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+# ATTENTIVE WEIGHTS GENERATION FOR FEW SHOTLEARNING VIA INFORMATION MAXIMIZATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Few shot image classification aims at learning a classifier from limited labeled data. Generating the classification weights has been applied in many metalearning approaches for few shot image classification due to its simplicity and effectiveness. However, we argue that it is difficult to generate the exact and universal classification weights for all the diverse query samples from very few training samples. In this work, we introduce Attentive Weights Generation for few shot learning via Information Maximization (AWGIM), which addresses current issues by two novel contributions. i) AWGIM generates different classification weights for different query samples by letting each of query samples attends to the whole support set. ii) To guarantee the generated weights adaptive to different query sample, we re-formulate the problem to maximize the lower bound of mutual information between generated weights and query as well as support data. As far as we can see, this is the first attempt to unify information maximization into few shot learning. Both two contributions are proved to be effective in the extensive experiments and we show that AWGIM is able to achieve state-of-the-art performance on benchmark datasets.
+
+# 1 INTRODUCTION
+
+While deep learning methods achieve great success in domains such as computer vision (He et al., 2016), natural language processing (Devlin et al., 2018), reinforcement learning (Silver et al., 2018), their hunger for large amount of labeled data limits the application scenarios where only a few data are available for training. Humans, in contrast, are able to learn from limited data, which is desirable for deep learning methods. Few shot learning is thus proposed to enable deep models to learn from very few samples (Fei-Fei et al., 2006).
+
+Meta learning is by far the most popular and promising approach for few shot problems (Vinyals et al., 2016; Finn et al., 2017; Snell et al., 2017; Ravi & Larochelle, 2016; Rusu et al., 2019). In meta learning approaches, the model extracts high level knowledge across different tasks so that it can adapt itself quickly to a new-coming task (Schmidhuber, 1987; Andrychowicz et al., 2016). There are several kinds of meta learning methods for few shot learning, such as gradient-based (Finn et al., 2017; Ravi & Larochelle, 2016) and metric-based (Snell et al., 2017; Sung et al., 2018). Weights generation, among these different methods, has shown effectiveness with simple formulation (Qi et al., 2018; Qiao et al., 2018; Gidaris & Komodakis, 2018; 2019). In general, weights generation methods learn to generate the classification weights for different tasks conditioned on the limited labeled data. However, fixed classification weights for different query samples within one task might be sub-optimal, due to the few shot challenge.
+
+We introduce Attentive Weights Generation for few shot learning via Information Maximization (AWGIM) in this work to address these limitations. In AWGIM, the classification weights are generated for each query sample specifically. This is done by two encoding paths where the query sample attends to the task context. However, we show in experiments that simple cross attention between query samples and support set fails to guarantee classification weights fitted to diverse query data since the query-specific information is lost during weights generation. Therefore, we propose to maximize the lower bound of mutual information between generated weights and query, support data. As far as we know, AWGIM is the first work introducing Variational Information Maximization in few shot learning. The induced computational overhead is minimal due to the nature of few shot problems. Furthermore, by maximizing the lower bound of mutual information, AWGIM gets rid of inner update without compromising performance.
+
+AWGIM is evaluated on two benchmark datasets and shows state-of-the-art performance. We also conducted detailed analysis to validate the contribution of each component in AWGIM.
+
+# 2 RELATED WORKS
+
+# 2.1 FEW SHOT LEARNING
+
+Learning from few labeled training data has received growing attentions recently. Most successful existing methods apply meta learning to solve this problem and can be divided into several categories. In the gradient-based approaches, an optimal initialization for all tasks is learned (Finn et al., 2017). Ravi & Larochelle (2016) learned a meta-learner LSTM directly to optimize the given fewshot classification task. Sun et al. (2019) learned the transformation for activations of each layer by gradients to better suit the current task.
+
+In the metric-based methods, a similarity metric between query and support samples is learned. (Koch et al., 2015; Vinyals et al., 2016; Snell et al., 2017; Sung et al., 2018; Li et al., 2019a). Spatial information or local image descriptors are also considered in some works to compute richer similarities (Lifchitz et al., 2019; Li et al., 2019b; Wertheimer & Hariharan, 2019).
+
+Generating the classification weights directly has been explored by some works. Gidaris & Komodakis (2018) generated classification weights as linear combinations of weights for base and novel classes. Similarly, Qiao et al. (2018) and Qi et al. (2018) both generated the classification weights from activations of a trained feature extractor. Graph neural network denoising autoencoders are used in (Gidaris & Komodakis, 2019). Munkhdalai & Yu (2017) proposed to generate “fast weights” from the loss gradient for each task. All these methods do not consider generating different weights for different query examples, nor maximizing the mutual information.
+
+There are some other methods for few-shot classification. Generative models are used to generate or hallucinate more data in (Zhang et al., 2018; Wang et al., 2018; Chen et al., 2019). Bertinetto et al. (2019) and Lee et al. (2019) used the closed-form solutions directly for few shot classification. Liu et al. (2019) integrated label propagation on a transductive graph to predict the query class label.
+
+# 2.2 ATTENTION
+
+Attention mechanism shows great success in computer vision (Xu et al., 2015; Parmar et al., 2018) and natural language processing (Bahdanau et al., 2015; Vaswani et al., 2017). It is effective in modeling the interaction between queries and key-value pairs from certain context. Based on the fact that keys and queries point to the same entities or not, people refer to attention as self attention or cross attention. In this work, we use both types of attention to encode the task and query-task information. The work most similar to ours is Attentive Neural Processes (Kim et al., 2019), which also employs self and cross attention. However, we are using attention for few-shot image classification via maximizing the mutual information. In stark contrast, Kim et al. (2019) worked on regression from the perspective of a stochastic process and the variational objective is optimized.
+
+# 2.3 MUTUAL INFORMATION
+
+Given two random variables $\mathbf { X }$ and $\mathsf { y }$ , mutual information $I ( { \bf x } ; { \bf y } )$ measures the decrease of uncertainty in one random variable when another is known. It is defined as the Kullback-Leibler divergence between joint distribution $p ( \mathbf { x } , \mathbf { y } )$ and product of marginal distributions $p ( \mathbf { x } ) \otimes p ( \mathbf { y } )$ ,
+
+$$
+I ( \mathbf { x } ; \mathbf { y } ) = D _ { \mathrm { K L } } ( p ( \mathbf { x } , \mathbf { y } ) \| p ( \mathbf { x } ) \otimes p ( \mathbf { y } ) ) .
+$$
+
+When x and y are independent, $p ( \mathbf { x } , \mathbf { y } ) = p ( \mathbf { x } ) \otimes p ( \mathbf { y } )$ so that $I ( { \bf x } , { \bf y } ) = 0$ , indicating that knowing $\mathbf { X }$ does not reveal any information about y. When y is a deterministic function of $\mathbf { X }$ , $I ( \mathbf { x } , \mathbf { y } )$ achieves its maximum value. Mutual information has been widely applied in applications such as Generative Adversarial Networks(Chen et al., 2016), self-supervised learning(Hjelm et al., 2019), visual question generation Krishna et al. (2019) and so on.
+
+![](images/1337a8d6253363eec5b99fdb1e3eb590ae641d0447bd8c18d057cba309ac3b81.jpg)  
+Figure 1: The overview of our proposed AWGIM. The input task is 5-way 1-shot with $\mathbf { X }$ as support set and $\hat { \bf x }$ as one query example. Different colors of the data in support set indicate different categories. The encoding process in contextual path produces context-aware support representations $\mathbf { X } ^ { c p }$ . Similarly, the attentive path enables the query sample $\hat { \bf x }$ to be equipped with task knowledge. Both paths are achieved by attention mechanism. $\hat { \mathbf { x } } ^ { a p }$ is repeated to concatenate with $\mathbf { X } ^ { c p }$ . The weight generator $g$ takes these concatenated representations as input to generate classification weights W specific for $\hat { \bf x }$ , denoted by the colorful matrix with slash. It can be used to predict the class label for $\hat { \bf x }$ and X. W is also used to reconstruct the inputs of the generator $g$ by two networks $r _ { 1 }$ and $r _ { 2 }$ . In this way, the lower bound of mutual information is maximized and $g$ is forced to generate classification weights sensitive to different query samples.
+
+# 3 PROPOSED METHOD
+
+In this section, we provide the problem formulation first. Then the proposed model is described in Sec. 3.3. The objective function, which maximizes the mutual information between certain variables, and theoretical analysis are provided in Sec. 3.4.
+
+# 3.1 PROBLEM FORMULATION
+
+Following many popular meta-learning methods for few shot classification, we formulate the problem under episodic training paradigm (Vinyals et al., 2016; Finn et al., 2017). One $N$ -way $K$ -shot task sampled from an unknown task distribution $P ( \tau )$ includes support set and query set:
+
+$$
+{ \mathcal { T } } = ( { \mathcal { S } } , { \mathcal { Q } } ) ,
+$$
+
+where $\mathcal { S } = \{ ( \mathbf { x } ^ { c _ { n } ; k } , \mathbf { y } ^ { c _ { n } ; k } ) | k = 1 , . . . , K ; n = 1 , . . . , N \}$ , $\mathcal { Q } = \{ ( \hat { \mathbf { x } } _ { 1 } , . . . , \hat { \mathbf { x } } _ { | \mathcal { Q } | } ) \}$ . Support set $s$ contains $N K$ labeled samples. Query set $\mathcal { Q }$ includes $\hat { \bf x }$ and we need to predict label $\hat { \mathbf { y } }$ for $\hat { \bf x }$ based on $s$ . During meta-training, the meta-loss is estimated on $\mathcal { Q }$ to optimize the model. During metatesting, the performance of meta-learning method is evaluated on $\mathcal { Q }$ , provided the labeled $s$ . The classes used in meta-training and meta-testing are disjoint so that the meta-learned model needs to learn the knowledge transferable across tasks and adapt itself quickly to novel tasks.
+
+Our proposed approach follows the general framework to generate the classification weights (Qi et al., 2018; Qiao et al., 2018; Rusu et al., 2019; Gidaris & Komodakis, 2018; 2019). In this framework, there is a feature extractor to output image feature embeddings. The meta-learner needs to generate the classification weights for different tasks.
+
+# 3.2 LATENT EMBEDDING OPTIMIZATION
+
+Latent Embedding Optimization (LEO) (Rusu et al., 2019) is one of the weights generation methods that is most related to our work. In LEO, the latent code $_ z$ is generated by $h$ conditioned on support set $s$ , described as $z = h ( S )$ . $h$ is instantiated as relation networks (Santoro et al., 2017). Classification weights $\pmb { w }$ can be decoded from $_ { z }$ with $l$ , $w = l ( z )$ . In the inner loop, we use $\pmb { w }$ to compute the loss (usually cross entropy) on the support set and then update $_ z$ :
+
+$$
+\begin{array} { r } { z ^ { \prime } = z - \eta \nabla _ { z } \mathcal { L } _ { \mathcal { S } } ( \pmb { w } ) , } \end{array}
+$$
+
+where $\mathcal { L } _ { S }$ indicates that the loss is evaluated on $s$ only. The updated latent code $z ^ { \prime }$ is used to decode new classification weights $\mathbf { \Delta } \mathbf { w ^ { \prime } }$ with generating function $l$ . $\mathbf { \Delta } _ { \mathbf { \ b { w } } ^ { \prime } }$ is adopted in the outer loop for query set $\mathcal { Q }$ and the objective function of LEO then can be written as
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } _ { \mathcal { Q } } ( w ^ { \prime } ) .
+$$
+
+Here $\theta$ stands for the parameters of $h$ and $l$ and we omit the regularization terms for clarity. LEO avoids updating high-dimensional $\textbf { \em w }$ in the inner loop by learning a lower-dimensional latent space, from which sampled $_ z$ can be used to generate $\pmb { w }$ . The most significant difference between LEO and AWGIM is that we do not need inner updates to adapt the model. Instead, AWGIM is a feedforward network trained to maximize the mutual information so that it fits to different tasks well. On the other hand, AWGIM learns to generate optimal classification weights for each query sample while LEO generates fixed weights conditioned on the support set within one task. In Section 3.4 we will show LEO can be casted as a special case of AWGIM under certain conditions.
+
+# 3.3 ATTENTIVE WEIGHTS GENERATION
+
+The framework of our proposed method is shown in Figure 1. Assume that we have a feature extractor, which can be a simple 4-layer Convnet or a deeper Resnet. All the images included in the sampled task $\tau$ are processed by this feature extractor and represented as $d$ -dimensional vectors afterwards, i.e., $\mathbf { x } ^ { c _ { n } ; k } , \hat { \mathbf { x } } \in \mathbb { R } ^ { d }$ . There are two paths to encode the task context and the individual query sample respectively, which are called contextual path and attentive path. The outputs of both paths are concatenated together as input to the generator for classification weights. Generated classification weights are used to not only predict the label of $\hat { \bf x }$ , but also maximize the lower bound of mutual information between itself and other variables, which will be discussed in the following section 3.4.
+
+# 3.3.1 CONTEXTUAL AND ATTENTIVE PATHS
+
+The encoding process includes two paths, namely the contextual path and attentive path. The contextual path aims at learning representations for only the support set with a multi-head self-attention network $f _ { s a } ^ { c p }$ (Vaswani et al., 2017). The outputs of contextual path ${ \bf X } ^ { c p } \in \mathbb { R } ^ { N K \times d _ { h } }$ 1 thus contain richer information about the task and can be used later for weights generation.
+
+Existing weights generation methods generate the classification weights conditioned on the support set only, which is equivalent to using contextual path. However, the classification weights generated in this way might be sub-optimal. This is because estimating the exact and universal classification weights from very few labeled data in the support set is difficult and sometimes impossible. The generated weights are usually in lack of adaptation to different query samples. We address this issue by introducing attentive path, where the individual query example attends to the task context and then is used to generate the classification weights. Therefore, the classification weights are adaptive to different query samples and aware of the task context as well.
+
+In the attentive path, a new multi-head self-attention network $f _ { s a } ^ { a p }$ on the support set is employed to encode the global task information. $f _ { s a } ^ { a p }$ is different from $f _ { s a } ^ { c p }$ in contextual path because the selfattention network in contextual path emphasizes on generating the classification weights. On the contrary, outputs of self-attention here plays the role of providing the V alue context for different query samples to attend in the following cross attention. Sharing the same self-attention networks might limit the expressiveness of learned representations in both paths. The cross attention network $f _ { c a } ^ { a p }$ applied on each query sample and task-aware support set is followed to produce $\hat { \bf X } ^ { a p } \in \mathbb { R } ^ { | \mathcal { Q } | \times d _ { h } }$ .
+
+We use multi-head attention with $h$ heads in both paths. In one attention block, we produce $h$ different sets of queries, keys and values. Multi-head attention is claimed to be able to learn more comprehensive and expressive representations from $h$ different subspaces (Vaswani et al., 2017; Voita et al., 2019). More details of these two paths can be found in A.2.
+
+# 3.3.2 WEIGHTS GENERATOR
+
+We replicate them afterwa ${ \bf X } ^ { c p } \in \mathbb { R } ^ { N K \times d _ { h } }$ anve $\hat { \bf X } ^ { a p } \in \mathbb { R } ^ { | \mathcal { Q } | \times d _ { h } }$ r $| \mathcal { Q } |$ $N K$ ectively and reshape. These two tensors $\pmb { \chi } ^ { c p } \in \mathbb { R } ^ { | \mathcal { Q } | \times N K \times d _ { h } }$ $\hat { \pmb { \mathsf { X } } } ^ { a p } \in \mathbb { R } ^ { | \mathcal { Q } | \times N K \times d _ { h } }$
+
+$d _ { h } < d$ is the hidden dimension. We use matrix form here to be consistent with the description in 3.3.2.
+
+are concatenated to become $\pmb { \chi } ^ { c p \oplus a p } \in \mathbb { R } ^ { | \mathcal { Q } | \times N K \times 2 d _ { h } }$ . $\mathbf { \pmb { \chi } } ^ { c p \oplus a p }$ can be interpreted that each query sample has its own latent representations for support set to generate specific classification weights, which are both aware of the task-context and adaptive to individual query sample.
+
+$\mathbf { X } ^ { c p \oplus a p }$ is decoded by the weights generator $g : \mathbb { R } ^ { 2 d _ { h } }  \mathbb { R } ^ { 2 d }$ . We assume that the classification weights follow Gaussian distribution with diagonal covariance. $g$ outputs the distribution parameters and we sample the weights from learned distribution during meta-training. The sampled classification weights are represented as $\pmb { \mathsf { W } } \in \mathbb { R } ^ { | \mathcal { Q } | \times N K \times d }$ . To reduce complexity, we compute the mean value on $K$ classification weights for each class to have ${ \pmb W } ^ { f i n a l } \in \mathbb { R } ^ { | \mathcal { Q } | \times N \times d }$ . Therefore, ith query sample has its specifican be computed by c classifica Wf inali,:,: tion weight matrix . The support data $\pmb { \mathsf { W } } _ { i , : , : } ^ { f i n a l } \in \mathbb { R } ^ { N \times d }$ . Th for rediction for query datatimes and reshaped as $\hat { \mathbf { X } } \mathbf { W } ^ { f i n a l \mathbf { T } }$ $\mathbf { X }$ $| \mathcal { Q } |$ $\pmb { \chi } _ { s } \in \mathbb { R } ^ { | \mathcal { Q } | \times N K \times d }$ . So the prediction for support data can also be computed as ${ \pmb x } _ { s } { \pmb w } ^ { f i n a l { \bf T } }$ .
+
+Besides the weights generator $g$ , we have another two decoders $r _ { 1 } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d _ { h } }$ and $r _ { 2 } : \mathbb { R } ^ { d } $ $\mathbb { R } ^ { d _ { h } }$ . They both take the generated weights $\boldsymbol { \mathsf { W } }$ as inputs and learn to reconstruct $\mathbf { X } ^ { c p }$ and ${ \hat { \mathbf { X } } } ^ { a p }$ respectively. The outputs of $r _ { 1 }$ and $r _ { 2 }$ are denoted as $\bar { \pmb { \chi } } _ { r e } ^ { c p } , \hat { \pmb { \chi } } _ { r e } ^ { a p } \in \mathbb { R } ^ { | \mathcal { Q } | \times N K \times d _ { h } }$ . The reason we are using reconstruction as auxiliary tasks will be discussed in following Sec. 3.4.
+
+# 3.4 INFORMATION MAXIMIZATION
+
+In this section, we perform the analysis for one query sample without loss of generality. The subscripts for classification weights are omitted for clarity. In general, we use $\displaystyle ( \mathbf { x } , \mathbf { y } )$ and $( \hat { \mathbf { x } } , \hat { \mathbf { y } } )$ to represent support and query samples respectively.
+
+Since the classification weights w generated from $g$ are encoded with attentive path and contextual path, it is expected that we can directly have the query-specific weights. However, we show in the experiments that simply doing this does not outperform a weight generator conditioned only on the $s$ significantly, which implies that the generated classification weights from two paths are not sensitive to different query samples. In other words, the information from attentive path is not kept well during the weights generation.
+
+To address this limitation, we propose to maximize the mutual information between generated weights w and support as well as query data. The objective function can be described as
+
+$$
+\operatorname* { m a x } I ( ( \hat { \mathbf { x } } , \hat { \mathbf { y } } ) ; \mathbf { w } ) + \sum _ { ( \mathbf { x } , \mathbf { y } ) \in S } I ( ( \mathbf { x } , \mathbf { y } ) ; \mathbf { w } )
+$$
+
+According to the chain rule of mutual information, we have
+
+$$
+I ( ( \hat { \mathbf { x } } , \hat { \mathbf { y } } ) ; \mathbf { w } ) = I ( \hat { \mathbf { x } } ; \mathbf { w } ) + I ( \hat { \mathbf { y } } ; \mathbf { w } | \hat { \mathbf { x } } ) .
+$$
+
+Equation 6 stands for both terms in 5. So the objective function can be written as
+
+$$
+\operatorname* { m a x } I ( \hat { \mathbf { x } } ; \mathbf { w } ) + I ( \hat { \mathbf { y } } ; \mathbf { w } | \hat { \mathbf { x } } ) + \sum _ { ( \mathbf { x } , \mathbf { y } ) \in \mathcal { S } } [ I ( \mathbf { x } ; \mathbf { w } ) + I ( \mathbf { y } ; \mathbf { w } | \mathbf { x } ) ] .
+$$
+
+Directly computing the mutual information in Equation 7 is intractable since the true posteriori distributions like $p ( \hat { \mathbf { y } } | \hat { \mathbf { x } } , \mathbf { w } )$ , $p ( \hat { \mathbf { x } } | \mathbf { w } )$ are still unknown. Therefore, we use Variational Information Maximization (Barber & Agakov, 2003; Chen et al., 2016) to compute the lower bound of Equation 5. We use $p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { w } )$ to approximate the true posteriori distribution, where $\theta$ represents the model parameters. As a result, we have
+
+$$
+\begin{array} { r c l } { I ( \hat { \mathbf { x } } ; \mathbf { w } ) } & { = } & { H ( \hat { \mathbf { x } } ) - H ( \hat { \mathbf { x } } | \mathbf { w } ) } \\ & { = } & { H ( \hat { \mathbf { x } } ) + \mathbb { E } _ { \mathbf { w } \sim p ( \mathbf { w } | \hat { \mathbf { x } } , S ) } [ \mathbb { E } _ { \hat { \mathbf { x } } \sim p ( \hat { \mathbf { x } } | \mathbf { w } ) } [ \log p ( \hat { \mathbf { x } } | \mathbf { w } ) ] ] } \\ & { = } & { H ( \hat { \mathbf { x } } ) + \mathbb { E } _ { \mathbf { w } \sim p ( \mathbf { w } | \hat { \mathbf { x } } , S ) } [ D _ { \mathrm { K L } } ( p ( \hat { \mathbf { x } } | \mathbf { w } ) | | p \theta ( \hat { \mathbf { x } } | \mathbf { w } ) ) + \mathbb { E } _ { \hat { \mathbf { x } } \sim p ( \hat { \mathbf { x } } | \mathbf { w } ) } [ \log p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { w } ) ] ] } \\ & { \geq } & { H ( \hat { \mathbf { x } } ) + \mathbb { E } _ { \mathbf { w } \sim p ( \mathbf { w } | \hat { \mathbf { x } } , S ) } [ \mathbb { E } _ { \hat { \mathbf { x } } \sim p ( \hat { \mathbf { x } } | \mathbf { w } ) } [ \log p \theta ( \hat { \mathbf { x } } | \mathbf { w } ) ] ] } \end{array}
+$$
+
+$H ( \cdot )$ is the entropy of a random variable. $H ( { \hat { \mathbf { x } } } )$ is a constant value for given data. We can maximize this lower bound as the proxy for the true mutual information.
+
+Similar to $I ( \hat { \mathbf { x } } ; \mathbf { w } )$
+
+$$
+I ( \hat { \mathbf { y } } ; \mathbf { w } | \hat { \mathbf { x } } ) \geq H ( \hat { \mathbf { y } } | \hat { \mathbf { x } } ) + \mathbb { E } _ { \mathbf { w } \sim p ( \mathbf { w } | \hat { \mathbf { x } } , S ) } [ \mathbb { E } _ { \hat { \mathbf { y } } \sim p ( \hat { \mathbf { y } } | \hat { \mathbf { x } } , \mathbf { w } ) } [ \log p _ { \theta } ( \hat { \mathbf { y } } | \hat { \mathbf { x } } , \mathbf { w } ) ] ] ,
+$$
+
+$$
+\begin{array} { r } { \displaystyle \sum _ { \mathbf { x } , \mathbf { y } ) \in S } I ( ( \mathbf { x } , \mathbf { y } ) ; \mathbf { w } ) \geq \displaystyle \sum _ { ( \mathbf { x } , \mathbf { y } ) \in S } H ( ( \mathbf { x } , \mathbf { y } ) ) + \mathbb { E } _ { \mathbf { w } \sim p ( \mathbf { w } | \hat { \mathbf { x } } , S ) } [ \mathbb { E } _ { ( \mathbf { x } , \mathbf { y } ) \sim p ( ( \mathbf { x } , \mathbf { y } ) | \mathbf { w } ) } [ \log p _ { \theta } ( \mathbf { x } | \mathbf { w } ) + \log p _ { \theta } ( \mathbf { y } | \mathbf { x } , \mathbf { w } ) ] , } \end{array}
+$$
+
+$p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { w } ) , p _ { \theta } ( \mathbf { x } , \mathbf { y } | \mathbf { w } )$ are used to approximate the true posteriori distribution $p ( \hat { \mathbf { x } } | \mathbf { w } )$ and $p ( \mathbf { x } , \mathbf { y } | \mathbf { w } )$ .
+
+Put the lower bounds back into Equation 7. Omit the constant entropy terms and the expectation subscripts for clarity, we have the new objective function as
+
+$$
+\operatorname* { m a x } _ { \theta } \mathbb { E } [ \log p _ { \theta } ( \hat { \mathbf { y } } | \hat { \mathbf { x } } , \mathbf { w } ) + \log p _ { \theta } ( \mathbf { y } | \mathbf { x } , \mathbf { w } ) + \log p _ { \theta } ( \mathbf { x } | \mathbf { w } ) + \log p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { w } ) ] .
+$$
+
+The first two terms are maximizing the log likelihood of label for both support and query data with respective to the network parameters, given the generated classification weights. This is equivalent to minimizing the cross entropy between prediction and ground-truth. We assume that $p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { w } )$ and $p _ { \theta } ( \mathbf { x } | \mathbf { w } )$ are Gaussian distributions. $r _ { 1 }$ and $r _ { 2 }$ are used to approximate the mean of these two Gaussian distributions. Therefore maximizing the log likelihood is equivalent to reconstruct $\mathbf { x } ^ { c p }$ and $\hat { \mathbf { x } } ^ { a p }$ with $L 2$ loss. Thus the loss function to train the network can be written as
+
+$$
+L = \mathrm { C E } ( \hat { \bf y } _ { p r e d } , \hat { \bf y } ) + \lambda _ { 1 } \sum _ { { \bf y } \in \mathcal { S } } \mathrm { C E } ( { \bf y } _ { p r e d } , { \bf y } ) + \lambda _ { 2 } \sum _ { { \bf x } ^ { c p } \in \mathcal { S } } | | { \bf x } ^ { c p } - { \bf x } _ { r e } ^ { c p } | | _ { 2 } + \lambda _ { 3 } | | \hat { \bf x } ^ { a p } - \hat { \bf x } _ { r e } ^ { a p } | | _ { 2 } .
+$$
+
+CE here stands for cross entropy. $\mathbf { x } ^ { c p }$ and $\hat { \mathbf { x } } ^ { a p }$ are the inputs to weights generator $g$ . $\mathbf { x } _ { r e } ^ { c p } \sim p _ { \theta } ( \mathbf { x } | \mathbf { w } )$ and $\hat { \mathbf { x } } _ { r e } ^ { a p } \sim p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { w } )$ are the reconstruction of $\mathbf { x } ^ { c p }$ and $\hat { \mathbf { x } } ^ { a p }$ . Since we convert the log likelihood in Equation 14 to mean square error or cross entropy in Equation 15 to optimize, the value of each term in Equation 15 is not equal to real log likelihood and we have to decide the weightage for each one. $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$ are thus hyper-parameters for trade-off of different terms. With the help of last three terms, the generated classification weights are forced to carry information about the support data and the specific query sample.
+
+In LEO (Rusu et al., 2019), the inner update loss is computed as cross entropy on support data. If we merge the inner update into outer loop, then the loss becomes the summation of first two terms in Equation 15. However, the weight generation in LEO does not involve specific query samples, thus making reconstructing $\hat { \mathbf { x } } ^ { a p }$ impossible. In this sense, LEO can be regarded as a special case of our proposed method, where (1) only contextual path exits and (2) $\lambda _ { 2 } = \lambda _ { 3 } = 0$ .
+
+# 3.5 COMPLEXITY ANALYSIS
+
+The encoding process in contextual path results in computational complexity $O ( ( N K ) ^ { 2 } )$ due to self-attention. Similarly, the computational complexity of attentive path is $O ( ( N K ) ^ { 2 } + | \mathcal { Q } | ( N K ) )$ . In total, the complexity is ${ \cal O } ( ( \bar { N } K ) ^ { 2 } + | \mathcal { Q } | ( \bar { N ^ { } } K ) )$ . However, because of the nature of few-shot learning problem, the value of $( N K ) ^ { 2 }$ is usually negligible. The value of $| \mathcal { Q } |$ depends on the setting and the cross attention can be implemented parallelly via matrix multiplication. Therefore, the induced computational overhead will be negligible. AWGIM avoids the inner update without compromising the performance, which furthers reduces both training and inference time significantly. The empirical evaluation is presented in A.3.4.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASETS AND PROTOCOLS
+
+We conduct experiments on miniImageNet (Vinyals et al., 2016) and tieredImageNet (Ren et al., 2018), two commonly used benchmark datasets, to compare with other methods and analyze our model. Both datasets are subsets of ILSVRC-12 dataset (Russakovsky et al., 2015). miniImageNet contains 100 randomly sampled classes with 600 images per class. We follow the train/test split in (Ravi & Larochelle, 2016), where 64 classes are used for meta-training, 16 for meta-validation and 20 for meta-testing. tieredImageNet is a larger dataset compared to miniImageNet. There are 608 classes and 779,165 images in total. They are selected from 34 higher level nodes in ImageNet (Deng et al., 2009) hierarchy. 351 classes from 20 high level nodes are used for meta-training, 97 from 6 nodes for meta-validation and 160 from 8 nodes for meta-testing.
+
+We use the image features in LEO (Rusu et al., 2019) provided by the authors 2. They trained a 28-layer Wide Residual Network (Zagoruyko & Komodakis, 2016) on the meta-training set. Each image then is represented by a 640 dimensional vector, which is used as the input to our model.
+
+For $N$ -way $K$ -shot experiments, we randomly sample $N$ classes from meta-training set and each of them contains $K$ samples as the support set and 15 as query set. Similar to other works, we train 5-way 1-shot and 5-shot models on two dataset. During meta-testing, 600 $N$ -way $K$ -shot tasks are sampled from meta-testing set and the average accuracy for query set is reported with $9 5 \%$ confidence interval, as done in recent works (Finn et al., 2017; Snell et al., 2017; Rusu et al., 2019).
+
+# 4.2 IMPLEMENTATION DETAILS
+
+We use TensorFlow (Abadi et al., 2016) to implement our method and the code will be made available. $d = 6 4 0$ is the dimension of feature embeddings. $d _ { h }$ is set to be 128. The number of heads $h$ in attention module is set to be 4. $g , r _ { 1 }$ and $r _ { 2 }$ are 2-layer MLPs with 256 hidden units. We decide $\lambda _ { 1 } = 1$ , $\lambda _ { 2 } = \lambda _ { 3 } = 0 . 0 0 1$ by meta-validation performance.
+
+Table 1: Accuracy comparison with other approaches on miniImageNet. The results are averaged on 600 tasks from meta-testing set with $9 5 \%$ confidence interval. Best results are highlighted.   
+
+<table><tr><td>Model</td><td>Feature Extractor</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>Matching Networks (Vinyals et al., 2016)</td><td>Conv-4</td><td>46.60</td><td>60.00</td></tr><tr><td>MAML(Finn et al.,2017)</td><td>Conv-4</td><td>48.70 ± 1.84%</td><td>63.11 ± 0.92%</td></tr><tr><td>Meta LSTM (Ravi &amp; Larochelle, 2016)</td><td>Conv-4</td><td>43.44 ± 0.77%</td><td>60.60 ± 0.71%</td></tr><tr><td>Prototypical Nets (Snell et al.,2017)</td><td>Conv-4</td><td>49.42 ± 0.78%</td><td>68.20 ± 0.66%</td></tr><tr><td>Relation Nets (Sung et al.,2018)</td><td>Conv-4</td><td>50.44 ± 0.82%</td><td>65.32 ± 0.70%</td></tr><tr><td>SNAIL (Mishra et al., 2018)</td><td>Resnets-12</td><td>55.71 ± 0.99%</td><td>68.88 ± 0.92%</td></tr><tr><td>TPN (Liu et al., 2019)</td><td>Resnets-12</td><td>59.46</td><td>75.65</td></tr><tr><td>MTL (Sun et al., 2019)</td><td>Resnets-12</td><td>61.20 ± 1.80%</td><td>75.50 ± 0.80</td></tr><tr><td>Dynamic (Gidaris &amp; Komodakis,2018)</td><td>WRN-28-10</td><td>60.06 ± 0.14%</td><td>76.39 ± 0.11%</td></tr><tr><td>Prediction (Qiao et al., 2018)</td><td>WRN-28-10</td><td>59.60 ± 0.41%</td><td>73.74 ± 0.19%</td></tr><tr><td>DAE-GNN (Gidaris &amp; Komodakis,2019)</td><td>WRN-28-10</td><td>62.96 ± 0.15%</td><td>78.85 ± 0.10%</td></tr><tr><td>LEO (Rusu et al., 2019)</td><td>WRN-28-10</td><td>61.76 ± 0.08%</td><td>77.59 ± 0.12%</td></tr><tr><td>AWGIM (ours)</td><td>WRN-28-10</td><td>63.12 ± 0.08%</td><td>78.40 ± 0.11%</td></tr></table>
+
+Table 2: Accuracy comparison with other approaches on tieredImageNet. The results are averaged on 600 tasks from meta-testing set with $9 5 \%$ confidence interval. Best results are highlighted.   
+
+<table><tr><td>Model</td><td>Feature Extractor</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>MAML (Finn et al., 2017)</td><td>Conv-4</td><td>51.67 ± 1.81%</td><td>70.30 ± 1.75%</td></tr><tr><td>Prototypical Nets (Snell et al., 2017)</td><td>Conv-4</td><td>53.31± 0.89%</td><td>72.69 ± 0.74%</td></tr><tr><td>Relation Nets (Sung et al., 2018)</td><td>Conv-4</td><td>54.48 ± 0.93%</td><td>71.32 ± 0.78%</td></tr><tr><td>TPN (Liu et al., 2019)</td><td>Conv-4</td><td>59.91 ± 0.96%</td><td>72.85 ± 0.74%</td></tr><tr><td>MetaOptNet (Lee et al., 2019)</td><td>Resnets-12</td><td>65.81 ± 0.74%</td><td>81.75 ± 0.53%</td></tr><tr><td>LEO (Rusu et al., 2019)</td><td>WRN-28-10</td><td>66.33 ± 0.05%</td><td>81.44 ± 0.09%</td></tr><tr><td>AWGIM (ours)</td><td>WRN-28-10</td><td>67.69 ± 0.11%</td><td>82.82 ± 0.13%</td></tr></table>
+
+ADAMW Loshchilov & Hutter (2017) is used to optimize the network with weight decay $1 \times 1 0 ^ { - 6 }$ . The initial learning rate is set to 0.0002 for 5-way 1-shot and 0.001 for 5-way 5-shot, which is decayed by 0.2 for every 15,000 iterations. We train the model for 50,000 iterations. Batch size is 64 for 5-way 1-shot and 32 for 5-way 5-shot. Similar to LEO (Rusu et al., 2019), we first train the model on meta-training set and choose the optimal hyper-parameters by validation results. Then we train the model on meta-training and meta-validation sets together using fixed hyper-parameters.
+
+# 4.3 COMPARISON WITH OTHER METHODS
+
+We compare the performance of our approach AWGIM on two datasets with several state-of-theart methods proposed in recent years. The results of MAML, Prototypical Nets, Relation Nets on tieredImageNet are evaluated by Liu et al. (2019). The results of Dynamic on miniImageNet with WRN-28-10 as the feature extractor is reported in (Gidaris & Komodakis, 2019). The other results are reported in the corresponding original papers. We also include the backbone network structure of the used feature extractor for reference. The results on miniImageNet and tieredImageNet are shown in Table 1 and 2 respectively.
+
+The top half parts of Table 1 and 2 display the methods belonging with different meta learning categories, such as metric-based(Matching Networks, Prototypical Nets), gradient-based (MAML, MTL), graph-based (TPN). The bottom part shows the classification weights generation approaches including Dynamic, Prediction, DAE-GNN, LEO and our proposed AWGIM.
+
+AWGIM can outperform all the methods in top parts of two table. Comparing with other classification weights generation methods in the bottom part, AWGIM still shows very competitive performance, namely the best on tieredImageNet and close to the state-of-the-art on miniImageNet. We note that all the classification weights generation methods are using WRN-28-10 as backbone network, which makes the comparison fair. In particular, AWGIM can outperform LEO in all settings.
+
+# 4.4 ANALYSIS
+
+Table 3: Analysis of our proposed AWGIM. In the top half, the attentive path is removed to compare with LEO. In the bottom part, ablation analysis with respective to different components is provided. We also shuffle the generated classification weights randomly to show that they are indeed optimal for different query samples.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">miniImageNet</td><td colspan="2">tieredImageNet</td></tr><tr><td>5-way 1-shot</td><td>5-way 5-shot</td><td> 5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>LEO</td><td>61.76 %</td><td>77.59 %</td><td>66.33%</td><td>81.44 %</td></tr><tr><td>Generator in LEO</td><td>60.33 %</td><td>74.53 %</td><td>65.17%</td><td>78.77 %</td></tr><tr><td>Generator conditioned on S only</td><td>61.02%</td><td>74.33%</td><td>66.22%</td><td>79.66%</td></tr><tr><td>Generator conditioned on S with IM</td><td>62.04%</td><td>77.54%</td><td>66.43%</td><td>81.73%</td></tr><tr><td>MLP encoding,入1 = 入2= 入3=0</td><td>58.95%</td><td>71.68%</td><td>63.92%</td><td>75.80%</td></tr><tr><td>MLP encoding</td><td>62.26%</td><td>76.91%</td><td>65.84%</td><td>79.24%</td></tr><tr><td>入1=λ2=λ3=0</td><td>61.61%</td><td>74.14%</td><td>65.65%</td><td>79.93%</td></tr><tr><td>入1=入2=0</td><td>62.06%</td><td>74.18%</td><td>65.85%</td><td>80.42%</td></tr><tr><td>入3=0</td><td>62.91%</td><td>77.88%</td><td>67.27%</td><td>81.67%</td></tr><tr><td>入1=0</td><td>62.19%</td><td>74.21%</td><td>66.82%</td><td>80.61%</td></tr><tr><td>2=入g=0</td><td>62.12%</td><td>77.65%</td><td>66.86%</td><td>81.03%</td></tr><tr><td>random shuffle in class</td><td>62.87%</td><td>77.48%</td><td>67.52%</td><td>82.55%</td></tr><tr><td>random shuffle between classes</td><td>61.20%</td><td>77.48%</td><td>66.55%</td><td>82.53%</td></tr><tr><td>AWGIM (ours)</td><td>63.12%</td><td>78.40%</td><td>67.69 %</td><td>82.82%</td></tr></table>
+
+We perform detailed analysis on AWGIM, shown in Table 3. We include the results of LEO Rusu et al. (2019) for reference. “Generator in LEO” means that there is no inner update in LEO. In the upper part of the table, we first studied the effect of attentive path. We implemented two generators including only the contextual path during encoding. “Generator conditioned on $s$ with IM” indicates that we add the cross entropy loss and reconstruction loss for support set. It can be observed that “Generator conditioned on $s$ only” is trained with cross entropy on query set, which is similar to “Generator in LEO” without inner update. It is able to achieve similar or slightly better results than “Generator in LEO”, which implies that self-attention is no worse than relation networks used in LEO to model task-context. With information maximization, our generator is able to obtain slightly better performance than LEO.
+
+The effect of attention is investigated by replacing the attention modules with 2-layer MLPs, which is shown as “MLP encoding”. More specifically, one MLP in contextual path is used for support set and another MLP in attentive path for query samples. We can see that even without attention to encode the task-contextual information, “MLP encoding” can achieve accuracy close to LEO, for the sake of information maximization. However, if we let $\lambda _ { 1 } = \lambda _ { 2 } = \lambda _ { 3 } = 0$ for MLP encoding, the performance drops significantly, which demonstrates the importance of maximizing the information.
+
+We conducted ablation analysis with respective to $\lambda _ { 1 } , \lambda _ { 2 }$ and $\lambda _ { 3 }$ to investigate the effect of information maximization. First, $\lambda _ { 1 }$ , $\lambda _ { 2 }$ and $\lambda _ { 3 }$ are all set to be 0. In this case, the accuracy is similar to “generator conditioned on $s$ only”, showing that the generated classification weights are not fitted for different query samples, even with the attentive path. It can also be observed that maximizing the mutual information between weights and support is more crucial since $\lambda _ { 1 } = \lambda _ { 2 } = 0$ degrades accuracy significantly, comparing with $\lambda _ { 3 } = 0$ . We further investigate the relative importance of the classification on support as well as reconstruction. $\lambda _ { 1 } = 0$ affects the performance noticeably. We conjecture that the support label prediction is more critical for information maximization.
+
+The classification weights are generated specifically for each query sample in AWGIM. To this point, we shuffle the classification weights between query samples within the same classes and between different classes as well to study whether the classification weights are adapted for different query samples. Assume there are T query samples per class in one task. Wf inal ∈ R|Q|×N×d can be reshaped into Wf inal $\mathbf { W } ^ { f i n a l } \in \mathbb { R } ^ { N \times T \times N \times \bar { d } }$ . Then we shuffle this weight tensor along the first and second axis randomly. The results are shown as “random shuffle between classes” and “random shuffle in class” in Table 3. For 5-way 1-shot experiments, the random shuffle between classes degrades the accuracy noticeably while the random shuffle in class dose not affect too much. This indicates that when the support data are very limited, the generated weights for query samples from the same class are very similar to each other while distinct for different classes. When there are more labeled data in support set, two kinds of random shuffle show very close or even the same results in 5-way 5-shot experiments, which are both worse than the original ones. This implies that the generated classification weights are more diverse and specific for each query sample in 5-way 5-shot setting. The possible reason is that larger support set provides more knowledge to estimate the optimal classification weights for each query example.
+
+More analysis is provided in Appendix A.3.
+
+# 5 CONCLUSION
+
+In this work, we introduce Attentive Weights Generation via Information Maximization (AWGIM) for few shot image classification. AWGIM learns to generate optimal classification weights for each query sample within the task by two encoding paths. To guarantee this, the lower bound of mutual information between generated weights and query, support data is maximized. As far as we know, AWGIM is the first work utilizing mutual information techniques for few shot learning. The effectiveness of AWGIM is demonstrated by state-of-the-art performance on two benchmark datasets and extensive analysis.
+
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+
+# A APPENDIX
+
+# A.1 MUTLI-HEAD ATTENTION IN AWGIM
+
+The multi-head attention can be described as
+
+$$
+M u l t i H e a d ( Q , K , V ) = C o n c a t ( h e a d _ { 1 } , . . . , h e a d _ { H } ) W ^ { O } ,
+$$
+
+$$
+h e a d _ { i } ( Q ^ { i } , K ^ { i } , V ^ { i } ) = A t t e n t i o n ( Q ^ { i } , K ^ { i } , V ^ { i } ) ,
+$$
+
+$$
+A t t e n t i o n ( Q , K , V ) = s o f t m a x ( \frac { Q K ^ { T } } { \sqrt { d _ { k } } } V ) ,
+$$
+
+$$
+Q ^ { i } = Q W _ { Q } ^ { i } , K ^ { i } = K W _ { K } ^ { i } , V ^ { i } = V W _ { V } ^ { i } ,
+$$
+
+Here $Q , K , V$ are query, key, value matrices. $W _ { Q } ^ { i } , W _ { K } ^ { i } , W _ { V } ^ { i }$ are the weight matrices for $i$ th head. $W ^ { O }$ is the weight matrix for output. $d _ { k }$ is the dimension of keys. Original $Q$ is added to the output of Equation 16 to stabilize the training as residual learning.
+
+# A.2 MODEL DETAILS
+
+# A.2.1 CONTEXTUAL PATH
+
+The encoding process in contextual path is realized by a simple multi-head self-attention network on support data. First, ${ \bf x } ^ { c _ { n } ; k }$ are mapped to a lower dimensional hidden space by a MLP $f _ { 1 } : \mathbb { R } ^ { d } $ $\mathbb { R } ^ { d _ { h } }$ to reduce the computation complexity. Then the low-dimensional representations $\mathbf { x } _ { h 1 } ^ { c _ { n } ; k }$ are processed by the -head self-attention network $f _ { c p } ^ { s a } : \mathbb { R } ^ { d _ { h } }  \mathbb { R } ^ { d _ { h } }$ ,
+
+$$
+{ \bf X } ^ { c p } = M u l t i H e a d A t t e n t i o n ( Q = { \bf X } _ { h 1 } , K = { \bf X } _ { h 1 } , V = { \bf X } _ { h 1 } ) .
+$$
+
+$\mathbf { X } _ { h 1 } \in \mathbb { R } ^ { N K \times d _ { h } }$ is the matrix where each row stands for one support sample $\mathbf { x } _ { h 1 } ^ { c _ { n } ; k }$ . For one $N$ -way $K$ -shot task, the outputs of $f _ { c p } ^ { s a }$ .
+
+# A.2.2 ATTENTIVE PATH
+
+The attentive path is instantiated by attention, similar to contextual path. First, a MLP $f _ { 2 } : \mathbb { R } ^ { d } $ $\mathbb { R } ^ { d _ { h } }$ is used to map both ${ \bf x } ^ { c _ { n } ; k }$ and $\hat { \bf x }$ to $\mathbf { x } _ { h 2 } ^ { c _ { n } ; k }$ and $\hat { \mathbf { x } } _ { h 2 }$ . Then we employ another $H$ -head selfattention network $f _ { a p } ^ { s a } : \mathbb { R } ^ { d _ { h } }  \mathbb { R } ^ { d _ { h } }$ on $\mathbf { x } _ { h 2 } ^ { c _ { n } ; k }$ to encode the global task information to each support sample,
+
+$$
+{ \bf X } ^ { a p } = M u l t i H e a d A t t e n t i o n ( Q = { \bf X } _ { h 2 } , K = { \bf X } _ { h 2 } , V = { \bf X } _ { h 2 } ) .
+$$
+
+The cross attention between query and context-aware support samples are computed as
+
+$$
+{ \hat { \bf X } } ^ { a p } = M u l t i H e a d A t t e n t i o n ( Q = { \hat { \bf X } } _ { h 2 } , K = { \bf X } _ { h 2 } , V = { \bf X } ^ { a p } ) .
+$$
+
+Here $\hat { \mathbf { X } } ^ { a p } \in \mathbb { R } ^ { | \mathcal { Q } | \times d _ { h } }$ is the matrix form of $\hat { \mathbf { x } } _ { q }$ , where each query sample is context-aware.
+
+# A.2.3 WEIGHT GENERATOR
+
+Assume $\mathbf { x } ^ { c p \oplus a p } = \mathbf { X } _ { i , j , : } ^ { c p \oplus a p } \in \mathbb { R } ^ { 2 d _ { h } }$ , where $i , j$ stands for $i$ th query sample and $j$ th support sample. $\mathbf { x } ^ { c p \oplus a p }$ is decoded by the weights generator $g : \mathbb { R } ^ { 2 d _ { h } }  \mathbb { R } ^ { 2 d }$ . We assume that the classification weights follow Gaussian distribution with diagonal covariance and we sample the weights from this distribution during meta-training, shown in Equation 23 and 24.
+
+$$
+\mu _ { \mathbf { w } } , \sigma _ { \mathbf { w } } = g ( \mathbf { z } )
+$$
+
+$$
+\mathbf { w } \sim \mathcal { N } ( \mu _ { \mathbf { w } } , \pmb { \Sigma } _ { \mathbf { w } } )
+$$
+
+# A.3 EXPERIMENTAL ANALYSIS
+
+# A.3.1 FEW SHOT REGRESSION
+
+AWGIM can be applied to few shot regression task by slight modification. During meta-training, we set the number of classes $N$ equal to 1 and adapt the cross entropy loss to mean square error. We use the data points $( x , y )$ as inputs to AWGIM and generate weight as well as bias parameters for a three layer MLP with hidden dimension 40. This is consistent with few shot regression experimental setting in LEO.
+
+The few shot regression tasks are constructed as either sinusoidal or linear regression tasks. For sinusoidal regression tasks, the amplitude range is [0.1, 5], phase range $[ 0 , 2 \pi ]$ , frequency range [0.5, 2.0]. For linear regression tasks, the slope range is $[ - 1 , 1 ]$ , intercept range $[ - 5 , 5 ]$ . Input $x$ is randomly sample from $[ - 5 , 5 ]$ . Gaussian noise with standard deviation 0.3 is added to $y$ during meta-training. We show some qualitative results in Figure 2. (a) and (b) are examples that can be tackled easily. For some non-trivial cases such as (c) and (d), AWGIM produces predictions slightly mixing with another regression family, despite that overall results are still faithful.
+
+![](images/df7812dbdfdc7c5baf047dcc3b50d4a47220543c760af8eaedde57f4917c7892.jpg)  
+Figure 2: 5-shot regression results for a multi-modal task distribution. Regression targets are plotted in red and prediction in black. 5 training samples per task are plotted with blue solid circles.
+
+# A.3.2 EFFECT OF MULTI-HEAD ATTENTION
+
+We replace the multi-head attention in the two paths with single-head attention and conduct the 5- way 1-shot and 5-way 5-shot experiments on miniImageNet dataset. The results are shown in Table 4. We can see clearly that multi-head attention improve the performance. In particular, for 5-way 1-shot experiment, single head attention gives results close to MLP encoding, which indicates that single head attention struggles when data are extremely scarce.
+
+Table 4: Accuracy results on miniImageNet with 4 heads or single head in attention networks.   
+
+<table><tr><td>Method</td><td>5-way 1 -shot</td><td> 5-way 5-shot</td></tr><tr><td>4 heads</td><td>63.12%</td><td>78.40%</td></tr><tr><td>single head</td><td>62.35%</td><td>77.75%</td></tr></table>
+
+# A.3.3 CONVERGENCE
+
+We compare AWGIM with LEO in terms of convergence speed. The batch size is set to be 16 for both methods. We use the hyper-parameters tuned by authors to train LEO. The accuracy of metavalidation set during meta-training on 5-way 1-shot miniImageNet is plotted, shown in Figure 3. we can see clearly that AWGIM converges faster than LEO and outperforms LEO except for the first few iterations.
+
+![](images/70012493ad6568c4a0bf7152014453a6a19cdde405da29751675aab5579ea992.jpg)  
+Figure 3: The meta-validation accuracy during meta-training.
+
+# A.3.4 INFERENCE TIME
+
+We measure the inference time of AWGIM to show that it induces minimal computational overhead. In comparison, we use “MLP encoding” in two paths, which has time complexity $O ( N K + | \mathcal { Q } | )$ . We use two set-ups on miniImageNet and the batch size is set to be 64. 100 batches are processed and we report the average consumed time for one batch. All these experiments on done with the same GPU and workstation. The results are shown in Table 5. It can be observed that the usage of self-attention and cross attention in AWGIM occurs negligible overhead, compared with MLP encoding. This is because the values of $N , K , | \mathcal { Q } |$ are all relatively small and matrix multiplication further can be processed very fast by GPU.
+
+Table 5: The comparison of inference time between AWGIM and MLP encoding.   
+
+<table><tr><td>Method</td><td>5-way 1 -shot</td><td>5-way 5-shot</td></tr><tr><td>AWGIM</td><td>0.036s</td><td>0.093s</td></tr><tr><td>MLP encoding</td><td>0.033s</td><td>0.093s</td></tr></table>
+
+# A.3.5 VISUALIZATION
+
+We visualize the generated classification weights by t-SNE (Maaten & Hinton, 2008). First we sample 400 tasks from meta-validation set of 5-way 1-shot miniImageNet experiment. Each task contains 5 query samples from 5 different classes. Thus in total there are $4 0 0 \times 5 \times 5 = 1 0 , 0 0 0$ weight vectors to visualize. As comparison, inputs to the generator $g$ are also plotted. The visualization results are shown in Figure 4. The inputs to $g$ are displayed in (a, b) and the generated classification weights in (c, d). From the comparison between (a) and (c), we can see the decoded weights for each class in (c) are clustered closer than (a) in general. Red and blue dots in (b, d) denotes the classification weights for two query samples from two classes within one task. It can be observed that $g$ can generate adapted weights for different query samples. This is consistent with Table 3, where the results of “random shuffle between classes” suggest that query samples from different class have distinct classification weights.
+
+![](images/7279dd6a6dd60341910d31f04f08cc86bd5dc1b2d43395c825b56188a42d47cb.jpg)  
+Figure 4: t-SNE visualization of the inputs to $g$ in (a, b) and the generated classification weights in (c, d). Blue and red dots in (b) and (d) are the classification weights for two query samples in the same task.

parse/train/BJxpIJHKwB/BJxpIJHKwB_content_list.json

@@ -0,0 +1,2121 @@
+[
+    {
+        "type": "text",
+        "text": "ATTENTIVE WEIGHTS GENERATION FOR FEW SHOTLEARNING VIA INFORMATION MAXIMIZATION",
+        "text_level": 1,
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+    {
+        "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,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Few shot image classification aims at learning a classifier from limited labeled data. Generating the classification weights has been applied in many metalearning approaches for few shot image classification due to its simplicity and effectiveness. However, we argue that it is difficult to generate the exact and universal classification weights for all the diverse query samples from very few training samples. In this work, we introduce Attentive Weights Generation for few shot learning via Information Maximization (AWGIM), which addresses current issues by two novel contributions. i) AWGIM generates different classification weights for different query samples by letting each of query samples attends to the whole support set. ii) To guarantee the generated weights adaptive to different query sample, we re-formulate the problem to maximize the lower bound of mutual information between generated weights and query as well as support data. As far as we can see, this is the first attempt to unify information maximization into few shot learning. Both two contributions are proved to be effective in the extensive experiments and we show that AWGIM is able to achieve state-of-the-art performance on benchmark datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            336,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While deep learning methods achieve great success in domains such as computer vision (He et al., 2016), natural language processing (Devlin et al., 2018), reinforcement learning (Silver et al., 2018), their hunger for large amount of labeled data limits the application scenarios where only a few data are available for training. Humans, in contrast, are able to learn from limited data, which is desirable for deep learning methods. Few shot learning is thus proposed to enable deep models to learn from very few samples (Fei-Fei et al., 2006). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Meta learning is by far the most popular and promising approach for few shot problems (Vinyals et al., 2016; Finn et al., 2017; Snell et al., 2017; Ravi & Larochelle, 2016; Rusu et al., 2019). In meta learning approaches, the model extracts high level knowledge across different tasks so that it can adapt itself quickly to a new-coming task (Schmidhuber, 1987; Andrychowicz et al., 2016). There are several kinds of meta learning methods for few shot learning, such as gradient-based (Finn et al., 2017; Ravi & Larochelle, 2016) and metric-based (Snell et al., 2017; Sung et al., 2018). Weights generation, among these different methods, has shown effectiveness with simple formulation (Qi et al., 2018; Qiao et al., 2018; Gidaris & Komodakis, 2018; 2019). In general, weights generation methods learn to generate the classification weights for different tasks conditioned on the limited labeled data. However, fixed classification weights for different query samples within one task might be sub-optimal, due to the few shot challenge. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We introduce Attentive Weights Generation for few shot learning via Information Maximization (AWGIM) in this work to address these limitations. In AWGIM, the classification weights are generated for each query sample specifically. This is done by two encoding paths where the query sample attends to the task context. However, we show in experiments that simple cross attention between query samples and support set fails to guarantee classification weights fitted to diverse query data since the query-specific information is lost during weights generation. Therefore, we propose to maximize the lower bound of mutual information between generated weights and query, support data. As far as we know, AWGIM is the first work introducing Variational Information Maximization in few shot learning. The induced computational overhead is minimal due to the nature of few shot problems. Furthermore, by maximizing the lower bound of mutual information, AWGIM gets rid of inner update without compromising performance. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "AWGIM is evaluated on two benchmark datasets and shows state-of-the-art performance. We also conducted detailed analysis to validate the contribution of each component in AWGIM. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            189,
+            354,
+            205
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 FEW SHOT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            222,
+            364,
+            236
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning from few labeled training data has received growing attentions recently. Most successful existing methods apply meta learning to solve this problem and can be divided into several categories. In the gradient-based approaches, an optimal initialization for all tasks is learned (Finn et al., 2017). Ravi & Larochelle (2016) learned a meta-learner LSTM directly to optimize the given fewshot classification task. Sun et al. (2019) learned the transformation for activations of each layer by gradients to better suit the current task. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the metric-based methods, a similarity metric between query and support samples is learned. (Koch et al., 2015; Vinyals et al., 2016; Snell et al., 2017; Sung et al., 2018; Li et al., 2019a). Spatial information or local image descriptors are also considered in some works to compute richer similarities (Lifchitz et al., 2019; Li et al., 2019b; Wertheimer & Hariharan, 2019). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Generating the classification weights directly has been explored by some works. Gidaris & Komodakis (2018) generated classification weights as linear combinations of weights for base and novel classes. Similarly, Qiao et al. (2018) and Qi et al. (2018) both generated the classification weights from activations of a trained feature extractor. Graph neural network denoising autoencoders are used in (Gidaris & Komodakis, 2019). Munkhdalai & Yu (2017) proposed to generate “fast weights” from the loss gradient for each task. All these methods do not consider generating different weights for different query examples, nor maximizing the mutual information. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There are some other methods for few-shot classification. Generative models are used to generate or hallucinate more data in (Zhang et al., 2018; Wang et al., 2018; Chen et al., 2019). Bertinetto et al. (2019) and Lee et al. (2019) used the closed-form solutions directly for few shot classification. Liu et al. (2019) integrated label propagation on a transductive graph to predict the query class label. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 ATTENTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Attention mechanism shows great success in computer vision (Xu et al., 2015; Parmar et al., 2018) and natural language processing (Bahdanau et al., 2015; Vaswani et al., 2017). It is effective in modeling the interaction between queries and key-value pairs from certain context. Based on the fact that keys and queries point to the same entities or not, people refer to attention as self attention or cross attention. In this work, we use both types of attention to encode the task and query-task information. The work most similar to ours is Attentive Neural Processes (Kim et al., 2019), which also employs self and cross attention. However, we are using attention for few-shot image classification via maximizing the mutual information. In stark contrast, Kim et al. (2019) worked on regression from the perspective of a stochastic process and the variational objective is optimized. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.3 MUTUAL INFORMATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Given two random variables $\\mathbf { X }$ and $\\mathsf { y }$ , mutual information $I ( { \\bf x } ; { \\bf y } )$ measures the decrease of uncertainty in one random variable when another is known. It is defined as the Kullback-Leibler divergence between joint distribution $p ( \\mathbf { x } , \\mathbf { y } )$ and product of marginal distributions $p ( \\mathbf { x } ) \\otimes p ( \\mathbf { y } )$ , ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/a5c5aceddcad5349f89ef0ff593cc9e307d3cc83ab1b05115265cea9c48d5a58.jpg",
+        "text": "$$\nI ( \\mathbf { x } ; \\mathbf { y } ) = D _ { \\mathrm { K L } } ( p ( \\mathbf { x } , \\mathbf { y } ) \\| p ( \\mathbf { x } ) \\otimes p ( \\mathbf { y } ) ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "When x and y are independent, $p ( \\mathbf { x } , \\mathbf { y } ) = p ( \\mathbf { x } ) \\otimes p ( \\mathbf { y } )$ so that $I ( { \\bf x } , { \\bf y } ) = 0$ , indicating that knowing $\\mathbf { X }$ does not reveal any information about y. When y is a deterministic function of $\\mathbf { X }$ , $I ( \\mathbf { x } , \\mathbf { y } )$ achieves its maximum value. Mutual information has been widely applied in applications such as Generative Adversarial Networks(Chen et al., 2016), self-supervised learning(Hjelm et al., 2019), visual question generation Krishna et al. (2019) and so on. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/1337a8d6253363eec5b99fdb1e3eb590ae641d0447bd8c18d057cba309ac3b81.jpg",
+        "image_caption": [
+            "Figure 1: The overview of our proposed AWGIM. The input task is 5-way 1-shot with $\\mathbf { X }$ as support set and $\\hat { \\bf x }$ as one query example. Different colors of the data in support set indicate different categories. The encoding process in contextual path produces context-aware support representations $\\mathbf { X } ^ { c p }$ . Similarly, the attentive path enables the query sample $\\hat { \\bf x }$ to be equipped with task knowledge. Both paths are achieved by attention mechanism. $\\hat { \\mathbf { x } } ^ { a p }$ is repeated to concatenate with $\\mathbf { X } ^ { c p }$ . The weight generator $g$ takes these concatenated representations as input to generate classification weights W specific for $\\hat { \\bf x }$ , denoted by the colorful matrix with slash. It can be used to predict the class label for $\\hat { \\bf x }$ and X. W is also used to reconstruct the inputs of the generator $g$ by two networks $r _ { 1 }$ and $r _ { 2 }$ . In this way, the lower bound of mutual information is maximized and $g$ is forced to generate classification weights sensitive to different query samples. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3 PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we provide the problem formulation first. Then the proposed model is described in Sec. 3.3. The objective function, which maximizes the mutual information between certain variables, and theoretical analysis are provided in Sec. 3.4. ",
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+    {
+        "type": "text",
+        "text": "3.1 PROBLEM FORMULATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Following many popular meta-learning methods for few shot classification, we formulate the problem under episodic training paradigm (Vinyals et al., 2016; Finn et al., 2017). One $N$ -way $K$ -shot task sampled from an unknown task distribution $P ( \\tau )$ includes support set and query set: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0de867923f89b0805ac52d436041277cdab8febfac60733b5ef9e8d07f3f4bde.jpg",
+        "text": "$$\n{ \\mathcal { T } } = ( { \\mathcal { S } } , { \\mathcal { Q } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            616,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { S } = \\{ ( \\mathbf { x } ^ { c _ { n } ; k } , \\mathbf { y } ^ { c _ { n } ; k } ) | k = 1 , . . . , K ; n = 1 , . . . , N \\}$ , $\\mathcal { Q } = \\{ ( \\hat { \\mathbf { x } } _ { 1 } , . . . , \\hat { \\mathbf { x } } _ { | \\mathcal { Q } | } ) \\}$ . Support set $s$ contains $N K$ labeled samples. Query set $\\mathcal { Q }$ includes $\\hat { \\bf x }$ and we need to predict label $\\hat { \\mathbf { y } }$ for $\\hat { \\bf x }$ based on $s$ . During meta-training, the meta-loss is estimated on $\\mathcal { Q }$ to optimize the model. During metatesting, the performance of meta-learning method is evaluated on $\\mathcal { Q }$ , provided the labeled $s$ . The classes used in meta-training and meta-testing are disjoint so that the meta-learned model needs to learn the knowledge transferable across tasks and adapt itself quickly to novel tasks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our proposed approach follows the general framework to generate the classification weights (Qi et al., 2018; Qiao et al., 2018; Rusu et al., 2019; Gidaris & Komodakis, 2018; 2019). In this framework, there is a feature extractor to output image feature embeddings. The meta-learner needs to generate the classification weights for different tasks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 LATENT EMBEDDING OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Latent Embedding Optimization (LEO) (Rusu et al., 2019) is one of the weights generation methods that is most related to our work. In LEO, the latent code $_ z$ is generated by $h$ conditioned on support set $s$ , described as $z = h ( S )$ . $h$ is instantiated as relation networks (Santoro et al., 2017). Classification weights $\\pmb { w }$ can be decoded from $_ { z }$ with $l$ , $w = l ( z )$ . In the inner loop, we use $\\pmb { w }$ to compute the loss (usually cross entropy) on the support set and then update $_ z$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1a8f1648970e8cb0965b908fe05c3d331b98ca8281d97df1aaa73f35151166b4.jpg",
+        "text": "$$\n\\begin{array} { r } { z ^ { \\prime } = z - \\eta \\nabla _ { z } \\mathcal { L } _ { \\mathcal { S } } ( \\pmb { w } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { S }$ indicates that the loss is evaluated on $s$ only. The updated latent code $z ^ { \\prime }$ is used to decode new classification weights $\\mathbf { \\Delta } \\mathbf { w ^ { \\prime } }$ with generating function $l$ . $\\mathbf { \\Delta } _ { \\mathbf { \\ b { w } } ^ { \\prime } }$ is adopted in the outer loop for query set $\\mathcal { Q }$ and the objective function of LEO then can be written as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2776c5a7aa8f78186817e0baf926d489cbb0a4f2d8bcf4b2c986d290fa80b5b8.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\mathcal { L } _ { \\mathcal { Q } } ( w ^ { \\prime } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            452,
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+            170
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here $\\theta$ stands for the parameters of $h$ and $l$ and we omit the regularization terms for clarity. LEO avoids updating high-dimensional $\\textbf { \\em w }$ in the inner loop by learning a lower-dimensional latent space, from which sampled $_ z$ can be used to generate $\\pmb { w }$ . The most significant difference between LEO and AWGIM is that we do not need inner updates to adapt the model. Instead, AWGIM is a feedforward network trained to maximize the mutual information so that it fits to different tasks well. On the other hand, AWGIM learns to generate optimal classification weights for each query sample while LEO generates fixed weights conditioned on the support set within one task. In Section 3.4 we will show LEO can be casted as a special case of AWGIM under certain conditions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 ATTENTIVE WEIGHTS GENERATION",
+        "text_level": 1,
+        "bbox": [
+            176,
+            299,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The framework of our proposed method is shown in Figure 1. Assume that we have a feature extractor, which can be a simple 4-layer Convnet or a deeper Resnet. All the images included in the sampled task $\\tau$ are processed by this feature extractor and represented as $d$ -dimensional vectors afterwards, i.e., $\\mathbf { x } ^ { c _ { n } ; k } , \\hat { \\mathbf { x } } \\in \\mathbb { R } ^ { d }$ . There are two paths to encode the task context and the individual query sample respectively, which are called contextual path and attentive path. The outputs of both paths are concatenated together as input to the generator for classification weights. Generated classification weights are used to not only predict the label of $\\hat { \\bf x }$ , but also maximize the lower bound of mutual information between itself and other variables, which will be discussed in the following section 3.4. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 CONTEXTUAL AND ATTENTIVE PATHS ",
+        "text_level": 1,
+        "bbox": [
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+            465,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The encoding process includes two paths, namely the contextual path and attentive path. The contextual path aims at learning representations for only the support set with a multi-head self-attention network $f _ { s a } ^ { c p }$ (Vaswani et al., 2017). The outputs of contextual path ${ \\bf X } ^ { c p } \\in \\mathbb { R } ^ { N K \\times d _ { h } }$ 1 thus contain richer information about the task and can be used later for weights generation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Existing weights generation methods generate the classification weights conditioned on the support set only, which is equivalent to using contextual path. However, the classification weights generated in this way might be sub-optimal. This is because estimating the exact and universal classification weights from very few labeled data in the support set is difficult and sometimes impossible. The generated weights are usually in lack of adaptation to different query samples. We address this issue by introducing attentive path, where the individual query example attends to the task context and then is used to generate the classification weights. Therefore, the classification weights are adaptive to different query samples and aware of the task context as well. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the attentive path, a new multi-head self-attention network $f _ { s a } ^ { a p }$ on the support set is employed to encode the global task information. $f _ { s a } ^ { a p }$ is different from $f _ { s a } ^ { c p }$ in contextual path because the selfattention network in contextual path emphasizes on generating the classification weights. On the contrary, outputs of self-attention here plays the role of providing the V alue context for different query samples to attend in the following cross attention. Sharing the same self-attention networks might limit the expressiveness of learned representations in both paths. The cross attention network $f _ { c a } ^ { a p }$ applied on each query sample and task-aware support set is followed to produce $\\hat { \\bf X } ^ { a p } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times d _ { h } }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use multi-head attention with $h$ heads in both paths. In one attention block, we produce $h$ different sets of queries, keys and values. Multi-head attention is claimed to be able to learn more comprehensive and expressive representations from $h$ different subspaces (Vaswani et al., 2017; Voita et al., 2019). More details of these two paths can be found in A.2. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3.2 WEIGHTS GENERATOR",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We replicate them afterwa ${ \\bf X } ^ { c p } \\in \\mathbb { R } ^ { N K \\times d _ { h } }$ anve $\\hat { \\bf X } ^ { a p } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times d _ { h } }$ r $| \\mathcal { Q } |$ $N K$ ectively and reshape. These two tensors $\\pmb { \\chi } ^ { c p } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times N K \\times d _ { h } }$ $\\hat { \\pmb { \\mathsf { X } } } ^ { a p } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times N K \\times d _ { h } }$ ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$d _ { h } < d$ is the hidden dimension. We use matrix form here to be consistent with the description in 3.3.2. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "are concatenated to become $\\pmb { \\chi } ^ { c p \\oplus a p } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times N K \\times 2 d _ { h } }$ . $\\mathbf { \\pmb { \\chi } } ^ { c p \\oplus a p }$ can be interpreted that each query sample has its own latent representations for support set to generate specific classification weights, which are both aware of the task-context and adaptive to individual query sample. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { X } ^ { c p \\oplus a p }$ is decoded by the weights generator $g : \\mathbb { R } ^ { 2 d _ { h } }  \\mathbb { R } ^ { 2 d }$ . We assume that the classification weights follow Gaussian distribution with diagonal covariance. $g$ outputs the distribution parameters and we sample the weights from learned distribution during meta-training. The sampled classification weights are represented as $\\pmb { \\mathsf { W } } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times N K \\times d }$ . To reduce complexity, we compute the mean value on $K$ classification weights for each class to have ${ \\pmb W } ^ { f i n a l } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times N \\times d }$ . Therefore, ith query sample has its specifican be computed by c classifica Wf inali,:,: tion weight matrix . The support data $\\pmb { \\mathsf { W } } _ { i , : , : } ^ { f i n a l } \\in \\mathbb { R } ^ { N \\times d }$ . Th for rediction for query datatimes and reshaped as $\\hat { \\mathbf { X } } \\mathbf { W } ^ { f i n a l \\mathbf { T } }$ $\\mathbf { X }$ $| \\mathcal { Q } |$ $\\pmb { \\chi } _ { s } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times N K \\times d }$ . So the prediction for support data can also be computed as ${ \\pmb x } _ { s } { \\pmb w } ^ { f i n a l { \\bf T } }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Besides the weights generator $g$ , we have another two decoders $r _ { 1 } : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { d _ { h } }$ and $r _ { 2 } : \\mathbb { R } ^ { d } $ $\\mathbb { R } ^ { d _ { h } }$ . They both take the generated weights $\\boldsymbol { \\mathsf { W } }$ as inputs and learn to reconstruct $\\mathbf { X } ^ { c p }$ and ${ \\hat { \\mathbf { X } } } ^ { a p }$ respectively. The outputs of $r _ { 1 }$ and $r _ { 2 }$ are denoted as $\\bar { \\pmb { \\chi } } _ { r e } ^ { c p } , \\hat { \\pmb { \\chi } } _ { r e } ^ { a p } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times N K \\times d _ { h } }$ . The reason we are using reconstruction as auxiliary tasks will be discussed in following Sec. 3.4. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 INFORMATION MAXIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we perform the analysis for one query sample without loss of generality. The subscripts for classification weights are omitted for clarity. In general, we use $\\displaystyle ( \\mathbf { x } , \\mathbf { y } )$ and $( \\hat { \\mathbf { x } } , \\hat { \\mathbf { y } } )$ to represent support and query samples respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Since the classification weights w generated from $g$ are encoded with attentive path and contextual path, it is expected that we can directly have the query-specific weights. However, we show in the experiments that simply doing this does not outperform a weight generator conditioned only on the $s$ significantly, which implies that the generated classification weights from two paths are not sensitive to different query samples. In other words, the information from attentive path is not kept well during the weights generation. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To address this limitation, we propose to maximize the mutual information between generated weights w and support as well as query data. The objective function can be described as ",
+        "bbox": [
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+            529,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fed913301f83fe1b959036702bbf395c27791c81e656dca3ca3caa5d335ae230.jpg",
+        "text": "$$\n\\operatorname* { m a x } I ( ( \\hat { \\mathbf { x } } , \\hat { \\mathbf { y } } ) ; \\mathbf { w } ) + \\sum _ { ( \\mathbf { x } , \\mathbf { y } ) \\in S } I ( ( \\mathbf { x } , \\mathbf { y } ) ; \\mathbf { w } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            561,
+            630,
+            598
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "According to the chain rule of mutual information, we have ",
+        "bbox": [
+            173,
+            602,
+            565,
+            616
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61fcd925fdf111ec33de41849440b37a34e656fe8c7fff9040a5524cccab468c.jpg",
+        "text": "$$\nI ( ( \\hat { \\mathbf { x } } , \\hat { \\mathbf { y } } ) ; \\mathbf { w } ) = I ( \\hat { \\mathbf { x } } ; \\mathbf { w } ) + I ( \\hat { \\mathbf { y } } ; \\mathbf { w } | \\hat { \\mathbf { x } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            621,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Equation 6 stands for both terms in 5. So the objective function can be written as ",
+        "bbox": [
+            173,
+            642,
+            705,
+            657
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0d1fc3c48557d62a5122d8ca90541e84ccdafb6203647ae01d4fed13aaad7dcd.jpg",
+        "text": "$$\n\\operatorname* { m a x } I ( \\hat { \\mathbf { x } } ; \\mathbf { w } ) + I ( \\hat { \\mathbf { y } } ; \\mathbf { w } | \\hat { \\mathbf { x } } ) + \\sum _ { ( \\mathbf { x } , \\mathbf { y } ) \\in \\mathcal { S } } [ I ( \\mathbf { x } ; \\mathbf { w } ) + I ( \\mathbf { y } ; \\mathbf { w } | \\mathbf { x } ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            660,
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+            696
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Directly computing the mutual information in Equation 7 is intractable since the true posteriori distributions like $p ( \\hat { \\mathbf { y } } | \\hat { \\mathbf { x } } , \\mathbf { w } )$ , $p ( \\hat { \\mathbf { x } } | \\mathbf { w } )$ are still unknown. Therefore, we use Variational Information Maximization (Barber & Agakov, 2003; Chen et al., 2016) to compute the lower bound of Equation 5. We use $p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { w } )$ to approximate the true posteriori distribution, where $\\theta$ represents the model parameters. As a result, we have ",
+        "bbox": [
+            174,
+            700,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1de098c1111875164014e5414138efe3365ce6eb8179c183f7e9443c072c973e.jpg",
+        "text": "$$\n\\begin{array} { r c l } { I ( \\hat { \\mathbf { x } } ; \\mathbf { w } ) } & { = } & { H ( \\hat { \\mathbf { x } } ) - H ( \\hat { \\mathbf { x } } | \\mathbf { w } ) } \\\\ & { = } & { H ( \\hat { \\mathbf { x } } ) + \\mathbb { E } _ { \\mathbf { w } \\sim p ( \\mathbf { w } | \\hat { \\mathbf { x } } , S ) } [ \\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p ( \\hat { \\mathbf { x } } | \\mathbf { w } ) } [ \\log p ( \\hat { \\mathbf { x } } | \\mathbf { w } ) ] ] } \\\\ & { = } & { H ( \\hat { \\mathbf { x } } ) + \\mathbb { E } _ { \\mathbf { w } \\sim p ( \\mathbf { w } | \\hat { \\mathbf { x } } , S ) } [ D _ { \\mathrm { K L } } ( p ( \\hat { \\mathbf { x } } | \\mathbf { w } ) | | p \\theta ( \\hat { \\mathbf { x } } | \\mathbf { w } ) ) + \\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p ( \\hat { \\mathbf { x } } | \\mathbf { w } ) } [ \\log p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { w } ) ] ] } \\\\ & { \\geq } & { H ( \\hat { \\mathbf { x } } ) + \\mathbb { E } _ { \\mathbf { w } \\sim p ( \\mathbf { w } | \\hat { \\mathbf { x } } , S ) } [ \\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p ( \\hat { \\mathbf { x } } | \\mathbf { w } ) } [ \\log p \\theta ( \\hat { \\mathbf { x } } | \\mathbf { w } ) ] ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            202,
+            773,
+            794,
+            852
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$H ( \\cdot )$ is the entropy of a random variable. $H ( { \\hat { \\mathbf { x } } } )$ is a constant value for given data. We can maximize this lower bound as the proxy for the true mutual information. ",
+        "bbox": [
+            173,
+            853,
+            825,
+            882
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Similar to $I ( \\hat { \\mathbf { x } } ; \\mathbf { w } )$ ",
+        "bbox": [
+            174,
+            888,
+            300,
+            904
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fde9fd0c0a9e113954b4199b3c2ec470ee20ba384175ea107bd0715fdc74e685.jpg",
+        "text": "$$\nI ( \\hat { \\mathbf { y } } ; \\mathbf { w } | \\hat { \\mathbf { x } } ) \\geq H ( \\hat { \\mathbf { y } } | \\hat { \\mathbf { x } } ) + \\mathbb { E } _ { \\mathbf { w } \\sim p ( \\mathbf { w } | \\hat { \\mathbf { x } } , S ) } [ \\mathbb { E } _ { \\hat { \\mathbf { y } } \\sim p ( \\hat { \\mathbf { y } } | \\hat { \\mathbf { x } } , \\mathbf { w } ) } [ \\log p _ { \\theta } ( \\hat { \\mathbf { y } } | \\hat { \\mathbf { x } } , \\mathbf { w } ) ] ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            909,
+            723,
+            926
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a0bb2ef83d73a2a8a8d61438c2a019e0852c7ce0628f9bd2e04ca041ba42746b.jpg",
+        "text": "$$\n\\begin{array} { r } { \\displaystyle \\sum _ { \\mathbf { x } , \\mathbf { y } ) \\in S } I ( ( \\mathbf { x } , \\mathbf { y } ) ; \\mathbf { w } ) \\geq \\displaystyle \\sum _ { ( \\mathbf { x } , \\mathbf { y } ) \\in S } H ( ( \\mathbf { x } , \\mathbf { y } ) ) + \\mathbb { E } _ { \\mathbf { w } \\sim p ( \\mathbf { w } | \\hat { \\mathbf { x } } , S ) } [ \\mathbb { E } _ { ( \\mathbf { x } , \\mathbf { y } ) \\sim p ( ( \\mathbf { x } , \\mathbf { y } ) | \\mathbf { w } ) } [ \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { w } ) + \\log p _ { \\theta } ( \\mathbf { y } | \\mathbf { x } , \\mathbf { w } ) ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            98,
+            839,
+            136
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "$p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { w } ) , p _ { \\theta } ( \\mathbf { x } , \\mathbf { y } | \\mathbf { w } )$ are used to approximate the true posteriori distribution $p ( \\hat { \\mathbf { x } } | \\mathbf { w } )$ and $p ( \\mathbf { x } , \\mathbf { y } | \\mathbf { w } )$ . ",
+        "bbox": [
+            169,
+            146,
+            823,
+            162
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Put the lower bounds back into Equation 7. Omit the constant entropy terms and the expectation subscripts for clarity, we have the new objective function as ",
+        "bbox": [
+            171,
+            167,
+            823,
+            196
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/87cdb9217a5cdc480b1ccc23dc8963ac875ed8caf39aaf4647fd020529f592ac.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta } \\mathbb { E } [ \\log p _ { \\theta } ( \\hat { \\mathbf { y } } | \\hat { \\mathbf { x } } , \\mathbf { w } ) + \\log p _ { \\theta } ( \\mathbf { y } | \\mathbf { x } , \\mathbf { w } ) + \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { w } ) + \\log p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { w } ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            207,
+            738,
+            231
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The first two terms are maximizing the log likelihood of label for both support and query data with respective to the network parameters, given the generated classification weights. This is equivalent to minimizing the cross entropy between prediction and ground-truth. We assume that $p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { w } )$ and $p _ { \\theta } ( \\mathbf { x } | \\mathbf { w } )$ are Gaussian distributions. $r _ { 1 }$ and $r _ { 2 }$ are used to approximate the mean of these two Gaussian distributions. Therefore maximizing the log likelihood is equivalent to reconstruct $\\mathbf { x } ^ { c p }$ and $\\hat { \\mathbf { x } } ^ { a p }$ with $L 2$ loss. Thus the loss function to train the network can be written as ",
+        "bbox": [
+            173,
+            239,
+            825,
+            324
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f142754583e0d6a4cbc852117c9d1b2d57bdc1588065af6efe1040e10029a1f1.jpg",
+        "text": "$$\nL = \\mathrm { C E } ( \\hat { \\bf y } _ { p r e d } , \\hat { \\bf y } ) + \\lambda _ { 1 } \\sum _ { { \\bf y } \\in \\mathcal { S } } \\mathrm { C E } ( { \\bf y } _ { p r e d } , { \\bf y } ) + \\lambda _ { 2 } \\sum _ { { \\bf x } ^ { c p } \\in \\mathcal { S } } | | { \\bf x } ^ { c p } - { \\bf x } _ { r e } ^ { c p } | | _ { 2 } + \\lambda _ { 3 } | | \\hat { \\bf x } ^ { a p } - \\hat { \\bf x } _ { r e } ^ { a p } | | _ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            332,
+            779,
+            366
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "CE here stands for cross entropy. $\\mathbf { x } ^ { c p }$ and $\\hat { \\mathbf { x } } ^ { a p }$ are the inputs to weights generator $g$ . $\\mathbf { x } _ { r e } ^ { c p } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { w } )$ and $\\hat { \\mathbf { x } } _ { r e } ^ { a p } \\sim p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { w } )$ are the reconstruction of $\\mathbf { x } ^ { c p }$ and $\\hat { \\mathbf { x } } ^ { a p }$ . Since we convert the log likelihood in Equation 14 to mean square error or cross entropy in Equation 15 to optimize, the value of each term in Equation 15 is not equal to real log likelihood and we have to decide the weightage for each one. $\\lambda _ { 1 } , \\lambda _ { 2 } , \\lambda _ { 3 }$ are thus hyper-parameters for trade-off of different terms. With the help of last three terms, the generated classification weights are forced to carry information about the support data and the specific query sample. ",
+        "bbox": [
+            173,
+            377,
+            825,
+            476
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In LEO (Rusu et al., 2019), the inner update loss is computed as cross entropy on support data. If we merge the inner update into outer loop, then the loss becomes the summation of first two terms in Equation 15. However, the weight generation in LEO does not involve specific query samples, thus making reconstructing $\\hat { \\mathbf { x } } ^ { a p }$ impossible. In this sense, LEO can be regarded as a special case of our proposed method, where (1) only contextual path exits and (2) $\\lambda _ { 2 } = \\lambda _ { 3 } = 0$ . ",
+        "bbox": [
+            174,
+            481,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.5 COMPLEXITY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            573,
+            382,
+            587
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The encoding process in contextual path results in computational complexity $O ( ( N K ) ^ { 2 } )$ due to self-attention. Similarly, the computational complexity of attentive path is $O ( ( N K ) ^ { 2 } + | \\mathcal { Q } | ( N K ) )$ . In total, the complexity is ${ \\cal O } ( ( \\bar { N } K ) ^ { 2 } + | \\mathcal { Q } | ( \\bar { N ^ { } } K ) )$ . However, because of the nature of few-shot learning problem, the value of $( N K ) ^ { 2 }$ is usually negligible. The value of $| \\mathcal { Q } |$ depends on the setting and the cross attention can be implemented parallelly via matrix multiplication. Therefore, the induced computational overhead will be negligible. AWGIM avoids the inner update without compromising the performance, which furthers reduces both training and inference time significantly. The empirical evaluation is presented in A.3.4. ",
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+        "type": "text",
+        "text": "4 EXPERIMENTS ",
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+        "text": "4.1 DATASETS AND PROTOCOLS ",
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+        "text": "We conduct experiments on miniImageNet (Vinyals et al., 2016) and tieredImageNet (Ren et al., 2018), two commonly used benchmark datasets, to compare with other methods and analyze our model. Both datasets are subsets of ILSVRC-12 dataset (Russakovsky et al., 2015). miniImageNet contains 100 randomly sampled classes with 600 images per class. We follow the train/test split in (Ravi & Larochelle, 2016), where 64 classes are used for meta-training, 16 for meta-validation and 20 for meta-testing. tieredImageNet is a larger dataset compared to miniImageNet. There are 608 classes and 779,165 images in total. They are selected from 34 higher level nodes in ImageNet (Deng et al., 2009) hierarchy. 351 classes from 20 high level nodes are used for meta-training, 97 from 6 nodes for meta-validation and 160 from 8 nodes for meta-testing. ",
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+        "text": "We use the image features in LEO (Rusu et al., 2019) provided by the authors 2. They trained a 28-layer Wide Residual Network (Zagoruyko & Komodakis, 2016) on the meta-training set. Each image then is represented by a 640 dimensional vector, which is used as the input to our model. ",
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+        "text": "For $N$ -way $K$ -shot experiments, we randomly sample $N$ classes from meta-training set and each of them contains $K$ samples as the support set and 15 as query set. Similar to other works, we train 5-way 1-shot and 5-shot models on two dataset. During meta-testing, 600 $N$ -way $K$ -shot tasks are sampled from meta-testing set and the average accuracy for query set is reported with $9 5 \\%$ confidence interval, as done in recent works (Finn et al., 2017; Snell et al., 2017; Rusu et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "4.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We use TensorFlow (Abadi et al., 2016) to implement our method and the code will be made available. $d = 6 4 0$ is the dimension of feature embeddings. $d _ { h }$ is set to be 128. The number of heads $h$ in attention module is set to be 4. $g , r _ { 1 }$ and $r _ { 2 }$ are 2-layer MLPs with 256 hidden units. We decide $\\lambda _ { 1 } = 1$ , $\\lambda _ { 2 } = \\lambda _ { 3 } = 0 . 0 0 1$ by meta-validation performance. ",
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+        "table_caption": [
+            "Table 1: Accuracy comparison with other approaches on miniImageNet. The results are averaged on 600 tasks from meta-testing set with $9 5 \\%$ confidence interval. Best results are highlighted. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Feature Extractor</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>Matching Networks (Vinyals et al., 2016)</td><td>Conv-4</td><td>46.60</td><td>60.00</td></tr><tr><td>MAML(Finn et al.,2017)</td><td>Conv-4</td><td>48.70 ± 1.84%</td><td>63.11 ± 0.92%</td></tr><tr><td>Meta LSTM (Ravi &amp; Larochelle, 2016)</td><td>Conv-4</td><td>43.44 ± 0.77%</td><td>60.60 ± 0.71%</td></tr><tr><td>Prototypical Nets (Snell et al.,2017)</td><td>Conv-4</td><td>49.42 ± 0.78%</td><td>68.20 ± 0.66%</td></tr><tr><td>Relation Nets (Sung et al.,2018)</td><td>Conv-4</td><td>50.44 ± 0.82%</td><td>65.32 ± 0.70%</td></tr><tr><td>SNAIL (Mishra et al., 2018)</td><td>Resnets-12</td><td>55.71 ± 0.99%</td><td>68.88 ± 0.92%</td></tr><tr><td>TPN (Liu et al., 2019)</td><td>Resnets-12</td><td>59.46</td><td>75.65</td></tr><tr><td>MTL (Sun et al., 2019)</td><td>Resnets-12</td><td>61.20 ± 1.80%</td><td>75.50 ± 0.80</td></tr><tr><td>Dynamic (Gidaris &amp; Komodakis,2018)</td><td>WRN-28-10</td><td>60.06 ± 0.14%</td><td>76.39 ± 0.11%</td></tr><tr><td>Prediction (Qiao et al., 2018)</td><td>WRN-28-10</td><td>59.60 ± 0.41%</td><td>73.74 ± 0.19%</td></tr><tr><td>DAE-GNN (Gidaris &amp; Komodakis,2019)</td><td>WRN-28-10</td><td>62.96 ± 0.15%</td><td>78.85 ± 0.10%</td></tr><tr><td>LEO (Rusu et al., 2019)</td><td>WRN-28-10</td><td>61.76 ± 0.08%</td><td>77.59 ± 0.12%</td></tr><tr><td>AWGIM (ours)</td><td>WRN-28-10</td><td>63.12 ± 0.08%</td><td>78.40 ± 0.11%</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [
+            "Table 2: Accuracy comparison with other approaches on tieredImageNet. The results are averaged on 600 tasks from meta-testing set with $9 5 \\%$ confidence interval. Best results are highlighted. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Feature Extractor</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>MAML (Finn et al., 2017)</td><td>Conv-4</td><td>51.67 ± 1.81%</td><td>70.30 ± 1.75%</td></tr><tr><td>Prototypical Nets (Snell et al., 2017)</td><td>Conv-4</td><td>53.31± 0.89%</td><td>72.69 ± 0.74%</td></tr><tr><td>Relation Nets (Sung et al., 2018)</td><td>Conv-4</td><td>54.48 ± 0.93%</td><td>71.32 ± 0.78%</td></tr><tr><td>TPN (Liu et al., 2019)</td><td>Conv-4</td><td>59.91 ± 0.96%</td><td>72.85 ± 0.74%</td></tr><tr><td>MetaOptNet (Lee et al., 2019)</td><td>Resnets-12</td><td>65.81 ± 0.74%</td><td>81.75 ± 0.53%</td></tr><tr><td>LEO (Rusu et al., 2019)</td><td>WRN-28-10</td><td>66.33 ± 0.05%</td><td>81.44 ± 0.09%</td></tr><tr><td>AWGIM (ours)</td><td>WRN-28-10</td><td>67.69 ± 0.11%</td><td>82.82 ± 0.13%</td></tr></table>",
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+        "type": "text",
+        "text": "ADAMW Loshchilov & Hutter (2017) is used to optimize the network with weight decay $1 \\times 1 0 ^ { - 6 }$ . The initial learning rate is set to 0.0002 for 5-way 1-shot and 0.001 for 5-way 5-shot, which is decayed by 0.2 for every 15,000 iterations. We train the model for 50,000 iterations. Batch size is 64 for 5-way 1-shot and 32 for 5-way 5-shot. Similar to LEO (Rusu et al., 2019), we first train the model on meta-training set and choose the optimal hyper-parameters by validation results. Then we train the model on meta-training and meta-validation sets together using fixed hyper-parameters. ",
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+        "text": "4.3 COMPARISON WITH OTHER METHODS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We compare the performance of our approach AWGIM on two datasets with several state-of-theart methods proposed in recent years. The results of MAML, Prototypical Nets, Relation Nets on tieredImageNet are evaluated by Liu et al. (2019). The results of Dynamic on miniImageNet with WRN-28-10 as the feature extractor is reported in (Gidaris & Komodakis, 2019). The other results are reported in the corresponding original papers. We also include the backbone network structure of the used feature extractor for reference. The results on miniImageNet and tieredImageNet are shown in Table 1 and 2 respectively. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The top half parts of Table 1 and 2 display the methods belonging with different meta learning categories, such as metric-based(Matching Networks, Prototypical Nets), gradient-based (MAML, MTL), graph-based (TPN). The bottom part shows the classification weights generation approaches including Dynamic, Prediction, DAE-GNN, LEO and our proposed AWGIM. ",
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+    {
+        "type": "text",
+        "text": "AWGIM can outperform all the methods in top parts of two table. Comparing with other classification weights generation methods in the bottom part, AWGIM still shows very competitive performance, namely the best on tieredImageNet and close to the state-of-the-art on miniImageNet. We note that all the classification weights generation methods are using WRN-28-10 as backbone network, which makes the comparison fair. In particular, AWGIM can outperform LEO in all settings. ",
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+    {
+        "type": "text",
+        "text": "4.4 ANALYSIS ",
+        "text_level": 1,
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+        "table_caption": [
+            "Table 3: Analysis of our proposed AWGIM. In the top half, the attentive path is removed to compare with LEO. In the bottom part, ablation analysis with respective to different components is provided. We also shuffle the generated classification weights randomly to show that they are indeed optimal for different query samples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">miniImageNet</td><td colspan=\"2\">tieredImageNet</td></tr><tr><td>5-way 1-shot</td><td>5-way 5-shot</td><td> 5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>LEO</td><td>61.76 %</td><td>77.59 %</td><td>66.33%</td><td>81.44 %</td></tr><tr><td>Generator in LEO</td><td>60.33 %</td><td>74.53 %</td><td>65.17%</td><td>78.77 %</td></tr><tr><td>Generator conditioned on S only</td><td>61.02%</td><td>74.33%</td><td>66.22%</td><td>79.66%</td></tr><tr><td>Generator conditioned on S with IM</td><td>62.04%</td><td>77.54%</td><td>66.43%</td><td>81.73%</td></tr><tr><td>MLP encoding,入1 = 入2= 入3=0</td><td>58.95%</td><td>71.68%</td><td>63.92%</td><td>75.80%</td></tr><tr><td>MLP encoding</td><td>62.26%</td><td>76.91%</td><td>65.84%</td><td>79.24%</td></tr><tr><td>入1=λ2=λ3=0</td><td>61.61%</td><td>74.14%</td><td>65.65%</td><td>79.93%</td></tr><tr><td>入1=入2=0</td><td>62.06%</td><td>74.18%</td><td>65.85%</td><td>80.42%</td></tr><tr><td>入3=0</td><td>62.91%</td><td>77.88%</td><td>67.27%</td><td>81.67%</td></tr><tr><td>入1=0</td><td>62.19%</td><td>74.21%</td><td>66.82%</td><td>80.61%</td></tr><tr><td>2=入g=0</td><td>62.12%</td><td>77.65%</td><td>66.86%</td><td>81.03%</td></tr><tr><td>random shuffle in class</td><td>62.87%</td><td>77.48%</td><td>67.52%</td><td>82.55%</td></tr><tr><td>random shuffle between classes</td><td>61.20%</td><td>77.48%</td><td>66.55%</td><td>82.53%</td></tr><tr><td>AWGIM (ours)</td><td>63.12%</td><td>78.40%</td><td>67.69 %</td><td>82.82%</td></tr></table>",
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+        "type": "text",
+        "text": "We perform detailed analysis on AWGIM, shown in Table 3. We include the results of LEO Rusu et al. (2019) for reference. “Generator in LEO” means that there is no inner update in LEO. In the upper part of the table, we first studied the effect of attentive path. We implemented two generators including only the contextual path during encoding. “Generator conditioned on $s$ with IM” indicates that we add the cross entropy loss and reconstruction loss for support set. It can be observed that “Generator conditioned on $s$ only” is trained with cross entropy on query set, which is similar to “Generator in LEO” without inner update. It is able to achieve similar or slightly better results than “Generator in LEO”, which implies that self-attention is no worse than relation networks used in LEO to model task-context. With information maximization, our generator is able to obtain slightly better performance than LEO. ",
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+    {
+        "type": "text",
+        "text": "The effect of attention is investigated by replacing the attention modules with 2-layer MLPs, which is shown as “MLP encoding”. More specifically, one MLP in contextual path is used for support set and another MLP in attentive path for query samples. We can see that even without attention to encode the task-contextual information, “MLP encoding” can achieve accuracy close to LEO, for the sake of information maximization. However, if we let $\\lambda _ { 1 } = \\lambda _ { 2 } = \\lambda _ { 3 } = 0$ for MLP encoding, the performance drops significantly, which demonstrates the importance of maximizing the information. ",
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+        "type": "text",
+        "text": "We conducted ablation analysis with respective to $\\lambda _ { 1 } , \\lambda _ { 2 }$ and $\\lambda _ { 3 }$ to investigate the effect of information maximization. First, $\\lambda _ { 1 }$ , $\\lambda _ { 2 }$ and $\\lambda _ { 3 }$ are all set to be 0. In this case, the accuracy is similar to “generator conditioned on $s$ only”, showing that the generated classification weights are not fitted for different query samples, even with the attentive path. It can also be observed that maximizing the mutual information between weights and support is more crucial since $\\lambda _ { 1 } = \\lambda _ { 2 } = 0$ degrades accuracy significantly, comparing with $\\lambda _ { 3 } = 0$ . We further investigate the relative importance of the classification on support as well as reconstruction. $\\lambda _ { 1 } = 0$ affects the performance noticeably. We conjecture that the support label prediction is more critical for information maximization. ",
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+    {
+        "type": "text",
+        "text": "The classification weights are generated specifically for each query sample in AWGIM. To this point, we shuffle the classification weights between query samples within the same classes and between different classes as well to study whether the classification weights are adapted for different query samples. Assume there are T query samples per class in one task. Wf inal ∈ R|Q|×N×d can be reshaped into Wf inal $\\mathbf { W } ^ { f i n a l } \\in \\mathbb { R } ^ { N \\times T \\times N \\times \\bar { d } }$ . Then we shuffle this weight tensor along the first and second axis randomly. The results are shown as “random shuffle between classes” and “random shuffle in class” in Table 3. For 5-way 1-shot experiments, the random shuffle between classes degrades the accuracy noticeably while the random shuffle in class dose not affect too much. This indicates that when the support data are very limited, the generated weights for query samples from the same class are very similar to each other while distinct for different classes. When there are more labeled data in support set, two kinds of random shuffle show very close or even the same results in 5-way 5-shot experiments, which are both worse than the original ones. This implies that the generated classification weights are more diverse and specific for each query sample in 5-way 5-shot setting. The possible reason is that larger support set provides more knowledge to estimate the optimal classification weights for each query example. ",
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+    {
+        "type": "text",
+        "text": "More analysis is provided in Appendix A.3. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we introduce Attentive Weights Generation via Information Maximization (AWGIM) for few shot image classification. AWGIM learns to generate optimal classification weights for each query sample within the task by two encoding paths. To guarantee this, the lower bound of mutual information between generated weights and query, support data is maximized. As far as we know, AWGIM is the first work utilizing mutual information techniques for few shot learning. The effectiveness of AWGIM is demonstrated by state-of-the-art performance on two benchmark datasets and extensive analysis. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 MUTLI-HEAD ATTENTION IN AWGIM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            483,
+            150
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The multi-head attention can be described as ",
+        "bbox": [
+            174,
+            161,
+            468,
+            176
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8afbc345a51adf63a2ba312dc71f4cfa00a3cc5df50506647b72d47660225ec6.jpg",
+        "text": "$$\nM u l t i H e a d ( Q , K , V ) = C o n c a t ( h e a d _ { 1 } , . . . , h e a d _ { H } ) W ^ { O } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            184,
+            692,
+            202
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d0b100deca88258dbd63f93aaeea767993505bc88714fd792cbf8beec0071f0.jpg",
+        "text": "$$\nh e a d _ { i } ( Q ^ { i } , K ^ { i } , V ^ { i } ) = A t t e n t i o n ( Q ^ { i } , K ^ { i } , V ^ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            212,
+            651,
+            232
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/182af2f52f35feabb7376fcf4d5db4194b9258b694a9cd87685a286f425cd3b2.jpg",
+        "text": "$$\nA t t e n t i o n ( Q , K , V ) = s o f t m a x ( \\frac { Q K ^ { T } } { \\sqrt { d _ { k } } } V ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            239,
+            648,
+            275
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2171eaaa8655f60350a45f7fdb72379d74297ef38001615123062a30cf21f233.jpg",
+        "text": "$$\nQ ^ { i } = Q W _ { Q } ^ { i } , K ^ { i } = K W _ { K } ^ { i } , V ^ { i } = V W _ { V } ^ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            282,
+            632,
+            303
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here $Q , K , V$ are query, key, value matrices. $W _ { Q } ^ { i } , W _ { K } ^ { i } , W _ { V } ^ { i }$ are the weight matrices for $i$ th head. $W ^ { O }$ is the weight matrix for output. $d _ { k }$ is the dimension of keys. Original $Q$ is added to the output of Equation 16 to stabilize the training as residual learning. ",
+        "bbox": [
+            173,
+            316,
+            825,
+            363
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 MODEL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            382,
+            334,
+            397
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2.1 CONTEXTUAL PATH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            410,
+            367,
+            424
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The encoding process in contextual path is realized by a simple multi-head self-attention network on support data. First, ${ \\bf x } ^ { c _ { n } ; k }$ are mapped to a lower dimensional hidden space by a MLP $f _ { 1 } : \\mathbb { R } ^ { d } $ $\\mathbb { R } ^ { d _ { h } }$ to reduce the computation complexity. Then the low-dimensional representations $\\mathbf { x } _ { h 1 } ^ { c _ { n } ; k }$ are processed by the -head self-attention network $f _ { c p } ^ { s a } : \\mathbb { R } ^ { d _ { h } }  \\mathbb { R } ^ { d _ { h } }$ , ",
+        "bbox": [
+            174,
+            434,
+            825,
+            496
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cf38f56688dd50a03b63ca1a3a93aed736836d470b8526704c8564a88b4c860.jpg",
+        "text": "$$\n{ \\bf X } ^ { c p } = M u l t i H e a d A t t e n t i o n ( Q = { \\bf X } _ { h 1 } , K = { \\bf X } _ { h 1 } , V = { \\bf X } _ { h 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            505,
+            715,
+            522
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { X } _ { h 1 } \\in \\mathbb { R } ^ { N K \\times d _ { h } }$ is the matrix where each row stands for one support sample $\\mathbf { x } _ { h 1 } ^ { c _ { n } ; k }$ . For one $N$ -way $K$ -shot task, the outputs of $f _ { c p } ^ { s a }$ . ",
+        "bbox": [
+            173,
+            534,
+            825,
+            565
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2.2 ATTENTIVE PATH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            582,
+            349,
+            597
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The attentive path is instantiated by attention, similar to contextual path. First, a MLP $f _ { 2 } : \\mathbb { R } ^ { d } $ $\\mathbb { R } ^ { d _ { h } }$ is used to map both ${ \\bf x } ^ { c _ { n } ; k }$ and $\\hat { \\bf x }$ to $\\mathbf { x } _ { h 2 } ^ { c _ { n } ; k }$ and $\\hat { \\mathbf { x } } _ { h 2 }$ . Then we employ another $H$ -head selfattention network $f _ { a p } ^ { s a } : \\mathbb { R } ^ { d _ { h } }  \\mathbb { R } ^ { d _ { h } }$ on $\\mathbf { x } _ { h 2 } ^ { c _ { n } ; k }$ to encode the global task information to each support sample, ",
+        "bbox": [
+            173,
+            606,
+            825,
+            669
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8787916c00021d49460bcd080a4e69848a2772678f0f486e7734e5ce4c4279d9.jpg",
+        "text": "$$\n{ \\bf X } ^ { a p } = M u l t i H e a d A t t e n t i o n ( Q = { \\bf X } _ { h 2 } , K = { \\bf X } _ { h 2 } , V = { \\bf X } _ { h 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            670,
+            715,
+            688
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The cross attention between query and context-aware support samples are computed as ",
+        "bbox": [
+            171,
+            700,
+            745,
+            715
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b5ceefbeb0428b082e6acf93b38a09c6687333a0b42dda38e489b18e3f51dbb0.jpg",
+        "text": "$$\n{ \\hat { \\bf X } } ^ { a p } = M u l t i H e a d A t t e n t i o n ( Q = { \\hat { \\bf X } } _ { h 2 } , K = { \\bf X } _ { h 2 } , V = { \\bf X } ^ { a p } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            723,
+            715,
+            743
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here $\\hat { \\mathbf { X } } ^ { a p } \\in \\mathbb { R } ^ { | \\mathcal { Q } | \\times d _ { h } }$ is the matrix form of $\\hat { \\mathbf { x } } _ { q }$ , where each query sample is context-aware. ",
+        "bbox": [
+            169,
+            753,
+            764,
+            770
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2.3 WEIGHT GENERATOR",
+        "text_level": 1,
+        "bbox": [
+            174,
+            786,
+            380,
+            801
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Assume $\\mathbf { x } ^ { c p \\oplus a p } = \\mathbf { X } _ { i , j , : } ^ { c p \\oplus a p } \\in \\mathbb { R } ^ { 2 d _ { h } }$ , where $i , j$ stands for $i$ th query sample and $j$ th support sample. $\\mathbf { x } ^ { c p \\oplus a p }$ is decoded by the weights generator $g : \\mathbb { R } ^ { 2 d _ { h } }  \\mathbb { R } ^ { 2 d }$ . We assume that the classification weights follow Gaussian distribution with diagonal covariance and we sample the weights from this distribution during meta-training, shown in Equation 23 and 24. ",
+        "bbox": [
+            173,
+            810,
+            825,
+            872
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0c933c6c2eed95051878f22e12d9223deb47d7728efda5fa0ecf8c1ded9be0c9.jpg",
+        "text": "$$\n\\mu _ { \\mathbf { w } } , \\sigma _ { \\mathbf { w } } = g ( \\mathbf { z } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            446,
+            880,
+            552,
+            897
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/914787c41dbb5cdc072aa30baae3b72825522e3c8f5f1dee246482e8e844b141.jpg",
+        "text": "$$\n\\mathbf { w } \\sim \\mathcal { N } ( \\mu _ { \\mathbf { w } } , \\pmb { \\Sigma } _ { \\mathbf { w } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            907,
+            562,
+            925
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 EXPERIMENTAL ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            401,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3.1 FEW SHOT REGRESSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            132,
+            401,
+            147
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "AWGIM can be applied to few shot regression task by slight modification. During meta-training, we set the number of classes $N$ equal to 1 and adapt the cross entropy loss to mean square error. We use the data points $( x , y )$ as inputs to AWGIM and generate weight as well as bias parameters for a three layer MLP with hidden dimension 40. This is consistent with few shot regression experimental setting in LEO. ",
+        "bbox": [
+            174,
+            160,
+            825,
+            229
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The few shot regression tasks are constructed as either sinusoidal or linear regression tasks. For sinusoidal regression tasks, the amplitude range is [0.1, 5], phase range $[ 0 , 2 \\pi ]$ , frequency range [0.5, 2.0]. For linear regression tasks, the slope range is $[ - 1 , 1 ]$ , intercept range $[ - 5 , 5 ]$ . Input $x$ is randomly sample from $[ - 5 , 5 ]$ . Gaussian noise with standard deviation 0.3 is added to $y$ during meta-training. We show some qualitative results in Figure 2. (a) and (b) are examples that can be tackled easily. For some non-trivial cases such as (c) and (d), AWGIM produces predictions slightly mixing with another regression family, despite that overall results are still faithful. ",
+        "bbox": [
+            173,
+            237,
+            825,
+            335
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/df7812dbdfdc7c5baf047dcc3b50d4a47220543c760af8eaedde57f4917c7892.jpg",
+        "image_caption": [
+            "Figure 2: 5-shot regression results for a multi-modal task distribution. Regression targets are plotted in red and prediction in black. 5 training samples per task are plotted with blue solid circles. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            386,
+            781,
+            720
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3.2 EFFECT OF MULTI-HEAD ATTENTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            827,
+            493,
+            840
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We replace the multi-head attention in the two paths with single-head attention and conduct the 5- way 1-shot and 5-way 5-shot experiments on miniImageNet dataset. The results are shown in Table 4. We can see clearly that multi-head attention improve the performance. In particular, for 5-way 1-shot experiment, single head attention gives results close to MLP encoding, which indicates that single head attention struggles when data are extremely scarce. ",
+        "bbox": [
+            174,
+            853,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/b35774e18aaaba69f80aa31a8b488a9e3337624d57822c24791f4c865a5b7a58.jpg",
+        "table_caption": [
+            "Table 4: Accuracy results on miniImageNet with 4 heads or single head in attention networks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>5-way 1 -shot</td><td> 5-way 5-shot</td></tr><tr><td>4 heads</td><td>63.12%</td><td>78.40%</td></tr><tr><td>single head</td><td>62.35%</td><td>77.75%</td></tr></table>",
+        "bbox": [
+            352,
+            128,
+            645,
+            184
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3.3 CONVERGENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            208,
+            341,
+            222
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We compare AWGIM with LEO in terms of convergence speed. The batch size is set to be 16 for both methods. We use the hyper-parameters tuned by authors to train LEO. The accuracy of metavalidation set during meta-training on 5-way 1-shot miniImageNet is plotted, shown in Figure 3. we can see clearly that AWGIM converges faster than LEO and outperforms LEO except for the first few iterations. ",
+        "bbox": [
+            173,
+            232,
+            825,
+            303
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/70012493ad6568c4a0bf7152014453a6a19cdde405da29751675aab5579ea992.jpg",
+        "image_caption": [
+            "Figure 3: The meta-validation accuracy during meta-training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            310,
+            333,
+            650,
+            530
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3.4 INFERENCE TIME ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            577,
+            351,
+            592
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We measure the inference time of AWGIM to show that it induces minimal computational overhead. In comparison, we use “MLP encoding” in two paths, which has time complexity $O ( N K + | \\mathcal { Q } | )$ . We use two set-ups on miniImageNet and the batch size is set to be 64. 100 batches are processed and we report the average consumed time for one batch. All these experiments on done with the same GPU and workstation. The results are shown in Table 5. It can be observed that the usage of self-attention and cross attention in AWGIM occurs negligible overhead, compared with MLP encoding. This is because the values of $N , K , | \\mathcal { Q } |$ are all relatively small and matrix multiplication further can be processed very fast by GPU. ",
+        "bbox": [
+            173,
+            601,
+            825,
+            713
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/8832d13a103fa54771879e563b1b44c4469a1b5472b96ac62306206e98011cd7.jpg",
+        "table_caption": [
+            "Table 5: The comparison of inference time between AWGIM and MLP encoding. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>5-way 1 -shot</td><td>5-way 5-shot</td></tr><tr><td>AWGIM</td><td>0.036s</td><td>0.093s</td></tr><tr><td>MLP encoding</td><td>0.033s</td><td>0.093s</td></tr></table>",
+        "bbox": [
+            339,
+            752,
+            656,
+            809
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3.5 VISUALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            829,
+            344,
+            844
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We visualize the generated classification weights by t-SNE (Maaten & Hinton, 2008). First we sample 400 tasks from meta-validation set of 5-way 1-shot miniImageNet experiment. Each task contains 5 query samples from 5 different classes. Thus in total there are $4 0 0 \\times 5 \\times 5 = 1 0 , 0 0 0$ weight vectors to visualize. As comparison, inputs to the generator $g$ are also plotted. The visualization results are shown in Figure 4. The inputs to $g$ are displayed in (a, b) and the generated classification weights in (c, d). From the comparison between (a) and (c), we can see the decoded weights for each class in (c) are clustered closer than (a) in general. Red and blue dots in (b, d) denotes the classification weights for two query samples from two classes within one task. It can be observed that $g$ can generate adapted weights for different query samples. This is consistent with Table 3, where the results of “random shuffle between classes” suggest that query samples from different class have distinct classification weights. ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/7279dd6a6dd60341910d31f04f08cc86bd5dc1b2d43395c825b56188a42d47cb.jpg",
+        "image_caption": [
+            "Figure 4: t-SNE visualization of the inputs to $g$ in (a, b) and the generated classification weights in (c, d). Blue and red dots in (b) and (d) are the classification weights for two query samples in the same task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            224,
+            810,
+            592
+        ],
+        "page_idx": 14
+    }
+]

parse/train/H1K6Tb-AZ/H1K6Tb-AZ.md

@@ -0,0 +1,199 @@
+# TESLA: TASK-WISE EARLY STOPPING AND LOSS AGGREGATION FOR DYNAMIC NEURAL NETWORK INFERENCE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+For inference operations in deep neural networks on end devices, it is desirable to deploy a single pre-trained neural network model, which can dynamically scale across a computation range without comprising accuracy. To achieve this goal, Incomplete Dot Product (IDP) has been proposed to use only a subset of terms in dot products during forward propagation. However, there are some limitations, including noticeable performance degradation in operating regions with low computational costs, and essential performance limitations since IDP uses hand-crafted profile coefficients. In this paper, we extend IDP by proposing new training algorithms involving a single profile, which may be trainable or pre-determined, to significantly improve the overall performance, especially in operating regions with low computational costs. Specifically, we propose the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm, which is showed in our 3-layer multilayer perceptron on MNIST that outperforms the original IDP by $3 2 \%$ when only $1 0 \%$ of dot products terms are used and achieves $9 4 . 7 \%$ accuracy on average. By introducing trainable profile coefficients, TESLA further improves the accuracy to $9 5 . 5 \%$ without specifying coefficients in advance. Besides, TESLA is applied to the VGG-16 model, which achieves $8 0 \%$ accuracy using only $2 0 \%$ of dot product terms on CIFAR-10 and also keeps $6 0 \%$ accuracy using only $3 0 \%$ of dot product terms on CIFAR-100, but the original IDP performs like a random guess in these two datasets at such low computation costs. Finally, we visualize the learned representations at different dot product percentages by class activation map and show that, by applying TESLA, the learned representations can adapt over a wide range of operation regions.
+
+# 1 INTRODUCTION
+
+Inference operations in deep neural networks on end devices, such as mobile phones, embedded sensors, IoT devices, etc., have recently received increasing attention including McMahan et al. (2016), Howard et al. (2017), and Teerapittayanon et al. (2017). In such applications, it is desirable to deploy a single pre-trained CNN model on end devices, while allowing multiple operating regions to meet different power consumption, latency, and accuracy requirements. To achieve this goal, McDanel et al. (2017a) proposed the incomplete dot product (IDP) operation, where only a subset of terms is used in dot products of forward propagation. From now on, $x \%$ dot product (DP), where 0 $\leq x \leq 1 0 0$ , means the $x \%$ of terms used in dot products. As illustrated in Figure 1, $5 0 \%$ DP means half of filters are used during forward propagation, and thus only half of the output channels are retained. To reduce the deviation induced by IDP, filters are prioritized from most important to the least important by pre-determined monotonically non-increasing profile coefficients (say, $\gamma _ { 1 } , . . . , \gamma _ { N } )$ during training. Therefore, IDP can be applied at inference time with dynamically-adjusted degrees of completeness (specified by the percentage of terms being used) to trade off accuracy slightly for lowered power consumption and reduced latency. Specifically, VGG-16 model with $5 0 \%$ DP achieves $7 0 \%$ in accuracy on the CIFAR-10 dataset compared to the standard network achieves only $3 5 \%$ accuracy when using the reduced channel set.
+
+While the original IDP design seems promising, there are two limitations. First, since the training process aims at optimizing the loss function computed using all weights of the model ( $1 0 0 \%$ DP), there will be a mismatch between training and testing. It is no surprise that inference performance significantly decreases in low DP percentages and thus narrow the dynamic computation range. To mitigate this problem, the original IDP design utilizes the multiple-profile training strategy, where different profiles can be specified to focus on different dot product ranges. In such a multipleprofile training process, however, certain subset of weights will be freezed in each training stage corresponding to the profile being focused, and hence the overall performance may not be fully optimized. Besides, each profile needs to maintain a separate first and last layer for adjusting to its own dot product range, resulting in additional memory overhead. The second limitation relates to the pre-determined nature of profile coefficients. While there are multiple ways to set the profile based on different dynamic range requirements, the original IDP design did not focus on finding a single ”best” profile that leads to the best performance. Instead, they use multiple hand-crafted profile coefficients, which make the system design less general among different applications, and hence may limit the overall performance of the system.
+
+![](images/b0dd91008c4e8fc441f3be52ecf73951a42e6e8f3ccdd53eb25a978714cc5aae.jpg)  
+Figure 1: Comparison between complete dot product (CDP) and incomplete dot product (IDP) where $X \%$ DP implies only $X \%$ of filters are used to compute the corresponding output channel. Since only $X \%$ filters are unused, the resulting output is an approximation of the output under CDP.
+
+To reduce the mismatch between training and testing performances, we propose the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm, in which multiple loss functions are computed in different DP percentages. By gradually aggregating these loss functions in decreasing order of DP percentages as the objective function to be optimized, TESLA significantly improves testing performances in low DP percentages without compromising accuracy in medium to high DP percentages. The loss functions can also be aggregated in random order of DP percentages to make a variant of TESLA, called Randomized TESLA (R-TESLA), which enables better performances under prespecified operating regions of end devices. Moreover, we relax the constraint of pre-determined profile coefficients and propose the alternate training procedure (ATP) to alternately train the profile coefficients along with weights of the model. By introducing trainable profile coefficients, customization among different applications can be achieved in a more generalized way, and the overall performance can also be further improved. This paper has made two major contributions: (1) We propose the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm and Randomized TESLA that can achieve dynamic scaling over a computation range in neural network inference without compromising accuracy. (2) We also propose the Alternate Training Procedure (ATP) that can learn the profile coefficients and the model weights simultaneously without the need of manual configuration of the profile coefficients.
+
+# 2 INCOMPLETE NEURAL NETWORKS
+
+Incomplete dot product (IDP) is a novel mechanism proposed by McDanel et al. (2017a) that can be applied to a hidden layer of MLPs or deep CNN models to dynamically lower the inference costs by computing only a subset of terms in dot products during forward propagation. By introducing a set of non-increasing coefficients $\gamma _ { i }$ , referred to as a profile, to the channels during training, the channels will be ordered implicitly in non-increasing order from the most important to the least important. By simply dropping out less important channels at inference time, it suffices to train and deploy a single network, while still supporting different levels of computation scaling without compromising accuracy significantly. In this section, we briefly introduce the main concepts of IDP.
+
+# 2.1 INCOMPLETE DOT PRODUCT OPERATION
+
+Mathematically, for an IDP fully-connected layer with input dimension $N$ and output dimension $M$ , the $j$ -th output component $y _ { j }$ is computed as
+
+$$
+y _ { j } = \sum _ { i = 1 } ^ { N } \gamma _ { i } w _ { j i } x _ { i } ,
+$$
+
+for $j \in \{ 1 , 2 , . . . , M \}$ , where $x _ { i }$ is the $i$ -th input component, $w _ { j i }$ is the weight corresponding to the $j$ -th output component and the $i$ -th input component, and $\gamma _ { i }$ is the $i$ -th profile coefficient.
+
+Similar expression can be derived for the IDP operation applied to a convolutional layer of CNN, as illustrated in Figure 1. For an IDP convolutional layer with number of input channels $N$ and number of output channels $M$ , the $j$ -th output channel ${ \bf y } _ { j }$ is computed as
+
+$$
+\mathbf { y } _ { j } = \gamma _ { j } \sum _ { i = 1 } ^ { N } \mathbf { f } _ { j i } * \mathbf { x } _ { i } ,
+$$
+
+for $j \in \{ 1 , 2 , . . . , M \}$ , where $\mathbf { f } _ { j i } * \mathbf { x } _ { i }$ denotes the convolution operation of the $i$ -th input channel $\mathbf { x } _ { i }$ and the $i$ -th channel of the $j$ -th filter $\mathbf { f } _ { j i }$ , and $\gamma _ { j }$ is the profile coefficient for the $j$ -th filter. Note that, instead of applying profile coefficients depthwise on each filter before convolution as is the case in the original IDP design, we multiply each $\gamma _ { j }$ to each output channel after a complete convolution to produce ${ \bf y } _ { j }$ . These two approaches, however, are equivalent with negligible difference induced by the first hidden layer. Since the output channels ${ \bf y } _ { j }$ ’s become input channels $\mathbf { x } _ { i }$ ’s to the next layer, applying $\gamma _ { j }$ ’s to ${ \bf y } _ { j }$ ’s is equivalent to applying them into the convolution operation in the next layer.
+
+To compute IDP with a target dot product percentage, a truncated version of Eq. 1 or Eq. 2 replaces the original computation to keep only a subset of the beginning terms. As for the case with all terms are kept, we refer to such operations as complete dot product (CDP) or $1 0 0 \%$ DP, interchangeably. Note that in the training process in the original IDP design, only CDP is used.
+
+# 2.2 MULTIPLE-PROFILE INCOMPLETE NEURAL NETWORKS
+
+In the work of McDanel et al. (2017a), several profile coefficients are proposed and applied in a pre-determined manner. When only a single profile is applied to the model, the trade-off between computation range and performance in high DP percentage regions is also demonstrated. Generally, the faster the profile coefficients decrease, the larger computation range can be achieved, at the expense of a performance degradation in high DP percentage regions. To cover a larger computation range while maintaining the performance in high DP percentage regions, McDanel et al. (2017a) further introduced the multiple-profile incomplete neural networks (MP-IDP), where different profiles can be specified to focus on different DP ranges. During training, all the specified profiles are applied in increasing order of their operating DP ranges. When a profile is applied, only weights corresponding to its operating DP range will be updated, leaving weights corresponding to lower DP percentages freezed since they have been trained in previous stages, and weights corresponding to higher IDP percentages set to zeros since they will be trained in later stages. In such a stage-by-stage training process, the overall performance may not be fully optimized.
+
+# 3 TASK-WISE EARLY STOPPING AND LOSS AGGREGATION
+
+As discussed in Section 2, in the original IDP design, CDP is used during training but IDP is applied at inference time. This mismatch leads to a noticeable degradation in inference performance, especially in low DP percentages. To mitigate this problem, we propose the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm. In this paper, a task is defined as the learning process that uses only a subset of weights determined by a DP percentage to learn the optimal representations. For example, a task of $5 0 \%$ DP implies that the first half of network weights are used for dot product computations and thus only these $5 0 \%$ of weights will be updated while conducting backpropagation. With TESLA, we can optimize a network by tasks with different DP percentages to support various levels of computation scaling and meanwhile reduce the mismatch between training and inference. The design of TESLA is described as follows.
+
+# 3.1 TASK-WISE EARLY STOPPING
+
+Since tasks with different DP percentages may have different learning difficulties and convergence rates, we apply an early stopping mechanism to automatically adjust the learning processes of tasks. Specifically, we keep all hyper-parameters unchanged except the numbers of epoches, which are controlled by the early stopping mechanism that halts the training process as long as the task performance has not been improved for a certain number of iterations. For example, considering two tasks, one using $7 0 \%$ DP (task 1) and the other using $4 0 \%$ DP (task 2), we first optimize task 1 and then switch to optimize task 2 until the optimization process of task 1 reaches the early stopping criterion. With this task-wise early stopping, we are able to optimize all the tasks sequentially, and each task initializes its model using the weights that have been optimized for all previous tasks. However, the weights used in task 2 is exactly a subset of weights used in task 1 such that the optimization process of task 2 may contaminate the well-trained weights for task 1. To reduce this unexpected disturbance while learning multiple tasks, some kinds of loss aggregation are needed to learn a new task without sacrificing the performance of all the past tasks too much.
+
+# Algorithm 1 Task-wise Early Stopping and Loss Aggregation, TESLA
+
+1: Input: a task set in decreasing order, $T = L _ { i }$ ; aggregation coefficient $\alpha$   
+2: Initialization: $L _ { 1 } ^ { o b j }  L _ { 1 }$ and $i \gets 1$   
+3: while $i \leq s i z e ( T )$ do   
+4: 5: $L _ { i + 1 } ^ { o b j }  \alpha \times L _ { i + 1 } + ( 1 - \alpha ) \times L _ { i } ^ { o b j }$ $L _ { i } ^ { o b j }$ pping criteria   
+6: i ← i + 1   
+7: end while
+
+# Algorithm 2 Randomized TESLA, R-TESLA
+
+1: Input: a task set in any order, $T = L _ { i }$ ; allowable epoch, max epoch; aggregation coefficient $\alpha$   
+2: Initialization: $L _ { 1 } ^ { o b j }  L _ { 1 }$ , $i \gets 0$ , and $n \gets 0$   
+3: while $n \leq$ max epoch do   
+4: optimize $L _ { i } ^ { o b j }$ until meeting early stopping criteria, which takes n epochs   
+5: 6: $L _ { i + 1 } ^ { o b j } \gets \alpha \times L _ { k } + ( 1 - \alpha ) \times L _ { i } ^ { o b j }$ $L _ { k }$   
+7: i ← i + 1   
+8: $n  n + n$ epcohs   
+9: end while
+
+# 3.2 TASK-WISE LOSS AGGREGATION
+
+Task-wise loss aggregation is therefore proposed to jointly learn the shared representation for all tasks. By considering one new task at a time, we add the loss of the new task into the current objective function and optimize the aggregated objective function such that tasks are optimized incrementally and jointly. The aggregated objective function can be expressed as
+
+$$
+L _ { 1 } ^ { o b j } = L _ { 1 } ~ \mathrm { a n d } ~ L _ { i + 1 } ^ { o b j } = \alpha \times L _ { i + 1 } + ( 1 - \alpha ) \times L _ { i } ^ { o b j } ~ , ~ \forall i = 1 , \cdots , N - 1
+$$
+
+where $\alpha$ is the aggregation coefficient shared by all subsequent tasks and greater $\alpha$ implies that we care more about the optimization of the new task. As a consequence, the objective function in the whole learning process is an affine combination of the losses of currently considered tasks. By
+
+![](images/0fd499cedc4c9afbb9bd09c480424ad51915337c4d0feb7009e3f509d3814295.jpg)  
+Figure 2: Network structures in study
+
+Table 1: Hyper-parameters in Experiments   
+
+<table><tr><td>Experiment</td><td>MLP on MNIST</td><td>VGG-16 onCIFAR-10</td><td>VGG-16 on CIFAR-100</td></tr><tr><td>Tasks at DP %</td><td>100,70,40,10</td><td>100,50,20</td><td>100,70,50,30</td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.004</td><td>0.004</td></tr><tr><td>Optimizer</td><td>Adam</td><td>SGD momentum=0.9</td><td>SGD momentum =0.9</td></tr><tr><td>Batch size</td><td>28</td><td>32</td><td>64</td></tr><tr><td>Aggregation coefficient</td><td>0.5</td><td>0.5</td><td>0.5</td></tr><tr><td>TESLA stopping criteria</td><td>not improve in4 epochs</td><td>not improve in 4 epochs</td><td>not improve in 4 epochs</td></tr><tr><td>R-TESLAstopping criteria</td><td># epochs over 50</td><td>#epochs over35</td><td>#epochs over35</td></tr><tr><td>Initial weights</td><td>random</td><td>pre-trained on ImageNet</td><td>pre-trained on ImageNet</td></tr></table>
+
+task-wise loss aggregation, these losses are aggregated incrementally and can be jointly optimized to learn a shared representation to be relevant to all tasks.
+
+# 3.3 TESLA AND RANDOMIZED TESLA
+
+Task-wise Early Stopping and Loss Aggregation, TESLA. We integrate task-wise early stopping and task-wise loss aggregation as TESLA to learn dynamic representations in neural networks. The entire training process optimizes all tasks in an arbitrary order. It is obvious that we have several options to order tasks in (i) increasing, (ii) decreasing, or (iii) random DP percentages. Recall that we add a non-increasing coefficients to prioritize terms in computing dot product, and thus the beginning terms, e.g. at $1 0 \%$ DP, are more important than the terms at last $1 0 \%$ terms. Therefore, discarding the terms from the end is less harmful to the optimized parameters, so TESLA is designed to optimize tasks in decreasing order of DP percentages. The TESLA algorithm is shown in Algorithm 1.
+
+Randomized Task-wise Early Stopping and Loss Aggregation, R-TESLA. Here Randomized means that tasks are optimized in random order. The benefits of R-TESLA is two fold. First, RTESLA provides an opportunity to turn attention back to optimize a task which had been halted before, and allows to finetune the weights, which may have been contaminated by other tasks. Second, unlike TESLA that optimizes each task only once, R-TESLA allows each task to be optimized for multiple times, which can be specified by a customized task distribution derived from the behavioral statistics of users or the specification of hardware design. The detailed procedures of R-TESLA are in Algorithm 2.
+
+# 3.4 TRAINABLE PROFILE COEFFICIENTS
+
+In this section we propose to learn profile coefficients along with weights of the model alternately. We initialize all coefficients as one and as long as any update of profile coefficients, we manually clip the coefficients to keep the non-increasing property. The alternate training procedure (ATP) relaxes the constraint of fixed coefficients and we demonstrate the feasibility of ATP in the experiment of the MLP model on MNIST dataset in Section 4.
+
+# 4 EXPERIMENTS
+
+In this section, we demonstrate the effectiveness of using TESLA and R-TESLA to learn dynamic representations in MLP and CNN models, with the widely-used datasets MNIST, CIFAR-10, and CIFAR-100. Figure 2 shows the network architectures in study. Note that while working on CIFAR100, the last fully connected layer of Figure 2(b) is replaced by a single 100-class classifier. Here we compare TESLA and R-TESLA with the original IDP design proposed by McDanel et al. (2017a) over a range of dynamic scaling during inference. All hyper-parameters and experiment settings are summarized in Table 1.
+
+# 4.1 MULTILAYER PERCEPTRONS
+
+First, we consider a 3-layer MLP model, in which the IDP operation is applied to the first hidden layer, as shown in Figure 2(a), and evaluate on the MNIST dataset. In this experiment, we define four tasks that optimize the model at $1 0 \%$ , $4 0 \%$ , $7 0 \%$ , and $1 0 0 \%$ DP, respectively. It is noteworthy that defining too many tasks in our experiment would not benefit much, since there must be a large amount of shared parameters among tasks which makes the model vulnerable to overfitting.
+
+![](images/1b204a0ffa4be7521a49124c4af28e44f0315059e93f44280d7eddbd5e8711ea.jpg)  
+Figure 3: Performance comparisons by a MLP model over the MNIST dataset
+
+TESLA versus original IDP. We compare TESLA and the original IDP design under various profiles. Figure 3(a) shows that at $2 0 \%$ DP, the original IDP achieves $8 0 \%$ , $6 3 \%$ and $5 5 \%$ accuracy for the harmonic, all-one, and linear profiles respectively but TESLA keeps at least $8 8 \%$ accuracy for all profiles at $2 0 \%$ DP and reaches average accuracy of $9 4 . 7 \%$ using the linear profile. Most importantly, compared to the original IDP, TESLA performs only about $1 \%$ worse in accuracy at $1 0 0 \%$ DP but gains a significant improvement from $5 0 \%$ to $9 0 \%$ in accuracy at $1 0 \%$ DP, which is an acceptable trade-off under practical applications.
+
+R-TESLA versus TESLA and original IDP. Figure 3(b) shows that R-TESLA outperforms the original IDP by a large margin and R-TESLA has comparable performance with TESLA in most cases. R-TESLA with the harmonic profile leads to the best average accuracy of $9 5 . 2 \%$ in this experiment. By observing the optimization progress, we find that TESLA achieves its best result after completing the last task thanks to its ordinal optimization. On the other hand, we cannot ensure that R-TESLA can make the ultimate model retains the best dynamic representations due to its random nature.
+
+Learn profile coefficients by ATP. Here we demonstrate the feasibility of learning profile coefficients along with weights. From Figure 3(c), with the help of trainable profile coefficients, both TESLA and R-TESLA further boost by $1 \%$ in average, and we also observe that the learned profile coefficients are similar to harmonic ones as shown in Figure 3(d). This may support why performance of harmonic coefficients is the best in the original IDP. By allowing coefficients to be trainable, it is no longer to require hand-crafted profile coefficients and determine the best profile coefficients by extensive experiments.
+
+# 4.2 CONVOLUTIONAL NEURAL NETWORKS
+
+We choose the known VGG-16 model pre-trained on ImageNet to evaluate over CIFAR-10 and CIFAR-100 dataset so that the last few dense layers are replaced by a 10-class classifier and a 100- class classifier respectively. Here we use the linear profile coefficients to compare: (i) the original IDP design, (ii) multiple-profile IDP design (MP-IDP) as proposed in McDanel et al. (2017a), (iii) TESLA, and (iv) R-TESLA. The experimental results are summarized below.
+
+VGG-16 on CIFAR-10. According to Figure 4(a), the performance of original IDP by all-one coefficients drops much faster than that by linear coefficients. Appling all-one coefficients is equivalent to using the original VGG-16 network; however, linear profile coefficients implicitly encourages networks to learn channel importance in order, and also brings about that pruning away later channels at different DP percentages does not hurt the performance that much. With the use of multiple profiles, MP-IDP does enlarge the computational range with an increase in accuracy to $7 5 \%$ at $5 0 \%$ DP. Furthermore, the proposed algorithms, TESLA and R-TESLA, boost the accuracy to reach $8 5 \%$ at $5 0 \%$ DP, and an even higher accuracy at $1 0 0 \%$ DP.
+
+Following the previous experiment, here we augment another new task of $2 0 \%$ DP and observe whether TESLA can leverage up the performance at low DP percentages by adding a task of a low DP percentage. Figure 4(b) shows that TESLA and R-TESLA greatly widens the computational ranges by making accuracy reaching $7 5 \%$ at $2 0 \%$ DP. We contribute this effect to applying TESLA and R-TESLA in decreasing order of dot product percentages so that the representation learned at $1 0 0 \%$ DP drives the training of representation at $5 0 \%$ DP, which also makes the representation much easier to be learned at $2 0 \%$ DP. Compared to TESLA and R-TESLA, MP-IDP trains models in increasing order of DP percentages and thus MP-IDP doesn’t see much improvement at lower IDP percentages although adding another task at $2 0 \%$ DP.
+
+![](images/fc8029bb88c608867e56b19ad9e18e324046ff3b51891264bc821bcb56eb9d01.jpg)  
+Figure 4: Performance comparisons by the VGG-16 model over the CIFAR-10 and CIFAR-100 dataset
+
+![](images/db98606cfff4d19e86ef988bb3e742d28530e3d0cbd05b2161071314c462df87.jpg)  
+Figure 5: CAMs at different DP percentages. Red colored text means wrong prediction and green colored text means correct prediction.
+
+![](images/d7f7b88869bdbd7652219e42dd81d852c03cd882246096e26ed8bc3e9aa459f6.jpg)  
+Figure 6: CAMs of a testing image that is correctly classified at all specified DP percentages.
+
+VGG-16 on CIFAR-100. To sufficiently illustrate the effectiveness of the proposed approaches, we evaluate over a larger dataset, CIFAR-100. Figure 4(c) shows the performance of TESLA and R-TESLA still keeps around $6 0 \%$ accuracy from $3 0 \%$ to $5 0 \%$ DP, which outperforms either original IDP or MP-IDP by a significant margin, which is consistent with the result of CIFAR-10. Specifically, both TESLA and R-TESLA sacrifice about $4 \%$ accuracy at $1 0 0 \%$ DP but gain a great improvements of $6 0 \%$ accuracy in low DP percentages.
+
+CAM visualization. We visualize what the model sees at different DP percentages by deploying the Class Activation Mapping (CAM) technique introduced by Zhou et al. (2016a). A resulting CAM indicates how much each location contribute to the final class prediction. In this stage, we replace the max-pooling layers with average-pooling layers and train the VGG-16 network with linear coefficients optimized at $2 0 \%$ , $5 0 \%$ , $1 0 0 \%$ DP. From CAMs at different DP percentages, we found that the network is easier to make wrong prediction at $1 0 \%$ and $3 0 \%$ DP but still makes correct prediction at $2 0 \%$ DP as shown in Figure 5 since the representations at $2 0 \%$ DP are optimized. This finding implies that we can specify any DP percentages to be optimized for satisfying custom requirements. Compared to Figure 6, we also notice that the CAMs at $1 0 \%$ DP are almost the same no matter the correctness of predictions, which indicates too limited capacity to capture meaningful patterns, and thus the network at $1 0 \%$ DP behaves like a random guess.
+
+# 5 RELATED WORK
+
+Our work is rooted from IDP proposed by McDanel et al. (2017a), which, in addition to MLPs and regular CNNs, can also be used in conjunction with other variants of convolutional layers, such as separable convolution layer Howard et al. (2017) and binary convolutional layer McDanel et al. (2017b). As discussed throughout this paper, our work extends the original IDP design by proposing new training algorithms involving a single profile, which may be trainable or pre-determined, to significantly improve the overall performance, especially in low DP percentages.
+
+Network pruning is a widely-studied area that also aims at compressing the CNN models. Early works of network pruning construct a threshold for dropping weights by information obtained from Hessian matrix or inverse Hessian matrix in LeCun et al. (1990); Hassibi & Stork (1993), which adds memory and computation costs. In most of the recent works, magnitude-based pruning and recovering are incorporated to compensate the potential loss incurred by inadequate pruning. For example, Guo et al. (2016) introduces the splicing operation to enable connection recovery, and Han et al. (2016) directly makes the network dense again. Li et al. (2016) also prune filters in CNNs based on magnitude, but the number of filters pruned away in each layer is decided by layer-wise sensitivity. Besides magnitude-based pruning, a Taylor expansion-based criterion is introduced in Molchanov et al. (2016) to approximate the change in the cost function induced by pruning. In addition to network pruning, some works focus on low-rank decomposition for network compression. For example, Denton et al. (2014) and Jaderberg et al. (2014) approximate the weight matrix into low-rank components by minimizing the reconstruction error. Yu et al. (2017) further decomposes the weight matrix into its low-rank and sparse component. Other works focus on grouping similar weights, such as quantization by Han et al. (2015), Gong et al. (2014), and Zhou et al. (2017) and weight sharing by Ullrich et al. (2017), aiming at reducing the level of redundancy and the required storage. Yet another approach introduces group sparsity regularizer to constrain the structure of the model in Wen et al. (2016), Zhou et al. (2016b), and Alvarez & Salzmann (2016).
+
+While all the above techniques are promising in reducing the size of the networks, none of them supports dynamic adjustment during inference as IDP does. Furthermore, most of the above techniques involve retraining the model iteratively, resulting in computational overhead. In our proposed work, the goal of efficient inference with dynamic adjustment can be readily fulfilled by training a single model at once, and the effectiveness is expected to be further improved by incorporating with other techniques listed above.
+
+# 6 CONCLUSION
+
+In this paper, we extend the idea of incomplete dot product (IDP) by proposing the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm to significantly improve the performance of neural networks with dynamically computation regions at inference time without significantly compromising accuracy. A task is defined as the learning process that uses only a subset of weights specified by a DP percentage to learn the optimal representations of the network. By introducing non-increasing profile coefficients to prioritize weights or filters during training, TESLA can be used to optimize multiple tasks in decreasing order of DP percentages by aggregating the their loss functions. Additionally, we propose Randomized TESLA (R-TESLA) which optimizes tasks in random order, and show that both TESLA and R-TESLA outperform original IDP and multiple-profile IDP significantly. The visualization of the class activation maps (CAMs) provide a strong evidence that the representations learned by TESLA allow dynamically scaling across a computation range to meet various power consumption, latency and accuracy requirements on end devices.
+
+# REFERENCES
+
+Jose M Alvarez and Mathieu Salzmann. Learning the number of neurons in deep networks. In Advances in Neural Information Processing Systems, pp. 2270–2278, 2016.
+
+Emily L Denton, Wojciech Zaremba, Joan Bruna, Yann LeCun, and Rob Fergus. Exploiting linear structure within convolutional networks for efficient evaluation. In Advances in Neural Information Processing Systems, pp. 1269–1277, 2014.
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+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015.
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+Song Han, Jeff Pool, Sharan Narang, Huizi Mao, Enhao Gong, Shijian Tang, Erich Elsen, Peter Vajda, Manohar Paluri, John Tran, et al. Dsd: Dense-sparse-dense training for deep neural networks. 2016.
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+Andrew G Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. Mobilenets: Efficient convolutional neural networks for mobile vision applications. arXiv preprint arXiv:1704.04861, 2017.
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+Max Jaderberg, Andrea Vedaldi, and Andrew Zisserman. Speeding up convolutional neural networks with low rank expansions. arXiv preprint arXiv:1405.3866, 2014.
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+Yann LeCun, John S. Denker, and Sara A. Solla. Optimal brain damage. In Advances in Neural Information Processing Systems 2, pp. 598–605. 1990.
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+Hao Li, Asim Kadav, Igor Durdanovic, Hanan Samet, and Hans Peter Graf. Pruning filters for efficient convnets. arXiv preprint arXiv:1608.08710, 2016.
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+Bradley McDanel, Surat Teerapittayanon, and HT Kung. Incomplete dot products for dynamic computation scaling in neural network inference. 2017a.
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+Bradley McDanel, Surat Teerapittayanon, and H.T. Kung. Embedded binarized neural networks. In Proceedings of the 2017 International Conference on Embedded Wireless Systems and Networks, 2017b.
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+H Brendan McMahan, Eider Moore, Daniel Ramage, and Blaise Aguera y Arcas. Federated learning of deep networks using model averaging. 2016.
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+Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz. Pruning convolutional neural networks for resource efficient inference. 2016.
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+Surat Teerapittayanon, Bradley McDanel, and HT Kung. Distributed deep neural networks over the cloud, the edge and end devices. In Distributed Computing Systems (ICDCS), 2017 IEEE 37th International Conference on, pp. 328–339. IEEE, 2017.
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+Karen Ullrich, Edward Meeds, and Max Welling. Soft weight-sharing for neural network compression. arXiv preprint arXiv:1702.04008, 2017.
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+Wei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Learning structured sparsity in deep neural networks. In Advances in Neural Information Processing Systems, pp. 2074–2082, 2016.
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+Xiyu Yu, Tongliang Liu, Xinchao Wang, and Dacheng Tao. On compressing deep models by low rank and sparse decomposition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7370–7379, 2017.
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+Aojun Zhou, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen. Incremental network quantization: Towards lossless cnns with low-precision weights. arXiv preprint arXiv:1702.03044, 2017.
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+Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba. Learning deep features for discriminative localization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2921–2929, 2016a.
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+        "type": "text",
+        "text": "TESLA: TASK-WISE EARLY STOPPING AND LOSS AGGREGATION FOR DYNAMIC NEURAL NETWORK INFERENCE ",
+        "text_level": 1,
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+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For inference operations in deep neural networks on end devices, it is desirable to deploy a single pre-trained neural network model, which can dynamically scale across a computation range without comprising accuracy. To achieve this goal, Incomplete Dot Product (IDP) has been proposed to use only a subset of terms in dot products during forward propagation. However, there are some limitations, including noticeable performance degradation in operating regions with low computational costs, and essential performance limitations since IDP uses hand-crafted profile coefficients. In this paper, we extend IDP by proposing new training algorithms involving a single profile, which may be trainable or pre-determined, to significantly improve the overall performance, especially in operating regions with low computational costs. Specifically, we propose the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm, which is showed in our 3-layer multilayer perceptron on MNIST that outperforms the original IDP by $3 2 \\%$ when only $1 0 \\%$ of dot products terms are used and achieves $9 4 . 7 \\%$ accuracy on average. By introducing trainable profile coefficients, TESLA further improves the accuracy to $9 5 . 5 \\%$ without specifying coefficients in advance. Besides, TESLA is applied to the VGG-16 model, which achieves $8 0 \\%$ accuracy using only $2 0 \\%$ of dot product terms on CIFAR-10 and also keeps $6 0 \\%$ accuracy using only $3 0 \\%$ of dot product terms on CIFAR-100, but the original IDP performs like a random guess in these two datasets at such low computation costs. Finally, we visualize the learned representations at different dot product percentages by class activation map and show that, by applying TESLA, the learned representations can adapt over a wide range of operation regions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Inference operations in deep neural networks on end devices, such as mobile phones, embedded sensors, IoT devices, etc., have recently received increasing attention including McMahan et al. (2016), Howard et al. (2017), and Teerapittayanon et al. (2017). In such applications, it is desirable to deploy a single pre-trained CNN model on end devices, while allowing multiple operating regions to meet different power consumption, latency, and accuracy requirements. To achieve this goal, McDanel et al. (2017a) proposed the incomplete dot product (IDP) operation, where only a subset of terms is used in dot products of forward propagation. From now on, $x \\%$ dot product (DP), where 0 $\\leq x \\leq 1 0 0$ , means the $x \\%$ of terms used in dot products. As illustrated in Figure 1, $5 0 \\%$ DP means half of filters are used during forward propagation, and thus only half of the output channels are retained. To reduce the deviation induced by IDP, filters are prioritized from most important to the least important by pre-determined monotonically non-increasing profile coefficients (say, $\\gamma _ { 1 } , . . . , \\gamma _ { N } )$ during training. Therefore, IDP can be applied at inference time with dynamically-adjusted degrees of completeness (specified by the percentage of terms being used) to trade off accuracy slightly for lowered power consumption and reduced latency. Specifically, VGG-16 model with $5 0 \\%$ DP achieves $7 0 \\%$ in accuracy on the CIFAR-10 dataset compared to the standard network achieves only $3 5 \\%$ accuracy when using the reduced channel set. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While the original IDP design seems promising, there are two limitations. First, since the training process aims at optimizing the loss function computed using all weights of the model ( $1 0 0 \\%$ DP), there will be a mismatch between training and testing. It is no surprise that inference performance significantly decreases in low DP percentages and thus narrow the dynamic computation range. To mitigate this problem, the original IDP design utilizes the multiple-profile training strategy, where different profiles can be specified to focus on different dot product ranges. In such a multipleprofile training process, however, certain subset of weights will be freezed in each training stage corresponding to the profile being focused, and hence the overall performance may not be fully optimized. Besides, each profile needs to maintain a separate first and last layer for adjusting to its own dot product range, resulting in additional memory overhead. The second limitation relates to the pre-determined nature of profile coefficients. While there are multiple ways to set the profile based on different dynamic range requirements, the original IDP design did not focus on finding a single ”best” profile that leads to the best performance. Instead, they use multiple hand-crafted profile coefficients, which make the system design less general among different applications, and hence may limit the overall performance of the system. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/b0dd91008c4e8fc441f3be52ecf73951a42e6e8f3ccdd53eb25a978714cc5aae.jpg",
+        "image_caption": [
+            "Figure 1: Comparison between complete dot product (CDP) and incomplete dot product (IDP) where $X \\%$ DP implies only $X \\%$ of filters are used to compute the corresponding output channel. Since only $X \\%$ filters are unused, the resulting output is an approximation of the output under CDP. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "To reduce the mismatch between training and testing performances, we propose the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm, in which multiple loss functions are computed in different DP percentages. By gradually aggregating these loss functions in decreasing order of DP percentages as the objective function to be optimized, TESLA significantly improves testing performances in low DP percentages without compromising accuracy in medium to high DP percentages. The loss functions can also be aggregated in random order of DP percentages to make a variant of TESLA, called Randomized TESLA (R-TESLA), which enables better performances under prespecified operating regions of end devices. Moreover, we relax the constraint of pre-determined profile coefficients and propose the alternate training procedure (ATP) to alternately train the profile coefficients along with weights of the model. By introducing trainable profile coefficients, customization among different applications can be achieved in a more generalized way, and the overall performance can also be further improved. This paper has made two major contributions: (1) We propose the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm and Randomized TESLA that can achieve dynamic scaling over a computation range in neural network inference without compromising accuracy. (2) We also propose the Alternate Training Procedure (ATP) that can learn the profile coefficients and the model weights simultaneously without the need of manual configuration of the profile coefficients. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 INCOMPLETE NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            862,
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+            877
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Incomplete dot product (IDP) is a novel mechanism proposed by McDanel et al. (2017a) that can be applied to a hidden layer of MLPs or deep CNN models to dynamically lower the inference costs by computing only a subset of terms in dot products during forward propagation. By introducing a set of non-increasing coefficients $\\gamma _ { i }$ , referred to as a profile, to the channels during training, the channels will be ordered implicitly in non-increasing order from the most important to the least important. By simply dropping out less important channels at inference time, it suffices to train and deploy a single network, while still supporting different levels of computation scaling without compromising accuracy significantly. In this section, we briefly introduce the main concepts of IDP. ",
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+    {
+        "type": "text",
+        "text": "2.1 INCOMPLETE DOT PRODUCT OPERATION",
+        "text_level": 1,
+        "bbox": [
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+            204,
+            500,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Mathematically, for an IDP fully-connected layer with input dimension $N$ and output dimension $M$ , the $j$ -th output component $y _ { j }$ is computed as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/277a5e77dcda76023ab234cbc1b96ba86d1f59eac2e8de423f950f4a5586de27.jpg",
+        "text": "$$\ny _ { j } = \\sum _ { i = 1 } ^ { N } \\gamma _ { i } w _ { j i } x _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            265,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for $j \\in \\{ 1 , 2 , . . . , M \\}$ , where $x _ { i }$ is the $i$ -th input component, $w _ { j i }$ is the weight corresponding to the $j$ -th output component and the $i$ -th input component, and $\\gamma _ { i }$ is the $i$ -th profile coefficient. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Similar expression can be derived for the IDP operation applied to a convolutional layer of CNN, as illustrated in Figure 1. For an IDP convolutional layer with number of input channels $N$ and number of output channels $M$ , the $j$ -th output channel ${ \\bf y } _ { j }$ is computed as ",
+        "bbox": [
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+            348,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b646fb82e63d26af9d1f7af0cfcc8505ac8bf79bb20fc34f7a63fc4082417b9d.jpg",
+        "text": "$$\n\\mathbf { y } _ { j } = \\gamma _ { j } \\sum _ { i = 1 } ^ { N } \\mathbf { f } _ { j i } * \\mathbf { x } _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for $j \\in \\{ 1 , 2 , . . . , M \\}$ , where $\\mathbf { f } _ { j i } * \\mathbf { x } _ { i }$ denotes the convolution operation of the $i$ -th input channel $\\mathbf { x } _ { i }$ and the $i$ -th channel of the $j$ -th filter $\\mathbf { f } _ { j i }$ , and $\\gamma _ { j }$ is the profile coefficient for the $j$ -th filter. Note that, instead of applying profile coefficients depthwise on each filter before convolution as is the case in the original IDP design, we multiply each $\\gamma _ { j }$ to each output channel after a complete convolution to produce ${ \\bf y } _ { j }$ . These two approaches, however, are equivalent with negligible difference induced by the first hidden layer. Since the output channels ${ \\bf y } _ { j }$ ’s become input channels $\\mathbf { x } _ { i }$ ’s to the next layer, applying $\\gamma _ { j }$ ’s to ${ \\bf y } _ { j }$ ’s is equivalent to applying them into the convolution operation in the next layer. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To compute IDP with a target dot product percentage, a truncated version of Eq. 1 or Eq. 2 replaces the original computation to keep only a subset of the beginning terms. As for the case with all terms are kept, we refer to such operations as complete dot product (CDP) or $1 0 0 \\%$ DP, interchangeably. Note that in the training process in the original IDP design, only CDP is used. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 MULTIPLE-PROFILE INCOMPLETE NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the work of McDanel et al. (2017a), several profile coefficients are proposed and applied in a pre-determined manner. When only a single profile is applied to the model, the trade-off between computation range and performance in high DP percentage regions is also demonstrated. Generally, the faster the profile coefficients decrease, the larger computation range can be achieved, at the expense of a performance degradation in high DP percentage regions. To cover a larger computation range while maintaining the performance in high DP percentage regions, McDanel et al. (2017a) further introduced the multiple-profile incomplete neural networks (MP-IDP), where different profiles can be specified to focus on different DP ranges. During training, all the specified profiles are applied in increasing order of their operating DP ranges. When a profile is applied, only weights corresponding to its operating DP range will be updated, leaving weights corresponding to lower DP percentages freezed since they have been trained in previous stages, and weights corresponding to higher IDP percentages set to zeros since they will be trained in later stages. In such a stage-by-stage training process, the overall performance may not be fully optimized. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 TASK-WISE EARLY STOPPING AND LOSS AGGREGATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As discussed in Section 2, in the original IDP design, CDP is used during training but IDP is applied at inference time. This mismatch leads to a noticeable degradation in inference performance, especially in low DP percentages. To mitigate this problem, we propose the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm. In this paper, a task is defined as the learning process that uses only a subset of weights determined by a DP percentage to learn the optimal representations. For example, a task of $5 0 \\%$ DP implies that the first half of network weights are used for dot product computations and thus only these $5 0 \\%$ of weights will be updated while conducting backpropagation. With TESLA, we can optimize a network by tasks with different DP percentages to support various levels of computation scaling and meanwhile reduce the mismatch between training and inference. The design of TESLA is described as follows. ",
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+    {
+        "type": "text",
+        "text": "3.1 TASK-WISE EARLY STOPPING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since tasks with different DP percentages may have different learning difficulties and convergence rates, we apply an early stopping mechanism to automatically adjust the learning processes of tasks. Specifically, we keep all hyper-parameters unchanged except the numbers of epoches, which are controlled by the early stopping mechanism that halts the training process as long as the task performance has not been improved for a certain number of iterations. For example, considering two tasks, one using $7 0 \\%$ DP (task 1) and the other using $4 0 \\%$ DP (task 2), we first optimize task 1 and then switch to optimize task 2 until the optimization process of task 1 reaches the early stopping criterion. With this task-wise early stopping, we are able to optimize all the tasks sequentially, and each task initializes its model using the weights that have been optimized for all previous tasks. However, the weights used in task 2 is exactly a subset of weights used in task 1 such that the optimization process of task 2 may contaminate the well-trained weights for task 1. To reduce this unexpected disturbance while learning multiple tasks, some kinds of loss aggregation are needed to learn a new task without sacrificing the performance of all the past tasks too much. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Task-wise Early Stopping and Loss Aggregation, TESLA ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "1: Input: a task set in decreasing order, $T = L _ { i }$ ; aggregation coefficient $\\alpha$   \n2: Initialization: $L _ { 1 } ^ { o b j }  L _ { 1 }$ and $i \\gets 1$   \n3: while $i \\leq s i z e ( T )$ do   \n4: 5: $L _ { i + 1 } ^ { o b j }  \\alpha \\times L _ { i + 1 } + ( 1 - \\alpha ) \\times L _ { i } ^ { o b j }$ $L _ { i } ^ { o b j }$ pping criteria   \n6: i ← i + 1   \n7: end while ",
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Randomized TESLA, R-TESLA ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "1: Input: a task set in any order, $T = L _ { i }$ ; allowable epoch, max epoch; aggregation coefficient $\\alpha$   \n2: Initialization: $L _ { 1 } ^ { o b j }  L _ { 1 }$ , $i \\gets 0$ , and $n \\gets 0$   \n3: while $n \\leq$ max epoch do   \n4: optimize $L _ { i } ^ { o b j }$ until meeting early stopping criteria, which takes n epochs   \n5: 6: $L _ { i + 1 } ^ { o b j } \\gets \\alpha \\times L _ { k } + ( 1 - \\alpha ) \\times L _ { i } ^ { o b j }$ $L _ { k }$   \n7: i ← i + 1   \n8: $n  n + n$ epcohs   \n9: end while ",
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+    {
+        "type": "text",
+        "text": "3.2 TASK-WISE LOSS AGGREGATION ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Task-wise loss aggregation is therefore proposed to jointly learn the shared representation for all tasks. By considering one new task at a time, we add the loss of the new task into the current objective function and optimize the aggregated objective function such that tasks are optimized incrementally and jointly. The aggregated objective function can be expressed as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f6970a875a1d8271302602c13ee6fdeb4520db67bccf40fbaa5f0a85f50969ab.jpg",
+        "text": "$$\nL _ { 1 } ^ { o b j } = L _ { 1 } ~ \\mathrm { a n d } ~ L _ { i + 1 } ^ { o b j } = \\alpha \\times L _ { i + 1 } + ( 1 - \\alpha ) \\times L _ { i } ^ { o b j } ~ , ~ \\forall i = 1 , \\cdots , N - 1\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha$ is the aggregation coefficient shared by all subsequent tasks and greater $\\alpha$ implies that we care more about the optimization of the new task. As a consequence, the objective function in the whole learning process is an affine combination of the losses of currently considered tasks. By ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/0fd499cedc4c9afbb9bd09c480424ad51915337c4d0feb7009e3f509d3814295.jpg",
+        "image_caption": [
+            "Figure 2: Network structures in study "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8c5e5c7beaa3e8ddea8f481bcbd682605f88d3a4d64c7902cb15c8ceb72837bb.jpg",
+        "table_caption": [
+            "Table 1: Hyper-parameters in Experiments "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Experiment</td><td>MLP on MNIST</td><td>VGG-16 onCIFAR-10</td><td>VGG-16 on CIFAR-100</td></tr><tr><td>Tasks at DP %</td><td>100,70,40,10</td><td>100,50,20</td><td>100,70,50,30</td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.004</td><td>0.004</td></tr><tr><td>Optimizer</td><td>Adam</td><td>SGD momentum=0.9</td><td>SGD momentum =0.9</td></tr><tr><td>Batch size</td><td>28</td><td>32</td><td>64</td></tr><tr><td>Aggregation coefficient</td><td>0.5</td><td>0.5</td><td>0.5</td></tr><tr><td>TESLA stopping criteria</td><td>not improve in4 epochs</td><td>not improve in 4 epochs</td><td>not improve in 4 epochs</td></tr><tr><td>R-TESLAstopping criteria</td><td># epochs over 50</td><td>#epochs over35</td><td>#epochs over35</td></tr><tr><td>Initial weights</td><td>random</td><td>pre-trained on ImageNet</td><td>pre-trained on ImageNet</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "task-wise loss aggregation, these losses are aggregated incrementally and can be jointly optimized to learn a shared representation to be relevant to all tasks. ",
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+    {
+        "type": "text",
+        "text": "3.3 TESLA AND RANDOMIZED TESLA ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Task-wise Early Stopping and Loss Aggregation, TESLA. We integrate task-wise early stopping and task-wise loss aggregation as TESLA to learn dynamic representations in neural networks. The entire training process optimizes all tasks in an arbitrary order. It is obvious that we have several options to order tasks in (i) increasing, (ii) decreasing, or (iii) random DP percentages. Recall that we add a non-increasing coefficients to prioritize terms in computing dot product, and thus the beginning terms, e.g. at $1 0 \\%$ DP, are more important than the terms at last $1 0 \\%$ terms. Therefore, discarding the terms from the end is less harmful to the optimized parameters, so TESLA is designed to optimize tasks in decreasing order of DP percentages. The TESLA algorithm is shown in Algorithm 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Randomized Task-wise Early Stopping and Loss Aggregation, R-TESLA. Here Randomized means that tasks are optimized in random order. The benefits of R-TESLA is two fold. First, RTESLA provides an opportunity to turn attention back to optimize a task which had been halted before, and allows to finetune the weights, which may have been contaminated by other tasks. Second, unlike TESLA that optimizes each task only once, R-TESLA allows each task to be optimized for multiple times, which can be specified by a customized task distribution derived from the behavioral statistics of users or the specification of hardware design. The detailed procedures of R-TESLA are in Algorithm 2. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 TRAINABLE PROFILE COEFFICIENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section we propose to learn profile coefficients along with weights of the model alternately. We initialize all coefficients as one and as long as any update of profile coefficients, we manually clip the coefficients to keep the non-increasing property. The alternate training procedure (ATP) relaxes the constraint of fixed coefficients and we demonstrate the feasibility of ATP in the experiment of the MLP model on MNIST dataset in Section 4. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we demonstrate the effectiveness of using TESLA and R-TESLA to learn dynamic representations in MLP and CNN models, with the widely-used datasets MNIST, CIFAR-10, and CIFAR-100. Figure 2 shows the network architectures in study. Note that while working on CIFAR100, the last fully connected layer of Figure 2(b) is replaced by a single 100-class classifier. Here we compare TESLA and R-TESLA with the original IDP design proposed by McDanel et al. (2017a) over a range of dynamic scaling during inference. All hyper-parameters and experiment settings are summarized in Table 1. ",
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+    {
+        "type": "text",
+        "text": "4.1 MULTILAYER PERCEPTRONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, we consider a 3-layer MLP model, in which the IDP operation is applied to the first hidden layer, as shown in Figure 2(a), and evaluate on the MNIST dataset. In this experiment, we define four tasks that optimize the model at $1 0 \\%$ , $4 0 \\%$ , $7 0 \\%$ , and $1 0 0 \\%$ DP, respectively. It is noteworthy that defining too many tasks in our experiment would not benefit much, since there must be a large amount of shared parameters among tasks which makes the model vulnerable to overfitting. ",
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+    {
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+        "img_path": "images/1b204a0ffa4be7521a49124c4af28e44f0315059e93f44280d7eddbd5e8711ea.jpg",
+        "image_caption": [
+            "Figure 3: Performance comparisons by a MLP model over the MNIST dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "TESLA versus original IDP. We compare TESLA and the original IDP design under various profiles. Figure 3(a) shows that at $2 0 \\%$ DP, the original IDP achieves $8 0 \\%$ , $6 3 \\%$ and $5 5 \\%$ accuracy for the harmonic, all-one, and linear profiles respectively but TESLA keeps at least $8 8 \\%$ accuracy for all profiles at $2 0 \\%$ DP and reaches average accuracy of $9 4 . 7 \\%$ using the linear profile. Most importantly, compared to the original IDP, TESLA performs only about $1 \\%$ worse in accuracy at $1 0 0 \\%$ DP but gains a significant improvement from $5 0 \\%$ to $9 0 \\%$ in accuracy at $1 0 \\%$ DP, which is an acceptable trade-off under practical applications. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "R-TESLA versus TESLA and original IDP. Figure 3(b) shows that R-TESLA outperforms the original IDP by a large margin and R-TESLA has comparable performance with TESLA in most cases. R-TESLA with the harmonic profile leads to the best average accuracy of $9 5 . 2 \\%$ in this experiment. By observing the optimization progress, we find that TESLA achieves its best result after completing the last task thanks to its ordinal optimization. On the other hand, we cannot ensure that R-TESLA can make the ultimate model retains the best dynamic representations due to its random nature. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Learn profile coefficients by ATP. Here we demonstrate the feasibility of learning profile coefficients along with weights. From Figure 3(c), with the help of trainable profile coefficients, both TESLA and R-TESLA further boost by $1 \\%$ in average, and we also observe that the learned profile coefficients are similar to harmonic ones as shown in Figure 3(d). This may support why performance of harmonic coefficients is the best in the original IDP. By allowing coefficients to be trainable, it is no longer to require hand-crafted profile coefficients and determine the best profile coefficients by extensive experiments. ",
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+    {
+        "type": "text",
+        "text": "4.2 CONVOLUTIONAL NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We choose the known VGG-16 model pre-trained on ImageNet to evaluate over CIFAR-10 and CIFAR-100 dataset so that the last few dense layers are replaced by a 10-class classifier and a 100- class classifier respectively. Here we use the linear profile coefficients to compare: (i) the original IDP design, (ii) multiple-profile IDP design (MP-IDP) as proposed in McDanel et al. (2017a), (iii) TESLA, and (iv) R-TESLA. The experimental results are summarized below. ",
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+    {
+        "type": "text",
+        "text": "VGG-16 on CIFAR-10. According to Figure 4(a), the performance of original IDP by all-one coefficients drops much faster than that by linear coefficients. Appling all-one coefficients is equivalent to using the original VGG-16 network; however, linear profile coefficients implicitly encourages networks to learn channel importance in order, and also brings about that pruning away later channels at different DP percentages does not hurt the performance that much. With the use of multiple profiles, MP-IDP does enlarge the computational range with an increase in accuracy to $7 5 \\%$ at $5 0 \\%$ DP. Furthermore, the proposed algorithms, TESLA and R-TESLA, boost the accuracy to reach $8 5 \\%$ at $5 0 \\%$ DP, and an even higher accuracy at $1 0 0 \\%$ DP. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following the previous experiment, here we augment another new task of $2 0 \\%$ DP and observe whether TESLA can leverage up the performance at low DP percentages by adding a task of a low DP percentage. Figure 4(b) shows that TESLA and R-TESLA greatly widens the computational ranges by making accuracy reaching $7 5 \\%$ at $2 0 \\%$ DP. We contribute this effect to applying TESLA and R-TESLA in decreasing order of dot product percentages so that the representation learned at $1 0 0 \\%$ DP drives the training of representation at $5 0 \\%$ DP, which also makes the representation much easier to be learned at $2 0 \\%$ DP. Compared to TESLA and R-TESLA, MP-IDP trains models in increasing order of DP percentages and thus MP-IDP doesn’t see much improvement at lower IDP percentages although adding another task at $2 0 \\%$ DP. ",
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+    {
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+        "img_path": "images/fc8029bb88c608867e56b19ad9e18e324046ff3b51891264bc821bcb56eb9d01.jpg",
+        "image_caption": [
+            "Figure 4: Performance comparisons by the VGG-16 model over the CIFAR-10 and CIFAR-100 dataset "
+        ],
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+        "img_path": "images/db98606cfff4d19e86ef988bb3e742d28530e3d0cbd05b2161071314c462df87.jpg",
+        "image_caption": [
+            "Figure 5: CAMs at different DP percentages. Red colored text means wrong prediction and green colored text means correct prediction. "
+        ],
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+        "image_caption": [
+            "Figure 6: CAMs of a testing image that is correctly classified at all specified DP percentages. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "VGG-16 on CIFAR-100. To sufficiently illustrate the effectiveness of the proposed approaches, we evaluate over a larger dataset, CIFAR-100. Figure 4(c) shows the performance of TESLA and R-TESLA still keeps around $6 0 \\%$ accuracy from $3 0 \\%$ to $5 0 \\%$ DP, which outperforms either original IDP or MP-IDP by a significant margin, which is consistent with the result of CIFAR-10. Specifically, both TESLA and R-TESLA sacrifice about $4 \\%$ accuracy at $1 0 0 \\%$ DP but gain a great improvements of $6 0 \\%$ accuracy in low DP percentages. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "CAM visualization. We visualize what the model sees at different DP percentages by deploying the Class Activation Mapping (CAM) technique introduced by Zhou et al. (2016a). A resulting CAM indicates how much each location contribute to the final class prediction. In this stage, we replace the max-pooling layers with average-pooling layers and train the VGG-16 network with linear coefficients optimized at $2 0 \\%$ , $5 0 \\%$ , $1 0 0 \\%$ DP. From CAMs at different DP percentages, we found that the network is easier to make wrong prediction at $1 0 \\%$ and $3 0 \\%$ DP but still makes correct prediction at $2 0 \\%$ DP as shown in Figure 5 since the representations at $2 0 \\%$ DP are optimized. This finding implies that we can specify any DP percentages to be optimized for satisfying custom requirements. Compared to Figure 6, we also notice that the CAMs at $1 0 \\%$ DP are almost the same no matter the correctness of predictions, which indicates too limited capacity to capture meaningful patterns, and thus the network at $1 0 \\%$ DP behaves like a random guess. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our work is rooted from IDP proposed by McDanel et al. (2017a), which, in addition to MLPs and regular CNNs, can also be used in conjunction with other variants of convolutional layers, such as separable convolution layer Howard et al. (2017) and binary convolutional layer McDanel et al. (2017b). As discussed throughout this paper, our work extends the original IDP design by proposing new training algorithms involving a single profile, which may be trainable or pre-determined, to significantly improve the overall performance, especially in low DP percentages. ",
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+    {
+        "type": "text",
+        "text": "Network pruning is a widely-studied area that also aims at compressing the CNN models. Early works of network pruning construct a threshold for dropping weights by information obtained from Hessian matrix or inverse Hessian matrix in LeCun et al. (1990); Hassibi & Stork (1993), which adds memory and computation costs. In most of the recent works, magnitude-based pruning and recovering are incorporated to compensate the potential loss incurred by inadequate pruning. For example, Guo et al. (2016) introduces the splicing operation to enable connection recovery, and Han et al. (2016) directly makes the network dense again. Li et al. (2016) also prune filters in CNNs based on magnitude, but the number of filters pruned away in each layer is decided by layer-wise sensitivity. Besides magnitude-based pruning, a Taylor expansion-based criterion is introduced in Molchanov et al. (2016) to approximate the change in the cost function induced by pruning. In addition to network pruning, some works focus on low-rank decomposition for network compression. For example, Denton et al. (2014) and Jaderberg et al. (2014) approximate the weight matrix into low-rank components by minimizing the reconstruction error. Yu et al. (2017) further decomposes the weight matrix into its low-rank and sparse component. Other works focus on grouping similar weights, such as quantization by Han et al. (2015), Gong et al. (2014), and Zhou et al. (2017) and weight sharing by Ullrich et al. (2017), aiming at reducing the level of redundancy and the required storage. Yet another approach introduces group sparsity regularizer to constrain the structure of the model in Wen et al. (2016), Zhou et al. (2016b), and Alvarez & Salzmann (2016). ",
+        "bbox": [
+            174,
+            227,
+            825,
+            476
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "While all the above techniques are promising in reducing the size of the networks, none of them supports dynamic adjustment during inference as IDP does. Furthermore, most of the above techniques involve retraining the model iteratively, resulting in computational overhead. In our proposed work, the goal of efficient inference with dynamic adjustment can be readily fulfilled by training a single model at once, and the effectiveness is expected to be further improved by incorporating with other techniques listed above. ",
+        "bbox": [
+            174,
+            483,
+            825,
+            565
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            589,
+            318,
+            606
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we extend the idea of incomplete dot product (IDP) by proposing the Task-wise Early Stopping and Loss Aggregation (TESLA) algorithm to significantly improve the performance of neural networks with dynamically computation regions at inference time without significantly compromising accuracy. A task is defined as the learning process that uses only a subset of weights specified by a DP percentage to learn the optimal representations of the network. By introducing non-increasing profile coefficients to prioritize weights or filters during training, TESLA can be used to optimize multiple tasks in decreasing order of DP percentages by aggregating the their loss functions. Additionally, we propose Randomized TESLA (R-TESLA) which optimizes tasks in random order, and show that both TESLA and R-TESLA outperform original IDP and multiple-profile IDP significantly. The visualization of the class activation maps (CAMs) provide a strong evidence that the representations learned by TESLA allow dynamically scaling across a computation range to meet various power consumption, latency and accuracy requirements on end devices. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
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parse/train/HkzSQhCcK7/HkzSQhCcK7.md

@@ -0,0 +1,322 @@
+# STCN: STOCHASTIC TEMPORAL CONVOLUTIONAL NETWORKS
+
+Emre Aksan & Otmar Hilliges   
+Department of Computer Science   
+ETH Zurich, Switzerland   
+{emre.aksan, otmar.hilliges}@inf.ethz.ch
+
+# ABSTRACT
+
+Convolutional architectures have recently been shown to be competitive on many sequence modelling tasks when compared to the de-facto standard of recurrent neural networks (RNNs), while providing computational and modeling advantages due to inherent parallelism. However, currently there remains a performance gap to more expressive stochastic RNN variants, especially those with several layers of dependent random variables. In this work, we propose stochastic temporal convolutional networks (STCNs), a novel architecture that combines the computational advantages of temporal convolutional networks (TCN) with the representational power and robustness of stochastic latent spaces. In particular, we propose a hierarchy of stochastic latent variables that captures temporal dependencies at different time-scales. The architecture is modular and flexible due to decoupling of deterministic and stochastic layers. We show that the proposed architecture achieves state of the art log-likelihoods across several tasks. Finally, the model is capable of predicting high-quality synthetic samples over a long-range temporal horizon in modeling of handwritten text.
+
+# 1 INTRODUCTION
+
+Generative modeling of sequence data requires capturing long-term dependencies and learning of correlations between output variables at the same time-step. Recurrent neural networks (RNNs) and its variants have been very successful in a vast number of problem domains which rely on sequential data. Recent work in audio synthesis, language modeling and machine translation tasks (Dauphin et al., 2016; Van Den Oord et al., 2016; Dieleman et al., 2018; Gehring et al., 2017) has demonstrated that temporal convolutional networks (TCNs) can also achieve at least competitive performance without relying on recurrence, and hence reducing the computational cost for training.
+
+Both RNNs and TCNs model the joint probability distribution over sequences by decomposing the distribution over discrete time-steps. In other words, such models are trained to predict the next step, given all previous time-steps. RNNs are able to model long-term dependencies by propagating information through their deterministic hidden state, acting as an internal memory. In contrast, TCNs leverage large receptive fields by stacking many dilated convolutions, allowing them to model even longer time scales up to the entire sequence length. It is noteworthy that there is no explicit temporal dependency between the model outputs and hence the computations can be performed in parallel. The TCN architecture also introduces a temporal hierarchy: the upper layers have access to longer input sub-sequences and learn representations at a larger time scale. The local information from the lower layers is propagated through the hierarchy by means of residual and skip connections (Van Den Oord et al., 2016; Bai et al., 2018).
+
+However, while TCN architectures have been shown to perform similar or better than standard recurrent architectures on particular tasks (Van Den Oord et al., 2016; Bai et al., 2018), there currently remains a performance gap to more recent stochastic RNN variants (Bayer & Osendorfer, 2014; Chung et al., 2015; Fabius & van Amersfoort, 2014; Fraccaro et al., 2016; Goyal et al., 2017; Shabanian et al., 2017). Following a similar approach to stochastic RNNs, Lai et al. (2018) present a significant improvement in the log-likelihood when a TCN model is coupled with latent variables, albeit at the cost of limited receptive field size.
+
+![](images/d359808562547faeee9a83d1af529c4304cb6f63f56df5ac6e6025a77b3485c8.jpg)  
+Figure 1: The computational graph of generative (left) and inference (right) models of STCN. The approximate posterior $q$ is conditioned on $\mathbf { d } _ { t }$ and is updated by the prior $p$ which is conditioned on the TCN representations of the previous time-step $\mathbf { d } _ { t - 1 }$ . The random latent variables at the upper layers have access to a long history while lower layers receive inputs from more recent time steps.
+
+In this work we propose a new approach for augmenting TCNs with random latent variables, that decouples deterministic and stochastic structures yet leverages the increased modeling capacity efficiently. Motivated by the simplicity and computational advantages of TCNs and the robustness and performance of stochastic RNNs, we introduce stochastic temporal convolutional networks (STCN) by incorporating a hierarchy of stochastic latent variables into TCNs which enables learning of representations at many timescales. However, due to the absence of an internal state in TCNs, introducing latent random variables analogously to stochastic RNNs is not feasible. Furthermore, defining conditional random variables across time-steps would result in breaking the parallelism of TCNs and is hence undesirable.
+
+In STCN the latent random variables are arranged in correspondence to the temporal hierarchy of the TCN blocks, effectively distributing them over the various timescales (see figure 1). Crucially, our hierarchical latent structure is designed to be a modular add-on for any temporal convolutional network architecture. Separating the deterministic and stochastic layers allows us to build STCNs without requiring modifications to the base TCN architecture, and hence retains the scalability of TCNs with respect to the receptive field. This conditioning of the latent random variables via different timescales is especially effective in the case of TCNs. We show this experimentally by replacing the TCN layers with stacked LSTM cells, leading to reduced performance compared to STCN.
+
+We propose two different inference networks. In the canonical configuration, samples from each latent variable are passed down from layer to layer and only one sample from the lowest layer is used to condition the prediction of the output. In the second configuration, called STCN-dense, we take inspiration from recent CNN architectures (Huang et al., 2017) and utilize samples from all latent random variables via concatenation before computing the final prediction.
+
+Our contributions can thus be summarized as: 1) We present a modular and scalable approach to augment temporal convolutional network models with effective stochastic latent variables. 2) We empirically show that the STCN-dense design prevents the model from ignoring latent variables in the upper layers (Zhao et al., 2017). 3) We achieve state-of-the-art log-likelihood performance, measured by ELBO, on the IAM-OnDB, Deepwriting, TIMIT and the Blizzard datasets. 4) Finally we show that the quality of the synthetic samples matches the significant quantitative improvements.
+
+# 2 BACKGROUND
+
+Auto-regressive models such as RNNs and TCNs factorize the joint probability of a variable-length sequence $\mathbf { x } = \{ x _ { 1 } , \dots , x _ { T } \}$ as a product of conditionals as follows:
+
+$$
+p _ { \theta } ( { \bf x } ) = \prod _ { t = 1 } ^ { T } p _ { \theta } ( x _ { t } | x _ { 1 : t - 1 } ) \quad ,
+$$
+
+where the joint distribution is parametrized by $\theta$ . The prediction at each time-step is conditioned on all previous observations. The observation model is frequently chosen to be a Gaussian or Gaussian mixture model (GMM) for real-valued data, and a categorical distribution for discrete-valued data.
+
+# 2.1 TEMPORAL CONVOLUTIONAL NETWORKS
+
+In TCNs the joint probabilities in Eq. (1) are parametrized by a stack of convolutional layers. Causal convolutions are the central building block of such models and are designed to be asymmetric such that the model has no access to future information. In order to produce outputs of the same size as the input, zero-padding is applied at every layer.
+
+In the absence of a state transition function, a large receptive field is crucial in capturing long-range dependencies. To avoid the need for vast numbers of causal convolution layers, typically dilated convolutions are used. Exponentially increasing the dilation factor results in an exponential growth of the receptive field size with depth (Yu & Koltun, 2015; Van Den Oord et al., 2016; Bai et al., 2018). In this work, without loss of generality, we use the building blocks of Wavenet (Van Den Oord et al., 2016) as gated activation units (van den Oord et al., 2016) have been reported to perform better.
+
+A deterministic TCN representation $d _ { t } ^ { l }$ at time-step $t$ and layer $l$ summarizes the input sequence $x _ { 1 : t }$
+
+$$
+d _ { t } ^ { l } = \mathbf { C o n v } ^ { ( l ) } ( d _ { t } ^ { l - 1 } , d _ { t - j } ^ { l - 1 } ) \quad \mathrm { a n d } \quad d _ { t } ^ { 1 } = \mathbf { C o n v } ^ { ( 1 ) } ( x _ { t } , x _ { t - j } ) \quad ,
+$$
+
+where the filter width is 2 and $j$ denotes the dilation step. In our work, the stochastic variables $z ^ { l } , l \ = \ 1 \ldots L$ are conditioned on TCN representations $d ^ { l }$ that are constructed by stacking $K$ Wavenet blocks over the previous $d ^ { l - 1 }$ (for details see Figure 4 in Appendix).
+
+# 2.2 NON-SEQUENTIAL LATENT VARIABLE MODELS
+
+VAEs (Kingma & Welling, 2013; Rezende et al., 2014) introduce a latent random variable $\mathbf { z }$ to learn the variations in the observed non-sequential data where the generation of the sample $\mathbf { X }$ is conditioned on the latent variable $\mathbf { z }$ . The joint probability distribution is defined as:
+
+$$
+\begin{array} { r } { p _ { \theta } ( \mathbf { x } , \mathbf { z } ) = p _ { \theta } ( \mathbf { x } | \mathbf { z } ) p _ { \theta } ( \mathbf { z } ) \quad , } \end{array}
+$$
+
+and parametrized by $\theta$ . Optimizing the marginal likelihood is intractable due to the non-linear mappings between $\mathbf { z }$ and $\mathbf { X }$ and the integration over $\mathbf { z }$ . Instead the VAE framework introduces an approximate posterior $q _ { \phi } ( { \bf z } | { \bf x } )$ and optimizes a lower-bound on the marginal likelihood:
+
+$$
+\begin{array} { r } { \log p _ { \theta } ( \mathbf { x } ) \geq - K L ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) | | p _ { \theta } ( \mathbf { z } ) ) + \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ \log p _ { \theta } ( \mathbf { x } | \mathbf { z } ) ] \quad , } \end{array}
+$$
+
+where $K L$ denotes the Kullback-Leibler divergence. Typically the prior $p _ { \boldsymbol { \theta } } ( \mathbf { z } )$ and the approximate $q _ { \phi } ( { \bf z } | { \bf x } )$ are chosen to be in simple parametric form, such as a Gaussian distribution with diagonal covariance, which allows for an analytical calculation of the $K L$ -term in Eq. (4).
+
+# 2.3 STOCHASTIC RNNS
+
+An RNN captures temporal dependencies by recursively processing each input, while updating an internal state $h _ { t }$ at each time-step via its state-transition function:
+
+$$
+h _ { t } = f ^ { ( h ) } ( x _ { t } , h _ { t - 1 } ) \quad ,
+$$
+
+where $f ^ { ( h ) }$ is a deterministic transition function such as LSTM (Hochreiter & Schmidhuber, 1997) or GRU (Cho et al., 2014) cells. The computation has to be sequential because $h _ { t }$ depends on $h _ { t - 1 }$ .
+
+The VAE framework has been extended for sequential data, where a latent variable $z _ { t }$ augments the RNN state $h _ { t }$ at each sequence step. The joint distribution $p _ { \boldsymbol { \theta } } ( \mathbf { x } , \mathbf { z } )$ is modeled via an auto-regressive model which results in the following factorization:
+
+$$
+p _ { \theta } ( \mathbf { x } , \mathbf { z } ) = \prod _ { t = 1 } ^ { T } p _ { \theta } ( x _ { t } | z _ { 1 : t } , x _ { 1 : t - 1 } ) p _ { \theta } ( z _ { t } | x _ { 1 : t - 1 } , z _ { 1 : t - 1 } ) \quad .
+$$
+
+In contrast to the fixed prior of VAEs, $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ , sequential variants define prior distributions conditioned on the RNN hidden state $\mathbf { h }$ and implicitly on the input sequence $\mathbf { X }$ (Chung et al., 2015).
+
+![](images/c0aec47397b8ede7adba3b46cdcc820188b2ad1e6dfda787b03aa04c1a251b89.jpg)  
+Figure 2: Graphical model view of generative models of STCN (left) and STCN-dense (middle), and the inference model (right), which is shared by both variants. Diamonds represent the outputs of deterministic dilated convolution blocks where the dependence of $d _ { t }$ on the past inputs is not shown for clarity (see Eq. (2)). $x _ { t }$ and $z _ { t }$ are observable inputs and latent random variables, respectively. The generative task is to predict the next step in the sequence, given all past steps. Note that in the STCN-dense variant the next step is conditioned on all latent variables $z _ { t } ^ { l }$ for $l = 1 \ldots L$ .
+
+# 3 STOCHASTIC TEMPORAL CONVOLUTIONAL NETWORKS
+
+The mechanics of STCNs are related to those of VRNNs and LVAEs. Intuitively, the RNN state $h _ { t }$ is replaced by temporally independent TCN layers $d _ { t } ^ { l }$ . In the absence of an internal state, we define hierarchical latent variables $\dot { \boldsymbol { z } } _ { t } ^ { l }$ that are conditioned vertically, i.e., in the same time-step, but independent horizontally, i.e., across time-steps. We follow a similar approach to LVAEs (Sønderby et al., 2016) in defining the hierarchy in a top-down fashion and in how we estimate the approximate posterior. The inference network first computes the approximate likelihood, and then this estimate is corrected by the prior, resulting in the approximate posterior. The TCN layers $\mathbf { d }$ are shared between the inference and generator networks, analogous to VRNNs (Chung et al., 2015).
+
+Figure 2 depicts the proposed STCN as a graphical model. STCNs consist of two main modules: the deterministic temporal convolutional network and the stochastic latent variable hierarchy. For a given input sequence $\mathbf { x } = \{ x _ { t } \} , t = 1 . . . T$ we first apply dilated convolutions over the entire sequence to compute a set of deterministic representations $\dot { d } _ { t } ^ { \check { l } } , l = 1 \ldots L$ . Here, $d _ { t } ^ { l }$ corresponds to the output of a block of dilated convolutions at layer $l$ and time-step $t$ . The output $d _ { t } ^ { l }$ is then used to update a set of random latent variables $z _ { t } ^ { l }$ arranged to correspond with different time-scales.
+
+To preserve the parallelism of TCNs, we do not introduce an explicit dependency between different time-steps. However, we suggest that conditioning a latent variable $z _ { t } ^ { l - \bar { 1 } }$ on the preceding variable $z _ { t } ^ { l }$ implicitly introduces temporal dependencies. Importantly, the random latent variables in the upper layer have access to a larger receptive field due to its deterministic input $d _ { t - 1 } ^ { l }$ , whereas latent random variables in lower layers are updated with different, more local information. However, the latent variable $z _ { t } ^ { l - 1 }$ may receive longer-range information from $z _ { t } ^ { l }$ .
+
+The generative and inference models are jointly trained by optimizing a step-wise variational lower bound on the log-likelihood (Kingma $\&$ Welling, 2013; Rezende et al., 2014). In the following sections we describe these components and build up the lower-bound for a single time-step $t$ .
+
+# 3.1 GENERATIVE MODEL
+
+Each sequence step $x _ { t }$ is generated from a set of latent variables $z _ { t }$ , split into layers as follows:
+
+$$
+p _ { \theta } \big ( z _ { t } \vert x _ { 1 : t - 1 } \big ) = p _ { \theta } \big ( z _ { t } ^ { L } \vert d _ { t - 1 } ^ { L } \big ) \prod _ { l = 1 } ^ { L - 1 } p _ { \theta } \big ( z _ { t } ^ { l } \vert z _ { t } ^ { l + 1 } , d _ { t - 1 } ^ { l } \big ) \quad ,
+$$
+
+$$
+p _ { \theta } ( z _ { t } ^ { l } | z _ { t } ^ { l + 1 } , d _ { t - 1 } ^ { l } ) = \mathcal { N } ( \mu _ { t , p } ^ { l } , \sigma _ { t , p } ^ { l } ) \quad \mathrm { a n d } \quad [ \mu _ { t , p } ^ { l } , \sigma _ { t , p } ^ { l } ] = f _ { p } ^ { ( l ) } ( z _ { t } ^ { l + 1 } , d _ { t - 1 } ^ { l } ) \quad .
+$$
+
+Here the prior is modeled by a Gaussian distribution with diagonal covariance, as is common in the VAE framework. The subscript $p$ denotes items of the generative distribution. For the inference distribution we use the subscript $q$ . The distributions are parameterized by a neural network $f _ { p } ^ { ( l ) }$ and conditioned on: (1) the $d _ { t - 1 } ^ { l }$ computed by the dilated convolutions from the previous time-step, and (2) a sample from the preceding level at the same time-step zl+1t . Please note that at inference time we draw samples from the approximate posterior distribution $z _ { t } ^ { l + 1 } \sim q _ { \phi } ( z _ { t } ^ { l + 1 } | \cdot )$ . The generative model, on the other hand, uses the prior $z _ { t } ^ { l + 1 } \sim p _ { \theta } ( z _ { t } ^ { l + 1 } | \cdot )$ .
+
+We propose two variants of the observation model. In the non-sequential scenario, the observations are defined to be conditioned on only the last latent variable in the hierarchy, i.e., $p _ { \theta } ( x _ { t } | z _ { t } ^ { 1 } )$ , following Sønderby et al. (2016); Gulrajani et al. (2016) and Rezende et al. (2014) our STCN variant uses the same observation model, allowing for an efficient optimization. However, latent units are likely to become inactive during training in this configuration (Burda et al., 2015; Bowman et al., 2015; Zhao et al., 2017) resulting in a loss of representational power.
+
+The latent variables at different layers are conditioned on different contexts due to the inputs $d _ { t } ^ { l }$ . Hence, the latent variables are expected to capture complementary aspects of the temporal context. To propagate the information all the way to the final prediction and to ensure that gradients flow through all layers, we take inspiration from Huang et al. (2017) and directly condition the output probability on samples from all latent variables. We call this variant of our architecture STCN-dense.
+
+The final predictions are then computed by the respective observation functions:
+
+$$
+p _ { \theta } ( x _ { t } | z _ { t } ) = f ^ { ( o ) } ( z _ { t } ^ { 1 } ) \quad \mathrm { a n d } \quad p _ { \theta } ^ { d e n s e } ( x _ { t } | z _ { t } ) = f ^ { ( o ) } ( z _ { t } ^ { 1 } , z _ { t } ^ { 2 } \dots z _ { t } ^ { L } ) \quad ,
+$$
+
+where $f ^ { ( o ) }$ corresponds to the output layer constructed by stacking 1D convolutions or Wavenet blocks depending on the dataset.
+
+# 3.2 INFERENCE MODEL
+
+In the original VAE framework the inference model is defined as a bottom-up process, where the latent variables are conditioned on the stochastic layer below. Furthermore, the parameterization of the prior and approximate posterior distributions are computed separately (Burda et al., 2015; Rezende et al., 2014). In contrast, Sønderby et al. (2016) propose a top-down dependency structure shared across the generative and inference models. From a probabilistic point of view, the approximate Gaussian likelihood, computed bottom-up by the inference model, is combined with the Gaussian prior, computed top-down from the generative model. We follow a similar procedure in computing the approximate posterior.
+
+First, the parameters of the approximate likelihood are computed for each stochastic layer $l$ :
+
+$$
+[ \hat { \mu } _ { t , q } ^ { l } , \hat { \sigma } _ { t , q } ^ { l } ] = f _ { q } ^ { ( l ) } ( z _ { t } ^ { l + 1 } , d _ { t } ^ { l } ) \quad ,
+$$
+
+followed by the downward pass, recursively computing the prior and approximate posterior by precision-weighted addition:
+
+$$
+\begin{array} { l } { { \sigma _ { t , q } ^ { l } = \frac { 1 } { ( \hat { \sigma } _ { t , q } ^ { l } ) ^ { - 2 } + ( \sigma _ { t , p } ^ { l } ) ^ { - 2 } } \quad , } } \\ { { \mu _ { t , q } ^ { l } = \sigma _ { t , q } ^ { l } ( \hat { \mu } _ { t , q } ^ { l } ( \hat { \sigma } _ { t , q } ^ { l } ) ^ { - 2 } + \mu _ { t , p } ^ { l } ( \sigma _ { t , p } ^ { l } ) ^ { - 2 } ) \quad . } } \end{array}
+$$
+
+Finally, the approximate posterior has the same decomposition as the prior (see Eq. (7)):
+
+$$
+q _ { \phi } ( z _ { t } | x _ { 1 : t } ) = q _ { \phi } ( z _ { t } ^ { L } | d _ { t } ^ { L } ) \prod _ { l = 1 } ^ { L - 1 } q _ { \phi } ( z _ { t } ^ { l } | z _ { t } ^ { l + 1 } , d _ { t } ^ { l } ) \quad ,
+$$
+
+$$
+\begin{array} { r } { q _ { \phi } ( z _ { t } ^ { l } | z _ { t } ^ { l + 1 } , d _ { t } ^ { l } ) = \mathcal { N } ( \mu _ { t , q } ^ { l } , \sigma _ { t , q } ^ { l } ) \quad . } \end{array}
+$$
+
+Note that the inference and generative network share the parameters of dilated convolutions $\mathrm { C o n v } ^ { ( l ) }$
+
+# 3.3 LEARNING
+
+The variational lower-bound on the log-likelihood at time-step $t$ can be defined as follows:
+
+$$
+\begin{array} { r l } & { \log p ( x _ { t } ) \geq \mathbb { E } _ { q _ { \phi } ( z _ { t } | x _ { t } ) } [ \log p \theta ( x _ { t } | z _ { t } ) ] - D _ { K L } ( q _ { \phi } ( z _ { t } | x _ { 1 : t } ) | | p \theta ( z _ { t } | x _ { 1 : t - 1 } ) ) } \\ & { \qquad = \mathbb { E } _ { q _ { \phi } ( z _ { t } ^ { 1 } \dots z _ { t } ^ { L } | x _ { t } ) } [ \log p \theta ( x _ { t } | z _ { t } ^ { 1 } \dots z _ { t } ^ { L } ) ] - D _ { K L } ( q _ { \phi } ( z _ { t } ^ { 1 } \dots z _ { t } ^ { L } | x _ { 1 : t } ) | | p \theta ( z _ { t } ^ { 1 } \dots z _ { t } ^ { L } | x _ { 1 : t - 1 } ) ) } \\ & { \mathcal { C } _ { t } ( \theta , \phi ; x _ { t } ) = \mathcal { L } _ { t } ^ { R e c o n } + \mathcal { L } _ { t } ^ { K L } . } \end{array}
+$$
+
+Using the decompositions from Eq. (7) and (12), the Kullback-Leibler divergence term becomes:
+
+$$
+\begin{array} { r l } { \displaystyle \mathcal { L } _ { t } ^ { K L } = - D _ { K L } \big ( q _ { \phi } ( \boldsymbol { z } _ { t } ^ { L } | \boldsymbol { d } _ { t } ^ { L } ) | | p _ { \theta } \big ( \boldsymbol { z } _ { t } ^ { L } | \boldsymbol { d } _ { t - 1 } ^ { L } \big ) \big ) } & { } \\ { \displaystyle - \sum _ { l = 1 } ^ { L - 1 } \mathbb { E } _ { q _ { \phi } ( \boldsymbol { z } _ { t } ^ { l + 1 } | \cdot ) } \big [ D _ { K L } \big ( q _ { \phi } \big ( \boldsymbol { z } _ { t } ^ { l } | \boldsymbol { z } _ { t } ^ { l + 1 } , \boldsymbol { d } _ { t } ^ { l } \big ) | | p _ { \theta } \big ( \boldsymbol { z } _ { t } ^ { l } | \boldsymbol { z } _ { t } ^ { l + 1 } , \boldsymbol { d } _ { t - 1 } ^ { l } \big ) \big ) \big ] } & { . } \end{array}
+$$
+
+The KL term is the same for the STCN and STCN-dense variants. The reconstruction term $\mathcal { L } _ { t } ^ { R e c o n }$ , however, is different. In STCN we only use samples from the lowest layer of the hierarchy, whereas in STCN-dense we use all latent samples in the observation model:
+
+$$
+\begin{array} { r l } { \mathcal { L } _ { t } ^ { R e c o n } = \mathbb { E } _ { q _ { \phi } ( z _ { t } ^ { 1 } \ldots z _ { t } ^ { L } | x _ { t } ) } [ \log p _ { \theta } ( x _ { t } | z _ { t } ^ { 1 } ) ] } & { { } , } \\ { \mathcal { L } _ { t } ^ { R e c o n - d e n s e } = \mathbb { E } _ { q _ { \phi } ( z _ { t } ^ { 1 } \ldots z _ { t } ^ { L } | x _ { t } ) } [ \log p _ { \theta } ( x _ { t } | z _ { t } ^ { 1 } \ldots z _ { t } ^ { L } ] } & { { } . } \end{array}
+$$
+
+In the dense variant, samples drawn from the latent variables $z _ { t } ^ { l }$ are carried over the dense connections. Similar to Maaløe et al. (2016), the expectation over $z _ { t } ^ { l }$ variables are computed by Monte Carlo sampling using the reparameterization trick (Kingma & Welling, 2013; Rezende et al., 2014).
+
+Please note that the computation of $\mathcal { L } _ { t } ^ { R e c o n - d e n s e }$ does not introduce any additional computational cost. In STCN, all latent variables have to be visited in terms of ancestral sampling in order to draw the latent sample $z _ { t } ^ { 1 }$ for the observation $x _ { t }$ . Similarly in STCN-dense, the same intermediate samples $z _ { t } ^ { l }$ are used in the prediction of $x _ { t }$ .
+
+One alternative option to use the latent samples could be to sum individual samples before feeding them into the observation model, i.e., $s u m ( \dot { [ } z _ { t } ^ { 1 } \dots z _ { t } ^ { L } ] )$ , (Maaløe et al., 2016). We empirically found that this does not work well in STCN-dense. Instead, we concatenate all samples $\left[ z _ { t } ^ { 1 } \circ \cdots \circ z _ { t } ^ { L } \right]$ analogously to DenseNet (Huang et al., 2017) and (Kaiser et al., 2018).
+
+# 4 EXPERIMENTS
+
+We evaluate the proposed variants STCN and STCN-dense both quantitatively and qualitatively on modeling of digital handwritten text and speech. We compare with vanilla TCNs, RNNs, VRNNs and state-of-the art models on the corresponding tasks.
+
+In our experiments we use two variants of the Wavenet model: (1) the original model proposed in (Van Den Oord et al., 2016) and (2) a variant that we augment with skip connections analogously to STCN-dense. This additional baseline evaluates the benefit of learning multi-scale representations in the deterministic setting. Details of the experimental setup are provided in the Appendix. Our code is available at https://ait.ethz.ch/projects/2019/stcn/.
+
+Handwritten text: The IAM-OnDB and Deepwriting datasets consist of digital handwriting sequences where each time-step contains real-valued $( x , y )$ pen coordinates and a binary pen-up event. The IAM-OnDB data is split and pre-processed as done in (Chung et al., 2015). Aksan et al. (2018) extend this dataset with additional samples and better pre-processing.
+
+Table 1 reveals that again both our variants outperform the vanilla variants of TCNs and RNNs on IAM-OnDB. While the stochastic VRNN and SWaveNet are competitive wrt to the STCN variant, both are outperformed by the STCN-dense version. The same relative ordering is maintained on the Deepwriting dataset, indicating that the proposed architecture is robust across datasets.
+
+![](images/b93a1887df7d73e452966757899db96ab3e51bdbb1aecfabae9ef2b6ba19969a.jpg)  
+Figure 3: (a) Handwriting samples from IAM-OnDB dataset. Generated samples from (b) VRNN, (c) SWaveNet and (d) our model STCN-dense. Each line corresponds to one sample.
+
+Table 1: Average log-likelihood per sequence on TIMIT, Blizzard, IAM-OnDB and Deepwriting datasets. (Normal) and (GMM) stand for unimodal Gaussian or multi-modal Gaussian Mixture Model (GMM) as the observation model (Graves, 2013; Chung et al., 2015). Asterisks ∗ indicate that we used our re-implementation only for the Deepwriting dataset.   
+
+<table><tr><td>Models</td><td>TIMIT</td><td>Blizzard</td><td>IAM-OnDB</td><td>Deepwriting</td></tr><tr><td>Wavenet (GMM)</td><td>30188</td><td>8190</td><td>1381</td><td>612</td></tr><tr><td>Wavenet-dense (GMM)</td><td>30636</td><td>8212</td><td>1380</td><td>642</td></tr><tr><td>RNN (GMM) Chung et al. (2015)</td><td>26643</td><td>7413</td><td>1358</td><td>528 *</td></tr><tr><td>VRNN (Normal) Chung et al.(2015)</td><td>~30235</td><td>~9516</td><td>≈1354</td><td>≥ 495 *</td></tr><tr><td>VRNN (GMM) Chung et al. (2015)</td><td>≈ 29604</td><td>~9392</td><td>≈1384</td><td>≥ 673 *</td></tr><tr><td>SRNN (Normal) Fraccaro et al. (2016)</td><td>≥ 60550</td><td>≥11991</td><td>n/a</td><td>n/a</td></tr><tr><td>Z-forcing (Normal) Goyal etal. (2017)</td><td>≥ 70469</td><td>≥ 15430</td><td>n/a</td><td>n/a</td></tr><tr><td>Var.Bi-LSTM (Normal) Shabanian et al.(2017)</td><td>≥ 73976</td><td>≥ 17319</td><td>n/a</td><td>n/a</td></tr><tr><td>SWaveNet (Normal) Lai etal. (2018)</td><td>≥ 72463</td><td>≥ 15708</td><td>≥1301</td><td>n/a</td></tr><tr><td>STCN (GMM)</td><td>≥ 69195</td><td>M 15800</td><td>≥ 1338</td><td>≥ 605</td></tr><tr><td>STCN-dense (GMM)</td><td>≥ 71386</td><td>≥ 16288</td><td>≥ 1796</td><td>≥ 797</td></tr><tr><td>STCN-dense-large (GMM)</td><td>≥ 77438</td><td>≥ 17670</td><td>n/a</td><td>n/a</td></tr></table>
+
+Fig. 3 compares generated handwriting samples. While all models produce consistent style, our model generates more natural looking samples. Note that the spacing between words is clearly visible and most of the letters are distinguishable.
+
+Speech modeling: TIMIT and Blizzard are standard benchmark dataset in speech modeling. The models are trained and tested on 200 dimensional real-valued amplitudes. We apply the same pre-processing as Chung et al. (2015). For this task we introduce STCN-dense-large, with increased model capacity. Here we use 512 instead of 256 convolution filters. Note that the total number of model parameters is comparable to SWaveNet and other SOA models.
+
+On TIMIT, STCN-dense (Table 1) significantly outperforms the vanilla TCN and RNN, and stochastic models. On the Blizzard dataset, our model is marginally better than the Variational Bi-LSTM. Note that the inference models of SRNN (Fraccaro et al., 2016), Z-forcing (Goyal et al., 2017), and Variational Bi-LSTM (Shabanian et al., 2017) receive future information by using backward RNN cells. Similarly, SWaveNet (Lai et al., 2018) applies causal convolutions in the backward direction. Hence, the latent variable can be expected to model future dynamics of the sequence. In contrast, our models have only access to information up to the current time-step. These results indicate that the STCN variants perform very well on the speech modeling task.
+
+Latent Space Analysis: Zhao et al. (2017) observe that in hierarchical latent variable models the upper layers have a tendency to become inactive, indicated by a low KL loss (Sønderby et al., 2016; Dieng et al., 2018). Table 2 shows the KL loss per latent variable and the corresponding log-likelihood measured by ELBO in our models. Across the datasets it can be observed that our models make use of many of the latent variables which may explain the strong performance across tasks in terms of log-likelihoods. Note that STCN uses a standard hierarchical structure. However, individual latent variables have different information context due to the corresponding TCN block’s receptive field. This observation suggests that the proposed combination of TCNs and stochastic variables is indeed effective. Furthermore, in STCN we see a similar utilization pattern of the $z$ variables across tasks, whereas STCN-dense may have more flexibility in modeling the temporal dependencies within the data due to its dense connections to the output layer.
+
+Table 2: KL-loss per latent variable computed over the entire test split. KL5 corresponds to the KL-loss of the top-most latent variable.   
+
+<table><tr><td>Dataset (Model)</td><td>ELBO</td><td>KL</td><td>KL1</td><td>KL2</td><td>KL3</td><td>KL4</td><td>KL5</td></tr><tr><td>IAM-OnDB (sTCN-dense)</td><td>≥ 1796.3</td><td>1653.9</td><td>17.9</td><td>1287.4</td><td>305.3</td><td>41.0</td><td>2.4</td></tr><tr><td>IAM-OnDB (sTCN)</td><td>≥ 1339.2</td><td>964.2</td><td>846.0</td><td>105.2</td><td>12.9</td><td>0.1</td><td>0.0</td></tr><tr><td>TIMIT (sTCN-dense)</td><td>≥ 71385.9</td><td>22297.5</td><td>16113.0</td><td>5641.6</td><td>529.0</td><td>8.3</td><td>5.7</td></tr><tr><td>TIMIT (STCN)</td><td>≥ 69194.9</td><td>23118.3</td><td>22275.5</td><td>487.2</td><td>355.5</td><td>0.0</td><td>0.0</td></tr></table>
+
+Replacing TCN with RNN: To better understand potential symergies between dilated CNNs and the proposed latent variable hierarchy, we perform an ablation study, isolating the effect of TCNs and the latent space. To this end the deterministic TCN blocks are replaced with LSTM cells by keeping the latent structure intact. We dub this condition LadderRNN. We use the TIMIT and IAM-OnDB datasets for evaluation. Table 3 summarizes performance measured by the ELBO.
+
+The most direct translation of the the STCN architecture into an RNN counterpart has 25 stacked LSTM cells with 256 units each. Similar to STCN, we use 5 stochastic layers (see Appendix 7.1). Note that stacking this many LSTM cells is unusual and resulted in instabilities during training. Hence, the performance is similar to vanilla RNNs. The second LadderRNN configuration uses 5 stacked LSTM cells with 512 units and a one-to-one mapping with the stochastic layers. On the TIMIT dataset, all LadderRNN configurations show a significant improvement. We also observe a pattern of improvement with densely connected latent variables.
+
+This experiments shows that the proposed modular latent variable design does allow for the usage of different building blocks. Even when attached to LSTM cells, it boosts the log-likelihood performance (see 5x512- LadderRNN), in particular when used with dense connections. However, the empirical results suggest that the densely connected latent hierarchy interacts particularly well with dilated CNNs. We suggest this is due to the hierarchical nature on both sides of the architecture. On both datasets STCN models achieved the best performance and significantly improve with dense connections. This supports our contribution of a latent variable hierarchy, which models different aspects of information from the input time-series.
+
+Table 3: ELBO of LadderRNN and STCN models using the same latent space configuration. The prefix of a model entries denote the number of RNN or TCN layers and unit size per layer. Models have similar number of trainable parameters.   
+
+<table><tr><td>Models</td><td>TIMIT</td><td>IAM-OnDB</td></tr><tr><td>25x256-LadderRNN (Normal)</td><td>≥ 28207</td><td>≥ 1305</td></tr><tr><td>25x256-LadderRNN-dense (Normal)</td><td>≥ 27413</td><td>&gt;I &gt;I 1278</td></tr><tr><td>25x256-LadderRNN (GMM)</td><td>≥ 24839</td><td>1381</td></tr><tr><td>25x256-LadderRNN-dense (GMM)</td><td>≥ 26240</td><td>≥ 1377</td></tr><tr><td>5x512-LadderRNN (Normal)</td><td>≥ 49770</td><td>≥ 1299</td></tr><tr><td>5x512-LadderRNN-dense (Normal)</td><td>M 48612</td><td>1374</td></tr><tr><td>5x512-LadderRNN (GMM)</td><td>M 47179</td><td>&gt;I &gt;I 1359</td></tr><tr><td>5x512-LadderRNN-dense (GMM)</td><td>≥ 50113</td><td>≥ 1581</td></tr><tr><td>25x256-STCN (Normal)</td><td>≥ 64913</td><td>≥ 1327</td></tr><tr><td>25x256-STCN-dense (Normal)</td><td>M 70294</td><td>N 1729</td></tr><tr><td>25x256-STCN (GMM)</td><td>M ：69195</td><td>≥ 1339</td></tr><tr><td>25x256-STCN-dense (GMM)</td><td>M 71386</td><td>≥ 1796</td></tr></table>
+
+# 5 RELATED WORK
+
+Rezende et al. (2014) propose Deep Latent Gaussian Models (DLGM) and Sønderby et al. (2016) propose the Ladder Variational Autoencoder (LVAE). In both models the latent variables are hierarchically defined and conditioned on the preceding stochastic layer. LVAEs improve upon DLGMs via implementation of a top-down hierarchy both in the generative and inference model. The approximate posterior is computed via a precisionweighted update of the approximate likelihood (i.e., the inference model) and prior (i.e., the generative model). Similarly, the PixelVAE (Gulrajani et al., 2016) incorporates a hierarchical latent space decomposition and uses an autoregressive decoder. Zhao et al. (2017) show under mild conditions that straightforward stacking of latent variables (as is done e.g. in LVAE and PixelVAE) can be ineffective, because the latent variables that are not directly conditioned on the observation variable become inactive.
+
+Due to the nature of the sequential problem domain, our approach differs in the crucial aspects that STCNs use dynamic, i.e., conditional, priors (Chung et al., 2015) at every level. Moreover, the hierarchy is not only implicitly defined by the network architecture but also explicitly defined by the information content, i.e., receptive field size. Dieng et al. (2018) both theoretically and empirically show that using skip connections from the latent variable to every layer of the decoder increases mutual information between the latent and observation variables. Similar to Dieng et al. (2018) in STCN-dense, we introduce skip connections from all latent variables to the output. In STCN the model is expected to encode and propagate the information through its hierarchy.
+
+Yang et al. (2017) suggest using autoregressive TCN decoders to remedy the posterior collapse problem observed in language modeling with LSTM decoders (Bowman et al., 2015). van den Oord et al. (2017) and Dieleman et al. (2018) use TCN decoders conditioned on discrete latent variables to model audio signals.
+
+Stochastic RNN architectures mostly vary in the way they employ the latent variable and parametrize the approximate posterior for variational inference. Chung et al. (2015) and Bayer & Osendorfer (2014) use the latent random variable to capture high-level information causing the variability observed in sequential data. Particularly Chung et al. (2015) shows that using a conditional prior rather than a standard Gaussian distribution is very effective in sequence modeling. In (Fraccaro et al., 2016; Goyal et al., 2017; Shabanian et al., 2017), the inference model, i.e., the approximate posterior, receives both the past and future summaries of the sequence from the hidden states of forward and backward RNN cells. The KL-divergence term in the objective enforces the model to learn predictive latent variables in order to capture the future states of the sequence.
+
+Lai et al. (2018)’s SWaveNet is most closely related to ours. SWaveNet also introduces latent variables into TCNs. However, in SWaveNet the deterministic and stochastic units are coupled which may prevent stacking of larger numbers of TCN blocks. Since the number of stacked dilated convolutions determines the receptive field size, this directly correlates with the model capacity. For example, the performance of SWaveNet on the IAM-OnDB dataset degrades after stacking more than 3 stochastic layers (Lai et al., 2018), limiting the model to a small receptive field. In contrast, we aim to preserve the flexibility of stacking dilated convolutions in the base TCN. In STCNs, the deterministic TCN units do not have any dependency on the stochastic variables (see Figure 1) and the ratio of stochastic to deterministic units can be adjusted, depending on the task.
+
+# 6 CONCLUSION
+
+In this paper we proposed STCNs, a novel auto-regressive model, combining the computational benefits of convolutional architectures and expressiveness of hierarchical stochastic latent spaces. We have shown the effectivness of the approach across several sequence modelling tasks and datasets. The proposed models are trained via optimization of the ELBO objective. Tighter lower bounds such as IWAE (Burda et al., 2015) or FIVO (Maddison et al., 2017) may further improve modeling performance. We leave this for future work.
+
+# ACKNOWLEDGEMENTS
+
+This work was supported in parts by the ERC grant OPTINT (StG-2016-717054). We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.
+
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+Alex Graves. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013.   
+Ishaan Gulrajani, Kundan Kumar, Faruk Ahmed, Adrien Ali Taiga, Francesco Visin, David Vazquez, and Aaron Courville. Pixelvae: A latent variable model for natural images. arXiv preprint arXiv:1611.05013, 2016.   
+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8):1735–1780, 1997.   
+Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In CVPR, volume 1, pp. 3, 2017.   
+Łukasz Kaiser, Aurko Roy, Ashish Vaswani, Niki Pamar, Samy Bengio, Jakob Uszkoreit, and Noam Shazeer. Fast decoding in sequence models using discrete latent variables. arXiv preprint arXiv:1803.03382, 2018.   
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.   
+Guokun Lai, Bohan Li, Guoqing Zheng, and Yiming Yang. Stochastic wavenet: A generative latent variable model for sequential data, 2018.   
+Lars Maaløe, Casper Kaae Sønderby, Søren Kaae Sønderby, and Ole Winther. Auxiliary deep generative models. arXiv preprint arXiv:1602.05473, 2016.   
+Chris J Maddison, John Lawson, George Tucker, Nicolas Heess, Mohammad Norouzi, Andriy Mnih, Arnaud Doucet, and Yee Teh. Filtering variational objectives. In Advances in Neural Information Processing Systems, pp. 6573–6583, 2017.   
+Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. arXiv preprint arXiv:1401.4082, 2014.   
+Samira Shabanian, Devansh Arpit, Adam Trischler, and Yoshua Bengio. Variational bi-lstms. arXiv preprint arXiv:1711.05717, 2017.   
+Casper Kaae Sønderby, Tapani Raiko, Lars Maaløe, Søren Kaae Sønderby, and Ole Winther. Ladder variational autoencoders. In Advances in neural information processing systems, pp. 3738–3746, 2016.   
+Aaron Van Den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalch- ¨ brenner, Andrew W Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. In SSW, pp. 125, 2016.   
+Aaron van den Oord, Nal Kalchbrenner, Lasse Espeholt, Oriol Vinyals, Alex Graves, et al. Conditional image ¨ generation with pixelcnn decoders. In Advances in Neural Information Processing Systems, pp. 4790–4798, 2016.   
+Aaron van den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In Advances in Neural Information Processing Systems, pp. 6306–6315, 2017.   
+Zichao Yang, Zhiting Hu, Ruslan Salakhutdinov, and Taylor Berg-Kirkpatrick. Improved variational autoencoders for text modeling using dilated convolutions. arXiv preprint arXiv:1702.08139, 2017.   
+Fisher Yu and Vladlen Koltun. Multi-scale context aggregation by dilated convolutions. arXiv preprint arXiv:1511.07122, 2015.   
+Shengjia Zhao, Jiaming Song, and Stefano Ermon. Learning hierarchical features from generative models. arXiv preprint arXiv:1702.08396, 2017.
+
+# 7 APPENDIX
+
+# 7.1 NETWORK DETAILS
+
+![](images/19aa4d2b81f05aacc2011ed44ab2ccd4481cbc37a564cfc54e2d0bf8b1253e60.jpg)  
+Figure 4: Generative model of STCN-dense architecture. Building blocks are highlighted. Note that the dependence of $d _ { t } ^ { l } , l = 1 \cdots L$ on past inputs is not visualized for clarity.
+
+The network architecture of the proposed model is illustrated in Fig. 4. We make only a small modification to the vanilla Wavenet architecture. Instead of using skip connections from Wavenet blocks, we only use the latent sample $z _ { t }$ in order to make a prediction of $x _ { t }$ . In STCN-dense configuration, $z _ { t }$ is the concatenation of all latent variables in the hierarchy, i.e., $\boldsymbol { z } _ { t } = [ \boldsymbol { z } _ { t } ^ { 1 } \circ \cdot \cdot \cdot \circ \boldsymbol { z } _ { t } ^ { L } ]$ , whereas in STCN only $z _ { t } ^ { 1 }$ is fed to the output layer.
+
+Each stochastic latent variable $z _ { t } ^ { l }$ (except the top-most $z _ { t } ^ { L }$ ) is conditioned on a deterministic TCN representation $d _ { t } ^ { l }$ and the preceding random variable $z _ { t } ^ { l + 1 }$ . The latent variables are calculated by using the latent layers $f _ { p } ^ { ( l ) }$ or $f _ { q } ^ { ( l ) }$ which are neural networks.
+
+We do not define a latent variable per TCN layer. Instead, the stochastic layers are uniformly distributed where each random variable is conditioned on a number of stacked TCN layers $\dot { d } _ { t } ^ { l }$ . We stack $K$ Wavenet blocks (see figure 4 left) with exponentially increasing dilation size.
+
+Observation Model: We use Normal or GMM distributions with 20 components to model real-valued data.   
+All Gaussian distributions have diagonal covariance matrix.
+
+Output layer $f ^ { ( o ) }$ : For the IAM-OnDB and Deepwriting datasets we use 1D convolutions with ReLU nonlinearity. We stack 5 of these layers with 256 filters and filter size 1.
+
+For TIMIT and Blizzard datasets Wavenet blocks in the output layer perform significantly better. We stack 5 Wavenet blocks with dilation size 1. For each convolution operation in the block we use 256 filters. The filter size of the dilated convolution is set to 2. The STCN-dense-large model is constructed by using 512 filters instead of 256.
+
+TCN blocks $d _ { t } ^ { l }$ : The number of Wavenet blocks is usually determined by the desired receptive field size.
+
+• For the handwriting datasets $K = 6$ and $L = 5$ . In total we have 30 Wavenet blocks where each convolution operation has 256 filters with size 2. • For speech datasets $K = 5$ and $L = 5$ . In total we have 25 Wavenet blocks where each convolution operation has 256 filters with size 2. The large model configuration uses 512 filters.
+
+Latent layers $f _ { p } ^ { ( l ) }$ and $f _ { q } ^ { ( l ) }$ : The number of stochastic layers per task is given by $L$ . We used [32, 16, 8, 5, 2] dimensional latent variables for the handwriting tasks. It is [256, 128, 64, 32, 16] for speech datasets. Note that the first entry of the list corresponds to $z ^ { 1 }$ .
+
+The mean and sigma parameters of the Normal distributions modeling the latent variables are calculated by the $f _ { p } ^ { ( l ) }$ and $f _ { q } ^ { ( l ) }$ networks. We stack $^ { 2 1 0 }$ convolutions with ReLU nonlinearity and filter size 1. The number of filters are the same as the number of Wavenet block filters for the corresponding task.
+
+Finally, we clamped the latent sigma predictions between 0.001 and 5.
+
+# 7.2 TRAINING DETAILS
+
+In all STCN experiments we applied KL annealing. In all tasks, the weight of the KL term is initialized with 0 and increased by $1 \times e ^ { - 4 }$ at every step until it reaches 1.
+
+The batch size was 20 for all datasets except for Blizzard where it was 128.
+
+We use the ADAM optimizer with its default parameters and exponentially decay the learning rate. For the handwriting datasets the learning rate was initialized with $5 \times e ^ { - 4 }$ and followed a decay rate of 0.94 over 1000 decay steps. On the speech datasets it was initialized with $1 \times e ^ { - 3 }$ and decayed with a rate of 0.98. We applied early stopping by measuring the ELBO performance on the validation splits.
+
+We implement STCN models in Tensorflow (Abadi et al., 2016). Our code and models achieving the SOA results are available at https://ait.ethz.ch/projects/2019/stcn/.
+
+# 7.3 DETAILED RESULTS
+
+Here we provide the extended results table with Normal observation model entries for available models.
+
+Table 4: Average log-likelihood per sequence on TIMIT, Blizzard, IAM-OnDB and Deepwriting datasets. (Normal) and (GMM) stand for unimodal Gaussian or multi-modal Gaussian Mixture Model (GMM) as the observation model (Graves, 2013; Chung et al., 2015). Asterisks ∗ indicate that we used our re-implementation only for the Deepwriting dataset.   
+
+<table><tr><td>Models</td><td>TIMIT</td><td>Blizzard</td><td>IAM-OnDB</td><td>Deepwriting</td></tr><tr><td>Wavenet (Normal)</td><td>-7443</td><td>3784</td><td>1053</td><td>337</td></tr><tr><td>Wavenet (GMM)</td><td>30188</td><td>8190</td><td>1381</td><td>612</td></tr><tr><td>Wavenet-dense (Normal)</td><td>-8579</td><td>3712</td><td>1030</td><td>323</td></tr><tr><td>Wavenet-dense (GMM)</td><td>30636</td><td>8212</td><td>1380</td><td>642</td></tr><tr><td>RNN (Normal) Chung et al. (2015)</td><td>-1900</td><td>3539</td><td>1016</td><td>363 *</td></tr><tr><td>RNN (GMM) Chung et al. (2015)</td><td>26643</td><td>7413</td><td>1358</td><td>528 *</td></tr><tr><td>VRNN (Normal)Chung et al. (2015)</td><td>~ 30235</td><td>~9516</td><td>≈1354</td><td>≥ 495 *</td></tr><tr><td>VRNN (GMM) Chung et a. (2015)</td><td>~ 29604</td><td>~9392</td><td>≈1384</td><td>≥673 *</td></tr><tr><td>SRNN (Normal) Fraccaro et al. (2016)</td><td>≥ 60550</td><td>≥ 11991</td><td>n/a</td><td>n/a</td></tr><tr><td>Z-forcing (Normal)Goyal etal. (2017)</td><td>≥ 70469</td><td>≥ 15430</td><td>n/a</td><td>n/a</td></tr><tr><td>Var.Bi-LSTM (Normal)Shabanian et al. (2017)</td><td>≥ 73976</td><td>≥ 17319</td><td>n/a</td><td>n/a</td></tr><tr><td>SWaveNet (Normal)Lai et al.(2018)</td><td>≥ 72463</td><td>M 15708</td><td>≥1301</td><td>n/a</td></tr><tr><td>STCN(Normal)</td><td>≥ 64913</td><td>M 13273</td><td>≥ 1327</td><td>≥ 575</td></tr><tr><td>STCN(GMM)</td><td>≥ 69195</td><td>≥15800</td><td>≥ 1338</td><td>≥ 605</td></tr><tr><td>STCN-dense(Normal)</td><td>≥ 70294</td><td>≥ 15950</td><td>≥ 1729</td><td>≥ 740</td></tr><tr><td>STCN-dense(GMM)</td><td>≥ 71386</td><td>M 16288</td><td>≥ 1796</td><td>≥ 797</td></tr><tr><td>STCN-dense-large (GMM)</td><td>≥ 77438</td><td>≥ 17670</td><td>n/a</td><td>n/a</td></tr></table>

parse/train/HkzSQhCcK7/HkzSQhCcK7_content_list.json

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+    {
+        "type": "text",
+        "text": "STCN: STOCHASTIC TEMPORAL CONVOLUTIONAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Emre Aksan & Otmar Hilliges   \nDepartment of Computer Science   \nETH Zurich, Switzerland   \n{emre.aksan, otmar.hilliges}@inf.ethz.ch ",
+        "bbox": [
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+            170,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional architectures have recently been shown to be competitive on many sequence modelling tasks when compared to the de-facto standard of recurrent neural networks (RNNs), while providing computational and modeling advantages due to inherent parallelism. However, currently there remains a performance gap to more expressive stochastic RNN variants, especially those with several layers of dependent random variables. In this work, we propose stochastic temporal convolutional networks (STCNs), a novel architecture that combines the computational advantages of temporal convolutional networks (TCN) with the representational power and robustness of stochastic latent spaces. In particular, we propose a hierarchy of stochastic latent variables that captures temporal dependencies at different time-scales. The architecture is modular and flexible due to decoupling of deterministic and stochastic layers. We show that the proposed architecture achieves state of the art log-likelihoods across several tasks. Finally, the model is capable of predicting high-quality synthetic samples over a long-range temporal horizon in modeling of handwritten text. ",
+        "bbox": [
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+            295,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
+            336,
+            546
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative modeling of sequence data requires capturing long-term dependencies and learning of correlations between output variables at the same time-step. Recurrent neural networks (RNNs) and its variants have been very successful in a vast number of problem domains which rely on sequential data. Recent work in audio synthesis, language modeling and machine translation tasks (Dauphin et al., 2016; Van Den Oord et al., 2016; Dieleman et al., 2018; Gehring et al., 2017) has demonstrated that temporal convolutional networks (TCNs) can also achieve at least competitive performance without relying on recurrence, and hence reducing the computational cost for training. ",
+        "bbox": [
+            174,
+            563,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Both RNNs and TCNs model the joint probability distribution over sequences by decomposing the distribution over discrete time-steps. In other words, such models are trained to predict the next step, given all previous time-steps. RNNs are able to model long-term dependencies by propagating information through their deterministic hidden state, acting as an internal memory. In contrast, TCNs leverage large receptive fields by stacking many dilated convolutions, allowing them to model even longer time scales up to the entire sequence length. It is noteworthy that there is no explicit temporal dependency between the model outputs and hence the computations can be performed in parallel. The TCN architecture also introduces a temporal hierarchy: the upper layers have access to longer input sub-sequences and learn representations at a larger time scale. The local information from the lower layers is propagated through the hierarchy by means of residual and skip connections (Van Den Oord et al., 2016; Bai et al., 2018). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, while TCN architectures have been shown to perform similar or better than standard recurrent architectures on particular tasks (Van Den Oord et al., 2016; Bai et al., 2018), there currently remains a performance gap to more recent stochastic RNN variants (Bayer & Osendorfer, 2014; Chung et al., 2015; Fabius & van Amersfoort, 2014; Fraccaro et al., 2016; Goyal et al., 2017; Shabanian et al., 2017). Following a similar approach to stochastic RNNs, Lai et al. (2018) present a significant improvement in the log-likelihood when a TCN model is coupled with latent variables, albeit at the cost of limited receptive field size. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/d359808562547faeee9a83d1af529c4304cb6f63f56df5ac6e6025a77b3485c8.jpg",
+        "image_caption": [
+            "Figure 1: The computational graph of generative (left) and inference (right) models of STCN. The approximate posterior $q$ is conditioned on $\\mathbf { d } _ { t }$ and is updated by the prior $p$ which is conditioned on the TCN representations of the previous time-step $\\mathbf { d } _ { t - 1 }$ . The random latent variables at the upper layers have access to a long history while lower layers receive inputs from more recent time steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work we propose a new approach for augmenting TCNs with random latent variables, that decouples deterministic and stochastic structures yet leverages the increased modeling capacity efficiently. Motivated by the simplicity and computational advantages of TCNs and the robustness and performance of stochastic RNNs, we introduce stochastic temporal convolutional networks (STCN) by incorporating a hierarchy of stochastic latent variables into TCNs which enables learning of representations at many timescales. However, due to the absence of an internal state in TCNs, introducing latent random variables analogously to stochastic RNNs is not feasible. Furthermore, defining conditional random variables across time-steps would result in breaking the parallelism of TCNs and is hence undesirable. ",
+        "bbox": [
+            174,
+            337,
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+            462
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In STCN the latent random variables are arranged in correspondence to the temporal hierarchy of the TCN blocks, effectively distributing them over the various timescales (see figure 1). Crucially, our hierarchical latent structure is designed to be a modular add-on for any temporal convolutional network architecture. Separating the deterministic and stochastic layers allows us to build STCNs without requiring modifications to the base TCN architecture, and hence retains the scalability of TCNs with respect to the receptive field. This conditioning of the latent random variables via different timescales is especially effective in the case of TCNs. We show this experimentally by replacing the TCN layers with stacked LSTM cells, leading to reduced performance compared to STCN. ",
+        "bbox": [
+            174,
+            469,
+            825,
+            582
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We propose two different inference networks. In the canonical configuration, samples from each latent variable are passed down from layer to layer and only one sample from the lowest layer is used to condition the prediction of the output. In the second configuration, called STCN-dense, we take inspiration from recent CNN architectures (Huang et al., 2017) and utilize samples from all latent random variables via concatenation before computing the final prediction. ",
+        "bbox": [
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+            588,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions can thus be summarized as: 1) We present a modular and scalable approach to augment temporal convolutional network models with effective stochastic latent variables. 2) We empirically show that the STCN-dense design prevents the model from ignoring latent variables in the upper layers (Zhao et al., 2017). 3) We achieve state-of-the-art log-likelihood performance, measured by ELBO, on the IAM-OnDB, Deepwriting, TIMIT and the Blizzard datasets. 4) Finally we show that the quality of the synthetic samples matches the significant quantitative improvements. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            767,
+            326,
+            784
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Auto-regressive models such as RNNs and TCNs factorize the joint probability of a variable-length sequence $\\mathbf { x } = \\{ x _ { 1 } , \\dots , x _ { T } \\}$ as a product of conditionals as follows: ",
+        "bbox": [
+            173,
+            799,
+            823,
+            828
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f283bf6c2ab7ccb6a6dc5445627057e98aaf482a231d227f809608e4c68f728e.jpg",
+        "text": "$$\np _ { \\theta } ( { \\bf x } ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( x _ { t } | x _ { 1 : t - 1 } ) \\quad ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            833,
+            596,
+            877
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where the joint distribution is parametrized by $\\theta$ . The prediction at each time-step is conditioned on all previous observations. The observation model is frequently chosen to be a Gaussian or Gaussian mixture model (GMM) for real-valued data, and a categorical distribution for discrete-valued data. ",
+        "bbox": [
+            174,
+            881,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 TEMPORAL CONVOLUTIONAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            506,
+            117
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In TCNs the joint probabilities in Eq. (1) are parametrized by a stack of convolutional layers. Causal convolutions are the central building block of such models and are designed to be asymmetric such that the model has no access to future information. In order to produce outputs of the same size as the input, zero-padding is applied at every layer. ",
+        "bbox": [
+            173,
+            131,
+            825,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the absence of a state transition function, a large receptive field is crucial in capturing long-range dependencies. To avoid the need for vast numbers of causal convolution layers, typically dilated convolutions are used. Exponentially increasing the dilation factor results in an exponential growth of the receptive field size with depth (Yu & Koltun, 2015; Van Den Oord et al., 2016; Bai et al., 2018). In this work, without loss of generality, we use the building blocks of Wavenet (Van Den Oord et al., 2016) as gated activation units (van den Oord et al., 2016) have been reported to perform better. ",
+        "bbox": [
+            173,
+            194,
+            825,
+            279
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A deterministic TCN representation $d _ { t } ^ { l }$ at time-step $t$ and layer $l$ summarizes the input sequence $x _ { 1 : t }$ ",
+        "bbox": [
+            168,
+            284,
+            823,
+            299
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb60277180df9f0e2489b7393563340e7f46fc72cda853a18a168a840d2aa7f1.jpg",
+        "text": "$$\nd _ { t } ^ { l } = \\mathbf { C o n v } ^ { ( l ) } ( d _ { t } ^ { l - 1 } , d _ { t - j } ^ { l - 1 } ) \\quad \\mathrm { a n d } \\quad d _ { t } ^ { 1 } = \\mathbf { C o n v } ^ { ( 1 ) } ( x _ { t } , x _ { t - j } ) \\quad ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            309,
+            699,
+            332
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the filter width is 2 and $j$ denotes the dilation step. In our work, the stochastic variables $z ^ { l } , l \\ = \\ 1 \\ldots L$ are conditioned on TCN representations $d ^ { l }$ that are constructed by stacking $K$ Wavenet blocks over the previous $d ^ { l - 1 }$ (for details see Figure 4 in Appendix). ",
+        "bbox": [
+            173,
+            339,
+            826,
+            383
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 NON-SEQUENTIAL LATENT VARIABLE MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            402,
+            545,
+            417
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "VAEs (Kingma & Welling, 2013; Rezende et al., 2014) introduce a latent random variable $\\mathbf { z }$ to learn the variations in the observed non-sequential data where the generation of the sample $\\mathbf { X }$ is conditioned on the latent variable $\\mathbf { z }$ . The joint probability distribution is defined as: ",
+        "bbox": [
+            174,
+            430,
+            825,
+            473
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b00a53d0785f1b7b832d4304ed10466f3f25fea1c7838cf18da40264b76b03f.jpg",
+        "text": "$$\n\\begin{array} { r } { p _ { \\theta } ( \\mathbf { x } , \\mathbf { z } ) = p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) p _ { \\theta } ( \\mathbf { z } ) \\quad , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            483,
+            588,
+            501
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and parametrized by $\\theta$ . Optimizing the marginal likelihood is intractable due to the non-linear mappings between $\\mathbf { z }$ and $\\mathbf { X }$ and the integration over $\\mathbf { z }$ . Instead the VAE framework introduces an approximate posterior $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ and optimizes a lower-bound on the marginal likelihood: ",
+        "bbox": [
+            174,
+            510,
+            825,
+            554
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f10fc4090c85987387686ad8f616d194fef06c0002dbbeb5bdc0fefe8669f9da.jpg",
+        "text": "$$\n\\begin{array} { r } { \\log p _ { \\theta } ( \\mathbf { x } ) \\geq - K L ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) | | p _ { \\theta } ( \\mathbf { z } ) ) + \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) ] \\quad , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            563,
+            699,
+            583
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $K L$ denotes the Kullback-Leibler divergence. Typically the prior $p _ { \\boldsymbol { \\theta } } ( \\mathbf { z } )$ and the approximate $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ are chosen to be in simple parametric form, such as a Gaussian distribution with diagonal covariance, which allows for an analytical calculation of the $K L$ -term in Eq. (4). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 STOCHASTIC RNNS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            656,
+            354,
+            670
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "An RNN captures temporal dependencies by recursively processing each input, while updating an internal state $h _ { t }$ at each time-step via its state-transition function: ",
+        "bbox": [
+            174,
+            684,
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+            712
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/247b27dcba34205e46be8ae8872f06dfb2cc8dfffc1a19caed5782867c2f9f2d.jpg",
+        "text": "$$\nh _ { t } = f ^ { ( h ) } ( x _ { t } , h _ { t - 1 } ) \\quad ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            722,
+            576,
+            742
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $f ^ { ( h ) }$ is a deterministic transition function such as LSTM (Hochreiter & Schmidhuber, 1997) or GRU (Cho et al., 2014) cells. The computation has to be sequential because $h _ { t }$ depends on $h _ { t - 1 }$ . ",
+        "bbox": [
+            171,
+            753,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The VAE framework has been extended for sequential data, where a latent variable $z _ { t }$ augments the RNN state $h _ { t }$ at each sequence step. The joint distribution $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } , \\mathbf { z } )$ is modeled via an auto-regressive model which results in the following factorization: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca6283fbe2fb450720f27de25beb58549f25a7888b9a7abaca0c4c673fdffa72.jpg",
+        "text": "$$\np _ { \\theta } ( \\mathbf { x } , \\mathbf { z } ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( x _ { t } | z _ { 1 : t } , x _ { 1 : t - 1 } ) p _ { \\theta } ( z _ { t } | x _ { 1 : t - 1 } , z _ { 1 : t - 1 } ) \\quad .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            842,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In contrast to the fixed prior of VAEs, $\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ , sequential variants define prior distributions conditioned on the RNN hidden state $\\mathbf { h }$ and implicitly on the input sequence $\\mathbf { X }$ (Chung et al., 2015). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/c0aec47397b8ede7adba3b46cdcc820188b2ad1e6dfda787b03aa04c1a251b89.jpg",
+        "image_caption": [
+            "Figure 2: Graphical model view of generative models of STCN (left) and STCN-dense (middle), and the inference model (right), which is shared by both variants. Diamonds represent the outputs of deterministic dilated convolution blocks where the dependence of $d _ { t }$ on the past inputs is not shown for clarity (see Eq. (2)). $x _ { t }$ and $z _ { t }$ are observable inputs and latent random variables, respectively. The generative task is to predict the next step in the sequence, given all past steps. Note that in the STCN-dense variant the next step is conditioned on all latent variables $z _ { t } ^ { l }$ for $l = 1 \\ldots L$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 STOCHASTIC TEMPORAL CONVOLUTIONAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            345,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The mechanics of STCNs are related to those of VRNNs and LVAEs. Intuitively, the RNN state $h _ { t }$ is replaced by temporally independent TCN layers $d _ { t } ^ { l }$ . In the absence of an internal state, we define hierarchical latent variables $\\dot { \\boldsymbol { z } } _ { t } ^ { l }$ that are conditioned vertically, i.e., in the same time-step, but independent horizontally, i.e., across time-steps. We follow a similar approach to LVAEs (Sønderby et al., 2016) in defining the hierarchy in a top-down fashion and in how we estimate the approximate posterior. The inference network first computes the approximate likelihood, and then this estimate is corrected by the prior, resulting in the approximate posterior. The TCN layers $\\mathbf { d }$ are shared between the inference and generator networks, analogous to VRNNs (Chung et al., 2015). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 depicts the proposed STCN as a graphical model. STCNs consist of two main modules: the deterministic temporal convolutional network and the stochastic latent variable hierarchy. For a given input sequence $\\mathbf { x } = \\{ x _ { t } \\} , t = 1 . . . T$ we first apply dilated convolutions over the entire sequence to compute a set of deterministic representations $\\dot { d } _ { t } ^ { \\check { l } } , l = 1 \\ldots L$ . Here, $d _ { t } ^ { l }$ corresponds to the output of a block of dilated convolutions at layer $l$ and time-step $t$ . The output $d _ { t } ^ { l }$ is then used to update a set of random latent variables $z _ { t } ^ { l }$ arranged to correspond with different time-scales. ",
+        "bbox": [
+            173,
+            496,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To preserve the parallelism of TCNs, we do not introduce an explicit dependency between different time-steps. However, we suggest that conditioning a latent variable $z _ { t } ^ { l - \\bar { 1 } }$ on the preceding variable $z _ { t } ^ { l }$ implicitly introduces temporal dependencies. Importantly, the random latent variables in the upper layer have access to a larger receptive field due to its deterministic input $d _ { t - 1 } ^ { l }$ , whereas latent random variables in lower layers are updated with different, more local information. However, the latent variable $z _ { t } ^ { l - 1 }$ may receive longer-range information from $z _ { t } ^ { l }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The generative and inference models are jointly trained by optimizing a step-wise variational lower bound on the log-likelihood (Kingma $\\&$ Welling, 2013; Rezende et al., 2014). In the following sections we describe these components and build up the lower-bound for a single time-step $t$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 GENERATIVE MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Each sequence step $x _ { t }$ is generated from a set of latent variables $z _ { t }$ , split into layers as follows: ",
+        "bbox": [
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+            767,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/487f6afda26af4a7f76fe6cf177c981f1515d2dd58f8e2495a136c4714786f5d.jpg",
+        "text": "$$\np _ { \\theta } \\big ( z _ { t } \\vert x _ { 1 : t - 1 } \\big ) = p _ { \\theta } \\big ( z _ { t } ^ { L } \\vert d _ { t - 1 } ^ { L } \\big ) \\prod _ { l = 1 } ^ { L - 1 } p _ { \\theta } \\big ( z _ { t } ^ { l } \\vert z _ { t } ^ { l + 1 } , d _ { t - 1 } ^ { l } \\big ) \\quad ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/441a796df9d580291b7d78a24f71f16bbc714a1bcedbc404359f5563dcf9c735.jpg",
+        "text": "$$\np _ { \\theta } ( z _ { t } ^ { l } | z _ { t } ^ { l + 1 } , d _ { t - 1 } ^ { l } ) = \\mathcal { N } ( \\mu _ { t , p } ^ { l } , \\sigma _ { t , p } ^ { l } ) \\quad \\mathrm { a n d } \\quad [ \\mu _ { t , p } ^ { l } , \\sigma _ { t , p } ^ { l } ] = f _ { p } ^ { ( l ) } ( z _ { t } ^ { l + 1 } , d _ { t - 1 } ^ { l } ) \\quad .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            839,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here the prior is modeled by a Gaussian distribution with diagonal covariance, as is common in the VAE framework. The subscript $p$ denotes items of the generative distribution. For the inference distribution we use the subscript $q$ . The distributions are parameterized by a neural network $f _ { p } ^ { ( l ) }$ and conditioned on: (1) the $d _ { t - 1 } ^ { l }$ computed by the dilated convolutions from the previous time-step, and (2) a sample from the preceding level at the same time-step zl+1t . Please note that at inference time we draw samples from the approximate posterior distribution $z _ { t } ^ { l + 1 } \\sim q _ { \\phi } ( z _ { t } ^ { l + 1 } | \\cdot )$ . The generative model, on the other hand, uses the prior $z _ { t } ^ { l + 1 } \\sim p _ { \\theta } ( z _ { t } ^ { l + 1 } | \\cdot )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            102,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We propose two variants of the observation model. In the non-sequential scenario, the observations are defined to be conditioned on only the last latent variable in the hierarchy, i.e., $p _ { \\theta } ( x _ { t } | z _ { t } ^ { 1 } )$ , following Sønderby et al. (2016); Gulrajani et al. (2016) and Rezende et al. (2014) our STCN variant uses the same observation model, allowing for an efficient optimization. However, latent units are likely to become inactive during training in this configuration (Burda et al., 2015; Bowman et al., 2015; Zhao et al., 2017) resulting in a loss of representational power. ",
+        "bbox": [
+            173,
+            156,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The latent variables at different layers are conditioned on different contexts due to the inputs $d _ { t } ^ { l }$ . Hence, the latent variables are expected to capture complementary aspects of the temporal context. To propagate the information all the way to the final prediction and to ensure that gradients flow through all layers, we take inspiration from Huang et al. (2017) and directly condition the output probability on samples from all latent variables. We call this variant of our architecture STCN-dense. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The final predictions are then computed by the respective observation functions: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffe041d904d311f63882192e6149185482df06268e26b6926bccae3fdbc1608b.jpg",
+        "text": "$$\np _ { \\theta } ( x _ { t } | z _ { t } ) = f ^ { ( o ) } ( z _ { t } ^ { 1 } ) \\quad \\mathrm { a n d } \\quad p _ { \\theta } ^ { d e n s e } ( x _ { t } | z _ { t } ) = f ^ { ( o ) } ( z _ { t } ^ { 1 } , z _ { t } ^ { 2 } \\dots z _ { t } ^ { L } ) \\quad ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            342,
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+            361
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $f ^ { ( o ) }$ corresponds to the output layer constructed by stacking 1D convolutions or Wavenet blocks depending on the dataset. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 INFERENCE MODEL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            410,
+            351,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the original VAE framework the inference model is defined as a bottom-up process, where the latent variables are conditioned on the stochastic layer below. Furthermore, the parameterization of the prior and approximate posterior distributions are computed separately (Burda et al., 2015; Rezende et al., 2014). In contrast, Sønderby et al. (2016) propose a top-down dependency structure shared across the generative and inference models. From a probabilistic point of view, the approximate Gaussian likelihood, computed bottom-up by the inference model, is combined with the Gaussian prior, computed top-down from the generative model. We follow a similar procedure in computing the approximate posterior. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "First, the parameters of the approximate likelihood are computed for each stochastic layer $l$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/317596939f6f4a16eb956bf4d9352c190fd9d3d7ee607591e59396b3e2f854eb.jpg",
+        "text": "$$\n[ \\hat { \\mu } _ { t , q } ^ { l } , \\hat { \\sigma } _ { t , q } ^ { l } ] = f _ { q } ^ { ( l ) } ( z _ { t } ^ { l + 1 } , d _ { t } ^ { l } ) \\quad ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            571,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "followed by the downward pass, recursively computing the prior and approximate posterior by precision-weighted addition: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/62f34dfd2796aca0afd912404c4bba5d8a862f5e61fd8af3e231a3cbc3836ca5.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\sigma _ { t , q } ^ { l } = \\frac { 1 } { ( \\hat { \\sigma } _ { t , q } ^ { l } ) ^ { - 2 } + ( \\sigma _ { t , p } ^ { l } ) ^ { - 2 } } \\quad , } } \\\\ { { \\mu _ { t , q } ^ { l } = \\sigma _ { t , q } ^ { l } ( \\hat { \\mu } _ { t , q } ^ { l } ( \\hat { \\sigma } _ { t , q } ^ { l } ) ^ { - 2 } + \\mu _ { t , p } ^ { l } ( \\sigma _ { t , p } ^ { l } ) ^ { - 2 } ) \\quad . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, the approximate posterior has the same decomposition as the prior (see Eq. (7)): ",
+        "bbox": [
+            168,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9e3190983c5176073ff4fcffd2953fd7109d99500340eee6334a424b74db02d.jpg",
+        "text": "$$\nq _ { \\phi } ( z _ { t } | x _ { 1 : t } ) = q _ { \\phi } ( z _ { t } ^ { L } | d _ { t } ^ { L } ) \\prod _ { l = 1 } ^ { L - 1 } q _ { \\phi } ( z _ { t } ^ { l } | z _ { t } ^ { l + 1 } , d _ { t } ^ { l } ) \\quad ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec3453744c3eab5b6b47a841a2827b4ab06d39be19cd49fd3fe2fc8bea02423d.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { \\phi } ( z _ { t } ^ { l } | z _ { t } ^ { l + 1 } , d _ { t } ^ { l } ) = \\mathcal { N } ( \\mu _ { t , q } ^ { l } , \\sigma _ { t , q } ^ { l } ) \\quad . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the inference and generative network share the parameters of dilated convolutions $\\mathrm { C o n v } ^ { ( l ) }$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The variational lower-bound on the log-likelihood at time-step $t$ can be defined as follows: ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b54f672128ae20d90ff8ea21cf67390facb2889559eed4cd7127a91806eafb34.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log p ( x _ { t } ) \\geq \\mathbb { E } _ { q _ { \\phi } ( z _ { t } | x _ { t } ) } [ \\log p \\theta ( x _ { t } | z _ { t } ) ] - D _ { K L } ( q _ { \\phi } ( z _ { t } | x _ { 1 : t } ) | | p \\theta ( z _ { t } | x _ { 1 : t - 1 } ) ) } \\\\ & { \\qquad = \\mathbb { E } _ { q _ { \\phi } ( z _ { t } ^ { 1 } \\dots z _ { t } ^ { L } | x _ { t } ) } [ \\log p \\theta ( x _ { t } | z _ { t } ^ { 1 } \\dots z _ { t } ^ { L } ) ] - D _ { K L } ( q _ { \\phi } ( z _ { t } ^ { 1 } \\dots z _ { t } ^ { L } | x _ { 1 : t } ) | | p \\theta ( z _ { t } ^ { 1 } \\dots z _ { t } ^ { L } | x _ { 1 : t - 1 } ) ) } \\\\ & { \\mathcal { C } _ { t } ( \\theta , \\phi ; x _ { t } ) = \\mathcal { L } _ { t } ^ { R e c o n } + \\mathcal { L } _ { t } ^ { K L } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Using the decompositions from Eq. (7) and (12), the Kullback-Leibler divergence term becomes: ",
+        "bbox": [
+            169,
+            102,
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+            119
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aa4dc9b41b4e7d261db9666c74056860681dbaa1cc794eda1f57b83197404870.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\displaystyle \\mathcal { L } _ { t } ^ { K L } = - D _ { K L } \\big ( q _ { \\phi } ( \\boldsymbol { z } _ { t } ^ { L } | \\boldsymbol { d } _ { t } ^ { L } ) | | p _ { \\theta } \\big ( \\boldsymbol { z } _ { t } ^ { L } | \\boldsymbol { d } _ { t - 1 } ^ { L } \\big ) \\big ) } & { } \\\\ { \\displaystyle - \\sum _ { l = 1 } ^ { L - 1 } \\mathbb { E } _ { q _ { \\phi } ( \\boldsymbol { z } _ { t } ^ { l + 1 } | \\cdot ) } \\big [ D _ { K L } \\big ( q _ { \\phi } \\big ( \\boldsymbol { z } _ { t } ^ { l } | \\boldsymbol { z } _ { t } ^ { l + 1 } , \\boldsymbol { d } _ { t } ^ { l } \\big ) | | p _ { \\theta } \\big ( \\boldsymbol { z } _ { t } ^ { l } | \\boldsymbol { z } _ { t } ^ { l + 1 } , \\boldsymbol { d } _ { t - 1 } ^ { l } \\big ) \\big ) \\big ] } & { . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            122,
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+            189
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The KL term is the same for the STCN and STCN-dense variants. The reconstruction term $\\mathcal { L } _ { t } ^ { R e c o n }$ , however, is different. In STCN we only use samples from the lowest layer of the hierarchy, whereas in STCN-dense we use all latent samples in the observation model: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/525d02d7e73e8bc457353e5d9744c9483be9878f8b3cdbd1066b710f92764f84.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathcal { L } _ { t } ^ { R e c o n } = \\mathbb { E } _ { q _ { \\phi } ( z _ { t } ^ { 1 } \\ldots z _ { t } ^ { L } | x _ { t } ) } [ \\log p _ { \\theta } ( x _ { t } | z _ { t } ^ { 1 } ) ] } & { { } , } \\\\ { \\mathcal { L } _ { t } ^ { R e c o n - d e n s e } = \\mathbb { E } _ { q _ { \\phi } ( z _ { t } ^ { 1 } \\ldots z _ { t } ^ { L } | x _ { t } ) } [ \\log p _ { \\theta } ( x _ { t } | z _ { t } ^ { 1 } \\ldots z _ { t } ^ { L } ] } & { { } . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In the dense variant, samples drawn from the latent variables $z _ { t } ^ { l }$ are carried over the dense connections. Similar to Maaløe et al. (2016), the expectation over $z _ { t } ^ { l }$ variables are computed by Monte Carlo sampling using the reparameterization trick (Kingma & Welling, 2013; Rezende et al., 2014). ",
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+    {
+        "type": "text",
+        "text": "Please note that the computation of $\\mathcal { L } _ { t } ^ { R e c o n - d e n s e }$ does not introduce any additional computational cost. In STCN, all latent variables have to be visited in terms of ancestral sampling in order to draw the latent sample $z _ { t } ^ { 1 }$ for the observation $x _ { t }$ . Similarly in STCN-dense, the same intermediate samples $z _ { t } ^ { l }$ are used in the prediction of $x _ { t }$ . ",
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+        "type": "text",
+        "text": "One alternative option to use the latent samples could be to sum individual samples before feeding them into the observation model, i.e., $s u m ( \\dot { [ } z _ { t } ^ { 1 } \\dots z _ { t } ^ { L } ] )$ , (Maaløe et al., 2016). We empirically found that this does not work well in STCN-dense. Instead, we concatenate all samples $\\left[ z _ { t } ^ { 1 } \\circ \\cdots \\circ z _ { t } ^ { L } \\right]$ analogously to DenseNet (Huang et al., 2017) and (Kaiser et al., 2018). ",
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+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "We evaluate the proposed variants STCN and STCN-dense both quantitatively and qualitatively on modeling of digital handwritten text and speech. We compare with vanilla TCNs, RNNs, VRNNs and state-of-the art models on the corresponding tasks. ",
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+        "text": "In our experiments we use two variants of the Wavenet model: (1) the original model proposed in (Van Den Oord et al., 2016) and (2) a variant that we augment with skip connections analogously to STCN-dense. This additional baseline evaluates the benefit of learning multi-scale representations in the deterministic setting. Details of the experimental setup are provided in the Appendix. Our code is available at https://ait.ethz.ch/projects/2019/stcn/. ",
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+        "text": "Handwritten text: The IAM-OnDB and Deepwriting datasets consist of digital handwriting sequences where each time-step contains real-valued $( x , y )$ pen coordinates and a binary pen-up event. The IAM-OnDB data is split and pre-processed as done in (Chung et al., 2015). Aksan et al. (2018) extend this dataset with additional samples and better pre-processing. ",
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+        "text": "Table 1 reveals that again both our variants outperform the vanilla variants of TCNs and RNNs on IAM-OnDB. While the stochastic VRNN and SWaveNet are competitive wrt to the STCN variant, both are outperformed by the STCN-dense version. The same relative ordering is maintained on the Deepwriting dataset, indicating that the proposed architecture is robust across datasets. ",
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+        "image_caption": [
+            "Figure 3: (a) Handwriting samples from IAM-OnDB dataset. Generated samples from (b) VRNN, (c) SWaveNet and (d) our model STCN-dense. Each line corresponds to one sample. "
+        ],
+        "image_footnote": [],
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+        "table_caption": [
+            "Table 1: Average log-likelihood per sequence on TIMIT, Blizzard, IAM-OnDB and Deepwriting datasets. (Normal) and (GMM) stand for unimodal Gaussian or multi-modal Gaussian Mixture Model (GMM) as the observation model (Graves, 2013; Chung et al., 2015). Asterisks ∗ indicate that we used our re-implementation only for the Deepwriting dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>TIMIT</td><td>Blizzard</td><td>IAM-OnDB</td><td>Deepwriting</td></tr><tr><td>Wavenet (GMM)</td><td>30188</td><td>8190</td><td>1381</td><td>612</td></tr><tr><td>Wavenet-dense (GMM)</td><td>30636</td><td>8212</td><td>1380</td><td>642</td></tr><tr><td>RNN (GMM) Chung et al. (2015)</td><td>26643</td><td>7413</td><td>1358</td><td>528 *</td></tr><tr><td>VRNN (Normal) Chung et al.(2015)</td><td>~30235</td><td>~9516</td><td>≈1354</td><td>≥ 495 *</td></tr><tr><td>VRNN (GMM) Chung et al. (2015)</td><td>≈ 29604</td><td>~9392</td><td>≈1384</td><td>≥ 673 *</td></tr><tr><td>SRNN (Normal) Fraccaro et al. (2016)</td><td>≥ 60550</td><td>≥11991</td><td>n/a</td><td>n/a</td></tr><tr><td>Z-forcing (Normal) Goyal etal. (2017)</td><td>≥ 70469</td><td>≥ 15430</td><td>n/a</td><td>n/a</td></tr><tr><td>Var.Bi-LSTM (Normal) Shabanian et al.(2017)</td><td>≥ 73976</td><td>≥ 17319</td><td>n/a</td><td>n/a</td></tr><tr><td>SWaveNet (Normal) Lai etal. (2018)</td><td>≥ 72463</td><td>≥ 15708</td><td>≥1301</td><td>n/a</td></tr><tr><td>STCN (GMM)</td><td>≥ 69195</td><td>M 15800</td><td>≥ 1338</td><td>≥ 605</td></tr><tr><td>STCN-dense (GMM)</td><td>≥ 71386</td><td>≥ 16288</td><td>≥ 1796</td><td>≥ 797</td></tr><tr><td>STCN-dense-large (GMM)</td><td>≥ 77438</td><td>≥ 17670</td><td>n/a</td><td>n/a</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Fig. 3 compares generated handwriting samples. While all models produce consistent style, our model generates more natural looking samples. Note that the spacing between words is clearly visible and most of the letters are distinguishable. ",
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+        "type": "text",
+        "text": "Speech modeling: TIMIT and Blizzard are standard benchmark dataset in speech modeling. The models are trained and tested on 200 dimensional real-valued amplitudes. We apply the same pre-processing as Chung et al. (2015). For this task we introduce STCN-dense-large, with increased model capacity. Here we use 512 instead of 256 convolution filters. Note that the total number of model parameters is comparable to SWaveNet and other SOA models. ",
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+        "type": "text",
+        "text": "On TIMIT, STCN-dense (Table 1) significantly outperforms the vanilla TCN and RNN, and stochastic models. On the Blizzard dataset, our model is marginally better than the Variational Bi-LSTM. Note that the inference models of SRNN (Fraccaro et al., 2016), Z-forcing (Goyal et al., 2017), and Variational Bi-LSTM (Shabanian et al., 2017) receive future information by using backward RNN cells. Similarly, SWaveNet (Lai et al., 2018) applies causal convolutions in the backward direction. Hence, the latent variable can be expected to model future dynamics of the sequence. In contrast, our models have only access to information up to the current time-step. These results indicate that the STCN variants perform very well on the speech modeling task. ",
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+        "text": "Latent Space Analysis: Zhao et al. (2017) observe that in hierarchical latent variable models the upper layers have a tendency to become inactive, indicated by a low KL loss (Sønderby et al., 2016; Dieng et al., 2018). Table 2 shows the KL loss per latent variable and the corresponding log-likelihood measured by ELBO in our models. Across the datasets it can be observed that our models make use of many of the latent variables which may explain the strong performance across tasks in terms of log-likelihoods. Note that STCN uses a standard hierarchical structure. However, individual latent variables have different information context due to the corresponding TCN block’s receptive field. This observation suggests that the proposed combination of TCNs and stochastic variables is indeed effective. Furthermore, in STCN we see a similar utilization pattern of the $z$ variables across tasks, whereas STCN-dense may have more flexibility in modeling the temporal dependencies within the data due to its dense connections to the output layer. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 2: KL-loss per latent variable computed over the entire test split. KL5 corresponds to the KL-loss of the top-most latent variable. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset (Model)</td><td>ELBO</td><td>KL</td><td>KL1</td><td>KL2</td><td>KL3</td><td>KL4</td><td>KL5</td></tr><tr><td>IAM-OnDB (sTCN-dense)</td><td>≥ 1796.3</td><td>1653.9</td><td>17.9</td><td>1287.4</td><td>305.3</td><td>41.0</td><td>2.4</td></tr><tr><td>IAM-OnDB (sTCN)</td><td>≥ 1339.2</td><td>964.2</td><td>846.0</td><td>105.2</td><td>12.9</td><td>0.1</td><td>0.0</td></tr><tr><td>TIMIT (sTCN-dense)</td><td>≥ 71385.9</td><td>22297.5</td><td>16113.0</td><td>5641.6</td><td>529.0</td><td>8.3</td><td>5.7</td></tr><tr><td>TIMIT (STCN)</td><td>≥ 69194.9</td><td>23118.3</td><td>22275.5</td><td>487.2</td><td>355.5</td><td>0.0</td><td>0.0</td></tr></table>",
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+        "type": "text",
+        "text": "Replacing TCN with RNN: To better understand potential symergies between dilated CNNs and the proposed latent variable hierarchy, we perform an ablation study, isolating the effect of TCNs and the latent space. To this end the deterministic TCN blocks are replaced with LSTM cells by keeping the latent structure intact. We dub this condition LadderRNN. We use the TIMIT and IAM-OnDB datasets for evaluation. Table 3 summarizes performance measured by the ELBO. ",
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+    {
+        "type": "text",
+        "text": "The most direct translation of the the STCN architecture into an RNN counterpart has 25 stacked LSTM cells with 256 units each. Similar to STCN, we use 5 stochastic layers (see Appendix 7.1). Note that stacking this many LSTM cells is unusual and resulted in instabilities during training. Hence, the performance is similar to vanilla RNNs. The second LadderRNN configuration uses 5 stacked LSTM cells with 512 units and a one-to-one mapping with the stochastic layers. On the TIMIT dataset, all LadderRNN configurations show a significant improvement. We also observe a pattern of improvement with densely connected latent variables. ",
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+    {
+        "type": "text",
+        "text": "This experiments shows that the proposed modular latent variable design does allow for the usage of different building blocks. Even when attached to LSTM cells, it boosts the log-likelihood performance (see 5x512- LadderRNN), in particular when used with dense connections. However, the empirical results suggest that the densely connected latent hierarchy interacts particularly well with dilated CNNs. We suggest this is due to the hierarchical nature on both sides of the architecture. On both datasets STCN models achieved the best performance and significantly improve with dense connections. This supports our contribution of a latent variable hierarchy, which models different aspects of information from the input time-series. ",
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+    {
+        "type": "table",
+        "img_path": "images/092fb0cfa50de0257e738c75953f96b0e886e905a34ddf72b3e3879b97a980c9.jpg",
+        "table_caption": [
+            "Table 3: ELBO of LadderRNN and STCN models using the same latent space configuration. The prefix of a model entries denote the number of RNN or TCN layers and unit size per layer. Models have similar number of trainable parameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>TIMIT</td><td>IAM-OnDB</td></tr><tr><td>25x256-LadderRNN (Normal)</td><td>≥ 28207</td><td>≥ 1305</td></tr><tr><td>25x256-LadderRNN-dense (Normal)</td><td>≥ 27413</td><td>&gt;I &gt;I 1278</td></tr><tr><td>25x256-LadderRNN (GMM)</td><td>≥ 24839</td><td>1381</td></tr><tr><td>25x256-LadderRNN-dense (GMM)</td><td>≥ 26240</td><td>≥ 1377</td></tr><tr><td>5x512-LadderRNN (Normal)</td><td>≥ 49770</td><td>≥ 1299</td></tr><tr><td>5x512-LadderRNN-dense (Normal)</td><td>M 48612</td><td>1374</td></tr><tr><td>5x512-LadderRNN (GMM)</td><td>M 47179</td><td>&gt;I &gt;I 1359</td></tr><tr><td>5x512-LadderRNN-dense (GMM)</td><td>≥ 50113</td><td>≥ 1581</td></tr><tr><td>25x256-STCN (Normal)</td><td>≥ 64913</td><td>≥ 1327</td></tr><tr><td>25x256-STCN-dense (Normal)</td><td>M 70294</td><td>N 1729</td></tr><tr><td>25x256-STCN (GMM)</td><td>M ：69195</td><td>≥ 1339</td></tr><tr><td>25x256-STCN-dense (GMM)</td><td>M 71386</td><td>≥ 1796</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Rezende et al. (2014) propose Deep Latent Gaussian Models (DLGM) and Sønderby et al. (2016) propose the Ladder Variational Autoencoder (LVAE). In both models the latent variables are hierarchically defined and conditioned on the preceding stochastic layer. LVAEs improve upon DLGMs via implementation of a top-down hierarchy both in the generative and inference model. The approximate posterior is computed via a precisionweighted update of the approximate likelihood (i.e., the inference model) and prior (i.e., the generative model). Similarly, the PixelVAE (Gulrajani et al., 2016) incorporates a hierarchical latent space decomposition and uses an autoregressive decoder. Zhao et al. (2017) show under mild conditions that straightforward stacking of latent variables (as is done e.g. in LVAE and PixelVAE) can be ineffective, because the latent variables that are not directly conditioned on the observation variable become inactive. ",
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+    {
+        "type": "text",
+        "text": "Due to the nature of the sequential problem domain, our approach differs in the crucial aspects that STCNs use dynamic, i.e., conditional, priors (Chung et al., 2015) at every level. Moreover, the hierarchy is not only implicitly defined by the network architecture but also explicitly defined by the information content, i.e., receptive field size. Dieng et al. (2018) both theoretically and empirically show that using skip connections from the latent variable to every layer of the decoder increases mutual information between the latent and observation variables. Similar to Dieng et al. (2018) in STCN-dense, we introduce skip connections from all latent variables to the output. In STCN the model is expected to encode and propagate the information through its hierarchy. ",
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+    {
+        "type": "text",
+        "text": "Yang et al. (2017) suggest using autoregressive TCN decoders to remedy the posterior collapse problem observed in language modeling with LSTM decoders (Bowman et al., 2015). van den Oord et al. (2017) and Dieleman et al. (2018) use TCN decoders conditioned on discrete latent variables to model audio signals. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Stochastic RNN architectures mostly vary in the way they employ the latent variable and parametrize the approximate posterior for variational inference. Chung et al. (2015) and Bayer & Osendorfer (2014) use the latent random variable to capture high-level information causing the variability observed in sequential data. Particularly Chung et al. (2015) shows that using a conditional prior rather than a standard Gaussian distribution is very effective in sequence modeling. In (Fraccaro et al., 2016; Goyal et al., 2017; Shabanian et al., 2017), the inference model, i.e., the approximate posterior, receives both the past and future summaries of the sequence from the hidden states of forward and backward RNN cells. The KL-divergence term in the objective enforces the model to learn predictive latent variables in order to capture the future states of the sequence. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Lai et al. (2018)’s SWaveNet is most closely related to ours. SWaveNet also introduces latent variables into TCNs. However, in SWaveNet the deterministic and stochastic units are coupled which may prevent stacking of larger numbers of TCN blocks. Since the number of stacked dilated convolutions determines the receptive field size, this directly correlates with the model capacity. For example, the performance of SWaveNet on the IAM-OnDB dataset degrades after stacking more than 3 stochastic layers (Lai et al., 2018), limiting the model to a small receptive field. In contrast, we aim to preserve the flexibility of stacking dilated convolutions in the base TCN. In STCNs, the deterministic TCN units do not have any dependency on the stochastic variables (see Figure 1) and the ratio of stochastic to deterministic units can be adjusted, depending on the task. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this paper we proposed STCNs, a novel auto-regressive model, combining the computational benefits of convolutional architectures and expressiveness of hierarchical stochastic latent spaces. We have shown the effectivness of the approach across several sequence modelling tasks and datasets. The proposed models are trained via optimization of the ELBO objective. Tighter lower bounds such as IWAE (Burda et al., 2015) or FIVO (Maddison et al., 2017) may further improve modeling performance. We leave this for future work. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This work was supported in parts by the ERC grant OPTINT (StG-2016-717054). We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES   \nMartin Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, Manjunath Kudlur, Josh Levenberg, Rajat Monga, Sherry Moore, Derek G. Murray, Benoit Steiner, Paul Tucker, Vijay Vasudevan, Pete Warden, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. Tensorflow: A system for large-scale machine learning. In 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), pp. 265–283, 2016. URL https://www.usenix.org/system/files/conference/osdi16/osdi16-abadi.pdf.   \nEmre Aksan, Fabrizio Pece, and Otmar Hilliges. DeepWriting: Making Digital Ink Editable via Deep Generative Modeling. In SIGCHI Conference on Human Factors in Computing Systems, CHI ’18, New York, NY, USA, 2018. ACM.   \nShaojie Bai, J Zico Kolter, and Vladlen Koltun. An empirical evaluation of generic convolutional and recurrent networks for sequence modeling. arXiv preprint arXiv:1803.01271, 2018.   \nJustin Bayer and Christian Osendorfer. Learning stochastic recurrent networks. arXiv preprint arXiv:1411.7610, 2014.   \nSamuel R Bowman, Luke Vilnis, Oriol Vinyals, Andrew M Dai, Rafal Jozefowicz, and Samy Bengio. Generating sentences from a continuous space. arXiv preprint arXiv:1511.06349, 2015.   \nYuri Burda, Roger Grosse, and Ruslan Salakhutdinov. Importance weighted autoencoders. arXiv preprint arXiv:1509.00519, 2015.   \nKyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger ¨ Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078, 2014.   \nJunyoung Chung, Kyle Kastner, Laurent Dinh, Kratarth Goel, Aaron C Courville, and Yoshua Bengio. A recurrent latent variable model for sequential data. 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Convolutional sequence to sequence learning. arXiv preprint arXiv:1705.03122, 2017.   \nAnirudh Goyal ALIAS PARTH Goyal, Alessandro Sordoni, Marc-Alexandre Cotˆ e, Nan Ke, and Yoshua Ben- ´ gio. Z-forcing: Training stochastic recurrent networks. In Advances in Neural Information Processing Systems, pp. 6713–6723, 2017.   \nAlex Graves. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013.   \nIshaan Gulrajani, Kundan Kumar, Faruk Ahmed, Adrien Ali Taiga, Francesco Visin, David Vazquez, and Aaron Courville. Pixelvae: A latent variable model for natural images. arXiv preprint arXiv:1611.05013, 2016.   \nSepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8):1735–1780, 1997.   \nGao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. 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Stochastic backpropagation and approximate inference in deep generative models. arXiv preprint arXiv:1401.4082, 2014.   \nSamira Shabanian, Devansh Arpit, Adam Trischler, and Yoshua Bengio. Variational bi-lstms. arXiv preprint arXiv:1711.05717, 2017.   \nCasper Kaae Sønderby, Tapani Raiko, Lars Maaløe, Søren Kaae Sønderby, and Ole Winther. Ladder variational autoencoders. In Advances in neural information processing systems, pp. 3738–3746, 2016.   \nAaron Van Den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalch- ¨ brenner, Andrew W Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. In SSW, pp. 125, 2016.   \nAaron van den Oord, Nal Kalchbrenner, Lasse Espeholt, Oriol Vinyals, Alex Graves, et al. Conditional image ¨ generation with pixelcnn decoders. In Advances in Neural Information Processing Systems, pp. 4790–4798, 2016.   \nAaron van den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In Advances in Neural Information Processing Systems, pp. 6306–6315, 2017.   \nZichao Yang, Zhiting Hu, Ruslan Salakhutdinov, and Taylor Berg-Kirkpatrick. Improved variational autoencoders for text modeling using dilated convolutions. arXiv preprint arXiv:1702.08139, 2017.   \nFisher Yu and Vladlen Koltun. Multi-scale context aggregation by dilated convolutions. arXiv preprint arXiv:1511.07122, 2015.   \nShengjia Zhao, Jiaming Song, and Stefano Ermon. Learning hierarchical features from generative models. arXiv preprint arXiv:1702.08396, 2017. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "7.1 NETWORK DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            135,
+            352,
+            148
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/19aa4d2b81f05aacc2011ed44ab2ccd4481cbc37a564cfc54e2d0bf8b1253e60.jpg",
+        "image_caption": [
+            "Figure 4: Generative model of STCN-dense architecture. Building blocks are highlighted. Note that the dependence of $d _ { t } ^ { l } , l = 1 \\cdots L$ on past inputs is not visualized for clarity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            172,
+            816,
+            361
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The network architecture of the proposed model is illustrated in Fig. 4. We make only a small modification to the vanilla Wavenet architecture. Instead of using skip connections from Wavenet blocks, we only use the latent sample $z _ { t }$ in order to make a prediction of $x _ { t }$ . In STCN-dense configuration, $z _ { t }$ is the concatenation of all latent variables in the hierarchy, i.e., $\\boldsymbol { z } _ { t } = [ \\boldsymbol { z } _ { t } ^ { 1 } \\circ \\cdot \\cdot \\cdot \\circ \\boldsymbol { z } _ { t } ^ { L } ]$ , whereas in STCN only $z _ { t } ^ { 1 }$ is fed to the output layer. ",
+        "bbox": [
+            173,
+            421,
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+            486
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Each stochastic latent variable $z _ { t } ^ { l }$ (except the top-most $z _ { t } ^ { L }$ ) is conditioned on a deterministic TCN representation $d _ { t } ^ { l }$ and the preceding random variable $z _ { t } ^ { l + 1 }$ . The latent variables are calculated by using the latent layers $f _ { p } ^ { ( l ) }$ or $f _ { q } ^ { ( l ) }$ which are neural networks. ",
+        "bbox": [
+            174,
+            491,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We do not define a latent variable per TCN layer. Instead, the stochastic layers are uniformly distributed where each random variable is conditioned on a number of stacked TCN layers $\\dot { d } _ { t } ^ { l }$ . We stack $K$ Wavenet blocks (see figure 4 left) with exponentially increasing dilation size. ",
+        "bbox": [
+            176,
+            542,
+            821,
+            582
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Observation Model: We use Normal or GMM distributions with 20 components to model real-valued data.   \nAll Gaussian distributions have diagonal covariance matrix. ",
+        "bbox": [
+            176,
+            588,
+            823,
+            614
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Output layer $f ^ { ( o ) }$ : For the IAM-OnDB and Deepwriting datasets we use 1D convolutions with ReLU nonlinearity. We stack 5 of these layers with 256 filters and filter size 1. ",
+        "bbox": [
+            174,
+            622,
+            820,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For TIMIT and Blizzard datasets Wavenet blocks in the output layer perform significantly better. We stack 5 Wavenet blocks with dilation size 1. For each convolution operation in the block we use 256 filters. The filter size of the dilated convolution is set to 2. The STCN-dense-large model is constructed by using 512 filters instead of 256. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "TCN blocks $d _ { t } ^ { l }$ : The number of Wavenet blocks is usually determined by the desired receptive field size. ",
+        "bbox": [
+            173,
+            713,
+            790,
+            727
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• For the handwriting datasets $K = 6$ and $L = 5$ . In total we have 30 Wavenet blocks where each convolution operation has 256 filters with size 2. • For speech datasets $K = 5$ and $L = 5$ . In total we have 25 Wavenet blocks where each convolution operation has 256 filters with size 2. The large model configuration uses 512 filters. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Latent layers $f _ { p } ^ { ( l ) }$ and $f _ { q } ^ { ( l ) }$ : The number of stochastic layers per task is given by $L$ . We used [32, 16, 8, 5, 2] dimensional latent variables for the handwriting tasks. It is [256, 128, 64, 32, 16] for speech datasets. Note that the first entry of the list corresponds to $z ^ { 1 }$ . ",
+        "bbox": [
+            176,
+            815,
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+            856
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The mean and sigma parameters of the Normal distributions modeling the latent variables are calculated by the $f _ { p } ^ { ( l ) }$ and $f _ { q } ^ { ( l ) }$ networks. We stack $^ { 2 1 0 }$ convolutions with ReLU nonlinearity and filter size 1. The number of filters are the same as the number of Wavenet block filters for the corresponding task. ",
+        "bbox": [
+            174,
+            862,
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+            904
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, we clamped the latent sigma predictions between 0.001 and 5. ",
+        "bbox": [
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+            910,
+            589,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "7.2 TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            351,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In all STCN experiments we applied KL annealing. In all tasks, the weight of the KL term is initialized with 0 and increased by $1 \\times e ^ { - 4 }$ at every step until it reaches 1. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            155
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The batch size was 20 for all datasets except for Blizzard where it was 128. ",
+        "bbox": [
+            174,
+            161,
+            617,
+            175
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use the ADAM optimizer with its default parameters and exponentially decay the learning rate. For the handwriting datasets the learning rate was initialized with $5 \\times e ^ { - 4 }$ and followed a decay rate of 0.94 over 1000 decay steps. On the speech datasets it was initialized with $1 \\times e ^ { - 3 }$ and decayed with a rate of 0.98. We applied early stopping by measuring the ELBO performance on the validation splits. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We implement STCN models in Tensorflow (Abadi et al., 2016). Our code and models achieving the SOA results are available at https://ait.ethz.ch/projects/2019/stcn/. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7.3 DETAILED RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+            281,
+            354,
+            296
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here we provide the extended results table with Normal observation model entries for available models. ",
+        "bbox": [
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+            306,
+            784,
+            320
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/dc66a0817ce8e640da19283d5f79e18c8a7a83d2f23a1303a1387efa848b1248.jpg",
+        "table_caption": [
+            "Table 4: Average log-likelihood per sequence on TIMIT, Blizzard, IAM-OnDB and Deepwriting datasets. (Normal) and (GMM) stand for unimodal Gaussian or multi-modal Gaussian Mixture Model (GMM) as the observation model (Graves, 2013; Chung et al., 2015). Asterisks ∗ indicate that we used our re-implementation only for the Deepwriting dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>TIMIT</td><td>Blizzard</td><td>IAM-OnDB</td><td>Deepwriting</td></tr><tr><td>Wavenet (Normal)</td><td>-7443</td><td>3784</td><td>1053</td><td>337</td></tr><tr><td>Wavenet (GMM)</td><td>30188</td><td>8190</td><td>1381</td><td>612</td></tr><tr><td>Wavenet-dense (Normal)</td><td>-8579</td><td>3712</td><td>1030</td><td>323</td></tr><tr><td>Wavenet-dense (GMM)</td><td>30636</td><td>8212</td><td>1380</td><td>642</td></tr><tr><td>RNN (Normal) Chung et al. (2015)</td><td>-1900</td><td>3539</td><td>1016</td><td>363 *</td></tr><tr><td>RNN (GMM) Chung et al. (2015)</td><td>26643</td><td>7413</td><td>1358</td><td>528 *</td></tr><tr><td>VRNN (Normal)Chung et al. (2015)</td><td>~ 30235</td><td>~9516</td><td>≈1354</td><td>≥ 495 *</td></tr><tr><td>VRNN (GMM) Chung et a. (2015)</td><td>~ 29604</td><td>~9392</td><td>≈1384</td><td>≥673 *</td></tr><tr><td>SRNN (Normal) Fraccaro et al. (2016)</td><td>≥ 60550</td><td>≥ 11991</td><td>n/a</td><td>n/a</td></tr><tr><td>Z-forcing (Normal)Goyal etal. (2017)</td><td>≥ 70469</td><td>≥ 15430</td><td>n/a</td><td>n/a</td></tr><tr><td>Var.Bi-LSTM (Normal)Shabanian et al. (2017)</td><td>≥ 73976</td><td>≥ 17319</td><td>n/a</td><td>n/a</td></tr><tr><td>SWaveNet (Normal)Lai et al.(2018)</td><td>≥ 72463</td><td>M 15708</td><td>≥1301</td><td>n/a</td></tr><tr><td>STCN(Normal)</td><td>≥ 64913</td><td>M 13273</td><td>≥ 1327</td><td>≥ 575</td></tr><tr><td>STCN(GMM)</td><td>≥ 69195</td><td>≥15800</td><td>≥ 1338</td><td>≥ 605</td></tr><tr><td>STCN-dense(Normal)</td><td>≥ 70294</td><td>≥ 15950</td><td>≥ 1729</td><td>≥ 740</td></tr><tr><td>STCN-dense(GMM)</td><td>≥ 71386</td><td>M 16288</td><td>≥ 1796</td><td>≥ 797</td></tr><tr><td>STCN-dense-large (GMM)</td><td>≥ 77438</td><td>≥ 17670</td><td>n/a</td><td>n/a</td></tr></table>",
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+    }
+]

parse/train/SkFAWax0-/SkFAWax0-_content_list.json

@@ -0,0 +1,1580 @@
+[
+    {
+        "type": "text",
+        "text": "VOICELOOP: VOICE FITTING AND SYNTHESIS VIA A PHONOLOGICAL LOOP ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yaniv Taigman, Lior Wolf, Adam Polyak and Eliya Nachmani ",
+        "bbox": [
+            183,
+            165,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Facebook AI Research {yaniv, wolf, adampolyak, eliyan} $@$ fb.com ",
+        "bbox": [
+            184,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            246,
+            544,
+            261
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present a new neural text to speech (TTS) method that is able to transform text to speech in voices that are sampled in the wild. Unlike other systems, our solution is able to deal with unconstrained voice samples and without requiring aligned phonemes or linguistic features. The network architecture is simpler than those in the existing literature and is based on a novel shifting buffer working memory. The same buffer is used for estimating the attention, computing the output audio, and for updating the buffer itself. The input sentence is encoded using a context-free lookup table that contains one entry per character or phoneme. The speakers are similarly represented by a short vector that can also be fitted to new identities, even with only a few samples. Variability in the generated speech is achieved by priming the buffer prior to generating the audio. Experimental results on several datasets demonstrate convincing capabilities, making TTS accessible to a wider range of applications. In order to promote reproducibility, we release our source code and models1. ",
+        "bbox": [
+            233,
+            276,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            502,
+            336,
+            518
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We study the task of mimicking a person’s voice based on samples that are captured in-the-wild. As far as we know, no other solution exists for this highly applicable learning problem. While the current systems are mostly based on carefully collected or curated audio samples, our method is able to employ the audio of public speeches (from youtube), despite a large amount of background noise and clapping and even with an inaccurate automatic transcript. Moreover, almost all in-the-wild videos contain multiple other speakers that become challenging voice sample outliers and, in some cases, the videos are shot with home equipment and are of reduced quality. ",
+        "bbox": [
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+            532,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our method, called VoiceLoop, is inspired by a working-memory model known as the phonological loop (Baddeley, 1986). The loop holds verbal information for short periods of time. It comprises both a phonological store, where information is constantly being replaced, and a rehearsal process, which maintains longer-term representations in the phonological store. ",
+        "bbox": [
+            174,
+            635,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In our method, we construct a phonological store by employing a shifting buffer that is best seen as a matrix $S ~ \\in \\mathbb { R } ^ { d \\times k }$ with columns $\\bar { S } [ 1 ] \\ldots S [ k ]$ . At every time point, all columns shift to the right $( S [ i + 1 ] = S [ i ]$ for $1 \\leq i < k \\}$ ), column $k$ is discarded, and a new representation vector $u$ is placed in the first position $( S [ 1 ] = u )$ ). $u$ is a function of four parameters, among which are the latest “spoken” output and the buffer $S$ itself. The buffer is, therefore, constantly refreshed with new information, similar to the phonological store, and the mechanism that creates the representations reuses the existing information in the buffer, thus creating long term dependencies. ",
+        "bbox": [
+            174,
+            694,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The two other input parameters of the network that computes the new representation $u$ are the identity of the speaker and the current attention-mediated context. The identity is captured by a learned embedding and is stored in a lookup table (for the individuals in the training set) or fitted (for new individuals). The usage of this embedding for the phonological store means that it influences the dynamic behavior of the store, the attention mechanism and the output process. Since the last process requires heavy personalization, it also receives the identity embedding directly. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a0456ecc35aa3e52127fdd48e495db69d8c185454a419ec899a6e62f7b9bc2e2.jpg",
+        "image_caption": [
+            "Figure 1: An overview of the VoiceLoop architecture. The reader combines the encoding of the sentence’s phonemes using the attention weights to create the current context. A new representation is created by a shallow network that receives the context, the speaker ID, the previous output, and the buffer. The new representation is inserted into the buffer and the earliest vector in the buffer is discarded. The output is obtained by another shallow network that receives the buffer and the speaker as inputs. Once trained, fitting a new voice is done by freezing the network, except for the speaker embedding. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            225,
+            104,
+            767,
+            375
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The input sentences in our system are represented as a list of phonemes. Each phoneme out of the 42 in the dictionary being employed, is encoded as a short vector. The encoding of an input sentence is the list of vectors which corresponds to its list of phonemes. The context, either through a Recurrent Neural Network (RNN) or triphones, is not used. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "At each time point, the encodings of the phonemes are weighted and then summed, using a vector of attention weights, to form the current context vector. As attention mechanism, we employ the Graves attention model (Graves, 2013), which ensures a monotonic increase in the position along the sequence of input phonemes. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A few properties of our methods stand out in the landscape of neural text to speech work: (i) Instead of conventional RNNs, we propose to employ a memory buffer. (ii) The same memory is shared between all processes and is repeatedly used to make all inferences. (iii) We employ shallow fully-connected networks for all computations. (iv) The input encoding part of the “reader” mechanism is extremely simple. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We hypothesize that these properties make our architecture more robust than existing methods and allow us to mimic speakers based on noisy and limited training data. Moreover, since the output is more directly linked to the inputs, we are able to fit new speakers using relatively short audio sequences coupled with automatically generated text. ",
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+    {
+        "type": "text",
+        "text": "Finally, the output of our system is deterministic, given its input. However, multiple intonations are readily generated by employing priming, which involves initializing the buffer $S$ prior to the synthesis process. ",
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+        "text": "Experimentally, we evaluate our method in two ways. For TTS quality, we follow the standard Mean Opinion Score (MOS) experiment done by Arik et al. (2017a). For speaker identification, we train a multi-class network which achieves near-perfect performance on a real validation set, and test it against generated ones. ",
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+        "type": "text",
+        "text": "2 PREVIOUS WORK ",
+        "text_level": 1,
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+        "text": "Text to speech (TTS) methods can be mostly classified into four families: rule-based, concatenative, statistical-parametric (mostly HMM based), and neural. HMM-based methods (Zen et al., 2009) require careful collection of the samples, or as recently attempted by Baljekar & Black, filtering of noisy samples for in-the-wild application. Concatenative methods are somewhat less restrictive but still require tens of minutes of clean and well transcribed samples from the target voice. Emerging neural methods may hold the (currently unrealized) promise of allowing the imitation of new speakers, based on limited and unconstrained samples captured in the wild. ",
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+        "type": "text",
+        "text": "Very recent neural TTS systems include the Deep Voice systems DV1 (Arik et al., 2017b) & DV2 (Arik et al., 2017a), WaveNet (Oord et al., 2016), Char2Wav (Sotelo et al., 2017), and Tacotron (Wang et al., 2017). The DV2 system is a well-engineered system, which includes specialized subsystems for segmenting phonemes, predicting phoneme duration, and predicting the fundamental frequency. Each subsystem includes stacked bidirectional recurrent networks, multilayer fully connected networks and many residual connections. This stands in stark contrast to our system, which employs a single shared memory, one output process, and shallow fully connected networks. ",
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+    {
+        "type": "text",
+        "text": "DV2 is the only other current method that models multiple speakers in a single network. However, in contrast to our results, there are three critical differences: (a) There are no in-the-wild experiments; (b) no fitting to a new speaker that did not appear in the training set is shown possible; and (c) the authors employ a large private set and delegate the attention problem to sub-systems, including strong ground-truth alignment between phonemes, waveforms and linguistic features. The linguistic features, which comprise of phone duration, syllable stress, number of syllables in a word and position of the current syllable in a phrase, are also used during inference for generating the samples (used in the subjective Mean Opinion Score tasks as well). In contrast, our method learns “where to read” from the input. Note that (a) and (b) are crucial capabilities in making TTS accessible to a wide range of applications, in particular when casually and efficiently modeling non-professional speakers. The need for professionally collected datasets and the lack of post-training fitting could be inherent to the DV2 architecture, since it has a large number of speaker-dependent modules, whereas we fit a new speaker in a single place. ",
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+    {
+        "type": "text",
+        "text": "The Tacotron system employs a multi-stage encoder-decoder architecture with multiple RNNs and a block called CBHG (Lee et al., 2016) components, with each CBHG containing multiple convolutional layers, a highway network (Srivastava et al., 2015), and a bidirectional GRU (Cho et al., 2014). The output is a synthesized spectrogram, from which the audio is reconstructed by the GriffinLim (Griffin & Lim, 1984) method. Trained on a large private training set recorded by a professional single speaker, the Tacotron system is able to read raw text (characters and not phonemes). While Tacotron was not trained for multiple speakers, Arik et al. (2017a) have done so and report a high level of sensitivity to the choice of parameters and a need to incorporate the input embedding in many network sites. The Char2Wav architecture employs RNNs for both the reader and the generator. As an attention mechanism, the Graves positional attention mechanism (Graves, 2013) is used. The same attention mechanism is used in our work. However, in our case, the parameters of the attention model are based on the shared memory store (the buffer). Similarly to our method, the network was also trained to predict vocoder features. In addition, for added quality, the vocoder was replaced by a SampleRNN network (Mehri et al., 2016). In contrast to the above mentioned systems, which employ RNNs, the WaveNet architecture is based on stacks of dilated convolutions, which are termed “causal” for not looking into the future. The output audio is generated sample by sample, which, at typical sampling rates of thousands of hertz, is too slow for current TTS applications. Wavenet has shown single-speaker TTS capabilities, but not multi-speaker. ",
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+    {
+        "type": "text",
+        "text": "Waveforms Synthesis There is currently no TTS method which can synthesize waveforms from scratch. WaveNet, DV1, DV2, Char2wav and Tacotron were all conditioned on top of lower level generators. Wavenet was conditioned on F0 vocoder features, as well as linguistic features extracted from separately trained RNN-based text representations. SampleRNNs were employed on top of vocoders. Tacotron synthesized spectograms from mel-spectograms, approximating waveforms using Griffin-Lim. As observed by DV2, small errors in the spectrogram generation result in unnatural (metallic) noise in the reconstruction. Further audio processing can be used to alleviate them, but to a limited extent. Better results were achieved (Arik et al., 2017a) by replacing Griffin-Lim with a Wavenet-like net conditioned on the generated spectogram and speaker. ",
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+    {
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+        "img_path": "images/6cf864e31b1b0670316e3b0767419d1264467b9af8c477619d33e381c48e9b79.jpg",
+        "table_caption": [
+            "Table 1: The components of the VoiceLoop model "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Symbol</td><td>Description</td><td>Computed as:</td></tr><tr><td>St ∈Rdxk 2aiaier ut∈Rd EeRdpxl zERds Kt,βt,Yt ∈RC Ht,O²,Y∈RC at∈Rl Ct ∈Rdp</td><td>buffer at time t new representation for the buffer embedding of the input sequence embedding of the current speaker attention model parameters attention GMM parameters attention vector at time t context vector at time t</td><td>St[1]= ut; St[i+1]= St-1[i] Nu([St-1,[ct +tanh(Fuz),Ot-1]l) E[i]=LUTp[si] LUTs[id] orSec. 3.2 Na(St-1) μt= μt-1+et,σ²=eβt,γt=sm(Yt） See Eq.3,4 Ct =Eαt</td></tr><tr><td>Ot ERd Nu :kd+dp+do→d Nrureees Na :kd →3c N:kd →do LUTp ∈Rdp×42 LUT∈RdsXN Fu:ds→dp F:ds→do</td><td>output vector at time t buffer update network attention network output network embedding of each phoneme embedding of the speakers projection of the speaker for update projection of the speaker for output</td><td>N(St)+Foz</td></tr><tr><td>d sraaeieetr k d ds C S1...St,1≤s≤42 1 N</td><td>dimensionality of the buffer capacity of the buffer dim.of the input embedding LUT dim.of the vocoder feature vector dim. of the speaker embedding # GMM component (attention model) input sequence length of the input sequence number of speakers in the training set</td><td>dp+do 20 256 63 dp 10</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Our system was designed with simplicity in mind in order to promote robustness and reproducibility. We focus on modeling the underlying generation process and do not integrate or condition explicitly for waveforms synthesis. Instead, we employ the WORLD (Morise et al., 2016) vocoder (D4C edition) for feature extraction and waveform synthesis. While this bounds the achievable quality, we also experimented with adding WaveNet and SampleRNN. However, the added performance did not seem to justify the extra effort, especially for in-the-wild voice training data, where we observed no improvement. ",
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+    {
+        "type": "text",
+        "text": "Differentiable Memory The differentiable buffer architecture, in which a new representation is added at every step, and the last vector added is discarded in a FIFO manner, is novel as far as we know. There are multiple other network models in the literature that are augmented by an external memory structure, e.g., (Joulin & Mikolov, 2015; Sukhbaatar et al., 2015; Graves et al., 2014). However, to our knowledge, our work is one of very few applications of such memory networks outside in practice. ",
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+    {
+        "type": "text",
+        "text": "Perhaps the closest model to our work is Stack RNN by Joulin & Mikolov (2015), in which the network is augmented with an infinite stack to which a state vector can be added (PUSH) or removed (POP) at every time step. Unlike our model, only the top of the stack is read each time. ",
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+    {
+        "type": "text",
+        "text": "3 THE ARCHITECTURE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The architecture of the VoiceLoop model is depicted in Fig. 1 and the components of the architecture are listed in Tab. 1. The forward pass of the network has four steps, which are run sequentially. Following a context-free encoding of the input sequence and an encoding of the speaker, the buffer at time t, $\\mathbf { \\bar { \\mathbf { } } } { S _ { t } } \\in \\mathbb { R } ^ { d \\times k }$ , plays a major role in all of the remaining steps and links between the other components of each step. It also carries the error signal from the output to the earlier steps. ",
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+        "text": "Step I: Encoding the speaker and the input sentence Every speaker is represented by a vector $z$ During training, the vectors of the training speakers are stored in a lookup table $L U T _ { s }$ which maps a running id number to a representation of dimensionality $d _ { s }$ . For new speakers, which are being fitted after the network was trained, the vector $z$ is computed by the straightforward optimization process described in Sec. 3.2. ",
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+        "type": "text",
+        "text": "The input sentence is converted to a sequence of phonemes $s _ { 1 } , s _ { 2 } , \\ldots , s _ { l }$ by employing the CMU pronouncing dictionary (Weide, 1998). The number of phonemes in this dictionary is 40, to which two items are added to indicate pauses of different lengths. Each $s _ { i }$ is then mapped separately to an encoding that is based on a trained lookup table $L U T _ { p }$ . This results in an encoding matrix $E$ of size $d _ { p } \\times l$ , where $d _ { p }$ is the size of the encoding, and $l$ is the sequence length. ",
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+        "text": "Step II: Computing the context Similar to (Sotelo et al., 2017; Chorowski et al., 2015), we employ the Graves Gaussian Mixture Model (GMM)-based monotonic attention mechanism. At each output time point $t = 1 , 2 , \\dots$ , the attention network $N _ { a }$ receives the buffer from the previous time step $S _ { t - 1 }$ as input and outputs the GMM priors $\\gamma _ { t }$ , shifts $\\kappa _ { t }$ , and log-variances $\\beta _ { t }$ . For a GMM with $c$ components, each of these is a vector in $\\mathbb { R } ^ { c }$ . $N _ { a }$ has one hidden layer, of dimensionality $\\textstyle { \\frac { d k } { 1 0 } }$ and a ReLU activation function for the hidden layer. ",
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+        "type": "text",
+        "text": "The attention is then computed as follows: ",
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+        "text": "$$\n\\gamma _ { t } ^ { \\prime } [ i ] = \\frac { e x p ( \\gamma _ { t } [ i ] ) } { \\sum _ { j } e x p ( \\gamma _ { t } [ j ] ) } , i = 1 , 2 , \\ldots , c\n$$",
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+        "type": "text",
+        "text": "i.e., the softmax function is applied to the priors. The means of the GMMs are increased: ",
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+        "text": "$$\n\\mu _ { t } = \\mu _ { t - 1 } + e x p ( \\kappa _ { t } ) ,\n$$",
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+    {
+        "type": "text",
+        "text": "and the variances are computed as $\\sigma _ { t } ^ { 2 } = e x p ( \\beta _ { t } )$ . For each GMM component $1 \\leq i \\leq c$ and each point along the input sequence $1 \\le j \\le l$ , we then compute: ",
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+        "img_path": "images/3e1a26ddf7d8cf84f4759521f7317ae65be3e944d2b3692075f8202338a61655.jpg",
+        "text": "$$\n\\phi [ i , j ] = \\frac { \\gamma _ { t } ^ { \\prime } [ i ] } { \\sqrt { 2 \\pi \\sigma _ { t } ^ { 2 } [ i ] } } e x p ( - \\frac { ( j - \\mu _ { t } [ i ] ) ^ { 2 } } { 2 \\sigma _ { t } ^ { 2 } [ i ] } )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The attention weights $\\alpha _ { t }$ are computed for each location in the sequence by summing along all $c$ components: ",
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+        "text": "$$\n\\alpha _ { t } [ j ] = \\sum _ { i = 1 } ^ { c } \\phi [ i , j ]\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The context vector $c _ { t }$ is then computed as weighted sum of the columns of the input sequence embedding matrix $E$ as $c _ { t } = E \\alpha _ { t }$ . The loss function of the entire model depends on the attention vector through this context vector. The GMM is differentiable with respect to mean, std and weight, and these are updated, during training, through backpropagation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Step III: Updating the buffer At each time step, a new representation vector $u$ of dimensionality $d$ is added to the buffer at the first location $S _ { t } [ 1 ]$ , the last column of the buffer from the previous time step $S _ { t - 1 } [ k ]$ is discarded, and the rest are copied $S _ { t } [ i + 1 ] = S _ { t - 1 } [ i ]$ for $i = 1 , \\ldots , k - 1$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our implementation, the number of features in the buffer $d$ is the sum of the dimensionality of the embedding of the phonemes $d _ { p }$ and the output’s dimensionality $d _ { o }$ . This choice was made so that a direct comparison to a buffer that does not employ an update network can be performed. In this case, $u$ is simply the concatenation of the current context vector $c _ { t }$ and the output from the previous time step $o _ { t - 1 }$ . It soon became very clear that this loop-less buffer update leads to poor results, emphasizing the role of using information of the buffer $S$ itself in the update process. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The vector $u$ is, therefore, computed using a shallow fully connected network $N _ { u }$ , with one hidden layer of a size that is the tenth of the input dimensionality and a ReLU activation function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The network receives as input the buffer $S _ { t - 1 }$ , the context vector $c _ { t }$ , and the previous output $o _ { t - 1 }$ The new vector $u$ is also made speaker dependent by adding a projection of the speaker embedding $z$ to the context vector. This projection is followed by a hyperbolic tangent activation function, in order to maintain scale. Therefore, ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/4d1499dbf109477c339244f2d0e0de045d330266468bbb5f5f7d1ca003a05e2e.jpg",
+        "text": "$$\nC _ { t } = [ c _ { t } + t a n h ( F _ { u } z ) , o _ { t - 1 } ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "image",
+        "img_path": "images/f6ad362c76523e8b3b0cd6dd526216f23c413d9f98cb5391787d3aaaf06b2497.jpg",
+        "image_caption": [
+            "Figure 2: Memory Location Significance. For each of the three networks $N _ { u }$ , $N _ { a }$ and $N _ { o }$ , we average the absolute values of the weights to the hidden layer across all hidden neurons and across the $d$ rows of the buffer. The result is a measure of the relative importance of each column of the buffer. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2857c03e47ac304e6ec3899e6b3f9029ac5b5091630e84d309c7a37e724eb299.jpg",
+        "text": "$$\nu = N _ { u } ( [ S _ { t - 1 } , C _ { t } ] ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $[ a , b ]$ is the concatenation of the two column vectors $a$ and $b$ to one column vector, or the concatenation of two matrices $a$ and $b$ side by side. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Another way in which we allow the speaker to influence the generated output is by initializing the buffer based on the speaker’s embedding. Specifically, in our implementation, the speaker embedding size $d _ { s }$ is the same as the phoneme embedding size $d _ { p }$ and we set the top part of the buffer $S _ { 0 }$ to be $z$ repeated $k$ times. The lower part of size $d _ { o } \\times k$ is set to zero. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Step IV: Generating the output The output is generated using a network $N _ { o }$ that is of the same architecture as $N _ { a }$ and $N _ { u }$ and a projection of the user by a learned matrix $F _ { o }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/a331857c96baef97373fb8ca6db8c11bd03633188946171b26537f17f3a15c43.jpg",
+        "text": "$$\no _ { t } = N _ { o } ( S _ { t } + F _ { o } z )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Memory Location Significance In order to better understand the behavior of the buffer, we consider the relative role of each buffer location $1 , 2 , \\ldots , k$ on the activations of $N _ { u } , N _ { a }$ , and $N _ { o }$ Specifically, we average the absolute values of the weights from the input (buffer elements) to the hidden layer. The averaging is performed across all $d$ features and ${ \\frac { d k } { \\frac { 1 0 } { \\alpha } } }$ hidden units, and provides one value per each location. As can be seen in Fig 2, the weights of the latest elements are more prominent, especially, as expected, for the output network $N _ { o }$ . However, even the rightmost column has a relative contribution that is at least one third of the leftmost column. This supports the utility of our buffer architecture, in which all memory locations are equal inputs to the downstream fully connected networks. ",
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+    {
+        "type": "text",
+        "text": "3.1 TRAINING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In our current implementation, the output is a vector of vocoder features of dimensionality $d _ { o } = 6 3$ . Similar to (Sotelo et al., 2017), these features were computed using the Merlin toolkit (Wu et al., 2016). During training, the output at each time frame $t$ is compared to the vocoder features of the ground truth data $Y _ { t }$ using the MSE loss: $\\frac { 1 } { d _ { o } } \\Vert Y _ { t } - o _ { t } \\Vert ^ { 2 }$ . This loss requires an exact temporal alignment of the input and the output sequence. However, human speech is not deterministic and one cannot expect a deterministic method to predict the ground truth. For example, even the same speaker cannot replicate her voice to completely remove the MSE loss since there is variability when repeating the same sentence. Teacher forcing solves this since it eliminates most of the drift and enforces a specific way of uttering the sentence. ",
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+    {
+        "type": "text",
+        "text": "In conventional teacher forcing, during training, the input to the network $N _ { u }$ is $Y _ { t - 1 }$ and not $o _ { t - 1 }$ This holds the danger of teaching the network to predict only one time frame ahead, which would create a drift in the output when run on test data. We, therefore, employ a variant of the teacher-forcing technique, which uses the following input to $N _ { u }$ as the previous output ",
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+        "img_path": "images/b58e93ef71edd3e793110b1ac9626debe9d466d5501b8681cd9522d74f31d8ce.jpg",
+        "text": "$$\n\\frac { o _ { t - 1 } + Y _ { t - 1 } } { 2 } + \\eta ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta$ is a random noise vector. When training starts, the predicted output $o _ { t - 1 }$ is by itself a source of noise. As training progresses, it becomes more similar to $Y _ { t - 1 }$ . However, the systematic difference between the two allows the network to better fit the situation that occurs at test time. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "During training, a forward pass on all of the output sequences is performed (without truncation), followed by a backward pass. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Efficiency The full model contains 9.3 million parameters and runs near real-time on an Intel Xeon E5 single-core CPU and 5 times faster when on M40 NVIDIA GPU, including vocoder CPU decoding. This was benchmarked with our publicly available python PyTorch implementation. Therefore, even without special optimizations, engineering VoiceLoop to run on a mobile client is possible, similar to existing non-neural TTS client solutions (e.g. Android’s text-to-speech APK). ",
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+    },
+    {
+        "type": "text",
+        "text": "3.2 FITTING A NEW PERSON ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Different people exhibit different patterns and present various mannerisms in their speech. Therefore, learning to fit these factors from a limited amount of speech is a challenging task. The goal of speaker mimicking TTS is to be able to mimic a new person based on a relatively short voice sample. Ideally, the new voice would be captured by the parameters of the speaker embedding $z$ , without the need to retrain the network. Naturally, enough variability in the population of the training speakers is needed in order to support this. To fit a new speaker, we are given voice samples and transcribed text. We then employ the training procedure, where the weights of all networks and projections $( N _ { a } , N _ { u } , N _ { o } , L U T _ { p } , F _ { u } , F _ { o } )$ are kept fixed and only vector $z$ is learned (using SGD) to form the embedding of the new speaker. ",
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+    {
+        "type": "text",
+        "text": "The same training procedure as detailed in Sec. 3.1 is employed for fitting a new person, including the application of teacher-forcing. We find that the fitting process is very stable with regards to voice characteristics such as pitch. We also noticed that the accent in the new sample needs to be relatively close to the accents presented in the training samples. See Sec. 3.2 for fitting experiments. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 GENERATING VARIABILITY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As mentioned, natural speech is not deterministic and each time a sentence is said, it is said in a different way. For simplicity, our method does not employ a random component, such as a variational autoencoder. However, we can generate different outputs by employing priming (Graves, 2013). In this technique, the initial buffer $S _ { 0 }$ is initialized based on an initial process in which another word or sentence is run through the system. One can expect that a sentence from the training set that is said in excitement, would paint the buffer differently than one that is flatter. Experimenting with this technique, demonstrates that we are indeed able to achieve the desired level of variability. However, the direct link between the nature of the priming sequence and the generated output is only anecdotal at this point. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We make use of multiple datasets. First, for comparing with existing single speaker techniques, we employ single speaker literature datasets. Second, we employ various subsets of the VCTK dataset (Veaux et al., 2017) for various multi-speaker training and/or fitting experiments. Third, we create a dataset that is composed from four to five public speeches of four public figures. The data was downloaded from youtube, where these speeches are publicly available and were automatically transcribed. Samples generated by our method are available on the project’s website https: //github.com/facebookresearch/loop. ",
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+    {
+        "type": "text",
+        "text": "The MOS measure for the proposed method was computed using the crowdMOS toolkit by P. Ribeiro et al. (2011) and Amazon Mechanical Turk. All samples were presented at 16kHz and the raters were told that they are presented with the results of the different algorithms. At least 20 raters participated in each such experiment, with $9 5 \\%$ confidence intervals. We restricted all experiments to North American raters. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.1 SINGLE SPEAKER EXPERIMENTS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The single speaker experiments took place on the LJ (Ito, 2017a), the Nancy corpus from the 2011 Blizzard Challenge (King & Karaiskos, 2011), and the English audiobook data for the 2013 Blizzard Challenge (King & Karaiskos, 2013). Our method was compared to the ground truth as well as to Char2Wav and to Tacotron. The Char2Wav system was trained by us using the authors’ implementation available at https://github.com/sotelo/parrot. The training of the Char2Wav model, in each experiment, was optimized by measuring the loss on the validation set, over the following hyperparameters: initial learning rate of $\\left[ 1 e - 2 , 1 e - 3 , 1 e - 4 \\right]$ , source noise standard deviation $( [ 1 , 2 , 4 ] )$ , batch-size ([16, 32, 64]) and the length of each training sample $( [ 1 0 e 2 , 1 0 e 4 ] )$ . ",
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+    {
+        "type": "text",
+        "text": "The Tacotron models were pretrained models available from the best public implementation we could find, which is by Ito (2017b). This re-implementation has models only for the LJ and the Nancy datasets. Note that Tacotron has raised a lot of attention and considerable effort was put by the community to replicate the paper’s results. However, there would very likely be a different choice of hyper-parameters between such re-implementations and the one of the authors. ",
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+    {
+        "type": "text",
+        "text": "The MOS scores are shown in Tab. 2. These were computed using the “same_sentence” option of crowdMOS, following DV2 (personal communication). As can be seen, our single speaker results are better than those of the other two algorithms across datasets, but still somewhat lower than the ground truth results. ",
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+    {
+        "type": "text",
+        "text": "It is interesting to note that on Blizzard 2011, our results are better than Tacotron (reimplementation) but not significantly better than Char2Wav, while on Blizzard 2013 it is significantly better than both. This can be attributed to the clean nature of Blizzard 2011, for which Char2Wav is robust enough, and demonstrates our method’s robustness to noise. ",
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+    },
+    {
+        "type": "text",
+        "text": "Tab. 3 presents Mel Cepstral Distortion (MCD) scores. This is an automatic, albeit limited, method of testing compatibility between two audio sequences. Since the sequences are not aligned, we employ MCD DTW, which uses dynamic time warping (DTW) to align the sequences. As can be seen, in this metric too, our method outperforms the baseline methods. The single except is Tacotron’s lower distortion on the LJ dataset. However, as shown in Tab. 2, Tacotron is not competitive on this dataset. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/dde9347ed6a4fd0ced328b3f82e42db9c7613ed759e60ff2a99a4acf78972c6e.jpg",
+        "table_caption": [
+            "Table 2: Single Speaker MOS Scores (Mean ± SD) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>LJ</td><td>Blizzard 2011</td><td>Blizzard 2013</td></tr><tr><td>Tacotron (re-impl)</td><td>2.06 ± 1.02</td><td>2.15 ± 1.10</td><td>N/A</td></tr><tr><td>Char2wav</td><td>3.42 ± 1.14</td><td>3.33 ± 1.06</td><td>2.03 ± 1.16</td></tr><tr><td>VoiceLoop</td><td>3.69 ± 1.04</td><td>3.38 ± 1.00</td><td>3.40 ± 1.03</td></tr><tr><td>Ground truth</td><td>4.60 ± 0.71</td><td>4.56 ± 0.67</td><td>4.80 ± 0.50</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/774d624c0dcf7e9441b212003602bc866dff43176c6f9dbfc56e8d8acc97a4db.jpg",
+        "table_caption": [
+            "Table 3: Single Speaker MCD Scores (Mean ± SD; lower is better) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>LJ</td><td>Blizzard 2011</td><td>Blizzard 2013</td></tr><tr><td>Tacotron (re-impl)</td><td>12.82 ± 1.41</td><td>14.60 ± 7.02</td><td>N/A</td></tr><tr><td>Char2wav</td><td>19.41 ± 5.15</td><td>13.97 ± 4.93</td><td>18.72 ± 6.41</td></tr><tr><td>VoiceLoop</td><td>14.42 ± 1.39</td><td>8.86 ±1.22</td><td>8.67 ± 1.26</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 MULTI-SPEAKER EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Multi-speaker experiments were performed on the VCTK dataset (Veaux et al., 2017). The 109 speakers were divided into four different nested subsets: 22 North American speakers, both male and females; and 65, 85 and 101 random selection of speakers, where the remaining eight speakers were left out for validation. Each subset was shuffled into train and test sets. Different models were trained to each of the subsets. Qualitatively, the models provide distinguished voices, and as can be seen in Fig. 3, the generated voice samples display a different dynamic behavior for different speakers. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/9ec1c8ec5ba0ce9ff107e31130d9098f2d9274b357e12f53f06e3803b35e9cd8.jpg",
+        "image_caption": [
+            "Figure 3: Top: The attention probabilities obtained when mimicking three different North American speakers from VCTK using the same sentence: “but there is no eye contact”. The $\\mathbf { X } ^ { } -$ -axis is the time along the generated audio. The y-axis depicts the sequence of phonemes. Dots indicate the maximal response along time for each phoneme, illustrating learned phoneme duration differences between identities (not given during training). Bottom: The 4-th Mel-cepstrum for the three generated sentences (dashed) as well as the ground-truth (solid) of the leftmost speaker. Best viewed in zoom. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a4261faa2c7db80b6dd6d3b430039060517ccfd011ec4e4e809751a1426e8a0.jpg",
+        "table_caption": [
+            "Table 4: Multi-speaker MOS scores (Mean ± SE) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>VCTK22</td><td>VCTK65</td><td>VCTK85</td><td>VCTK101</td></tr><tr><td>Char2wav</td><td>2.84 ± 1.20</td><td>2.85 ± 1.19</td><td>2.76 ± 1.19</td><td>2.66 ± 1.16</td></tr><tr><td>VoiceLoop</td><td>3.57 ± 1.08</td><td>3.40 ± 1.00</td><td>3.13 ± 1.17</td><td>3.33 ± 1.10</td></tr><tr><td>GT</td><td>4.61 ± 0.75</td><td>4.59 ± 0.72</td><td>4.64 ± 0.64</td><td>4.63 ± 0.66</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we employ the author’s implementation of Char2Wav mentioned above as baseline. Note that while the Char2Wav paper did not present multi-speaker results, the open implementation is more general and includes this option. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Sentences from the test set of VCTK are employed for testing. Following DV2 (private communication), the MOS results were computed using the “diff_sentences” option of the crowdMOS toolkit, and are depicted in Tab. 4. As can be seen, our multi-speaker method shows a considerable advantage over the Char2Wav system across all VCTK subsets, but is not as good as the ground truth. These results are consistent with the MCD scores as reported in Tab. 5. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Speaker Identification The capability of the system to generate distinguished voices that match the original voices was tested, as was done in DV2, using a speaker classifier. We train a multi-class convolutional network on the ground-truth training set of multiple speakers, and test on the generated ones. The network gets as input an arbitrary size of vocoder samples, performs five convolutional layers of 3x3 filters over 32 batch-normalized channels, followed by max-pooling, average pooling over time, two fully-connected layers, and ending with a softmax of the number of classes tested. All intermediate layers were linearly rectified. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The identification results are shown in Tab. 6. The VoiceLoop results are more accurate than the results on the VCTK test split, despite using the same text. This might indicate that the voices generated are more similar to the training voices than the natural variability that is present in the dataset. The Char2Wav results are considerably lower. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/010c74ff84df050071e3ea31dc8c5aa7ddb3b399105aa3e26e887d02f9df6723.jpg",
+        "table_caption": [
+            "Table 5: Multi-speaker MCD scores (Mean ± SE; lower is better) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>VCTK22</td><td>VCTK65</td><td>VCTK85</td><td>VCTK101</td></tr><tr><td>Char2wav</td><td>15.71 ± 1.82</td><td>15.1 ± 1.45</td><td>15.23 ± 1.49</td><td>15.06 ± 1.32</td></tr><tr><td>VoiceLoop</td><td>13.74 ± 0.98</td><td>14.1 ± 0.94</td><td>14.16 ± 0.87</td><td>14.22 ± 0.88</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/a601e689467eb39377d023985de8b7df3c710be5dc31a5e17be03bd2958ff888.jpg",
+        "table_caption": [
+            "Table 6: Multi-Speaker Identification Top-1 Accuracy (%) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>VCTK85</td><td>VCTK101</td></tr><tr><td>VCTK test split</td><td>98.25</td><td>97.16</td></tr><tr><td>Char2Wav on test split sentences</td><td>75.70</td><td>81.63</td></tr><tr><td>VoiceLoop on test split sentences</td><td>100</td><td>99.76</td></tr></table>",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.3 NEW SPEAKER FITTING EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our system is the only published system that is capable of post-training fitting of new speakers. In order to experiment with this capability, we employ the VoiceLoop model trained on VCTK85 and experiment on the remaining 16 speakers one by one, where only the speaker embedding $z$ gets updated. While TTS systems typically require several hours of data to model a single speaker (Zen et al., 2009), our fitting set contains only 23.65 minutes per speaker on average. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/91ceb44e61dcff3043cbbf2aa4dd6e1394d679764444c8de47326ccc710dba71.jpg",
+        "image_caption": [
+            "Figure 4: Fitting new speaker embeddings to an existing VoiceLoop model. The graph plots top1 identification accuracy with respect to a sample set length (in minutes) per speaker. Scores were averaged over 5 splits each. The “Full training” horizontal line is the top-1 accuracy for the corresponding speakers, when trained together with the model from scratch. The leftmost datapoint is for two sentences (about 10sec) per speaker. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As described in 3.2, we randomly initialize a new embedding for every new speaker and update only its weights during back-propagation on the fitting data. The newly fitted speakers achieve ${ \\bf 3 . 0 8 \\pm }$ 0.95 MOS, suggesting that the generation mechanism has not deteriorated below a “fair” level by the new entries. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Similar to the multi-speaker case, we train classifiers for the corresponding identities on ground-truth data, but test on the fitted ones, achieving $8 7 . 6 \\%$ top-1 identification accuracy. Despite lower rates than those in Tab. 6, generations of fitted identities are still reasonably discriminative. We conjecture that training VoiceLoop on a larger set of speakers (e.g. LibriSpeech Panayotov et al. (2015)) will be able to represent unseen identities better. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Fitting Data Size The performance of fitting a new identity clearly relies on the length of the sample that is available for that speaker. In order to understand the influence of the sample size, we repeated the above fitting process for the 16 speakers, but capped the available fitting data per speaker. Specifically, we experimented with a maximal amount of training data of 1, 5, 10, 15 and 20 minutes of voice for each speaker. Instead of cutting the last sentence in the middle, it was removed in case that the threshold was crossed. We repeated this fitting process 5 times, each time fitting a different set of samples at a particular limit. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 4 we report identification accuracies for each limit. Surprisingly, even with two sentences per speaker, totaling about 10 seconds in average, we can fit a new speaker into VoiceLoop such that the speaker is identifiable at $6 4 . 4 \\%$ top-1 identification rate. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "4.4 IN THE WILD EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            416,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the flexibility of our method, we downloaded several publicly available videos from youtube. We picked four different known speakers (see samples page), and for each we retrieved the top four to five results, provided that they are longer than 20 minutes. We extracted the audio and its associated (youtube’s) automatically transcribed text. The total amount of data is 6.2 hours, which we then segmented into 8000 segments. Each segment length is around three seconds, similar to the datasets used in the experiments above. Both the data and its corresponding text are noisy: some of the samples include panel discussions and others with questions from various reporters. Sometimes, microphone echo was observed, or relatively low quality audio originated from mobile video conference sessions. We then trained on this data a VoiceLoop model from scratch, using exactly the same training procedure used by the other experiments. This achieved MOS is ${ \\bf 2 . 9 7 \\bar { \\pm } }$ 1.03, and top-1 accuracy of $9 5 . 8 1 \\%$ . ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We also demonstrate priming (Sec. 3.3) on this dataset. Even for the same speaker, multiple intonations can be generated by initializing $S _ { 0 }$ in different ways. This capability is depicted in Fig. 5 and in the samples page. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Employing web-based in-the-wild training data means that the network is trained on mixed data that contains both speech and other sources. For example, our samples contain a considerable amount of clapping and laughs. Moreover, public speeches contain a larger than usual amount of dramatic prosody and methodological pauses (the same is also true with audiobooks). As our experiments show, our method is mostly robust to these, since it is able to model the voices despite of these difficulties and without replicating the background noises in the synthesized output. The baseline model of Char2Wav was not able to properly model the voices of the youtube dataset and presented clapping sounds in its output. ",
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+    {
+        "type": "text",
+        "text": "The architectural simplicity of our system is likely to be the reason for its robustness. Another advantage that stems directly from it, is its computational efficiency. Based on a few shallow networks and on an iterative process that does not consider future samples, our method can generate voice on mobile devices in speeds far exceeding real-time. For comparison, deep voice (Arik et al., 2017b) is posed as a real time neural TTS system, and it achieves a rate of up to 2.7 times real-time on a Intel Xeon E5-2660 v3 Haswell CPU, running 6 concurrent threads (GPU does not provide speedup for the inference of the deep voice system). ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/5d1173b5d35905582c3853492c246eded508e9ae4fac068b49997462f06f90db.jpg",
+        "image_caption": [
+            "Figure 5: Same input, different intonations. A single in the wild speaker saying the sentence “priming is done like that ”, where each time $S _ { 0 }$ is initialized differently. (a) Without priming. (b) Priming with the word “I\". (c) Priming with the word “had”. (d) Priming with the word “must”. (e) Priming with the word “bye”. The figure shows the raw waveform, spectrogram, and F0 estimation (include voicedness) in the first, second and third rows respectively. From the spectrogram plots we can observe different duration for some phonemes. The F0 estimation of (c) and (d) shows that the speaker talks in higher tone while in (b) and (e) we can observe lower tone of the speaker. This demonstrates how priming changes the intonations of the model outputs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            127,
+            761,
+            473
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The link we form to the model of Baddeley (1986) is by way of analogy and, to be clear, does not imply that we implement this model as is. Specifically, by phonological features, we mean a joint (mixed) representation, in memory, of sound based information and language based information, which is a unique characteristic of our model in comparison to previous work. The short term memory in Baddleley’s model is analog to our buffer and the analog to the rehearsal mechanism is the recursive way in which our buffer is updated. Namely, the new element in the buffer $( u )$ is calculated based on the entire buffer. As noted in Sec. 3, without this dependency on the buffer, our model becomes completely ineffective. ",
+        "bbox": [
+            173,
+            683,
+            825,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "While we employ the loop-updated buffer for the task of speech synthesis, the model is quite general. For example, we have employed the buffer for machine translation from English to French using a dot product based attention model (Bahdanau et al., 2014). The discrete nature of the output means that an output embedding had to be added, but the overall structure remained the same. The performance seemed at least similar to the baseline RNN attention model. However, no attempt has yet been made to achieve state of the art results on existing benchmarks. Surprisingly, relatively large buffer sizes (9) seem to produce better results, despite the input and the output being relatively short. Staying in the realm of voice, the buffer model can be readily used to form a transformation in the other direction (from speech to text), and applied to audio denoising. ",
+        "bbox": [
+            173,
+            797,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            318,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We present a new memory architecture that serves as an effective working memory module. Building on this, we are able to present a neural TTS solution of an architecture that is less complex than those found in the recent literature. It also does not require any alignment between phonemes and acoustics or linguistic features as inputs. Using the new architecture, we are able to present, for the first time as far as we know, multi-speaker TTS that is based on unconstrained samples collected from public speeches. Our work also presents a unique ability to fit new speakers (post-training), which is demonstrated even for very limited sample size. ",
+        "bbox": [
+            173,
+            130,
+            825,
+            228
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
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parse/train/tjdHCnPqoo/tjdHCnPqoo.md

@@ -0,0 +1,381 @@
+# Is Automated Topic Model Evaluation Broken?: The Incoherence of Coherence
+
+# Alexander Hoyle∗
+
+Pranav Goel∗
+
+Denis Peskov∗
+
+Computer Science
+
+Andrew Hian-Cheong∗
+
+Jordan Boyd-Graber
+
+Philip Resnik
+
+CS, iSchool, UMIACS, LSC UMIACS, Lingusitics
+
+University of Maryland {hoyle,pgoel1,dpeskov,andrewhc,jbg,resnik}@cs.umd.edu
+
+# Abstract
+
+Topic model evaluation, like evaluation of other unsupervised methods, can be contentious. However, the field has coalesced around automated estimates of topic coherence, which rely on the frequency of word co-occurrences in a reference corpus. Contemporary neural topic models surpass classical ones according to these metrics. At the same time, topic model evaluation suffers from a validation gap: automated coherence, developed for classical models, has not been validated using human experimentation for neural models. In addition, a meta-analysis of topic modeling literature reveals a substantial standardization ${ g a p }$ in automated topic modeling benchmarks. To address the validation gap, we compare automated coherence with the two most widely accepted human judgment tasks: topic rating and word intrusion. To address the standardization gap, we systematically evaluate a dominant classical model and two state-of-the-art neural models on two commonly used datasets. Automated evaluations declare a winning model when corresponding human evaluations do not, calling into question the validity of fully automatic evaluations independent of human judgments.
+
+# 1 Revisiting Topic Model Evaluation
+
+Topic models are a machine learning technique widely used outside computer science, including political science (Grimmer and Stewart, 2013; Isoaho et al., 2021), social and cultural studies (Mohr and Bogdanov, 2013), digital humanities (Meeks and Weingart, 2012), and bioinformatics (Liu et al., 2016). Typically, topic model users are domain experts trying to identify global categories or themes present in a document collection (Boyd-Graber et al., 2017). This practice constitutes a computer-assisted form of content analysis (Krippendorff, 2004; Chuang et al., 2014), also related to distant reading in literary studies (Underwood, 2017). In general, topic models help humans understand large corpora.2
+
+Evaluation of topic models has vacillated between automated and human-centered. While real-world users of topic models evaluate outputs based on their specific needs, topic model developers have gravitated toward generalized, automated proxies of human judgment to help inform rapid iteration of models (Doogan and Buntine, 2021). Initially, models were evaluated with held-out perplexity, but it disagrees with human interpretability (Chang et al., 2009). Consequently, the field adopted automated coherence metrics like normalized pointwise mutual information (NPMI), a measure of word relatedness that does correlate with topic interpretability (Section 2.2; Newman et al., 2010; Aletras and Stevenson, 2013; Lau et al., 2014). The balance shifted towards automated coherence.
+
+Human evaluations have been abandoned by topic model developers in the years since automated coherence metrics were adopted. In a thorough meta-analysis of contemporary topic model methods papers, none conduct systematic human evaluations (Section 3). Instead, they rely solely on automated metrics for model comparison.3 However, current neural topic models are a far cry from the classical models that substantiated the original correlations—manifestly, topics produced by neural models are often qualitatively distinct from those of classical models (e.g., Table 1).4 This validation gap raises the question of whether automated metrics are still consistent with human judgments of topic quality.
+
+<table><tr><td></td><td colspan="3">Classical</td><td colspan="3">Neural</td></tr><tr><td></td><td>station line bridge railway trains</td><td>album band music song released</td><td>tropical storm hurricane cyclone depression</td><td>tropical landfall cyclone utc weakening</td><td>spore basidia spores mycologist hyphae</td><td>manhattan_project los_alamos_laboratory robert_oppenheimer enrico_fermi physicist</td></tr><tr><td>NPMI</td><td>0.274</td><td>0.285</td><td>0.394</td><td>0.446</td><td>0.456</td><td>0.470</td></tr></table>
+
+Table 1: The first three columns are the highest-NPMI topics for a classical topic model (LDA estimated via Gibbs sampling using Mallet, McCallum, 2002; Griffiths and Steyvers, 2004). The next three are counterparts from a neural model (our D-VAE reimplementation, Burkhardt and Kramer, 2019). Models are trained on Wikitext (Merity et al., 2017) with fifty topics, and NPMI is estimated over the top five words in each topic using a 4.6M-document reference Wikipedia corpus. The mean top-five NPMI over all topics is 0.156 for the classical and 0.256 for the neural model.
+
+Moreover, we should always be cautious when extrapolating outside the range of data that was used to establish a relationship between variables. As an example, a neural model in Hoyle et al. (2020) produces much larger NPMI values than those used to determine human correlations in the original Lau et al. (2014) study; the implicit assumption is that greater NPMI corresponds to more humaninterpretable topics. Finally, a myopic focus on a presumed proxy for human preferences can produce low-quality results (Stiennon et al., 2020). Does Goodharts’ law—“when a measure becomes a target, it ceases to be a good measure” (Strathern, 1997)—apply to automated metrics of topic models?
+
+Another challenge for automated evaluation, whether of classical or neural topic models, is widespread inconsistency (Section 3). Researchers frequently fail to specify the information needed to calculate automated metrics or diverge from the practices that underpin human correlations. Furthermore, evaluation datasets, preprocessing, and hyperparameter optimization vary dramatically, even within a given paper. This standardization gap likely limits the generalizability and reliability of topic model developers’ findings.
+
+We address the standardization and validation gaps in topic model evaluation:
+
+1. We present a meta-analysis of neural topic model evaluation (Section 3);   
+2. we develop standardized, pre-processed versions of two widely-used English-language evaluation datasets, along with a transparent end-to-end code pipeline for reproduction of results (Section 4.1)5;   
+3. we optimize three topic models—one classical and two neural—using identical preprocessing, model selection criteria, and hyperparameter tuning (Section 4.2);   
+4. we evaluate these models using human ratings and word intrusion tasks (Section 5); and   
+5. we provide new evaluations of the correlation between automated and human evaluations (Section 6).
+
+Our findings challenge the validity of fully-automated evaluations as currently practiced: automated evaluation declares winners between models when the corresponding human evaluations cannot.
+
+# 2 Operationalizing Topic Coherence
+
+A topic model is a probabilistic generative model of text that uses latent topics to summarize a larger collection of documents. The most influential variant, latent Dirichlet allocation (Blei et al., 2003, LDA), assumes that $K$ latent topics are distributions over word types, $\beta _ { k }$ , and that the documents $\mathcal { D }$ are admixtures over the topics, $\theta _ { d }$ . Users often evaluate model outputs globally, focusing on the most probable $N$ words of each topic, and locally, considering the most probable topics for each document.
+
+While techniques for topic modeling have progressed from variational inference (Blei et al., 2003) to Gibbs sampling (Griffiths and Steyvers, 2004) to deep generative approaches (Srivastava and Sutton, 2017; Wang et al., 2020b), the core goal discussed in Section 1, obtaining human-understandable categories, remains central. The latest wave of methods, neural topic models (NTM), use continuous word representations and gradient optimization to fit parameters. These models claim to produce more interpretable topics than other prior methods, including LDA.
+
+Those claims are supported by improvements on automated measures of topic coherence.
+
+# 2.1 Human Metrics of Topic Coherence
+
+Like the concept of interpretability, that of real-world coherence is “simultaneously important and slippery” (Lipton, 2018). We will not attempt to formalize it here—though see discussion in Section 7. For present purposes, the term has its roots in Latin cohaerere, “to stick together,” and we will think of coherence as an intangible sense, available to human readers, that a set of terms, when viewed together, enable human recognition of an identifiable category.6 We review two human ratings of topic quality: direct ratings and intrusion.
+
+Rating Raters see a topic and then give the topic a quality score, conventionally on a three-point ordinal scale (Newman et al., 2010; Mimno et al., 2011; Aletras and Stevenson, 2013, inter alia).
+
+Intrusion Chang et al. (2009) devise the word intrusion task as a behavioral way to assess topic coherence. The core idea is that when the top words in a topic identify a coherent latent category, it is easier to identify words that do not belong to that category. Operationally, each topic is represented as its top words plus one “intruder” word which has a low probability of belonging to that topic, but a high probability of belonging to a different topic. Topic coherence is then judged by how well human annotators detect the “intruder” word.
+
+# 2.2 NPMI: The Standard Automated Topic Model Coherence Evaluation
+
+Using the word intrusion task, Chang et al. (2009) showed that perplexity—the original topic model evaluation metric—negatively correlates with human evaluations of topic quality. This finding revealed a need for an automated measurement of topic coherence: an automated metric can measure model quality without expensive, time-consuming, and difficult-to-reproduce human experiments.
+
+Lau et al. (2014) find some metrics that positively correlate with human intrusion and rating scores, particularly when aggregating scores over all topics from a given model. Because of that validation, the prevailing evaluation for model comparison is pairwise normalized pointwise mutual information. NPMI scores topics highly if the top $N$ words—summed over all pairs $w _ { i }$ and $w _ { j }$ —have high joint probability $P ( w _ { j } , w _ { i } )$ compared to their marginal probability:7
+
+$$
+\sum _ { j = 2 } ^ { N } \sum _ { i = 1 } ^ { j - 1 } \frac { l o g \frac { P ( w _ { j } , w _ { i } ) } { P ( w _ { i } ) P ( w _ { j } ) } } { - l o g P ( w _ { i } , w _ { j } ) } .
+$$
+
+The probabilities are estimated using word co-occurrence counts from a reference corpus for a specific context window (which can range from ten words to the entire document). As a result, the choice of reference corpus determines the strength of human correlation (Lau et al., 2014; Röder et al., 2015).
+
+<table><tr><td>Experimentation</td><td>Count</td><td></td></tr><tr><td>Preprocessing</td><td></td><td></td></tr><tr><td>Inconsistent over datasets</td><td>12</td><td>(30%)</td></tr><tr><td>Ambiguous preprocessing</td><td>9</td><td>(23%)</td></tr><tr><td>Model comparisons</td><td></td><td></td></tr><tr><td>All models tuned</td><td>5</td><td>(13%)</td></tr><tr><td>Unclearh.paramsearch</td><td>16</td><td>(40%)</td></tr><tr><td>UnclearLDA baseline,if used</td><td>7</td><td>(24%)</td></tr><tr><td>Recent baseline (w/in 2 yrs)</td><td>31</td><td>(78%)</td></tr><tr><td>Multiple runs /sig. testing</td><td>11</td><td>(28%)</td></tr></table>
+
+Table 2: Meta-analysis of forty neural topic modeling papers (denominator may change, as not all conditions are applicable). No recent neural topic modeling papers use human evaluations of coherence, and the metrics and models are difficult to replicate.   
+
+<table><tr><td>Evaluation</td><td colspan="2">Count</td></tr><tr><td>Number of human evaluations AutomatedCoherence</td><td>0</td><td>(0%)</td></tr><tr><td>Metric NPMI</td><td>26</td><td>(72%)</td></tr><tr><td>Other</td><td>22</td><td>(61%)</td></tr><tr><td>Explicit implementation</td><td>22</td><td>(61%)</td></tr><tr><td>Explicit ref. corpus</td><td>10</td><td>(28%)</td></tr><tr><td>Perplexity w/o coherence</td><td>3</td><td>(8%)</td></tr></table>
+
+A measurement is valid to the extent that it measures what it is intended to measure in the real world. Historically, automated coherence has been validated using human judgements from either crowdworkers (Newman et al., 2010; Aletras and Stevenson, 2013) or experts (Mimno et al., 2011). However, correlations based on classical models may not be applicable for NTMs. Our skepticism is motivated by theory, as neural word representations are intimately connected to NPMI, as explicitly used by Aletras and Stevenson (2013) and which produce similar NPMI scores as Lau et al. (2014). Levy and Goldberg (2014) show that multiple representations create factorizations of PMI matrices. Topic models that have access to these rich representations (e.g. Dieng et al., 2020, and others) could thus create topics with good NPMI scores without explaining the corpus well to a user. In contrast to classical topic models, no one has investigated the validity of NPMI evaluation for NTMs.
+
+Given this lacuna, we conduct experiments aimed at validating that automated topic evaluations still correlate with human judgments of neural topic model quality. We compare against two common human evaluations of individual topic quality: direct rating and intrusion. Human evaluations, like automated topic modeling, lack standardization, which we address in Section 5.
+
+# 3 A Meta-Analysis of Neural Topic Modeling
+
+We survey the neural topic modeling (NTM) literature to assess the state of evaluation in contemporary topic model development. First, we take all references made by an existing, comprehensive survey of NTMs (Zhao et al., 2021b), from which we select (a) modeling papers which (b) mention topic interpretability and (c) compare models’ topics with an existing baseline. This yields forty models, which all claim superior topic coherence. We examine data processing steps, hyperparameter tuning, baseline selection, and automated coherence calculations. Table 2 summarizes our results and Appendix A.1 enumerates the papers.
+
+Our analysis reveals variance in all areas. Preprocessing, which can significantly affect model quality and automated metrics, is often $( 3 0 \% )$ inconsistent across datasets within the same paper. When preprocessing is consistent, authors omit details necessary to fully replicate the pipeline. These issues imply that automated metrics for the same baselines and source datasets vary across papers. Compounding the problem, researchers often train their models on different datasets from those used to establish the relationships between human annotations and automated metrics; Doogan and Buntine (2021) find that the same metrics may not predict interpretability in new domains. Mirroring findings from Dodge et al. (2019), $40 \%$ of papers fail to clearly specify their model tuning procedure, often even the metric used for model selection.
+
+Calculation of automated coherence metrics is equally fraught. As discussed in Section 2.2, a complete specification for NPMI involves several pieces of information, including the reference corpus used to estimate joint word probabilities, the co-occurrence window size, and the number of words selected from the head of the topic distribution. Three out of four papers fail to explicitly indicate the reference corpus; even when we can assume the input corpus is used (13 cases), it remains uncertain whether authors use, e.g., a held-out set or the training documents themselves. For the $61 \%$ that specify the implementation of their coherence metric (by pointing to a code repository or writing out the formula), some of these factors may still be in question. For instance, six authors reference Lau et al. (2014) and the supporting code,8 but the implications are ambiguous: the original paper suggests a large corpus from the same source as the training data, but the repository script defaults to Wikipedia. In other cases, authors use bespoke implementations, which creates room for errors, or deviate from the settings used in human experiments. For example, several papers use a document-wide context window with NPMI, which has not been correlated with human judgments.
+
+Last, even $i f$ automated evaluations are consistent, all claims of coherence improvement depend on the validity results in Lau et al. (2014) generalizing to neural topic models.
+
+# 4 Closing the Standardization Gap for Topic Models
+
+Our human evaluation of topic model outputs serves multiple purposes: (a) establishing whether NTMs show improved coherence over a classical baseline and (b) re-evaluating the efficacy and reliability of automated coherence metrics. In addition, a key goal is (c) to provide a standardized preprocessing pipeline to support head-to-head comparisons as new methods are developed.9
+
+We identify two commonly-used datasets, which we in turn process using a standard pipeline. We then estimate topic models on each dataset following a computationally fair hyperparameter search. Our standardization efforts are similar to concurrent work by Terragni et al. (2021); the main differences are that we (a) mandate consistent preprocessing between training and reference corpora, (b) support multi-word expressions during vocabulary creation (see below), and (c) support distributed hyperparameter searches.
+
+# 4.1 Datasets and Preprocessing
+
+Following Chang et al. (2009), we use English articles from Wikipedia and the New York Times (Table 7). For Wikipedia, we use Wikitext-103 (WIKI, Merity et al., 2017), and for the Times, we subsample roughly $15 \%$ of documents from LDC2008T19 (NYT, Sandhaus, 2008), making it an order of magnitude larger than WIKI. To compute reference counts, we use a 4.6M document Wikipedia dump from September 2017 and the full 1.8M document LDC2008T19 set, processed identically to the training data.
+
+We use SpaCy (Honnibal et al., 2020) to tokenize and identify entities in the text. We create new tokens for detected entities of the form New_York_City, per Krasnashchok and Jouili (2018). Schofield and Mimno (2016) find that lemmatization and word-stemming can hurt English topic interpretability, so we do not lemmatize. To maintain a roughly equal vocabulary size over datasets, we use a power-law relationship of corpus size (c.f. Zipf, 1949) to rule out tokens occurring in fewer than a given number of documents.10 In addition to a standard stopword list, we define corpus-specific stopwords as tokens appearing in more than $90 \%$ of documents. See Appendix A.2 for complete preprocessing details.
+
+# 4.2 Models
+
+We evaluate one venerable classical model and two newer neural models:
+
+Gibbs-LDA As a strong classical baseline, we use the widely-loved Mallet (McCallum, 2002) implementation of Gibbs-sampling for LDA (Griffiths and Steyvers, 2004). Mallet produces topics of (qualitatively) competitive quality to neural models (Srivastava and Sutton, 2017).
+
+Dirichlet-VAE We reimplement Dirichlet-VAE (Burkhardt and Kramer, 2019), a state-of-the-art NTM. For simplicitly, we use pathwise gradients for the Dirichlet (Jankowiak and Obermeyer, 2018), rather than the rejection sampling variational inference of the authors’ primary variant.11 DirichletVAE is a wholesale improvement on one of the first successful NTMs, the popular ProdLDA (Srivastava and Sutton, 2017), and is competitive against recent models on automated coherence. The generative
+
+![](images/b78bf2833c66347f64516d418b096007d4f44b4af7667f0dc917e03dd66a25a5.jpg)
+
+Figure 1: The word intrusion task presented to crowdworkers (the ratings task is in Appendix A.4).
+
+model is simple and retains a broad similarity to LDA. The primary difference is that it does not constrain the estimated topic-word distributions to the simplex.
+
+ETM Thanks to their improved flexiblity, many NTMs incorporate external word representations, on the premise that large-scale, general language knowledge improves topic quality (Bianchi et al., 2021; Hoyle et al., 2020). The Embedded Topic Model (Dieng et al., 2020) is a popular NTM that relies on word embeddings in its generative model.12
+
+We maintain a fixed computational budget per model following the exhortation of Dodge et al. (2019) and use a random set of 164 hyperparameter settings across datasets for each model type.13 We train models for a variable number of steps (a hyperparameter); to calculate automated coherence for the model, we use the topics produced at the last step. For human evaluations, we select the models that maximize NPMI, estimated using the reference corpus with a ten-word window over the top ten topic words, per Lau et al. (2014). We follow the recommendation of Dieng et al. (2020) and learn skip-gram embeddings on the training corpus for ETM (experiments with external pretrained embeddings did not yield substantially different results). As in Hoyle et al. (2020), we eliminate models with highly redundant topics, a known degeneracy of NTMs (Burkhardt and Kramer, 2019): (a) models in which any of the top five words of one topic overlap with another and (b) models that have a topic uniqueness score (Nan et al., 2019) above 0.7. Ranges for hyperparameters and other details are in Appendix A.3.
+
+# 5 Human Evaluations of Topic Quality
+
+We use the ratings and word intrusion tasks from Section 2.2 as human evaluations of topic quality. We recruit crowdworkers using Prolific.co, an online panel provider and collect data with the Qualtrics survey platform. We pay workers 2.5 USD per ratings survey and 3 USD per word intrusion survey, equivalent to 15 USD/hour.
+
+In order to draw meaningful conclusions from human annotations, we require an adequate number of participants to ensure acceptable statistical power. However, Card et al. (2020) show that many NLP experiments, including those relying on human evaluation, are insufficiently powered to detect model differences at reported levels. Adopting a straightforward generative model of annotations (Appendix A.5), we select enough crowdworkers per task to ensure sufficient statistical power (at least $1 - \beta = 0 . 9 ,$ ) to obtain significance at $\alpha = 0 . 0 5$ , resulting in a minimum of fifteen crowdworkers per topic for both tasks. On this criterion, both Chang et al. (2009) and thus Lau et al. (2014), with eight annotators, are underpowered.
+
+For each of our two datasets, we generate fifty topics each from the three models in Section 4.2. In the word intrusion task, we sample five of the top ten topic words plus one intruder; for the ratings task, we present the top ten words in order (Figure 4). We separate the datasets for each task and randomly sample 40 of the 150 topics. In the ratings task, we include an additional sixteen synthetic poor-quality topics to help calibrate scores and filter out low-quality respondents.14
+
+![](images/d4b91800d9e7b7ff7997c7159fa9a556a5d2a977b8f0c42eea33a90c0248c366.jpg)  
+Figure 2: While automated evaluations (here, NPMI) suggest a clear winner between models, human evaluation is more nuanced. Human judgments exhibit greater variability over a smaller range of values. Colored circles correspond to pairwise one-tailed significance tests between model scores at $\alpha = 0 . 0 5$ ; for example, the rightmost orange circle at bottom right shows that human intrusion ratings for D-VAE are significantly higher than ETM for topics derived from Wikipedia.
+
+Phrasing of questions closely follows the wording used by Chang et al. (2009), and crowdworkers received detailed instructions with examples (Appendix A.4) before responding to items.15 As topics can be esoteric (e.g., last columns of Table 1), we ask crowdworkers about their familiarity with the words in each question. We speculate that this question can help protect against spurious low scores for otherwise coherent topics, as real-world users of topic models are usually familiar with domain-specific terminology (see further discussion in Section 7).
+
+# 6 Human Judgment Differs From Automated Metrics
+
+We compare human judgments to automated methods on topics estimated using our three models.
+
+# 6.1 Human Assessment
+
+To establish model differences using human ratings, we use pairwise significance tests: a proportion test for the intrusion scores, a $U$ test (Mann and Whitney, 1947) for the ratings, and a $t$ -test for automated metrics (Figure 2), using one-tailed tests for each pair in both directions. Although D-VAE fares better on the intrusion task, evaluation using ratings favors G-LDA.16
+
+Our human evaluation results are consistent with past iterations of the ratings and word intrusion tasks for topic models. Mimno et al. (2011) report an average of 2.36 on the ratings task on a dataset of medical paper abstracts.17 Our ratings means are 2.5 to 2.8 across all variations (Figure 2). Our word intrusion means range from 0.7 to 0.8, which is comparable to the roughly 0.8 accuracy on the LDA model evaluated in Chang et al. (2009). Median time taken on the tasks was 8–9 minutes.
+
+Table 3: Spearman correlation coefficients between mean human scores and automated metrics. Underlined values have overlapping bootstrapped $9 5 \%$ confidence intervals with that of the largest value in each row. “Concatenated” refers to correlations computed on a concatenation of values for the NYT and WIKI items. “Val” is a small held-out set of $15 \%$ of the training corpus. Using the more data-appropriate logistic and ordered probit regressions for word intrusion and ratings data leads to different conclusions about relative metric strength (Appendix Table 10). CIs are estimated using 1,000 samples.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Ref. Corpus→ Train Corpus↓</td><td colspan="4">NPMI (10-token window)</td><td colspan="4">Cu (110-token window)</td></tr><tr><td>NYT</td><td>WIKI</td><td>Train</td><td>Val</td><td>NYT</td><td>WIKI</td><td>Train</td><td>Val</td></tr><tr><td>Intrusion</td><td>NYT</td><td>0.27</td><td>0.43</td><td>0.27</td><td>0.24</td><td>0.34</td><td>0.45</td><td>0.35</td><td>0.34</td></tr><tr><td></td><td>WIKI</td><td>0.34</td><td>0.36</td><td>0.39</td><td>0.17</td><td>0.32</td><td>0.34</td><td>0.34</td><td>0.20</td></tr><tr><td></td><td>Concatenated</td><td>0.29</td><td>0.40</td><td>0.32</td><td>0.17</td><td>0.32</td><td>0.40</td><td>0.35</td><td>0.24</td></tr><tr><td>Rating</td><td>NYT</td><td>0.37</td><td>0.48</td><td>0.37</td><td>0.39</td><td>0.41</td><td>0.46</td><td>0.44</td><td>0.45</td></tr><tr><td></td><td>WIKI</td><td>0.34</td><td>0.41</td><td>0.44</td><td>0.28</td><td>0.32</td><td>0.40</td><td>0.40</td><td>0.34</td></tr><tr><td></td><td>Concatenated</td><td>0.37</td><td>0.44</td><td>0.41</td><td>0.35</td><td>0.38</td><td>0.42</td><td>0.42</td><td>0.42</td></tr></table>
+
+Following Aletras and Stevenson (2013), we calculate inter-annotator agreement with the mean Spearman correlation between each respondent’s score per topic and the average of other respondent scores, obtaining a value of 0.75 (compare to their value of 0.7 on the NYT corpus). Additionally, we include synthetic poor-quality topics (footnote 14)—correctly identified by annotators—and we monitor the duration taken for the survey to hedge against insincere submissions.
+
+# 6.2 Automated Metrics
+
+NPMI declares D-VAE the unequivocal victor among the three models (with G-LDA a clear second), a very different story from the human judgments. To understand the relationship between automated metrics and human ratings, we estimate the Spearman correlation between the two sets of values for each task and dataset for metric variants (Table 3). Although previous studies have used mean human ratings over topics, this decision obscures the inherent variance of the human ratings and leads to overconfident estimates. We therefore construct $9 5 \%$ confidence intervals by resampling ratings, with replacement, equal to the number of annotators per task (Table 3). We estimate NPMI with the standard 10-word window and $C _ { v }$ (Röder et al., 2015) with the recommended 110-word window.18 The Wikipedia corpus appears to be best correlated with human judgments, even for the models trained on the NYT corpus—this contradicts Lau et al. (2014), where within-domain data have the highest correlations.
+
+While all correlation coefficients are statistically significant, the strength of the correlation alone does not justify their use in model selection, as is standard in the NTM literature (Section 3). In particular, the inherent uncertainty of human judgments means that it is difficult to determine when an increase in a model’s mean automated coherence implies a significant improvement in the corresponding human scores.19
+
+As noted above (Figure 2), automated metrics exaggerate model differences compared to human judgments. To help clarify the utility of automated metrics for model selection, we ask how often an automated metric incorrectly asserts that one model is superior to another. To do so, we generate a bootstrapped estimate of the false discovery rate of each model. First, for each dataset, we randomly sample two independent sets of $K = 5 0$ topics (without replacement) from the original pool of 150, along with their corresponding automated and human scores (resampled with replacement, as in Table 3). Treating the two sampled sets as outputs from two different models, we compute pairwise significance tests between each set for both the $K$ automated metrics and $K \times M$ human scores (using a proportions $z$ -test for the intrusion scores and $t$ -tests for all other values). After repeating this process for $N = 1 0 0 0$ iterations, we report the proportion of significant differences detected using
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Ref. Corpus→ Train Corpus ↓</td><td colspan="3">NPMI (10-token window)</td><td colspan="3">Cu (110-token window)</td></tr><tr><td>NYT</td><td>WIKI</td><td>Train</td><td>NYT</td><td>WIKI</td><td>Train</td></tr><tr><td>Intrusion</td><td>NYT</td><td>46/53</td><td>34/48</td><td>48/50</td><td>35/38</td><td>30 /29</td><td>34/35</td></tr><tr><td rowspan="4">Rating</td><td>WIKI</td><td>44/76</td><td>33/78</td><td>33/75</td><td>45/48</td><td>38/49</td><td>37 /45</td></tr><tr><td>Concatenated</td><td>42/67</td><td>40/66</td><td>41/64</td><td>36/46</td><td>31/44</td><td>30/45</td></tr><tr><td>NYT</td><td>45/50</td><td>45/51</td><td>41/47</td><td>27/29</td><td>26/26</td><td>21/26</td></tr><tr><td>WIKI</td><td>40/73</td><td>31/73</td><td>33/71</td><td>38/40</td><td>31/40</td><td>28 /34</td></tr><tr><td></td><td>Concatenated</td><td>39 /66</td><td>36/66</td><td>37/62</td><td>31/38</td><td>28/38</td><td>19 /36</td></tr></table>
+
+Table 4: False discovery rate (1−precision, lower is better) and false omission rate of significant model differences when using automated metrics; automated metrics often overstate meaningful model differences. Bolded values are those with the lowest geometric mean of FDR and FOR. We sample two independent sets of 50 topics along with their human scores and automated metrics; these sets act as the outputs of two “models”. We then compute significance tests between sets (per Figure 2) on both the automated scores and human scores. A false positive occurs when one set has significantly larger automated scores despite no meaningful difference in actual human scores. Estimates are over 1,000 samples.
+
+![](images/7c93a79f520ef9c7705d69318a21f9cbb9c645542c5025b7c7b1f9c5756e0d16.jpg)  
+Figure 3: Mean human evaluation on the ratings and word intrusion tasks, after filtering out respondents who reported a lack of familiarity with the topic words. When filtering, D-VAE scores improve, highlighting its tendency to produce esoteric topics.
+
+the predicted scores despite equivalent human scores (after correcting for the probability of type I errors, $\alpha = 0 . 0 5$ ).20 Even the best-performing automated metrics predict significant differences absent a meaningful human effect roughly one-fifth of the time (Table 4).
+
+These results suggest that automated metrics alone may be inadequate for model comparison.
+
+# 6.3 Explaining the discrepancy
+
+One reason for the discrepancy between human judgments and automated metrics is that metrics favor more esoteric topics. Specifically, there is a significant negative correlation between a topic’s NPMI or $C _ { v }$ and the share of respondents reporting familiarity with topic words (Pearson’s $\rho = - 0 . 2 9 )$ . And while D-VAE achieves the highest automated metric scores of the three models, it produces topics with the fewest familiar words: respondents report familiarity with terms over $90 \%$ of the time on both tasks for G-LDA and ETM, but they do so only $70 \%$ of the time for D-VAE. This difference suggests that the topics selected by D-VAE are narrower in scope than those of the other models. As shown in Figure 3, removing item annotations where respondents indicate unfamiliarity causes both accuracy in the word intrusion task and the ratio of “Very related” terms in the ratings task for D-VAE to increase substantially.
+
+Qualitatively, this result is apparent when examining topics with a high NPMI but low humans ratings. In Table 5, the top rows consists of financial terms that frequently appear together in NYT articles,
+
+<table><tr><td>Data</td><td>Model</td><td>Topic</td><td>NPMI</td><td>Rat.</td><td>Int.</td></tr><tr><td>NYT</td><td>D-VAE</td><td>inc 6mo earns otc rev qtr 9mo nyse outst dec</td><td>0.56</td><td>1.60</td><td>0.77</td></tr><tr><td>WIKI</td><td>D-VAE</td><td>Waterline conning turrets boilers amidships aft knots armament guns mounts</td><td>0.33</td><td>1.93</td><td>0.65</td></tr><tr><td>NYT</td><td>G-LDA</td><td>bedroom room bath taxes year market listed kitchen broker weeks</td><td>0.30</td><td>2.00</td><td>0.23</td></tr><tr><td>NYT</td><td>D-VAE</td><td>condolences mourns mourn board_of_directors heartfelt deepest esteemed</td><td>0.38</td><td>2.60</td><td>0.23</td></tr><tr><td>NYT</td><td>D-VAE</td><td>shareholders earnings federated mci shares takeover new_york_stock_exchange</td><td>0.18</td><td>3.00</td><td>0.81</td></tr><tr><td>WIKI</td><td>D-VAE</td><td>continental_army expedition militia frigate musket frigates muskets skirmish</td><td>0.11</td><td>3.00</td><td>0.69</td></tr><tr><td>NYT</td><td>D-VAE</td><td>medicaid medicare hospitals welfare uninsured patients</td><td>0.13</td><td>2.80</td><td>0.96</td></tr><tr><td>NYT</td><td>G-LDA</td><td>city mayor state new_york new_york_city officials county yesterday governor</td><td>0.09</td><td>2.53</td><td>1.00</td></tr></table>
+
+Table 5: Topics with the largest human–NPMI discrepancies; top half are topics where NPMI is high and human preferences are low, bottom half is the reverse. NPMI favors esoteric and corpus-specific topics. NPMI is calculated with a 10-token sliding window over the in-domain reference corpus, Rat. is the average 3-point rating for a topic, and Int. refers to the percentage of annotators who identify the intruder word.
+
+and the second row contains rare terms about boating—arguably both are reasonable topics for their respective corpora. We can also see instances where words are qualitatively very related (bottom half of table), but that NPMI fails to score high—perhaps because these words, while related, may not frequently appear together within a ten-word sliding window (Equation 1).
+
+Even for familiar words, some topics may be sensible in the context of the specific corpus, despite their component words lacking an immediately obvious semantic relationship. For example, the topic words in the third and fourth rows appear somewhat unrelated (e.g., “taxes” and “bedroom” in the third row), but they are in fact characteristic of common document types in the New York Times: real estate listings and obituaries. Topics like these render the word intrusion task more difficult: only $23 \%$ of crowdworkers identified the intruder for both topics.
+
+Furthermore, using term familiarity as a proxy for domain expertise does not address the key problems with topic model evaluation: even after filtering out respondents who are not familiar with topic terms, automated metrics still overstate model differences (Appendix A.7). The problems with topic model evaluation may therefore extend to our choice of human evaluations as well.
+
+# 7 So. . . is Automated Topic Modeling Evaluation Broken?
+
+To the extent that our experimentation accurately represents current practice, our results do suggest that topic model evaluation—both automated and human—is overdue for a careful reconsideration. In this, we agree with Doogan and Buntine (2021), who write that “coherence measures designed for older models [. . . ] may be incompatible with newer models” and instead argue for evaluation paradigms centered on corpus exploration and labeling. The right starting point for this reassessment is the recognition that both automated and human evaluations are abstractions of a real-world problem. The familiar use of precision-at-10 in information retrieval, for example, corresponds to a user who is only willing to consider the top ten retrieved documents. In future work, we intend to explore automated metrics that better approximate the preferences of real-world topic model users.
+
+One primary use of topic models is in computer-assisted content analysis. In that context, rather than taking a methods-driven approach to evaluation, it would make sense to take a needs-driven approach.21 Generic evaluation of topic models using domain-general corpora like NYT needs to be revisited, since there is no such thing as a “generic” corpus for content analysis, nor a generic analyst. Content analysis can be formulated in a broad way, as Krippendorff (2004) has shown, but its actual application is always in a domain, by people familiar with that domain. This fact stands in tension with the desirable practicalities of general corpora and crowdworker annotation, and the field will need to address this tension. We have identified “coherence” as calling out a latent concept in the mind of a reader. It follows that we must think about who the relevant human readers are and the conceptual spaces that matter to them.
+
+# Acknowledgements
+
+This material is based upon work supported by the National Science Foundation under Grants 2031736, 2008761, 1822494, ARLIS, and by an Amazon Research Award. We thank Sweta Agrawal for her suggestion to conduct a meta-analysis. We owe much appreciation to Dallas Card for his keen advice on power analyses. Thanks to Frank Fineis for help on several statistical questions, as well as Shuo Chen for his suggestions regarding the false discovery rate calculations. Finally, we thank Caitie Doogan for her helpful comments on the clarity of argumentation, as well as our anonymous reviewers.
+
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+        "text": "Abstract ",
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+        "type": "text",
+        "text": "Topic model evaluation, like evaluation of other unsupervised methods, can be contentious. However, the field has coalesced around automated estimates of topic coherence, which rely on the frequency of word co-occurrences in a reference corpus. Contemporary neural topic models surpass classical ones according to these metrics. At the same time, topic model evaluation suffers from a validation gap: automated coherence, developed for classical models, has not been validated using human experimentation for neural models. In addition, a meta-analysis of topic modeling literature reveals a substantial standardization ${ g a p }$ in automated topic modeling benchmarks. To address the validation gap, we compare automated coherence with the two most widely accepted human judgment tasks: topic rating and word intrusion. To address the standardization gap, we systematically evaluate a dominant classical model and two state-of-the-art neural models on two commonly used datasets. Automated evaluations declare a winning model when corresponding human evaluations do not, calling into question the validity of fully automatic evaluations independent of human judgments. ",
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+        "text": "1 Revisiting Topic Model Evaluation ",
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+        "type": "text",
+        "text": "Topic models are a machine learning technique widely used outside computer science, including political science (Grimmer and Stewart, 2013; Isoaho et al., 2021), social and cultural studies (Mohr and Bogdanov, 2013), digital humanities (Meeks and Weingart, 2012), and bioinformatics (Liu et al., 2016). Typically, topic model users are domain experts trying to identify global categories or themes present in a document collection (Boyd-Graber et al., 2017). This practice constitutes a computer-assisted form of content analysis (Krippendorff, 2004; Chuang et al., 2014), also related to distant reading in literary studies (Underwood, 2017). In general, topic models help humans understand large corpora.2 ",
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+        "text": "Evaluation of topic models has vacillated between automated and human-centered. While real-world users of topic models evaluate outputs based on their specific needs, topic model developers have gravitated toward generalized, automated proxies of human judgment to help inform rapid iteration of models (Doogan and Buntine, 2021). Initially, models were evaluated with held-out perplexity, but it disagrees with human interpretability (Chang et al., 2009). Consequently, the field adopted automated coherence metrics like normalized pointwise mutual information (NPMI), a measure of word relatedness that does correlate with topic interpretability (Section 2.2; Newman et al., 2010; Aletras and Stevenson, 2013; Lau et al., 2014). The balance shifted towards automated coherence. ",
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+        "text": "Human evaluations have been abandoned by topic model developers in the years since automated coherence metrics were adopted. In a thorough meta-analysis of contemporary topic model methods papers, none conduct systematic human evaluations (Section 3). Instead, they rely solely on automated metrics for model comparison.3 However, current neural topic models are a far cry from the classical models that substantiated the original correlations—manifestly, topics produced by neural models are often qualitatively distinct from those of classical models (e.g., Table 1).4 This validation gap raises the question of whether automated metrics are still consistent with human judgments of topic quality. ",
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+            "Table 1: The first three columns are the highest-NPMI topics for a classical topic model (LDA estimated via Gibbs sampling using Mallet, McCallum, 2002; Griffiths and Steyvers, 2004). The next three are counterparts from a neural model (our D-VAE reimplementation, Burkhardt and Kramer, 2019). Models are trained on Wikitext (Merity et al., 2017) with fifty topics, and NPMI is estimated over the top five words in each topic using a 4.6M-document reference Wikipedia corpus. The mean top-five NPMI over all topics is 0.156 for the classical and 0.256 for the neural model. "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Classical</td><td colspan=\"3\">Neural</td></tr><tr><td></td><td>station line bridge railway trains</td><td>album band music song released</td><td>tropical storm hurricane cyclone depression</td><td>tropical landfall cyclone utc weakening</td><td>spore basidia spores mycologist hyphae</td><td>manhattan_project los_alamos_laboratory robert_oppenheimer enrico_fermi physicist</td></tr><tr><td>NPMI</td><td>0.274</td><td>0.285</td><td>0.394</td><td>0.446</td><td>0.456</td><td>0.470</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Moreover, we should always be cautious when extrapolating outside the range of data that was used to establish a relationship between variables. As an example, a neural model in Hoyle et al. (2020) produces much larger NPMI values than those used to determine human correlations in the original Lau et al. (2014) study; the implicit assumption is that greater NPMI corresponds to more humaninterpretable topics. Finally, a myopic focus on a presumed proxy for human preferences can produce low-quality results (Stiennon et al., 2020). Does Goodharts’ law—“when a measure becomes a target, it ceases to be a good measure” (Strathern, 1997)—apply to automated metrics of topic models? ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Another challenge for automated evaluation, whether of classical or neural topic models, is widespread inconsistency (Section 3). Researchers frequently fail to specify the information needed to calculate automated metrics or diverge from the practices that underpin human correlations. Furthermore, evaluation datasets, preprocessing, and hyperparameter optimization vary dramatically, even within a given paper. This standardization gap likely limits the generalizability and reliability of topic model developers’ findings. ",
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+    {
+        "type": "text",
+        "text": "We address the standardization and validation gaps in topic model evaluation: ",
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+    {
+        "type": "text",
+        "text": "1. We present a meta-analysis of neural topic model evaluation (Section 3);   \n2. we develop standardized, pre-processed versions of two widely-used English-language evaluation datasets, along with a transparent end-to-end code pipeline for reproduction of results (Section 4.1)5;   \n3. we optimize three topic models—one classical and two neural—using identical preprocessing, model selection criteria, and hyperparameter tuning (Section 4.2);   \n4. we evaluate these models using human ratings and word intrusion tasks (Section 5); and   \n5. we provide new evaluations of the correlation between automated and human evaluations (Section 6). ",
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+        "type": "text",
+        "text": "Our findings challenge the validity of fully-automated evaluations as currently practiced: automated evaluation declares winners between models when the corresponding human evaluations cannot. ",
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+        "text": "2 Operationalizing Topic Coherence ",
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+    {
+        "type": "text",
+        "text": "A topic model is a probabilistic generative model of text that uses latent topics to summarize a larger collection of documents. The most influential variant, latent Dirichlet allocation (Blei et al., 2003, LDA), assumes that $K$ latent topics are distributions over word types, $\\beta _ { k }$ , and that the documents $\\mathcal { D }$ are admixtures over the topics, $\\theta _ { d }$ . Users often evaluate model outputs globally, focusing on the most probable $N$ words of each topic, and locally, considering the most probable topics for each document. ",
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+    {
+        "type": "text",
+        "text": "While techniques for topic modeling have progressed from variational inference (Blei et al., 2003) to Gibbs sampling (Griffiths and Steyvers, 2004) to deep generative approaches (Srivastava and Sutton, 2017; Wang et al., 2020b), the core goal discussed in Section 1, obtaining human-understandable categories, remains central. The latest wave of methods, neural topic models (NTM), use continuous word representations and gradient optimization to fit parameters. These models claim to produce more interpretable topics than other prior methods, including LDA. ",
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+        "text": "Those claims are supported by improvements on automated measures of topic coherence. ",
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+        "type": "text",
+        "text": "2.1 Human Metrics of Topic Coherence ",
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+        "text": "Like the concept of interpretability, that of real-world coherence is “simultaneously important and slippery” (Lipton, 2018). We will not attempt to formalize it here—though see discussion in Section 7. For present purposes, the term has its roots in Latin cohaerere, “to stick together,” and we will think of coherence as an intangible sense, available to human readers, that a set of terms, when viewed together, enable human recognition of an identifiable category.6 We review two human ratings of topic quality: direct ratings and intrusion. ",
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+        "text": "Rating Raters see a topic and then give the topic a quality score, conventionally on a three-point ordinal scale (Newman et al., 2010; Mimno et al., 2011; Aletras and Stevenson, 2013, inter alia). ",
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+        "text": "Intrusion Chang et al. (2009) devise the word intrusion task as a behavioral way to assess topic coherence. The core idea is that when the top words in a topic identify a coherent latent category, it is easier to identify words that do not belong to that category. Operationally, each topic is represented as its top words plus one “intruder” word which has a low probability of belonging to that topic, but a high probability of belonging to a different topic. Topic coherence is then judged by how well human annotators detect the “intruder” word. ",
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+        "text": "2.2 NPMI: The Standard Automated Topic Model Coherence Evaluation ",
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+        "text": "Using the word intrusion task, Chang et al. (2009) showed that perplexity—the original topic model evaluation metric—negatively correlates with human evaluations of topic quality. This finding revealed a need for an automated measurement of topic coherence: an automated metric can measure model quality without expensive, time-consuming, and difficult-to-reproduce human experiments. ",
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+        "text": "Lau et al. (2014) find some metrics that positively correlate with human intrusion and rating scores, particularly when aggregating scores over all topics from a given model. Because of that validation, the prevailing evaluation for model comparison is pairwise normalized pointwise mutual information. NPMI scores topics highly if the top $N$ words—summed over all pairs $w _ { i }$ and $w _ { j }$ —have high joint probability $P ( w _ { j } , w _ { i } )$ compared to their marginal probability:7 ",
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+        "img_path": "images/58511c557f8003ea8644d8ab42e1a4179b4b6dfa84d05be6f22241676bd16a7d.jpg",
+        "text": "$$\n\\sum _ { j = 2 } ^ { N } \\sum _ { i = 1 } ^ { j - 1 } \\frac { l o g \\frac { P ( w _ { j } , w _ { i } ) } { P ( w _ { i } ) P ( w _ { j } ) } } { - l o g P ( w _ { i } , w _ { j } ) } .\n$$",
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+        "text": "The probabilities are estimated using word co-occurrence counts from a reference corpus for a specific context window (which can range from ten words to the entire document). As a result, the choice of reference corpus determines the strength of human correlation (Lau et al., 2014; Röder et al., 2015). ",
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+        "table_body": "<table><tr><td>Experimentation</td><td>Count</td><td></td></tr><tr><td>Preprocessing</td><td></td><td></td></tr><tr><td>Inconsistent over datasets</td><td>12</td><td>(30%)</td></tr><tr><td>Ambiguous preprocessing</td><td>9</td><td>(23%)</td></tr><tr><td>Model comparisons</td><td></td><td></td></tr><tr><td>All models tuned</td><td>5</td><td>(13%)</td></tr><tr><td>Unclearh.paramsearch</td><td>16</td><td>(40%)</td></tr><tr><td>UnclearLDA baseline,if used</td><td>7</td><td>(24%)</td></tr><tr><td>Recent baseline (w/in 2 yrs)</td><td>31</td><td>(78%)</td></tr><tr><td>Multiple runs /sig. testing</td><td>11</td><td>(28%)</td></tr></table>",
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+        "table_caption": [
+            "Table 2: Meta-analysis of forty neural topic modeling papers (denominator may change, as not all conditions are applicable). No recent neural topic modeling papers use human evaluations of coherence, and the metrics and models are difficult to replicate. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Evaluation</td><td colspan=\"2\">Count</td></tr><tr><td>Number of human evaluations AutomatedCoherence</td><td>0</td><td>(0%)</td></tr><tr><td>Metric NPMI</td><td>26</td><td>(72%)</td></tr><tr><td>Other</td><td>22</td><td>(61%)</td></tr><tr><td>Explicit implementation</td><td>22</td><td>(61%)</td></tr><tr><td>Explicit ref. corpus</td><td>10</td><td>(28%)</td></tr><tr><td>Perplexity w/o coherence</td><td>3</td><td>(8%)</td></tr></table>",
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+        "type": "text",
+        "text": "A measurement is valid to the extent that it measures what it is intended to measure in the real world. Historically, automated coherence has been validated using human judgements from either crowdworkers (Newman et al., 2010; Aletras and Stevenson, 2013) or experts (Mimno et al., 2011). However, correlations based on classical models may not be applicable for NTMs. Our skepticism is motivated by theory, as neural word representations are intimately connected to NPMI, as explicitly used by Aletras and Stevenson (2013) and which produce similar NPMI scores as Lau et al. (2014). Levy and Goldberg (2014) show that multiple representations create factorizations of PMI matrices. Topic models that have access to these rich representations (e.g. Dieng et al., 2020, and others) could thus create topics with good NPMI scores without explaining the corpus well to a user. In contrast to classical topic models, no one has investigated the validity of NPMI evaluation for NTMs. ",
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+        "text": "Given this lacuna, we conduct experiments aimed at validating that automated topic evaluations still correlate with human judgments of neural topic model quality. We compare against two common human evaluations of individual topic quality: direct rating and intrusion. Human evaluations, like automated topic modeling, lack standardization, which we address in Section 5. ",
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+        "text": "3 A Meta-Analysis of Neural Topic Modeling ",
+        "text_level": 1,
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+        "text": "We survey the neural topic modeling (NTM) literature to assess the state of evaluation in contemporary topic model development. First, we take all references made by an existing, comprehensive survey of NTMs (Zhao et al., 2021b), from which we select (a) modeling papers which (b) mention topic interpretability and (c) compare models’ topics with an existing baseline. This yields forty models, which all claim superior topic coherence. We examine data processing steps, hyperparameter tuning, baseline selection, and automated coherence calculations. Table 2 summarizes our results and Appendix A.1 enumerates the papers. ",
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+        "type": "text",
+        "text": "Our analysis reveals variance in all areas. Preprocessing, which can significantly affect model quality and automated metrics, is often $( 3 0 \\% )$ inconsistent across datasets within the same paper. When preprocessing is consistent, authors omit details necessary to fully replicate the pipeline. These issues imply that automated metrics for the same baselines and source datasets vary across papers. Compounding the problem, researchers often train their models on different datasets from those used to establish the relationships between human annotations and automated metrics; Doogan and Buntine (2021) find that the same metrics may not predict interpretability in new domains. Mirroring findings from Dodge et al. (2019), $40 \\%$ of papers fail to clearly specify their model tuning procedure, often even the metric used for model selection. ",
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+        "type": "text",
+        "text": "Calculation of automated coherence metrics is equally fraught. As discussed in Section 2.2, a complete specification for NPMI involves several pieces of information, including the reference corpus used to estimate joint word probabilities, the co-occurrence window size, and the number of words selected from the head of the topic distribution. Three out of four papers fail to explicitly indicate the reference corpus; even when we can assume the input corpus is used (13 cases), it remains uncertain whether authors use, e.g., a held-out set or the training documents themselves. For the $61 \\%$ that specify the implementation of their coherence metric (by pointing to a code repository or writing out the formula), some of these factors may still be in question. For instance, six authors reference Lau et al. (2014) and the supporting code,8 but the implications are ambiguous: the original paper suggests a large corpus from the same source as the training data, but the repository script defaults to Wikipedia. In other cases, authors use bespoke implementations, which creates room for errors, or deviate from the settings used in human experiments. For example, several papers use a document-wide context window with NPMI, which has not been correlated with human judgments. ",
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+        "type": "text",
+        "text": "Last, even $i f$ automated evaluations are consistent, all claims of coherence improvement depend on the validity results in Lau et al. (2014) generalizing to neural topic models. ",
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+    {
+        "type": "text",
+        "text": "4 Closing the Standardization Gap for Topic Models ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our human evaluation of topic model outputs serves multiple purposes: (a) establishing whether NTMs show improved coherence over a classical baseline and (b) re-evaluating the efficacy and reliability of automated coherence metrics. In addition, a key goal is (c) to provide a standardized preprocessing pipeline to support head-to-head comparisons as new methods are developed.9 ",
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+        "text": "We identify two commonly-used datasets, which we in turn process using a standard pipeline. We then estimate topic models on each dataset following a computationally fair hyperparameter search. Our standardization efforts are similar to concurrent work by Terragni et al. (2021); the main differences are that we (a) mandate consistent preprocessing between training and reference corpora, (b) support multi-word expressions during vocabulary creation (see below), and (c) support distributed hyperparameter searches. ",
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+        "text": "4.1 Datasets and Preprocessing ",
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+        "type": "text",
+        "text": "Following Chang et al. (2009), we use English articles from Wikipedia and the New York Times (Table 7). For Wikipedia, we use Wikitext-103 (WIKI, Merity et al., 2017), and for the Times, we subsample roughly $15 \\%$ of documents from LDC2008T19 (NYT, Sandhaus, 2008), making it an order of magnitude larger than WIKI. To compute reference counts, we use a 4.6M document Wikipedia dump from September 2017 and the full 1.8M document LDC2008T19 set, processed identically to the training data. ",
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+    {
+        "type": "text",
+        "text": "We use SpaCy (Honnibal et al., 2020) to tokenize and identify entities in the text. We create new tokens for detected entities of the form New_York_City, per Krasnashchok and Jouili (2018). Schofield and Mimno (2016) find that lemmatization and word-stemming can hurt English topic interpretability, so we do not lemmatize. To maintain a roughly equal vocabulary size over datasets, we use a power-law relationship of corpus size (c.f. Zipf, 1949) to rule out tokens occurring in fewer than a given number of documents.10 In addition to a standard stopword list, we define corpus-specific stopwords as tokens appearing in more than $90 \\%$ of documents. See Appendix A.2 for complete preprocessing details. ",
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+        "text": "4.2 Models ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate one venerable classical model and two newer neural models: ",
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+        "type": "text",
+        "text": "Gibbs-LDA As a strong classical baseline, we use the widely-loved Mallet (McCallum, 2002) implementation of Gibbs-sampling for LDA (Griffiths and Steyvers, 2004). Mallet produces topics of (qualitatively) competitive quality to neural models (Srivastava and Sutton, 2017). ",
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+        "type": "text",
+        "text": "Dirichlet-VAE We reimplement Dirichlet-VAE (Burkhardt and Kramer, 2019), a state-of-the-art NTM. For simplicitly, we use pathwise gradients for the Dirichlet (Jankowiak and Obermeyer, 2018), rather than the rejection sampling variational inference of the authors’ primary variant.11 DirichletVAE is a wholesale improvement on one of the first successful NTMs, the popular ProdLDA (Srivastava and Sutton, 2017), and is competitive against recent models on automated coherence. The generative ",
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+        "type": "text",
+        "text": "Figure 1: The word intrusion task presented to crowdworkers (the ratings task is in Appendix A.4). ",
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+        "type": "text",
+        "text": "model is simple and retains a broad similarity to LDA. The primary difference is that it does not constrain the estimated topic-word distributions to the simplex. ",
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+        "text": "ETM Thanks to their improved flexiblity, many NTMs incorporate external word representations, on the premise that large-scale, general language knowledge improves topic quality (Bianchi et al., 2021; Hoyle et al., 2020). The Embedded Topic Model (Dieng et al., 2020) is a popular NTM that relies on word embeddings in its generative model.12 ",
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+        "type": "text",
+        "text": "We maintain a fixed computational budget per model following the exhortation of Dodge et al. (2019) and use a random set of 164 hyperparameter settings across datasets for each model type.13 We train models for a variable number of steps (a hyperparameter); to calculate automated coherence for the model, we use the topics produced at the last step. For human evaluations, we select the models that maximize NPMI, estimated using the reference corpus with a ten-word window over the top ten topic words, per Lau et al. (2014). We follow the recommendation of Dieng et al. (2020) and learn skip-gram embeddings on the training corpus for ETM (experiments with external pretrained embeddings did not yield substantially different results). As in Hoyle et al. (2020), we eliminate models with highly redundant topics, a known degeneracy of NTMs (Burkhardt and Kramer, 2019): (a) models in which any of the top five words of one topic overlap with another and (b) models that have a topic uniqueness score (Nan et al., 2019) above 0.7. Ranges for hyperparameters and other details are in Appendix A.3. ",
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+        "type": "text",
+        "text": "5 Human Evaluations of Topic Quality ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We use the ratings and word intrusion tasks from Section 2.2 as human evaluations of topic quality. We recruit crowdworkers using Prolific.co, an online panel provider and collect data with the Qualtrics survey platform. We pay workers 2.5 USD per ratings survey and 3 USD per word intrusion survey, equivalent to 15 USD/hour. ",
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+        "type": "text",
+        "text": "In order to draw meaningful conclusions from human annotations, we require an adequate number of participants to ensure acceptable statistical power. However, Card et al. (2020) show that many NLP experiments, including those relying on human evaluation, are insufficiently powered to detect model differences at reported levels. Adopting a straightforward generative model of annotations (Appendix A.5), we select enough crowdworkers per task to ensure sufficient statistical power (at least $1 - \\beta = 0 . 9 ,$ ) to obtain significance at $\\alpha = 0 . 0 5$ , resulting in a minimum of fifteen crowdworkers per topic for both tasks. On this criterion, both Chang et al. (2009) and thus Lau et al. (2014), with eight annotators, are underpowered. ",
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+        "type": "text",
+        "text": "For each of our two datasets, we generate fifty topics each from the three models in Section 4.2. In the word intrusion task, we sample five of the top ten topic words plus one intruder; for the ratings task, we present the top ten words in order (Figure 4). We separate the datasets for each task and randomly sample 40 of the 150 topics. In the ratings task, we include an additional sixteen synthetic poor-quality topics to help calibrate scores and filter out low-quality respondents.14 ",
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+        "img_path": "images/d4b91800d9e7b7ff7997c7159fa9a556a5d2a977b8f0c42eea33a90c0248c366.jpg",
+        "image_caption": [
+            "Figure 2: While automated evaluations (here, NPMI) suggest a clear winner between models, human evaluation is more nuanced. Human judgments exhibit greater variability over a smaller range of values. Colored circles correspond to pairwise one-tailed significance tests between model scores at $\\alpha = 0 . 0 5$ ; for example, the rightmost orange circle at bottom right shows that human intrusion ratings for D-VAE are significantly higher than ETM for topics derived from Wikipedia. "
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+        "type": "text",
+        "text": "Phrasing of questions closely follows the wording used by Chang et al. (2009), and crowdworkers received detailed instructions with examples (Appendix A.4) before responding to items.15 As topics can be esoteric (e.g., last columns of Table 1), we ask crowdworkers about their familiarity with the words in each question. We speculate that this question can help protect against spurious low scores for otherwise coherent topics, as real-world users of topic models are usually familiar with domain-specific terminology (see further discussion in Section 7). ",
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+        "type": "text",
+        "text": "6 Human Judgment Differs From Automated Metrics ",
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+        "type": "text",
+        "text": "We compare human judgments to automated methods on topics estimated using our three models. ",
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+        "text": "6.1 Human Assessment ",
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+        "type": "text",
+        "text": "To establish model differences using human ratings, we use pairwise significance tests: a proportion test for the intrusion scores, a $U$ test (Mann and Whitney, 1947) for the ratings, and a $t$ -test for automated metrics (Figure 2), using one-tailed tests for each pair in both directions. Although D-VAE fares better on the intrusion task, evaluation using ratings favors G-LDA.16 ",
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+        "type": "text",
+        "text": "Our human evaluation results are consistent with past iterations of the ratings and word intrusion tasks for topic models. Mimno et al. (2011) report an average of 2.36 on the ratings task on a dataset of medical paper abstracts.17 Our ratings means are 2.5 to 2.8 across all variations (Figure 2). Our word intrusion means range from 0.7 to 0.8, which is comparable to the roughly 0.8 accuracy on the LDA model evaluated in Chang et al. (2009). Median time taken on the tasks was 8–9 minutes. ",
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+        "table_caption": [
+            "Table 3: Spearman correlation coefficients between mean human scores and automated metrics. Underlined values have overlapping bootstrapped $9 5 \\%$ confidence intervals with that of the largest value in each row. “Concatenated” refers to correlations computed on a concatenation of values for the NYT and WIKI items. “Val” is a small held-out set of $15 \\%$ of the training corpus. Using the more data-appropriate logistic and ordered probit regressions for word intrusion and ratings data leads to different conclusions about relative metric strength (Appendix Table 10). CIs are estimated using 1,000 samples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Ref. Corpus→ Train Corpus↓</td><td colspan=\"4\">NPMI (10-token window)</td><td colspan=\"4\">Cu (110-token window)</td></tr><tr><td>NYT</td><td>WIKI</td><td>Train</td><td>Val</td><td>NYT</td><td>WIKI</td><td>Train</td><td>Val</td></tr><tr><td>Intrusion</td><td>NYT</td><td>0.27</td><td>0.43</td><td>0.27</td><td>0.24</td><td>0.34</td><td>0.45</td><td>0.35</td><td>0.34</td></tr><tr><td></td><td>WIKI</td><td>0.34</td><td>0.36</td><td>0.39</td><td>0.17</td><td>0.32</td><td>0.34</td><td>0.34</td><td>0.20</td></tr><tr><td></td><td>Concatenated</td><td>0.29</td><td>0.40</td><td>0.32</td><td>0.17</td><td>0.32</td><td>0.40</td><td>0.35</td><td>0.24</td></tr><tr><td>Rating</td><td>NYT</td><td>0.37</td><td>0.48</td><td>0.37</td><td>0.39</td><td>0.41</td><td>0.46</td><td>0.44</td><td>0.45</td></tr><tr><td></td><td>WIKI</td><td>0.34</td><td>0.41</td><td>0.44</td><td>0.28</td><td>0.32</td><td>0.40</td><td>0.40</td><td>0.34</td></tr><tr><td></td><td>Concatenated</td><td>0.37</td><td>0.44</td><td>0.41</td><td>0.35</td><td>0.38</td><td>0.42</td><td>0.42</td><td>0.42</td></tr></table>",
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+        "text": "Following Aletras and Stevenson (2013), we calculate inter-annotator agreement with the mean Spearman correlation between each respondent’s score per topic and the average of other respondent scores, obtaining a value of 0.75 (compare to their value of 0.7 on the NYT corpus). Additionally, we include synthetic poor-quality topics (footnote 14)—correctly identified by annotators—and we monitor the duration taken for the survey to hedge against insincere submissions. ",
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+        "type": "text",
+        "text": "6.2 Automated Metrics ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "NPMI declares D-VAE the unequivocal victor among the three models (with G-LDA a clear second), a very different story from the human judgments. To understand the relationship between automated metrics and human ratings, we estimate the Spearman correlation between the two sets of values for each task and dataset for metric variants (Table 3). Although previous studies have used mean human ratings over topics, this decision obscures the inherent variance of the human ratings and leads to overconfident estimates. We therefore construct $9 5 \\%$ confidence intervals by resampling ratings, with replacement, equal to the number of annotators per task (Table 3). We estimate NPMI with the standard 10-word window and $C _ { v }$ (Röder et al., 2015) with the recommended 110-word window.18 The Wikipedia corpus appears to be best correlated with human judgments, even for the models trained on the NYT corpus—this contradicts Lau et al. (2014), where within-domain data have the highest correlations. ",
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+        "type": "text",
+        "text": "While all correlation coefficients are statistically significant, the strength of the correlation alone does not justify their use in model selection, as is standard in the NTM literature (Section 3). In particular, the inherent uncertainty of human judgments means that it is difficult to determine when an increase in a model’s mean automated coherence implies a significant improvement in the corresponding human scores.19 ",
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+    {
+        "type": "text",
+        "text": "As noted above (Figure 2), automated metrics exaggerate model differences compared to human judgments. To help clarify the utility of automated metrics for model selection, we ask how often an automated metric incorrectly asserts that one model is superior to another. To do so, we generate a bootstrapped estimate of the false discovery rate of each model. First, for each dataset, we randomly sample two independent sets of $K = 5 0$ topics (without replacement) from the original pool of 150, along with their corresponding automated and human scores (resampled with replacement, as in Table 3). Treating the two sampled sets as outputs from two different models, we compute pairwise significance tests between each set for both the $K$ automated metrics and $K \\times M$ human scores (using a proportions $z$ -test for the intrusion scores and $t$ -tests for all other values). After repeating this process for $N = 1 0 0 0$ iterations, we report the proportion of significant differences detected using ",
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+        "type": "table",
+        "img_path": "images/8f7dda6e58503aa97e761ea9d08c64722af057daf158dda1d1115ceca6f020e5.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Ref. Corpus→ Train Corpus ↓</td><td colspan=\"3\">NPMI (10-token window)</td><td colspan=\"3\">Cu (110-token window)</td></tr><tr><td>NYT</td><td>WIKI</td><td>Train</td><td>NYT</td><td>WIKI</td><td>Train</td></tr><tr><td>Intrusion</td><td>NYT</td><td>46/53</td><td>34/48</td><td>48/50</td><td>35/38</td><td>30 /29</td><td>34/35</td></tr><tr><td rowspan=\"4\">Rating</td><td>WIKI</td><td>44/76</td><td>33/78</td><td>33/75</td><td>45/48</td><td>38/49</td><td>37 /45</td></tr><tr><td>Concatenated</td><td>42/67</td><td>40/66</td><td>41/64</td><td>36/46</td><td>31/44</td><td>30/45</td></tr><tr><td>NYT</td><td>45/50</td><td>45/51</td><td>41/47</td><td>27/29</td><td>26/26</td><td>21/26</td></tr><tr><td>WIKI</td><td>40/73</td><td>31/73</td><td>33/71</td><td>38/40</td><td>31/40</td><td>28 /34</td></tr><tr><td></td><td>Concatenated</td><td>39 /66</td><td>36/66</td><td>37/62</td><td>31/38</td><td>28/38</td><td>19 /36</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 4: False discovery rate (1−precision, lower is better) and false omission rate of significant model differences when using automated metrics; automated metrics often overstate meaningful model differences. Bolded values are those with the lowest geometric mean of FDR and FOR. We sample two independent sets of 50 topics along with their human scores and automated metrics; these sets act as the outputs of two “models”. We then compute significance tests between sets (per Figure 2) on both the automated scores and human scores. A false positive occurs when one set has significantly larger automated scores despite no meaningful difference in actual human scores. Estimates are over 1,000 samples. ",
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+        "img_path": "images/7c93a79f520ef9c7705d69318a21f9cbb9c645542c5025b7c7b1f9c5756e0d16.jpg",
+        "image_caption": [
+            "Figure 3: Mean human evaluation on the ratings and word intrusion tasks, after filtering out respondents who reported a lack of familiarity with the topic words. When filtering, D-VAE scores improve, highlighting its tendency to produce esoteric topics. "
+        ],
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+    {
+        "type": "text",
+        "text": "the predicted scores despite equivalent human scores (after correcting for the probability of type I errors, $\\alpha = 0 . 0 5$ ).20 Even the best-performing automated metrics predict significant differences absent a meaningful human effect roughly one-fifth of the time (Table 4). ",
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+    {
+        "type": "text",
+        "text": "These results suggest that automated metrics alone may be inadequate for model comparison. ",
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+    {
+        "type": "text",
+        "text": "6.3 Explaining the discrepancy ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "One reason for the discrepancy between human judgments and automated metrics is that metrics favor more esoteric topics. Specifically, there is a significant negative correlation between a topic’s NPMI or $C _ { v }$ and the share of respondents reporting familiarity with topic words (Pearson’s $\\rho = - 0 . 2 9 )$ . And while D-VAE achieves the highest automated metric scores of the three models, it produces topics with the fewest familiar words: respondents report familiarity with terms over $90 \\%$ of the time on both tasks for G-LDA and ETM, but they do so only $70 \\%$ of the time for D-VAE. This difference suggests that the topics selected by D-VAE are narrower in scope than those of the other models. As shown in Figure 3, removing item annotations where respondents indicate unfamiliarity causes both accuracy in the word intrusion task and the ratio of “Very related” terms in the ratings task for D-VAE to increase substantially. ",
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+    {
+        "type": "text",
+        "text": "Qualitatively, this result is apparent when examining topics with a high NPMI but low humans ratings. In Table 5, the top rows consists of financial terms that frequently appear together in NYT articles, ",
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+        "table_body": "<table><tr><td>Data</td><td>Model</td><td>Topic</td><td>NPMI</td><td>Rat.</td><td>Int.</td></tr><tr><td>NYT</td><td>D-VAE</td><td>inc 6mo earns otc rev qtr 9mo nyse outst dec</td><td>0.56</td><td>1.60</td><td>0.77</td></tr><tr><td>WIKI</td><td>D-VAE</td><td>Waterline conning turrets boilers amidships aft knots armament guns mounts</td><td>0.33</td><td>1.93</td><td>0.65</td></tr><tr><td>NYT</td><td>G-LDA</td><td>bedroom room bath taxes year market listed kitchen broker weeks</td><td>0.30</td><td>2.00</td><td>0.23</td></tr><tr><td>NYT</td><td>D-VAE</td><td>condolences mourns mourn board_of_directors heartfelt deepest esteemed</td><td>0.38</td><td>2.60</td><td>0.23</td></tr><tr><td>NYT</td><td>D-VAE</td><td>shareholders earnings federated mci shares takeover new_york_stock_exchange</td><td>0.18</td><td>3.00</td><td>0.81</td></tr><tr><td>WIKI</td><td>D-VAE</td><td>continental_army expedition militia frigate musket frigates muskets skirmish</td><td>0.11</td><td>3.00</td><td>0.69</td></tr><tr><td>NYT</td><td>D-VAE</td><td>medicaid medicare hospitals welfare uninsured patients</td><td>0.13</td><td>2.80</td><td>0.96</td></tr><tr><td>NYT</td><td>G-LDA</td><td>city mayor state new_york new_york_city officials county yesterday governor</td><td>0.09</td><td>2.53</td><td>1.00</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 5: Topics with the largest human–NPMI discrepancies; top half are topics where NPMI is high and human preferences are low, bottom half is the reverse. NPMI favors esoteric and corpus-specific topics. NPMI is calculated with a 10-token sliding window over the in-domain reference corpus, Rat. is the average 3-point rating for a topic, and Int. refers to the percentage of annotators who identify the intruder word. ",
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+    {
+        "type": "text",
+        "text": "and the second row contains rare terms about boating—arguably both are reasonable topics for their respective corpora. We can also see instances where words are qualitatively very related (bottom half of table), but that NPMI fails to score high—perhaps because these words, while related, may not frequently appear together within a ten-word sliding window (Equation 1). ",
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+    {
+        "type": "text",
+        "text": "Even for familiar words, some topics may be sensible in the context of the specific corpus, despite their component words lacking an immediately obvious semantic relationship. For example, the topic words in the third and fourth rows appear somewhat unrelated (e.g., “taxes” and “bedroom” in the third row), but they are in fact characteristic of common document types in the New York Times: real estate listings and obituaries. Topics like these render the word intrusion task more difficult: only $23 \\%$ of crowdworkers identified the intruder for both topics. ",
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+    {
+        "type": "text",
+        "text": "Furthermore, using term familiarity as a proxy for domain expertise does not address the key problems with topic model evaluation: even after filtering out respondents who are not familiar with topic terms, automated metrics still overstate model differences (Appendix A.7). The problems with topic model evaluation may therefore extend to our choice of human evaluations as well. ",
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+    {
+        "type": "text",
+        "text": "7 So. . . is Automated Topic Modeling Evaluation Broken? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To the extent that our experimentation accurately represents current practice, our results do suggest that topic model evaluation—both automated and human—is overdue for a careful reconsideration. In this, we agree with Doogan and Buntine (2021), who write that “coherence measures designed for older models [. . . ] may be incompatible with newer models” and instead argue for evaluation paradigms centered on corpus exploration and labeling. The right starting point for this reassessment is the recognition that both automated and human evaluations are abstractions of a real-world problem. The familiar use of precision-at-10 in information retrieval, for example, corresponds to a user who is only willing to consider the top ten retrieved documents. In future work, we intend to explore automated metrics that better approximate the preferences of real-world topic model users. ",
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+    {
+        "type": "text",
+        "text": "One primary use of topic models is in computer-assisted content analysis. In that context, rather than taking a methods-driven approach to evaluation, it would make sense to take a needs-driven approach.21 Generic evaluation of topic models using domain-general corpora like NYT needs to be revisited, since there is no such thing as a “generic” corpus for content analysis, nor a generic analyst. Content analysis can be formulated in a broad way, as Krippendorff (2004) has shown, but its actual application is always in a domain, by people familiar with that domain. This fact stands in tension with the desirable practicalities of general corpora and crowdworker annotation, and the field will need to address this tension. We have identified “coherence” as calling out a latent concept in the mind of a reader. It follows that we must think about who the relevant human readers are and the conceptual spaces that matter to them. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This material is based upon work supported by the National Science Foundation under Grants 2031736, 2008761, 1822494, ARLIS, and by an Amazon Research Award. We thank Sweta Agrawal for her suggestion to conduct a meta-analysis. We owe much appreciation to Dallas Card for his keen advice on power analyses. Thanks to Frank Fineis for help on several statistical questions, as well as Shuo Chen for his suggestions regarding the false discovery rate calculations. Finally, we thank Caitie Doogan for her helpful comments on the clarity of argumentation, as well as our anonymous reviewers. ",
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