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+# VARIATIONAL LOSSY AUTOENCODER
+
+Xi Chen†‡, Diederik P. Kingma‡, Tim Salimans‡, Yan Duan†‡, Prafulla Dhariwal‡,   
+John Schulman†‡, Ilya Sutskever‡, Pieter Abbeel†‡   
+† UC Berkeley, Department of Electrical Engineering and Computer Science   
+‡ OpenAI   
+{peter,dpkingma,tim,rocky,prafulla,joschu,ilyasu,pieter}@openai.com
+
+# ABSTRACT
+
+Representation learning seeks to expose certain aspects of observed data in a learned representation that’s amenable to downstream tasks like classification. For instance, a good representation for 2D images might be one that describes only global structure and discards information about detailed texture. In this paper, we present a simple but principled method to learn such global representations by combining Variational Autoencoder (VAE) with neural autoregressive models such as RNN, MADE and PixelRNN/CNN. Our proposed VAE model allows us to have control over what the global latent code can learn and by designing the architecture accordingly, we can force the global latent code to discard irrelevant information such as texture in 2D images, and hence the VAE only “autoencodes” data in a lossy fashion. In addition, by leveraging autoregressive models as both prior distribution $p ( z )$ and decoding distribution $p ( x | z )$ , we can greatly improve generative modeling performance of VAEs, achieving new state-of-the-art results on MNIST, OMNIGLOT and Caltech-101 Silhouettes density estimation tasks as well as competitive results on CIFAR10.
+
+# 1 INTRODUCTION
+
+A key goal of representation learning is to identify and disentangle the underlying causal factors of the data, so that it becomes easier to understand the data, to classify it, or to perform other tasks (Bengio et al., 2013). For image data this often means that we are interested in uncovering the “global structure” that captures the content of an image (for example, the identity of objects present in the image) and its “style”, but that we are typically less interested in the local and high frequency sources of variation such as the specific textures or white noise patterns.
+
+A popular approach for learning representations is to fit a probabilistic latent variable model, an approach also known as analysis-by-synthesis (Yuille & Kersten, 2006; Nair et al., 2008). By learning a generative model of the data with the appropriate hierarchical structure of latent variables, it is hoped that the model will somehow uncover and untangle those causal sources of variations that we happen to be interested in. However, without further assumptions, representation learning via generative modeling is ill-posed: there are many different possible generative models with different (or no) kinds of latent variables that all encode the same probability density function on our observed data. Thus, the results we empirically get using this approach are highly dependent on the specific architectural and modeling choices that are made. Moreover, the objective that we optimize is often completely disconnected from the goal of learning a good representation: An autoregressive model of the data may achieve the same log-likelihood as a variational autoencoder (VAE) (Kingma & Welling, 2013), but the structure learned by the two models is completely different: the latter typically has a clear hierarchy of latent variables, while the autoregressive model has no stochastic latent variables at all (although it is conceivable that the deterministic hidden units of the autoregressive models will have meaningful and useful representations). For this reason, autoregressive models have thus far not been popular for the purpose of learning representations, even though they are extremely powerful as generative models (see e.g. van den Oord et al., 2016a).
+
+A natural question becomes: is it possible to have a model that is a powerful density estimator and at the same time has the right hierarchical structure for representation learning? A potential solution would be to use a hybrid model that has both the latent variable structure of a VAE, as well as the powerful recurrence of an autoregressive model. However, earlier attempts at combining these two kinds of models have run into the problem that the autoregressive part of the model ends up explaining all structure in the data, while the latent variables are not used (Fabius & van Amersfoort, 2014; Chung et al., 2015; Bowman et al., 2015; Serban et al., 2016; Fraccaro et al., 2016; Xu & Sun, 2016). Bowman et al. (2015) noted that weakening the autoregressive part of the model by, for example, dropout can encourage the latent variables to be used. We analyze why weakening is necessary, and we propose a principled solution that takes advantage of this property to control what kind of information goes into latent variables. The model we propose performs well as a density estimator, as evidenced by state-of-the-art log-likelihood results on MNIST, OMNIGLOT and Caltech-101, and also has a structure that is uniquely suited for learning interesting global representations of data.
+
+# 2 VAES DO NOT AUTOENCODE IN GENERAL
+
+A VAE is frequently interpreted as a regularized autoencoder (Kingma & Welling, 2013; Zhang et al., 2016), but the conditions under which it is guaranteed to autoencode (reconstruction being close to original datapoint) are not discussed. In this section, we discuss the often-neglected fact that VAEs do not always autoencode and give explicit reasons why previous attempts to apply VAE in sequence modeling found that the latent code is generally not used unless the decoder is weakened (Bowman et al., 2015; Serban et al., 2016; Fraccaro et al., 2016). The understanding of when VAE does autoencode will be an essential building piece for VLAE.
+
+# 2.1 TECHNICAL BACKGROUND
+
+Let $\mathbf { x }$ be observed variables, $\mathbf { z }$ latent variables and let $p ( \mathbf { x } , \mathbf { z } )$ be the parametric model of their joint distribution, called the generative model defined over the variables. Given a dataset ${ \bf X } =$ $\mathbf { \bar { \{ x } } ^ { 1 } , . . . , x  ^ { N } \}$ we wish to perform maximum likelihood learning of its parameters:
+
+$$
+\log p ( \mathbf { X } ) = \sum _ { i = 1 } ^ { N } \log p ( \mathbf { x } ^ { ( i ) } ) ,
+$$
+
+but in general this marginal likelihood is intractable to compute or differentiate directly for flexible generative models that have high-dimensional latent variables and flexible priors and likelihoods. A solution is to introduce $q ( \mathbf { z } | \mathbf { x } )$ , a parametric inference model defined over the latent variables, and optimize the variational lower bound on the marginal log-likelihood of each observation $\mathbf { x }$ :
+
+$$
+\log p ( \mathbf { x } ) \geq \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } ) } \left[ \log p ( \mathbf { x } , \mathbf { z } ) - \log q ( \mathbf { z } | \mathbf { x } ) \right] = \mathcal { L } ( \mathbf { x } ; \boldsymbol { \theta } )
+$$
+
+where $\theta$ indicates the parameters of $p$ and $q$ models.
+
+There are various ways to optimize the lower bound $\mathcal { L } ( \mathbf { x } ; \theta )$ ; for continuous $\mathbf { z }$ it can be done efficiently through a re-parameterization of $q ( \mathbf { z } | \mathbf { x } )$ (Kingma & Welling, 2013; Rezende et al., 2014).
+
+This way of optimizing the variational lower bound with a parametric inference network and reparameterization of continuous latent variables is usually called VAE. The “autoencoding” terminology comes from the fact that the lower bound $\mathcal { L } ( \mathbf { x } ; \theta )$ can be re-arranged:
+
+$$
+\begin{array} { r l } & { \mathcal { L } ( \mathbf { x } ; \theta ) = \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } ) } \left[ \log p ( \mathbf { x } , \mathbf { z } ) - \log q ( \mathbf { z } | \mathbf { x } ) \right] } \\ & { \qquad = \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } ) } \left[ \log p ( \mathbf { x } | \mathbf { z } ) \right] - D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } ) ) } \end{array}
+$$
+
+where the first term can be seen as the expectation of negative reconstruction error and the KL divergence term can be seen as a regularizer, which as a whole could be seen as a regularized autoencoder loss with $q ( \mathbf { z } | \mathbf { x } )$ being the encoder and $p ( \mathbf { x } | \mathbf { z } )$ being the decoder. In the context of 2D images modeling, the decoding distribution $p ( \mathbf { x } | \mathbf { z } )$ is usually chosen to be a simple factorized distribution, i.e. $\begin{array} { r } { p ( \mathbf { \bar { x } } | \mathbf { z } ) = \prod _ { i } p ( \mathbf { \bar { x } } _ { i } | \mathbf { z } ) } \end{array}$ , and this setup often yields a sharp decoding distribution $p ( \mathbf { x } | \mathbf { z } )$ that tends to reconstruct original datapoint $\mathbf { x }$ exactly.
+
+# 2.2 BITS-BACK CODING AND INFORMATION PREFERENCE
+
+It’s straightforward to see that having a more powerful $p ( \mathbf { x } | \mathbf { z } )$ will make VAE’s marginal generative distribution $\begin{array} { r } { p ( { \bf x } ) = \int _ { \bf z } p ( { \bf z } ) p ( { \bf x } | { \bf z } ) \bar { d } { \bf z } } \end{array}$ more expressive. This idea has been explored extensively in previous work applying VAE to sequence modeling (Fabius & van Amersfoort, 2014; Chung et al., 2015; Bowman et al., 2015; Serban et al., 2016; Fraccaro et al., 2016; Xu & Sun, 2016), where the decoding distribution is a powerful RNN with autoregressive dependency, i.e., $p ( \mathbf { x } | \mathbf { z } ) =$ $\begin{array} { r } { \prod _ { i } p ( \mathbf { x } _ { i } | \mathbf { z } , \mathbf { x } _ { < i } ) } \end{array}$ . Since RNNs are universal function approximators and any joint distribution over $\mathbf { x }$ admits an autoregressive factorization, the RNN autoregressive decoding distribution can in theory represent any probability distribution even without dependence on $\mathbf { z }$ .
+
+However, previous attempts have found it hard to benefit from VAE when using an expressive decoding distribution $p ( \mathbf { x } | \mathbf { z } )$ . Indeed it’s documented in detail by Bowman et al. (2015) that in most cases when an RNN autoregressive decoding distribution is used, the latent code $\mathbf { z }$ is completely ignored and the model regresses to be a standard unconditional RNN autoregressive distribution that doesn’t depend on the latent code. This phenomenon is commonly attributed to “optimization challenges” of VAE in the literature (Bowman et al., 2015; Serban et al., 2016; Kaae Sønderby et al., 2016) because early in the training the approximate posterior $q ( \mathbf { z } | \mathbf { x } )$ carries little information about datapoint $\mathbf { x }$ and hence it’s easy for the model to just set the approximate posterior to be the prior to avoid paying any regularization cost $D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } ) )$ .
+
+Here we present a simple but often-neglected observation that this phenomenon arises not just due to optimization challenges and instead even if we can solve the optimization problems exactly, the latent code should still be ignored at optimum for most practical instances of VAE that have intractable true posterior distributions and sufficiently powerful decoders. It is easiest to understand this observation from a Bits-Back Coding perspective of VAE.
+
+It is well-known that Bits-Back Coding is an information-theoretic view of Variational Inference (Hinton & Van Camp, 1993; Honkela & Valpola, 2004) and specific links have been established between Bits-Back Coding and the Helmholtz Machine/VAE (Hinton & Zemel, 1994; Gregor et al., 2013). Here we briefly relate VAE to Bits-Back Coding for self-containedness:
+
+First recall that the goal of designing an efficient coding protocol is to minimize the expected code length of communicating x. To explain Bits-Back Coding, let’s first consider a more naive coding scheme. VAE can be seen as a way to encode data in a two-part code: $p ( \mathbf { z } )$ and $p ( \mathbf { x } | \mathbf { z } )$ , where $\mathbf { z }$ can be seen as the essence/structure of a datum and is encoded first and then the modeling error (deviation from $\mathbf { z }$ ’s structure) is encoded next. The expected code length under this naive coding scheme for a given data distribution is hence:
+
+$$
+\mathcal { C } _ { \mathrm { n a i v e } } ( \mathbf { x } ) = \mathbb { E } _ { \mathbf { x } \sim \mathrm { d a t a } , \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } ) } \left[ - \log p ( \mathbf { z } ) - \log p ( \mathbf { x } | \mathbf { z } ) \right]
+$$
+
+This coding scheme is, however, inefficient. Bits-Back Coding improves on it by noticing that the encoder distribution $q ( \mathbf { z } | \mathbf { x } )$ can be used to transmit additional information, up to $H ( q ( \mathbf { z } | \mathbf { x } ) )$ expected nats, as long as the receiver also has access to $q ( \mathbf { z } | \mathbf { x } )$ . The decoding scheme works as follows: a receiver first decodes $\mathbf { z }$ from $p ( \mathbf { z } )$ , then decodes $\mathbf { x }$ from $p ( \mathbf { x } | \mathbf { z } )$ and, by running the same approximate posterior that the sender is using, decodes a secondary message from $q ( \mathbf { z } | \mathbf { x } )$ . Hence, to properly measure the code length of VAE’s two-part code, we need to subtract the extra information from $\mathbf { \bar { \rho } } _ { \mathbf { q } } ( \mathbf { z } | \mathbf { x } )$ . Using Bit-Back Coding, the expected code length equates to the negative variational lower bound or the so-called Helmholtz variational free energy, which means minimizing code length is equivalent to maximizing the variational lower bound:
+
+$$
+\begin{array} { r l } & { { \mathcal { C } } _ { \mathrm { B i t s B a c k } } ( \mathbf { x } ) = \mathbb { E } _ { \mathbf { x } \sim \mathrm { d a t a } , \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } ) } \left[ \log q ( \mathbf { z } | \mathbf { x } ) - \log p ( \mathbf { z } ) - \log p ( \mathbf { x } | \mathbf { z } ) \right] } \\ & { \qquad = \mathbb { E } _ { \mathbf { x } \sim \mathrm { d a t a } } \left[ - { \mathcal { L } } ( \mathbf { x } ) \right] } \end{array}
+$$
+
+Casting the problem of optimizing VAE into designing an efficient coding scheme easily allows us to reason when the latent code $\mathbf { z }$ will be used: the latent code z will be used when the two-part code is an efficient code. Recalling that the lower-bound of expected code length for data is given by the Shannon entropy of data generation distribution: $\mathcal { H } ( \mathrm { d a t a } ) = \mathbb { E } _ { \boldsymbol { x } \sim \mathrm { d } }$ ata $[ - \log p \mathrm { d a t a } ( x ) ]$ , we can analyze VAE’s coding efficiency:
+
+$$
+\begin{array} { r l } & { \mathcal { C } _ { \mathrm { B i t s B a c k } } ( \mathbf { x } ) = \mathbb { E } _ { \mathbf { x } \sim \mathrm { d a t a } , \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } ) } \left[ \log q ( \mathbf { z } | \mathbf { x } ) - \log p ( \mathbf { z } ) - \log p ( \mathbf { x } | \mathbf { z } ) \right] } \\ & { \qquad = \mathbb { E } _ { \mathbf { x } \sim \mathrm { d a t a } } \left[ - \log p ( \mathbf { x } ) + D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } | \mathbf { x } ) ) \right] } \\ & { \qquad \geq \mathbb { E } _ { \mathbf { x } \sim \mathrm { d a t a } } \left[ - \log p _ { \mathrm { d a t a } } ( \mathbf { x } ) + D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } | \mathbf { x } ) ) \right] } \\ & { \qquad = \mathcal { H } ( \mathrm { d a t a } ) + \mathbb { E } _ { \mathbf { x } \sim \mathrm { d a t a } } \left[ D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } | \mathbf { x } ) ) \right] } \end{array}
+$$
+
+Since Kullback Leibler divergence is always non-negative, we know that using the two-part code derived from VAE suffers at least an extra code length of $D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } | \mathbf { x } ) )$ nats for using a posterior that’s not precise. Many previous works in Variational Inference have designed flexible approximate posteriors to better approximate true posterior (Salimans et al., 2014; Rezende & Mohamed, 2015; Tran et al., 2015; Kingma et al., 2016). Improved posterior approximations have shown to be effective in improving variational inference but none of the existing methods are able to completely close the gap between approximate posterior and true posterior. This leads us to believe that for most practical models, at least in the near future, the extra coding cost $D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } | \mathbf { x } ) )$ will exist and will not be negligible.
+
+Once we understand the inefficiency of the Bits-Back Coding mechanism, it’s simple to realize why sometimes the latent code $z$ is not used: if the $p ( \mathbf { x } | \mathbf { z } )$ could model $p _ { \mathrm { d a t a } } ( \mathbf { x } )$ without using information from $\mathbf { z }$ , then it will not use $\mathbf { z }$ , in which case the true posterior $p ( \mathbf { z } | \mathbf { x } )$ is simply the prior $p ( \mathbf { z } )$ and it’s usually easy to set $q ( \mathbf { z } | \mathbf { x } )$ to be $p ( \mathbf { z } )$ to avoid incurring an extra cost $D _ { K L } \bar { ( { q } ( { \bf { z } } | { \bf { x } } ) ) } | | p ( { \bf { z } } | { \bf { x } } ) ) \bar { }$ . And it’s exactly the case when a powerful decoding distribution is used like an RNN autoregressive distribution, which given enough capacity is able to model arbitrarily complex distributions. Hence there exists a preference of information when a VAE is optimized: information that can be modeled locally by decoding distribution $p ( \mathbf { x } | \mathbf { z } )$ without access to z will be encoded locally and only the remainder will be encoded in $\mathbf { z }$ .
+
+We note that one common way to encourage putting information into the code is to use a factorized decoder $\begin{array} { r } { p ( \mathbf { x } | \mathbf { z } ) = \prod _ { i } p ( \mathbf { x } _ { i } | \mathbf { z } ) } \end{array}$ but so long as there is one dimension $\mathbf { x } _ { j }$ that’s independent of all other dimensions for true data distribution, $p _ { \mathrm { d a t a } } ( \mathbf { x } ) = p _ { \mathrm { d a t a } } ( \mathbf { x } _ { j } ) p _ { \mathrm { d a t a } } ( \mathbf { \check { x } } _ { \neq j } )$ , then the latent code doesn’t contain all the information about $\mathbf { x }$ since at least $\mathbf { x } _ { j }$ will be modeled locally by factorized $p ( \mathbf { x } | \mathbf { z } )$ . This kind of independence structure rarely exists in images so common VAEs that have factorized decoder autoencode almost exactly. Other techniques to encourage the usage of the latent code include annealing the relative weight of of $D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } ) )$ in the variational lower bound (Bowman et al., 2015; Kaae Sønderby et al., 2016) or the use of free bits (Kingma et al., 2016), which can serve the dual purpose of smoothing the optimization landscape and canceling out part of the Bits-Back Code inefficiency $D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) \bar { | } | p ( \mathbf { z } | \mathbf { x } ) )$ .
+
+# 3 VARIATIONAL LOSSY AUTOENCODER
+
+The discussion in Section 2.2 suggests that autoregressive models cannot be combined with VAE since information will be preferred to be modeled by autoregressive models. Nevertheless, in this section, we present two complementary classes of improvements to VAE that utilize autoregressive models fruitfully to explicitly control representation learning and improve density estimation.
+
+# 3.1 LOSSY CODE VIA EXPLICIT INFORMATION PLACEMENT
+
+Even though the information preference property of VAE might suggest that one should always use the full autoregressive models to achieve a better code length/log-likelihood, especially when slow data generation is not a concern, we argue that this information preference property can be exploited to turn the VAE into a powerful representation learning method that gives us fine-grained control over the kind of information that gets included in the learned representation.
+
+When we try to learn a lossy compression/representation of data, we can simply construct a decoding distribution that’s capable of modeling the part of information that we don’t want the lossy representation to capture, but, critically, that’s incapable of modelling the information that we do want the lossy representation to capture.
+
+For instance, if we are interested in learning a global representation for 2D images that doesn’t encode information about detailed texture, we can construct a specific factorization of the autoregressive distribution such that it has a small local receptive field as decoding distribution, e.g., $\begin{array} { r } { \bar { p } _ { \mathrm { l o c a l } } ( \mathbf { x } | \mathbf { z } ) = \prod _ { i } p ( \mathbf { x } _ { i } | \mathbf { z } , \mathbf { x } _ { \mathrm { W i n d o w A r o u n d } ( i ) } ) } \end{array}$ . Notice that, as long as xWindowAround(i) is smaller than $\mathbf { x } _ { < i }$ , $p _ { \mathrm { l o c a l } } ( \mathbf { x } | \mathbf { z } )$ won’t be able to represent arbitrarily complex distribution over $\mathbf { x }$ without dependence on $\mathbf { z }$ since the receptive field is limited such that not all distributions over $\mathbf { x }$ admit such factorizations. In particular, the receptive field window can be a small rectangle adjacent to a pixel $\mathbf { x } _ { i }$ and in this case long-range dependency will be encoded in the latent code $\mathbf { z }$ . On the other hand, if the true data distribution admits such factorization for a given datum $\mathbf { x }$ and dimension $i$ , i.e.
+
+$p _ { \mathrm { d a t a } } \bigl ( \mathbf { x } _ { i } | \mathbf { x } _ { \mathrm { W i n d o w A r o u n d } ( i ) } \bigr ) = p _ { \mathrm { d a t a } } \bigl ( \mathbf { x } _ { i } | \mathbf { x } _ { < i } \bigr )$ , then the information preference property discussed in Section 2.2 will apply here, which means that all the information will be encoded in local autoregressive distribution for $\mathbf { x } _ { i }$ . Local statistics of 2D images like texture will likely be modeled completely by a small local window, whereas global structural information of an images like shapes of objects is long-range dependency that can only be communicated through latent code $\mathbf { z }$ . Therefore we have given an example VAE that will produce a lossy compression of 2D images carrying exclusively global information that can’t be modeled locally.
+
+Notice that a global representation is only one of many possible lossy representations that we can construct using this information preference property. For instance, the conditional of an autoregressive distribution might depend on a heavily down-sampled receptive field so that it can only model long-range pattern whereas local high-frequency statistics need to be encoded into the latent code. Hence we have demonstrated that we can achieve explicit placement of information by constraining the receptive field/factorization of an autoregressive distribution that’s used as decoding distribution.
+
+We want to additionally emphasize the information preference property is an asymptotic view in a sense that it only holds when the variational lowerbound can be optimized well. Thus, we are not proposing an alternative to techniques like free bits Kingma et al. (2016) or KL annealing, and indeed they are still useful methods to smooth the optimization problem and used in this paper’s experiments.
+
+# 3.2 LEARNED PRIOR WITH AUTOREGRESSIVE FLOW
+
+Inefficiency in Bits-Back Coding, i.e., the mismatch between approximate posterior and true posterior, can be exploited to construct a lossy code but it’s still important to minimize such inefficiency to improve overall modeling performance/coding efficiency. We propose to parametrize the prior distribution $p ( \mathbf { z } ; \theta )$ with an autoregressive model and show that a type of autoregressive latent code can in theory reduce inefficiency in Bits-Back coding.
+
+It is well-known that limited approximate posteriors impede learning and therefore various expressive posterior approximations have been proposed to improve VAE’s density estimation performance (Turner et al., 2008; Mnih & Gregor, 2014; Salimans et al., 2014; Rezende & Mohamed, 2015; Kingma et al., 2016). One such class of approximate posteriors that has been shown to attain good empirical performance is based on the idea of Normalizing Flow, which is to apply an invertible mapping to a simple random variable, for example a factorized Gaussian as commonly used for $q ( \mathbf { z } | \mathbf { x } )$ , in order to obtain a complicated random variable. For an invertible transformation between a simple distribution $\mathbf { y }$ and a more flexible $\mathbf { z }$ , we know from the change-of-variable technique that $\begin{array} { r } { \log q ( \mathbf { z } | \mathbf { x } ) = \log q ( \mathbf { y } | \mathbf { x } ) - \log \operatorname* { d e t } \frac { d \mathbf { z } } { d \mathbf { v } } } \end{array}$ and using $q ( \mathbf { z } | \mathbf { x } )$ as approximate posterior will decrease the coding efficiency gap $D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | \dot { p } ( \mathbf { z } | \mathbf { x } ) )$ provided the transformation is sufficiently expressive.
+
+Kingma et al. (2016) introduced Inverse Autoregressive Flow, which is a powerful class of such invertible mappings that have simple determinant: $\begin{array} { r } { z _ { i } = \frac { y _ { i } - \mu _ { i } \left( y _ { 1 : i - 1 } \right) } { \sigma _ { i } \left( y _ { 1 : i - 1 } \right) } } \end{array}$ , where $\mu _ { i } ( . ) \in \mathcal { R } , \sigma _ { i } ( . ) \in \mathcal { R } ^ { + }$ are general functions that can be parametrized by expressive neural networks, such as MADE and PixelCNN variants (Germain et al., 2015; van den Oord et al., 2016a). Inverse autoregressive flow is the inverse/whitening of autoregressive flow: $y _ { i } = z _ { i } \sigma _ { i } ( y _ { 1 : i - 1 } ) + \mu _ { i } ( y _ { 1 : i - 1 } )$ . We refer interested readers to (Rezende & Mohamed, 2015; Kingma et al., 2016) for in-depth discussions on related topics.
+
+In this paper, we propose to parametrize our learnable prior as an autoregressive flow from some simple noise source like spherical Gaussian. Next, we show that using latent code transformed by autoregressive flow (AF) is equivalent to using inverse autoregressive flow (IAF) approximate posterior, which explains why it can similarly improve Bits-Back Coding efficiency. Moreover, compared with an IAF posterior, an AF prior has a more expressive generative model that essentially “comes for free”.
+
+For an autoregressive flow $f$ , some continuous noise source $\epsilon$ is transformed into latent code $\mathbf { z }$ : $\mathbf { z } = f ( \epsilon )$ . Assuming the density function for noise source is $u ( \epsilon )$ , we similarly know that $\log p ( \mathbf { z } ) =$ $\begin{array} { r } { \log u ( \epsilon ) + \log \operatorname* { d e t } \frac { d \epsilon } { d \mathbf { z } } } \end{array}$
+
+Simply re-arranging the variational lowerbound for using AF prior reveals that having an AF latent code $\mathbf { z }$ is equivalent to using an IAF posterior for $\epsilon$ that we can interpret as the new latent code:
+
+$$
+\begin{array} { r l } & { \mathcal { L } ( \mathbf { x } ; \theta ) = \mathbb { E } _ { \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } ) } \left[ \log p ( \mathbf { x } | \mathbf { z } ) + \log p ( \mathbf { z } ) - \log q ( \mathbf { z } | \mathbf { x } ) \right] } \\ & { \quad \quad \quad = \mathbb { E } _ { \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } ) , \epsilon = f ^ { - 1 } ( \mathbf { z } ) } \left[ \log p ( \mathbf { x } | f ( \epsilon ) ) + \log u ( \epsilon ) + \log \operatorname* { d e t } \frac { d \epsilon } { d \mathbf { z } } - \log q ( \mathbf { z } | \mathbf { x } ) \right] } \\ & { \quad \quad \quad = \mathbb { E } _ { \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } ) , \epsilon = f ^ { - 1 } ( \mathbf { z } ) } \left[ \log p ( \mathbf { x } | f ( \epsilon ) ) + \log u ( \epsilon ) - \underbrace { ( \log q ( \mathbf { z } | \mathbf { x } ) - \log \operatorname* { d e t } \frac { d \epsilon } { d \mathbf { z } } ) } _ { \mathrm { L A F P o s t e r i o r } } \right] } \end{array}
+$$
+
+AF prior is the same as IAF posterior along the encoder path, $f ^ { - 1 } ( q ( \mathbf { z } | \mathbf { x } ) )$ , but differs along the decoder/generator path: IAF posterior has a shorter decoder path $p ( \mathbf { x } | \mathbf { z } )$ whereas AF prior has a deeper decoder path $p ( \mathbf { x } | f ( \epsilon ) )$ . The crucial observation is that AF prior and IAF posterior have the same computation cost under the expectation of $\mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } )$ , so using AF prior makes the model more expressive at no training time cost.
+
+# 4 EXPERIMENTS
+
+In this paper, we evaluate VLAE on 2D images and leave extensions to other forms of data to future work. For the rest of the section, we define a VLAE model as a VAE that uses AF prior and autoregressive decoder. We choose to implement conditional distribution $p ( \mathbf { x } | \mathbf { z } )$ with a smallreceptive-field PixelCNN (van den Oord et al., 2016a), which has been proved to be a scalable autoregressive model.
+
+For evaluation, we use binary image datasets that are commonly used for density estimation tasks: MNIST (LeCun et al., 1998) (both statically binarized 1 and dynamically binarized version (Burda et al., 2015a)), OMNIGLOT (Lake et al., 2013; Burda et al., 2015a) and Caltech-101 Silhouettes (Marlin et al., 2010). All datasets uniformly consist of $2 8 \mathrm { { x } 2 8 }$ binary images, which allow us to use a unified architecture. VAE networks used in binary image datasets are simple variants of ResNet VAEs described in (Salimans et al., 2014; Kingma et al., 2016). For the decoder, we use a variant of PixelCNN that has 6 layers of masked convolution with filter size 3, which means the window of dependency, xWindowAround(i), is limited to a small local patch. During training, ”free bits” (Kingma et al., 2016) is used improve optimization stability. Experimental setup and hyperparameters are detailed in the appendix. Reported marginal NLL is estimated using Importance Sampling with 4096 samples.
+
+We designed experiments to answer the following questions:
+
+• Can VLAE learn lossy codes that encode global statistics? • Does using AF priors improves upon using IAF posteriors as predicted by theory? • Does using autoregressive decoding distributions improve density estimation performance?
+
+# 4.1 LOSSY COMPRESSION
+
+First we are interested in whether VLAE can learn a lossy representation/compression of data by using the PixelCNN decoder to model local statistics. We trained VLAE model on Statically Binarized MNIST and the converged model has $\mathbb { E } [ D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } ) ) ] = 1 3 . 3$ nats = 19.2 bits, which is the number of bits it uses on average to encode/compress one MNIST image. By comparison, an identical VAE model with factorized decoding distribution will uses on average 37.3 bits in latent code, and this thus indicates that VLAE can learn a lossier compression than a VAE with regular factorized conditional distribution.
+
+The next question is whether VLAE’s lossy compression encodes global statistics and discards local statistics. In Fig 1a, we visualize original images $\mathbf { x } _ { \mathrm { d a t a } }$ and one random “decompression” xdecompressed from VLAE: $\mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } _ { \mathrm { d a t a } } )$ , xdecompressed $\sim p ( \mathbf { x } | \mathbf { z } )$ . We observe that none of the decompressions is an exact reconstruction of the original image but instead the global structure of the image was encoded in the lossy code $\mathbf { z }$ and regenerated. Also worth noting is that local statistics are not preserved but a new set of likely local statistics are generated in the decompressed images: the binary masks are usually different and local styles like stroke width are sometimes slightly different.
+
+![](images/581a9d551d664aabdf904e8791f6618ada526a3da8745712ac099b9b89b79b11.jpg)  
+Figure 1: Statically Binarized MNIST
+
+However, we remark that the lossy code z doesn’t always capture the kind of global information that we care about and it’s dependent on the type of constraint we put on the decoder. For instance, in Fig 4b, we show decompressions for OMNIGLOT dataset, which has more meaningful variations in small patches than MNIST, and we can observe that semantics are not preserved in some cases. This highlights the need to specify the type of statistics we care about in a representation, which will be different across tasks and datasets, and design decoding distribution accordingly.
+
+![](images/506353593b6cf7e2101d467217f2fb78982898e2c374034a9412bb6d1d364e90.jpg)  
+Figure 2: OMNIGLOT
+
+# 4.2 DENSITY ESTIMATION
+
+Next we investigate whether leveraging autoregressive models as latent distribution $p ( \mathbf { z } )$ and as decoding distribution $p ( \mathbf { x } | \mathbf { z } )$ would improve density estimation performance.
+
+To verify whether AF prior is able to improve upon IAF posterior alone, it’s desirable to test this model without using autoregressive decoder but instead using the conventional independent Bernoulli distribution for $p ( \mathbf { x } | \mathbf { z } )$ . Hence we use the best performing model from Kingma et al.
+
+Table 1: Statically Binarized MNIST   
+
+<table><tr><td>Model</td><td>NLL Test</td></tr><tr><td>Normalizing flows (Rezende &amp;Mohamed,2015)</td><td>85.10</td></tr><tr><td>DRAW (Gregor et al., 2015)</td><td>&lt;80.97</td></tr><tr><td>Discrete VAE (Rolfe,2016)</td><td>81.01</td></tr><tr><td>PixelRNN (van den Oord et al., 2016a)</td><td>79.20</td></tr><tr><td>IAF VAE (Kingma et al., 2016)</td><td>79.88</td></tr><tr><td>AFVAE</td><td>79.30</td></tr><tr><td>VLAE</td><td>79.03</td></tr></table>
+
+(2016) on statically binarized MNIST and make the single modification of replacing the original IAF posterior with an equivalent AF prior, removing the context. As seen in Table 1, VAE with AF prior is outperforming VAE with an equivalent IAF posterior, indicating that the deeper generative model from AF prior is beneficial. A similar gain carries over when an autoregressive decoder is used: on statically binarized MNIST, using AF prior instead of IAF posterior reduces train NLL by 0.8 nat and test NLL by 0.6 nat.
+
+Next we evaluate whether using autoregressive decoding distribution can improve performance and we show in Table 1 that a VLAE model, with AF prior and PixelCNN conditional, is able to outperform a VAE with just AF prior and achieves new state-of-the-art results on statically binarized MNIST.
+
+In addition, we hypothesize that the separation of different types of information, the modeling global structure in latent code and local statistics in PixelCNN, likely has some form of good inductive biases for 2D images. In order to evaluate if VLAE is an expressive density estimator with good inductive biases, we will test a single VLAE model, with the same network architecture, on all binary datasets. We choose hyperparameters manually on statically binarized MNIST and use the same hyperparameters to evaluate on dynamically binarized MNIST, OMNIGLOT and Caltech-101 Silhouettes. We also note that better performance can be obtained if we individually tune hyperparameters for each dataset. As a concrete demonstration, we report the performance of a fine-tuned VLAE on OMNIGLOT dataset in Table 3.
+
+Table 2: Dynamically binarized MNIST   
+
+<table><tr><td>Model</td><td>NLL Test</td></tr><tr><td>Convolutional VAE +HVI (Salimans etal., 2014)</td><td>81.94</td></tr><tr><td>DLGM 2hl + IWAE (Burda et al., 2015a)</td><td>82.90</td></tr><tr><td>Discrete VAE (Rolfe, 2016)</td><td>80.04</td></tr><tr><td>LVAE (Kaae Sonderby et al., 2016)</td><td>81.74</td></tr><tr><td>DRAW + VGP (Tran et al., 2015)</td><td>&lt; 79.88</td></tr><tr><td>IAF VAE (Kingma et al., 2016)</td><td>79.10</td></tr><tr><td>Unconditional Decoder</td><td>87.55</td></tr><tr><td>VLAE</td><td>78.53</td></tr></table>
+
+Table 3: OMNIGLOT. [1] (Burda et al., 2015a), [2] (Burda et al., 2015b), [3] (Gregor et al., 2015), [4] (Gregor et al., 2016),   
+
+<table><tr><td>Model</td><td>NLL Test</td></tr><tr><td>VAE [1]</td><td>106.31</td></tr><tr><td>IWAE [1]</td><td>103.38</td></tr><tr><td>RBM(500 hidden) [2]</td><td>100.46</td></tr><tr><td>DRAW[3]</td><td>&lt;96.50</td></tr><tr><td>Conv DRAW [4]</td><td>&lt;91.00</td></tr><tr><td>Unconditional Decoder</td><td>95.02</td></tr><tr><td>VLAE</td><td>90.98</td></tr><tr><td>VLAE (fine-tuned)</td><td>89.83</td></tr></table>
+
+Table 4: Caltech-101 Silhouettes. [1] (Bornschein & Bengio, 2014), [2] (Cho et al., 2011), [3] (Du et al., 2015), [4] (Rolfe, 2016), [5] (Goessling & Amit, 2015),
+
+<table><tr><td>Model</td><td>NLL Test</td></tr><tr><td>RWS SBN[1]</td><td>113.3</td></tr><tr><td>RBM[2] NAIS NADE [3]</td><td>107.8 100.0</td></tr><tr><td>Discrete VAE [4]</td><td>97.6</td></tr><tr><td>SpARN[5]</td><td>88.48</td></tr><tr><td>Unconditional Decoder VLAE</td><td>89.26</td></tr></table>
+
+As seen in Table 2,3,4, with the same set of hyperparameters tuned on statically binarized MNIST, VLAE is able to perform well on the rest of datasets, significantly exceeding previous state-ofthe-art results on dynamically binarized MNIST and Caltech-101 Silhouettes and tying statistically with best previous result on OMNIGLOT. In order to isolate the effect of expressive PixelCNN as decoder, we also report performance of the same PixelCNN trained without VAE part under the name “Unconditional Decoder”.
+
+# 4.3 NATURAL IMAGES: CIFAR10
+
+In addition to binary image datasets, we have applied VLAE to the CIFAR10 dataset of natural images. Density estimation of CIFAR10 images has been a challenging benchmark problem used by many recent generative models and hence is great task to position VLAE among existing methods.
+
+We investigated using ResNet (He et al., 2016) and DenseNet (Huang et al., 2016) as building blocks for VAE networks and observed that DenseNet reduces overfitting. We also propose a new optimization technique that blends the advantages of KL annealing (Serban et al., 2016) and ”free bits” (Kingma et al., 2016) to stabilize learning on this challenging dataset. Detailed experimental setup is described in Appendix.
+
+VLAE is compared to other methods on CIFAR10 in Table 5. We show that VLAE models attain new state-of-the-art performance among other variationally trained latent-variable models. DenseNet VLAE model also outperforms most other tractable likelihood models including Gated PixelCNN and PixelRNN and has results only slightly worse than currently unarchived state-of-the-art Pixel$\mathrm { C N N + + }$ .
+
+Table 5: CIFAR10. Likelihood for VLAE is approximated with 512 importance samples. [1] (van den Oord et al., 2016a), [2] (Dinh et al., 2014), [3] (van den Oord & Schrauwen, 2014), [4] (Dinh et al., 2016), [5] (van den Oord et al., 2016b), [6] (Salimans et al., 2017), [7] (Sohl-Dickstein et al., 2015), [8] (Gregor et al., 2016), [9] (Kingma et al., 2016)   
+
+<table><tr><td>Method</td><td>bits/dim ≤</td></tr><tr><td>Resultswith tractable likelihood models: Uniform distribution [1]</td><td>8.00</td></tr><tr><td>Multivariate Gaussian [1]</td><td>4.70</td></tr><tr><td>NICE [2] Deep GMMs [3]</td><td>4.48 4.00</td></tr><tr><td>Real NVP [4]</td><td>3.49</td></tr><tr><td>PixelCNN[1]</td><td>3.14</td></tr><tr><td>Gated PixelCNN [5]</td><td>3.03</td></tr><tr><td>PixelRNN[1] PixelCNN++ [6]</td><td>3.00</td></tr><tr><td>Resultswith variationallytrained latent-variablemodels:</td><td>2.92</td></tr><tr><td>Deep Diffusion [7]</td><td></td></tr><tr><td>Convolutional DRAW[8]</td><td>5.40</td></tr><tr><td>ResNetVAE with IAF[9]</td><td>3.58</td></tr><tr><td>ResNetVLAE</td><td>3.11</td></tr><tr><td>DenseNetVLAE</td><td>3.04</td></tr></table>
+
+We also investigate learning lossy codes on CIFAR10 images. To illustrate how does the receptive field size of PixelCNN decoder influence properties of learned latent codes, we show visualizations of similar VLAE models with receptive fields of different sizes. Specifically we say a receptive field, xWindowAround(i), has size $A \mathbf { x } B$ when a pixel $x _ { i }$ can depend on the rectangle block of size $A \mathbf { x } B$ immediately on top of $x _ { i }$ as well as the $\textstyle \left\lceil { \frac { A - 1 } { 2 } } \right\rceil$ pixels immediately to the left of $x _ { i }$ . We use this notation to refer to different types of PixelCNN decoders in Figure 3.
+
+From (a)-(c) in Figure 3, we can see that larger receptive fields progressively make autoregressive decoders capture more structural information. In (a), a smaller receptive field tends to preserve rather detailed shape information in the lossy code whereas the latent code only retains rough shape in (c) with a larger receptive field.
+
+![](images/03d275bd9dc480d78267d9b3390b7ede544ee5236bf055cc58ba8f7b4da7bc86.jpg)  
+Figure 3: CIFAR10: Original test-set images (left) and “decompressioned” versions from VLAE’s lossy code (right) with different types of receptive fields
+
+It’s interesting to also note that in (a)-(c), oftentimes color information is partially omitted from latent codes and one explanation can be that color is very predictable locally. However, color information can be important to preserve if our task is, for example, object classification. To demonstrate how we can encode color information in the lossy code, we can choose to make PixelCNN decoder depend only on images’ grayscale versions. In other words, instead of choosing the decoder to be $\begin{array} { r } { \dot { p } _ { \mathrm { l o c a l } } ( \mathbf { x } | \mathbf { \bar { z } } ) = \prod _ { i } ^ { } p ( \mathbf { x } _ { i } | \mathbf { z } } \end{array}$ , xWindowAround(i)), we use a decoder of the form $\begin{array} { r } { p _ { \mathrm { l o c a l } } ( \mathbf { x } | \mathbf { z } ) ~ { = } ~ \prod _ { i } p ( \mathbf { x } _ { i } | \mathbf { z } , \mathrm { G r a y s c a l e } ( \mathbf { x } _ { \mathrm { W i n d o w A r o u n d } ( i ) } ) ) , } \end{array}$ ). In (d) of Figure 3, we visualize lossy codes for a VLAE that has the same receptive field size as (c) but uses a “grayscale receptive field”. We note that the lossy codes in (d) encode roughly the same structural information as those in (c) but generally generate objects that are more recognizable due to the preservation of color information. This serves as one example of how we can design the lossy latent code carefully to encode what’s important and what’s not.
+
+# 5 RELATED WORK
+
+We investigate a fusion between variational autoencoders with continuous latent variables (Kingma & Welling, 2013; Rezende et al., 2014) and neural autoregressive models. For autoregression, we specifically apply a novel type of architecture where autoregression is realised through a carefully constructed deep convolutional network, introduced in the PixelCNN model for images (van den Oord et al., 2016a,b). These family of convolutional autoregressive models was further explored, and extended, for audio in WaveNet (Oord et al., 2016), video in Video Pixel Networks (Kalchbrenner et al., 2016b) and language in ByteNet (Kalchbrenner et al., 2016a).
+
+The combination of latent variables with expressive decoder was previously explored using recurrent networks mainly in the context of language modeling (Chung et al., 2015; Bowman et al., 2015; Serban et al., 2016; Fraccaro et al., 2016; Xu & Sun, 2016). Bowman et al. (2015) has also proposed to weaken an otherwise too expressive decoder by dropout to force some information into latent codes.
+
+Concurrent with our work, PixelVAE (Gulrajani et al., 2016) also explored using conditional PixelCNN as a VAE’s decoder and has obtained impressive density modeling results through the use of multiple levels of stochastic units.
+
+Using autoregressive model on latent code was explored in the context of discrete latent variables in DARN (Gregor et al., 2013). Kingma et al. (2016), Kaae Sønderby et al. (2016), Gregor et al. (2016) and Salimans (2016) explored VAE architecture with an explicitly deep autoregressive prior for continuous latent variables, but the autoregressive data likelihood is intractable in those architectures and needs to inferred variationally. In contrast, we use multiple steps of autoregressive flows that has exact likelihood and analyze the effect of using expressive latent code.
+
+Optimization challenges for using (all levels of) continuous latent code were discussed before and practical solutions were proposed (Bowman et al., 2015; Kaae Sønderby et al., 2016; Kingma et al., 2016). In this paper, we present a complementary perspective on when/how should the latent code be used by appealing to a Bits-Back interpretation of VAE.
+
+Learning a lossy compressor with latent variable model has been investigated with ConvDRAW (Gregor et al., 2016). It learns a hierarchy of latent variables and just using high-level latent variables will result in a lossy compression that performs similarly to JPEG. Our model similarly learns a lossy compressor but it uses an autoregressive model to explicitly control what kind of information should be lost in compression.
+
+# 6 CONCLUSION
+
+In this paper, we analyze the condition under which the latent code in VAE should be used, i.e. when does VAE autoencode, and use this observation to design a VAE model that’s a lossy compressor of observed data. At modeling level, we propose two complementary improvements to VAE that are shown to have good empirical performance.
+
+VLAE has the appealing properties of controllable representation learning and improved density estimation performance but these properties come at a cost: compared with VAE models that have simple prior and decoder, VLAE is slower at generation due to the sequential nature of autoregressive model.
+
+Moving forward, we believe it’s exciting to extend this principle of learning lossy codes to other forms of data, in particular those that have a temporal aspect like audio and video. Another promising direction is to design representations that contain only information for downstream tasks and utilize those representations to improve semi-supervised learning.
+
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+
+# APPENDIX
+
+# A DETAILED EXPERIMENT SETUP FOR BINARY IMAGES
+
+For VAE’s encoder and decoder, we use the same ResNet (He et al., 2015) VAE architecture as the one used in IAF MNIST experiment (Kingma et al., 2016). The only difference is that the decoder network now, instead of outputing a $2 8 x 2 8 \mathrm { x } 1$ spatial feature map to specify the mean of a factorized bernoulli distribution, outputs a $2 8 \mathbf { x } 2 8 \mathbf { x } 4$ spatial feature map that’s concatenated with the original binary image channel-wise, forming a $2 8 \mathrm { x } 2 8 \mathrm { x } 5$ feature map that’s then fed through a typical masked PixelCNN (van den Oord et al., 2016a). As such even though the PixelCNN conditions on the latent code, we don’t call it a Conditional PixelCNN because it doesn’t use the specific architecture that was proposed in van den Oord et al. (2016b). For the PixelCNN, it has 6 masked convolution layers with $1 2 3 \mathrm { x } 3 $ filters organized in ResNet blocks, and it has 4 additional 1x1 convolution ResNet block between every other masked convolution layer to increase processing capacity since it employs fewer masked convolutions than usual. All the masked convolution layer have their weights tied to reduce overfitting on statically binarized MNIST, and untying the weights will increase performance for other datasets. Experiments are tuned on the validation set and then final experiment was run with train and validation set, with performance evaluated with test set. Exponential Linear Units (Clevert et al., 2015) are used as activation functions in both VAE network and PixelCNN network. Weight normalization is everywhere with data-dependent initialization (Salimans & Kingma, 2016).
+
+A latent code of dimension 64 was used. For AF prior, it’s implemented with MADE (Germain et al., 2015) as detailed in Kingma et al. (2016). We used 4 steps of autoregressive flow and each flow is implemented by a 3-layer MADE that has 640 hidden units and uses Relu (Nair & Hinton, 2010) as activation functions. Differing from the practice of Kingma et al. (2016), we use mean-only autoregressive flow, which we found to be more numerically stable.
+
+In terms of training, Adamax (Kingma & Ba, 2014) was used with a learning rate of 0.002. 0.01 nats/data-dim free bits (Kingma et al., 2016) was found to be effective in dealing with the problem of all the latent code being ignored early in training. Polyak averaging (Polyak & Juditsky, 1992) was used to compute the final parameters, with $\alpha = 0 . 9 9 8$ .
+
+All experiments are implemented using TensorFlow (Abadi et al., 2016).
+
+# B ADDITIONAL EXPERIMENT SETUP FOR CIFAR10
+
+Latent codes are represented by 16 feature maps of size 8x8, and this choice of spatial stochastic units are inspired by ResNet IAF VAE (Kingma et al., 2016). Prior distribution is factorized Gaussian noise transformed by 6 autoregressive flows, each of which is implemented by a PixelCNN (van den Oord et al., 2016a) with 2 hidden layers and 128 feature maps. Between every other autoregressive flow, the ordering of stochastic units is reversed.
+
+ResNet VLAE has the following structure for encoder: 2 ResNet blocks, Conv w/ stride $^ { = 2 }$ , 2 ResNet blocks, Conv w/ stride $^ { = 2 }$ , 3 ResNet blocks, 1x1 convolution and has a symmetric decoder. Channel size $= 4 8$ for $3 2 \mathrm { x } 3 2 $ feature maps and 96 for other feature maps. DenseNet VLAE follows a similar structure: replacing 2 ResNet blocks with one DenseNet block of 3 steps and each step produces a certain number of feature maps such that at the end of a block, the concatenated feature maps is slightly more than the ResNet VLAE at the same stage.
+
+Conditional $\mathrm { P i x e l C N N + + }$ (Salimans et al., 2017) is used as the decoder. Specifically the channelautoregressive variant is used to ensure there is sufficient capacity even when the receptive field is small. Specifically, the decoder PixelCNN has 4 blocks of 64 feature maps where each block is conditioned on previous blocks with Gated ResNet connections and hence the PixelCNN decoders we use are shallow but very wide. For $4 \mathbf { x } 2$ receptive field experiment, we use 1 layer of vertical stack convolutions and 2 layers of horizontal stack convolutions; for 5x3 receptive field experiment, we use 2 layers of vertical stack convolutions and 2 layers of horizontal stack convolutions; For 5x3 receptive field experiment, we use 2 layers of vertical stack convolutions and 2 layers of horizontal stack convolutions; For $7 \mathrm { x 4 }$ receptive field experiment, we use 3 layers of vertical stack convolutions and 3 layers of horizontal stack convolutions; for $7 \mathrm { x } 4$ Grayscale experiment, we transform RGB
+
+images into gray-scale images via this specific transformation: $( 0 . 2 9 9 * R ) + ( 0 . 5 8 7 G ) + ( 0 . 1 1 4 B )$ .   
+Best density estimation result is obtained with $7 \mathrm { x } 4$ receptive field experiments.
+
+# C SOFT FREE BITS
+
+”Free bits” was a technique proposed in (Kingma et al., 2016) where $K$ groups of stochastic units are encouraged to be used through the following surrogate objective:
+
+$$
+\widetilde { \mathcal { L } } _ { \lambda } = \mathbb { E } _ { \mathbf { x } \sim \mathcal { M } } \left[ \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } ) } \left[ \log p ( \mathbf { x } | \mathbf { z } ) \right] \right] - \sum _ { j = 1 } ^ { K } \operatorname* { m a x i m u m } ( \lambda , \mathbb { E } _ { \mathbf { x } \sim \mathcal { M } } \left[ D _ { K L } ( q ( \mathbf { z } _ { j } | \mathbf { x } ) | | p ( \mathbf { z } _ { j } ) ) \right] )
+$$
+
+This technique is easy to use since it’s usually easy to determine the minimum number of bits/nats, $\lambda$ , stochastic units need to encode. Choosing $\lambda$ is hence easier than setting a fixed KL annealing schedule (Serban et al., 2016).
+
+On the other hand, Kl annealing has the benefit of the surrogate objective will smoothly become the true objective, the variational lower bound where as ”free bits” has a sharp transition at the boundary. Therefore, we propose to still use $\lambda$ as hyperparameter to specify at least $\lambda$ nats should be used but try to change the optimization objective as slowly as possible:
+
+$$
+\mathcal { L } _ { \mathrm { S o f t F r e e B i t s } } ( \mathbf { x } ; \theta ) = \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } ) } \left[ \log p ( \mathbf { x } | \mathbf { z } ) \right] - \gamma D _ { K L } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } ) )
+$$
+
+where $0 < \gamma \leq 1$
+
+And we make the optimization smoother by changing $\gamma$ slowly online to make sure at least $\lambda$ nats are used: when Kl is too much higher than $\lambda$ (we experimented wide range of thresholds from $3 \%$ to $3 0 \%$ , all of which yield improved results, and we tend to use $5 \%$ us a threshold), $\gamma$ is increased, and when Kl lower than $\lambda$ , $\gamma$ is decreased to encourage information flow.
+
+We found it sufficient to increase/decrease at $1 0 \%$ increment and didn’t further tune this parameter.
+
+# D AUTOREGRESSIVE DECODER WITHOUT AUTOREGRESSIVE PRIOR
+
+In this section, we investigate the scenario of just using an autoregressive decoder without using an autoregressive prior. We compare the exact same model in three configurations: 1) using smallreceptive-field PixelCNN as an unconditional density estimator; 2) using small-receptive-field as a decoder in a VAE with Gaussian latent variables; 3) replacing Gaussian latent variables with autoregressive flow latent variables in 2).
+
+Table 1: Ablation on Dynamically binarized MNIST   
+
+<table><tr><td>Model</td><td>NLL Test</td><td>KL</td></tr><tr><td>Unconditional PixelCNN</td><td>87.55</td><td>0</td></tr><tr><td>PixelCNNDecoder+Gaussian Prior</td><td>79.48</td><td>10.60</td></tr><tr><td>PixelCNN Decoder+ AF Prior</td><td>78.94</td><td>11.73</td></tr></table>
+
+In Table 1, we can observe that each step of modification improves density estimation performance. In addition, using an autoregressive latent code makes the latent code transmit more information as shown in the difference of $\mathbb { E } [ D _ { K L } ( q ( { \mathbf { z } } | { \mathbf { x } } ) | | p ( { \mathbf { z } } ) ) ]$ .
+
+# E CIFAR10 GENERATED SAMPLES
+
+# REFERENCES
+
+Martın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016.
+
+![](images/f291dbad13edec3ff9c63635bcb14588a89abd0d94e339a44b41b6c41a847df8.jpg)  
+Figure 4: CIFAR10: Generated samples for different models
+
+Djork-Arne Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast and accurate deep network ´ learning by Exponential Linear Units (ELUs). arXiv preprint arXiv:1511.07289, 2015.
+
+Mathieu Germain, Karol Gregor, Iain Murray, and Hugo Larochelle. Made: Masked autoencoder for distribution estimation. arXiv preprint arXiv:1502.03509, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. arXiv preprint arXiv:1512.03385, 2015.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Diederik P Kingma, Tim Salimans, and Max Welling. Improving variational inference with inverse autoregressive flow. arXiv preprint arXiv:1606.04934, 2016.
+
+Vinod Nair and Geoffrey E Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th International Conference on Machine Learning (ICML-10), pp. 807–814, 2010.
+
+Boris T Polyak and Anatoli B Juditsky. Acceleration of stochastic approximation by averaging. SIAM Journal on Control and Optimization, 30(4):838–855, 1992.
+
+Tim Salimans and Diederik P Kingma. Weight normalization: A simple reparameterization to accelerate training of deep neural networks. arXiv preprint arXiv:1602.07868, 2016.
+
+Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. Pixelcnn $^ { + + }$ : Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517, 2017.
+
+Iulian Vlad Serban, Alessandro Sordoni, Ryan Lowe, Laurent Charlin, Joelle Pineau, Aaron Courville, and Yoshua Bengio. A hierarchical latent variable encoder-decoder model for generating dialogues. arXiv preprint arXiv:1605.06069, 2016.
+
+Aaron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. arXiv preprint arXiv:1601.06759, 2016a.
+
+Aaron van den Oord, Nal Kalchbrenner, Oriol Vinyals, Lasse Espeholt, Alex Graves, and Koray Kavukcuoglu. Conditional image generation with pixelcnn decoders. arXiv preprint arXiv:1606.05328, 2016b.

parse/train/HkxTwkrKDB/HkxTwkrKDB.md

@@ -0,0 +1,350 @@
+# ON UNIVERSAL EQUIVARIANT SET NETWORKS
+
+Nimrod Segol & Yaron Lipman   
+Department of Computer Science and Applied Mathematics   
+Weizmann Institute of Science   
+Rehovot, Israel   
+{nimrod.segol,yaron.lipman}@weizmann.ac.il
+
+# ABSTRACT
+
+Using deep neural networks that are either invariant or equivariant to permutations in order to learn functions on unordered sets has become prevalent. The most popular, basic models are DeepSets (Zaheer et al., 2017) and PointNet (Qi et al., 2017). While known to be universal for approximating invariant functions, DeepSets and PointNet are not known to be universal when approximating equivariant set functions. On the other hand, several recent equivariant set architectures have been proven equivariant universal (Sannai et al., 2019; Keriven & Peyre, 2019), however ´ these models either use layers that are not permutation equivariant (in the standard sense) and/or use higher order tensor variables which are less practical. There is, therefore, a gap in understanding the universality of popular equivariant set models versus theoretical ones.
+
+In this paper we close this gap by proving that: (i) PointNet is not equivariant universal; and (ii) adding a single linear transmission layer makes PointNet universal. We call this architecture PointNetST and argue it is the simplest permutation equivariant universal model known to date. Another consequence is that DeepSets is universal, and also PointNetSeg, a popular point cloud segmentation network (used e.g., in (Qi et al., 2017)) is universal.
+
+The key theoretical tool used to prove the above results is an explicit characterization of all permutation equivariant polynomial layers. Lastly, we provide numerical experiments validating the theoretical results and comparing different permutation equivariant models.
+
+# 1 INTRODUCTION
+
+Many interesting tasks in machine learning can be described by functions $\pmb { F }$ that take as input a set, $\textbf { \textit { X } } = \mathbf { \beta } ( \pmb { x } _ { 1 } , \dots , \pmb { x } _ { n } )$ , and output some per-element features or values, $F ( X ) \ =$ $\bar { ( } F ( X ) _ { 1 } , \ldots , F ( X ) _ { n } )$ . Permutation equivariance is the property required of $\pmb { F }$ so it is welldefined. Namely, it assures that reshuffling the elements in $\boldsymbol { X }$ and applying $\pmb { F }$ results in the same output, reshuffled in the same manner. For example, if $\tilde { \textbf { \textit { X } } } = \mathbf { \Gamma } ( { \bf x } _ { 2 } , { \bf x } _ { 1 } , { \bf x } _ { 3 } , \ldots , { \bf x } _ { n } )$ then $\pmb { F } ( \tilde { \pmb { X } } ) = ( \pmb { F } ( \pmb { X } ) _ { 2 } , \pmb { F } ( \pmb { X } ) _ { 1 } , \pmb { F } ( \pmb { X } ) _ { 3 } , \dots , \pmb { F } ( \pmb { X } ) _ { n } ) .$ .
+
+Building neural networks that are permutation equivariant by construction proved extremely useful in practice. Arguably the most popular models are DeepSets Zaheer et al. (2017) and PointNet Qi et al. (2017). These models enjoy small number of parameters, low memory footprint and computational efficiency along with high empirical expressiveness. Although both DeepSets and PointNet are known to be invariant universal (i.e., can approximate arbitrary invariant continuous functions) they are not known to be equivariant universal (i.e., can approximate arbitrary equivariant continuous functions).
+
+On the other hand, several researchers have suggested theoretical permutation equivariant models and proved they are equivariant universal. Sannai et al. (2019) builds a universal equivariant network by taking $n$ copies of $( n - 1 )$ -invariant networks and combines them with a layer that is not permutation invariant in the standard (above mentioned) sense. Keriven & Peyre (2019) solves a more general ´ problem of building networks that are equivariant universal over arbitrary high order input tensors $\mathbb { R } ^ { n ^ { d } }$ (including graphs); their construction, however, uses higher order tensors as hidden variables which is of less practical value. Yarotsky (2018) proves that neural networks constructed using a finite set of invariant and equivariant polynomial layers are also equivariant universal, however his network is not explicit (i.e., the polynomials are not characterized for the equivariant case) and also of less practical interest due to the high degree polynomial layers.
+
+In this paper we close the gap between the practical and theoretical permutation equivariant constructions and prove:
+
+# Theorem 1.
+
+(i) PointNet is not equivariant universal.   
+(ii) Adding a single linear transmission layer (i.e., $X \mapsto \mathbf { 1 1 } ^ { T } X ,$ ) to PointNet makes it equivariant universal.   
+(iii) Using ReLU activation the minimal width required for universal permutation equivariant network satisfies $\begin{array} { r } { \omega \le k _ { o u t } + k _ { i n } + \binom { n + k _ { i n } } { k _ { i n } } } \end{array}$ .
+
+This theorem suggests that, arguably, PointNet with an addition of a single linear layer is the simplest universal equivariant network, able to learn arbitrary continuous equivariant functions of sets. An immediate corollary of this theorem is
+
+Corollary 1. DeepSets and PointNetSeg are universal.
+
+PointNetSeg is a network used often for point cloud segmentation (e.g., in Qi et al. (2017)). One of the benefit of our result is that it provides a simple characterization of universal equivariant architectures that can be used in the network design process to guarantee universality.
+
+The theoretical tool used for the proof of Theorem 1 is an explicit characterization of the permutation equivariant polynomials over sets of vectors in $\mathbb { R } ^ { k }$ using power-sum multi-symmetric polynomials. We prove:
+
+Theorem 2. Let $P : \mathbb { R } ^ { n \times k }  \mathbb { R } ^ { n \times l }$ be a permutation equivariant polynomial map. Then,
+
+$$
+P ( X ) = \sum _ { | \alpha | \leq n } b _ { \alpha } q _ { \alpha } ^ { T } ,
+$$
+
+where $\pmb { b } _ { \alpha } = ( \pmb { x } _ { 1 } ^ { \alpha } , \dots , \pmb { x } _ { n } ^ { \alpha } ) ^ { T }$ , $\pmb { q } _ { \alpha } = ( q _ { \alpha , 1 } , \dots , q _ { \alpha , l } ) ^ { T }$ , where $q _ { \alpha , j } = q _ { \alpha , j } ( s _ { 1 } , \ldots , s _ { t } )$ , $t = \binom { n + k } { k }$ , are polynomials; $\begin{array} { r } { s _ { j } ( \pmb { X } ) = \sum _ { i = 1 } ^ { n } \pmb { x } _ { i } ^ { \alpha _ { j } } } \end{array}$ k are the power-sum multi-symmetric polynomials. On the other hand every polynomial map $_ { P }$ satisfying Equation is equivariant.
+
+This theorem, which extends Proposition 2.27 in Golubitsky & Stewart (2002) to sets of vectors using multivariate polynomials, lends itself to expressing arbitrary equivariant polynomials as a composition of entry-wise continuous functions and a single linear transmission, which in turn facilitates the proof of Theorem 1.
+
+We conclude the paper by numerical experiments validating the theoretical results and testing several permutation equivariant networks for the tasks of set classification and regression.
+
+# 2 PRELIMINARIES
+
+Equivariant maps. Vectors $\pmb { x } \in \mathbb { R } ^ { k }$ are by default column vectors; $\mathbf { 0 } , \mathbf { 1 }$ are the all zero and all one vectors/tensors; $e _ { i }$ is the $i$ -th standard basis vector; $\pmb { I }$ is the identity matrix; all dimensions are inferred from context or mentioned explicitly. We represent a set of $n$ vectors in $\mathbb { R } ^ { k }$ as a matrix $\ b { X } \in \mathbb { R } ^ { n \times k }$ and denote $\pmb { X } = ( \pmb { x } _ { 1 } , \pmb { x } _ { 2 } , \dots , \pmb { x } _ { n } ) ^ { T }$ , where $\mathbf { { \boldsymbol { x } } } _ { i } \in \mathbb { R } ^ { k }$ , $i \in [ n ]$ , are the columns of $\boldsymbol { X }$ . We denote by $S _ { n }$ the permutation group of $[ n ]$ ; its action on $\boldsymbol { X }$ is defined by $\boldsymbol { \sigma } \cdot \pmb { X } = ( \pmb { x } _ { \sigma ^ { - 1 } ( 1 ) } , \pmb { x } _ { \sigma ^ { - 1 } ( 2 ) } , \dots , \pmb { x } _ { \sigma ^ { - 1 } ( n ) } ) ^ { T }$ $\sigma \in S _ { n }$ . That is, $\sigma$ is reshuffling the rows of $\boldsymbol { X }$ . The natural class of maps assigning a value or feature vector to every element in an input set is permutation equivariant maps:
+
+Definition 1. A map $F : \mathbb { R } ^ { n \times k }  \mathbb { R } ^ { n \times l }$ satisfying $\pmb { F } ( \sigma \cdot \pmb { X } ) = \sigma \cdot \pmb { F } ( \pmb { X } )$ for all $\sigma \in S _ { n }$ and $\ b { X } \in \mathbb { R } ^ { n \times d }$ is called permutation equivariant.
+
+Power-sum multi-symmetric polynomials. Given a vector $z = ( z _ { 1 } , \ldots , z _ { n } ) \in \mathbb { R } ^ { n }$ the powersum symmetric polynomials $\begin{array} { r } { s _ { j } ( z ) = \sum _ { i = 1 } ^ { n } z _ { i } ^ { j } } \end{array}$ , with $j \in [ n ]$ , uniquely characterize $_ { z }$ up to permuting its entries. In other words, for $z , y \in \mathbb { R } ^ { n }$ we have $\pmb { y } = \sigma \cdot \pmb { z }$ for some $\sigma \in S _ { n }$ if and only if $s _ { j } ( \pmb { y } ) =$ $s _ { j } ( z )$ for all $j \in [ n ]$ . An equivalent property is that every $S _ { n }$ invariant polynomial $p$ can be expressed as a polynomial in the power-sum symmetric polynomials, i.e., $p ( z ) = q ( s _ { 1 } ( z ) , \ldots , s _ { n } ( z ) )$ , see Rydh (2007) Corollary 8.4, Briand (2004) Theorem 3. This fact was previously used in Zaheer et al. (2017) to prove that DeepSets is universal for invariant functions. We extend this result to equivariant functions and the multi-feature (sets of vectors) case.
+
+For a vector $\pmb { x } \in \mathbb { R } ^ { k }$ and a multi-index vector $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { k } ) \in \mathbb { N } ^ { k }$ we define $\pmb { x } ^ { \alpha } = x _ { 1 } ^ { \alpha _ { 1 } } \cdot \cdot \cdot x _ { k } ^ { \alpha _ { k } }$ and $\begin{array} { r } { | \alpha | = \sum _ { i \in [ k ] } \alpha _ { i } } \end{array}$ . A generalization of the power-sum symmetric polynomials to matrices exists and is called power-sum multi-symmetric polynomials, defined with a bit of notation abuse: $\begin{array} { r } { s _ { \alpha } ( { \pmb X } ) = \sum _ { i = 1 } ^ { n } { \pmb x } _ { i } ^ { \alpha } } \end{array}$ , where $\alpha \in \mathbb { N } ^ { k }$ is a multi-index satisfying $| { \boldsymbol { \alpha } } | \leq n$ . Note that the number of power-sum multi-symmetric polynomials acting on $\ b { X } \in \mathbb { R } ^ { n \times k }$ is $t = \binom { n + k } { k }$ . For notation simplicity let $\alpha _ { 1 } , \ldots , \alpha _ { t }$ be a list of all $\alpha \in \mathbb { N } ^ { k }$ with $| \alpha | \le n$ . Then we index the collection of power-sum multi-symmetric polynomials as $s _ { 1 } , \ldots , s _ { t }$ .
+
+Similarly to the vector case the numbers $s _ { j } ( X )$ , $j \in [ t ]$ characterize $\boldsymbol { X }$ up to permutation of its rows. That is $Y = \sigma \cdot X$ for some $\sigma \in S _ { n }$ iff $s _ { j } ( \dot { Y } ) = s _ { j } ( Y )$ for all $j \in [ t ]$ . Furthermore, every $S _ { n }$ invariant polynomial $p : \mathbb { R } ^ { n \times k }  \mathbb { R }$ can be expressed as a polynomial in the power-sum multi-symmetric polynomials (see (Rydh, 2007) corollary 8.4), i.e.,
+
+$$
+p ( { \pmb X } ) = q ( s _ { 1 } ( { \pmb X } ) , \ldots , s _ { t } ( { \pmb X } ) ) ,
+$$
+
+These polynomials were recently used to encode multi-sets in Maron et al. (2019).
+
+# 3 EQUIVARIANT MULTI-SYMMETRIC POLYNOMIAL LAYERS
+
+In this section we develop the main theoretical tool of this paper, namely, a characterization of all permutation equivariant polynomial layers. As far as we know, these layers were not fully characterized before.
+
+Theorem 2 provides an explicit representation of arbitrary permutation equivariant polynomial maps $P : \mathbb { R } ^ { n \times k ^ { \ast } }  \mathbb { R } ^ { n \times l }$ using the basis of power-sum multi-symmetric polynomials, $s _ { i } ( X )$ . The particular use of power-sum polynomials $s _ { i } ( X )$ has the advantage it can be encoded efficiently using a neural network: as we will show $s _ { i } ( X )$ can be approximated using a PointNet with a single linear transmission layer. This allows approximating an arbitrary equivariant polynomial map using PointNet with a single linear transmission layer.
+
+A version of this theorem for vectors instead of matrices (i.e., the case of $k = 1$ ) appears as Proposition 2.27 in Golubitsky & Stewart (2002); we extend their proof to matrices, which is the relevant scenario for ML applications as it allows working with sets of vectors. For $k = 1$ Theorem 2 reduces to the following form: $\begin{array} { r } { \pmb { p } ( \pmb { x } ) _ { i } = \sum _ { a \leq n } p _ { a } \big ( s _ { 1 } ( \pmb { x } ) , \ldots , s _ { n } ( \pmb { x } ) \big ) x _ { i } ^ { a } } \end{array}$ with $\begin{array} { r } { s _ { j } ( \pmb { x } ) = \sum _ { i } x _ { i } ^ { j } } \end{array}$ . For matrices the monomial $x _ { i } ^ { k }$ is replaced by $\mathbf { \mathscr { x } } _ { i } ^ { \alpha }$ for a multi-index $\alpha$ and the power-sum symmetric polynomials are replaced by the power-sum multi-symmetric polynomials.
+
+First, note that it is enough to prove Theorem 1 for $l = 1$ and apply it to every column of $_ { r }$ . Hence, we deal with a vector of polynomials $p : \mathbb { R } ^ { n \times k }  \mathbb { R } ^ { n }$ and need to prove it can be expressed as $\begin{array} { r } { p = \sum _ { | \alpha | \leq n } { b _ { \alpha } q _ { \alpha } } } \end{array}$ , for $S _ { n }$ invariant polynomial $q _ { \alpha }$ .
+
+Given a polynomial $p ( X )$ and the cyclic permutation $\sigma ^ { - 1 } = ( 1 2 3 \cdot \cdot \cdot n )$ the following operation, taking a polynomial to a vector of polynomials, is useful in characterizing equivariant polynomial maps:
+
+$$
+\lceil p \rceil ( \boldsymbol { X } ) = \left( \begin{array} { c } { p ( \boldsymbol { X } ) } \\ { p ( \sigma \cdot \boldsymbol { X } ) } \\ { p ( \sigma ^ { 2 } \cdot \boldsymbol { X } ) } \\ { \vdots } \\ { p ( \sigma ^ { n - 1 } \cdot \boldsymbol { X } ) } \end{array} \right)
+$$
+
+Theorem 2 will be proved using the following two lemmas:
+
+Lemma 1. Let $p : \mathbb { R } ^ { n \times k }  \mathbb { R } ^ { n }$ be an equivariant polynomial map. Then, there exists a polynomial $p : \mathbb { R } ^ { n \times k }  \mathbb { R } ,$ , invariant to $S _ { n - 1 }$ (permuting the last $n - 1$ rows of $\boldsymbol { X }$ ) so that $p = \lceil p \rceil$ .
+
+Proof. Equivariance of $\pmb { p }$ means that for all $\sigma \in S _ { n }$ it holds that ${ \boldsymbol { \sigma } } \cdot { \boldsymbol { p } } ( { \boldsymbol { X } } ) = { \boldsymbol { p } } ( { \boldsymbol { \sigma } } \cdot { \boldsymbol { X } } )$
+
+$$
+\sigma \cdot p ( X ) = p ( \sigma \cdot X ) .
+$$
+
+Choosing an arbitrary permutation $\sigma \in \operatorname { s t a b } ( 1 ) < S _ { n }$ , namely a permutation satisfying $\sigma ( 1 ) = 1$ , and observing the first row in Equation 4 we get $p _ { 1 } ( { \pmb X } ) = p _ { 1 } ( { \sigma } \cdot { \pmb X } ) = p _ { 1 } ( { \pmb x } _ { 1 } , { \pmb x } _ { \sigma ^ { - 1 } ( 2 ) } , \dots , { \pmb x } _ { \sigma ^ { - 1 } ( n ) } )$ . Since this is true for all $\sigma \in \mathrm { s t a b } ( 1 ) p _ { 1 }$ is $S _ { n - 1 }$ invariant. Next, applying $\sigma = ( 1 i )$ to Equation 4 and observing the first row again we get $p _ { i } ( { \cal X } ) = p _ { 1 } ( \pmb { x } _ { i } , . . . , \pmb { x } _ { 1 } , . . . )$ . Using the invariance of $p _ { 1 }$ to $S _ { n - 1 }$ we get $\bar { \boldsymbol { p } } = \lceil p _ { 1 } \rceil$ . □
+
+Lemma 2. Let $p : \mathbb { R } ^ { n \times k }  \mathbb { R }$ be a polynomial invariant to $S _ { n - 1 }$ (permuting the last $n - 1$ rows of $X$ ) then
+
+$$
+p ( { \pmb X } ) = \sum _ { | \alpha | \leq n } { \pmb x } _ { 1 } ^ { \alpha } q _ { \alpha } ( { \pmb X } ) ,
+$$
+
+where $q _ { \alpha }$ are $S _ { n }$ invariant.
+
+Proof. Expanding $p$ with respect to $\scriptstyle { \mathbf { \mathscr { x } } } _ { 1 }$ we get
+
+$$
+p ( \pmb { X } ) = \sum _ { | \alpha | \leq m } \pmb { x } _ { 1 } ^ { \alpha } p _ { \alpha } ( \pmb { x } _ { 2 } , \ldots , \pmb { x } _ { n } ) ,
+$$
+
+for some $\textit { m } \in  { \mathbb { N } }$ . We first claim $p _ { \alpha }$ are $S _ { n - 1 }$ invariant. Indeed, note that if $p ( \boldsymbol { X } ) \ =$ $p ( \pmb { x } _ { 1 } , \pmb { x } _ { 2 } , \ldots , \pmb { x } _ { n } )$ is $S _ { n - 1 }$ invariant, i.e., invariant to permutations of $\pmb { x } _ { 2 } , \ldots , \pmb { x } _ { n }$ , then also its derivative s ∂|β|∂xβ p(X) are Sn−1 permutation invariant, for all β ∈ Nk. Taking the derivative ∂|β|/∂xβ1 on both sides of Equation 6 we get that $p _ { \beta }$ is $S _ { n - 1 }$ equivariant.
+
+For brevity denote $p = p _ { \alpha }$ . Since $p$ is $S _ { n - 1 }$ invariant it can be expressed as a polynomial in the powersum multi-symmetric polynomials, i.e., $p ( \pmb { x } _ { 2 } , \ldots , \pmb { x } _ { n } ) = r ( s _ { 1 } ( \pmb { x } _ { 2 } , \ldots , \pmb { x } _ { n } ) , \ldots , s _ { t } ( \pmb { x } _ { 2 } , \ldots , \pmb { x } _ { n } ) )$ . Note that $s _ { i } ( { \pmb x } _ { 2 } , \ldots , { \pmb x } _ { n } ) = s _ { i } ( { \pmb X } ) - { \pmb x } _ { 1 } ^ { \alpha _ { i } }$ and therefore
+
+$$
+p ( \pmb { x } _ { 2 } , \ldots , \pmb { x } _ { n } ) = r ( s _ { 1 } ( \pmb { X } ) - \pmb { x } _ { 1 } ^ { \alpha _ { 1 } } , \ldots , s _ { t } ( \pmb { X } ) - \pmb { x } _ { 1 } ^ { \alpha _ { t } } ) .
+$$
+
+Since $r$ is a polynomial, expanding its monomials in $s _ { i } ( X )$ and $\pmb { x } _ { 1 } ^ { \alpha }$ shows $p$ can be expressed as $\begin{array} { r } { p = \sum _ { | \alpha | \leq m ^ { \prime } } \bar { \pmb { x } } _ { 1 } ^ { \alpha } \tilde { p } _ { \alpha } } \end{array}$ , where $m ^ { \prime } \in \mathbb { N }$ , and $\tilde { p } _ { \alpha }$ are $S _ { n }$ invariant (as multiplication of invariant $S _ { n }$ polynomials $s _ { i } ( X ) \rangle$ . Plugging this in Equation 6 we get Equation 5, possibly with the sum over some $n ^ { \prime } > n$ . It remains to show $n ^ { \prime }$ can be taken to be at-most $n$ . This is proved in Corollary 5 in Briand (2004) □
+
+Proof. (Theorem 2) Given an equivariant $\pmb { p }$ as above, use Lemma 1 to write $p = \lceil p \rceil$ where $p ( X )$ is invariant to permuting the last $n - 1$ rows of $\boldsymbol { X }$ . Use Lemma 2 to write $\begin{array} { r } { p ( \pmb { X } ) = \sum _ { | \alpha | \leq n } \pmb { x } _ { 1 } ^ { \alpha } q _ { \alpha } ( \pmb { X } ) } \end{array}$ , where $q _ { \alpha }$ are $S _ { n }$ invariant. We get,
+
+$$
+p = \lceil p \rceil = \sum _ { | \alpha | \leq n } b _ { \alpha } q _ { \alpha } .
+$$
+
+The converse direction is immediate after noting that $b _ { \alpha }$ are equivariant and $q _ { \alpha }$ are invariant.
+
+# 4 UNIVERSALITY OF SET EQUIVARIANT NEURAL NETWORKS
+
+We consider equivariant deep neural networks $f : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R } ^ { n \times k _ { o u t } } ,$ ,
+
+$$
+{ \pmb F } ( { \pmb X } ) = { \pmb L } _ { m } \circ \nu \circ \cdots \circ \nu \circ { \pmb L } _ { 1 } ( { \pmb X } ) ,
+$$
+
+where $L _ { i } : \mathbb { R } ^ { n \times k _ { i } }  \mathbb { R } ^ { n \times k _ { i + 1 } }$ are affine equivariant transformations, and $\nu$ is an entry-wise nonlinearity (e.g., ReLU). We define the width of the network to be $\omega = \mathrm { m a x } _ { i } k _ { i }$ ; note that this definition is different from the one used for standard MLP where the width would be $n \omega$ , see e.g., Hanin & Sellke (2017). Zaheer et al. (2017) proved that affine equivariant $\mathbf { { L } } _ { i }$ are of the form
+
+$$
+L _ { i } ( X ) = X A + \frac { 1 } { n } \mathbf { 1 1 } ^ { T } X B + \mathbf { 1 } c ^ { T } ,
+$$
+
+where $A , B \in \mathbb { R } ^ { k _ { i } \times k _ { i + 1 } }$ , and $\boldsymbol { c } \in \mathbb { R } ^ { k _ { i + 1 } }$ are the layer’s trainable parameters; we call the linear transformation $\textstyle X \mapsto { \frac { 1 } { n } } \mathbf { 1 } \mathbf { 1 } ^ { T } X B$ a linear transmission layer.
+
+We now define the equivariant models considered in this paper: The DeepSets (Zaheer et al., 2017) architecture is Equation 7 with the choice of layers as in Equation 8. Taking $B = 0$ in all layers is the PointNet architecture (Qi et al., 2017). PointNetST is an equivariant model of the form Equation 7 with layers as in Equation 8 where only a single layer $\mathbf { { { L } } } _ { i }$ has a non-zero $\textbf {  { B } }$ . The PointNetSeg (Qi et al., 2017) architecture is PointNet composed with an invariant max layer, namely $\begin{array} { r } { \operatorname* { m a x } ( \boldsymbol { F } ( \boldsymbol { X } ) ) _ { j } = \operatorname* { m a x } _ { i \in [ n ] } \boldsymbol { F } ( \boldsymbol { X } ) _ { i , j } } \end{array}$ and then concatenating it with the input $\boldsymbol { X }$ , i.e., $[ X , \mathbf { 1 } \mathbf { m a x } ( F ( X ) ) ]$ , and feeding is as input to another PointNet $G$ , that is $G ( [ X , \mathbf { 1 } \operatorname* { m a x } ( F ( X ) ) ] )$ .
+
+We will prove PointNetST is permutation equivariant universal and therefore arguably the simplest permutation equivariant universal model known to date.
+
+Universality of equivariant deep networks is defined next.
+
+Definition 2. Permutation equivariant universality1 of a model $F : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R } ^ { n \times k _ { o u t } }$ means that for every permutation equivariant continuous function $H : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R } ^ { n \times k _ { o u t } }$ defined over the cube $\overset { \cdot } { K } \overset { \cdot } { = } [ 0 , 1 ] ^ { n \times k _ { i n } } \subset \overset { \cdot } { \mathbb { R } } ^ { n \times k _ { i n } }$ , and $\epsilon > 0$ there exists a choice of m (i.e., network depth), $k _ { i }$ (i.e., network width) and the trainable parameters of $\pmb { F }$ so that $\| H ( X ) - F ( X ) \| _ { \infty } < \epsilon$ for all $X \in K$ .
+
+Proof. (Theorem $^ { l }$ ) Fact $( i )$ , namely that PointNet is not equivariant universal is a consequence of the following simple lemma:
+
+Lemma 3. Let $\pmb { h } = ( h _ { 1 } , \ldots , h _ { n } ) ^ { T } : \mathbb { R } ^ { n }  \mathbb { R } ^ { n }$ be the equivariant linear function defined by $h ( \pmb { x } ) = \mathbf { 1 } \mathbf { 1 } ^ { T } \pmb { x }$ . There is no $f : \mathbb { R } \to \mathbb { R }$ so that $\begin{array} { r } { | h _ { i } ( { \pmb x } ) - f ( x _ { i } ) | < \frac { 1 } { 2 } } \end{array}$ for all $i \in [ n ]$ and $\pmb { x } \in [ 0 , 1 ] ^ { n }$ .
+
+Proof. Assume such $f$ exists. Let $\pmb { e } _ { 1 } = ( 1 , 0 , \ldots , 0 ) ^ { T } \in \mathbb { R } ^ { n }$ . Then,
+
+$$
+1 = | h _ { 2 } ( e _ { 1 } ) - h _ { 2 } ( { \bf 0 } ) | \leq | h _ { 2 } ( e _ { 1 } ) - f ( 0 ) | + | f ( 0 ) - h _ { 2 } ( { \bf 0 } ) | < 1
+$$
+
+reaching a contradiction.
+
+To prove $( i i )$ we first reduce the problem from the class of all continuous equivariant functions to the class of equivariant polynomials. This is justified by the following lemma.
+
+Lemma 4. Equivariant polynomials $P : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R } ^ { n \times k _ { o u t } }$ are dense in the space of continuous equivariant functions $F : \dot { \mathbb { R } ^ { n \times k _ { i n } } }  \mathbb { R } ^ { n \times k _ { o u t } }$ over the cube $K$ .
+
+Proof. Take an arbitrary $\epsilon > 0$ . Consider the function $f _ { i j } : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R }$ , which denotes the $( i , j )$ -th output entry of $\pmb { F }$ . By the Stone-Weierstrass Theorem there exists a polynomial $p _ { i j } : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R }$ such that $\left\| f _ { i j } ( X ) - p _ { i j } ( X ) \right\| _ { \infty } \leq \epsilon$ for all $X \in K$ . Consider the polynomial map $P : \mathbb { R } ^ { n \times k _ { i n } } $ $\mathbb { R } ^ { n \times k _ { o u t } }$ defined by $( P ) _ { i j } = p _ { i j }$ . $_ { P }$ is in general not equivariant. To finish the proof we will symmetrize $_ { r }$ :
+
+$$
+\begin{array} { r l } { \bigg | F ( \pmb X ) - \displaystyle \frac { 1 } { n ! } \sum _ { \sigma \in S _ { n } } \sigma \cdot \pmb P ( \sigma ^ { - 1 } \cdot \pmb X ) \bigg | \Bigg | _ { \infty } = \bigg | \bigg | \frac { 1 } { n ! } \displaystyle \sum _ { \sigma \in S _ { n } } \sigma \cdot \pmb F ( \sigma ^ { - 1 } \cdot \pmb X ) - \frac { 1 } { n ! } \displaystyle \sum _ { \sigma \in S _ { n } } \sigma \cdot \pmb P ( \sigma ^ { - 1 } \cdot \pmb X ) \bigg | \bigg | _ { \infty } } & { } \\ { = \bigg | \bigg | \frac { 1 } { n ! } \displaystyle \sum _ { \sigma \in S _ { n } } \sigma \cdot \big ( F ( \sigma ^ { - 1 } \cdot \pmb X ) - \pmb P ( \sigma ^ { - 1 } \cdot \pmb X ) \big ) \bigg | \bigg | _ { \infty } } & { } \\ { \leq \frac { 1 } { n ! } \displaystyle \sum _ { \sigma \in S _ { n } } \epsilon = \epsilon , } \end{array}
+$$
+
+where in the first equality we used the fact that $\pmb { F }$ is equivariant. This concludes the proof since $\begin{array} { r l } { \sum _ { \sigma \in S _ { n } } \sigma \cdot P ( \sigma ^ { - 1 } \cdot \overline { { X } } ) } & { { } } \end{array}$ is an equivariant polynomial map. □
+
+Now, according to Theorem 2 an arbitrary equivariant polynomial $P : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R } ^ { n \times k _ { o u t } }$ can be written as $\begin{array} { r } { P \ = \ \sum _ { | \alpha | \leq n } { b } _ { \alpha } ( X ) \dot { q _ { \alpha } } ( \dot { X } ) ^ { T } } \end{array}$ , where $\bar { b _ { \alpha } } ( \mathbf { X } ) \ = \ \left\lceil \mathbf { x } _ { 1 } ^ { \alpha } \right\rceil \ \in \ \mathbb { R } ^ { n }$ and $\begin{array} { r l } { q _ { \alpha } } & { { } = } \end{array}$ $( q _ { \alpha , 1 } , \dots , q _ { \alpha , k _ { o u t } } ) \in \mathbb { R } ^ { k _ { o u t } }$ are invariant polynomials. Remember that every $S _ { n }$ invariant polynomial can be expressed as a polynomial in the $t = \binom { n + k _ { i n } } { k _ { i n } }$ power-sum multi-symmetric polynomials $\begin{array} { r } { s _ { j } ( { \pmb X } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } { \pmb x } _ { i } ^ { \alpha _ { j } } , j \in [ t ] } \end{array}$ (we use the $1 / n$ normalized version for a bit more simplicity later on). We can therefore write $_ { r }$ as composition of three maps:
+
+$$
+P = Q \circ L \circ B ,
+$$
+
+where $\pmb { B } : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R } ^ { n \times t }$ is defined by
+
+$$
+B ( X ) = ( b ( x _ { 1 } ) , \dots , b ( x _ { n } ) ) ^ { T } ,
+$$
+
+${ \pmb b } ( { \pmb x } ) = ( { \pmb x } ^ { \alpha _ { 1 } } , \ldots , { \pmb x } ^ { \alpha _ { t } } ) ; L$ is defined as in Equation 8 with $\boldsymbol { B } = [ \mathbf { 0 } , I ]$ and $\pmb { A } = [ e _ { 1 } , \ldots , e _ { k _ { i n } } , \pmb { 0 } ]$ , where $\ b { I } \in \mathbb { R } ^ { t \times t }$ the identity matrix and $\boldsymbol { e } _ { i } \in \mathbb { R } ^ { t }$ represents the standard basis (as-usual). We assume $\boldsymbol { \alpha } _ { j } = \boldsymbol { e } _ { j } \in \mathbb { R } ^ { k _ { i n } }$ , for $j \in [ \bar { k } _ { i n } ]$ . Note that the output of $\pmb { L }$ is of the form
+
+$$
+\pmb { L } ( \pmb { B } ( \pmb { X } ) ) = ( \pmb { X } , \pmb { 1 } s _ { 1 } ( \pmb { X } ) , \pmb { 1 } s _ { 2 } ( \pmb { X } ) , \dots , \pmb { 1 } s _ { t } ( \pmb { X } ) ) .
+$$
+
+Finally, $Q : \mathbb { R } ^ { n \times ( k _ { i n } + t ) }  \mathbb { R } ^ { n \times k _ { o u t } }$ is defined by
+
+$$
+Q ( X , \mathbf { 1 } s _ { 1 } , \ldots , \mathbf { 1 } s _ { t } ) = ( q ( x _ { 1 } , s _ { 1 } , \ldots , s _ { t } ) , \ldots , q ( x _ { n } , s _ { 1 } , \ldots , s _ { t } ) ) ^ { T } ,
+$$
+
+The decomposition in Equation 9 of $_ { r }$ suggests that replacing $Q , B$ with Multi-Layer Perceptrons (MLPs) would lead to a universal permutation equivariant network consisting of PointNet with a single linear transmission layer, namely PointNetST.
+
+The $\pmb { F }$ approximating $_ { P }$ will be defined as
+
+$$
+\pmb { F } = \pmb { \Psi } \circ \pmb { L } \circ \pmb { \Phi } ,
+$$
+
+![](images/796c0b9d0b7cceb714674120d0a995bcd66e93c12cd3a844d159a94be243f91d.jpg)  
+Figure 1: The construction of the universal network (PointNetST).
+
+where $\Phi : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R } ^ { n \times t }$ and $\Psi : \mathbb { R } ^ { n \times ( t + k _ { i n } ) } $ $\mathbb { R } ^ { n \times k _ { o u t } }$ are both of PointNet architecture, namely there exist MLPs $\phi : \mathbb { R } ^ { k _ { i n } }  \mathbb { R } ^ { t }$ and $\psi : \mathbb { R } ^ { t + k _ { i n } } \stackrel { \cdot } {  }$ $\mathbb { R } ^ { k _ { o u t } }$ so that $\Phi ( X ) = ( \phi ( { \pmb x } _ { 1 } ) , \dots , \phi ( { \pmb x } _ { n } ) ) ^ { T }$ and $\pmb { \Psi } ( \pmb { X } ) = ( \pmb { \psi } ( \pmb { x } _ { 1 } ) , \dots , \pmb { \psi } ( \pmb { x } _ { n } ) ) ^ { T }$ . See Figure 1 for an illustration of ${ \pmb F } .$
+
+To build the MLPs $\phi , \psi$ we will first construct $\psi$ to approximate $\pmb q$ , that is, we use the universality of MLPS (see (Hornik, 1991; Sonoda & Murata, 2017; Hanin & Sellke, 2017)) to construct $\psi$ so that $\begin{array} { r } { \| \psi ( \pmb { x } , s _ { 1 } , . . . , s _ { t } ) - \pmb { q } ( \pmb { x } , s _ { 1 } , . . . , s _ { t } ) \| _ { \infty } < \frac { \epsilon } { 2 } } \end{array}$ for all $( \pmb { x } , s _ { 1 } , \dots , s _ { t } ) \in [ 0 , 1 ] ^ { k _ { i n } + t }$ . Furthermore, as $\psi$ over $[ 0 , 1 ] ^ { k _ { i n } + t }$ is uniformly continuous, let $\delta$ be such that if $z , z ^ { \prime } \in [ 0 , 1 ] ^ { k _ { i n } + t } , \| z - z ^ { \prime } \| _ { \infty } < \delta$ then $\| \dot { \psi ( z ) } - \psi ( z ^ { \prime } ) \| _ { \infty } < \frac { \epsilon } { 2 }$ . Now, we use universality again to construct $\phi$ approximating $^ { b }$ , that is we take $\phi$ so that $\| \phi ( \pmb { x } ) - \pmb { b } ( \pmb { x } ) \| _ { \infty } < \delta$ for all $\pmb { x } \in [ 0 , 1 ] ^ { k _ { i n } }$ .
+
+$$
+{ \begin{array} { r l } & { F ( { \boldsymbol { X } } ) - P ( { \boldsymbol { X } } ) \| _ { \infty } \leq \| \Psi ( { \boldsymbol { L } } ( \Phi ( { \boldsymbol { X } } ) ) ) - \Psi ( { \boldsymbol { L } } ( B ( { \boldsymbol { X } } ) ) ) \| _ { \infty } + \| \Psi ( { \boldsymbol { L } } ( B ( { \boldsymbol { X } } ) ) ) - Q ( { \boldsymbol { L } } ( B ( { \boldsymbol { X } } ) ) ) \| _ { \infty } } \\ & { \qquad = \operatorname { e r r } _ { 1 } + \operatorname { e r r } _ { 2 } } \end{array} }
+$$
+
+First, $\left\| L ( \Phi ( X ) ) - L ( B ( X ) ) \right\| _ { \infty } < \delta$ for all $X \in K$ and therefore $\mathrm { e r r } _ { 1 } < \frac { \epsilon } { 2 }$ . Second, note that if $X \in K$ then $B ( X ) \in [ 0 , 1 ] ^ { n \times t }$ and $L ( B ( X ) ) \in [ 0 , 1 ] ^ { n \times ( k _ { i n } + t ) }$ . Therefore by construction of $\psi$ we have $\mathrm { e r r } _ { 2 } < \frac { \epsilon } { 2 }$ .
+
+To prove $( i i i )$ we use the result in Hanin & Sellke (2017) (see Theorem 1) bounding the width of an MLP approximating a function $f : [ 0 , 1 ] ^ { d _ { i n } } \to \mathbb { R } ^ { d _ { o u t } }$ by $d _ { i n } + d _ { o u t }$ . Therefore, the width of the MLP $\phi$ is bounded by $k _ { i n } + t$ , where the width of the MLP $\psi$ is bounded by $t + k _ { i n } + k _ { o u t }$ , proving the bound. □
+
+We can now prove Cororllary 1.
+
+# Proof. (Corollary 1)
+
+The fact that the DeepSets model is equivariant universal is immediate. Indeed, The PointNetST model can be obtained from the DeepSets model by setting $B = 0$ in all but one layer, with $\textbf {  { B } }$ as in Equation 8.
+
+For the PointNetSeg model note that by Theorem 1 in $\mathrm { Q i }$ et al. (2017) every invariant function $f : \mathbb { R } ^ { n \times k _ { i n } }  \mathbb { R } ^ { t }$ can be approximated by a network of the form $\Psi ( \operatorname* { m a x } ( F ( X ) ) )$ , where $( F ( X )$ is a PointNet model and $\Psi$ is an MLP. In particular, for every $\varepsilon > 0$ there exists such $\mathbfcal { F } , \Phi$ for which $\| \Psi ( \operatorname* { m a x } ( F ( X ) ) ) - ( s _ { 1 } ( X ) , \ldots , s _ { t } ( X ) ) \| _ { \infty } < \varepsilon$ for every $\boldsymbol { X } \in [ 0 , 1 ] ^ { n \times k _ { i n } }$ where $s _ { 1 } ( X ) , \ldots , s _ { t } ( X )$ are the power-sum multi-symmetric polynomials. It follows that we can use PointNetSeg to approximate ${ \bf 1 } ( s _ { 1 } ( X ) , \dots , s _ { t } ( \dot { X } ) )$ . The rest of the proof closely resembles the proof of Theorem 1.
+
+Graph neural networks with constructed adjacency. One approach sometimes applied to learning from sets of vectors is to define an adjacency matrix (e.g., by thresholding distances of node feature vectors) and apply a graph neural network to the resulting graph (e.g., Wang et al. (2019), Li et al. (2019)). Using the common message passing paradigm (Gilmer et al., 2017) in this case boils to layers of the form: $\begin{array} { r } { \pmb { L } ( \pmb { X } ) _ { i } = \psi ( \pmb { x } _ { i } , \sum _ { j \in N _ { i } } \tilde { \phi } ( \pmb { x } _ { i } , \pmb { x } _ { j } ) ) = \psi ( \pmb { x } _ { i } , \sum _ { j \in [ n ] } N ( \pmb { x } _ { i } , \pmb { x } _ { j } ) \phi ( \pmb { x } _ { i } , \pmb { x } _ { j } ) ) } \end{array}$ , where $\phi , \psi$ are MLPs, $N _ { i }$ is the index set of neighbors of node $i$ , and $N ( \pmb { x } _ { i } , \pmb { x } _ { j } )$ is the indicator function for the edge $( i , j )$ . If $N$ can be approximated by a continuous function, which is the case at least in the $L _ { 2 }$ sense for a finite set of vectors, then since $\pmb { L }$ is also equivariant it follows from Theorem 1 that such a network can be approximated (again, at least in $L _ { 2 }$ norm) arbitrarily well by any universal equivariant network such as PointNetST or DeepSets.
+
+We tested the ability of a DeepSets model with varying depth and width to approximate a single graph convolution layer. We found that a DeepSets model with a small number of layers can approximate a graph convolution layer reasonably well. For details see Appendix A.
+
+# 5 EXPERIMENTS
+
+We conducted experiments in order to validate our theoretical observations. We compared the results of several equivariant models, as well as baseline (full) MLP, on three equivariant learning tasks: a classification task (knapsack) and two regression tasks (squared norm and Fiedler eigen vector). For all tasks we compare results of 7 different models: DeepSets, PointNet, PointNetSeg, PointNetST, PointNetQT and GraphNet. PointNetQT is PointNet with a single quadratic equivariant transmission layer as defined in Appendix B. GraphNet is similar to the graph convolution network in Kipf & Welling (2016) and is defined explicitly in Appendix B. We generated the adjacency matrices for GraphNet by taking 10 nearest neighbors of each set element. In all experiments we used a network of the form Equation 7 with $m = 6$ depth and varying width, fixed across all layers. 2
+
+Equivariant classification. For classification, we chose to learn the multidimensional knapsack problem, which is known to be NP-hard. We are given a set of 4-vectors, represented by $\pmb { X } \in \bar { \mathbb { R } } ^ { n \times 4 }$ and our goal is to learn the equivariant classification function $f : \mathbb { R } ^ { n \times 4 } \stackrel { \cdot } { \to } \left\{ 0 , 1 \right\} ^ { n }$ defined by the following optimization problem:
+
+$$
+\begin{array} { r l } { f ( \pmb { X } ) = \arg \underset { z } { \operatorname* { m a x } } } & { \displaystyle \sum _ { i = 1 } ^ { n } x _ { i 1 } z _ { i } } \\ { \mathrm { s . t . } } & { \displaystyle \sum _ { i = 1 } ^ { n } x _ { i j } z _ { i } \leq w _ { j } , \qquad j = 2 , 3 , 4 } \\ & { z _ { i } \in \{ 0 , 1 \} , \qquad i \in [ n ] } \end{array}
+$$
+
+Intuitively, given a set of vectors $\ b { X } \in \mathbb { R } ^ { n \times 4 }$ , $( \boldsymbol { X } ) _ { i j } = x _ { i j }$ , where each row represents an element in a set, our goal is to find a subset maximizing the value while satisfying budget constraints. The first column of $\boldsymbol { X }$ defines the value of each element, and the three other columns the costs.
+
+To evaluate the success of a trained model we record the percentage of sets for which the predicted subset is such that all the budget constrains are satisfied and the value is within $1 0 \%$ of the optimal value. In Appendix C we detail how we generated this dataset.
+
+Equivariant regression. The first equivariant function we considered for regression is the function $\begin{array} { r } { f ( \mathbf { X } ) = \mathbf { 1 } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { k } ( X _ { i , j } - \frac { 1 } { 2 } ) ^ { 2 } } \end{array}$ . Hanin & Sellke (2017) showed this function cannot be approximated by MLPs of small width. We drew $1 0 k$ training examples and $1 k$ test examples i.i.d. from a $\begin{array} { r } { \mathcal { N } \big ( \frac { 1 } { 2 } , 1 \big ) } \end{array}$ distribution (per entry of $\boldsymbol { X }$ ).
+
+![](images/6904d861b888584598cf9bb58d2ad82b12289646750b069e73dd4fd09ba854b8.jpg)  
+Figure 2: Classification and regression tasks with permutation equivariant models. All the universal permutation equivariant models perform similarly, while the equivariant non-universal PointNet demonstrates reduced performace consistently; MLP baseline (with the same number of parameters as the equivariant models) performs poorly.
+
+The second equivariant function we considered is defined on point clouds $\boldsymbol { X } \in \mathbb { R } ^ { n \times 3 }$ . For each point cloud we computed a graph by connecting every point to its 10 nearest neighbors. We then computed the absolute value of the first non trivial eigenvector of the graph Laplacian. We used the ModelNet dataset ( $\mathrm { W u }$ et al., 2015) which contains $\sim 9 k$ training meshes and $\sim 2 k$ test meshes. The point clouds are generated by randomly sampling 512 points from each mesh.
+
+Result summary. Figure 2 summarizes train and test accuracy of the 6 models after training (training details in Appendix C) as a function of the network width $\omega$ . We have tested $1 5 \omega$ values equidistant in [5, $\textstyle { \frac { n k _ { i n } } { 2 } } \int$ .
+
+As can be seen in the graphs, in all three datasets the equivariant universal models (PointNetST, PointNetQT , DeepSets, PointNetSeg) achieved comparable accuracy. PointNet, which is not equivariant universal, consistently achieved inferior performance compared to the universal models, as expected by the theory developed in this paper. The non-equivariant MLP, although universal, used the same width (i.e., same number of parameters) as the equivariant models and was able to over-fit only on one train set (the quadratic function); its performance on the test sets was inferior by a large margin to the equivariant models. We also note that in general the GraphNet model achieved comparable results to the equivariant universal models but was still outperformed by the DeepSets model.
+
+An interesting point is that although the width used in the experiments in much smaller than the bound $k _ { o u t } + \bar { k _ { i n } } + \binom { n + k _ { i n } } { k _ { i n } }$ established by Theorem 1, the universal models are still able to learn well the functions we tested on. This raises the question of the tightness of this bound, which we leave to future work.
+
+# 6 CONCLUSIONS
+
+In this paper we analyze several set equivariant neural networks and compare their approximation power. We show that while vanilla PointNet (Qi et al., 2017) is not equivariant universal, adding a single linear transmission layer makes it equivariant universal. Our proof strategy is based on a characterization of polynomial equivariant functions. As a corollary we show that the DeepSets model (Zaheer et al., 2017) and PointNetSeg (Qi et al., 2017) are equivariant universal. Experimentally, we tested the different models on several classification and regression tasks finding that adding a single linear transmitting layer to PointNet makes a significant positive impact on performance.
+
+# 7 ACKNOWLEDGEMENTS
+
+This research was supported in part by the European Research Council (ERC Consolidator Grant, LiftMatch 771136) and the Israel Science Foundation (Grant No. 1830/17).
+
+# REFERENCES
+
+Emmanuel Briand. When is the algebra of multisymmetric polynomials generated by the elementary multisymmetric polynomials. Beitrge zur Algebra und Geometrie, 45, 01 2004.
+
+Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. CoRR, 2017.
+
+Martin Golubitsky and Ian Stewart. The symmetry perspective. 2002.
+
+Boris Hanin and Mark Sellke. Approximating continuous functions by relu nets of minimal width. CoRR, abs/1710.11278, 2017. URL http://arxiv.org/abs/1710.11278.
+
+Kurt Hornik. Approximation capabilities of multilayer feedforward networks. Neural networks, 4(2): 251–257, 1991.
+
+Nicolas Keriven and Gabriel Peyre. Universal invariant and equivariant graph neural networks. ´ arXiv preprint arXiv:1905.04943, 2019.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016.
+
+Guohao Li, Matthias Mller, Ali Thabet, and Bernard Ghanem. Deepgcns: Can gcns go as deep as cnns? In The IEEE International Conference on Computer Vision (ICCV), 2019.
+
+Haggai Maron, Heli Ben-Hamu, Hadar Serviansky, and Yaron Lipman. Provably powerful graph networks. CoRR, abs/1905.11136, 2019. URL http://arxiv.org/abs/1905.11136.
+
+Silvano Martello and Paolo Toth. Knapsack Problems: Algorithms and Computer Implementations. John Wiley & Sons, Inc., New York, NY, USA, 1990. ISBN 0-471-92420-2.
+
+Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch. In NIPS Autodiff Workshop, 2017.
+
+Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 652–660, 2017.
+
+David Rydh. A minimal set of generators for the ring of multisymmetric functions. In Annales de l’institut Fourier, volume 57, pp. 1741–1769, 2007.
+
+Akiyoshi Sannai, Yuuki Takai, and Matthieu Cordonnier. Universal approximations of permutation invariant/equivariant functions by deep neural networks. arXiv preprint arXiv:1903.01939, 2019.
+
+Sho Sonoda and Noboru Murata. Neural network with unbounded activation functions is universal approximator. Applied and Computational Harmonic Analysis, 43(2):233–268, 2017.
+
+Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E. Sarma, Michael M. Bronstein, and Justin M. Solomon. Dynamic graph cnn for learning on point clouds. ACM Transactions on Graphics (TOG), 2019.
+
+Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015.
+
+Dmitry Yarotsky. Universal approximations of invariant maps by neural networks. arXiv preprint arXiv:1804.10306, 2018.
+
+Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan R Salakhutdinov, and Alexander J Smola. Deep sets. In Advances in neural information processing systems, pp. 3391–3401, 2017.
+
+# A APPROXIMATING GRAPH CONVOLUTION LAYER WITH DEEPSETS
+
+To test the ability of an equivariant universal model to approximate a graph convolution layer, we conducted an experiment where we applied a single graph convolution layer (see Appendix B for a full description of the graph convolution layers used in this paper) with 3 in features and 10 out features. We constructed a knn graph by taking 10 neighbors. We sampled 1000 examples in $\mathbb { R } ^ { 1 0 0 \times 3 }$ i.i.d from a $\begin{array} { r } { \mathcal { N } ( \frac { 1 } { 2 } , 1 ) } \end{array}$ distribution (per entry of $\boldsymbol { X }$ ). The results are summarized in Figure 3. We regressed to the output of a graph convolution layer using the smooth $L _ { 1 }$ loss.
+
+![](images/4977f9c73f73c91a611a3996fb5c3ae4b489ae9271643b0516748e47e4b99792.jpg)  
+Figure 3: Using DeepSets to regress to a single graph convolution layer. In the left image we varied the depth and took a constant width of 200. In the right we varied the width and took a fixed depth of 6 layers. The $y$ axes are in log scale. Note that even a 2 layer DeepSets can approximate a graph convolution layer up to an error of $\sim 0 . 0 1$ .
+
+# B DESCRIPTION OF LAYERS
+
+# B.1 QUADRATIC LAYER
+
+One potential application of Theorem 2 is augmenting an equivariant neural network (Equation 7) with equivariant polynomial layers $P : \mathbb { R } ^ { n \times k }  \mathbb { R } ^ { n \times l }$ of some maximal degree $d$ . This can be done in the following way: look for all solutions to $\alpha , \beta _ { 1 } , \beta _ { 2 } , \ldots \in \mathbb { N } ^ { k }$ so that $\begin{array} { r } { \left| \alpha \right| + \sum _ { i } \left| \beta _ { i } \right| \leq d } \end{array}$ . Any solution to this equation will give a basis element of the form $\begin{array} { r } { \pmb { p } ( \pmb { X } ) = \lceil \pmb { x } _ { 1 } ^ { \alpha } \rceil \prod _ { j } \left( \sum _ { i = 1 } ^ { n } \pmb { x } _ { i } ^ { \beta _ { j } } \right) } \end{array}$ .
+
+In the paper we tested PointNetQT, an architecture that adds to PointNet a single quadratic equivariant layer. We opted to use only the quadratic transmission operators: For a matrix $\bar { \boldsymbol { X } } \in \mathbb { R } ^ { n \times k }$ we define $\boldsymbol { L } ( \boldsymbol { X } ) \in \mathbb { R } ^ { \tilde { n } \times k }$ as follows:
+
+$$
+\begin{array} { r } { L ( X ) = X W _ { 1 } + 1 \mathbf { 1 } ^ { T } X W _ { 2 } + ( \mathbf { 1 } \mathbf { 1 } ^ { T } X ) \odot ( \mathbf { 1 } \mathbf { 1 } ^ { T } X ) W _ { 3 } + ( X \odot X ) W _ { 4 } + ( \mathbf { 1 } \mathbf { 1 } ^ { T } X ) \odot X W _ { 5 } , } \end{array}
+$$
+
+where $\odot$ is a point-wise multiplication and $W _ { i } \in \mathbb { R } ^ { n \times k } , i \in [ 5 ]$ are the learnable parameters.
+
+# B.2 GRAPH CONVOLUTION LAYER
+
+We implement a graph convolution layers as follows
+
+$$
+L ( X ) = B X W _ { 2 } + X W _ { 1 } + \mathbf { 1 } c ^ { T }
+$$
+
+with $W _ { 1 } , W _ { 2 } , c$ learnable parameters. The matrix $\textbf {  { B } }$ is defined as in Kipf & Welling (2016) from the knn graph of the set. $\stackrel { \cdot } { B } = D ^ { - \frac { 1 } { 2 } } A D ^ { - \frac { 1 } { 2 } }$ where $_ { D }$ is the degree matrix of the graph and $\pmb { A }$ is the adjacency matrix of the graph with added self-connections.
+
+# C IMPLEMENTATION DETAILS
+
+Knapsack data generation. We constructed a dataset of $1 0 k$ training examples and $1 k$ test examples consisting of $5 0 \times 4$ matrices. We took $w _ { 1 } = 1 0 0$ , $w _ { 2 } = 8 0$ , $w _ { 3 } = 5 0$ . To generate $\pmb { X } \in \mathbb { R } ^ { 5 0 \times 4 }$ , we draw an integer uniformly at random between 1 and 100 and randomly choose 50 integers between 1 as the first column of $\boldsymbol { X }$ . We also randomly chose an integer between 1 and 25 and then randomly chose 150 integers in that range as the three last columns of $\boldsymbol { X }$ . The labels for each input $\boldsymbol { X }$ were computed by a standard dynamic programming approach, see Martello & Toth (1990).
+
+Optimization. We implemented the experiments in Pytorch Paszke et al. (2017) with the Adam Kingma & Ba (2014) optimizer for learning. For the classification we used the cross entropy loss and trained for 150 epochs with learning rate 0.001, learning rate decay of 0.5 every 100 epochs and batch size 32. For the quadratic function regression we trained for 150 epochs with leaning rate of 0.001, learning rate decay 0.1 every 50 epochs and batch size 64; for the regression to the leading eigen vector we trained for 50 epochs with leaning rate of 0.001 and batch size 32. To regress to the output of a single graph convolution layer we trained for 200 epochs with leaning rate of 0.001 and batch size 32.

parse/train/HkxTwkrKDB/HkxTwkrKDB_content_list.json

@@ -0,0 +1,1789 @@
+[
+    {
+        "type": "text",
+        "text": "ON UNIVERSAL EQUIVARIANT SET NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            98,
+            743,
+            122
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nimrod Segol & Yaron Lipman   \nDepartment of Computer Science and Applied Mathematics   \nWeizmann Institute of Science   \nRehovot, Israel   \n{nimrod.segol,yaron.lipman}@weizmann.ac.il ",
+        "bbox": [
+            184,
+            147,
+            589,
+            218
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            255,
+            544,
+            270
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Using deep neural networks that are either invariant or equivariant to permutations in order to learn functions on unordered sets has become prevalent. The most popular, basic models are DeepSets (Zaheer et al., 2017) and PointNet (Qi et al., 2017). While known to be universal for approximating invariant functions, DeepSets and PointNet are not known to be universal when approximating equivariant set functions. On the other hand, several recent equivariant set architectures have been proven equivariant universal (Sannai et al., 2019; Keriven & Peyre, 2019), however ´ these models either use layers that are not permutation equivariant (in the standard sense) and/or use higher order tensor variables which are less practical. There is, therefore, a gap in understanding the universality of popular equivariant set models versus theoretical ones. ",
+        "bbox": [
+            232,
+            286,
+            766,
+            439
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we close this gap by proving that: (i) PointNet is not equivariant universal; and (ii) adding a single linear transmission layer makes PointNet universal. We call this architecture PointNetST and argue it is the simplest permutation equivariant universal model known to date. Another consequence is that DeepSets is universal, and also PointNetSeg, a popular point cloud segmentation network (used e.g., in (Qi et al., 2017)) is universal. ",
+        "bbox": [
+            232,
+            443,
+            766,
+            526
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The key theoretical tool used to prove the above results is an explicit characterization of all permutation equivariant polynomial layers. Lastly, we provide numerical experiments validating the theoretical results and comparing different permutation equivariant models. ",
+        "bbox": [
+            233,
+            529,
+            766,
+            583
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            606,
+            336,
+            622
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many interesting tasks in machine learning can be described by functions $\\pmb { F }$ that take as input a set, $\\textbf { \\textit { X } } = \\mathbf { \\beta } ( \\pmb { x } _ { 1 } , \\dots , \\pmb { x } _ { n } )$ , and output some per-element features or values, $F ( X ) \\ =$ $\\bar { ( } F ( X ) _ { 1 } , \\ldots , F ( X ) _ { n } )$ . Permutation equivariance is the property required of $\\pmb { F }$ so it is welldefined. Namely, it assures that reshuffling the elements in $\\boldsymbol { X }$ and applying $\\pmb { F }$ results in the same output, reshuffled in the same manner. For example, if $\\tilde { \\textbf { \\textit { X } } } = \\mathbf { \\Gamma } ( { \\bf x } _ { 2 } , { \\bf x } _ { 1 } , { \\bf x } _ { 3 } , \\ldots , { \\bf x } _ { n } )$ then $\\pmb { F } ( \\tilde { \\pmb { X } } ) = ( \\pmb { F } ( \\pmb { X } ) _ { 2 } , \\pmb { F } ( \\pmb { X } ) _ { 1 } , \\pmb { F } ( \\pmb { X } ) _ { 3 } , \\dots , \\pmb { F } ( \\pmb { X } ) _ { n } ) .$ . ",
+        "bbox": [
+            173,
+            637,
+            825,
+            727
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Building neural networks that are permutation equivariant by construction proved extremely useful in practice. Arguably the most popular models are DeepSets Zaheer et al. (2017) and PointNet Qi et al. (2017). These models enjoy small number of parameters, low memory footprint and computational efficiency along with high empirical expressiveness. Although both DeepSets and PointNet are known to be invariant universal (i.e., can approximate arbitrary invariant continuous functions) they are not known to be equivariant universal (i.e., can approximate arbitrary equivariant continuous functions). ",
+        "bbox": [
+            174,
+            732,
+            825,
+            830
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, several researchers have suggested theoretical permutation equivariant models and proved they are equivariant universal. Sannai et al. (2019) builds a universal equivariant network by taking $n$ copies of $( n - 1 )$ -invariant networks and combines them with a layer that is not permutation invariant in the standard (above mentioned) sense. Keriven & Peyre (2019) solves a more general ´ problem of building networks that are equivariant universal over arbitrary high order input tensors $\\mathbb { R } ^ { n ^ { d } }$ (including graphs); their construction, however, uses higher order tensors as hidden variables which is of less practical value. Yarotsky (2018) proves that neural networks constructed using a finite set of invariant and equivariant polynomial layers are also equivariant universal, however his network is not explicit (i.e., the polynomials are not characterized for the equivariant case) and also of less practical interest due to the high degree polynomial layers. ",
+        "bbox": [
+            174,
+            837,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper we close the gap between the practical and theoretical permutation equivariant constructions and prove: ",
+        "bbox": [
+            173,
+            166,
+            825,
+            195
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            199,
+            254,
+            213
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "(i) PointNet is not equivariant universal.   \n(ii) Adding a single linear transmission layer (i.e., $X \\mapsto \\mathbf { 1 1 } ^ { T } X ,$ ) to PointNet makes it equivariant universal.   \n(iii) Using ReLU activation the minimal width required for universal permutation equivariant network satisfies $\\begin{array} { r } { \\omega \\le k _ { o u t } + k _ { i n } + \\binom { n + k _ { i n } } { k _ { i n } } } \\end{array}$ . ",
+        "bbox": [
+            176,
+            223,
+            826,
+            320
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This theorem suggests that, arguably, PointNet with an addition of a single linear layer is the simplest universal equivariant network, able to learn arbitrary continuous equivariant functions of sets. An immediate corollary of this theorem is ",
+        "bbox": [
+            176,
+            328,
+            823,
+            369
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1. DeepSets and PointNetSeg are universal. ",
+        "bbox": [
+            173,
+            373,
+            534,
+            388
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "PointNetSeg is a network used often for point cloud segmentation (e.g., in Qi et al. (2017)). One of the benefit of our result is that it provides a simple characterization of universal equivariant architectures that can be used in the network design process to guarantee universality. ",
+        "bbox": [
+            174,
+            398,
+            826,
+            443
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The theoretical tool used for the proof of Theorem 1 is an explicit characterization of the permutation equivariant polynomials over sets of vectors in $\\mathbb { R } ^ { k }$ using power-sum multi-symmetric polynomials. We prove: ",
+        "bbox": [
+            174,
+            448,
+            826,
+            491
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Let $P : \\mathbb { R } ^ { n \\times k }  \\mathbb { R } ^ { n \\times l }$ be a permutation equivariant polynomial map. Then, ",
+        "bbox": [
+            171,
+            493,
+            764,
+            510
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/408ed6cba4bf59ec14e87f1699f44517f0ca3807ae5cbae65c6b789a1ab112d3.jpg",
+        "text": "$$\nP ( X ) = \\sum _ { | \\alpha | \\leq n } b _ { \\alpha } q _ { \\alpha } ^ { T } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            513,
+            573,
+            550
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { b } _ { \\alpha } = ( \\pmb { x } _ { 1 } ^ { \\alpha } , \\dots , \\pmb { x } _ { n } ^ { \\alpha } ) ^ { T }$ , $\\pmb { q } _ { \\alpha } = ( q _ { \\alpha , 1 } , \\dots , q _ { \\alpha , l } ) ^ { T }$ , where $q _ { \\alpha , j } = q _ { \\alpha , j } ( s _ { 1 } , \\ldots , s _ { t } )$ , $t = \\binom { n + k } { k }$ , are polynomials; $\\begin{array} { r } { s _ { j } ( \\pmb { X } ) = \\sum _ { i = 1 } ^ { n } \\pmb { x } _ { i } ^ { \\alpha _ { j } } } \\end{array}$ k are the power-sum multi-symmetric polynomials. On the other hand every polynomial map $_ { P }$ satisfying Equation is equivariant. ",
+        "bbox": [
+            173,
+            563,
+            825,
+            609
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This theorem, which extends Proposition 2.27 in Golubitsky & Stewart (2002) to sets of vectors using multivariate polynomials, lends itself to expressing arbitrary equivariant polynomials as a composition of entry-wise continuous functions and a single linear transmission, which in turn facilitates the proof of Theorem 1. ",
+        "bbox": [
+            173,
+            619,
+            825,
+            676
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We conclude the paper by numerical experiments validating the theoretical results and testing several permutation equivariant networks for the tasks of set classification and regression. ",
+        "bbox": [
+            174,
+            683,
+            820,
+            712
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            717,
+            339,
+            733
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Equivariant maps. Vectors $\\pmb { x } \\in \\mathbb { R } ^ { k }$ are by default column vectors; $\\mathbf { 0 } , \\mathbf { 1 }$ are the all zero and all one vectors/tensors; $e _ { i }$ is the $i$ -th standard basis vector; $\\pmb { I }$ is the identity matrix; all dimensions are inferred from context or mentioned explicitly. We represent a set of $n$ vectors in $\\mathbb { R } ^ { k }$ as a matrix $\\ b { X } \\in \\mathbb { R } ^ { n \\times k }$ and denote $\\pmb { X } = ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } , \\dots , \\pmb { x } _ { n } ) ^ { T }$ , where $\\mathbf { { \\boldsymbol { x } } } _ { i } \\in \\mathbb { R } ^ { k }$ , $i \\in [ n ]$ , are the columns of $\\boldsymbol { X }$ . We denote by $S _ { n }$ the permutation group of $[ n ]$ ; its action on $\\boldsymbol { X }$ is defined by $\\boldsymbol { \\sigma } \\cdot \\pmb { X } = ( \\pmb { x } _ { \\sigma ^ { - 1 } ( 1 ) } , \\pmb { x } _ { \\sigma ^ { - 1 } ( 2 ) } , \\dots , \\pmb { x } _ { \\sigma ^ { - 1 } ( n ) } ) ^ { T }$ $\\sigma \\in S _ { n }$ . That is, $\\sigma$ is reshuffling the rows of $\\boldsymbol { X }$ . The natural class of maps assigning a value or feature vector to every element in an input set is permutation equivariant maps: ",
+        "bbox": [
+            173,
+            747,
+            825,
+            847
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. A map $F : \\mathbb { R } ^ { n \\times k }  \\mathbb { R } ^ { n \\times l }$ satisfying $\\pmb { F } ( \\sigma \\cdot \\pmb { X } ) = \\sigma \\cdot \\pmb { F } ( \\pmb { X } )$ for all $\\sigma \\in S _ { n }$ and $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ is called permutation equivariant. ",
+        "bbox": [
+            174,
+            848,
+            823,
+            878
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Power-sum multi-symmetric polynomials. Given a vector $z = ( z _ { 1 } , \\ldots , z _ { n } ) \\in \\mathbb { R } ^ { n }$ the powersum symmetric polynomials $\\begin{array} { r } { s _ { j } ( z ) = \\sum _ { i = 1 } ^ { n } z _ { i } ^ { j } } \\end{array}$ , with $j \\in [ n ]$ , uniquely characterize $_ { z }$ up to permuting its entries. In other words, for $z , y \\in \\mathbb { R } ^ { n }$ we have $\\pmb { y } = \\sigma \\cdot \\pmb { z }$ for some $\\sigma \\in S _ { n }$ if and only if $s _ { j } ( \\pmb { y } ) =$ $s _ { j } ( z )$ for all $j \\in [ n ]$ . An equivalent property is that every $S _ { n }$ invariant polynomial $p$ can be expressed as a polynomial in the power-sum symmetric polynomials, i.e., $p ( z ) = q ( s _ { 1 } ( z ) , \\ldots , s _ { n } ( z ) )$ , see Rydh (2007) Corollary 8.4, Briand (2004) Theorem 3. This fact was previously used in Zaheer et al. (2017) to prove that DeepSets is universal for invariant functions. We extend this result to equivariant functions and the multi-feature (sets of vectors) case. ",
+        "bbox": [
+            173,
+            892,
+            823,
+            926
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            102,
+            825,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a vector $\\pmb { x } \\in \\mathbb { R } ^ { k }$ and a multi-index vector $\\alpha = ( \\alpha _ { 1 } , \\ldots , \\alpha _ { k } ) \\in \\mathbb { N } ^ { k }$ we define $\\pmb { x } ^ { \\alpha } = x _ { 1 } ^ { \\alpha _ { 1 } } \\cdot \\cdot \\cdot x _ { k } ^ { \\alpha _ { k } }$ and $\\begin{array} { r } { | \\alpha | = \\sum _ { i \\in [ k ] } \\alpha _ { i } } \\end{array}$ . A generalization of the power-sum symmetric polynomials to matrices exists and is called power-sum multi-symmetric polynomials, defined with a bit of notation abuse: $\\begin{array} { r } { s _ { \\alpha } ( { \\pmb X } ) = \\sum _ { i = 1 } ^ { n } { \\pmb x } _ { i } ^ { \\alpha } } \\end{array}$ , where $\\alpha \\in \\mathbb { N } ^ { k }$ is a multi-index satisfying $| { \\boldsymbol { \\alpha } } | \\leq n$ . Note that the number of power-sum multi-symmetric polynomials acting on $\\ b { X } \\in \\mathbb { R } ^ { n \\times k }$ is $t = \\binom { n + k } { k }$ . For notation simplicity let $\\alpha _ { 1 } , \\ldots , \\alpha _ { t }$ be a list of all $\\alpha \\in \\mathbb { N } ^ { k }$ with $| \\alpha | \\le n$ . Then we index the collection of power-sum multi-symmetric polynomials as $s _ { 1 } , \\ldots , s _ { t }$ . ",
+        "bbox": [
+            173,
+            193,
+            826,
+            300
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Similarly to the vector case the numbers $s _ { j } ( X )$ , $j \\in [ t ]$ characterize $\\boldsymbol { X }$ up to permutation of its rows. That is $Y = \\sigma \\cdot X$ for some $\\sigma \\in S _ { n }$ iff $s _ { j } ( \\dot { Y } ) = s _ { j } ( Y )$ for all $j \\in [ t ]$ . Furthermore, every $S _ { n }$ invariant polynomial $p : \\mathbb { R } ^ { n \\times k }  \\mathbb { R }$ can be expressed as a polynomial in the power-sum multi-symmetric polynomials (see (Rydh, 2007) corollary 8.4), i.e., ",
+        "bbox": [
+            174,
+            305,
+            826,
+            364
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8e87796432ee8b2ca8d6dc360ba47ecb8ce152ef6c426f799513bc01d93bc76b.jpg",
+        "text": "$$\np ( { \\pmb X } ) = q ( s _ { 1 } ( { \\pmb X } ) , \\ldots , s _ { t } ( { \\pmb X } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            371,
+            604,
+            388
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "These polynomials were recently used to encode multi-sets in Maron et al. (2019). ",
+        "bbox": [
+            174,
+            395,
+            710,
+            411
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 EQUIVARIANT MULTI-SYMMETRIC POLYNOMIAL LAYERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            421,
+            679,
+            436
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section we develop the main theoretical tool of this paper, namely, a characterization of all permutation equivariant polynomial layers. As far as we know, these layers were not fully characterized before. ",
+        "bbox": [
+            173,
+            452,
+            826,
+            494
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 provides an explicit representation of arbitrary permutation equivariant polynomial maps $P : \\mathbb { R } ^ { n \\times k ^ { \\ast } }  \\mathbb { R } ^ { n \\times l }$ using the basis of power-sum multi-symmetric polynomials, $s _ { i } ( X )$ . The particular use of power-sum polynomials $s _ { i } ( X )$ has the advantage it can be encoded efficiently using a neural network: as we will show $s _ { i } ( X )$ can be approximated using a PointNet with a single linear transmission layer. This allows approximating an arbitrary equivariant polynomial map using PointNet with a single linear transmission layer. ",
+        "bbox": [
+            173,
+            501,
+            825,
+            585
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A version of this theorem for vectors instead of matrices (i.e., the case of $k = 1$ ) appears as Proposition 2.27 in Golubitsky & Stewart (2002); we extend their proof to matrices, which is the relevant scenario for ML applications as it allows working with sets of vectors. For $k = 1$ Theorem 2 reduces to the following form: $\\begin{array} { r } { \\pmb { p } ( \\pmb { x } ) _ { i } = \\sum _ { a \\leq n } p _ { a } \\big ( s _ { 1 } ( \\pmb { x } ) , \\ldots , s _ { n } ( \\pmb { x } ) \\big ) x _ { i } ^ { a } } \\end{array}$ with $\\begin{array} { r } { s _ { j } ( \\pmb { x } ) = \\sum _ { i } x _ { i } ^ { j } } \\end{array}$ . For matrices the monomial $x _ { i } ^ { k }$ is replaced by $\\mathbf { \\mathscr { x } } _ { i } ^ { \\alpha }$ for a multi-index $\\alpha$ and the power-sum symmetric polynomials are replaced by the power-sum multi-symmetric polynomials. ",
+        "bbox": [
+            173,
+            592,
+            825,
+            683
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "First, note that it is enough to prove Theorem 1 for $l = 1$ and apply it to every column of $_ { r }$ . Hence, we deal with a vector of polynomials $p : \\mathbb { R } ^ { n \\times k }  \\mathbb { R } ^ { n }$ and need to prove it can be expressed as $\\begin{array} { r } { p = \\sum _ { | \\alpha | \\leq n } { b _ { \\alpha } q _ { \\alpha } } } \\end{array}$ , for $S _ { n }$ invariant polynomial $q _ { \\alpha }$ . ",
+        "bbox": [
+            173,
+            688,
+            825,
+            732
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a polynomial $p ( X )$ and the cyclic permutation $\\sigma ^ { - 1 } = ( 1 2 3 \\cdot \\cdot \\cdot n )$ the following operation, taking a polynomial to a vector of polynomials, is useful in characterizing equivariant polynomial maps: ",
+        "bbox": [
+            176,
+            739,
+            823,
+            782
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a28d7a3d8a8a81d2f62000e7d18b0d111c24e0ee3d19db234f204bd8598b566.jpg",
+        "text": "$$\n\\lceil p \\rceil ( \\boldsymbol { X } ) = \\left( \\begin{array} { c } { p ( \\boldsymbol { X } ) } \\\\ { p ( \\sigma \\cdot \\boldsymbol { X } ) } \\\\ { p ( \\sigma ^ { 2 } \\cdot \\boldsymbol { X } ) } \\\\ { \\vdots } \\\\ { p ( \\sigma ^ { n - 1 } \\cdot \\boldsymbol { X } ) } \\end{array} \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            781,
+            594,
+            866
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 will be proved using the following two lemmas: ",
+        "bbox": [
+            173,
+            876,
+            562,
+            891
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Let $p : \\mathbb { R } ^ { n \\times k }  \\mathbb { R } ^ { n }$ be an equivariant polynomial map. Then, there exists a polynomial $p : \\mathbb { R } ^ { n \\times k }  \\mathbb { R } ,$ , invariant to $S _ { n - 1 }$ (permuting the last $n - 1$ rows of $\\boldsymbol { X }$ ) so that $p = \\lceil p \\rceil$ . ",
+        "bbox": [
+            169,
+            895,
+            828,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proof. Equivariance of $\\pmb { p }$ means that for all $\\sigma \\in S _ { n }$ it holds that ${ \\boldsymbol { \\sigma } } \\cdot { \\boldsymbol { p } } ( { \\boldsymbol { X } } ) = { \\boldsymbol { p } } ( { \\boldsymbol { \\sigma } } \\cdot { \\boldsymbol { X } } )$ ",
+        "bbox": [
+            173,
+            102,
+            740,
+            118
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/87784881a93b079290cfd355984632f75ff04c0c9151adc9d3bcbfcd30fe0259.jpg",
+        "text": "$$\n\\sigma \\cdot p ( X ) = p ( \\sigma \\cdot X ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            423,
+            125,
+            573,
+            142
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Choosing an arbitrary permutation $\\sigma \\in \\operatorname { s t a b } ( 1 ) < S _ { n }$ , namely a permutation satisfying $\\sigma ( 1 ) = 1$ , and observing the first row in Equation 4 we get $p _ { 1 } ( { \\pmb X } ) = p _ { 1 } ( { \\sigma } \\cdot { \\pmb X } ) = p _ { 1 } ( { \\pmb x } _ { 1 } , { \\pmb x } _ { \\sigma ^ { - 1 } ( 2 ) } , \\dots , { \\pmb x } _ { \\sigma ^ { - 1 } ( n ) } )$ . Since this is true for all $\\sigma \\in \\mathrm { s t a b } ( 1 ) p _ { 1 }$ is $S _ { n - 1 }$ invariant. Next, applying $\\sigma = ( 1 i )$ to Equation 4 and observing the first row again we get $p _ { i } ( { \\cal X } ) = p _ { 1 } ( \\pmb { x } _ { i } , . . . , \\pmb { x } _ { 1 } , . . . )$ . Using the invariance of $p _ { 1 }$ to $S _ { n - 1 }$ we get $\\bar { \\boldsymbol { p } } = \\lceil p _ { 1 } \\rceil$ . □ ",
+        "bbox": [
+            173,
+            148,
+            826,
+            222
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. Let $p : \\mathbb { R } ^ { n \\times k }  \\mathbb { R }$ be a polynomial invariant to $S _ { n - 1 }$ (permuting the last $n - 1$ rows of $X$ ) then ",
+        "bbox": [
+            169,
+            227,
+            828,
+            256
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f9a01434984bffbfbe09f3a2a1f59bc357edbac62119fe596066a2e76f5f9899.jpg",
+        "text": "$$\np ( { \\pmb X } ) = \\sum _ { | \\alpha | \\leq n } { \\pmb x } _ { 1 } ^ { \\alpha } q _ { \\alpha } ( { \\pmb X } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            255,
+            583,
+            291
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $q _ { \\alpha }$ are $S _ { n }$ invariant. ",
+        "bbox": [
+            174,
+            294,
+            351,
+            309
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. Expanding $p$ with respect to $\\scriptstyle { \\mathbf { \\mathscr { x } } } _ { 1 }$ we get ",
+        "bbox": [
+            174,
+            324,
+            477,
+            339
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/921182b337f1ae158d32ddc815b4b2ce31e7477cdd8961d872f63572473fa7e4.jpg",
+        "text": "$$\np ( \\pmb { X } ) = \\sum _ { | \\alpha | \\leq m } \\pmb { x } _ { 1 } ^ { \\alpha } p _ { \\alpha } ( \\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            344,
+            614,
+            382
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for some $\\textit { m } \\in  { \\mathbb { N } }$ . We first claim $p _ { \\alpha }$ are $S _ { n - 1 }$ invariant. Indeed, note that if $p ( \\boldsymbol { X } ) \\ =$ $p ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n } )$ is $S _ { n - 1 }$ invariant, i.e., invariant to permutations of $\\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n }$ , then also its derivative s ∂|β|∂xβ p(X) are Sn−1 permutation invariant, for all β ∈ Nk. Taking the derivative ∂|β|/∂xβ1 on both sides of Equation 6 we get that $p _ { \\beta }$ is $S _ { n - 1 }$ equivariant. ",
+        "bbox": [
+            173,
+            388,
+            826,
+            455
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For brevity denote $p = p _ { \\alpha }$ . Since $p$ is $S _ { n - 1 }$ invariant it can be expressed as a polynomial in the powersum multi-symmetric polynomials, i.e., $p ( \\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n } ) = r ( s _ { 1 } ( \\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n } ) , \\ldots , s _ { t } ( \\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n } ) )$ . Note that $s _ { i } ( { \\pmb x } _ { 2 } , \\ldots , { \\pmb x } _ { n } ) = s _ { i } ( { \\pmb X } ) - { \\pmb x } _ { 1 } ^ { \\alpha _ { i } }$ and therefore ",
+        "bbox": [
+            173,
+            460,
+            826,
+            503
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/92afd39b333491b84f91ea1b8c08a8a5449b2572600f23ef19f95a4b91046c7b.jpg",
+        "text": "$$\np ( \\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n } ) = r ( s _ { 1 } ( \\pmb { X } ) - \\pmb { x } _ { 1 } ^ { \\alpha _ { 1 } } , \\ldots , s _ { t } ( \\pmb { X } ) - \\pmb { x } _ { 1 } ^ { \\alpha _ { t } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            510,
+            678,
+            527
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since $r$ is a polynomial, expanding its monomials in $s _ { i } ( X )$ and $\\pmb { x } _ { 1 } ^ { \\alpha }$ shows $p$ can be expressed as $\\begin{array} { r } { p = \\sum _ { | \\alpha | \\leq m ^ { \\prime } } \\bar { \\pmb { x } } _ { 1 } ^ { \\alpha } \\tilde { p } _ { \\alpha } } \\end{array}$ , where $m ^ { \\prime } \\in \\mathbb { N }$ , and $\\tilde { p } _ { \\alpha }$ are $S _ { n }$ invariant (as multiplication of invariant $S _ { n }$ polynomials $s _ { i } ( X ) \\rangle$ . Plugging this in Equation 6 we get Equation 5, possibly with the sum over some $n ^ { \\prime } > n$ . It remains to show $n ^ { \\prime }$ can be taken to be at-most $n$ . This is proved in Corollary 5 in Briand (2004) □ ",
+        "bbox": [
+            173,
+            534,
+            825,
+            607
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. (Theorem 2) Given an equivariant $\\pmb { p }$ as above, use Lemma 1 to write $p = \\lceil p \\rceil$ where $p ( X )$ is invariant to permuting the last $n - 1$ rows of $\\boldsymbol { X }$ . Use Lemma 2 to write $\\begin{array} { r } { p ( \\pmb { X } ) = \\sum _ { | \\alpha | \\leq n } \\pmb { x } _ { 1 } ^ { \\alpha } q _ { \\alpha } ( \\pmb { X } ) } \\end{array}$ , where $q _ { \\alpha }$ are $S _ { n }$ invariant. We get, ",
+        "bbox": [
+            173,
+            622,
+            826,
+            667
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b270621a4ff571131dfd95048cfba2d3583b19b7d378eedda0e7cb405128c335.jpg",
+        "text": "$$\np = \\lceil p \\rceil = \\sum _ { | \\alpha | \\leq n } b _ { \\alpha } q _ { \\alpha } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            672,
+            576,
+            710
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The converse direction is immediate after noting that $b _ { \\alpha }$ are equivariant and $q _ { \\alpha }$ are invariant. ",
+        "bbox": [
+            169,
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+            781,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 UNIVERSALITY OF SET EQUIVARIANT NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            758,
+            691,
+            775
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We consider equivariant deep neural networks $f : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R } ^ { n \\times k _ { o u t } } ,$ , ",
+        "bbox": [
+            173,
+            789,
+            640,
+            805
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1293ce23c9bb3c9772d7a1ac5aade50df0506822efe083b2aa4459ae5d24a398.jpg",
+        "text": "$$\n{ \\pmb F } ( { \\pmb X } ) = { \\pmb L } _ { m } \\circ \\nu \\circ \\cdots \\circ \\nu \\circ { \\pmb L } _ { 1 } ( { \\pmb X } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            811,
+            620,
+            829
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $L _ { i } : \\mathbb { R } ^ { n \\times k _ { i } }  \\mathbb { R } ^ { n \\times k _ { i + 1 } }$ are affine equivariant transformations, and $\\nu$ is an entry-wise nonlinearity (e.g., ReLU). We define the width of the network to be $\\omega = \\mathrm { m a x } _ { i } k _ { i }$ ; note that this definition is different from the one used for standard MLP where the width would be $n \\omega$ , see e.g., Hanin & Sellke (2017). Zaheer et al. (2017) proved that affine equivariant $\\mathbf { { L } } _ { i }$ are of the form ",
+        "bbox": [
+            173,
+            835,
+            826,
+            892
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c6dd17f4f12c883d16a561b7311e994dd3c8460a4ac6a192ecf696d9b6db4215.jpg",
+        "text": "$$\nL _ { i } ( X ) = X A + \\frac { 1 } { n } \\mathbf { 1 1 } ^ { T } X B + \\mathbf { 1 } c ^ { T } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            898,
+            622,
+            929
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $A , B \\in \\mathbb { R } ^ { k _ { i } \\times k _ { i + 1 } }$ , and $\\boldsymbol { c } \\in \\mathbb { R } ^ { k _ { i + 1 } }$ are the layer’s trainable parameters; we call the linear transformation $\\textstyle X \\mapsto { \\frac { 1 } { n } } \\mathbf { 1 } \\mathbf { 1 } ^ { T } X B$ a linear transmission layer. ",
+        "bbox": [
+            171,
+            102,
+            823,
+            132
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now define the equivariant models considered in this paper: The DeepSets (Zaheer et al., 2017) architecture is Equation 7 with the choice of layers as in Equation 8. Taking $B = 0$ in all layers is the PointNet architecture (Qi et al., 2017). PointNetST is an equivariant model of the form Equation 7 with layers as in Equation 8 where only a single layer $\\mathbf { { { L } } } _ { i }$ has a non-zero $\\textbf {  { B } }$ . The PointNetSeg (Qi et al., 2017) architecture is PointNet composed with an invariant max layer, namely $\\begin{array} { r } { \\operatorname* { m a x } ( \\boldsymbol { F } ( \\boldsymbol { X } ) ) _ { j } = \\operatorname* { m a x } _ { i \\in [ n ] } \\boldsymbol { F } ( \\boldsymbol { X } ) _ { i , j } } \\end{array}$ and then concatenating it with the input $\\boldsymbol { X }$ , i.e., $[ X , \\mathbf { 1 } \\mathbf { m a x } ( F ( X ) ) ]$ , and feeding is as input to another PointNet $G$ , that is $G ( [ X , \\mathbf { 1 } \\operatorname* { m a x } ( F ( X ) ) ] )$ . ",
+        "bbox": [
+            173,
+            138,
+            826,
+            239
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We will prove PointNetST is permutation equivariant universal and therefore arguably the simplest permutation equivariant universal model known to date. ",
+        "bbox": [
+            174,
+            244,
+            823,
+            273
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Universality of equivariant deep networks is defined next. ",
+        "bbox": [
+            174,
+            280,
+            552,
+            295
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. Permutation equivariant universality1 of a model $F : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R } ^ { n \\times k _ { o u t } }$ means that for every permutation equivariant continuous function $H : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R } ^ { n \\times k _ { o u t } }$ defined over the cube $\\overset { \\cdot } { K } \\overset { \\cdot } { = } [ 0 , 1 ] ^ { n \\times k _ { i n } } \\subset \\overset { \\cdot } { \\mathbb { R } } ^ { n \\times k _ { i n } }$ , and $\\epsilon > 0$ there exists a choice of m (i.e., network depth), $k _ { i }$ (i.e., network width) and the trainable parameters of $\\pmb { F }$ so that $\\| H ( X ) - F ( X ) \\| _ { \\infty } < \\epsilon$ for all $X \\in K$ . ",
+        "bbox": [
+            173,
+            297,
+            826,
+            356
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. (Theorem $^ { l }$ ) Fact $( i )$ , namely that PointNet is not equivariant universal is a consequence of the following simple lemma: ",
+        "bbox": [
+            173,
+            366,
+            823,
+            396
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3. Let $\\pmb { h } = ( h _ { 1 } , \\ldots , h _ { n } ) ^ { T } : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { n }$ be the equivariant linear function defined by $h ( \\pmb { x } ) = \\mathbf { 1 } \\mathbf { 1 } ^ { T } \\pmb { x }$ . There is no $f : \\mathbb { R } \\to \\mathbb { R }$ so that $\\begin{array} { r } { | h _ { i } ( { \\pmb x } ) - f ( x _ { i } ) | < \\frac { 1 } { 2 } } \\end{array}$ for all $i \\in [ n ]$ and $\\pmb { x } \\in [ 0 , 1 ] ^ { n }$ . ",
+        "bbox": [
+            173,
+            398,
+            825,
+            430
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. Assume such $f$ exists. Let $\\pmb { e } _ { 1 } = ( 1 , 0 , \\ldots , 0 ) ^ { T } \\in \\mathbb { R } ^ { n }$ . Then, ",
+        "bbox": [
+            173,
+            445,
+            617,
+            463
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f1d4f26a55ce5167f85c41a52485e8f57594cbf9bce6b8fcb9b103bca67975b5.jpg",
+        "text": "$$\n1 = | h _ { 2 } ( e _ { 1 } ) - h _ { 2 } ( { \\bf 0 } ) | \\leq | h _ { 2 } ( e _ { 1 } ) - f ( 0 ) | + | f ( 0 ) - h _ { 2 } ( { \\bf 0 } ) | < 1\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            468,
+            710,
+            486
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "reaching a contradiction. ",
+        "bbox": [
+            174,
+            492,
+            338,
+            507
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To prove $( i i )$ we first reduce the problem from the class of all continuous equivariant functions to the class of equivariant polynomials. This is justified by the following lemma. ",
+        "bbox": [
+            173,
+            523,
+            823,
+            553
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 4. Equivariant polynomials $P : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R } ^ { n \\times k _ { o u t } }$ are dense in the space of continuous equivariant functions $F : \\dot { \\mathbb { R } ^ { n \\times k _ { i n } } }  \\mathbb { R } ^ { n \\times k _ { o u t } }$ over the cube $K$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. Take an arbitrary $\\epsilon > 0$ . Consider the function $f _ { i j } : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R }$ , which denotes the $( i , j )$ -th output entry of $\\pmb { F }$ . By the Stone-Weierstrass Theorem there exists a polynomial $p _ { i j } : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R }$ such that $\\left\\| f _ { i j } ( X ) - p _ { i j } ( X ) \\right\\| _ { \\infty } \\leq \\epsilon$ for all $X \\in K$ . Consider the polynomial map $P : \\mathbb { R } ^ { n \\times k _ { i n } } $ $\\mathbb { R } ^ { n \\times k _ { o u t } }$ defined by $( P ) _ { i j } = p _ { i j }$ . $_ { P }$ is in general not equivariant. To finish the proof we will symmetrize $_ { r }$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6db409ab68aa117b4810fc98c3f7408cfcbb21e299a298494d683ad4e5cc9afe.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\bigg | F ( \\pmb X ) - \\displaystyle \\frac { 1 } { n ! } \\sum _ { \\sigma \\in S _ { n } } \\sigma \\cdot \\pmb P ( \\sigma ^ { - 1 } \\cdot \\pmb X ) \\bigg | \\Bigg | _ { \\infty } = \\bigg | \\bigg | \\frac { 1 } { n ! } \\displaystyle \\sum _ { \\sigma \\in S _ { n } } \\sigma \\cdot \\pmb F ( \\sigma ^ { - 1 } \\cdot \\pmb X ) - \\frac { 1 } { n ! } \\displaystyle \\sum _ { \\sigma \\in S _ { n } } \\sigma \\cdot \\pmb P ( \\sigma ^ { - 1 } \\cdot \\pmb X ) \\bigg | \\bigg | _ { \\infty } } & { } \\\\ { = \\bigg | \\bigg | \\frac { 1 } { n ! } \\displaystyle \\sum _ { \\sigma \\in S _ { n } } \\sigma \\cdot \\big ( F ( \\sigma ^ { - 1 } \\cdot \\pmb X ) - \\pmb P ( \\sigma ^ { - 1 } \\cdot \\pmb X ) \\big ) \\bigg | \\bigg | _ { \\infty } } & { } \\\\ { \\leq \\frac { 1 } { n ! } \\displaystyle \\sum _ { \\sigma \\in S _ { n } } \\epsilon = \\epsilon , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            683,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where in the first equality we used the fact that $\\pmb { F }$ is equivariant. This concludes the proof since $\\begin{array} { r l } { \\sum _ { \\sigma \\in S _ { n } } \\sigma \\cdot P ( \\sigma ^ { - 1 } \\cdot \\overline { { X } } ) } & { { } } \\end{array}$ is an equivariant polynomial map. □ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Now, according to Theorem 2 an arbitrary equivariant polynomial $P : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R } ^ { n \\times k _ { o u t } }$ can be written as $\\begin{array} { r } { P \\ = \\ \\sum _ { | \\alpha | \\leq n } { b } _ { \\alpha } ( X ) \\dot { q _ { \\alpha } } ( \\dot { X } ) ^ { T } } \\end{array}$ , where $\\bar { b _ { \\alpha } } ( \\mathbf { X } ) \\ = \\ \\left\\lceil \\mathbf { x } _ { 1 } ^ { \\alpha } \\right\\rceil \\ \\in \\ \\mathbb { R } ^ { n }$ and $\\begin{array} { r l } { q _ { \\alpha } } & { { } = } \\end{array}$ $( q _ { \\alpha , 1 } , \\dots , q _ { \\alpha , k _ { o u t } } ) \\in \\mathbb { R } ^ { k _ { o u t } }$ are invariant polynomials. Remember that every $S _ { n }$ invariant polynomial can be expressed as a polynomial in the $t = \\binom { n + k _ { i n } } { k _ { i n } }$ power-sum multi-symmetric polynomials $\\begin{array} { r } { s _ { j } ( { \\pmb X } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } { \\pmb x } _ { i } ^ { \\alpha _ { j } } , j \\in [ t ] } \\end{array}$ (we use the $1 / n$ normalized version for a bit more simplicity later on). We can therefore write $_ { r }$ as composition of three maps: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d40de6d1ee37ec97892d0e5784aee987ccf174355a366b5ff1d95ca47fd7139.jpg",
+        "text": "$$\nP = Q \\circ L \\circ B ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { B } : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R } ^ { n \\times t }$ is defined by ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ff639040e5756e14326d0e01bd17d481938c2bef2190735d97c6d555b734b705.jpg",
+        "text": "$$\nB ( X ) = ( b ( x _ { 1 } ) , \\dots , b ( x _ { n } ) ) ^ { T } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "${ \\pmb b } ( { \\pmb x } ) = ( { \\pmb x } ^ { \\alpha _ { 1 } } , \\ldots , { \\pmb x } ^ { \\alpha _ { t } } ) ; L$ is defined as in Equation 8 with $\\boldsymbol { B } = [ \\mathbf { 0 } , I ]$ and $\\pmb { A } = [ e _ { 1 } , \\ldots , e _ { k _ { i n } } , \\pmb { 0 } ]$ , where $\\ b { I } \\in \\mathbb { R } ^ { t \\times t }$ the identity matrix and $\\boldsymbol { e } _ { i } \\in \\mathbb { R } ^ { t }$ represents the standard basis (as-usual). We assume $\\boldsymbol { \\alpha } _ { j } = \\boldsymbol { e } _ { j } \\in \\mathbb { R } ^ { k _ { i n } }$ , for $j \\in [ \\bar { k } _ { i n } ]$ . Note that the output of $\\pmb { L }$ is of the form ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a1665dbcb95601a2d302ffacd03f4da8e91eab260b6d07452d27144ab4ee6dc0.jpg",
+        "text": "$$\n\\pmb { L } ( \\pmb { B } ( \\pmb { X } ) ) = ( \\pmb { X } , \\pmb { 1 } s _ { 1 } ( \\pmb { X } ) , \\pmb { 1 } s _ { 2 } ( \\pmb { X } ) , \\dots , \\pmb { 1 } s _ { t } ( \\pmb { X } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            289,
+            668,
+            306
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, $Q : \\mathbb { R } ^ { n \\times ( k _ { i n } + t ) }  \\mathbb { R } ^ { n \\times k _ { o u t } }$ is defined by ",
+        "bbox": [
+            173,
+            311,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7c3753fb1b29f6b4957ee71fadbfb31d115711a74edf8e423c3e96076bd9b1d6.jpg",
+        "text": "$$\nQ ( X , \\mathbf { 1 } s _ { 1 } , \\ldots , \\mathbf { 1 } s _ { t } ) = ( q ( x _ { 1 } , s _ { 1 } , \\ldots , s _ { t } ) , \\ldots , q ( x _ { n } , s _ { 1 } , \\ldots , s _ { t } ) ) ^ { T } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            334,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The decomposition in Equation 9 of $_ { r }$ suggests that replacing $Q , B$ with Multi-Layer Perceptrons (MLPs) would lead to a universal permutation equivariant network consisting of PointNet with a single linear transmission layer, namely PointNetST. ",
+        "bbox": [
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+            382,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The $\\pmb { F }$ approximating $_ { P }$ will be defined as ",
+        "bbox": [
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+            459,
+            455,
+            473
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a03151f232b06cdcdbd6e44992b42ea0437edcc36a83abcaae4173d71c55798e.jpg",
+        "text": "$$\n\\pmb { F } = \\pmb { \\Psi } \\circ \\pmb { L } \\circ \\pmb { \\Phi } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            481,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/796c0b9d0b7cceb714674120d0a995bcd66e93c12cd3a844d159a94be243f91d.jpg",
+        "image_caption": [
+            "Figure 1: The construction of the universal network (PointNetST). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            531,
+            381,
+            818,
+            494
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\Phi : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R } ^ { n \\times t }$ and $\\Psi : \\mathbb { R } ^ { n \\times ( t + k _ { i n } ) } $ $\\mathbb { R } ^ { n \\times k _ { o u t } }$ are both of PointNet architecture, namely there exist MLPs $\\phi : \\mathbb { R } ^ { k _ { i n } }  \\mathbb { R } ^ { t }$ and $\\psi : \\mathbb { R } ^ { t + k _ { i n } } \\stackrel { \\cdot } {  }$ $\\mathbb { R } ^ { k _ { o u t } }$ so that $\\Phi ( X ) = ( \\phi ( { \\pmb x } _ { 1 } ) , \\dots , \\phi ( { \\pmb x } _ { n } ) ) ^ { T }$ and $\\pmb { \\Psi } ( \\pmb { X } ) = ( \\pmb { \\psi } ( \\pmb { x } _ { 1 } ) , \\dots , \\pmb { \\psi } ( \\pmb { x } _ { n } ) ) ^ { T }$ . See Figure 1 for an illustration of ${ \\pmb F } .$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            544,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To build the MLPs $\\phi , \\psi$ we will first construct $\\psi$ to approximate $\\pmb q$ , that is, we use the universality of MLPS (see (Hornik, 1991; Sonoda & Murata, 2017; Hanin & Sellke, 2017)) to construct $\\psi$ so that $\\begin{array} { r } { \\| \\psi ( \\pmb { x } , s _ { 1 } , . . . , s _ { t } ) - \\pmb { q } ( \\pmb { x } , s _ { 1 } , . . . , s _ { t } ) \\| _ { \\infty } < \\frac { \\epsilon } { 2 } } \\end{array}$ for all $( \\pmb { x } , s _ { 1 } , \\dots , s _ { t } ) \\in [ 0 , 1 ] ^ { k _ { i n } + t }$ . Furthermore, as $\\psi$ over $[ 0 , 1 ] ^ { k _ { i n } + t }$ is uniformly continuous, let $\\delta$ be such that if $z , z ^ { \\prime } \\in [ 0 , 1 ] ^ { k _ { i n } + t } , \\| z - z ^ { \\prime } \\| _ { \\infty } < \\delta$ then $\\| \\dot { \\psi ( z ) } - \\psi ( z ^ { \\prime } ) \\| _ { \\infty } < \\frac { \\epsilon } { 2 }$ . Now, we use universality again to construct $\\phi$ approximating $^ { b }$ , that is we take $\\phi$ so that $\\| \\phi ( \\pmb { x } ) - \\pmb { b } ( \\pmb { x } ) \\| _ { \\infty } < \\delta$ for all $\\pmb { x } \\in [ 0 , 1 ] ^ { k _ { i n } }$ . ",
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a92de332d8fe41ad3b1499709a7400e7f57adef21c8a6dde891531b78eebacaf.jpg",
+        "text": "$$\n{ \\begin{array} { r l } & { F ( { \\boldsymbol { X } } ) - P ( { \\boldsymbol { X } } ) \\| _ { \\infty } \\leq \\| \\Psi ( { \\boldsymbol { L } } ( \\Phi ( { \\boldsymbol { X } } ) ) ) - \\Psi ( { \\boldsymbol { L } } ( B ( { \\boldsymbol { X } } ) ) ) \\| _ { \\infty } + \\| \\Psi ( { \\boldsymbol { L } } ( B ( { \\boldsymbol { X } } ) ) ) - Q ( { \\boldsymbol { L } } ( B ( { \\boldsymbol { X } } ) ) ) \\| _ { \\infty } } \\\\ & { \\qquad = \\operatorname { e r r } _ { 1 } + \\operatorname { e r r } _ { 2 } } \\end{array} }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            674,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "First, $\\left\\| L ( \\Phi ( X ) ) - L ( B ( X ) ) \\right\\| _ { \\infty } < \\delta$ for all $X \\in K$ and therefore $\\mathrm { e r r } _ { 1 } < \\frac { \\epsilon } { 2 }$ . Second, note that if $X \\in K$ then $B ( X ) \\in [ 0 , 1 ] ^ { n \\times t }$ and $L ( B ( X ) ) \\in [ 0 , 1 ] ^ { n \\times ( k _ { i n } + t ) }$ . Therefore by construction of $\\psi$ we have $\\mathrm { e r r } _ { 2 } < \\frac { \\epsilon } { 2 }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To prove $( i i i )$ we use the result in Hanin & Sellke (2017) (see Theorem 1) bounding the width of an MLP approximating a function $f : [ 0 , 1 ] ^ { d _ { i n } } \\to \\mathbb { R } ^ { d _ { o u t } }$ by $d _ { i n } + d _ { o u t }$ . Therefore, the width of the MLP $\\phi$ is bounded by $k _ { i n } + t$ , where the width of the MLP $\\psi$ is bounded by $t + k _ { i n } + k _ { o u t }$ , proving the bound. □ ",
+        "bbox": [
+            173,
+            767,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We can now prove Cororllary 1. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof. (Corollary 1) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            859,
+            312,
+            875
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The fact that the DeepSets model is equivariant universal is immediate. Indeed, The PointNetST model can be obtained from the DeepSets model by setting $B = 0$ in all but one layer, with $\\textbf {  { B } }$ as in Equation 8. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For the PointNetSeg model note that by Theorem 1 in $\\mathrm { Q i }$ et al. (2017) every invariant function $f : \\mathbb { R } ^ { n \\times k _ { i n } }  \\mathbb { R } ^ { t }$ can be approximated by a network of the form $\\Psi ( \\operatorname* { m a x } ( F ( X ) ) )$ , where $( F ( X )$ is a PointNet model and $\\Psi$ is an MLP. In particular, for every $\\varepsilon > 0$ there exists such $\\mathbfcal { F } , \\Phi$ for which $\\| \\Psi ( \\operatorname* { m a x } ( F ( X ) ) ) - ( s _ { 1 } ( X ) , \\ldots , s _ { t } ( X ) ) \\| _ { \\infty } < \\varepsilon$ for every $\\boldsymbol { X } \\in [ 0 , 1 ] ^ { n \\times k _ { i n } }$ where $s _ { 1 } ( X ) , \\ldots , s _ { t } ( X )$ are the power-sum multi-symmetric polynomials. It follows that we can use PointNetSeg to approximate ${ \\bf 1 } ( s _ { 1 } ( X ) , \\dots , s _ { t } ( \\dot { X } ) )$ . The rest of the proof closely resembles the proof of Theorem 1. ",
+        "bbox": [
+            173,
+            102,
+            825,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Graph neural networks with constructed adjacency. One approach sometimes applied to learning from sets of vectors is to define an adjacency matrix (e.g., by thresholding distances of node feature vectors) and apply a graph neural network to the resulting graph (e.g., Wang et al. (2019), Li et al. (2019)). Using the common message passing paradigm (Gilmer et al., 2017) in this case boils to layers of the form: $\\begin{array} { r } { \\pmb { L } ( \\pmb { X } ) _ { i } = \\psi ( \\pmb { x } _ { i } , \\sum _ { j \\in N _ { i } } \\tilde { \\phi } ( \\pmb { x } _ { i } , \\pmb { x } _ { j } ) ) = \\psi ( \\pmb { x } _ { i } , \\sum _ { j \\in [ n ] } N ( \\pmb { x } _ { i } , \\pmb { x } _ { j } ) \\phi ( \\pmb { x } _ { i } , \\pmb { x } _ { j } ) ) } \\end{array}$ , where $\\phi , \\psi$ are MLPs, $N _ { i }$ is the index set of neighbors of node $i$ , and $N ( \\pmb { x } _ { i } , \\pmb { x } _ { j } )$ is the indicator function for the edge $( i , j )$ . If $N$ can be approximated by a continuous function, which is the case at least in the $L _ { 2 }$ sense for a finite set of vectors, then since $\\pmb { L }$ is also equivariant it follows from Theorem 1 that such a network can be approximated (again, at least in $L _ { 2 }$ norm) arbitrarily well by any universal equivariant network such as PointNetST or DeepSets. ",
+        "bbox": [
+            173,
+            237,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We tested the ability of a DeepSets model with varying depth and width to approximate a single graph convolution layer. We found that a DeepSets model with a small number of layers can approximate a graph convolution layer reasonably well. For details see Appendix A. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            443,
+            328,
+            458
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conducted experiments in order to validate our theoretical observations. We compared the results of several equivariant models, as well as baseline (full) MLP, on three equivariant learning tasks: a classification task (knapsack) and two regression tasks (squared norm and Fiedler eigen vector). For all tasks we compare results of 7 different models: DeepSets, PointNet, PointNetSeg, PointNetST, PointNetQT and GraphNet. PointNetQT is PointNet with a single quadratic equivariant transmission layer as defined in Appendix B. GraphNet is similar to the graph convolution network in Kipf & Welling (2016) and is defined explicitly in Appendix B. We generated the adjacency matrices for GraphNet by taking 10 nearest neighbors of each set element. In all experiments we used a network of the form Equation 7 with $m = 6$ depth and varying width, fixed across all layers. 2 ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Equivariant classification. For classification, we chose to learn the multidimensional knapsack problem, which is known to be NP-hard. We are given a set of 4-vectors, represented by $\\pmb { X } \\in \\bar { \\mathbb { R } } ^ { n \\times 4 }$ and our goal is to learn the equivariant classification function $f : \\mathbb { R } ^ { n \\times 4 } \\stackrel { \\cdot } { \\to } \\left\\{ 0 , 1 \\right\\} ^ { n }$ defined by the following optimization problem: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f16f00ee8a4cbd99cb7d8488d95a59a6e1db9f5c80f8855b25e2339935df53bb.jpg",
+        "text": "$$\n\\begin{array} { r l } { f ( \\pmb { X } ) = \\arg \\underset { z } { \\operatorname* { m a x } } } & { \\displaystyle \\sum _ { i = 1 } ^ { n } x _ { i 1 } z _ { i } } \\\\ { \\mathrm { s . t . } } & { \\displaystyle \\sum _ { i = 1 } ^ { n } x _ { i j } z _ { i } \\leq w _ { j } , \\qquad j = 2 , 3 , 4 } \\\\ & { z _ { i } \\in \\{ 0 , 1 \\} , \\qquad i \\in [ n ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            657,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, given a set of vectors $\\ b { X } \\in \\mathbb { R } ^ { n \\times 4 }$ , $( \\boldsymbol { X } ) _ { i j } = x _ { i j }$ , where each row represents an element in a set, our goal is to find a subset maximizing the value while satisfying budget constraints. The first column of $\\boldsymbol { X }$ defines the value of each element, and the three other columns the costs. ",
+        "bbox": [
+            173,
+            765,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the success of a trained model we record the percentage of sets for which the predicted subset is such that all the budget constrains are satisfied and the value is within $1 0 \\%$ of the optimal value. In Appendix C we detail how we generated this dataset. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Equivariant regression. The first equivariant function we considered for regression is the function $\\begin{array} { r } { f ( \\mathbf { X } ) = \\mathbf { 1 } \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { k } ( X _ { i , j } - \\frac { 1 } { 2 } ) ^ { 2 } } \\end{array}$ . Hanin & Sellke (2017) showed this function cannot be approximated by MLPs of small width. We drew $1 0 k$ training examples and $1 k$ test examples i.i.d. from a $\\begin{array} { r } { \\mathcal { N } \\big ( \\frac { 1 } { 2 } , 1 \\big ) } \\end{array}$ distribution (per entry of $\\boldsymbol { X }$ ). ",
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+    {
+        "type": "image",
+        "img_path": "images/6904d861b888584598cf9bb58d2ad82b12289646750b069e73dd4fd09ba854b8.jpg",
+        "image_caption": [
+            "Figure 2: Classification and regression tasks with permutation equivariant models. All the universal permutation equivariant models perform similarly, while the equivariant non-universal PointNet demonstrates reduced performace consistently; MLP baseline (with the same number of parameters as the equivariant models) performs poorly. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            101,
+            810,
+            369
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            477,
+            823,
+            506
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The second equivariant function we considered is defined on point clouds $\\boldsymbol { X } \\in \\mathbb { R } ^ { n \\times 3 }$ . For each point cloud we computed a graph by connecting every point to its 10 nearest neighbors. We then computed the absolute value of the first non trivial eigenvector of the graph Laplacian. We used the ModelNet dataset ( $\\mathrm { W u }$ et al., 2015) which contains $\\sim 9 k$ training meshes and $\\sim 2 k$ test meshes. The point clouds are generated by randomly sampling 512 points from each mesh. ",
+        "bbox": [
+            174,
+            513,
+            825,
+            584
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Result summary. Figure 2 summarizes train and test accuracy of the 6 models after training (training details in Appendix C) as a function of the network width $\\omega$ . We have tested $1 5 \\omega$ values equidistant in [5, $\\textstyle { \\frac { n k _ { i n } } { 2 } } \\int$ . ",
+        "bbox": [
+            173,
+            598,
+            825,
+            643
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As can be seen in the graphs, in all three datasets the equivariant universal models (PointNetST, PointNetQT , DeepSets, PointNetSeg) achieved comparable accuracy. PointNet, which is not equivariant universal, consistently achieved inferior performance compared to the universal models, as expected by the theory developed in this paper. The non-equivariant MLP, although universal, used the same width (i.e., same number of parameters) as the equivariant models and was able to over-fit only on one train set (the quadratic function); its performance on the test sets was inferior by a large margin to the equivariant models. We also note that in general the GraphNet model achieved comparable results to the equivariant universal models but was still outperformed by the DeepSets model. ",
+        "bbox": [
+            173,
+            650,
+            825,
+            775
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "An interesting point is that although the width used in the experiments in much smaller than the bound $k _ { o u t } + \\bar { k _ { i n } } + \\binom { n + k _ { i n } } { k _ { i n } }$ established by Theorem 1, the universal models are still able to learn well the functions we tested on. This raises the question of the tightness of this bound, which we leave to future work. ",
+        "bbox": [
+            174,
+            781,
+            825,
+            840
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            847,
+            330,
+            863
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper we analyze several set equivariant neural networks and compare their approximation power. We show that while vanilla PointNet (Qi et al., 2017) is not equivariant universal, adding a single linear transmission layer makes it equivariant universal. Our proof strategy is based on a characterization of polynomial equivariant functions. As a corollary we show that the DeepSets model (Zaheer et al., 2017) and PointNetSeg (Qi et al., 2017) are equivariant universal. Experimentally, we tested the different models on several classification and regression tasks finding that adding a single linear transmitting layer to PointNet makes a significant positive impact on performance. ",
+        "bbox": [
+            174,
+            868,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            157,
+            398,
+            172
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This research was supported in part by the European Research Council (ERC Consolidator Grant, LiftMatch 771136) and the Israel Science Foundation (Grant No. 1830/17). ",
+        "bbox": [
+            174,
+            183,
+            825,
+            210
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "text": "Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015. ",
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+    {
+        "type": "text",
+        "text": "Dmitry Yarotsky. Universal approximations of invariant maps by neural networks. arXiv preprint arXiv:1804.10306, 2018. ",
+        "bbox": [
+            173,
+            161,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan R Salakhutdinov, and Alexander J Smola. Deep sets. In Advances in neural information processing systems, pp. 3391–3401, 2017. ",
+        "bbox": [
+            174,
+            205,
+            826,
+            248
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A APPROXIMATING GRAPH CONVOLUTION LAYER WITH DEEPSETS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            281,
+            745,
+            297
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "To test the ability of an equivariant universal model to approximate a graph convolution layer, we conducted an experiment where we applied a single graph convolution layer (see Appendix B for a full description of the graph convolution layers used in this paper) with 3 in features and 10 out features. We constructed a knn graph by taking 10 neighbors. We sampled 1000 examples in $\\mathbb { R } ^ { 1 0 0 \\times 3 }$ i.i.d from a $\\begin{array} { r } { \\mathcal { N } ( \\frac { 1 } { 2 } , 1 ) } \\end{array}$ distribution (per entry of $\\boldsymbol { X }$ ). The results are summarized in Figure 3. We regressed to the output of a graph convolution layer using the smooth $L _ { 1 }$ loss. ",
+        "bbox": [
+            173,
+            314,
+            826,
+            398
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/4977f9c73f73c91a611a3996fb5c3ae4b489ae9271643b0516748e47e4b99792.jpg",
+        "image_caption": [
+            "Figure 3: Using DeepSets to regress to a single graph convolution layer. In the left image we varied the depth and took a constant width of 200. In the right we varied the width and took a fixed depth of 6 layers. The $y$ axes are in log scale. Note that even a 2 layer DeepSets can approximate a graph convolution layer up to an error of $\\sim 0 . 0 1$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            434,
+            761,
+            585
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B DESCRIPTION OF LAYERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            691,
+            421,
+            707
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B.1 QUADRATIC LAYER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            724,
+            352,
+            739
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "One potential application of Theorem 2 is augmenting an equivariant neural network (Equation 7) with equivariant polynomial layers $P : \\mathbb { R } ^ { n \\times k }  \\mathbb { R } ^ { n \\times l }$ of some maximal degree $d$ . This can be done in the following way: look for all solutions to $\\alpha , \\beta _ { 1 } , \\beta _ { 2 } , \\ldots \\in \\mathbb { N } ^ { k }$ so that $\\begin{array} { r } { \\left| \\alpha \\right| + \\sum _ { i } \\left| \\beta _ { i } \\right| \\leq d } \\end{array}$ . Any solution to this equation will give a basis element of the form $\\begin{array} { r } { \\pmb { p } ( \\pmb { X } ) = \\lceil \\pmb { x } _ { 1 } ^ { \\alpha } \\rceil \\prod _ { j } \\left( \\sum _ { i = 1 } ^ { n } \\pmb { x } _ { i } ^ { \\beta _ { j } } \\right) } \\end{array}$ . ",
+        "bbox": [
+            173,
+            752,
+            825,
+            819
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In the paper we tested PointNetQT, an architecture that adds to PointNet a single quadratic equivariant layer. We opted to use only the quadratic transmission operators: For a matrix $\\bar { \\boldsymbol { X } } \\in \\mathbb { R } ^ { n \\times k }$ we define $\\boldsymbol { L } ( \\boldsymbol { X } ) \\in \\mathbb { R } ^ { \\tilde { n } \\times k }$ as follows: ",
+        "bbox": [
+            174,
+            825,
+            825,
+            867
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84b57d5672b1ba8a81212a3a3269ba0483fbeaec8308d9330a34a7a67f43c3bd.jpg",
+        "text": "$$\n\\begin{array} { r } { L ( X ) = X W _ { 1 } + 1 \\mathbf { 1 } ^ { T } X W _ { 2 } + ( \\mathbf { 1 } \\mathbf { 1 } ^ { T } X ) \\odot ( \\mathbf { 1 } \\mathbf { 1 } ^ { T } X ) W _ { 3 } + ( X \\odot X ) W _ { 4 } + ( \\mathbf { 1 } \\mathbf { 1 } ^ { T } X ) \\odot X W _ { 5 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            878,
+            813,
+            897
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ is a point-wise multiplication and $W _ { i } \\in \\mathbb { R } ^ { n \\times k } , i \\in [ 5 ]$ are the learnable parameters. ",
+        "bbox": [
+            176,
+            907,
+            787,
+            925
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B.2 GRAPH CONVOLUTION LAYER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            426,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We implement a graph convolution layers as follows ",
+        "bbox": [
+            173,
+            128,
+            517,
+            145
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be69056d3c81cf894ef8d8b957fdd85f5e2293a2bb55a3844bb220e1c03c0a3b.jpg",
+        "text": "$$\nL ( X ) = B X W _ { 2 } + X W _ { 1 } + \\mathbf { 1 } c ^ { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            150,
+            616,
+            170
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "with $W _ { 1 } , W _ { 2 } , c$ learnable parameters. The matrix $\\textbf {  { B } }$ is defined as in Kipf & Welling (2016) from the knn graph of the set. $\\stackrel { \\cdot } { B } = D ^ { - \\frac { 1 } { 2 } } A D ^ { - \\frac { 1 } { 2 } }$ where $_ { D }$ is the degree matrix of the graph and $\\pmb { A }$ is the adjacency matrix of the graph with added self-connections. ",
+        "bbox": [
+            173,
+            176,
+            826,
+            220
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            241,
+            436,
+            257
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Knapsack data generation. We constructed a dataset of $1 0 k$ training examples and $1 k$ test examples consisting of $5 0 \\times 4$ matrices. We took $w _ { 1 } = 1 0 0$ , $w _ { 2 } = 8 0$ , $w _ { 3 } = 5 0$ . To generate $\\pmb { X } \\in \\mathbb { R } ^ { 5 0 \\times 4 }$ , we draw an integer uniformly at random between 1 and 100 and randomly choose 50 integers between 1 as the first column of $\\boldsymbol { X }$ . We also randomly chose an integer between 1 and 25 and then randomly chose 150 integers in that range as the three last columns of $\\boldsymbol { X }$ . The labels for each input $\\boldsymbol { X }$ were computed by a standard dynamic programming approach, see Martello & Toth (1990). ",
+        "bbox": [
+            173,
+            271,
+            826,
+            356
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Optimization. We implemented the experiments in Pytorch Paszke et al. (2017) with the Adam Kingma & Ba (2014) optimizer for learning. For the classification we used the cross entropy loss and trained for 150 epochs with learning rate 0.001, learning rate decay of 0.5 every 100 epochs and batch size 32. For the quadratic function regression we trained for 150 epochs with leaning rate of 0.001, learning rate decay 0.1 every 50 epochs and batch size 64; for the regression to the leading eigen vector we trained for 50 epochs with leaning rate of 0.001 and batch size 32. To regress to the output of a single graph convolution layer we trained for 200 epochs with leaning rate of 0.001 and batch size 32. ",
+        "bbox": [
+            173,
+            371,
+            825,
+            483
+        ],
+        "page_idx": 10
+    }
+]

parse/train/HyXBcYg0b/HyXBcYg0b.md

@@ -0,0 +1,254 @@
+# RESIDUAL GATED GRAPH CONVNETS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Graph-structured data such as social networks, functional brain networks, gene regulatory networks, communications networks have brought the interest in generalizing deep learning techniques to graph domains. In this paper, we are interested to design neural networks for graphs with variable length in order to solve learning problems such as vertex classification, graph classification, graph regression, and graph generative tasks. Most existing works have focused on recurrent neural networks (RNNs) to learn meaningful representations of graphs, and more recently new convolutional neural networks (ConvNets) have been introduced. In this work, we want to compare rigorously these two fundamental families of architectures to solve graph learning tasks. We review existing graph RNN and ConvNet architectures, and propose natural extension of LSTM and ConvNet to graphs with arbitrary size. Then, we design a set of analytically controlled experiments on two basic graph problems, i.e. subgraph matching and graph clustering, to test the different architectures. Numerical results show that the proposed graph ConvNets are $3 . 1 7 \%$ more accurate and $1 . 5 – 4 \mathrm { x }$ faster than graph RNNs. Graph ConvNets are also $36 \%$ more accurate than variational (non-learning) techniques. Finally, the most effective graph ConvNet architecture uses gated edges and residuality. Residuality plays an essential role to learn multi-layer architectures as they provide a $10 \%$ gain of performance.
+
+# 1 INTRODUCTION
+
+Convolutional neural networks of LeCun et al. (1998) and recurrent neural networks of Hochreiter & Schmidhuber (1997) are deep learning architectures that have been applied with great success to computer vision (CV) and natural language processing (NLP) tasks. Such models require the data domain to be regular, such as 2D or 3D Euclidean grids for CV and 1D line for NLP. Beyond CV and NLP, data does not usually lie on regular domains but on heterogeneous graph domains. Users on social networks, functional time series on brain structures, gene DNA on regulatory networks, IP packets on telecommunication networks are a a few examples to motivate the development of new neural network techniques that can be applied to graphs. One possible classification of these techniques is to consider neural network architectures with fixed length graphs and variable length graphs.
+
+In the case of graphs with fixed length, a family of convolutional neural networks has been developed on spectral graph theory by Chung (1997). The early work of Bruna et al. (2013) proposed to formulate graph convolutional operations in the spectral domain with the graph Laplacian, as an analogy of the Euclidean Fourier transform as proposed by Hammond et al. (2011). This work was extended by Henaff et al. (2015) to smooth spectral filters for spatial localization. Defferrard et al. (2016) used Chebyshev polynomials to achieve linear complexity for sparse graphs, Levie et al. (2017) applied Cayley polynomials to focus on narrow-band frequencies, and Monti et al. (2017b) dealt with multiple (fixed) graphs. Finally, Kipf & Welling (2017) simplified the spectral convnets architecture using 1-hop filters to solve the semi-supervised clustering task. For related works, see also the works of Bronstein et al. (2017b), Bronstein et al. (2017a) and references therein.
+
+For graphs with variable length, a generic formulation was proposed by Gori et al. (2005); Scarselli et al. (2009) based on recurrent neural networks. The authors defined a multilayer perceptron of a vanilla RNN. This work was extended by Li et al. (2016) using a GRU architecture and a hidden state that captures the average information in local neighborhoods of the graph. The work of Sukhbaatar et al. (2016) introduced a vanilla graph ConvNet and used this new architecture to solve learning communication tasks. Marcheggiani & Titov (2017) introduced an edge gating mechanism in graph ConvNets for semantic role labeling. Finally, Bruna & Li (2017) designed a network to learn nonlinear approximations of the power of graph Laplacian operators, and applied it to the unsupervised graph clustering problem. Other works for drugs design, computer graphics and vision are presented by Duvenaud et al. (2015); Boscaini et al. (2016); Monti et al. (2017a).
+
+In this work, we study the two fundamental classes of neural networks, RNNs and ConvNets, in the context of graphs with arbitrary length. Section 2 reviews the existing techniques. Section 3 presents the new graph NN models. Section 4 reports the numerical experiments.
+
+# 2 NEURAL NETWORKS FOR GRAPHS WITH ARBITRARY LENGTH
+
+# 2.1 RECURRENT NEURAL NETWORKS
+
+Generic formulation. Consider a standard RNN for word prediction in natural language processing. Let $h _ { i }$ be the feature vector associated with word $i$ in the sequence. In a regular vanilla RNN, $h _ { i }$ is computed with the feature vector $h _ { j }$ from the previous step and the current word $x _ { i }$ , so we have:
+
+$$
+h _ { i } = f _ { \mathrm { V R N N } } \left( \begin{array} { l } { x _ { i } \ , \ \{ h _ { j } : j = i - 1 \} } \end{array} \right)
+$$
+
+The notion of neighborhood for regular RNNs is the previous step in the sequence. For graphs, the notion of neighborhood is given by the graph structure. If $h _ { i }$ stands for the feature vector of vertex $i$ , then the most generic version of a feature vector for a graph RNN is
+
+$$
+h _ { i } = f _ { \mathrm { G - R N N } } ( \begin{array} { l } { x _ { i } } \end{array} , \begin{array} { l } { \{ h _ { j } : j  i \} } \end{array} )
+$$
+
+where $x _ { i }$ refers to a data vector and $\{ h _ { j } : j  i \}$ denotes the set of feature vectors of the neighboring vertices. Observe that the set $\{ h _ { j } \}$ is unordered, meaning that $h _ { i }$ is intrinsic, i.e. invariant by vertex re-indexing (no vertex matching between graphs is required). Other properties of $f _ { \mathrm { G - R N N } }$ are locality as only neighbors of vertex $i$ are considered, weight sharing, and such vector is independent of the graph length. In summary, to define a feature vector in a graph RNN, one needs a mapping $f$ that takes as input an unordered set of vectors $\{ h _ { j } \}$ , i.e. the feature vectors of all neighboring vertices, and a data vector $x _ { i }$ , Figure 1(a).
+
+We refer to the mapping $f _ { \mathrm { G - R N N } }$ as the neighborhood transfer function in graph RNNs. In a regular RNN, each neighbor as a distinct position relatively to the current word (1 position left from the center). In a graph, if the edges are not weighted or annotated, neighbors are not distinguishable. The only vertex which is special is the center vertex around which the neighborhood is built. This explains the generic formulation of Eq. (1). This type of formalism for deep learning for graphs with variable length is described by Scarselli et al. (2009); Gilmer et al. (2017); Bronstein et al. (2017a) with slightly different terminology and notations.
+
+Graph Neural Networks of Scarselli et al. (2009). The earliest work of graph RNNs for arbitrary graphs was introduced by Gori et al. (2005); Scarselli et al. (2009). The authors proposed to use a vanilla RNN with a multilayer perceptron to define the feature vector $h _ { i }$ :
+
+$$
+h _ { i } = f _ { \mathrm { G \cdot V R N } } ( x _ { i } , \{ h _ { j } : j  i \} ) = \sum _ { j  i } \mathcal { C } _ { \mathrm { G \cdot V R N } } ( x _ { i } , h _ { j } )
+$$
+
+with
+
+$$
+\mathcal { C } _ { \mathrm { G - V R N N } } ( x _ { i } , h _ { j } ) = A \sigma ( B \sigma ( U x _ { i } + V h _ { j } ) ) ,
+$$
+
+and $\sigma$ is the sigmoid function, $A , B , U , V$ are the weight parameters to learn.
+
+Minimization of Eq. (2) does not hold a closed-form solution as the dependence computational graph of the model is not a directed acyclic graph (DAG). Scarselli et al. (2009) proposed a fixed-point iterative scheme: for $t = 0 , 1 , 2 , \ldots$ .
+
+$$
+h _ { i } ^ { t + 1 } = \sum _ { j  i } { \mathcal { C } } ( x _ { i } , h _ { j } ^ { t } ) , \quad h _ { i } ^ { t = 0 } = 0 \ \forall i .
+$$
+
+The iterative scheme is guaranteed to converge as long as the mapping is contractive, which can be a strong assumption. Besides, a large number of iterations can be computational expensive.
+
+![](images/ae08cb64e7e1f123a452ccae4f2f714f7c91898685153b01f5b0bf6018a27c4c.jpg)  
+Figure 1: Generic feature representation $h _ { i }$ of vertex $i$ on a graph RNN (a) and a graph convNet (b).
+
+Gated Graph Neural Networks of Li et al. (2016). In this work, the authors use the gated recurrent units (GRU) of Chung et al. (2014):
+
+$$
+h _ { i } = f _ { \mathrm { G - G R U } } ( x _ { i } , \{ h _ { j } : j  i \} ) = \mathcal { C } _ { \mathrm { G - G R U } } ( x _ { i } , \sum _ { j  i } h _ { j } )
+$$
+
+As the minimization of Eq. (4) does not have an analytical solution, Li et al. (2016) designed the following iterative scheme:
+
+$$
+\begin{array} { r c l l } { { { h } _ { i } ^ { t + 1 } } } & { { = } } & { { \displaystyle { \mathcal C } _ { \mathrm { G - G R U } } ( h _ { i } ^ { t } , \bar { h } _ { i } ^ { t } ) , } } & { { \displaystyle h _ { i } ^ { t = 0 } = x _ { i } \forall i , } } \\ { { \mathrm { w h e r e } \quad \bar { h } _ { i } ^ { t } } } & { { = } } & { { \displaystyle \sum _ { j \to i } h _ { j } ^ { t } , } } \end{array}
+$$
+
+and $\mathcal { C } _ { \mathrm { G - G R U } } ( h _ { i } ^ { t } , \bar { h } _ { i } ^ { t } )$ is equal to
+
+$$
+\begin{array} { r c l } { { z _ { i } ^ { t + 1 } } } & { { = } } & { { \sigma ( U _ { z } h _ { i } ^ { t } + V _ { z } \bar { h } _ { i } ^ { t } ) } } \\ { { r _ { i } ^ { t + 1 } } } & { { = } } & { { \sigma ( U _ { r } h _ { i } ^ { t } + V _ { r } \bar { h } _ { i } ^ { t } ) } } \\ { { \tilde { h } _ { i } ^ { t + 1 } } } & { { = } } & { { \operatorname { t a n h } \bigl ( U _ { h } ( h _ { i } ^ { t } \odot r _ { i } ^ { t + 1 } ) + V _ { h } \bar { h } _ { i } ^ { t } \bigr ) } } \\ { { h _ { i } ^ { t + 1 } } } & { { = } } & { { ( 1 - z _ { i } ^ { t + 1 } ) \odot h _ { i } ^ { t } + z _ { i } ^ { t + 1 } \odot \tilde { h } _ { i } ^ { t + 1 } , } } \end{array}
+$$
+
+where $\odot$ is the Hadamard point-wise multiplication operator. This model was used for NLP tasks by Li et al. (2016) and also in quantum chemistry by Gilmer et al. (2017) for fast organic molecule properties estimation, for which standard techniques (DFT) require expensive computational time.
+
+Tree-Structured LSTM of Tai et al. (2015). The authors extended the original LSTM model of Hochreiter $\&$ Schmidhuber (1997) to a tree-graph structure:
+
+$$
+h _ { i } = f _ { \mathrm { T L S T M } } \left( x _ { i } , \left\{ h _ { j } : j \in C ( i ) \right\} \right) = \mathcal { C } _ { \mathrm { T L S T M } } ( x _ { i } , h _ { i } , \sum _ { j \in C ( i ) } h _ { j } ) ,
+$$
+
+where $C ( i )$ refers the set of children of node $i$ . $\begin{array} { r } { \mathcal { C } _ { \mathrm { T - L S T M } } ( x _ { i } , h _ { i } , \sum _ { j \in C ( i ) } h _ { j } ) } \end{array}$ is equal to
+
+$$
+\begin{array} { r c l } { \displaystyle \bar { h } _ { i } } & { = } & { \displaystyle \sum _ { j \in { \cal C } ( i ) } h _ { j } } \\ { \displaystyle i _ { i } } & { = } & { \sigma ( U _ { i } x _ { i } + V _ { i } \bar { h } _ { i } ) } \\ { \displaystyle o _ { i } } & { = } & { \sigma ( U _ { o } x _ { i } + V _ { o } \bar { h } _ { i } ) } \\ { \displaystyle \tilde { c } _ { i } } & { = } & { \mathrm { t a n h } \big ( U _ { c } x _ { i } + V _ { c } \bar { h } _ { i } \big ) } \\ { \displaystyle f _ { i j } } & { = } & { \sigma \big ( U _ { f } x _ { i } + V _ { f } h _ { j } \big ) } \\ { \displaystyle c _ { i } } & { = } & { i _ { i } \odot \tilde { c } _ { i } + { \displaystyle \sum _ { i \in { \cal C } ( i ) } f _ { i j } } \odot c _ { j } } \\ { \displaystyle h _ { i } } & { = } & { o _ { i } \odot \mathrm { t a n h } \big ( c _ { i } \big ) } \end{array}
+$$
+
+Unlike the works of Scarselli et al. (2009); Li et al. (2016), Tree-LSTM does not require an iterative process to update its feature vector $h _ { i }$ as the tree structure is also a DAG as original LSTM. Consequently, the feature representation (5) can be updated with a recurrent formula. Nevertheless, a tree is a special case of graphs, and such recurrence formula cannot be directly applied to arbitrary graph structure. A key property of this model is the function $f _ { i j }$ which acts as a gate on the edge from neighbor $j$ to vertex $i$ . Given the task, the gate will close to let the information flow from neighbor $j$ to vertex $i$ , or it will open to stop it. It seems to be an essential property for learning systems on graphs as some neighbors can be irrelevant. For example, for the community detection task, the graph neural network should learn which neighbors to communicate (same community) and which neighbors to ignore (different community). In different contexts, Dauphin et al. (2017) added a gated mechanism inside the regular ConvNets in order to improve language modeling for translation tasks, and van den Oord et al. (2016) considered a gated unit with the convolutional layers after activation, and used it for image generation.
+
+# 2.2 CONVOLUTIONAL NEURAL NETWORKS
+
+Generic formulation. Consider now a classical ConvNet for computer vision. Let $h _ { i j } ^ { \ell }$ denote the feature vector at layer $\ell$ associated with pixel $( i , j )$ . In a regular ConvNet, $h _ { i j } ^ { \ell + 1 }$ is obtained by applying a non linear transformation to the feature vectors $h _ { i ^ { \prime } j ^ { \prime } } ^ { \ell }$ for all pixels $( i ^ { \prime } , j ^ { \prime } )$ in a neighborhood of pixel $( i , j )$ . For example, with $3 \times 3$ filters, we would have:
+
+$$
+h _ { i j } ^ { \ell + 1 } = f _ { \mathrm { C N N } } ^ { \ell } \left( \left\{ h _ { i ^ { \prime } j ^ { \prime } } ^ { \ell } : | i - i ^ { \prime } | \leq 1 \mathrm { ~ a n d ~ } | j - j ^ { \prime } | \leq 1 \right\} \right)
+$$
+
+In the above, the notation $\{ h _ { i ^ { \prime } j ^ { \prime } } ^ { \ell } : | i - i ^ { \prime } | \le 1$ and $| j - j ^ { \prime } | \leq 1 \}$ denote the concatenation of all feature vectors $h _ { i ^ { \prime } j ^ { \prime } } ^ { \ell }$ belonging to the $3 \times 3$ neighborhood of vertex $( i , j )$ . In ConvNets, the notion of neighborhood is given by the euclidian distance. As previously noticed, for graphs, the notion of neighborhood is given by the graph structure. Thus, the most generic version of a feature vector $h _ { i }$ at vertex $i$ for a graph ConvNet is
+
+$$
+h _ { i } ^ { \ell + 1 } = f _ { \scriptscriptstyle \mathrm { G - C N N } } ( h _ { i } ^ { \ell } , \{ h _ { j } ^ { \ell } : j  i \} )
+$$
+
+where $\{ h _ { j } ^ { \ell } : j  i \}$ denotes the set of feature vectors of the neighboring vertices. In other words, to define a graph ConvNet, one needs a mapping $f _ { \mathrm { G - C N N } }$ taking as input a vector $h _ { i } ^ { \ell }$ (the feature vector of the center vertex) as well as an unordered set of vectors $\{ h _ { j } ^ { \ell } \}$ (the feature vectors of all neighboring vertices), see Figure 1(b). We also refer to the mapping $f _ { \mathrm { G - C N N } }$ as the neighborhood transfer function. In a regular ConvNet, each neighbor as a distinct position relatively to the center pixel (for example 1 pixel up and 1 pixel left from the center). As for graph RNNs, the only vertex which is special for graph ConvNets is the center vertex around which the neighborhood is built.
+
+CommNets of Sukhbaatar et al. (2016). The authors introduced one of the simplest instantiations of a graph ConvNet with the following neighborhood transfer function:
+
+$$
+h _ { i } ^ { \ell + 1 } = f _ { \mathrm { G \cdot v C N N } } ^ { \ell } \left( h _ { i } ^ { \ell } , \frac { } { } \{ h _ { j } ^ { \ell } : j \to i \} \right) = \mathrm { R e L U } \left( U ^ { \ell } h _ { i } ^ { \ell } + V ^ { \ell } \sum _ { j \to i } h _ { j } ^ { \ell } \right) ,
+$$
+
+where $\ell$ denotes the layer level, and ReLU is the rectified linear unit. We will refer to this architecture as the vanilla graph ConvNet. Sukhbaatar et al. (2016) used this graph neural network to learn the communication between multiple agents to solve multiple tasks like traffic control.
+
+Syntactic Graph Convolutional Networks of Marcheggiani $\pmb { \& }$ Titov (2017). The authors proposed the following transfer function:
+
+$$
+h _ { i } ^ { \ell + 1 } = f _ { \mathrm { s - G C N } } ^ { \ell } \left( \frac { } { } \{ h _ { j } ^ { \ell } : j \to i \} \ \right) = \mathrm { R e L U } \left( \sum _ { j \to i } \eta _ { i j } \odot V ^ { \ell } h _ { j } ^ { \ell } \right)
+$$
+
+where $\eta _ { i j }$ act as edge gates, and are computed by:
+
+$$
+\eta _ { i j } = \sigma \left( A ^ { \ell } h _ { i } ^ { \ell } + B ^ { \ell } h _ { j } ^ { \ell } \right) .
+$$
+
+These gated edges are very similar in spirit to the Tree-LSTM proposed in Tai et al. (2015). We believe this mechanism to be important for graphs, as they will be able to learn what edges are important for the graph learning task to be solved.
+
+# 3 MODELS
+
+Proposed Graph LSTM. First, we propose to extend the Tree-LSTM of Tai et al. (2015) to arbitrary graphs and multiple layers:
+
+$$
+{ h } _ { i } ^ { \ell + 1 } = f _ { \mathrm { G \mathrm { \tiny { L S T M } } } } ^ { \ell } ( x _ { i } ^ { \ell } , \{ h _ { j } ^ { \ell } : j  i \} ) = \mathcal { C } _ { \mathrm { G \mathrm { \tiny { L S T M } } } } ( x _ { i } ^ { \ell } , h _ { i } ^ { \ell } , \sum _ { j  i } h _ { j } ^ { \ell } , c _ { i } ^ { \ell } )
+$$
+
+As there is no recurrent formula is the general case of graphs, we proceed as Scarselli et al. (2009) and use an iterative process to solve Eq. (10): At layer $\ell$ , for $t = 0 , 1 , . . . , T$
+
+$$
+\begin{array} { r l } { \bar { h } _ { \dot { i } } ^ { \ell , t } } & { = \displaystyle \sum _ { j  i } h _ { j } ^ { \ell , t } , } \\ { { \dot { a } _ { i } ^ { \ell , t + 1 } } } & { = \sigma ( U _ { i } ^ { \ell } x _ { i } ^ { \ell } + V _ { i } ^ { \ell } { \bar { h } _ { i } ^ { \ell , t } } ) } \\ { { \dot { o } _ { i } ^ { \ell , t + 1 } } } & { = \sigma ( U _ { i } ^ { \ell } x _ { i } ^ { \ell } + V _ { i } ^ { \ell } { \bar { h } _ { i } ^ { \ell , t } } ) } \\ { { \dot { c } _ { i } ^ { \ell , t + 1 } } } & { = \mathrm { t a n h } ( { \bar { c } _ { i } ^ { \ell } x _ { i } ^ { \ell } + \bar { h } _ { i } ^ { \ell } \bar { h } _ { i } ^ { \ell , t } } ) } \\ { { \dot { f } _ { i } ^ { \ell , t + 1 } } } & { = \sigma ( U _ { i } ^ { \ell } x _ { i } ^ { \ell } + V _ { i } ^ { \ell } { \bar { h } _ { i } ^ { \ell , t } } ) } \\ { { \dot { c } _ { i } ^ { \ell , t + 1 } } } & { = { \dot { \iota } _ { i } ^ { \ell , t + 1 } } \odot { \bar { c } _ { i } ^ { \ell } } ^ { t + 1 } + \displaystyle \sum _ { j  i } f _ { i j } ^ { \ell , t + 1 } \odot { c _ { j } ^ { \ell , t + 1 } } } \\ { { \dot { h } _ { \dot { i } } ^ { \ell , t + 1 } } } & { = { \dot { \sigma } _ { i } ^ { \ell , t + 1 } } \odot \mathrm { t a n h } ( { c } _ { i } ^ { \ell } t ^ { + 1 } ) } \end{array}
+$$
+
+and initial conditions: $\begin{array} { r c l } { { h _ { i } ^ { \ell , t = 0 } } } & { { = } } & { { c _ { i } ^ { \ell , t = 0 } = 0 , \forall i , \ell } } \\ { { x _ { i } ^ { \ell } } } & { { = } } & { { h _ { i } ^ { \ell - 1 , T } , x _ { i } ^ { \ell = 0 } = x _ { i } , \forall i , \ell } } \end{array}$
+
+In other words, the vector $h _ { i } ^ { \ell + 1 }$ is computed by running the model from $t = 0 , . . , T$ at layer $\ell$ . It produces the vector $h _ { i } ^ { \ell , t = T }$ which becomes $h _ { i } ^ { \ell + 1 }$ and also the input $x _ { i } ^ { \ell + 1 }$ for the next layer. The proposed Graph LSTM model differs from Liang et al. (2016); Peng et al. (2017) mostly because the cell $\mathcal { C } _ { \mathrm { G - L S T M } }$ in these previous models is not iterated over multiple times $T$ , which reduces the performance of Graph LSTM (see numerical experiments on Figure 4).
+
+Proposed Gated Graph ConvNets. We leverage the vanilla graph ConvNet architecture of Sukhbaatar et al. (2016), Eq.(7), and the edge gating mechanism of Marcheggiani $\&$ Titov (2017), Eq.(8), by considering the following model:
+
+$$
+h _ { i } ^ { \ell + 1 } = f _ { \mathrm { G \ell { \mathrm { G C N N } } } } ^ { \ell } ( h _ { i } ^ { \ell } , \{ h _ { j } ^ { \ell } : j  i \} ) = \mathrm { R e L U } ( U ^ { \ell } h _ { i } ^ { \ell } + \sum _ { j  i } \eta _ { i j } \odot V ^ { \ell } h _ { j } ^ { \ell } )
+$$
+
+where $h _ { i } ^ { \ell = 0 } = x _ { i } , \forall i$ , and the edge gates $\eta _ { i j }$ are defined in Eq. (9). This model is the most generic formulation of a graph ConvNet (because it uses both the feature vector $h _ { i } ^ { \ell }$ of the center vertex and the feature vectors $h _ { j } ^ { \ell }$ of neighboring vertices) with the edge gating property.
+
+Residual Gated Graph ConvNets. In addition, we formulate a multi-layer gated graph ConvNet using residual networks (ResNets) introduced by He et al. (2016). This boils down to add the identity operator between successive convolutional layers:
+
+$$
+\begin{array} { r l r } { h _ { i } ^ { \ell + 1 } } & { = } & { f ^ { \ell } \left( \ h _ { i } ^ { \ell } , \ \{ h _ { j } ^ { \ell } : j \to i \} \ \right) + h _ { i } ^ { \ell } . } \end{array}
+$$
+
+As we will see, such multi-layer strategy work very well for graph neural networks.
+
+# 4 EXPERIMENTS
+
+# 4.1 SUBGRAPH MATCHING
+
+We consider the subgraph matching problem presented by Scarselli et al. (2009), see Figure 2(a).   
+The goal is to find the vertices of a given subgraph $P$ in larger graphs $G _ { k }$ with variable sizes.
+
+![](images/fe9266c226259a8fd9f2ac74589691f443597b836426fd524d2758931935a96a.jpg)  
+Figure 2: Graph learning tasks.
+
+Identifying similar localized patterns in different graphs is one of the most basic tasks for graph neural networks. The subgraph $P$ and larger graph $G _ { k }$ are generated with the stochastic block model (SBM), see for example Abbe (2017). A SBM is a random graph which assigns communities to each node as follows: any two vertices are connected with the probability $p$ if they belong to the same community, or they are connected with the probability $q$ if they belong to different communities. For all experiments, we generate a subgraph $P$ of 20 nodes with a SBM $q = 0 . 5$ , and the signal on $P$ is generated with a uniform random distribution with a vocabulary of size 3, i.e. $\{ 0 , 1 , 2 \}$ . Larger graphs $G _ { k }$ are composed of 10 communities with sizes randomly generated between 15 and 25. The SBM of each community is $p = 0 . 5$ . The value of $q$ , which acts as the noise level, is 0.1, unless otherwise specified. Besides, the signal on $G _ { k }$ is also randomly generated between $\{ 0 , 1 , 2 \}$ . Inputs of all neural networks are the graphs with variable size, and outputs are vertex classification vectors of input graphs. Finally, the output of neural networks are simple fully connected layers from the hidden states.
+
+All reported results are averaged over 5 trails. We run 5 algorithms; Gated Graph Neural Networks of Li et al. (2016), CommNets of Sukhbaatar et al. (2016), SyntacticNets of Marcheggiani & Titov (2017), and the proposed Graph LSTM and Gated ConvNets from Section 3. We upgrade the existing models of Li et al. (2016); Sukhbaatar et al. (2016); Marcheggiani & Titov (2017) with a multilayer version for Li et al. (2016) and using ResNets for all three architectures. We also use the batch normalization technique of Ioffe & Szegedy (2015) to speed up learning convergence for our algorithms, and also for Li et al. (2016); Sukhbaatar et al. (2016); Marcheggiani & Titov (2017). The learning schedule is as follows: the maximum number of iterations, or equivalently the number of randomly generated graphs with the attached subgraph is 5,000 and the learning rate is decreased by a factor 1.25 if the loss averaged over 100 iterations does not decrease. The loss is the cross-entropy with 2 classes (the subgraph $P$ class and the class of the larger graph $G _ { k }$ ) respectively weighted by their sizes. The accuracy is the average of the diagonal of the normalized confusion matrix w.r.t. the cluster sizes (the confusion matrix measures the number of nodes correctly and badly classified for each class). We also report the time for a batch of 100 generated graphs. The choice of the architectures will be given for each experiment. All algorithms are optimized as follow. We fix a budget of parameters of $B = 1 0 0 K$ and a number of layers $L = 6$ . The number of hidden neurons $H$ for each layer is automatically computed. Then we manually select the optimizer and learning rate for each architecture that best minimize the loss. For this task, Li et al. (2016); Sukhbaatar et al. (2016); Marcheggiani & Titov (2017) and our gated ConvNets work well with Adam and learning rate 0.00075. Graph LSTM uses SGD with learning rate 0.075. Besides, the value of inner iterative steps $T$ for graph LSTM and Li et al. (2016) is 3.
+
+The first experiment focuses on shallow graph neural networks, i.e. with a single layer $L = 1$ . We also vary the level of noise, that is the probability $q$ in the SBM that connects two vertices in two different communities (the higher $q$ the more mixed are the communities). The hyper-parameters are selected as follows. Besides $L = 1$ , the budget is $B = 1 0 0 K$ and the number of hidden neurons $H$ is automatically computed for each architecture to satisfy the budget. First row of Figure 3 reports the accuracy and time for the five algorithms and for different levels of noise $q = \{ 0 . \bar { 1 } , 0 . 2 , 0 . \bar { 3 } 5 , 0 . 5 \}$ . RNN architectures are plotted in dashed lines and ConvNet architectures in solid lines. For shallow networks, all RNN architectures (graph LSTM and Li et al. (2016)) performs much better, but they also take more time than the graph ConvNets architectures we propose, as well as Sukhbaatar et al.
+
+(2016); Marcheggiani & Titov (2017). As expected, all algorithms performances decrease when the noise increases.
+
+The second experiment demonstrates the importance of having multiple layers compared to shallow networks. We vary the number of layers $\bar { L } = \{ 1 , 2 , 4 , 6 , 1 \bar { 0 } \}$ and we fix the number of hidden neurons to $H \ : = \ : 5 0$ . Notice that the budget is not the same for all architectures. Second row of Figure 3 reports the accuracy and time w.r.t. $L$ (middle figure is a zoom in the left figure). All models clearly benefit with more layers, but RNN-based architectures see their performances decrease for a large number of layers. The ConvNet architectures benefit from large $L$ values, with the proposed graph ConvNet performing slightly better than Sukhbaatar et al. (2016); Marcheggiani & Titov (2017). Besides, all ConvNet models are faster than RNN models.
+
+In the third experiment, we evaluate the algorithms for different budgets of parameters $B \ =$ $\{ 2 5 K , 5 0 K , 7 5 K , 1 0 0 K , 1 5 0 K \}$ . For this experiment, we fix the number of layers $L \ = \ 6$ and the number of neurons $H$ is automatically computed given the budget $B$ . The results are reported in the third row of Figure 3. For this task, the proposed graph ConvNet best performs for a large budget, while being faster than RNNs.
+
+![](images/dcde2c4f276a50992f1d845a91368215338f8db5149ca1d2eb732f34dd415995.jpg)  
+Figure 3: Subgraph matching: First row studies shallow networks w.r.t. noise. Second row investigates multilayer graph networks. Third row reports graph architectures w.r.t. budget.
+
+We also show the influence of hyper-parameter $T$ for Li et al. (2016) and the proposed graph LSTM. We fix $H = 5 0$ , $L = 3$ and $B = 1 0 0 K$ . Figure 4 reports the results for $T = \{ 1 , 2 , 3 , 4 , 6 \}$ . The $T$ value has an undesirable impact on the performance of graph LSTM. Multi-layer Li et al. (2016) is not really influenced by $T$ . Finally, the computational time naturally increases with larger $T$ values.
+
+# 4.2 SEMI-SUPERVISED CLUSTERING
+
+In this section, we consider the semi-supervised clustering problem, see Figure 2(b). This is also a standard task in network science. For this work, it consists in finding 10 communities on a graph given 1 single label for each community. This problem is more discriminative w.r.t. to the architectures than the previous single pattern matching problem where there were only 2 clusters to find (i.e. $50 \%$ random chance). For clustering, we have 10 clusters (around $10 \%$ random chance). As in the previous section, we use SBM to generate graphs of communities with variable length. The size for each community is randomly generated between 5 and 25, and the label is randomly selected in each community. Probability $p$ is 0.5, and $q$ depends on the experiment. For this task, Li et al. (2016); Sukhbaatar et al. (2016); Marcheggiani & Titov (2017) and the proposed gated ConvNets work well with Adam and learning rate 0.00075. Graph LSTM uses SGD with learning rate 0.0075. The value of $T$ for graph LSTM and Li et al. (2016) is 3.
+
+![](images/29fc1a6ccd0fa8c50620d388786c8fec23357cf50224fe7466b1f70c4f0ff43f.jpg)  
+Figure 4: Influence of hyper-parameter $T$ on RNN architectures. Left figure is for graph matching, middle figure for semi-supervised clustering, and right figure are the batch time for the clustering task (same trend for matching).
+
+The same set of experiments as in the previous task are reported in Figure 5. ConvNet architectures get clearly better than RNNs when the number of layers increase (middle row), with the proposed Gated ConvNet outperforming the other architectures. For a fixed number of layers $L = 6$ , our graph ConvNets and Marcheggiani & Titov (2017) best perform for all budgets, while paying a reasonable computational cost.
+
+![](images/ff4244af7cedb9a88d147a5dea9aab32820c1913899132c1e396441ff789676f.jpg)  
+Figure 5: Semi-supervised clustering: First row reports shallow networks w.r.t. noise $q$ . Second row shows multilayer graph networks w.r.t. $L$ . Third row is about graph architectures w.r.t. budget $B$ .
+
+Next, we report the learning speed of the models. We fix $L \ = \ 6$ , $B \ = \ 1 0 0 K$ with $H$ being automatically computed to satisfy the budget. Figure 6 reports the accuracy w.r.t. time. The ConvNet architectures converge faster than RNNs, in particular for the semi-supervised task.
+
+![](images/c2aa29c01817800597511abc623b80c55cc6ca7fb741d397a4b0118a11a0e63b.jpg)  
+Figure 6: Learning speed of RNN and ConvNet architectures. Left figure is for graph matching and right figure semi-supervised clustering.
+
+To close this study, we are interested in comparing learning based approaches to non-learning variational ones. To this aim, we solve the variational Dirichlet problem with labeled and unlabelled data as proposed by Grady (2006). We run 100 experiments and report an average accuracy of $4 5 . 3 7 \%$ using the same setting as the learning techniques (one label per class). The performance of the best learning model is $82 \%$ . Learning techniques produce better performances with a different paradigm as they use training data with ground truth, while variational techniques do not use such information. The downside is the need to see 2000 training graphs to get to $82 \%$ . However, when the training is done, the test complexity of these learning techniques is $O ( E )$ , where $E$ is the number of edges in the graph. This is an advantage over the variational Dirichlet model that solves a sparse linear system of equations with complexity ${ \cal O } ( E ^ { 3 / 2 } )$ , see Lipton et al. (1979).
+
+# 5 CONCLUSION
+
+This work explores the choice of graph neural network architectures for solving learning tasks with graphs of variable length. We developed analytically controlled experiments for two fundamental graph learning problems, that are subgraph matching and graph clustering. Numerical experiments showed that graph ConvNets had a monotonous increase of accuracy when the network gets deeper, unlike graph RNNs for which performance decreases for a large number of layers. This led us to consider the most generic formulation of gated graph ConvNets, Eq. (11). We also explored the benefit of residuality for graphs, Eq. (12). Without residuality, existing graph neural networks are not able to stack more than a few layers. This makes this property essential for graph neural networks, which receive a $10 \%$ boost of accuracy when more than 6 layers were stacked. Future work will focus on solving domain-specific problems in chemistry, physics, and neuroscience.
+
+# REFERENCES
+
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+D. Boscaini, J. Masci, E. Rodola, and M. M. Bronstein. Learning shape correspondence with \` anisotropic convolutional neural networks. In Proc. NIPS, 2016.   
+M. Bronstein, X. Bresson, A. Szlam, J. Bruna, and Y. LeCun. Tutorial on Geometric Deep Learning on Graphs and Manifolds. Conference on Computer Vision and Pattern Recognition (CVPR), 2017a.   
+M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam, and P. Vandergheynst. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 2017b.   
+J. Bruna and X. Li. Community Detection with Graph Neural Networks. arXiv preprint arXiv:1705.08415, 2017.   
+J. Bruna, W. Zaremba, A. Szlam, and Y. LeCun. Spectral Networks and Deep Locally Connected Networks on Graphs. arXiv:1312.6203, 2013.   
+F. R. K. Chung. Spectral Graph Theory, volume 92. American Mathematical Society, 1997.   
+J. Chung, C. Gulcehre, K. Cho, and Y. Bengio. Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling. arXiv preprint arXiv:1412.3555, 2014.   
+Y. Dauphin, A. Fan, M. Auli, and D. Grangier. Language Modeling with Gated Convolutional Networks. International Conference on Machine Learning (ICML), 2017.   
+M. Defferrard, X. Bresson, and P. Vandergheynst. Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering. Advances in Neural Information Processing Systems (NIPS), pp. 3844–3852, 2016.   
+D. K. Duvenaud, D. Maclaurin, J. Iparraguirre, R. Bombarell, T. Hirzel, A. Aspuru-Guzik, and R. P Adams. Convolutional networks on graphs for learning molecular fingerprints. In Proc. NIPS, 2015.   
+J. Gilmer, S. Schoenholz, P. Riley, O. Vinyals, and G. Dahl. Neural Message Passing for Quantum Chemistry. arXiv preprint arXiv:1704.01212, 2017.   
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+M. Henaff, J. Bruna, and Y. LeCun. Deep Convolutional Networks on Graph-Structured Data. arXiv:1506.05163, 2015.   
+S. Hochreiter and J. Schmidhuber. Long Short-Term Memory. Neural Computation, 9(8):1735– 1780, 1997.   
+S. Ioffe and C. Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. International Conference on Machine Learning (ICML), pp. 448–456, 2015.   
+T. Kipf and M. Welling. Semi-Supervised Classification with Graph Convolutional Networks. International Conference on Learning Representations (ICLR), 2017.   
+Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-Based Learning Applied to Document Recognition. In Proceedings of the IEEE, 86(11), pp. 2278–2324, 1998.   
+R. Levie, F. Monti, X. Bresson, and M.M. Bronstein. CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters. arXiv preprint arXiv:1705.07664, 2017.   
+Y. Li, D. Tarlow, M. Brockschmidt, and R. Zemel. Gated Graph Sequence Neural Networks. International Conference on Learning Representations (ICLR), 2016.   
+X. Liang, X. Shen, J. Feng, L. Lin, and S. Yan. Semantic Object Parsing with graph LSTM. In European Conference on Computer Vision (ECCV), pp. 125–143, 2016.   
+R. Lipton, D. Rose, and R. Tarjan. Generalized Nested Dissection. SIAM Journal on Numerical Analysis, 16(2):346–358, 1979.   
+D. Marcheggiani and I. Titov. Encoding Sentences with Graph Convolutional Networks for Semantic Role Labeling. arXiv preprint arXiv:1703.04826, 2017.   
+F. Monti, D. Boscaini, J. Masci, E. Rodola, J. Svoboda, and M. M. Bronstein. Geometric deep \` learning on graphs and manifolds using mixture model CNNs. In Proc. CVPR, 2017a.   
+F. Monti, M.M. Bronstein, and X. Bresson. Geometric Matrix Completion with Recurrent MultiGraph Neural Networks. Advances in Neural Information Processing Systems (NIPS), 2017b.   
+N. Peng, H. Poon, C. Quirk, K. Toutanova, and W.T. Yih. Cross-Sentence N-ary Relation Extraction with Graph LSTMs. Transactions of the Association for Computational Linguistics, 5:101–115, 2017.   
+F. Scarselli, M. Gori, A. Tsoi, M. Hagenbuchner, and G. Monfardini. The Graph Neural Network Model. IEEE Transactions on Neural Networks, 20(1):61–80, 2009.   
+S. Sukhbaatar, A Szlam, and R. Fergus. Learning Multiagent Communication with Backpropagation. Advances in Neural Information Processing Systems (NIPS), pp. 2244–2252, 2016.   
+K. Tai, R. Socher, and C. Manning. Improved Semantic Representations from Tree-Structured Long Short-Term Memory Networks. Association for Computational Linguistics (ACL), pp. 1556– 1566, 2015.   
+van den Oord, N. Kalchbrenner, L. Espeholt, O. Vinyals, A. Graves, and K. Kavukcuoglu. Conditional Image Generation with PixelCNN Decoders. In Advances in Neural Information Processing Systems (NIPS), pp. 4790–4798, 2016.

parse/train/HyXBcYg0b/HyXBcYg0b_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "RESIDUAL GATED GRAPH CONVNETS ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph-structured data such as social networks, functional brain networks, gene regulatory networks, communications networks have brought the interest in generalizing deep learning techniques to graph domains. In this paper, we are interested to design neural networks for graphs with variable length in order to solve learning problems such as vertex classification, graph classification, graph regression, and graph generative tasks. Most existing works have focused on recurrent neural networks (RNNs) to learn meaningful representations of graphs, and more recently new convolutional neural networks (ConvNets) have been introduced. In this work, we want to compare rigorously these two fundamental families of architectures to solve graph learning tasks. We review existing graph RNN and ConvNet architectures, and propose natural extension of LSTM and ConvNet to graphs with arbitrary size. Then, we design a set of analytically controlled experiments on two basic graph problems, i.e. subgraph matching and graph clustering, to test the different architectures. Numerical results show that the proposed graph ConvNets are $3 . 1 7 \\%$ more accurate and $1 . 5 – 4 \\mathrm { x }$ faster than graph RNNs. Graph ConvNets are also $36 \\%$ more accurate than variational (non-learning) techniques. Finally, the most effective graph ConvNet architecture uses gated edges and residuality. Residuality plays an essential role to learn multi-layer architectures as they provide a $10 \\%$ gain of performance. ",
+        "bbox": [
+            233,
+            239,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
+            334,
+            546
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional neural networks of LeCun et al. (1998) and recurrent neural networks of Hochreiter & Schmidhuber (1997) are deep learning architectures that have been applied with great success to computer vision (CV) and natural language processing (NLP) tasks. Such models require the data domain to be regular, such as 2D or 3D Euclidean grids for CV and 1D line for NLP. Beyond CV and NLP, data does not usually lie on regular domains but on heterogeneous graph domains. Users on social networks, functional time series on brain structures, gene DNA on regulatory networks, IP packets on telecommunication networks are a a few examples to motivate the development of new neural network techniques that can be applied to graphs. One possible classification of these techniques is to consider neural network architectures with fixed length graphs and variable length graphs. ",
+        "bbox": [
+            174,
+            561,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the case of graphs with fixed length, a family of convolutional neural networks has been developed on spectral graph theory by Chung (1997). The early work of Bruna et al. (2013) proposed to formulate graph convolutional operations in the spectral domain with the graph Laplacian, as an analogy of the Euclidean Fourier transform as proposed by Hammond et al. (2011). This work was extended by Henaff et al. (2015) to smooth spectral filters for spatial localization. Defferrard et al. (2016) used Chebyshev polynomials to achieve linear complexity for sparse graphs, Levie et al. (2017) applied Cayley polynomials to focus on narrow-band frequencies, and Monti et al. (2017b) dealt with multiple (fixed) graphs. Finally, Kipf & Welling (2017) simplified the spectral convnets architecture using 1-hop filters to solve the semi-supervised clustering task. For related works, see also the works of Bronstein et al. (2017b), Bronstein et al. (2017a) and references therein. ",
+        "bbox": [
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+            708,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For graphs with variable length, a generic formulation was proposed by Gori et al. (2005); Scarselli et al. (2009) based on recurrent neural networks. The authors defined a multilayer perceptron of a vanilla RNN. This work was extended by Li et al. (2016) using a GRU architecture and a hidden state that captures the average information in local neighborhoods of the graph. The work of Sukhbaatar et al. (2016) introduced a vanilla graph ConvNet and used this new architecture to solve learning communication tasks. Marcheggiani & Titov (2017) introduced an edge gating mechanism in graph ConvNets for semantic role labeling. Finally, Bruna & Li (2017) designed a network to learn nonlinear approximations of the power of graph Laplacian operators, and applied it to the unsupervised graph clustering problem. Other works for drugs design, computer graphics and vision are presented by Duvenaud et al. (2015); Boscaini et al. (2016); Monti et al. (2017a). ",
+        "bbox": [
+            174,
+            854,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            103,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we study the two fundamental classes of neural networks, RNNs and ConvNets, in the context of graphs with arbitrary length. Section 2 reviews the existing techniques. Section 3 presents the new graph NN models. Section 4 reports the numerical experiments. ",
+        "bbox": [
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+            180,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 NEURAL NETWORKS FOR GRAPHS WITH ARBITRARY LENGTH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            243,
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+            258
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 RECURRENT NEURAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            273,
+            446,
+            289
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Generic formulation. Consider a standard RNN for word prediction in natural language processing. Let $h _ { i }$ be the feature vector associated with word $i$ in the sequence. In a regular vanilla RNN, $h _ { i }$ is computed with the feature vector $h _ { j }$ from the previous step and the current word $x _ { i }$ , so we have: ",
+        "bbox": [
+            173,
+            299,
+            825,
+            342
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/958918586b03fa654db319fb2b44b1df37b3b4047ac21ca115a6ecd81ff933fc.jpg",
+        "text": "$$\nh _ { i } = f _ { \\mathrm { V R N N } } \\left( \\begin{array} { l } { x _ { i } \\ , \\ \\{ h _ { j } : j = i - 1 \\} } \\end{array} \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            348,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The notion of neighborhood for regular RNNs is the previous step in the sequence. For graphs, the notion of neighborhood is given by the graph structure. If $h _ { i }$ stands for the feature vector of vertex $i$ , then the most generic version of a feature vector for a graph RNN is ",
+        "bbox": [
+            174,
+            371,
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+            414
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/16794c50f835dc7fc203041f303fe1dc374da25c09ca89e9f35f7afc39bd4b48.jpg",
+        "text": "$$\nh _ { i } = f _ { \\mathrm { G - R N N } } ( \\begin{array} { l } { x _ { i } } \\end{array} , \\begin{array} { l } { \\{ h _ { j } : j  i \\} } \\end{array} )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            420,
+            609,
+            438
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $x _ { i }$ refers to a data vector and $\\{ h _ { j } : j  i \\}$ denotes the set of feature vectors of the neighboring vertices. Observe that the set $\\{ h _ { j } \\}$ is unordered, meaning that $h _ { i }$ is intrinsic, i.e. invariant by vertex re-indexing (no vertex matching between graphs is required). Other properties of $f _ { \\mathrm { G - R N N } }$ are locality as only neighbors of vertex $i$ are considered, weight sharing, and such vector is independent of the graph length. In summary, to define a feature vector in a graph RNN, one needs a mapping $f$ that takes as input an unordered set of vectors $\\{ h _ { j } \\}$ , i.e. the feature vectors of all neighboring vertices, and a data vector $x _ { i }$ , Figure 1(a). ",
+        "bbox": [
+            173,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We refer to the mapping $f _ { \\mathrm { G - R N N } }$ as the neighborhood transfer function in graph RNNs. In a regular RNN, each neighbor as a distinct position relatively to the current word (1 position left from the center). In a graph, if the edges are not weighted or annotated, neighbors are not distinguishable. The only vertex which is special is the center vertex around which the neighborhood is built. This explains the generic formulation of Eq. (1). This type of formalism for deep learning for graphs with variable length is described by Scarselli et al. (2009); Gilmer et al. (2017); Bronstein et al. (2017a) with slightly different terminology and notations. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Graph Neural Networks of Scarselli et al. (2009). The earliest work of graph RNNs for arbitrary graphs was introduced by Gori et al. (2005); Scarselli et al. (2009). The authors proposed to use a vanilla RNN with a multilayer perceptron to define the feature vector $h _ { i }$ : ",
+        "bbox": [
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+            651,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b649bebc96924662e3b0bd5ac207347a6d42b4332133a35af723139942a6aabb.jpg",
+        "text": "$$\nh _ { i } = f _ { \\mathrm { G \\cdot V R N } } ( x _ { i } , \\{ h _ { j } : j  i \\} ) = \\sum _ { j  i } \\mathcal { C } _ { \\mathrm { G \\cdot V R N } } ( x _ { i } , h _ { j } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            700,
+            661,
+            736
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "with ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d109cfb8d178f93c16927e0e85e024c9aeacb88c0aa147e3427896981516e96b.jpg",
+        "text": "$$\n\\mathcal { C } _ { \\mathrm { G - V R N N } } ( x _ { i } , h _ { j } ) = A \\sigma ( B \\sigma ( U x _ { i } + V h _ { j } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            762,
+            619,
+            780
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "and $\\sigma$ is the sigmoid function, $A , B , U , V$ are the weight parameters to learn. ",
+        "bbox": [
+            174,
+            785,
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+            801
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Minimization of Eq. (2) does not hold a closed-form solution as the dependence computational graph of the model is not a directed acyclic graph (DAG). Scarselli et al. (2009) proposed a fixed-point iterative scheme: for $t = 0 , 1 , 2 , \\ldots$ . ",
+        "bbox": [
+            174,
+            806,
+            825,
+            849
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b489c08f6aa03d3388437662fb12832c60d41c5c1416881e265ab1d43c4348bf.jpg",
+        "text": "$$\nh _ { i } ^ { t + 1 } = \\sum _ { j  i } { \\mathcal { C } } ( x _ { i } , h _ { j } ^ { t } ) , \\quad h _ { i } ^ { t = 0 } = 0 \\ \\forall i .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            854,
+            611,
+            890
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The iterative scheme is guaranteed to converge as long as the mapping is contractive, which can be a strong assumption. Besides, a large number of iterations can be computational expensive. ",
+        "bbox": [
+            173,
+            895,
+            823,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/ae08cb64e7e1f123a452ccae4f2f714f7c91898685153b01f5b0bf6018a27c4c.jpg",
+        "image_caption": [
+            "Figure 1: Generic feature representation $h _ { i }$ of vertex $i$ on a graph RNN (a) and a graph convNet (b). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            114,
+            794,
+            306
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Gated Graph Neural Networks of Li et al. (2016). In this work, the authors use the gated recurrent units (GRU) of Chung et al. (2014): ",
+        "bbox": [
+            173,
+            358,
+            823,
+            387
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/06eec6349d9d46888e94f499833a59eac1aa985e768f2cfc2eec67202ce578f3.jpg",
+        "text": "$$\nh _ { i } = f _ { \\mathrm { G - G R U } } ( x _ { i } , \\{ h _ { j } : j  i \\} ) = \\mathcal { C } _ { \\mathrm { G - G R U } } ( x _ { i } , \\sum _ { j  i } h _ { j } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            390,
+            653,
+            425
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As the minimization of Eq. (4) does not have an analytical solution, Li et al. (2016) designed the following iterative scheme: ",
+        "bbox": [
+            176,
+            428,
+            823,
+            455
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27c4086e8ab2da494e096402bc85eefd9c0b600b046ffc8ac6d9504a97c2e380.jpg",
+        "text": "$$\n\\begin{array} { r c l l } { { { h } _ { i } ^ { t + 1 } } } & { { = } } & { { \\displaystyle { \\mathcal C } _ { \\mathrm { G - G R U } } ( h _ { i } ^ { t } , \\bar { h } _ { i } ^ { t } ) , } } & { { \\displaystyle h _ { i } ^ { t = 0 } = x _ { i } \\forall i , } } \\\\ { { \\mathrm { w h e r e } \\quad \\bar { h } _ { i } ^ { t } } } & { { = } } & { { \\displaystyle \\sum _ { j \\to i } h _ { j } ^ { t } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            457,
+            658,
+            513
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and $\\mathcal { C } _ { \\mathrm { G - G R U } } ( h _ { i } ^ { t } , \\bar { h } _ { i } ^ { t } )$ is equal to ",
+        "bbox": [
+            174,
+            516,
+            367,
+            534
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b3a0e4fa2dd28c57e6a22258f137fb8d6921f5c3baf90aeb8be5c6ba75645d4.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { z _ { i } ^ { t + 1 } } } & { { = } } & { { \\sigma ( U _ { z } h _ { i } ^ { t } + V _ { z } \\bar { h } _ { i } ^ { t } ) } } \\\\ { { r _ { i } ^ { t + 1 } } } & { { = } } & { { \\sigma ( U _ { r } h _ { i } ^ { t } + V _ { r } \\bar { h } _ { i } ^ { t } ) } } \\\\ { { \\tilde { h } _ { i } ^ { t + 1 } } } & { { = } } & { { \\operatorname { t a n h } \\bigl ( U _ { h } ( h _ { i } ^ { t } \\odot r _ { i } ^ { t + 1 } ) + V _ { h } \\bar { h } _ { i } ^ { t } \\bigr ) } } \\\\ { { h _ { i } ^ { t + 1 } } } & { { = } } & { { ( 1 - z _ { i } ^ { t + 1 } ) \\odot h _ { i } ^ { t } + z _ { i } ^ { t + 1 } \\odot \\tilde { h } _ { i } ^ { t + 1 } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ is the Hadamard point-wise multiplication operator. This model was used for NLP tasks by Li et al. (2016) and also in quantum chemistry by Gilmer et al. (2017) for fast organic molecule properties estimation, for which standard techniques (DFT) require expensive computational time. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Tree-Structured LSTM of Tai et al. (2015). The authors extended the original LSTM model of Hochreiter $\\&$ Schmidhuber (1997) to a tree-graph structure: ",
+        "bbox": [
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+            666,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/83184bc10330b916590da43a12e108a85233aea92617bd6390e8d5b5e61a66e3.jpg",
+        "text": "$$\nh _ { i } = f _ { \\mathrm { T L S T M } } \\left( x _ { i } , \\left\\{ h _ { j } : j \\in C ( i ) \\right\\} \\right) = \\mathcal { C } _ { \\mathrm { T L S T M } } ( x _ { i } , h _ { i } , \\sum _ { j \\in C ( i ) } h _ { j } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            699,
+            687,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $C ( i )$ refers the set of children of node $i$ . $\\begin{array} { r } { \\mathcal { C } _ { \\mathrm { T - L S T M } } ( x _ { i } , h _ { i } , \\sum _ { j \\in C ( i ) } h _ { j } ) } \\end{array}$ is equal to ",
+        "bbox": [
+            174,
+            737,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3edba823aeef57583d6143e984e8fcac162789bb85bba448eede530cc2a28f2e.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\displaystyle \\bar { h } _ { i } } & { = } & { \\displaystyle \\sum _ { j \\in { \\cal C } ( i ) } h _ { j } } \\\\ { \\displaystyle i _ { i } } & { = } & { \\sigma ( U _ { i } x _ { i } + V _ { i } \\bar { h } _ { i } ) } \\\\ { \\displaystyle o _ { i } } & { = } & { \\sigma ( U _ { o } x _ { i } + V _ { o } \\bar { h } _ { i } ) } \\\\ { \\displaystyle \\tilde { c } _ { i } } & { = } & { \\mathrm { t a n h } \\big ( U _ { c } x _ { i } + V _ { c } \\bar { h } _ { i } \\big ) } \\\\ { \\displaystyle f _ { i j } } & { = } & { \\sigma \\big ( U _ { f } x _ { i } + V _ { f } h _ { j } \\big ) } \\\\ { \\displaystyle c _ { i } } & { = } & { i _ { i } \\odot \\tilde { c } _ { i } + { \\displaystyle \\sum _ { i \\in { \\cal C } ( i ) } f _ { i j } } \\odot c _ { j } } \\\\ { \\displaystyle h _ { i } } & { = } & { o _ { i } \\odot \\mathrm { t a n h } \\big ( c _ { i } \\big ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unlike the works of Scarselli et al. (2009); Li et al. (2016), Tree-LSTM does not require an iterative process to update its feature vector $h _ { i }$ as the tree structure is also a DAG as original LSTM. Consequently, the feature representation (5) can be updated with a recurrent formula. Nevertheless, a tree is a special case of graphs, and such recurrence formula cannot be directly applied to arbitrary graph structure. A key property of this model is the function $f _ { i j }$ which acts as a gate on the edge from neighbor $j$ to vertex $i$ . Given the task, the gate will close to let the information flow from neighbor $j$ to vertex $i$ , or it will open to stop it. It seems to be an essential property for learning systems on graphs as some neighbors can be irrelevant. For example, for the community detection task, the graph neural network should learn which neighbors to communicate (same community) and which neighbors to ignore (different community). In different contexts, Dauphin et al. (2017) added a gated mechanism inside the regular ConvNets in order to improve language modeling for translation tasks, and van den Oord et al. (2016) considered a gated unit with the convolutional layers after activation, and used it for image generation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 CONVOLUTIONAL NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Generic formulation. Consider now a classical ConvNet for computer vision. Let $h _ { i j } ^ { \\ell }$ denote the feature vector at layer $\\ell$ associated with pixel $( i , j )$ . In a regular ConvNet, $h _ { i j } ^ { \\ell + 1 }$ is obtained by applying a non linear transformation to the feature vectors $h _ { i ^ { \\prime } j ^ { \\prime } } ^ { \\ell }$ for all pixels $( i ^ { \\prime } , j ^ { \\prime } )$ in a neighborhood of pixel $( i , j )$ . For example, with $3 \\times 3$ filters, we would have: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ad4690e46e078896abcd26979e22b6cbb52ecd5a52651d473ea380dd57a5ba3.jpg",
+        "text": "$$\nh _ { i j } ^ { \\ell + 1 } = f _ { \\mathrm { C N N } } ^ { \\ell } \\left( \\left\\{ h _ { i ^ { \\prime } j ^ { \\prime } } ^ { \\ell } : | i - i ^ { \\prime } | \\leq 1 \\mathrm { ~ a n d ~ } | j - j ^ { \\prime } | \\leq 1 \\right\\} \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the above, the notation $\\{ h _ { i ^ { \\prime } j ^ { \\prime } } ^ { \\ell } : | i - i ^ { \\prime } | \\le 1$ and $| j - j ^ { \\prime } | \\leq 1 \\}$ denote the concatenation of all feature vectors $h _ { i ^ { \\prime } j ^ { \\prime } } ^ { \\ell }$ belonging to the $3 \\times 3$ neighborhood of vertex $( i , j )$ . In ConvNets, the notion of neighborhood is given by the euclidian distance. As previously noticed, for graphs, the notion of neighborhood is given by the graph structure. Thus, the most generic version of a feature vector $h _ { i }$ at vertex $i$ for a graph ConvNet is ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dfb7333e56efaaa43214936d88dfea5300b3115e708aa7ead5bb7f64478b32c7.jpg",
+        "text": "$$\nh _ { i } ^ { \\ell + 1 } = f _ { \\scriptscriptstyle \\mathrm { G - C N N } } ( h _ { i } ^ { \\ell } , \\{ h _ { j } ^ { \\ell } : j  i \\} )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\{ h _ { j } ^ { \\ell } : j  i \\}$ denotes the set of feature vectors of the neighboring vertices. In other words, to define a graph ConvNet, one needs a mapping $f _ { \\mathrm { G - C N N } }$ taking as input a vector $h _ { i } ^ { \\ell }$ (the feature vector of the center vertex) as well as an unordered set of vectors $\\{ h _ { j } ^ { \\ell } \\}$ (the feature vectors of all neighboring vertices), see Figure 1(b). We also refer to the mapping $f _ { \\mathrm { G - C N N } }$ as the neighborhood transfer function. In a regular ConvNet, each neighbor as a distinct position relatively to the center pixel (for example 1 pixel up and 1 pixel left from the center). As for graph RNNs, the only vertex which is special for graph ConvNets is the center vertex around which the neighborhood is built. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "CommNets of Sukhbaatar et al. (2016). The authors introduced one of the simplest instantiations of a graph ConvNet with the following neighborhood transfer function: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/488a58ca0282db0cd449863cdea620d7d8a496103a7e3929ebc176b25c7a7115.jpg",
+        "text": "$$\nh _ { i } ^ { \\ell + 1 } = f _ { \\mathrm { G \\cdot v C N N } } ^ { \\ell } \\left( h _ { i } ^ { \\ell } , \\frac { } { } \\{ h _ { j } ^ { \\ell } : j \\to i \\} \\right) = \\mathrm { R e L U } \\left( U ^ { \\ell } h _ { i } ^ { \\ell } + V ^ { \\ell } \\sum _ { j \\to i } h _ { j } ^ { \\ell } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            659,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\ell$ denotes the layer level, and ReLU is the rectified linear unit. We will refer to this architecture as the vanilla graph ConvNet. Sukhbaatar et al. (2016) used this graph neural network to learn the communication between multiple agents to solve multiple tasks like traffic control. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Syntactic Graph Convolutional Networks of Marcheggiani $\\pmb { \\& }$ Titov (2017). The authors proposed the following transfer function: ",
+        "bbox": [
+            173,
+            760,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d26bfa224d95620ce441fc620e2d95c4c5f580176c8687f9823d3b85f6a7f25.jpg",
+        "text": "$$\nh _ { i } ^ { \\ell + 1 } = f _ { \\mathrm { s - G C N } } ^ { \\ell } \\left( \\frac { } { } \\{ h _ { j } ^ { \\ell } : j \\to i \\} \\ \\right) = \\mathrm { R e L U } \\left( \\sum _ { j \\to i } \\eta _ { i j } \\odot V ^ { \\ell } h _ { j } ^ { \\ell } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta _ { i j }$ act as edge gates, and are computed by: ",
+        "bbox": [
+            176,
+            843,
+            501,
+            858
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3f228339b043e592099203b4789ce5d36522e69cf4205f89cfd7e11855644da.jpg",
+        "text": "$$\n\\eta _ { i j } = \\sigma \\left( A ^ { \\ell } h _ { i } ^ { \\ell } + B ^ { \\ell } h _ { j } ^ { \\ell } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "These gated edges are very similar in spirit to the Tree-LSTM proposed in Tai et al. (2015). We believe this mechanism to be important for graphs, as they will be able to learn what edges are important for the graph learning task to be solved. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            281,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposed Graph LSTM. First, we propose to extend the Tree-LSTM of Tai et al. (2015) to arbitrary graphs and multiple layers: ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ba4ade9bbb845014b88538d241daca3e26e8b37fd92139bb2582e84a38b55082.jpg",
+        "text": "$$\n{ h } _ { i } ^ { \\ell + 1 } = f _ { \\mathrm { G \\mathrm { \\tiny { L S T M } } } } ^ { \\ell } ( x _ { i } ^ { \\ell } , \\{ h _ { j } ^ { \\ell } : j  i \\} ) = \\mathcal { C } _ { \\mathrm { G \\mathrm { \\tiny { L S T M } } } } ( x _ { i } ^ { \\ell } , h _ { i } ^ { \\ell } , \\sum _ { j  i } h _ { j } ^ { \\ell } , c _ { i } ^ { \\ell } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As there is no recurrent formula is the general case of graphs, we proceed as Scarselli et al. (2009) and use an iterative process to solve Eq. (10): At layer $\\ell$ , for $t = 0 , 1 , . . . , T$ ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8785d7d6ee4df39fcb3fadf8d2592dbf9c9e0bdc0f8a7baa6769175ad1092485.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\bar { h } _ { \\dot { i } } ^ { \\ell , t } } & { = \\displaystyle \\sum _ { j  i } h _ { j } ^ { \\ell , t } , } \\\\ { { \\dot { a } _ { i } ^ { \\ell , t + 1 } } } & { = \\sigma ( U _ { i } ^ { \\ell } x _ { i } ^ { \\ell } + V _ { i } ^ { \\ell } { \\bar { h } _ { i } ^ { \\ell , t } } ) } \\\\ { { \\dot { o } _ { i } ^ { \\ell , t + 1 } } } & { = \\sigma ( U _ { i } ^ { \\ell } x _ { i } ^ { \\ell } + V _ { i } ^ { \\ell } { \\bar { h } _ { i } ^ { \\ell , t } } ) } \\\\ { { \\dot { c } _ { i } ^ { \\ell , t + 1 } } } & { = \\mathrm { t a n h } ( { \\bar { c } _ { i } ^ { \\ell } x _ { i } ^ { \\ell } + \\bar { h } _ { i } ^ { \\ell } \\bar { h } _ { i } ^ { \\ell , t } } ) } \\\\ { { \\dot { f } _ { i } ^ { \\ell , t + 1 } } } & { = \\sigma ( U _ { i } ^ { \\ell } x _ { i } ^ { \\ell } + V _ { i } ^ { \\ell } { \\bar { h } _ { i } ^ { \\ell , t } } ) } \\\\ { { \\dot { c } _ { i } ^ { \\ell , t + 1 } } } & { = { \\dot { \\iota } _ { i } ^ { \\ell , t + 1 } } \\odot { \\bar { c } _ { i } ^ { \\ell } } ^ { t + 1 } + \\displaystyle \\sum _ { j  i } f _ { i j } ^ { \\ell , t + 1 } \\odot { c _ { j } ^ { \\ell , t + 1 } } } \\\\ { { \\dot { h } _ { \\dot { i } } ^ { \\ell , t + 1 } } } & { = { \\dot { \\sigma } _ { i } ^ { \\ell , t + 1 } } \\odot \\mathrm { t a n h } ( { c } _ { i } ^ { \\ell } t ^ { + 1 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            252,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and initial conditions: $\\begin{array} { r c l } { { h _ { i } ^ { \\ell , t = 0 } } } & { { = } } & { { c _ { i } ^ { \\ell , t = 0 } = 0 , \\forall i , \\ell } } \\\\ { { x _ { i } ^ { \\ell } } } & { { = } } & { { h _ { i } ^ { \\ell - 1 , T } , x _ { i } ^ { \\ell = 0 } = x _ { i } , \\forall i , \\ell } } \\end{array}$ ",
+        "bbox": [
+            289,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In other words, the vector $h _ { i } ^ { \\ell + 1 }$ is computed by running the model from $t = 0 , . . , T$ at layer $\\ell$ . It produces the vector $h _ { i } ^ { \\ell , t = T }$ which becomes $h _ { i } ^ { \\ell + 1 }$ and also the input $x _ { i } ^ { \\ell + 1 }$ for the next layer. The proposed Graph LSTM model differs from Liang et al. (2016); Peng et al. (2017) mostly because the cell $\\mathcal { C } _ { \\mathrm { G - L S T M } }$ in these previous models is not iterated over multiple times $T$ , which reduces the performance of Graph LSTM (see numerical experiments on Figure 4). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposed Gated Graph ConvNets. We leverage the vanilla graph ConvNet architecture of Sukhbaatar et al. (2016), Eq.(7), and the edge gating mechanism of Marcheggiani $\\&$ Titov (2017), Eq.(8), by considering the following model: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f79613cca2a80c1389b870eb5624a2756a588ca0e80d033b67e6188bc6a7a5b.jpg",
+        "text": "$$\nh _ { i } ^ { \\ell + 1 } = f _ { \\mathrm { G \\ell { \\mathrm { G C N N } } } } ^ { \\ell } ( h _ { i } ^ { \\ell } , \\{ h _ { j } ^ { \\ell } : j  i \\} ) = \\mathrm { R e L U } ( U ^ { \\ell } h _ { i } ^ { \\ell } + \\sum _ { j  i } \\eta _ { i j } \\odot V ^ { \\ell } h _ { j } ^ { \\ell } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            619,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $h _ { i } ^ { \\ell = 0 } = x _ { i } , \\forall i$ , and the edge gates $\\eta _ { i j }$ are defined in Eq. (9). This model is the most generic formulation of a graph ConvNet (because it uses both the feature vector $h _ { i } ^ { \\ell }$ of the center vertex and the feature vectors $h _ { j } ^ { \\ell }$ of neighboring vertices) with the edge gating property. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Residual Gated Graph ConvNets. In addition, we formulate a multi-layer gated graph ConvNet using residual networks (ResNets) introduced by He et al. (2016). This boils down to add the identity operator between successive convolutional layers: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a40c6d54fbf0c1d113b044dec32a0843947c7ada32433aaada15586c0eb45c30.jpg",
+        "text": "$$\n\\begin{array} { r l r } { h _ { i } ^ { \\ell + 1 } } & { = } & { f ^ { \\ell } \\left( \\ h _ { i } ^ { \\ell } , \\ \\{ h _ { j } ^ { \\ell } : j \\to i \\} \\ \\right) + h _ { i } ^ { \\ell } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As we will see, such multi-layer strategy work very well for graph neural networks. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 SUBGRAPH MATCHING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We consider the subgraph matching problem presented by Scarselli et al. (2009), see Figure 2(a).   \nThe goal is to find the vertices of a given subgraph $P$ in larger graphs $G _ { k }$ with variable sizes. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/fe9266c226259a8fd9f2ac74589691f443597b836426fd524d2758931935a96a.jpg",
+        "image_caption": [
+            "Figure 2: Graph learning tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Identifying similar localized patterns in different graphs is one of the most basic tasks for graph neural networks. The subgraph $P$ and larger graph $G _ { k }$ are generated with the stochastic block model (SBM), see for example Abbe (2017). A SBM is a random graph which assigns communities to each node as follows: any two vertices are connected with the probability $p$ if they belong to the same community, or they are connected with the probability $q$ if they belong to different communities. For all experiments, we generate a subgraph $P$ of 20 nodes with a SBM $q = 0 . 5$ , and the signal on $P$ is generated with a uniform random distribution with a vocabulary of size 3, i.e. $\\{ 0 , 1 , 2 \\}$ . Larger graphs $G _ { k }$ are composed of 10 communities with sizes randomly generated between 15 and 25. The SBM of each community is $p = 0 . 5$ . The value of $q$ , which acts as the noise level, is 0.1, unless otherwise specified. Besides, the signal on $G _ { k }$ is also randomly generated between $\\{ 0 , 1 , 2 \\}$ . Inputs of all neural networks are the graphs with variable size, and outputs are vertex classification vectors of input graphs. Finally, the output of neural networks are simple fully connected layers from the hidden states. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "All reported results are averaged over 5 trails. We run 5 algorithms; Gated Graph Neural Networks of Li et al. (2016), CommNets of Sukhbaatar et al. (2016), SyntacticNets of Marcheggiani & Titov (2017), and the proposed Graph LSTM and Gated ConvNets from Section 3. We upgrade the existing models of Li et al. (2016); Sukhbaatar et al. (2016); Marcheggiani & Titov (2017) with a multilayer version for Li et al. (2016) and using ResNets for all three architectures. We also use the batch normalization technique of Ioffe & Szegedy (2015) to speed up learning convergence for our algorithms, and also for Li et al. (2016); Sukhbaatar et al. (2016); Marcheggiani & Titov (2017). The learning schedule is as follows: the maximum number of iterations, or equivalently the number of randomly generated graphs with the attached subgraph is 5,000 and the learning rate is decreased by a factor 1.25 if the loss averaged over 100 iterations does not decrease. The loss is the cross-entropy with 2 classes (the subgraph $P$ class and the class of the larger graph $G _ { k }$ ) respectively weighted by their sizes. The accuracy is the average of the diagonal of the normalized confusion matrix w.r.t. the cluster sizes (the confusion matrix measures the number of nodes correctly and badly classified for each class). We also report the time for a batch of 100 generated graphs. The choice of the architectures will be given for each experiment. All algorithms are optimized as follow. We fix a budget of parameters of $B = 1 0 0 K$ and a number of layers $L = 6$ . The number of hidden neurons $H$ for each layer is automatically computed. Then we manually select the optimizer and learning rate for each architecture that best minimize the loss. For this task, Li et al. (2016); Sukhbaatar et al. (2016); Marcheggiani & Titov (2017) and our gated ConvNets work well with Adam and learning rate 0.00075. Graph LSTM uses SGD with learning rate 0.075. Besides, the value of inner iterative steps $T$ for graph LSTM and Li et al. (2016) is 3. ",
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+    },
+    {
+        "type": "text",
+        "text": "The first experiment focuses on shallow graph neural networks, i.e. with a single layer $L = 1$ . We also vary the level of noise, that is the probability $q$ in the SBM that connects two vertices in two different communities (the higher $q$ the more mixed are the communities). The hyper-parameters are selected as follows. Besides $L = 1$ , the budget is $B = 1 0 0 K$ and the number of hidden neurons $H$ is automatically computed for each architecture to satisfy the budget. First row of Figure 3 reports the accuracy and time for the five algorithms and for different levels of noise $q = \\{ 0 . \\bar { 1 } , 0 . 2 , 0 . \\bar { 3 } 5 , 0 . 5 \\}$ . RNN architectures are plotted in dashed lines and ConvNet architectures in solid lines. For shallow networks, all RNN architectures (graph LSTM and Li et al. (2016)) performs much better, but they also take more time than the graph ConvNets architectures we propose, as well as Sukhbaatar et al. ",
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+    {
+        "type": "text",
+        "text": "(2016); Marcheggiani & Titov (2017). As expected, all algorithms performances decrease when the noise increases. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The second experiment demonstrates the importance of having multiple layers compared to shallow networks. We vary the number of layers $\\bar { L } = \\{ 1 , 2 , 4 , 6 , 1 \\bar { 0 } \\}$ and we fix the number of hidden neurons to $H \\ : = \\ : 5 0$ . Notice that the budget is not the same for all architectures. Second row of Figure 3 reports the accuracy and time w.r.t. $L$ (middle figure is a zoom in the left figure). All models clearly benefit with more layers, but RNN-based architectures see their performances decrease for a large number of layers. The ConvNet architectures benefit from large $L$ values, with the proposed graph ConvNet performing slightly better than Sukhbaatar et al. (2016); Marcheggiani & Titov (2017). Besides, all ConvNet models are faster than RNN models. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the third experiment, we evaluate the algorithms for different budgets of parameters $B \\ =$ $\\{ 2 5 K , 5 0 K , 7 5 K , 1 0 0 K , 1 5 0 K \\}$ . For this experiment, we fix the number of layers $L \\ = \\ 6$ and the number of neurons $H$ is automatically computed given the budget $B$ . The results are reported in the third row of Figure 3. For this task, the proposed graph ConvNet best performs for a large budget, while being faster than RNNs. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/dcde2c4f276a50992f1d845a91368215338f8db5149ca1d2eb732f34dd415995.jpg",
+        "image_caption": [
+            "Figure 3: Subgraph matching: First row studies shallow networks w.r.t. noise. Second row investigates multilayer graph networks. Third row reports graph architectures w.r.t. budget. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also show the influence of hyper-parameter $T$ for Li et al. (2016) and the proposed graph LSTM. We fix $H = 5 0$ , $L = 3$ and $B = 1 0 0 K$ . Figure 4 reports the results for $T = \\{ 1 , 2 , 3 , 4 , 6 \\}$ . The $T$ value has an undesirable impact on the performance of graph LSTM. Multi-layer Li et al. (2016) is not really influenced by $T$ . Finally, the computational time naturally increases with larger $T$ values. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.2 SEMI-SUPERVISED CLUSTERING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider the semi-supervised clustering problem, see Figure 2(b). This is also a standard task in network science. For this work, it consists in finding 10 communities on a graph given 1 single label for each community. This problem is more discriminative w.r.t. to the architectures than the previous single pattern matching problem where there were only 2 clusters to find (i.e. $50 \\%$ random chance). For clustering, we have 10 clusters (around $10 \\%$ random chance). As in the previous section, we use SBM to generate graphs of communities with variable length. The size for each community is randomly generated between 5 and 25, and the label is randomly selected in each community. Probability $p$ is 0.5, and $q$ depends on the experiment. For this task, Li et al. (2016); Sukhbaatar et al. (2016); Marcheggiani & Titov (2017) and the proposed gated ConvNets work well with Adam and learning rate 0.00075. Graph LSTM uses SGD with learning rate 0.0075. The value of $T$ for graph LSTM and Li et al. (2016) is 3. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "image",
+        "img_path": "images/29fc1a6ccd0fa8c50620d388786c8fec23357cf50224fe7466b1f70c4f0ff43f.jpg",
+        "image_caption": [
+            "Figure 4: Influence of hyper-parameter $T$ on RNN architectures. Left figure is for graph matching, middle figure for semi-supervised clustering, and right figure are the batch time for the clustering task (same trend for matching). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The same set of experiments as in the previous task are reported in Figure 5. ConvNet architectures get clearly better than RNNs when the number of layers increase (middle row), with the proposed Gated ConvNet outperforming the other architectures. For a fixed number of layers $L = 6$ , our graph ConvNets and Marcheggiani & Titov (2017) best perform for all budgets, while paying a reasonable computational cost. ",
+        "bbox": [
+            173,
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/ff4244af7cedb9a88d147a5dea9aab32820c1913899132c1e396441ff789676f.jpg",
+        "image_caption": [
+            "Figure 5: Semi-supervised clustering: First row reports shallow networks w.r.t. noise $q$ . Second row shows multilayer graph networks w.r.t. $L$ . Third row is about graph architectures w.r.t. budget $B$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            541,
+            800,
+            877
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Next, we report the learning speed of the models. We fix $L \\ = \\ 6$ , $B \\ = \\ 1 0 0 K$ with $H$ being automatically computed to satisfy the budget. Figure 6 reports the accuracy w.r.t. time. The ConvNet architectures converge faster than RNNs, in particular for the semi-supervised task. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c2aa29c01817800597511abc623b80c55cc6ca7fb741d397a4b0118a11a0e63b.jpg",
+        "image_caption": [
+            "Figure 6: Learning speed of RNN and ConvNet architectures. Left figure is for graph matching and right figure semi-supervised clustering. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            172,
+            715,
+            279
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To close this study, we are interested in comparing learning based approaches to non-learning variational ones. To this aim, we solve the variational Dirichlet problem with labeled and unlabelled data as proposed by Grady (2006). We run 100 experiments and report an average accuracy of $4 5 . 3 7 \\%$ using the same setting as the learning techniques (one label per class). The performance of the best learning model is $82 \\%$ . Learning techniques produce better performances with a different paradigm as they use training data with ground truth, while variational techniques do not use such information. The downside is the need to see 2000 training graphs to get to $82 \\%$ . However, when the training is done, the test complexity of these learning techniques is $O ( E )$ , where $E$ is the number of edges in the graph. This is an advantage over the variational Dirichlet model that solves a sparse linear system of equations with complexity ${ \\cal O } ( E ^ { 3 / 2 } )$ , see Lipton et al. (1979). ",
+        "bbox": [
+            174,
+            344,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            507,
+            318,
+            523
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work explores the choice of graph neural network architectures for solving learning tasks with graphs of variable length. We developed analytically controlled experiments for two fundamental graph learning problems, that are subgraph matching and graph clustering. Numerical experiments showed that graph ConvNets had a monotonous increase of accuracy when the network gets deeper, unlike graph RNNs for which performance decreases for a large number of layers. This led us to consider the most generic formulation of gated graph ConvNets, Eq. (11). We also explored the benefit of residuality for graphs, Eq. (12). Without residuality, existing graph neural networks are not able to stack more than a few layers. This makes this property essential for graph neural networks, which receive a $10 \\%$ boost of accuracy when more than 6 layers were stacked. Future work will focus on solving domain-specific problems in chemistry, physics, and neuroscience. ",
+        "bbox": [
+            173,
+            539,
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+            678
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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American Mathematical Society, 1997.   \nJ. Chung, C. Gulcehre, K. Cho, and Y. Bengio. Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling. arXiv preprint arXiv:1412.3555, 2014.   \nY. Dauphin, A. Fan, M. Auli, and D. Grangier. Language Modeling with Gated Convolutional Networks. International Conference on Machine Learning (ICML), 2017.   \nM. Defferrard, X. Bresson, and P. Vandergheynst. Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering. Advances in Neural Information Processing Systems (NIPS), pp. 3844–3852, 2016.   \nD. K. Duvenaud, D. Maclaurin, J. Iparraguirre, R. Bombarell, T. Hirzel, A. Aspuru-Guzik, and R. P Adams. Convolutional networks on graphs for learning molecular fingerprints. In Proc. NIPS, 2015.   \nJ. Gilmer, S. Schoenholz, P. Riley, O. Vinyals, and G. Dahl. Neural Message Passing for Quantum Chemistry. arXiv preprint arXiv:1704.01212, 2017.   \nM. Gori, G. Monfardini, and F. Scarselli. 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parse/train/_4VxORHq-0g/_4VxORHq-0g.md

@@ -0,0 +1,219 @@
+# Dynamic Distillation Network for Cross-Domain Few-Shot Recognition with Unlabeled Data
+
+Ashraful Islam Rensselaer Polytechnic Institute islama6@rpi.edu
+
+Chun-Fu Chen MIT-IBM Watson AI Lab chenrich@us.ibm.com
+
+Rameswar Panda MIT-IBM Watson AI Lab rpanda@ibm.com
+
+Leonid KarlinskyIBM Researchleonidka@il.ibm.com
+
+Rogerio FerisIBM Researchrsferis@us.ibm.com
+
+Richard J. Radke Rensselaer Polytechnic Institute rjradke@ecse.rpi.edu
+
+# Abstract
+
+Most existing works in few-shot learning rely on meta-learning the network on a large base dataset which is typically from the same domain as the target dataset. We tackle the problem of cross-domain few-shot learning where there is a large shift between the base and target domain. The problem of cross-domain few-shot recognition with unlabeled target data is largely unaddressed in the literature. STARTUP was the first method that tackles this problem using self-training. However, it uses a fixed teacher pretrained on a labeled base dataset to create soft labels for the unlabeled target samples. As the base dataset and unlabeled dataset are from different domains, projecting the target images in the class-domain of the base dataset with a fixed pretrained model might be sub-optimal. We propose a simple dynamic distillation-based approach to facilitate unlabeled images from the novel/base dataset. We impose consistency regularization by calculating predictions from the weakly-augmented versions of the unlabeled images from a teacher network and matching it with the strongly augmented versions of the same images from a student network. The parameters of the teacher network are updated as exponential moving average of the parameters of the student network. We show that the proposed network learns representation that can be easily adapted to the target domain even though it has not been trained with target-specific classes during the pretraining phase. Our model outperforms the current state-of-the art method by $\bar { 4 . 4 \% }$ for 1-shot and $3 . 6 \%$ for 5-shot classification in the BSCD-FSL benchmark, and also shows competitive performance on traditional in-domain few-shot learning task. Our code is available at: https://git.io/Jilgs.
+
+# 1 Introduction
+
+The tremendous success of deep learning in visual recognition tasks is, to a great extent, attributed to the availability of large scale labeled datasets. While humans can recognize an object by looking only at a few examples, modern deep neural networks require hundreds or thousands of images for each category to achieve human-level visual recognition capability. This has led to the research on few-shot learning which aims at learning from a much smaller dataset. In a typical few-shot learning setting, there are two stages: meta-training and meta-testing. In the meta-training stage, a base dataset with labeled images is provided to train the model. In the meta-testing stage, the learned model is quickly adapted to a set of novel classes with only a few examples per class (the support set) and evaluated on a set of test images from the same novel classes (the query set). The base classes and novel classes are typically disjoint, but the images are obtained from the same domain. However, in many real world settings, training the model on a base dataset from the same domain as the target dataset is difficult and infeasible. Guo et al. [7] proposed a cross-domain few-shot benchmark, BSCD-FSL, which contains datasets from extremely different domains. In this benchmark, the meta-training is done on a labeled source dataset, and the few-shot evaluation is performed on a target dataset which is from different domain than the source dataset. The benchmark shows that traditional pretraining and finetuning outperforms more complicated meta-learning based few-shot learning methods by a significant margin.
+
+![](images/dbee5802125f4bfd45fbd04dea03e627fa4cb3b948e5122506897507063753e4.jpg)  
+Figure 1: Problem setup. (Left) In typical few-shot learning task, a model is trained on a base dataset first during meta-training stage. In meta-testing stage, a few examples from novel classes, referred to as support set, are provided, and the network predicts the categories of different samples from the same classes as support set. The base dataset and the target dataset generally come from the same domain with disjoint categories. (Middle) In cross-domain few-shot learning, there is a domain gap between the base dataset and the target dataset. For example, in the figure, the base dataset contains natural images from miniImageNet [29], and the target dataset consists of satellite images from EuroSAT dataset [8]. (Right) Our setting is similar to cross-domain few-shot learning setup. However, additional unlabeled images are also available during meta-training stage. Although the unlabeled dataset comes from the same domain as the target dataset, it does not contain any images either from the support set or query set.
+
+In the real-world scenarios, the target domain should have many unlabeled images, and it might be beneficial to use the unlabeled data to learn more target domain specific representations. We hypothesize that using both labeled base data and unlabeled target data during training provides a common embedding for both base and target domain. Then the natural question could be - why not use the unlabeled target data only, it might provide more target-specific representation. One issue with this approach is that self-supervised learning generally requires a large amount of unlabeled data to work, and, as pointed out by Phoo and Hariharan [18], plain self-supervised learning struggles to outperform the naive transfer learning baseline in few-shot learning setup. Secondly, it has been shown that combining supervised and unsupervised learning during training provides more transferable representation [9]. We argue that similar conclusion holds for cross-domain few-shot learning, i.e., combining supervised and unsupervised loss provides better representation for the downstream task. Figure 1 illustrates our experimental setup in contrast to traditional few-shot learning or cross-domain few-shot learning setup. We show that labeled images from the base dataset are still important to learn generic image features, and images from the target domain, even if unlabeled, can help developing more target domain specific representations.
+
+Figure 2 illustrates our approach. Our goal is to train a feature extractor which will be used to evaluate few-shot learning performance on the target dataset. We propose a dynamic distillationbased approach to this end. The student network consists of an encoder $f _ { s }$ and classifier $g _ { s }$ , and the teacher network shares similar architecture as the student network (denoted as $f _ { t }$ and $g _ { t }$ ). The classifier $g _ { s }$ is a linear layer that predicts the class-logits of the samples from the base dataset. We calculate a supervised cross-entropy loss between the student’s predictions and ground-truth labels on the base dataset. For the unlabeled target data, we compute the teacher’s prediction for a weaklyaugmented version of an image and the student’s prediction for a strongly augmented version of the same image, and optimize a distillation loss to match the predictions. We also apply sharpening in the teacher prediction to encourage low-entropy prediction from the student. Both the supervised loss and distillation loss are used to learn the student’s weights. The teacher network is updated as a moving average of the student network. During few-shot evaluation, we only use the student encoder $f _ { s }$ as a feature extractor, learn a classifier head on the labeled support images consisting of few examples per category, and calculate the class predictions of the query images.
+
+Our main contributions are:
+
+• We propose a simple method for few-shot learning across extreme domain difference. • We use dynamic distillation based approach that uses both labeled source data and unlabeled target data to learn a better representation for the few-shot evaluation on the target domain. • Our method significantly outperforms the current state of the art in the BSCD-FSL benchmark with unlabeled images by $4 . 4 \%$ for 1-shot and $3 . 6 \%$ for 5-shot classification in terms of average top-1 accuracy. It even shows superior performance for in-domain few-shot classification on miniImageNet and tieredImageNet datasets.
+
+# 2 Related Work
+
+Few-shot classification Few-shot learning methods can be divided into three broad categories - generative [36], metric-base [21, 28, 23] and adaptation-based [10, 14]. Early work on few-shot learning was based on meta-learning [29, 23, 10, 13]. Matching Networks [29] uses cosine similarities on feature vectors produced by independently learned feature extractors, while Relation Networks [23] learn its own similarity metric. MAML [10] learns good initialization parameters that can be quickly adapted to a new task. Prototypical Networks [21] learn a feature extractor that is used to calculate distances between features of test images and the mean features of support images. MetaOptNet [14] uses a discriminatively trained linear predictor to learn representations for few-shot learning.
+
+Self-training Self-training trains a student model that mimicks the predictions of a teacher model. Self-training can improves ImageNet classification [33]. It is also a dominant approach in semisupervised learning, where the teacher network is used to create pseudo [35, 34] or soft labels [33] for a huge set of unlabeled images, and the student network is trained to mimic the teacher.
+
+Semi-supervised Learning Our method is inspired from recent developments in semi-supervised learning. Both [16] and [27] uses supervised cross-entropy loss with unsupervised regularization loss. Pseudo-labeling based approaches first train a model on a labeled dataset, use the trained model to create pseudo-labels of the unlabeled samples, and retrain the model with both labeled and pseudo-labeled samples [12, 1]. FixMatch [22] proposes a simplified model which simultaneously optimizes cross-entropy loss on the labeled samples and generates pseudo labels using the model’s prediction on weakly-augmented unlabeled images. If the pseudo labels are confident enough, the model is trained to predict the pseudo labels with a strongly augmented version of the same images. We adopt a similar approach by imposing consistency regularization. However, while FixMatch is a semi-supervised technique where the unlabeled data is assumed to be from the same domain, our approach is applicable to the cross-domain few-shot learning problem. We also calculates the prediction from a mean teacher network instead of using the same network as FixMatch.
+
+Cross-domain few-shot learning Guo et al. [7] proposed a cross-domain few-shot learning benchmark, and noted that existing state-of-the-art approaches fail to achieve good accuracy on this benchmark. One potential solution could be to use an unlabeled dataset from the target to learn representations that are adaptable to a completely different domain. Many approaches also explored few-shot learning with unlabeled data [11, 15, 19]; however, most of these works still assume a smaller gap between the base and target domains. Our method shares some similarity with the recently developed STARTUP [18] method for cross-domain few-shot learning. STARTUP also uses unlabeled data for learning a better representation. However, STARTUP uses a fixed pretrained model to produce pseudo labels for the unlabeled samples, and then train the network with the labeled base dataset and pseudo-labeled target dataset. Additionally, STARTUP also incorporates a self-supervised contrastive loss on the unlabeled images to improve accuracy, where our method does not require additional contrastive loss. Actually, we argue that our distillation loss works like a self-supervised non-contrastive loss, similar to BYOL [6], for which we might not need to add any extra self-supervised loss. We propose a dynamic distillation approach, where the parameters of the teacher network are updated during training. We obtain the prediction for the weakly-augmented version of an unlabeled image from the teacher network, and optimizes the model such that the prediction of the strongly augmented version of the same image obtained from the student network matches that of teacher network. Note that FixMatch [22] also uses similar consistency regularization loss for semi-supervised learning. To our knowledge, we are the first to use consistency regularization and dynamic distillation for cross-domain few-shot learning.
+
+![](images/ed07d4f711cf566e714bb944dc36fee82b96495882cb354e92aa82e147cd71b2.jpg)  
+Figure 2: Diagram of our approach. Given labeled base data and unlabeled target data, our goal is to train a feature extractor which will be used to evaluate few-shot learning performance on the target dataset. The student network consists of an encoder $f _ { s }$ and classifier $g _ { s }$ , and the teacher network share similar architecture as the student network. We use the labeled base dataset to optimize the supervised cross-entropy loss. For a target image, we compute the teacher’s prediction for a weakly-augmented and student’s prediction for a strongly augmented version of the image, and optimize the distillation loss to match the predictions. We also apply sharpening in the teacher prediction to encourage low-entropy prediction from the student. Both the supervised loss and distillation loss are used to learn student’s weights. The teacher network is updated as a moving average of the student network. During few-shot evaluation, we simply learn a new classifier header on the few-shot support images, and evaluate on the query images.
+
+# 3 Methodology
+
+# 3.1 Preliminary
+
+Few-shot Learning Formulation A few-shot learning task consists of a support set $S$ , which containing $K$ data points from $N$ classes for $N$ -way $K$ -shot task, and a query $Q = \bar { \{ x _ { i } \} } _ { i = 1 } ^ { m }$ consisting of data points only from the $N$ classes of the support set. The goal is to classify the query points with the help of the labeled support set. In the typical few-shot learning setting, (1) an embedding is learned from the base/source dataset $\mathcal { D } _ { S }$ , (2) a linear classifier is learned on top of the fixed embedding on the support set, and (3) the classifications of the query data points are determined.
+
+Cross-domain Few-shot Learning The difference between the typical few-shot learning setup and cross-domain few-shot learning is that the base/source dataset is drawn from a very different domain than the target domain. Additionally, in our setting, we are provided unlabeled data points $\mathcal { D } _ { U } = \{ x _ { i } \} _ { i = 1 } ^ { N _ { U } }$ from the tarbase dataset domain. The unlabeled, and an unlabeled set aset contains more classes than the support, we need to learn an embedding that can $\mathcal { D } _ { S }$ $\mathcal { D } _ { U }$ extract a representation that can be used for few-shot learning evaluation in the target-domain.
+
+# 3.2 Proposed Method
+
+Encoder We facilitate knowledge distillation to train our base encoder on both source datatset and target dataset. Denote the embedding network as $f _ { s }$ that embeds an input image $x$ to a ${ \mathrm { d } } \cdot$ -dimensional vector $f _ { s } ( x )$ . We add a classifier header $g _ { s }$ on top of $f _ { s }$ , which predicts $n _ { c }$ logits from the embeddings, where $n _ { c }$ is the total number of classes in the base dataset. Since the labels of the data points of the base dataset are provided, we calculate the supervised cross-entropy loss:
+
+$$
+l _ { \mathrm { C E } } ( y , p ) = H ( y , p )
+$$
+
+where $p = \mathsf { S o f t m a x } ( g _ { s } ( f _ { s } ( x ) ) )$ , and $H ( a , b ) = - a \log b$ .
+
+Dynamic distillation Additionally, we also have a teacher encoder $f _ { t }$ and teacher classifier $g _ { t }$ . The task of the teacher network is to produce pseudo labels for the unlabeled images. Given an image $x _ { i }$ from the unlabeled set $\mathcal { D } _ { U }$ , we compute the model’s prediction $p _ { i } ^ { w }$ from a weakly-augmented version (denoted as $x _ { i } ^ { w }$ ) and $p _ { i } ^ { s }$ from a strongly-augmented version (denoted as $\boldsymbol { x } _ { i } ^ { s }$ ) of the unlabeled image. The prediction of the weakly-augmented version is produced from the teacher network, which serves as a soft-target for the strongly-augmented images. We use the student network to get the prediction $p _ { i } ^ { s }$ for the strongly-augmented images. Specifically,
+
+$$
+p _ { i } ^ { s } = \mathsf { S o f t m a x } \big ( g _ { s } \big ( f _ { s } ( x _ { i } ^ { s } ) \big ) \big ) ; \quad p _ { i } ^ { w } = \mathsf { S o f t m a x } \big ( g _ { t } \big ( f _ { t } ( x _ { i } ^ { w } ) \big ) / \tau \big )
+$$
+
+where $\tau$ is a sharpening parameter. Note that we do not let gradient pass through the teacher network. We calculate the distillation loss by the cross-entropy function
+
+$$
+l _ { U } ( p _ { i } ^ { w } , p _ { i } ^ { s } ) = H ( p _ { i } ^ { w } , p _ { i } ^ { s } )
+$$
+
+Eq. 3 works like a consistency regularizer so that the network predicts similar scores for different augmented versions of the image. We can also consider Eq. 3 as self-supervised loss, similar to BYOL [6] or DINO [2]. However, one major difference is that - in BYOL or DINO the projection head is a random linear layer, where, we are using the supervised classification head as the projection head.
+
+The total loss function is:
+
+$$
+\mathcal { L } = \frac { 1 } { { { N _ { S } } } } \sum _ { ( { x _ { i } } , { y _ { i } } ) \in { \mathcal { D } _ { S } } } { l _ { \mathrm { { C E } } } } ( y , p ) + \lambda \frac { 1 } { { { N _ { U } } } } \sum _ { { x _ { i } } \in { \mathcal { D } _ { U } } } { l _ { U } } ( p _ { i } ^ { w } , p _ { i } ^ { s } )
+$$
+
+where $\lambda$ is a hyper-parameter. The loss function is used to update the parameters of the student network. For the teacher network, we use mean teacher approach [24], i.e., we update the teacher weights $\theta _ { t }$ from the student weights $\theta _ { s }$ by: $\theta _ { t } = m \theta _ { t } + ( 1 - m ) \theta _ { s }$ , where $m$ is the momentum parameter. Note that when $m = 1$ , we are essentially using fixed teacher, and when $m = 0$ , the teacher and student share the same model. When $m > 0$ , the teacher network is a moving average of the student network denoting the distillation process as dynamic. Please refer to the Appendix for PyTorch-like pseudo-code of our method.
+
+# 4 Experiments
+
+# 4.1 Experimental Setup
+
+Dataset We follow the evaluation protocol of the BSCD-FSL benchmark [7], which contains novel data from CropDisease [17], EuroSAT [8], ISIC [4], and ChestX [32]. The base dataset is miniImageNet [29], which contains 100 classes from ImageNet dataset [5] where each class has 600 images. The novel datasets are chosen based on increasing dissimilarity from the miniImageNet dataset. More details about the datasets are provided in the Appendix. Following [18], we randomly sample $20 \%$ of the data from each novel dataset to construct the unlabeled set $\mathcal { D } _ { U }$ , and the remaining images are used for evaluation, where we perform 5-way 1-shot and 5-way 5-shot classification. For evaluation metric, we report top-1 accuracy and $9 5 \%$ confidence interval over 600 runs. We also report evaluation results on the larger tieredImageNet dataset [19].
+
+Implementation details We use ResNet-10 as the backbone network [7, 18]. Our pretraining has two steps. In the first step, we train our network only on the miniImageNet dataset for 200 epochs. We use SGD with momentum 0.9, weight decay 1e-4, learning rate 0.01, batch size 32, and the cosine learning rate scheduler. In the next step, we use the miniImageNet-pretrained network, and use both the base dataset and the unlabeled dataset to optimize the loss function in Eq. 4 for 60 epochs. During training, we increase the weight of distillation loss $\lambda$ from 0 to 1 until 40 epochs using cosine scheduling. The sharpening temperature and teacher momentum parameter are set to 0.1 and 0.99 respectively. For the base images and weakly-augmented unlabeled images, we use the random-resize-crop, horizontal flip and normalization augmentations. For strong augmentation, we additionally use the color jitter, Gaussian blur, and random gray scale transformations. The other hyperparameters are kept the same. Refer to the supplementary for more details about hyper-parameter selection. For few-shot evaluation, we learn a logistic regression classifier on the support set, and evaluate on the query set.
+
+Table 1: 5-way 1-shot and 5-shot scores on the BSCD-FSL benchmark datasets. The mean and $9 5 \%$ confidence interval of 600 runs are reported. The ∗ indicates that the numbers are reported from [7] where no unlabeled data is used. The † are the numbers reported from [18], which uses $20 \%$ of the original set as the unlabeled dataset. We also use similar number of unlabeled images as [18]; however, the splits might be different for random sampling.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">1-shot</td><td colspan="4">5-shot</td></tr><tr><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>MAML*</td><td></td><td></td><td></td><td></td><td>71.70±.72</td><td>78.05±.70</td><td>40.13±.58</td><td>23.48±.48</td></tr><tr><td>ProtoNet*</td><td></td><td></td><td></td><td></td><td>73.29±.71</td><td>79.72±.79</td><td>39.57±.57</td><td>24.05±1.01</td></tr><tr><td>MetaOpt*</td><td></td><td></td><td></td><td></td><td>64.44±.73</td><td>68.41±.73</td><td>36.28±.50</td><td>22.53±.91</td></tr><tr><td>STARTUPt</td><td>63.88±.84</td><td>75.93±.80</td><td>32.66±.60</td><td>23.09±.43</td><td>82.29±.60</td><td>93.02±.45</td><td>47.22±.61</td><td>26.94±.45</td></tr><tr><td>ProtoNet</td><td>55.32±.88</td><td>52.94±.81</td><td>29.58±.57</td><td>21.32±.37</td><td>76.92±.67</td><td>81.84±.68</td><td>42.49±.58</td><td>24.72±.43</td></tr><tr><td>MatchingNet</td><td>54.88±.90</td><td>46.86±.88</td><td>27.37±.51</td><td>20.65±.29</td><td>68.00±.68</td><td>63.94±.84</td><td>33.96±.54</td><td>22.62±.36</td></tr><tr><td>Transfer</td><td>58.14±.83</td><td>68.78±.84</td><td>32.12±.59</td><td>22.60±.39</td><td>80.09±.61</td><td>89.79±.52</td><td>43.88±.57</td><td>26.51±.43</td></tr><tr><td>SimCLR(Base)</td><td>58.28±.90</td><td>61.58±.88</td><td>32.43±.56</td><td>22.37±.42</td><td>80.83±.64</td><td>83.44±.61</td><td>44.04±.55</td><td>26.63±.46</td></tr><tr><td>SimCLR</td><td>62.63±.87</td><td>69.22±.93</td><td>31.45±.59</td><td>23.59±.44</td><td>82.76±.59</td><td>89.31±.53</td><td>42.18±.54</td><td>29.56±.49</td></tr><tr><td>STARTUP</td><td>64.32±.88</td><td>74.45±.86</td><td>31.73±.57</td><td>22.27±.41</td><td>83.58±.60</td><td>92.41±.47</td><td>45.73±.62</td><td>26.21±.46</td></tr><tr><td>Transfer+SimCLR</td><td>63.91±.83</td><td>70.35±.85</td><td>31.67±.55</td><td>23.72±.44</td><td>85.78±.51</td><td>91.10±.49</td><td>45.97±.54</td><td>29.45±.10</td></tr><tr><td> Ours</td><td>73.14±.84</td><td>82.14±.78</td><td>34.66±.58</td><td>23.38±.43</td><td>89.07±.47</td><td>95.54±.38</td><td>49.36±.59</td><td>28.31±.46</td></tr></table>
+
+# 4.2 Main Results
+
+Comparison to State-of-the-arts Table 1 shows the performance comparison of our approach with other methods on the BSCD-FSL benchmark. All models are trained on the miniImageNet dataset. “Transfer” denotes the baseline trained by cross-entropy loss on the base dataset. “SimCLR” is trained only on the unlabeled images. As noted by [9], self-supervised contrastive learning learns better transferable representation for a different downstream domain. Hence, we also create a “SimCLR(Base)” baseline that trains the encoder by optimizing self-supervised contrastive loss on the base dataset. “Transfer+SimCLR” refers to the model which is trained with supervised cross-entropy loss from the base dataset and self-supervised contrastive loss from the unlabeled target dataset, which has been reported to show superior transferability across domain [9]. STARTUP [18] is trained with three losses: cross-entropy loss on the base dataset, KL-divergence loss on the unlabeled dataset similar, and self-supervised contrastive loss on unlabeled images. STARTUP, Transfer+SimCLR and our approach use both the base dataset and additional unlabeled dataset during the representation learning phase.
+
+Our method outperforms all meta-learning-based approaches by a significant margin at all settings. Moreover, compared to Transfer, we achieve more than ${ \sim } 5 . 5 \%$ improvement for 5-shot classification on average. The performance improvement on 1-shot is more significant; we achieve $7 . 9 \%$ improvement on average.
+
+We outperform STARTUP by $5 . 5 \%$ on EuroSAT, $3 . 1 \%$ on CropDisease, $2 . 1 \%$ on ChestX, and $3 . 6 \%$ on ISIC for 5-way 5-shot classification. The performance improvement is also quite significant for 1-shot classification; specifically, our method achieves $8 . 8 \%$ improvement on EuroSAT and $7 . 7 \%$ improvement on CropDisease dataset over STARTUP. We only perform slightly worse on the ChestX dataset. For ChestX, it seems that pure unsupervised learning performs pretty well. Considering that our method does not use any self-supervised training or distillation, the performance improvement is impressive. Note that STARTUP uses a fixed teacher to extract pseudo-labels for the unlabeled images, whereas we extract pseudo labels from the weakly-augmented images from the same network that is being trained. In that sense, our model works like a dynamic teacher, where the pseudo labels get more refined as training progress. We hypothesize that the superior performance might be attributed to the dynamic approach of our model over STARTUP.
+
+Table 2: 5-way 1-shot and 5-shot scores on the BSCD-FSL benchmark datasets when tieredImageNet is used as base dataset. All models use ResNet-18 for backbone [25]. The mean and $9 5 \%$ confidence interval of 600 runs are reported. For EuroSAT and CropDisease dataset, our method achieves significant performance improvement over other models. The improvement for 1-shot learning is huge (over $7 \%$ for EuroSAT and $10 \%$ for CropDisease).   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">1-shot</td><td colspan="4">5-shot</td></tr><tr><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>Transfer</td><td>58.07±.86</td><td>69.94±.87</td><td>29.76±.55</td><td>22.46±.41</td><td>81.34±.53</td><td>90.12±.49</td><td>41.27±.58</td><td>26.33±.45</td></tr><tr><td>SimCLR(Base)</td><td>62.14±.89</td><td>62.45±.90</td><td>31.03±.55</td><td>22.28±.40</td><td>81.85±.59</td><td>84.11±.60</td><td>42.91±.55</td><td>25.96±.44</td></tr><tr><td>STARTUP</td><td>64.32±.87</td><td>70.09±.86</td><td>29.73±.51</td><td>22.10±.40</td><td>85.19±.50</td><td>90.81±.49</td><td>43.55±.56</td><td>26.03±.44</td></tr><tr><td>Transfer+SimCLR</td><td>58.08±.83</td><td>71.25±.89</td><td>31.71±.55</td><td>23.81±.46</td><td>86.08±.47</td><td>91.31±.49</td><td>45.08±.56</td><td>30.26±.50</td></tr><tr><td>Ours</td><td>72.15±.75</td><td>84.41±.75</td><td>33.87±.56</td><td>22.70±.42</td><td>89.44±.42</td><td>95.90±.34</td><td>47.21±.56</td><td>27.67±.46</td></tr></table>
+
+Table 3: Few-shot evaluation on the same domain in terms of 5-way 5-shot and 5-way 1-shot accuracy on miniImageNet and tieredImageNet datasets. We use ResNet-10 backbone for miniImageNet and ResNet-18 backbone for tieredImageNet.   
+
+<table><tr><td></td><td colspan="2">miniImageNet</td><td colspan="2">tieredImageNet</td></tr><tr><td></td><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>ProtoNet</td><td>51.06±.83</td><td>73.49±.63</td><td></td><td></td></tr><tr><td>MatchingNet</td><td>52.34±.81</td><td>67.28±.67</td><td>1</td><td>=</td></tr><tr><td>Transfer</td><td>53.40±.80</td><td>74.26±.64</td><td>58.61±.97</td><td>81.42±.65</td></tr><tr><td>Transfer+SimCLR</td><td>51.63±.82</td><td>74.65±.60</td><td>61.33±.96</td><td>82.89±.65</td></tr><tr><td>STARTUP</td><td>51.68±.84</td><td>74.05±.66</td><td>60.92±.96</td><td>82.11±.64</td></tr><tr><td>Ours</td><td>53.71±.83</td><td>76.02±.61</td><td>69.00±.96</td><td>85.93±.60</td></tr></table>
+
+Results with tieredImageNet base data tieredImageNet [19] is a subset of ImageNet dataset with 608 classes. The classes are grouped into 34 super-categories, from which 20 training categories (351 classes), 6 validation categories (97 classes), and 8 testing categories (160 classes) are selected. We use a larger backbone ResNet-18 for meta-training with tieredImageNet [25]. Table 2 shows the performance comparison with other models. We get similar conclusion as we get from Table 1. Particularly, we get $6 . 7 2 \%$ average improvement for 1-shot and $3 . 6 6 \%$ average improvement for 5-shot over STARTUP. However, we do not see much better accuracy than miniImageNet pretrained models even though we are using a much larger base dataset, which suggests that the size of base dataset is not as important as other transfer learning task in cross domain few-shot learning.
+
+Few-shot performance on similar domain It has been shown that self-training improves ImageNet classification [33]. Given the distillation approach of our model, one could expect that it might improve the few-shot accuracy even when the target data come from the same domain. Note that STARTUP does not improve the in-domain accuracy [18], even though it is also a self-training based model. We evaluate on miniImageNet and tieredImageNet dataset in terms of 5-way 1-shot and 5-way 5-shot performance. We use the official training split as base dataset, and the unlabeled target data are obtained from $20 \%$ of the novel (test) set and the rest are used for evaluation. For backbone, we use ResNet-10 for miniImageNet and ResNet-18 for tieredImageNet. Table 3 reports the results for in-domain few-shot performance. We see that our method achieves the best performance among the baselines. Particularly, for 1-shot learning in tieredImageNet, our model outperforms the best one by $7 . 7 \%$ . We infer that our method can be safely applied to few-shot learning task when the domain gap between base and target dataset is small, which is in contrast with STARTUP that does not show improvement over “Transfer" for few-shot learning on similar domain.
+
+# 4.3 Analysis
+
+Effect of dynamic distillation To understand how distillation helps to learn better representation, we use the pretrained models to extract features of the target dataset. Then we use KMeans algorithm to create clusters from the features. The number of clusters in the KMeans is set to be the number of classes of the target dataset. In Table
+
+Table 4: V-measure cluster score $( \% )$ [20] on the KMeans clustering of the extracted features with the ground-truth clustering. The backbone is ResNet-10 pretrained on the miniImageNet dataset and/or the unlabeled target dataset.   
+
+<table><tr><td></td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>Transfer</td><td>57.01</td><td>62.58</td><td>14.67</td><td>2.45</td></tr><tr><td>SimCLR</td><td>60.06</td><td>62.02</td><td>12.12</td><td>3.84</td></tr><tr><td>STARTUP</td><td>62.02</td><td>69.50</td><td>14.05</td><td>2.71</td></tr><tr><td>Ours</td><td>69.58</td><td>73.27</td><td>14.32</td><td>3.32</td></tr></table>
+
+4, the V-measurement cluster scores [20] between the KMeans clusters and original ground-truth are shown. The V-score has $100 \%$ value when there is maximum agreement between ground-truth and predicted clusters, and $0 \%$ when there’s no agreement. Table 4 shows that our method achieves higher v-scores for EuroSAT and CropDisease dataset, and the v-scores for ISIC and ChestX are also very competitive. It suggests that our model learns a good clustering of the target data even when we are not using any target labels. The clustering is much better when the domain gap is not extreme.
+
+Fig. 3 shows t-SNE plots [26] from 10 representative classes from the CropDisease and EuroSAT datasets. We compare the embeddings extracted from “Transfer” and our approach. We see that our method creates better grouping on the embeddings of the target datasets, even though we do not use any labels for the target dataset during pretraining.
+
+![](images/36501101abeac70a3ef2f8ddf896c10e3853579dfdaab8e0e092e6c1fa408902.jpg)  
+Figure 3: t-SNE plot of 10 classes from CropDisease (a & b) and EuroSAT (c & d) test sets with features obtained from Transfer and our method.
+
+Comparison with self-supervised learning If we ignore the supervised loss, our model has similarity with self-supervised non-contrastive loss similar to BYOL or DINO. However, the projection head we are using to calculate the final predictions of the two different views of an unlabeled image is the same classification head that is used to predict the classification logits of the labeled base samples. In Table 5, “Ours (distillation head)” represents the model where we use separate projection head for the predictions of the unlabeled images. We see that separate projection head performs much worse. We found that the distillation loss is simply converging to a trivial solution in this case. To discourage trivial solution, we add recently developed tricks in self-supervised learning, namely, centering and strong augmentation - which turns the unlabeled branch similar to ResNet DINO [2]. “Ours (DINO head)” achieves better accuracy than “Ours (distillation head)”, suggesting that it alleviates the issue of trivial solution. However, our original method still achieves a significant $3 . 0 7 \%$ more improvement. It is interesting to note that a separate projection head causes trivial solution for the unlabeled images, whereas our model does not converge to trivial solution with similar settings. We infer that using a supervised classifier linear layer as the projection head can solve the issue of trivial solution for the self-supervised learning to some extent without requiring extra tricks like BatchNorm [6] or centering $I 2 I$ . Additionally, it provides a better clustering of the unlabeled features, even if the unlabeled samples come from different domain than the labeled samples. Table 5 also shows results for “Ours $^ +$ SimCLR”, which simply adds a SimCLR loss for the unlabeled samples. It performs slightly better only in ChestX dataset. On average, the performance is similar to “Ours”, which signifies that there is no clear benefit using a self-supervised contrastive loss to achieve better transferability for our method. Note that STARTUP comes to a different conclusion reporting that adding SimCLR loss consistently improves the performance.
+
+Table 5: Our method with self-supervised approaches. The evaluation is performed on BSCD-FSL benchmark in terms of 5-way 5-shot accuracy $( \% )$ . “Ours (distillation head)” refers to the model where we use a separate projection head for the distillation loss, which achieves much worse scores. In “Ours $^ +$ DINO”, we use a separate projection head with centering and strong augmentation as in DINO [2]. It achieves better performance than naive transfer, but still under-performs in comparison to our approach. “Ours $^ +$ SimCLR” simply adds a self-supervised contrastive loss for the unlabeled samples. See Appendix for more details.   
+
+<table><tr><td></td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>Ours</td><td>89.07</td><td>95.54</td><td>49.36</td><td>28.31</td></tr><tr><td>Ours (distillation head)</td><td>80.06</td><td>89.31</td><td>46.63</td><td>25.29</td></tr><tr><td>Ours (DINO head)</td><td>85.74</td><td>90.55</td><td>46.24</td><td>25.42</td></tr><tr><td>Ours+SimCLR</td><td>88.48</td><td>93.80</td><td>49.10</td><td>29.45</td></tr></table>
+
+Experiment of few-shot classification on fine-grained dataset CUB [30] contains 200 classes and 11,788 images of different bird species. ‘miniImageNet- $\cdot >$ CUB’ is an interesting experiment to show the transferability of different models to a fine-grained dataset. We report the results in Table 6 in terms of 5-way 5-shot scores. For CUB, we found that vanilla Transfer performs surprisingly well (also reported by [3]), and adding SimCLR with Transfer (Transfer+SimCLR) actually decreases the accuracy. Wallace and Hariharan [31] also experimented with different self-supervised methods on smaller domain and found that all of them under-perform for fine-grained task [2]. However, our method still performs the best, demonstrating the effectiveness of our approach in a fine-grained downstream task.
+
+Table 6: Experiment on mini-ImageNet $\bullet >$ CUB. All the scores are reproduced by us.   
+
+<table><tr><td>ProtoNet</td><td>Transfer</td><td>SimCLR</td><td>Transfer+SimCLR</td><td>STARTUP</td><td>Ours</td></tr><tr><td>63.19</td><td>68.72</td><td>62.84</td><td>67.82</td><td>66.10</td><td>69.50</td></tr></table>
+
+Experiment of distillation with unlabeled datasets from different domain Table 7 reports the few-shot accuracy when our model is trained on different unlabeled datasets. The best accuracy is achieved when the unlabeled data and target data are from the same domain. Even if the unlabeled data consists of images from multiple domains including the target domain (denoted as “Ours-all”), it still significantly under-performs the base model.
+
+Table 7: Effect of unlabeled datasets from a different domain than the target dataset in terms of 5-way 5-shot accuracy $( \% )$ . “Ours-X” denotes that we use base and “X” dataset during pretraining. “Ours-all” denotes that we use unlabeled images from all four target datasets.   
+
+<table><tr><td></td><td>EuroSAT</td><td>CropDisease</td><td>ChestX</td><td>ISIC</td></tr><tr><td>Ours-EuroSAT</td><td>89.07</td><td>90.43</td><td>26.02</td><td>46.82</td></tr><tr><td>Ours-CropDisease</td><td>81.86</td><td>95.54</td><td>26.17</td><td>45.12</td></tr><tr><td>Ours-ISIC</td><td>81.94</td><td>89.69</td><td>26.70</td><td>49.36</td></tr><tr><td>Ours-ChestX</td><td>81.87</td><td>90.38</td><td>28.31</td><td>45.20</td></tr><tr><td>Ours-All</td><td>82.75</td><td>91.31</td><td>26.01</td><td>46.43</td></tr></table>
+
+Effect of data augmentation On the unlabeled images, we apply two types of augmentation: weak augmentation to extract pseudo labels and strong augmentation to impose consistency regularization. This setting is denoted as “weak-strong” (w-s), where ‘weak’ (w) augmentation is applied to the image that is fed into the teacher network and ‘strong’ (s) augmentation is applied to the image that is fed into the student network. We also show results with “weak-weak” (w-w), “strong-weak” (s-w) and “strong-strong” (s-s) augmentation settings in Figure 4a. Both “weak-weak” and “strong-strong” perform worse than the other augmentations. We also note that in self-supervised learning, generally strong augmentation is applied to all training images to get good performance. In our experiment, we find that applying weak augmentation in one of the image pairs improves the performance.
+
+More unlabeled data To measure the effect of amount of unlabeled dataset during pretraining, we divide the target dataset by $80 \%$ and $20 \%$ splits. The $20 \%$ split is used for evaluation. From the $80 \%$ split, we vary the amount of unlabeled data and then pretrain with source and the unlabeled set. Fig. 4b shows the average 5-shot accuracy for different amounts of the unlabeled dataset during the pretraining phase. As expected, more information from the unlabeled dataset helps to learn better representations on the target domain. However, the performance saturates later, and we get diminishing return for more unlabeled data. It also signifies that there are scopes to improve the performance by using more unlabeled data denoting potential future research direction.
+
+# 4.4 Addition Ablation Studies
+
+We perform several ablation studies of different components of our approach. All scores are reported for 5-way 5-shot evaluation. More ablations are provided in the Appendix.
+
+Longer training We use the base dataset pretrained network as initialization for our network, and then train on both base dataset and unlabeled dataset for 60 epochs. Figure $_ \mathrm { 4 c }$ reports average 5-way 5-shot few-shot on the BSCD-FSL benchmark performance for our pretrained with longer training epoch. Training for more epochs can result in minor improvement $( 0 . 2 \% )$ .
+
+![](images/f1b5c962ab2ccd9c9213cde32790f32a6824e26df90bce206252a13bf3f3f81d.jpg)  
+Figure 4: Ablation studies for (a) data augmentation on unlabeled images, (b) effect of longer training on base and unlabeled data, (c) amount of unlabeled data. The Y-axis represents average top1 accuracy $( \% )$ on the four benchmark datasets for 5-shot classification for 600 episodes. We use miniImageNet for labeled source dataset and ResNet-10 as backbone.
+
+Table 8: Ablation studies on different settings. Mean over 600 runs.   
+
+<table><tr><td></td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>Ours</td><td>89.07</td><td>95.54</td><td>49.36</td><td>28.31</td></tr><tr><td>Ours(w/o base)</td><td>82.11</td><td>90.52</td><td>39.76</td><td>26.83</td></tr><tr><td>Ours(1-step)</td><td>86.16</td><td>87.28</td><td>46.10</td><td>25.11</td></tr></table>
+
+Effect of base dataset Here, we perform experiments without the first term in Eq. 4, i.e., we train the network on the base dataset first and then re-train only on the unlabeled images (without joint training on the base dataset), denoted as “Ours (w/o base)”. Table 8 shows that the performance of “Ours (w/o base)” is poor, suggesting that the representation related to the labeled base dataset is still helpful to the target domain.
+
+Training without pretrained model on base dataset We perform 2-step training during the representation learning phase - we first train the model on mini-IN only, and then jointly train on mini-IN and unlabeled images. In Table 8, “Ours(1-step)” denotes training on the unlabeled images and mini-IN from scratch for 300 epochs, which performs worse than 2-step training. Our assumption is that the proposed model is also like self-training where a well-trained teacher is needed.
+
+# 5 Conclusion
+
+We introduced a novel approach to utilize unlabeled data from the target domain for cross-domain fewshot learning. Experiments show that our method achieves state-of-the-art results in the BSCD-FSL benchmark for both 1-shot and 5-shot classification. Our model also outperforms other approaches in the same-domain few-shot learning. Future work can be focused on applying our approach in each episode during meta-testing so that the model can learn more category-specific representations.
+
+# 6 Broader Impact
+
+The approach tackles a practical problem of the existing few-shot learning setup that the base dataset and the novel samples generally come from different domain. Our work uses the unlabeled samples from the target dataset to learn more target specific representation. Like any other machine learning tool, the final impact depends on the intention of the people or institution applying it. However, it can be useful in drug discovery or medical image analysis where labeled dataset is limited.
+
+# 7 Acknowledgments
+
+This material is based upon work supported by the U.S. Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Grant Award 2013-ST-061- ED0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Department of Homeland Security.
+
+# References
+
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+        "text": "Abstract ",
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+        "text": "Most existing works in few-shot learning rely on meta-learning the network on a large base dataset which is typically from the same domain as the target dataset. We tackle the problem of cross-domain few-shot learning where there is a large shift between the base and target domain. The problem of cross-domain few-shot recognition with unlabeled target data is largely unaddressed in the literature. STARTUP was the first method that tackles this problem using self-training. However, it uses a fixed teacher pretrained on a labeled base dataset to create soft labels for the unlabeled target samples. As the base dataset and unlabeled dataset are from different domains, projecting the target images in the class-domain of the base dataset with a fixed pretrained model might be sub-optimal. We propose a simple dynamic distillation-based approach to facilitate unlabeled images from the novel/base dataset. We impose consistency regularization by calculating predictions from the weakly-augmented versions of the unlabeled images from a teacher network and matching it with the strongly augmented versions of the same images from a student network. The parameters of the teacher network are updated as exponential moving average of the parameters of the student network. We show that the proposed network learns representation that can be easily adapted to the target domain even though it has not been trained with target-specific classes during the pretraining phase. Our model outperforms the current state-of-the art method by $\\bar { 4 . 4 \\% }$ for 1-shot and $3 . 6 \\%$ for 5-shot classification in the BSCD-FSL benchmark, and also shows competitive performance on traditional in-domain few-shot learning task. Our code is available at: https://git.io/Jilgs. ",
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+        "text": "1 Introduction ",
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+        "text": "The tremendous success of deep learning in visual recognition tasks is, to a great extent, attributed to the availability of large scale labeled datasets. While humans can recognize an object by looking only at a few examples, modern deep neural networks require hundreds or thousands of images for each category to achieve human-level visual recognition capability. This has led to the research on few-shot learning which aims at learning from a much smaller dataset. In a typical few-shot learning setting, there are two stages: meta-training and meta-testing. In the meta-training stage, a base dataset with labeled images is provided to train the model. In the meta-testing stage, the learned model is quickly adapted to a set of novel classes with only a few examples per class (the support set) and evaluated on a set of test images from the same novel classes (the query set). The base classes and novel classes are typically disjoint, but the images are obtained from the same domain. However, in many real world settings, training the model on a base dataset from the same domain as the target dataset is difficult and infeasible. Guo et al. [7] proposed a cross-domain few-shot benchmark, BSCD-FSL, which contains datasets from extremely different domains. In this benchmark, the meta-training is done on a labeled source dataset, and the few-shot evaluation is performed on a target dataset which is from different domain than the source dataset. The benchmark shows that traditional pretraining and finetuning outperforms more complicated meta-learning based few-shot learning methods by a significant margin. ",
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+            "Figure 1: Problem setup. (Left) In typical few-shot learning task, a model is trained on a base dataset first during meta-training stage. In meta-testing stage, a few examples from novel classes, referred to as support set, are provided, and the network predicts the categories of different samples from the same classes as support set. The base dataset and the target dataset generally come from the same domain with disjoint categories. (Middle) In cross-domain few-shot learning, there is a domain gap between the base dataset and the target dataset. For example, in the figure, the base dataset contains natural images from miniImageNet [29], and the target dataset consists of satellite images from EuroSAT dataset [8]. (Right) Our setting is similar to cross-domain few-shot learning setup. However, additional unlabeled images are also available during meta-training stage. Although the unlabeled dataset comes from the same domain as the target dataset, it does not contain any images either from the support set or query set. "
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+        "text": "In the real-world scenarios, the target domain should have many unlabeled images, and it might be beneficial to use the unlabeled data to learn more target domain specific representations. We hypothesize that using both labeled base data and unlabeled target data during training provides a common embedding for both base and target domain. Then the natural question could be - why not use the unlabeled target data only, it might provide more target-specific representation. One issue with this approach is that self-supervised learning generally requires a large amount of unlabeled data to work, and, as pointed out by Phoo and Hariharan [18], plain self-supervised learning struggles to outperform the naive transfer learning baseline in few-shot learning setup. Secondly, it has been shown that combining supervised and unsupervised learning during training provides more transferable representation [9]. We argue that similar conclusion holds for cross-domain few-shot learning, i.e., combining supervised and unsupervised loss provides better representation for the downstream task. Figure 1 illustrates our experimental setup in contrast to traditional few-shot learning or cross-domain few-shot learning setup. We show that labeled images from the base dataset are still important to learn generic image features, and images from the target domain, even if unlabeled, can help developing more target domain specific representations. ",
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+        "text": "Figure 2 illustrates our approach. Our goal is to train a feature extractor which will be used to evaluate few-shot learning performance on the target dataset. We propose a dynamic distillationbased approach to this end. The student network consists of an encoder $f _ { s }$ and classifier $g _ { s }$ , and the teacher network shares similar architecture as the student network (denoted as $f _ { t }$ and $g _ { t }$ ). The classifier $g _ { s }$ is a linear layer that predicts the class-logits of the samples from the base dataset. We calculate a supervised cross-entropy loss between the student’s predictions and ground-truth labels on the base dataset. For the unlabeled target data, we compute the teacher’s prediction for a weaklyaugmented version of an image and the student’s prediction for a strongly augmented version of the same image, and optimize a distillation loss to match the predictions. We also apply sharpening in the teacher prediction to encourage low-entropy prediction from the student. Both the supervised loss and distillation loss are used to learn the student’s weights. The teacher network is updated as a moving average of the student network. During few-shot evaluation, we only use the student encoder $f _ { s }$ as a feature extractor, learn a classifier head on the labeled support images consisting of few examples per category, and calculate the class predictions of the query images. ",
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+        "text": "Our main contributions are: ",
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+        "text": "• We propose a simple method for few-shot learning across extreme domain difference. • We use dynamic distillation based approach that uses both labeled source data and unlabeled target data to learn a better representation for the few-shot evaluation on the target domain. • Our method significantly outperforms the current state of the art in the BSCD-FSL benchmark with unlabeled images by $4 . 4 \\%$ for 1-shot and $3 . 6 \\%$ for 5-shot classification in terms of average top-1 accuracy. It even shows superior performance for in-domain few-shot classification on miniImageNet and tieredImageNet datasets. ",
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+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "text": "Few-shot classification Few-shot learning methods can be divided into three broad categories - generative [36], metric-base [21, 28, 23] and adaptation-based [10, 14]. Early work on few-shot learning was based on meta-learning [29, 23, 10, 13]. Matching Networks [29] uses cosine similarities on feature vectors produced by independently learned feature extractors, while Relation Networks [23] learn its own similarity metric. MAML [10] learns good initialization parameters that can be quickly adapted to a new task. Prototypical Networks [21] learn a feature extractor that is used to calculate distances between features of test images and the mean features of support images. MetaOptNet [14] uses a discriminatively trained linear predictor to learn representations for few-shot learning. ",
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+        "text": "Self-training Self-training trains a student model that mimicks the predictions of a teacher model. Self-training can improves ImageNet classification [33]. It is also a dominant approach in semisupervised learning, where the teacher network is used to create pseudo [35, 34] or soft labels [33] for a huge set of unlabeled images, and the student network is trained to mimic the teacher. ",
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+        "text": "Semi-supervised Learning Our method is inspired from recent developments in semi-supervised learning. Both [16] and [27] uses supervised cross-entropy loss with unsupervised regularization loss. Pseudo-labeling based approaches first train a model on a labeled dataset, use the trained model to create pseudo-labels of the unlabeled samples, and retrain the model with both labeled and pseudo-labeled samples [12, 1]. FixMatch [22] proposes a simplified model which simultaneously optimizes cross-entropy loss on the labeled samples and generates pseudo labels using the model’s prediction on weakly-augmented unlabeled images. If the pseudo labels are confident enough, the model is trained to predict the pseudo labels with a strongly augmented version of the same images. We adopt a similar approach by imposing consistency regularization. However, while FixMatch is a semi-supervised technique where the unlabeled data is assumed to be from the same domain, our approach is applicable to the cross-domain few-shot learning problem. We also calculates the prediction from a mean teacher network instead of using the same network as FixMatch. ",
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+        "type": "text",
+        "text": "Cross-domain few-shot learning Guo et al. [7] proposed a cross-domain few-shot learning benchmark, and noted that existing state-of-the-art approaches fail to achieve good accuracy on this benchmark. One potential solution could be to use an unlabeled dataset from the target to learn representations that are adaptable to a completely different domain. Many approaches also explored few-shot learning with unlabeled data [11, 15, 19]; however, most of these works still assume a smaller gap between the base and target domains. Our method shares some similarity with the recently developed STARTUP [18] method for cross-domain few-shot learning. STARTUP also uses unlabeled data for learning a better representation. However, STARTUP uses a fixed pretrained model to produce pseudo labels for the unlabeled samples, and then train the network with the labeled base dataset and pseudo-labeled target dataset. Additionally, STARTUP also incorporates a self-supervised contrastive loss on the unlabeled images to improve accuracy, where our method does not require additional contrastive loss. Actually, we argue that our distillation loss works like a self-supervised non-contrastive loss, similar to BYOL [6], for which we might not need to add any extra self-supervised loss. We propose a dynamic distillation approach, where the parameters of the teacher network are updated during training. We obtain the prediction for the weakly-augmented version of an unlabeled image from the teacher network, and optimizes the model such that the prediction of the strongly augmented version of the same image obtained from the student network matches that of teacher network. Note that FixMatch [22] also uses similar consistency regularization loss for semi-supervised learning. To our knowledge, we are the first to use consistency regularization and dynamic distillation for cross-domain few-shot learning. ",
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+        "img_path": "images/ed07d4f711cf566e714bb944dc36fee82b96495882cb354e92aa82e147cd71b2.jpg",
+        "image_caption": [
+            "Figure 2: Diagram of our approach. Given labeled base data and unlabeled target data, our goal is to train a feature extractor which will be used to evaluate few-shot learning performance on the target dataset. The student network consists of an encoder $f _ { s }$ and classifier $g _ { s }$ , and the teacher network share similar architecture as the student network. We use the labeled base dataset to optimize the supervised cross-entropy loss. For a target image, we compute the teacher’s prediction for a weakly-augmented and student’s prediction for a strongly augmented version of the image, and optimize the distillation loss to match the predictions. We also apply sharpening in the teacher prediction to encourage low-entropy prediction from the student. Both the supervised loss and distillation loss are used to learn student’s weights. The teacher network is updated as a moving average of the student network. During few-shot evaluation, we simply learn a new classifier header on the few-shot support images, and evaluate on the query images. "
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+        "text": "3 Methodology ",
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+        "text": "3.1 Preliminary ",
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+        "text": "Few-shot Learning Formulation A few-shot learning task consists of a support set $S$ , which containing $K$ data points from $N$ classes for $N$ -way $K$ -shot task, and a query $Q = \\bar { \\{ x _ { i } \\} } _ { i = 1 } ^ { m }$ consisting of data points only from the $N$ classes of the support set. The goal is to classify the query points with the help of the labeled support set. In the typical few-shot learning setting, (1) an embedding is learned from the base/source dataset $\\mathcal { D } _ { S }$ , (2) a linear classifier is learned on top of the fixed embedding on the support set, and (3) the classifications of the query data points are determined. ",
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+        "text": "Cross-domain Few-shot Learning The difference between the typical few-shot learning setup and cross-domain few-shot learning is that the base/source dataset is drawn from a very different domain than the target domain. Additionally, in our setting, we are provided unlabeled data points $\\mathcal { D } _ { U } = \\{ x _ { i } \\} _ { i = 1 } ^ { N _ { U } }$ from the tarbase dataset domain. The unlabeled, and an unlabeled set aset contains more classes than the support, we need to learn an embedding that can $\\mathcal { D } _ { S }$ $\\mathcal { D } _ { U }$ extract a representation that can be used for few-shot learning evaluation in the target-domain. ",
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+        "text": "3.2 Proposed Method ",
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+        "text": "Encoder We facilitate knowledge distillation to train our base encoder on both source datatset and target dataset. Denote the embedding network as $f _ { s }$ that embeds an input image $x$ to a ${ \\mathrm { d } } \\cdot$ -dimensional vector $f _ { s } ( x )$ . We add a classifier header $g _ { s }$ on top of $f _ { s }$ , which predicts $n _ { c }$ logits from the embeddings, where $n _ { c }$ is the total number of classes in the base dataset. Since the labels of the data points of the base dataset are provided, we calculate the supervised cross-entropy loss: ",
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+        "img_path": "images/431f6d8eabe201c41b7b7063963f40b324063a243fbdba3e1dc285b9999d8fab.jpg",
+        "text": "$$\nl _ { \\mathrm { C E } } ( y , p ) = H ( y , p )\n$$",
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+        "type": "text",
+        "text": "where $p = \\mathsf { S o f t m a x } ( g _ { s } ( f _ { s } ( x ) ) )$ , and $H ( a , b ) = - a \\log b$ . ",
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+        "text": "Dynamic distillation Additionally, we also have a teacher encoder $f _ { t }$ and teacher classifier $g _ { t }$ . The task of the teacher network is to produce pseudo labels for the unlabeled images. Given an image $x _ { i }$ from the unlabeled set $\\mathcal { D } _ { U }$ , we compute the model’s prediction $p _ { i } ^ { w }$ from a weakly-augmented version (denoted as $x _ { i } ^ { w }$ ) and $p _ { i } ^ { s }$ from a strongly-augmented version (denoted as $\\boldsymbol { x } _ { i } ^ { s }$ ) of the unlabeled image. The prediction of the weakly-augmented version is produced from the teacher network, which serves as a soft-target for the strongly-augmented images. We use the student network to get the prediction $p _ { i } ^ { s }$ for the strongly-augmented images. Specifically, ",
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+    {
+        "type": "equation",
+        "img_path": "images/cb3d2737460962fb371472adfe292680b3b903edef522612f8a396d828524dee.jpg",
+        "text": "$$\np _ { i } ^ { s } = \\mathsf { S o f t m a x } \\big ( g _ { s } \\big ( f _ { s } ( x _ { i } ^ { s } ) \\big ) \\big ) ; \\quad p _ { i } ^ { w } = \\mathsf { S o f t m a x } \\big ( g _ { t } \\big ( f _ { t } ( x _ { i } ^ { w } ) \\big ) / \\tau \\big )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\tau$ is a sharpening parameter. Note that we do not let gradient pass through the teacher network. We calculate the distillation loss by the cross-entropy function ",
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+        "type": "equation",
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+        "text": "$$\nl _ { U } ( p _ { i } ^ { w } , p _ { i } ^ { s } ) = H ( p _ { i } ^ { w } , p _ { i } ^ { s } )\n$$",
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+        "text": "Eq. 3 works like a consistency regularizer so that the network predicts similar scores for different augmented versions of the image. We can also consider Eq. 3 as self-supervised loss, similar to BYOL [6] or DINO [2]. However, one major difference is that - in BYOL or DINO the projection head is a random linear layer, where, we are using the supervised classification head as the projection head. ",
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+        "text": "The total loss function is: ",
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+        "type": "equation",
+        "img_path": "images/1f2eb5795f5320b50fa7d90cd1a78ae3d828019be31825adab5ed37eb7bc7c12.jpg",
+        "text": "$$\n\\mathcal { L } = \\frac { 1 } { { { N _ { S } } } } \\sum _ { ( { x _ { i } } , { y _ { i } } ) \\in { \\mathcal { D } _ { S } } } { l _ { \\mathrm { { C E } } } } ( y , p ) + \\lambda \\frac { 1 } { { { N _ { U } } } } \\sum _ { { x _ { i } } \\in { \\mathcal { D } _ { U } } } { l _ { U } } ( p _ { i } ^ { w } , p _ { i } ^ { s } )\n$$",
+        "text_format": "latex",
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+        "text": "where $\\lambda$ is a hyper-parameter. The loss function is used to update the parameters of the student network. For the teacher network, we use mean teacher approach [24], i.e., we update the teacher weights $\\theta _ { t }$ from the student weights $\\theta _ { s }$ by: $\\theta _ { t } = m \\theta _ { t } + ( 1 - m ) \\theta _ { s }$ , where $m$ is the momentum parameter. Note that when $m = 1$ , we are essentially using fixed teacher, and when $m = 0$ , the teacher and student share the same model. When $m > 0$ , the teacher network is a moving average of the student network denoting the distillation process as dynamic. Please refer to the Appendix for PyTorch-like pseudo-code of our method. ",
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+        "text": "4 Experiments ",
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+        "text": "4.1 Experimental Setup ",
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+        "text": "Dataset We follow the evaluation protocol of the BSCD-FSL benchmark [7], which contains novel data from CropDisease [17], EuroSAT [8], ISIC [4], and ChestX [32]. The base dataset is miniImageNet [29], which contains 100 classes from ImageNet dataset [5] where each class has 600 images. The novel datasets are chosen based on increasing dissimilarity from the miniImageNet dataset. More details about the datasets are provided in the Appendix. Following [18], we randomly sample $20 \\%$ of the data from each novel dataset to construct the unlabeled set $\\mathcal { D } _ { U }$ , and the remaining images are used for evaluation, where we perform 5-way 1-shot and 5-way 5-shot classification. For evaluation metric, we report top-1 accuracy and $9 5 \\%$ confidence interval over 600 runs. We also report evaluation results on the larger tieredImageNet dataset [19]. ",
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+        "text": "Implementation details We use ResNet-10 as the backbone network [7, 18]. Our pretraining has two steps. In the first step, we train our network only on the miniImageNet dataset for 200 epochs. We use SGD with momentum 0.9, weight decay 1e-4, learning rate 0.01, batch size 32, and the cosine learning rate scheduler. In the next step, we use the miniImageNet-pretrained network, and use both the base dataset and the unlabeled dataset to optimize the loss function in Eq. 4 for 60 epochs. During training, we increase the weight of distillation loss $\\lambda$ from 0 to 1 until 40 epochs using cosine scheduling. The sharpening temperature and teacher momentum parameter are set to 0.1 and 0.99 respectively. For the base images and weakly-augmented unlabeled images, we use the random-resize-crop, horizontal flip and normalization augmentations. For strong augmentation, we additionally use the color jitter, Gaussian blur, and random gray scale transformations. The other hyperparameters are kept the same. Refer to the supplementary for more details about hyper-parameter selection. For few-shot evaluation, we learn a logistic regression classifier on the support set, and evaluate on the query set. ",
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+    {
+        "type": "table",
+        "img_path": "images/833f2cf690ed57a28da01f5bc777c15ebbc2be12c1dcf29482c351a0e33fa5ec.jpg",
+        "table_caption": [
+            "Table 1: 5-way 1-shot and 5-shot scores on the BSCD-FSL benchmark datasets. The mean and $9 5 \\%$ confidence interval of 600 runs are reported. The ∗ indicates that the numbers are reported from [7] where no unlabeled data is used. The † are the numbers reported from [18], which uses $20 \\%$ of the original set as the unlabeled dataset. We also use similar number of unlabeled images as [18]; however, the splits might be different for random sampling. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"4\">1-shot</td><td colspan=\"4\">5-shot</td></tr><tr><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>MAML*</td><td></td><td></td><td></td><td></td><td>71.70±.72</td><td>78.05±.70</td><td>40.13±.58</td><td>23.48±.48</td></tr><tr><td>ProtoNet*</td><td></td><td></td><td></td><td></td><td>73.29±.71</td><td>79.72±.79</td><td>39.57±.57</td><td>24.05±1.01</td></tr><tr><td>MetaOpt*</td><td></td><td></td><td></td><td></td><td>64.44±.73</td><td>68.41±.73</td><td>36.28±.50</td><td>22.53±.91</td></tr><tr><td>STARTUPt</td><td>63.88±.84</td><td>75.93±.80</td><td>32.66±.60</td><td>23.09±.43</td><td>82.29±.60</td><td>93.02±.45</td><td>47.22±.61</td><td>26.94±.45</td></tr><tr><td>ProtoNet</td><td>55.32±.88</td><td>52.94±.81</td><td>29.58±.57</td><td>21.32±.37</td><td>76.92±.67</td><td>81.84±.68</td><td>42.49±.58</td><td>24.72±.43</td></tr><tr><td>MatchingNet</td><td>54.88±.90</td><td>46.86±.88</td><td>27.37±.51</td><td>20.65±.29</td><td>68.00±.68</td><td>63.94±.84</td><td>33.96±.54</td><td>22.62±.36</td></tr><tr><td>Transfer</td><td>58.14±.83</td><td>68.78±.84</td><td>32.12±.59</td><td>22.60±.39</td><td>80.09±.61</td><td>89.79±.52</td><td>43.88±.57</td><td>26.51±.43</td></tr><tr><td>SimCLR(Base)</td><td>58.28±.90</td><td>61.58±.88</td><td>32.43±.56</td><td>22.37±.42</td><td>80.83±.64</td><td>83.44±.61</td><td>44.04±.55</td><td>26.63±.46</td></tr><tr><td>SimCLR</td><td>62.63±.87</td><td>69.22±.93</td><td>31.45±.59</td><td>23.59±.44</td><td>82.76±.59</td><td>89.31±.53</td><td>42.18±.54</td><td>29.56±.49</td></tr><tr><td>STARTUP</td><td>64.32±.88</td><td>74.45±.86</td><td>31.73±.57</td><td>22.27±.41</td><td>83.58±.60</td><td>92.41±.47</td><td>45.73±.62</td><td>26.21±.46</td></tr><tr><td>Transfer+SimCLR</td><td>63.91±.83</td><td>70.35±.85</td><td>31.67±.55</td><td>23.72±.44</td><td>85.78±.51</td><td>91.10±.49</td><td>45.97±.54</td><td>29.45±.10</td></tr><tr><td> Ours</td><td>73.14±.84</td><td>82.14±.78</td><td>34.66±.58</td><td>23.38±.43</td><td>89.07±.47</td><td>95.54±.38</td><td>49.36±.59</td><td>28.31±.46</td></tr></table>",
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+        "text": "4.2 Main Results ",
+        "text_level": 1,
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+        "text": "Comparison to State-of-the-arts Table 1 shows the performance comparison of our approach with other methods on the BSCD-FSL benchmark. All models are trained on the miniImageNet dataset. “Transfer” denotes the baseline trained by cross-entropy loss on the base dataset. “SimCLR” is trained only on the unlabeled images. As noted by [9], self-supervised contrastive learning learns better transferable representation for a different downstream domain. Hence, we also create a “SimCLR(Base)” baseline that trains the encoder by optimizing self-supervised contrastive loss on the base dataset. “Transfer+SimCLR” refers to the model which is trained with supervised cross-entropy loss from the base dataset and self-supervised contrastive loss from the unlabeled target dataset, which has been reported to show superior transferability across domain [9]. STARTUP [18] is trained with three losses: cross-entropy loss on the base dataset, KL-divergence loss on the unlabeled dataset similar, and self-supervised contrastive loss on unlabeled images. STARTUP, Transfer+SimCLR and our approach use both the base dataset and additional unlabeled dataset during the representation learning phase. ",
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+    {
+        "type": "text",
+        "text": "Our method outperforms all meta-learning-based approaches by a significant margin at all settings. Moreover, compared to Transfer, we achieve more than ${ \\sim } 5 . 5 \\%$ improvement for 5-shot classification on average. The performance improvement on 1-shot is more significant; we achieve $7 . 9 \\%$ improvement on average. ",
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+    {
+        "type": "text",
+        "text": "We outperform STARTUP by $5 . 5 \\%$ on EuroSAT, $3 . 1 \\%$ on CropDisease, $2 . 1 \\%$ on ChestX, and $3 . 6 \\%$ on ISIC for 5-way 5-shot classification. The performance improvement is also quite significant for 1-shot classification; specifically, our method achieves $8 . 8 \\%$ improvement on EuroSAT and $7 . 7 \\%$ improvement on CropDisease dataset over STARTUP. We only perform slightly worse on the ChestX dataset. For ChestX, it seems that pure unsupervised learning performs pretty well. Considering that our method does not use any self-supervised training or distillation, the performance improvement is impressive. Note that STARTUP uses a fixed teacher to extract pseudo-labels for the unlabeled images, whereas we extract pseudo labels from the weakly-augmented images from the same network that is being trained. In that sense, our model works like a dynamic teacher, where the pseudo labels get more refined as training progress. We hypothesize that the superior performance might be attributed to the dynamic approach of our model over STARTUP. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f900298a059783d6dbbd56533fc5289e8a35373ad30af22a37f3a5f411e3846a.jpg",
+        "table_caption": [
+            "Table 2: 5-way 1-shot and 5-shot scores on the BSCD-FSL benchmark datasets when tieredImageNet is used as base dataset. All models use ResNet-18 for backbone [25]. The mean and $9 5 \\%$ confidence interval of 600 runs are reported. For EuroSAT and CropDisease dataset, our method achieves significant performance improvement over other models. The improvement for 1-shot learning is huge (over $7 \\%$ for EuroSAT and $10 \\%$ for CropDisease). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"4\">1-shot</td><td colspan=\"4\">5-shot</td></tr><tr><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>Transfer</td><td>58.07±.86</td><td>69.94±.87</td><td>29.76±.55</td><td>22.46±.41</td><td>81.34±.53</td><td>90.12±.49</td><td>41.27±.58</td><td>26.33±.45</td></tr><tr><td>SimCLR(Base)</td><td>62.14±.89</td><td>62.45±.90</td><td>31.03±.55</td><td>22.28±.40</td><td>81.85±.59</td><td>84.11±.60</td><td>42.91±.55</td><td>25.96±.44</td></tr><tr><td>STARTUP</td><td>64.32±.87</td><td>70.09±.86</td><td>29.73±.51</td><td>22.10±.40</td><td>85.19±.50</td><td>90.81±.49</td><td>43.55±.56</td><td>26.03±.44</td></tr><tr><td>Transfer+SimCLR</td><td>58.08±.83</td><td>71.25±.89</td><td>31.71±.55</td><td>23.81±.46</td><td>86.08±.47</td><td>91.31±.49</td><td>45.08±.56</td><td>30.26±.50</td></tr><tr><td>Ours</td><td>72.15±.75</td><td>84.41±.75</td><td>33.87±.56</td><td>22.70±.42</td><td>89.44±.42</td><td>95.90±.34</td><td>47.21±.56</td><td>27.67±.46</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/502ad74eee0364a3b52db292cbbe82ce6c04aaba10e5d3baed5a4f0364e90963.jpg",
+        "table_caption": [
+            "Table 3: Few-shot evaluation on the same domain in terms of 5-way 5-shot and 5-way 1-shot accuracy on miniImageNet and tieredImageNet datasets. We use ResNet-10 backbone for miniImageNet and ResNet-18 backbone for tieredImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">miniImageNet</td><td colspan=\"2\">tieredImageNet</td></tr><tr><td></td><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>ProtoNet</td><td>51.06±.83</td><td>73.49±.63</td><td></td><td></td></tr><tr><td>MatchingNet</td><td>52.34±.81</td><td>67.28±.67</td><td>1</td><td>=</td></tr><tr><td>Transfer</td><td>53.40±.80</td><td>74.26±.64</td><td>58.61±.97</td><td>81.42±.65</td></tr><tr><td>Transfer+SimCLR</td><td>51.63±.82</td><td>74.65±.60</td><td>61.33±.96</td><td>82.89±.65</td></tr><tr><td>STARTUP</td><td>51.68±.84</td><td>74.05±.66</td><td>60.92±.96</td><td>82.11±.64</td></tr><tr><td>Ours</td><td>53.71±.83</td><td>76.02±.61</td><td>69.00±.96</td><td>85.93±.60</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results with tieredImageNet base data tieredImageNet [19] is a subset of ImageNet dataset with 608 classes. The classes are grouped into 34 super-categories, from which 20 training categories (351 classes), 6 validation categories (97 classes), and 8 testing categories (160 classes) are selected. We use a larger backbone ResNet-18 for meta-training with tieredImageNet [25]. Table 2 shows the performance comparison with other models. We get similar conclusion as we get from Table 1. Particularly, we get $6 . 7 2 \\%$ average improvement for 1-shot and $3 . 6 6 \\%$ average improvement for 5-shot over STARTUP. However, we do not see much better accuracy than miniImageNet pretrained models even though we are using a much larger base dataset, which suggests that the size of base dataset is not as important as other transfer learning task in cross domain few-shot learning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Few-shot performance on similar domain It has been shown that self-training improves ImageNet classification [33]. Given the distillation approach of our model, one could expect that it might improve the few-shot accuracy even when the target data come from the same domain. Note that STARTUP does not improve the in-domain accuracy [18], even though it is also a self-training based model. We evaluate on miniImageNet and tieredImageNet dataset in terms of 5-way 1-shot and 5-way 5-shot performance. We use the official training split as base dataset, and the unlabeled target data are obtained from $20 \\%$ of the novel (test) set and the rest are used for evaluation. For backbone, we use ResNet-10 for miniImageNet and ResNet-18 for tieredImageNet. Table 3 reports the results for in-domain few-shot performance. We see that our method achieves the best performance among the baselines. Particularly, for 1-shot learning in tieredImageNet, our model outperforms the best one by $7 . 7 \\%$ . We infer that our method can be safely applied to few-shot learning task when the domain gap between base and target dataset is small, which is in contrast with STARTUP that does not show improvement over “Transfer\" for few-shot learning on similar domain. ",
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+    {
+        "type": "text",
+        "text": "4.3 Analysis ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Effect of dynamic distillation To understand how distillation helps to learn better representation, we use the pretrained models to extract features of the target dataset. Then we use KMeans algorithm to create clusters from the features. The number of clusters in the KMeans is set to be the number of classes of the target dataset. In Table ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/77ca027ff69828cc474beacc5a7ee57897566181bcb26679d2ab19b53f7ebcab.jpg",
+        "table_caption": [
+            "Table 4: V-measure cluster score $( \\% )$ [20] on the KMeans clustering of the extracted features with the ground-truth clustering. The backbone is ResNet-10 pretrained on the miniImageNet dataset and/or the unlabeled target dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>Transfer</td><td>57.01</td><td>62.58</td><td>14.67</td><td>2.45</td></tr><tr><td>SimCLR</td><td>60.06</td><td>62.02</td><td>12.12</td><td>3.84</td></tr><tr><td>STARTUP</td><td>62.02</td><td>69.50</td><td>14.05</td><td>2.71</td></tr><tr><td>Ours</td><td>69.58</td><td>73.27</td><td>14.32</td><td>3.32</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4, the V-measurement cluster scores [20] between the KMeans clusters and original ground-truth are shown. The V-score has $100 \\%$ value when there is maximum agreement between ground-truth and predicted clusters, and $0 \\%$ when there’s no agreement. Table 4 shows that our method achieves higher v-scores for EuroSAT and CropDisease dataset, and the v-scores for ISIC and ChestX are also very competitive. It suggests that our model learns a good clustering of the target data even when we are not using any target labels. The clustering is much better when the domain gap is not extreme. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Fig. 3 shows t-SNE plots [26] from 10 representative classes from the CropDisease and EuroSAT datasets. We compare the embeddings extracted from “Transfer” and our approach. We see that our method creates better grouping on the embeddings of the target datasets, even though we do not use any labels for the target dataset during pretraining. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/36501101abeac70a3ef2f8ddf896c10e3853579dfdaab8e0e092e6c1fa408902.jpg",
+        "image_caption": [
+            "Figure 3: t-SNE plot of 10 classes from CropDisease (a & b) and EuroSAT (c & d) test sets with features obtained from Transfer and our method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparison with self-supervised learning If we ignore the supervised loss, our model has similarity with self-supervised non-contrastive loss similar to BYOL or DINO. However, the projection head we are using to calculate the final predictions of the two different views of an unlabeled image is the same classification head that is used to predict the classification logits of the labeled base samples. In Table 5, “Ours (distillation head)” represents the model where we use separate projection head for the predictions of the unlabeled images. We see that separate projection head performs much worse. We found that the distillation loss is simply converging to a trivial solution in this case. To discourage trivial solution, we add recently developed tricks in self-supervised learning, namely, centering and strong augmentation - which turns the unlabeled branch similar to ResNet DINO [2]. “Ours (DINO head)” achieves better accuracy than “Ours (distillation head)”, suggesting that it alleviates the issue of trivial solution. However, our original method still achieves a significant $3 . 0 7 \\%$ more improvement. It is interesting to note that a separate projection head causes trivial solution for the unlabeled images, whereas our model does not converge to trivial solution with similar settings. We infer that using a supervised classifier linear layer as the projection head can solve the issue of trivial solution for the self-supervised learning to some extent without requiring extra tricks like BatchNorm [6] or centering $I 2 I$ . Additionally, it provides a better clustering of the unlabeled features, even if the unlabeled samples come from different domain than the labeled samples. Table 5 also shows results for “Ours $^ +$ SimCLR”, which simply adds a SimCLR loss for the unlabeled samples. It performs slightly better only in ChestX dataset. On average, the performance is similar to “Ours”, which signifies that there is no clear benefit using a self-supervised contrastive loss to achieve better transferability for our method. Note that STARTUP comes to a different conclusion reporting that adding SimCLR loss consistently improves the performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/7b65b152f60e6d492646e3bbf7de1a55bbfca3755eca716edc9421f45f47c11b.jpg",
+        "table_caption": [
+            "Table 5: Our method with self-supervised approaches. The evaluation is performed on BSCD-FSL benchmark in terms of 5-way 5-shot accuracy $( \\% )$ . “Ours (distillation head)” refers to the model where we use a separate projection head for the distillation loss, which achieves much worse scores. In “Ours $^ +$ DINO”, we use a separate projection head with centering and strong augmentation as in DINO [2]. It achieves better performance than naive transfer, but still under-performs in comparison to our approach. “Ours $^ +$ SimCLR” simply adds a self-supervised contrastive loss for the unlabeled samples. See Appendix for more details. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>Ours</td><td>89.07</td><td>95.54</td><td>49.36</td><td>28.31</td></tr><tr><td>Ours (distillation head)</td><td>80.06</td><td>89.31</td><td>46.63</td><td>25.29</td></tr><tr><td>Ours (DINO head)</td><td>85.74</td><td>90.55</td><td>46.24</td><td>25.42</td></tr><tr><td>Ours+SimCLR</td><td>88.48</td><td>93.80</td><td>49.10</td><td>29.45</td></tr></table>",
+        "bbox": [
+            274,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Experiment of few-shot classification on fine-grained dataset CUB [30] contains 200 classes and 11,788 images of different bird species. ‘miniImageNet- $\\cdot >$ CUB’ is an interesting experiment to show the transferability of different models to a fine-grained dataset. We report the results in Table 6 in terms of 5-way 5-shot scores. For CUB, we found that vanilla Transfer performs surprisingly well (also reported by [3]), and adding SimCLR with Transfer (Transfer+SimCLR) actually decreases the accuracy. Wallace and Hariharan [31] also experimented with different self-supervised methods on smaller domain and found that all of them under-perform for fine-grained task [2]. However, our method still performs the best, demonstrating the effectiveness of our approach in a fine-grained downstream task. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/cab554749bd3468d3b6841f444df823bf85a36c29897bf4bd44a1f61216fe728.jpg",
+        "table_caption": [
+            "Table 6: Experiment on mini-ImageNet $\\bullet >$ CUB. All the scores are reproduced by us. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ProtoNet</td><td>Transfer</td><td>SimCLR</td><td>Transfer+SimCLR</td><td>STARTUP</td><td>Ours</td></tr><tr><td>63.19</td><td>68.72</td><td>62.84</td><td>67.82</td><td>66.10</td><td>69.50</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Experiment of distillation with unlabeled datasets from different domain Table 7 reports the few-shot accuracy when our model is trained on different unlabeled datasets. The best accuracy is achieved when the unlabeled data and target data are from the same domain. Even if the unlabeled data consists of images from multiple domains including the target domain (denoted as “Ours-all”), it still significantly under-performs the base model. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/471f337e2af6502563bad5a2dfb040be4611d7d90f542f1ca0b940507958f4c6.jpg",
+        "table_caption": [
+            "Table 7: Effect of unlabeled datasets from a different domain than the target dataset in terms of 5-way 5-shot accuracy $( \\% )$ . “Ours-X” denotes that we use base and “X” dataset during pretraining. “Ours-all” denotes that we use unlabeled images from all four target datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>EuroSAT</td><td>CropDisease</td><td>ChestX</td><td>ISIC</td></tr><tr><td>Ours-EuroSAT</td><td>89.07</td><td>90.43</td><td>26.02</td><td>46.82</td></tr><tr><td>Ours-CropDisease</td><td>81.86</td><td>95.54</td><td>26.17</td><td>45.12</td></tr><tr><td>Ours-ISIC</td><td>81.94</td><td>89.69</td><td>26.70</td><td>49.36</td></tr><tr><td>Ours-ChestX</td><td>81.87</td><td>90.38</td><td>28.31</td><td>45.20</td></tr><tr><td>Ours-All</td><td>82.75</td><td>91.31</td><td>26.01</td><td>46.43</td></tr></table>",
+        "bbox": [
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+            424,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Effect of data augmentation On the unlabeled images, we apply two types of augmentation: weak augmentation to extract pseudo labels and strong augmentation to impose consistency regularization. This setting is denoted as “weak-strong” (w-s), where ‘weak’ (w) augmentation is applied to the image that is fed into the teacher network and ‘strong’ (s) augmentation is applied to the image that is fed into the student network. We also show results with “weak-weak” (w-w), “strong-weak” (s-w) and “strong-strong” (s-s) augmentation settings in Figure 4a. Both “weak-weak” and “strong-strong” perform worse than the other augmentations. We also note that in self-supervised learning, generally strong augmentation is applied to all training images to get good performance. In our experiment, we find that applying weak augmentation in one of the image pairs improves the performance. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "More unlabeled data To measure the effect of amount of unlabeled dataset during pretraining, we divide the target dataset by $80 \\%$ and $20 \\%$ splits. The $20 \\%$ split is used for evaluation. From the $80 \\%$ split, we vary the amount of unlabeled data and then pretrain with source and the unlabeled set. Fig. 4b shows the average 5-shot accuracy for different amounts of the unlabeled dataset during the pretraining phase. As expected, more information from the unlabeled dataset helps to learn better representations on the target domain. However, the performance saturates later, and we get diminishing return for more unlabeled data. It also signifies that there are scopes to improve the performance by using more unlabeled data denoting potential future research direction. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.4 Addition Ablation Studies ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We perform several ablation studies of different components of our approach. All scores are reported for 5-way 5-shot evaluation. More ablations are provided in the Appendix. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Longer training We use the base dataset pretrained network as initialization for our network, and then train on both base dataset and unlabeled dataset for 60 epochs. Figure $_ \\mathrm { 4 c }$ reports average 5-way 5-shot few-shot on the BSCD-FSL benchmark performance for our pretrained with longer training epoch. Training for more epochs can result in minor improvement $( 0 . 2 \\% )$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/f1b5c962ab2ccd9c9213cde32790f32a6824e26df90bce206252a13bf3f3f81d.jpg",
+        "image_caption": [
+            "Figure 4: Ablation studies for (a) data augmentation on unlabeled images, (b) effect of longer training on base and unlabeled data, (c) amount of unlabeled data. The Y-axis represents average top1 accuracy $( \\% )$ on the four benchmark datasets for 5-shot classification for 600 episodes. We use miniImageNet for labeled source dataset and ResNet-10 as backbone. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b159bc939ec8c1b384cbda64938faa7c8576c1585a92ca1a04ea3c7c774a82e.jpg",
+        "table_caption": [
+            "Table 8: Ablation studies on different settings. Mean over 600 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>EuroSAT</td><td>CropDisease</td><td>ISIC</td><td>ChestX</td></tr><tr><td>Ours</td><td>89.07</td><td>95.54</td><td>49.36</td><td>28.31</td></tr><tr><td>Ours(w/o base)</td><td>82.11</td><td>90.52</td><td>39.76</td><td>26.83</td></tr><tr><td>Ours(1-step)</td><td>86.16</td><td>87.28</td><td>46.10</td><td>25.11</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Effect of base dataset Here, we perform experiments without the first term in Eq. 4, i.e., we train the network on the base dataset first and then re-train only on the unlabeled images (without joint training on the base dataset), denoted as “Ours (w/o base)”. Table 8 shows that the performance of “Ours (w/o base)” is poor, suggesting that the representation related to the labeled base dataset is still helpful to the target domain. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training without pretrained model on base dataset We perform 2-step training during the representation learning phase - we first train the model on mini-IN only, and then jointly train on mini-IN and unlabeled images. In Table 8, “Ours(1-step)” denotes training on the unlabeled images and mini-IN from scratch for 300 epochs, which performs worse than 2-step training. Our assumption is that the proposed model is also like self-training where a well-trained teacher is needed. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We introduced a novel approach to utilize unlabeled data from the target domain for cross-domain fewshot learning. Experiments show that our method achieves state-of-the-art results in the BSCD-FSL benchmark for both 1-shot and 5-shot classification. Our model also outperforms other approaches in the same-domain few-shot learning. Future work can be focused on applying our approach in each episode during meta-testing so that the model can learn more category-specific representations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 Broader Impact ",
+        "text_level": 1,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The approach tackles a practical problem of the existing few-shot learning setup that the base dataset and the novel samples generally come from different domain. Our work uses the unlabeled samples from the target dataset to learn more target specific representation. Like any other machine learning tool, the final impact depends on the intention of the people or institution applying it. However, it can be useful in drug discovery or medical image analysis where labeled dataset is limited. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "7 Acknowledgments ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This material is based upon work supported by the U.S. Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Grant Award 2013-ST-061- ED0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Department of Homeland Security. ",
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+    },
+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
+            174,
+            90,
+            266,
+            106
+        ],
+        "page_idx": 10
+    },
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+        "type": "text",
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Wang, Y. Peng, L. Lu, Z. Lu, M. Bagheri, and R. M. Summers. Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2097–2106, 2017.   \n[33] Q. Xie, M.-T. Luong, E. Hovy, and Q. V. Le. Self-training with noisy student improves imagenet classification. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10687–10698, 2020.   \n[34] Q. Xu, T. Likhomanenko, J. Kahn, A. Hannun, G. Synnaeve, and R. Collobert. Iterative pseudo-labeling for speech recognition. arXiv preprint arXiv:2005.09267, 2020.   \n[35] I. Z. Yalniz, H. Jégou, K. Chen, M. Paluri, and D. Mahajan. Billion-scale semi-supervised learning for image classification. arXiv preprint arXiv:1905.00546, 2019.   \n[36] R. Zhang, T. Che, Z. Ghahramani, Y. Bengio, and Y. Song. 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parse/train/ryxnY3NYPS/ryxnY3NYPS.md

@@ -0,0 +1,335 @@
+# DIVERSE TRAJECTORY FORECASTING WITH DETERMINANTAL POINT PROCESSES
+
+Ye Yuan, Kris M. Kitani Robotics Institute Carnegie Mellon University {yyuan2,kkitani}@cs.cmu.edu
+
+# ABSTRACT
+
+The ability to forecast a set of likely yet diverse possible future behaviors of an agent (e.g., future trajectories of a pedestrian) is essential for safety-critical perception systems (e.g., autonomous vehicles). In particular, a set of possible future behaviors generated by the system must be diverse to account for all possible outcomes in order to take necessary safety precautions. It is not sufficient to maintain a set of the most likely future outcomes because the set may only contain perturbations of a dominating single outcome (major mode). While generative models such as variational autoencoders (VAEs) have been shown to be a powerful tool for learning a distribution over future trajectories, randomly drawn samples from the learned implicit likelihood model may not be diverse – the likelihood model is derived from the training data distribution and the samples will concentrate around the major mode of the data. In this work, we propose to learn a diversity sampling function (DSF) that generates a diverse yet likely set of future trajectories. The DSF maps forecasting context features to a set of latent codes which can be decoded by a generative model (e.g., VAE) into a set of diverse trajectory samples. Concretely, the process of identifying the diverse set of samples is posed as DSF parameter estimation. To learn the parameters of the DSF, the diversity of the trajectory samples is evaluated by a diversity loss based on a determinantal point process (DPP). Gradient descent is performed over the DSF parameters, which in turn moves the latent codes of the sample set to find an optimal set of diverse yet likely trajectories. Our method is a novel application of DPPs to optimize a set of items (forecasted trajectories) in continuous space. We demonstrate the diversity of the trajectories produced by our approach on both low-dimensional 2D trajectory data and high-dimensional human motion data. (Video1)
+
+# 1 INTRODUCTION
+
+Forecasting future trajectories of vehicles and human has many useful applications in autonomous driving, virtual reality and assistive living. What makes trajectory forecasting challenging is that the future is uncertain and multi-modal – vehicles can choose different routes and people can perform different future actions. In many safety-critical applications, it is important to consider a diverse set of possible future trajectories, even those that are less likely, so that necessary preemptive actions can be taken. For example, an autonomous vehicle should understand that a neighboring car can merge into its lane even though the car is most likely to keep driving straight. To address this requirement, we need to take a generative approach to trajectory forecasting that can fully characterize the multimodal distribution of future trajectories. To capture all modes of a data distribution, variational autoencoders (VAEs) are well-suited generative models. However, random samples from a learned VAE model with Gaussian latent codes are not guaranteed to be diverse for two reasons. First, the sampling procedure is stochastic and the VAE samples can fail to cover some minor modes even with a large number of samples. Second, since VAE sampling is based on the implicit likelihood function encoded in the training data, if most of the training data is centered around a specific mode while other modes have less data (Fig. 1 (a)), the VAE samples will reflect this bias and concentrate around the major mode (Fig. 1 (b)). To tackle this problem, we propose to learn a diversity sampling function (DSF) that can reliably generate a diverse set of trajectory samples (Fig. 1 (c)).
+
+![](images/935dc525723c5a63173fac98ebc77b977ebc640ac583ddbd40861f52edb857bf.jpg)  
+Figure 1: A toy trajectory forecasting example. (a) The three modes (pink, blue, purple) of the future trajectory distribution are shown in both the trajectory space and the latent space of a learned VAE model. The data distribution is imbalanced, where the blue mode has most data and covers most of the latent space. (b) Random samples from the VAE only cover the major (blue) mode. (c) Our proposed DSF generates a diverse set of future trajectories covering all three modes.
+
+The proposed DSF is a deterministic parameterized function that maps forecasting context features (e.g., past trajectories) to a set of latent codes. The latent codes are decoded by the VAE docoder into a set of future trajectory samples, denoted as the DSF samples. In order to optimize the DSF, we formulate a diversity loss based on a determinantal point process (DPP) (Macchi, 1975) to evaluate the diversity of the DSF samples. The DPP defines the probability of choosing a random subset from the set of trajectory samples. It models the negative correlations between samples: the inclusion of a sample reduces the probability of including a similar sample. This makes the DPP an ideal tool for modeling the diversity within a set. In particular, we use the expected cardinality of the DPP as the diversity measure, which is defined as the expected size of a random subset drawn from the set of trajectory samples according to the DPP. Intuitively, since the DPP inhibits selection of similar samples, if the set of trajectory samples is more diverse, the random subset is more likely to select more samples from the set. The expected cardinality of the DPP is easy to compute and differentiable, which allows us to use it as the objective to optimize the DSF to enable diverse trajectory sampling.
+
+Our contributions are as follows: (1) We propose a new forecasting approach that learns a diversity sampling function to produce a diverse set of future trajectories; (2) We propose a novel application of DPPs to optimize a set of items (trajectories) in continuous space with a DPP-based diversity measure; (3) Experiments on synthetic data and human motion validate that our method can reliably generate a more diverse set of future trajectories compared to state-of-the-art generative models.
+
+# 2 RELATED WORK
+
+Trajectory Forecasting has recently received significant attention from the vision community. A large portion of previous work focuses on forecasting 2D future trajectories for pedestrians (Kitani et al., 2012; Ma et al., 2017; Ballan et al., 2016; Xie et al., 2013) or vehicles (Jain et al., 2016a). Some approaches use deterministic trajectory modeling and only forecast one future trajectory (Alahi et al., 2016; Yagi et al., 2018; Robicquet et al., 2016). As there are often multiple plausible future trajectories, several approaches have tried to forecast distributions over trajectories (Lee et al., 2017; Galceran et al., 2015; Gupta et al., 2018). Recently, Rhinehart et al. (2018; 2019) propose a generative model that can accurately forecast multi-modal trajectories for vehicles. Soo Park et al. (2016) also use egocentric videos to predict the future trajectories of the camera wearer. Some work has investigated forecasting higher dimensional trajectories such as the 3D fullbody pose sequence of human motions. Most existing work takes a deterministic approach and forecasts only one possible future motion from past 3D poses (Fragkiadaki et al., 2015; Butepage et al., 2017; Li et al., 2017; Jain et al., 2016b), static images (Chao et al., 2017; Kanazawa et al., 2018) or egocentric videos (Yuan and Kitani, 2019). Differently, some probabilistic approaches (Habibie et al., 2017; Yan et al., 2018) use conditional variational autoencoders (cVAEs) to generate multiple future motions. In constrast to previous work, our approach can generate a diverse set of future motions with a limited number of samples.
+
+Diverse Solutions have been sought after in a number of problems in computer vision and machine learning. A branch of these methods aiming for diversity stems from the M-Best MAP problem (Nilsson, 1998; Seroussi and Golmard, 1994), including diverse M-Best solutions (Batra et al., 2012) and multiple choice learning (Guzman-Rivera et al., 2012; Lee et al., 2016). Alternatively, previous work has used submodular function maximization to select a diverse subset of garments from fashion images (Hsiao and Grauman, 2018). Determinantal point processes (DPPs) (Macchi, 1975; Kulesza et al., 2012) are efficient probabilistic models that can measure both the diversity and quality of items in a subset, which makes it a natural choice for the diverse subset selection problem. DPPs have been applied for document and video summarization (Kulesza and Taskar, 2011; Gong et al., 2014), recommendation systems (Gillenwater et al., 2014), object detection (Azadi et al., 2017), and grasp clustering (Huang et al., 2015). Elfeki et al. (2018) have also used DPPs to mitigate mode collapse in generative adversarial networks (GANs). The work most related ours is (Gillenwater et al., 2014), which also uses the cardinality of DPPs as a proxy for user engagement. However, there are two important differences between our approach and theirs. First, the context is different as they use the cardinality for a subset selection problem while we apply the cardinality as an objective of a continuous optimization problem in the setting of generative models. Second, their main motivation behind using the cardinality is that it aligns better with the user engagement semantics, while our motivation is that using cardinality as a diversity loss for deep neural networks is more stable due to its tolerance of similar trajectories, which are often produced by deep neural networks during stochastic gradient descent.
+
+# 3 BACKGROUND
+
+# 3.1 VARIATIONAL AUTOENCODERS
+
+The aim of multi-modal trajectory forecasting is to learn a generative model over future trajectories. Variational autoencoders (VAEs) are a popular choice of generative models for trajectory forecasting (Lee et al., 2017; Walker et al., 2016) because it can effectively capture all possible future trajectories by explicitly mapping each data point to a latent code. VAEs model the joint distribution $p _ { \theta } \mathbf { \bar { ( x , z ) } } = p \mathbf { \bar { ( z ) } } \bar { p _ { \theta } } ( \mathbf { x } | \mathbf { z } )$ of each data sample $\mathbf { x }$ (e.g., a future trajectory) and its corresponding latent code $\mathbf { z }$ , where $p ( \mathbf { z } )$ denotes some prior distribution (e.g., Gaussians) and $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ denotes the conditional likelihood model. To calculate the marginal likelihood $p _ { \theta } ( \mathbf { x } ) = p _ { \theta } ( \mathbf { x } , \mathbf { z } ) / p _ { \theta } ( \mathbf { z } | \mathbf { x } )$ , one needs to compute the posterior distribution $p _ { \theta } ( \mathbf { z } | \mathbf { x } )$ which is typically intractable. To tackle this, VAEs use variational inference (Jordan et al., 1999) which introduces an approximate posterior $q _ { \phi } ( { \bf z } | { \bf x } )$ and decomposes the marginal log-likelihood as
+
+$$
+\begin{array} { r } { \log p _ { \theta } ( \mathbf { x } ) = \mathrm { K L } \left( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) \| p _ { \theta } ( \mathbf { z } | \mathbf { x } ) \right) + \mathcal { L } ( \mathbf { x } ; \theta , \phi ) , } \end{array}
+$$
+
+where $\mathcal { L } ( \mathbf { x } ; \theta , \phi )$ is the evidence lower bound (ELBO) defined as
+
+$$
+\begin{array} { r } { \mathcal { L } ( \mathbf { x } ; \theta , \phi ) = \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } \left[ \log p _ { \theta } ( \mathbf { x } | \mathbf { z } ) \right] - \mathrm { K L } \left( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) \| p ( \mathbf { z } ) \right) . } \end{array}
+$$
+
+During training, VAEs jointly optimize the recognition model (encoder) $q _ { \phi } ( { \bf z } | { \bf x } )$ and the likelihood model (decoder) $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ to maximize the ELBO. In the context of multi-modal trajectory forecasting, one can generate future trajectories from $p ( \mathbf { x } )$ by drawing a latent code $\mathbf { z }$ from the prior $p ( \mathbf { z } )$ and decoding $\mathbf { z }$ with the decoder $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ to produce a corresponding future trajectory $\mathbf { x }$ .
+
+# 3.2 DETERMINANTAL POINT PROCESSES
+
+Our core technical innovation is a method to learn a diversity sampling function (DSF) that can generate a diverse set of future trajectories. To achieve this, we must equip ourselves with a tool to evaluate the diversity of a set of trajectories. To this end, we make use of determinantal point processes (DPPs) to model the diversity within a set. DPPs promote diversity within a set because the inclusion of one item makes the inclusion of a similar item less likely if the set is sampled according to a DPP.
+
+Formally, given a set of items (e.g., data points) $\mathcal { Y } = \{ \mathbf { x } _ { 1 } , \dotsc , \mathbf { x } _ { N } \}$ , a point process $\mathcal { P }$ on the ground set $\mathcal { V }$ is a probability measure on $2 ^ { y }$ , i.e., the set of all subsets of $\mathcal { V }$ . $\mathcal { P }$ is called a determinantal point process if a random subset $\mathbf { Y }$ drawn according to $\mathcal { P }$ has
+
+$$
+\mathcal { P } _ { \mathbf { L } } ( Y = Y ) = \frac { \operatorname* { d e t } \left( \mathbf { L } _ { Y } \right) } { \sum _ { Y \subseteq \mathcal { V } } \operatorname* { d e t } \left( \mathbf { L } _ { Y } \right) } = \frac { \operatorname* { d e t } \left( \mathbf { L } _ { Y } \right) } { \operatorname* { d e t } ( \mathbf { L } + \mathbf { I } ) } ,
+$$
+
+where $Y \subseteq \mathcal { V }$ , I is the identity matrix, $\mathbf { L } \in \mathbb { R } ^ { N \times N }$ is the DPP kernel, a symmetric positive semidefinite matrix, and $\mathbf { L } _ { Y } \in \mathbb { R } ^ { | Y | \times | Y | }$ is a submatrix of $\mathbf { L }$ indexed by elements of $Y$ .
+
+The DPP kernel $\mathbf { L }$ is typically constructed by a similarity matrix S, where $\mathbf { S } _ { i j }$ defines the similarity between two items $\mathbf { x } _ { i }$ and $\mathbf { x } _ { j }$ . If we use the inner product as the similarity measure, $\mathbf { L }$ can be written in the form of a Gram matrix $\mathbf { L } = \mathbf { S } = \mathbf { X } ^ { T } \mathbf { X }$ where $\mathbf { X }$ is the stacked feature matrix of $\mathcal { V }$ . As a property of the Gram matrix, det $\left( { \bf L } _ { Y } \right)$ equals the squared volume spanned by vectors $\mathbf { x } _ { i } \in Y$ . With this geometric interpretation in mind, one can observe that diverse sets are more probable because their features are more orthogonal, thus spanning a larger volume.
+
+In addition to set diversity encoded in the similarity matrix S, it is also convenient to introduce a quality vector $\mathbf { r } = [ r _ { 1 } , \dots , r _ { N } ]$ to weigh each item according to some unary metric. For example, the quality weight might be derived from the likelihood of an item. To capture both diversity and quality of a subset, the DPP kernel $\mathbf { L }$ is often decomposed in the more general form:
+
+$$
+\mathbf { L } = \mathrm { D i a g } ( \mathbf { r } ) \cdot \mathbf { S } \cdot \mathrm { D i a g } ( \mathbf { r } ) .
+$$
+
+With this decomposition, we can see that both the quality vector $\mathbf { r }$ and similarity matrix S contribute to the DPP probability of a subset $Y$ :
+
+$$
+\mathcal { P } _ { L } ( \pmb { Y } = Y ) \propto \operatorname* { d e t } \left( \mathbf { L } _ { Y } \right) = \left( \prod _ { \mathbf { x } _ { i } \in Y } r _ { i } ^ { 2 } \right) \operatorname* { d e t } \left( \mathbf { S } _ { Y } \right) .
+$$
+
+Due to its ability to capture the global diversity and quality of a set of items, we choose DPPs as the probabilistic approach to evaluate and optimize the diversity of the future trajectories drawn by our proposed diversity sampling function.
+
+# 4 APPROACH
+
+Safety-critical applications often require that the system can maintain a diverse set of outcomes covering all modes of a predictive distribution and not just the most likely one. To address this requirement, we propose to learn a diversity sampling function (DSF) to draw deterministic trajectory samples by generating a set of latent codes in the latent space of a conditional variational autoencoder (cVAE) and decoding them into trajectories using the cVAE decoder. The DSF trajectory samples are evaluated with a DPP-based diversity loss, which in turn optimizes the parameters of the DSF for more diverse trajectory samples.
+
+Formally, the future trajectory $\mathbf { x } \in \mathbb { R } ^ { T \times D }$ is a random variable denoting a $D$ dimensional feature over a future time horizon $T$ (e.g., a vehicle trajectory or a sequence of humanoid poses). The context $\boldsymbol { \psi } = \{ { \bf h } , { \bf f } \}$ provides the information to infer the future trajectory $\mathbf { x }$ , and it contains the past trajectory $\mathbf { \bar { h } } \in \mathbf { \mathbb { R } } ^ { \mathbf { \bar { H } } \times D }$ of last $H$ time steps and optionally other side information f , such as an obstacle map. In the following, we first describe how we learn the future trajectory model $p _ { \theta } ( \mathbf { x } | \psi )$ with a cVAE. Then, we introduce the DSF and the DPP-based diversity loss used to optimize the DSF.
+
+# 4.1 LEARNING A CVAE FOR FUTURE TRAJECTORIES
+
+In order to generate a diverse set of future trajectory samples, we need to learn a generative trajectory forecasting model $p _ { \theta } ( \mathbf { x } | \psi )$ that can cover all modes of the data distribution. Here we use cVAEs (other proper generative models can also be used), which explicitly map data $\mathbf { x }$ with the encoder $q _ { \phi } ( { \bf z } | { \bf x } , \psi )$ to its corresponding latent code $\mathbf { z }$ and reconstruct the data from the latent code using the decoder $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } , \boldsymbol { \psi } )$ . By maintaining this one-on-one mapping between the data and the latent code, cVAEs attempt to capture all modes of the data. As discussed in Sec. 3.1, cVAEs jointly optimize the encoder and decoder to maximize the variational lower bound:
+
+$$
+\mathcal { L } ( \mathbf { x } , \psi ; \theta , \phi ) = \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } , \psi ) } \left[ \log p _ { \theta } ( \mathbf { x } | \mathbf { z } , \psi ) \right] - \mathrm { K L } \left( q _ { \phi } ( \mathbf { z } | \mathbf { x } , \psi ) \| p ( \mathbf { z } ) \right) .
+$$
+
+We use multivariate Gaussians for the prior, encoder and decoder: $p ( \mathbf { z } ) = \mathcal { N } ( \mathbf { z } ; \mathbf { 0 } , \mathbf { I } )$ , $q _ { \phi } ( { \bf z } | { \bf x } , \psi ) =$ $\mathcal { N } ( \mathbf { z } ; \pmb { \mu } , \pmb { \sigma } ^ { 2 } \mathbf { I } )$ , and $p _ { \theta } ( \mathbf { x } | \mathbf { z } , \psi ) = \mathcal { N } ( \mathbf { x } ; \tilde { \mathbf { x } } , \alpha \mathbf { I } )$ . Both the encoder and decoder are implemented as neural networks. The encoder network $f _ { \phi }$ outputs the parameters of the posterior distribution: $( \mu , \sigma ) = f _ { \phi } ( { \bf x } , \psi )$ . The decoder network $g _ { \theta }$ outputs the reconstructed future trajectory $\tilde { \bf x }$ :
+
+$\tilde { \mathbf { x } } = g _ { \theta } ( \mathbf { z } , \psi )$ . Detailed network architectures are given in Appendix B.1. Based on the Gaussian parameterization of the cVAE, the objective in Eq. 6 can be rewritten as
+
+$$
+\mathcal { L } _ { c v a e } ( \mathbf { x } , \psi ; \theta , \phi ) = - \frac { 1 } { V } \sum _ { v = 1 } ^ { V } \Vert \tilde { \mathbf { x } } _ { v } - \mathbf { x } \Vert ^ { 2 } + \beta \cdot \frac { 1 } { D _ { z } } \sum _ { j = 1 } ^ { D _ { z } } \left( 1 + 2 \log \sigma _ { j } - \mu _ { j } ^ { 2 } - \sigma _ { j } ^ { 2 } \right) ,
+$$
+
+where we take $V$ samples from the posterior $q _ { \phi } ( { \bf z } | { \bf x } , \psi )$ , $D _ { z }$ is the number of latent dimensions, and $\beta = 1 / \alpha$ is a weighting factor. The training procedure for the $\operatorname { c V A E }$ is detailed in Alg. 2 (Appendix A). Once the cVAE model is trained, sampling from the learned future trajectory model $p _ { \theta } ( \mathbf { x } | \psi )$ is efficient: we can sample a latent code $\mathbf { z }$ according to the prior $p ( \mathbf { z } )$ and use the decoder $p _ { \theta } ( \mathbf { x } | \mathbf { z } , \psi )$ to decode it into a future trajectory $\mathbf { x }$ .
+
+# Algorithm 1 Training the diversity sampling function (DSF) $S _ { \gamma } ( \psi )$
+
+1: Input: Training data $\{ \mathbf { x } ^ { ( i ) } , \boldsymbol { \psi } ^ { ( i ) } \} _ { i = 1 } ^ { M }$ , cVAE decoder network $g _ { \boldsymbol { \theta } } ( \mathbf { z } , \boldsymbol { \psi } )$   
+2: Output: DSF $S _ { \gamma } ( \psi )$   
+3: Initialize $\gamma$ randomly   
+4: while not converged do   
+5: for each $\psi ^ { ( i ) }$ do   
+6: Generate latent codes $\mathcal { Z } = \{ \mathbf { z } _ { 1 } , \ldots , \mathbf { z } _ { N } \}$ with the DSF $S _ { \gamma } ( \psi )$   
+7: Generate the trajectory ground set $\mathcal { Y } = \{ \mathbf { x } _ { 1 } , \dotsc , \mathbf { x } _ { N } \}$ with the decoder $g _ { \boldsymbol { \theta } } ( \mathbf { z } , \boldsymbol { \psi } )$   
+8: Compute the similarity matrix S and quality vector r with Eq. 8 and 9   
+9: Compute the DPP kernel $\mathbf { L } ( \gamma ) = \mathrm { D i a g } ( \mathbf { r } ) \cdot \mathbf { S } \cdot \mathrm { D i a g } ( \mathbf { r } )$   
+10: Calculate the diversity loss Ldiverse   
+11: Update $\gamma$ with the gradient ∇Ldiverse   
+12: end for   
+13: end while
+
+# 4.2 DIVERSITY SAMPLING FUNCTION (DSF)
+
+As mentioned before, randomly sampling from the learned cVAE model according to the implicit likelihood function $p _ { \theta } ( \mathbf { x } | \psi )$ , i.e., sampling latent codes from the prior $p ( \mathbf { z } )$ , does not guarantee that the trajectory samples are diverse: major modes (those having more data) with higher likelihood will produce most of the samples while minor modes with lower likelihood will have almost no sample. This prompts us to devise a new sampling strategy that can reliably generate a diverse set of samples covering both major and minor modes. We propose to learn a diversity sampling function (DSF) $S _ { \gamma } ( \psi )$ that maps context $\psi$ to a set of latent codes $\mathcal { Z } = \{ \mathbf { z } _ { 1 } , \ldots , \mathbf { z } _ { N } \}$ . The DSF is implemented as a $\gamma$ -parameterized neural network which takes $\psi$ as input and outputs a vector of length $N \cdot D _ { z }$ (see Appendix B.1 for network details). The latent codes $\mathcal { Z }$ are decoded into a diverse set of future trajectories $\mathcal { Y } = \{ \mathbf { x } _ { 1 } , \dotsc , \mathbf { x } _ { N } \}$ , which are denoted as the DSF trajectory samples. We note that $N$ is the sampling budget. To solve for the parameters of the DSF, we propose a diversity loss based on a DPP defined over $\mathcal { V }$ . In this section, we first describe how the DPP kernel $\mathbf { L }$ is defined, which involves the construction of the similarity matrix S and quality vector $\mathbf { r }$ . We then discuss how we use the DPP kernel $\mathbf { L }$ to formulate a diversity loss to optimize the parameters of the DSF.
+
+Recall that the DPP kernel is defined as $\mathbf { L } = \mathrm { D i a g } ( \mathbf { r } ) \cdot \mathbf { S } \cdot \mathrm { D i a g } ( \mathbf { r } )$ , where $\mathbf { r }$ defines the quality of each trajectory and $\mathbf { S }$ measures the similarity between two trajectories. The DPP kernel $\mathbf { L } ( \gamma )$ is a function of $\gamma$ as it is defined over the ground set $\mathcal { V }$ output by the DSF $S _ { \gamma } ( \psi )$ .
+
+Similarity. We measure the similarity $\mathbf { S } _ { i j }$ between two trajectories $\mathbf { x } _ { i }$ and $\mathbf { x } _ { j }$ as
+
+$$
+\mathbf { S } _ { i j } = \exp \left( - k \cdot d _ { \mathbf { x } } ^ { 2 } ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) \right) ,
+$$
+
+where $d _ { \mathbf { x } }$ is the Euclidean distance and $k$ is a scaling factor. This similarity design ensures that $0 \leq \mathbf { S } _ { i j } \leq 1$ and $\mathbf { S } _ { i i } = 1$ . It also makes $\mathbf { S }$ positive definite since the Gaussian kernel we use is a positive definite kernel.
+
+Quality. It may be tempting to use $p ( \mathbf { x } | \boldsymbol { \psi } )$ to define the quality of each trajectory sample. However, this likelihood-based measure will clearly favor major modes that have higher probabilities, making it less likely to generate samples from minor modes. This motivates us to design a quality metric that
+
+treats all modes equally. To this end, unlike the similarity metric which is defined in the trajectory space, the quality of each sample is measured in the latent space and is defined as
+
+$$
+r _ { i } = \left\{ \begin{array} { l l } { \omega , } & { \mathrm { i f ~ } \| \mathbf { z } _ { i } \| \leq R } \\ { \omega \exp \left( - \mathbf { z } _ { i } ^ { T } \mathbf { z } _ { i } + R ^ { 2 } \right) , } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+Geometrically, let $R$ be the radius of a sphere $\Phi$ containing most samples from the Gaussian prior $p ( \mathbf { z } )$ . We treat samples inside $\Phi$ equally and only penalize samples outside $\Phi$ . In this way, samples from major modes are not preferred over those from minor modes as long as they are inside $\Phi$ , while samples far away from the data manifold are heavily penalized as they are outside $\Phi$ . The radius $R$ is determined by where $\rho$ percent of the Gaussian samples lie within, and we set $\rho = 9 0$ . To compute $R$ , we use the percentage point function of the chi-squared distribution which models the distribution over the sum of squares of independent standard normal variables. The base quality $\omega$ is a hyperparameter which we set to 1 during training in our experiments. At test time, we can use a larger $\omega$ to encourage the DPP to select more items from the ground set $\mathcal { V }$ . The hyperparameter $\rho$ (or $R$ ) allows for the trade-off between diversity and quality. When $R$ is small, the quality metric is reduced to a pure likelihood-based metric (proportional to the latent likelihood), which will prefer samples with high likelihood and result in a less diverse sample set. When $R$ is large, most samples will have the same quality, and the resulting samples will be highly diverse but less likely. In practice, the choice of $R$ should be application dependent, as one could imagine autonomous vehicles would need to consider more diverse scenarios including those less likely ones to ensure robustness. We note that after the diverse samples are obtained, it is possible to reassign the quality score for each sample based on its likelihood to allow users to prioritize more likely samples.
+
+Diversity Loss. To optimize the DSF $S _ { \gamma } ( \psi )$ , we need to define a diversity loss that measures the diversity of the trajectory ground set $\mathcal { V }$ generated by $S _ { \gamma } ( \psi )$ . An obvious choice for the diversity loss would be the negative log likelihood $- \log \mathcal { P } _ { \mathbf { L } ( \gamma ) } ( \boldsymbol { Y } = \mathcal { Y } ) = - \log \operatorname* { d e t } ( \mathbf { L } ( \gamma ) ) + \log \operatorname* { d e t } ( \mathbf { L } ( \gamma ) + \mathbf { I } )$ . However, there is a problem with directly using the DPP log likelihood. The log likelihood heavily penalizes repeated items: if two trajectories inside $\mathcal { V }$ are very similar, their corresponding rows in $\mathbf { L }$ will be almost identical, making $\operatorname* { d e t } ( \mathbf { L } ( \gamma ) ) = \lambda _ { 1 } \lambda _ { 2 } \dots \lambda _ { N } \approx 0$ ( $\lambda _ { n }$ is the $n$ -th eigenvalue). In practice, if the number of modes in the trajectory distribution $p ( \mathbf { x } | \boldsymbol { \psi } )$ is smaller than $| \mathcal { V } |$ , $\mathcal { V }$ will always have similar trajectories, thus making $\operatorname* { d e t } ( \mathbf { L } ( \gamma ) )$ always close to zero. In such cases, optimizing the negative log likelihood causes numerical issues, which is observed in our early experiments.
+
+Instead, the expected cardinality of the DPP is a better measure for the diversity of $\mathcal { V }$ , which is defined as $\mathbb { E } _ { Y \sim \mathcal { P } _ { \mathbf { L } ( \gamma ) } } [ | Y | ]$ . Intuitively, since the DPP discourages selection of similar items, if $\mathcal { V }$ is more diverse, a random subset $\mathbf { Y }$ drawn according to the DPP is more likely to select more items from $\mathcal { V }$ , thus having larger cardinality. The expected cardinality can be computed as (Eq. 15 and 34 in Kulesza et al. (2012)):
+
+$$
+\mathbb { E } [ | \boldsymbol { Y } | ] = \sum _ { n = 1 } ^ { N } \frac { \lambda _ { n } } { \lambda _ { n } + 1 } = \mathrm { t r } \left( \mathbf { I } - ( \mathbf { L } ( \boldsymbol { \gamma } ) + \mathbf { I } ) ^ { - 1 } \right) .
+$$
+
+The main advantage of the expected cardinality is that it is well defined even when the ground set $\mathcal { V }$ has duplicated items, since it does not require all eigenvalues of $\mathbf { L }$ to be non-zero as the log likelihood does. Thus, our diversity loss is defined as $\bar { \mathcal { L } } _ { d i v e r s e } ^ { \bar { } } ( \gamma ) = - \mathrm { t r } \left( \mathbf { I } - ( \mathbf { L } ( \gamma ) + \mathbf { I } ) ^ { - 1 } \right)$ . The training procedure for $S _ { \gamma } ( \psi )$ is outlined in Alg. 1.
+
+Inference. At test time, given current context $\psi$ ,we use the learned DSF $S _ { \gamma } ( \psi )$ to generate the future trajectory ground set $\mathcal { V }$ . In some cases, $\mathcal { V }$ may still contain some trajectories that are similar to others. In order to obtain a diverse set of trajectories without repetition, we aim to perform MAP inference for the DPP to find the most diverse subset $Y ^ { * } = \arg \operatorname* { m a x } _ { Y \in \mathcal { Y } } \mathcal { P } _ { \mathbf { L } ( \gamma ) } ( Y )$ . A useful property of DPPs is that the log-probability function is submodular (Gillenwater et al., 2012). Even though submodular maximization is NP-hard, we use a greedy algorithm (Nemhauser et al., 1978) which is a popular optimization procedure that works well in practice. As outlined in Alg. 3, the output set $Y _ { f }$ is initialized as $\varnothing$ , and at each iteration, the trajectory which maximizes the log probability
+
+$$
+\begin{array} { r } { \mathbf { x } ^ { * } = \underset { \mathbf { x } \in \mathcal { V } \backslash Y _ { f } } { \arg \operatorname* { m a x } } ~ \log \operatorname* { d e t } \left( \mathbf { L } _ { Y _ { f } \cup \{ \mathbf { x } \} } \right) } \end{array}
+$$
+
+is added to $Y _ { f }$ , until the marginal gain becomes negative or $Y _ { f } = y$ .
+
+# 5 EXPERIMENTS
+
+The primary focus of our experiments is to answer the following questions: (1) Are trajectory samples generated with our diversity sampling function more diverse than samples from the cVAE and other baselines? (2) How does our method perform on both balanced and imbalanced data? (3) Is our method general enough to perform well on both low-dimensional and high-dimensional tasks?
+
+Metrics. A problem with trajectory forecasting evaluation is that in real data each context $\psi ^ { ( i ) }$ usually only has one future trajectory $\mathbf { x } ^ { ( i ) }$ , which means we only have one sample from a multi-modal distribution. Let us consider a scenario of three data examples {x(i), ψ(i)}3 as shown in Fig. 2 (red, purple, blue). The contexts (past trajectories) of the three examples are instances of the same trajectory but they are slightly different due to noise. As these three contexts have the same semantic meaning, they should share the future trajectories, e.g., the purple and blue future trajectories are also valid for the red context. If we evaluate each example $( \mathbf { x } ^ { ( i ) } , \psi ^ { ( i ) } )$ only with its own future trajectory $\mathbf { x } ^ { ( i ) }$ , a method can achieve high scores by only forecasting the mode corresponding to $\mathbf { x } ^ { ( i ) }$ and dropping other modes. This is undesirable because we want a good method to capture all modes of the future trajectory distribution, not just a single mode. To allow for multi-modal evaluation, we propose collecting multiple future trajectories for each example by clustering examples with similar contexts. Specifically, we augment each data example $\bar { ( \mathbf { x } ^ { ( i ) } , \psi ^ { ( i ) } ) }$ with a future trajectory set $\mathcal { X } ^ { ( i ) } = \{ \mathbf { x } ^ { ( j ) } | \| \boldsymbol { \psi } ^ { ( j ) } - \boldsymbol { \psi } ^ { ( i ) } \| \leq \varepsilon$ , $j = 1 , \dots , M \}$ and metrics are calculated based on $\chi ^ { ( i ) }$ instead of $\mathbf { x } ^ { ( i ) }$ , i.e., we compute metrics for each $\mathbf { x } \in \mathcal { X } ^ { ( i ) }$ and average the results.
+
+![](images/339718b64625c26a09d2391fc59d2fb39d3ea27dbcb5c3732aa26b7ef6c03c8e.jpg)  
+Figure 2: In real data, contexts (past trajectories) are seldom the same due to noise.
+
+We use the following metrics for evaluation: (1) Average Displacement Error (ADE): average mean square error (MSE) over all time steps between the ground truth future trajectory $\mathbf { x }$ and the closest sample $\tilde { \bf x }$ in the forecasted set of trajectories $Y _ { f }$ . (2) Final Displacement Error (FDE): MSE between the final ground truth position $\mathbf { x } ^ { T }$ and the closest sample’s final position $\tilde { \mathbf { x } } ^ { T }$ . (3) Average Self Distance (ASD): average $L 2$ distance over all time steps between a forecasted sample $\tilde { \mathbf { x } } _ { i }$ and its closest neighbor $\tilde { \mathbf { x } } _ { j }$ in $Y _ { f }$ . (4) Final Self Distance (FSD): $L 2$ distance between the final position of a sample $\tilde { \mathbf { x } } _ { i } ^ { T }$ and its closest neighbor’s final position $\tilde { \mathbf { x } } _ { j } ^ { T }$ . The ADE and FDE are common metrics used in prior work on trajectory forecasting (Alahi et al., 2016; Lee et al., 2017; Rhinehart et al., 2018; Gupta et al., 2018). However, these two metrics do not penalize repeated samples. To address this, we introduce two new metrics ASD and FSD to evaluate the similarity between samples in the set of forecasted trajectories. Larger ASD and FSD means the forecasted trajectories are more non-repetitive and diverse.
+
+Baselines. We compare our Diversity Sampler Function (DSF) with the following baselines: (1) cVAE: a method that follows the original sampling scheme of cVAE by sampling latent codes from a Gaussian prior $p ( \mathbf { z } )$ . (2) MCL: an approach that uses multiple choice learning (Lee et al., 2016) to optimize the sampler $S _ { \gamma } ( \psi )$ with the following loss: $\begin{array} { r } { \mathcal { L } _ { \mathrm { m c l } } \bar { \mathbf { \Psi } } = \operatorname* { m i n } _ { \tilde { \mathbf { x } } \in \mathcal { Y } } \| \tilde { \mathbf { x } } - \mathbf { x } \| ^ { 2 } } \end{array}$ , where $\mathbf { x }$ is the ground truth future trajectory. (3) R2P2: a method proposed in (Rhinehart et al., 2018) that uses a reparametrized pushforward policy to improve modeling of multi-modal distributions for vehicle trajectories. (4) cGAN: generative adversarial networks (Goodfellow et al., 2014) conditioned on the forecasting context. We implement all baselines using similar networks and perform hyperparameter search for each method for fair comparisons. For methods whose samples are stochastic, we use 10 random seeds and report the average results for all metrics.
+
+# 5.1 SYNTHETIC 2D TRAJECTORY DATA
+
+We first use synthetic data to evaluate our method’s performance for low-dimensional tasks. We design a virtual 2D traffic scene where a vehicle comes to a crossroad and can choose three different future routes: forward, left, and right. We consider two types of synthetic data: (1) Balanced data, which means the probabilities of the vehicle choosing one of the three routes are the same; (2)
+
+![](images/e449598c237031babc8a911ea804edb59e2cc40f470548bd63bb814a445de844.jpg)  
+Figure 3: Qualitative results on synthetic data for both balanced and imbalanced data distribution when $N = 1 0$ . Blue represents the past trajectory and red represents forecasted future trajectories.
+
+<table><tr><td></td><td colspan="4">Balanced Data</td><td colspan="4">Imbalanced Data</td></tr><tr><td>Method</td><td>ADE↓</td><td>FDE↓</td><td>ASD ↑</td><td>FSD ↑</td><td>ADE↓</td><td>FDE↓</td><td>ASD ↑</td><td>FSD ↑</td></tr><tr><td>DSF (Ours)</td><td>0.182</td><td>0.344</td><td>0.147</td><td>0.340</td><td>0.198</td><td>0.371</td><td>0.207</td><td>0.470</td></tr><tr><td>cVAE</td><td>0.262</td><td>0.518</td><td>0.022</td><td>0.050</td><td>0.332</td><td>0.662</td><td>0.021</td><td>0.050</td></tr><tr><td>MCL</td><td>0.276</td><td>0.548</td><td>0.012</td><td>0.030</td><td>0.457</td><td>0.938</td><td>0.005</td><td>0.010</td></tr><tr><td>R2P2</td><td>0.211</td><td>0.361</td><td>0.047</td><td>0.080</td><td>0.393</td><td>0.776</td><td>0.019</td><td>0.030</td></tr><tr><td>cGAN</td><td>0.808</td><td>1.619</td><td>0.018</td><td>0.010</td><td>1.784</td><td>3.744</td><td>0.006</td><td>0.001</td></tr></table>
+
+Table 1: Quantitative results on synthetic data (numbers scaled by 10) when $N = 1 0$ .
+
+Imbalanced data, where the probabilities of the vehicle going forward, left and right are 0.8, 0.1, 0.1, respectively. We synthesize trajectory data by simulating the vehicle’s behavior and adding Gaussian noise to vehicle velocities. Each data example $( \mathbf { x } ^ { ( i ) } , \bar { \psi } ^ { ( i ) } )$ contains future trajectories of 3 steps and past trajectories of 2 steps. We also add an obstacle map around the current position to the context $\bar { \psi ^ { ( i ) } }$ . In total, we have around 1100 training examples and 1000 test examples. Please refer to Appendix B for more implementation details.
+
+Table 1 summarizes the quantitative results for both balanced and imbalanced data when the sampling budget $N$ is 10. We can see that our method DSF outperforms the baselines in all metrics in both test settings. As shown in Fig. 3, our method generates more diverse trajectories and is less affected by the imbalanced data distribution. The trajectory samples of our method are also less repetitive, a feature afforded by our DPP formulation. Fig. 4 shows how ADE changes as a function of the sampling budget $N$ .
+
+# 5.2 DIVERSE HUMAN MOTION FORECASTING
+
+To further evaluate our method’s performance for more complex and high-dimensional tasks, we apply our method to forecast future human motions (pose sequences). We use motion capture to obtain 10 motion sequences including different types of motions such as walking, turning, jogging, bending, and crouching. Each sequence is about 1 minute long and each pose consists of 59 joint angles. We use past 3 poses
+
+<table><tr><td>Method</td><td>ADE↓</td><td>FDE↓</td><td>ASD个</td><td>FSD↑</td></tr><tr><td>DSF (Ours)</td><td>0.259</td><td>0.421</td><td>0.115</td><td>0.282</td></tr><tr><td>cVAE</td><td>0.332</td><td>0.642</td><td>0.034</td><td>0.098</td></tr><tr><td>MCL</td><td>0.344</td><td>0.674</td><td>0.036</td><td>0.122</td></tr><tr><td>cGAN</td><td>0.652</td><td>1.296</td><td>0.001</td><td>0.003</td></tr></table>
+
+Table 2: Quantitative results on for human motion forecasting when $N = 1 0$ .
+
+(0.1s) to forecast next 30 poses (1s). There are around 9400 training examples and 2000 test examples where we use different sequences for training and testing. More implementation details can be found in Appendix B.
+
+We present quantitative results in Table 2 and we can see that our approach outperforms other methods in all metrics. As the dynamics model used in R2P2 (Rhinehart et al., 2018) does not generalize well for high-dimensional human motion, we find the model fails to converge and we do not compare with it in this experiment. Fig. 4 shows that our method achieves large improvement when the sampling budget is big $\mathrm { \Delta } N = 5 0 \mathrm { \Omega }$ ). We also present qualitative results in Fig. 5, where we show the starting pose and the final pose of all 10 forecasted motion samples for each method. We can clearly see that our method generates more diverse future human motions than the baselines. Please refer to Appendix C and our video for additional qualitative results.
+
+![](images/ff9fe6900c65b3c7340feda04c9935bdbcf6b74f4b07a3fe374531d9c2ae1f9a.jpg)  
+Figure 4: ADE vs. $N$ for synthetic data and human motion forecasting. cGAN is not shown in this plot as it is much worse than other methods due to mode collapse.
+
+![](images/1df4f19ed6c9ea796a9731ebe271c06940cbbe9aaa4215f855d8629862c36a0c.jpg)  
+Figure 5: Qualitative results for human motion forecasting when $N = 1 0$ . The left shows the starting pose, and the right shows for each method the final pose of all 10 forecasted motion samples.
+
+# 5.3 ADDITIONAL EXPERIMENTS WITH DIVERSITY-BASED BASELINES
+
+In this section, we perform additional experiments on a large human motion dataset (3.6 million frames), Human3.6M (Ionescu et al., 2013), to evaluate the generalization ability of our approach. We predict future motion of 2 seconds based on observed motion of 0.5 seconds. Please refer to Appendix B.3 for implementation details. We also use a new selection of baselines including several variants of our method (DSF) and the cVAE to validate several design choices of our method, including the choice of the expected cardinality over the negative log likelihood (NLL) of the DPP as the diversity loss. Specifically, we use the following new baselines: (1) DSF-NLL: a variant of DSF that uses NLL as the diversity loss instead of the expected cardinality. (2) DSF-COS: a DSF variant that uses cosine similarity to build the similarity matrix S for the DPP kernel L. (3) DSF-NLL: a variant of the cVAE that samples 100 latent codes and performs DPP MAP inference on the latent codes to obtain a diverse set of latent codes, which are then decoded into trajectory samples.
+
+We present quantitative results in Table 3 when the number of samples $N$ is 10 and 50. The baseline DSF-COS is able to achieve very high diversity (ASD and FSD) but its samples are overly diverse and have poor quality which is indicated by the large ADE and FDE. Compared with DSF-NLL, our method achieves better diversity (ASD and FSD) and similar ADE and FDE when the number of samples is small $N = 1 0$ ). For a larger number of samples $\mathrm { ~ N ~ } = 5 0 $ ), NLL becomes unstable even with a large $\epsilon$ (1e-3) added to the diagonal. This behavior of NLL, i.e., stable for small $N$ but unstable for large $N$ , matches our intuition that NLL becomes unstable when samples become similar (as discussed in Sec. 4.2), because when there are more samples, it is easier to have similar samples during the SGD updates of the DSF network. The baseline cVAE-LDPP also performs worse than DSF in all metrics even though it is able to outperfom the cVAE. We believe the reason is that diversity in sample space may not be well reflected in the latent space due to the non-linear mapping from latent codes to samples induced by deep neural networks.
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">N=10</td><td colspan="4">N= 50</td></tr><tr><td>ADE↓</td><td>FDE↓</td><td>ASD↑</td><td>FSD ↑</td><td>ADE↓</td><td>FDE↓</td><td>ASD↑</td><td>FSD ↑</td></tr><tr><td>DSF (Ours)</td><td>0.340</td><td>0.521</td><td>0.381</td><td>0.621</td><td>0.236</td><td>0.306</td><td>0.313</td><td>0.415</td></tr><tr><td>DSF-NLL</td><td>0.335</td><td>0.514</td><td>0.343</td><td>0.496</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>DSF-COS</td><td>2.588</td><td>1.584</td><td>5.093</td><td>5.718</td><td>0.978</td><td>0.891</td><td>2.007</td><td>1.968</td></tr><tr><td>cVAE</td><td>0.363</td><td>0.549</td><td>0.235</td><td>0.360</td><td>0.276</td><td>0.369</td><td>0.160</td><td>0.220</td></tr><tr><td>cVAE-LDPP</td><td>0.373</td><td>0.554</td><td>0.280</td><td>0.426</td><td>0.277</td><td>0.365</td><td>0.176</td><td>0.240</td></tr></table>
+
+Table 3: Quantitative results on Human3.6M (Ionescu et al., 2013) for $N = 1 0$ and $N = 5 0$ . X means the method is unable to learn a model due to numerical instability.
+
+# 6 CONCLUSION
+
+We proposed a novel forecasting approach using a DSF to optimize over the sample space of a generative model. Our method learns the DSF with a DPP-based diversity measure to generate a diverse set of trajectories. The diversity measure is a novel application of DPPs to optimize a set of items in continuous space. Experiments have shown that our approach can generate more diverse vehicle trajectories and human motions compared to state-of-the-art baseline forecasting approaches.
+
+Acknowledgment. This project was sponsored in part by JST CREST (JPMJCR14E1), NSF NRI (1637927) and IARPA (D17PC00340).
+
+# REFERENCES
+
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+
+# A ALGORITHMS
+
+# Algorithm 2 Training the cVAE
+
+1: Input: Training data $\{ \mathbf { x } ^ { ( i ) } , \boldsymbol { \psi } ^ { ( i ) } \} _ { i = 1 } ^ { M }$   
+2: Output: cVAE encoder network $f _ { \phi } (  { \mathbf { x } } , \psi )$ and decoder network $g _ { \boldsymbol { \theta } } ( \mathbf { z } , \boldsymbol { \psi } )$   
+3: Initialize $\phi$ and $\theta$ randomly   
+4: while not converged do   
+5: for each $( \mathbf { x } ^ { ( i ) } , \psi ^ { ( i ) } )$ do   
+6: Compute parameters $( \mu , \sigma )$ of the posterior distribution $q _ { \phi } ( { \bf z } | { \bf x } , \psi )$ using $f _ { \phi } ( \mathbf { x } , \psi )$   
+7: Sample $V$ Gaussian noises $\{ \epsilon _ { 1 } , \ldots , \epsilon _ { V } \}$ from $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$   
+8: Transform noises to latent samples from $q _ { \phi } ( { \bf z } | { \bf x } , \psi )$ : $\mathbf { z } _ { v } = \pmb { \mu } + \pmb { \sigma } \odot \pmb { \epsilon } _ { v }$   
+9: Decode latent samples into reconstructed trajectories $\big \{ \tilde { \mathbf { x } } _ { 1 } , \dots , \tilde { \mathbf { x } } _ { V } \big \}$ using $g _ { \boldsymbol { \theta } } ( \mathbf { z } , \boldsymbol { \psi } )$   
+10: Calculate the cVAE loss $\mathcal { L } _ { c v a e }$ according to Eq. 6   
+11: Update $\phi$ and $\theta$ with $\nabla _ { \phi } \mathcal { L } _ { c v a e }$ and $\nabla _ { \theta } \mathcal { L } _ { c v a e }$   
+12: end for   
+13: end while
+
+Algorithm 3 Inference with the DSF $S _ { \gamma } ( \psi )$
+
+1: Input: Context $\psi$ , DSF $S _ { \gamma } ( \psi )$ , cVAE decoder network $g _ { \boldsymbol { \theta } } ( \mathbf { z } , \boldsymbol { \psi } )$   
+2: Output: Forecasted trajectory set $Y _ { f }$   
+3: Generate latent codes $\mathcal { Z } = \{ \mathbf { z } _ { 1 } , \ldots , \mathbf { z } _ { N } \}$ with the DSF $S _ { \gamma } ( \psi )$   
+4: Generate the trajectory ground set $\mathcal { Y } = \{ \mathbf { x } _ { 1 } , \dotsc , \mathbf { x } _ { N } \}$ with the decoder $g _ { \boldsymbol { \theta } } ( \mathbf { z } , \boldsymbol { \psi } )$   
+5: Compute the DPP kernel $\mathbf { L } = \mathrm { D i a g } ( \mathbf { r } ) \cdot \mathbf { S } \cdot \mathrm { D i a g } ( \mathbf { r } )$   
+6: $Y _ { f } \gets \emptyset , U \gets y$   
+7: while $U$ is not empty do   
+8: $\mathbf { x } ^ { * }  \arg \operatorname* { m a x } _ { \mathbf { x } \in U }$ $\log \operatorname* { d e t } \left( \mathbf { L } _ { Y _ { f } \cup \{ \mathbf { x } \} } \right)$   
+9: $\mathbf { | } \mathbf { f } \log \operatorname* { d e t } \left( \mathbf { L } _ { Y _ { f } \cup \{ \mathbf { x } ^ { * } \} } \right) - \log \operatorname* { d e t } \left( \mathbf { L } _ { Y _ { f } } \right) < 0$ then   
+10: break   
+11: end if   
+12: $\begin{array} { l } { Y _ { f }  Y _ { f } \cup \{ \mathbf { x } ^ { * } \} } \\ { U  U \setminus \{ \mathbf { x } ^ { * } \} } \end{array}$   
+13:   
+14: end while
+
+![](images/32e613102435caf57bf44ee931dab86ab62337b0e3a87d0e948258d656b47358.jpg)  
+Figure 6: Network architectures for synthetic data and human motion. Top: for synthetic data, we use a CNN to process the obstacle map f and directly flatten trajectories $\mathbf { x }$ and $\mathbf { h }$ into vectors. The reconstructed trajectory $\tilde { \bf x }$ is decoded with an MLP. Bottom: for human motion, we use Bi-LSTMs to extract temporal features for $\mathbf { x }$ and $\mathbf { h }$ and decode the reconstructed trajectory $\tilde { \bf x }$ with a forward LSTM.
+
+# B.1 NETWORK ARCHITECTURES
+
+Synthetic data. Fig. 6 (Top) shows the network architecture for synthetic data. The number of latent dimensions is 2. By default, we use ReLU activation for all networks. The future trajectory $\mathbf { x } \in \mathbb { R } ^ { 3 \times 2 }$ consists of 3 future positions of the vehicle. The context $\psi$ contains past trajectories $\textbf { h } \in \mathbb { R } ^ { 2 \times 2 }$ of 2 time steps and a obstacle map $\textbf { f } \in \{ 0 , 1 \} ^ { 2 8 \times 2 8 }$ spanning a $4 \times 4$ area around the current position of the vehicle (the road width is 2). For the encoder, we use a convolutional neural network (CNN) with three 32-channel convolutional layers to process f. The first two layers have kernel size 4 and stride 2 while the last layer has kernel size 6 and stride 1. The obtained CNN features are concatenated with flattened $\mathbf { x }$ and $\mathbf { h }$ into a unified feature, which is feed into a multilayer perceptron (MLP). The MLP has one 128-dim hidden layer and two heads outputing the mean $\pmb { \mu }$ and variance $\sigma$ of the latent distribution. For the decoder, we concatenate the CNN feature from f with the latent code $\textbf { z } \in \mathbb { R } ^ { 2 }$ and flattened $\mathbf { h }$ into a unified feature. The feature is passed through an MLP with one 128-dim hidden layer which outputs the reconstructed future trajectory $\tilde { \mathbf { x } } \in \breve { \mathbb { R } } ^ { 3 \times 2 }$ . For the diversity sampler function (DSF), we concatenate the CNN feature from f with the flattened h and pass it through an MLP with one 128-dim hidden layer to obtain a set of latent codes $\left\{ \mathbf { z } _ { 1 } , \ldots , \mathbf { z } _ { N } \right\}$ which are represented by a vector of length $2 N$ .
+
+Human motion. Fig. 6 (Bottom) shows the network architecture for synthetic data. The number of latent dimensions is 8. The future trajectory $\mathbf { x } \in \mathbb { R } ^ { 3 0 \times 5 9 }$ consists of future poses of 30 time steps (1s). The context $\psi$ contains past poses $\mathbf { h } \in \mathbb { R } ^ { 3 \times 5 9 }$ of 3 time steps (0.1s). Each pose consists of 59 joint angles. For the encoder, we use two 128-dim bidirectional LSTMs (Bi-LSTMs) and mean pooling to obtain the temporal features for $\mathbf { x }$ and $\mathbf { h }$ . We then concatenate the temporal features into a unified feature and feed it into an MLP with two hidden layers (300, 200) and two heads to obtain the mean $\pmb { \mu }$ and variance $\sigma$ of the latent distribution. For the decoder, we reuse the Bi-LSTM of the encoder for the context h and a 128-dim forward LSTM to decode the future trajectory $\tilde { \bf x }$ . At each time step $t$ , the forward LSTM takes as input the previous pose $\tilde { \mathbf { x } } ^ { t - 1 }$ $\mathbf { h } ^ { H }$ for $t = 0$ ), the latent code $\mathbf { z } \in \mathbb { R } ^ { 8 }$ and the temporal features from h, and outputs a 128-dim feature. The feature is then passed through an MLP with two hidden layers (300, 200) to generate the reconstructed pose $\tilde { \mathbf { x } } ^ { t }$ . For the DSF, we use a different 128-dim Bi-LSTM to obtain the temporal feature for $\mathbf { h }$ , which is feed into an MLP with a 128-dim hidden layer to produce a set of latent codes $\left\{ \mathbf { z } _ { 1 } , \ldots , \mathbf { z } _ { N } \right\}$ which are represented by a vector of length $8 N$ .
+
+# B.2 TRAINING AND EVALUATION
+
+When training the cVAE model using Eq. 7, we take $V = 1$ sample from the posterior $q _ { \phi } ( \mathbf { z } | \mathbf { x } , \psi )$ . The weighting factor $\beta$ for the KL term is set to 0.1 for synthetic data and 1e-4 for human motion. We use Adam (Kingma and Ba, 2014) to jointly optimize the encoder and decoder. The learning rate is set to 1e-4 and we use a mini batch size of 32 for synthetic data. We optimize the model for 500 epochs for synthetic data and 100 epochs for human motion.
+
+When training the DSF, the scale factor $k$ for the similarity matrix S is set to 1 for synthetic data and 1e-2 for human motions. For both synthetic data and human motions, we use Adam with learning rate 1e-4 to optimize the DSF for 20 epochs.
+
+Recall that in the metrics section (Sec. 5.1), we need the grouping threshold $\varepsilon$ to build the ground truth future trajectory set $\mathcal { X } ^ { ( i ) } = \{ \mathbf { x } ^ { ( j ) } \vert \Vert \psi ^ { ( j ) } - \psi ^ { ( i ) } \Vert \leq \varepsilon$ , $j = 1 , \dots , M \}$ . For synthetic data, $\varepsilon$ is set to 0.1 and we only use past trajectories $\mathbf { h }$ to compute the distance between contexts. For human motion, $\varepsilon$ is set to 0.5.
+
+B.3 IMPLEMENTATION DETAILS FOR EXPERIMENTS ON HUMAN3.6M
+
+Following previous work (Martinez et al., 2017; Pavlakos et al., 2017; Pavllo et al., 2019), we convert the motion sequences in the dataset into sequences of 3D joint positions, and adopt a 17-joint skeleton. We train on five subjects (S1, S5, S6, S7, S8), and test on two subjects (S9 and S11).
+
+We use the same network architectures (Fig.6 (Bottom)) in this experiment as the one used in the human motion forecasting experiment above. The number of latent dimensions is 128. When training the cVAE model, the weighting factor $\beta$ is set to 0.1. We sample 5000 training examples every epoch and optimize the cVAE for 500 epochs using Adam and a learning rate of 1e-4. We set the batch size to 64 for the optimization.
+
+The scale factor $k$ for the similarity matrix S of the DPP kernel is set to 5. When learning the DSF, we use a batch size of 64 and sample 1000 training examples every epoch and optimize the DSF for 20 epochs using Adam and a learning rate of 1e-3.
+
+When computing the metrics, we set the grouping threshold $\varepsilon$ to 0.1.
+
+# C ADDITIONAL VISUALIZATION
+
+We also show additional qualitative results for human motion forecasting in Fig. 7. The quality and diversity of the forecasted motions are best seen in our video2.
+
+# The final pose of 10 forecasted motions
+
+![](images/f3e399ae3bc56208ba1dccca639da93a1d6e9a7fe8064fe2b76e19f24f16ec80.jpg)  
+Figure 7: Additional visualization for human motion forecasting. The left shows the starting pose, and on the right we show for each method the final pose of 10 forecasted motion samples.

parse/train/ryxnY3NYPS/ryxnY3NYPS_content_list.json

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+    {
+        "type": "text",
+        "text": "DIVERSE TRAJECTORY FORECASTING WITH DETERMINANTAL POINT PROCESSES ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Ye Yuan, Kris M. Kitani Robotics Institute Carnegie Mellon University {yyuan2,kkitani}@cs.cmu.edu ",
+        "bbox": [
+            183,
+            170,
+            447,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The ability to forecast a set of likely yet diverse possible future behaviors of an agent (e.g., future trajectories of a pedestrian) is essential for safety-critical perception systems (e.g., autonomous vehicles). In particular, a set of possible future behaviors generated by the system must be diverse to account for all possible outcomes in order to take necessary safety precautions. It is not sufficient to maintain a set of the most likely future outcomes because the set may only contain perturbations of a dominating single outcome (major mode). While generative models such as variational autoencoders (VAEs) have been shown to be a powerful tool for learning a distribution over future trajectories, randomly drawn samples from the learned implicit likelihood model may not be diverse – the likelihood model is derived from the training data distribution and the samples will concentrate around the major mode of the data. In this work, we propose to learn a diversity sampling function (DSF) that generates a diverse yet likely set of future trajectories. The DSF maps forecasting context features to a set of latent codes which can be decoded by a generative model (e.g., VAE) into a set of diverse trajectory samples. Concretely, the process of identifying the diverse set of samples is posed as DSF parameter estimation. To learn the parameters of the DSF, the diversity of the trajectory samples is evaluated by a diversity loss based on a determinantal point process (DPP). Gradient descent is performed over the DSF parameters, which in turn moves the latent codes of the sample set to find an optimal set of diverse yet likely trajectories. Our method is a novel application of DPPs to optimize a set of items (forecasted trajectories) in continuous space. We demonstrate the diversity of the trajectories produced by our approach on both low-dimensional 2D trajectory data and high-dimensional human motion data. (Video1) ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+            176,
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+            336,
+            651
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Forecasting future trajectories of vehicles and human has many useful applications in autonomous driving, virtual reality and assistive living. What makes trajectory forecasting challenging is that the future is uncertain and multi-modal – vehicles can choose different routes and people can perform different future actions. In many safety-critical applications, it is important to consider a diverse set of possible future trajectories, even those that are less likely, so that necessary preemptive actions can be taken. For example, an autonomous vehicle should understand that a neighboring car can merge into its lane even though the car is most likely to keep driving straight. To address this requirement, we need to take a generative approach to trajectory forecasting that can fully characterize the multimodal distribution of future trajectories. To capture all modes of a data distribution, variational autoencoders (VAEs) are well-suited generative models. However, random samples from a learned VAE model with Gaussian latent codes are not guaranteed to be diverse for two reasons. First, the sampling procedure is stochastic and the VAE samples can fail to cover some minor modes even with a large number of samples. Second, since VAE sampling is based on the implicit likelihood function encoded in the training data, if most of the training data is centered around a specific mode while other modes have less data (Fig. 1 (a)), the VAE samples will reflect this bias and concentrate around the major mode (Fig. 1 (b)). To tackle this problem, we propose to learn a diversity sampling function (DSF) that can reliably generate a diverse set of trajectory samples (Fig. 1 (c)). ",
+        "bbox": [
+            174,
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/935dc525723c5a63173fac98ebc77b977ebc640ac583ddbd40861f52edb857bf.jpg",
+        "image_caption": [
+            "Figure 1: A toy trajectory forecasting example. (a) The three modes (pink, blue, purple) of the future trajectory distribution are shown in both the trajectory space and the latent space of a learned VAE model. The data distribution is imbalanced, where the blue mode has most data and covers most of the latent space. (b) Random samples from the VAE only cover the major (blue) mode. (c) Our proposed DSF generates a diverse set of future trajectories covering all three modes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "The proposed DSF is a deterministic parameterized function that maps forecasting context features (e.g., past trajectories) to a set of latent codes. The latent codes are decoded by the VAE docoder into a set of future trajectory samples, denoted as the DSF samples. In order to optimize the DSF, we formulate a diversity loss based on a determinantal point process (DPP) (Macchi, 1975) to evaluate the diversity of the DSF samples. The DPP defines the probability of choosing a random subset from the set of trajectory samples. It models the negative correlations between samples: the inclusion of a sample reduces the probability of including a similar sample. This makes the DPP an ideal tool for modeling the diversity within a set. In particular, we use the expected cardinality of the DPP as the diversity measure, which is defined as the expected size of a random subset drawn from the set of trajectory samples according to the DPP. Intuitively, since the DPP inhibits selection of similar samples, if the set of trajectory samples is more diverse, the random subset is more likely to select more samples from the set. The expected cardinality of the DPP is easy to compute and differentiable, which allows us to use it as the objective to optimize the DSF to enable diverse trajectory sampling. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are as follows: (1) We propose a new forecasting approach that learns a diversity sampling function to produce a diverse set of future trajectories; (2) We propose a novel application of DPPs to optimize a set of items (trajectories) in continuous space with a DPP-based diversity measure; (3) Experiments on synthetic data and human motion validate that our method can reliably generate a more diverse set of future trajectories compared to state-of-the-art generative models. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Trajectory Forecasting has recently received significant attention from the vision community. A large portion of previous work focuses on forecasting 2D future trajectories for pedestrians (Kitani et al., 2012; Ma et al., 2017; Ballan et al., 2016; Xie et al., 2013) or vehicles (Jain et al., 2016a). Some approaches use deterministic trajectory modeling and only forecast one future trajectory (Alahi et al., 2016; Yagi et al., 2018; Robicquet et al., 2016). As there are often multiple plausible future trajectories, several approaches have tried to forecast distributions over trajectories (Lee et al., 2017; Galceran et al., 2015; Gupta et al., 2018). Recently, Rhinehart et al. (2018; 2019) propose a generative model that can accurately forecast multi-modal trajectories for vehicles. Soo Park et al. (2016) also use egocentric videos to predict the future trajectories of the camera wearer. Some work has investigated forecasting higher dimensional trajectories such as the 3D fullbody pose sequence of human motions. Most existing work takes a deterministic approach and forecasts only one possible future motion from past 3D poses (Fragkiadaki et al., 2015; Butepage et al., 2017; Li et al., 2017; Jain et al., 2016b), static images (Chao et al., 2017; Kanazawa et al., 2018) or egocentric videos (Yuan and Kitani, 2019). Differently, some probabilistic approaches (Habibie et al., 2017; Yan et al., 2018) use conditional variational autoencoders (cVAEs) to generate multiple future motions. In constrast to previous work, our approach can generate a diverse set of future motions with a limited number of samples. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Diverse Solutions have been sought after in a number of problems in computer vision and machine learning. A branch of these methods aiming for diversity stems from the M-Best MAP problem (Nilsson, 1998; Seroussi and Golmard, 1994), including diverse M-Best solutions (Batra et al., 2012) and multiple choice learning (Guzman-Rivera et al., 2012; Lee et al., 2016). Alternatively, previous work has used submodular function maximization to select a diverse subset of garments from fashion images (Hsiao and Grauman, 2018). Determinantal point processes (DPPs) (Macchi, 1975; Kulesza et al., 2012) are efficient probabilistic models that can measure both the diversity and quality of items in a subset, which makes it a natural choice for the diverse subset selection problem. DPPs have been applied for document and video summarization (Kulesza and Taskar, 2011; Gong et al., 2014), recommendation systems (Gillenwater et al., 2014), object detection (Azadi et al., 2017), and grasp clustering (Huang et al., 2015). Elfeki et al. (2018) have also used DPPs to mitigate mode collapse in generative adversarial networks (GANs). The work most related ours is (Gillenwater et al., 2014), which also uses the cardinality of DPPs as a proxy for user engagement. However, there are two important differences between our approach and theirs. First, the context is different as they use the cardinality for a subset selection problem while we apply the cardinality as an objective of a continuous optimization problem in the setting of generative models. Second, their main motivation behind using the cardinality is that it aligns better with the user engagement semantics, while our motivation is that using cardinality as a diversity loss for deep neural networks is more stable due to its tolerance of similar trajectories, which are often produced by deep neural networks during stochastic gradient descent. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 VARIATIONAL AUTOENCODERS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The aim of multi-modal trajectory forecasting is to learn a generative model over future trajectories. Variational autoencoders (VAEs) are a popular choice of generative models for trajectory forecasting (Lee et al., 2017; Walker et al., 2016) because it can effectively capture all possible future trajectories by explicitly mapping each data point to a latent code. VAEs model the joint distribution $p _ { \\theta } \\mathbf { \\bar { ( x , z ) } } = p \\mathbf { \\bar { ( z ) } } \\bar { p _ { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ of each data sample $\\mathbf { x }$ (e.g., a future trajectory) and its corresponding latent code $\\mathbf { z }$ , where $p ( \\mathbf { z } )$ denotes some prior distribution (e.g., Gaussians) and $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ denotes the conditional likelihood model. To calculate the marginal likelihood $p _ { \\theta } ( \\mathbf { x } ) = p _ { \\theta } ( \\mathbf { x } , \\mathbf { z } ) / p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } )$ , one needs to compute the posterior distribution $p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } )$ which is typically intractable. To tackle this, VAEs use variational inference (Jordan et al., 1999) which introduces an approximate posterior $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ and decomposes the marginal log-likelihood as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ec1fbf517bc6fb24c237e11ac6d9e27fd8870f4bf5e8d716fd8e9f4bd8567a7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\log p _ { \\theta } ( \\mathbf { x } ) = \\mathrm { K L } \\left( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) \\| p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } ) \\right) + \\mathcal { L } ( \\mathbf { x } ; \\theta , \\phi ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } ( \\mathbf { x } ; \\theta , \\phi )$ is the evidence lower bound (ELBO) defined as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ff419cc69a05b6afdccae25e1cae76f5921fa218fa0915f9e3bb50886f69f825.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\mathbf { x } ; \\theta , \\phi ) = \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } \\left[ \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) \\right] - \\mathrm { K L } \\left( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) \\| p ( \\mathbf { z } ) \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "During training, VAEs jointly optimize the recognition model (encoder) $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ and the likelihood model (decoder) $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ to maximize the ELBO. In the context of multi-modal trajectory forecasting, one can generate future trajectories from $p ( \\mathbf { x } )$ by drawing a latent code $\\mathbf { z }$ from the prior $p ( \\mathbf { z } )$ and decoding $\\mathbf { z }$ with the decoder $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ to produce a corresponding future trajectory $\\mathbf { x }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 DETERMINANTAL POINT PROCESSES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our core technical innovation is a method to learn a diversity sampling function (DSF) that can generate a diverse set of future trajectories. To achieve this, we must equip ourselves with a tool to evaluate the diversity of a set of trajectories. To this end, we make use of determinantal point processes (DPPs) to model the diversity within a set. DPPs promote diversity within a set because the inclusion of one item makes the inclusion of a similar item less likely if the set is sampled according to a DPP. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Formally, given a set of items (e.g., data points) $\\mathcal { Y } = \\{ \\mathbf { x } _ { 1 } , \\dotsc , \\mathbf { x } _ { N } \\}$ , a point process $\\mathcal { P }$ on the ground set $\\mathcal { V }$ is a probability measure on $2 ^ { y }$ , i.e., the set of all subsets of $\\mathcal { V }$ . $\\mathcal { P }$ is called a determinantal point process if a random subset $\\mathbf { Y }$ drawn according to $\\mathcal { P }$ has ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6533b73068af982a9307337becd886458b84788ca77aa30f20b2e1451b062e92.jpg",
+        "text": "$$\n\\mathcal { P } _ { \\mathbf { L } } ( Y = Y ) = \\frac { \\operatorname* { d e t } \\left( \\mathbf { L } _ { Y } \\right) } { \\sum _ { Y \\subseteq \\mathcal { V } } \\operatorname* { d e t } \\left( \\mathbf { L } _ { Y } \\right) } = \\frac { \\operatorname* { d e t } \\left( \\mathbf { L } _ { Y } \\right) } { \\operatorname* { d e t } ( \\mathbf { L } + \\mathbf { I } ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $Y \\subseteq \\mathcal { V }$ , I is the identity matrix, $\\mathbf { L } \\in \\mathbb { R } ^ { N \\times N }$ is the DPP kernel, a symmetric positive semidefinite matrix, and $\\mathbf { L } _ { Y } \\in \\mathbb { R } ^ { | Y | \\times | Y | }$ is a submatrix of $\\mathbf { L }$ indexed by elements of $Y$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The DPP kernel $\\mathbf { L }$ is typically constructed by a similarity matrix S, where $\\mathbf { S } _ { i j }$ defines the similarity between two items $\\mathbf { x } _ { i }$ and $\\mathbf { x } _ { j }$ . If we use the inner product as the similarity measure, $\\mathbf { L }$ can be written in the form of a Gram matrix $\\mathbf { L } = \\mathbf { S } = \\mathbf { X } ^ { T } \\mathbf { X }$ where $\\mathbf { X }$ is the stacked feature matrix of $\\mathcal { V }$ . As a property of the Gram matrix, det $\\left( { \\bf L } _ { Y } \\right)$ equals the squared volume spanned by vectors $\\mathbf { x } _ { i } \\in Y$ . With this geometric interpretation in mind, one can observe that diverse sets are more probable because their features are more orthogonal, thus spanning a larger volume. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In addition to set diversity encoded in the similarity matrix S, it is also convenient to introduce a quality vector $\\mathbf { r } = [ r _ { 1 } , \\dots , r _ { N } ]$ to weigh each item according to some unary metric. For example, the quality weight might be derived from the likelihood of an item. To capture both diversity and quality of a subset, the DPP kernel $\\mathbf { L }$ is often decomposed in the more general form: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b7b4bb9fe3b12a9903ff005c2d686b7b8d451754ddadce27e42c96ce8efa217d.jpg",
+        "text": "$$\n\\mathbf { L } = \\mathrm { D i a g } ( \\mathbf { r } ) \\cdot \\mathbf { S } \\cdot \\mathrm { D i a g } ( \\mathbf { r } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With this decomposition, we can see that both the quality vector $\\mathbf { r }$ and similarity matrix S contribute to the DPP probability of a subset $Y$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/238e44358b528cd8cc2b7d6fc9ea97821a020fdc15eda85e3587f8ec418bd058.jpg",
+        "text": "$$\n\\mathcal { P } _ { L } ( \\pmb { Y } = Y ) \\propto \\operatorname* { d e t } \\left( \\mathbf { L } _ { Y } \\right) = \\left( \\prod _ { \\mathbf { x } _ { i } \\in Y } r _ { i } ^ { 2 } \\right) \\operatorname* { d e t } \\left( \\mathbf { S } _ { Y } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Due to its ability to capture the global diversity and quality of a set of items, we choose DPPs as the probabilistic approach to evaluate and optimize the diversity of the future trajectories drawn by our proposed diversity sampling function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 APPROACH",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Safety-critical applications often require that the system can maintain a diverse set of outcomes covering all modes of a predictive distribution and not just the most likely one. To address this requirement, we propose to learn a diversity sampling function (DSF) to draw deterministic trajectory samples by generating a set of latent codes in the latent space of a conditional variational autoencoder (cVAE) and decoding them into trajectories using the cVAE decoder. The DSF trajectory samples are evaluated with a DPP-based diversity loss, which in turn optimizes the parameters of the DSF for more diverse trajectory samples. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Formally, the future trajectory $\\mathbf { x } \\in \\mathbb { R } ^ { T \\times D }$ is a random variable denoting a $D$ dimensional feature over a future time horizon $T$ (e.g., a vehicle trajectory or a sequence of humanoid poses). The context $\\boldsymbol { \\psi } = \\{ { \\bf h } , { \\bf f } \\}$ provides the information to infer the future trajectory $\\mathbf { x }$ , and it contains the past trajectory $\\mathbf { \\bar { h } } \\in \\mathbf { \\mathbb { R } } ^ { \\mathbf { \\bar { H } } \\times D }$ of last $H$ time steps and optionally other side information f , such as an obstacle map. In the following, we first describe how we learn the future trajectory model $p _ { \\theta } ( \\mathbf { x } | \\psi )$ with a cVAE. Then, we introduce the DSF and the DPP-based diversity loss used to optimize the DSF. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1 LEARNING A CVAE FOR FUTURE TRAJECTORIES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to generate a diverse set of future trajectory samples, we need to learn a generative trajectory forecasting model $p _ { \\theta } ( \\mathbf { x } | \\psi )$ that can cover all modes of the data distribution. Here we use cVAEs (other proper generative models can also be used), which explicitly map data $\\mathbf { x }$ with the encoder $q _ { \\phi } ( { \\bf z } | { \\bf x } , \\psi )$ to its corresponding latent code $\\mathbf { z }$ and reconstruct the data from the latent code using the decoder $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } , \\boldsymbol { \\psi } )$ . By maintaining this one-on-one mapping between the data and the latent code, cVAEs attempt to capture all modes of the data. As discussed in Sec. 3.1, cVAEs jointly optimize the encoder and decoder to maximize the variational lower bound: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/2f3ef2260cbdd96087b092432902ccc8261b3c989b20dc1bb1142b8ea1f7c6fd.jpg",
+        "text": "$$\n\\mathcal { L } ( \\mathbf { x } , \\psi ; \\theta , \\phi ) = \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } , \\psi ) } \\left[ \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\psi ) \\right] - \\mathrm { K L } \\left( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } , \\psi ) \\| p ( \\mathbf { z } ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use multivariate Gaussians for the prior, encoder and decoder: $p ( \\mathbf { z } ) = \\mathcal { N } ( \\mathbf { z } ; \\mathbf { 0 } , \\mathbf { I } )$ , $q _ { \\phi } ( { \\bf z } | { \\bf x } , \\psi ) =$ $\\mathcal { N } ( \\mathbf { z } ; \\pmb { \\mu } , \\pmb { \\sigma } ^ { 2 } \\mathbf { I } )$ , and $p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\psi ) = \\mathcal { N } ( \\mathbf { x } ; \\tilde { \\mathbf { x } } , \\alpha \\mathbf { I } )$ . Both the encoder and decoder are implemented as neural networks. The encoder network $f _ { \\phi }$ outputs the parameters of the posterior distribution: $( \\mu , \\sigma ) = f _ { \\phi } ( { \\bf x } , \\psi )$ . The decoder network $g _ { \\theta }$ outputs the reconstructed future trajectory $\\tilde { \\bf x }$ : ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$\\tilde { \\mathbf { x } } = g _ { \\theta } ( \\mathbf { z } , \\psi )$ . Detailed network architectures are given in Appendix B.1. Based on the Gaussian parameterization of the cVAE, the objective in Eq. 6 can be rewritten as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1490d3af934f10fe629455d1671b34d7fab50bb6685642fca23670463012316e.jpg",
+        "text": "$$\n\\mathcal { L } _ { c v a e } ( \\mathbf { x } , \\psi ; \\theta , \\phi ) = - \\frac { 1 } { V } \\sum _ { v = 1 } ^ { V } \\Vert \\tilde { \\mathbf { x } } _ { v } - \\mathbf { x } \\Vert ^ { 2 } + \\beta \\cdot \\frac { 1 } { D _ { z } } \\sum _ { j = 1 } ^ { D _ { z } } \\left( 1 + 2 \\log \\sigma _ { j } - \\mu _ { j } ^ { 2 } - \\sigma _ { j } ^ { 2 } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we take $V$ samples from the posterior $q _ { \\phi } ( { \\bf z } | { \\bf x } , \\psi )$ , $D _ { z }$ is the number of latent dimensions, and $\\beta = 1 / \\alpha$ is a weighting factor. The training procedure for the $\\operatorname { c V A E }$ is detailed in Alg. 2 (Appendix A). Once the cVAE model is trained, sampling from the learned future trajectory model $p _ { \\theta } ( \\mathbf { x } | \\psi )$ is efficient: we can sample a latent code $\\mathbf { z }$ according to the prior $p ( \\mathbf { z } )$ and use the decoder $p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\psi )$ to decode it into a future trajectory $\\mathbf { x }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Training the diversity sampling function (DSF) $S _ { \\gamma } ( \\psi )$ ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1: Input: Training data $\\{ \\mathbf { x } ^ { ( i ) } , \\boldsymbol { \\psi } ^ { ( i ) } \\} _ { i = 1 } ^ { M }$ , cVAE decoder network $g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\boldsymbol { \\psi } )$   \n2: Output: DSF $S _ { \\gamma } ( \\psi )$   \n3: Initialize $\\gamma$ randomly   \n4: while not converged do   \n5: for each $\\psi ^ { ( i ) }$ do   \n6: Generate latent codes $\\mathcal { Z } = \\{ \\mathbf { z } _ { 1 } , \\ldots , \\mathbf { z } _ { N } \\}$ with the DSF $S _ { \\gamma } ( \\psi )$   \n7: Generate the trajectory ground set $\\mathcal { Y } = \\{ \\mathbf { x } _ { 1 } , \\dotsc , \\mathbf { x } _ { N } \\}$ with the decoder $g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\boldsymbol { \\psi } )$   \n8: Compute the similarity matrix S and quality vector r with Eq. 8 and 9   \n9: Compute the DPP kernel $\\mathbf { L } ( \\gamma ) = \\mathrm { D i a g } ( \\mathbf { r } ) \\cdot \\mathbf { S } \\cdot \\mathrm { D i a g } ( \\mathbf { r } )$   \n10: Calculate the diversity loss Ldiverse   \n11: Update $\\gamma$ with the gradient ∇Ldiverse   \n12: end for   \n13: end while ",
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+    {
+        "type": "text",
+        "text": "4.2 DIVERSITY SAMPLING FUNCTION (DSF) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As mentioned before, randomly sampling from the learned cVAE model according to the implicit likelihood function $p _ { \\theta } ( \\mathbf { x } | \\psi )$ , i.e., sampling latent codes from the prior $p ( \\mathbf { z } )$ , does not guarantee that the trajectory samples are diverse: major modes (those having more data) with higher likelihood will produce most of the samples while minor modes with lower likelihood will have almost no sample. This prompts us to devise a new sampling strategy that can reliably generate a diverse set of samples covering both major and minor modes. We propose to learn a diversity sampling function (DSF) $S _ { \\gamma } ( \\psi )$ that maps context $\\psi$ to a set of latent codes $\\mathcal { Z } = \\{ \\mathbf { z } _ { 1 } , \\ldots , \\mathbf { z } _ { N } \\}$ . The DSF is implemented as a $\\gamma$ -parameterized neural network which takes $\\psi$ as input and outputs a vector of length $N \\cdot D _ { z }$ (see Appendix B.1 for network details). The latent codes $\\mathcal { Z }$ are decoded into a diverse set of future trajectories $\\mathcal { Y } = \\{ \\mathbf { x } _ { 1 } , \\dotsc , \\mathbf { x } _ { N } \\}$ , which are denoted as the DSF trajectory samples. We note that $N$ is the sampling budget. To solve for the parameters of the DSF, we propose a diversity loss based on a DPP defined over $\\mathcal { V }$ . In this section, we first describe how the DPP kernel $\\mathbf { L }$ is defined, which involves the construction of the similarity matrix S and quality vector $\\mathbf { r }$ . We then discuss how we use the DPP kernel $\\mathbf { L }$ to formulate a diversity loss to optimize the parameters of the DSF. ",
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+        "type": "text",
+        "text": "Recall that the DPP kernel is defined as $\\mathbf { L } = \\mathrm { D i a g } ( \\mathbf { r } ) \\cdot \\mathbf { S } \\cdot \\mathrm { D i a g } ( \\mathbf { r } )$ , where $\\mathbf { r }$ defines the quality of each trajectory and $\\mathbf { S }$ measures the similarity between two trajectories. The DPP kernel $\\mathbf { L } ( \\gamma )$ is a function of $\\gamma$ as it is defined over the ground set $\\mathcal { V }$ output by the DSF $S _ { \\gamma } ( \\psi )$ . ",
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+        "text": "Similarity. We measure the similarity $\\mathbf { S } _ { i j }$ between two trajectories $\\mathbf { x } _ { i }$ and $\\mathbf { x } _ { j }$ as ",
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+        "text": "$$\n\\mathbf { S } _ { i j } = \\exp \\left( - k \\cdot d _ { \\mathbf { x } } ^ { 2 } ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) \\right) ,\n$$",
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+        "type": "text",
+        "text": "where $d _ { \\mathbf { x } }$ is the Euclidean distance and $k$ is a scaling factor. This similarity design ensures that $0 \\leq \\mathbf { S } _ { i j } \\leq 1$ and $\\mathbf { S } _ { i i } = 1$ . It also makes $\\mathbf { S }$ positive definite since the Gaussian kernel we use is a positive definite kernel. ",
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+        "type": "text",
+        "text": "Quality. It may be tempting to use $p ( \\mathbf { x } | \\boldsymbol { \\psi } )$ to define the quality of each trajectory sample. However, this likelihood-based measure will clearly favor major modes that have higher probabilities, making it less likely to generate samples from minor modes. This motivates us to design a quality metric that ",
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+        "type": "text",
+        "text": "treats all modes equally. To this end, unlike the similarity metric which is defined in the trajectory space, the quality of each sample is measured in the latent space and is defined as ",
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+    {
+        "type": "equation",
+        "img_path": "images/ad9d27827398aaade5f0221be67f37a4bd7751c0610862e7740a0033568d703d.jpg",
+        "text": "$$\nr _ { i } = \\left\\{ \\begin{array} { l l } { \\omega , } & { \\mathrm { i f ~ } \\| \\mathbf { z } _ { i } \\| \\leq R } \\\\ { \\omega \\exp \\left( - \\mathbf { z } _ { i } ^ { T } \\mathbf { z } _ { i } + R ^ { 2 } \\right) , } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "Geometrically, let $R$ be the radius of a sphere $\\Phi$ containing most samples from the Gaussian prior $p ( \\mathbf { z } )$ . We treat samples inside $\\Phi$ equally and only penalize samples outside $\\Phi$ . In this way, samples from major modes are not preferred over those from minor modes as long as they are inside $\\Phi$ , while samples far away from the data manifold are heavily penalized as they are outside $\\Phi$ . The radius $R$ is determined by where $\\rho$ percent of the Gaussian samples lie within, and we set $\\rho = 9 0$ . To compute $R$ , we use the percentage point function of the chi-squared distribution which models the distribution over the sum of squares of independent standard normal variables. The base quality $\\omega$ is a hyperparameter which we set to 1 during training in our experiments. At test time, we can use a larger $\\omega$ to encourage the DPP to select more items from the ground set $\\mathcal { V }$ . The hyperparameter $\\rho$ (or $R$ ) allows for the trade-off between diversity and quality. When $R$ is small, the quality metric is reduced to a pure likelihood-based metric (proportional to the latent likelihood), which will prefer samples with high likelihood and result in a less diverse sample set. When $R$ is large, most samples will have the same quality, and the resulting samples will be highly diverse but less likely. In practice, the choice of $R$ should be application dependent, as one could imagine autonomous vehicles would need to consider more diverse scenarios including those less likely ones to ensure robustness. We note that after the diverse samples are obtained, it is possible to reassign the quality score for each sample based on its likelihood to allow users to prioritize more likely samples. ",
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+        "type": "text",
+        "text": "Diversity Loss. To optimize the DSF $S _ { \\gamma } ( \\psi )$ , we need to define a diversity loss that measures the diversity of the trajectory ground set $\\mathcal { V }$ generated by $S _ { \\gamma } ( \\psi )$ . An obvious choice for the diversity loss would be the negative log likelihood $- \\log \\mathcal { P } _ { \\mathbf { L } ( \\gamma ) } ( \\boldsymbol { Y } = \\mathcal { Y } ) = - \\log \\operatorname* { d e t } ( \\mathbf { L } ( \\gamma ) ) + \\log \\operatorname* { d e t } ( \\mathbf { L } ( \\gamma ) + \\mathbf { I } )$ . However, there is a problem with directly using the DPP log likelihood. The log likelihood heavily penalizes repeated items: if two trajectories inside $\\mathcal { V }$ are very similar, their corresponding rows in $\\mathbf { L }$ will be almost identical, making $\\operatorname* { d e t } ( \\mathbf { L } ( \\gamma ) ) = \\lambda _ { 1 } \\lambda _ { 2 } \\dots \\lambda _ { N } \\approx 0$ ( $\\lambda _ { n }$ is the $n$ -th eigenvalue). In practice, if the number of modes in the trajectory distribution $p ( \\mathbf { x } | \\boldsymbol { \\psi } )$ is smaller than $| \\mathcal { V } |$ , $\\mathcal { V }$ will always have similar trajectories, thus making $\\operatorname* { d e t } ( \\mathbf { L } ( \\gamma ) )$ always close to zero. In such cases, optimizing the negative log likelihood causes numerical issues, which is observed in our early experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Instead, the expected cardinality of the DPP is a better measure for the diversity of $\\mathcal { V }$ , which is defined as $\\mathbb { E } _ { Y \\sim \\mathcal { P } _ { \\mathbf { L } ( \\gamma ) } } [ | Y | ]$ . Intuitively, since the DPP discourages selection of similar items, if $\\mathcal { V }$ is more diverse, a random subset $\\mathbf { Y }$ drawn according to the DPP is more likely to select more items from $\\mathcal { V }$ , thus having larger cardinality. The expected cardinality can be computed as (Eq. 15 and 34 in Kulesza et al. (2012)): ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ed697b7fda2924fdc8d22c805530d31ca8851093385b6100cbf3c2dc5abd30ca.jpg",
+        "text": "$$\n\\mathbb { E } [ | \\boldsymbol { Y } | ] = \\sum _ { n = 1 } ^ { N } \\frac { \\lambda _ { n } } { \\lambda _ { n } + 1 } = \\mathrm { t r } \\left( \\mathbf { I } - ( \\mathbf { L } ( \\boldsymbol { \\gamma } ) + \\mathbf { I } ) ^ { - 1 } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The main advantage of the expected cardinality is that it is well defined even when the ground set $\\mathcal { V }$ has duplicated items, since it does not require all eigenvalues of $\\mathbf { L }$ to be non-zero as the log likelihood does. Thus, our diversity loss is defined as $\\bar { \\mathcal { L } } _ { d i v e r s e } ^ { \\bar { } } ( \\gamma ) = - \\mathrm { t r } \\left( \\mathbf { I } - ( \\mathbf { L } ( \\gamma ) + \\mathbf { I } ) ^ { - 1 } \\right)$ . The training procedure for $S _ { \\gamma } ( \\psi )$ is outlined in Alg. 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Inference. At test time, given current context $\\psi$ ,we use the learned DSF $S _ { \\gamma } ( \\psi )$ to generate the future trajectory ground set $\\mathcal { V }$ . In some cases, $\\mathcal { V }$ may still contain some trajectories that are similar to others. In order to obtain a diverse set of trajectories without repetition, we aim to perform MAP inference for the DPP to find the most diverse subset $Y ^ { * } = \\arg \\operatorname* { m a x } _ { Y \\in \\mathcal { Y } } \\mathcal { P } _ { \\mathbf { L } ( \\gamma ) } ( Y )$ . A useful property of DPPs is that the log-probability function is submodular (Gillenwater et al., 2012). Even though submodular maximization is NP-hard, we use a greedy algorithm (Nemhauser et al., 1978) which is a popular optimization procedure that works well in practice. As outlined in Alg. 3, the output set $Y _ { f }$ is initialized as $\\varnothing$ , and at each iteration, the trajectory which maximizes the log probability ",
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+        "type": "equation",
+        "img_path": "images/83355f582f7d5ed3bceaae804996b784c59b2117c3656f0a6ddd912f455b8187.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { x } ^ { * } = \\underset { \\mathbf { x } \\in \\mathcal { V } \\backslash Y _ { f } } { \\arg \\operatorname* { m a x } } ~ \\log \\operatorname* { d e t } \\left( \\mathbf { L } _ { Y _ { f } \\cup \\{ \\mathbf { x } \\} } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "is added to $Y _ { f }$ , until the marginal gain becomes negative or $Y _ { f } = y$ . ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The primary focus of our experiments is to answer the following questions: (1) Are trajectory samples generated with our diversity sampling function more diverse than samples from the cVAE and other baselines? (2) How does our method perform on both balanced and imbalanced data? (3) Is our method general enough to perform well on both low-dimensional and high-dimensional tasks? ",
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+    {
+        "type": "text",
+        "text": "Metrics. A problem with trajectory forecasting evaluation is that in real data each context $\\psi ^ { ( i ) }$ usually only has one future trajectory $\\mathbf { x } ^ { ( i ) }$ , which means we only have one sample from a multi-modal distribution. Let us consider a scenario of three data examples {x(i), ψ(i)}3 as shown in Fig. 2 (red, purple, blue). The contexts (past trajectories) of the three examples are instances of the same trajectory but they are slightly different due to noise. As these three contexts have the same semantic meaning, they should share the future trajectories, e.g., the purple and blue future trajectories are also valid for the red context. If we evaluate each example $( \\mathbf { x } ^ { ( i ) } , \\psi ^ { ( i ) } )$ only with its own future trajectory $\\mathbf { x } ^ { ( i ) }$ , a method can achieve high scores by only forecasting the mode corresponding to $\\mathbf { x } ^ { ( i ) }$ and dropping other modes. This is undesirable because we want a good method to capture all modes of the future trajectory distribution, not just a single mode. To allow for multi-modal evaluation, we propose collecting multiple future trajectories for each example by clustering examples with similar contexts. Specifically, we augment each data example $\\bar { ( \\mathbf { x } ^ { ( i ) } , \\psi ^ { ( i ) } ) }$ with a future trajectory set $\\mathcal { X } ^ { ( i ) } = \\{ \\mathbf { x } ^ { ( j ) } | \\| \\boldsymbol { \\psi } ^ { ( j ) } - \\boldsymbol { \\psi } ^ { ( i ) } \\| \\leq \\varepsilon$ , $j = 1 , \\dots , M \\}$ and metrics are calculated based on $\\chi ^ { ( i ) }$ instead of $\\mathbf { x } ^ { ( i ) }$ , i.e., we compute metrics for each $\\mathbf { x } \\in \\mathcal { X } ^ { ( i ) }$ and average the results. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/339718b64625c26a09d2391fc59d2fb39d3ea27dbcb5c3732aa26b7ef6c03c8e.jpg",
+        "image_caption": [
+            "Figure 2: In real data, contexts (past trajectories) are seldom the same due to noise. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "We use the following metrics for evaluation: (1) Average Displacement Error (ADE): average mean square error (MSE) over all time steps between the ground truth future trajectory $\\mathbf { x }$ and the closest sample $\\tilde { \\bf x }$ in the forecasted set of trajectories $Y _ { f }$ . (2) Final Displacement Error (FDE): MSE between the final ground truth position $\\mathbf { x } ^ { T }$ and the closest sample’s final position $\\tilde { \\mathbf { x } } ^ { T }$ . (3) Average Self Distance (ASD): average $L 2$ distance over all time steps between a forecasted sample $\\tilde { \\mathbf { x } } _ { i }$ and its closest neighbor $\\tilde { \\mathbf { x } } _ { j }$ in $Y _ { f }$ . (4) Final Self Distance (FSD): $L 2$ distance between the final position of a sample $\\tilde { \\mathbf { x } } _ { i } ^ { T }$ and its closest neighbor’s final position $\\tilde { \\mathbf { x } } _ { j } ^ { T }$ . The ADE and FDE are common metrics used in prior work on trajectory forecasting (Alahi et al., 2016; Lee et al., 2017; Rhinehart et al., 2018; Gupta et al., 2018). However, these two metrics do not penalize repeated samples. To address this, we introduce two new metrics ASD and FSD to evaluate the similarity between samples in the set of forecasted trajectories. Larger ASD and FSD means the forecasted trajectories are more non-repetitive and diverse. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Baselines. We compare our Diversity Sampler Function (DSF) with the following baselines: (1) cVAE: a method that follows the original sampling scheme of cVAE by sampling latent codes from a Gaussian prior $p ( \\mathbf { z } )$ . (2) MCL: an approach that uses multiple choice learning (Lee et al., 2016) to optimize the sampler $S _ { \\gamma } ( \\psi )$ with the following loss: $\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { m c l } } \\bar { \\mathbf { \\Psi } } = \\operatorname* { m i n } _ { \\tilde { \\mathbf { x } } \\in \\mathcal { Y } } \\| \\tilde { \\mathbf { x } } - \\mathbf { x } \\| ^ { 2 } } \\end{array}$ , where $\\mathbf { x }$ is the ground truth future trajectory. (3) R2P2: a method proposed in (Rhinehart et al., 2018) that uses a reparametrized pushforward policy to improve modeling of multi-modal distributions for vehicle trajectories. (4) cGAN: generative adversarial networks (Goodfellow et al., 2014) conditioned on the forecasting context. We implement all baselines using similar networks and perform hyperparameter search for each method for fair comparisons. For methods whose samples are stochastic, we use 10 random seeds and report the average results for all metrics. ",
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+    {
+        "type": "text",
+        "text": "5.1 SYNTHETIC 2D TRAJECTORY DATA ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We first use synthetic data to evaluate our method’s performance for low-dimensional tasks. We design a virtual 2D traffic scene where a vehicle comes to a crossroad and can choose three different future routes: forward, left, and right. We consider two types of synthetic data: (1) Balanced data, which means the probabilities of the vehicle choosing one of the three routes are the same; (2) ",
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+    {
+        "type": "image",
+        "img_path": "images/e449598c237031babc8a911ea804edb59e2cc40f470548bd63bb814a445de844.jpg",
+        "image_caption": [
+            "Figure 3: Qualitative results on synthetic data for both balanced and imbalanced data distribution when $N = 1 0$ . Blue represents the past trajectory and red represents forecasted future trajectories. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/85a29e0bd05c9c12c4fcf226bd838183289224d0032ee3e31664f3c626e93033.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Quantitative results on synthetic data (numbers scaled by 10) when $N = 1 0$ . "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Balanced Data</td><td colspan=\"4\">Imbalanced Data</td></tr><tr><td>Method</td><td>ADE↓</td><td>FDE↓</td><td>ASD ↑</td><td>FSD ↑</td><td>ADE↓</td><td>FDE↓</td><td>ASD ↑</td><td>FSD ↑</td></tr><tr><td>DSF (Ours)</td><td>0.182</td><td>0.344</td><td>0.147</td><td>0.340</td><td>0.198</td><td>0.371</td><td>0.207</td><td>0.470</td></tr><tr><td>cVAE</td><td>0.262</td><td>0.518</td><td>0.022</td><td>0.050</td><td>0.332</td><td>0.662</td><td>0.021</td><td>0.050</td></tr><tr><td>MCL</td><td>0.276</td><td>0.548</td><td>0.012</td><td>0.030</td><td>0.457</td><td>0.938</td><td>0.005</td><td>0.010</td></tr><tr><td>R2P2</td><td>0.211</td><td>0.361</td><td>0.047</td><td>0.080</td><td>0.393</td><td>0.776</td><td>0.019</td><td>0.030</td></tr><tr><td>cGAN</td><td>0.808</td><td>1.619</td><td>0.018</td><td>0.010</td><td>1.784</td><td>3.744</td><td>0.006</td><td>0.001</td></tr></table>",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Imbalanced data, where the probabilities of the vehicle going forward, left and right are 0.8, 0.1, 0.1, respectively. We synthesize trajectory data by simulating the vehicle’s behavior and adding Gaussian noise to vehicle velocities. Each data example $( \\mathbf { x } ^ { ( i ) } , \\bar { \\psi } ^ { ( i ) } )$ contains future trajectories of 3 steps and past trajectories of 2 steps. We also add an obstacle map around the current position to the context $\\bar { \\psi ^ { ( i ) } }$ . In total, we have around 1100 training examples and 1000 test examples. Please refer to Appendix B for more implementation details. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Table 1 summarizes the quantitative results for both balanced and imbalanced data when the sampling budget $N$ is 10. We can see that our method DSF outperforms the baselines in all metrics in both test settings. As shown in Fig. 3, our method generates more diverse trajectories and is less affected by the imbalanced data distribution. The trajectory samples of our method are also less repetitive, a feature afforded by our DPP formulation. Fig. 4 shows how ADE changes as a function of the sampling budget $N$ . ",
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+    {
+        "type": "text",
+        "text": "5.2 DIVERSE HUMAN MOTION FORECASTING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To further evaluate our method’s performance for more complex and high-dimensional tasks, we apply our method to forecast future human motions (pose sequences). We use motion capture to obtain 10 motion sequences including different types of motions such as walking, turning, jogging, bending, and crouching. Each sequence is about 1 minute long and each pose consists of 59 joint angles. We use past 3 poses ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/fb02cbb95e546c3f878e1cbba7e67bba57b9432eca118496759ad48386cc849c.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Quantitative results on for human motion forecasting when $N = 1 0$ . "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>ADE↓</td><td>FDE↓</td><td>ASD个</td><td>FSD↑</td></tr><tr><td>DSF (Ours)</td><td>0.259</td><td>0.421</td><td>0.115</td><td>0.282</td></tr><tr><td>cVAE</td><td>0.332</td><td>0.642</td><td>0.034</td><td>0.098</td></tr><tr><td>MCL</td><td>0.344</td><td>0.674</td><td>0.036</td><td>0.122</td></tr><tr><td>cGAN</td><td>0.652</td><td>1.296</td><td>0.001</td><td>0.003</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(0.1s) to forecast next 30 poses (1s). There are around 9400 training examples and 2000 test examples where we use different sequences for training and testing. More implementation details can be found in Appendix B. ",
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+        "type": "text",
+        "text": "We present quantitative results in Table 2 and we can see that our approach outperforms other methods in all metrics. As the dynamics model used in R2P2 (Rhinehart et al., 2018) does not generalize well for high-dimensional human motion, we find the model fails to converge and we do not compare with it in this experiment. Fig. 4 shows that our method achieves large improvement when the sampling budget is big $\\mathrm { \\Delta } N = 5 0 \\mathrm { \\Omega }$ ). We also present qualitative results in Fig. 5, where we show the starting pose and the final pose of all 10 forecasted motion samples for each method. We can clearly see that our method generates more diverse future human motions than the baselines. Please refer to Appendix C and our video for additional qualitative results. ",
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+        "image_caption": [
+            "Figure 4: ADE vs. $N$ for synthetic data and human motion forecasting. cGAN is not shown in this plot as it is much worse than other methods due to mode collapse. "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 5: Qualitative results for human motion forecasting when $N = 1 0$ . The left shows the starting pose, and the right shows for each method the final pose of all 10 forecasted motion samples. "
+        ],
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+        "text": "5.3 ADDITIONAL EXPERIMENTS WITH DIVERSITY-BASED BASELINES",
+        "text_level": 1,
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+        "text": "In this section, we perform additional experiments on a large human motion dataset (3.6 million frames), Human3.6M (Ionescu et al., 2013), to evaluate the generalization ability of our approach. We predict future motion of 2 seconds based on observed motion of 0.5 seconds. Please refer to Appendix B.3 for implementation details. We also use a new selection of baselines including several variants of our method (DSF) and the cVAE to validate several design choices of our method, including the choice of the expected cardinality over the negative log likelihood (NLL) of the DPP as the diversity loss. Specifically, we use the following new baselines: (1) DSF-NLL: a variant of DSF that uses NLL as the diversity loss instead of the expected cardinality. (2) DSF-COS: a DSF variant that uses cosine similarity to build the similarity matrix S for the DPP kernel L. (3) DSF-NLL: a variant of the cVAE that samples 100 latent codes and performs DPP MAP inference on the latent codes to obtain a diverse set of latent codes, which are then decoded into trajectory samples. ",
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+        "text": "We present quantitative results in Table 3 when the number of samples $N$ is 10 and 50. The baseline DSF-COS is able to achieve very high diversity (ASD and FSD) but its samples are overly diverse and have poor quality which is indicated by the large ADE and FDE. Compared with DSF-NLL, our method achieves better diversity (ASD and FSD) and similar ADE and FDE when the number of samples is small $N = 1 0$ ). For a larger number of samples $\\mathrm { ~ N ~ } = 5 0 $ ), NLL becomes unstable even with a large $\\epsilon$ (1e-3) added to the diagonal. This behavior of NLL, i.e., stable for small $N$ but unstable for large $N$ , matches our intuition that NLL becomes unstable when samples become similar (as discussed in Sec. 4.2), because when there are more samples, it is easier to have similar samples during the SGD updates of the DSF network. The baseline cVAE-LDPP also performs worse than DSF in all metrics even though it is able to outperfom the cVAE. We believe the reason is that diversity in sample space may not be well reflected in the latent space due to the non-linear mapping from latent codes to samples induced by deep neural networks. ",
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+        "type": "table",
+        "img_path": "images/7b462bade4920a5874ad6f7e15079e28b92b9cbb06447af97005cb10b61821bc.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Quantitative results on Human3.6M (Ionescu et al., 2013) for $N = 1 0$ and $N = 5 0$ . X means the method is unable to learn a model due to numerical instability. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">N=10</td><td colspan=\"4\">N= 50</td></tr><tr><td>ADE↓</td><td>FDE↓</td><td>ASD↑</td><td>FSD ↑</td><td>ADE↓</td><td>FDE↓</td><td>ASD↑</td><td>FSD ↑</td></tr><tr><td>DSF (Ours)</td><td>0.340</td><td>0.521</td><td>0.381</td><td>0.621</td><td>0.236</td><td>0.306</td><td>0.313</td><td>0.415</td></tr><tr><td>DSF-NLL</td><td>0.335</td><td>0.514</td><td>0.343</td><td>0.496</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>DSF-COS</td><td>2.588</td><td>1.584</td><td>5.093</td><td>5.718</td><td>0.978</td><td>0.891</td><td>2.007</td><td>1.968</td></tr><tr><td>cVAE</td><td>0.363</td><td>0.549</td><td>0.235</td><td>0.360</td><td>0.276</td><td>0.369</td><td>0.160</td><td>0.220</td></tr><tr><td>cVAE-LDPP</td><td>0.373</td><td>0.554</td><td>0.280</td><td>0.426</td><td>0.277</td><td>0.365</td><td>0.176</td><td>0.240</td></tr></table>",
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+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We proposed a novel forecasting approach using a DSF to optimize over the sample space of a generative model. Our method learns the DSF with a DPP-based diversity measure to generate a diverse set of trajectories. The diversity measure is a novel application of DPPs to optimize a set of items in continuous space. Experiments have shown that our approach can generate more diverse vehicle trajectories and human motions compared to state-of-the-art baseline forecasting approaches. ",
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+        "text": "Acknowledgment. This project was sponsored in part by JST CREST (JPMJCR14E1), NSF NRI (1637927) and IARPA (D17PC00340). ",
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+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            412,
+            285,
+            428
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A. Alahi, K. Goel, V. Ramanathan, A. Robicquet, L. Fei-Fei, and S. Savarese. Social lstm: Human trajectory prediction in crowded spaces. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 961–971, 2016.   \nS. Azadi, J. Feng, and T. Darrell. Learning detection with diverse proposals. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7149–7157, 2017.   \nL. Ballan, F. Castaldo, A. Alahi, F. Palmieri, and S. Savarese. Knowledge transfer for scene-specific motion prediction. In European Conference on Computer Vision, pages 697–713. Springer, 2016.   \nD. Batra, P. Yadollahpour, A. Guzman-Rivera, and G. Shakhnarovich. Diverse m-best solutions in markov random fields. In European Conference on Computer Vision, pages 1–16. Springer, 2012.   \nJ. Butepage, M. J. Black, D. Kragic, and H. Kjellstrom. Deep representation learning for human motion prediction and classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 6158–6166, 2017.   \nY.-W. Chao, J. Yang, B. Price, S. Cohen, and J. Deng. Forecasting human dynamics from static images. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 548–556, 2017.   \nM. Elfeki, C. Couprie, M. Riviere, and M. Elhoseiny. Gdpp: Learning diverse generations using determinantal point process. arXiv preprint arXiv:1812.00068, 2018.   \nK. Fragkiadaki, S. Levine, P. Felsen, and J. Malik. Recurrent network models for human dynamics. In Proceedings of the IEEE International Conference on Computer Vision, pages 4346–4354, 2015.   \nE. Galceran, A. G. Cunningham, R. M. Eustice, and E. Olson. Multipolicy decision-making for autonomous driving via changepoint-based behavior prediction. In Robotics: Science and Systems, volume 1, 2015.   \nJ. Gillenwater, A. Kulesza, and B. Taskar. Near-optimal map inference for determinantal point processes. In Advances in Neural Information Processing Systems, pages 2735–2743, 2012.   \nJ. A. Gillenwater, A. Kulesza, E. Fox, and B. Taskar. Expectation-maximization for learning determinantal point processes. In Advances in Neural Information Processing Systems, pages 3149–3157, 2014.   \nB. Gong, W.-L. Chao, K. Grauman, and F. Sha. Diverse sequential subset selection for supervised video summarization. In Advances in Neural Information Processing Systems, pages 2069–2077, 2014.   \nI. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In Advances in neural information processing systems, pages 2672–2680, 2014.   \nA. Gupta, J. Johnson, L. Fei-Fei, S. Savarese, and A. Alahi. Social gan: Socially acceptable trajectories with generative adversarial networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2255–2264, 2018.   \nA. Guzman-Rivera, D. Batra, and P. Kohli. Multiple choice learning: Learning to produce multiple structured outputs. In Advances in Neural Information Processing Systems, pages 1799–1807, 2012.   \nI. Habibie, D. Holden, J. Schwarz, J. Yearsley, and T. Komura. A recurrent variational autoencoder for human motion synthesis. BMVC17, 2017.   \nW.-L. Hsiao and K. Grauman. Creating capsule wardrobes from fashion images. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7161–7170, 2018.   \nD.-A. Huang, M. Ma, W.-C. Ma, and K. M. Kitani. How do we use our hands? discovering a diverse set of common grasps. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 666–675, 2015.   \nC. Ionescu, D. Papava, V. Olaru, and C. Sminchisescu. Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments. IEEE transactions on pattern analysis and machine intelligence, 36(7):1325–1339, 2013.   \nA. Jain, A. Singh, H. S. Koppula, S. Soh, and A. Saxena. Recurrent neural networks for driver activity anticipation via sensory-fusion architecture. In Robotics and Automation (ICRA), 2016 IEEE International Conference on, pages 3118–3125. IEEE, 2016a.   \nA. Jain, A. R. Zamir, S. Savarese, and A. Saxena. Structural-rnn: Deep learning on spatio-temporal graphs. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5308–5317, 2016b.   \nM. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to variational methods for graphical models. Machine learning, 37(2):183–233, 1999.   \nA. Kanazawa, J. Zhang, P. Felsen, and J. Malik. Learning 3d human dynamics from video. arXiv preprint arXiv:1812.01601, 2018.   \nD. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   \nK. M. Kitani, B. D. Ziebart, J. A. Bagnell, and M. Hebert. Activity forecasting. In European Conference on Computer Vision, pages 201–214. Springer, 2012.   \nA. Kulesza and B. Taskar. k-dpps: Fixed-size determinantal point processes. In Proceedings of the 28th International Conference on Machine Learning (ICML-11), pages 1193–1200, 2011.   \nA. Kulesza, B. Taskar, et al. Determinantal point processes for machine learning. Foundations and Trend $\\textsuperscript { \\textregistered }$ in Machine Learning, 5(2–3):123–286, 2012.   \nN. Lee, W. Choi, P. Vernaza, C. B. Choy, P. H. Torr, and M. Chandraker. Desire: Distant future prediction in dynamic scenes with interacting agents. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 336–345, 2017.   \nS. Lee, S. P. S. Prakash, M. Cogswell, V. Ranjan, D. Crandall, and D. Batra. Stochastic multiple choice learning for training diverse deep ensembles. In Advances in Neural Information Processing Systems, pages 2119–2127, 2016.   \nZ. Li, Y. Zhou, S. Xiao, C. He, Z. Huang, and H. Li. Auto-conditioned recurrent networks for extended complex human motion synthesis. arXiv preprint arXiv:1707.05363, 2017.   \nW.-C. Ma, D.-A. Huang, N. Lee, and K. M. Kitani. Forecasting interactive dynamics of pedestrians with fictitious play. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 774–782, 2017.   \nO. Macchi. The coincidence approach to stochastic point processes. Advances in Applied Probability, 7(1): 83–122, 1975.   \nJ. Martinez, R. Hossain, J. Romero, and J. J. Little. A simple yet effective baseline for 3d human pose estimation. In Proceedings of the IEEE International Conference on Computer Vision, pages 2640–2649, 2017.   \nG. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing submodular set functionsi. Mathematical programming, 14(1):265–294, 1978.   \nD. Nilsson. An efficient algorithm for finding the m most probable configurationsin probabilistic expert systems. Statistics and computing, 8(2):159–173, 1998.   \nG. Pavlakos, X. Zhou, K. G. Derpanis, and K. Daniilidis. Coarse-to-fine volumetric prediction for single-image 3d human pose. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7025–7034, 2017.   \nD. Pavllo, C. Feichtenhofer, D. Grangier, and M. Auli. 3d human pose estimation in video with temporal convolutions and semi-supervised training. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7753–7762, 2019.   \nN. Rhinehart, K. M. Kitani, and P. Vernaza. R2p2: A reparameterized pushforward policy for diverse, precise generative path forecasting. In Proceedings of the European Conference on Computer Vision (ECCV), pages 772–788, 2018.   \nN. Rhinehart, R. McAllister, K. Kitani, and S. Levine. Precog: Prediction conditioned on goals in visual multi-agent settings. arXiv preprint arXiv:1905.01296, 2019.   \nA. Robicquet, A. Sadeghian, A. Alahi, and S. Savarese. Learning social etiquette: Human trajectory understanding in crowded scenes. In European conference on computer vision, pages 549–565. Springer, 2016.   \nB. Seroussi and J.-L. Golmard. An algorithm directly finding the k most probable configurations in bayesian networks. International Journal of Approximate Reasoning, 11(3):205–233, 1994.   \nH. Soo Park, J.-J. Hwang, Y. Niu, and J. Shi. Egocentric future localization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4697–4705, 2016.   \nJ. Walker, C. Doersch, A. Gupta, and M. Hebert. An uncertain future: Forecasting from static images using variational autoencoders. In European Conference on Computer Vision, pages 835–851. Springer, 2016.   \nD. Xie, S. Todorovic, and S.-C. Zhu. Inferring ”dark matter” and ”dark energy” from videos. 2013 IEEE International Conference on Computer Vision, pages 2224–2231, 2013.   \nT. Yagi, K. Mangalam, R. Yonetani, and Y. Sato. Future person localization in first-person videos. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.   \nX. Yan, A. Rastogi, R. Villegas, K. Sunkavalli, E. Shechtman, S. Hadap, E. Yumer, and H. Lee. Mt-vae: Learning motion transformations to generate multimodal human dynamics. In Proceedings of the European Conference on Computer Vision (ECCV), pages 265–281, 2018.   \nY. Yuan and K. Kitani. Ego-pose estimation and forecasting as real-time pd control. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), pages 10082–10092, 2019. ",
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+        "text": "A ALGORITHMS ",
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+        "text": "Algorithm 2 Training the cVAE ",
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+        "text": "1: Input: Training data $\\{ \\mathbf { x } ^ { ( i ) } , \\boldsymbol { \\psi } ^ { ( i ) } \\} _ { i = 1 } ^ { M }$   \n2: Output: cVAE encoder network $f _ { \\phi } (  { \\mathbf { x } } , \\psi )$ and decoder network $g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\boldsymbol { \\psi } )$   \n3: Initialize $\\phi$ and $\\theta$ randomly   \n4: while not converged do   \n5: for each $( \\mathbf { x } ^ { ( i ) } , \\psi ^ { ( i ) } )$ do   \n6: Compute parameters $( \\mu , \\sigma )$ of the posterior distribution $q _ { \\phi } ( { \\bf z } | { \\bf x } , \\psi )$ using $f _ { \\phi } ( \\mathbf { x } , \\psi )$   \n7: Sample $V$ Gaussian noises $\\{ \\epsilon _ { 1 } , \\ldots , \\epsilon _ { V } \\}$ from $\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$   \n8: Transform noises to latent samples from $q _ { \\phi } ( { \\bf z } | { \\bf x } , \\psi )$ : $\\mathbf { z } _ { v } = \\pmb { \\mu } + \\pmb { \\sigma } \\odot \\pmb { \\epsilon } _ { v }$   \n9: Decode latent samples into reconstructed trajectories $\\big \\{ \\tilde { \\mathbf { x } } _ { 1 } , \\dots , \\tilde { \\mathbf { x } } _ { V } \\big \\}$ using $g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\boldsymbol { \\psi } )$   \n10: Calculate the cVAE loss $\\mathcal { L } _ { c v a e }$ according to Eq. 6   \n11: Update $\\phi$ and $\\theta$ with $\\nabla _ { \\phi } \\mathcal { L } _ { c v a e }$ and $\\nabla _ { \\theta } \\mathcal { L } _ { c v a e }$   \n12: end for   \n13: end while ",
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+        "text": "Algorithm 3 Inference with the DSF $S _ { \\gamma } ( \\psi )$ ",
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+        "text": "1: Input: Context $\\psi$ , DSF $S _ { \\gamma } ( \\psi )$ , cVAE decoder network $g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\boldsymbol { \\psi } )$   \n2: Output: Forecasted trajectory set $Y _ { f }$   \n3: Generate latent codes $\\mathcal { Z } = \\{ \\mathbf { z } _ { 1 } , \\ldots , \\mathbf { z } _ { N } \\}$ with the DSF $S _ { \\gamma } ( \\psi )$   \n4: Generate the trajectory ground set $\\mathcal { Y } = \\{ \\mathbf { x } _ { 1 } , \\dotsc , \\mathbf { x } _ { N } \\}$ with the decoder $g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\boldsymbol { \\psi } )$   \n5: Compute the DPP kernel $\\mathbf { L } = \\mathrm { D i a g } ( \\mathbf { r } ) \\cdot \\mathbf { S } \\cdot \\mathrm { D i a g } ( \\mathbf { r } )$   \n6: $Y _ { f } \\gets \\emptyset , U \\gets y$   \n7: while $U$ is not empty do   \n8: $\\mathbf { x } ^ { * }  \\arg \\operatorname* { m a x } _ { \\mathbf { x } \\in U }$ $\\log \\operatorname* { d e t } \\left( \\mathbf { L } _ { Y _ { f } \\cup \\{ \\mathbf { x } \\} } \\right)$   \n9: $\\mathbf { | } \\mathbf { f } \\log \\operatorname* { d e t } \\left( \\mathbf { L } _ { Y _ { f } \\cup \\{ \\mathbf { x } ^ { * } \\} } \\right) - \\log \\operatorname* { d e t } \\left( \\mathbf { L } _ { Y _ { f } } \\right) < 0$ then   \n10: break   \n11: end if   \n12: $\\begin{array} { l } { Y _ { f }  Y _ { f } \\cup \\{ \\mathbf { x } ^ { * } \\} } \\\\ { U  U \\setminus \\{ \\mathbf { x } ^ { * } \\} } \\end{array}$   \n13:   \n14: end while ",
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+        "image_caption": [
+            "Figure 6: Network architectures for synthetic data and human motion. Top: for synthetic data, we use a CNN to process the obstacle map f and directly flatten trajectories $\\mathbf { x }$ and $\\mathbf { h }$ into vectors. The reconstructed trajectory $\\tilde { \\bf x }$ is decoded with an MLP. Bottom: for human motion, we use Bi-LSTMs to extract temporal features for $\\mathbf { x }$ and $\\mathbf { h }$ and decode the reconstructed trajectory $\\tilde { \\bf x }$ with a forward LSTM. "
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+        "text": "B.1 NETWORK ARCHITECTURES ",
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+        "text": "Synthetic data. Fig. 6 (Top) shows the network architecture for synthetic data. The number of latent dimensions is 2. By default, we use ReLU activation for all networks. The future trajectory $\\mathbf { x } \\in \\mathbb { R } ^ { 3 \\times 2 }$ consists of 3 future positions of the vehicle. The context $\\psi$ contains past trajectories $\\textbf { h } \\in \\mathbb { R } ^ { 2 \\times 2 }$ of 2 time steps and a obstacle map $\\textbf { f } \\in \\{ 0 , 1 \\} ^ { 2 8 \\times 2 8 }$ spanning a $4 \\times 4$ area around the current position of the vehicle (the road width is 2). For the encoder, we use a convolutional neural network (CNN) with three 32-channel convolutional layers to process f. The first two layers have kernel size 4 and stride 2 while the last layer has kernel size 6 and stride 1. The obtained CNN features are concatenated with flattened $\\mathbf { x }$ and $\\mathbf { h }$ into a unified feature, which is feed into a multilayer perceptron (MLP). The MLP has one 128-dim hidden layer and two heads outputing the mean $\\pmb { \\mu }$ and variance $\\sigma$ of the latent distribution. For the decoder, we concatenate the CNN feature from f with the latent code $\\textbf { z } \\in \\mathbb { R } ^ { 2 }$ and flattened $\\mathbf { h }$ into a unified feature. The feature is passed through an MLP with one 128-dim hidden layer which outputs the reconstructed future trajectory $\\tilde { \\mathbf { x } } \\in \\breve { \\mathbb { R } } ^ { 3 \\times 2 }$ . For the diversity sampler function (DSF), we concatenate the CNN feature from f with the flattened h and pass it through an MLP with one 128-dim hidden layer to obtain a set of latent codes $\\left\\{ \\mathbf { z } _ { 1 } , \\ldots , \\mathbf { z } _ { N } \\right\\}$ which are represented by a vector of length $2 N$ . ",
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+        "text": "Human motion. Fig. 6 (Bottom) shows the network architecture for synthetic data. The number of latent dimensions is 8. The future trajectory $\\mathbf { x } \\in \\mathbb { R } ^ { 3 0 \\times 5 9 }$ consists of future poses of 30 time steps (1s). The context $\\psi$ contains past poses $\\mathbf { h } \\in \\mathbb { R } ^ { 3 \\times 5 9 }$ of 3 time steps (0.1s). Each pose consists of 59 joint angles. For the encoder, we use two 128-dim bidirectional LSTMs (Bi-LSTMs) and mean pooling to obtain the temporal features for $\\mathbf { x }$ and $\\mathbf { h }$ . We then concatenate the temporal features into a unified feature and feed it into an MLP with two hidden layers (300, 200) and two heads to obtain the mean $\\pmb { \\mu }$ and variance $\\sigma$ of the latent distribution. For the decoder, we reuse the Bi-LSTM of the encoder for the context h and a 128-dim forward LSTM to decode the future trajectory $\\tilde { \\bf x }$ . At each time step $t$ , the forward LSTM takes as input the previous pose $\\tilde { \\mathbf { x } } ^ { t - 1 }$ $\\mathbf { h } ^ { H }$ for $t = 0$ ), the latent code $\\mathbf { z } \\in \\mathbb { R } ^ { 8 }$ and the temporal features from h, and outputs a 128-dim feature. The feature is then passed through an MLP with two hidden layers (300, 200) to generate the reconstructed pose $\\tilde { \\mathbf { x } } ^ { t }$ . For the DSF, we use a different 128-dim Bi-LSTM to obtain the temporal feature for $\\mathbf { h }$ , which is feed into an MLP with a 128-dim hidden layer to produce a set of latent codes $\\left\\{ \\mathbf { z } _ { 1 } , \\ldots , \\mathbf { z } _ { N } \\right\\}$ which are represented by a vector of length $8 N$ . ",
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "B.2 TRAINING AND EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "When training the cVAE model using Eq. 7, we take $V = 1$ sample from the posterior $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } , \\psi )$ . The weighting factor $\\beta$ for the KL term is set to 0.1 for synthetic data and 1e-4 for human motion. We use Adam (Kingma and Ba, 2014) to jointly optimize the encoder and decoder. The learning rate is set to 1e-4 and we use a mini batch size of 32 for synthetic data. We optimize the model for 500 epochs for synthetic data and 100 epochs for human motion. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "When training the DSF, the scale factor $k$ for the similarity matrix S is set to 1 for synthetic data and 1e-2 for human motions. For both synthetic data and human motions, we use Adam with learning rate 1e-4 to optimize the DSF for 20 epochs. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Recall that in the metrics section (Sec. 5.1), we need the grouping threshold $\\varepsilon$ to build the ground truth future trajectory set $\\mathcal { X } ^ { ( i ) } = \\{ \\mathbf { x } ^ { ( j ) } \\vert \\Vert \\psi ^ { ( j ) } - \\psi ^ { ( i ) } \\Vert \\leq \\varepsilon$ , $j = 1 , \\dots , M \\}$ . For synthetic data, $\\varepsilon$ is set to 0.1 and we only use past trajectories $\\mathbf { h }$ to compute the distance between contexts. For human motion, $\\varepsilon$ is set to 0.5. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.3 IMPLEMENTATION DETAILS FOR EXPERIMENTS ON HUMAN3.6M ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Following previous work (Martinez et al., 2017; Pavlakos et al., 2017; Pavllo et al., 2019), we convert the motion sequences in the dataset into sequences of 3D joint positions, and adopt a 17-joint skeleton. We train on five subjects (S1, S5, S6, S7, S8), and test on two subjects (S9 and S11). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use the same network architectures (Fig.6 (Bottom)) in this experiment as the one used in the human motion forecasting experiment above. The number of latent dimensions is 128. When training the cVAE model, the weighting factor $\\beta$ is set to 0.1. We sample 5000 training examples every epoch and optimize the cVAE for 500 epochs using Adam and a learning rate of 1e-4. We set the batch size to 64 for the optimization. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The scale factor $k$ for the similarity matrix S of the DPP kernel is set to 5. When learning the DSF, we use a batch size of 64 and sample 1000 training examples every epoch and optimize the DSF for 20 epochs using Adam and a learning rate of 1e-3. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "When computing the metrics, we set the grouping threshold $\\varepsilon$ to 0.1. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL VISUALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We also show additional qualitative results for human motion forecasting in Fig. 7. The quality and diversity of the forecasted motions are best seen in our video2. ",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The final pose of 10 forecasted motions ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/f3e399ae3bc56208ba1dccca639da93a1d6e9a7fe8064fe2b76e19f24f16ec80.jpg",
+        "image_caption": [
+            "Figure 7: Additional visualization for human motion forecasting. The left shows the starting pose, and on the right we show for each method the final pose of 10 forecasted motion samples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    }
+]