Datasets:

zai-org
/

RPC-Bench

+# ENCODED PRIOR SLICED WASSERSTEIN AUTOENCODER FOR LEARNING LATENT MANIFOLD REPRESENTATIONS
+Anonymous authors Paper under double-blind review
+# ABSTRACT
+While variational autoencoders have been successful in a variety of tasks, the use of conventional Gaussian or Gaussian mixture priors are limited in their ability to encode underlying structure of data in the latent representation. In this work, we introduce an Encoded Prior Sliced Wasserstein AutoEncoder (EPSWAE) wherein an additional prior-encoder network facilitates learns an embedding of the data manifold which preserves topological and geometric properties of the data, thus improving the structure of latent space. The autoencoder and prior-encoder networks are iteratively trained using the Sliced Wasserstein (SW) distance, which efficiently measures the distance between two arbitrary sampleable distributions without being constrained to a specific form as in the KL divergence, and without requiring expensive adversarial training. To improve the representation, we use (1) a structural consistency term in the loss that encourages isometry between feature space and latent space and (2) a nonlinear variant of the SW distance which averages over random nonlinear shearing. The effectiveness of the learned manifold encoding is best explored by traversing the latent space through interpolations along geodesics which generate samples that lie on the manifold and hence are advantageous compared to standard Euclidean interpolation. To this end, we introduce a graph-based algorithm for interpolating along network-geodesics in latent space by maximizing the density of samples along the path while minimizing total energy. We use the 3D-spiral data to show that the prior does indeed encode the geometry underlying the data and to demonstrate the advantages of the network-algorithm for interpolation. Additionally, we apply our framework to MNIST, and CelebA datasets, and show that outlier generations, latent representations, and geodesic interpolations are comparable to the state of the art.
+# 1 INTRODUCTION
+Generative models have the potential to capture rich representations of data and use them to generate realistic outputs. In particular, Variational AutoEncoders (VAEs) (Kingma & Welling, 2014) can capture important properties of high-dimensional data in their latent embeddings, and sample from a prior distribution to generate realistic images. Whille VAEs have been very successful in a variety of tasks, the use of a simplistic standard normal prior is known to cause problems such as under-fitting and over-regularization, and fails to use the network’s entire modeling capacity (Burda et al., 2016). Gaussian or Gaussian mixture model (GMM) priors are also limited in their ability to represent geometric and topological properties of the underlying data manifold. High-dimensional data can typically be modeled as lying on or near an embedded low-dimensional, nonlinear manifold (Fefferman et al., 2016). Learning improved latent representations of this nonlinear manifold is an important problem, for which a more flexible prior may be desirable.
+Conventional variational inference uses Kullback-Leibler (KL) divergence as a measure of distance between the posterior and the prior, restricting the prior distribution to cases that have tractable approximations of the KL divergence. Many works such as Guo et al. (2020); Tomczak & Welling (2018); Rezende & Mohamed (2015) etc. have investigated the use of more complicated priors (notably GMMs) which lead to improved latent representation and generation compared to a single Gaussion prior. Alternate approaches such as adversarial training learn arbitrary priors by using a discriminator network to compute a divergence (Wang et al., 2020), however they are reported to be harder to train and are computationally expensive.
+In this work, we introduce the Encoded Prior Sliced Wasserstein AutoEncoder (EPSWAE), which consists of a conventional autoencoder architecture and an additional prior-encoder network that learns an unconstrained prior distribution that encodes the geometry and topology of any data manifold. We use a type of Sliced Wasserstein (SW) distance (Bonnotte, 2013; Bonneel et al., 2015), a concept from optimal transport theory that is a simple and convenient alternative to the KL divergence for any sampleable distributions. A Sliced Wasserstein AutoEncoder (SWAE) that regularizes an autoencoder using SW distance was proposed in Kolouri et al. (2018a). Several works improve the SW distance through additional optimizations (Deshpande et al., 2019; Chen et al., 2020b; Deshpande et al., 2018), and show improved generation, however involve additional training and use a fixed (usually Gaussian) prior. Kolouri et al. (2019) presents a comparison between max-SW distance, polynomial generalized SW distances, and their combinations. In contrast, we use a simple and efficient nonlinear shearing which requires no additional optimization.
+Additionally, we introduce a structural consistency term that encourages the latent space to be isometric to the feature space, which is typically measured at the output of the convolutional layers of the data encoder. Variants of this penalty have previously been used to encourage isometry between the latent space and data space (Yu et al., 2013; Benaim & Wolf, 2017; Sainburg et al., 2018). The structural consistency term further encourages the prior to match the encoded data manifold by preserving feature-isometry, which in turn is expected to assist with encoding the geometry of the data manifold , thus leading to improved latent representations.
+A key contribution of our work is the graph-based geodesic-interpolation algorithm. Conventionally, VAEs use Euclidean interpolation between two points in latent space. However, since manifolds typically have curvature, this is an unintuitive distance metric that can lead to unrealistic intermediate points. Our goal is to learn a true representation of the underlying data manifold, hence it is natural to interpolate along the manifold geodesics in latent space. Several works such as Shao et al. (2018); Miolane & Holmes (2020b) endow the latent space with a Riemannian geometry and measure corresponding distances, however these are difficult and involve explicitly solving expensive ordinary differential equations. In this work, we introduce ‘network-geodesics’, a graph-based method for interpolating along a manifold in latent space, that maximizes sample density along paths while minimizing total energy. This involves first generating a distance graph between samples from the prior. Then this network is non-uniformly thresholded such that the set of allowable paths from a given sample traverse high density regions through short hops. Lastly, we use a shortest path algorithm like Dijkstra’s algorithm (Dijkstra, 1959) to identify the lowest ‘energy’ path between two samples through the allowable paths. Since the prior is trained to learn the data manifold, the resulting network-geodesic curves give a notion of distance on the manifold and can be used to generate realistic interpolation points with relatively few prior samples.
+The novel contributions of this work are:
+• We introduce a novel architecture, EPSWAE, that consists of a prior-encoder network that is efficiently trained (without expensive adversarial methods) to generate a prior that encodes the geometric and topological properties of data. We introduce a novel graph-based method for interpolating along network-geodesics in latent space through maximizing sample density while minimizing total energy. We show that it generates natural interpolations through realistic images.
+• Improvements to the latent space representation are obtained by using a structural consistency term in the loss that encourages isometry between feature space and latent space and by using a simple and efficient nonlinear variant of the SW distance.
+# 2 BACKGROUND AND RELATED WORK
+Several works have attempted to increase the complexity of the prior in order to obtain better latent representations. Most data can mathematically be thought of as living on a high dimensional manifold. In an image dataset, for instance, if images in high dimensional pixel space are effectively parametrized using a small number of continuous variables, they will lie on or near a low dimensional manifold (Lu et al., 1998). Many works such as Weinberger & Saul (2006); Rahimi et al.
+(2005) have investigated image and video manifolds. Encoding and exploiting the topological and geometric properties of such data is a question of increasing interest.
+Some representative works are Dilokthanakul et al. (2016) which shows improved unsupervised clustering of latent space by using GMM priors, Takahashi et al. (2019) which uses the density ratio trick to calculate the KL divergence using implicit priors, Rezende & Mohamed (2015), which maps a Gaussian prior by a series of explicit transformations, Guo et al. (2020) which learns a GMM prior using an approximate ELBO, Goyal et al. (2017) which uses a hierarchical Bayesian framework, and VampPrior Tomczak & Welling (2018) which learns a two-layer hierarchical GMM from aggregated posteriors. Yin & Zhou (2018); Molchanov et al. (2019); Liutkus et al. (2019) expand the variational family to incorporate implicit variational distributions while providing exact theoretical support.
+The KL divergence is tractable only for Gaussian distributions, however the Wasserstein distance of 1D projections (also called Sliced Wasserstein (SW) distance) has a closed-form for any arbitrary distribution (Kolouri et al., 2018b). Wasserstein distances (Villani, 2003) have several of the same properties as KL divergence, but often lead to better sampling (Gulrajani et al., 2017; Tolstikhin et al., 2018), and have been used in several machine learning applications (e.g. Arjovsky et al. (2017); Tolstikhin et al. (2018); Kolouri et al. (2018b)).
+A Sliced Wasserstein AutoEncoder (SWAE) that regularizes an autoencoder using SW distance was proposed in Kolouri et al. (2018a). There exist several variants of SW distance that have been successful at improving generative ability - for example, Deshpande et al. (2018); Chen et al. (2020b) train discriminator-like networks to perform nonlinear transformations (instead of purely linear projections) whereas Deshpande et al. (2019) introduces max-SW distance, which looks for the best linear projection, generalized in Nguyen et al. (2020) to finding optimal distributions of projections. Wu et al. (2019) uses a different optimization approach to computing the SW distance based on the Kantorovich dual formulation. All of the above use some form of additional training. In contrast, we use a simple and efficient nonlinear shearing which requires no additional optimization and is sufficient for improving latent representation, however, our method can easily incorporate other SW distances if desired.
+Alternatively, many works such as Wang et al. (2020); Makhzani et al. (2015); Arjovsky et al. (2017); Sainburg et al. (2018) replace the KL divergence with an adversarial loss, however adversarial methods tend to be significantly more expensive and difficult to optimize. In higher dimensions, using a discriminator network as in adversarial approaches is a natural way of implicitly computing an equivalent of the Wasserstein-1 distance (Arjovsky et al., 2017; Tolstikhin et al., 2018). However, using the SW distance is much simpler and more efficient. Adversarial training for interpolation in Sainburg et al. (2018) uses a structural consistency term that encourages relative distances in data space to be preserved in latent space. Similar distance-preserving terms have also been used successfully in Yu et al. (2013); Benaim & Wolf (2017), however Euclidean distances in data space can be a poor measure of the natural geometry of the data. In our work, we preserve relative distances in latent space and feature space to improve latent encoding, where features are extracted at the output of the last convolutional layer in the data encoder.
+Several manifold learning techniques such as Dollar et al. (2007); Weinberger & Saul (2006) com-´ pute embeddings of high-dimensional data but are less general and lack generative or interpolation abilities. The hyperspherical VAE Davidson et al. (2018) outperforms the standard VAE for data residing on a hyperspherical manifold. Miolane & Holmes (2020a) formulates a Riemannian VAE, however computing its ELBO is challenging. Along similar lines, Arvanitidis et al. (2017) shows that under certain conditions, a Riemannian metric is naturally induced in the latent space, and uses it to compute geodesics. In Chen et al. (2020a), a flat manifold is approximated by penalizing curvature in latent space, and geodesics are defined through Euclidean distances on the flat manifold.
+There exists limited work on integrating graphical structures with generative models (Kipf & Welling, 2016). Hadjeres et al. (2017) studies monophonic music using a latent space geodesic regularization that allows interpolations in the latent space to be more meaningful, giving a notion of geodesic distance. Several approaches such as Tenenbaum et al. (2000); Bernstein et al. (2000); Klein & Zachmann (2004); Memoli & Sapiro (2005); Luo & Hu (2020) approximate geodesics on ´ point clouds by, for instance, building a nearest-neighbor network on the manifold and applying a shortest path algorithm, however, these often require many samples in order generate reasonable approximations and aren’t robust to noise (Sober et al., 2020). Inspired by this, we introduce an energy-based network algorithm to identify network-geodesics in latent space with relatively few, noisy samples.
+# 3 EPSWAE
+# 3.1 NONLINEAR SLICED WASSERSTEIN DISTANCE
+Wasserstein distances are a natural metric for measuring distances between probability distributions, however they xare difficult to compute in dimensions two and higher. The Sliced Wasserstein distance $d _ { S W }$ averages over the Wasserstein distance of 1D projections (see Appendix A for derivation). Several works (e.g. Kolouri et al. (2019); Deshpande et al. (2019)) discuss why a conventional SW distance may be sub-optimal as a large number of linear projections may be required to distinguish two distributions. In this work we use a Nonlinear Sliced Wasserstein distance $( d _ { N S W } )$ , an averaging procedure over (random) nonlinear 1D projections, between two distributions $\mu$ and $\nu$ defined as:
+$$
+d _ { N S W } ( \mu , \nu ) = \mathbb { E } d _ { S W } ( \mathcal { N } _ { * } ^ { \zeta , \gamma } \mu , \mathcal { N } _ { * } ^ { \zeta , \gamma } \nu ) \approx \frac { 1 } { L } \sum _ { \ell = 1 } ^ { L } d _ { S W } ( \mathcal { N } _ { * } ^ { \zeta _ { \ell } , \gamma _ { \ell } } \mu , \mathcal { N } _ { * } ^ { \zeta _ { \ell } , \gamma _ { \ell } } \nu )
+$$
+where $L$ is the number of nonlinear transformations, and $\zeta , \gamma$ are chosen to be normal random variables with mean 0, and variance matching that of $\mu$ , and $\mathcal { N }$ is defined below. We define the “push-forward” (also known as “random variable transform”) as follows: given any function $f :$ $X  Z$ and probability measure $\mu$ , we define $f _ { * } \mu ( A ) = \mu ( f ^ { - 1 } ( A ) )$ for all $A \subset Z$ measurable. Most importantly, to generate samples of $z \sim f _ { * } \mu$ one simply generates $x \sim \mu$ and defines $z =$ $f ( x )$ .
+The nonlinear transformations are related to a special case of the generalized SW distance (Kolouri et al., 2019) (with the difference that we choose distribution-dependent shear frequencies that are most likely to produce deformations which highlight differences in the measures, thus breaking the homogeneity condition H2 in Kolouri et al. (2019)), and are given by
+$$
+\begin{array} { r } { \mathcal { N } ^ { \zeta _ { \ell } , \gamma _ { \ell } } ( x ) = x + \zeta _ { \ell } \sin ( \gamma _ { \ell } \cdot x ) . } \end{array}
+$$
+In all our experiments, we use $L = 5$ nonlinear transformations with 50 linear 1D projections each. Several works such as Deshpande et al. (2018; 2019); Wu et al. (2019); Nguyen et al. (2020); Chen et al. (2020b) improve the SW distance through some form of optimization or training an additional discriminator-like network, and show improved generative results (with Deshpande et al. (2018) being the most similar to our method). We did not implement these other variants, however they could easily be used in conjunction with our method. Our choice of nonlinearity is motivated by certain tail-behavior considerations (such as boundedness and non-saturation of nonlinearity), along with computational efficiency, and for our goals, a simple nonlinearity was sufficient. See section 4.2 and Appendix E for computational time and loss comparisons with some other nonlinearities discussed in Kolouri et al. (2019). For further discussion on NSW distance and derivation see Appendix A.
+# 3.2 ALGORITHM DETAILS
+In the Encoded Prior Sliced Wasserstein AutoEncoder (EPSWAE) (see Fig. 1 for schematic), let’s define the data encoder as $\Psi _ { E }$ , the decoder $\Psi _ { D }$ , and the prior-encoder as $\Psi _ { P E }$ , each with parameters $\phi _ { E } , \phi _ { D } , \phi _ { P E }$ respectively. Input samples $\mathbf { x } ^ { ( j ) } \sim P _ { X }$ , where $P _ { X }$ is the probability distribution of the input data, are passed through $\Psi _ { E }$ to generate posterior samples $\mathbf { z } ^ { ( j ) } \sim ( \Psi _ { E } ) _ { * } P _ { X }$ by setting $\mathbf { z } ^ { ( j ) } =$ $\Psi _ { E } ( \mathbf { x } ^ { ( j ) } )$ . Similarly, prior-encoder input samples $\xi ^ { ( j ) } \sim \mu$ , where $\mu$ is the probability distribution of the input to the prior-encoder (chosen to be a mixture of Gaussians) from a distribution, are passed through $\Psi _ { P E }$ to generate prior samples $\mathbf { y } ^ { ( j ) } \sim ( \Psi _ { P E } ) _ { * } \mu$ by setting $\mathbf { y } ^ { ( j ) } = \Psi _ { P E } ( \xi ^ { ( j ) } )$ . The priorencoder network and the autoencoder (data encoder and decoder) network are trained iteratively in a two step process:
+1. Given a minibatch, parameters of the autoencoder $( \phi _ { E } , \phi _ { D } )$ are trained for $k _ { 1 }$ steps while parameters of the prior encoder $( \phi _ { P E } )$ are fixed. The loss function for the autoencoder consists of the reconstruction error, the NSW distance, and a Feature Structural Consistency (FSC) term $\mathcal { L } _ { F S C }$ (given in Eqn. 4):
+$$
+\mathcal { L } _ { A E } = \alpha \mathbb { E } _ { { \mathbf { x } } \sim P _ { X } } \mathcal { L } _ { r e c } \left( { \mathbf { x } } , \bar { \mathbf { x } } \right) + \beta d _ { N S W } ( ( \Psi _ { E } ) _ { * } P _ { X } , ( \Psi _ { P E } ) _ { * } \mu ) + \kappa \mathcal { L } _ { F S C } ,
+$$
+where $\bar { \mathbf { x } } = \Psi _ { D } \big ( \Psi _ { E } ( \mathbf { x } ) \big )$ is the autoencoder output. Note that $d _ { N S W }$ is efficiently computed just from samples of the distributions.
+In this measure theoretic notation, $( \Psi _ { E } ) _ { * } P _ { X }$ is the posterior distribution (denoted as $q _ { \phi _ { E } } ( z | x )$ in Bayesian literature), and $\left( \Psi _ { P E } \right) _ { * } \mu$ is the prior distribution (denoted as $p _ { \phi _ { P E } } ( z )$ in Bayesian literature). Similar to $\beta$ -VAE ((Higgins et al., 2016)), the hyperparameters $\alpha , \beta , \kappa$ are tuneable; in our experiments they are fixed, but it could be advantageous to have them vary over training epochs.
+$\mathcal { L } _ { F S C }$ encourages relative distances in feature space to be preserved in latent space, where features are extracted at the output of the last convolutional layer in the encoder (or at input data if no convolutional layers are present). $\mathcal { L } _ { F S C }$ (adapted from Sainburg et al. (2018)) for two point clouds $F = [ f _ { 1 } , \dots f _ { N } ]$ in feature space and the corresponding points $Z = [ z _ { 1 } , \dotsc z _ { N } ]$ in latent space in a minibatch of size $N$ is given by:
+$$
+\mathcal { L } _ { F S C } = \frac { 1 } { N ^ { 2 } } \sum _ { \ell , j = 1 } ^ { N } \left( \log \left( 1 + \frac { \| f _ { j } - f _ { \ell } \| ^ { 2 } } { \frac { 1 } { N ^ { 2 } } \sum _ { m , n } \| f _ { m } - f _ { n } \| ^ { 2 } } \right) - \log \left( 1 + \frac { \| z _ { j } - z _ { \ell } \| ^ { 2 } } { \frac { 1 } { N ^ { 2 } } \sum _ { m , n } \| z _ { m } - z _ { n } \| ^ { 2 } } \right) \right) ^ { 2 } .
+$$
+2. In the second step of minimization, the parameters of the prior-encoder $( \Psi _ { P E } )$ are trained for $k _ { 2 }$ steps while parameters of the autoencoder $( \phi _ { E } , \phi _ { D } )$ are fixed. The loss function for the prior-encoder consists of the NSW distance between the prior and posterior:
+$$
+\mathcal { L } _ { P E } ( \mathbf { x } ) = d _ { N S W } ( ( \Psi _ { E } ) _ { * } P _ { X } , ( \Psi _ { P E } ) _ { * } \mu ) .
+$$
+The pseudocode is outlined in Appendix B. Additional architecture and training details for the experiments in this paper are in Appendix C.
+# 3.3 INTERPOLATION AND APPROXIMATE GEODESICS
+EPSWAE attempts to learn a representation of the embedded manifold geometry that the data lies along. The question arises: how does one make use of this representation? A natural way of interpolating data which lies on a manifold is through geodesics or paths on the manifold which traverse dense regions. Here we present a network algorithm for efficiently approximating networkgeodesics with relatively few, noisy samples from the prior, by encouraging connections through high density regions of latent space while minimizing total energy.
+1. Gather samples of the posterior $( \Psi _ { E } ) _ { * } P _ { X }$ , i.e., $\Psi _ { E } ( x )$ with $x$ is a minibatch of data. Additional prior samples can be used to supplement this if desired;
+2. For each sample $j$ , compute the average Euclidean distance $c _ { j }$ of the $\mathbf { k }$ -nearest neighbors;
+3. Generate a thresholded network: Let samples be represented as nodes. For a given threshold value $t$ , sample $j$ is connected to sample $i$ with edge weight $d ^ { h } ( i , j )$ if $d ( \bar { i } , j ) < t \cdot c _ { j }$ . Here $h$ is the energy parameter in the edge weight (chosen to be 1 or 2 in experiments, with $h = 2$ encouraging shorter hops). Sample-specific thresholding, i.e., thresholding dependent on $c _ { j }$ , increases the probability of a ‘central’ node having a connection. Thus, this encourages (1) paths through high density regions, i.e., high-degree nodes, and (2) traversal of latent space through short hops as a consequence of localization of paths, i.e., paths existing only between nearby points due to thresholding.
+4. Continue increasing $t$ until the graph is connected, i.e., there exists a path, direct or indirect, from every sample to every other sample. Then, use Dijkstra’s algorithm (Dijkstra,
+1959) to identify network-geodesics with least total energy through allowable paths on the thresholded network.
+# 4 EXPERIMENTS
+We run experiments using EPSWAE on three datasets. First, we use a 3D spiral data, where the latent space can be visualized easily, to demonstrate that the learned prior captures the geometry of the 3D spiral. We also present here the advantages of the network-geodesic interpolation over linear interpolation. Then we present interpolation and generation results on the MNIST (LeCun et al., 2010) and CelebA (Liu et al., 2018) datasets.
+![](images/1080f389b94013a36d2b12c45bc3a31dcf6f2cc36e625c650cc56aa48c2c035f.jpg)
+Figure 1: (a) is a schematic of the EPSWAE architecture. The red arrows indicate calculation of the loss terms. The prior-encoder generates a prior in latent space. (b) is simple example of a shape where interpolating between points A and B ‘through the manifold’ (red dashed line through point C) is desired, since linear interpolation (black line) leads out of the concave hull.
+In all of the experiments, the prior-encoder $\Psi _ { P E }$ consists of three fully connected hidden layers with ReLU activation. A sampleable distribution with dimension larger than the dimension of latent space is input to the prior-encoder which yields the prior. In principle, samples from any distribution can be the input $\mu$ to the prior-encoder. In the artificial data set we use a normal distribution, whereas for MNIST and CelebA we use a mixture of 10 Gaussians with random i.i.d means (from Gaussian with $\sigma = 2$ ). The schematic of the architecture is presented in Fig. 1(a). Note that the specifics of the layers shown in this schematic are for the image datasets MNIST and CelebA, the 3D Spiral data does not have convolutional layers. Details of the architecture, training, and hyperparameters for each dataset are given in Appendix C. The optimizer Adam (Kingma & Ba, 2014) with a learning rate of 0.001 was used for learning in both networks: the prior-encoder and the autoencoder.
+# 4.2 LEARNING A LATENT MANIFOLD
+We consider a 3D spiral randomly embedded to 40D space with noise as follows: (a) The formula for the spiral is given by $( x ( t ) , \dot { y } ( t ) , z ( t ) ) = ( t \cos 7 \dot { t } \pi / 4 , t \sin 7 t \pi / 4 , 2 t )$ , (b) a random $4 0 \times 3$ matrix is generated with i.i.d normal entries to map the spiral into $\mathbb { R } ^ { 4 0 }$ , and (c) Gaussian noise with standard deviation 0.1 is added. The input to the prior-encoder is a Gaussian in $\mathbb { R } ^ { 4 0 }$ , and the latent space is $\mathbb { R } ^ { 3 }$ . For these tests, the data encoder, prior-encoder, and decoder all consist of three Fully Connected (FC) layers with 40 nodes each and ReLU activations.
+![](images/2f673d71ac16e3c5724bbd2ea1c56de2f69ceb694860f1ab89cfb1dbc5534610.jpg)
+Figure 2: (a) side view and (b) top view of samples of the prior (green) and posterior (blue) in 3D latent space generated by EPSWAE after 100 epochs on the high dimensional input. The red line shows the interpolation along network-geodesics between two prior samples. The brown line shows the trajectory of linear interpolation between two points on the spiral.
+As seen in Figure 2, after training, the posterior matches the spiral almost exactly and the prior learns the geometry of the spiral. A Nonlinear Sliced Wasserstein (NSW) distance and structural consistency term (where latent space is isometric with data space since convolutional layers aren’t used) are used in the loss in Eqns. 3 and 5. A comparison highlighting the effects of these terms is shown in Fig. 5 in the Appendix D. We observe that the use of the NSW distance (as opposed to linear SW distance) improves the accuracy of the learned prior.
+Manifold interpolation (in red in Fig. 2) uses the network-algorithm outlined in section 3.3. The interpolation is seen to have the desired form on the manifold, i.e., it approximates geodesics on the manifold. The larger the number of samples of the prior, the smoother the corresponding interpolation. In contrast, linear interpolation (brown line) between two points on the spiral does not capture the geometry at all, and goes through regions that are largely empty and untrained. In higher dimensional datasets, this may result in unrealistic interpolation. In order to traverse the manifold, both the prior-encoder (that allows the prior to capture the geometry of the spiral), and the networkinterpolation algorithm are equally essential. Comparisons with baselines SWAE (Kolouri et al., 2018a) and VAE (Kingma & Welling, 2014) are shown in Appendix Fig. 6, and EPSWAE is seen to significantly outperform them in learning an improved latent representation.
+Computational cost The choice of nonlinearity was, in part, motivated by computational efficiency and simplicity, i.e., not requiring additional optimizations every evaluation or the training of a discriminator-like network to select optimal transformations. Our sinusoidal shear nonlinearity is closely related to a special case of the generalized SW distance defined in Kolouri et al. (2019) (see discussion in section 3.1 for differences ). Here, we present a comparison of computational time with two polynomial-type nonlinearities discussed in Kolouri et al. (2019) - cubic and quintic. Computations done on a laptop with Intel Core i7-10710 CPU, 16GB RAM, and computed over 1000 runs. In practice, the losses were indistinguishable with choice of nonlinearity (see Appendix E), whereas the sine shear is slightly less expensive (see Table 1). However, the complexity of cubic and quintic nonlinearities are exponential in dimension and hence are impractical for larger data.
+Table 1: Comparison of computational time for nonlinearities in SW distance.
+<table><tr><td>sine-shear NSW</td><td></td><td>cubic NSW</td><td>quintic NSW</td></tr><tr><td>Computational Cost</td><td>0.0050s ±0.0006s</td><td>0.0054s±0.0007s</td><td>0.0060s ± 0.0008s</td></tr></table>
+In order to demonstrate one of the advantages of using a prior encoder in improving latent structure, we consider generation with an increased probability of sampling from outliers, i.e., by increasing the standard deviation of the distribution that feeds into the prior encoder. Studying generation from outliers may seem nonstandard, however, it is informative in the study of how data is encoded in the latent space. In Fig. 3 compare our results with the baseline SWAE (Kolouri et al., 2018a) in order to assess the effect of the prior encoder in improving latent structure. It is worth noting that while there exist several recent works that build upon SWAE for better generation, they do not attempt to solve the same problem as ours (learning latent manifolds). As seen in Fig. 3, the behavior of SWAE and EPSWAE at outliers is very different. For large $\sigma$ , SWAE tends to generate unrealistic faces with distorted colorations, whereas EPSWAE is more likely to generate realistic faces (albiet with increased mode collapse at large $\sigma$ ). We see that EPSWAE encodes coherent information in a large region of latent space as a consequence of nonlinear transformations through the prior encoder. Since we are not competing with state of the art in generation here, we use downsized images and a small, unsophisticated model compared to state of the art methods.
+Both SWAE and EPSWAE use the same data encoder and decoder, and are independently optimized: the encoder is (after downsizing CelebA images to $6 4 x 6 4$ ) Conv (3,16,3) BatchNorm $ \mathrm { { R e L u } }  \mathrm { { M a x P o o l } } ( 2 , 2 )  \mathrm { { C o n v } ( 1 6 , 3 2 , 3 )  \mathrm { { B a t c h N o r m }  \mathrm { { R e L u } }  \mathrm { { M a x P o o l } } } }$ $( 2 , 2 ) $ Conv $( 3 2 , 6 4 , 3 ) ~  ~ \mathrm { { B a t c h N o r m } ~  ~ \mathrm { { R e L u } ~ f o l } }$ llowed by two FC layers of 512 and 256 nodes respectively with a leaky ReLu nonlinearity and the decoder is the reverse (replace Conv layers with Conv-transpose and MaxPool with Upsample). The prior-encoder consists of three FC layers from $1 8 6 \mathrm { D }  1 2 8 \mathrm { D }$ latent space. See Appendix C for more details and hyperparameter values.
+![](images/a141f17f4a600ea3cb98c5d0b41f009bc32335b404616544c9c12c3bb04fe332.jpg)
+Figure 3: Images generated after 100 epochs from prior samples in (a) EPSWAE (b) Baseline SWAE at increasing standard deviations $\sigma$ , i.e., progressively more ‘outlying’.
+We also present a comparison with SWAE on CelebA generation from samples around the mean in Appendix G, and find that EPSWAE sees a marginal improvement in the quality of faces. Generation on MNIST using EPSWAE (and comparisons with SWAE (Kolouri et al., 2018a) and VAE (Kingma & Welling, 2014)) are shown in Appendix F. MNIST results are obtained with $5 D$ latent space. EPSWAE generation is more natural, whereas both other baselines generate some bloated and unidentifiable numbers.
+Table 2: FID score comparison generated with 10000 samples (lower is better)
+<table><tr><td>SWAE</td><td>EPSWAE (bl)</td><td>EPSWAE(FSConly)</td><td>EPSWAE (NSW only)</td><td>ESPWAE (FSC +NSW)</td></tr><tr><td>178.04</td><td>161.55</td><td>162.14</td><td>161.06</td><td>157.86</td></tr></table>
+For completeness, in Table 4.3 we present Frechet Inception Distance (FID) (Salimans et al., 2016) scores with 10000 images with $\sigma = 1$ (code from Seitzer (2020)). It is important to note that one can have good generation, despite learning a poor latent representation (and vice-versa), so scores like FID may not be the best way to evaluate the latent structure. However, comparison between SWAE and EPSWAE-baseline(bl) indicate that the prior-encoder significantly improves generation.
+Computational cost : The computational cost of the different terms in the EPSWAE loss on the CelebA dataset is as follows: (1) FSC computation: $0 . 0 0 0 6 6 s \pm 0 . 0 0 0 1 s$ , (2) linear SW distance computation: $0 . 0 1 5 6 1 s \pm 0 . 0 0 0 5 s$ (3) sine-shear NSW computation: $0 . 0 1 5 0 s \pm 0 . 0 0 0 7 s$ (with same number of total projections). All computations for CelebA were done on a workstation with one NVIDIA Tesla P100 GPU, two Intel Xeon E5-26660v4 CPUs, and 64Gb RAM.
+# 4.4 INTERPOLATION
+In this paper, we introduce a graph-based algorithm that interpolates over network-geodesics in curved latent manifolds, thus capturing properties of the geometry. Here we show that we obtain smooth interpolation using the network-geodesics algorithm, in contrast with linear interpolations that often tend to generate unrealistic intermediate images.
+The start and end points correspond to real images (posterior samples). Figure 4(a) presents a comparison of linear interpolation (top) vs interpolation through network-geodesics (bottom) described in section 3.3 on CelebA. One can see that linear interpolations often go through unrealistic images (this corresponds to regions in latent space where training is limited), whereas geodesic interpolations go through more realistic and less blurry faces. This also serves as evidence, that the latent space for CelebA contains some natural structure that can be exploited.
+![](images/e21025a64aaacbf7ebc4c9f2b01d82d7eb850cd87447f989d37491e740f8e177.jpg)
+Figure 4: (a) Comparison of linear vs geodesic interpolation using EPSWAE. The top row of each row-pair is linear, and the bottom is geodesic-interpolation. (b) Randomly chosen instances for geodesic interpolations for MNIST and CelebA. The first and last images are reconstructions of real data, and the interpolations traverse through samples of the prior using the network-geodesic algorithm. A total of 400 samples are used in all cases.
+Figure 4 (b) shows examples of interpolations on MNIST and CelebA datasets. MNIST interpolations are smooth and intuitive, for instance, in the bottom left MNIST panel in Fig. 4(b), the top part of a ‘7’ first changes to a $\mathbf { \nabla } ^ { 6 } 9 ^ { \bullet }$ naturally, followed by transformation of the $\cdot 9 ^ { \cdot }$ to a $\cdot _ { 4 } ,$ . Interpolation along network-geodesics ensures that reconstructions of intermediate samples are realistic. Interpolations on CelebA are smooth and pass through intermediate images that could arguably pass for celebrities these days. The state of the art interpolations along network-geodesics on the manifold indicate that the learned prior does indeed encode the data manifold. For all interpolations shown in Fig. 4, we use energy parameter $h = 2$ . Comparisons between interpolations corresponding to $h = 1$ and $h = 2$ are presented in Appendix 10. In experiments, energy parameter is not found to have a significant impact on the quality of interpolations. Additionally, comparisons of linear interpolation between equivalent networks of EPSWAE and SWAE are presented in Appendix I.
+# 5 CONCLUSION
+We introduce the Encoded Prior Sliced Wasserstein AutoEncoder (EPSWAE) that learns improved latent representations through training an encoded prior to approximate an embedding of the data manifold that preserves geometric and topological properties. The learning of an arbitrary shaped prior is facilitated by the use of the Sliced Wasserstein distance, which can be efficiently computed from samples only. We use a nonlinear variant of SW distance to capture differences between two distributions more efficiently, and employ a feature structural consistency term to improve the latent space representation. Finally, we introduce an energy-based algorithm to identify networkgeodesics in latent space that maximize path density while minimizing total energy. We demonstrate EPSWAE’s ability to learn the geometry and topology of a 3D spiral from a noisy 40D embedding. We also show that our model embeds information in large regions of latent space, leading to better outlier generation. Lastly, we show that our geodesic interpolation results on MNIST and CelebA are efficient and comparable to state of the art techniques. Our code is publicly available.
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+# A BACKGROUND ON THE SLICED WASSERSTEIN DISTANCE
+Wasserstein distances provide a natural metric for measuring distances between probability distributions based on optimal transport, i.e., the cost of deforming one probability distribution into another. For a measurable cost function $c ( x , y )$ for $x , y \in \mathbb { R } ^ { d }$ , and probability distributions $\mu$ and $\nu$ on $\mathbb { R } ^ { d }$ , we define the $p$ -Wasserstein distance between the distributions as
+$$
+d _ { W , p } ( \mu , \nu ) = \operatorname* { i n f } _ { \Gamma \in \Pi ( \mu , \nu ) } \left( \int _ { \mathbb { R } ^ { d } \times \mathbb { R } ^ { d } } c ^ { p } ( x , y ) \mathrm { d } \Gamma ( x , y ) \right) ^ { 1 / p } ,
+$$
+where $\Pi$ is the set of all joint distributions with marginals $\mu$ and $\nu$ (see Villani (2003)).
+This Wasserstein distance between probability distributions is extremely difficult and computationally intensive in dimensions two and higher, i.e., $d \geq 2$ . However, in dimension one, there exists a closed-form solution (Villani, 2003). A simple algorithm for computing the 1D Wasserstein distance is given as follows (see for example (Kolouri et al., 2018a)): (a) Generate $N$ one dimensional i.i.d samples $x _ { j } \sim \mu$ , $y _ { j } \sim \nu$ ; (b) sort each list $X = [ x _ { 1 } , \ldots , x _ { N } ] , Y = [ y _ { 1 } , \ldots , y _ { N } ]$ into ascending order denoted by $\tilde { X } , \tilde { Y }$ respectively; (c) define the approximation to the 1D p-Wasserstein distance
+$$
+d _ { W , p } ( \mu , \nu ) \approx \left( \frac { 1 } { N } \sum _ { j = 1 } ^ { N } c ^ { p } ( \tilde { x _ { j } } , \tilde { y _ { j } } ) \right) ^ { 1 / p } ,
+$$
+where $\tilde { x } _ { j } \in \tilde { X } , \tilde { y } _ { j } \in \tilde { Y }$ . The Sliced Wasserstein (SW) distance defines a metric on probability measures (Bonnotte, 2013) which provides an alternative to Eqn. 6 by exploiting the computational feasibility of the 1D Wasserstein distance in Eqn. 7. It involves averaging over one-dimensional orthogonal projections $\pi ^ { \theta } x : = ( \theta \cdot x ) \theta$ as follows:
+$$
+d _ { S W } ( \mu , \nu ) = \biggl ( \int _ { \mathbb { S } ^ { d - 1 } } d _ { W , p } ^ { p } ( \pi _ { * } ^ { \theta } \mu , \pi _ { * } ^ { \theta } \nu ) d S ( \theta ) \biggr ) ^ { 1 / p } ,
+$$
+The SW distance has seen a variety of implementations (Bonneel et al., 2015; Kolouri et al., 2018b;
+Lee et al., 2019).
+The integral in Eqn. 8 can be easily approximated by sampling $M$ random one-dimensional vectors $\theta _ { k }$ uniformly on $\mathbb { S } ^ { d - 1 }$ and computing
+$$
+d _ { S W } ( \mu , \nu ) \approx \left( \frac { 1 } { M } \sum _ { k = 1 } ^ { M } d _ { W , p } ^ { p } ( \pi _ { * } ^ { \theta _ { k } } \mu , \pi _ { * } ^ { \theta _ { k } } \nu ) \right) ^ { 1 / p } .
+$$
+Several works (for example Nguyen et al. (2020); Deshpande et al. (2019)) have discussed reasons why this often does not result in the best computational method. Linear projections may be suboptimal for extracting information about the differences between $\mu$ and $\nu$ , since a large number of linear projections may be required to get an accurate approximation for $d _ { S W }$ . Several works have suggested possible methods for improving the effectiveness of the SW distance (Chen et al., 2020b; Kolouri et al., 2019; Nguyen et al., 2020; Deshpande et al., 2019). In contrast, we use a Nonlinear Sliced Wasserstein (NSW) distance, an averaging procedure over (random) nonlinear transformations. For our goals, the choice of nonlinearity was motivated by the following considerations: (1) a bounded non-linearity would be beneficial since unbounded non-linearities (such as cubic polynomials) have a pronounced deformation on the tails of a measure and may excessively weight outliers. (2) A sigmoid is another potential candidate, but it saturates at high values and we want the non-linearity to be similarly effective everywhere. (3) The use of polynomials has been discussed in Kolouri et al. (2019), however, a full set of all higher order polynomials have exponential complexity and may be prohibitively expensive in high latent dimensions. We compared our method to cubic and quintic polynomials for the simple 3D spiral and found that the choice of the nonlinearity did not have a significant effect on performance, with our choice being the fastest (see section 4.2 and Appendix E for computational time and loss comparisons). In principle, other ensembles of nonlinear transformations could be used.
+# B PSEUDOCODE
+# Algorithm 1 Training EPSWAE
+1: while not converged do
+2: Update the autoencoder $\Psi _ { E } , \Psi _ { D }$ :
+3: for $k _ { 1 }$ substeps do
+4: Sample minibatch from data $\{ \mathbf { x } ^ { ( 1 ) } , . . . , \mathbf { x } ^ { ( N ) } \}$ with $\mathbf { x } ^ { ( j ) } \sim P _ { X }$
+5: Compute feature extractor samples $\mathbf { f } ^ { ( j ) } = \Psi _ { F E } ( \mathbf { x } ^ { ( j ) } )$
+6: Compute posterior samples $\mathbf { z } ^ { ( j ) } = \Psi _ { E } ( \mathbf { x } ^ { ( j ) } )$
+7: Compute decoded output $\mathbf { x } _ { r e c o n } ^ { ( j ) } = \Psi _ { D } ( ( \mathbf { z } ) ^ { j } )$
+8: Generate samples $\{ \xi ^ { ( 1 ) } , . . . \xi ^ { ( J ) } \}$ with $\xi ^ { ( j ) } \sim \mu$
+9: Compute prior samples $\mathbf { y } ^ { ( j ) } = \Psi _ { P E } ( \xi ^ { ( j ) } )$
+10: 11: Compute reconstruction error: Compute NSW distance: LNSW distance $\begin{array} { r } { \mathcal { L } _ { r e c o n } = \frac { 1 } { J } \sum _ { j = 1 } ^ { J } d ( \mathbf { x } ^ { ( j ) } , \mathbf { x } _ { r e c o n } ^ { ( j ) } ) } \end{array}$ $\begin{array} { r } { = d _ { N S W } \big ( \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \delta _ { \mathbf { z } ^ { ( j ) } } , \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \delta _ { \mathbf { y } ^ { ( j ) } } \big ) } \end{array}$
+12: Compute FSC loss: $\mathcal { L } _ { F S C } = d _ { F S C } ( \{ \mathbf { f } ^ { ( 1 ) } , . . . , \mathbf { f } ^ { ( J ) } \} , \{ \mathbf { z } ^ { ( 1 ) } , . . . , \mathbf { z } ^ { ( J ) } \} )$
+13: Compute autoencoder loss $\mathcal { L } _ { A E } = \alpha \mathcal { L } _ { r e c o n } + \beta \mathcal { L } _ { N S W d i s t a n c e } + \kappa \mathcal { L } _ { F S C }$
+14: Compute gradients of $\mathcal { L } _ { A E }$ wrt to $\phi _ { E } , \phi _ { D }$
+15: Update $\phi _ { E } , \phi _ { D }$
+16: end for
+17: Update the prior-encoder $\Psi _ { P E }$ :
+18: for $k _ { 2 }$ substeps do
+19: Sample minibatch from data $\{ \mathbf { x } ^ { ( 1 ) } , . . . , \mathbf { x } ^ { ( J ) } \}$ with $\mathbf { x } ^ { ( j ) } \sim P _ { X }$
+20: Compute posterior samples $\mathbf { z } ^ { ( j ) } = \Psi _ { E } ( \mathbf { x } ^ { ( j ) } )$
+21: Generate samples $\{ \xi ^ { ( 1 ) } , . . . \xi ^ { ( J ) } \}$ with $\xi ^ { ( j ) } \sim \mu$
+22: 23: Compute prior samples Compute prior-encoder $\mathbf { y } ^ { ( j ) } = \Psi _ { P E } ( \xi ^ { ( j ) } )$ $\begin{array} { r } { \mathcal { L } _ { P E } = d _ { N S W } \big ( \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \delta _ { \mathbf { z } ^ { ( j ) } } , \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \delta _ { \mathbf { y } ^ { ( j ) } } \big ) } \end{array}$
+24: Compute gradients of $\mathcal { L } _ { P E }$ wrt to $\phi _ { P E }$
+25: Update $\phi _ { P E }$
+26: end for
+27: end while
+# C ARCHITECTURE AND TRAINING DETAILS
+For all datasets, the prior-encoder $\Psi _ { P E }$ consists of three fully connected hidden layers and ReLU activations. For all datasets, the autoencoder and prior-encoder losses (given in Eqns 3 and 5) are trained iteratively using the optimizer Adam (Kingma & Welling, 2014) with a learning rate of 0.001. We experimented with both $p = 1$ and $p = 2$ (corresponding to $\mathsf { p }$ -Wasserstein) in the SW distance and did not find any significant differences; all results in this paper use $p = 2$ . For each calculation of the NSW distance, $L = 5$ random nonlinear transformations were taken followed by $M = 5 0$ one dimensional projections per transformation. Data-specific model and parameter details are given below.
+3D Spiral dataset: The input to the prior-encoder is a 40D Gaussian, and the latent space is 3D. The input to the autoencoder is a 40D embedding of a 3D spiral manifold with $10 \%$ noise. The dataset consists of 10000 samples, and a batch size of 100 was used. The prior-encoder, the data encoder, and the decoder consist of three Fully Connected (FC) layers with 40 nodes each and ReLU activations. The reconstruction loss is given by the Mean Square Error and $\alpha = 1 , \beta = 0 . 1 , \kappa = 0 . 0 1$ in Eqn 3. In the absence of convolutional layers, the FSC term encourages the pairwise distances of the minibatch in latent space to be similar to the pairwise distances of the minibatch in the data space. The prior-encoder is trained $k _ { 1 } = 2$ times for each training of the autoencoder $k _ { 1 } = 1$ . Power of distance $h = 2$ is used to compute edge weights for computing network-geodesics.
+MNIST dataset: The input to the prior-encoder is a 40 dimensional mixture of 10 Gaussians, and the latent space is 5 dimensional. The data encoder takes MNIST images as input using a batch size of 100, and consists of the following layers Con $\ v \ ( 1 , 1 0 , 3 ) \ \to \ \mathrm { B a t c h N o r m } \ \to \ \mathrm { R e L u } \to \ \mathrm { M a x P o o l }$ (2,2) → Conv (10,16,3) BatchNorm ReLu followed by two FC layers of 512 and 256 nodes respectively with a leaky ReLu nonlinearity. The encoder outputs a 5 dimensional latent representation. The decoder consists of the reverse, i.e., three fully connected layers of size 256, 512, and 1936 nodes respectively. This is followed by ConvTranspose $( 1 6 , 1 0 , 3 ) $ LeakyReLu Upsample(2,2) ConvTranspose $( 1 0 , 1 , 3 ) $ Sigmoid. The decoder output has size $2 8 ~ \times$ 28. The prior-encoder is trained $k _ { 1 } = 1$ times for each training of the autoencoder $k _ { 2 } = 1$ . The reconstruction loss is given by the Binary Cross Entropy and $\alpha = 1 , \beta = 0 . 1 , \kappa = 0 . 0 0 1$ in Eqn. 3. The FSC loss encourages the pairwise distances of the minibatch in latent space to be similar to the pairwise distances of the minibatch in feature space computed at the output of the last convolutional layer in the data encoder. Energy parameter $h = 2$ is used to compute edge weights for computing network-geodesics.
+CelebA dataset: The input to the prior-encoder is a 186 dimensional mixture of 10 Gaussians, and the latent space is 128 dimensional. We ran experiments with different latent dimensions and found that the generation ability didn’t vary significantly as a function of latent dimension. We trade-off image quality of the input images for computational speed by downsizing the input and employ a fairly simple network compared to state of the art computer vision architectures that use CelebA. The data encoder takes CelebA images (of size $2 1 8 \times 1 7 8 \times 3 )$ and downsizes them to size $6 4 \times 6 4 \times 3$ . The data encoder consists of the following layers Conv (3,16,3) BatchNorm Re $\mathrm { L u } \to \mathrm { M a x P o o l } ( 2 , 2 ) \to \mathrm { C o n v } ( 1 6 , 3 2 , 3 ) \to \mathrm { B a t c h N o r m } \to \mathrm { R e L u } \to \mathrm { M a x P o o l } ( 2 , 2 ) \to \mathrm { C o n v } ( 1 6 , 3 2 , 3 ) \to \mathrm { C o n v } ( 1 0 , 2 ) .$ $( 2 , 2 ) \to \mathbf { C o n v }$ $( 3 2 , 6 4 , 3 )  \mathrm { B a t c h N o r m }  \mathrm { R e }$ Lu followed by two FC layers of 512 and 256 nodes respectively with a leaky ReLu nonlinearity. The encoder outputs a 128 dimensional latent representation. As in the case of MNIST, the decoder consists of the reverse, with convolutions replaced by Convolution Transpose, and MaxPool replaced by Upsample. The output of the decoder is passed through a sigmoid nonlinearity and is of size $6 4 \times 6 4 \times 3$ . The prior-encoder was trained $k _ { 1 } = 1$ times for each training of the autoencoder. The reconstruction loss is given by the Binary Cross Entropy and $\alpha = 5 0 0 , \beta = 5 0 , \kappa = 0 . 0 5$ in Eqn. 3. The FSC loss encourages the pairwise distances of the minibatch in latent space to be similar to the pairwise distances of the minibatch in feature space, i.e., computed at the output of the last convolutional layer in the encoder. Energy parameter $h = 2$ is used to compute edge weights for computing network-geodesics.
+# D SPIRAL BASELINE COMPARISONS
+![](images/ff01431f6c397dda57f86eef94d1020200695e2e547994c386d55decc8c9156f.jpg)
+Figure 5: Comparisons of the EPSWAE model with different loss terms. Top panels present top views, and bottom panels present the corresponding side views. The red curves show interpolation between two randomly selected samples using the network-geodesic algorithm. All figures are generated after 100 epochs with a $\mathrm { l r } { = } 0 . 0 1$ , and batch size $= 1 0 0$ . $k _ { 1 } = 1 , k _ { 2 } = 2$ .
+Figure 5 shows the effects of the different loss terms in EPSWAE on the geometry of the learned prior and posterior. The position and orientation of the spiral in the 3D plots are random and the views in the image are hand-chosen to be equivalent. (a) shows latent space and interpolations (red) using EPSWAE with a linear Sliced Wasserstein distance in the loss, and no structural consistency term. (b) shows that adding a structural consistency term doesn’t remarkably improve the quality of the manifold learned, however, consistent with other experiments, it seems to improves interpolation slightly. Note here that since we don’t use convolutional layers for the 3D spiral, the stuctural consistency term preserves distance in latent space corresponding to distances in data space. (c) shows that employing the NSW distance term significantly improves the learned structure in latent space. The improvements resulting from incorporation of the NSW and structural consistency terms as seen in these visualizations of the 3D spiral lead us to use both loss terms (as in (d)) on all results in the main paper.
+![](images/bbc3c8de28f787cbd7a0b99539c39be2ac30b08a7b3ad77d85e27bd8b383c4bc.jpg)
+Figure 6: Comparison of EPSWAE with baselines SWAE (Kolouri et al., 2018a) and VAE (Kingma & Welling, 2014). All figures are generated after 100 epochs with a $\mathrm { l r } { = } 0 . 0 1$ , and batch size $= 1 0 0$ . For EPSWAE, $k _ { 1 } = 1 , k _ { 2 } = 2$ , $\alpha = 1 , \beta = 0 . 1$ , $\kappa = 0 . 0 0 1$ .
+Figure 6 compares the EPSWAE model with baselines VAE and SWAE. The VAE uses KL divergence in the loss, which constrains the prior to be sampled from a Gaussian distribution. While the vanilla SWAE uses the SW distance, the prior remains a Gaussian, leading to an unnatural embedding of the data manifold. In contrast, EPSWAE results in a significantly better learned prior as a consequence of the nonlinear SW distance and the prior-encoder network trained explicitly to improve the latent representation.
+# E COMPARISON WITH OTHER SLICED WASSERSTEIN NONLINEARITIES
+Here, we compare our choice of nonlinearity in the SW loss (sinusoidal shear) with some other common nonlinearities. While there exist several methods to improve the SW distance (e.g. maxSW (Deshpande et al., 2019) which has seen several subsequent variations), these involve additional training, and hence aren’t considered (although many such versions of SW distance could be used in conjunction with out method). Instead we consider the polynomial generalized Radon transforms described in (Kolouri et al., 2019). The computational cost of a single polynomial generalized Radon transform is ${ \mathcal { O } } ( d ^ { k } )$ for dimension $d$ and $k ^ { t h }$ order of polynomial, and hence we limit our investigation to cubic and quintic polynomials. We show results on the artificial 3D Spiral dataset trained upto 100 epochs with $\alpha = 1$ , $\beta \ : = \ : 0 . 1$ , $\kappa = 0 . 0 1$ . We see in Fig. 7 that the choice of nonlinearity does not have a significant effect on the loss, however, as seen in Table4.2, the sineshear has slightly lower computational cost.
+![](images/a23e5f118fbb68e03ecfd96e11e03d1f150081d3a42c6154d83856a72989b4b5.jpg)
+Figure 7: Loss as a function of training epochs for linear SW, cubic nonlinearity, quintic nonlinearity and our (sine-shear) NSW distance. A full set of cubic and quintic functions terms are considered for computing the generalized Radon transform in the SW distance. $L = 5$ nonlinear transforms with $M = 5 0$ linear 1D projections each taken for all cases.
+# F MNIST GENERATION RESULTS
+![](images/500dcb810a72852600f992a470033b71d68c0d83f3d72e92ded8f749e117415c.jpg)
+Figure 8: Comparison between (a) EPSWAE (b) SWAE and (c) VAE for generation on MNIST. for all three, batch size $= 2 0 0$ , epochs $_ { \mathrm { \scriptsize = } 1 0 0 }$ , $\mathrm { l r } = 0 . 0 0 1$ , for EPSWAE $k _ { 1 } = 2$ , $k _ { 1 } = 1$ , $\alpha = 1 , \beta =$ $0 . 1 , \kappa = 0 . 0 0 1$ .
+Figures 8 (a,b,c) shows generation on the MNIST dataset after 100 epochs on EPSWAE, baseline SWAE(Kolouri et al., 2018a), and baseline VAE (Kingma & Welling, 2014) respectively. All networks EPSWAE and baselines SWAE and VAE use equivalent architectures (outlined in section of Appendix C), and hyperparameters are optimized individually. As seen in the figure, we observe that EPSWAE generated samples were consistently found to have a lower fraction of ‘false’ digits, i.e., digits that are unrealistic.
+# G CELEBA GENERATION RESULTS
+![](images/6b8244130d046c4c720cb444a8c031583d0a5908cd8ecabb113ce6145c5b1f64.jpg)
+Figure 9: Images generated from prior samples in (a) EPSWAE (b) Baseline SWAE. Batch size $=$ 200, epochs $_ { \mathrm { { s } = 1 0 0 } }$ , $\mathrm { l r } = 0 . 0 0 1$ . $\alpha = 5 0 0 , \beta = 5 0 , \kappa = 0 . 0 5$ .
+CelebA raw images are downsized to $6 4 \times 6 4$ pixels, and a simple architecture is employed (see Appendix C). This serves as a proof of principle for a better latent representation and interpolation, without a high computational cost. Naturally, using a more sophisticated network (for instance ResNet, VGG etc.) without downsizing the data would yield higher quality images at a computational cost.
+Figure 9 shows generation on the CelebA dataset after 100 epochs with (a) EPSWAE and (b) baseline SWAE (right) respectively. Both employ equivalent architectures and take downsized images as input. As seen in the figure, EPSWAE generated images are more realistic. Note that this comparison is for equivalent training epochs and architectural details, and does not make claims about the quality of SWAE images with longer training.
+# H EFFECT OF ENERGY PARAMETER ON CELEBA INTERPOLATIONS
+Additional CelebA interpolations are shown in Fig. 10. The length of interpolations is automatically selected by the network algorithm. In some cases, a linearly interpolated point is added between every two samples on the network-geodesic to make smoother transitions; this would be unnecessary with enough samples of the prior, however it could be impractical in high latent dimension to generated sufficiently many prior samples. Note that even if only one intermediate point is selected by the network geodesic, the generated interpolation could be significantly different from a purely linear interpolation (an illustration is provided in Figure 4.1 - the direct linear interpolation from A to B is significantly different from the interpolation through the point C). Fig. 10 presents a comparison between energy parameter $h = 1$ and $h = 2$ . The energy parameter determines the power of the distance metric used to compute the edge weight. One can think of the higher value $h = 2$ ) as corresponding to stronger connections between points, and encouraging shorter hops. In practice, as seen from the figure, there is no clear advantage to choosing a specific value of energy parameter $h$ .
+![](images/806c76f199305fdf25fa33ab629027b8ee4a493ac1b440fcf59bd62a75602096.jpg)
+Figure 10: Additional CelebA interpolations. The first and last images are reconstructions of real data, and the interpolations traverse through samples of the prior using the network-geodesic algorithm. (a) Top three panels show interpolations for energy parameter $h = 1$ , and (b) bottom three panels show interpolations for energy parameter $h = 2$ . Latent space is 128 dimensional. Hyperparameters are the same as those used in the paper. A total of a 400 samples in latent space are used.
+![](images/ddb55a4bd659d746b29c13afa5a163fa391dae953b795f18ff08b6b686ddae49.jpg)
+Figure 11: Comparisons of linear interpolation between EPSWAE and SWAE. The first and last images are reconstructions of randomly picked real data (a) shows long interpolation (over two lines), (b,c) show shorter interpolations. Hyperparameters are the same as those used in the paper. Latent space is 128 dimensional. A total of a 400 samples in latent space are used.
+We compare here linear interpolations on the CelebA dataset for MNIST and CelebA. The autoencoder networks used are identical and outlined in section C. In contrast, EPSWAE uses the additional prior encoder. Both models are independently optimized, and the interpolations are linear. As seen in Fig. 11, interpolations in EPSWAE are more realistically identifiable as ’faces’ than SWAE, which mixes up the features and generates blurry intermediate images. This is suggestive that the prior-encoder may play a role in improving latent representation.

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+# MERLOT: Multimodal Neural Script Knowledge Models
+Rowan Zellers♠ Ximing $\mathbf { L } \mathbf { u } ^ { \mathsf { \pmb { \alpha } } \odot }$ Jack Hessel♥
+Youngjae ${ \bf Y } { \bf u } ^ { \bigcirc }$ Jae Sung Park♠ Jize $\mathbf { C a o ^ { \otimes \bigcirc } }$ Ali Farhadi♠ Yejin Choi♠♥
+♠Paul G. Allen School of Computer Science & Engineering, University of Washington ♥Allen Institute for Artificial Intelligence https://rowanzellers.com/merlot
+# Abstract
+As humans, we understand events in the visual world contextually, performing multimodal reasoning across time to make inferences about the past, present, and future. We introduce MERLOT, a model that learns multimodal script knowledge by watching millions of YouTube videos with transcribed speech – in an entirely label-free, self-supervised manner. By pretraining with a mix of both framelevel (spatial) and video-level (temporal) objectives, our model not only learns to match images to temporally corresponding words, but also to contextualize what is happening globally over time. As a result, MERLOT exhibits strong out-of-the-box representations of temporal commonsense, and achieves state-ofthe-art performance on 12 different video QA datasets when finetuned. It also transfers well to the world of static images, allowing models to reason about the dynamic context behind visual scenes. On Visual Commonsense Reasoning, $M E R \backslash \mathrm { { O T } }$ answers questions correctly with $8 0 . 6 \%$ accuracy, outperforming stateof-the-art models of similar size by over $3 \%$ , even those that make heavy use of auxiliary supervised data (like object bounding boxes).
+Ablation analyses demonstrate the complementary importance of: 1) training on videos versus static images; 2) scaling the magnitude and diversity of the pretraining video corpus; and 3) using diverse objectives that encourage full-stack multimodal reasoning, from the recognition to cognition level.
+![](images/6749f9c91c9554a9b75db291a758189c47ce465ee4928dc0f88a92043898682a.jpg)
+Figure 1: Multimodal Event Representation Learning Over Time. We learn representations of multimodal script knowledge from 6 million YouTube videos. These representations can then be applied to a variety of downstream tasks that require commonsense or temporal visual reasoning.
+# 1 Introduction
+The human capacity for commonsense reasoning is shaped by how we experience causes and effects over time. Consider the still image of people dining at a restaurant in the bottom right of Figure 1: while a literal, concrete description like “people sitting at a table eating" might be technically correct for the static scene, it doesn’t capture the richer temporal, commonsense inferences that are nonetheless obvious: before sitting down, the people had to meet up, agree where to go, and enter the restaurant; at present, the man is pointing because the server just came to the table, and she might want to know whose food is whose; and after, it is likely the server will return to the kitchen to help another table.
+Teaching machines this type of script knowledge [95] is a significant challenge in no small part because enumerating all facts, inferences, and counterfactuals is prohibitive. As a result, the highest performing models on vision-and-language tasks, including Visual Commonsense Reasoning (VCR) (where Figure 1’s scene originates from), learn about the visual world exclusively through static images paired with literal captions [108, 22, 69, 75, 119, 36]. Though some captions might hint at the past and future, it is not obvious that even training on, e.g., 400M literal image/text pairs [89] will result in models capable of temporal reasoning.
+In this paper, we introduce MERLOT, short for Multimodal Event Representation Learning Over Time. MERLOT is a model that learns commonsense representations of multimodal events by selfsupervised pretraining over 6M unlabelled YouTube videos. With the goal of learning multimodal reasoning capacity beyond static images/literal captions, we train MERLOT to a) match individual video frames with contextualized representations of the associated transcripts, and to b), contextualize those frame-level representations over time by “unmasking" distant word-level corruptions [27] and reordering scrambled video frames.
+We validate our model on a diverse suite of video tasks, requiring both recognition- and cognition-level reasoning across long and short timescales; when finetuned, MERLOT achieves a new state-of-theart on 12 such tasks. Additionally, we show that our script-knowledge representations transfer to the single image domain. On Visual Commonsense Reasoning (VCR; [123]), our model achieves particularly strong performance, outperforming models that require heavy visual supervision (in the form of object detection bounding boxes, or images paired with pristine captions).
+Beyond finetuning, we show both quantitatively and qualitatively that MERLOT has a strong outof-the-box understanding of everyday events and situations. Given a scrambled visual story, [50, 2], MERLOT can sort image sequences to match captions which tell a globally coherent narrative. Despite considerable domain shift from videos to static images, MERLOT outperforms strong baselines like CLIP [89] and UNITER [22], which independently match images to text and thus cannot reason over long-term contexts as effectively. This capacity for temporal coherence emerges during pretraining: analysis of MERLOT’s attention patterns (Figure 11) show that regions attend to captions that are distant in time (and vice versa), allowing it perform cross-modal coreference to piece together a holistic view of situations.
+Finally, ablations of MERLOT show that 1) pretraining works better when we train on videos rather than still images, aided crucially by our strategy of corrupting highly visual words in the masked language modeling task, 2) using a diverse set of videos covering many aspects of everyday situations improves downstream performance compared to curated instructional video corpora [107, 80] which both cover a smaller slice of the visual world (confirming hypotheses from past work [47]); and 3) MERLOT’s performance does not saturate even after many epochs of training on the pretraining corpus we curated, YT-Temporal-180M, as it continues to improve performance simply with more pretraining. The combination of these results suggests that learning full-stack visual reasoning and multimodal world knowledge from video data is a promising path forward for future research.
+In summary, our main contributions are:
+1. MERLOT a performant end-to-end vision and language model, that learns powerful multimodal world representations from videos and their transcripts – using no labeled data. 2. YT-Temporal-180M, a diverse corpus of frames/ASR derived from a filtered set of 6M diverse YouTube videos, which we show greatly aids performance, and 3. A set of experiments/ablations demonstrating the strong performance of MERLOT on a set of 14 tasks, spanning finetuning and zero-shot transfer, and images and videos.
+At rowanzellers.com/merlot, we have released code, data, and models for public research use.
+# 2 Related Work
+# 2.1 Joint representations of written text and images
+There is a long history of work on learning joint text-image representations [14]. Recently, several papers have proposed “Visual BERT” models [108, 22, 8, 69, 75, 119, 36], trained on image captioning datasets such as MSCOCO [71]. In general, features are extracted using Anderson et al. [10]’s frozen object detector, which was originally trained on Visual Genome [60]. Some exceptions are Zhang et al. [125], who use an even larger object detector trained on more labeled data; Kim et al. [57], who use an ImageNet-pretrained backbone [26], and Shen et al. [100], who study a CLIP backbone [89] pretrained on web image-caption pairs.
+Overall, these approaches all learn visual representations of static images, and rely on significant human annotation in doing so (e.g. through literal image descriptions). Instead, our approach learns dynamic visual representations purely from videos – their frames, and a transcript of what is said – thus using no human annotation.
+# 2.2 Learning from videos, with automatic speech recognition (ASR) transcripts
+Prior works have used web videos with ASR to build weakly-supervised object detectors [87], action detectors/classifiers [120, 6, 62, 84], instruction aligners [77, 5, 19], video captioners [96, 46, 86, 101], and visual reference resolvers [49]. Of late, works have sought to learn multimodal representations transferable to many tasks from uncurated sets of (usually how-to) videos [80, 106, 107, 81, 127, 9, 7, 4]; generally these are applied to video understanding tasks like activity recognition. One challenge is designing an appropriate objective for learning video-level representations. Lei et al. [67]’s ClipBERT model learns vision-language representations from image captions, which more literally describe image content versus the longer ASR transcripts we consider. Tang et al. [109] use a pretrained dense image captioner [59] to provide auxiliary labels for web how-to videos. Both approaches use (supervised) ResNets pretrained on ImageNet [43] as their visual backbones. MERLOT is trained using a combination of objectives requiring no manual supervision; it nonetheless outperforms both prior approaches on downstream tasks.
+# 2.3 Temporal ordering and forecasting
+There has been a large body of work on analyzing ‘what happens next’ in videos [58]. Some modeling choices include using pixels [34, 113], graphs [11], euclidean distance using sensors [3], or studying cycle consistency across time [32]. In addition to extrapolation, past work has studied deshuffling objectives in videos [82, 115], though this has mostly been limited to the visual modality. In contrast to these papers, our goal is learning multimodal script knowledge representations: using both language and vision as complementary views into the world, instead of just tracking what changes on-screen.
+# 3 MERLOT: Multimodal Event Representation Learning Over Time
+We now present our unified model for learning script knowledge through web videos; including our pretraining dataset, architecture, and objectives.
+# 3.1 YT-Temporal-180M
+We collect YT-Temporal-180M, a dataset for learning multimodal script knowledge, derived from 6 million public YouTube videos. Our YT-Temporal-180M intentionally spans many domains, datasets, and topics. We began with 27 million candidate video IDs (which we then filtered), including instructional videos from HowTo100M [80], lifestyle vlogs of everyday events from the VLOG dataset [35], and YouTube’s auto-suggested videos for popular topics like ‘science’ or ‘home improvement.’ Our intent (in making the corpus as diverse as possible) was to encourage the model to learn about a broad range of objects, actions, and scenes [47]: we will later show through an ablation that limiting our pretraining to only instructional videos indeed hurts performance downstream.
+We filtered videos using the YouTube API, which provides access to videos themselves, their ASR track (automatically transcribed speech tokens), and other metadata. We discard videos 1) without an English ASR track; 2) that are over 20 minutes long; 3) that belong to visually “ungrounded" categories like video game commentaries; and 4) that have thumbnails unlikely to contain objects, according to a lightweight image classifier. We add punctuation to the ASR by applying a sequenceto-sequence model trained to add punctuation to sentences/paragraphs from news articles. Full details of the scraping and filtering are in Appendix A.
+![](images/b964db22d299f764a08ad5f6ee7e0cecb1bf83b0c516a48d6e6d60757d05b965.jpg)
+Figure 2: Left: MERLOT learns to match contextualized captions with their corresponding video frames. Right: the same image encoding is provided, along with (masked) word embeddings, into a joint vision-language Transformer model; it then unmasks ground words (like ‘saw’ in this example) and puts scrambled video frames into the correct order.
+Each video $\nu$ might contain thousands of frames. In this work, we represent a video $\nu$ as a sequence of consecutive video segments $\left\{ \boldsymbol { s } _ { t } \right\}$ . Each segment $\mathbf { \boldsymbol { s } } _ { t }$ consists of:
+a. an image frame $\mathbf { } I _ { t }$ , extracted from the middle timestep of the segment, b. the words ${ \pmb w } _ { t }$ spoken during the segment, with a total length of $L$ tokens.
+To split the videos into segments, we byte-pair-encode (BPE; [97, 88]) each video transcript and align tokens with YouTube’s word-level timestamps. This enables us to split the videos into segments of $L { = } 3 2$ BPE tokens each (Appendix A.4); our final dataset has 180 million segments of this form.
+# 3.2 MERLOT Architecture
+A diagram of $M { \mathrm { E R } } [ \mathcal { O } ]$ is given in Figure 2. MERLOT takes a sequence of video frames $\left\{ \boldsymbol { s } _ { t } \right\}$ as input. We encode each frame $\scriptstyle { I _ { t } }$ using an image encoder, embed the words ${ \pmb w } _ { t }$ using a learned embedding, and jointly encode both using a Transformer [112]. After pretraining, the architecture can be applied to a variety of vision-and-language tasks with minimal modification. For video QA, for example, we pass several video frames to the image encoder, the question to the text encoder, and extract a single vector representation from the CLS token position. For each task, we learn a lightweight classification head mapping from this hidden state to the task’s label space; specific modeling/optimization details are given in Appendix E.2.
+Image encoder. We train our image encoder end-to-end, alongside the rest of the model, from random initialization (thus without learning from supervised data). While most performant visionand-language models pre-extract features from a (supervised) object detector [108, 69, 75, 22, 68], for the sake of pre-training efficiency we use a grid-based hybrid ResNet/Vision Transformer.1
+Specifically: our encoder uses a ResNet-50 backbone, followed by a 12-layer, 768-dimensional Vision Transformer [43, 112, 31]. We made additional modifications that improve efficiency, including: 1) we trained on smaller, widescreen images of size $1 9 2 \mathrm { x } 3 5 2$ (because most YouTube videos are widescreen) using a patch size of 16x16 pixels; 2) we mirror [31]’s alterations of removing the C5 block in ResNet-50; and 3) we save compute further by average-pooling the final-layer region cells using a kernel size of $2 \times 2$ . With these modifications, our image encoder requires 40 gigaFLOPs for a forward pass, which is $2 \%$ of the 2 teraFLOPs required for the Faster-RCNN.
+In summary: given an image of size $W \times H$ , the image encoder will output a $W / 3 2 \times H / 3 2$ feature map, along with two CLS hidden states: one for pooling a global representation of the image, and another for pretraining (Task 1.).
+Joint Vision-Language Encoder. The joint encoder is a 12-layer, 768-dimensional Transformer [112], mirroring the RoBERTa base architecture [72]; we initialize it with pretrained RoBERTa weights. To compute joint representations, we first embed the tokens $\{ \pmb { w } _ { t } \}$ via lookup, and then add position embeddings to both language and vision components (i.e., $\{ I _ { t } \} )$ . The position embeddings differ between different segments, so as to distinguish between images and captions at different timesteps. Finally, we pass the independent visual and textual feature maps to our joint encoder.
+The tokens ${ \pmb w } _ { t }$ in each segment begin with a CLS token; recall that the feature maps for each frame $\mathbf { } I _ { t }$ start with one as well. At those positions, we will later pool final-layer hidden-state representations, for use in pretraining along with downstream tasks.
+# 3.3 Pretraining Tasks and Objectives
+We use the following three objectives to pretrain $M { \mathrm { E R } } [ \mathcal { O } ]$ , that cover ‘full-stack’ visual reasoning – from recognition subtasks (like object detection) that operate at the frame level, to more ‘cognitive’ tasks that operate at the video level.
+1. Contrastive frame-transcript matching [126, 89]. We want to ensure that the underlying image encoder produces helpful image representations. Thus, we use the video transcript to compute a ‘language-only’ representation of each video segment; and use a contrastive loss to maximize its similarity to corresponding representations from the image encoder.2
+Unlike what is the case for many image captions, the words ${ \pmb w } _ { t }$ in each segment are often not sufficient to describe the gist of $\mathbf { } I _ { t }$ , or even what the key objects might be – for that, video-level contextualization is often required. We thus pass the entire transcript into the language-only encoder, which then extracts hidden states for each segment at the segment-level CLS tokens.
+Given matching representations for each frame $\scriptstyle { I _ { t } }$ and caption ${ \pmb w } _ { t }$ as positive examples, the negative examples come from all other frame-caption pairs in the batch – whether or not they come from the same video. We project both of these representations into a size-768 hidden state which is then unit-L2-normalized, and compute an all-pairs dot-product between all image and text representations. We divide these logits by a temperature of $\tau = 0 . 0 5$ , and then apply a pairwise cross entropy loss to encourage matching captions and frames.
+2. (Attention) Masked Language Modeling When providing words into the joint vision-andlanguage encoder, we randomly replace $20 \%$ with a MASK token, a random word, or the same word; MERLOT must then reconstruct the correct word with a cross-entropy loss, following [27].
+This approach is commonly used by ‘visual BERT’ models in the image captioning domain, where captions are concise, and thus the identity of masked concrete words is difficult for models to recover given language context alone. However, we observed qualitatively that videos break these assumptions: people tend to ramble, and often mention key objects multiple times. Thus, applying vanilla BERT-style masking often causes ungrounded fillers like ‘umm’ or ‘yeah’ to get masked, while the (repeated) names of important objects are often partially masked, penalizing the learning of multimodal representations.
+We introduce a simple solution to this problem, that we call attention masking: we use attention weights from a language-only transformer (introduced in the previous objective) as a heuristic for which words are grounded. $50 \%$ of the time, we mask out a random token; the other $50 \%$ of the time, we mask out one of the top $20 \%$ most-attended-to-tokens. We then apply SpanBERT masking [54], randomly corrupting the following or preceding tokens with an average length of 0.5 tokens in each direction; this makes it harder for models to over-rely on BPE artifacts. We show in ablations that this improves performance.
+Table 1: Results on VCR [123]. We compare against SOTA models of the same ‘base’ size as ours (12-layer vision-and-language Transformers). MERLOT performs best on all metrics.
+<table><tr><td></td><td>|Q→A QA→R Q→AR</td><td></td></tr><tr><td>ViLBERT[75]</td><td>73.3</td><td>74.6 54.8</td></tr><tr><td>Unicoder-VL [68]</td><td>73.4 74.4</td><td>54.9</td></tr><tr><td>VLBERT[69]</td><td>73.8 74.4 75.0</td><td>55.2</td></tr><tr><td>UNITER [22]</td><td>77.2</td><td>58.2</td></tr><tr><td>VILLA [36]</td><td>79.1</td><td>60.6</td></tr><tr><td>ERNIE-ViL [119]</td><td>76.4 77.0 80.3</td><td>62.1</td></tr><tr><td>MERIOT (base-sized)</td><td>80.6</td><td>80.4 65.1</td></tr></table>
+Table 2: Results unscrambling SIND visual stories[50, 2]. Captions are provided in the correct order; models must arrange the images temporally. $M { \mathrm { E R U O T } }$ performs best on all metrics by reasoning over the entire story, instead of independently matching images with captions.
+<table><tr><td>Spearman (↑)</td><td>Pairwise acc (1）</td><td>Distance (）</td></tr><tr><td>CLIP [89]</td><td>78.7</td><td>.638</td></tr><tr><td>UNITER [22]</td><td>75.2</td><td>.745</td></tr><tr><td>MERLOT</td><td>.733</td><td>.498</td></tr></table>
+3. Temporal Reordering. We have the model order the image frames in a video, forcing it to explicitly learn temporal reasoning and giving it an interface to measure such temporal reasoning. Here, $40 \%$ of the time, we randomly pick an integer $i$ between 2 and $N$ (the number of segments provided to the joint encoder). Then we randomly scramble $i$ video frames chosen at random, by replacing the segment-level position embeddings (e.g. [image_t]) for that frame with a random and unique position embedding, e.g. [image_unk_0]). These random position embeddings are learned, and separate from the ‘unshuffled’ position embeddings. This allows the model to order each ‘shuffled’ frame conditioned on frames provided in the correct order (if any).
+To compute the reordering loss, we extract hidden states from each frame at the CLS token position. For each pair of frames, we concatenate their hidden states $h _ { t _ { i } }$ and $h _ { t _ { j } }$ and pass the result through a two-layer MLP, predicting if $t _ { i } < t _ { j }$ or $t _ { i } > t _ { j }$ . We optimize this using a cross-entropy loss.
+# 3.4 Pretraining MERLOT
+We pretrain our model for 40 epochs over our video dataset. We preprocess the dataset into examples with sequences of $N { = } 1 6$ video segments each, each containing up to $L { = } 3 2$ BPE tokens.3 The language-only encoder computes contrastive representations given this entire sequence, its total length is thus 512 tokens. To save memory, we provide the joint vision-language encoder 4 groups of $N = 4$ segments each. At an image training resolution of $1 9 2 \times 3 5 2$ , the joint model’s sequence length is 396 tokens. To combine the losses, we multiply the contrastive loss by a coefficient of 0.25, which we found scaled its gradient magnitudes to roughly the same magnitude as the Mask LM loss.
+We train the model using a v3-1024 TPU pod, at a batch size of 1024 sequences (or 16k segments) in total. This pretraining process on this hardware takes 30 hours. We provide additional information about hyperparameters and experimental setup in Appendix E.1.
+# 4 Experiments: Transferring MERLOT to Downstream Tasks
+In this section, we explore $M { \mathrm { E R U O T } }$ on 14 different tasks, covering vision-language reasoning on static images as well as videos; we present analysis and ablations to dig deeper into our performance.
+# 4.1 Image tasks
+VCR. We consider VCR [123], a task and dataset where models must answer commonsense visual questions about images. These questions, about e.g. ‘what might happen next’ or ‘what are people’s intentions,’ force MERLOT to transfer video-level understanding to the world of single images.
+VCR provides additional ‘referring expression’ information to models in the form of bounding boxes around named entities. For example, if Person1 is referenced in the question, the location of Person1 is also given in the image. We provide this information to models by drawing (in pixel space) a colored highlight around the referenced entity (Appendix E.3.1), this differs from prior works (that integrate these entities into detection architectures).
+<table><tr><td>Tasks</td><td>|Split</td><td>Vid.Length ActBERT[127]</td><td></td><td>ClipBERT8x2[67]</td><td>SOTA</td><td>MERLOT</td></tr><tr><td>MSRVTT-QA</td><td>Test</td><td>Short</td><td>=</td><td>37.4</td><td>41.5 [118]</td><td>43.1</td></tr><tr><td>MSR-VTT-MC</td><td>Test</td><td>Short</td><td>88.2</td><td>-</td><td>88.2 [127]</td><td>90.9</td></tr><tr><td>TGIF-Action</td><td>Test</td><td>Short</td><td>-</td><td>82.8</td><td>82.8[67]</td><td>94.0</td></tr><tr><td>TGIF-Transition</td><td>Test</td><td>Short</td><td>-</td><td>87.8</td><td>87.8[67]</td><td>96.2</td></tr><tr><td>TGIF-Frame QA</td><td>Test</td><td>Short</td><td></td><td>60.3</td><td>60.3 [67]</td><td>69.5</td></tr><tr><td>LSMDC-FiB QA</td><td>Test</td><td>Short</td><td>48.6</td><td></td><td>48.6 [127]</td><td>52.9</td></tr><tr><td>LSMDC-MC</td><td>Test</td><td>Short</td><td>-</td><td>- -</td><td>73.5 [121]</td><td>81.7</td></tr><tr><td>ActivityNetQA</td><td>Test</td><td>Long</td><td></td><td></td><td>38.9 [118]</td><td>41.4</td></tr><tr><td>Drama-QA</td><td>Val</td><td>Long</td><td></td><td></td><td>81.0 [56]</td><td>81.4</td></tr><tr><td>TVQA</td><td>Test</td><td>Long</td><td></td><td></td><td>76.2 [56]</td><td>78.7</td></tr><tr><td>TVQA+</td><td>Test</td><td>Long</td><td></td><td></td><td>76.2 [56]</td><td>80.9</td></tr><tr><td>VLEP</td><td>Test</td><td>Long</td><td></td><td>=</td><td>67.5 [66]</td><td>68.4</td></tr></table>
+Table 3: Comparison with state-of-the-art methods on video reasoning tasks. MERLOT outperforms state of the art methods in 12 downstream tasks that involve short and long videos.
+Our results on the three VCR settings, in comparison to other models at the same (‘base’) scale, are given in Table 1. Our model outperforms these other models, that all learn from exclusively static images (paired with captions and supervised object detections).
+Unsupervised ordering of Visual Stories. To probe our model’s ability to do out-of-the-box commonsense reasoning over events in images, we next consider the Visual Storytelling dataset [50, 74]. Each story in this dataset contains five images and captions in a certain order; the order tells a joint narrative between the captions and the images. Past work has considered unshuffling imagecaption pairs [2], but we take a slightly different approach in this work to avoid language-only biases, which can rely on discursive clues to order text [27, 102]. In our formulation, models are given the captions in sorted order, and must match frames to the captions. Our formulation disarms language-only baselines, while still allowing us to quantify MERLOT’s capacity for commonsense temporal reasoning.
+We compare MERLOT with two strong out-of-the-box baselines for text-image matching: CLIP [89], which encodes each caption and image separately and computes similarity through a dot product, and UNITER [22] which jointly represents each image/caption pair, and is trained in part using a ‘textimage matching’ objective. We use our temporal reordering loss to find the most probable ordering of the video frames (Appendix E.1.1); for CLIP and UNITER we compute a maximum-weight bipartite matching [63] over the pairwise image-text similarity scores.
+Results over 5K stories are given in Table 2. MERLOT’s performance in comparison to the algorithms trained from image-literal caption pairs suggests that, with no fine-tuning, our model has strong capability to reason about past and future events expressed in collections of temporal visual stories.
+# 4.2 Video Reasoning
+We report results on 12 video reasoning tasks: TVQA [64], TVQA $( + )$ [65], VLEP [66], MSRVTT-QA [117], MSRVTT-Multichoice [121], LSMDC-Multichoice, LSMDC fill-in-the-blank QA [110, 92], ActivityNetQA [122, 45], TGIFQA [52], and DramaQA [23]. We apply MERLOT to these tasks in the same way. We sample a sequence of 5 to 7 still frames from each video clip, initialize new parameters only to map the model’s pooled CLS hidden state into the output labels, and finetune MERLOT with a softmax cross entropy loss; see Appendix E.2 for details.
+As shown in Table 3, for all these datasets MERLOT sets a new state-of-the-art. Given the diversity of tasks and the strengths of the comparison models, these results provide strong evidence that MERLOT learned strong multimodal and temporal representations.
+# 4.3 Ablations
+We present ablations over VCR and TVQA $^ +$ to study the effect of several modeling decisions.
+<table><tr><td>Training setup</td><td>|VCR TVQA+</td></tr><tr><td>One segment (N=1)</td><td>173.8 75.2</td></tr><tr><td>One segment, attention masking</td><td>73.5 74.5</td></tr><tr><td>Four segments</td><td>74.1 73.3</td></tr><tr><td>YFour segments,attention masking</td><td>75.2 75.8</td></tr></table>
+(a) Context helps together with attention masking. Pretraining on more segments at once improves performance, but more context can encourage language-only representation learning. Attention masking counteracts this, giving an additional 1 point boost.
+<table><tr><td>Training setup</td><td>|VCR TVQA+</td></tr><tr><td>No contrastive V-L loss</td><td>57.5 67.6</td></tr><tr><td>No temporal ordering loss</td><td>75.5 75.6</td></tr><tr><td>All losses</td><td>75.2 75.8</td></tr></table>
+<table><tr><td></td><td>VCR</td></tr><tr><td>No boxes</td><td>74.8</td></tr><tr><td>□ Drawn-on boxes</td><td>79.4</td></tr></table>
+(b) Contrastive $\mathbf { V } { + } \mathbf { L }$ loss is crucial. Removing it makes performance drop significantly; the temporal ordering loss is not as important for downstream finetuning.
+(c) Drawing on bounding boxes helps, suggesting that our model uses it to decode the ‘referring expression’ information (e.g. person1).
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>VCR</td></tr><tr><td rowspan=1 colspan=1>Conceptual U COCOHowTo100MYYT-Temporal-180M</td><td rowspan=1 colspan=1>58.966.375.2</td></tr><tr><td rowspan=1 colspan=1>HowTo100M-sized YT-Temporal-180MYTT180M,rawASR</td><td rowspan=1 colspan=1>72.872.8</td></tr></table>
+(d) Diverse (video) data is important. Applying our architecture to caption data leads to poor results. Our model performs better on HowTo100M, yet still below our (more diverse) YT-Temporal-180M, even when controlled for size. Using raw ASR (vs. denoised ASR) reduces performance.
+(e) Training for longer helps, with performance increasing monotonically over training iterations.
+<table><tr><td># epochs</td><td>VCR</td></tr><tr><td>5 epochs</td><td>75.2</td></tr><tr><td>10 epochs</td><td>75.9</td></tr><tr><td>20 epochs 30 epochs</td><td>77.0 78.5</td></tr><tr><td></td><td></td></tr><tr><td>40 epochs</td><td>79.4</td></tr></table>
+Table 4: Ablation study on the validation set of VCR question answering $Q  A ,$ and $\mathrm { T V Q A + }$ , in accuraty $( \% )$ . We put a $\boldsymbol { \Xi }$ next to the configurations we chose for ${ \cal M } \mathrm { E R } \bar { \backslash } \bar { \mathrm { O T } }$ .
+Context size. Table 4a shows the effect of varying the number of segments $N$ given to the joint vision-and-language encoder during pretraining. In the first two rows, we provide only a single video segment $\scriptstyle { N = 1 } )$ ) to the model.4 In this limited regime, we find that our ‘attention masking’ approach (preferential masking of tokens that were highly attended-to by the contrastive language-only encoder) does not outperform a strong baseline of masking spans randomly [54]. Yet, when we expand the sequence length to $N { = } 4$ segments/128 tokens, our masking becomes more effective, improving by 1 point over the baseline. This supports our hypothesis (Section 3.3.2.) that text-only shortcuts become increasingly viable with length, and that our attention-masking approach counteracts them.5
+Losses. In Table 4b, we ablate the losses. We find that the contrastive frame-transcript matching loss is crucial to performance, suggesting that an explicit objective is critical for the (randomly initialized) image backbone to learn visual representations. The temporal ordering loss appears less critical for downstream tasks; it helps for TVQA but performance drops slightly for VCR. Thus, we find that it helps primarily as an interface by which we can query the model about temporal events (i.e. for the story ordering experiments); the model might be learning this information from other objectives.
+Drawing bounding boxes. Table $_ \mathrm { 4 c }$ shows the effects of providing grounding information to VCR models by drawing boxes. Performance drops $5 \%$ when they are removed, suggesting that they help.
+Dataset source. In Table 4d, we investigate pretraining $M { \mathrm { E R U O T } }$ on two datasets beyond YTTemporal-180M. First, we train on 3 million static image-caption pairs from Conceptual Captions [99] combined with MSCOCO [71]; for fair comparison, we train for the same number of steps as 5 epochs on our dataset. The resulting model achieves $5 8 . 9 \%$ accuracy on VCR. We suspect this might be due to 1) a smaller context window (Table 4a), and 2) overfitting (5 epochs on YT-Temporal180M corresponds to 300 epochs on the caption data). Because our vision pipeline is trained from scratch, the scale of the curated/supervised image pairing corpora is a concern.
+We next investigate the impact of video selection, comparing YT-Temporal-180M with HowTo100M [80]. To control for number of videos, we train for an equivalent amount of steps: 5 epochs on our dataset, 30 epochs on HowTo100M, and likewise 30 epochs on a ‘HowTo100M-sized YTTemporal-180M’. Using diverse YT-Temporal-180M data vs. only instructional videos improves VCR performance by 6.5 points. This suggests that the how-to domain is limited in terms of visual phenomena covered, and that other domains (like web dramas and VLOGs) provide helpful signal for tasks like VCR [47]. Using all the data gives an additional 2.4-point performance boost.
+![](images/d21ef969d14a99791dac2ae4f816537611708005113b98ba661fe4d6d6264d72.jpg)
+Figure 3: Zero-shot story ordering (same setup as Table 2). MERLOT performs temporal commonsense reasoning accross frames. In the first row, it uses ‘the old man’ mentioned to identify the ‘kids’ as parent-aged; in the second, it identifies riding a merry-go-round as an activity that takes a while.
+Last, we investigate our choice to preprocess the YouTube ASR text with a language model (adding punctuation, etc); using ‘raw ASR’ instead of this preprocessing reduces performance by 2.4 points.
+Pretraining longer. Last, in Table 4e, we investigate the effect of pretraining MERLOT for longer. The performance increases monotonically and doesn’t begin to plateau, which suggests that had we pretrained MERLOT for even longer, its performance could improve even further.
+# 4.4 Qualitative examples
+In Figure 3, we show two qualitative examples of MERLOT’s zero-shot story ordering capability. More examples (and a comparison with the best-scoring baseline, CLIP [89]) are in Appendix C.2. The examples here show that MERLOT has a strong understanding of events, transcending individual frames. In the first row, it orders the story correctly, performing vision-and-language coreference across several frames (e.g. frames and captions 2 and 3 use ‘he’ to refer to ‘the old man’ only mentioned in the first caption). Without resolving this coreference (establishing the subject as an elderly family member), it seems unlikely that anyone would describe the adults in frame (3) as ‘kids.’ Investigating the attention patterns of MERLOT (Appendix C.3) backs up this claim; they show that MERLOT frequently addresses video tasks by merging attention across (distant) video segments.
+MERLOT gets the second row ‘wrong’, but for an interesting reason. It reverses the order of frames (3) and (4), which groups the merry-go-round pictures together – even though caption (3) mentions a barn. This seems to capture the temporal commonsense intuition that people might ride a merry-go-round for a while, i.e., it is not an atomic event [25].
+# 5 Conclusion, Limitations, and Broader Impacts
+We introduced Multimodal Event Representation Learning Over Time (MERLOT). We trained the model through a combination of self-supervised objectives on 6M YouTube videos, in service of learning powerful multimodal representations that go beyond single frames. The model achieves strong performance on tasks requiring event-level reasoning over videos and static images. We hope that MERLOT can inspire future work for learning vision+language representations in a more human-like fashion compared to learning from literal captions and their corresponding images.
+There are several potential limitations of MERLOT that would make for promising avenues of future work, including: 1) exploring finer-grained temporal reasoning pretraining objectives vs. frame ordering e.g., a temporal frame localization within transcripts; and 2) learning multilingually from non-English videos and communities on YouTube.
+Like other pretraining work, MERLOT risks some potential negative impacts. We discuss these in more detail below, in addition to the steps we took to reduce these harms.
+# 5.1 Data collection and privacy.
+As with other corpora gathered from the web used for pretraining data, YT-Temporal-180M contains publicly available content posted by users. We thus shaped our data gathering and release strategy to minimize inherent privacy and consent harms (Appendix A.5). Perhaps most importantly, we plan to only share video IDs for download, following a release strategy from prior work [1, 80] and giving users the right to opt out of not just YouTube, but our dataset as well.
+# 5.2 Social biases.
+The curation choices we made in this work could cause the model to exhibit undesirable social biases – for this reason, along with others, we do not advocate for deployed use-cases. For example, $30 \%$ of the data selected for by our filtering pipeline was local broadcast news (uploaded to YouTube). Including these news videos seems to perform better than filtering them out and only using how-to videos (Table 4b), however, there are risks when training on them. Local broadcast news (at least in the US) dedicates significant time to covering crime, sometimes in a racist and sensationalized manner [38, 29, 44]. Indeed, running a topic model over our data identifies several ‘crime’ categories (Appendix B). Past work has shown correlation between watching local news and having more explicit racialized beliefs about crime [28]; it seems likely therefore that training models on this data could teach them learn the same racist patterns.
+Additionally, there are inherent social biases on YouTube – and treating these videos as equivalent to ‘the world’ [111] can embed hegemonic perspectives [42, 114, 13]. Most popular YouTubers are men [30] and video practices emerging on YouTube are often gendered [83]. YouTube also has problems with hate, including radical alt-right and ‘alt-lite’ content [90]. These problems – as with other problems in representation and power – are themselves amplified by the ‘YouTube algorithm’ [15] that recommends content to users. Though we downloaded videos independently of YouTube’s recommender system, by filtering based on what content has views, we are implicitly filtering based on this algorithm. The dynamics of YouTube (i.e., which videos get popular/monetized) influence the style and content of videos that get made and uploaded to the platform; this in turn shapes and is shaped by culture more broadly [104].
+# 5.3 Dual use.
+The video QA tasks that we studied carry risk of dual use, through possible downstream applications like surveillance [91, 128]. It seems unlikely that purely technological fixes and defenses – which themselves can be problematic [40] – could resolve these dynamics. Studying how well video-level pretraining enables surveillance applications might be an important avenue for future work, if only to inform stakeholders and policymakers about these risks.
+# 5.4 Energy consumption.
+The pretraining that we used in this work was expensive upfront [105]. Our results suggest that scaling up the amount of data and compute that we used might yield additional performance gains – but at increased environmental cost. To pretrain more efficiently, we used a much more lightweight architecture (in terms of FLOPs) than is standard for today’s vision and language models. We hope that our public release of the model (for research use) can further amortize this cost.
+# 5.5 Synthesizing these risks.
+With these issues in mind, we release MERLOT and YT-Temporal-180M for researchers. We view our work, and our research artifacts, to be part of a larger conversation on the limits of pretrained ‘foundation models’ [17]. These models have broad impact to real-world areas like healthcare, law, and education. At the same time, these models have significant risks, including the harms that we outlined. We believe that further academic research into this video-and-language pretraining paradigm is important – especially to probe its limits and possible harms. We hope that our paper, code, and data release can contribute to this direction.
+# Acknowledgements and Funding Transparency Statement
+We thank the anonymous reviewers for their helpful feedback that improved this work, along with Oren Etzioni and Gabriel Ilharco. Thanks also to Zak Stone and the Google Cloud TPU team for providing access to the TPU machines used for conducting experiments, and for help with the computing infrastructure. Last, but not least, thanks to all the YouTubers who share interesting videos with the world. This work was funded by DARPA MCS program through NIWC Pacific (N66001-19-2-4031), and the Allen Institute for AI.
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+# REPRESENTING ENTROPY : A SHORT PROOF OF THE EQUIVALENCE BETWEEN SOFT Q-LEARNING AND POLICY GRADIENTS
+Anonymous authors Paper under double-blind review
+# ABSTRACT
+Two main families of reinforcement learning algorithms, Q-learning and policy gradients, have recently been proven to be equivalent when using a softmax relaxation on one part, and an entropic regularization on the other. We relate this result to the well-known convex duality of Shannon entropy and the softmax function. Such a result is also known as the Donsker-Varadhan formula. This provides a short proof of the equivalence. We then interpret this duality further, and use ideas of convex analysis to prove a new policy inequality relative to soft Q-learning.
+# 1 INTRODUCTION AND SETTING
+Deep reinforcement learning as a research field is currently undergoing tremendous growth, largely due to empirical successes brought about by scaling the technique to real-world examples such as Atari games and Go. Historically, two main families of algorithms have existed:
+• $Q$ -learning (V. Mnih (2015)) proposes to iteratively refine estimates of a family of scalar action-value functions. These represent the reward expected after undertaking a given action, so as to be able to act greedily (or $\epsilon$ -greedily) with respect to those numbers; • Policy gradients (V. Mnih & Kavukcuoglu. (2016)), looks to maximize the expected reward by improving policies to favor high-reward actions. In general, the target loss function is regularized by the addition of an entropic functional for the policy. This makes policies more diffuse and less likely to yield degenerate results.
+A critical step in the theoretical understanding of the field has been a smooth relaxation of the greedy max operation involved in selecting actions, turned into a Boltzmann softmax O. Nachum & Schuurmans. (2017b.). This new context has lead to a breakthrough this year J. Schulman & Abbeel. (2017) with the proof of the equivalence of both methods of Q-learning and policy gradients. While that result is extremely impressive in its unification, we argue that it is critical to look additionally at the fundamental reasons as to why it occurs. We believe that the convexity of the entropy functional used for policy regularization is at the root of the phenomenon, and that (Lagrangian) duality can be exploited as well, either yielding faster proofs, or further understanding. The contributions of our paper are as follows:
+1. We show how convex duality expedites the proof of the equivalence between soft Qlearning and softmax entropic policy gradients - heuristically in the general case, rigorously in the bandit case.
+2. We introduce a transportation inequality that relates the expected optimality gap of any policy with its Kullback-Leibler divergence to the optimal policy.
+We describe our notations here. Abusing notation heavily by identifying measures with their densities as in $d \pi ( a | s ) = \pi ( a | s ) d a$ , if we note as either $r ( s , a )$ or $r ( a , s )$ the reward obtained by taking action $a$ in state $s$ , the expected reward expands as:
+$$
+K _ { r } ( \pi ) = \mathbb { E } _ { \pi } { \big [ } r ( s , a ) { \big ] } = \int _ { \mathbb { A } } r ( s , a ) d \pi ( a | s )
+$$
+$K _ { r }$ is a linear functional of $\pi$ . Adding Shannon entropic regularization1 improves numerical stability of the algorithm, and prevents early convergence to degenerate solutions. Noting regularization strength $\beta$ , the objective becomes a free energy functional, named by analogy with a similar quantity in statistical mechanics:
+$$
+J ( \pi ) = \int _ { \mathbb { A } } r ( s , a ) d \pi ( a | s ) - \beta \int _ { \mathbb { A } } \log \pi ( a | s ) d \pi ( a | s )
+$$
+Crucially, viewed as a functional of $\pi$ , $J$ is convex and is the sum of two parts
+$$
+J ( \pi ) = K _ { r } ( \pi ) - \beta H ( \pi ) , \quad H ( \pi ) = \int _ { \mathbb { A } } \log \pi ( a | s ) d \pi ( a | s )
+$$
+# 2 THE GIBBS VARIATIONAL PRINCIPLE FOR POLICY EVALUATION
+# 2.1 LEGENDRE TRANSFORM AND POLICY ENTROPY
+Here we are interested in the optimal value of the policy functional $J$ , achieved for an optimal policy $\pi ^ { * }$ . We hence look for $J ^ { * } = J ( \pi ^ { * } ) = \operatorname* { s u p } _ { \pi \in \mathbb { P } } J ( \pi )$ . In the one step-one state bandit setting we are in, this is in fact almost the same as deriving the state-value function.
+The principles of convex duality Bauschke & Combettes. (2011); Ziebart. (2010); G. Neu & Jonsson. (2017) yield a useful representation. Non-regularized empirical rewards in equation 1 can be seen as the standard inner product in Hilbert space $L ^ { 2 }$ . We therefore equate inner product, expectation and integral over A. Writing $J ^ { * }$ as
+$$
+J ^ { * } = \operatorname* { s u p } _ { \pi \in \mathbb { P } } J ( \pi ) = \operatorname* { s u p } _ { \pi \in \mathbb { P } } \quad \langle r ( s , a ) , \pi ( a | s ) \rangle - \beta H ( \pi )
+$$
+with $H$ the entropy functional defined above, we recover exactly the definition of the LegendreFenchel transformation, or convex conjugate, of $\beta \cdot H$ . The word convex applies to the entropy functional, and doesn’t make any assumptions on the rewards $r ( s , a )$ , other that they be well-behaved enough to be integrable in $a$ .
+The Legendre transform inverts derivatives. A simple calculation shows that the formal convex conjugate of $f : t  t \log t $ is $f ^ { * } : p \to e ^ { ( p - 1 ) }$ - this because their respective derivatives log and exp are reciprocal. We can apply this to $f ( \pi ( a | s ) ) = \pi ( a | s ) \log \pi ( a | s )$ , and then this relationship can also be integrated in $a$ . Hence the dual Legendre representation of the entropy functional $H$ is known. The Gibbs variational principle states that, taking $\beta = 1 / \lambda$ as the inverse temperature parameter, and for each Borelian (measurable) test function $\Phi \in C ^ { b } ( \dot { \mathbb { A } } )$ :
+$$
+\forall \Phi \in C ^ { b } ( \mathbb { A } ) , \quad \operatorname* { s u p } _ { \pi \in \mathbb { P } } \Big [ \int _ { \mathbb { A } } \Phi d \pi - \frac { 1 } { \lambda } H ( \pi ) \Big ] = \frac { 1 } { \lambda } \log \int _ { \mathbb { A } } e ^ { \lambda \Phi } d a
+$$
+or in shorter notation, for each real random variable $X$ with exponential moments,
+$$
+\forall X \in \mathbb { P } , \quad \operatorname* { s u p } _ { \pi \in \mathbb { P } } \quad \mathbb { E } _ { \pi } ( X ) - \frac { 1 } { \lambda } H ( \pi ) = \frac { 1 } { \lambda } \log \mathbb { E } ( e ^ { \lambda X } )
+$$
+We can prove a stronger result. If $\mu$ is a reference measure (or policy), and we now consider the relative entropy (or Kullback-Leibler divergence) with respect to $\mu$ , $H _ { \mu } ( \cdot )$ , instead of the entropy $H ( \cdot )$ , then the Gibbs variational principle still holds (Villani. (2008), chapter 22). This result regarding dual representation formulas for entropy is important and in fact found in several areas of science:
+• as above, in thermodynamics, where it is named the Gibbs variational principle; • in large deviations, this also known as the Donsker-Varadhan variational formula Dembo & Zeitouni. (2010);
+• in statistics, it is the well-known duality between maximum entropy and maximum likelihood estimation Altun & Smola. (2006); • finally, the theory of information geometry Amari. (2016) groups all three views and posits that there exists a general, dually flat Riemannian information manifold.
+The general form of the result is as follows. For each $\Phi$ representing a rewards function $r ( s , a )$ or an estimator of it:
+$$
+\forall \Phi \in C ^ { b } ( \mathbb { A } ) , \quad \operatorname* { s u p } _ { \pi \in \mathbb { P } } \Big [ \int _ { \mathbb { A } } \Phi d \pi - \frac { 1 } { \lambda } H _ { \mu } ( \pi ) \Big ] = \frac { 1 } { \lambda } \log \int _ { \mathbb { A } } e ^ { \lambda \Phi } d \mu
+$$
+and the supremum is reached for the measure $\pi ^ { * } \in \mathbb { P }$ defined by its Radon-Nikodym derivative equal to the Gibbs-Boltzmann measure yielding an energy policy:
+$$
+\frac { d \pi ^ { * } } { d \mu } = \frac { 1 } { Z } e ^ { \Phi }
+$$
+In the special case where $\mu$ is the Lebesgue measure on a bounded domain (that is, the uniform policy), we find back the result 5 above, up to a constant irrelevant for maximization. In the general case, the mathematically inclined reader will also see this as a rephrasing of the fact the Bregman divergence associated with Shannon entropy is the Kullback-Leibler divergence. For completeness’ sake, we provide here its full proof :
+Proposition 1. Donsker-Varadhan variational formula. Let $G$ be a bounded measurable function on $\mathcal { A }$ and $\pi$ , $\tilde { \pi }$ be probability measures on $\mathcal { A }$ , with $\pi$ absolutely continuous w.r.t. $\tilde { \pi }$ . Then
+$$
+\int _ { A } G d \pi - \tau D _ { \mathrm { K L } } [ \pi \| \tilde { \pi } ] = \ln \int _ { A } e ^ { G / \tau } d \tilde { \pi } - \tau D _ { \mathrm { K L } } [ \pi \| \pi ^ { * } ]
+$$
+where $\pi ^ { * }$ is a probability measure defined by the Radon-Nikodym derivative:
+$$
+\frac { d \pi ^ { * } } { d \tilde { \pi } } = \frac { e ^ { G / \tau } } { \int _ { A } e ^ { G / \tau } d \tilde { \pi } }
+$$
+Proof.
+$$
+\begin{array} { r l } { \int _ { A } G d \pi - \tau { \cal P } _ { \mathrm { K L } } [ \pi ] [ \tilde { \pi } ] = \int _ { A } G d \pi - \tau \int _ { A } ( \ln \frac { d \pi } { d \pi } ) d \pi } & { } \\ & { = \int _ { A } G d \pi - \tau \int _ { A } \left( \ln \frac { d \pi } { d \pi } \right) d \pi - \tau \int _ { A } ( \ln \frac { d \pi ^ { * } } { d \pi } ) d \pi } \\ & { = \int _ { A } \left( G - \tau ( \ln \frac { d \pi ^ { * } } { d \pi } ) \right) \partial \pi - \tau { \cal P } _ { \mathrm { K L } } [ \pi ] [ \pi ^ { * } ] } \\ & { = \int _ { A } \left( G - \tau ( \ln \frac { e ^ { G / \tau } } { \int _ { A } \epsilon ^ { G / \tau } d \pi } ) \right) d \pi - \tau { \cal P } _ { \mathrm { K L } } [ \pi ] [ \pi ^ { * } ] } \\ & { = \int _ { A } \left( \ln \int _ { A } \epsilon ^ { G / \tau } d \tilde { \pi } \right) d \pi - \tau { \cal D } _ { \mathrm { K L } } [ \pi ] [ \pi ^ { * } ] } \\ & { = \ln \int _ { A } \epsilon ^ { G / \tau } d \tilde { \pi } - \tau { \cal D } _ { \mathrm { K L } } [ \pi ] [ \pi ^ { * } ] } \end{array}
+$$
+Proposition 2. Corollary :
+$$
+\operatorname* { m a x } _ { \pi } \left[ \int _ { \cal A } G d \pi - \tau D _ { \mathrm { K L } } [ \pi | | \tilde { \pi } ] \right] = \ln \int _ { \cal A } e ^ { G / \tau } d \tilde { \pi }
+$$
+and the maximum is attained uniquely by $\pi ^ { * }$ .
+Proof. $D _ { \mathrm { K L } } [ \pi \lVert \pi ^ { * } ] \geq 0$ , and $D _ { \mathrm { K L } } [ \pi \lVert \boldsymbol { \pi } ^ { * } ] = 0$ if and only if $\pi = \pi ^ { * }$ .
+The link with reinforcement learning is made by picking $\Phi = r ( s , a )$ , $\pi = \pi ( a | s )$ , $\lambda = 1 / \beta$ , and by recalling the implicit dependency of the right member on $s$ but not on $\pi$ at optimality, so that we can write
+$$
+J ^ { * } = V ^ { * } ( s ) = \beta \cdot \log \int _ { \mathbb { A } } e ^ { r ( s , a ) / \beta } d \mu ( a )
+$$
+which is the definition of the one-step soft Bellman operator at optimum R. Fox & Tishby. (2015); O. Nachum & Schuurmans. (2017b.); T. Haarnoja & Levine. (2017). Note that here $V ^ { * } ( s )$ depends on the reference measure $\mu$ which is used to pick actions frequency - we can be off-policy, in which case $V ^ { * }$ is only a pseudo state-value function.
+# 2.2 PROVING SOFT Q-LEARNING EQUIVALENCE
+In this simplified one-step setting, this provides a short and direct proof that in expectation, and trained to optimality, soft Q-learning and policy gradients ascent yield the same result J. Schulman & Abbeel. (2017). Standard Q-learning is the special case $\beta \to 0 , \quad \lambda \to \infty$ where by the Laplace principle we recover $V ( s ) \to \operatorname* { m a x } _ { \mathbb { A } } r ( s , a )$ ; that is, the zero-temperature limit, with no entropy regularization. For simplicity of exposition, we have restricted so far to the proof in the bandit setting; now we extend it to the general case.
+First by inserting $V ^ { * } ( s ) = \operatorname* { s u p } _ { \pi } V ^ { \pi } ( s )$ in the representation formulas above applied to $r ( s , a ) + \gamma V ^ { * } ( s ^ { \prime } )$ , so that
+$$
+V ^ { * } ( s ) = \operatorname* { s u p } _ { \pi } \left[ \mathbb { E } _ { \pi } [ r ( s , a ) + \gamma V ^ { * } ( s ^ { \prime } ) ] - \beta H ( \pi ) \right] = \beta \cdot \log \int _ { \mathbb { A } } e ^ { \frac { r ( s , a ) + \gamma V ^ { * } ( s ^ { \prime } ) } { \beta } } d a
+$$
+The proof in the general case will then be finished if we assume that we could apply the Bellman optimality principle not to the hard-max, but to the soft-max operator. This requires proving that the soft-Bellman operator admits a unique fixed point, which is the above. By the Brouwer fixed point theorem, it is enough to prove that it is a contraction, or at least non-expansive (we assume that the discount factor $\gamma < 1$ to that end). We do so below, noting that this result has been shown many times in the literature, for instance in O. Nachum & Schuurmans. (2017b.). Refining the soft-Bellman operator just like above, but in the multi-step case, by the expression
+$$
+( B ^ { * } V ) ( s ) = \beta \cdot \log \int _ { a } e ^ { \frac { r ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } \mid s , a } ( V ( s ^ { \prime } ) ) } { \beta } } d a
+$$
+we get the:
+# Proposition 3. Nonexpansiveness of the soft-Bellman operator for the supremum norm $\| f \| _ { \infty }$
+$$
+\begin{array} { r } { \left. B ^ { * } V ^ { ( 1 ) } - B ^ { * } V ^ { ( 2 ) } \right. _ { \infty } < \lVert V ^ { ( 1 ) } - V ^ { ( 2 ) } \rVert _ { \infty } } \end{array}
+$$
+Proof. Let us consider two state-value functions $V ^ { ( 1 ) } ( s )$ and $V ^ { ( 2 ) } ( s )$ along with the associated action-value functions $Q ^ { ( 1 ) } ( s , a )$ and $Q ^ { ( 2 ) } ( s , a )$ . Besides, denote MDP transition probability by $p ( s ^ { \prime } | s , a )$ . Then :
+$$
+\begin{array} { r l } { \left\| { \boldsymbol B } ^ { * } { \boldsymbol V } ^ { ( 1 ) } - { \boldsymbol B } ^ { * } { \boldsymbol V } ^ { ( 2 ) } \right\| _ { \infty } = \underset { s } { \operatorname* { m a x } } \left| ( { \boldsymbol B } ^ { * } { \boldsymbol V } ^ { ( 1 ) } ) ( s ) - ( { \boldsymbol B } ^ { * } { \boldsymbol V } ^ { ( 2 ) } ) ( s ) \right| } & { } \\ { \ } & { \leq \underset { s } { \operatorname* { m a x } } \underset { a } { \operatorname* { m a x } } \left| Q ^ { ( 1 ) } ( s , a ) - Q ^ { ( 2 ) } ( s , a ) \right| } \\ { \ } & { = \gamma \underset { s } { \operatorname* { m a x } } \underset { a } { \operatorname* { m a x } } \left| \mathbb { E } _ { s ^ { \prime } \mid s , a } \big [ { \boldsymbol V } ^ { ( 1 ) } ( s ^ { \prime } ) - { \boldsymbol V } ^ { ( 2 ) } ( s ^ { \prime } ) \big ] \right| } \\ { \ } & { \leq \gamma \underset { s } { \operatorname* { m a x } } \underset { a } { \operatorname* { m a x } } \left\| { p ( s ^ { \prime } \mid s , a ) } \right\| _ { 1 } \left\| { \boldsymbol V } ^ { ( 1 ) } - { \boldsymbol V } ^ { ( 2 ) } \right\| _ { \infty } ~ \mathrm { b y ~ H \ddot { o } l d e r ' s ~ i n e q u a l i t y } } \\ { \ } & { = \gamma \| { \boldsymbol V } ^ { ( 1 ) } - { \boldsymbol V } ^ { ( 2 ) } \| _ { \infty } < \| { \boldsymbol V } ^ { ( 1 ) } - { \boldsymbol V } ^ { ( 2 ) } \| _ { \infty } } \end{array}
+$$
+# 2.3 INTERPRETATION
+In summary, the program of the proof was as below :
+1. Write down the entropy-regularised policy gradient functional, and apply the DonskerVaradhan formula to it.
+2. Write down the resulting softmax Bellman operator as a solution to the sup maximization - this obviously also proves existence.
+3. Show that the softmax operator, just like the hard max, is still a contraction for the max norm, hence prove uniqueness of the solution by fixed point theorem.
+The above also shows formally that, should we discretize the action space A to replace integration over actions by finite sums, any strong estimator ${ \hat { r } } ( s , a )$ of $r ( s , a )$ , applied to the partition function of rewards $\begin{array} { r } { \frac { 1 } { \lambda } \log \sum _ { a } e ^ { \lambda r \left( s , a \right) } } \end{array}$ , could be used for Q-learning-like iterations. This is because strong convergence would imply weak convergence (especially convergence of the characteristic function, via Levy’s continuity theorem), and hence convergence towards the log-sum-exp cumulant generative function above. Different estimators ${ \hat { r } } ( s , a )$ lead to different algorithms. When the MDP and the rewards function $r$ are not known, the parameterised critic choice $\hat { r } ( s , a ) \approx Q _ { w } ( s , a )$ recovers Nachum’s Path Consistency Learning O. Nachum & Schuurmans. (2017b.;c). O’Donoghue’s PGQ method B. O’Donoghue & Mnih. (2016) can be seen as a control variate balancing of the two terms in 7. In theory, the rewards distribution could be also recovered simply by varying $\lambda$ (or $\beta$ ), for instance by inverse Laplace transform.
+# 3 POLICY OPTIMALITY GAP AND TEMPERATURE ANNEALING
+In this section, we propose an inequality that relates the optimality gap of a policy - by how much that policy is sub-optimal on average - to the Kullback-Leibler divergence between the current policy and the optimum. The proof draws on ideas of convex analysis and Legendre transormation exposed earlier in the context of soft Q-learning.
+Let us assume that $X$ is a real-valued bounded random variable. We denote $\operatorname* { s u p } | X | \leq M$ with $M$ constant. Furthermore we assume that $X$ is centered, that is, $\mathbb { E } [ X ] = 0$ . This can always be achieved just by picking $Y = X - \mathbb { E } [ X ]$ .
+Then, by the Hoeffding inequality :
+$$
+\log \mathbb { E } ( e ^ { \beta X } ) \leq K \frac { \beta ^ { 2 } } { 2 }
+$$
+with $K$ a positive real constant, i.e., the variable $X$ is sub-Gaussian, so that its cumulant generating function grows less than quadratically. By taking a Legendre transformation and inverting it, we get that for any pair of measures $\mathbb { P }$ and $\mathbb { Q }$ that are mutually absolutely continuous, one has
+$$
+\begin{array} { r } { \mathbb { E } _ { \mathbb { Q } } \mathopen { } \mathclose \bgroup \left( X \aftergroup \egroup \right) - \mathbb { E } _ { \mathbb { P } } \mathopen { } \mathclose \bgroup \left( X \aftergroup \egroup \right) \leq \sqrt { 2 K \cdot D _ { K L } \mathopen { } \mathclose \bgroup \left( \mathbb { Q } \aftergroup \egroup \right| \mathopen { } \mathclose \bgroup \left| \mathbb { P } \aftergroup \egroup \right) } } \end{array}
+$$
+which by specializing $\mathbb { Q }$ to be the measure associated to $\mathbb { P } ^ { * }$ the optimal policy, $\mathbb { P } _ { \theta }$ the current parameterized policy, and $X$ an advantage return $r$ :
+$$
+\mathbb { E } _ { \mathbb { P } ^ { * } } \left( r \right) \leq \mathbb { E } _ { \mathbb { P } _ { \theta } } \left( r \right) + \sqrt { 2 K } \sqrt { D _ { K L } \left( \mathbb { P } ^ { * } | | \mathbb { P } _ { \theta } \right) }
+$$
+By the same logic, any upper bound on $\log \mathbb { E } \left( e ^ { \beta X } \right)$ can give us information about $\mathbb { E } _ { \mathbb { Q } } ( X ) - \mathbb { E } _ { \mathbb { P } } ( X )$ . This enables us to relate the size of Kullback-Leibler trust regions to the amount by which our policy could be improved. In fact by combining the entropy duality formula with the Legendre transformation, one easily proves the below :
+Proposition 4. Let $X$ a real-valued integrable random variable, and $f$ a convex and differentiable function such that $f ( 0 ) = f ^ { \prime } ( 0 ) = 0$ . Then with $f ^ { * } : x  f ^ { * } ( x ) = \operatorname* { s u p } ( \beta x - f ( \beta ) )$ the Legendre transformation of $f$ , $f ^ { * - 1 }$ its reciprocal, and $\mathbb { P }$ and $\mathbb { Q }$ any two mutually absolutely continuous measures, one has the equivalence:
+$$
+\begin{array} { r } { \log \mathbb { E } _ { \mathbb { P } } \big ( e ^ { \beta ( X - \mathbb { E } _ { \mathcal { F } } ( X ) ) } \big ) \leq f ( \beta ) \quad \iff \quad \mathbb { E } _ { \mathbb { Q } } ( X ) - \mathbb { E } _ { \mathbb { P } } ( X ) \leq f ^ { * - 1 } \big [ D _ { K L } \big ( \mathbb { Q } | | \mathbb { P } \big ) \big ] } \end{array}
+$$
+Proof. By Donsker-Varadhan formula, one has that the equivalence is proven if and only if
+$$
+\mathbb { E } _ { \mathbb { Q } } ( X ) - \mathbb { E } _ { \mathbb { P } } ( X ) \leq \operatorname* { i n f } _ { \beta } { \big [ } { \frac { f ( \beta ) + D _ { K L } ( \mathbb { Q } | | \mathbb { P } ) } { \beta } } { \big ] }
+$$
+but this right term is easily proven to be nothing but
+$$
+f ^ { * - 1 } ( D _ { K L } ( \mathbb { Q } | | \mathbb { P } ) )
+$$
+the inverse of the Legendre transformation of $f$ applied to $D _ { K L } ( \mathbb { Q } | | \mathbb { P } )$ .
+This also opens up the possibility of using various softmax temperatures $\beta _ { i }$ in practical algorithms in order to estimate $f$ . Finally, note that if $\mathbb { P } _ { \theta }$ is a parameterized softmax policy associated with action-value functions $Q _ { \theta } ( a , s )$ and temperature $\beta$ , then because $\mathbb { P } ^ { * }$ is proportional to $e ^ { - r ( a , s ) / \beta }$ , one readily has
+$$
+D _ { K L } \left( \mathbb { P } ^ { * } | | \mathbb { P } _ { \theta } \right) = \frac { 1 } { \beta } \Big [ \mathbb { E } ( Q _ { \theta } ) - \mathbb { E } ( r ) \Big ]
+$$
+which can easily be inserted in the inequality above for the special case $\mathbb { Q } = \mathbb { P } ^ { * }$ .
+# 4 RELATED WORK
+Entropic reinforcement learning has appeared early in the literature with two different motivations. The view of exploration with a self-information intrinsic reward was pioneered by Tishby, and developed in Ziebart’s PhD. thesis Ziebart. (2010). It was rediscovered recently that within the asynchronous actor-critic framework, entropic regularization is crucial to ensure convergence in practice V. Mnih & Kavukcuoglu. (2016). Furthermore, the idea of taking steepest KL divergence steps as a practical reinforcement learning method per se was adopted by Schulman J. Schulman & Abbeel. (2015a.). The Lagrangian duality view was pioneered in a practical context with O’Donoghue’s PGQ algorithm B. O’Donoghue & Mnih. (2016), and followed by the development of soft Q-learning jointly in R. Fox & Tishby. (2015) and in Nachum et al. O. Nachum & Schuurmans. (2017b.). The key common development in these works has been to make entropic regularization recursively follow the Bellman equation, rather than naively regularizing one-step policies G. Neu & Jonsson. (2017). Schulman thereafter proposed a general proof of the equivalence, in the limit, of policy gradient and soft Q-learning methods J. Schulman & Abbeel. (2017), but the proof does not explicitly make the connection with convex duality and the expeditive justification it yields in the one-step case. Applying the Gibbs/Donsker-Varadhan variational formula to entropy in a machine learning context is, however, not new; see for instance Altun and Smola Altun & Smola. (2006). Some of the convex optimization results they invoke, including proximal stepping, can be found in the complete treatment by Bauschke Bauschke & Combettes. (2011). In the context of neural networks, convex analysis and partial differential equation methods are covered by Chaudhari P. Chaudhari & Carlier. (2017).
+# 5 FURTHER WORK
+Using dual formulas for the entropy functional in reinforcement learning has vast potential ramifications. One avenue of research will be to interpret our findings in a large deviations framework - the log-sum-exp cumulant generative function being an example of rate function governing fluctuations of the tail of empirical n-step returns. Smart drift change techniques could lead to significant variance reduction for Monte-Carlo rollout estimators. We also hope to exploit further concentration inequalities in order to provide more bounds for the state value function. Finally, a complete theory of the one-to-one correspondence between convex approximation algorithms and reinforcement learning methods is still lacking to date. We hope to be able to contribute in this direction through further work.
+# REFERENCES
+Y. Altun and A. Smola. Unifying divergence minimization and statistical inference via convex duality. COLT, 19th Annual Conference on Learning Theory, 2006.
+S. Amari. Information Geometry and Its Applications. Springer, Applied Mathematical Sciences., 2016.
+K. Kavukcuoglu B. O’Donoghue, R. Munos and V. Mnih. Pgq $:$ Combining policy gradient and q-learning. arXiv preprint arXiv:1611.01626, 2016.
+H. H. Bauschke and P. L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer-Verlag, New York., 2011.
+A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Springer, Applications of Mathematics, 38., 2010.
+V. Gomez G. Neu and A. Jonsson. A unified view of entropy-regularized markov decision processes. arXiv preprint arXiv:1705.07798, 2017.
+P. Moritz M. I. Jordan J. Schulman, S. Levine and P. Abbeel. High-dimensional continuous control using generalized advantage estimation. arXiv preprint arXiv:1502.05477, 2015a.
+X. Chen J. Schulman and P. Abbeel. Equivalence between policy gradients and soft q-learning. arXiv preprint arXiv:1704.06440, 2017.
+K. Xu O. Nachum, M. Norouzi and D. Schuurmans. Bridging the gap between value and policy based reinforcement learning. arXiv preprint arXiv:1702.08892, 2017b.
+K. Xu O. Nachum, M. Norouzi and D. Schuurmans. Trust-pcl: An off-policy trust region method for continuous control. arXiv preprint arXiv:1707.01891, 2017c.
+S. Osher S. Soatto P. Chaudhari, A. Oberman and G. Carlier. Deep relaxation: partial differential equations for optimizing deep neural networks. arXiv preprint arXiv:1704.04932, 2017.
+A. Pakman R. Fox and N. Tishby. Taming the noise in reinforcement learning via soft updates. arXiv preprint arXiv:1512.08562, 2015.
+P. Abbeel T. Haarnoja, H. Tang and S. Levine. Reinforcement learning with deep energy-based policies. arXiv preprint arXiv:1702.08165, 2017.
+D. Silver A. A. Rusu J. Veness M. G. Bellemare A. Graves M. Riedmiller A. K. Fidjeland G. Ostrovski et al. V. Mnih, K. Kavukcuoglu. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015.
+M. Mirza A. Graves T. P Lillicrap T. Harley D. Silver V. Mnih, A. Puigdomenech Badia and K. Kavukcuoglu. Asynchronous methods for deep reinforcement learning. arXiv preprint arXiv:1602.01783, 2016.
+C. Villani. Optimal Transport : Old and New. Grundlehren der mathematischen Wissenschaften, volume 338., 2008.
+B. D. Ziebart. Modeling purposeful adaptive behavior with the principle of maximum causal entropy. PhD Thesis., 2010.

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+    {
+        "type": "text",
+        "text": "REPRESENTING ENTROPY : A SHORT PROOF OF THE EQUIVALENCE BETWEEN SOFT Q-LEARNING AND POLICY GRADIENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            797,
+            171
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            195,
+            398,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            261,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Two main families of reinforcement learning algorithms, Q-learning and policy gradients, have recently been proven to be equivalent when using a softmax relaxation on one part, and an entropic regularization on the other. We relate this result to the well-known convex duality of Shannon entropy and the softmax function. Such a result is also known as the Donsker-Varadhan formula. This provides a short proof of the equivalence. We then interpret this duality further, and use ideas of convex analysis to prove a new policy inequality relative to soft Q-learning. ",
+        "bbox": [
+            232,
+            292,
+            766,
+            390
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION AND SETTING ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            417,
+            454,
+            434
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep reinforcement learning as a research field is currently undergoing tremendous growth, largely due to empirical successes brought about by scaling the technique to real-world examples such as Atari games and Go. Historically, two main families of algorithms have existed: ",
+        "bbox": [
+            176,
+            449,
+            825,
+            492
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• $Q$ -learning (V. Mnih (2015)) proposes to iteratively refine estimates of a family of scalar action-value functions. These represent the reward expected after undertaking a given action, so as to be able to act greedily (or $\\epsilon$ -greedily) with respect to those numbers; • Policy gradients (V. Mnih & Kavukcuoglu. (2016)), looks to maximize the expected reward by improving policies to favor high-reward actions. In general, the target loss function is regularized by the addition of an entropic functional for the policy. This makes policies more diffuse and less likely to yield degenerate results. ",
+        "bbox": [
+            215,
+            503,
+            825,
+            608
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A critical step in the theoretical understanding of the field has been a smooth relaxation of the greedy max operation involved in selecting actions, turned into a Boltzmann softmax O. Nachum & Schuurmans. (2017b.). This new context has lead to a breakthrough this year J. Schulman & Abbeel. (2017) with the proof of the equivalence of both methods of Q-learning and policy gradients. While that result is extremely impressive in its unification, we argue that it is critical to look additionally at the fundamental reasons as to why it occurs. We believe that the convexity of the entropy functional used for policy regularization is at the root of the phenomenon, and that (Lagrangian) duality can be exploited as well, either yielding faster proofs, or further understanding. The contributions of our paper are as follows: ",
+        "bbox": [
+            173,
+            619,
+            825,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. We show how convex duality expedites the proof of the equivalence between soft Qlearning and softmax entropic policy gradients - heuristically in the general case, rigorously in the bandit case.   \n2. We introduce a transportation inequality that relates the expected optimality gap of any policy with its Kullback-Leibler divergence to the optimal policy. ",
+        "bbox": [
+            210,
+            757,
+            825,
+            834
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We describe our notations here. Abusing notation heavily by identifying measures with their densities as in $d \\pi ( a | s ) = \\pi ( a | s ) d a$ , if we note as either $r ( s , a )$ or $r ( a , s )$ the reward obtained by taking action $a$ in state $s$ , the expected reward expands as: ",
+        "bbox": [
+            178,
+            845,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae6ae1d42cf0aed2406ee5cef85d9c4c6460d1cece32c0c4cc30df429391c8db.jpg",
+        "text": "$$\nK _ { r } ( \\pi ) = \\mathbb { E } _ { \\pi } { \\big [ } r ( s , a ) { \\big ] } = \\int _ { \\mathbb { A } } r ( s , a ) d \\pi ( a | s )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            895,
+            642,
+            928
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$K _ { r }$ is a linear functional of $\\pi$ . Adding Shannon entropic regularization1 improves numerical stability of the algorithm, and prevents early convergence to degenerate solutions. Noting regularization strength $\\beta$ , the objective becomes a free energy functional, named by analogy with a similar quantity in statistical mechanics: ",
+        "bbox": [
+            174,
+            102,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/82272d25e05bce0b10d447dcca3dd9051c59d84724a54097853a942aad8f092a.jpg",
+        "text": "$$\nJ ( \\pi ) = \\int _ { \\mathbb { A } } r ( s , a ) d \\pi ( a | s ) - \\beta \\int _ { \\mathbb { A } } \\log \\pi ( a | s ) d \\pi ( a | s )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            165,
+            673,
+            198
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Crucially, viewed as a functional of $\\pi$ , $J$ is convex and is the sum of two parts ",
+        "bbox": [
+            173,
+            212,
+            686,
+            227
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/51cb5d15e68db5783e3ffb12aa67ea8183afaea8dc8ae2d82d3c921537d18c7e.jpg",
+        "text": "$$\nJ ( \\pi ) = K _ { r } ( \\pi ) - \\beta H ( \\pi ) , \\quad H ( \\pi ) = \\int _ { \\mathbb { A } } \\log \\pi ( a | s ) d \\pi ( a | s )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            233,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 THE GIBBS VARIATIONAL PRINCIPLE FOR POLICY EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            284,
+            728,
+            301
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 LEGENDRE TRANSFORM AND POLICY ENTROPY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            316,
+            540,
+            330
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Here we are interested in the optimal value of the policy functional $J$ , achieved for an optimal policy $\\pi ^ { * }$ . We hence look for $J ^ { * } = J ( \\pi ^ { * } ) = \\operatorname* { s u p } _ { \\pi \\in \\mathbb { P } } J ( \\pi )$ . In the one step-one state bandit setting we are in, this is in fact almost the same as deriving the state-value function. ",
+        "bbox": [
+            173,
+            342,
+            823,
+            386
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The principles of convex duality Bauschke & Combettes. (2011); Ziebart. (2010); G. Neu & Jonsson. (2017) yield a useful representation. Non-regularized empirical rewards in equation 1 can be seen as the standard inner product in Hilbert space $L ^ { 2 }$ . We therefore equate inner product, expectation and integral over A. Writing $J ^ { * }$ as ",
+        "bbox": [
+            173,
+            391,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/960886da7649b9f1133c799d252cc93845352c0dff762259cc5430b05e24f698.jpg",
+        "text": "$$\nJ ^ { * } = \\operatorname* { s u p } _ { \\pi \\in \\mathbb { P } } J ( \\pi ) = \\operatorname* { s u p } _ { \\pi \\in \\mathbb { P } } \\quad \\langle r ( s , a ) , \\pi ( a | s ) \\rangle - \\beta H ( \\pi )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            455,
+            669,
+            481
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "with $H$ the entropy functional defined above, we recover exactly the definition of the LegendreFenchel transformation, or convex conjugate, of $\\beta \\cdot H$ . The word convex applies to the entropy functional, and doesn’t make any assumptions on the rewards $r ( s , a )$ , other that they be well-behaved enough to be integrable in $a$ . ",
+        "bbox": [
+            173,
+            488,
+            826,
+            545
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The Legendre transform inverts derivatives. A simple calculation shows that the formal convex conjugate of $f : t  t \\log t $ is $f ^ { * } : p \\to e ^ { ( p - 1 ) }$ - this because their respective derivatives log and exp are reciprocal. We can apply this to $f ( \\pi ( a | s ) ) = \\pi ( a | s ) \\log \\pi ( a | s )$ , and then this relationship can also be integrated in $a$ . Hence the dual Legendre representation of the entropy functional $H$ is known. The Gibbs variational principle states that, taking $\\beta = 1 / \\lambda$ as the inverse temperature parameter, and for each Borelian (measurable) test function $\\Phi \\in C ^ { b } ( \\dot { \\mathbb { A } } )$ : ",
+        "bbox": [
+            173,
+            551,
+            825,
+            637
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf5339e47d620075d1dd4d9e6d82d257837754baa24bad0633b488d69086d8a0.jpg",
+        "text": "$$\n\\forall \\Phi \\in C ^ { b } ( \\mathbb { A } ) , \\quad \\operatorname* { s u p } _ { \\pi \\in \\mathbb { P } } \\Big [ \\int _ { \\mathbb { A } } \\Phi d \\pi - \\frac { 1 } { \\lambda } H ( \\pi ) \\Big ] = \\frac { 1 } { \\lambda } \\log \\int _ { \\mathbb { A } } e ^ { \\lambda \\Phi } d a\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            654,
+            700,
+            689
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "or in shorter notation, for each real random variable $X$ with exponential moments, ",
+        "bbox": [
+            174,
+            699,
+            712,
+            715
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6726d431b8206d6576b11831b3f2fc92293259f1faebcef0b35b843b41e7b150.jpg",
+        "text": "$$\n\\forall X \\in \\mathbb { P } , \\quad \\operatorname* { s u p } _ { \\pi \\in \\mathbb { P } } \\quad \\mathbb { E } _ { \\pi } ( X ) - \\frac { 1 } { \\lambda } H ( \\pi ) = \\frac { 1 } { \\lambda } \\log \\mathbb { E } ( e ^ { \\lambda X } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            720,
+            678,
+            755
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We can prove a stronger result. If $\\mu$ is a reference measure (or policy), and we now consider the relative entropy (or Kullback-Leibler divergence) with respect to $\\mu$ , $H _ { \\mu } ( \\cdot )$ , instead of the entropy $H ( \\cdot )$ , then the Gibbs variational principle still holds (Villani. (2008), chapter 22). This result regarding dual representation formulas for entropy is important and in fact found in several areas of science: ",
+        "bbox": [
+            176,
+            768,
+            826,
+            827
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• as above, in thermodynamics, where it is named the Gibbs variational principle; • in large deviations, this also known as the Donsker-Varadhan variational formula Dembo & Zeitouni. (2010); ",
+        "bbox": [
+            215,
+            837,
+            825,
+            886
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• in statistics, it is the well-known duality between maximum entropy and maximum likelihood estimation Altun & Smola. (2006); • finally, the theory of information geometry Amari. (2016) groups all three views and posits that there exists a general, dually flat Riemannian information manifold. ",
+        "bbox": [
+            210,
+            103,
+            826,
+            166
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The general form of the result is as follows. For each $\\Phi$ representing a rewards function $r ( s , a )$ or an estimator of it: ",
+        "bbox": [
+            173,
+            176,
+            825,
+            205
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e24c2b0c8afc4a22c7448b917769252dc4a5ec367d0aafb6a640f2d2e19a4d4.jpg",
+        "text": "$$\n\\forall \\Phi \\in C ^ { b } ( \\mathbb { A } ) , \\quad \\operatorname* { s u p } _ { \\pi \\in \\mathbb { P } } \\Big [ \\int _ { \\mathbb { A } } \\Phi d \\pi - \\frac { 1 } { \\lambda } H _ { \\mu } ( \\pi ) \\Big ] = \\frac { 1 } { \\lambda } \\log \\int _ { \\mathbb { A } } e ^ { \\lambda \\Phi } d \\mu\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            209,
+            707,
+            243
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and the supremum is reached for the measure $\\pi ^ { * } \\in \\mathbb { P }$ defined by its Radon-Nikodym derivative equal to the Gibbs-Boltzmann measure yielding an energy policy: ",
+        "bbox": [
+            174,
+            257,
+            823,
+            286
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35431e5c11b5a8ccb425e403753fbf6c545c65ab6fef534ae50fae033ef60c44.jpg",
+        "text": "$$\n\\frac { d \\pi ^ { * } } { d \\mu } = \\frac { 1 } { Z } e ^ { \\Phi }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            454,
+            292,
+            544,
+            327
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the special case where $\\mu$ is the Lebesgue measure on a bounded domain (that is, the uniform policy), we find back the result 5 above, up to a constant irrelevant for maximization. In the general case, the mathematically inclined reader will also see this as a rephrasing of the fact the Bregman divergence associated with Shannon entropy is the Kullback-Leibler divergence. For completeness’ sake, we provide here its full proof : ",
+        "bbox": [
+            173,
+            332,
+            826,
+            402
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Donsker-Varadhan variational formula. Let $G$ be a bounded measurable function on $\\mathcal { A }$ and $\\pi$ , $\\tilde { \\pi }$ be probability measures on $\\mathcal { A }$ , with $\\pi$ absolutely continuous w.r.t. $\\tilde { \\pi }$ . Then ",
+        "bbox": [
+            173,
+            406,
+            823,
+            435
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1bf1e53e2b722dadc2ae5d0f07d959bbeec21989f53eb64e2b7e9756675c9e6a.jpg",
+        "text": "$$\n\\int _ { A } G d \\pi - \\tau D _ { \\mathrm { K L } } [ \\pi \\| \\tilde { \\pi } ] = \\ln \\int _ { A } e ^ { G / \\tau } d \\tilde { \\pi } - \\tau D _ { \\mathrm { K L } } [ \\pi \\| \\pi ^ { * } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            441,
+            686,
+            476
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\pi ^ { * }$ is a probability measure defined by the Radon-Nikodym derivative: ",
+        "bbox": [
+            174,
+            481,
+            676,
+            497
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bcffad462181079216d5e3c1ee39cd498f2ea775b332049361fb24da66d766c4.jpg",
+        "text": "$$\n\\frac { d \\pi ^ { * } } { d \\tilde { \\pi } } = \\frac { e ^ { G / \\tau } } { \\int _ { A } e ^ { G / \\tau } d \\tilde { \\pi } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            433,
+            502,
+            565,
+            541
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proof. ",
+        "bbox": [
+            173,
+            555,
+            217,
+            570
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e42181688ad3f2a6044cfedbb6e30687b432de50055dd86885df07b292311383.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\int _ { A } G d \\pi - \\tau { \\cal P } _ { \\mathrm { K L } } [ \\pi ] [ \\tilde { \\pi } ] = \\int _ { A } G d \\pi - \\tau \\int _ { A } ( \\ln \\frac { d \\pi } { d \\pi } ) d \\pi } & { } \\\\ & { = \\int _ { A } G d \\pi - \\tau \\int _ { A } \\left( \\ln \\frac { d \\pi } { d \\pi } \\right) d \\pi - \\tau \\int _ { A } ( \\ln \\frac { d \\pi ^ { * } } { d \\pi } ) d \\pi } \\\\ & { = \\int _ { A } \\left( G - \\tau ( \\ln \\frac { d \\pi ^ { * } } { d \\pi } ) \\right) \\partial \\pi - \\tau { \\cal P } _ { \\mathrm { K L } } [ \\pi ] [ \\pi ^ { * } ] } \\\\ & { = \\int _ { A } \\left( G - \\tau ( \\ln \\frac { e ^ { G / \\tau } } { \\int _ { A } \\epsilon ^ { G / \\tau } d \\pi } ) \\right) d \\pi - \\tau { \\cal P } _ { \\mathrm { K L } } [ \\pi ] [ \\pi ^ { * } ] } \\\\ & { = \\int _ { A } \\left( \\ln \\int _ { A } \\epsilon ^ { G / \\tau } d \\tilde { \\pi } \\right) d \\pi - \\tau { \\cal D } _ { \\mathrm { K L } } [ \\pi ] [ \\pi ^ { * } ] } \\\\ & { = \\ln \\int _ { A } \\epsilon ^ { G / \\tau } d \\tilde { \\pi } - \\tau { \\cal D } _ { \\mathrm { K L } } [ \\pi ] [ \\pi ^ { * } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            577,
+            751,
+            787
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. Corollary : ",
+        "bbox": [
+            174,
+            815,
+            352,
+            830
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/305ff9c814f7a6cf917e41787943b796d6509ad2def83c3d7520d7315fbf8294.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi } \\left[ \\int _ { \\cal A } G d \\pi - \\tau D _ { \\mathrm { K L } } [ \\pi | | \\tilde { \\pi } ] \\right] = \\ln \\int _ { \\cal A } e ^ { G / \\tau } d \\tilde { \\pi }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            837,
+            658,
+            872
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and the maximum is attained uniquely by $\\pi ^ { * }$ . ",
+        "bbox": [
+            174,
+            878,
+            468,
+            892
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proof. $D _ { \\mathrm { K L } } [ \\pi \\lVert \\pi ^ { * } ] \\geq 0$ , and $D _ { \\mathrm { K L } } [ \\pi \\lVert \\boldsymbol { \\pi } ^ { * } ] = 0$ if and only if $\\pi = \\pi ^ { * }$ . ",
+        "bbox": [
+            173,
+            907,
+            620,
+            925
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The link with reinforcement learning is made by picking $\\Phi = r ( s , a )$ , $\\pi = \\pi ( a | s )$ , $\\lambda = 1 / \\beta$ , and by recalling the implicit dependency of the right member on $s$ but not on $\\pi$ at optimality, so that we can write ",
+        "bbox": [
+            174,
+            102,
+            823,
+            145
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7296b72ce5f6339cc7ed0cf33077e3870f369fcb93e2b4ab8aaf53e15d9ced3.jpg",
+        "text": "$$\nJ ^ { * } = V ^ { * } ( s ) = \\beta \\cdot \\log \\int _ { \\mathbb { A } } e ^ { r ( s , a ) / \\beta } d \\mu ( a )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            145,
+            633,
+            179
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which is the definition of the one-step soft Bellman operator at optimum R. Fox & Tishby. (2015); O. Nachum & Schuurmans. (2017b.); T. Haarnoja & Levine. (2017). Note that here $V ^ { * } ( s )$ depends on the reference measure $\\mu$ which is used to pick actions frequency - we can be off-policy, in which case $V ^ { * }$ is only a pseudo state-value function. ",
+        "bbox": [
+            173,
+            183,
+            825,
+            239
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 PROVING SOFT Q-LEARNING EQUIVALENCE",
+        "text_level": 1,
+        "bbox": [
+            174,
+            258,
+            517,
+            273
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this simplified one-step setting, this provides a short and direct proof that in expectation, and trained to optimality, soft Q-learning and policy gradients ascent yield the same result J. Schulman & Abbeel. (2017). Standard Q-learning is the special case $\\beta \\to 0 , \\quad \\lambda \\to \\infty$ where by the Laplace principle we recover $V ( s ) \\to \\operatorname* { m a x } _ { \\mathbb { A } } r ( s , a )$ ; that is, the zero-temperature limit, with no entropy regularization. For simplicity of exposition, we have restricted so far to the proof in the bandit setting; now we extend it to the general case. ",
+        "bbox": [
+            173,
+            285,
+            826,
+            371
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "First by inserting $V ^ { * } ( s ) = \\operatorname* { s u p } _ { \\pi } V ^ { \\pi } ( s )$ in the representation formulas above applied to $r ( s , a ) + \\gamma V ^ { * } ( s ^ { \\prime } )$ , so that ",
+        "bbox": [
+            173,
+            376,
+            743,
+            406
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/479f13f5228a422ceea70c0d1f8b67e4e319a86858104088a811792917cea3bb.jpg",
+        "text": "$$\nV ^ { * } ( s ) = \\operatorname* { s u p } _ { \\pi } \\left[ \\mathbb { E } _ { \\pi } [ r ( s , a ) + \\gamma V ^ { * } ( s ^ { \\prime } ) ] - \\beta H ( \\pi ) \\right] = \\beta \\cdot \\log \\int _ { \\mathbb { A } } e ^ { \\frac { r ( s , a ) + \\gamma V ^ { * } ( s ^ { \\prime } ) } { \\beta } } d a\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            414,
+            761,
+            457
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proof in the general case will then be finished if we assume that we could apply the Bellman optimality principle not to the hard-max, but to the soft-max operator. This requires proving that the soft-Bellman operator admits a unique fixed point, which is the above. By the Brouwer fixed point theorem, it is enough to prove that it is a contraction, or at least non-expansive (we assume that the discount factor $\\gamma < 1$ to that end). We do so below, noting that this result has been shown many times in the literature, for instance in O. Nachum & Schuurmans. (2017b.). Refining the soft-Bellman operator just like above, but in the multi-step case, by the expression ",
+        "bbox": [
+            173,
+            464,
+            825,
+            564
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f2bcc556c85832643a3a418d46cf4bb20a7d879a7117e87f6490ec402ee9938.jpg",
+        "text": "$$\n( B ^ { * } V ) ( s ) = \\beta \\cdot \\log \\int _ { a } e ^ { \\frac { r ( s , a ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } \\mid s , a } ( V ( s ^ { \\prime } ) ) } { \\beta } } d a\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            571,
+            650,
+            607
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "we get the: ",
+        "bbox": [
+            173,
+            623,
+            246,
+            637
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. Nonexpansiveness of the soft-Bellman operator for the supremum norm $\\| f \\| _ { \\infty }$ ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            642,
+            810,
+            659
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf306841e15aee5c51ed9bacd863bdef09bcd39ae4095053e1572a33e0f90444.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left. B ^ { * } V ^ { ( 1 ) } - B ^ { * } V ^ { ( 2 ) } \\right. _ { \\infty } < \\lVert V ^ { ( 1 ) } - V ^ { ( 2 ) } \\rVert _ { \\infty } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            666,
+            642,
+            695
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let us consider two state-value functions $V ^ { ( 1 ) } ( s )$ and $V ^ { ( 2 ) } ( s )$ along with the associated action-value functions $Q ^ { ( 1 ) } ( s , a )$ and $Q ^ { ( 2 ) } ( s , a )$ . Besides, denote MDP transition probability by $p ( s ^ { \\prime } | s , a )$ . Then : ",
+        "bbox": [
+            173,
+            717,
+            825,
+            762
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0ec93395f8bacb0ed7cc6a649ea443dc5157e110ffc1ced1efa159266c4ac62.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\left\\| { \\boldsymbol B } ^ { * } { \\boldsymbol V } ^ { ( 1 ) } - { \\boldsymbol B } ^ { * } { \\boldsymbol V } ^ { ( 2 ) } \\right\\| _ { \\infty } = \\underset { s } { \\operatorname* { m a x } } \\left| ( { \\boldsymbol B } ^ { * } { \\boldsymbol V } ^ { ( 1 ) } ) ( s ) - ( { \\boldsymbol B } ^ { * } { \\boldsymbol V } ^ { ( 2 ) } ) ( s ) \\right| } & { } \\\\ { \\ } & { \\leq \\underset { s } { \\operatorname* { m a x } } \\underset { a } { \\operatorname* { m a x } } \\left| Q ^ { ( 1 ) } ( s , a ) - Q ^ { ( 2 ) } ( s , a ) \\right| } \\\\ { \\ } & { = \\gamma \\underset { s } { \\operatorname* { m a x } } \\underset { a } { \\operatorname* { m a x } } \\left| \\mathbb { E } _ { s ^ { \\prime } \\mid s , a } \\big [ { \\boldsymbol V } ^ { ( 1 ) } ( s ^ { \\prime } ) - { \\boldsymbol V } ^ { ( 2 ) } ( s ^ { \\prime } ) \\big ] \\right| } \\\\ { \\ } & { \\leq \\gamma \\underset { s } { \\operatorname* { m a x } } \\underset { a } { \\operatorname* { m a x } } \\left\\| { p ( s ^ { \\prime } \\mid s , a ) } \\right\\| _ { 1 } \\left\\| { \\boldsymbol V } ^ { ( 1 ) } - { \\boldsymbol V } ^ { ( 2 ) } \\right\\| _ { \\infty } ~ \\mathrm { b y ~ H \\ddot { o } l d e r ' s ~ i n e q u a l i t y } } \\\\ { \\ } & { = \\gamma \\| { \\boldsymbol V } ^ { ( 1 ) } - { \\boldsymbol V } ^ { ( 2 ) } \\| _ { \\infty } < \\| { \\boldsymbol V } ^ { ( 1 ) } - { \\boldsymbol V } ^ { ( 2 ) } \\| _ { \\infty } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            770,
+            805,
+            904
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 INTERPRETATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            336,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In summary, the program of the proof was as below : ",
+        "bbox": [
+            173,
+            128,
+            521,
+            145
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. Write down the entropy-regularised policy gradient functional, and apply the DonskerVaradhan formula to it.   \n2. Write down the resulting softmax Bellman operator as a solution to the sup maximization - this obviously also proves existence.   \n3. Show that the softmax operator, just like the hard max, is still a contraction for the max norm, hence prove uniqueness of the solution by fixed point theorem. ",
+        "bbox": [
+            212,
+            154,
+            825,
+            247
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The above also shows formally that, should we discretize the action space A to replace integration over actions by finite sums, any strong estimator ${ \\hat { r } } ( s , a )$ of $r ( s , a )$ , applied to the partition function of rewards $\\begin{array} { r } { \\frac { 1 } { \\lambda } \\log \\sum _ { a } e ^ { \\lambda r \\left( s , a \\right) } } \\end{array}$ , could be used for Q-learning-like iterations. This is because strong convergence would imply weak convergence (especially convergence of the characteristic function, via Levy’s continuity theorem), and hence convergence towards the log-sum-exp cumulant generative function above. Different estimators ${ \\hat { r } } ( s , a )$ lead to different algorithms. When the MDP and the rewards function $r$ are not known, the parameterised critic choice $\\hat { r } ( s , a ) \\approx Q _ { w } ( s , a )$ recovers Nachum’s Path Consistency Learning O. Nachum & Schuurmans. (2017b.;c). O’Donoghue’s PGQ method B. O’Donoghue & Mnih. (2016) can be seen as a control variate balancing of the two terms in 7. In theory, the rewards distribution could be also recovered simply by varying $\\lambda$ (or $\\beta$ ), for instance by inverse Laplace transform. ",
+        "bbox": [
+            173,
+            256,
+            826,
+            411
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 POLICY OPTIMALITY GAP AND TEMPERATURE ANNEALING",
+        "text_level": 1,
+        "bbox": [
+            174,
+            433,
+            694,
+            446
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we propose an inequality that relates the optimality gap of a policy - by how much that policy is sub-optimal on average - to the Kullback-Leibler divergence between the current policy and the optimum. The proof draws on ideas of convex analysis and Legendre transormation exposed earlier in the context of soft Q-learning. ",
+        "bbox": [
+            174,
+            462,
+            825,
+            518
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let us assume that $X$ is a real-valued bounded random variable. We denote $\\operatorname* { s u p } | X | \\leq M$ with $M$ constant. Furthermore we assume that $X$ is centered, that is, $\\mathbb { E } [ X ] = 0$ . This can always be achieved just by picking $Y = X - \\mathbb { E } [ X ]$ . ",
+        "bbox": [
+            173,
+            523,
+            825,
+            568
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Then, by the Hoeffding inequality : ",
+        "bbox": [
+            174,
+            573,
+            405,
+            588
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6a3db94a8ecad1b93c0939c5225b04dd26122e401c3300d7e7229c12e9a73e4.jpg",
+        "text": "$$\n\\log \\mathbb { E } ( e ^ { \\beta X } ) \\leq K \\frac { \\beta ^ { 2 } } { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            590,
+            570,
+            623
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with $K$ a positive real constant, i.e., the variable $X$ is sub-Gaussian, so that its cumulant generating function grows less than quadratically. By taking a Legendre transformation and inverting it, we get that for any pair of measures $\\mathbb { P }$ and $\\mathbb { Q }$ that are mutually absolutely continuous, one has ",
+        "bbox": [
+            173,
+            631,
+            825,
+            674
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/96134c49961a91f178e3129963c86e916d85b5441a880ddb0f9ccb33c500c59a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\mathbb { Q } } \\mathopen { } \\mathclose \\bgroup \\left( X \\aftergroup \\egroup \\right) - \\mathbb { E } _ { \\mathbb { P } } \\mathopen { } \\mathclose \\bgroup \\left( X \\aftergroup \\egroup \\right) \\leq \\sqrt { 2 K \\cdot D _ { K L } \\mathopen { } \\mathclose \\bgroup \\left( \\mathbb { Q } \\aftergroup \\egroup \\right| \\mathopen { } \\mathclose \\bgroup \\left| \\mathbb { P } \\aftergroup \\egroup \\right) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            676,
+            637,
+            702
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which by specializing $\\mathbb { Q }$ to be the measure associated to $\\mathbb { P } ^ { * }$ the optimal policy, $\\mathbb { P } _ { \\theta }$ the current parameterized policy, and $X$ an advantage return $r$ : ",
+        "bbox": [
+            173,
+            710,
+            826,
+            739
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/68c343a4a74422211d2003417752ae9f719cea3c253a0fb96c1394e60271079a.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathbb { P } ^ { * } } \\left( r \\right) \\leq \\mathbb { E } _ { \\mathbb { P } _ { \\theta } } \\left( r \\right) + \\sqrt { 2 K } \\sqrt { D _ { K L } \\left( \\mathbb { P } ^ { * } | | \\mathbb { P } _ { \\theta } \\right) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            755,
+            643,
+            779
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By the same logic, any upper bound on $\\log \\mathbb { E } \\left( e ^ { \\beta X } \\right)$ can give us information about $\\mathbb { E } _ { \\mathbb { Q } } ( X ) - \\mathbb { E } _ { \\mathbb { P } } ( X )$ . This enables us to relate the size of Kullback-Leibler trust regions to the amount by which our policy could be improved. In fact by combining the entropy duality formula with the Legendre transformation, one easily proves the below : ",
+        "bbox": [
+            173,
+            787,
+            826,
+            844
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4. Let $X$ a real-valued integrable random variable, and $f$ a convex and differentiable function such that $f ( 0 ) = f ^ { \\prime } ( 0 ) = 0$ . Then with $f ^ { * } : x  f ^ { * } ( x ) = \\operatorname* { s u p } ( \\beta x - f ( \\beta ) )$ the Legendre transformation of $f$ , $f ^ { * - 1 }$ its reciprocal, and $\\mathbb { P }$ and $\\mathbb { Q }$ any two mutually absolutely continuous measures, one has the equivalence: ",
+        "bbox": [
+            173,
+            847,
+            825,
+            904
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/55780191548dc707fafb0ee934642af3ace539eead3b16c00d54820fd54639e2.jpg",
+        "text": "$$\n\\begin{array} { r } { \\log \\mathbb { E } _ { \\mathbb { P } } \\big ( e ^ { \\beta ( X - \\mathbb { E } _ { \\mathcal { F } } ( X ) ) } \\big ) \\leq f ( \\beta ) \\quad \\iff \\quad \\mathbb { E } _ { \\mathbb { Q } } ( X ) - \\mathbb { E } _ { \\mathbb { P } } ( X ) \\leq f ^ { * - 1 } \\big [ D _ { K L } \\big ( \\mathbb { Q } | | \\mathbb { P } \\big ) \\big ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            906,
+            756,
+            926
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. By Donsker-Varadhan formula, one has that the equivalence is proven if and only if ",
+        "bbox": [
+            171,
+            103,
+            769,
+            118
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/da39e690095322bf8aceeb2d8a867cf5b60784f08ad41e867b9b1458b08545f1.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathbb { Q } } ( X ) - \\mathbb { E } _ { \\mathbb { P } } ( X ) \\leq \\operatorname* { i n f } _ { \\beta } { \\big [ } { \\frac { f ( \\beta ) + D _ { K L } ( \\mathbb { Q } | | \\mathbb { P } ) } { \\beta } } { \\big ] }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            125,
+            651,
+            159
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "but this right term is easily proven to be nothing but ",
+        "bbox": [
+            173,
+            164,
+            514,
+            179
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/489129890a4ae356ce42c4f94e532e4758d2ea396f81b8f674ce7b4d69c6be9b.jpg",
+        "text": "$$\nf ^ { * - 1 } ( D _ { K L } ( \\mathbb { Q } | | \\mathbb { P } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            184,
+            563,
+            204
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "the inverse of the Legendre transformation of $f$ applied to $D _ { K L } ( \\mathbb { Q } | | \\mathbb { P } )$ . ",
+        "bbox": [
+            174,
+            209,
+            640,
+            226
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This also opens up the possibility of using various softmax temperatures $\\beta _ { i }$ in practical algorithms in order to estimate $f$ . Finally, note that if $\\mathbb { P } _ { \\theta }$ is a parameterized softmax policy associated with action-value functions $Q _ { \\theta } ( a , s )$ and temperature $\\beta$ , then because $\\mathbb { P } ^ { * }$ is proportional to $e ^ { - r ( a , s ) / \\beta }$ , one readily has ",
+        "bbox": [
+            174,
+            239,
+            825,
+            296
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e18863a25d70644042d793708ddea6f61c0906df6dd58f191abdd08dfaee0828.jpg",
+        "text": "$$\nD _ { K L } \\left( \\mathbb { P } ^ { * } | | \\mathbb { P } _ { \\theta } \\right) = \\frac { 1 } { \\beta } \\Big [ \\mathbb { E } ( Q _ { \\theta } ) - \\mathbb { E } ( r ) \\Big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            295,
+            620,
+            328
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "which can easily be inserted in the inequality above for the special case $\\mathbb { Q } = \\mathbb { P } ^ { * }$ . ",
+        "bbox": [
+            174,
+            329,
+            700,
+            344
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            364,
+            341,
+            381
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Entropic reinforcement learning has appeared early in the literature with two different motivations. The view of exploration with a self-information intrinsic reward was pioneered by Tishby, and developed in Ziebart’s PhD. thesis Ziebart. (2010). It was rediscovered recently that within the asynchronous actor-critic framework, entropic regularization is crucial to ensure convergence in practice V. Mnih & Kavukcuoglu. (2016). Furthermore, the idea of taking steepest KL divergence steps as a practical reinforcement learning method per se was adopted by Schulman J. Schulman & Abbeel. (2015a.). The Lagrangian duality view was pioneered in a practical context with O’Donoghue’s PGQ algorithm B. O’Donoghue & Mnih. (2016), and followed by the development of soft Q-learning jointly in R. Fox & Tishby. (2015) and in Nachum et al. O. Nachum & Schuurmans. (2017b.). The key common development in these works has been to make entropic regularization recursively follow the Bellman equation, rather than naively regularizing one-step policies G. Neu & Jonsson. (2017). Schulman thereafter proposed a general proof of the equivalence, in the limit, of policy gradient and soft Q-learning methods J. Schulman & Abbeel. (2017), but the proof does not explicitly make the connection with convex duality and the expeditive justification it yields in the one-step case. Applying the Gibbs/Donsker-Varadhan variational formula to entropy in a machine learning context is, however, not new; see for instance Altun and Smola Altun & Smola. (2006). Some of the convex optimization results they invoke, including proximal stepping, can be found in the complete treatment by Bauschke Bauschke & Combettes. (2011). In the context of neural networks, convex analysis and partial differential equation methods are covered by Chaudhari P. Chaudhari & Carlier. (2017). ",
+        "bbox": [
+            173,
+            395,
+            826,
+            674
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 FURTHER WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            694,
+            341,
+            710
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Using dual formulas for the entropy functional in reinforcement learning has vast potential ramifications. One avenue of research will be to interpret our findings in a large deviations framework - the log-sum-exp cumulant generative function being an example of rate function governing fluctuations of the tail of empirical n-step returns. Smart drift change techniques could lead to significant variance reduction for Monte-Carlo rollout estimators. We also hope to exploit further concentration inequalities in order to provide more bounds for the state value function. Finally, a complete theory of the one-to-one correspondence between convex approximation algorithms and reinforcement learning methods is still lacking to date. We hope to be able to contribute in this direction through further work. ",
+        "bbox": [
+            174,
+            726,
+            826,
+            851
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            872,
+            285,
+            887
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Y. Altun and A. Smola. Unifying divergence minimization and statistical inference via convex duality. COLT, 19th Annual Conference on Learning Theory, 2006. ",
+        "bbox": [
+            176,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "S. Amari. Information Geometry and Its Applications. Springer, Applied Mathematical Sciences., 2016.   \nK. Kavukcuoglu B. O’Donoghue, R. Munos and V. Mnih. Pgq $:$ Combining policy gradient and q-learning. arXiv preprint arXiv:1611.01626, 2016.   \nH. H. Bauschke and P. L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer-Verlag, New York., 2011.   \nA. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Springer, Applications of Mathematics, 38., 2010.   \nV. Gomez G. Neu and A. Jonsson. A unified view of entropy-regularized markov decision processes. arXiv preprint arXiv:1705.07798, 2017.   \nP. Moritz M. I. Jordan J. Schulman, S. Levine and P. Abbeel. High-dimensional continuous control using generalized advantage estimation. arXiv preprint arXiv:1502.05477, 2015a.   \nX. Chen J. Schulman and P. Abbeel. Equivalence between policy gradients and soft q-learning. arXiv preprint arXiv:1704.06440, 2017.   \nK. Xu O. Nachum, M. Norouzi and D. Schuurmans. Bridging the gap between value and policy based reinforcement learning. arXiv preprint arXiv:1702.08892, 2017b.   \nK. Xu O. Nachum, M. Norouzi and D. Schuurmans. Trust-pcl: An off-policy trust region method for continuous control. arXiv preprint arXiv:1707.01891, 2017c.   \nS. Osher S. Soatto P. Chaudhari, A. Oberman and G. Carlier. Deep relaxation: partial differential equations for optimizing deep neural networks. arXiv preprint arXiv:1704.04932, 2017.   \nA. Pakman R. Fox and N. Tishby. Taming the noise in reinforcement learning via soft updates. arXiv preprint arXiv:1512.08562, 2015.   \nP. Abbeel T. Haarnoja, H. Tang and S. Levine. Reinforcement learning with deep energy-based policies. arXiv preprint arXiv:1702.08165, 2017.   \nD. Silver A. A. Rusu J. Veness M. G. Bellemare A. Graves M. Riedmiller A. K. Fidjeland G. Ostrovski et al. V. Mnih, K. Kavukcuoglu. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015.   \nM. Mirza A. Graves T. P Lillicrap T. Harley D. Silver V. Mnih, A. Puigdomenech Badia and K. Kavukcuoglu. Asynchronous methods for deep reinforcement learning. arXiv preprint arXiv:1602.01783, 2016.   \nC. Villani. Optimal Transport : Old and New. Grundlehren der mathematischen Wissenschaften, volume 338., 2008.   \nB. D. Ziebart. Modeling purposeful adaptive behavior with the principle of maximum causal entropy. PhD Thesis., 2010. ",
+        "bbox": [
+            169,
+            90,
+            828,
+            736
+        ],
+        "page_idx": 6
+    }
+]

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+# Automated Segmentation of Knee Bone and Cartilage combining Statistical Shape Knowledge and Convolutional Neural Networks
+Data from the Osteoarthritis Initiative
+Felix Ambellana,∗, Alexander Tacka,∗, Moritz $\mathrm { E h l k e } ^ { b , a }$ , Stefan Zachowa,b aZuse Institute Berlin, b1000shapes GmbH, Berlin, Germany {ambellan, tack, ehlke, zachow}@zib.de
+# Abstract
+We present a method for the automated segmentation of knee bones and cartilage from magnetic resonance imaging, that combines a priori knowledge of anatomical shape with Convolutional Neural Networks (CNNs). The proposed approach incorporates 3D Statistical Shape Models (SSMs) as well as 2D and 3D CNNs to achieve a robust and accurate segmentation of even highly pathological knee structures. The method is evaluated on data of the MICCAI grand challenge “Segmentation of Knee Images 2010”. For the first time an accuracy equivalent to the inter-observer variability of human readers has been achieved in this challenge. Moreover, the quality of the proposed method is thoroughly assessed using various measures for 507 manual segmentations of bone and cartilage, and 88 additional manual segmentations of cartilage. Our method yields sub-voxel accuracy. In conclusion, combining of anatomical knowledge using SSMs with localized classification via CNNs results in a state-of-the-art segmentation method.
+# 1 Introduction
+Knee osteoarthritis (OA) is a chronic, degenerative joint disease affecting a significant fraction of the human population [1]. Due to the rising average life expectancy, an increasing obesity and interest in an active lifestyle, research to understand and prevent OA will become even more important. Magnetic Resonance Imaging (MRI) is commonly used to assess knee joint degeneration, especially of the femoral bone (FB), tibial bone (TB), and the respective femoral and tibial cartilage (FC,TC). Quantitative image-based biomarkers from MRI, e.g. the apparent bone volume divided by total bone tissue volume [2] or the cartilage volume [3], already show potential for diagnosis of OA, treatment planning, and prognostic purposes. However, the determination of such biomarkers requires the segmentation of bone and/or cartilage.
+Clinical studies with a large number of subjects are required to extract quantitative image-based biomarkers indicating pathogenesis of OA and to evaluate the efficacy of therapeutic approaches. Precise segmentations are also a prerequisite for computer-based surgical planning of interventions affecting the knee. Manual segmentation of the knee joint is, however, tedious, subjective, and labor-intensive, which renders the analysis of larger cohorts impractical. Thus, since many years, the performance and quality of automated methods have been improved and new methods are being developed.
+Earlier methods often employed Statistical Shape Models (SSMs) [4] to segment knee bones and cartilage. In 2010, Vincent et al. [5] presented a method based on an active appearance model, which was created using a minimum description length approach to optimize correspondences. In the same year, Seim et al. [6] presented a method that utilizes SSMs for bone segmentation and a multi-object graph optimization for cartilage segmentation. SSM-based methods employ anatomical knowledge via geometric priors, which allows for a robust segmentation even in the presence of artifacts or low image contrast. Such methods usually require heuristically designed models of appearance, to adjust the SSM to the image data. Often, appearance models are manually fine-tuned to one specific image modality and cannot be easily generalized to cope with differing ones. To alleviate this problem, Mukhopadhyay et al. [7] proposed to derive appearance models directly from the image data via joint dictionary learning.
+Meanwhile, Convolutional Neural Networks (CNNs) have been employed successfully for segmentation tasks on medical image data, but only few of these methods address the domain of musculoskeletal research. In 2013, Prasoon et al. [8] presented an approach for tibial cartilage segmentation from MRIs using three 2D CNNs. Each CNN independently classifies foreground and background pixels from slices in either the axial, coronal, and sagittal image planes of the 3D MRI dataset. In a similar fashion, Liu et al. [9] in 2017 applied 2D U-Nets [10] as well as the 2D CNN architecture “SegNet” in combination with 3D simplex deformable modeling to obtain 3D segmentations from MRIs. Both methods train 2D convolutional filters from individual slices in the 3D MRI data, since the memory consumption of deep 3D CNNs is often excessive at the scale of full-resolution 3D medical datasets. Consequently, the image information available to the CNNs is strictly localized and lacks context w.r.t. the surrounding voxel intensities in neighboring slices. This is in contrast to previous SSM-based approaches, where 3D anatomical shape information regularizes the segmentation outcome across several slices.
+Our aim in this work is to improve on the segmentation accuracy of previous approaches by combining the strengths of SSM-based and CNN-based segmentation methods. Another aim is to segment large cohort data automatically, e.g. the databases of the Osteoarthritis Initiative2 (OAI) or the Study of Health in Pomerania3. We propose a method that utilizes 2D CNNs as well as localized 3D CNNs to incorporate as much context information as possible into the segmentation process. SSMs are integrated into our segmentation workflow to support decision making in areas of low confidence through a voting scheme. The segmentation accuracy is validated based on a large pool of diverse datasets from the MICCAI “Segmentation of Knee Images $2 0 1 0 ^ { \circ \circ { 4 } }$ [11] (SKI10) challenge and the OAI, as described in section 3. By utilizing SSMs as anatomical shape prior for regularization and CNNs for learning descriptors of local appearance, our method robustly segments varying MRI sequences, even when the images show subjects with severe OA grades.
+# 2 Automated segmentation of bone and cartilage
+Our aim is to establish an automated method that produces highly accurate segmentations of the knee and is robust against pathological data, imaging artifacts, as well as the varying image appearance in different MRI sequences. For this purpose, we consecutively apply 2D CNNs, sub-regional 3D CNNs and SSM-based regularization for both steps, yielding accurate bone segmentations. Given the margins of the bones, we then extract subvolumes along the femoral condyles and tibial plateaus and segment the cartilage in these regions using 3D CNNs (cf. Fig. 1). The first step CNN-2D creates initial segmentation masks of FB and TB. The second step SSM adjustment regularizes the results of step $C N N { - } 2 D$ by fitting SSMs to these masks. The third step CNN-3D is a refinement step that employs 3D CNNs to segment small MRI subvolumes at the bone surfaces as given by the preceding SSM adjustment. The fourth step SSM postprocessing uses regions pre-defined on SSMs to regularize the results of CNN-3D. After bone segmentation is finished the FC and TC are segmented using 3D CNNs. Each step is performed separately for femur and tibia. Thus, CNNs and SSMs are developed independently and individually for both structures. In the following, details are given for all steps of our segmentation pipeline.
+# 2.1 CNN-2D
+The first step CNN-2D is inspired by Liu et al. [9] and applies a variant of the 2D U-Net (Fig. 2 left) for the independent, slice-wise segmentation of the image data. Training is carried out using the (slice-wise) DICE similarity coefficient (cf. Appendix A) as a loss function. The number of input channels of the 2D CNN was extended compared to the standard architecture in order to improve spatial consistency of segmentation results between individual slices of the MRI stack. Eight adjacent slices on both sides of the one that is to be segmented are additionally supplied resulting in 17 channels in total. Note that while this method provides additional context information to the CNN, it cannot substitute for the true volumetric input as processed by 3D CNNs, since the additional channels are only directly visible to the first convolutional layer. The memory requirements are, however, significantly reduced compared to 3D U-Nets with a similar architecture.
+![](images/1f5c7357ba01b1897ae843e21dc001d2ee425795b4fdd0446130da83dd4acdb4.jpg)
+Figure 1: Proposed pipeline for knee bone and cartilage segmentation.
+# 2.2 SSM adjustment
+Due to our observation that segmentations of the previous step show inaccuracies in areas of low intensity contrast or imaging artifacts (cf. Fig. 3 left column and Fig. 4 upper left column) we decided to add a regularization step. The aim of the SSM adjustment step is to regularize and to fill holes and notches in the segmentation mask through statistical knowledge about the global variation of anatomical shape. For this purpose, an SSM (cf. Section 3) is fitted to the segmentation results from the CNN-2D stage. The output is guaranteed to be anatomically plausible (i.e. within the shape span of the SSM) and given as one connected component. Further details on SSMs, their construction and adjustment can be found in [12].
+The SSM matching procedure is as follows: In order to initialize the SSM considering the side of the knee (left/right) a template mesh of the condyle region is fitted to the mask, one time affinely and a second time additionally mirrored along the epi-condyle axis. The actual knee side is detected according to the lower distance between template and data. The shape modes of the SSM and similarity transformation are adjusted iteratively to fit the vertex positions of the SSM to the mask:
+$$
+\operatorname * { a r g m i n } _ { v \left( b ^ { i + 1 } , T ^ { i + 1 } \right) } \left\| \left( v \left( b ^ { i } , T ^ { i } \right) + \Delta v ^ { i } \right) - v \left( b ^ { i + 1 } , T ^ { i + 1 } \right) \right\| , i \gets i + 1 ,
+$$
+where $\Delta v ^ { i }$ is the displacement along the normals of the vertices resulting from the $i$ -th step, s.t. they are placed as close as possible to the interface between segmentation mask (intensity value $= 1$ ) and background (intensity value ${ } = 0$ ). The vertex positions, obtained from the SSM w.r.t. shape weights $b ^ { i }$ and transformation $T ^ { i }$ , are denoted by $v \left( b ^ { \dot { i } } , T ^ { i } \right)$ .
+# 2.3 CNN-3D
+SSMs, as utilized in the previous stage of the pipeline, cannot express osteophytic details completely since these deformations are highly patient-specific and might not be derived from the training cohort.
+![](images/caf94137143c329d4f79454dd8648663c4e483836be754a90fc8b7ac1e20e696.jpg)
+Figure 2: Architectures of the 2D and 3D U-Nets employed in this work.
+We approach this issue in the CNN-3D step by employing 3D U-Nets (Fig. 2 right) with local input fields to segment MRI subvolumes along the bone contours. This step is carried out utilizing the same architecture, but individual training for every structure to capture anatomical details of bone tissue as well as cartilage. Similar to the 2D-U-Nets, the networks are trained via a loss function based on the DICE similarity coefficient. The loss is, however, defined on volumetric subvolumes in the MRI rather than individual 2D slices, which generally leads to better classification accuracy in local regions of the knee compared to the 2D slicing approach. We set the dimensions of the subvolumes to $6 4 \mathrm { x } 6 4 \mathrm { x } 1 6$ voxels, compromising between the input fields size and the memory consumption of the 3D CNNs. To get a feasible number $( \approx 1 0 0 0 )$ of subvolumes they are randomly sampled following a Poisson distribution for the FB and TB segmentation along the outline of the adjusted SSM’s mask. The subvolumes for FC segmentation are extracted along the outline of the condyle region in a similar manner. For TC segmentation, the natural (almost) planar conditions of the tibial plateaus are utilized and subvolumes are sampled at the superior margin of each sagittal slice in the TB mask, s.t. the in-plane overlap of subvolumes is half its size. A visualization is given in Appendix C.
+Since the subvolumes are partially overlapping, conflicting labels might be assigned in overlapping regions. A voting scheme solves this ambiguity by summarizing voxel-wise decisions in a voting mask $V$ (2). The outcome of the SSM adjustment stage $( G )$ is thereby taken into account, biasing the FB and TB segmentations towards the SSM shape in case votes conflict. The contribution of the SSM is set to zero for the segmentation of cartilage.
+Voting can be formularized as follows: Let $\mathbf { x } = ( i , j , k )$ be a (global) index triplet of an image voxel. Let further be $I _ { s }$ the mapping from local indices of subvolume ${ \mathrm { ~ \it ~ B ~ } } _ { s }$ to global indices in the image, and let $\mathrm { I m } ( I _ { s } )$ be the set of all global indices covered by $B _ { s }$ . The voting mask is computed as:
+$$
+V ( \mathbf { x } ) = \omega \cdot G ( \mathbf { x } ) + \sum _ { \mathbf { x } \in \mathrm { I m } ( I _ { s } ) } \Big ( 2 \cdot B _ { s } \big ( I _ { s } ^ { - 1 } ( \mathbf { x } ) \big ) - 1 \Big ) ,
+$$
+with factor $\omega$ set to 25, weighting the contribution of SSM and CNNs roughly equal. The higher $\omega$ is chosen, the more trust is put into the SSM-based regularization. Using this voting mask, majority voting is applied to generate the segmentation mask $\breve { U }$ :
+$$
+U ( \mathbf { x } ) = \left\{ \begin{array} { l l } { 1 , } & { \mathrm { i f } V ( \mathbf { x } ) \geq \tau } \\ { 0 , } & { \mathrm { i f } V ( \mathbf { x } ) < \tau , } \end{array} \right.
+$$
+where the threshold parameter $\tau$ has a fixed value of 1, since this way, a majority of CNN-classified subvolumes have to up-vote a voxel in order to include it in the final result, if it is not already captured by the SSM.
+# 2.4 SSM postprocessing
+Bone segmentation is finalized in the SSM postprocessing step. The idea behind SSM-based postprocessing as utilized in our approach is to remove wrongly classified foreground labels from the FB and TB segmentations that are located outside the typical range of osteophytic growth. Foreground labels are excluded from the segmentation mask depending on their surface distance to the SSM. We found that a conservative regularization after processing the subvolumes with the proposed voting method (CNN-3D stage) helps to remove segmentation errors due to the localized nature of the 3D CNN-based classification and low image contrast or noise (cf. Fig. 4, central and right column).
+We identified regions on the SSMs of the FB and TB that show higher or lower inter-patient variability in shape (cf. Appendix B). Regions of higher variability are typically associated with osteophytic growth. To identify these regions in the segmentation, SSMs are matched to the masks obtained from the CNN-3D stage and the distance between foreground voxels and fitted surface is calculated, s.t. every voxel is assigned a distance value either to areas of low or high variability on the SSMs. This leads to a maximum distance in areas of low variability $d _ { l v }$ and to a maximum distance in areas of high variability $d _ { h v }$ .
+If $d _ { h v } \geq d _ { l v }$ , the postprocessing is terminated since no unexpected deviation is identified. Otherwise, the matched surface is converted to a mask denoted by $\tilde { U }$ and the set difference $D = U \setminus \tilde { U }$ is calculated. D naturally consists of its 3D connected components $D _ { j }$ . Every component is considered a candidate for removal, if there exists a voxel $x \in D _ { j }$ , s.t. its distance to the surface $d _ { \mathbf { x } }$ is realized in the area of low variability and additionally $d _ { \mathbf { x } } \geq d _ { h v }$ and $d _ { \mathbf { x } } \ \geq \ t o l$ hold for an empirically determined tolerance of $5 . 5 \mathrm { m m }$ . All other components describe either osteophytic growth or normal morphological deviation. For components that fulfill the former rule, every voxel that realizes its distance in the area of low variability is removed from the mask. See Appendix B for a detailed schematic description.
+# 3 Experiments and Results
+We evaluated the accuracy of our method on three different datasets SKI10, OAI Imorphics and OAI ZIB (Table 1) employing volume-based and distance-based measures (cf. Appendix A).
+# 3.1 MRI datasets
+Dataset SKI10 consists of MRIs from the MICCAI SKI10 grand challenge. These data is divided into 60 training, 40 validation, and 50 submission images. All scans were acquired for surgery planning of partial or complete knee replacement, and thus show a high degree of pathological deformities in the knee region.
+Dataset OAI Imorphics consists of MRI sequences from the OAI database with manual segmentations supplied by Imorphics $\left( \mathrm { N } = 8 8 \right)$ ). The dataset contains only cases of moderate and severe OA.
+Dataset OAI ZIB consists of additional data from the OAI database for which manual segmentations were carried out thoroughly by experienced users at Zuse Institute Berlin $\mathrm { { N } } = 5 0 7 { \mathrm { { } } }$ ) starting from automatic segmentations employing [6]. The data cover the full spectrum of OA grades, with a strong tendency towards severe cases.
+Table 1: Summary of the datasets used for training and validation. Images were acquired either once per patient (baseline) or twice with an additional 12-month follow-up $( 1 2 \mathrm { m } )$ .
+<table><tr><td></td><td>SKI10</td><td>OAI Imorphics</td><td>OAI ZIB</td></tr><tr><td>MRI scanner</td><td>GE,Siemens,Philips,Toshiba,Hitachi. Mostly1.5T, some 3T,a few 1T</td><td>Siemens 3T Trio</td><td>Siemens 3T Trio</td></tr><tr><td>MRI sequence</td><td>Many (T1, T2, GRE, Spoiled-GRE) partly with fat suppression</td><td>DESS</td><td>DESS</td></tr><tr><td>Acquisition plane</td><td>sagittal</td><td>sagittal</td><td>sagittal</td></tr><tr><td>Image resolution [mm]</td><td>0.39×0.39× 1.0</td><td>0.36×0.36×0.7</td><td>0.36×0.36×0.7</td></tr><tr><td>Manual segmentations</td><td>bones and cartilage</td><td>cartilage</td><td>bones and cartilage</td></tr><tr><td>Number of subjects</td><td>60 training 40 validation 50 submission</td><td>88</td><td>507</td></tr><tr><td>Sex (male;female)</td><td>n.a.</td><td>(45,43)</td><td>(262,245)</td></tr><tr><td>Age [years]</td><td>n.a.</td><td>61.24±9.98</td><td>61.87±9.33</td></tr><tr><td>BMI [kg/m²]</td><td>n.a.</td><td>31.06±4.61</td><td>29.27±4.52</td></tr><tr><td>rOA grade (0,1,2,3,4)</td><td>n.a.</td><td>(0,0,15,56,17)</td><td>(60,77,61,151,158)</td></tr><tr><td>timepoints</td><td>baseline</td><td>baseline,12m</td><td>baseline</td></tr></table>
+# 3.2 Experimental setup
+The employed SSMs consist of 15,172 vertices and 30,220 faces (FB), and 16,244 vertices and 32,351 faces (TB) independent of the dataset. Construction was done following [6]. For the SKI10 dataset, training of CNNs and construction of SSMs is carried out using the 60 training cases. Our method is evaluated for the validation and the submission cases separately. Two-fold cross-validation studies are performed for datasets OAI ZIB and OAI Imorphics. For OAI ZIB, decomposition is done by random choice (253/254). For OAI Imorphics the cohort’s subject ids are sorted numerically and split into upper and lower half (44/44). However, since no manual segmentations of bones are available for the OAI Imorphics dataset, the SSMs built from the SKI10 training data are employed. The CNNs are trained using the OAI Imorphics baseline data only. Thus, the OAI Imorphics $1 2 \mathrm { m }$ follow-up data is exclusively used for evaluation still within the cross-validation setting.
+# 3.3 Measures of segmentation accuracy
+The accuracy of our method is evaluated using the DICE Similarity Coefficient (DSC), average surface distance (ASD), root mean square distance (RSD), maximum distance (MSD), volume difference (VD), and volume overlap error (VOE). All these measures are symmetric apart from VD that is considered relative to the manual segmentation. Volumetric measures (Appendix A, 3) are suitable for assessing the segmentation results globally. However, volume-based measures provide limited sensitivity to errors on the boundaries of the segmentation if the segmented volume is relatively large. We therefore also include surface distance measures (Appendix A, 4) in the evaluation, which are sensitive to segmentation errors on the anatomical boundary.
+# 3.4 Results
+Table 2 summarizes our results for the SKI10 validation dataset. Our method reaches a total score of $7 3 . 6 { \pm } 7 . 6 $ in terms of the SKI10 metrics [11]. This is a notable improvement w.r.t scores reported in previous publications (Table 3). Our method achieves a total score of 75.73 on the SKI10 submit data and is currently ranked first5.
+The results for the OAI Imorphics dataset are shown in Table 4. For FC the DSC is $8 9 . 4 \%$ for baseline and $8 9 . 1 \%$ for $1 2 \mathrm { m }$ . For medial tibial cartilage (MTC) and lateral tibial cartilage (LTC) the DSC is $8 6 . 1 \%$ resp. $9 0 . 4 \%$ for baseline, and $8 5 . 8 \%$ resp. $9 0 . 0 \%$ for $1 2 \mathrm { m }$ . Again, the ASD is smaller than the image resolution $( < 0 . 3 6 \mathrm { m m } )$ for both, FC and TC.
+Table 5 summarizes the segmentation accuracy for the OAI ZIB dataset. The DSC is $9 8 . 5 \%$ for FB, $9 8 . 5 \%$ for TB, $8 9 . 9 \%$ for FC, and $8 5 . 6 \%$ for TC. The ASD is smaller than the image resolution for bone as well as for cartilage $( < 0 . 3 6 \mathrm { m m } )$ .
+Computation for the whole segmentation pipeline (end to end) was $9 \mathrm { m } ~ 2 2 \mathrm { s }$ on a consumer-grade workstation (CPU: Intel Xeon E5-2650 v3, 2.30GHz; GPU: GeForce GTX $9 8 0 \mathrm { T i }$ ). Implementation of CNNs was done employing Keras with Theano-backend6. All calculations regarding SSMs were carried out using Amira ZIB Edition7
+Table 2: Segmentation accuracy for the SKI10 validation dataset.
+<table><tr><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1>ASD (mm)</td><td rowspan=2 colspan=1>RSD (mm)</td><td rowspan=2 colspan=1>VD (%)</td><td></td></tr><tr><td rowspan=1 colspan=1>VOE (%)</td></tr><tr><td rowspan=1 colspan=1>FBTBFCTC</td><td rowspan=1 colspan=1>0.43± 0.130.37 ± 0.11</td><td rowspan=1 colspan=1>0.75± 0.280.63 ± 0.26</td><td rowspan=1 colspan=1>7.18 ± 10.514.29 ± 12.34</td><td rowspan=1 colspan=1>20.99 ± 5.0819.06 ± 5.18</td></tr></table>
+Total score as computed employing the SKI10 metrics: 73.6 ± 7.6
+# 4 Discussion and Conclusion
+We presented a novel fully automated segmentation method for knee bone and cartilage by combining the advantages of SSM-based regularization with CNN-based classification of voxel intensities. The method was evaluated using datasets from the SKI10 challenge as well as from the OAI database. Accuracy was evaluated using volume-based and distance-based measures to provide a transparent analysis w.r.t global and local level of detail. The proposed method consistently achieved high segmentation accuracy, despite severely arthritic knees and various different MRI sequences. For the first time, a total score greater than 75 was reached on the SKI10 submission data, which is comparable to the inter-observer variability of two expert readers [11]. In our experience, the automated method reduces the time effort for an accurate segmentation of knee bones and cartilage at least by a factor of six compared to manual segmentations by an experienced reader $\mathrm { ( > l h ) }$ . However, large scale databases for studying the OA disease, such as provided by the OAI, can contain 50.000 or more MRIs. Using our implementation, it would take 43 weeks to segment the full OAI database on a singe computational node. We therefore aim at reducing the computational time of the algorithm further as well as distributing the work-load over several nodes. Our goal is to segment the full OAI database and make the results available to the public in the near future.
+![](images/0d5c3166ce58e7835f95e33bac36ed1928b38c9cc06e0d09958f02cd04676f9f.jpg)
+Figure 3: Segmentation of osteophytic regions in different stages (colored contours). CNN-2D stage (left) is error prone, SSM adjustment (middle) smoothly regularizes and CNN-3D (right) segments osteophytes precisely.
+![](images/89b552d45e95bfe9dd555306396e6f1bfe3a6d44909c411e212a8be6a9b59ba5.jpg)
+Figure 4: Left column: An image artifact results in an erroneous CNN-2D classification (top). An anatomically plausible segmentation (bottom) is restored using SSM adjustment regularization. Middle and right column: CNN-3D segmentation introduces errors in the shaft region due to the subvolumes’ locality and missing image contrast (top). SSM postprocessing corrects in an anatomically plausible manner (bottom).
+Table 3: SKI10 validation data: Our approach yields the best of all published results as of April 2018.
+<table><tr><td rowspan=1 colspan=1>Imorphics [5]</td><td rowspan=1 colspan=1>ZIB (2010) [6]</td><td rowspan=1 colspan=1>BioMedIA[13]</td><td rowspan=1 colspan=1>Liu et al. [9]</td><td rowspan=1 colspan=1>Biomediq [14]</td><td rowspan=1 colspan=1>ZIB (2018)</td></tr><tr><td rowspan=1 colspan=1>52.3 ± 8.6</td><td rowspan=1 colspan=1>54.4±8.8</td><td rowspan=1 colspan=1>56.5 ± 9.2</td><td rowspan=1 colspan=1>64.1 ± 9.5</td><td rowspan=1 colspan=1>67.1 ± 8.0</td><td rowspan=1 colspan=1>73.6 ± 7.6</td></tr></table>
+Table 4: Segmentation accuracy for the OAI Imorphics dataset.
+<table><tr><td colspan="3">DSC (%)</td><td>ASD (mm)</td><td>RSD (mm)</td><td>MSD (mm)</td><td>VD (%)</td><td>VOE (%)</td></tr><tr><td rowspan="3">00m</td><td>FC</td><td>89.4 ± 2.41</td><td>0.19 ± 0.08</td><td>0.50± 0.30</td><td>6.65± 2.99</td><td>7.0± 6.04</td><td>19.1 ± 3.88</td></tr><tr><td>MTC</td><td>86.1 ± 5.33</td><td>0.26 ± 0.23</td><td>0.63 ± 0.55</td><td>5.16 ± 2.93</td><td>8.0 ± 17.15</td><td>24.1 ± 7.74</td></tr><tr><td>LTC</td><td>90.4 ± 2.42</td><td>0.17 ± 0.06</td><td>0.41 ± 0.16</td><td>3.93 ± 2.07</td><td>6.9 ± 7.14</td><td>17.5 ± 3.96</td></tr><tr><td rowspan="3">12m</td><td>FC</td><td>89.1 ± 2.41</td><td>0.20 ± 0.09</td><td>0.53 ± 0.33</td><td>6.86 ± 3.16</td><td>7.6± 6.78</td><td>19.6 ± 3.86</td></tr><tr><td>MTC</td><td>85.8 ± 5.00</td><td>0.28 ± 0.22</td><td>0.67 ± 0.55</td><td>5.25 ± 3.06</td><td>6.6 ± 16.38</td><td>24.5 ± 7.37</td></tr><tr><td>LTC</td><td>90.0 ± 2.57</td><td>0.18 ± 0.06</td><td>0.44 ± 0.19</td><td>4.08 ± 2.11</td><td>7.2 ± 7.74</td><td>18.1 ± 4.16</td></tr></table>
+Table 5: Segmentation accuracy on the OAI ZIB dataset.
+<table><tr><td></td><td>DSC (%)</td><td>ASD (mm)</td><td>RSD (mm)</td><td>MSD (mm)</td><td>VD (%)</td><td>VOE (%)</td></tr><tr><td>FB</td><td>98.5 ± 3.02</td><td>0.17 ± 0.05</td><td>0.35± 0.09</td><td>2.93± 1.24</td><td>-0.09± 0.87</td><td>2.8± 0.58</td></tr><tr><td>TB</td><td>98.5 ± 3.25</td><td>0.18 ± 0.06</td><td>0.37 ± 0.18</td><td>3.16 ± 2.03</td><td>-0.03 ± 0.82</td><td>2.9 ± 0.63</td></tr><tr><td>FC</td><td>89.9 ± 3.60</td><td>0.16 ± 0.07</td><td>0.38 ± 0.17</td><td>5.35 ± 2.50</td><td>1.5 ± 5.87</td><td>18.1 ± 5.90</td></tr><tr><td>TC</td><td>85.6 ± 4.54</td><td>0.23 ± 0.12</td><td>0.60 ± 0.38</td><td>6.35 ± 4.36</td><td>-1.0 ± 11.92</td><td>24.9 ± 6.79</td></tr></table>
+A promising line of future work is to investigate approaches that couple SSMs and CNNs more directly, e.g. by introducing learned appearance from CNNs to an SSM segmentation framework.
+# Acknowledgments
+We would like to thank Heiko Ramm (née Seim, 1000shapes GmbH) for valuable insights into SSM-based segmentation methods. We further would like to thank Irene Ziska, Agnieszka Putyra, and Robert Joachimsky for creating the manual segmentations for our OAI ZIB dataset by thoroughly correcting automated presegmentations. The authors gratefully acknowledge the financial support by the German federal ministry of education and research (BMBF) research network on musculoskeletal diseases, grant no. 01EC1408B (Overload/PrevOP) and grant no. 01EC1406E (TOKMIS). The Osteoarthritis Initiative is a public-private partnership comprised of five contracts (N01-AR-2-2258; N01-AR-2-2259; N01-AR-2-2260; N01-AR-2-2261; N01-AR-2-2262) funded by the National Institutes of Health, a branch of the Department of Health and Human Services, and conducted by the OAI Study Investigators. Private funding partners include Merck Research Laboratories; Novartis Pharmaceuticals Corporation, GlaxoSmithKline; and Pfizer, Inc. Private sector funding for the OAI is managed by the Foundation for the National Institutes of Health. This manuscript was prepared using an OAI public use data set and does not necessarily reflect the opinions or views of the OAI investigators, the NIH, or the private funding partners.
+# References
+[1] Reva C Lawrence, David T Felson, Charles G Helmick, Lesley M Arnold, Hyon Choi, Richard A Deyo, Sherine Gabriel, Rosemarie Hirsch, Marc C Hochberg, Gene G Hunder, et al. Estimates of the prevalence of arthritis and other rheumatic conditions in the united states: Part ii. Arthritis & Rheumatology, 58(1): 26–35, 2008. [2] Felix Eckstein, Deborah Burstein, and Thomas M Link. Quantitative MRI of cartilage and bone: degenerative changes in osteoarthritis. NMR in Biomedicine, 19(7):822–854, 2006. [3] Felix Eckstein, Flavia Cicuttini, Jean Pierre Raynauld, John C Waterton, and Charles Peterfy. Magnetic resonance imaging (MRI) of articular cartilage in knee osteoarthritis (OA): morphological assessment. Osteoarthritis and Cartilage, 14:46–75, 2006. [4] Tobias Heimann and Hans-Peter Meinzer. Statistical shape models for 3d medical image segmentation: a review. Medical Image Analysis, 13(4):543–563, 2009. [5] Graham Vincent, Chris Wolstenholme, Ian Scott, and Mike Bowes. Fully automatic segmentation of the knee joint using active appearance models. In MICCAI Workshop Medical Image Analysis for the Clinic, pages 224 – 230, 2010. [6] Heiko Seim, Dagmar Kainmueller, Hans Lamecker, Matthias Bindernagel, Jana Malinowski, and Stefan Zachow. Model-based auto-segmentation of knee bones and cartilage in MRI data. In MICCAI Workshop Medical Image Analysis for the Clinic, pages 215 – 223, 2010. [7] Anirban Mukhopadhyay, Oscar Salvador Morillo Victoria, Stefan Zachow, and Hans Lamecker. Robust and accurate appearance models based on joint dictionary learning data from the osteoarthritis initiative. In International Workshop on Patch-based Techniques in Medical Imaging, pages 25–33. Springer, 2016. [8] Adhish Prasoon, Kersten Petersen, Christian Igel, François Lauze, Erik Bjørnager Dam, and Mads Nielsen. Deep feature learning for knee cartilage segmentation using a triplanar convolutional neural network. In International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), pages 246 – 253, 2013.
+[9] Fang Liu, Zhaoye Zhou, Hyungseok Jang, Alexey Samsonov, Gengyan Zhao, and Richard Kijowski. Deep convolutional neural network and 3d deformable approach for tissue segmentation in musculoskeletal magnetic resonance imaging. Magnetic Resonance in Medicine, 79(4):2379–2391, 2017.
+[10] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), pages 234–241, 2015.
+[11] Tobias Heimann, Bryan J Morrison, Martin A Styner, Marc Niethammer, and Simon K Warfield. Segmentation of knee images: a grand challenge. In MICCAI Workshop on Medical Image Analysis for the Clinic, pages 207–214, 2010.
+[12] Dagmar Kainmueller. Deformable Meshes for Medical Image Segmentation: Accurate Automatic Segmentation of Anatomical Structures. Springer, 2014.
+[13] Zehan Wang, Claire Donoghue, and Daniel Rueckert. Patch-based segmentation without registration: application to knee MRI. In MICCAI Workshop Machine Learning in Medical Imaging, pages 98 – 105, 2013.
+[14] Erik B Dam, Martin Lillholm, Joselene Marques, and Mads Nielsen. Automatic segmentation of high-and low-field knee MRIs using knee image quantification with data from the osteoarthritis initiative. Journal of Medical Imaging, 2(2):024001 – 024001, 2015.
+# Appendix A: Employed surface-based and volume-based distance measures
+$$
+\begin{array} { l } { { \mathrm { D S C } = 1 0 0 \cdot \frac { 2 \displaystyle | \boldsymbol { B } \cap \boldsymbol { A } | } { \displaystyle | \boldsymbol { B } \rvert + \displaystyle | \boldsymbol { A } | } , \ \mathrm { V O E } = 1 0 0 \cdot \left( 1 - \frac { \displaystyle \mathrm { D S C } } { \displaystyle 2 0 0 - \displaystyle \mathrm { D S C } } \right) , \ \mathrm { V D } = 1 0 0 \cdot \frac { \displaystyle | \boldsymbol { B } | - \displaystyle | \boldsymbol { A } | } { \displaystyle | \boldsymbol { A } | } , } } \\ { { \mathrm { A S D } = \frac { 1 } { \displaystyle n _ { \partial A } + n _ { \partial B } } \left( \sum _ { i = 1 } ^ { n _ { \partial A } } \operatorname* { m i n } _ { b \in \partial B } \| a _ { i } - b \| _ { 2 } + \sum _ { j = 1 } ^ { n _ { \partial B } } \operatorname* { m i n } _ { a \in \partial A } \| b _ { j } - a \| _ { 2 } \right) , \ ~ } } \\ { { \mathrm { R S D } = \sqrt { \displaystyle \frac { 1 } { n _ { \partial A } + n _ { \partial B } } \left( \sum _ { i = 1 } ^ { n _ { \partial A } } \operatorname* { m i n } _ { b \in \partial B } \| a _ { i } - b \| _ { 2 } ^ { 2 } + \sum _ { j = 1 } ^ { n _ { \partial B } } \operatorname* { m i n } _ { a \in \partial A } \| b _ { j } - a \| _ { 2 } ^ { 2 } \right) } , \ ~ } } \\ { { \mathrm { M S D } = \operatorname* { m a x } \left( \operatorname* { m a x } _ { a \in \partial A } \operatorname* { m i n } _ { b \in \partial B } \| a - b \| _ { 2 } , \operatorname* { m a x } _ { b \in \partial B } \operatorname* { m i n } _ { a \in \partial A } \| b - a \| _ { 2 } \right) . } } \end{array}
+$$
+Within the above, $A$ denotes the set of manually segmented (ground-truth) voxels and $B$ denotes the segmentation result from the automated method; $\partial A$ and $\partial B$ represent the boundary of $A$ and $B$ . The boundary contains every voxel having at least one neighbor that is not part of the respective segmentation mask. The number of voxels on the boundary $\partial A , \partial B$ is written as $n _ { \partial A } , n _ { \partial B }$ . Lastly $| \cdot |$ denotes a volume and $\lVert \cdot \rVert _ { 2 }$ the usual Euclidean norm.
+![](images/ea35e55d45ce0c215386f8683675d1aafee3d56993fc16036018dcbb9f855d8b.jpg)
+Appendix B: SSM postprocessing schematic visualization
+Figure 5: Predefined regions of high (yellow) and low (blue) variability on SSM surfaces of FB (left) and TB (middle). Right: Schematic visualization of a typical postprocessing situation. Distances are computed from 3D CNN mask (red) to SSM surface. The cross-section depicted shows (w.l.o.g.) the realization of $d _ { h v }$ in component $A$ . Component $B$ is completely removed since there exists one $x \in B$ s.t. $d _ { x } > d _ { h v }$ and all elements of $B$ realize their closest distance in the area of low variability.
+# Appendix C: Sampling for CNN-3D – visualization
+![](images/edebfd34044c75a5ffd7cb1518cb568fe1f7fdeb8bfb2a72392fe0df561ee4d4.jpg)
+Figure 6: CNN-3D segmentation step: OAI subject ID 9793168. Exemplary sampling of subvolumes for (f.l.t.r.) FB (1052 points), TB (825 points), FC (845 points) and TC (228 points).

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+        "text": "Automated Segmentation of Knee Bone and Cartilage combining Statistical Shape Knowledge and Convolutional Neural Networks ",
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+        "text": "We present a method for the automated segmentation of knee bones and cartilage from magnetic resonance imaging, that combines a priori knowledge of anatomical shape with Convolutional Neural Networks (CNNs). The proposed approach incorporates 3D Statistical Shape Models (SSMs) as well as 2D and 3D CNNs to achieve a robust and accurate segmentation of even highly pathological knee structures. The method is evaluated on data of the MICCAI grand challenge “Segmentation of Knee Images 2010”. For the first time an accuracy equivalent to the inter-observer variability of human readers has been achieved in this challenge. Moreover, the quality of the proposed method is thoroughly assessed using various measures for 507 manual segmentations of bone and cartilage, and 88 additional manual segmentations of cartilage. Our method yields sub-voxel accuracy. In conclusion, combining of anatomical knowledge using SSMs with localized classification via CNNs results in a state-of-the-art segmentation method. ",
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+        "type": "text",
+        "text": "1 Introduction ",
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+        "text": "Knee osteoarthritis (OA) is a chronic, degenerative joint disease affecting a significant fraction of the human population [1]. Due to the rising average life expectancy, an increasing obesity and interest in an active lifestyle, research to understand and prevent OA will become even more important. Magnetic Resonance Imaging (MRI) is commonly used to assess knee joint degeneration, especially of the femoral bone (FB), tibial bone (TB), and the respective femoral and tibial cartilage (FC,TC). Quantitative image-based biomarkers from MRI, e.g. the apparent bone volume divided by total bone tissue volume [2] or the cartilage volume [3], already show potential for diagnosis of OA, treatment planning, and prognostic purposes. However, the determination of such biomarkers requires the segmentation of bone and/or cartilage. ",
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+        "text": "Clinical studies with a large number of subjects are required to extract quantitative image-based biomarkers indicating pathogenesis of OA and to evaluate the efficacy of therapeutic approaches. Precise segmentations are also a prerequisite for computer-based surgical planning of interventions affecting the knee. Manual segmentation of the knee joint is, however, tedious, subjective, and labor-intensive, which renders the analysis of larger cohorts impractical. Thus, since many years, the performance and quality of automated methods have been improved and new methods are being developed. ",
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+        "text": "Earlier methods often employed Statistical Shape Models (SSMs) [4] to segment knee bones and cartilage. In 2010, Vincent et al. [5] presented a method based on an active appearance model, which was created using a minimum description length approach to optimize correspondences. In the same year, Seim et al. [6] presented a method that utilizes SSMs for bone segmentation and a multi-object graph optimization for cartilage segmentation. SSM-based methods employ anatomical knowledge via geometric priors, which allows for a robust segmentation even in the presence of artifacts or low image contrast. Such methods usually require heuristically designed models of appearance, to adjust the SSM to the image data. Often, appearance models are manually fine-tuned to one specific image modality and cannot be easily generalized to cope with differing ones. To alleviate this problem, Mukhopadhyay et al. [7] proposed to derive appearance models directly from the image data via joint dictionary learning. ",
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+        "text": "Meanwhile, Convolutional Neural Networks (CNNs) have been employed successfully for segmentation tasks on medical image data, but only few of these methods address the domain of musculoskeletal research. In 2013, Prasoon et al. [8] presented an approach for tibial cartilage segmentation from MRIs using three 2D CNNs. Each CNN independently classifies foreground and background pixels from slices in either the axial, coronal, and sagittal image planes of the 3D MRI dataset. In a similar fashion, Liu et al. [9] in 2017 applied 2D U-Nets [10] as well as the 2D CNN architecture “SegNet” in combination with 3D simplex deformable modeling to obtain 3D segmentations from MRIs. Both methods train 2D convolutional filters from individual slices in the 3D MRI data, since the memory consumption of deep 3D CNNs is often excessive at the scale of full-resolution 3D medical datasets. Consequently, the image information available to the CNNs is strictly localized and lacks context w.r.t. the surrounding voxel intensities in neighboring slices. This is in contrast to previous SSM-based approaches, where 3D anatomical shape information regularizes the segmentation outcome across several slices. ",
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+        "text": "Our aim in this work is to improve on the segmentation accuracy of previous approaches by combining the strengths of SSM-based and CNN-based segmentation methods. Another aim is to segment large cohort data automatically, e.g. the databases of the Osteoarthritis Initiative2 (OAI) or the Study of Health in Pomerania3. We propose a method that utilizes 2D CNNs as well as localized 3D CNNs to incorporate as much context information as possible into the segmentation process. SSMs are integrated into our segmentation workflow to support decision making in areas of low confidence through a voting scheme. The segmentation accuracy is validated based on a large pool of diverse datasets from the MICCAI “Segmentation of Knee Images $2 0 1 0 ^ { \\circ \\circ { 4 } }$ [11] (SKI10) challenge and the OAI, as described in section 3. By utilizing SSMs as anatomical shape prior for regularization and CNNs for learning descriptors of local appearance, our method robustly segments varying MRI sequences, even when the images show subjects with severe OA grades. ",
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+        "type": "text",
+        "text": "2 Automated segmentation of bone and cartilage ",
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+        "text": "Our aim is to establish an automated method that produces highly accurate segmentations of the knee and is robust against pathological data, imaging artifacts, as well as the varying image appearance in different MRI sequences. For this purpose, we consecutively apply 2D CNNs, sub-regional 3D CNNs and SSM-based regularization for both steps, yielding accurate bone segmentations. Given the margins of the bones, we then extract subvolumes along the femoral condyles and tibial plateaus and segment the cartilage in these regions using 3D CNNs (cf. Fig. 1). The first step CNN-2D creates initial segmentation masks of FB and TB. The second step SSM adjustment regularizes the results of step $C N N { - } 2 D$ by fitting SSMs to these masks. The third step CNN-3D is a refinement step that employs 3D CNNs to segment small MRI subvolumes at the bone surfaces as given by the preceding SSM adjustment. The fourth step SSM postprocessing uses regions pre-defined on SSMs to regularize the results of CNN-3D. After bone segmentation is finished the FC and TC are segmented using 3D CNNs. Each step is performed separately for femur and tibia. Thus, CNNs and SSMs are developed independently and individually for both structures. In the following, details are given for all steps of our segmentation pipeline. ",
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+        "text": "2.1 CNN-2D ",
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+        "text": "The first step CNN-2D is inspired by Liu et al. [9] and applies a variant of the 2D U-Net (Fig. 2 left) for the independent, slice-wise segmentation of the image data. Training is carried out using the (slice-wise) DICE similarity coefficient (cf. Appendix A) as a loss function. The number of input channels of the 2D CNN was extended compared to the standard architecture in order to improve spatial consistency of segmentation results between individual slices of the MRI stack. Eight adjacent slices on both sides of the one that is to be segmented are additionally supplied resulting in 17 channels in total. Note that while this method provides additional context information to the CNN, it cannot substitute for the true volumetric input as processed by 3D CNNs, since the additional channels are only directly visible to the first convolutional layer. The memory requirements are, however, significantly reduced compared to 3D U-Nets with a similar architecture. ",
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+        "image_caption": [
+            "Figure 1: Proposed pipeline for knee bone and cartilage segmentation. "
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+        "text": "2.2 SSM adjustment ",
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+        "text": "Due to our observation that segmentations of the previous step show inaccuracies in areas of low intensity contrast or imaging artifacts (cf. Fig. 3 left column and Fig. 4 upper left column) we decided to add a regularization step. The aim of the SSM adjustment step is to regularize and to fill holes and notches in the segmentation mask through statistical knowledge about the global variation of anatomical shape. For this purpose, an SSM (cf. Section 3) is fitted to the segmentation results from the CNN-2D stage. The output is guaranteed to be anatomically plausible (i.e. within the shape span of the SSM) and given as one connected component. Further details on SSMs, their construction and adjustment can be found in [12]. ",
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+        "text": "The SSM matching procedure is as follows: In order to initialize the SSM considering the side of the knee (left/right) a template mesh of the condyle region is fitted to the mask, one time affinely and a second time additionally mirrored along the epi-condyle axis. The actual knee side is detected according to the lower distance between template and data. The shape modes of the SSM and similarity transformation are adjusted iteratively to fit the vertex positions of the SSM to the mask: ",
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+        "img_path": "images/950088b23e9a36139c19c56071810d5939b198d104275cc26b13bab88de26a44.jpg",
+        "text": "$$\n\\operatorname * { a r g m i n } _ { v \\left( b ^ { i + 1 } , T ^ { i + 1 } \\right) } \\left\\| \\left( v \\left( b ^ { i } , T ^ { i } \\right) + \\Delta v ^ { i } \\right) - v \\left( b ^ { i + 1 } , T ^ { i + 1 } \\right) \\right\\| , i \\gets i + 1 ,\n$$",
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+        "type": "text",
+        "text": "where $\\Delta v ^ { i }$ is the displacement along the normals of the vertices resulting from the $i$ -th step, s.t. they are placed as close as possible to the interface between segmentation mask (intensity value $= 1$ ) and background (intensity value ${ } = 0$ ). The vertex positions, obtained from the SSM w.r.t. shape weights $b ^ { i }$ and transformation $T ^ { i }$ , are denoted by $v \\left( b ^ { \\dot { i } } , T ^ { i } \\right)$ . ",
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+        "text": "2.3 CNN-3D ",
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+        "text": "SSMs, as utilized in the previous stage of the pipeline, cannot express osteophytic details completely since these deformations are highly patient-specific and might not be derived from the training cohort. ",
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+        "image_caption": [
+            "Figure 2: Architectures of the 2D and 3D U-Nets employed in this work. "
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+        "text": "We approach this issue in the CNN-3D step by employing 3D U-Nets (Fig. 2 right) with local input fields to segment MRI subvolumes along the bone contours. This step is carried out utilizing the same architecture, but individual training for every structure to capture anatomical details of bone tissue as well as cartilage. Similar to the 2D-U-Nets, the networks are trained via a loss function based on the DICE similarity coefficient. The loss is, however, defined on volumetric subvolumes in the MRI rather than individual 2D slices, which generally leads to better classification accuracy in local regions of the knee compared to the 2D slicing approach. We set the dimensions of the subvolumes to $6 4 \\mathrm { x } 6 4 \\mathrm { x } 1 6$ voxels, compromising between the input fields size and the memory consumption of the 3D CNNs. To get a feasible number $( \\approx 1 0 0 0 )$ of subvolumes they are randomly sampled following a Poisson distribution for the FB and TB segmentation along the outline of the adjusted SSM’s mask. The subvolumes for FC segmentation are extracted along the outline of the condyle region in a similar manner. For TC segmentation, the natural (almost) planar conditions of the tibial plateaus are utilized and subvolumes are sampled at the superior margin of each sagittal slice in the TB mask, s.t. the in-plane overlap of subvolumes is half its size. A visualization is given in Appendix C. ",
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+        "type": "text",
+        "text": "Since the subvolumes are partially overlapping, conflicting labels might be assigned in overlapping regions. A voting scheme solves this ambiguity by summarizing voxel-wise decisions in a voting mask $V$ (2). The outcome of the SSM adjustment stage $( G )$ is thereby taken into account, biasing the FB and TB segmentations towards the SSM shape in case votes conflict. The contribution of the SSM is set to zero for the segmentation of cartilage. ",
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+        "text": "Voting can be formularized as follows: Let $\\mathbf { x } = ( i , j , k )$ be a (global) index triplet of an image voxel. Let further be $I _ { s }$ the mapping from local indices of subvolume ${ \\mathrm { ~ \\it ~ B ~ } } _ { s }$ to global indices in the image, and let $\\mathrm { I m } ( I _ { s } )$ be the set of all global indices covered by $B _ { s }$ . The voting mask is computed as: ",
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+        "img_path": "images/cefdc42a5e28c7a458882efedbde459ada5fc66681812da3e0769c585e8fea0b.jpg",
+        "text": "$$\nV ( \\mathbf { x } ) = \\omega \\cdot G ( \\mathbf { x } ) + \\sum _ { \\mathbf { x } \\in \\mathrm { I m } ( I _ { s } ) } \\Big ( 2 \\cdot B _ { s } \\big ( I _ { s } ^ { - 1 } ( \\mathbf { x } ) \\big ) - 1 \\Big ) ,\n$$",
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+        "text": "with factor $\\omega$ set to 25, weighting the contribution of SSM and CNNs roughly equal. The higher $\\omega$ is chosen, the more trust is put into the SSM-based regularization. Using this voting mask, majority voting is applied to generate the segmentation mask $\\breve { U }$ : ",
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+        "text": "$$\nU ( \\mathbf { x } ) = \\left\\{ \\begin{array} { l l } { 1 , } & { \\mathrm { i f } V ( \\mathbf { x } ) \\geq \\tau } \\\\ { 0 , } & { \\mathrm { i f } V ( \\mathbf { x } ) < \\tau , } \\end{array} \\right.\n$$",
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+        "type": "text",
+        "text": "where the threshold parameter $\\tau$ has a fixed value of 1, since this way, a majority of CNN-classified subvolumes have to up-vote a voxel in order to include it in the final result, if it is not already captured by the SSM. ",
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+        "text": "2.4 SSM postprocessing ",
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+        "text": "Bone segmentation is finalized in the SSM postprocessing step. The idea behind SSM-based postprocessing as utilized in our approach is to remove wrongly classified foreground labels from the FB and TB segmentations that are located outside the typical range of osteophytic growth. Foreground labels are excluded from the segmentation mask depending on their surface distance to the SSM. We found that a conservative regularization after processing the subvolumes with the proposed voting method (CNN-3D stage) helps to remove segmentation errors due to the localized nature of the 3D CNN-based classification and low image contrast or noise (cf. Fig. 4, central and right column). ",
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+        "text": "We identified regions on the SSMs of the FB and TB that show higher or lower inter-patient variability in shape (cf. Appendix B). Regions of higher variability are typically associated with osteophytic growth. To identify these regions in the segmentation, SSMs are matched to the masks obtained from the CNN-3D stage and the distance between foreground voxels and fitted surface is calculated, s.t. every voxel is assigned a distance value either to areas of low or high variability on the SSMs. This leads to a maximum distance in areas of low variability $d _ { l v }$ and to a maximum distance in areas of high variability $d _ { h v }$ . ",
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+        "text": "If $d _ { h v } \\geq d _ { l v }$ , the postprocessing is terminated since no unexpected deviation is identified. Otherwise, the matched surface is converted to a mask denoted by $\\tilde { U }$ and the set difference $D = U \\setminus \\tilde { U }$ is calculated. D naturally consists of its 3D connected components $D _ { j }$ . Every component is considered a candidate for removal, if there exists a voxel $x \\in D _ { j }$ , s.t. its distance to the surface $d _ { \\mathbf { x } }$ is realized in the area of low variability and additionally $d _ { \\mathbf { x } } \\geq d _ { h v }$ and $d _ { \\mathbf { x } } \\ \\geq \\ t o l$ hold for an empirically determined tolerance of $5 . 5 \\mathrm { m m }$ . All other components describe either osteophytic growth or normal morphological deviation. For components that fulfill the former rule, every voxel that realizes its distance in the area of low variability is removed from the mask. See Appendix B for a detailed schematic description. ",
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+        "type": "text",
+        "text": "3 Experiments and Results ",
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+        "text": "We evaluated the accuracy of our method on three different datasets SKI10, OAI Imorphics and OAI ZIB (Table 1) employing volume-based and distance-based measures (cf. Appendix A). ",
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+        "text": "3.1 MRI datasets ",
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+        "text": "Dataset SKI10 consists of MRIs from the MICCAI SKI10 grand challenge. These data is divided into 60 training, 40 validation, and 50 submission images. All scans were acquired for surgery planning of partial or complete knee replacement, and thus show a high degree of pathological deformities in the knee region. ",
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+        "text": "Dataset OAI Imorphics consists of MRI sequences from the OAI database with manual segmentations supplied by Imorphics $\\left( \\mathrm { N } = 8 8 \\right)$ ). The dataset contains only cases of moderate and severe OA. ",
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+        "text": "Dataset OAI ZIB consists of additional data from the OAI database for which manual segmentations were carried out thoroughly by experienced users at Zuse Institute Berlin $\\mathrm { { N } } = 5 0 7 { \\mathrm { { } } }$ ) starting from automatic segmentations employing [6]. The data cover the full spectrum of OA grades, with a strong tendency towards severe cases. ",
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+        "table_caption": [
+            "Table 1: Summary of the datasets used for training and validation. Images were acquired either once per patient (baseline) or twice with an additional 12-month follow-up $( 1 2 \\mathrm { m } )$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>SKI10</td><td>OAI Imorphics</td><td>OAI ZIB</td></tr><tr><td>MRI scanner</td><td>GE,Siemens,Philips,Toshiba,Hitachi. Mostly1.5T, some 3T,a few 1T</td><td>Siemens 3T Trio</td><td>Siemens 3T Trio</td></tr><tr><td>MRI sequence</td><td>Many (T1, T2, GRE, Spoiled-GRE) partly with fat suppression</td><td>DESS</td><td>DESS</td></tr><tr><td>Acquisition plane</td><td>sagittal</td><td>sagittal</td><td>sagittal</td></tr><tr><td>Image resolution [mm]</td><td>0.39×0.39× 1.0</td><td>0.36×0.36×0.7</td><td>0.36×0.36×0.7</td></tr><tr><td>Manual segmentations</td><td>bones and cartilage</td><td>cartilage</td><td>bones and cartilage</td></tr><tr><td>Number of subjects</td><td>60 training 40 validation 50 submission</td><td>88</td><td>507</td></tr><tr><td>Sex (male;female)</td><td>n.a.</td><td>(45,43)</td><td>(262,245)</td></tr><tr><td>Age [years]</td><td>n.a.</td><td>61.24±9.98</td><td>61.87±9.33</td></tr><tr><td>BMI [kg/m²]</td><td>n.a.</td><td>31.06±4.61</td><td>29.27±4.52</td></tr><tr><td>rOA grade (0,1,2,3,4)</td><td>n.a.</td><td>(0,0,15,56,17)</td><td>(60,77,61,151,158)</td></tr><tr><td>timepoints</td><td>baseline</td><td>baseline,12m</td><td>baseline</td></tr></table>",
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+        "text": "3.2 Experimental setup ",
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+        "text": "The employed SSMs consist of 15,172 vertices and 30,220 faces (FB), and 16,244 vertices and 32,351 faces (TB) independent of the dataset. Construction was done following [6]. For the SKI10 dataset, training of CNNs and construction of SSMs is carried out using the 60 training cases. Our method is evaluated for the validation and the submission cases separately. Two-fold cross-validation studies are performed for datasets OAI ZIB and OAI Imorphics. For OAI ZIB, decomposition is done by random choice (253/254). For OAI Imorphics the cohort’s subject ids are sorted numerically and split into upper and lower half (44/44). However, since no manual segmentations of bones are available for the OAI Imorphics dataset, the SSMs built from the SKI10 training data are employed. The CNNs are trained using the OAI Imorphics baseline data only. Thus, the OAI Imorphics $1 2 \\mathrm { m }$ follow-up data is exclusively used for evaluation still within the cross-validation setting. ",
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+    {
+        "type": "text",
+        "text": "3.3 Measures of segmentation accuracy ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The accuracy of our method is evaluated using the DICE Similarity Coefficient (DSC), average surface distance (ASD), root mean square distance (RSD), maximum distance (MSD), volume difference (VD), and volume overlap error (VOE). All these measures are symmetric apart from VD that is considered relative to the manual segmentation. Volumetric measures (Appendix A, 3) are suitable for assessing the segmentation results globally. However, volume-based measures provide limited sensitivity to errors on the boundaries of the segmentation if the segmented volume is relatively large. We therefore also include surface distance measures (Appendix A, 4) in the evaluation, which are sensitive to segmentation errors on the anatomical boundary. ",
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+    {
+        "type": "text",
+        "text": "3.4 Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Table 2 summarizes our results for the SKI10 validation dataset. Our method reaches a total score of $7 3 . 6 { \\pm } 7 . 6 $ in terms of the SKI10 metrics [11]. This is a notable improvement w.r.t scores reported in previous publications (Table 3). Our method achieves a total score of 75.73 on the SKI10 submit data and is currently ranked first5. ",
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+        "type": "text",
+        "text": "The results for the OAI Imorphics dataset are shown in Table 4. For FC the DSC is $8 9 . 4 \\%$ for baseline and $8 9 . 1 \\%$ for $1 2 \\mathrm { m }$ . For medial tibial cartilage (MTC) and lateral tibial cartilage (LTC) the DSC is $8 6 . 1 \\%$ resp. $9 0 . 4 \\%$ for baseline, and $8 5 . 8 \\%$ resp. $9 0 . 0 \\%$ for $1 2 \\mathrm { m }$ . Again, the ASD is smaller than the image resolution $( < 0 . 3 6 \\mathrm { m m } )$ for both, FC and TC. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 5 summarizes the segmentation accuracy for the OAI ZIB dataset. The DSC is $9 8 . 5 \\%$ for FB, $9 8 . 5 \\%$ for TB, $8 9 . 9 \\%$ for FC, and $8 5 . 6 \\%$ for TC. The ASD is smaller than the image resolution for bone as well as for cartilage $( < 0 . 3 6 \\mathrm { m m } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Computation for the whole segmentation pipeline (end to end) was $9 \\mathrm { m } ~ 2 2 \\mathrm { s }$ on a consumer-grade workstation (CPU: Intel Xeon E5-2650 v3, 2.30GHz; GPU: GeForce GTX $9 8 0 \\mathrm { T i }$ ). Implementation of CNNs was done employing Keras with Theano-backend6. All calculations regarding SSMs were carried out using Amira ZIB Edition7 ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/f9a598defbdb843a0b1f49675fb53ff4243ee9ed14a5bf7f7d678fde727c1c88.jpg",
+        "table_caption": [
+            "Table 2: Segmentation accuracy for the SKI10 validation dataset. "
+        ],
+        "table_footnote": [
+            "Total score as computed employing the SKI10 metrics: 73.6 ± 7.6 "
+        ],
+        "table_body": "<table><tr><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1>ASD (mm)</td><td rowspan=2 colspan=1>RSD (mm)</td><td rowspan=2 colspan=1>VD (%)</td><td></td></tr><tr><td rowspan=1 colspan=1>VOE (%)</td></tr><tr><td rowspan=1 colspan=1>FBTBFCTC</td><td rowspan=1 colspan=1>0.43± 0.130.37 ± 0.11</td><td rowspan=1 colspan=1>0.75± 0.280.63 ± 0.26</td><td rowspan=1 colspan=1>7.18 ± 10.514.29 ± 12.34</td><td rowspan=1 colspan=1>20.99 ± 5.0819.06 ± 5.18</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 Discussion and Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We presented a novel fully automated segmentation method for knee bone and cartilage by combining the advantages of SSM-based regularization with CNN-based classification of voxel intensities. The method was evaluated using datasets from the SKI10 challenge as well as from the OAI database. Accuracy was evaluated using volume-based and distance-based measures to provide a transparent analysis w.r.t global and local level of detail. The proposed method consistently achieved high segmentation accuracy, despite severely arthritic knees and various different MRI sequences. For the first time, a total score greater than 75 was reached on the SKI10 submission data, which is comparable to the inter-observer variability of two expert readers [11]. In our experience, the automated method reduces the time effort for an accurate segmentation of knee bones and cartilage at least by a factor of six compared to manual segmentations by an experienced reader $\\mathrm { ( > l h ) }$ . However, large scale databases for studying the OA disease, such as provided by the OAI, can contain 50.000 or more MRIs. Using our implementation, it would take 43 weeks to segment the full OAI database on a singe computational node. We therefore aim at reducing the computational time of the algorithm further as well as distributing the work-load over several nodes. Our goal is to segment the full OAI database and make the results available to the public in the near future. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0d5c3166ce58e7835f95e33bac36ed1928b38c9cc06e0d09958f02cd04676f9f.jpg",
+        "image_caption": [
+            "Figure 3: Segmentation of osteophytic regions in different stages (colored contours). CNN-2D stage (left) is error prone, SSM adjustment (middle) smoothly regularizes and CNN-3D (right) segments osteophytes precisely. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/89b552d45e95bfe9dd555306396e6f1bfe3a6d44909c411e212a8be6a9b59ba5.jpg",
+        "image_caption": [
+            "Figure 4: Left column: An image artifact results in an erroneous CNN-2D classification (top). An anatomically plausible segmentation (bottom) is restored using SSM adjustment regularization. Middle and right column: CNN-3D segmentation introduces errors in the shaft region due to the subvolumes’ locality and missing image contrast (top). SSM postprocessing corrects in an anatomically plausible manner (bottom). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/99ff313d8d376f9759313d464f0ec463d8d98d83b7a9660f922ebe252b90b1c8.jpg",
+        "table_caption": [
+            "Table 3: SKI10 validation data: Our approach yields the best of all published results as of April 2018. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Imorphics [5]</td><td rowspan=1 colspan=1>ZIB (2010) [6]</td><td rowspan=1 colspan=1>BioMedIA[13]</td><td rowspan=1 colspan=1>Liu et al. [9]</td><td rowspan=1 colspan=1>Biomediq [14]</td><td rowspan=1 colspan=1>ZIB (2018)</td></tr><tr><td rowspan=1 colspan=1>52.3 ± 8.6</td><td rowspan=1 colspan=1>54.4±8.8</td><td rowspan=1 colspan=1>56.5 ± 9.2</td><td rowspan=1 colspan=1>64.1 ± 9.5</td><td rowspan=1 colspan=1>67.1 ± 8.0</td><td rowspan=1 colspan=1>73.6 ± 7.6</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0a2dfa41ce37bb2fab5cb6572f4fa67e4877d25d1b29f5ba0e60821d738b263c.jpg",
+        "table_caption": [
+            "Table 4: Segmentation accuracy for the OAI Imorphics dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">DSC (%)</td><td>ASD (mm)</td><td>RSD (mm)</td><td>MSD (mm)</td><td>VD (%)</td><td>VOE (%)</td></tr><tr><td rowspan=\"3\">00m</td><td>FC</td><td>89.4 ± 2.41</td><td>0.19 ± 0.08</td><td>0.50± 0.30</td><td>6.65± 2.99</td><td>7.0± 6.04</td><td>19.1 ± 3.88</td></tr><tr><td>MTC</td><td>86.1 ± 5.33</td><td>0.26 ± 0.23</td><td>0.63 ± 0.55</td><td>5.16 ± 2.93</td><td>8.0 ± 17.15</td><td>24.1 ± 7.74</td></tr><tr><td>LTC</td><td>90.4 ± 2.42</td><td>0.17 ± 0.06</td><td>0.41 ± 0.16</td><td>3.93 ± 2.07</td><td>6.9 ± 7.14</td><td>17.5 ± 3.96</td></tr><tr><td rowspan=\"3\">12m</td><td>FC</td><td>89.1 ± 2.41</td><td>0.20 ± 0.09</td><td>0.53 ± 0.33</td><td>6.86 ± 3.16</td><td>7.6± 6.78</td><td>19.6 ± 3.86</td></tr><tr><td>MTC</td><td>85.8 ± 5.00</td><td>0.28 ± 0.22</td><td>0.67 ± 0.55</td><td>5.25 ± 3.06</td><td>6.6 ± 16.38</td><td>24.5 ± 7.37</td></tr><tr><td>LTC</td><td>90.0 ± 2.57</td><td>0.18 ± 0.06</td><td>0.44 ± 0.19</td><td>4.08 ± 2.11</td><td>7.2 ± 7.74</td><td>18.1 ± 4.16</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/488270e31c8b10269a2f5459c1bea65e2825b297b964d0e704f196651f48d429.jpg",
+        "table_caption": [
+            "Table 5: Segmentation accuracy on the OAI ZIB dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>DSC (%)</td><td>ASD (mm)</td><td>RSD (mm)</td><td>MSD (mm)</td><td>VD (%)</td><td>VOE (%)</td></tr><tr><td>FB</td><td>98.5 ± 3.02</td><td>0.17 ± 0.05</td><td>0.35± 0.09</td><td>2.93± 1.24</td><td>-0.09± 0.87</td><td>2.8± 0.58</td></tr><tr><td>TB</td><td>98.5 ± 3.25</td><td>0.18 ± 0.06</td><td>0.37 ± 0.18</td><td>3.16 ± 2.03</td><td>-0.03 ± 0.82</td><td>2.9 ± 0.63</td></tr><tr><td>FC</td><td>89.9 ± 3.60</td><td>0.16 ± 0.07</td><td>0.38 ± 0.17</td><td>5.35 ± 2.50</td><td>1.5 ± 5.87</td><td>18.1 ± 5.90</td></tr><tr><td>TC</td><td>85.6 ± 4.54</td><td>0.23 ± 0.12</td><td>0.60 ± 0.38</td><td>6.35 ± 4.36</td><td>-1.0 ± 11.92</td><td>24.9 ± 6.79</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "A promising line of future work is to investigate approaches that couple SSMs and CNNs more directly, e.g. by introducing learned appearance from CNNs to an SSM segmentation framework. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            630,
+            303,
+            645
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Heiko Ramm (née Seim, 1000shapes GmbH) for valuable insights into SSM-based segmentation methods. We further would like to thank Irene Ziska, Agnieszka Putyra, and Robert Joachimsky for creating the manual segmentations for our OAI ZIB dataset by thoroughly correcting automated presegmentations. The authors gratefully acknowledge the financial support by the German federal ministry of education and research (BMBF) research network on musculoskeletal diseases, grant no. 01EC1408B (Overload/PrevOP) and grant no. 01EC1406E (TOKMIS). The Osteoarthritis Initiative is a public-private partnership comprised of five contracts (N01-AR-2-2258; N01-AR-2-2259; N01-AR-2-2260; N01-AR-2-2261; N01-AR-2-2262) funded by the National Institutes of Health, a branch of the Department of Health and Human Services, and conducted by the OAI Study Investigators. Private funding partners include Merck Research Laboratories; Novartis Pharmaceuticals Corporation, GlaxoSmithKline; and Pfizer, Inc. Private sector funding for the OAI is managed by the Foundation for the National Institutes of Health. This manuscript was prepared using an OAI public use data set and does not necessarily reflect the opinions or views of the OAI investigators, the NIH, or the private funding partners. ",
+        "bbox": [
+            174,
+            655,
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            266,
+            106
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[1] Reva C Lawrence, David T Felson, Charles G Helmick, Lesley M Arnold, Hyon Choi, Richard A Deyo, Sherine Gabriel, Rosemarie Hirsch, Marc C Hochberg, Gene G Hunder, et al. Estimates of the prevalence of arthritis and other rheumatic conditions in the united states: Part ii. Arthritis & Rheumatology, 58(1): 26–35, 2008. [2] Felix Eckstein, Deborah Burstein, and Thomas M Link. Quantitative MRI of cartilage and bone: degenerative changes in osteoarthritis. NMR in Biomedicine, 19(7):822–854, 2006. [3] Felix Eckstein, Flavia Cicuttini, Jean Pierre Raynauld, John C Waterton, and Charles Peterfy. Magnetic resonance imaging (MRI) of articular cartilage in knee osteoarthritis (OA): morphological assessment. Osteoarthritis and Cartilage, 14:46–75, 2006. [4] Tobias Heimann and Hans-Peter Meinzer. Statistical shape models for 3d medical image segmentation: a review. Medical Image Analysis, 13(4):543–563, 2009. [5] Graham Vincent, Chris Wolstenholme, Ian Scott, and Mike Bowes. Fully automatic segmentation of the knee joint using active appearance models. In MICCAI Workshop Medical Image Analysis for the Clinic, pages 224 – 230, 2010. [6] Heiko Seim, Dagmar Kainmueller, Hans Lamecker, Matthias Bindernagel, Jana Malinowski, and Stefan Zachow. Model-based auto-segmentation of knee bones and cartilage in MRI data. In MICCAI Workshop Medical Image Analysis for the Clinic, pages 215 – 223, 2010. [7] Anirban Mukhopadhyay, Oscar Salvador Morillo Victoria, Stefan Zachow, and Hans Lamecker. Robust and accurate appearance models based on joint dictionary learning data from the osteoarthritis initiative. In International Workshop on Patch-based Techniques in Medical Imaging, pages 25–33. Springer, 2016. [8] Adhish Prasoon, Kersten Petersen, Christian Igel, François Lauze, Erik Bjørnager Dam, and Mads Nielsen. Deep feature learning for knee cartilage segmentation using a triplanar convolutional neural network. In International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), pages 246 – 253, 2013.   \n[9] Fang Liu, Zhaoye Zhou, Hyungseok Jang, Alexey Samsonov, Gengyan Zhao, and Richard Kijowski. Deep convolutional neural network and 3d deformable approach for tissue segmentation in musculoskeletal magnetic resonance imaging. Magnetic Resonance in Medicine, 79(4):2379–2391, 2017.   \n[10] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), pages 234–241, 2015.   \n[11] Tobias Heimann, Bryan J Morrison, Martin A Styner, Marc Niethammer, and Simon K Warfield. Segmentation of knee images: a grand challenge. In MICCAI Workshop on Medical Image Analysis for the Clinic, pages 207–214, 2010.   \n[12] Dagmar Kainmueller. Deformable Meshes for Medical Image Segmentation: Accurate Automatic Segmentation of Anatomical Structures. Springer, 2014.   \n[13] Zehan Wang, Claire Donoghue, and Daniel Rueckert. Patch-based segmentation without registration: application to knee MRI. In MICCAI Workshop Machine Learning in Medical Imaging, pages 98 – 105, 2013.   \n[14] Erik B Dam, Martin Lillholm, Joselene Marques, and Mads Nielsen. Automatic segmentation of high-and low-field knee MRIs using knee image quantification with data from the osteoarthritis initiative. Journal of Medical Imaging, 2(2):024001 – 024001, 2015. ",
+        "bbox": [
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+            108,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Appendix A: Employed surface-based and volume-based distance measures ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            88,
+            803,
+            107
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cd429bcf9a3d7021897e001ae95e444461e2c14c700503db76b12b01f73f4ef.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\mathrm { D S C } = 1 0 0 \\cdot \\frac { 2 \\displaystyle | \\boldsymbol { B } \\cap \\boldsymbol { A } | } { \\displaystyle | \\boldsymbol { B } \\rvert + \\displaystyle | \\boldsymbol { A } | } , \\ \\mathrm { V O E } = 1 0 0 \\cdot \\left( 1 - \\frac { \\displaystyle \\mathrm { D S C } } { \\displaystyle 2 0 0 - \\displaystyle \\mathrm { D S C } } \\right) , \\ \\mathrm { V D } = 1 0 0 \\cdot \\frac { \\displaystyle | \\boldsymbol { B } | - \\displaystyle | \\boldsymbol { A } | } { \\displaystyle | \\boldsymbol { A } | } , } } \\\\ { { \\mathrm { A S D } = \\frac { 1 } { \\displaystyle n _ { \\partial A } + n _ { \\partial B } } \\left( \\sum _ { i = 1 } ^ { n _ { \\partial A } } \\operatorname* { m i n } _ { b \\in \\partial B } \\| a _ { i } - b \\| _ { 2 } + \\sum _ { j = 1 } ^ { n _ { \\partial B } } \\operatorname* { m i n } _ { a \\in \\partial A } \\| b _ { j } - a \\| _ { 2 } \\right) , \\ ~ } } \\\\ { { \\mathrm { R S D } = \\sqrt { \\displaystyle \\frac { 1 } { n _ { \\partial A } + n _ { \\partial B } } \\left( \\sum _ { i = 1 } ^ { n _ { \\partial A } } \\operatorname* { m i n } _ { b \\in \\partial B } \\| a _ { i } - b \\| _ { 2 } ^ { 2 } + \\sum _ { j = 1 } ^ { n _ { \\partial B } } \\operatorname* { m i n } _ { a \\in \\partial A } \\| b _ { j } - a \\| _ { 2 } ^ { 2 } \\right) } , \\ ~ } } \\\\ { { \\mathrm { M S D } = \\operatorname* { m a x } \\left( \\operatorname* { m a x } _ { a \\in \\partial A } \\operatorname* { m i n } _ { b \\in \\partial B } \\| a - b \\| _ { 2 } , \\operatorname* { m a x } _ { b \\in \\partial B } \\operatorname* { m i n } _ { a \\in \\partial A } \\| b - a \\| _ { 2 } \\right) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            125,
+            785,
+            277
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Within the above, $A$ denotes the set of manually segmented (ground-truth) voxels and $B$ denotes the segmentation result from the automated method; $\\partial A$ and $\\partial B$ represent the boundary of $A$ and $B$ . The boundary contains every voxel having at least one neighbor that is not part of the respective segmentation mask. The number of voxels on the boundary $\\partial A , \\partial B$ is written as $n _ { \\partial A } , n _ { \\partial B }$ . Lastly $| \\cdot |$ denotes a volume and $\\lVert \\cdot \\rVert _ { 2 }$ the usual Euclidean norm. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/ea35e55d45ce0c215386f8683675d1aafee3d56993fc16036018dcbb9f855d8b.jpg",
+        "image_caption": [
+            "Appendix B: SSM postprocessing schematic visualization ",
+            "Figure 5: Predefined regions of high (yellow) and low (blue) variability on SSM surfaces of FB (left) and TB (middle). Right: Schematic visualization of a typical postprocessing situation. Distances are computed from 3D CNN mask (red) to SSM surface. The cross-section depicted shows (w.l.o.g.) the realization of $d _ { h v }$ in component $A$ . Component $B$ is completely removed since there exists one $x \\in B$ s.t. $d _ { x } > d _ { h v }$ and all elements of $B$ realize their closest distance in the area of low variability. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            404,
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+            554
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Appendix C: Sampling for CNN-3D – visualization ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            660,
+            598,
+            679
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/edebfd34044c75a5ffd7cb1518cb568fe1f7fdeb8bfb2a72392fe0df561ee4d4.jpg",
+        "image_caption": [
+            "Figure 6: CNN-3D segmentation step: OAI subject ID 9793168. Exemplary sampling of subvolumes for (f.l.t.r.) FB (1052 points), TB (825 points), FC (845 points) and TC (228 points). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    }
+]

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+# Relative Uncertainty Learning for Facial Expression Recognition
+Yuhang Zhang, Chengrui Wang, Weihong Deng Beijing University of Posts and Telecommunications zyhzyh@bupt.edu.cn, crwang@bupt.edu.cn, whdeng@bupt.edu.cn
+# Abstract
+In facial expression recognition (FER), the uncertainties introduced by inherent noises like ambiguous facial expressions and inconsistent labels raise concerns about the credibility of recognition results. To quantify these uncertainties and achieve good performance under noisy data, we regard uncertainty as a relative concept and propose an innovative uncertainty learning method called Relative Uncertainty Learning (RUL). Rather than assuming Gaussian uncertainty distributions for all datasets, RUL builds an extra branch to learn uncertainty from the relative difficulty of samples by feature mixup. Specifically, we use uncertainties as weights to mix facial features and design an add-up loss to encourage uncertainty learning. It is easy to implement and adds little or no extra computation overhead. Extensive experiments show that RUL outperforms state-of-the-art FER uncertainty learning methods in both realworld and synthetic noisy FER datasets. Besides, RUL also works well on other datasets such as CIFAR and Tiny ImageNet. The code is available at https://github.com/zyh-uaiaaaa/Relative-Uncertainty-Learning.
+# 1 Introduction
+Although well-trained models can give high-confidence inferences directly, the inference results may actually be wrong and cause serious consequences [22]. Therefore, the concept of uncertainty is widely used in measuring how well the model trusts its inference results, which has become a research hotspot in trustworthy machine learning [3, 11, 10, 22, 26, 16]. It is expected that a reliable model assigns high level uncertainty to its erroneous predictions so that humans can intervene to avoid lots of disasters. As described by Kiureghian et al. [25], the uncertainties in machine learning can be split into aleatoric uncertainty and epistemic uncertainty, which are also called data uncertainty and model uncertainty. In this paper, we focus on quantifying data uncertainty as model uncertainty can be eliminated by introducing more training data. Concretely, data uncertainty in facial expression recognition (FER) is mainly caused by ambiguous facial expressions and the subjectiveness of annotators, which blocks the improvement of recognition performance [47].
+Recent studies propose a variety of methods to solve the uncertainty learning problem in the face recognition field. PFE [37] introduces a new branch to learn the uncertainty of face recognition and recognizes faces by measuring the similarity between two Gaussian distributions. Chang et al. [4] further propose DUL to learn feature and uncertainty simultaneously in order to improve facial feature learning. When it comes to FER, SCN [47] uses a fully-connected layer to learn an importance weight for each image and suppresses uncertainties according to the learned weights. However, the strong learning ability of neural networks will deteriorate the uncertainty learning branch. One solution is adding regularization to force the model to learn uncertainty, which needs to be tuned carefully, otherwise, it would lead to underfitting. For example, SCN uses margin loss to keep the gap between certain and uncertain images, but choosing a suitable gap value is non-trivial. DUL assumes features to follow a Gaussian distribution, but the real distribution of the dataset is not always following such an assumption. Therefore, it is still a challenge to learn uncertainty distributions of real datasets without deteriorating feature learning.
+To address the problem, we perceive data uncertainty from a different point of view. We realize that data uncertainty – the difficulty to classify a sample correctly is actually a relative concept. More specifically, the difficulty of classification is a person’s subjective feeling that comes from comparison. One can only know whether a sample is easy or not according to a reference. Inspired by this idea, we propose an innovative method called Relative Uncertainty Learning (RUL) to help deep learning models to learn uncertainty for each sample. Concretely, we build a new branch to model the uncertainties of input images and utilize the uncertainties as weights to mix two features of different labels. Through an add-up loss function, the model is encouraged to recognize two expressions simultaneously from the mixed features, which enables the model to learn uncertainty comparatively while minimizing the total loss. As discussed in Section 3.3, the uncertainty learning branch will assign large uncertainty values to uncertain facial expression images while small uncertainty values to certain images.
+The main contributions of this work are as follows:
+• We perceive uncertainty from a new perspective and propose an innovative uncertainty learning method to learn uncertainty from the relative difficulty of two samples. • We get state-of-the-art performance in both real-world and synthetic noisy FER datasets. • RUL does not need prior knowledge of the dataset uncertainty distribution and can be easily applied to different classification tasks with low computation cost.
+# 2 Related Work
+Uncertainty learning in the face field Uncertainties can be mainly categorized into model uncertainty which accounts for uncertainty in the model parameters and data uncertainty which is the noise inherent in the dataset [22]. In the face recognition field, uncertainty learning is important for learning discriminative feature embeddings of noisy face images. Because noisy face images are usually out of the cluster and have large variances in the latent embedding space, which might cause wrong recognition [37]. There have been several proposals to solve the uncertainty learning problem in face recognition tasks [13, 51, 23, 37, 4]. Some of them [13, 51, 23] have tried to analyze and learn face representations using model uncertainty. Gong et al. [13] estimate the capacity of a given face representation by explicitly accounting for the manifold structure, model uncertainty and data variability. Khan et al. [23] present a framework for class imbalance learning based on model uncertainty. Zafar et al. [51] cope with false positives through employing model uncertainty to improve the efficacy of face recognition systems. Thereinto the methods considering data uncertainty learning. Shi et al. [37] use PFE to learn a variance for each fixed feature and then measure the likelihood of each positive face pair of $( x _ { i } , x _ { j } )$ sharing the same distribution of latent embedding $\mathsf { p } ( z _ { i }$ $z _ { j } .$ ). Chang et al. [4] further propose to learn feature and uncertainty simultaneously by encoding the latent feature embedding as a Gaussian distribution. In the facial expression recognition task, Wang et al. [47] try to learn an attention weight for each image, then utilize it to weigh cross-entropy loss in order to suppress the influence of noisy samples. She et al. [36] propose to use several branches to model the latent label distribution of facial expression images and use cosine similarity to capture the uncertain images.
+Noisy dataset training With the enhancement of deep learning network capabilities, the datasets for facial expression recognition (FER) are getting larger and larger. For the large-scale FER datasets [43, 54, 7, 1, 33, 9, 27], due to the ambiguity of expressions and the subjectiveness of human annotators, it is extremely hard to get high-quality labels. Training with label noise has been studied for a long time. The basic idea is to enable the model to access the quality of the labels or estimate the noise distribution using a small set of clean dataset [5, 28, 39, 45]. Li et al. [28] design a distillation framework to use information from both a small clean dataset and label relations in knowledge graphs to prevent the model learning from noisy labels. Veit et al. [45] use a small amount of clean annotations to reduce the noisy samples before fine-tuning the network using both the clean set and the full set. Other methods do not use a small clean dataset, but usually assume extra constraints on the noisy samples. Natarajan et al. [34] design a specific loss function for randomly flipped labels. Goldberger et al. [12] explicitly model the noises in datasets by a softmax layer which connects the correct labels to the noisy ones. Without collecting a clean set of dataset or using prior knowledge of noise distribution, our method learns uncertainty through the relativity of the given dataset and suppresses the most uncertain samples to get a good performance under label noise.
+![](images/f588edd04c2b28be0867d1c6d8fe5e4e828fce71f2980f01e36577a7b3670eb5.jpg)
+Figure 1: Overview of our Relative Uncertainty Learning method. We use two branches to learn facial features $\mu$ and uncertainty vectors $\sigma$ simultaneously. Then we normalize uncertainty vectors with their counterparts sampled from shuffled indexes. We compute the mean of the uncertainty vectors as the uncertainty values. Next, we mix the facial features according to the mean uncertainty values. Note that the easy sample will get a small uncertainty value because its facial feature is very obvious for recognition. In contrast, the ambiguous sample tends to get a large uncertainty value in order to make it easy for the model to recognize its corresponding label from the mixed features. Finally, we use the add-up loss to encourage the model to recognize the two expressions from the mixed feature, which implicitly forces the model to learn reliable uncertainty values.
+# 3 Proposed Method
+In order to reduce the negative impact of the strong learning ability of neural networks and learn uncertainty values closer to the real dataset uncertainty distribution, we propose a simple and effective method named Relative Uncertainty Learning (RUL). In this section, we first provide the motivation behind our method. Then we demonstrate the details of RUL. Finally, we explain why RUL is effective in uncertainty learning.
+# 3.1 Motivation
+The uncertainty learning methods mentioned before suffer from the strong learning ability of deep neural networks more or less. For example, the introduced variances in DUL are equivalent to some noises being added to feature points, so the model will learn very small variances for all feature points making the estimation of uncertainty less reliable. SCN also learns large importance weights for noisy samples to obtain small cross-entropy loss values as these noisy images will be memorized by the networks after training for several epochs.
+In order to prevent the strong learning ability of neural networks from degrading the uncertainty learning branch, we perceive uncertainty from a relative point of view. We find that learning uncertainty through comparison forms a natural regularization because if the model assigns a very small uncertainty value for one image of the comparison pair, then it means the counterpart image gets a large uncertainty value which prevents the model learning small uncertainty values for all images. Learning uncertainty through relativity also conforms to human cognition – it is very difficult for a person to define which sample is certain and which sample is uncertain without using one image as a reference. In the following section, we thoroughly illustrate the details of our proposed RUL.
+# 3.2 Overview of Relative Uncertainty Learning
+In this section, we illustrate the structure of our proposed RUL. The pipeline of RUL is shown in Figure 1.
+Two Branches Model Given a batch of face images $\boldsymbol { X }$ , assuming the conventional feature extraction module $f _ { \theta } ( X )$ can be decomposed as
+$$
+f _ { \theta } ( \boldsymbol { X } ) = f _ { \theta } ^ { ( l ) } ( f _ { \theta } ^ { ( l - 1 ) } ( \boldsymbol { X } ) ) ,
+$$
+we first extract general facial features $F _ { g }$ from the penultimate feature extraction layer denoted as $\pmb { F } _ { g } = f _ { \theta } ^ { ( l - 1 ) } ( \pmb { X } )$ [50]. Then, in order to learn facial features and uncertainty values simultaneously, we drop the layer indexed by $l$ , i.e., $f _ { \theta } ^ { ( l ) }$ and build two separate branches $f _ { \theta _ { \mu } } ( f _ { \theta } ^ { ( l - 1 ) } ( X ) )$ and $f _ { \theta _ { \sigma } } ( f _ { \theta } ^ { ( l - 1 ) } ( { \pmb X } ) )$ to produce facial features $\pmb { F } = [ \pmb { \mu } _ { 1 } , \pmb { \mu } _ { 2 } , . . . , \pmb { \mu } _ { N } ] \in \mathbb { R } ^ { D \times N }$ and uncertainty vectors $U = [ \pmb { \sigma } _ { 1 } , \pmb { \sigma } _ { 2 } , . . . , \pmb { \sigma } _ { N } ] \in \mathbb { R } ^ { D \times N } . \ D , N$ denote output dimension and batch size separately.
+Relative Uncertainty Learning Inspired by the relativity of the uncertainty concept and the mixup method [53, 42, 46], we mix two different facial features according to their uncertainty values which enables the FER model to learn uncertainty through the relativity of different samples. First, we mix a mini-batch with their shuffled ones which results in the facial images of the whole dataset being compared with each other during the training process because of the randomness introduced by the dataloader and the shuffle of mini-batches. Before mixing features, we first normalize the uncertainty vectors element-wise to compare two features with each other following Equation (2)
+$$
+\hat { \pmb { \sigma } } _ { i } , \hat { \pmb { \sigma } } _ { j } = \frac { \pmb { \sigma } _ { i } } { \pmb { \sigma } _ { i } + \pmb { \sigma } _ { j } } , \frac { \pmb { \sigma } _ { j } } { \pmb { \sigma } _ { i } + \pmb { \sigma } _ { j } } ,
+$$
+$\sigma _ { i }$ and $\pmb { \sigma } _ { j }$ are the uncertainty vectors for image $i$ and its counterpart image $j$ sampled from the shuffled indexes, note that they have different labels. This operation not only forms a natural regularization as the model can not learn small uncertainty values for both features, but also benefits the training process as extreme uncertainty values might make the training process unstable or even the loss can not converge. As mentioned by DUL [4], the mean of the predicted uncertainty vectors $\hat { \pmb { \sigma } } _ { i } \in \mathbb { R } ^ { D }$ can be viewed as an approximated measurement of the estimated uncertainty for image $i$ , we also use the mean of $\hat { \pmb { \sigma } } _ { i }$ as learned uncertainty value for image $i$ denoted as $\hat { \sigma } _ { i _ { m e a n } } \in \mathbb { R }$ . Then we get an uncertainty value for each input image denoted as $\pmb { U } = [ \hat { \sigma } _ { 1 _ { m e a n } } , \hat { \sigma } _ { 2 _ { m e a n } } , . . . , \hat { \sigma } _ { N _ { m e a n } } ] \in \mathbb { R } ^ { 1 \times N }$ We then mix features according to their uncertainty values following Equation (3)
+$$
+\tilde { \mu } = \hat { \sigma } _ { i _ { m e a n } } \mu _ { i } + \hat { \sigma } _ { j _ { m e a n } } \mu _ { j } ,
+$$
+$\pmb { \mu _ { i } }$ and $\mu _ { j }$ represent embedding features of the two images, $\hat { \sigma } _ { i _ { m e a n } }$ and σˆjmea represent their corresponding uncertainty values. We introduce negligible computation overhead as we only mix facial features according to their learned uncertainty values in the training phase. Furthermore, the mix feature part will be removed for deployment. Therefore, our network is end-to-end and adds no extra cost on inference.
+Classification Loss We design an add-up loss to encourage the uncertainty learning branch to learn different uncertainty values for different facial images. Specifically, add-up loss requires the FER model to recognize two expressions of input images simultaneously from mixed features. Since $\tilde { \mu }$ is the mixed feature, we feed $\tilde { \mu }$ to the classifier:
+$$
+L _ { t o t a l } = - \frac { 1 } { N } \sum _ { i , j } ^ { N } ( \log \frac { e ^ { W _ { y _ { i } } \tilde { \mu } } } { \sum _ { c } ^ { C } e ^ { W _ { c } \tilde { \mu } } } + \log \frac { e ^ { W _ { y _ { j } } \tilde { \mu } } } { \sum _ { c } ^ { C } e ^ { W _ { c } \tilde { \mu } } } ) .
+$$
+The loss function adds up the loss of recognizing $\tilde { \mu }$ as class $i$ and as class $j$ . $y _ { i } , y _ { i }$ means $l a b e l _ { i }$ , $l a b e l _ { j }$ , $W _ { c }$ is the $\mathrm { c }$ -th classifier and $C$ means the total number of expression classes. The loss function forces the model to recognize two expressions equally from the mixed features. When mixing two facial features, there will be a relatively easy facial feature for expression recognition, and the other will be relatively hard. For simplicity, we note $\pmb { \mu } _ { i }$ as the easy facial feature. Our model will learn a small uncertainty value for $\pmb { \mu _ { i } }$ to mix with $\mu _ { j }$ as $\pmb { \mu _ { i } }$ is easy to recognize, the model can already get a small classification loss with its label $y _ { i }$ after training for several epochs, even $\pmb { \mu } _ { i }$ only takes up a small amount of the mixed feature $\tilde { \mu }$ . While for the hard image $j$ , it needs to take up a large amount of the mixed feature $\tilde { \mu }$ to enable the model to find useful feature to get a small classification loss with label $y _ { j }$ .
+![](images/bb86495aca805b5ec9de1ef69062e89d9f0ffe5cb3cc1b0fda07d46f5dc5930d.jpg)
+Figure 2: An experiment to show why RUL is effective in uncertainty learning. We mix different features with different weights and print the losses of recognizing the mixed feature to different labels. Visually, the face with anger expression is the certain one, and the face with sad expression is the uncertain one. In the process of feature mixup, RUL will assign a larger weight to the uncertain one to minimize the total loss.
+# 3.3 Why RUL is Effective
+In this section, we illustrate why RUL is effective in uncertainty learning. For visualization simplicity, we plot the mixed images instead of the mixed features in Figure 2. At the start of the training, RUL might learn similar uncertainty values for all images, which is shown in the left part of Figure 2. As the certain facial expression image with anger label in Figure 2 can be easily recognized from the mixed feature, the model can already get a small loss value with anger label while gets a large loss value with a sad label. In order to get the minimum total loss, the model needs to recognize the sad expression from the mixed feature more effectively. As training progress, the mix weight (uncertainty value) of the uncertain sad image will increase to enable the model to recognize the sad expression from the mixed feature. Thus, RUL will learn larger uncertainty values for uncertain images.
+# 4 Experiments
+In this section, we first describe three in-the-wild FER datasets and our implementation details. We then evaluate our proposed method on these datasets. We compare our method with state-of-the-art uncertainty learning methods on the accuracy, noisy label training, and accuracy with rejection [41]. Furthermore, we provide qualitative and quantitative analysis to show what is the meaning of our learned uncertainty values. Last, we compare our method to other state-of-the-art FER methods and also demonstrate that RUL is applicable to datasets in different fields. We also carry out an ablation study to find the effectiveness of the mix part in RUL.
+# 4.1 Datasets
+To evaluate the effectiveness of our model, we select three widely used in-the-wild FER datasets. Compared with FER datasets collected in the laboratory, they have more noises like noisy labels and ambiguous expressions. So they are more complex, variable, and challenging for expression recognition tasks. The details are as follows.
+RAF-DB [27] is a crowdsourced facial expression dataset that contains 29672 facial images annotated with basic or compound expressions by 40 trained annotators. In this work, we use facial images with 6 basic expressions [8] and neutral expression, including 12271 images as training data and 3068 images as test data.
+FER2013 [14] consists of 35,887 grayscale $4 8 \mathbf { x } 4 8$ pixel images in total, with 28,709 training samples, 3,589 public test samples, and 3,589 private test samples. These facial images are classified into 7 emotion classes, same as RAF-DB.
+AffectNet [33] is currently the largest FER dataset, including 440,000 images. The images are collected from the Internet by querying the major search engines with 1,250 emotion-related keywords. There are around 280,000 training images and 4000 testing images annotated by human. They are annotated to 8 emotion classes (7 emotion classes same as RAF-DB and Contempt class).
+# 4.2 Implementation Details
+Given a batch of images from RAF-DB, we first resize them to $2 2 4 \times 2 2 4$ pixels and use ResNet18 [18], which is pretrained on Ms-Celeb-1M [17] as backbone. The two branches have the same structure as BatchNorm2d [20]-Dropout [38]-Flatten-FC-BatchNorm1d. We set dropout rate as 0.4, output dimension as 64. Note that the second branch outputs $\log \sigma$ instead of $\pmb { \sigma }$ in order to stabilize the training following [4]. The model is trained in an end-to-end manner with a single GTX 1080ti GPU for 70 epochs with batch size of 64. We also utilize an Adam optimizer [24] with weight decay of 0.0001. The learning rate is initialized as 0.0002 except the last fully connected layer for classification, which is 0.002. We use ExponentialLR [30] learning rate scheduler with gamma of 0.9 to decrease the learning rate after each epoch. The setting under AffectNet dataset is the same as RAF-DB. As for FER2013, we use the same setting as [49] as the image in FER2013 is $4 8 \mathbf { x } 4 8$ pixel grayscale image.
+# 4.3 Evaluation on Label Noises
+Both ambiguous facial expressions and inconsistent annotations will cause label noises in FER datasets. We carry out extensive experiments to demonstrate the improvement of RUL compared with state-of-the-art noise-tolerant FER methods. Following SCN [47], we randomly choose $10 \%$ , $20 \%$ , $30 \%$ of training data and flip their labels to other expression categories randomly. To make a fair comparison with SCN, we also consider the most uncertain samples in the training process as samples with label noises and if the maximum prediction probability is higher than the one of the given label with a threshold (set to 0.2 in all cases), we choose to rely on the FER model and change the label to the index of the maximum prediction probability.
+We run all experiments three times and compute the mean and standard variance of the results. As shown in Table 1, RUL outperforms SCN and DUL under all circumstances. RUL improves test accuracy by $4 . 3 0 \%$ , $4 . 3 2 \%$ , $4 . 6 0 \%$ with noise ratio of $10 \%$ , $20 \%$ and $30 \%$ on RAF-DB, $1 . 0 1 \%$ , $1 . 5 6 \%$ , $2 . 0 1 \%$ on FER2013 and $1 . 9 4 \%$ , $2 . 0 1 \%$ , $4 . 0 5 \%$ on AffectNet compared with SCN. Experiment results show that the benefit from RUL becomes more obvious as the noise ratio increases up which means RUL is more robust to label noises. Note that we do not use pretrained model when training in FER2013 to find how the pretrained model affects the experiment results. From the results we can draw the conclusion that using pretrained model can better deal with label noise as RUL improves the accuracy most when using pretrained models which aligns to the finding in [19]. The results also demonstrate that RUL can learn meaningful uncertainty values as we use uncertainty values to guide the model to find the most uncertain training samples and suppress label noises.
+Table 1: Test accuracy $( \% )$ on RAF-DB, FER2013 and AffectNet with synthetic noisy labels.
+<table><tr><td>Method</td><td>Noisy (%)</td><td>RAF-DB</td><td>FER2013</td><td>AffectNet</td></tr><tr><td>Baseline</td><td>10</td><td>80.43±0.72</td><td>69.25±0.18</td><td>56.85±0.14</td></tr><tr><td>SCN [47]</td><td>10</td><td>81.92±0.69</td><td>69.28±0.05</td><td>58.72±0.20</td></tr><tr><td>DUL [4]</td><td>10</td><td>85.08±0.21</td><td>69.43±0.27</td><td>58.26±0.10</td></tr><tr><td>RUL</td><td>10</td><td>86.22±0.29</td><td>70.29±0.28</td><td>60.66±0.13</td></tr><tr><td>Baseline</td><td>20</td><td>78.01±0.29</td><td>64.87±0.32</td><td>54.74±0.62</td></tr><tr><td>SCN [47]</td><td>20</td><td>80.02±0.32</td><td>66.30±0.49</td><td>56.35±0.61</td></tr><tr><td>DUL [4]</td><td>20</td><td>81.95±0.32</td><td>65.55±0.31</td><td>56.25±0.09</td></tr><tr><td>RUL</td><td>20</td><td>84.34±0.29</td><td>67.86±0.35</td><td>58.36±0.28</td></tr><tr><td>Baseline</td><td>30</td><td>75.12±0.78</td><td>62.52±0.56</td><td>51.46±0.52</td></tr><tr><td>SCN [47]</td><td>30</td><td>77.46±0.64</td><td>62.61±0.82</td><td>52.60±0.86</td></tr><tr><td>DUL [4]</td><td>30</td><td>78.90±0.80</td><td>60.98±0.34</td><td>55.09±0.32</td></tr><tr><td>RUL</td><td>30</td><td>82.06±0.44</td><td>64.62±0.39</td><td>56.65±0.13</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr></table>
+Table 2: Test accuracy $( \% )$ versus rejection rate results of different uncertainty learning methods on RAF-DB and FER2013. The best performance among each column is shown in bold form.
+<table><tr><td rowspan="2">Method</td><td colspan="4">RAF-DB</td><td colspan="4">FER2013 Private Testset</td></tr><tr><td>0rejection</td><td>10% rejection</td><td>20% rejection</td><td>30% rejection</td><td>0rejection</td><td>10% rejection</td><td>20% rejection</td><td>30% rejection</td></tr><tr><td>SCN [47]</td><td>87.35</td><td>86.85</td><td>86.63</td><td>87.28</td><td>72.67</td><td>72.82</td><td>74.29</td><td>75.80</td></tr><tr><td>CONF [6]</td><td>88.30</td><td>91.78</td><td>93.97</td><td>95.90</td><td>72.69</td><td>73.00</td><td>73.11</td><td>72.97</td></tr><tr><td>DUL [4]</td><td>88.04</td><td>90.11</td><td>92.58</td><td>94.50</td><td>73.67</td><td>76.01</td><td>79.10</td><td>82.25</td></tr><tr><td>RUL</td><td>88.98</td><td>92.72</td><td>95.40</td><td>97.35</td><td>73.75</td><td>77.24</td><td>80.98</td><td>84.75</td></tr></table>
+# 4.4 Test with Rejection.
+In order to illustrate the quality of our learned uncertainty values more intuitively, we utilize an evaluation metric similar to [41, 2, 15, 40] named accuracy versus rejection rate. The metric shows a test accuracy over the fraction of unconsidered facial expression images. Specifically, based on the predicted uncertainty values, these unconsidered images are the most uncertain images to recognize, and the test accuracy is calculated on the remaining images. The metric indicates good uncertainty estimation when the test accuracy increases consistently when increasing the ratio of unconsidered images. We also compare with the out-of-distribution detection method [6] noted as CONF. CONF uses labels as hints to help indicate when the network tends to give wrong predictions. We show the test accuracy in Table 2.We also plot the accuracy versus rejection rate curve in the supplementary material. RUL gets the best performance in all cases, which illustrates that the uncertainty values learned by RUL are more related to the recognition confidence than other methods. Another interesting finding is that CONF performs pretty well on RAF-DB while degrades on FER2013. It may be explained by the following reason that we use pretrained model on RAF-DB while training from scratch on FER2013. CONF needs a strong backbone to make good use of the information from labels while our method can perform well in both ways.
+# 4.5 Understand learned uncertainty values.
+In this section, we visualize the learned features of different uncertainty learning methods and provide qualitative and quantitative analysis to show the meaning of our learned uncertainty values.
+Visualization of learned features. We use t-SNE [44] to visualize the learned feature distributions of different uncertainty learning methods to show the effectiveness of RUL. The results are shown in Figure 3. It is shown that the comparison of different facial expressions encourages intra-class compactness and inter-class seperability of the learned features. We believe that this is because RUL needs to recognize both expressions from the mixed feature, and it will be forced to learn the most discriminative feature that can tell an expression image apart from all the other expression images compared with it. We plot the uncertainty distribution of images with high uncertainty and images with low uncertainty seperately in the supplementary material. The results show that the samples with high uncertainty learned by RUL congregate at the center of the figure. They tend to contain ambiguous expressions like neutral (class 6), which can be easily confused with other expressions. However, the samples with small uncertainty are far from each other which are very easy to be recognized. The feature visualization demonstrates that RUL can learn feature distribution which reflects the uncertainty distribution of the dataset.
+![](images/98527bb515e01b3103c175347f1861e4367ba73e31e87c21f3c94d6f1a6ddfbd.jpg)
+Figure 3: The learned feature distribution of SCN, DUL and RUL on RAF-DB dataset.(0:Surprise, 1:Fear, 2:Disgust, 3:Happy, 4:Sad, 5:Angry, 6:Neutral)
+Visualization of uncertainty values. We display some expression images in testset with their estimated uncertainty values in Figure 4. The first row shows that images with large uncertainty values are more likely to contain ambiguous expressions. They are easy to be misclassified (The wrong predicted labels are marked at the bottom of the first row images in red color). In contrast, the second row shows that images with small uncertainty values tend to contain obvious facial features, and they are easy to be rightly classified. This illustrates that our learned uncertainty values can represent the difficulty of expression images as well as the model’s confidence of its predictions. That property brings us benefits that are important for the development of safe AI systems. If the model knows when it is prone to be wrong then it can turn to human for help, which might avoid disasters in fields with high accuracy requirements.
+![](images/01832afead1c1a2c8c1d4d6918a0ce70f2408fe148735c6ff7419b6c6ee21171.jpg)
+Figure 4: The learned uncertainty values for different images. Images with large and small uncertainty values are shown in the first and second rows respectively. Uncertainty values are marked in the upper left corner, and predicted labels are marked at the bottom. Note that RUL learns large uncertainty values for images with ambiguous expressions while small uncertainty values for images with obvious expressions.
+Distribution of uncertainty values. RUL can learn different uncertainty distributions based on different datasets instead of constraining uncertainty distributions of all datasets to the diagonal multivariate normal Gaussian distribution like DUL. We show the learned uncertainty distributions of different datasets in Figure 5 and Figure 6. In order to make a comparison, we normalize RUL learned uncertainty values to [0,1]. Note that larger uncertainty values mean the images are more uncertain. We cut off at 400 on the y-axis in Figure 5 as SCN learns too many uncertainty values close to 0, which indicates bad performance. The results demonstrate that RUL learns large uncertainty values for more samples while the other two methods learn relative small uncertainty values (smaller than 0.4 on RAF-DB and smaller than 0.6 on FER2013) for all samples, which means RUL can reduce the negative influence from the strong learning ability of networks mentioned in Section 3.1. Furthermore, we also randomly sample several images which are hard for human to recognize and find that only RUL learns large uncertainty values for them.
+![](images/e06ac79931055ee3030c90c7d1abc3216d797a82699964c978f4005cc8c96a52.jpg)
+Figure 5: The learned uncertainty distribution of RAF-DB. Uncertainty values are marked at the top of the image, different colors represent different methods.
+![](images/a12c8dc28d5a8776c35eb56723a7ab06e83759d88b550e74370030e82d9db42e.jpg)
+Figure 6: The learned uncertainty distribution of FER2013. Uncertainty values are marked at the top of the image, different colors represent different methods.
+Table 3: Comparison on RAF-DB
+<table><tr><td>Method</td><td>Acc.(%)</td></tr><tr><td>DLP-CNN [27]</td><td>84.22</td></tr><tr><td>gaCNN [29]</td><td>85.07</td></tr><tr><td>IPA2LT[52]</td><td>86.77</td></tr><tr><td>RAN [48]</td><td>86.90</td></tr><tr><td>SCN [47]</td><td>87.03</td></tr><tr><td>RUL</td><td>88.98</td></tr></table>
+Table 4: Comparison on FER2013
+<table><tr><td>Method</td><td>Acc.(%)</td></tr><tr><td>Conv+Inception [32]</td><td>66.40</td></tr><tr><td>Bag of Words [21]</td><td>67.40</td></tr><tr><td>Deep-Emotion [31]</td><td>70.02</td></tr><tr><td>VGG [35]</td><td>72.70</td></tr><tr><td>SCN [47]</td><td>72.67</td></tr><tr><td>RUL</td><td>73.75</td></tr></table>
+Table 5: Comparison of different structures.
+<table><tr><td>Method</td><td>Acc.(%)</td></tr><tr><td>mix output loss attention RUL</td><td>88.36 88.75 88.98</td></tr><tr><td>Method</td><td>10% rejection</td></tr><tr><td>mix output loss attention RUL</td><td>92.17 89.32 92.72</td></tr></table>
+Table 6: The test accuracy versus rejection rate of different uncertainty learning methods on CIFAR-10 and ImageNet-100.
+<table><tr><td rowspan="2">Method</td><td colspan="4">CIFAR-10</td><td colspan="4">ImageNet-100</td></tr><tr><td>0rejection</td><td>10% rejection</td><td>20% rejection</td><td>30% rejection</td><td>0 rejection</td><td>10% rejection</td><td>20% rejection</td><td>30% rejection</td></tr><tr><td>CONF [6]</td><td>93.47</td><td>96.84</td><td>98.90</td><td>99.56</td><td>76.48</td><td>76.46</td><td>76.36</td><td>76.21</td></tr><tr><td>SCN [47]</td><td>95.37</td><td>94.87</td><td>94.24</td><td>94.14</td><td>75.49</td><td>74.61</td><td>74.49</td><td>74.99</td></tr><tr><td>DUL [4]</td><td>95.24</td><td>98.16</td><td>99.22</td><td>99.49</td><td>76.03</td><td>75.80</td><td>79.64</td><td>82.86</td></tr><tr><td>RUL</td><td>95.58</td><td>98.60</td><td>99.34</td><td>99.44</td><td>77.74</td><td>77.65</td><td>82.48</td><td>86.74</td></tr></table>
+# 4.6 Comparison with state-of-the-art FER methods.
+Although we aim to deal with uncertainty learning problems in FER, our RUL method still gets better test accuracy comparing to a series of state-of-the-art FER methods according to the results in Table 3 and Table 4.
+Note that though RUL uses no extra data for co-training, it still achieves better performance. We speculate that this is because the comparison of different FER images in the training process implicitly introduces the data augmentation effect. Meanwhile, as uncertain images will get larger weights to mix with certain images, the FER model will pay more attention to learning the uncertain images after it fits the certain images, which improves the model’s ability to recognize difficult facial expressions. We provide the confusion matrix of different methods on RAF-DB in the supplementary material. As mentioned by [23], the FER model tends to make more mistakes in classes with fewer samples, such as disgust and fear. However, the confusion matrices show that RUL increases test accuracy mainly in these two classes, which means RUL can better recognize difficult facial expressions.
+We also replace the Gaussian variances which are the learned uncertainty values in DUL with RUL learned uncertainty values to guide the DUL feature learning branch. Though RUL does not constrain the uncertainty distribution to a Gaussian distribution, it still improves the performance of DUL from $8 8 . 1 7 \%$ to $8 8 . 7 5 \%$ , which means the uncertainty learned by RUL can better reflect the uncertainty distribution of the dataset.
+# 4.7 Other Experiments.
+Experiments on CIFAR-10 and ImageNet-100. To show RUL is effective not only on FER tasks, we also carry out experiments on CIAFR-10 and ImageNet-100. Table 6 shows that RUL could still improve the performance of the deep learning model on other classification tasks. The test accuracy increases along with the rejection of the most uncertain test samples, which illustrates that RUL can still learn meaningful uncertainty values on datasets besides FER tasks.
+Ablation study. In order to show that the mixed feature part is useful for uncertainty learning, we implement a mix output method and a loss attention method to make a comparison. Specifically, we do not mix facial features in the mix output method. Instead, we classify two images separately and mix the logits according to uncertainty values. Results are shown at Table 5. RUL performs better than mix output in both the test accuracy and accuracy with $10 \%$ rejection, reflecting that mixed features may contain more relativity information than mixed logits. Loss attention method means we skip the mix part and use the uncertainty values to directly weight the loss values. Table 5 shows that RUL outperforms the loss attention method, which illustrates that RUL is effective not only because uncertainty values affect loss values but also due to the interaction of different facial features during the training process.
+# 5 Conclusion
+In this paper, we propose a novel and effective uncertainty learning method called Relative Uncertainty Learning for uncertainty quantification and noisy label training. Unlike traditional uncertainty learning methods, which use uncertainty values to weigh the loss and then add regularization to force the network to learn uncertainty, we view uncertainty as a relative concept and learn uncertainty through the relativity of different samples. Extensive experiments have shown that RUL can learn meaningful uncertainty values, which reflect the difficulty of samples to be rightly classified. RUL can also help the FER model achieve state-of-the-art performance in both real-world and synthetic noisy FER datasets. We further demonstrate that RUL is also useful in other classification tasks.
+# Acknowledgments and Disclosure of Funding
+This work was supported by the National Natural Science Foundation of China under Grants No. 61871052
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+        "type": "text",
+        "text": "Relative Uncertainty Learning for Facial Expression Recognition ",
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+        "text": "Yuhang Zhang, Chengrui Wang, Weihong Deng Beijing University of Posts and Telecommunications zyhzyh@bupt.edu.cn, crwang@bupt.edu.cn, whdeng@bupt.edu.cn ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "In facial expression recognition (FER), the uncertainties introduced by inherent noises like ambiguous facial expressions and inconsistent labels raise concerns about the credibility of recognition results. To quantify these uncertainties and achieve good performance under noisy data, we regard uncertainty as a relative concept and propose an innovative uncertainty learning method called Relative Uncertainty Learning (RUL). Rather than assuming Gaussian uncertainty distributions for all datasets, RUL builds an extra branch to learn uncertainty from the relative difficulty of samples by feature mixup. Specifically, we use uncertainties as weights to mix facial features and design an add-up loss to encourage uncertainty learning. It is easy to implement and adds little or no extra computation overhead. Extensive experiments show that RUL outperforms state-of-the-art FER uncertainty learning methods in both realworld and synthetic noisy FER datasets. Besides, RUL also works well on other datasets such as CIFAR and Tiny ImageNet. The code is available at https://github.com/zyh-uaiaaaa/Relative-Uncertainty-Learning. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Although well-trained models can give high-confidence inferences directly, the inference results may actually be wrong and cause serious consequences [22]. Therefore, the concept of uncertainty is widely used in measuring how well the model trusts its inference results, which has become a research hotspot in trustworthy machine learning [3, 11, 10, 22, 26, 16]. It is expected that a reliable model assigns high level uncertainty to its erroneous predictions so that humans can intervene to avoid lots of disasters. As described by Kiureghian et al. [25], the uncertainties in machine learning can be split into aleatoric uncertainty and epistemic uncertainty, which are also called data uncertainty and model uncertainty. In this paper, we focus on quantifying data uncertainty as model uncertainty can be eliminated by introducing more training data. Concretely, data uncertainty in facial expression recognition (FER) is mainly caused by ambiguous facial expressions and the subjectiveness of annotators, which blocks the improvement of recognition performance [47]. ",
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+        "text": "Recent studies propose a variety of methods to solve the uncertainty learning problem in the face recognition field. PFE [37] introduces a new branch to learn the uncertainty of face recognition and recognizes faces by measuring the similarity between two Gaussian distributions. Chang et al. [4] further propose DUL to learn feature and uncertainty simultaneously in order to improve facial feature learning. When it comes to FER, SCN [47] uses a fully-connected layer to learn an importance weight for each image and suppresses uncertainties according to the learned weights. However, the strong learning ability of neural networks will deteriorate the uncertainty learning branch. One solution is adding regularization to force the model to learn uncertainty, which needs to be tuned carefully, otherwise, it would lead to underfitting. For example, SCN uses margin loss to keep the gap between certain and uncertain images, but choosing a suitable gap value is non-trivial. DUL assumes features to follow a Gaussian distribution, but the real distribution of the dataset is not always following such an assumption. Therefore, it is still a challenge to learn uncertainty distributions of real datasets without deteriorating feature learning. ",
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+        "text": "To address the problem, we perceive data uncertainty from a different point of view. We realize that data uncertainty – the difficulty to classify a sample correctly is actually a relative concept. More specifically, the difficulty of classification is a person’s subjective feeling that comes from comparison. One can only know whether a sample is easy or not according to a reference. Inspired by this idea, we propose an innovative method called Relative Uncertainty Learning (RUL) to help deep learning models to learn uncertainty for each sample. Concretely, we build a new branch to model the uncertainties of input images and utilize the uncertainties as weights to mix two features of different labels. Through an add-up loss function, the model is encouraged to recognize two expressions simultaneously from the mixed features, which enables the model to learn uncertainty comparatively while minimizing the total loss. As discussed in Section 3.3, the uncertainty learning branch will assign large uncertainty values to uncertain facial expression images while small uncertainty values to certain images. ",
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+        "text": "The main contributions of this work are as follows: ",
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+        "text": "• We perceive uncertainty from a new perspective and propose an innovative uncertainty learning method to learn uncertainty from the relative difficulty of two samples. • We get state-of-the-art performance in both real-world and synthetic noisy FER datasets. • RUL does not need prior knowledge of the dataset uncertainty distribution and can be easily applied to different classification tasks with low computation cost. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "text": "Uncertainty learning in the face field Uncertainties can be mainly categorized into model uncertainty which accounts for uncertainty in the model parameters and data uncertainty which is the noise inherent in the dataset [22]. In the face recognition field, uncertainty learning is important for learning discriminative feature embeddings of noisy face images. Because noisy face images are usually out of the cluster and have large variances in the latent embedding space, which might cause wrong recognition [37]. There have been several proposals to solve the uncertainty learning problem in face recognition tasks [13, 51, 23, 37, 4]. Some of them [13, 51, 23] have tried to analyze and learn face representations using model uncertainty. Gong et al. [13] estimate the capacity of a given face representation by explicitly accounting for the manifold structure, model uncertainty and data variability. Khan et al. [23] present a framework for class imbalance learning based on model uncertainty. Zafar et al. [51] cope with false positives through employing model uncertainty to improve the efficacy of face recognition systems. Thereinto the methods considering data uncertainty learning. Shi et al. [37] use PFE to learn a variance for each fixed feature and then measure the likelihood of each positive face pair of $( x _ { i } , x _ { j } )$ sharing the same distribution of latent embedding $\\mathsf { p } ( z _ { i }$ $z _ { j } .$ ). Chang et al. [4] further propose to learn feature and uncertainty simultaneously by encoding the latent feature embedding as a Gaussian distribution. In the facial expression recognition task, Wang et al. [47] try to learn an attention weight for each image, then utilize it to weigh cross-entropy loss in order to suppress the influence of noisy samples. She et al. [36] propose to use several branches to model the latent label distribution of facial expression images and use cosine similarity to capture the uncertain images. ",
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+        "text": "Noisy dataset training With the enhancement of deep learning network capabilities, the datasets for facial expression recognition (FER) are getting larger and larger. For the large-scale FER datasets [43, 54, 7, 1, 33, 9, 27], due to the ambiguity of expressions and the subjectiveness of human annotators, it is extremely hard to get high-quality labels. Training with label noise has been studied for a long time. The basic idea is to enable the model to access the quality of the labels or estimate the noise distribution using a small set of clean dataset [5, 28, 39, 45]. Li et al. [28] design a distillation framework to use information from both a small clean dataset and label relations in knowledge graphs to prevent the model learning from noisy labels. Veit et al. [45] use a small amount of clean annotations to reduce the noisy samples before fine-tuning the network using both the clean set and the full set. Other methods do not use a small clean dataset, but usually assume extra constraints on the noisy samples. Natarajan et al. [34] design a specific loss function for randomly flipped labels. Goldberger et al. [12] explicitly model the noises in datasets by a softmax layer which connects the correct labels to the noisy ones. Without collecting a clean set of dataset or using prior knowledge of noise distribution, our method learns uncertainty through the relativity of the given dataset and suppresses the most uncertain samples to get a good performance under label noise. ",
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+        "image_caption": [
+            "Figure 1: Overview of our Relative Uncertainty Learning method. We use two branches to learn facial features $\\mu$ and uncertainty vectors $\\sigma$ simultaneously. Then we normalize uncertainty vectors with their counterparts sampled from shuffled indexes. We compute the mean of the uncertainty vectors as the uncertainty values. Next, we mix the facial features according to the mean uncertainty values. Note that the easy sample will get a small uncertainty value because its facial feature is very obvious for recognition. In contrast, the ambiguous sample tends to get a large uncertainty value in order to make it easy for the model to recognize its corresponding label from the mixed features. Finally, we use the add-up loss to encourage the model to recognize the two expressions from the mixed feature, which implicitly forces the model to learn reliable uncertainty values. "
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+        "text": "3 Proposed Method ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to reduce the negative impact of the strong learning ability of neural networks and learn uncertainty values closer to the real dataset uncertainty distribution, we propose a simple and effective method named Relative Uncertainty Learning (RUL). In this section, we first provide the motivation behind our method. Then we demonstrate the details of RUL. Finally, we explain why RUL is effective in uncertainty learning. ",
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+    {
+        "type": "text",
+        "text": "3.1 Motivation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The uncertainty learning methods mentioned before suffer from the strong learning ability of deep neural networks more or less. For example, the introduced variances in DUL are equivalent to some noises being added to feature points, so the model will learn very small variances for all feature points making the estimation of uncertainty less reliable. SCN also learns large importance weights for noisy samples to obtain small cross-entropy loss values as these noisy images will be memorized by the networks after training for several epochs. ",
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+        "type": "text",
+        "text": "In order to prevent the strong learning ability of neural networks from degrading the uncertainty learning branch, we perceive uncertainty from a relative point of view. We find that learning uncertainty through comparison forms a natural regularization because if the model assigns a very small uncertainty value for one image of the comparison pair, then it means the counterpart image gets a large uncertainty value which prevents the model learning small uncertainty values for all images. Learning uncertainty through relativity also conforms to human cognition – it is very difficult for a person to define which sample is certain and which sample is uncertain without using one image as a reference. In the following section, we thoroughly illustrate the details of our proposed RUL. ",
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+    {
+        "type": "text",
+        "text": "3.2 Overview of Relative Uncertainty Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we illustrate the structure of our proposed RUL. The pipeline of RUL is shown in Figure 1. ",
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+        "type": "text",
+        "text": "Two Branches Model Given a batch of face images $\\boldsymbol { X }$ , assuming the conventional feature extraction module $f _ { \\theta } ( X )$ can be decomposed as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f55df34872a4efae04f230a3965856d9eacd9f6bacef94c4607313d01ea87637.jpg",
+        "text": "$$\nf _ { \\theta } ( \\boldsymbol { X } ) = f _ { \\theta } ^ { ( l ) } ( f _ { \\theta } ^ { ( l - 1 ) } ( \\boldsymbol { X } ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "we first extract general facial features $F _ { g }$ from the penultimate feature extraction layer denoted as $\\pmb { F } _ { g } = f _ { \\theta } ^ { ( l - 1 ) } ( \\pmb { X } )$ [50]. Then, in order to learn facial features and uncertainty values simultaneously, we drop the layer indexed by $l$ , i.e., $f _ { \\theta } ^ { ( l ) }$ and build two separate branches $f _ { \\theta _ { \\mu } } ( f _ { \\theta } ^ { ( l - 1 ) } ( X ) )$ and $f _ { \\theta _ { \\sigma } } ( f _ { \\theta } ^ { ( l - 1 ) } ( { \\pmb X } ) )$ to produce facial features $\\pmb { F } = [ \\pmb { \\mu } _ { 1 } , \\pmb { \\mu } _ { 2 } , . . . , \\pmb { \\mu } _ { N } ] \\in \\mathbb { R } ^ { D \\times N }$ and uncertainty vectors $U = [ \\pmb { \\sigma } _ { 1 } , \\pmb { \\sigma } _ { 2 } , . . . , \\pmb { \\sigma } _ { N } ] \\in \\mathbb { R } ^ { D \\times N } . \\ D , N$ denote output dimension and batch size separately. ",
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+        "text": "Relative Uncertainty Learning Inspired by the relativity of the uncertainty concept and the mixup method [53, 42, 46], we mix two different facial features according to their uncertainty values which enables the FER model to learn uncertainty through the relativity of different samples. First, we mix a mini-batch with their shuffled ones which results in the facial images of the whole dataset being compared with each other during the training process because of the randomness introduced by the dataloader and the shuffle of mini-batches. Before mixing features, we first normalize the uncertainty vectors element-wise to compare two features with each other following Equation (2) ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/15f0df4845def8bfb6cfb2b3b76f01229115de69df4862700b17da503fec78fa.jpg",
+        "text": "$$\n\\hat { \\pmb { \\sigma } } _ { i } , \\hat { \\pmb { \\sigma } } _ { j } = \\frac { \\pmb { \\sigma } _ { i } } { \\pmb { \\sigma } _ { i } + \\pmb { \\sigma } _ { j } } , \\frac { \\pmb { \\sigma } _ { j } } { \\pmb { \\sigma } _ { i } + \\pmb { \\sigma } _ { j } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$\\sigma _ { i }$ and $\\pmb { \\sigma } _ { j }$ are the uncertainty vectors for image $i$ and its counterpart image $j$ sampled from the shuffled indexes, note that they have different labels. This operation not only forms a natural regularization as the model can not learn small uncertainty values for both features, but also benefits the training process as extreme uncertainty values might make the training process unstable or even the loss can not converge. As mentioned by DUL [4], the mean of the predicted uncertainty vectors $\\hat { \\pmb { \\sigma } } _ { i } \\in \\mathbb { R } ^ { D }$ can be viewed as an approximated measurement of the estimated uncertainty for image $i$ , we also use the mean of $\\hat { \\pmb { \\sigma } } _ { i }$ as learned uncertainty value for image $i$ denoted as $\\hat { \\sigma } _ { i _ { m e a n } } \\in \\mathbb { R }$ . Then we get an uncertainty value for each input image denoted as $\\pmb { U } = [ \\hat { \\sigma } _ { 1 _ { m e a n } } , \\hat { \\sigma } _ { 2 _ { m e a n } } , . . . , \\hat { \\sigma } _ { N _ { m e a n } } ] \\in \\mathbb { R } ^ { 1 \\times N }$ We then mix features according to their uncertainty values following Equation (3) ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f161e344dba45c0e204a522241be65641ae75a5588785d77f0b641ab30e8c6c.jpg",
+        "text": "$$\n\\tilde { \\mu } = \\hat { \\sigma } _ { i _ { m e a n } } \\mu _ { i } + \\hat { \\sigma } _ { j _ { m e a n } } \\mu _ { j } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "$\\pmb { \\mu _ { i } }$ and $\\mu _ { j }$ represent embedding features of the two images, $\\hat { \\sigma } _ { i _ { m e a n } }$ and σˆjmea represent their corresponding uncertainty values. We introduce negligible computation overhead as we only mix facial features according to their learned uncertainty values in the training phase. Furthermore, the mix feature part will be removed for deployment. Therefore, our network is end-to-end and adds no extra cost on inference. ",
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+        "text": "Classification Loss We design an add-up loss to encourage the uncertainty learning branch to learn different uncertainty values for different facial images. Specifically, add-up loss requires the FER model to recognize two expressions of input images simultaneously from mixed features. Since $\\tilde { \\mu }$ is the mixed feature, we feed $\\tilde { \\mu }$ to the classifier: ",
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+    {
+        "type": "equation",
+        "img_path": "images/a299abf204bcce2be61cb46767e5cfd655172e5adf7abf8dd89af0cb600eac04.jpg",
+        "text": "$$\nL _ { t o t a l } = - \\frac { 1 } { N } \\sum _ { i , j } ^ { N } ( \\log \\frac { e ^ { W _ { y _ { i } } \\tilde { \\mu } } } { \\sum _ { c } ^ { C } e ^ { W _ { c } \\tilde { \\mu } } } + \\log \\frac { e ^ { W _ { y _ { j } } \\tilde { \\mu } } } { \\sum _ { c } ^ { C } e ^ { W _ { c } \\tilde { \\mu } } } ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The loss function adds up the loss of recognizing $\\tilde { \\mu }$ as class $i$ and as class $j$ . $y _ { i } , y _ { i }$ means $l a b e l _ { i }$ , $l a b e l _ { j }$ , $W _ { c }$ is the $\\mathrm { c }$ -th classifier and $C$ means the total number of expression classes. The loss function forces the model to recognize two expressions equally from the mixed features. When mixing two facial features, there will be a relatively easy facial feature for expression recognition, and the other will be relatively hard. For simplicity, we note $\\pmb { \\mu } _ { i }$ as the easy facial feature. Our model will learn a small uncertainty value for $\\pmb { \\mu _ { i } }$ to mix with $\\mu _ { j }$ as $\\pmb { \\mu _ { i } }$ is easy to recognize, the model can already get a small classification loss with its label $y _ { i }$ after training for several epochs, even $\\pmb { \\mu } _ { i }$ only takes up a small amount of the mixed feature $\\tilde { \\mu }$ . While for the hard image $j$ , it needs to take up a large amount of the mixed feature $\\tilde { \\mu }$ to enable the model to find useful feature to get a small classification loss with label $y _ { j }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/bb86495aca805b5ec9de1ef69062e89d9f0ffe5cb3cc1b0fda07d46f5dc5930d.jpg",
+        "image_caption": [
+            "Figure 2: An experiment to show why RUL is effective in uncertainty learning. We mix different features with different weights and print the losses of recognizing the mixed feature to different labels. Visually, the face with anger expression is the certain one, and the face with sad expression is the uncertain one. In the process of feature mixup, RUL will assign a larger weight to the uncertain one to minimize the total loss. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.3 Why RUL is Effective ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we illustrate why RUL is effective in uncertainty learning. For visualization simplicity, we plot the mixed images instead of the mixed features in Figure 2. At the start of the training, RUL might learn similar uncertainty values for all images, which is shown in the left part of Figure 2. As the certain facial expression image with anger label in Figure 2 can be easily recognized from the mixed feature, the model can already get a small loss value with anger label while gets a large loss value with a sad label. In order to get the minimum total loss, the model needs to recognize the sad expression from the mixed feature more effectively. As training progress, the mix weight (uncertainty value) of the uncertain sad image will increase to enable the model to recognize the sad expression from the mixed feature. Thus, RUL will learn larger uncertainty values for uncertain images. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we first describe three in-the-wild FER datasets and our implementation details. We then evaluate our proposed method on these datasets. We compare our method with state-of-the-art uncertainty learning methods on the accuracy, noisy label training, and accuracy with rejection [41]. Furthermore, we provide qualitative and quantitative analysis to show what is the meaning of our learned uncertainty values. Last, we compare our method to other state-of-the-art FER methods and also demonstrate that RUL is applicable to datasets in different fields. We also carry out an ablation study to find the effectiveness of the mix part in RUL. ",
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+    {
+        "type": "text",
+        "text": "4.1 Datasets ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To evaluate the effectiveness of our model, we select three widely used in-the-wild FER datasets. Compared with FER datasets collected in the laboratory, they have more noises like noisy labels and ambiguous expressions. So they are more complex, variable, and challenging for expression recognition tasks. The details are as follows. ",
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+    {
+        "type": "text",
+        "text": "RAF-DB [27] is a crowdsourced facial expression dataset that contains 29672 facial images annotated with basic or compound expressions by 40 trained annotators. In this work, we use facial images with 6 basic expressions [8] and neutral expression, including 12271 images as training data and 3068 images as test data. ",
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+    {
+        "type": "text",
+        "text": "FER2013 [14] consists of 35,887 grayscale $4 8 \\mathbf { x } 4 8$ pixel images in total, with 28,709 training samples, 3,589 public test samples, and 3,589 private test samples. These facial images are classified into 7 emotion classes, same as RAF-DB. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "AffectNet [33] is currently the largest FER dataset, including 440,000 images. The images are collected from the Internet by querying the major search engines with 1,250 emotion-related keywords. There are around 280,000 training images and 4000 testing images annotated by human. They are annotated to 8 emotion classes (7 emotion classes same as RAF-DB and Contempt class). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given a batch of images from RAF-DB, we first resize them to $2 2 4 \\times 2 2 4$ pixels and use ResNet18 [18], which is pretrained on Ms-Celeb-1M [17] as backbone. The two branches have the same structure as BatchNorm2d [20]-Dropout [38]-Flatten-FC-BatchNorm1d. We set dropout rate as 0.4, output dimension as 64. Note that the second branch outputs $\\log \\sigma$ instead of $\\pmb { \\sigma }$ in order to stabilize the training following [4]. The model is trained in an end-to-end manner with a single GTX 1080ti GPU for 70 epochs with batch size of 64. We also utilize an Adam optimizer [24] with weight decay of 0.0001. The learning rate is initialized as 0.0002 except the last fully connected layer for classification, which is 0.002. We use ExponentialLR [30] learning rate scheduler with gamma of 0.9 to decrease the learning rate after each epoch. The setting under AffectNet dataset is the same as RAF-DB. As for FER2013, we use the same setting as [49] as the image in FER2013 is $4 8 \\mathbf { x } 4 8$ pixel grayscale image. ",
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+    {
+        "type": "text",
+        "text": "4.3 Evaluation on Label Noises ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Both ambiguous facial expressions and inconsistent annotations will cause label noises in FER datasets. We carry out extensive experiments to demonstrate the improvement of RUL compared with state-of-the-art noise-tolerant FER methods. Following SCN [47], we randomly choose $10 \\%$ , $20 \\%$ , $30 \\%$ of training data and flip their labels to other expression categories randomly. To make a fair comparison with SCN, we also consider the most uncertain samples in the training process as samples with label noises and if the maximum prediction probability is higher than the one of the given label with a threshold (set to 0.2 in all cases), we choose to rely on the FER model and change the label to the index of the maximum prediction probability. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We run all experiments three times and compute the mean and standard variance of the results. As shown in Table 1, RUL outperforms SCN and DUL under all circumstances. RUL improves test accuracy by $4 . 3 0 \\%$ , $4 . 3 2 \\%$ , $4 . 6 0 \\%$ with noise ratio of $10 \\%$ , $20 \\%$ and $30 \\%$ on RAF-DB, $1 . 0 1 \\%$ , $1 . 5 6 \\%$ , $2 . 0 1 \\%$ on FER2013 and $1 . 9 4 \\%$ , $2 . 0 1 \\%$ , $4 . 0 5 \\%$ on AffectNet compared with SCN. Experiment results show that the benefit from RUL becomes more obvious as the noise ratio increases up which means RUL is more robust to label noises. Note that we do not use pretrained model when training in FER2013 to find how the pretrained model affects the experiment results. From the results we can draw the conclusion that using pretrained model can better deal with label noise as RUL improves the accuracy most when using pretrained models which aligns to the finding in [19]. The results also demonstrate that RUL can learn meaningful uncertainty values as we use uncertainty values to guide the model to find the most uncertain training samples and suppress label noises. ",
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+    {
+        "type": "table",
+        "img_path": "images/19de574506c77b23a72f2f7afbe6feb132e4e91f09244e2a3f6d3f2ffa5945fb.jpg",
+        "table_caption": [
+            "Table 1: Test accuracy $( \\% )$ on RAF-DB, FER2013 and AffectNet with synthetic noisy labels. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Noisy (%)</td><td>RAF-DB</td><td>FER2013</td><td>AffectNet</td></tr><tr><td>Baseline</td><td>10</td><td>80.43±0.72</td><td>69.25±0.18</td><td>56.85±0.14</td></tr><tr><td>SCN [47]</td><td>10</td><td>81.92±0.69</td><td>69.28±0.05</td><td>58.72±0.20</td></tr><tr><td>DUL [4]</td><td>10</td><td>85.08±0.21</td><td>69.43±0.27</td><td>58.26±0.10</td></tr><tr><td>RUL</td><td>10</td><td>86.22±0.29</td><td>70.29±0.28</td><td>60.66±0.13</td></tr><tr><td>Baseline</td><td>20</td><td>78.01±0.29</td><td>64.87±0.32</td><td>54.74±0.62</td></tr><tr><td>SCN [47]</td><td>20</td><td>80.02±0.32</td><td>66.30±0.49</td><td>56.35±0.61</td></tr><tr><td>DUL [4]</td><td>20</td><td>81.95±0.32</td><td>65.55±0.31</td><td>56.25±0.09</td></tr><tr><td>RUL</td><td>20</td><td>84.34±0.29</td><td>67.86±0.35</td><td>58.36±0.28</td></tr><tr><td>Baseline</td><td>30</td><td>75.12±0.78</td><td>62.52±0.56</td><td>51.46±0.52</td></tr><tr><td>SCN [47]</td><td>30</td><td>77.46±0.64</td><td>62.61±0.82</td><td>52.60±0.86</td></tr><tr><td>DUL [4]</td><td>30</td><td>78.90±0.80</td><td>60.98±0.34</td><td>55.09±0.32</td></tr><tr><td>RUL</td><td>30</td><td>82.06±0.44</td><td>64.62±0.39</td><td>56.65±0.13</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/10538db339ab83c07ee5149159224f1a84ab897b53832ea9caca73cf84babeeb.jpg",
+        "table_caption": [
+            "Table 2: Test accuracy $( \\% )$ versus rejection rate results of different uncertainty learning methods on RAF-DB and FER2013. The best performance among each column is shown in bold form. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">RAF-DB</td><td colspan=\"4\">FER2013 Private Testset</td></tr><tr><td>0rejection</td><td>10% rejection</td><td>20% rejection</td><td>30% rejection</td><td>0rejection</td><td>10% rejection</td><td>20% rejection</td><td>30% rejection</td></tr><tr><td>SCN [47]</td><td>87.35</td><td>86.85</td><td>86.63</td><td>87.28</td><td>72.67</td><td>72.82</td><td>74.29</td><td>75.80</td></tr><tr><td>CONF [6]</td><td>88.30</td><td>91.78</td><td>93.97</td><td>95.90</td><td>72.69</td><td>73.00</td><td>73.11</td><td>72.97</td></tr><tr><td>DUL [4]</td><td>88.04</td><td>90.11</td><td>92.58</td><td>94.50</td><td>73.67</td><td>76.01</td><td>79.10</td><td>82.25</td></tr><tr><td>RUL</td><td>88.98</td><td>92.72</td><td>95.40</td><td>97.35</td><td>73.75</td><td>77.24</td><td>80.98</td><td>84.75</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 Test with Rejection. ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to illustrate the quality of our learned uncertainty values more intuitively, we utilize an evaluation metric similar to [41, 2, 15, 40] named accuracy versus rejection rate. The metric shows a test accuracy over the fraction of unconsidered facial expression images. Specifically, based on the predicted uncertainty values, these unconsidered images are the most uncertain images to recognize, and the test accuracy is calculated on the remaining images. The metric indicates good uncertainty estimation when the test accuracy increases consistently when increasing the ratio of unconsidered images. We also compare with the out-of-distribution detection method [6] noted as CONF. CONF uses labels as hints to help indicate when the network tends to give wrong predictions. We show the test accuracy in Table 2.We also plot the accuracy versus rejection rate curve in the supplementary material. RUL gets the best performance in all cases, which illustrates that the uncertainty values learned by RUL are more related to the recognition confidence than other methods. Another interesting finding is that CONF performs pretty well on RAF-DB while degrades on FER2013. It may be explained by the following reason that we use pretrained model on RAF-DB while training from scratch on FER2013. CONF needs a strong backbone to make good use of the information from labels while our method can perform well in both ways. ",
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+        "type": "text",
+        "text": "4.5 Understand learned uncertainty values. ",
+        "text_level": 1,
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+        "text": "In this section, we visualize the learned features of different uncertainty learning methods and provide qualitative and quantitative analysis to show the meaning of our learned uncertainty values. ",
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+        "type": "text",
+        "text": "Visualization of learned features. We use t-SNE [44] to visualize the learned feature distributions of different uncertainty learning methods to show the effectiveness of RUL. The results are shown in Figure 3. It is shown that the comparison of different facial expressions encourages intra-class compactness and inter-class seperability of the learned features. We believe that this is because RUL needs to recognize both expressions from the mixed feature, and it will be forced to learn the most discriminative feature that can tell an expression image apart from all the other expression images compared with it. We plot the uncertainty distribution of images with high uncertainty and images with low uncertainty seperately in the supplementary material. The results show that the samples with high uncertainty learned by RUL congregate at the center of the figure. They tend to contain ambiguous expressions like neutral (class 6), which can be easily confused with other expressions. However, the samples with small uncertainty are far from each other which are very easy to be recognized. The feature visualization demonstrates that RUL can learn feature distribution which reflects the uncertainty distribution of the dataset. ",
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+        "image_caption": [
+            "Figure 3: The learned feature distribution of SCN, DUL and RUL on RAF-DB dataset.(0:Surprise, 1:Fear, 2:Disgust, 3:Happy, 4:Sad, 5:Angry, 6:Neutral) "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Visualization of uncertainty values. We display some expression images in testset with their estimated uncertainty values in Figure 4. The first row shows that images with large uncertainty values are more likely to contain ambiguous expressions. They are easy to be misclassified (The wrong predicted labels are marked at the bottom of the first row images in red color). In contrast, the second row shows that images with small uncertainty values tend to contain obvious facial features, and they are easy to be rightly classified. This illustrates that our learned uncertainty values can represent the difficulty of expression images as well as the model’s confidence of its predictions. That property brings us benefits that are important for the development of safe AI systems. If the model knows when it is prone to be wrong then it can turn to human for help, which might avoid disasters in fields with high accuracy requirements. ",
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+        "img_path": "images/01832afead1c1a2c8c1d4d6918a0ce70f2408fe148735c6ff7419b6c6ee21171.jpg",
+        "image_caption": [
+            "Figure 4: The learned uncertainty values for different images. Images with large and small uncertainty values are shown in the first and second rows respectively. Uncertainty values are marked in the upper left corner, and predicted labels are marked at the bottom. Note that RUL learns large uncertainty values for images with ambiguous expressions while small uncertainty values for images with obvious expressions. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Distribution of uncertainty values. RUL can learn different uncertainty distributions based on different datasets instead of constraining uncertainty distributions of all datasets to the diagonal multivariate normal Gaussian distribution like DUL. We show the learned uncertainty distributions of different datasets in Figure 5 and Figure 6. In order to make a comparison, we normalize RUL learned uncertainty values to [0,1]. Note that larger uncertainty values mean the images are more uncertain. We cut off at 400 on the y-axis in Figure 5 as SCN learns too many uncertainty values close to 0, which indicates bad performance. The results demonstrate that RUL learns large uncertainty values for more samples while the other two methods learn relative small uncertainty values (smaller than 0.4 on RAF-DB and smaller than 0.6 on FER2013) for all samples, which means RUL can reduce the negative influence from the strong learning ability of networks mentioned in Section 3.1. Furthermore, we also randomly sample several images which are hard for human to recognize and find that only RUL learns large uncertainty values for them. ",
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+        "type": "image",
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+        "image_caption": [
+            "Figure 5: The learned uncertainty distribution of RAF-DB. Uncertainty values are marked at the top of the image, different colors represent different methods. "
+        ],
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+    {
+        "type": "image",
+        "img_path": "images/a12c8dc28d5a8776c35eb56723a7ab06e83759d88b550e74370030e82d9db42e.jpg",
+        "image_caption": [
+            "Figure 6: The learned uncertainty distribution of FER2013. Uncertainty values are marked at the top of the image, different colors represent different methods. "
+        ],
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+    {
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+        "img_path": "images/6d1d68beff8464f4b80331f0abafb0e4bf8bd6e02f38b30b0b656074d5ad88db.jpg",
+        "table_caption": [
+            "Table 3: Comparison on RAF-DB "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Acc.(%)</td></tr><tr><td>DLP-CNN [27]</td><td>84.22</td></tr><tr><td>gaCNN [29]</td><td>85.07</td></tr><tr><td>IPA2LT[52]</td><td>86.77</td></tr><tr><td>RAN [48]</td><td>86.90</td></tr><tr><td>SCN [47]</td><td>87.03</td></tr><tr><td>RUL</td><td>88.98</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/695867461c1167eef2d1dc6574a7ceeb5c3ddfc3d353d37ac2c755b91570664c.jpg",
+        "table_caption": [
+            "Table 4: Comparison on FER2013 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Acc.(%)</td></tr><tr><td>Conv+Inception [32]</td><td>66.40</td></tr><tr><td>Bag of Words [21]</td><td>67.40</td></tr><tr><td>Deep-Emotion [31]</td><td>70.02</td></tr><tr><td>VGG [35]</td><td>72.70</td></tr><tr><td>SCN [47]</td><td>72.67</td></tr><tr><td>RUL</td><td>73.75</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/c656f1a63b6cede02d7bde69b17c805dbecfaebf9c8792b3f0bb3f1e1cf0437b.jpg",
+        "table_caption": [
+            "Table 5: Comparison of different structures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Acc.(%)</td></tr><tr><td>mix output loss attention RUL</td><td>88.36 88.75 88.98</td></tr><tr><td>Method</td><td>10% rejection</td></tr><tr><td>mix output loss attention RUL</td><td>92.17 89.32 92.72</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/87c8e9c7f648465ff4f400510661cca18ac3e11d85f60d10e1e25d7734c8772b.jpg",
+        "table_caption": [
+            "Table 6: The test accuracy versus rejection rate of different uncertainty learning methods on CIFAR-10 and ImageNet-100. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">CIFAR-10</td><td colspan=\"4\">ImageNet-100</td></tr><tr><td>0rejection</td><td>10% rejection</td><td>20% rejection</td><td>30% rejection</td><td>0 rejection</td><td>10% rejection</td><td>20% rejection</td><td>30% rejection</td></tr><tr><td>CONF [6]</td><td>93.47</td><td>96.84</td><td>98.90</td><td>99.56</td><td>76.48</td><td>76.46</td><td>76.36</td><td>76.21</td></tr><tr><td>SCN [47]</td><td>95.37</td><td>94.87</td><td>94.24</td><td>94.14</td><td>75.49</td><td>74.61</td><td>74.49</td><td>74.99</td></tr><tr><td>DUL [4]</td><td>95.24</td><td>98.16</td><td>99.22</td><td>99.49</td><td>76.03</td><td>75.80</td><td>79.64</td><td>82.86</td></tr><tr><td>RUL</td><td>95.58</td><td>98.60</td><td>99.34</td><td>99.44</td><td>77.74</td><td>77.65</td><td>82.48</td><td>86.74</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.6 Comparison with state-of-the-art FER methods. ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Although we aim to deal with uncertainty learning problems in FER, our RUL method still gets better test accuracy comparing to a series of state-of-the-art FER methods according to the results in Table 3 and Table 4. ",
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+    {
+        "type": "text",
+        "text": "Note that though RUL uses no extra data for co-training, it still achieves better performance. We speculate that this is because the comparison of different FER images in the training process implicitly introduces the data augmentation effect. Meanwhile, as uncertain images will get larger weights to mix with certain images, the FER model will pay more attention to learning the uncertain images after it fits the certain images, which improves the model’s ability to recognize difficult facial expressions. We provide the confusion matrix of different methods on RAF-DB in the supplementary material. As mentioned by [23], the FER model tends to make more mistakes in classes with fewer samples, such as disgust and fear. However, the confusion matrices show that RUL increases test accuracy mainly in these two classes, which means RUL can better recognize difficult facial expressions. ",
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+        "type": "text",
+        "text": "We also replace the Gaussian variances which are the learned uncertainty values in DUL with RUL learned uncertainty values to guide the DUL feature learning branch. Though RUL does not constrain the uncertainty distribution to a Gaussian distribution, it still improves the performance of DUL from $8 8 . 1 7 \\%$ to $8 8 . 7 5 \\%$ , which means the uncertainty learned by RUL can better reflect the uncertainty distribution of the dataset. ",
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+    {
+        "type": "text",
+        "text": "4.7 Other Experiments. ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Experiments on CIFAR-10 and ImageNet-100. To show RUL is effective not only on FER tasks, we also carry out experiments on CIAFR-10 and ImageNet-100. Table 6 shows that RUL could still improve the performance of the deep learning model on other classification tasks. The test accuracy increases along with the rejection of the most uncertain test samples, which illustrates that RUL can still learn meaningful uncertainty values on datasets besides FER tasks. ",
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+        "type": "text",
+        "text": "Ablation study. In order to show that the mixed feature part is useful for uncertainty learning, we implement a mix output method and a loss attention method to make a comparison. Specifically, we do not mix facial features in the mix output method. Instead, we classify two images separately and mix the logits according to uncertainty values. Results are shown at Table 5. RUL performs better than mix output in both the test accuracy and accuracy with $10 \\%$ rejection, reflecting that mixed features may contain more relativity information than mixed logits. Loss attention method means we skip the mix part and use the uncertainty values to directly weight the loss values. Table 5 shows that RUL outperforms the loss attention method, which illustrates that RUL is effective not only because uncertainty values affect loss values but also due to the interaction of different facial features during the training process. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we propose a novel and effective uncertainty learning method called Relative Uncertainty Learning for uncertainty quantification and noisy label training. Unlike traditional uncertainty learning methods, which use uncertainty values to weigh the loss and then add regularization to force the network to learn uncertainty, we view uncertainty as a relative concept and learn uncertainty through the relativity of different samples. Extensive experiments have shown that RUL can learn meaningful uncertainty values, which reflect the difficulty of samples to be rightly classified. RUL can also help the FER model achieve state-of-the-art performance in both real-world and synthetic noisy FER datasets. We further demonstrate that RUL is also useful in other classification tasks. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This work was supported by the National Natural Science Foundation of China under Grants No. 61871052 ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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