally divided into two main   
+24 categories. The first one is to estimate the noise transition matrix [41, 44, 15, 11], which denotes the   
+25 probabilities that clean labels flip into noisy labels. However, the noise transition matrix is hard to be   
+26 estimated accurately, especially when the number of classes is large [65]. The second approach is   
+27 sample selection, which is our focus in this paper. This approach is based on selecting possibly clean   
+28 examples from a mini-batch for training [12, 62, 50, 65, 23, 50, 51]. Intuitively, if we can exploit less   
+29 noisy data for network parameter updates, the network will be more robust.   
+30 A major question in sample selection is what criteria can be used to select possibly clean examples.   
+31 At the present stage, the selection based on the small-loss criteria is the most common method, and   
+32 has been verified to be effective in many circumstances [12, 16, 65, 52, 62]. Specifically, since   
+33 deep networks learn patterns first [2], they would first memorize training data of clean labels and   
+34 then those of noisy labels with the assumption that clean labels are of the majority in a noisy class.   
+35 Small-loss examples can thus be regarded as clean examples with high probability. Therefore, in   
+36 each iteration, prior methods [12, 52] select the small-loss examples based on the predictions of the   
+37 current network for robust training.   
+38 However, such a selection procedure is debatable, since it arguably does not consider uncertainty   
+39 in selection. The uncertainty comes from two aspects. First, this procedure has uncertainty about   
+40 small-loss examples. Specifically, the procedure uses limited time intervals and only exploits the   
+41 losses provided by the current predictions. For this reason, the estimation for the noisy class posterior   
+42 is unstable [63], which causes the network predictions to be equally unstable. It thus takes huge risks   
+43 to only use losses provided by the current predictions (Figure 1, left). Once wrong selection is made,   
+44 the inferiority of accumulated errors will arise [65]. Second, this procedure has uncertainty about   
+45 large-loss examples. To be specific, deep networks learn easy examples at the beginning of training,   
+46 but ignore some clean examples with large losses. Nevertheless, such examples are always critical for   
+47 generalization. For instance, when learning with imbalanced data, distinguishing the examples with   
+48 non-dominant labels are more pivotal during training [35]. Deep networks often give large losses to   
+49 such examples (Figure 1, right). Therefore, when learning under the realistic scenes, e.g., learning   
+50 with noisy imbalanced data, prior sample selection methods cannot address such an issue well.   
+51 To relieve the above issues, we study the uncertainty of losses in the sample selection procedure to   
+52 combat noisy labels. To reduce the uncertainty of small-loss examples, we extend time intervals and   
+53 utilize the mean of training losses at different training iterations. In consideration of the bad influence   
+54 of mislabeled data on training losses, we build two robust mean estimators from the perspectives of   
+55 soft truncation and hard truncation w.r.t. the truncation level, respectively. Soft truncation makes the   
+56 mean estimation more robust by holistically changing the behavior of losses. Hard truncation makes   
+57 the mean estimation more robust by locally removing outliers from losses. To reduce the uncertainty   
+58 of large-loss examples, we encourage networks to pick the sample that has not been selected in a   
+59 conservative way. Furthermore, to address the two issues simultaneously, we derive concentration   
+60 inequalities [5] for robust mean estimation and further employ statistical confidence bounds [3] to   
+61 consider the number of times an example was selected during training.   
+62 The study of uncertainty of losses in learning with noisy labels can be justified as follows. In statistical   
+63 learning, it is known that uncertainty is related to the quality of data [48]. Philosophically, we need   
+64 variety decrease for selected data and variety search for unselected data, which share a common   
+65 objective, i.e., reduce the uncertainty of data to improve generalization [37]. This is our original   
+66 intention, since noisy labels could bring more uncertainty because of the low quality of noisy data.   
+67 Nevertheless, due to the harm of noisy labels for generalization, we need to strike a good balance   
+68 between variety decrease and search. Technically, our method is specially designed for handling   
+69 noisy labels, which robustly uses network predictions and conservatively seeks less selected examples   
+70 meanwhile to reduce the uncertainty of losses and then generalize well.   
+71 Before delving into details, we clearly emphasize our contributions in two folds. First, we reveal prior   
+72 sample selection criteria in learning with noisy labels have some potential weaknesses and discuss   
+73 them in detail. The new selection criteria are then proposed with detailed theoretical analyses. Second,   
+74 we experimentally validate the proposed method on both synthetic noisy balanced/imbalanced datasets   
+75 and real-world noisy datasets, on which it achieves superior robustness compared with the state  
+76 of-the-art methods in learning with noisy labels. The rest of the paper is organized as follows. In   
+77 Section 2, we propose our robust learning paradigm step by step. Experimental results are discussed   
+78 in Section 3. The conclusion is given in Section 4.   
+80 In this section, we first introduce the problem setting and some background (Section 2.1). Then we   
+81 discuss how to exploit training losses at different iterations (Section 2.2). Finally, we introduce the   
+82 proposed method, which exploits training losses at different iterations more robustly and encourages   
+83 networks to pick the sample that is less selected but could be correctly labeled (Section 2.3).
+
+![](images/6fa2f40d55f35e96d285db061a4328bb3951bb24a7112e73fcd38ba7e9bfc641.jpg)  
+Figure 1: Illustrations of uncertainty of losses. Experiments are conducted on the imbalanced noisy MNIST dataset. Left: uncertainty of small-loss examples. At the beginning of training (Epochs 1 and 2), due to the instability of the current prediction, the network gives a larger loss to the clean example and does not select it for updates. If we consider the mean of training losses at different epochs, the clean example can be equipped with a smaller loss and then selected for updates. Right: uncertainty of large-loss examples. Since the deep network learns easy examples at the beginning of training, it gives a large loss to clean imbalanced data with non-dominant labels, which causes such data unable to be selected and severely influence generalization.
+
+# 84 2.1 Preliminaries
+
+85 Let $\mathcal { X }$ and $\mathcal { V }$ be the input and output spaces. Consider a $k$ -class classification problem, i.e., $\mathcal { V } = [ k ]$ ,   
+86 where $[ k ] = \{ 1 , \dots , k \}$ . In learning with noisy labels, the training data are all sampled from a   
+87 corrupted distribution on $\mathcal { X } \times \mathcal { V }$ . We are given a sample with noisy labels, i.e., $\tilde { S } = \{ ( \mathbf { x } , \tilde { y } ) \}$ , where   
+88 $\tilde { y }$ is the noisy label. The aim is to learn a robust classifier that could assign clean labels to test data by   
+89 only exploiting a training sample with noisy labels.   
+90 Let $f : \mathcal { X } \to \mathbb { R } ^ { k }$ be the classifier with learnable parameters w. At the $i$ -th iteration during training,   
+91 the parameters of the classifier $f$ can be denoted as $\mathbf { w } _ { i }$ . Let $\ell : \mathbb { R } ^ { k } \times \mathcal { Y }  \mathbb { R }$ be a surrogate loss   
+92 function for $k$ -class classification. We exploit the softmax cross entropy loss in this paper. Given an   
+93 arbitrary training example $( \mathbf { x } , \tilde { y } )$ , at the $i$ -th iteration, we can obtain a loss $\ell _ { i }$ , i.e., $\ell _ { i } = \ell ( f ( \mathbf { w } _ { i } ; \mathbf { x } ) , \tilde { y } )$   
+94 Hence, until the $t$ -th iteration, we can obtain a training loss set $L _ { t }$ about the example $( \mathbf { x } , \tilde { y } )$ , i.e.,   
+95 $L _ { t } = \{ \ell _ { 1 } , \ldots , \ell _ { t } \}$ .   
+96 In this paper, we assume that the training losses in $L _ { t }$ conform to a Markov process, which is to   
+97 represent a changing system under the assumption that future states only depend on the current state   
+98 (the Markov property) [43]. More specifically, at the $i$ -th iteration, if we exploit an optimization   
+99 algorithm for parameter updates (e.g., the stochastic gradient descent algorithm [4]) and omit other   
+100 dependencies (e.g., $\tilde { S }$ ), we will have $P ( \mathbf { w } _ { i } | \mathbf { w } _ { i - 1 } , \ldots , \mathbf { w } _ { 0 } ) = P ( \mathbf { w } _ { i } | \mathbf { w } _ { i - 1 } )$ , which means that the   
+101 future state of the classifier $f$ only depends on the current state. Furthermore, given a training example   
+102 and the parameters of the classifier $f$ , we can determine the loss of the training example as discussed.   
+103 Therefore, the training losses in $L _ { t }$ will also conform to a Markov process.
+
+# 2.2 Extended Time Intervals
+
+105 As limited time interval cannot address the instability issue of the estimation for the noisy class   
+106 posterior well [42], we extend time intervals and exploit the training losses at different training   
+107 iterations for sample selection. One straightforward idea is to use the mean of training losses at   
+108 different training iterations. Hence, the selection criterion could be
+
+$$
+\tilde { \mu } = \frac { 1 } { t } \sum _ { i = 1 } ^ { t } \ell _ { i } .
+$$
+
+109 It is intuitive and reasonable to use such a selection criterion for sample selection, since the operation   
+110 of averaging can mitigate the risks caused by the unstable estimation for the noisy class posterior,   
+111 following better generalization. Nevertheless, such a method could arguably achieve suboptimal   
+112 classification performance for learning with noisy labels. The main reason is that, due to the great   
+113 harm of mislabeled data, part of training losses are with too large uncertainty and could be seen as   
+114 outliers. Therefore, it could be biased to use the mean of training losses consisting of such outliers   
+115 [10], which further influences sample selection. More evaluations for our claims are provided in   
+116 Section 3.
+
+# 117 2.3 Robust Mean Estimation and Conservative Search
+
+118 We extend time intervals and meanwhile exploit the training losses at different training iterations more   
+119 robustly. Specifically, we build two robust mean estimators from the perspectives of soft truncation   
+120 and hard truncation [7]. Note that for specific tasks, it is feasible to decide the types of robust mean   
+121 estimation with statistical tests based on some assumptions [8]. We leave the analysis as future work.   
+122 Two distribution-free robust mean estimators are introduced as follows.   
+123 Soft truncation. We extend a classical M-estimator from [7] and exploit the widest possible choice of   
+124 the influence function. More specifically, give a random variable $X$ , let us consider a non-decreasing
+
+125 influence function $\psi : \mathbb { R } \to \mathbb { R }$ such that
+
+$$
+\psi ( X ) = \log ( 1 + X + X ^ { 2 } / 2 ) , X \geq 0 .
+$$
+
+126 The choice of $\psi$ is inspired by the Taylor expansion of the exponential function, which can make the   
+127 estimation results more robust by reducing the side effect of extremum holistically. The illustration   
+128 for this influence function is provided in Appendix A.1. For our task, given the observations on   
+129 training losses, i.e., $L _ { t } = \{ \ell _ { 1 } , \ldots , \ell _ { t } \}$ , we estimate the mean robustly as follows:
+
+$$
+\tilde { \mu } _ { s } = \frac { 1 } { t } \sum _ { i = 1 } ^ { t } \psi ( \ell _ { i } ) .
+$$
+
+130 We term the above robust mean estimator (3) the soft estimator.
+
+131 Hard truncation. We propose a new robust mean estimator based on hard truncation. Specifically,   
+132 given the observations on training losses $L _ { t }$ , we first exploit the $\mathbf { K }$ -nearest neighbor (KNN) algorithm   
+133 [27] to remove some underlying outliers in $L _ { t }$ . The number of outliers is denoted by $t _ { \mathrm { o } } ( t _ { \mathrm { o } } < t )$ , which   
+134 can be adaptively determined as discussed in [70]. Note that we can also employ other algorithms,   
+135 e.g., principal component analysis [45] and the local outlier factor [6], to identify underlying outliers   
+136 in $L _ { t }$ . The main reason we employ KNN is because of its relatively low computation costs [70].   
+137 The truncated loss observations on training losses are denoted by $L _ { t - t _ { \mathrm { o } } }$ . We then utilize $L _ { t - t _ { \mathrm { o } } }$ for   
+138 the mean estimation. As the potential outliers are removed with high probability, the robustness of   
+139 the estimation results will be enhanced. We denote such an estimated mean as ${ \tilde { \mu } } _ { h }$ . We have
+
+$$
+\tilde { \mu } _ { h } = \frac { 1 } { t - t _ { \mathrm { o } } } \sum _ { \ell _ { i } \in L _ { t - t _ { 0 } } } \ell _ { i } .
+$$
+
+140 The corresponding estimator (4) is termed the hard estimator.
+
+141 We derive concentration inequalities for the soft and hard estimators respectively. The search strategy   
+142 for less selected examples and overall selection criterion are then provided. Note that we do not need   
+143 to explicitly quantify the mean of training losses. We only need to sort the training examples based   
+144 on the proposed selection criterion and then use the selected examples for robust training.
+
+145 Theorem 1. Let $Z _ { n } = \{ z _ { 1 } , \cdots , z _ { n } \}$ be an observation set with mean $\mu _ { z }$ and variance $\sigma ^ { 2 }$ . By exploiting the non-decreasing influence function 46 $\psi ( z ) = \log ( 1 + z + z ^ { 2 } / 2 )$ . For any $\epsilon > 0$ , we have
+
+$$
+\left| \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \psi ( z _ { i } ) - \mu _ { z } \right| \leq \frac { \sigma ^ { 2 } ( n + \frac { \sigma ^ { 2 } \log ( \epsilon ^ { - 1 } ) } { n ^ { 2 } } ) } { n - \sigma ^ { 2 } } ,
+$$
+
+147 with probability at least $1 - 2 \epsilon$
+
+148 Proof can be found in Appendix A.1.
+
+149 Theorem 2. Let $Z _ { n } = \{ z _ { 1 } , \ldots , z _ { n } \}$ be a (not necessarily time homogeneous) Markov chain with   
+150 mean $\mu _ { z }$ , taking values in a Polish state space $\Lambda _ { 1 } \times \ldots \times \Lambda _ { n }$ , and with a minimal mixing time $\tau _ { \mathrm { m i n } }$ .   
+151 The truncated set with hard truncation is denoted by $Z _ { n _ { o } }$ , with $n _ { o } < n$ . If $| z _ { i } |$ is upper bounded by $Z$ .   
+152 For any $\epsilon _ { 1 } > 0$ and $\epsilon _ { 2 } > 0$ , we have
+
+$$
+\left| \frac { 1 } { n - n _ { o } } \sum _ { z _ { i } \in Z _ { n } \setminus Z _ { n _ { o } } } - \mu _ { z } \right| \leq \frac { 1 } { n - n _ { o } } \left( 2 Z \sqrt { 2 \tau _ { \mathrm { m i n } } \log \frac { 2 } { \epsilon _ { 1 } } } + \frac { 2 Z n _ { o } } { n } \sqrt { 2 \tau _ { \mathrm { m i n } } \log \frac { 2 n } { \epsilon _ { 2 } } } \right) ,
+$$
+
+153 with probability at least $1 - \epsilon _ { 1 } - \epsilon _ { 2 }$
+
+154 Proof can be found in Appendix A.2. For our task, let the training loss be upper-bounded by $L$ . The   
+155 value of $L$ can be determined easily by training networks on noisy datasets and observing the loss   
+156 distribution [1].   
+157 Conservative search and selection criteria. In this paper, we will use the concentration inequalities   
+158 (5) and (6) to present conservative search and the overall sample selection criterion. Specifically,   
+159 we exploit their lower bounds and consider the selected number of examples during training. The   
+160 selection of the examples that are less selected is encouraged.
+
+# Algorithm 1 CNLCU Algorithm.
+
+1: Input $\theta _ { 1 }$ and $\theta _ { 2 }$ , learning rate $\eta$ , fixed $\tau$ , epoch $T _ { k }$ and $T _ { \mathrm { m a x } }$ , iteration $t _ { \mathrm { m a x } }$ ; for $T = 1 , 2 , \dots , T _ { \mathrm { m a x } }$ do
+
+2: Shuffle training dataset $\tilde { S }$ ;
+
+# end
+
+8: Update $\begin{array} { r } { R ( T ) = 1 - \operatorname* { m i n } \left\{ \frac { T } { T _ { k } } \tau , \tau \right\} } \end{array}$
+
+# end
+
+9: Output $\theta _ { 1 }$ and $\theta _ { 2 }$
+
+Denote the number of times one example was selected by 161 $n _ { t } ( n _ { t } ~ \leq ~ t )$ . Let $\begin{array} { r } { \epsilon = \frac { 1 } { 2 t } } \end{array}$ . For the 162 circumstance with soft truncation, the selection criterion is
+
+$$
+\ell _ { s } ^ { \star } = \tilde { \mu } _ { s } - \frac { \sigma ^ { 2 } ( t + \frac { \sigma ^ { 2 } \log ( 2 t ) } { t ^ { 2 } } ) } { n _ { t } - \sigma ^ { 2 } } .
+$$
+
+Let 163 $\begin{array} { r } { \epsilon _ { 1 } = \epsilon _ { 2 } = \frac { 1 } { 2 t } } \end{array}$ , for the situation with hard truncation, by rewriting (6), the selection criterion is
+
+$$
+\ell _ { h } ^ { \star } = \tilde { \mu } _ { h } - \frac { 2 \sqrt { 2 \tau _ { \mathrm { m i n } } } L ( t + \sqrt { 2 } t _ { \mathrm { o } } ) } { ( t - t _ { \mathrm { o } } ) \sqrt { t } } \sqrt { \frac { \log ( 4 t ) } { n _ { t } } } .
+$$
+
+164 Note that we directly replace $t$ with $n _ { t }$ . If an example is rarely selected during training, $n _ { t }$ will be far   
+165 less than $n$ , which causes the lower bounds to change drastically. Hence, we do not use the mean of   
+166 all training losses, but use the mean of training losses in fixed-length time intervals. More details   
+167 about this can be checked in Section 3.   
+168 For the selection criteria (7) and (8), we can see that they consist of two terms and have one term   
+169 with a minus sign. The first term in Eq. (7) (or Eq. (8)) is to reduce the uncertainty of small-loss   
+170 examples, where we use robust mean estimation on training losses. The second term, i.e., the   
+171 statistical confidence bound, is to encourage the network to choose the less selected examples (with a   
+172 small $n _ { t }$ ). The two terms are constraining and balanced with $\sigma ^ { 2 }$ or $\tau _ { \mathrm { m i n } }$ . To avoid introducing strong   
+173 assumptions on the underlying distribution of losses [8], we tune $\sigma$ and $\tau _ { \mathrm { m i n } }$ with a noisy validation   
+174 set. For the mislabeled data, although the model has high uncertainties on them (i.e., a small $n _ { t }$ )   
+175 and tends to pick them, the overfitting to the mislabeled data is harmful. Also, the mislabeled data   
+176 and clean data are rather hard to distinguish in some cases as discussed. Thus, we should search   
+177 underlying clean data in a conservative way. In this paper, we initialize $\sigma$ and $\tau _ { \mathrm { m i n } }$ with small values.   
+178 This way can reduce the adverse effects of mislabeled data and meanwhile select the clean examples   
+179 with large losses, which helps generalize. More evaluations will be presented in Section 3.   
+180 The overall procedure of the proposed method, which combats noisy labels by concerning uncertainty   
+181 (CNLCU), is provided in Algorithm 1. CNLCU works in a mini-batch manner since all deep learning   
+182 training methods are based on stochastic gradient descent. Following [12], we exploit two networks   
+183 with parameters $\theta _ { 1 }$ and $\theta _ { 2 }$ respectively to teach each other. Specifically, when a mini-batch $\bar { S }$ is   
+184 formed (Step 3), we let two networks select a small proportion of examples in this mini-batch with   
+185 Eq. (7) or (8) (Step 4 and Step 5). The number of instances is controlled by the function $R ( T )$ , and   
+186 two networks only select $R ( T )$ percentage of examples out of the mini-batch. The value of $R ( T )$   
+187 should be larger at the beginning of training, and be smaller when the number of epochs goes large,   
+188 which can make better use of memorization effects of deep networks [12] for sample selection. Then,   
+189 the selected instances are fed into its peer network for parameter updates (Step 6 and Step 7).
+
+# 190 3 Experiments
+
+191 In this section, we evaluate the robustness of our proposed method to noisy labels with comprehensive   
+192 experiments on the synthetic balanced noisy datasets (Section 3.1), synthetic imbalanced noisy   
+193 datasets (Section 3.2), and real-world noisy dataset (Section 3.3).
+
+Datasets. We verify the effectiveness of our method on the manually corrupted version of the following datasets: MNIST [22], $F$ -MNIST [58], CIFAR-10 [21], and CIFAR-100 [21], because these datasets are popularly used for the evaluation of learning with noisy labels in the literature [12, 65, 54, 23]. The four datasets are class-balanced. The important statistics of the used synthetic datasets are summarized in Appendix B.1.
+
+Generating noisy labels. We consider broad types of label noise: (1). Symmetric noise (abbreviated as Sym.) [53, 31, 26]. (2) Asymmetric noise (abbreviated as Asym.) [32, 57, 52]. (3) Pairflip noise (abbreviated as Pair.) [12, 65, 71]. (4). Tridiagonal noise (abbreviated as Trid.) [68]. (5). Instance noise (abbreviated as Ins.) [9, 56]. The noise rate is set to $20 \%$ and $40 \%$ to ensure clean labels are diagonally dominant [32]. More details about above noise are provided in Appendix B.1. We leave out $10 \%$ of noisy training examples as a validation set.
+
+Baselines. We compare the proposed method (Algorithm 1) with following methods which focus on sample selection, and implement all methods with default parameters by PyTorch, and conduct all the experiments on NVIDIA Titan Xp GPUs. (1). S2E [62], which properly controls the sample selection process so that deep networks can better benefit from the memorization effects. (2). MentorNet [16], which learns a curriculum to filter out noisy data. We use self-paced MentorNet in this paper. (3). Co-teaching [12], which trains two networks simultaneously and cross-updates parameters of peer networks. (4). SIGUA [13], which exploits stochastic integrated gradient underweighted ascent to handle noisy labels. We use self-teaching SIGUA in this paper. (5). JoCor [52], which reduces the diversity of networks to improve robustness. Other types of baselines such as adding regularization are provided in Appendix B.2. Note that we do not compare the proposed method with some stateof-the-art methods, e.g., SELF [39] and DivideMix [24]. It is because their proposed methods are aggregations of multiple techniques. We mainly focus on sample selectionin in learning with noisy labels. Therefore, the comparison is not fair. Here, we term our methods with soft truncation and hard truncation as CNLCU-S and CNLCU-H respectively.
+
+Network structure and optimizer. For MNIST, $F$ -MNIST, and CIFAR-10, we use a 9-layer CNN structure from [12]. Due to the limited space, the experimental details on CIFAR-100 are provided in Appendix B.3. All network structures we used here are standard test beds for weakly-supervised learning. For all experiments, the Adam optimizer [20] (momentum $_ { 1 = 0 . 9 }$ ) is used with an initial learning rate of 0.001, and the batch size is set to 128 and we run 200 epochs. We linearly decay learning rate to zero from 80 to 200 epochs as did in [12]. We take two networks with the same architecture but different initializations as two classifiers as did in [12, 65, 52], since even with the same network and optimization method, different initializations can lead to different local optimal [12]. The details of network structures can be checked in Appendix C.
+
+For the hyper-parameters $\sigma ^ { 2 }$ and $\tau _ { \mathrm { m i n } }$ , we determine them in the range $\{ 1 0 ^ { - 1 } , 1 0 ^ { - 2 } , 1 0 ^ { - 3 } , 1 0 ^ { - 4 } \}$ with a noisy validation set. Here, we assume the noise level $\tau$ is known and set $R ( T ) = 1 -$ $\operatorname* { m i n } \{ \frac { T } { T _ { k } } \tau , \tau \}$ with ${ \mathit { T } } _ { k } { = } 1 0$ . If $\tau$ is not known in advanced, it can be inferred using validation sets [29, 66]. As for performance measurement, we use test accuracy, i.e., test accuracy $=$ (# of correct prediction) / (# of testing). All experiments are repeated five times. We report the mean and standard deviation of experimental results.
+
+Experimental results. The experimental results about test accuracy are provided in Table 1, 2, and 3. Specifically, for MNIST, as can be seen, our proposed methods, i.e., CNLCU-S and CNLCU-H, produce the best results in the vast majority of cases. In some cases such as asymmetric noise, the baseline S2E outperforms ours, which benefits the accurate estimation for the number of selected small-loss examples. For $F$ -MNIST, the training data becomes complicated. S2E cannot achieve the accurate estimation in such situation and thus has no great performance like it got on MNIST. Our methods achieve varying degrees of lead over baselines. For CIFAR-10, our methods once again outperforms all the baseline methods. Although some baseline, e.g., Co-teaching, can work well in some cases, experimental results show that it cannot handle various noise types. In contrast, the proposed methods achieve superior robustness against broad noise types. The results mean that our methods can be better applied to actual scenarios, where the noise is diversiform.
+
+46 Ablation study. We first conduct the ablation study to analyze the sensitivity of the length of time intervals. In order to avoid too dense figures, we exploit MNIST and $F$ -MNIST with the mentioned 8 noise settings as representative examples. For CNLCU-S, the length of time intervals is chosen in
+
+<table><tr><td>Noise type</td><td colspan="2">Sym.</td><td colspan="2">Asym.</td><td colspan="2">Pair.</td><td colspan="2">Trid.</td><td colspan="2">Ins.</td></tr><tr><td>Method/Noise ratio</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td></tr><tr><td rowspan="2">S2E</td><td>98.46</td><td>95.62</td><td>99.05</td><td>98.45</td><td>98.56</td><td>94.22</td><td>99.02</td><td>97.23</td><td>97.93</td><td>94.02</td></tr><tr><td>±0.06</td><td>±0.91</td><td>±0.02</td><td>±0.26</td><td>±0.32</td><td>±0.79</td><td>±0.09</td><td>±1.26</td><td>±1.26</td><td>±2.39</td></tr><tr><td rowspan="2">MentorNet</td><td>95.04</td><td>92.08</td><td>96.32</td><td>90.86</td><td>93.19</td><td>90.93</td><td>96.42</td><td>93.28</td><td>94.65</td><td>90.11</td></tr><tr><td>±0.03</td><td>±0.42</td><td>±0.17</td><td>±0.97</td><td>±0.17</td><td>±1.54</td><td>±0.09</td><td>±1.37</td><td>±0.73</td><td>±1.26</td></tr><tr><td rowspan="2">Co-teaching</td><td>97.53</td><td>95.62</td><td>98.25</td><td>95.08</td><td>96.05</td><td>94.16</td><td>98.05</td><td>96.18</td><td>97.96</td><td>95.02</td></tr><tr><td>±0.12</td><td>±0.30</td><td>±0.08</td><td>±0.43</td><td>±0.96</td><td>±1.37</td><td>±0.06</td><td>±0.85</td><td>±0.09</td><td>±0.39</td></tr><tr><td rowspan="2">SIGUA</td><td>92.31</td><td>91.88</td><td>93.96</td><td>62.59</td><td>93.77</td><td>86.22</td><td>94.92</td><td>83.46</td><td>92.90</td><td>86.34</td></tr><tr><td>±1.10</td><td>±0.92</td><td>±0.82</td><td>±0.15</td><td>±1.40</td><td>±1.75</td><td>±0.83</td><td>±2.98</td><td>±1.82</td><td>±3.51</td></tr><tr><td rowspan="2">JoCor</td><td>98.42</td><td>98.04</td><td>98.05</td><td>94.55</td><td>98.01</td><td>96.85</td><td>98.45</td><td>96.98</td><td>98.62</td><td>96.07</td></tr><tr><td>±0.14</td><td>±0.07</td><td>±0.37</td><td>±1.08</td><td>±0.19</td><td>±0.43</td><td>±0.17</td><td>±0.25</td><td>±0.06</td><td>±0.31</td></tr><tr><td rowspan="2">CNLCU-S</td><td>98.82</td><td>98.31</td><td>98.93</td><td>97.67</td><td>98.86</td><td>97.71</td><td>99.09</td><td>98.02</td><td>98.77</td><td>97.78</td></tr><tr><td>±0.03</td><td>士0.05</td><td>±0.06</td><td>±0.22</td><td>±0.06</td><td>±0.64</td><td>±0.04</td><td>±0.17</td><td>±0.08</td><td>±0.25</td></tr><tr><td rowspan="2">CNLCU-H</td><td>98.70</td><td>98.24</td><td>99.01 </td><td>98.01</td><td>98.44</td><td>97.37</td><td>98.89</td><td>97.92</td><td>98.74</td><td>97.42</td></tr><tr><td>±0.06</td><td>±0.06</td><td>±0.04</td><td>±0.03</td><td>±0.19</td><td>±0.32</td><td>±0.15</td><td>±0.05</td><td>±0.16</td><td>±0.39</td></tr></table>
+
+Table 1: Test accuracy $( \% )$ on MNIST over the last ten epochs. The best two results are in bold.
+
+<table><tr><td>Noise type</td><td colspan="2">Sym.</td><td colspan="2">Asym.</td><td colspan="2">Pair.</td><td colspan="2">Trid.</td><td colspan="2">Ins.</td></tr><tr><td>Method/Noise ratio</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td></tr><tr><td rowspan="2">S2E</td><td>89.99</td><td>75.32</td><td>89.00</td><td>81.03</td><td>88.66</td><td>67.09</td><td>89.53</td><td>77.29</td><td>88.65</td><td>79.35</td></tr><tr><td>±2.07</td><td>±5.84</td><td>±0.95</td><td>±1.93</td><td>±1.32</td><td>±4.03</td><td>±2.63</td><td>±3.97</td><td>±2.12</td><td>±3.04</td></tr><tr><td rowspan="2">MentorNet</td><td>90.37</td><td>86.53</td><td>89.69</td><td>67.21</td><td>87.92</td><td>83.70</td><td>88.74</td><td>85.63</td><td>87.52</td><td>83.27</td></tr><tr><td>±0.17</td><td>士0.65</td><td>±0.19</td><td>±2.94</td><td>±1.08</td><td>±0.49</td><td>±0.33</td><td>±0.59</td><td>±0.15</td><td>±1.42</td></tr><tr><td rowspan="2">Co-teaching</td><td>91.48</td><td>88.80</td><td>91.03</td><td>68.07</td><td>90.77</td><td>86.91</td><td>91.24</td><td>89.18</td><td>90.60</td><td>87.90</td></tr><tr><td>±0.10</td><td>±0.29</td><td>±0.14</td><td>±4.58</td><td>±0.23</td><td>±0.71</td><td>±0.11</td><td>±0.36</td><td>±0.12</td><td>±0.45</td></tr><tr><td rowspan="2">SIGUA</td><td>87.64</td><td>87.23</td><td>76.97</td><td>45.96</td><td>69.59</td><td>68.93</td><td>79.97</td><td>76.14</td><td>76.92</td><td>74.89</td></tr><tr><td>±1.29</td><td>±0.72</td><td>±2.59</td><td>±3.40</td><td>±5.75</td><td>±2.80</td><td>±3.23</td><td>±4.24</td><td>±5.09</td><td>士4.84</td></tr><tr><td rowspan="2">JoCor</td><td>91.97</td><td>89.96</td><td>90.95</td><td>79.79</td><td>91.52</td><td>87.40</td><td>92.01</td><td>89.42</td><td>91.43</td><td>87.59</td></tr><tr><td>±0.13</td><td>±0.19</td><td>±0.21</td><td>±2.39</td><td>士0.24</td><td>±0.58</td><td>±0.17</td><td>士0.33</td><td>±0.71</td><td>±0.94</td></tr><tr><td rowspan="2">CNLCU-S</td><td>92.37</td><td>91.45</td><td>92.57</td><td>83.14</td><td>92.04</td><td>88.20</td><td>92.24</td><td>90.08</td><td>91.69</td><td>89.02</td></tr><tr><td>±0.15</td><td>±0.28</td><td>±0.15</td><td>±1.77</td><td>±0.26</td><td>±0.44</td><td>±0.17</td><td>±0.34</td><td>±0.10</td><td>±1.02</td></tr><tr><td rowspan="2">CNLCU-H</td><td>92.42</td><td>91.60</td><td>92.60</td><td>82.69</td><td>91.70</td><td>87.70</td><td>92.33</td><td>90.22</td><td>91.50</td><td>88.79</td></tr><tr><td>±0.21</td><td>±0.19</td><td>±0.18</td><td>±0.43</td><td>±0.18</td><td>±0.69</td><td>±0.26</td><td>±0.71</td><td>±0.21</td><td>±1.22</td></tr></table>
+
+Table 2: Test accuracy on F-MNIST over the last ten epochs. The best two results are in bold.
+
+249 the range from 3 to 8. For CNLCU-H, the length of time intervals is chosen in the range from 10 to   
+250 15. Note that the reason for their different lengths is that their different mechanisms. Specifically,   
+251 CNLCU-S holistically changes the behavior of losses, but does not remove any loss from the loss set.   
+252 We thus do not need too long length of time intervals. As a comparison, CNLCU-H needs to remove   
+253 some outliers from the loss set as discussed. The length should be longer to guarantee the number of   
+254 examples available for robust mean estimation. The experimental results are provided in Appendix   
+255 B.4, which show the proposed CNLCU-S and CNLCU-H are robust to the choices of the length of   
+256 time intervals. Such robustness to hyperparameters means our methods can be applied in practice and   
+257 does not need too much effect to tune the hyperparameters.   
+258 Furthermore, since our methods concern uncertainty from two aspects, i.e., the uncertainty from both   
+259 small-loss and large-loss examples, we conduct experiments to analyze each part of our methods.   
+260 Also, as mentioned, we compare robust mean estimation with non-robust mean estimation when   
+261 learning with noisy labels. More details are provided in Appendix B.4.
+
+# 3.2 Experiments on Synthetic Imbalanced Noisy Datasets
+
+Experimental setup. We exploit MNIST and $F$ -MNIST. For these two datasets, we reduce the number of training examples along with the labels from $\mathbf { \bar { \theta } } ^ { 6 } 0 ^ { 9 }$ to $" 4 > "$ to $1 \%$ of previous numbers. We term such synthetic imbalanced noisy datasets as IM-MNIST and IM-F-MNIST respectively. This setting aims to simulate the extremely imbalanced circumstance, which is common in practice. Moreover, we exploit asymmetric noise, since these types of noise can produce more imbalanced case [41, 32]. Other settings such as the network structure and optimizer are the same as those in experiments on synthetic balanced noisy datasets.
+
+<table><tr><td>Noise type</td><td colspan="2">Sym.</td><td colspan="2">Asym.</td><td colspan="2">Pair.</td><td colspan="2">Trid.</td><td colspan="2">Ins.</td></tr><tr><td>Method/Noise ratio</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td></tr><tr><td rowspan="2">S2E</td><td>80.78</td><td>69.72</td><td>84.03</td><td>75.04</td><td>81.72</td><td>61.50</td><td>81.44</td><td>64.39</td><td>79.89</td><td>62.42</td></tr><tr><td>±0.88</td><td>±3.94</td><td>±1.01</td><td>±1.24</td><td>±0.93</td><td>±4.63</td><td>±0.59</td><td>±2.82</td><td>±0.26</td><td>±3.11</td></tr><tr><td rowspan="2">MentorNet</td><td>80.92</td><td>74.67</td><td>80.37</td><td>71.69</td><td>77.98</td><td>69.39</td><td>78.02</td><td>71.56</td><td>77.02</td><td>68.17</td></tr><tr><td>±0.48</td><td>±1.17</td><td>±0.26</td><td>±1.06</td><td>±0.31</td><td>±1.73</td><td>±0.29</td><td>±0.93</td><td>±0.71</td><td>±2.52</td></tr><tr><td rowspan="2">Co-teaching</td><td>82.35</td><td>77.96</td><td>83.87</td><td>73.43</td><td>80.94</td><td>72.81</td><td>81.17</td><td>74.37</td><td>79.92</td><td>73.29</td></tr><tr><td>±0.16</td><td>±0.39</td><td>±0.24</td><td>±0.62</td><td>±0.46</td><td>±0.92</td><td>±0.60</td><td>士0.64</td><td>±0.57</td><td>±1.62</td></tr><tr><td rowspan="2">SIGUA</td><td>78.19</td><td>77.67</td><td>75.14</td><td>52.76</td><td>74.41</td><td>61.91</td><td>75.75</td><td>74.05</td><td>74.34</td><td>67.98</td></tr><tr><td>±0.22</td><td>±0.41</td><td>±0.36</td><td>±0.68</td><td>±0.81</td><td>±5.27</td><td>±0.53</td><td>±0.41</td><td>±0.39</td><td>±1.34</td></tr><tr><td rowspan="2">JoCor</td><td>80.96</td><td>76.65</td><td>81.39</td><td>69.92</td><td>80.33</td><td>71.62</td><td>79.03</td><td>74.33</td><td>78.21</td><td>71.46</td></tr><tr><td>±0.25</td><td>±0.43</td><td>±0.74</td><td>±1.63</td><td>±0.20</td><td>±1.05</td><td>±0.13</td><td>±1.09</td><td>±0.34</td><td>±1.27</td></tr><tr><td rowspan="2">CNLCU-S</td><td>83.03</td><td>78.25</td><td>85.06</td><td>75.34</td><td>83.16</td><td>73.19</td><td>82.77</td><td>74.37</td><td>82.03</td><td>73.67</td></tr><tr><td>±0.21</td><td>±0.70</td><td>±0.17</td><td>±0.32</td><td>±0.25</td><td>±1.25</td><td>±0.32</td><td>±1.37</td><td>士0.37</td><td>±1.09</td></tr><tr><td rowspan="2">CNLCU-H</td><td>83.03</td><td>78.33</td><td>84.95</td><td>75.29</td><td>83.39</td><td>73.40</td><td>82.52</td><td>74.79</td><td>-81.93</td><td>73.58</td></tr><tr><td>±0.47</td><td>±0.50</td><td>±0.27</td><td>±0.80</td><td>±0.68</td><td>±1.53</td><td>±0.71</td><td>±1.13</td><td>±0.25</td><td>±1.39</td></tr></table>
+
+Table 3: Test accuracy $( \%$ ) on CIFAR-10 over the last ten epochs. The best two results are in bold.
+
+270 As for performance measurements, we use test accuracy. In addition, we exploit the selected ratio of   
+271 training examples with the imbalanced classes, i.e., selected ratio=(# of selected imbalanced labels /   
+272 # of all selected labels). Intuitively, a higher selected ratio means the proposed method can make   
+273 better use of training examples with the imbalanced classes, following better generalization [18].
+
+Experimental results. The test accuracy achieved on IM-MNIST and IM-F-MNIST is presented in Figure 2. Recall the experimental results in Table 1 and 2, we can see that the imbalanced issue is catastrophic to the sample selection approach when learning with noisy labels. For IM-MNIST, as can be seen, all the baselines have serious overfitting in the early stages of training. The curves of test accuracy drop dramatically. As a comparison, the proposed CNLCU-S and CNLCU-H can give a try to large-loss but less selected data which are possible to be clean but equipped with imbalanced labels. Therefore, our methods always outperform baselines clearly. In the case of Asym. $10 \%$ , our methods achieve nearly $30 \%$ lead over baselines. For IM-F-MNIST, we can also see that our methods perform well and always achieve about $5 \%$ lead over all the baselines. Note that due to the huge challenge of this task, some baseline, e.g., S2E, has a large error bar. In addition, the baseline SIGUA performs badly. It is because SIGUA exploits stochastic integrated gradient underweighted ascent on large-loss examples, which makes the examples with imbalanced classes more difficult to be selected than them in other sample selection methods.
+
+The selected ratio achieved on IM-MNIST and IM-F-MNIST is presented in Table 4. The results explain well why our methods perform better on synthetic imbalanced noisy datasets, i.e., our methods can make better use of training examples with the imbalanced classes. Note that since we give a try to large-loss but less selected data in a conservative way, the selected ratio is still far away from the class prior probability on the test set, i.e., $10 \%$ . However, a little improvement of the selection ratio can bring a considerable improvement of test accuracy. These results tell us that, in the sample selection approach when learning with noisy labels, improving the selected ratio of training examples with the imbalanced classes is challenging but promising for generalization. This practical problem deserves to be studied in depth.
+
+# 3.3 Experiments on Real-world Noisy Datasets
+
+Experimental setup. To verify the efficacy of our methods in the real-world scenario, we conduct experiments on the noisy dataset Clothing1M [59]. Specifically, for experiments on Clothing1M, we use the 1M images with noisy labels for training and 10k clean data for test respectively. Note that we do not use the $5 0 \mathrm { k }$ clean training data in all the experiments. For preprocessing, we resize the image to $2 5 6 \times 2 5 6$ , crop the middle $2 2 4 \times 2 2 4$ as input, and perform normalization. The experiments on Clothing1M are performed once due to the huge computational cost. We leave $10 \%$ noisy training data as a validation set for model selection. Note that we do not exploit the resampling trick during training [24]. Here, Best denotes the test accuracy of the epoch where the validation accuracy was optimal. Last denotes test accuracy of the last epoch. For the experiments on Clothing1M, we use a ResNet-18 pretrained on ImageNet as did in [52]. We also use the Adam optimizer and set the batch size to 64. During the training stage, we run 15 epochs in total and set the learning rate $8 \times 1 0 ^ { - 4 }$ , $5 \times 1 0 ^ { - 4 }$ , and $5 \times 1 0 ^ { - 5 }$ for 5 epochs each.
+
+<table><tr><td>Dataset</td><td colspan="4">IM-MNIST</td><td colspan="4">IM-F-MNIST</td></tr><tr><td>Method/Noiseratio</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td></tr><tr><td>S2E</td><td>0.13 ±0.12</td><td>0.11 ±0.05</td><td>0.09 ±0.02</td><td>0.05 ±0.01</td><td>0.13 ±0.04</td><td>0.17 ±0.03</td><td>0.16 ±0.02</td><td>0.12 ±0.04</td></tr><tr><td>MentorNet</td><td>0.10 ±0.02</td><td>0.15 ±0.02</td><td>0.12 ±0.03</td><td>0.13 ±0.02</td><td>0.12 ±0.01</td><td>0.15 ±0.03</td><td>0.09 ±0.01</td><td>0.14 ±0.02</td></tr><tr><td>Co-teaching</td><td>0.09 ±0.03</td><td>0.07 ±0.02</td><td>0.05 ±0.01</td><td>0.12 ±0.01</td><td>0.17 ±0.05</td><td>0.04 ±0.00</td><td>0.13 ±0.04</td><td>0.07 ±0.01</td></tr><tr><td>SIGUA</td><td>0.04 ±0.00</td><td>0.04 ±0.00</td><td>0.01 ±0.00</td><td>0.02 ±0.00</td><td>0.03 ±0.00</td><td>0.02 ±0.00</td><td>0.04 ±0.00</td><td>0.00 ±0.00</td></tr><tr><td>JoCor</td><td>0.11 ±0.04</td><td>0.08 ±0.01</td><td>0.07 ±0.03</td><td>0.06 ±0.02</td><td>0.05 ±0.01</td><td>0.13 ±0.04</td><td>0.13 ±0.03</td><td>0.07 ±0.02</td></tr><tr><td>CNLCU-S</td><td>0.60 ±0.11</td><td>0.37 ±0.09</td><td>0.39 ±0.04</td><td>0.38 ±0.06</td><td>0.35 ±0.03</td><td>0.39 ±0.04</td><td>0.36 ±0.03</td><td>0.30 ±0.02</td></tr><tr><td>CNLCU-H</td><td>0.57 ±0.13</td><td>0.32 ±0.01</td><td>0.37 ±0.07</td><td>0.32&#x27; ±0.05</td><td>0.34 ±0.02</td><td>0.35 ±0.06</td><td>0.32</td><td>&#x27;0.28&#x27; ±0.03</td></tr></table>
+
+Table 4: Selected ratio ( $\overline { { \mathcal { \vert } } }$ ) on IM-MNIST and IM-F-MNIST. The best two results are in bold.
+
+![](images/345618b16a2a497cb1770a7a6b812aef02c9db41ecbd901f24eafd6809680d9e.jpg)  
+Figure 2: Test accuracy vs. number of epochs on IM-MNIST and IM- $F$ -MNIST. The error bar for standard deviation in each figure has been shaded.
+
+309 Experimental results. The results on Clothing1M are provided in Table 5. Specifically, the proposed   
+310 methods get better results than state-of-the-art methods on Best, which achieve an improvement of   
+311 $+ 1 . 2 8 \%$ and $+ 0 . 9 9 \%$ over the best baseline JoCor. Likewise, the proposed methods outperform all the   
+312 baselines on Last. We achieve an improvement of $+ 1 . 0 1 \%$ and $+ 0 . 5 4 \%$ over JoCor. Note that the   
+313 results are a bit lower than some state-of-art methods, e.g., [64] and [46], because of the following   
+314 reasons. (1). We follow [52] and use ResNet-18 as a backbone. The state-of-art methods [64, 46]   
+315 use ResNet-50 as a backbone. Our aim is to make the experimental results directly comparable with   
+316 previous papers [52] in the same area. (2). We only focus on the sample selection approach and do   
+317 not employ other advanced techniques, e.g., introducing the prior distribution [46] and combining   
+semi-supervised learning [24, 39, 28].
+
+<table><tr><td>Methods</td><td>S2E</td><td>MentorNet</td><td>Co-teaching</td><td>SIGUA</td><td>JoCor</td><td>CNLCU-S</td><td>CNLCU-H</td></tr><tr><td>Best</td><td>67.34</td><td>68.36</td><td>69.37</td><td>62.89</td><td>70.09</td><td>71.37</td><td>71.08</td></tr><tr><td>Last</td><td>65.90</td><td>67.42</td><td>68.62</td><td>58.73</td><td>69.75</td><td>70.76</td><td>70.29</td></tr></table>
+
+Table 5: Test accuracy (%) on Clothing1M. The best two results are in bold.
+
+# 318319 4 Conclusion
+
+In this paper, we focus on promoting the prior sample selection in learning with noisy labels, which starts from concerning the uncertainty of losses during training. We robustly use the training losses at different iterations to reduce the uncertainty of small-loss examples, and adopt confidence interval estimation to reduce the uncertainty of large-loss examples. Experiments are conducted on benchmark datasets, demonstrating the effectiveness of our method. We believe that this paper opens up new possibilities in the topics of using sample selection to handle noisy labels, especially in improving the robustness of models on imbalanced noisy datasets.
+
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+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code and instructions are provided in the supplemental material. The used datasets can be publicly downloaded. Besides, the code for generating noisy labels is provided.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 3.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section 3.1 and 3.2.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See “Baselines” in Section 3.1.
+
+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] We use MNIST, $F$ -MNIST, CIFAR-10, CIFAR-100, and Clothing1M in this paper. We cite the creators, which can be checked in Section 3.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(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]

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+        "type": "text",
+        "text": "Sample Selection with Uncertainty of Losses for Learning with Noisy Labels ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "1 In learning with noisy labels, the sample selection approach is very popular, which   \n2 regards small-loss data as correctly labeled during training. However, losses are   \n3 generated on-the-fly based on the model being trained with noisy labels, and thus   \n4 large-loss data are likely but not certainly to be incorrect. There are actually   \n5 two possibilities of a large-loss data point: (a) it is mislabeled, and then its loss   \n6 decreases slower than other data, since deep neural networks “learn patterns first”;   \n7 (b) it belongs to an underrepresented group of data and has not been selected yet. In   \n8 this paper, we incorporate the uncertainty of losses by adopting interval estimation   \n9 instead of point estimation of losses, where lower bounds of the confidence intervals   \n10 of losses derived from distribution-free concentration inequalities, but not losses   \n11 themselves, are used for sample selection. In this way, we also give large-loss but   \n12 less selected data a try; then, we can better distinguish between the cases (a) and   \n13 (b) by seeing if the losses effectively decrease with the uncertainty after the try. As   \n14 a result, we can better explore underrepresented data that are correctly labeled but   \n15 seem to be mislabeled at first glance. Experiments demonstrate that the proposed   \n16 method is superior to baselines and robust to a broad range of label noise types. ",
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+        "text": "17 1 Introduction ",
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+        "text": "18 Learning with noisy labels is one of the most challenging problems in weakly-supervised learning,   \n19 since noisy labels are ubiquitous in the real world [36, 65, 40, 1, 61]. For instance, both crowdsourcing   \n20 and web crawling yield large numbers of noisy labels everyday [12]. Noisy labels can severely impair   \n21 the performance of deep neural networks with strong memorization capacities [67, 69, 42, 30].   \n22 To reduce the influence of noisy labels, a lot of approaches have been recently proposed [38, 29, 31,   \n23 68, 71, 55, 56, 46, 33, 25, 34, 47, 60, 49, 19, 17, 14]. They can be generally divided into two main   \n24 categories. The first one is to estimate the noise transition matrix [41, 44, 15, 11], which denotes the   \n25 probabilities that clean labels flip into noisy labels. However, the noise transition matrix is hard to be   \n26 estimated accurately, especially when the number of classes is large [65]. The second approach is   \n27 sample selection, which is our focus in this paper. This approach is based on selecting possibly clean   \n28 examples from a mini-batch for training [12, 62, 50, 65, 23, 50, 51]. Intuitively, if we can exploit less   \n29 noisy data for network parameter updates, the network will be more robust.   \n30 A major question in sample selection is what criteria can be used to select possibly clean examples.   \n31 At the present stage, the selection based on the small-loss criteria is the most common method, and   \n32 has been verified to be effective in many circumstances [12, 16, 65, 52, 62]. Specifically, since   \n33 deep networks learn patterns first [2], they would first memorize training data of clean labels and   \n34 then those of noisy labels with the assumption that clean labels are of the majority in a noisy class.   \n35 Small-loss examples can thus be regarded as clean examples with high probability. Therefore, in   \n36 each iteration, prior methods [12, 52] select the small-loss examples based on the predictions of the   \n37 current network for robust training.   \n38 However, such a selection procedure is debatable, since it arguably does not consider uncertainty   \n39 in selection. The uncertainty comes from two aspects. First, this procedure has uncertainty about   \n40 small-loss examples. Specifically, the procedure uses limited time intervals and only exploits the   \n41 losses provided by the current predictions. For this reason, the estimation for the noisy class posterior   \n42 is unstable [63], which causes the network predictions to be equally unstable. It thus takes huge risks   \n43 to only use losses provided by the current predictions (Figure 1, left). Once wrong selection is made,   \n44 the inferiority of accumulated errors will arise [65]. Second, this procedure has uncertainty about   \n45 large-loss examples. To be specific, deep networks learn easy examples at the beginning of training,   \n46 but ignore some clean examples with large losses. Nevertheless, such examples are always critical for   \n47 generalization. For instance, when learning with imbalanced data, distinguishing the examples with   \n48 non-dominant labels are more pivotal during training [35]. Deep networks often give large losses to   \n49 such examples (Figure 1, right). Therefore, when learning under the realistic scenes, e.g., learning   \n50 with noisy imbalanced data, prior sample selection methods cannot address such an issue well.   \n51 To relieve the above issues, we study the uncertainty of losses in the sample selection procedure to   \n52 combat noisy labels. To reduce the uncertainty of small-loss examples, we extend time intervals and   \n53 utilize the mean of training losses at different training iterations. In consideration of the bad influence   \n54 of mislabeled data on training losses, we build two robust mean estimators from the perspectives of   \n55 soft truncation and hard truncation w.r.t. the truncation level, respectively. Soft truncation makes the   \n56 mean estimation more robust by holistically changing the behavior of losses. Hard truncation makes   \n57 the mean estimation more robust by locally removing outliers from losses. To reduce the uncertainty   \n58 of large-loss examples, we encourage networks to pick the sample that has not been selected in a   \n59 conservative way. Furthermore, to address the two issues simultaneously, we derive concentration   \n60 inequalities [5] for robust mean estimation and further employ statistical confidence bounds [3] to   \n61 consider the number of times an example was selected during training.   \n62 The study of uncertainty of losses in learning with noisy labels can be justified as follows. In statistical   \n63 learning, it is known that uncertainty is related to the quality of data [48]. Philosophically, we need   \n64 variety decrease for selected data and variety search for unselected data, which share a common   \n65 objective, i.e., reduce the uncertainty of data to improve generalization [37]. This is our original   \n66 intention, since noisy labels could bring more uncertainty because of the low quality of noisy data.   \n67 Nevertheless, due to the harm of noisy labels for generalization, we need to strike a good balance   \n68 between variety decrease and search. Technically, our method is specially designed for handling   \n69 noisy labels, which robustly uses network predictions and conservatively seeks less selected examples   \n70 meanwhile to reduce the uncertainty of losses and then generalize well.   \n71 Before delving into details, we clearly emphasize our contributions in two folds. First, we reveal prior   \n72 sample selection criteria in learning with noisy labels have some potential weaknesses and discuss   \n73 them in detail. The new selection criteria are then proposed with detailed theoretical analyses. Second,   \n74 we experimentally validate the proposed method on both synthetic noisy balanced/imbalanced datasets   \n75 and real-world noisy datasets, on which it achieves superior robustness compared with the state  \n76 of-the-art methods in learning with noisy labels. The rest of the paper is organized as follows. In   \n77 Section 2, we propose our robust learning paradigm step by step. Experimental results are discussed   \n78 in Section 3. The conclusion is given in Section 4.   \n80 In this section, we first introduce the problem setting and some background (Section 2.1). Then we   \n81 discuss how to exploit training losses at different iterations (Section 2.2). Finally, we introduce the   \n82 proposed method, which exploits training losses at different iterations more robustly and encourages   \n83 networks to pick the sample that is less selected but could be correctly labeled (Section 2.3). ",
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+            "Figure 1: Illustrations of uncertainty of losses. Experiments are conducted on the imbalanced noisy MNIST dataset. Left: uncertainty of small-loss examples. At the beginning of training (Epochs 1 and 2), due to the instability of the current prediction, the network gives a larger loss to the clean example and does not select it for updates. If we consider the mean of training losses at different epochs, the clean example can be equipped with a smaller loss and then selected for updates. Right: uncertainty of large-loss examples. Since the deep network learns easy examples at the beginning of training, it gives a large loss to clean imbalanced data with non-dominant labels, which causes such data unable to be selected and severely influence generalization. "
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+        "type": "text",
+        "text": "84 2.1 Preliminaries ",
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+        "text": "85 Let $\\mathcal { X }$ and $\\mathcal { V }$ be the input and output spaces. Consider a $k$ -class classification problem, i.e., $\\mathcal { V } = [ k ]$ ,   \n86 where $[ k ] = \\{ 1 , \\dots , k \\}$ . In learning with noisy labels, the training data are all sampled from a   \n87 corrupted distribution on $\\mathcal { X } \\times \\mathcal { V }$ . We are given a sample with noisy labels, i.e., $\\tilde { S } = \\{ ( \\mathbf { x } , \\tilde { y } ) \\}$ , where   \n88 $\\tilde { y }$ is the noisy label. The aim is to learn a robust classifier that could assign clean labels to test data by   \n89 only exploiting a training sample with noisy labels.   \n90 Let $f : \\mathcal { X } \\to \\mathbb { R } ^ { k }$ be the classifier with learnable parameters w. At the $i$ -th iteration during training,   \n91 the parameters of the classifier $f$ can be denoted as $\\mathbf { w } _ { i }$ . Let $\\ell : \\mathbb { R } ^ { k } \\times \\mathcal { Y }  \\mathbb { R }$ be a surrogate loss   \n92 function for $k$ -class classification. We exploit the softmax cross entropy loss in this paper. Given an   \n93 arbitrary training example $( \\mathbf { x } , \\tilde { y } )$ , at the $i$ -th iteration, we can obtain a loss $\\ell _ { i }$ , i.e., $\\ell _ { i } = \\ell ( f ( \\mathbf { w } _ { i } ; \\mathbf { x } ) , \\tilde { y } )$   \n94 Hence, until the $t$ -th iteration, we can obtain a training loss set $L _ { t }$ about the example $( \\mathbf { x } , \\tilde { y } )$ , i.e.,   \n95 $L _ { t } = \\{ \\ell _ { 1 } , \\ldots , \\ell _ { t } \\}$ .   \n96 In this paper, we assume that the training losses in $L _ { t }$ conform to a Markov process, which is to   \n97 represent a changing system under the assumption that future states only depend on the current state   \n98 (the Markov property) [43]. More specifically, at the $i$ -th iteration, if we exploit an optimization   \n99 algorithm for parameter updates (e.g., the stochastic gradient descent algorithm [4]) and omit other   \n100 dependencies (e.g., $\\tilde { S }$ ), we will have $P ( \\mathbf { w } _ { i } | \\mathbf { w } _ { i - 1 } , \\ldots , \\mathbf { w } _ { 0 } ) = P ( \\mathbf { w } _ { i } | \\mathbf { w } _ { i - 1 } )$ , which means that the   \n101 future state of the classifier $f$ only depends on the current state. Furthermore, given a training example   \n102 and the parameters of the classifier $f$ , we can determine the loss of the training example as discussed.   \n103 Therefore, the training losses in $L _ { t }$ will also conform to a Markov process. ",
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+        "type": "text",
+        "text": "2.2 Extended Time Intervals ",
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+        "text": "105 As limited time interval cannot address the instability issue of the estimation for the noisy class   \n106 posterior well [42], we extend time intervals and exploit the training losses at different training   \n107 iterations for sample selection. One straightforward idea is to use the mean of training losses at   \n108 different training iterations. Hence, the selection criterion could be ",
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+        "text": "$$\n\\tilde { \\mu } = \\frac { 1 } { t } \\sum _ { i = 1 } ^ { t } \\ell _ { i } .\n$$",
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+        "text": "109 It is intuitive and reasonable to use such a selection criterion for sample selection, since the operation   \n110 of averaging can mitigate the risks caused by the unstable estimation for the noisy class posterior,   \n111 following better generalization. Nevertheless, such a method could arguably achieve suboptimal   \n112 classification performance for learning with noisy labels. The main reason is that, due to the great   \n113 harm of mislabeled data, part of training losses are with too large uncertainty and could be seen as   \n114 outliers. Therefore, it could be biased to use the mean of training losses consisting of such outliers   \n115 [10], which further influences sample selection. More evaluations for our claims are provided in   \n116 Section 3. ",
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+        "type": "text",
+        "text": "117 2.3 Robust Mean Estimation and Conservative Search ",
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+        "text": "118 We extend time intervals and meanwhile exploit the training losses at different training iterations more   \n119 robustly. Specifically, we build two robust mean estimators from the perspectives of soft truncation   \n120 and hard truncation [7]. Note that for specific tasks, it is feasible to decide the types of robust mean   \n121 estimation with statistical tests based on some assumptions [8]. We leave the analysis as future work.   \n122 Two distribution-free robust mean estimators are introduced as follows.   \n123 Soft truncation. We extend a classical M-estimator from [7] and exploit the widest possible choice of   \n124 the influence function. More specifically, give a random variable $X$ , let us consider a non-decreasing ",
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+        "type": "text",
+        "text": "125 influence function $\\psi : \\mathbb { R } \\to \\mathbb { R }$ such that ",
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+        "img_path": "images/7e66459d120dc863a24c148784f8665dbef433a9ec1452812b16116115fc36a7.jpg",
+        "text": "$$\n\\psi ( X ) = \\log ( 1 + X + X ^ { 2 } / 2 ) , X \\geq 0 .\n$$",
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+        "text": "126 The choice of $\\psi$ is inspired by the Taylor expansion of the exponential function, which can make the   \n127 estimation results more robust by reducing the side effect of extremum holistically. The illustration   \n128 for this influence function is provided in Appendix A.1. For our task, given the observations on   \n129 training losses, i.e., $L _ { t } = \\{ \\ell _ { 1 } , \\ldots , \\ell _ { t } \\}$ , we estimate the mean robustly as follows: ",
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+        "text": "$$\n\\tilde { \\mu } _ { s } = \\frac { 1 } { t } \\sum _ { i = 1 } ^ { t } \\psi ( \\ell _ { i } ) .\n$$",
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+        "text": "130 We term the above robust mean estimator (3) the soft estimator. ",
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+        "text": "131 Hard truncation. We propose a new robust mean estimator based on hard truncation. Specifically,   \n132 given the observations on training losses $L _ { t }$ , we first exploit the $\\mathbf { K }$ -nearest neighbor (KNN) algorithm   \n133 [27] to remove some underlying outliers in $L _ { t }$ . The number of outliers is denoted by $t _ { \\mathrm { o } } ( t _ { \\mathrm { o } } < t )$ , which   \n134 can be adaptively determined as discussed in [70]. Note that we can also employ other algorithms,   \n135 e.g., principal component analysis [45] and the local outlier factor [6], to identify underlying outliers   \n136 in $L _ { t }$ . The main reason we employ KNN is because of its relatively low computation costs [70].   \n137 The truncated loss observations on training losses are denoted by $L _ { t - t _ { \\mathrm { o } } }$ . We then utilize $L _ { t - t _ { \\mathrm { o } } }$ for   \n138 the mean estimation. As the potential outliers are removed with high probability, the robustness of   \n139 the estimation results will be enhanced. We denote such an estimated mean as ${ \\tilde { \\mu } } _ { h }$ . We have ",
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+        "text": "$$\n\\tilde { \\mu } _ { h } = \\frac { 1 } { t - t _ { \\mathrm { o } } } \\sum _ { \\ell _ { i } \\in L _ { t - t _ { 0 } } } \\ell _ { i } .\n$$",
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+        "type": "text",
+        "text": "140 The corresponding estimator (4) is termed the hard estimator. ",
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+        "text": "141 We derive concentration inequalities for the soft and hard estimators respectively. The search strategy   \n142 for less selected examples and overall selection criterion are then provided. Note that we do not need   \n143 to explicitly quantify the mean of training losses. We only need to sort the training examples based   \n144 on the proposed selection criterion and then use the selected examples for robust training. ",
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+        "text": "145 Theorem 1. Let $Z _ { n } = \\{ z _ { 1 } , \\cdots , z _ { n } \\}$ be an observation set with mean $\\mu _ { z }$ and variance $\\sigma ^ { 2 }$ . By exploiting the non-decreasing influence function 46 $\\psi ( z ) = \\log ( 1 + z + z ^ { 2 } / 2 )$ . For any $\\epsilon > 0$ , we have ",
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+        "text": "$$\n\\left| \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\psi ( z _ { i } ) - \\mu _ { z } \\right| \\leq \\frac { \\sigma ^ { 2 } ( n + \\frac { \\sigma ^ { 2 } \\log ( \\epsilon ^ { - 1 } ) } { n ^ { 2 } } ) } { n - \\sigma ^ { 2 } } ,\n$$",
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+        "type": "text",
+        "text": "147 with probability at least $1 - 2 \\epsilon$ ",
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+        "type": "text",
+        "text": "148 Proof can be found in Appendix A.1. ",
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+        "text": "149 Theorem 2. Let $Z _ { n } = \\{ z _ { 1 } , \\ldots , z _ { n } \\}$ be a (not necessarily time homogeneous) Markov chain with   \n150 mean $\\mu _ { z }$ , taking values in a Polish state space $\\Lambda _ { 1 } \\times \\ldots \\times \\Lambda _ { n }$ , and with a minimal mixing time $\\tau _ { \\mathrm { m i n } }$ .   \n151 The truncated set with hard truncation is denoted by $Z _ { n _ { o } }$ , with $n _ { o } < n$ . If $| z _ { i } |$ is upper bounded by $Z$ .   \n152 For any $\\epsilon _ { 1 } > 0$ and $\\epsilon _ { 2 } > 0$ , we have ",
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+        "text": "$$\n\\left| \\frac { 1 } { n - n _ { o } } \\sum _ { z _ { i } \\in Z _ { n } \\setminus Z _ { n _ { o } } } - \\mu _ { z } \\right| \\leq \\frac { 1 } { n - n _ { o } } \\left( 2 Z \\sqrt { 2 \\tau _ { \\mathrm { m i n } } \\log \\frac { 2 } { \\epsilon _ { 1 } } } + \\frac { 2 Z n _ { o } } { n } \\sqrt { 2 \\tau _ { \\mathrm { m i n } } \\log \\frac { 2 n } { \\epsilon _ { 2 } } } \\right) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "153 with probability at least $1 - \\epsilon _ { 1 } - \\epsilon _ { 2 }$ ",
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+        "text": "154 Proof can be found in Appendix A.2. For our task, let the training loss be upper-bounded by $L$ . The   \n155 value of $L$ can be determined easily by training networks on noisy datasets and observing the loss   \n156 distribution [1].   \n157 Conservative search and selection criteria. In this paper, we will use the concentration inequalities   \n158 (5) and (6) to present conservative search and the overall sample selection criterion. Specifically,   \n159 we exploit their lower bounds and consider the selected number of examples during training. The   \n160 selection of the examples that are less selected is encouraged. ",
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+        "type": "text",
+        "text": "Algorithm 1 CNLCU Algorithm. ",
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+        "text": "1: Input $\\theta _ { 1 }$ and $\\theta _ { 2 }$ , learning rate $\\eta$ , fixed $\\tau$ , epoch $T _ { k }$ and $T _ { \\mathrm { m a x } }$ , iteration $t _ { \\mathrm { m a x } }$ ; for $T = 1 , 2 , \\dots , T _ { \\mathrm { m a x } }$ do ",
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+        "text": "2: Shuffle training dataset $\\tilde { S }$ ; ",
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+        "text": "end ",
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+        "text": "8: Update $\\begin{array} { r } { R ( T ) = 1 - \\operatorname* { m i n } \\left\\{ \\frac { T } { T _ { k } } \\tau , \\tau \\right\\} } \\end{array}$ ",
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+        "text": "end ",
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+        "type": "text",
+        "text": "9: Output $\\theta _ { 1 }$ and $\\theta _ { 2 }$ ",
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+    {
+        "type": "text",
+        "text": "Denote the number of times one example was selected by 161 $n _ { t } ( n _ { t } ~ \\leq ~ t )$ . Let $\\begin{array} { r } { \\epsilon = \\frac { 1 } { 2 t } } \\end{array}$ . For the 162 circumstance with soft truncation, the selection criterion is ",
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+        "text": "$$\n\\ell _ { s } ^ { \\star } = \\tilde { \\mu } _ { s } - \\frac { \\sigma ^ { 2 } ( t + \\frac { \\sigma ^ { 2 } \\log ( 2 t ) } { t ^ { 2 } } ) } { n _ { t } - \\sigma ^ { 2 } } .\n$$",
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+        "type": "text",
+        "text": "Let 163 $\\begin{array} { r } { \\epsilon _ { 1 } = \\epsilon _ { 2 } = \\frac { 1 } { 2 t } } \\end{array}$ , for the situation with hard truncation, by rewriting (6), the selection criterion is ",
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+        "img_path": "images/d9e0e72952babc70de72b58eb168bf9c192b7a556f0830f75b753eef7e1114d5.jpg",
+        "text": "$$\n\\ell _ { h } ^ { \\star } = \\tilde { \\mu } _ { h } - \\frac { 2 \\sqrt { 2 \\tau _ { \\mathrm { m i n } } } L ( t + \\sqrt { 2 } t _ { \\mathrm { o } } ) } { ( t - t _ { \\mathrm { o } } ) \\sqrt { t } } \\sqrt { \\frac { \\log ( 4 t ) } { n _ { t } } } .\n$$",
+        "text_format": "latex",
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+        "text": "164 Note that we directly replace $t$ with $n _ { t }$ . If an example is rarely selected during training, $n _ { t }$ will be far   \n165 less than $n$ , which causes the lower bounds to change drastically. Hence, we do not use the mean of   \n166 all training losses, but use the mean of training losses in fixed-length time intervals. More details   \n167 about this can be checked in Section 3.   \n168 For the selection criteria (7) and (8), we can see that they consist of two terms and have one term   \n169 with a minus sign. The first term in Eq. (7) (or Eq. (8)) is to reduce the uncertainty of small-loss   \n170 examples, where we use robust mean estimation on training losses. The second term, i.e., the   \n171 statistical confidence bound, is to encourage the network to choose the less selected examples (with a   \n172 small $n _ { t }$ ). The two terms are constraining and balanced with $\\sigma ^ { 2 }$ or $\\tau _ { \\mathrm { m i n } }$ . To avoid introducing strong   \n173 assumptions on the underlying distribution of losses [8], we tune $\\sigma$ and $\\tau _ { \\mathrm { m i n } }$ with a noisy validation   \n174 set. For the mislabeled data, although the model has high uncertainties on them (i.e., a small $n _ { t }$ )   \n175 and tends to pick them, the overfitting to the mislabeled data is harmful. Also, the mislabeled data   \n176 and clean data are rather hard to distinguish in some cases as discussed. Thus, we should search   \n177 underlying clean data in a conservative way. In this paper, we initialize $\\sigma$ and $\\tau _ { \\mathrm { m i n } }$ with small values.   \n178 This way can reduce the adverse effects of mislabeled data and meanwhile select the clean examples   \n179 with large losses, which helps generalize. More evaluations will be presented in Section 3.   \n180 The overall procedure of the proposed method, which combats noisy labels by concerning uncertainty   \n181 (CNLCU), is provided in Algorithm 1. CNLCU works in a mini-batch manner since all deep learning   \n182 training methods are based on stochastic gradient descent. Following [12], we exploit two networks   \n183 with parameters $\\theta _ { 1 }$ and $\\theta _ { 2 }$ respectively to teach each other. Specifically, when a mini-batch $\\bar { S }$ is   \n184 formed (Step 3), we let two networks select a small proportion of examples in this mini-batch with   \n185 Eq. (7) or (8) (Step 4 and Step 5). The number of instances is controlled by the function $R ( T )$ , and   \n186 two networks only select $R ( T )$ percentage of examples out of the mini-batch. The value of $R ( T )$   \n187 should be larger at the beginning of training, and be smaller when the number of epochs goes large,   \n188 which can make better use of memorization effects of deep networks [12] for sample selection. Then,   \n189 the selected instances are fed into its peer network for parameter updates (Step 6 and Step 7). ",
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+        "type": "text",
+        "text": "190 3 Experiments ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "191 In this section, we evaluate the robustness of our proposed method to noisy labels with comprehensive   \n192 experiments on the synthetic balanced noisy datasets (Section 3.1), synthetic imbalanced noisy   \n193 datasets (Section 3.2), and real-world noisy dataset (Section 3.3). ",
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+    {
+        "type": "text",
+        "text": "Datasets. We verify the effectiveness of our method on the manually corrupted version of the following datasets: MNIST [22], $F$ -MNIST [58], CIFAR-10 [21], and CIFAR-100 [21], because these datasets are popularly used for the evaluation of learning with noisy labels in the literature [12, 65, 54, 23]. The four datasets are class-balanced. The important statistics of the used synthetic datasets are summarized in Appendix B.1. ",
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+    {
+        "type": "text",
+        "text": "Generating noisy labels. We consider broad types of label noise: (1). Symmetric noise (abbreviated as Sym.) [53, 31, 26]. (2) Asymmetric noise (abbreviated as Asym.) [32, 57, 52]. (3) Pairflip noise (abbreviated as Pair.) [12, 65, 71]. (4). Tridiagonal noise (abbreviated as Trid.) [68]. (5). Instance noise (abbreviated as Ins.) [9, 56]. The noise rate is set to $20 \\%$ and $40 \\%$ to ensure clean labels are diagonally dominant [32]. More details about above noise are provided in Appendix B.1. We leave out $10 \\%$ of noisy training examples as a validation set. ",
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+    {
+        "type": "text",
+        "text": "Baselines. We compare the proposed method (Algorithm 1) with following methods which focus on sample selection, and implement all methods with default parameters by PyTorch, and conduct all the experiments on NVIDIA Titan Xp GPUs. (1). S2E [62], which properly controls the sample selection process so that deep networks can better benefit from the memorization effects. (2). MentorNet [16], which learns a curriculum to filter out noisy data. We use self-paced MentorNet in this paper. (3). Co-teaching [12], which trains two networks simultaneously and cross-updates parameters of peer networks. (4). SIGUA [13], which exploits stochastic integrated gradient underweighted ascent to handle noisy labels. We use self-teaching SIGUA in this paper. (5). JoCor [52], which reduces the diversity of networks to improve robustness. Other types of baselines such as adding regularization are provided in Appendix B.2. Note that we do not compare the proposed method with some stateof-the-art methods, e.g., SELF [39] and DivideMix [24]. It is because their proposed methods are aggregations of multiple techniques. We mainly focus on sample selectionin in learning with noisy labels. Therefore, the comparison is not fair. Here, we term our methods with soft truncation and hard truncation as CNLCU-S and CNLCU-H respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Network structure and optimizer. For MNIST, $F$ -MNIST, and CIFAR-10, we use a 9-layer CNN structure from [12]. Due to the limited space, the experimental details on CIFAR-100 are provided in Appendix B.3. All network structures we used here are standard test beds for weakly-supervised learning. For all experiments, the Adam optimizer [20] (momentum $_ { 1 = 0 . 9 }$ ) is used with an initial learning rate of 0.001, and the batch size is set to 128 and we run 200 epochs. We linearly decay learning rate to zero from 80 to 200 epochs as did in [12]. We take two networks with the same architecture but different initializations as two classifiers as did in [12, 65, 52], since even with the same network and optimization method, different initializations can lead to different local optimal [12]. The details of network structures can be checked in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "For the hyper-parameters $\\sigma ^ { 2 }$ and $\\tau _ { \\mathrm { m i n } }$ , we determine them in the range $\\{ 1 0 ^ { - 1 } , 1 0 ^ { - 2 } , 1 0 ^ { - 3 } , 1 0 ^ { - 4 } \\}$ with a noisy validation set. Here, we assume the noise level $\\tau$ is known and set $R ( T ) = 1 -$ $\\operatorname* { m i n } \\{ \\frac { T } { T _ { k } } \\tau , \\tau \\}$ with ${ \\mathit { T } } _ { k } { = } 1 0$ . If $\\tau$ is not known in advanced, it can be inferred using validation sets [29, 66]. As for performance measurement, we use test accuracy, i.e., test accuracy $=$ (# of correct prediction) / (# of testing). All experiments are repeated five times. We report the mean and standard deviation of experimental results. ",
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+        "type": "text",
+        "text": "Experimental results. The experimental results about test accuracy are provided in Table 1, 2, and 3. Specifically, for MNIST, as can be seen, our proposed methods, i.e., CNLCU-S and CNLCU-H, produce the best results in the vast majority of cases. In some cases such as asymmetric noise, the baseline S2E outperforms ours, which benefits the accurate estimation for the number of selected small-loss examples. For $F$ -MNIST, the training data becomes complicated. S2E cannot achieve the accurate estimation in such situation and thus has no great performance like it got on MNIST. Our methods achieve varying degrees of lead over baselines. For CIFAR-10, our methods once again outperforms all the baseline methods. Although some baseline, e.g., Co-teaching, can work well in some cases, experimental results show that it cannot handle various noise types. In contrast, the proposed methods achieve superior robustness against broad noise types. The results mean that our methods can be better applied to actual scenarios, where the noise is diversiform. ",
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+        "type": "text",
+        "text": "46 Ablation study. We first conduct the ablation study to analyze the sensitivity of the length of time intervals. In order to avoid too dense figures, we exploit MNIST and $F$ -MNIST with the mentioned 8 noise settings as representative examples. For CNLCU-S, the length of time intervals is chosen in ",
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+    {
+        "type": "table",
+        "img_path": "images/895798a95ea64e2cdc42ba9e4f7d93e51e478ab1f5586de91c522f4214b4d561.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Test accuracy $( \\% )$ on MNIST over the last ten epochs. The best two results are in bold. "
+        ],
+        "table_body": "<table><tr><td>Noise type</td><td colspan=\"2\">Sym.</td><td colspan=\"2\">Asym.</td><td colspan=\"2\">Pair.</td><td colspan=\"2\">Trid.</td><td colspan=\"2\">Ins.</td></tr><tr><td>Method/Noise ratio</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td></tr><tr><td rowspan=\"2\">S2E</td><td>98.46</td><td>95.62</td><td>99.05</td><td>98.45</td><td>98.56</td><td>94.22</td><td>99.02</td><td>97.23</td><td>97.93</td><td>94.02</td></tr><tr><td>±0.06</td><td>±0.91</td><td>±0.02</td><td>±0.26</td><td>±0.32</td><td>±0.79</td><td>±0.09</td><td>±1.26</td><td>±1.26</td><td>±2.39</td></tr><tr><td rowspan=\"2\">MentorNet</td><td>95.04</td><td>92.08</td><td>96.32</td><td>90.86</td><td>93.19</td><td>90.93</td><td>96.42</td><td>93.28</td><td>94.65</td><td>90.11</td></tr><tr><td>±0.03</td><td>±0.42</td><td>±0.17</td><td>±0.97</td><td>±0.17</td><td>±1.54</td><td>±0.09</td><td>±1.37</td><td>±0.73</td><td>±1.26</td></tr><tr><td rowspan=\"2\">Co-teaching</td><td>97.53</td><td>95.62</td><td>98.25</td><td>95.08</td><td>96.05</td><td>94.16</td><td>98.05</td><td>96.18</td><td>97.96</td><td>95.02</td></tr><tr><td>±0.12</td><td>±0.30</td><td>±0.08</td><td>±0.43</td><td>±0.96</td><td>±1.37</td><td>±0.06</td><td>±0.85</td><td>±0.09</td><td>±0.39</td></tr><tr><td rowspan=\"2\">SIGUA</td><td>92.31</td><td>91.88</td><td>93.96</td><td>62.59</td><td>93.77</td><td>86.22</td><td>94.92</td><td>83.46</td><td>92.90</td><td>86.34</td></tr><tr><td>±1.10</td><td>±0.92</td><td>±0.82</td><td>±0.15</td><td>±1.40</td><td>±1.75</td><td>±0.83</td><td>±2.98</td><td>±1.82</td><td>±3.51</td></tr><tr><td rowspan=\"2\">JoCor</td><td>98.42</td><td>98.04</td><td>98.05</td><td>94.55</td><td>98.01</td><td>96.85</td><td>98.45</td><td>96.98</td><td>98.62</td><td>96.07</td></tr><tr><td>±0.14</td><td>±0.07</td><td>±0.37</td><td>±1.08</td><td>±0.19</td><td>±0.43</td><td>±0.17</td><td>±0.25</td><td>±0.06</td><td>±0.31</td></tr><tr><td rowspan=\"2\">CNLCU-S</td><td>98.82</td><td>98.31</td><td>98.93</td><td>97.67</td><td>98.86</td><td>97.71</td><td>99.09</td><td>98.02</td><td>98.77</td><td>97.78</td></tr><tr><td>±0.03</td><td>士0.05</td><td>±0.06</td><td>±0.22</td><td>±0.06</td><td>±0.64</td><td>±0.04</td><td>±0.17</td><td>±0.08</td><td>±0.25</td></tr><tr><td rowspan=\"2\">CNLCU-H</td><td>98.70</td><td>98.24</td><td>99.01 </td><td>98.01</td><td>98.44</td><td>97.37</td><td>98.89</td><td>97.92</td><td>98.74</td><td>97.42</td></tr><tr><td>±0.06</td><td>±0.06</td><td>±0.04</td><td>±0.03</td><td>±0.19</td><td>±0.32</td><td>±0.15</td><td>±0.05</td><td>±0.16</td><td>±0.39</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Test accuracy on F-MNIST over the last ten epochs. The best two results are in bold. "
+        ],
+        "table_body": "<table><tr><td>Noise type</td><td colspan=\"2\">Sym.</td><td colspan=\"2\">Asym.</td><td colspan=\"2\">Pair.</td><td colspan=\"2\">Trid.</td><td colspan=\"2\">Ins.</td></tr><tr><td>Method/Noise ratio</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td></tr><tr><td rowspan=\"2\">S2E</td><td>89.99</td><td>75.32</td><td>89.00</td><td>81.03</td><td>88.66</td><td>67.09</td><td>89.53</td><td>77.29</td><td>88.65</td><td>79.35</td></tr><tr><td>±2.07</td><td>±5.84</td><td>±0.95</td><td>±1.93</td><td>±1.32</td><td>±4.03</td><td>±2.63</td><td>±3.97</td><td>±2.12</td><td>±3.04</td></tr><tr><td rowspan=\"2\">MentorNet</td><td>90.37</td><td>86.53</td><td>89.69</td><td>67.21</td><td>87.92</td><td>83.70</td><td>88.74</td><td>85.63</td><td>87.52</td><td>83.27</td></tr><tr><td>±0.17</td><td>士0.65</td><td>±0.19</td><td>±2.94</td><td>±1.08</td><td>±0.49</td><td>±0.33</td><td>±0.59</td><td>±0.15</td><td>±1.42</td></tr><tr><td rowspan=\"2\">Co-teaching</td><td>91.48</td><td>88.80</td><td>91.03</td><td>68.07</td><td>90.77</td><td>86.91</td><td>91.24</td><td>89.18</td><td>90.60</td><td>87.90</td></tr><tr><td>±0.10</td><td>±0.29</td><td>±0.14</td><td>±4.58</td><td>±0.23</td><td>±0.71</td><td>±0.11</td><td>±0.36</td><td>±0.12</td><td>±0.45</td></tr><tr><td rowspan=\"2\">SIGUA</td><td>87.64</td><td>87.23</td><td>76.97</td><td>45.96</td><td>69.59</td><td>68.93</td><td>79.97</td><td>76.14</td><td>76.92</td><td>74.89</td></tr><tr><td>±1.29</td><td>±0.72</td><td>±2.59</td><td>±3.40</td><td>±5.75</td><td>±2.80</td><td>±3.23</td><td>±4.24</td><td>±5.09</td><td>士4.84</td></tr><tr><td rowspan=\"2\">JoCor</td><td>91.97</td><td>89.96</td><td>90.95</td><td>79.79</td><td>91.52</td><td>87.40</td><td>92.01</td><td>89.42</td><td>91.43</td><td>87.59</td></tr><tr><td>±0.13</td><td>±0.19</td><td>±0.21</td><td>±2.39</td><td>士0.24</td><td>±0.58</td><td>±0.17</td><td>士0.33</td><td>±0.71</td><td>±0.94</td></tr><tr><td rowspan=\"2\">CNLCU-S</td><td>92.37</td><td>91.45</td><td>92.57</td><td>83.14</td><td>92.04</td><td>88.20</td><td>92.24</td><td>90.08</td><td>91.69</td><td>89.02</td></tr><tr><td>±0.15</td><td>±0.28</td><td>±0.15</td><td>±1.77</td><td>±0.26</td><td>±0.44</td><td>±0.17</td><td>±0.34</td><td>±0.10</td><td>±1.02</td></tr><tr><td rowspan=\"2\">CNLCU-H</td><td>92.42</td><td>91.60</td><td>92.60</td><td>82.69</td><td>91.70</td><td>87.70</td><td>92.33</td><td>90.22</td><td>91.50</td><td>88.79</td></tr><tr><td>±0.21</td><td>±0.19</td><td>±0.18</td><td>±0.43</td><td>±0.18</td><td>±0.69</td><td>±0.26</td><td>±0.71</td><td>±0.21</td><td>±1.22</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "249 the range from 3 to 8. For CNLCU-H, the length of time intervals is chosen in the range from 10 to   \n250 15. Note that the reason for their different lengths is that their different mechanisms. Specifically,   \n251 CNLCU-S holistically changes the behavior of losses, but does not remove any loss from the loss set.   \n252 We thus do not need too long length of time intervals. As a comparison, CNLCU-H needs to remove   \n253 some outliers from the loss set as discussed. The length should be longer to guarantee the number of   \n254 examples available for robust mean estimation. The experimental results are provided in Appendix   \n255 B.4, which show the proposed CNLCU-S and CNLCU-H are robust to the choices of the length of   \n256 time intervals. Such robustness to hyperparameters means our methods can be applied in practice and   \n257 does not need too much effect to tune the hyperparameters.   \n258 Furthermore, since our methods concern uncertainty from two aspects, i.e., the uncertainty from both   \n259 small-loss and large-loss examples, we conduct experiments to analyze each part of our methods.   \n260 Also, as mentioned, we compare robust mean estimation with non-robust mean estimation when   \n261 learning with noisy labels. More details are provided in Appendix B.4. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.2 Experiments on Synthetic Imbalanced Noisy Datasets ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experimental setup. We exploit MNIST and $F$ -MNIST. For these two datasets, we reduce the number of training examples along with the labels from $\\mathbf { \\bar { \\theta } } ^ { 6 } 0 ^ { 9 }$ to $\" 4 > \"$ to $1 \\%$ of previous numbers. We term such synthetic imbalanced noisy datasets as IM-MNIST and IM-F-MNIST respectively. This setting aims to simulate the extremely imbalanced circumstance, which is common in practice. Moreover, we exploit asymmetric noise, since these types of noise can produce more imbalanced case [41, 32]. Other settings such as the network structure and optimizer are the same as those in experiments on synthetic balanced noisy datasets. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2353673c1904139182aa171853551168be1af9601ecaa55377f1888914e8b5b2.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Test accuracy $( \\%$ ) on CIFAR-10 over the last ten epochs. The best two results are in bold. "
+        ],
+        "table_body": "<table><tr><td>Noise type</td><td colspan=\"2\">Sym.</td><td colspan=\"2\">Asym.</td><td colspan=\"2\">Pair.</td><td colspan=\"2\">Trid.</td><td colspan=\"2\">Ins.</td></tr><tr><td>Method/Noise ratio</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td></tr><tr><td rowspan=\"2\">S2E</td><td>80.78</td><td>69.72</td><td>84.03</td><td>75.04</td><td>81.72</td><td>61.50</td><td>81.44</td><td>64.39</td><td>79.89</td><td>62.42</td></tr><tr><td>±0.88</td><td>±3.94</td><td>±1.01</td><td>±1.24</td><td>±0.93</td><td>±4.63</td><td>±0.59</td><td>±2.82</td><td>±0.26</td><td>±3.11</td></tr><tr><td rowspan=\"2\">MentorNet</td><td>80.92</td><td>74.67</td><td>80.37</td><td>71.69</td><td>77.98</td><td>69.39</td><td>78.02</td><td>71.56</td><td>77.02</td><td>68.17</td></tr><tr><td>±0.48</td><td>±1.17</td><td>±0.26</td><td>±1.06</td><td>±0.31</td><td>±1.73</td><td>±0.29</td><td>±0.93</td><td>±0.71</td><td>±2.52</td></tr><tr><td rowspan=\"2\">Co-teaching</td><td>82.35</td><td>77.96</td><td>83.87</td><td>73.43</td><td>80.94</td><td>72.81</td><td>81.17</td><td>74.37</td><td>79.92</td><td>73.29</td></tr><tr><td>±0.16</td><td>±0.39</td><td>±0.24</td><td>±0.62</td><td>±0.46</td><td>±0.92</td><td>±0.60</td><td>士0.64</td><td>±0.57</td><td>±1.62</td></tr><tr><td rowspan=\"2\">SIGUA</td><td>78.19</td><td>77.67</td><td>75.14</td><td>52.76</td><td>74.41</td><td>61.91</td><td>75.75</td><td>74.05</td><td>74.34</td><td>67.98</td></tr><tr><td>±0.22</td><td>±0.41</td><td>±0.36</td><td>±0.68</td><td>±0.81</td><td>±5.27</td><td>±0.53</td><td>±0.41</td><td>±0.39</td><td>±1.34</td></tr><tr><td rowspan=\"2\">JoCor</td><td>80.96</td><td>76.65</td><td>81.39</td><td>69.92</td><td>80.33</td><td>71.62</td><td>79.03</td><td>74.33</td><td>78.21</td><td>71.46</td></tr><tr><td>±0.25</td><td>±0.43</td><td>±0.74</td><td>±1.63</td><td>±0.20</td><td>±1.05</td><td>±0.13</td><td>±1.09</td><td>±0.34</td><td>±1.27</td></tr><tr><td rowspan=\"2\">CNLCU-S</td><td>83.03</td><td>78.25</td><td>85.06</td><td>75.34</td><td>83.16</td><td>73.19</td><td>82.77</td><td>74.37</td><td>82.03</td><td>73.67</td></tr><tr><td>±0.21</td><td>±0.70</td><td>±0.17</td><td>±0.32</td><td>±0.25</td><td>±1.25</td><td>±0.32</td><td>±1.37</td><td>士0.37</td><td>±1.09</td></tr><tr><td rowspan=\"2\">CNLCU-H</td><td>83.03</td><td>78.33</td><td>84.95</td><td>75.29</td><td>83.39</td><td>73.40</td><td>82.52</td><td>74.79</td><td>-81.93</td><td>73.58</td></tr><tr><td>±0.47</td><td>±0.50</td><td>±0.27</td><td>±0.80</td><td>±0.68</td><td>±1.53</td><td>±0.71</td><td>±1.13</td><td>±0.25</td><td>±1.39</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "270 As for performance measurements, we use test accuracy. In addition, we exploit the selected ratio of   \n271 training examples with the imbalanced classes, i.e., selected ratio=(# of selected imbalanced labels /   \n272 # of all selected labels). Intuitively, a higher selected ratio means the proposed method can make   \n273 better use of training examples with the imbalanced classes, following better generalization [18]. ",
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+        "text": "Experimental results. The test accuracy achieved on IM-MNIST and IM-F-MNIST is presented in Figure 2. Recall the experimental results in Table 1 and 2, we can see that the imbalanced issue is catastrophic to the sample selection approach when learning with noisy labels. For IM-MNIST, as can be seen, all the baselines have serious overfitting in the early stages of training. The curves of test accuracy drop dramatically. As a comparison, the proposed CNLCU-S and CNLCU-H can give a try to large-loss but less selected data which are possible to be clean but equipped with imbalanced labels. Therefore, our methods always outperform baselines clearly. In the case of Asym. $10 \\%$ , our methods achieve nearly $30 \\%$ lead over baselines. For IM-F-MNIST, we can also see that our methods perform well and always achieve about $5 \\%$ lead over all the baselines. Note that due to the huge challenge of this task, some baseline, e.g., S2E, has a large error bar. In addition, the baseline SIGUA performs badly. It is because SIGUA exploits stochastic integrated gradient underweighted ascent on large-loss examples, which makes the examples with imbalanced classes more difficult to be selected than them in other sample selection methods. ",
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+        "text": "The selected ratio achieved on IM-MNIST and IM-F-MNIST is presented in Table 4. The results explain well why our methods perform better on synthetic imbalanced noisy datasets, i.e., our methods can make better use of training examples with the imbalanced classes. Note that since we give a try to large-loss but less selected data in a conservative way, the selected ratio is still far away from the class prior probability on the test set, i.e., $10 \\%$ . However, a little improvement of the selection ratio can bring a considerable improvement of test accuracy. These results tell us that, in the sample selection approach when learning with noisy labels, improving the selected ratio of training examples with the imbalanced classes is challenging but promising for generalization. This practical problem deserves to be studied in depth. ",
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+        "text": "3.3 Experiments on Real-world Noisy Datasets ",
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+        "text": "Experimental setup. To verify the efficacy of our methods in the real-world scenario, we conduct experiments on the noisy dataset Clothing1M [59]. Specifically, for experiments on Clothing1M, we use the 1M images with noisy labels for training and 10k clean data for test respectively. Note that we do not use the $5 0 \\mathrm { k }$ clean training data in all the experiments. For preprocessing, we resize the image to $2 5 6 \\times 2 5 6$ , crop the middle $2 2 4 \\times 2 2 4$ as input, and perform normalization. The experiments on Clothing1M are performed once due to the huge computational cost. We leave $10 \\%$ noisy training data as a validation set for model selection. Note that we do not exploit the resampling trick during training [24]. Here, Best denotes the test accuracy of the epoch where the validation accuracy was optimal. Last denotes test accuracy of the last epoch. For the experiments on Clothing1M, we use a ResNet-18 pretrained on ImageNet as did in [52]. We also use the Adam optimizer and set the batch size to 64. During the training stage, we run 15 epochs in total and set the learning rate $8 \\times 1 0 ^ { - 4 }$ , $5 \\times 1 0 ^ { - 4 }$ , and $5 \\times 1 0 ^ { - 5 }$ for 5 epochs each. ",
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+            "Table 4: Selected ratio ( $\\overline { { \\mathcal { \\vert } } }$ ) on IM-MNIST and IM-F-MNIST. The best two results are in bold. "
+        ],
+        "table_body": "<table><tr><td>Dataset</td><td colspan=\"4\">IM-MNIST</td><td colspan=\"4\">IM-F-MNIST</td></tr><tr><td>Method/Noiseratio</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td></tr><tr><td>S2E</td><td>0.13 ±0.12</td><td>0.11 ±0.05</td><td>0.09 ±0.02</td><td>0.05 ±0.01</td><td>0.13 ±0.04</td><td>0.17 ±0.03</td><td>0.16 ±0.02</td><td>0.12 ±0.04</td></tr><tr><td>MentorNet</td><td>0.10 ±0.02</td><td>0.15 ±0.02</td><td>0.12 ±0.03</td><td>0.13 ±0.02</td><td>0.12 ±0.01</td><td>0.15 ±0.03</td><td>0.09 ±0.01</td><td>0.14 ±0.02</td></tr><tr><td>Co-teaching</td><td>0.09 ±0.03</td><td>0.07 ±0.02</td><td>0.05 ±0.01</td><td>0.12 ±0.01</td><td>0.17 ±0.05</td><td>0.04 ±0.00</td><td>0.13 ±0.04</td><td>0.07 ±0.01</td></tr><tr><td>SIGUA</td><td>0.04 ±0.00</td><td>0.04 ±0.00</td><td>0.01 ±0.00</td><td>0.02 ±0.00</td><td>0.03 ±0.00</td><td>0.02 ±0.00</td><td>0.04 ±0.00</td><td>0.00 ±0.00</td></tr><tr><td>JoCor</td><td>0.11 ±0.04</td><td>0.08 ±0.01</td><td>0.07 ±0.03</td><td>0.06 ±0.02</td><td>0.05 ±0.01</td><td>0.13 ±0.04</td><td>0.13 ±0.03</td><td>0.07 ±0.02</td></tr><tr><td>CNLCU-S</td><td>0.60 ±0.11</td><td>0.37 ±0.09</td><td>0.39 ±0.04</td><td>0.38 ±0.06</td><td>0.35 ±0.03</td><td>0.39 ±0.04</td><td>0.36 ±0.03</td><td>0.30 ±0.02</td></tr><tr><td>CNLCU-H</td><td>0.57 ±0.13</td><td>0.32 ±0.01</td><td>0.37 ±0.07</td><td>0.32&#x27; ±0.05</td><td>0.34 ±0.02</td><td>0.35 ±0.06</td><td>0.32</td><td>&#x27;0.28&#x27; ±0.03</td></tr></table>",
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+        "image_caption": [
+            "Figure 2: Test accuracy vs. number of epochs on IM-MNIST and IM- $F$ -MNIST. The error bar for standard deviation in each figure has been shaded. "
+        ],
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+        "text": "309 Experimental results. The results on Clothing1M are provided in Table 5. Specifically, the proposed   \n310 methods get better results than state-of-the-art methods on Best, which achieve an improvement of   \n311 $+ 1 . 2 8 \\%$ and $+ 0 . 9 9 \\%$ over the best baseline JoCor. Likewise, the proposed methods outperform all the   \n312 baselines on Last. We achieve an improvement of $+ 1 . 0 1 \\%$ and $+ 0 . 5 4 \\%$ over JoCor. Note that the   \n313 results are a bit lower than some state-of-art methods, e.g., [64] and [46], because of the following   \n314 reasons. (1). We follow [52] and use ResNet-18 as a backbone. The state-of-art methods [64, 46]   \n315 use ResNet-50 as a backbone. Our aim is to make the experimental results directly comparable with   \n316 previous papers [52] in the same area. (2). We only focus on the sample selection approach and do   \n317 not employ other advanced techniques, e.g., introducing the prior distribution [46] and combining   \nsemi-supervised learning [24, 39, 28]. ",
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+        "table_footnote": [
+            "Table 5: Test accuracy (%) on Clothing1M. The best two results are in bold. "
+        ],
+        "table_body": "<table><tr><td>Methods</td><td>S2E</td><td>MentorNet</td><td>Co-teaching</td><td>SIGUA</td><td>JoCor</td><td>CNLCU-S</td><td>CNLCU-H</td></tr><tr><td>Best</td><td>67.34</td><td>68.36</td><td>69.37</td><td>62.89</td><td>70.09</td><td>71.37</td><td>71.08</td></tr><tr><td>Last</td><td>65.90</td><td>67.42</td><td>68.62</td><td>58.73</td><td>69.75</td><td>70.76</td><td>70.29</td></tr></table>",
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+        "text": "318319 4 Conclusion ",
+        "text_level": 1,
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+        "text": "In this paper, we focus on promoting the prior sample selection in learning with noisy labels, which starts from concerning the uncertainty of losses during training. We robustly use the training losses at different iterations to reduce the uncertainty of small-loss examples, and adopt confidence interval estimation to reduce the uncertainty of large-loss examples. Experiments are conducted on benchmark datasets, demonstrating the effectiveness of our method. We believe that this paper opens up new possibilities in the topics of using sample selection to handle noisy labels, especially in improving the robustness of models on imbalanced noisy datasets. ",
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