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+# WEAKLY SUPERVISED DISENTANGLEMENTWITH GUARANTEES
+
+Rui $\mathbf { S h u } ^ { \dagger } \cdot$ ∗, Yining Chen†, Abhishek Kumar‡, Stefano Ermon†& Ben Poole‡
+
+†Stanford University, $^ \ddag$ Google Brain †{ruishu,cynnjjs,ermon}@stanford.edu ‡{abhishk,pooleb}@google.com
+
+# ABSTRACT
+
+Learning disentangled representations that correspond to factors of variation in real-world data is critical to interpretable and human-controllable machine learning. Recently, concerns about the viability of learning disentangled representations in a purely unsupervised manner has spurred a shift toward the incorporation of weak supervision. However, there is currently no formalism that identifies when and how weak supervision will guarantee disentanglement. To address this issue, we provide a theoretical framework to assist in analyzing the disentanglement guarantees (or lack thereof) conferred by weak supervision when coupled with learning algorithms based on distribution matching. We empirically verify the guarantees and limitations of several weak supervision methods (restricted labeling, match-pairing, and rank-pairing), demonstrating the predictive power and usefulness of our theoretical framework.
+
+# 1 INTRODUCTION
+
+Many real-world datasets can be intuitively described via a data-generating process that first samples an underlying set of interpretable factors, and then—conditional on those factors—generates an observed data point. For example, in image generation, one might first sample the object identity and pose, and then render an image with the object in the correct pose. The goal of disentangled representation learning is to learn a representation where each dimension of the representation corresponds to a distinct factor of variation in the dataset (Bengio et al., 2013). Learning such representations that align with the underlying factors of variation may be critical to the development of machine learning models that are explainable or human-controllable (Gilpin et al., 2018; Lee et al., 2019; Klys et al., 2018).
+
+In recent years, disentanglement research has focused on the learning of such representations in an unsupervised fashion, using only independent samples from the data distribution without access to the true factors of variation (Higgins et al., 2017; Chen et al., 2018a; Kim & Mnih, 2018; Esmaeili et al., 2018). However, Locatello et al. (2019) demonstrated that many existing methods for the unsupervised learning of disentangled representations are brittle, requiring careful supervision-based hyperparameter tuning. To build robust disentangled representation learning methods that do not require large amounts of supervised data, recent work has turned to forms of weak supervision (Chen & Batmanghelich, 2019; Gabbay & Hoshen, 2019). Weak supervision can allow one to build models that have interpretable representations even when human labeling is challenging (e.g., hair style in face generation, or style in music generation). While existing methods based on weaklysupervised learning demonstrate empirical gains, there is no existing formalism for describing the theoretical guarantees conferred by different forms of weak supervision (Kulkarni et al., 2015; Reed et al., 2015; Bouchacourt et al., 2018).
+
+In this paper, we present a comprehensive theoretical framework for weakly supervised disentanglement, and evaluate our framework on several datasets. Our contributions are several-fold.
+
+2. We propose a set of definitions for disentanglement that can handle correlated factors and are inspired by many existing definitions in the literature (Higgins et al., 2018; Suter et al., 2018; Ridgeway & Mozer, 2018).   
+3. Using these definitions, we provide a conceptually useful and theoretically rigorous calculus of disentanglement.   
+4. We apply our theoretical framework of disentanglement to analyze three notable classes of weak supervision methods (restricted labeling, match pairing, and rank pairing). We show that although certain weak supervision methods (e.g., style-labeling in style-content disentanglement) do not guarantee disentanglement, our calculus can determine whether disentanglement is guaranteed when multiple sources of weak supervision are combined.   
+5. Finally, we perform extensive experiments to systematically and empirically verify our predicted guarantees.1
+
+# 2 FROM UNSUPERVISED TO WEAKLY SUPERVISED DISTRIBUTION MATCHING
+
+Our goal in disentangled representation learning is to identify a latent-variable generative model whose latent variables correspond to ground truth factors of variation in the data. To identify the role that weak supervision plays in providing guarantees on disentanglement, we first formalize the model families we are considering, the forms of weak supervision, and finally the metrics we will use to evaluate and prove components of disentanglement.
+
+We consider data-generating processes where $S \in \mathbb { R } ^ { n }$ are the factors of variation, with distribution $p ^ { * } ( s )$ , and $X \in \mathbb { R } ^ { m }$ is the observed data point which is a deterministic function of $S$ , i.e., $X =$ $g ^ { * } ( S )$ . Many existing algorithms in unsupervised learning of disentangled representations aim to learn a latent-variable model with prior $p ( z )$ and generator $g$ , where $g ( Z ) \stackrel { d } { = } g ^ { * } ( S )$ . However, simply matching the marginal distribution over data is not enough: the learned latent variables $Z$ and the true generating factors $S$ could still be entangled with each other (Locatello et al., 2019).
+
+To address the failures of unsupervised learning of disentangled representations, we leverage weak supervision, where information about the data-generating process is conveyed through additional observations. By performing distribution matching on an augmented space (instead of just on the observation $X$ ), we can provide guarantees on learned representations.
+
+![](images/3bc732258f298fc18743f9cfe85e1617044cdd5ea0e0ee805dfae5303dc9cbee.jpg)  
+Figure 1: Augmented data distributions derived from weak supervision. Shaded nodes denote observed quantities, and unshaded nodes represent unobserved (latent) variables.
+
+We consider three practical forms of weak supervision: restricted labeling, match pairing, and rank pairing. All of these forms of supervision can be thought of as augmented forms of the original joint distribution, where we partition the latent variables in two $S = ( S _ { I } , S _ { \backslash I } )$ , and either observe a subset of the latent variables or share latents between multiple samples. A visualization of these augmented distributions is presented in Figure 1, and below we detail each form of weak supervision.
+
+In restricted labeling, we observe a subset of the ground truth factors, $S _ { I }$ in addition to $X$ . This allows us to perform distribution matching on $p ^ { * } ( s _ { I } , x )$ , the joint distribution over data and observed factors, instead of just the data, $p ^ { * } ( x )$ , as in unsupervised learning. This form of supervision is often leveraged in style-content disentanglement, where labels are available for content but not style (Kingma et al., 2014; Narayanaswamy et al., 2017; Chen et al., 2018b; Gabbay & Hoshen, 2019).
+
+Match Pairing uses paired data, $( x , x ^ { \prime } )$ that share values for a known subset of factors, $I$ . For many data modalities, factors of variation may be difficult to explicitly label. Instead, it may be easier to collect pairs of samples that share the same underlying factor (e.g., collecting pairs of images of different people wearing the same glasses is easier than defining labels for style of glasses). Match pairing is a weaker form of supervision than restricted labeling, as the learning algorithm no longer depends on the underlying value $s _ { I }$ , and only on the indices of shared factors $I$ . Several variants of match pairing have appeared in the literature (Kulkarni et al., 2015; Bouchacourt et al., 2018; Ridgeway & Mozer, 2018), but typically focus on groups of observations in contrast to the paired setting we consider in this paper.
+
+Rank Pairing is another form of paired data generation where the pairs $( x , x ^ { \prime } )$ are generated in an i.i.d. fashion, and an additional indicator variable $y$ is observed that determines whether the corresponding latent $s _ { i }$ is greater than $s _ { i } ^ { \prime }$ : $y = \mathbf { 1 } \left\{ s _ { i } \geq s _ { i } ^ { \prime } \right\}$ . Such a form of supervision is effective when it is easier to compare two samples with respect to an underlying factor than to directly collect labels (e.g., comparing two object sizes versus providing a ruler measurement of an object). Although supervision via ranking features prominently in the metric learning literature (McFee & Lanckriet, 2010; Wang et al., 2014), our focus in this paper will be on rank pairing in the context of disentanglement guarantees.
+
+For each form of weak supervision, we can train generative models with the same structure as in Figure 1, using data sampled from the ground truth model and a distribution matching objective. For example, for match pairing, we train a generative model $( p ( z ) , g )$ such that the paired random variable $( \bar { g } ( Z _ { I } , Z _ { \backslash I } ) , g ( \bar { Z } _ { I } , Z _ { \backslash I } ^ { \bar { \prime } } ) )$ from the generator matches the distribution of the corresponding paired random variable $( g ^ { * } ( { \cal S } _ { I } , { \cal S } _ { \setminus I } ) , g ^ { * } ( { \cal S } _ { I } , { \cal S } _ { \setminus I } ^ { \prime } ) )$ from the augmented data distribution.
+
+# 3 DEFINING DISENTANGLEMENT
+
+To identify the role that weak supervision plays in providing guarantees on disentanglement, we introduce a set of definitions that are consistent with our intuitions about what constitutes “disentanglement” and amenable to theoretical analysis. Our new definitions decompose disentanglement into two distinct concepts: consistency and restrictiveness. Different forms of weak supervision can enable consistency or restrictiveness on subsets of factors, and in Section 4 we build up a calculus of disentanglement from these primitives. We discuss the relationship to prior definitions of disentanglement in Appendix A.
+
+![](images/e997020c65d700a0ac576b16c110e7f4be88dfe3f0000e1cea1c02234869390f.jpg)  
+3.1 DECOMPOSING DISENTANGLEMENT INTO CONSISTENCY AND RESTRICTIVENESS   
+Figure 2: Illustration of disentanglement, consistency, and restrictiveness of $z _ { 1 }$ with respect to the factor of variation size. Each image of a shape represents the decoding $g ( z _ { 1 : 3 } )$ by the generative model. Each column denotes a fixed choice of $z _ { 1 }$ . Each row denotes a fixed choice of $\left( z _ { 2 } , z _ { 3 } \right)$ . A demonstration of consistency versus restrictiveness on models from disentanglement lib is available in Appendix B.
+
+To ground our discussion of disentanglement, we consider an oracle that generates shapes with factors of variation for size $( S _ { 1 } )$ , shape $( S _ { 2 } )$ , and color $( S _ { 3 } )$ . How can we determine whether $Z _ { 1 }$ of our generative model “disentangles” the concept of size? Intuitively, one way to check whether $Z _ { 1 }$ of the generative model disentangles size $( S _ { 1 } )$ is to visually inspect what happens as we vary $Z _ { 1 }$ , $Z _ { 2 }$ , and $Z _ { 3 }$ , and see whether the resulting visualizations are consistent with Figure 2a. In doing so, our visual inspection checks for two properties:
+
+1. When $Z _ { 1 }$ is fixed, the size $( S _ { 1 } )$ of the generated object never changes.
+
+2. When only $Z _ { 1 }$ is changed, the change is restricted to the size $( S _ { 1 } )$ of the generated object, meaning that there is no change in $S _ { j }$ for $j \neq 1$ .
+
+We argue that disentanglement decomposes into these two properties, which we refer to as generator consistency and generator restrictiveness. Next, we formalize these two properties.
+
+Let $\mathcal { H }$ be a hypothesis class of generative models from which we assume the true data-generating function is drawn. Each element of the hypothesis class $\mathcal { H }$ is a tuple $( p ( s ) , g , e )$ , where $p ( s )$ describes the distribution over factors of variation, the generator $g$ is a function that maps from the factor space $S \in \mathbb { R } ^ { n }$ to the observation space $\mathcal { X } \in \mathbb { R } ^ { m }$ , and the encoder $e$ is a function that maps from ${ \mathcal { X } } \to S$ . $S$ and $X$ can consist of both discrete and continuous random variables. We impose a few mild assumptions on $\mathcal { H }$ (see Appendix I.1). Notably, we assume every factor of variation is exactly recoverable from the observation $X$ , i.e. $e ( g ( S ) ) = { \dot { S } }$ .
+
+Given an oracle model $h ^ { \ast } = ( p ^ { \ast } , g ^ { \ast } , e ^ { \ast } ) \in \mathcal { H }$ , we would like to learn a model $h = ( p , g , e ) \in \mathcal { H }$ whose latent variables disentangle the latent variables in $h ^ { * }$ . We refer to the latent-variables in the oracle $h ^ { * }$ as $S$ and the alternative model $h$ ’s latent variables as $Z$ . If we further restrict $h$ to only those models where $g ( Z ) \stackrel { d } { = } g ^ { * } ( S )$ are equal in distribution, it is natural to align $Z$ and $S$ via $S = e ^ { * } \circ g ( Z )$ . Under this relation between $Z$ and $S$ , our goal is to construct definitions that describe whether the latent code $Z _ { i }$ disentangles the corresponding factor $S _ { i }$ .
+
+Generator Consistency. Let $I$ denote a set of indices and $p _ { I }$ denote the generating process
+
+$$
+\begin{array} { c } { { z _ { I } \sim p ( z _ { I } ) } } \\ { { z _ { \backslash I } , z _ { \backslash I } ^ { \prime } \overset { \mathrm { i i d } } { \sim } p ( z _ { \backslash I } \mid z _ { I } ) . } } \end{array}
+$$
+
+This generating process samples $Z _ { I }$ once and then conditionally samples $Z _ { I }$ twice in an i.i.d. fashion. We say that $Z _ { I }$ is consistent with $S _ { I }$ if
+
+$$
+\begin{array} { r } { \mathbb { E } _ { p _ { I } } \| e _ { I } ^ { * } \circ g ( z _ { I } , z _ { \setminus I } ) - e _ { I } ^ { * } \circ g ( z _ { I } , z _ { \setminus I } ^ { \prime } ) \| ^ { 2 } = 0 , } \end{array}
+$$
+
+where $e _ { I } ^ { * }$ is the oracle encoder restricted to the indices $I$
+
+Intuitively, Equation (3) states that, for any fixed choice of $Z _ { I }$ , resampling of $Z _ { \backslash I }$ will not influence the oracle’s measurement of the factors $S _ { I }$ . In other words, $S _ { I }$ is invariant to changes in $Z _ { \backslash I }$ . An illustration of a generative model where $Z _ { 1 }$ is consistent with size $( S _ { 1 } )$ is provided in Figure 2b. A notable property of our definition is that the prescribed sampling process $p _ { I }$ does not require the underlying factors of variation to be statistically independent. We characterize this property in contrast to previous definitions of disentanglement in Appendix A.
+
+Generator Restrictiveness. Let $p _ { \setminus I }$ denote the generating process
+
+$$
+\begin{array} { c } { { z _ { \backslash I } \sim p ( z _ { \backslash I } ) } } \\ { { z _ { I } , z _ { I } ^ { \prime } \overset { \mathrm { i i d } } { \sim } p ( z _ { I } \mid z _ { \backslash I } ) . } } \end{array}
+$$
+
+We say that $Z _ { I }$ is restricted to $S _ { I }$ if
+
+$$
+\begin{array} { r } { \mathbb { E } _ { p \setminus I } \| e \setminus ^ { * } \circ g ( z _ { I } , z _ { \setminus I } ) - e _ { \setminus I } ^ { * } \circ g ( z _ { I } ^ { \prime } , z _ { \setminus I } ) \| ^ { 2 } = 0 . } \end{array}
+$$
+
+Equation (6) states that, for any fixed choice of $Z _ { \backslash I }$ , resampling of $Z _ { I }$ will not influence the oracle’s measurement of the factors $S _ { \backslash I }$ . In other words, $\operatorname { \dot { \cal S } } _ { \lfloor I \rfloor }$ is invariant to changes in $Z _ { I }$ . Thus, changing $Z _ { I }$ is restricted to modifying only $S _ { I }$ . An illustration of a generative model where $Z _ { 1 }$ is restricted to size $( S _ { 1 } )$ is provided in Figure 2c.
+
+Generator Disentanglement. We now say that $Z _ { I }$ disentangles $S _ { I }$ if $Z _ { I }$ is consistent with and restricted to $S _ { I }$ . If we denote consistency and restrictiveness via Boolean functions $C ( I )$ and $R ( I )$ , we can now concisely state that
+
+$$
+D ( I ) : = C ( I ) \land R ( I ) ,
+$$
+
+where $D ( I )$ denotes whether $Z _ { I }$ disentangles $S _ { I }$ . An illustration of a generative model where $Z _ { 1 }$ disentangles size $( S _ { 1 } )$ is provided in Figure 2a. Note that while size increases monotonically with $Z _ { 1 }$ in the schematic figure, we wish to clarify that monotonicity is unrelated to the concepts of consistency and restrictiveness.
+
+# 3.2 RELATION TO BIJECTIVITY-BASED DEFINITION OF DISENTANGLEMENT
+
+Under our mild assumptions on $\mathcal { H }$ , distribution matching on $g ( Z ) \stackrel { d } { = } g ( S )$ combined with generator disentanglement on factor $I$ implies the existence of two invertible functions $f _ { I }$ and $f _ { \backslash I }$ such that the alignment via $S = e ^ { * } \circ g ( Z )$ decomposes into
+
+$$
+\biggl [ S _ { I } \biggr ] = \biggl [ f _ { I } \bigl ( Z _ { I } \bigr ) \biggr ] .
+$$
+
+This expression highlights the connection between disentanglement and invariance, whereby $S _ { I }$ is only influenced by $Z _ { I }$ , and $S _ { \backslash I }$ is only influenced by $Z _ { \backslash I }$ . However, such a bijectivity-based definition of disentanglement does not naturally expose the underlying primitives of consistency and restrictiveness, which we shall demonstrate in our theory and experiments to be valuable concepts for describing disentanglement guarantees under weak supervision.
+
+# 3.3 ENCODER-BASED DEFINITIONS FOR DISENTANGLEMENT
+
+Our proposed definitions are asymmetric—measuring the behavior of a generative model against an oracle encoder. So far, we have chosen to present the definitions from the perspective of a learned generator $( p , g )$ measured against an oracle encoder $e ^ { * }$ . In this sense, they are generator-based definitions. We can also develop a parallel set of definitions for encoder-based consistency, restrictiveness, and disentanglement within our framework simply by using an oracle generator $( p ^ { * } , g ^ { * } )$ measured against a learned encoder $e$ . Below, we present the encoder-based perspective on consistency.
+
+Encoder Consistency. Let $p _ { I } ^ { * }$ denote the generating process
+
+$$
+\begin{array} { c } { { s _ { I } \sim p ^ { * } ( s _ { I } ) } } \\ { { s _ { \backslash I } , s _ { \backslash I } ^ { \prime } \stackrel { \mathrm { i i d } } { \sim } p ^ { * } ( s _ { \backslash I } , \mid s _ { I } ) . } } \end{array}
+$$
+
+This generating process samples $S _ { I }$ once and then conditionally samples $S _ { I }$ twice in an i.i.d. fashion. We say that $S _ { I }$ is consistent with $Z _ { I }$ if
+
+$$
+\begin{array} { r } { \mathbb { E } _ { p _ { I } ^ { * } } \| e _ { I } \circ g ^ { * } ( s _ { I } , s _ { \setminus I } ) - e _ { I } \circ g ^ { * } ( s _ { I } , s _ { \setminus I } ^ { \prime } ) \| ^ { 2 } = 0 . } \end{array}
+$$
+
+We now make two important observations. First, a valuable trait of our encoder-based definitions is that one can check for encoder consistency $/$ restrictiveness $/$ disentanglement as long as one has access to match pairing data from the oracle generator. This is in contrast to the existing disentanglement definitions and metrics, which require access to the ground truth factors (Higgins et al., 2017; Kumar et al., 2018; Kim & Mnih, 2018; Chen et al., 2018a; Suter et al., 2018; Ridgeway & Mozer, 2018; Eastwood & Williams, 2018). The ability to check for our definitions in a weakly supervised fashion is the key to why we can develop a theoretical framework using the language of consistency and restrictiveness. Second, encoder-based definitions are tractable to measure when testing on synthetic data, since the synthetic data directly serves the role of the oracle generator. As such, while we develop our theory to guarantee both generator-based and the encoder-based disentanglement, all of our measurements in the experiments will be conducted with respect to a learned encoder.
+
+We make three remarks on notations. First, $D ( i ) : = D ( \{ i \} )$ . Second, $D ( \emptyset )$ evaluates to true. Finally, $D ( I )$ is implicitly dependent on either $( p , g , e ^ { * } )$ (generator-based) or $( p ^ { * } , g ^ { * } , e )$ (encoderbased). Where important, we shall make this dependency explicit (e.g., let $D ( I ; p , g , e ^ { * } )$ denote generator-based disentanglement). We apply these conventions to $C$ and $R$ analogously.
+
+# 4 A CALCULUS OF DISENTANGLEMENT
+
+There are several interesting relationships between restrictiveness and consistency. First, by definition, $C ( I )$ is equivalent to $\bar { \boldsymbol { R } } ( \backslash I )$ . Second, we can see from Figures 2b and $2 \mathrm { c }$ that $C ( I )$ and $R ( I )$ do not imply each other. Based on these observations and given that consistency and restrictiveness operate over subsets of the random variables, a natural question that arises is whether consistency or restrictiveness over certain sets of variables imply additional properties over other sets of variables.
+
+We develop a calculus for discovering implied relationships between learned latent variables $Z$ and ground truth factors of variation $S$ given known relationships as follows.
+
+<table><tr><td>Calculus of Disentanglement</td></tr><tr><td>Consistency and Restrictiveness C(I)  R(I) R(I)  C(I) C(I) ←→ R(\I)</td></tr><tr><td>Union Rules</td></tr><tr><td>C(i)∧C(J) =&gt; C(Iu J) R(I) ^ R(J) =→ R(I U J) Intersection Rules</td></tr><tr><td>C(I) ^ C(J) =→ C(I n J) R(I) ∧ R(J) =→ R(I n J)</td></tr><tr><td>Full Disentanglement</td></tr><tr><td>∧=1C(i） →A²=1D(i） ∧=1 R(i) →N=1D(i)</td></tr></table>
+
+Our calculus provides a theoretically rigorous procedure for reasoning about disentanglement. In particular, it is no longer necessary to prove whether the supervision method of interest satisfies consistency and restrictiveness for each and every factor. Instead, it suffices to show that a supervision method guarantees consistency or restrictiveness for a subset of factors, and then combine multiple supervision methods via the calculus to guarantee full disentanglement. We can additionally use the calculus to uncover consistency or restrictiveness on individual factors when weak supervision is available only for a subset of variables. For example, achieving consistency on $S _ { 1 , 2 }$ and $S _ { 2 , 3 }$ implies consistency on the intersection $S _ { 2 }$ . Furthermore, we note that these rules are agnostic to using generator or encoder-based definitions. We defer the complete proofs to Appendix I.2.
+
+# 5 FORMALIZING WEAK SUPERVISION WITH GUARANTEES
+
+In this section, we address the question of whether disentanglement arises from the supervision method or model inductive bias. This challenge was first put forth by Locatello et al. (2019), who noted that unsupervised disentanglement is heavily reliant on model inductive bias. As we transition toward supervised approaches, it is crucial that we formalize what it means for disentanglement to be guaranteed by weak supervision.
+
+Sufficiency for Disentanglement. Let $\mathcal { P }$ denote a family of augmented distributions. We say that a weak supervision method $\mathbf { S } : \mathcal { H } \to \mathcal { P }$ is sufficient for learning a generator whose latent codes $Z _ { I }$ disentangle the factors $S _ { I }$ if there exists a learning algorithm $\mathcal { A } : \mathcal { P }  \mathcal { H }$ such that for any choice of $( p ^ { * } ( s ) , \bar { g ^ { * } } , e ^ { * } ) \in \mathcal { H }$ , the procedure $\mathcal { A } \circ \mathbf { S } ( p ^ { \ast } ( s ) , \mathbf { \bar { { g } } } ^ { \ast } , \mathbf { \bar { { e } } } ^ { \ast } )$ returns a model $( p ( z ) , g , e )$ for which both $D ( I ; p , g , e ^ { * } )$ and $D ( I ; p ^ { * } , g ^ { * } , e )$ hold, and $g ( Z ) \stackrel { d } { = } g ^ { * } ( S )$ .
+
+The key insight of this definition is that we force the strategy and learning algorithm pair $( \mathbf { S } , { \mathcal { A } } )$ to handle all possible oracles drawn from the hypothesis class $\mathcal { H }$ . This prevents the exploitation of model inductive bias, since any bias from the learning algorithm $\mathcal { A }$ toward a reduced hypothesis class $\hat { \mathcal { H } } \subset \mathcal { H }$ will result in failure to handle oracles in the complementary hypothesis class $\mathcal { \bar { H } } \backslash \mathcal { \hat { H } }$ .
+
+The distribution matching requirement $g ( Z ) \stackrel { d } { = } g ^ { * } ( S )$ ensures latent code informativeness, i.e., preventing trivial solutions where the latent code is uninformative (see Proposition 6 for formal statement). Intuitively, distribution matching paired with a deterministic generator guarantees invertibility of the learned generator and encoder, enforcing that $Z _ { I }$ cannot encode less information than $S _ { I }$ (e.g., only encoding age group instead of numerical age) and vice versa.
+
+# 6 ANALYSIS OF WEAK SUPERVISION METHODS
+
+We now apply our theoretical framework to three practical weak supervision methods: restricted labeling, match pairing, and rank pairing. Our main theoretical findings are that: (1) these methods can be applied in a targeted manner to provide single factor consistency or restrictiveness guarantees; (2) by enforcing consistency (or restrictiveness) on all factors, we can learn models with strong disentanglement performance. Correspondingly, Figure 3 and Figure 5 are our main experimental results, demonstrating that these theoretical guarantees have predictive power in practice.
+
+# 6.1 THEORETICAL GUARANTEES FROM WEAK SUPERVISION
+
+We prove that if a training algorithm successfully matches the generated distribution to data distribution generated via restricted labeling, match pairing, or rank pairing of factors $S _ { I }$ , then $Z _ { I }$ is guaranteed to be consistent with $S _ { I }$ :
+
+Theorem 1. Given any oracle $( p ^ { \ast } ( s ) , g ^ { \ast } , e ^ { \ast } ) \in \mathcal { H }$ , consider the distribution-matching algorithm $\mathcal { A }$ that selects a model $( p ( z ) , g , e ) \in \mathcal { H }$ such that:
+
+1. $( g ^ { * } ( S ) , S _ { I } ) \overset { d } { = } ( g ( Z ) , Z _ { I } )$ (Restricted Labeling); or   
+2. $\left( g ^ { \ast } ( S _ { I } , S _ { \setminus I } ) , g ^ { \ast } ( S _ { I } , S _ { \setminus I } ^ { \prime } ) \right) \stackrel { d } { = } \left( g ( Z _ { I } , Z _ { \setminus I } ) , g ( Z _ { I } , Z _ { \setminus I } ^ { \prime } ) \right) ( M a t c h P a i r i n g ) ; o r$   
+3. $\begin{array} { r } { ( g ^ { * } ( S ) , g ^ { * } ( S ^ { \prime } ) , \mathbf { 1 } \left\{ S _ { I } \leq S _ { I } ^ { \prime } \right\} ) \stackrel { d } { = } ( g ( Z ) , g ( Z ^ { \prime } ) , \mathbf { 1 } \left\{ Z _ { I } \leq Z _ { I } ^ { \prime } \right\} ) ( R a n k ~ P a i r i n g ) . } \end{array}$
+
+Then $( p , g )$ satisfies $C ( I ; p , g , e ^ { * } )$ and $e$ satisfies $C ( I ; p ^ { * } , g ^ { * } , e )$ .
+
+Theorem 1 states that distribution-matching under restricted labeling, match pairing, or rank pairing of $S _ { I }$ guarantees both generator and encoder consistency for the learned generator and encoder respectively. We note that while the complement rule $C ( I ) \implies R ( \backslash I )$ further guarantees that $Z _ { \backslash I }$ is restricted to $S _ { \backslash I }$ , we can prove that the same supervision does not guarantee that $Z _ { I }$ is restricted to $S _ { I }$ (Theorem 2). However, if we additionally have restricted labeling for $S _ { \backslash I }$ , or match pairing for $S _ { \backslash I }$ , then we can see from the calculus that we will have guaranteed $R ( I ) \land { \dot { C } } ( I )$ , thus implying disentanglement of factor $I$ . We also note that while restricted labeling and match pairing can be applied on a set of factors at once (i.e. $| I | \geq 1 )$ , rank pairing is restricted to one-dimensional factors for which an ordering exists. In the experiments below, we empirically verify the theoretical guarantees provided in Theorem 1.
+
+# 6.2 EXPERIMENTS
+
+We conducted experiments on five prominent datasets in the disentanglement literature: Shapes3D (Kim & Mnih, 2018), dSprites (Higgins et al., 2017), Scream-dSprites (Locatello et al., 2019), SmallNORB (LeCun et al., 2004), and Cars3D (Reed et al., 2015). Since some of the underlying factors are treated as nuisance variables in SmallNORB and Scream-dSprites, we show in Appendix C that our theoretical framework can be easily adapted accordingly to handle such situations. We use generative adversarial networks (GANs, Goodfellow et al. (2014)) for learning $( p , g )$ but any distribution matching algorithm (e.g., maximum likelihood training in tractable models, or VI in latent-variable models) could be applied. Our results are collected over a broad range of hyperparameter configurations (see Appendix H for details).
+
+Since existing quantitative metrics of disentanglement all measure the performance of an encoder with respect to the true data generator, we trained an encoder post-hoc to approximately invert the learned generator, and measured all quantitative metrics (e.g., mutual information gap) on the encoder. Our theory assumes that the learned generator must be invertible. While this is not true for conventional GANs, our empirical results show that this is not an issue in practice (see Appendix G).
+
+We present three sets of experimental results: (1) Single-factor experiments, where we show that our theory can be applied in a targeted fashion to guarantee consistency or restrictiveness of a single factor. (2) Consistency versus restrictiveness experiments, where we show the extent to which single-factor consistency and restrictiveness are correlated even when the models are only trained to maximize one or the other. (3) Full disentanglement experiments, where we apply our theory to fully disentangle all factors. A more extensive set of experiments can be found in the Appendix.
+
+# 6.2.1 SINGLE-FACTOR CONSISTENCY AND RESTRICTIVENESS
+
+We empirically verify that single-factor consistency or restrictiveness can be achieved with the supervision methods of interest. Note there are two special cases of match pairing: one where $S _ { i }$ is the only factor that is shared between $x$ and $x ^ { \prime }$ and one where $S _ { i }$ is the only factor that is changed. We distinguish these two conditions as share pairing and change pairing, respectively. Theorem 1 shows that restricted labeling, share pairing, and rank pairing of the $i ^ { \mathrm { { t h } } }$ factor are each sufficient supervision strategies for guaranteeing consistency on $S _ { i }$ . Change pairing at $S _ { i }$ is equivalent to share pairing at $S _ { \backslash i }$ ; the complement rule $C ( I ) \Longleftrightarrow R ( \backslash I )$ allows us to conclude that change pairing guarantees restrictiveness. The first four heatmaps in Figure 3 show the results for restricted labeling, share pairing, change pairing, and rank pairing. The numbers shown in the heatmap are the normalized consistency and restrictiveness scores. We define the normalized consistency score as
+
+![](images/21d297e22f0f61e437df048c5c87528bf8bdddbd14ebbb34addb4348f71b27c9.jpg)  
+Figure 3: Heatmap visualization of ablation studies that measure either single-factor consistency or single-factor restrictiveness as a function of various supervision methods, conducted on Shapes3D. Our theory predicts the diagonal components to achieve the highest scores. Note that share pairing, change pairing, and change pair intersection are special cases of match pairing.
+
+$$
+\widetilde { c } ( I ; p ^ { * } , g ^ { * } , e ) = 1 - \frac { \mathbb { E } _ { p _ { I } ^ { * } } \| e _ { I } \circ g ^ { * } ( s _ { I } , s _ { \setminus I } ) - e _ { I } \circ g ^ { * } ( s _ { I } , s _ { \setminus I } ^ { \prime } ) \| ^ { 2 } } { \mathbb { E } _ { s , s ^ { \prime } \sim p ^ { * } } \| e _ { I } \circ g ^ { * } ( s ) - e _ { I } \circ g ^ { * } ( s ^ { \prime } ) \| ^ { 2 } } .
+$$
+
+This score is bounded on the interval $[ 0 , 1 ]$ (a consequence of Lemma 1) and is maximal when $C ( I : p ^ { * } , g ^ { * } , e )$ is satisfied. This normalization procedure is similar in spirit to the Interventional Robustness score in Suter et al. (2018). The normalized restrictiveness score $\tilde { r }$ can be analogously defined. In practice, we estimate this score via Monte Carlo estimation.
+
+The final heatmap in Figure 3 demonstrates the calculus of intersection. In practice, it may be easier to acquire paired data where multiple factors change simultaneously. If we have access to two kinds of datasets, one where $S _ { I }$ are changed and one where $S _ { J }$ are changed, our calculus predicts that training on both datasets will guarantee restrictiveness on $S _ { I \cap J }$ . The final heatmap shows six such intersection settings and measures the normalized restrictiveness score; in all but one setting, the results are consistent with our theory. We show in Figure 7 that this inconsistency is attributable to the failure of the GAN to distribution-match due to sensitivity to a specific hyperparameter.
+
+# 6.2.2 CONSISTENCY VERSUS RESTRICTIVENESS
+
+![](images/e1ef17f15ccc98853df5b85791a305d59b0905e1c3c5011306b5c61cec87594b.jpg)  
+Figure 4: Correlation plot and scatterplots demonstrating the empirical relationship between $\tilde { c } ( i )$ and $\tilde { r } ( i )$ across all 864 models trained on Shapes3D.
+
+We now determine the extent to which consistency and restrictiveness are correlated in practice. In Figure 4, we collected all 864 Shapes3D models that we trained in Section 6.2.1 and measured the consistency and restrictiveness of each model on each factor, providing both the correlation plot and scatterplots of $\tilde { c } ( i )$ versus $\tilde { r } ( i )$ . Since the models trained in Section 6.2.1 only ever targeted the consistency or restrictiveness of a single factor, and since our calculus demonstrates that consistency and restrictiveness do not imply each other, one might a priori expect to find no correlation in Figure 4. Our results show that the correlation is actually quite strong. Since this correlation is not guaranteed by our choice of weak supervision, it is necessarily a consequence of model inductive bias. We believe this correlation between consistency and restrictiveness to have been a general source of confusion in the disentanglement literature, causing many to either observe or believe that restricted labeling or share pairing on $S _ { i }$ (which only guarantees consistency) is sufficient for disentangling $S _ { i }$ (Kingma et al., 2014; Chen & Batmanghelich, 2019; Gabbay & Hoshen, 2019; Narayanaswamy et al., 2017). It remains an open question why consistency and restrictiveness are so strongly correlated when training existing models on real-world data.
+
+# 6.2.3 FULL DISENTANGLEMENT
+
+![](images/0a5e29532a19f3efca5e4de06ea9851e17b17a86b6f0367d3ccfe9f819a974ef.jpg)  
+Figure 5: Disentanglement performance of a vanilla GAN, share pairing GAN, change pairing GAN, rank pairing GAN, and fully-labeled GAN, as measured by the mutual information gap across several datasets. A comprehensive set of performance evaluations on existing disentanglement metrics is available in Figure 13.
+
+If we have access to share / change / rank-pairing data for each factor, our calculus states that it is possible to guarantee full disentanglement. We trained our generative model on either complete share pairing, complete change pairing, or complete rank pairing, and measured disentanglement performance via the discretized mutual information gap (Chen et al., 2018a; Locatello et al., 2019). As negative and positive controls, we also show the performance of an unsupervised GAN and a fully-supervised GAN where the latents are fixed to the ground truth factors of variation. Our results in Figure 5 empirically verify that combining single-factor weak supervision datasets leads to consistently high disentanglement scores.
+
+# 7 CONCLUSION
+
+In this work, we construct a theoretical framework to rigorously analyze the disentanglement guarantees of weak supervision algorithms. Our paper clarifies several important concepts, such as consistency and restrictiveness, that have been hitherto confused or overlooked in the existing literature, and provides a formalism that precisely distinguishes when disentanglement arises from supervision versus model inductive bias. Through our theory and a comprehensive set of experiments, we demonstrated the conditions under which various supervision strategies guarantee disentanglement. Our work establishes several promising directions for future research. First, we hope that our formalism and experiments inspire greater theoretical and scientific scrutiny of the inductive biases present in existing models. Second, we encourage the search for other learning algorithms (besides distribution-matching) that may have theoretical guarantees when paired with the right form of supervision. Finally, we hope that our framework enables the theoretical analysis of other promising weak supervision methods.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank James Brofos and Honglin Yuan for their insightful discussions on the theoretical analysis in this paper, and Aditya Grover and Hung H. Bui for their helpful feedback during the course of this project.
+
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+
+# APPENDIX
+
+Our appendix consists of nine sections. We provide a brief summary of each section below.
+
+Appendix A: We elaborate on the connections between existing definitions of disentanglement and our definitions of consistency / restrictiveness / disentanglement. In particular, we highlight three notable properties of our definitions not present in many existing definitions.
+
+Appendix B: We evaluate our consistency and restrictiveness metrics on the 10800 models in the disentanglement lib, and identify models where consistency and restrictiveness are not correlated.
+
+Appendix C: We adapt our definitions to be able to handle nuisance variables. We do so through a simple modification of the definition of restrictiveness.
+
+Appendix D: We show several additional single-factor experiments. We first address one of the results in the main text that is not consistent with our theory, and explain why it can be attributed to hyperparameter sensitivity. We next unwrap the heatmaps into more informative boxplots.
+
+Appendix E: We provide an additional suite of consistency versus restrictiveness experiments by comparing the effects of training with share pairing (which guarantees consistency), change pairing (which guarantees restrictiveness), and both.
+
+Appendix F: We provide full disentanglement results on all five datasets as measured according to six different metrics of disentanglement found in the literature.
+
+Appendix G: We show visualizations of a weakly supervised generative model trained to achieve full disentanglement.
+
+Appendix H: We describe the set of hyperparameter configurations used in all our experiments.
+
+Appendix I: We provide the complete set of assumptions and proofs for our theoretical framework.
+
+# A CONNECTIONS TO EXISTING DEFINITIONS
+
+Numerous definitions of disentanglement are present in the literature (Higgins et al., 2017; 2018; Kim & Mnih, 2018; Suter et al., 2018; Ridgeway & Mozer, 2018; Eastwood & Williams, 2018; Chen et al., 2018a). We mostly defer to the terminology suggested by Ridgeway & Mozer (2018), which decomposes disentanglement into modularity, compactness, and explicitness. Modularity means a latent code $Z _ { i }$ is predictive of at most one factor of variation $S _ { j }$ . Compactness means a factor of variation $S _ { i }$ is predicted by at most one latent code $Z _ { j }$ . And explicitness means a factor of variation $S _ { j }$ is predicted by the latent codes via a simple transformation (e.g. linear). Similar to Eastwood & Williams (2018); Higgins et al. (2018), we suggest a further decomposition of Ridgeway & Mozer (2018)’s explicitness into latent code informativeness and latent code simplicity. In this paper, we omit latent code simplicity from consideration. Since informativeness of the latent code is already enforced by our requirement that $g ( Z )$ is equal in distribution to $g ^ { * } ( S )$ (see Proposition 6), we focus on comparing our proposed concepts of consistency and restrictiveness to modularity and compactness. We make note of three important distinctions.
+
+Restrictiveness is not synonymous with either modularity or compactness. In Figure 2c, it is evident the factor of variation size is not predictable any individual $Z _ { i }$ (conversely, $Z _ { 1 }$ is not predictable from any individual factor $S _ { i }$ ). As such, $Z _ { 1 }$ is neither a modular nor compact representation of size, despite being restricted to size. To our knowledge, no existing quantitative definition of disentanglement (or its decomposition) specifically measures restrictiveness.
+
+Consistency and restrictiveness are invariant to statistically dependent factors of variation. Many existing definitions of disentanglement are instantiated by measuring the mutual information between $Z$ and $S$ . For example, Ridgeway & Mozer (2018) defines that a latent code $Z _ { i }$ to be “ideally modular” if it has high mutual information with a single factor $S _ { j }$ and zero mutual information with all other factors $S _ { \backslash j }$ . This presents a issue when the true factors of variation themselves are statistically dependent; even if $Z _ { 1 } = S _ { 1 }$ , the latent code $Z _ { 1 }$ would violate modularity if $S _ { 1 }$ itself has positive mutual information with $S _ { 2 }$ . Consistency and restrictiveness circumvent this issue by relying on conditional resampling. Consistency, for example, only measures the extent to which $S _ { I }$ is invariant to resampling of $Z _ { \backslash I }$ when conditioned on $Z _ { I }$ and is thus achieved as long as $s _ { I }$ is a function of only $z _ { I }$ —irrespective of whether $s _ { I }$ and $s _ { \backslash I }$ are statistically dependent. In this regard, our definitions draw inspiration from Suter et al. (2018)’s intervention-based definition but replaces the need for counterfactual reasoning with the simpler conditional sampling. Because we do not assume the factors of variation are statistically independent, our theoretical analysis is also distinct from the closely-related match pairing analysis in Gresele et al. (2019).
+
+Consistency and restrictiveness arise in weak supervision guarantees. One of our goals is to propose definitions that are amenable to theoretical analysis. As we can see in Section 4, consistency and restrictiveness serve as the core primitive concepts that we use to describe disentanglement guarantees conferred by various forms of weak supervision.
+
+# B EVALUATING CONSISTENCY AND RESTRICTIVENESS ON DISENTANGLEMENT-LIB MODELS
+
+To better understand the empirical relationship between consistency and restrictiveness, we calculated the normalized consistency and restrictiveness scores on the suite of 12800 models from disentanglement lib for each ground-truth factor. By using the normalized consistency and restrictiveness scores as probes, we were able to identify models that achieve high consistency but low restrictiveness (and vice versa). In Fig. 6, we highlight two models that are either consistent or restrictive for object color on the Shapes3D dataset.
+
+![](images/6f2b1f5a3b506c24ae2b878e58f1d8260796c7d714e7f24a7af3b898bd7ace9d.jpg)  
+Figure 6: Visualization of two models from disentanglement lib (model ids 11964 and 12307), matching the schematic in Fig. 2. For each panel, we visualize an interpolation along a single latent across rows, with each row corresponding to a fixed set of values for all other factors. In Fig. 6a, we can see that this factor consistenly represents object color, i.e. each column of images has the same object color, but as we move along rows we see that other factors change as well, e.g. object type, thus this factor is not restricted to object color. In Fig. 6b, we see that varying the factor along each row results in changes to object color but to no other attributes. However if we look across columns, we see that the representation of color changes depending on the setting of other factors, thus this factor is not consistent for object color.
+
+# C HANDLING NUISANCE VARIABLES
+
+Our theoretical framework can handle nuisance variables, i.e., variables we cannot measure or perform weak supervision on. It may be impossible to label, or provide match-pairing on that factor of variation. For example, while many features of an image are measurable (such as brightness and coloration), we may not be able to measure certain factors of variation or generate data pairs where these factors are kept constant. In this case, we can let one additional variable $\eta$ act as nuisance variable that captures all additional sources of variation / stochasticity.
+
+Formally, suppose the full set of true factors is $S \cup \{ \eta \} \in \mathbb { R } ^ { n + 1 }$ . We define $\eta$ -consistency $C _ { \eta } ( I ) =$ $C ( I )$ and $\eta$ -restrictiveness $R _ { \eta } ( I ) = R ( I \cup \{ \eta \} )$ . This captures our intuition that, with nuisance variable, for consistency, we still want changes to $Z _ { \backslash I } \cup \{ \eta \}$ to not modify $S _ { I }$ ; for restrictiveness, we want changes to $Z _ { I } \cup \{ \eta \}$ to only modify $S _ { I } \cup \{ \eta \}$ . We define $\eta$ -disentanglement as $D _ { \eta } ( I ) =$ $C _ { \eta } ( I ) \land R _ { \eta } ( I )$ .
+
+All of our calculus still holds where we substitute $C _ { \eta } ( I ) , R _ { \eta } ( I ) , D _ { \eta } ( I )$ for $C ( I ) , R ( I ) , D ( I )$ ; we prove one of the new full disentanglement rule as an illustration:
+
+Proposition 1. ${ \textstyle \bigwedge } _ { i = 1 } ^ { n } C _ { \eta } ( i ) \Longleftrightarrow { \textstyle \bigwedge } _ { i = 1 } ^ { n } D _ { \eta } ( i )$
+
+Proof. On the one hand, ${ \textstyle \bigwedge } _ { i = 1 } ^ { n } C _ { \eta } ( i ) \Longleftrightarrow { \textstyle \bigwedge } _ { i = 1 } ^ { n } C ( i ) \implies C ( 1 : n ) \implies R ( \eta )$ . On the other hand, ${ \textstyle \bigwedge } _ { i = 1 } ^ { n } C ( i ) \implies { \textstyle \bigwedge } _ { i = 1 } ^ { n } D ( i ) \implies { \textstyle \bigwedge } _ { i = 1 } ^ { n } R ( i )$ . Therefore $L H S ~ \Longrightarrow ~ \forall i \in [ n ] , R ( i ) \land$ $R ( \eta ) \implies R _ { \eta } ( i )$ . The reverse direction is trivial. □
+
+In (Locatello et al., 2019), the “instance” factor in SmallNORB and the background image factor in Scream-dSprites are treated as nuisance variables. By Proposition 1, as long as we perform weak supervision on all of the non-nuisance variables (via sharing-pairing, say) to guarantee their consistency with respect to the corresponding true factor of variation, we still have guaranteed full disentanglement despite the existence of nuisance variable and the fact that we cannot measure or perform weak supervision on nuisance variable.
+
+# D SINGLE-FACTOR EXPERIMENTS
+
+![](images/48a87f89dbfca3f29bbf65d4bbe6f5752ac51d0833554f39018a86158b7b5131.jpg)  
+Figure 7: This is the same plot as Figure 7, but where we restrict our hyperparameter sweep to always set extra dense $=$ False. See Appendix H for details about hyperparameter sweep.
+
+![](images/ca6e86c7e2ec81fca4fe615089f8acc3f757ee8c762a8d90a64540102cee47af.jpg)  
+Figure 8: Restricted pairing guarantees consistency. Each plot shows the normalized consistency score of each model for each factor of variation. Our theory predicts each boxplot highlighted in red to achieve the highest consistency. Due to the prevalence of restricted pairing in the existing literature, we chose to only conduct the single-factor restricted labeling experiment on Shapes3D.
+
+![](images/8761608da3d21b88e495463d6518501b3c9e91bf752ff63cfe2052c913fb0841.jpg)  
+Figure 9: Change pairing guarantees restrictiveness. Each plot shows normalized restrictiveness score of each model for each factor of variation (row) across different datasets (columns). Different colors indicate models trained with change pairing on different factors. The appropriatelysupervised model for each factor is marked in red.
+
+![](images/c8c53ab7dee87bdd8e6de334a6509e96697e236d3347ef818450dfdf6dca5676.jpg)  
+Figure 10: Share pairing guarantees consistency. Each plot shows normalized consistency score of each model for each factor of variation (row) across different datasets (columns). Different colors indicate models trained with share pairing on different factors. The appropriately-supervised model for each factor is marked in red.
+
+![](images/c2eb042b59e67e75ee5931756a312298a790e10524976331c3e96187a8419789.jpg)  
+Figure 11: Rank pairing guarantees consistency. Each plot shows normalized consistency score of each model for each factor of variation (row) across different datasets (columns). Different colors indicate models trained with rank pairing on different factors. The appropriately-supervised model for each factor is marked in red.
+
+![](images/85c439216fa7ed20c7b92361d2df941c4bcdd7fa62cd43b5601e9fc0990e3e27.jpg)  
+Figure 12: Normalized consistency vs. restrictiveness score of different models on each factor (row) across different datasets (columns). In many of the plots, we see that models trained via changesharing (blue) achieve higher restrictiveness; models trained via share-sharing (orange) achieve higher consistency; models trained via both techniques (green) simultaneously achieve restrictiveness and consistency in most cases.
+
+![](images/cf98c6b3673bda6e6de1b731e7b3207228b3b46930a71455fe97d894802b52c2.jpg)  
+Figure 13: Disentanglement performance of a vanilla GAN, share pairing GAN, change pairing GAN, rank pairing GAN, and fully-labeled GAN, as measured by multiple disentanglement metrics in existing literature (rows) across multiple datasets (columns). According to almost all metrics, our weakly supervised models surpass the baseline, and in some cases, even outperform the fully-labeled model.
+
+![](images/7df62580510bc29231379ea1f32178b810604d9bad0148be27c6d1807696208a.jpg)  
+Figure 14: Performance of a vanilla GAN (blue), share pairing GAN (orange), change pairing GAN (green), rank pairing GAN (red), and fully-labeled GAN (purple), as measured by normalized consistency score of each factor (rows) across multiple datasets (columns). Factors $\{ 3 , 4 , 5 \}$ in the first column shows that distribution matching to all six change / share pairing datasets is particularly challenging for the models when trained on certain hyperparameter choices. However, since consistency and restrictiveness can be measured in weakly supervised settings, it suffices to use these metrics for hyperparameter selection. We see in Figure 16 and Appendix G that using consistency and restrictiveness for hyperparameter selection serves as a viable weakly-supervised surrogate for existing fully-supervised disentanglement metrics.
+
+![](images/ec84da5777dc6cf08c3bd0e4f1bf19ffccc1fd93e6fb95860c649cdd1a05d9c5.jpg)  
+Figure 15: Performance of a vanilla GAN (blue), share pairing GAN (orange), change pairing GAN (green), rank pairing GAN (red), and fully-labeled GAN (purple), as measured by normalized restrictiveness score of each factor (rows) across multiple datasets (columns). Since restrictiveness and consistency are complementary, we see that the anomalies in Figure 14 are reflected in the complementary factors in this figure.
+
+![](images/22c22f5f6392c4e78b4bc2795bf262536e7ced74395364b2b3372271233eeed0.jpg)  
+Figure 16: Scatterplot of existing disentanglement metrics versus average normalized consistency and restrictiveness. Whereas existing disentanglement metrics are fully-supervised, it is possible to measure average normalized consistency and restrictiveness with weakly supervised data (sharepairing and match-pairing respectively), making it viable to perform hyperparameter tuning under weakly supervised conditions.
+
+# G FULL DISENTANGLEMENT VISUALIZATIONS
+
+As a demonstration of the weakly-supervised generative models, we visualize our best-performing match-pairing generative models (as selected according to the normalized consistency score averaged across all the factors). Recall from Figures 2a to 2c that, to visually check for consistency and restrictiveness, it is important that we not only ablate a single factor (across the column), but also show that the factor stays consistent (down the row). Each block of $3 \times 1 2$ images in Figures 17 to 21 checks for disentanglement of the corresponding factor. Each row is constructed by random sampling of $Z _ { \backslash i }$ and then ablating $Z _ { i }$ .
+
+![](images/7d096c9e453a3b9680429bd4633d8b83b292c35baf7aad0ecc629893ec35844b.jpg)  
+Figure 17: Cars3D. Ground truth factors: elevation, azimuth, object type.
+
+![](images/17cc6a81520a9cab0d45677aef8c64fee41a2456e33ca054776d9e4e3146157a.jpg)
+
+![](images/0b64db58fa0dd01441e39382138bba785be612c3cb25b3689bb1ec583a2da8b5.jpg)
+
+![](images/375b1617ad053ec92a4b773b88cdcf38b1bcf19c1ccbb9ad838c5edf0710b119.jpg)
+
+![](images/9925e71b6e0ecee7e32c2b11c305514edb17dfec65dc18a2b7fb3c05a78ba977.jpg)
+
+![](images/6eb7fa067c0ca8ace2214fb2b77b6dd5b6ed6e62c477e72a71e1e94038746898.jpg)  
+Figure 18: Shapes3D. Ground truth factors: floor color, wall color, object color, object size, object type, and azimuth.
+
+![](images/e6532d62c7e5b292d87aa4d9d9882dc6f7ac89f6a05f8b6b4aa18cd10a0efecb.jpg)  
+Figure 19: dSprites. Ground truth factors: shape, scale, orientation, X-position, Y-position.
+
+![](images/ee97f70b86110cad600d3063fec2e0c34b096179e9a24e21e44bad3f79129744.jpg)  
+Figure 20: Scream-dSprites. Ground truth factors: shape, scale, orientation, X-position, Y-position.
+
+![](images/4b7cfa4c56b7a7a611740799590a95ca65c1d6592873545098d3320234453969.jpg)
+
+![](images/0b755c8216c41ba29ad1ab6b698ae353abb92f0c7639c391d0f681be96365d7f.jpg)
+
+![](images/fdb16dfdb4f65189212fb308ad10293cfb43e6934d061f683c6268363b9102cd.jpg)
+
+![](images/27e5e9cbc771586d486247e29cad1e027ee98b4911f97cfc859cc6730d8bcb88.jpg)  
+Figure 21: SmallNORB. Ground truth factors: category, elevation, azimuth, lighting condition.
+
+# H HYPERPARAMETERS
+
+Table 1: We trained a probablistic Gaussian encoder to approximately invert the generative model. The encoder is not trained jointly with the generator, but instead trained separately from the generative model (i.e. encoder gradient does not backpropagate to generative model). During training, the encoder is only exposed to data generated by the learned generative model.
+
+Table 2: Generative model architecture.   
+
+<table><tr><td>Encoder</td></tr><tr><td>4 × 4 spectral norm conv. 32. lReLU 4 × 4 spectral norm conv. 32. IReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64. lReLU flatten</td></tr><tr><td>128 spectral norm dense. lReLU</td></tr><tr><td>2 × z-dim spectral norm dense</td></tr></table>
+
+<table><tr><td>Generator</td></tr><tr><td>128 dense.ReLU.batchnorm. 1024 dense.ReLU.batchnorm. 4 × 4 × 64 reshape.</td></tr><tr><td>4 × 4 conv. 64.lReLU.batchnorm.</td></tr><tr><td>4 × 4 conv. 32.lReLU.batchnorm.</td></tr><tr><td>4 × 4 conv. 32.lReLU.batchnorm. 4 × 4 conv. 3. sigmoid</td></tr></table>
+
+Table 3: Discriminator used for restricted labeling. Parts in red are part of hyperparameter search.   
+
+<table><tr><td rowspan=1 colspan=1>Discriminator Body</td></tr><tr><td rowspan=1 colspan=1>4 × 4 spectral norm conv. 32 × width. lReLU4 × 4 spectral norm conv. 32 × width. lReLU4 × 4 spectral norm conv. 64 × width. lReLU4 × 4 spectral norm conv. 64 × width. lReLUflatten if extra dense: 128 × width spectral norm dense. IReLU</td></tr><tr><td rowspan=1 colspan=1>Discriminator Auxiliary Channel for Label</td></tr><tr><td rowspan=1 colspan=1>128 × width spectral norm dense. IReLUIf extra dense: 128 × width spectral norm dense. lReLU</td></tr><tr><td rowspan=1 colspan=1>Discriminator head</td></tr><tr><td rowspan=1 colspan=1>concatenate body and auxiliary.128 × width spectral norm dense. iReLU128 × width spectral norm dense.lReLU1 spectral norm dense with bias.</td></tr></table>
+
+Table 4: Discriminator used for match pairing. We use a projection discriminator (Miyato & Koyama, 2018) and thus have an unconditional and conditional head. Parts in red are part of hyperparameter search.   
+
+<table><tr><td>Discriminator Body Applied Separately to x and x&#x27;</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. IReLU</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. IReLU 4 × 4 spectral norm conv. 64 × width. IReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU flatten</td></tr><tr><td> If extra dense: 128 × width spectral norm dense. lReLU</td></tr><tr><td>concatenate the pair. 128 × width spectral norm dense. lReLU</td></tr><tr><td>128 × width spectral norm dense. lReLU</td></tr><tr><td></td></tr><tr><td>Unconditional Head</td></tr><tr><td>1 spectral norm dense with bias Conditional Head</td></tr></table>
+
+Table 5: Discriminator used for rank pairing. For rank-pairing, we use a special variant of the projection discriminator, where the conditional logit is computed via taking the difference between the two pairs and multiplying by $y \in \{ - 1 , + 1 \}$ . The discriminator is thus implicitly taking on the role of an adversarially trained encoder that checks for violations of the ranking rule in the embedding space. Parts in red are part of hyperparameter search.   
+
+<table><tr><td>Discriminator Body Applied Separately to x and x&#x27;</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. lReLU 4 × 4 spectral norm conv. 32 × width. IReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU flatten</td></tr><tr><td>If extra dense: 128 × width spectral norm dense. lReLU concatenate the pair.</td></tr><tr><td>Unconditional Head Applied Separately to x and x&#x27;</td></tr><tr><td>1 spectral norm dense with bias.</td></tr><tr><td>Conditional Head Applied Separately to x and x&#x27;</td></tr><tr><td>y-dim spectral norm dense.</td></tr></table>
+
+For all models, we use the Adam optimizer with $\beta _ { 1 } = 0 . 5 , \beta _ { 2 } = 0 . 9 9 9$ and set the generator learning rate to $1 \times 1 0 ^ { - 3 }$ . We use a batch size of 64 and set the leaky ReLU negative slope to 0.2.
+
+To demonstrate some degree of robustness to hyperparameter choices, we considered five different ablations:
+
+1. Width multiplier on the discriminator network $( \{ 1 , 2 \} )$   
+2. Whether to add an extra fully-connected layer to the discriminator $( \{ \mathrm { T r u e } , \mathrm { F a l s e } \} )$ .   
+3. Whether to add a bias term to the head $\langle \mathrm { T r u e } , \mathrm { F a l s e } \rangle \rangle$ .   
+4. Whether to use two-time scale learning rate by setting encoder+discriminator learning rate   
+multipler to $( \{ 1 , 2 \} )$ .   
+5. Whether to use the default PyTorch or Keras initialization scheme in all models.
+
+As such, each of our experimental setting trains a total of 32 distinct models. The only exception is the intersection experiments where we fixed the width multiplier to 1.
+
+To give a sense of the scale of our experimental setup, note that the 864 models in Figure 4 originate as follows:
+
+1. 32 hyperparameter conditions $\times 6$ restricted labeling conditions.   
+2. 32 hyperparameter conditions $\times 6$ match pairing conditions.   
+3. 32 hyperparameter conditions $\times 6$ share pairing conditions.   
+4. 32 hyperparameter conditions $\times 6$ rank pairing conditions.   
+5. 16 hyperparameter conditions $\times 6$ intersection conditions.
+
+# I PROOFS
+
+# I.1 ASSUMPTIONS ON $\mathcal { H }$
+
+Assumption 1. Let $D \subseteq [ n ]$ indexes discrete random variables $S _ { D }$ . Assume that the remaining random variables $S _ { C } = S _ { \backslash D }$ have probability density function $p ( s _ { C } | s _ { D } )$ for any set of values $s _ { D }$ where $p ( S _ { D } = s _ { D } ) > 0$ .
+
+Assumption 2. Without loss of generality, suppose $S _ { 1 : n } = [ S _ { C } , S _ { D } ]$ is ordered by concatenating the continuous variables with the discrete variables. Let $\bar { \mathcal { B } ( s _ { D } ) } \ : = \ : [ \mathrm { i n t } ( \mathrm { s u p p } ( p ( s _ { C } \ : | \ : s _ { D } ) ) ) , s _ { D } ]$ denote the interior of the support of the continuous conditional distribution of $S _ { C }$ concatenated with its conditioning variable $s _ { D }$ drawn from $S _ { D }$ . With a slight abuse of notation, let $B ( S ) =$ $\textstyle \bigcup _ { s _ { D } : p ( s _ { D } > 0 ) } B ( s _ { D } )$ . We assume $B ( S )$ is zig-zag connected, i.e., for any $I , J \subseteq [ n ]$ , for any two points $s _ { 1 : n } , s _ { 1 : n } ^ { \prime } \in B ( S )$ that only differ in coordinates in $I \cup J$ , there exists a path $\{ s _ { 1 : n } ^ { t } \} _ { t = 0 : T }$ contained in $B ( S )$ such that
+
+$$
+\begin{array} { r l } & { s _ { 1 : n } ^ { 0 } = s _ { 1 : n } } \\ & { ~ s _ { 1 : n } ^ { T } = s _ { 1 : n } ^ { \prime } } \\ & { \forall 0 \leq t < T , \mathrm { e i t h e r } s _ { \setminus \ I } ^ { t } = s _ { \setminus I } ^ { t + 1 } \mathrm { o r } s _ { \setminus J } ^ { t } = s _ { \setminus J } ^ { t + 1 } , } \end{array}
+$$
+
+Intuitively, this assumption allows transition from $s _ { 1 : n }$ to $s _ { 1 : n } ^ { \prime }$ via a series of modifications that are only in $I$ or only in $J$ . Note that zig-zag connectedness is necessary for restrictiveness union (Proposition 3) and consistency intersection (Proposition 4). Fig. 22 gives examples where restrictiveness union is not satisfied when zig-zag connectedness is violated.
+
+Assumption 3. For arbitrary coordinate $j \in [ m ]$ of $g$ that maps to a continuous variable $X _ { j }$ , we assume that $g _ { j } ( s )$ is continuous at $s$ , $\forall s \in B ( S )$ ; For arbitrary coordinate $j \in [ m ]$ of $g$ that maps to a discrete variable $X _ { j }$ , $\forall s _ { D }$ where $p ( \boldsymbol { s } _ { D } ) > 0$ , we assume that $g _ { j } ( s )$ is constant over each connected component of $\operatorname { i n t } ( \operatorname { s u p p } ( p ( s _ { C } \mid s _ { D } ) )$ .
+
+Define $B ( X )$ analogously to $B ( S )$ . Symmetrically, for arbitrary coordinate $i \in [ n ]$ of $e$ that maps to a continuous variable $S _ { i }$ , we assume that $e _ { i } ( x )$ is continuous at $x$ $\ u , \forall x \in B ( X )$ ; For arbitrary coordinate $i \in [ n ]$ of $e$ that maps to a discrete $S _ { i }$ , $\forall x _ { D }$ where $p ( x _ { D } ) > 0$ , we assume that $e _ { i } ( x )$ is constant over each connected component of $\operatorname { i n t } ( \operatorname { s u p p } ( p ( x _ { C } \mid x _ { D } ) )$ .
+
+Assumption 4. Assume that every factor of variation is recoverable from the observation $\mathcal { X }$ . Formally, $( p , g , e )$ satisfies the following property
+
+$$
+\mathbb { E } _ { p ( s _ { 1 : n } ) } \| e \circ g ( s _ { 1 : n } ) - s _ { 1 : n } \| ^ { 2 } = 0 .
+$$
+
+# I.2 CALCULUS OF DISENTANGLEMENT
+
+# I.2.1 EXPECTED-NORM REDUCTION LEMMA
+
+Lemma 1. Let $x , y$ be two random variables with distribution $p$ , $f ( x , y )$ be arbitrary function. Then
+
+$$
+\begin{array} { r } { \mathbb { E } _ { x \sim p ( x ) } \mathbb { E } _ { y , y ^ { \prime } \sim p ( y | x ) } \| f ( x , y ) - f ( x , y ^ { \prime } ) \| ^ { 2 } \leq \mathbb { E } _ { ( x , y ) , ( x ^ { \prime } , y ^ { \prime } ) \sim p ( x , y ) } \| f ( x , y ) - f ( x ^ { \prime } , y ^ { \prime } ) \| ^ { 2 } . } \end{array}
+$$
+
+Proof. Assume w.l.o.g that $\mathbb { E } _ { ( x , y ) \sim p ( x , y ) } f ( x , y ) = 0$
+
+$$
+\begin{array} { r l } & { L H S = 2 \mathbb { E } _ { ( x , y ) \sim p ( x , y ) } \| f ( x , y ) \| ^ { 2 } - 2 \mathbb { E } _ { x \sim p ( x ) } \mathbb { E } _ { y , y ^ { \prime } \sim p ( y | x ) } f ( x , y ) ^ { T } f ( x , y ^ { \prime } ) } \\ & { \qquad = 2 \mathbb { E } _ { ( x , y ) \sim p ( x , y ) } \| f ( x , y ) \| ^ { 2 } - 2 \mathbb { E } _ { x \sim p ( x ) } \mathbb { E } _ { y \sim p ( y | x ) } f ( x , y ) ^ { T } \mathbb { E } _ { y ^ { \prime } \sim p ( y | x ) } f ( x , y ^ { \prime } ) } \\ & { \qquad = 2 \mathbb { E } _ { ( x , y ) \sim p ( x , y ) } \| f ( x , y ) \| ^ { 2 } - 2 \mathbb { E } _ { x \sim p ( x ) } \| \mathbb { E } _ { y \sim p ( y | x ) } f ( x , y ) \| ^ { 2 } } \\ & { \qquad \le 2 \mathbb { E } _ { ( x , y ) \sim p ( x , y ) } \| f ( x , y ) \| ^ { 2 } } \\ & { \qquad = 2 \mathbb { E } _ { ( x , y ) \sim p ( x , y ) } \| f ( x , y ) \| ^ { 2 } - 2 \| \mathbb { E } _ { ( x , y ) \sim p ( x , y ) } f ( x , y ) \| ^ { 2 } } \\ & { \qquad = 2 \mathbb { E } _ { ( x , y ) \sim p ( x , y ) } \| f ( x , y ) \| ^ { 2 } - 2 \mathbb { E } _ { ( x , y ) , ( x ^ { \prime } , y ^ { \prime } ) \sim p ( x , y ) } f ( x , y ) ^ { T } f ( x ^ { \prime } , y ^ { \prime } ) } \\ & { \qquad = R H S . } \end{array}
+$$
+
+# I.2.2 CONSISTENCY UNION
+
+Let $L = I \cap J , K = \backslash ( I \cup J )$ , $M = I - L , N = J - L .$ .   
+Proposition 2. $C ( I ) \land C ( J ) \implies C ( I \cup J )$ .
+
+Proof.
+
+$$
+\begin{array} { r } { C ( I ) \implies \mathbb { E } _ { z _ { M } , z _ { L } } \mathbb { E } _ { z _ { N } , z _ { N } ^ { \prime } , z _ { K } , z _ { K } ^ { \prime } } \lVert r _ { I } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) - r _ { I } \circ G ( z _ { M } , z _ { L } , z _ { N } ^ { \prime } , z _ { K } ^ { \prime } ) \rVert ^ { 2 } = 0 . } \end{array}
+$$
+
+For any fixed value of $z _ { M } , z _ { L }$ ,
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { z _ { N } , z _ { N } ^ { \prime } , z _ { K } , z _ { K } ^ { \prime } } \| r _ { I } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) - r _ { I } \circ G ( z _ { M } , z _ { L } , z _ { N } ^ { \prime } , z _ { K } ^ { \prime } ) \| ^ { 2 } } \\ & { \geq \mathbb { E } _ { z _ { N } } \mathbb { E } _ { z _ { K } , z _ { K } ^ { \prime } } \| r _ { I } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) - r _ { I } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ^ { \prime } ) \| ^ { 2 } . } \end{array}
+$$
+
+by plugging in $x = z _ { N } , y = z _ { K }$ into Lemma 1. Therefore
+
+$$
+\begin{array} { r } { C ( I ) \implies \mathbb { E } _ { z _ { M } , z _ { L } , z _ { N } } \mathbb { E } _ { z _ { K } , z _ { K } ^ { \prime } } \| r _ { I } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) - r _ { I } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ^ { \prime } ) \| ^ { 2 } = 0 . } \end{array}
+$$
+
+Similarly we have
+
+$$
+\begin{array} { r l } & { C ( J ) \implies \mathbb { E } _ { z _ { M } , z _ { L } , z _ { N } } \mathbb { E } _ { z _ { K } , z _ { K } ^ { \prime } } \| r _ { J } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) - r _ { J } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ^ { \prime } ) \| ^ { 2 } = 0 } \\ & { \qquad \implies \mathbb { E } _ { z _ { M } , z _ { L } , z _ { N } } \mathbb { E } _ { z _ { K } , z _ { K } ^ { \prime } } \| r _ { N } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) - r _ { N } \circ G ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ^ { \prime } ) \| ^ { 2 } = 0 . } \end{array}
+$$
+
+As $I \cap N = \emptyset$ , $I \cup N = I \cup J$ , adding the above two equations gives us $C ( I \cup J )$ .
+
+# I.2.3 RESTRICTIVENESS UNION
+
+![](images/3d939682e796ffbde633b301011b7c88c4e1e688ad23241eb9d8cc33261b4bce.jpg)  
+Figure 22: Zig-zag connectedness is necessary for restriveness union. Here $n = m = 3$ . Colored areas indicate the support of $p ( z _ { 1 } , z _ { 2 } )$ ; the marked numbers indicate the measurement of $s _ { 3 }$ given $( z _ { 1 } , z _ { 2 } )$ . Left two panels satisfy zig-zag connectedness (the paths are marked in gray) while the right two do not (indeed $R ( 1 ) \wedge R ( \bar { 2 } ) \not \Rightarrow R ( \bar { \{ 1 , 2 \} } ) )$ . In the right-most panel, any zig-zag path connecting two points from blue and orange areas has to pass through boundary of the support (disallowed).
+
+Similarly define index sets $L , K , M , N$
+
+Proposition 3. Under assumptions specified in Appendix I.1, $R ( I ) \land R ( J ) \implies R ( I \cup J )$ .
+
+Proof. Denote $f = e _ { K } ^ { * } \circ g$ . We claim that
+
+$$
+\begin{array} { r l } & { R ( I ) \Longleftrightarrow \mathbb { E } _ { z _ { \backslash I } } \mathbb { E } _ { z _ { I } , z _ { I } ^ { \prime } } \Vert f ( z _ { I } , z _ { \backslash I } ) - f ( z _ { I } ^ { \prime } , z _ { \backslash I } ) \Vert ^ { 2 } = 0 . } \\ & { \qquad \Longleftrightarrow \forall ( z _ { I } , z _ { \backslash I } ) , ( z _ { I } ^ { \prime } , z _ { \backslash I } ) \in \mathcal { B } ( Z ) , f ( z _ { I } , z _ { \backslash I } ) = f ( z _ { I } ^ { \prime } , z _ { \backslash I } ) . } \end{array}
+$$
+
+We first prove the backward direction: When we draw $z _ { \backslash I } \sim p ( z _ { \backslash I } ) , z _ { I } , z _ { I } ^ { \prime } \sim p ( z _ { I } | z _ { \backslash I } )$ , let $E _ { 1 }$ denote the event that $( z _ { I } , z _ { \backslash I } ) \notin B ( Z )$ , and $E _ { 2 }$ denote the event that $( z _ { I } ^ { \prime } , z _ { \backslash I } ) \notin B ( Z )$ . Reorder the indices of $( z _ { I } , z _ { \backslash I } )$ as $( z _ { C } , z _ { D } )$ . The probability that $( z _ { I } , z _ { \backslash I } ) \notin B ( Z )$ (i.e., $z _ { C }$ is on the boundary of $B ( z _ { D } ) )$ ) is 0. Therefore $\operatorname* { P r } [ E _ { 1 } ] = \operatorname* { P r } [ E _ { 2 } ] = 0$ . Therefore $\operatorname* { P r } [ E _ { 1 } \cup E _ { 2 } ] \leq \operatorname* { P r } [ E _ { 1 } ] + \operatorname* { P r } [ E _ { 2 } ] = 0$ , i.e., with probability 1, $\| f ( z _ { I } , z _ { \setminus I } ) - f ( z _ { I } ^ { \prime } , z _ { \setminus I } ) \| ^ { 2 } = 0$ .
+
+Now we prove the forward direction: Assume for the sake of contradiction that $\exists ( z _ { I } , z _ { \backslash I } ) , ( \bar { z } _ { I } ^ { \prime } , z _ { \backslash I } ) \in \mathcal { B } ( Z )$ such that $f ( z _ { I } , z _ { \backslash I } ) < f ( z _ { I } ^ { \prime } , z _ { \backslash I } )$ . Denote $U = I \cap D$ , $V = I \cap C$ , $W = \backslash I \cap D$ , $Q = \backslash I \cap C$ . We have $f ( z _ { U } , z _ { V } , z _ { W } , z _ { Q } ) < f ( z _ { U } ^ { \prime } , z _ { V } ^ { \prime } , z _ { W } , z _ { Q } )$ . Since $f$ is continuous (or constant) at $( z _ { U } , z _ { V } , z _ { W } , z _ { Q } )$ in the interior of $B ( [ z _ { U } , \bar { z } _ { W } ] )$ , and $f$ is also continuous (or constant) at $\left( z _ { U } ^ { \prime } , z _ { V } ^ { \prime } , z _ { W } , z _ { Q } \right)$ in the interior of $B \big ( \big [ z _ { U } ^ { \prime } , z _ { W } \big ] \big )$ , we can draw open balls of radius $r > 0$ around each point, i.e., $B _ { r } ( \tilde { z } _ { V } , z _ { Q } ) \subset B ( [ z _ { U } , z _ { W } ] )$ and $\vec { B _ { r } } ( z _ { V } ^ { \prime } , z _ { Q } ) \subset \vec { B } ( [ \bar { z } _ { U } ^ { \prime } , z _ { W } ] )$ , where
+
+$$
+\begin{array} { r } { \forall ( z _ { V } ^ { * } , z _ { Q } ^ { * } ) \in B _ { r } ( z _ { V } , z _ { Q } ) , \forall ( z _ { V } ^ { \Delta } , z _ { Q } ^ { \Delta } ) \in B _ { r } ( z _ { V } ^ { \prime } , z _ { Q } ) , f ( z _ { U } , z _ { V } ^ { * } , z _ { W } , z _ { Q } ^ { * } ) < f ( z _ { U } ^ { \prime } , z _ { V } ^ { \Delta } , z _ { W } , z _ { Q } ^ { \Delta } ) . } \end{array}
+$$
+
+When we draw $z _ { \backslash I } \sim p ( z _ { \backslash I } ) , z _ { I } , z _ { I } ^ { \prime } \sim p ( z _ { I } | z _ { \backslash I } )$ , let $C$ denote the event that $\left( z _ { I } , z _ { \backslash I } \right) \ =$ $( z _ { V } ^ { \ast } , z _ { U } , z _ { Q } ^ { \# } , z _ { W } )$ , $\begin{array} { r l r } { ( z _ { I } ^ { \prime } , z _ { \backslash I } ) } & { { } = } & { ( z _ { V } ^ { \Delta } , z _ { U } ^ { \prime } , z _ { Q } ^ { \# } , z _ { W } ) } \end{array}$ where $\begin{array} { r l r } { ( z _ { V } ^ { * } , z _ { Q } ^ { \# } ) } & { { } \in } & { B _ { r } ( z _ { V } , z _ { Q } ) } \end{array}$ and $\begin{array} { r l r } { ( z _ { V } ^ { \Delta } , z _ { Q } ^ { \# } ) } & { { } \in } & { B _ { r } ( z _ { V } ^ { \prime } , z _ { Q } ) } \end{array}$ . Since both balls have positive volume, $\mathrm { P r } [ C ] ~ > ~ 0$ . However, $\| f ( z _ { I } , z _ { \setminus I } ) - f ( z _ { I } ^ { \prime } , z _ { \setminus I } ) \| ^ { 2 } > 0$ whenever event $C$ happens, which contradicts $R ( I )$ . Therefore $\forall ( z _ { I } , z _ { \setminus I } ) , ( z _ { I } ^ { \prime } , z _ { \setminus I } ) \in \mathcal { B } ( Z ) , f ( z _ { I } , z _ { \setminus I } ) = f ( z _ { I } ^ { \prime } , z _ { \setminus I } )$ .
+
+We have shown that
+
+$$
+\begin{array} { r } { \{ ( I ) \Longleftrightarrow \forall ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) , ( z _ { M } ^ { \prime } , z _ { L } ^ { \prime } , z _ { N } , z _ { K } ) \in \mathcal { B } ( Z ) , f ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) = f ( z _ { M } ^ { \prime } , z _ { L } ^ { \prime } , z _ { N } , z _ { K } ) . } \end{array}
+$$
+
+Similarly
+
+$$
+\begin{array} { r l r } & { } & { R ( J ) \Longleftrightarrow \forall ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) , ( z _ { M } , z _ { L } ^ { \prime } , z _ { N } ^ { \prime } , z _ { K } ) \in \mathcal { B } ( Z ) , f ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) = f ( z _ { M } , z _ { L } ^ { \prime } , z _ { N } ^ { \prime } , z _ { K } ) } \\ & { } & { \mathrm { ~ } \mathrm ( 3 4 ) } \\ & { } &  \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm   \end{array}
+$$
+
+Let the zig-zag path between $\left( z _ { M } , z _ { L } , z _ { N } , z _ { K } \right)$ and $\begin{array} { r l r } { ( z _ { M } ^ { \prime } , z _ { L } ^ { \prime } , z _ { N } ^ { \prime } , z _ { K } ) } & { { } \in } & { B ( Z ) } \end{array}$ be $\{ ( z _ { M } ^ { t } , z _ { L } ^ { t } , z _ { N } ^ { t } , z _ { K } ) \} _ { t = 0 } ^ { T }$ . Repeatedly applying the equivalent conditions of $R ( I )$ and $R ( J )$ gives us
+
+$$
+\begin{array} { r } { { ^ { \mathrm { r } } ( z _ { M } , z _ { L } , z _ { N } , z _ { K } ) } = f ( z _ { M } ^ { 1 } , z _ { L } ^ { 1 } , z _ { N } ^ { 1 } , z _ { K } ) = \cdot \cdot \cdot = f ( z _ { M } ^ { T - 1 } , z _ { L } ^ { T - 1 } , z _ { N } ^ { T - 1 } , z _ { K } ) = f ( z _ { M } ^ { \prime } , z _ { L } ^ { \prime } , z _ { N } ^ { \prime } , z _ { K } ) . } \end{array}
+$$
+
+# I.3 CONSISTENCY AND RESTIVENESS INTERSECTION
+
+Proposition 4. Under the same assumptions as restrictiveness union, $C ( I ) \land C ( J ) \implies C ( I \cap J )$
+
+Proof.
+
+$$
+\begin{array} { r } { C ( I ) \wedge C ( J ) \implies R ( \backslash I ) \wedge R ( \backslash J ) } \\ { \implies R ( \backslash I \cup \backslash J ) } \\ { \implies C ( \backslash ( \backslash J \cup \backslash J ) ) } \\ { \implies C ( I \cap J ) . } \end{array}
+$$
+
+Proposition 5. $R ( I ) \land R ( J ) \implies R ( I \cap J )$ .
+
+Proof is analogous to Proposition 4.
+
+# I.4 DISTRIBUTION MATCHING GUARANTEES LATENT CODE INFORMATIVENESS
+
+Proposition 6. If $( p ^ { * } , g ^ { * } , e ^ { * } ) \in \mathcal { H }$ , and $( p , g , e ) \in \mathcal { H }$ , and $g ^ { * } ( S ) \ { \stackrel { d } { = } } \ g ( Z )$ , then there exists a continuous function $r$ such that
+
+$$
+\mathbb { E } _ { p ( s _ { 1 : n } ) } \| r \circ e \circ g ^ { * } ( s ) - s \| = 0 .
+$$
+
+Proof. We show that $r = e ^ { * } \circ g$ satisfies Proposition 6. By Assumption 4,
+
+$$
+\begin{array} { r } { \mathbb { E } _ { s } \| e ^ { * } \circ g ^ { * } ( s ) - s \| ^ { 2 } = 0 . } \\ { \mathbb { E } _ { z } \| e \circ g ( z ) - z \| ^ { 2 } = 0 . } \end{array}
+$$
+
+By the same reasoning as in the proof of Proposition 3,
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { s } \| e ^ { * } \circ g ^ { * } ( s ) - s \| ^ { 2 } = 0 \implies \forall s \in \mathcal { B } ( S ) , e ^ { * } \circ g ^ { * } ( s ) = s . } \\ & { \quad \mathbb { E } _ { z } \| e \circ g ( z ) - z \| ^ { 2 } = 0 \implies \forall z \in \mathcal { B } ( Z ) , e \circ g ( z ) = z . } \end{array}
+$$
+
+Let $s \sim p ( s )$ . We claim that $\mathrm { P r } [ E _ { 1 } ] = 1$ , where $E _ { 1 }$ denote the event that $\exists z \in B ( Z )$ such that $g ^ { * } ( s ) = g ( z )$ . Suppose to the contrary that there is a measure-non-zero set $S \subseteq s u p p ( p ( s ) )$ such that $\forall s \in S$ , no $z \in B ( Z )$ satisfies $g ^ { * } ( s ) = g ( z )$ . Let ${ \mathcal { X } } = \{ g ( s ) : s \in { \mathcal { S } } \}$ . As $g ^ { * } ( S ) \stackrel { d } { = } g ( Z )$ , $\operatorname* { P r } _ { s } [ g ^ { * } ( s ) \in \mathcal { X } ] = \operatorname* { P r } _ { z } [ g ( z ) \in \mathcal { X } ] > 0$ . Therefore $\exists \mathcal { Z } \subseteq s u p p ( p ( z ) ) - B ( Z )$ such that $\mathcal { X } \subseteq \{ g ( z ) :$ $z \in { \mathcal { Z } } ) \}$ . But $s u p p ( p ( z ) ) - B ( Z )$ has measure 0. Contradiction.
+
+When we draw $s$ , let $E _ { 2 }$ denote the event that $s \in B ( S )$ . $\mathrm { P r } [ E _ { 2 } ] = 1$ , so $\operatorname* { P r } [ E _ { 1 } \land E _ { 2 } ] = 1$ . When $E _ { 1 } \wedge E _ { 2 }$ happens, $e ^ { * } \circ g \circ e \circ g ^ { * } ( s ) = e ^ { * } \circ g \circ e \circ g ( z ) = e ^ { * } \circ g ( z ) = e ^ { * } \circ g ^ { * } ( s ) = s$ . Therefore
+
+$$
+\mathbb { E } _ { s } \| e ^ { * } \circ g \circ e \circ g ^ { * } ( s ) - s \| = 0 .
+$$
+
+# I.5 WEAK SUPERVISION GUARANTEE
+
+Theorem 1. Given any oracle $( p ^ { \ast } ( s ) , g ^ { \ast } , e ^ { \ast } ) \in \mathcal { H }$ , consider the distribution-matching algorithm $\mathcal { A }$ that selects a model $( p ( z ) , g , e ) \in \mathcal { H }$ such that:
+
+$$
+\left( g ^ { \ast } ( S _ { I } , S _ { \setminus I } ) , g ^ { \ast } ( S _ { I } , S _ { \setminus I } ^ { \prime } ) \right) \stackrel { d } { = } \left( g ( Z _ { I } , Z _ { \setminus I } ) , g ( Z _ { I } , Z _ { \setminus I } ^ { \prime } ) \right) ( M a t c h P a i r i n g ) ; o r
+$$
+
+$$
+\begin{array} { r } { ( g ^ { * } ( S ) , g ^ { * } ( S ^ { \prime } ) , \mathbf { 1 } \left\{ S _ { I } \leq S _ { I } ^ { \prime } \right\} ) \stackrel { d } { = } ( g ( Z ) , g ( Z ^ { \prime } ) , \mathbf { 1 } \left\{ Z _ { I } \leq Z _ { I } ^ { \prime } \right\} ) ( R a n k ~ P a i r i n g ) . } \end{array}
+$$
+
+Then $( p , g )$ satisfies $C ( I ; p , g , e ^ { * } )$ and e satisfies $C ( I ; p ^ { * } , g ^ { * } , e )$ .
+
+Proof. We prove the three cases separately:
+
+1. Since $( x _ { d } , s _ { I } ) \overset { d } { = } ( x _ { g } , z _ { I } )$ , consider the measurable function
+
+$$
+f ( a , b ) = \| e _ { I } ^ { * } ( a ) - b \| ^ { 2 } .
+$$
+
+We have
+
+$$
+\begin{array} { r } { \mathbb { E } \| e _ { I } ^ { * } ( x _ { d } ) - s _ { I } \| ^ { 2 } = \mathbb { E } \| e _ { I } ^ { * } ( x _ { g } ) - z _ { I } \| ^ { 2 } = 0 . } \end{array}
+$$
+
+By the same reasoning as in the proof of Proposition 3,
+
+$$
+\begin{array} { r } { \mathbb { E } _ { z } \| e _ { I } ^ { * } \circ g ( z ) - z _ { I } \| ^ { 2 } = 0 \implies \forall z \in \mathcal { B } ( Z ) , e _ { I } ^ { * } \circ g ( z ) = z _ { I } . } \end{array}
+$$
+
+Therefore
+
+$$
+\begin{array} { r } { \mathbb { E } _ { z _ { I } } \mathbb { E } _ { z _ { \searrow I } , z _ { \searrow I } ^ { \prime } } \Vert e _ { I } ^ { * } \circ g ( z _ { I } , z _ { \searrow I } ) - e _ { I } ^ { * } \circ g ( z _ { I } , z _ { \searrow I } ^ { \prime } ) \Vert ^ { 2 } = 0 . } \end{array}
+$$
+
+i.e., $g$ satisfies $C ( I ; p , g , e ^ { * } )$ . By symmetry, $e$ satisfies $C ( I ; p ^ { * } , g ^ { * } , e )$ .
+
+2.
+
+$$
+\begin{array} { r l r } & { } & { \left( g ^ { * } ( S _ { I } , S _ { \setminus I } ) , g ^ { * } ( S _ { I } , S _ { \setminus I } ^ { \prime } ) \right) \stackrel { d } { = } \left( g ( Z _ { I } , Z _ { \setminus I } ) , g ( Z _ { I } , Z _ { \setminus I } ^ { \prime } ) \right) \ ( 5 1 ) \qquad } \\ & { \implies \lVert e _ { I } ^ { * } \circ g ^ { * } ( S _ { I } , S _ { \setminus I } ) - e _ { I } ^ { * } \circ g ^ { * } ( S _ { I } , S _ { \setminus I } ^ { \prime } ) \rVert ^ { 2 } \stackrel { d } { = } \lVert e _ { I } ^ { * } \circ g ( Z _ { I } , Z _ { \setminus I } ) - e _ { I } ^ { * } \circ g ( Z _ { I } , Z _ { \setminus I } ^ { \prime } ) \rVert ^ { 2 } } \end{array}
+$$
+
+$$
+\begin{array} { r l r } & { } & { \implies \mathbb { E } _ { z _ { I } } \mathbb { E } _ { z _ { \setminus I } , z _ { \setminus I } ^ { \prime } } \lVert e _ { I } ^ { * } \circ g ( z _ { I } , z _ { \setminus I } ) - e _ { I } ^ { * } \circ g ( z _ { I } , z _ { \setminus I } ^ { \prime } ) \rVert ^ { 2 } = 0 . } \end{array}
+$$
+
+So $g$ satisfies $C ( I ; p , g , e ^ { * } )$ . By symmetry, $e$ satisfies $C ( I ; p ^ { * } , g ^ { * } , e )$ .
+
+3. Let $I = \{ i \}$ , $f = e _ { I } ^ { * } { \circ } g$ . Distribution matching implies that, with probability 1 over random draws of $Z , Z ^ { \prime }$ , the following event $P$ happens:
+
+$$
+Z _ { I } < = Z _ { I } ^ { \prime } \Longrightarrow f ( Z ) < = f ( Z ^ { \prime } ) .
+$$
+
+$$
+\mathbb { E } _ { z , z ^ { \prime } } \mathbf { 1 } [ \lnot P ] = 0 .
+$$
+
+Let $W = \backslash I \cap D , Q = \backslash I \cap \backslash D$ . We showed in the proof of Proposition 3 that
+
+$$
+C ( I ) \Longleftrightarrow \forall ( z _ { I } , z _ { W } , z _ { Q } ) , ( z _ { I } , z _ { W } ^ { \prime } , z _ { Q } ^ { \prime } ) \in \mathcal { B } ( Z ) , f ( z _ { I } , z _ { W } , z _ { Q } ) = f ( z _ { I } , z _ { W } ^ { \prime } , z _ { Q } ^ { \prime } ) .
+$$
+
+We prove by contradiction. Suppose $\exists ( z _ { I } , z _ { W } , z _ { Q } ) , ( z _ { I } , z _ { W } ^ { \prime } , z _ { Q } ^ { \prime } ) \ \in \ B ( Z )$ such that $f ( z _ { I } , z _ { W } , z _ { Q } ) < f ( z _ { I } , z _ { W } ^ { \prime } , z _ { Q } ^ { \prime } )$ .
+
+(a) Case $1 \colon Z _ { I }$ is discrete.
+
+Since $f$ is constant both at $( z _ { I } , z _ { W } , z _ { Q } )$ in the interior of $B ( \left[ z _ { I } , z _ { W } \right] )$ , and at $( z _ { I } , z _ { W } ^ { \prime } , z _ { Q } ^ { \prime } )$ in the interior of $B ( [ z _ { I } , z _ { W } ^ { \prime } ] )$ , we can draw open balls of radius $r > 0$ around each point, i.e., $B _ { r } ( z _ { Q } ) \subset B ( [ z _ { I } , z _ { W } ] )$ and $B _ { r } ( z _ { Q } ^ { \prime } ) \subset B ( [ z _ { I } , z _ { W } ^ { \prime } ] )$ , where
+
+$$
+\forall z _ { Q } ^ { * } \in B _ { r } ( z _ { Q } ) , \forall z _ { Q } ^ { \Delta } \in B _ { r } ( z _ { Q } ^ { \prime } ) , f ( z _ { I } , z _ { W } , z _ { Q } ^ { * } ) < f ( z _ { I } , z _ { W } ^ { \prime } , z _ { Q } ^ { \Delta } ) .
+$$
+
+When we draw $z , z ^ { \prime } \sim p ( z )$ , let $C$ denote the event that this specific value of $z _ { I }$ is picked for both $z , z ^ { \prime }$ , and we picked $z _ { \backslash I } \in B _ { r } ( z _ { Q } ^ { \prime } )$ , $z _ { \backslash I } ^ { \prime } \in B _ { r } ( \bar { z } _ { Q } )$ . Since both balls have positive volume, $\mathrm { P r } [ C ] > 0$ . However, $P$ does not happen whenever event $C$ happens, since $z _ { I } = z _ { I } ^ { \prime }$ but ${ \bar { f } } ( z ) > f ( z ^ { \prime } )$ , which contradicts $\mathrm { P r } [ P ] = 1$ .
+
+(b) Case 2: $z _ { I }$ is continuous. Similar to case 1, we can draw open balls of radius $r > 0$ around each point, i.e., $B _ { r } ( z _ { I } , z _ { Q } ) \subset B ( z _ { W } )$ and $B _ { r } ( z _ { I } , \bar { z } _ { Q } ^ { \prime } ) \subset B ( z _ { W } ^ { \prime } )$ , where
+
+$$
+\forall ( z _ { I } ^ { \star } , z _ { Q } ^ { \star } ) \in B _ { r } ( z _ { I } , z _ { Q } ) , \forall ( z _ { I } ^ { \Delta } , z _ { Q } ^ { \Delta } ) \in B _ { r } ( z _ { I } , z _ { Q } ^ { \prime } ) , f ( z _ { I } ^ { \star } , z _ { W } , z _ { Q } ^ { \star } ) < f ( z _ { I } ^ { \Delta } , z _ { W } ^ { \prime } , z _ { Q } ^ { \Delta } ) .
+$$
+
+Let $H ^ { 1 } = \{ ( z _ { I } ^ { * } , z _ { Q } ^ { * } ) \in B _ { r } ( z _ { I } , z _ { Q } ) : z _ { I } ^ { * } > = z _ { I } \}$ , $H ^ { 2 } = \{ ( z _ { I } ^ { \Delta } , z _ { Q } ^ { \Delta } ) \in B _ { r } ( z _ { I } , z _ { Q } ^ { \prime } ) :$ $z _ { I } ^ { \Delta } ~ < = ~ z _ { I } \}$ . When we draw $z , z ^ { \prime } \sim p ( z )$ , let $C$ denote the event that we picked $z ^ { \bar { \prime } } \in H ^ { 1 } \times \{ z _ { W } \}$ , $z \in H ^ { 2 } \times \{ z _ { W } ^ { \prime } \}$ . Since $\dot { H } ^ { 1 } , H ^ { 2 }$ have positive volume, $\mathrm { P r } [ C ] > 0$ However, $P$ does not happen whenever event $C$ happens, since $z _ { I } < = z _ { I } ^ { \prime }$ but $f ( z ) >$ $f ( z ^ { \prime } )$ , which contradicts $\mathrm { P r } [ P ] = 1$ .
+
+Therefore we showed
+
+$$
+\forall ( z _ { I } , z _ { W } , z _ { Q } ) , ( z _ { I } , z _ { W } ^ { \prime } , z _ { Q } ^ { \prime } ) \in \mathcal { B } ( Z ) , f ( z _ { I } , z _ { W } , z _ { Q } ) = f ( z _ { I } , z _ { W } ^ { \prime } , z _ { Q } ^ { \prime } ) ,
+$$
+
+i.e., $g$ satisfies $C ( I ; p , g , e ^ { * } )$ . By symmetry, $e$ satisfies $C ( I ; p ^ { * } , g ^ { * } , e )$ .
+
+# I.6 WEAK SUPERVISION IMPOSSIBILITY RESULT
+
+Theorem 2. Weak supervision via restricted labeling, match pairing, or ranking on $s _ { I }$ is not sufficient for learning a generative model whose latent code $Z _ { I }$ is restricted to $S _ { I }$ .
+
+Proof. We construct the following counterexample. Let $n = m = 3$ and $I = \{ 1 \}$ . The data generation process is $s _ { 1 } \sim \mathrm { u n i f } ( [ 0 , 2 \pi ) )$ , $( s _ { 2 } , s _ { 3 } ) \sim \operatorname { \bar { u n i f } } ( \{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 1 \} )$ ), $\bar { g ^ { * } } ( s ) = [ s _ { 1 } , s _ { 2 } , s _ { 3 } ]$ . Consider a generator $z _ { 1 } \sim \mathrm { u n i f } ( [ 0 , 2 \pi ) )$ , $( s _ { 2 } , s _ { 3 } ) \ \sim \ \mathrm { u n i f } ( \{ ( x , y ) \ : \ x ^ { 2 } + y ^ { 2 } \ \leq \ 1 \} )$ , $g ( z ) =$ $[ z _ { 1 } , \cos { ( z _ { 1 } ) } z _ { 2 } - \sin { ( z _ { 1 } ) } z _ { 3 } , \sin { ( z _ { 1 } ) } z _ { 2 } + \cos { ( z _ { 1 } ) } z _ { 3 } ]$ . Then $\left( x _ { d } , s _ { I } \right) \overset { d } { = } \left( x _ { g } , z _ { I } \right)$ but $R ( I ; p , g , e ^ { * } )$ and $R ( I ; p ^ { * } , g ^ { * } , e )$ does not hold. The same counterexample is applicable for match pairing and rank pairing. □

parse/train/SJlHwkBYDH/SJlHwkBYDH.md

@@ -0,0 +1,347 @@
+# NESTEROV ACCELERATED GRADIENT AND SCALE INVARIANCE FOR ADVERSARIAL ATTACKS
+
+Jiadong Lin & Chuanbiao Song & Kun He ∗ School of Computer Science and Technology Huazhong University of Science and Technology Wuhan, 430074, China {jdlin,cbsong,brooklet60}@hust.edu.cn
+
+Liwei Wang   
+School of Electronics Engineering   
+and Computer Sciences, Peking University   
+Peking, China   
+wanglw@cis.pku.edu.cn
+
+John E. Hopcroft Department of Computer Science Cornell University, NY 14853, USA jeh@cs.cornell.edu
+
+# ABSTRACT
+
+Deep learning models are vulnerable to adversarial examples crafted by applying human-imperceptible perturbations on benign inputs. However, under the blackbox setting, most existing adversaries often have a poor transferability to attack other defense models. In this work, from the perspective of regarding the adversarial example generation as an optimization process, we propose two new methods to improve the transferability of adversarial examples, namely Nesterov Iterative Fast Gradient Sign Method (NI-FGSM) and Scale-Invariant attack Method (SIM). NI-FGSM aims to adapt Nesterov accelerated gradient into the iterative attacks so as to effectively look ahead and improve the transferability of adversarial examples. While SIM is based on our discovery on the scale-invariant property of deep learning models, for which we leverage to optimize the adversarial perturbations over the scale copies of the input images so as to avoid “overfitting” on the white-box model being attacked and generate more transferable adversarial examples. NI-FGSM and SIM can be naturally integrated to build a robust gradient-based attack to generate more transferable adversarial examples against the defense models. Empirical results on ImageNet dataset demonstrate that our attack methods exhibit higher transferability and achieve higher attack success rates than state-of-the-art gradient-based attacks.
+
+# 1 INTRODUCTION
+
+Deep learning models have been shown to be vulnerable to adversarial examples (Goodfellow et al., 2014; Szegedy et al., 2014), which are generated by applying human-imperceptible perturbations on benign input to result in the misclassification. In addition, adversarial examples have an intriguing property of transferability, where adversarial examples crafted by the current model can also fool other unknown models. As adversarial examples can help identify the robustness of models (Arnab et al., 2018), as well as improve the robustness of models by adversarial training (Goodfellow et al., 2014), learning how to generate adversarial examples with high transferability is important and has gained increasing attentions in the literature (Liu et al., 2016; Dong et al., 2018; Xie et al., 2019; Dong et al., 2019; Wang et al., 2019).
+
+Several gradient-based attacks have been proposed to generate adversarial examples, such as onestep attacks (Goodfellow et al., 2014) and iterative attacks (Kurakin et al., 2016; Dong et al., 2018). Under the white-box setting, with the knowledge of the current model, existing attacks can achieve high success rates. However, they often exhibit low success rates under the black-box setting, especially for models with defense mechanism, such as adversarial training (Madry et al., 2018; Song et al., 2019) and input modification (Liao et al., 2018; Xie et al., 2018). Under the black-box setting, most existing attacks fail to generate robust adversarial examples against defense models.
+
+In this work, by regarding the adversarial example generation process as an optimization process, we propose two new methods to improve the transferability of adversarial examples: Nesterov Iterative Fast Gradient Sign Method (NI-FGSM) and Scale-Invariant attack Method (SIM).
+
+• Inspired by the fact that Nesterov accelerated gradient (Nesterov, 1983) is superior to momentum for conventionally optimization (Sutskever et al., 2013), we adapt Nesterov accelerated gradient into the iterative gradient-based attack, so as to effectively look ahead and improve the transferability of adversarial examples. We expect that NI-FGSM could replace the momentum iterative gradient-based method (Dong et al., 2018) in the gradient accumulating portion and yield higher performance.   
+• Besides, we discover that deep learning models have the scale-invariant property, and propose a Scale-Invariant attack Method (SIM) to improve the transferability of adversarial examples by optimizing the adversarial perturbations over the scale copies of the input images. SIM can avoid “overfitting” on the white-box model being attacked and generate more transferable adversarial examples against other black-box models.   
+• We found that combining our NI-FGSM and SIM with existing gradient-based attack methods (e.g., diverse input method (Xie et al., 2019)) can further boost the attack success rates of adversarial examples.
+
+Extensive experiments on the ImageNet dataset (Russakovsky et al., 2015) show that our methods attack both normally trained models and adversarially trained models with higher attack success rates than existing baseline attacks. Our best attack method, SI-NI-TI-DIM (Scale-Invariant Nesterov Iterative FGSM integrated with translation-invariant diverse input method), reaches an average success rate of $9 3 . 5 \%$ against adversarially trained models under the black-box setting. For further demonstration, we evaluate our methods by attacking the latest robust defense methods (Liao et al., 2018; Xie et al., 2018; Liu et al., 2019; Jia et al., 2019; Cohen et al., 2019). The results show that our attack methods can generate adversarial examples with higher transferability than state-of-theart gradient-based attacks.
+
+# 2 PRELIMINARY
+
+# 2.1 NOTATION
+
+Let $x$ and $y ^ { t r u e }$ be a benign image and the corresponding true label, respectively. Let ${ J } ( x , y ^ { t r u e } )$ be the loss function of the classifier (e.g. the cross-entropy loss). Let $x ^ { a d \hat { v } }$ be the adversarial example of the benign image $x$ . The goal of the non-targeted adversaries is to search an adversarial example $x ^ { a d v }$ to maximize the loss $\mathsf { \bar { J } } ( x ^ { a d v } , y ^ { t r u e } )$ in the $\ell _ { p }$ norm bounded perturbations. To align with previous works, we focus on $p = \infty$ in this work to measure the distortion between $x ^ { a d v }$ and $x$ . That is $\left\| x ^ { a d v } - x \right\| _ { \infty } \leq \epsilon$ , where $\epsilon$ is the magnitude of adversarial perturbations.
+
+# 2.2 ATTACK METHODS
+
+Several attack methods have been proposed to generate adversarial examples. Here we provide a brief introduction.
+
+Fast Gradient Sign Method (FGSM). FGSM (Goodfellow et al., 2014) generates an adversarial example $x ^ { a d v }$ by maximizing the loss function $J ( x ^ { a d v } , y ^ { t r u e } )$ with one-step update as:
+
+$$
+\begin{array} { r } { x ^ { a d v } = x + \epsilon \cdot \mathrm { s i g n } ( \nabla _ { x } J ( x , y ^ { t r u e } ) ) , } \end{array}
+$$
+
+where $\mathrm { s i g n } ( \cdot )$ function restricts the perturbation in the $L _ { \infty }$ norm bound.
+
+Iterative Fast Gradient Sign Method (I-FGSM). Kurakin et al. (2016) extend FGSM to an iterative version by applying FGSM with a small step size $\alpha$ :
+
+$$
+\begin{array} { r } { \boldsymbol { x } _ { 0 } = \boldsymbol { x } , ~ \boldsymbol { x } _ { t + 1 } ^ { a d v } = \mathrm { C l i p } _ { x } ^ { \epsilon } \{ \boldsymbol { x } _ { t } ^ { a d v } + \alpha \cdot \mathrm { s i g n } ( \nabla _ { x } J ( \boldsymbol { x } _ { t } ^ { a d v } , y ^ { t r u e } ) ) \} , } \end{array}
+$$
+
+where $\mathrm { C l i p } _ { x } ^ { \epsilon } ( \cdot )$ function restricts generated adversarial examples to be within the $\epsilon$ -ball of $x$
+
+Projected Gradient Descent (PGD). PGD attack (Madry et al., 2018) is a strong iterative variant of FGSM. It consists of a random start within the allowed norm ball and then follows by running several iterations of I-FGSM to generate adversarial examples.
+
+Momentum Iterative Fast Gradient Sign Method (MI-FGSM). Dong et al. (2018) integrate momentum into the iterative attack and lead to a higher transferability for adversarial examples. Their update procedure is formalized as follows:
+
+$$
+\begin{array} { r l } & { g _ { t + 1 } = \mu \cdot g _ { t } + \frac { \nabla _ { x } J ( x _ { t } ^ { a d v } , y ^ { t r u e } ) } { \left\| \nabla _ { x } J ( x _ { t } ^ { a d v } , y ^ { t r u e } ) \right\| _ { 1 } } , } \\ & { x _ { t + 1 } ^ { a d v } = \mathrm { C l i p } _ { x } ^ { \epsilon } \{ x _ { t } ^ { a d v } + \alpha \cdot \mathrm { s i g n } ( g _ { t + 1 } ) \} , } \end{array}
+$$
+
+where $g _ { t }$ is the accumulated gradient at iteration $t$ , and $\mu$ is the decay factor of $g _ { t }$
+
+Diverse Input Method (DIM). Xie et al. (2019) optimize the adversarial perturbations over the diverse transformation of the input image at each iteration. The transformations include the random resizing and the random padding. DIM can be naturally integrated into other gradient-based attacks to further improve the transferability of adversarial examples.
+
+Translation-Invariant Method (TIM). Instead of optimizing the adversarial perturbations on a single image, Dong et al. (2019) use a set of translated images to optimize the adversarial perturbations. They further develop an efficient algorithm to calculate the gradients by convolving the gradient at untranslated images with a kernel matrix. TIM can also be naturally integrated with other gradientbased attack methods. The combination of TIM and DIM, namely TI-DIM, is the current strongest black-box attack method.
+
+Carlini & Wagner attack (C&W). C&W attack (Carlini & Wagner, 2017) is an optimization-based method which directly optimizes the distance between the benign examples and the adversarial examples by solving:
+
+$$
+\operatorname * { a r g m i n } _ { x ^ { a d v } } \ \left\| x ^ { a d v } - x \right\| _ { p } - c \cdot J ( x ^ { a d v } , y ^ { t r u e } ) .
+$$
+
+It is a powerful method to find adversarial examples while minimizing perturbations for white-box attacks, but it lacks the transferability for black-box attacks.
+
+# 2.3 DEFENSE METHODS
+
+Various defense methods have been proposed to against adversarial examples, which can fall into the following two categories.
+
+Adversarial Training. One popular and promising defense method is adversarial training (Goodfellow et al., 2014; Szegedy et al., 2014; Zhai et al., 2019; Song et al., 2020), which augments the training data by the adversarial examples in the training process. Madry et al. (2018) develop a successful adversarial training method, which leverages the projected gradient descent (PGD) attack to generate adversarial examples. However, this method is difficult to scale to large-scale datasets (Kurakin et al., 2017). Tramr et al. (2018) propose ensemble adversarial training by augmenting the training data with perturbations transferred from various models , so as to further improve the robustness against the black-box attacks. Currently, adversarial training is still one of the best techniques to defend against adversarial attacks.
+
+Input Modification. The second category of defense methods aims to mitigate the effects of adversarial perturbations by modifying the input data. Guo et al. (2018) discover that there exists a range of image transformations, which have the potential to remove adversarial perturbations while preserving the visual information of the images. Xie et al. (2018) mitigate the adversarial effects through random transformations. Liao et al. (2018) propose high-level representation guided denoiser to purify the adversarial examples. Liu et al. (2019) propose a JPEG-based defensive compression framework to rectify adversarial examples without impacting classification accuracy on benign data. Jia et al. (2019) leverage an end-to-end image compression model to defend adversarial examples. Although these defense methods perform well in practice, they can not tell whether the model is truly robust to adversarial perturbations. Cohen et al. (2019) use randomized smoothing to obtain an ImageNet classifier with certified adversarial robustness.
+
+# 3 METHODOLOGY
+
+# 3.1 MOTIVATION
+
+Similar with the process of training neural networks, the process of generating adversarial examples can also be viewed as an optimization problem. In the optimizing phase, the white-box model being attacked to optimize the adversarial examples can be viewed as the training data on the training process. And the adversarial examples can be viewed as the training parameters of the model. Then in the testing phase, the black-box models to evaluate the adversarial examples can be viewed as the testing data of the model.
+
+From the perspective of the optimization, the transferability of the adversarial examples is similar with the generalization ability of the trained models (Dong et al., 2018). Thus, we can migrate the methods used to improve the generalization of models to the generation of adversarial examples, so as to improving the transferability of adversarial examples.
+
+Many methods have been proposed to improve the generalization ability of the deep learning models, which can be split to two aspects: (1) better optimization algorithm, such as Adam optimizer(Kingma & Ba, 2014); (2) data augmentation (Simonyan & Zisserman, 2014). Correspondingly, the methods to improve the transferability of adversarial examples can also be split to two aspects: (1) better optimization algorithm, such as MI-FGSM, which applies the idea of momentum; (2) model augmentation (i.e., ensemble attack on multiple models), such as the work of Dong et al. (2018), which considers to attack multiple models simultaneously. Based on above analysis, we aim to improve the transferability of adversarial examples by applying the idea of Nesterov accelerated gradient for optimization and using a set of scaled images to achieve model augmentation.
+
+# 3.2 NESTEROV ITERATIVE FAST GRADIENT SIGN METHOD
+
+Nesterov Accelerated Gradient (NAG) (Nesterov, 1983) is a slight variation of normal gradient descent, which can speed up the training process and improve the convergence significantly. NAG can be viewed as an improved momentum method, which can be expressed as:
+
+$$
+\begin{array} { r l } & { v _ { t + 1 } = \mu \cdot v _ { t } + \nabla _ { \theta _ { t } } J ( \theta _ { t } - \alpha \cdot \mu \cdot v _ { t } ) , } \\ & { \qquad \theta _ { t + 1 } = \theta _ { t } - \alpha \cdot v _ { t + 1 } . } \end{array}
+$$
+
+Typical gradient-based iterative attacks (e.g., I-FGSM) greedily perturb the images in the direction of the sign of the gradient at each iteration, which usually falls into poor local maxima, and shows weak transferability than single-step attacks (e.g., FGSM). Dong et al. (2018) show that adopting momentum (Polyak, 1964) into attacks can stabilize the update directions, which helps to escape from poor local maxima and improve the transferability. Compared to momentum, beyond stabilize the update directions, the anticipatory update of NAG gives previous accumulated gradient a correction that helps to effectively look ahead. Such looking ahead property of NAG can help us escape from poor local maxima easier and faster, resulting in the improvement on transferability.
+
+We integrate NAG into the iterative gradient-based attack to leverage the looking ahead property of NAG and build a robust adversarial attack, which we refer to as NI-FGSM (Nesterov Iterative Fast Gradient Sign Method). Specifically, we make a jump in the direction of previous accumulated gradients before computing the gradients in each iteration. Start with $g _ { 0 } = 0$ , the update procedure of NI-FGSM can be formalized as follows:
+
+$$
+x _ { t } ^ { n e s } = x _ { t } ^ { a d v } + \alpha \cdot \mu \cdot g _ { t } ,
+$$
+
+$$
+g _ { t + 1 } = \mu \cdot g _ { t } + \frac { \nabla _ { x } J ( x _ { t } ^ { n e s } , y ^ { t r u e } ) } { \| \nabla _ { x } J ( x _ { t } ^ { n e s } , y ^ { t r u e } ) \| _ { 1 } } ,
+$$
+
+$$
+\begin{array} { r } { x _ { t + 1 } ^ { a d v } = \mathrm { C l i p } _ { x } ^ { \epsilon } \{ x _ { t } ^ { a d v } + \alpha \cdot \mathrm { s i g n } ( g _ { t + 1 } ) \} , } \end{array}
+$$
+
+where $g _ { t }$ denotes the accumulated gradients at the iteration $t$ , and $\mu$ denotes the decay factor of $g _ { t }$ .
+
+# 3.3 SCALE-INVARIANT ATTACK METHOD
+
+Besides considering a better optimization algorithm for the adversaries, we can also improve the transferability of adversarial examples by model augmentation. We first introduce a formal definition of loss-preserving transformation and model augmentation as follows.
+
+Definition 1 Loss-preserving Transformation. Given an input $x$ with its ground-truth label $y ^ { t r u e }$ and a classifier $f ( x ) : x \in \mathcal { X } \to y \in \mathcal { Y }$ with the cross-entropy loss $J ( x , y )$ , if there exists an input transformation $\tau ( \cdot )$ that satisfies $J ( \mathcal { T } ( x ) , y ^ { t r u e } ) \approx J ( x , y ^ { t r u e } )$ for any $x \in \mathcal { X }$ , we say $\tau ( \cdot )$ is $a$ loss-preserving transformation.
+
+Definition 2 Model Augmentation. Given an input $x$ with its ground-truth label $y ^ { t r u e }$ and a model $f ( x ) : x \in \mathcal { X } \to y \in \mathcal { Y }$ with the cross-entropy loss $J ( x , y )$ , if there exists a loss-preserving transformation $\tau ( \cdot )$ , then we derive a new model by $f ^ { \prime } ( x ) = f ( \mathcal { T } ( x ) )$ from the original model $f$ . we define such derivation of models as model augmentation.
+
+Intuitively, similar to the generalization of models that can be improved by feeding more training data, the transferability of adversarial examples can be improved by attacking more models simultaneously. Dong et al. (2018) enhance the gradient-based attack by attacking an ensemble of models. However, their approach requires training a set of different models to attack, which has a large computational cost. Instead, in this work, we derive an ensemble of models from the original model by model augmentation, which is a simple way of obtaining multiple models via the loss-preserving transformation.
+
+To get the loss-preserving transformation, we discover that deep neural networks might have the scale-invariant property, besides the translation invariance. Specifically, the loss values are similar for the original and the scaled images on the same model, which is empirically validated in Section 4.2. Thus, the scale transformation can be served as a model augmentation method. Driven by the above analysis, we propose a Scale-Invariant attack Method (SIM), which optimizes the adversarial perturbations over the scale copies of the input image:
+
+$$
+\begin{array} { c } { { \displaystyle \arg \operatorname* { m a x } _ { \boldsymbol { x } ^ { a d v } } \frac { 1 } { m } \sum _ { i = 0 } ^ { m } J ( S _ { i } ( x ^ { a d v } ) , y ^ { t r u e } ) , } } \\ { { \mathrm { s . t . } \left\| x ^ { a d v } - x \right\| _ { \infty } \leq \epsilon , } } \end{array}
+$$
+
+where $S _ { i } ( x ) = x / 2 ^ { i }$ denotes the scale copy of the input image $x$ with the scale factor $1 / 2 ^ { i }$ , and $m$ denotes the number of the scale copies. With SIM, instead of training a set of models to attack, we can effectively achieve ensemble attacks on multiple models by model augmentation. More importantly, it can help avoid “overfitting” on the white-box model being attacked and generate more transferable adversarial examples.
+
+# 3.4 ATTACK ALGORITHM
+
+For the gradient processing of crafting adversarial examples, NI-FGSM introduces a better optimization algorithm to stabilize and correct the update directions at each iteration. For the ensemble attack of crafting adversarial examples, SIM introduces model augmentation to derive multiple models to attack from a single model. Thus, NI-FGSM and SIM can be naturally combined to build a stronger attack, which we refer to as SI-NI-FGSM (Scale-Invariant Nesterov Iterative Fast Gradient Sign Method). The algorithm of SI-NI-FGSM attack is summarized in Algorithm 1.
+
+In addition, SI-NI-FGSM can be integrated with DIM (Diverse Input Method), TIM (TranslationInvariant Method) and TI-DIM (Translation-Invariant with Diverse Input Method) as SI-NI-DIM, SINI-TIM and SI-NI-TI-DIM, respectively, to further boost the transferability of adversarial examples. The detailed algorithms for these attack methods are provided in Appendix A.
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, we provide experimental evidence on the advantage of the proposed methods. We first provide experimental setup, followed by the exploration of the scale-invariance property for deep learning models. We then compare the results of the proposed methods with baseline methods in Section 4.3 and 4.4 on both normally trained models and adversarially trained models. Beyond the defense models based on adversarial training, we also quantify the effectiveness of the proposed methods on other advanced defense in Section 4.5. Additional discussions, the comparison between NI-FGSM and MI-FGSM and the comparison with classic attacks, are in Section 4.6. Code is available at https://github.com/JHL-HUST/SI-NI-FGSM.
+
+# Algorithm 1 SI-NI-FGSM
+
+Input: A clean example $x$ with ground-truth label $y ^ { t r u e }$ ; a classifier $f$ with loss function $J$ ; Input: Perturbation size $\epsilon$ ; maximum iterations $T$ ; number of scale copies $m$ and decay factor $\mu$ Output: An adversarial example $x ^ { a d v }$
+
+1: $\alpha = \epsilon / T$   
+2: $g _ { 0 } = { \dot { 0 } } ; x _ { 0 } ^ { a d v } = x$   
+3: for $t = 0$ to $T - 1$ do   
+4: $g = 0$   
+5: Get $\boldsymbol { x } _ { t } ^ { n e s }$ by Eq.(6) $\triangleright$ make a jump in the direction of previous accumulated gradients   
+6: for $i = 0$ to $m - 1$ do $\triangleright$ sum the gradients over the scale copies of the input image   
+7: Get the gradients by $\nabla _ { x } J ( S _ { i } ( x _ { t } ^ { n e s } ) , y ^ { t r u e } )$   
+8: Sum the gradients as $g = g + \nabla _ { x } J ( S _ { i } ( x _ { t } ^ { n e s } ) , y ^ { t r u e } )$   
+9: Get average gradients as $\begin{array} { r } { g = \frac { 1 } { m } \cdot g } \end{array}$   
+10: Update gt+1 by gt+1 = µ · gt + gkgk   
+11: Update $x _ { t + 1 } ^ { a d v }$ by Eq.(8)   
+12: return xadv $x ^ { a d v } = x _ { T } ^ { a d v }$
+
+# 4.1 EXPERIMENTAL SETUP
+
+Dataset. We randomly choose 1000 images belonging to the 1000 categories from ILSVRC 2012 validation set, which are almost correctly classified by all the testing models.
+
+Models. For normally trained models, we consider Inception-v3 (Inc-v3) (Szegedy et al., 2016), Inception-v4 (Inc-v4), Inception-Resnet-v2 (IncRes-v2) (Szegedy et al., 2017) and Resnet-v2-101 (Res-101) (He et al., 2016). For adversarially trained models, we consider Inc- $\mathbf { \nabla \cdot v 3 _ { e n s 3 } }$ , Inc- $\mathbf { \nabla \cdot v } 3 _ { \mathrm { e n s 4 } }$ and IncRes- $\mathbf { \nabla \cdot v } 2 _ { \mathrm { e n s } }$ (Tramr et al., 2018).
+
+Additionally, we include other advanced defense models: high-level representation guided denoiser (HGD) (Liao et al., 2018), random resizing and padding (R&P) (Xie et al., 2018), NIPS- $. \mathrm { r } 3 ^ { 1 }$ , feature distillation (FD) (Liu et al., 2019), purifying perturbations via image compression model (Comdefend) (Jia et al., 2019) and randomized smoothing (RS) (Cohen et al., 2019).
+
+Baselines. We integrate our methods with DIM (Xie et al., 2019), TIM, and TI-DIM (Dong et al., 2019), to show the performance improvement of SI-NI-FGSM over these baselines. Denote our SI-NI-FGSM integrated with other attacks as SI-NI-DIM, SI-NI-TIM, and SI-NI-TIM-DIM, respectively.
+
+Hyper-parameters. For the hyper-parameters, we follow the settings in (Dong et al., 2018) with the maximum perturbation as $\epsilon = 1 6$ , number of iteration $T = 1 0$ , and step size $\alpha = 1 . 6$ . For MI-FGSM, we adopt the default decay factor $\mu = 1 . 0$ . For DIM, the transformation probability is set to 0.5. For TIM, we adopt the Gaussian kernel and the size of the kernel is set to $7 \times 7$ . For our SI-NI-FGSM, the number of scale copies is set to $m = 5$ .
+
+# 4.2 SCALE-INVARIANT PROPERTY
+
+To validate the scale-invariant property of deep neural networks, we randomly choose 1,000 original images from ImageNet dataset and keep the scale size in the range of [0.1, 2.0] with a step size 0.1. Then we feed the scaled images into the testing models, including Inc-v3, Inc-v4, IncRes-2, and Res-101, to get the average loss over 1,000 images.
+
+As shown in Figure 1, we can easily observe that the loss curves are smooth and stable when the scale size is in range [0.1, 1.3]. That is, the loss values are very similar for the original and scaled images. So we assume that the scale-invariant property of deep models is held within [0.1, 1.3], and we leverage the scale-invariant property to optimize the adversarial perturbations over the scale copies of the input images.
+
+![](images/7a7b0e5c2c1f2f0191bc39ec4a8ac4b070e2feced4fd9556cccca37367fd1c02.jpg)  
+Figure 1: The average losses for Inc-v3, Inc-v4, IncRes-v2 and Res-101 at each scale size. The results are averaged over 1000 images.
+
+# 4.3 ATTACKING A SINGLE MODEL
+
+In this subsection, we integrate our SI-NI-FGSM with TIM, DIM and TI-DIM, respectively, and compare the black-box attack success rates of our extensions with the baselines under single model setting. As shown in Table 1, our extension methods consistently outperform the baseline attacks by $1 0 \% \sim 3 5 \%$ under the black-box setting, and achieve nearly $100 \%$ success rates under the white-box setting. It indicates that SI-NI-FGSM can serve as a powerful approach to improve the transferability of adversarial examples.
+
+# 4.4 ATTACKING AN ENSEMBLE OF MODELS
+
+Following the work of (Liu et al., 2016), we consider to show the performance of our methods by attacking multiple models simultaneously. Specifically, we attack an ensemble of normally trained models (including Inc-v3, Inc-v4, IncRes-v2 and Res-101) with equal ensemble weights using TIM, SI-NI-TIM, DIM, SI-NI-DIM, TI-DIM and SI-NI-TI-DIM, respectively.
+
+As shown in Table 2, our methods improve the attack success rates across all experiments over the baselines. In general, our methods consistently outperform the baseline attacks by $1 0 \% \sim 3 0 \%$ under the black-box setting. Especially, SI-NI-TI-DIM, the extension by combining SI-NI-FGSM with TI-DIM, can fool the adversarially trained models with a high average success rate of $9 3 . 5 \%$ . It indicates that these advanced adversarially trained models provide little robustness guarantee under the black-box attack of SI-NI-TI-DIM.
+
+# 4.5 ATTACKING OTHER ADVANCED DEFENSE MODELS
+
+Besides normally trained models and adversarially trained models, we consider to quantify the effectiveness of our methods on other advanced defenses, including the top-3 defense solutions in the NIPS competition (high-level representation guided denoiser (HGD, rank-1) (Liao et al., 2018), random resizing and padding (R&P, rank-2) (Xie et al., 2018) and the rank-3 submission (NIPS-r3), and three recently proposed defense methods (feature distillation (FD) (Liu et al., 2019), purifying perturbations via image compression model (Comdefend) (Jia et al., 2019) and randomized smoothing (RS) (Cohen et al., 2019)).
+
+We compare our SI-NI-TI-DIM with MI-FGSM (Dong et al., 2018), which is the top-1 attack solution in the NIPS 2017 competition, and TI-DIM (Dong et al., 2019), which is state-of-the-art attack. We first generate adversarial examples on the ensemble models, including Inc-v3, Inc-v4, IncResv2, and Res-101 by using MI-FGSM, TI-DIM, and SI-NI-TI-DIM, respectively. Then, we evaluate the adversarial examples by attacking these defenses.
+
+As shown in Table 3, our method SI-NI-TI-DIM achieves an average attack success rate of $9 0 . 3 \%$ surpassing state-of-the-art attacks by a large margin of $1 4 . 7 \%$ . By solely depending on the trans
+
+# (a) Comparison of TIM and the SI-NI-TIM extension.
+
+Table 1: Attack success rates $( \% )$ of adversarial attacks against seven models under singlemodel setting. The adversarial examples are crafted on Inc-v3, Inc-v4, IncRes-v2, and Res-101 respectively. \* indicates the white-box attacks.   
+
+<table><tr><td>Model</td><td>Attack</td><td>Inc-v3</td><td>Inc-v4</td><td>IncRes-v2</td><td>Res-101</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td></tr><tr><td rowspan="2">Inc-v3</td><td>TIM</td><td>100.0*</td><td>47.8</td><td>42.8</td><td>39.5</td><td>24.0</td><td>21.4</td><td>12.9</td></tr><tr><td>SI-NI-TIM (Ours)</td><td>100.0*</td><td>77.2</td><td>75.8</td><td>66.5</td><td>51.8</td><td>45.9</td><td>33.5</td></tr><tr><td rowspan="2">Inc-v4</td><td>TIM</td><td>58.5</td><td>99.6*</td><td>47.5</td><td>43.2</td><td>25.7</td><td>23.3</td><td>17.3</td></tr><tr><td>SI-NI-TIM (Ours)</td><td>83.5</td><td>100.0*</td><td>76.6</td><td>68.9</td><td>57.8</td><td>54.3</td><td>42.9</td></tr><tr><td rowspan="2">IncRes-v2</td><td>TIM</td><td>62.0</td><td>56.2</td><td>97.5*</td><td>51.3</td><td>32.8</td><td>27.9</td><td>21.9</td></tr><tr><td>SI-NI-TIM(Ours)</td><td>86.4</td><td>83.2</td><td>99.5*</td><td>77.2</td><td>66.1</td><td>60.2</td><td>57.1</td></tr><tr><td rowspan="2">Res-101</td><td>TIM</td><td>59.0</td><td>53.6</td><td>51.8</td><td>99.3*</td><td>36.8</td><td>32.2</td><td>23.5</td></tr><tr><td>SI-NI-TIM (Ours)</td><td>78.3</td><td>74.1</td><td>73.0</td><td>99.8*</td><td>58.9</td><td>53.9</td><td>43.1</td></tr></table>
+
+(b) Comparison of DIM and the SI-NI-DIM extension.   
+
+<table><tr><td>Model</td><td>Attack</td><td>Inc-v3</td><td>Inc-v4</td><td>IncRes-v2</td><td>Res-101</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td></tr><tr><td rowspan="2">Inc-v3</td><td>DIM</td><td>98.7*</td><td>67.7</td><td>62.9</td><td>54.0</td><td>20.5</td><td>18.4</td><td>9.7</td></tr><tr><td>SI-NI-DIM(Ours)</td><td>99.6*</td><td>84.7</td><td>81.7</td><td>75.4</td><td>36.9</td><td>34.6</td><td>20.2</td></tr><tr><td rowspan="2">Inc-v4</td><td>DIM</td><td>70.7</td><td>98.0*</td><td>63.2</td><td>55.9</td><td>21.9</td><td>22.3</td><td>11.9</td></tr><tr><td>SI-NI-DIM (Ours)</td><td>89.7</td><td>99.3*</td><td>84.5</td><td>78.5</td><td>47.6</td><td>45.0</td><td>28.9</td></tr><tr><td rowspan="2">IncRes-v2</td><td>DIM</td><td>69.1</td><td>63.9</td><td>93.6*</td><td>57.4</td><td>29.4</td><td>24.0</td><td>17.3</td></tr><tr><td>SI-NI-DIM (Ours)</td><td>89.7</td><td>86.4</td><td>99.1*</td><td>81.2</td><td>55.0</td><td>48.2</td><td>38.1</td></tr><tr><td rowspan="2">Res-101</td><td>DIM</td><td>75.9</td><td>70.0</td><td>71.0</td><td>98.3*</td><td>36.0</td><td>32.4</td><td>19.3</td></tr><tr><td>SI-NI-DIM(Ours)</td><td>88.7</td><td>84.2</td><td>84.4</td><td>99.3*</td><td>52.4</td><td>48.0</td><td>33.2</td></tr></table>
+
+(c) Comparison of TI-DIM and the SI-NI-TI-DIM extension.   
+
+<table><tr><td>Model</td><td>Attack</td><td>Inc-v3</td><td>Inc-v4</td><td>IncRes-v2</td><td>Res-101</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td></tr><tr><td rowspan="2">Inc-v3</td><td>TI-DIM</td><td>98.5*</td><td>66.1</td><td>63.0</td><td>56.1</td><td>38.6</td><td>34.9</td><td>22.5</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>99.6*</td><td>85.5</td><td>80.9</td><td>75.7</td><td>61.5</td><td>56.9</td><td>40.7</td></tr><tr><td rowspan="2">Inc-v4</td><td>TI-DIM</td><td>72.5</td><td>97.8*</td><td>63.4</td><td>54.5</td><td>38.1</td><td>35.2</td><td>25.3</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>88.1</td><td>99.3*</td><td>83.7</td><td>77.0</td><td>65.0</td><td>63.1</td><td>49.4</td></tr><tr><td rowspan="2">IncRes-v2</td><td>TI-DIM</td><td>73.2</td><td>67.5</td><td>92.4*</td><td>61.3</td><td>46.4</td><td>40.2</td><td>35.8</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>89.6</td><td>87.0</td><td>99.1*</td><td>83.9</td><td>74.0</td><td>67.9</td><td>63.7</td></tr><tr><td rowspan="2">Res-101</td><td>TI-DIM</td><td>74.9</td><td>69.8</td><td>70.5</td><td>98.7*</td><td>52.6</td><td>49.1</td><td>37.8</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>86.4</td><td>82.6</td><td>84.6</td><td>99.0*</td><td>72.6</td><td>66.8</td><td>56.4</td></tr></table>
+
+Table 2: Attack success rates $( \% )$ of adversarial attacks against seven models under multimodel setting. \* indicates the white-box models being attacked.   
+
+<table><tr><td>Attack</td><td>Inc-v3*</td><td>Inc-v4*</td><td>IncRes-v2*</td><td>Res-101*</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td></tr><tr><td>TIM</td><td>99.9</td><td>99.3</td><td>99.3</td><td>99.8</td><td>71.6</td><td>67.0</td><td>53.2</td></tr><tr><td>SI-NI-TIM(Ours)</td><td>100.0</td><td>100.0</td><td>100.0</td><td>100.0</td><td>93.2</td><td>90.1</td><td>84.5</td></tr><tr><td>DIM</td><td>99.7</td><td>99.2</td><td>98.9</td><td>98.9</td><td>66.4</td><td>60.9</td><td>41.6</td></tr><tr><td>SI-NI-DIM (Ours)</td><td>100.0</td><td>100.0</td><td>100.0</td><td>99.9</td><td>88.2</td><td>85.1</td><td>69.7</td></tr><tr><td>TI-DIM</td><td>99.6</td><td>98.8</td><td>98.8</td><td>98.9</td><td>85.2</td><td>80.2</td><td>73.3</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>99.9</td><td>99.9</td><td>99.9</td><td>99.9</td><td>96.0</td><td>94.3</td><td>90.3</td></tr></table>
+
+ferability of adversarial examples and attacking on the normally trained models, SI-NI-TI-DIM can fool the adversarially trained models and other advanced defense mechanism, raising a new security issue for the development of more robust deep learning models. Some adversarial examples generated by SI-NI-TI-DIM are shown in Appendix B.
+
+Table 3: Attack success rates $( \% )$ of adversarial attacks against the advanced defense methods.   
+
+<table><tr><td>Attack</td><td>HGD</td><td>R&amp;P</td><td>NIPS-r3</td><td>FD</td><td>ComDefend</td><td>RS</td><td>Average</td></tr><tr><td>MI-FGSM</td><td>36.9</td><td>29.3</td><td>40.8</td><td>51.6</td><td>47.5</td><td>27.1</td><td>38.9</td></tr><tr><td>TI-DIM</td><td>84.8</td><td>75.3</td><td>80.7</td><td>83.5</td><td>79.1</td><td>50.3</td><td>75.6</td></tr><tr><td>SI-NI-TI-DIM (Ours)</td><td>96.1</td><td>91.3</td><td>94.4</td><td>95.4</td><td>93.3</td><td>71.4</td><td>90.3</td></tr></table>
+
+# 4.6 FURTHER ANALYSIS
+
+NI-FGSM vs. MI-FGSM. We perform additional analysis for the difference between NI-FGSM with MI-FGSM (Dong et al., 2018). The adversarial examples are crafted on Inc-v3 with various number of iterations ranging from 4 to 16, and then transfer to attack Inc-v4 and IncRes-v2. As shown in Figure 2, NI-FGSM yields higher attack success rates than MI-FGSM with the same number of iterations. In another view, NI-FGSM needs fewer number of iterations to gain the same attack success rate of MI-FGSM. The results not only indicate that NI-FGSM has a better transferability, but also demonstrate that with the property of looking ahead, NI-FGSM can accelerate the generation of adversarial examples.
+
+![](images/beff38286a98867ed12e1c3c6e0befc7110d9eebcfcd4d74e1818e65be7e46bc.jpg)  
+Figure 2: Attack success rates $( \% )$ of NI-FGSM and MI-FGSM on various number of iterations. The adversarial examples are crafted on Inc-v3 model against (a) Inc-v3 model, (b) Inc-v4 model and (c) IncRes-v2 model.
+
+Comparison with classic attacks. We consider to make addition comparison with classic attacks, including FGSM (Goodfellow et al., 2014), I-FGSM (Kurakin et al., 2016), PGD (Madry et al., 2018) and C&W (Carlini & Wagner, 2017). As shown in Table 4, our methods achieve $100 \%$ attack success rate which is the same as C&W under the white-box setting, and significantly outperform other methods under the black-box setting.
+
+Table 4: Attack success rates $( \% )$ of adversarial attacks against the models. The adversarial examples are crafted on Inc-v3 using FGSM, I-FGSM, PGD, C&W, NI-FGSM, and SI-NI-FGSM. \* indicates the white-box model being attacked.   
+
+<table><tr><td>Attack</td><td>Inc-v3*</td><td>Inc-v4</td><td>IncRes-v2</td><td>Res-101</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td><td>Average</td></tr><tr><td>FGSM</td><td>67.1</td><td>26.7</td><td>25.0</td><td>24.4</td><td>10.5</td><td>10.0</td><td>4.5</td><td>24.0</td></tr><tr><td>I-FGSM</td><td>99.9</td><td>20.7</td><td>18.5</td><td>15.3</td><td>3.6</td><td>5.8</td><td>2.9</td><td>23.8</td></tr><tr><td>PGD</td><td>99.5</td><td>17.3</td><td>15.1</td><td>13.1</td><td>6.1</td><td>5.6</td><td>3.1</td><td>20.9</td></tr><tr><td>C&amp;W</td><td>100.0</td><td>18.4</td><td>16.2</td><td>14.3</td><td>3.8</td><td>4.7</td><td>2.7</td><td>22.9</td></tr><tr><td>NI-FGSM(Ours)</td><td>100.0</td><td>52.6</td><td>51.4</td><td>41.0</td><td>12.9</td><td>12.8</td><td>6.4</td><td>39.6</td></tr><tr><td>SI-NI-FGSM(Ours)</td><td>100.0</td><td>76.0</td><td>73.3</td><td>67.6</td><td>31.6</td><td>30.0</td><td>17.4</td><td>56.6</td></tr></table>
+
+# 5 CONCLUSION AND FUTURE WORK
+
+In this work, we propose two new attack methods, namely Nesterov Iterative Fast Gradient Sign Method (NI-FGSM) and Scale-Invariant attack Method (SIM), to improve the transferability of adversarial examples. NI-FGSM aims to adopt Nesterov accelerated gradient method into the gradientbased attack, and SIM aims to achieve model augmentation by leveraging the scale-invariant property of models. NI-FGSM and SIM can be naturally combined to build a robust attack, namely SINI-FGSM. Moreover, by integrating SI-NI-FGSM with the baseline attacks, we can further improve the transferability of adversarial examples. Extensive experiments demonstrate that our methods not only yield higher success rates on adversarially trained models but also break other strong defense mechanism.
+
+Our work of NI-FGSM suggests that other momentum methods (e.g. Adam) may also be helpful to build a strong attack, which will be our future work, and the key is how to migrate the optimization method to the gradient-based iterative attack. Our work also shows that deep neural networks have the scale-invariant property, which we utilized to design the SIM to improve the attack transferability. However, it is not clear why the scale-invariant property holds. Possibly it is due to the batch normalization at each convolutional layer, that may mitigate the impact of the scale change. We will also explore the reason more thoroughly in our future work.
+
+# ACKNOWLEDGMENTS
+
+This work is supported by the Fundamental Research Funds for the Central Universities (2019kfyXKJC021) and Microsoft Research Asia.
+
+# REFERENCES
+
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+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In International Conference on Learning Representations, 2018.
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+Yurii Nesterov. A method for unconstrained convex minimization problem with the rate of convergence o $( 1 / \mathrm { k } ^ { \cdot } 2 )$ . In Doklady AN USSR, volume 269, pp. 543–547, 1983.
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+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211–252, 2015.
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+
+# A DETAILS OF THE ALGORITHMS
+
+The algorithm of SI-NI-TI-DIM attack is summarized in Algorithm 2. We can get the SI-NI-DIM attack algorithm by removing Step 10 of Algorithm 2, and get the SI-NI-TIM attack algorithm by removing $T ( \cdot ; p )$ in Step 7 of Algorithm 2.
+
+# Algorithm 2 SI-NI-TI-DIM
+
+Input: A clean example $x$ with ground-truth label $y ^ { t r u e }$ ; a classifier $f$ with loss function $J$ ; Input: Perturbation size $\epsilon$ ; maximum iterations $T$ ; number of scale copies $m$ and decay factor $\mu$ . Output: An adversarial example $x ^ { a d v }$
+
+1: $\alpha = \epsilon / T$   
+2 : g0 = 0; xadv0 = x   
+3: for $t = 0$ to $T - 1$ do   
+4: $g = 0$   
+5: Get $\boldsymbol { x } _ { t } ^ { n e s }$ by Eq.(6) $\triangleright$ make a jump in the direction of previous accumulated gradients   
+6: for $i = 0$ to $m - 1$ do $\triangleright$ sum the gradients over the scale copies of the input image   
+7: Get the gradients by $\nabla _ { \boldsymbol { x } } J ( T ( S _ { i } ( x _ { t } ^ { n e s } ) ; p ) , y ^ { t r u e } ) \ \circ$ apply random resizing and padding   
+to the inputs with the probability $p$   
+8: Sum the gradients as $g = \bar { g } + \nabla _ { x } J ( T ( S _ { i } ( x _ { t } ^ { n e s } ) ; p ) , y ^ { t r u e } )$   
+9: Get average gradients as $\begin{array} { r } { g = \frac { 1 } { m } \cdot g } \end{array}$   
+10: Convolve the gradients by $g = W * g \ \triangleright$ convolve gradient with the pre-defined kernel W   
+11: Update gt+1 by gt+1 = µ · gt + gkgk   
+12: Update $x _ { t + 1 } ^ { a d v }$ by Eq.(8)
+
+13: return $x ^ { a d v } = x _ { T } ^ { a d v }$
+
+# B VISUALIZATION OF ADVERSARIAL EXAMPLES
+
+We visualize 12 randomly selected benign images and their corresponding adversarial images in Figure 3. The adversarial images are crafted on the ensemble models, including Inc-v3, Inc-v4, IncRes-v2 and Res-101, using the proposed SI-NI-TI-DIM. We see that these generated adversarial perturbations are human imperceptible.
+
+![](images/0263936b72439ee188ead0f4050d98be89136e2374cb18d3b6e6cf2bddbe4389.jpg)  
+Figure 3: Visualization of randomly picked benign images and their corresponding adversarial images, crafted on the ensemble models using SI-NI-TI-DIM.

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+    {
+        "type": "text",
+        "text": "NESTEROV ACCELERATED GRADIENT AND SCALE INVARIANCE FOR ADVERSARIAL ATTACKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jiadong Lin & Chuanbiao Song & Kun He ∗ School of Computer Science and Technology Huazhong University of Science and Technology Wuhan, 430074, China {jdlin,cbsong,brooklet60}@hust.edu.cn ",
+        "bbox": [
+            184,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Liwei Wang   \nSchool of Electronics Engineering   \nand Computer Sciences, Peking University   \nPeking, China   \nwanglw@cis.pku.edu.cn ",
+        "bbox": [
+            562,
+            170,
+            843,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "John E. Hopcroft Department of Computer Science Cornell University, NY 14853, USA jeh@cs.cornell.edu ",
+        "bbox": [
+            184,
+            261,
+            423,
+            316
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            354,
+            544,
+            369
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning models are vulnerable to adversarial examples crafted by applying human-imperceptible perturbations on benign inputs. However, under the blackbox setting, most existing adversaries often have a poor transferability to attack other defense models. In this work, from the perspective of regarding the adversarial example generation as an optimization process, we propose two new methods to improve the transferability of adversarial examples, namely Nesterov Iterative Fast Gradient Sign Method (NI-FGSM) and Scale-Invariant attack Method (SIM). NI-FGSM aims to adapt Nesterov accelerated gradient into the iterative attacks so as to effectively look ahead and improve the transferability of adversarial examples. While SIM is based on our discovery on the scale-invariant property of deep learning models, for which we leverage to optimize the adversarial perturbations over the scale copies of the input images so as to avoid “overfitting” on the white-box model being attacked and generate more transferable adversarial examples. NI-FGSM and SIM can be naturally integrated to build a robust gradient-based attack to generate more transferable adversarial examples against the defense models. Empirical results on ImageNet dataset demonstrate that our attack methods exhibit higher transferability and achieve higher attack success rates than state-of-the-art gradient-based attacks. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            665,
+            336,
+            680
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning models have been shown to be vulnerable to adversarial examples (Goodfellow et al., 2014; Szegedy et al., 2014), which are generated by applying human-imperceptible perturbations on benign input to result in the misclassification. In addition, adversarial examples have an intriguing property of transferability, where adversarial examples crafted by the current model can also fool other unknown models. As adversarial examples can help identify the robustness of models (Arnab et al., 2018), as well as improve the robustness of models by adversarial training (Goodfellow et al., 2014), learning how to generate adversarial examples with high transferability is important and has gained increasing attentions in the literature (Liu et al., 2016; Dong et al., 2018; Xie et al., 2019; Dong et al., 2019; Wang et al., 2019). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Several gradient-based attacks have been proposed to generate adversarial examples, such as onestep attacks (Goodfellow et al., 2014) and iterative attacks (Kurakin et al., 2016; Dong et al., 2018). Under the white-box setting, with the knowledge of the current model, existing attacks can achieve high success rates. However, they often exhibit low success rates under the black-box setting, especially for models with defense mechanism, such as adversarial training (Madry et al., 2018; Song et al., 2019) and input modification (Liao et al., 2018; Xie et al., 2018). Under the black-box setting, most existing attacks fail to generate robust adversarial examples against defense models. ",
+        "bbox": [
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+        "page_idx": 0
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In this work, by regarding the adversarial example generation process as an optimization process, we propose two new methods to improve the transferability of adversarial examples: Nesterov Iterative Fast Gradient Sign Method (NI-FGSM) and Scale-Invariant attack Method (SIM). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Inspired by the fact that Nesterov accelerated gradient (Nesterov, 1983) is superior to momentum for conventionally optimization (Sutskever et al., 2013), we adapt Nesterov accelerated gradient into the iterative gradient-based attack, so as to effectively look ahead and improve the transferability of adversarial examples. We expect that NI-FGSM could replace the momentum iterative gradient-based method (Dong et al., 2018) in the gradient accumulating portion and yield higher performance.   \n• Besides, we discover that deep learning models have the scale-invariant property, and propose a Scale-Invariant attack Method (SIM) to improve the transferability of adversarial examples by optimizing the adversarial perturbations over the scale copies of the input images. SIM can avoid “overfitting” on the white-box model being attacked and generate more transferable adversarial examples against other black-box models.   \n• We found that combining our NI-FGSM and SIM with existing gradient-based attack methods (e.g., diverse input method (Xie et al., 2019)) can further boost the attack success rates of adversarial examples. ",
+        "bbox": [
+            179,
+            189,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Extensive experiments on the ImageNet dataset (Russakovsky et al., 2015) show that our methods attack both normally trained models and adversarially trained models with higher attack success rates than existing baseline attacks. Our best attack method, SI-NI-TI-DIM (Scale-Invariant Nesterov Iterative FGSM integrated with translation-invariant diverse input method), reaches an average success rate of $9 3 . 5 \\%$ against adversarially trained models under the black-box setting. For further demonstration, we evaluate our methods by attacking the latest robust defense methods (Liao et al., 2018; Xie et al., 2018; Liu et al., 2019; Jia et al., 2019; Cohen et al., 2019). The results show that our attack methods can generate adversarial examples with higher transferability than state-of-theart gradient-based attacks. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARY ",
+        "text_level": 1,
+        "bbox": [
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+            542,
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+            559
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 NOTATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Let $x$ and $y ^ { t r u e }$ be a benign image and the corresponding true label, respectively. Let ${ J } ( x , y ^ { t r u e } )$ be the loss function of the classifier (e.g. the cross-entropy loss). Let $x ^ { a d \\hat { v } }$ be the adversarial example of the benign image $x$ . The goal of the non-targeted adversaries is to search an adversarial example $x ^ { a d v }$ to maximize the loss $\\mathsf { \\bar { J } } ( x ^ { a d v } , y ^ { t r u e } )$ in the $\\ell _ { p }$ norm bounded perturbations. To align with previous works, we focus on $p = \\infty$ in this work to measure the distortion between $x ^ { a d v }$ and $x$ . That is $\\left\\| x ^ { a d v } - x \\right\\| _ { \\infty } \\leq \\epsilon$ , where $\\epsilon$ is the magnitude of adversarial perturbations. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 ATTACK METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Several attack methods have been proposed to generate adversarial examples. Here we provide a brief introduction. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Fast Gradient Sign Method (FGSM). FGSM (Goodfellow et al., 2014) generates an adversarial example $x ^ { a d v }$ by maximizing the loss function $J ( x ^ { a d v } , y ^ { t r u e } )$ with one-step update as: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b4b071f2139dcc486fcc233881f6095f00899bb06a52db3f773910d26db09453.jpg",
+        "text": "$$\n\\begin{array} { r } { x ^ { a d v } = x + \\epsilon \\cdot \\mathrm { s i g n } ( \\nabla _ { x } J ( x , y ^ { t r u e } ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            800,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { s i g n } ( \\cdot )$ function restricts the perturbation in the $L _ { \\infty }$ norm bound. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Iterative Fast Gradient Sign Method (I-FGSM). Kurakin et al. (2016) extend FGSM to an iterative version by applying FGSM with a small step size $\\alpha$ : ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46b7226047f9e9502a64040e824fd3f13334a9a396c9cc4c4c43434fa341194f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\boldsymbol { x } _ { 0 } = \\boldsymbol { x } , ~ \\boldsymbol { x } _ { t + 1 } ^ { a d v } = \\mathrm { C l i p } _ { x } ^ { \\epsilon } \\{ \\boldsymbol { x } _ { t } ^ { a d v } + \\alpha \\cdot \\mathrm { s i g n } ( \\nabla _ { x } J ( \\boldsymbol { x } _ { t } ^ { a d v } , y ^ { t r u e } ) ) \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            882,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { C l i p } _ { x } ^ { \\epsilon } ( \\cdot )$ function restricts generated adversarial examples to be within the $\\epsilon$ -ball of $x$ ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Projected Gradient Descent (PGD). PGD attack (Madry et al., 2018) is a strong iterative variant of FGSM. It consists of a random start within the allowed norm ball and then follows by running several iterations of I-FGSM to generate adversarial examples. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Momentum Iterative Fast Gradient Sign Method (MI-FGSM). Dong et al. (2018) integrate momentum into the iterative attack and lead to a higher transferability for adversarial examples. Their update procedure is formalized as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/83053f12b0a474579110c6eb78b2f0be0b3980c7f4e10a4555f3843cad3c7e8b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { g _ { t + 1 } = \\mu \\cdot g _ { t } + \\frac { \\nabla _ { x } J ( x _ { t } ^ { a d v } , y ^ { t r u e } ) } { \\left\\| \\nabla _ { x } J ( x _ { t } ^ { a d v } , y ^ { t r u e } ) \\right\\| _ { 1 } } , } \\\\ & { x _ { t + 1 } ^ { a d v } = \\mathrm { C l i p } _ { x } ^ { \\epsilon } \\{ x _ { t } ^ { a d v } + \\alpha \\cdot \\mathrm { s i g n } ( g _ { t + 1 } ) \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            205,
+            629,
+            266
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $g _ { t }$ is the accumulated gradient at iteration $t$ , and $\\mu$ is the decay factor of $g _ { t }$ ",
+        "bbox": [
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+            272,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Diverse Input Method (DIM). Xie et al. (2019) optimize the adversarial perturbations over the diverse transformation of the input image at each iteration. The transformations include the random resizing and the random padding. DIM can be naturally integrated into other gradient-based attacks to further improve the transferability of adversarial examples. ",
+        "bbox": [
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+            292,
+            825,
+            349
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Translation-Invariant Method (TIM). Instead of optimizing the adversarial perturbations on a single image, Dong et al. (2019) use a set of translated images to optimize the adversarial perturbations. They further develop an efficient algorithm to calculate the gradients by convolving the gradient at untranslated images with a kernel matrix. TIM can also be naturally integrated with other gradientbased attack methods. The combination of TIM and DIM, namely TI-DIM, is the current strongest black-box attack method. ",
+        "bbox": [
+            174,
+            356,
+            825,
+            440
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Carlini & Wagner attack (C&W). C&W attack (Carlini & Wagner, 2017) is an optimization-based method which directly optimizes the distance between the benign examples and the adversarial examples by solving: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d6113649875aa91d28a93011eb256ac5de328427253ec01ff7a86816f4b34344.jpg",
+        "text": "$$\n\\operatorname * { a r g m i n } _ { x ^ { a d v } } \\ \\left\\| x ^ { a d v } - x \\right\\| _ { p } - c \\cdot J ( x ^ { a d v } , y ^ { t r u e } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            491,
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+            520
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It is a powerful method to find adversarial examples while minimizing perturbations for white-box attacks, but it lacks the transferability for black-box attacks. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 DEFENSE METHODS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            577,
+            356,
+            590
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Various defense methods have been proposed to against adversarial examples, which can fall into the following two categories. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Adversarial Training. One popular and promising defense method is adversarial training (Goodfellow et al., 2014; Szegedy et al., 2014; Zhai et al., 2019; Song et al., 2020), which augments the training data by the adversarial examples in the training process. Madry et al. (2018) develop a successful adversarial training method, which leverages the projected gradient descent (PGD) attack to generate adversarial examples. However, this method is difficult to scale to large-scale datasets (Kurakin et al., 2017). Tramr et al. (2018) propose ensemble adversarial training by augmenting the training data with perturbations transferred from various models , so as to further improve the robustness against the black-box attacks. Currently, adversarial training is still one of the best techniques to defend against adversarial attacks. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Input Modification. The second category of defense methods aims to mitigate the effects of adversarial perturbations by modifying the input data. Guo et al. (2018) discover that there exists a range of image transformations, which have the potential to remove adversarial perturbations while preserving the visual information of the images. Xie et al. (2018) mitigate the adversarial effects through random transformations. Liao et al. (2018) propose high-level representation guided denoiser to purify the adversarial examples. Liu et al. (2019) propose a JPEG-based defensive compression framework to rectify adversarial examples without impacting classification accuracy on benign data. Jia et al. (2019) leverage an end-to-end image compression model to defend adversarial examples. Although these defense methods perform well in practice, they can not tell whether the model is truly robust to adversarial perturbations. Cohen et al. (2019) use randomized smoothing to obtain an ImageNet classifier with certified adversarial robustness. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            341,
+            118
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 MOTIVATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            132,
+            305,
+            147
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similar with the process of training neural networks, the process of generating adversarial examples can also be viewed as an optimization problem. In the optimizing phase, the white-box model being attacked to optimize the adversarial examples can be viewed as the training data on the training process. And the adversarial examples can be viewed as the training parameters of the model. Then in the testing phase, the black-box models to evaluate the adversarial examples can be viewed as the testing data of the model. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From the perspective of the optimization, the transferability of the adversarial examples is similar with the generalization ability of the trained models (Dong et al., 2018). Thus, we can migrate the methods used to improve the generalization of models to the generation of adversarial examples, so as to improving the transferability of adversarial examples. ",
+        "bbox": [
+            173,
+            248,
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+            305
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Many methods have been proposed to improve the generalization ability of the deep learning models, which can be split to two aspects: (1) better optimization algorithm, such as Adam optimizer(Kingma & Ba, 2014); (2) data augmentation (Simonyan & Zisserman, 2014). Correspondingly, the methods to improve the transferability of adversarial examples can also be split to two aspects: (1) better optimization algorithm, such as MI-FGSM, which applies the idea of momentum; (2) model augmentation (i.e., ensemble attack on multiple models), such as the work of Dong et al. (2018), which considers to attack multiple models simultaneously. Based on above analysis, we aim to improve the transferability of adversarial examples by applying the idea of Nesterov accelerated gradient for optimization and using a set of scaled images to achieve model augmentation. ",
+        "bbox": [
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+            311,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 NESTEROV ITERATIVE FAST GRADIENT SIGN METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            452,
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+            468
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Nesterov Accelerated Gradient (NAG) (Nesterov, 1983) is a slight variation of normal gradient descent, which can speed up the training process and improve the convergence significantly. NAG can be viewed as an improved momentum method, which can be expressed as: ",
+        "bbox": [
+            176,
+            478,
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+            521
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3f9b90e69325c50a859dbdd6b7f0acf0a5e140c76fb33bb0f65903393d1fb20.jpg",
+        "text": "$$\n\\begin{array} { r l } & { v _ { t + 1 } = \\mu \\cdot v _ { t } + \\nabla _ { \\theta _ { t } } J ( \\theta _ { t } - \\alpha \\cdot \\mu \\cdot v _ { t } ) , } \\\\ & { \\qquad \\theta _ { t + 1 } = \\theta _ { t } - \\alpha \\cdot v _ { t + 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            522,
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+            559
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Typical gradient-based iterative attacks (e.g., I-FGSM) greedily perturb the images in the direction of the sign of the gradient at each iteration, which usually falls into poor local maxima, and shows weak transferability than single-step attacks (e.g., FGSM). Dong et al. (2018) show that adopting momentum (Polyak, 1964) into attacks can stabilize the update directions, which helps to escape from poor local maxima and improve the transferability. Compared to momentum, beyond stabilize the update directions, the anticipatory update of NAG gives previous accumulated gradient a correction that helps to effectively look ahead. Such looking ahead property of NAG can help us escape from poor local maxima easier and faster, resulting in the improvement on transferability. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We integrate NAG into the iterative gradient-based attack to leverage the looking ahead property of NAG and build a robust adversarial attack, which we refer to as NI-FGSM (Nesterov Iterative Fast Gradient Sign Method). Specifically, we make a jump in the direction of previous accumulated gradients before computing the gradients in each iteration. Start with $g _ { 0 } = 0$ , the update procedure of NI-FGSM can be formalized as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae882f6c0ed7c46039b01fa610c9a7f9a9927e11a993e8dc3686a6d5681bdd6e.jpg",
+        "text": "$$\nx _ { t } ^ { n e s } = x _ { t } ^ { a d v } + \\alpha \\cdot \\mu \\cdot g _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/105dc7a0ebe4c4796cc7859bc3db0f1bfb588483804d40eb4308565442a16671.jpg",
+        "text": "$$\ng _ { t + 1 } = \\mu \\cdot g _ { t } + \\frac { \\nabla _ { x } J ( x _ { t } ^ { n e s } , y ^ { t r u e } ) } { \\| \\nabla _ { x } J ( x _ { t } ^ { n e s } , y ^ { t r u e } ) \\| _ { 1 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            773,
+            627,
+            808
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/99ff5bee7a05bd0393e1a9d6d6faf749f462895cd95868c3c9c25f62356d1601.jpg",
+        "text": "$$\n\\begin{array} { r } { x _ { t + 1 } ^ { a d v } = \\mathrm { C l i p } _ { x } ^ { \\epsilon } \\{ x _ { t } ^ { a d v } + \\alpha \\cdot \\mathrm { s i g n } ( g _ { t + 1 } ) \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
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+            625,
+            827
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $g _ { t }$ denotes the accumulated gradients at the iteration $t$ , and $\\mu$ denotes the decay factor of $g _ { t }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 SCALE-INVARIANT ATTACK METHOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            856,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Besides considering a better optimization algorithm for the adversaries, we can also improve the transferability of adversarial examples by model augmentation. We first introduce a formal definition of loss-preserving transformation and model augmentation as follows. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 Loss-preserving Transformation. Given an input $x$ with its ground-truth label $y ^ { t r u e }$ and a classifier $f ( x ) : x \\in \\mathcal { X } \\to y \\in \\mathcal { Y }$ with the cross-entropy loss $J ( x , y )$ , if there exists an input transformation $\\tau ( \\cdot )$ that satisfies $J ( \\mathcal { T } ( x ) , y ^ { t r u e } ) \\approx J ( x , y ^ { t r u e } )$ for any $x \\in \\mathcal { X }$ , we say $\\tau ( \\cdot )$ is $a$ loss-preserving transformation. ",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 2 Model Augmentation. Given an input $x$ with its ground-truth label $y ^ { t r u e }$ and a model $f ( x ) : x \\in \\mathcal { X } \\to y \\in \\mathcal { Y }$ with the cross-entropy loss $J ( x , y )$ , if there exists a loss-preserving transformation $\\tau ( \\cdot )$ , then we derive a new model by $f ^ { \\prime } ( x ) = f ( \\mathcal { T } ( x ) )$ from the original model $f$ . we define such derivation of models as model augmentation. ",
+        "bbox": [
+            173,
+            171,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, similar to the generalization of models that can be improved by feeding more training data, the transferability of adversarial examples can be improved by attacking more models simultaneously. Dong et al. (2018) enhance the gradient-based attack by attacking an ensemble of models. However, their approach requires training a set of different models to attack, which has a large computational cost. Instead, in this work, we derive an ensemble of models from the original model by model augmentation, which is a simple way of obtaining multiple models via the loss-preserving transformation. ",
+        "bbox": [
+            174,
+            239,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To get the loss-preserving transformation, we discover that deep neural networks might have the scale-invariant property, besides the translation invariance. Specifically, the loss values are similar for the original and the scaled images on the same model, which is empirically validated in Section 4.2. Thus, the scale transformation can be served as a model augmentation method. Driven by the above analysis, we propose a Scale-Invariant attack Method (SIM), which optimizes the adversarial perturbations over the scale copies of the input image: ",
+        "bbox": [
+            173,
+            344,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e63ac71b3ddecf63daaa109cdf6682c8cf7bd91f67dbebe358084bfd481c507c.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\displaystyle \\arg \\operatorname* { m a x } _ { \\boldsymbol { x } ^ { a d v } } \\frac { 1 } { m } \\sum _ { i = 0 } ^ { m } J ( S _ { i } ( x ^ { a d v } ) , y ^ { t r u e } ) , } } \\\\ { { \\mathrm { s . t . } \\left\\| x ^ { a d v } - x \\right\\| _ { \\infty } \\leq \\epsilon , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            433,
+            617,
+            498
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $S _ { i } ( x ) = x / 2 ^ { i }$ denotes the scale copy of the input image $x$ with the scale factor $1 / 2 ^ { i }$ , and $m$ denotes the number of the scale copies. With SIM, instead of training a set of models to attack, we can effectively achieve ensemble attacks on multiple models by model augmentation. More importantly, it can help avoid “overfitting” on the white-box model being attacked and generate more transferable adversarial examples. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 ATTACK ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            588,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the gradient processing of crafting adversarial examples, NI-FGSM introduces a better optimization algorithm to stabilize and correct the update directions at each iteration. For the ensemble attack of crafting adversarial examples, SIM introduces model augmentation to derive multiple models to attack from a single model. Thus, NI-FGSM and SIM can be naturally combined to build a stronger attack, which we refer to as SI-NI-FGSM (Scale-Invariant Nesterov Iterative Fast Gradient Sign Method). The algorithm of SI-NI-FGSM attack is summarized in Algorithm 1. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In addition, SI-NI-FGSM can be integrated with DIM (Diverse Input Method), TIM (TranslationInvariant Method) and TI-DIM (Translation-Invariant with Diverse Input Method) as SI-NI-DIM, SINI-TIM and SI-NI-TI-DIM, respectively, to further boost the transferability of adversarial examples. The detailed algorithms for these attack methods are provided in Appendix A. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            781,
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+            797
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide experimental evidence on the advantage of the proposed methods. We first provide experimental setup, followed by the exploration of the scale-invariance property for deep learning models. We then compare the results of the proposed methods with baseline methods in Section 4.3 and 4.4 on both normally trained models and adversarially trained models. Beyond the defense models based on adversarial training, we also quantify the effectiveness of the proposed methods on other advanced defense in Section 4.5. Additional discussions, the comparison between NI-FGSM and MI-FGSM and the comparison with classic attacks, are in Section 4.6. Code is available at https://github.com/JHL-HUST/SI-NI-FGSM. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 SI-NI-FGSM ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: A clean example $x$ with ground-truth label $y ^ { t r u e }$ ; a classifier $f$ with loss function $J$ ; Input: Perturbation size $\\epsilon$ ; maximum iterations $T$ ; number of scale copies $m$ and decay factor $\\mu$ Output: An adversarial example $x ^ { a d v }$ ",
+        "bbox": [
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+            127,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: $\\alpha = \\epsilon / T$   \n2: $g _ { 0 } = { \\dot { 0 } } ; x _ { 0 } ^ { a d v } = x$   \n3: for $t = 0$ to $T - 1$ do   \n4: $g = 0$   \n5: Get $\\boldsymbol { x } _ { t } ^ { n e s }$ by Eq.(6) $\\triangleright$ make a jump in the direction of previous accumulated gradients   \n6: for $i = 0$ to $m - 1$ do $\\triangleright$ sum the gradients over the scale copies of the input image   \n7: Get the gradients by $\\nabla _ { x } J ( S _ { i } ( x _ { t } ^ { n e s } ) , y ^ { t r u e } )$   \n8: Sum the gradients as $g = g + \\nabla _ { x } J ( S _ { i } ( x _ { t } ^ { n e s } ) , y ^ { t r u e } )$   \n9: Get average gradients as $\\begin{array} { r } { g = \\frac { 1 } { m } \\cdot g } \\end{array}$   \n10: Update gt+1 by gt+1 = µ · gt + gkgk   \n11: Update $x _ { t + 1 } ^ { a d v }$ by Eq.(8)   \n12: return xadv $x ^ { a d v } = x _ { T } ^ { a d v }$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+            390,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Dataset. We randomly choose 1000 images belonging to the 1000 categories from ILSVRC 2012 validation set, which are almost correctly classified by all the testing models. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Models. For normally trained models, we consider Inception-v3 (Inc-v3) (Szegedy et al., 2016), Inception-v4 (Inc-v4), Inception-Resnet-v2 (IncRes-v2) (Szegedy et al., 2017) and Resnet-v2-101 (Res-101) (He et al., 2016). For adversarially trained models, we consider Inc- $\\mathbf { \\nabla \\cdot v 3 _ { e n s 3 } }$ , Inc- $\\mathbf { \\nabla \\cdot v } 3 _ { \\mathrm { e n s 4 } }$ and IncRes- $\\mathbf { \\nabla \\cdot v } 2 _ { \\mathrm { e n s } }$ (Tramr et al., 2018). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Additionally, we include other advanced defense models: high-level representation guided denoiser (HGD) (Liao et al., 2018), random resizing and padding (R&P) (Xie et al., 2018), NIPS- $. \\mathrm { r } 3 ^ { 1 }$ , feature distillation (FD) (Liu et al., 2019), purifying perturbations via image compression model (Comdefend) (Jia et al., 2019) and randomized smoothing (RS) (Cohen et al., 2019). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Baselines. We integrate our methods with DIM (Xie et al., 2019), TIM, and TI-DIM (Dong et al., 2019), to show the performance improvement of SI-NI-FGSM over these baselines. Denote our SI-NI-FGSM integrated with other attacks as SI-NI-DIM, SI-NI-TIM, and SI-NI-TIM-DIM, respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Hyper-parameters. For the hyper-parameters, we follow the settings in (Dong et al., 2018) with the maximum perturbation as $\\epsilon = 1 6$ , number of iteration $T = 1 0$ , and step size $\\alpha = 1 . 6$ . For MI-FGSM, we adopt the default decay factor $\\mu = 1 . 0$ . For DIM, the transformation probability is set to 0.5. For TIM, we adopt the Gaussian kernel and the size of the kernel is set to $7 \\times 7$ . For our SI-NI-FGSM, the number of scale copies is set to $m = 5$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 SCALE-INVARIANT PROPERTY ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To validate the scale-invariant property of deep neural networks, we randomly choose 1,000 original images from ImageNet dataset and keep the scale size in the range of [0.1, 2.0] with a step size 0.1. Then we feed the scaled images into the testing models, including Inc-v3, Inc-v4, IncRes-2, and Res-101, to get the average loss over 1,000 images. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 1, we can easily observe that the loss curves are smooth and stable when the scale size is in range [0.1, 1.3]. That is, the loss values are very similar for the original and scaled images. So we assume that the scale-invariant property of deep models is held within [0.1, 1.3], and we leverage the scale-invariant property to optimize the adversarial perturbations over the scale copies of the input images. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/7a7b0e5c2c1f2f0191bc39ec4a8ac4b070e2feced4fd9556cccca37367fd1c02.jpg",
+        "image_caption": [
+            "Figure 1: The average losses for Inc-v3, Inc-v4, IncRes-v2 and Res-101 at each scale size. The results are averaged over 1000 images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 ATTACKING A SINGLE MODEL",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we integrate our SI-NI-FGSM with TIM, DIM and TI-DIM, respectively, and compare the black-box attack success rates of our extensions with the baselines under single model setting. As shown in Table 1, our extension methods consistently outperform the baseline attacks by $1 0 \\% \\sim 3 5 \\%$ under the black-box setting, and achieve nearly $100 \\%$ success rates under the white-box setting. It indicates that SI-NI-FGSM can serve as a powerful approach to improve the transferability of adversarial examples. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 ATTACKING AN ENSEMBLE OF MODELS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Following the work of (Liu et al., 2016), we consider to show the performance of our methods by attacking multiple models simultaneously. Specifically, we attack an ensemble of normally trained models (including Inc-v3, Inc-v4, IncRes-v2 and Res-101) with equal ensemble weights using TIM, SI-NI-TIM, DIM, SI-NI-DIM, TI-DIM and SI-NI-TI-DIM, respectively. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 2, our methods improve the attack success rates across all experiments over the baselines. In general, our methods consistently outperform the baseline attacks by $1 0 \\% \\sim 3 0 \\%$ under the black-box setting. Especially, SI-NI-TI-DIM, the extension by combining SI-NI-FGSM with TI-DIM, can fool the adversarially trained models with a high average success rate of $9 3 . 5 \\%$ . It indicates that these advanced adversarially trained models provide little robustness guarantee under the black-box attack of SI-NI-TI-DIM. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5 ATTACKING OTHER ADVANCED DEFENSE MODELS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Besides normally trained models and adversarially trained models, we consider to quantify the effectiveness of our methods on other advanced defenses, including the top-3 defense solutions in the NIPS competition (high-level representation guided denoiser (HGD, rank-1) (Liao et al., 2018), random resizing and padding (R&P, rank-2) (Xie et al., 2018) and the rank-3 submission (NIPS-r3), and three recently proposed defense methods (feature distillation (FD) (Liu et al., 2019), purifying perturbations via image compression model (Comdefend) (Jia et al., 2019) and randomized smoothing (RS) (Cohen et al., 2019)). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We compare our SI-NI-TI-DIM with MI-FGSM (Dong et al., 2018), which is the top-1 attack solution in the NIPS 2017 competition, and TI-DIM (Dong et al., 2019), which is state-of-the-art attack. We first generate adversarial examples on the ensemble models, including Inc-v3, Inc-v4, IncResv2, and Res-101 by using MI-FGSM, TI-DIM, and SI-NI-TI-DIM, respectively. Then, we evaluate the adversarial examples by attacking these defenses. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 3, our method SI-NI-TI-DIM achieves an average attack success rate of $9 0 . 3 \\%$ surpassing state-of-the-art attacks by a large margin of $1 4 . 7 \\%$ . By solely depending on the trans",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(a) Comparison of TIM and the SI-NI-TIM extension. ",
+        "text_level": 1,
+        "bbox": [
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+            155,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d3b1d5285c7f03d977bf9f3d462dc3221e780ee38d36a11c28e3b0a7fa80a70.jpg",
+        "table_caption": [
+            "Table 1: Attack success rates $( \\% )$ of adversarial attacks against seven models under singlemodel setting. The adversarial examples are crafted on Inc-v3, Inc-v4, IncRes-v2, and Res-101 respectively. \\* indicates the white-box attacks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Attack</td><td>Inc-v3</td><td>Inc-v4</td><td>IncRes-v2</td><td>Res-101</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td></tr><tr><td rowspan=\"2\">Inc-v3</td><td>TIM</td><td>100.0*</td><td>47.8</td><td>42.8</td><td>39.5</td><td>24.0</td><td>21.4</td><td>12.9</td></tr><tr><td>SI-NI-TIM (Ours)</td><td>100.0*</td><td>77.2</td><td>75.8</td><td>66.5</td><td>51.8</td><td>45.9</td><td>33.5</td></tr><tr><td rowspan=\"2\">Inc-v4</td><td>TIM</td><td>58.5</td><td>99.6*</td><td>47.5</td><td>43.2</td><td>25.7</td><td>23.3</td><td>17.3</td></tr><tr><td>SI-NI-TIM (Ours)</td><td>83.5</td><td>100.0*</td><td>76.6</td><td>68.9</td><td>57.8</td><td>54.3</td><td>42.9</td></tr><tr><td rowspan=\"2\">IncRes-v2</td><td>TIM</td><td>62.0</td><td>56.2</td><td>97.5*</td><td>51.3</td><td>32.8</td><td>27.9</td><td>21.9</td></tr><tr><td>SI-NI-TIM(Ours)</td><td>86.4</td><td>83.2</td><td>99.5*</td><td>77.2</td><td>66.1</td><td>60.2</td><td>57.1</td></tr><tr><td rowspan=\"2\">Res-101</td><td>TIM</td><td>59.0</td><td>53.6</td><td>51.8</td><td>99.3*</td><td>36.8</td><td>32.2</td><td>23.5</td></tr><tr><td>SI-NI-TIM (Ours)</td><td>78.3</td><td>74.1</td><td>73.0</td><td>99.8*</td><td>58.9</td><td>53.9</td><td>43.1</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/851922680914381b9b58f0f219eaedbc71969cacad6196a281a4a31655766dc1.jpg",
+        "table_caption": [
+            "(b) Comparison of DIM and the SI-NI-DIM extension. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Attack</td><td>Inc-v3</td><td>Inc-v4</td><td>IncRes-v2</td><td>Res-101</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td></tr><tr><td rowspan=\"2\">Inc-v3</td><td>DIM</td><td>98.7*</td><td>67.7</td><td>62.9</td><td>54.0</td><td>20.5</td><td>18.4</td><td>9.7</td></tr><tr><td>SI-NI-DIM(Ours)</td><td>99.6*</td><td>84.7</td><td>81.7</td><td>75.4</td><td>36.9</td><td>34.6</td><td>20.2</td></tr><tr><td rowspan=\"2\">Inc-v4</td><td>DIM</td><td>70.7</td><td>98.0*</td><td>63.2</td><td>55.9</td><td>21.9</td><td>22.3</td><td>11.9</td></tr><tr><td>SI-NI-DIM (Ours)</td><td>89.7</td><td>99.3*</td><td>84.5</td><td>78.5</td><td>47.6</td><td>45.0</td><td>28.9</td></tr><tr><td rowspan=\"2\">IncRes-v2</td><td>DIM</td><td>69.1</td><td>63.9</td><td>93.6*</td><td>57.4</td><td>29.4</td><td>24.0</td><td>17.3</td></tr><tr><td>SI-NI-DIM (Ours)</td><td>89.7</td><td>86.4</td><td>99.1*</td><td>81.2</td><td>55.0</td><td>48.2</td><td>38.1</td></tr><tr><td rowspan=\"2\">Res-101</td><td>DIM</td><td>75.9</td><td>70.0</td><td>71.0</td><td>98.3*</td><td>36.0</td><td>32.4</td><td>19.3</td></tr><tr><td>SI-NI-DIM(Ours)</td><td>88.7</td><td>84.2</td><td>84.4</td><td>99.3*</td><td>52.4</td><td>48.0</td><td>33.2</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/bb2702bc21016b6c191ad1f81a32dc91f4e39b97ea700dc9636c4827acedd4da.jpg",
+        "table_caption": [
+            "(c) Comparison of TI-DIM and the SI-NI-TI-DIM extension. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Attack</td><td>Inc-v3</td><td>Inc-v4</td><td>IncRes-v2</td><td>Res-101</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td></tr><tr><td rowspan=\"2\">Inc-v3</td><td>TI-DIM</td><td>98.5*</td><td>66.1</td><td>63.0</td><td>56.1</td><td>38.6</td><td>34.9</td><td>22.5</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>99.6*</td><td>85.5</td><td>80.9</td><td>75.7</td><td>61.5</td><td>56.9</td><td>40.7</td></tr><tr><td rowspan=\"2\">Inc-v4</td><td>TI-DIM</td><td>72.5</td><td>97.8*</td><td>63.4</td><td>54.5</td><td>38.1</td><td>35.2</td><td>25.3</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>88.1</td><td>99.3*</td><td>83.7</td><td>77.0</td><td>65.0</td><td>63.1</td><td>49.4</td></tr><tr><td rowspan=\"2\">IncRes-v2</td><td>TI-DIM</td><td>73.2</td><td>67.5</td><td>92.4*</td><td>61.3</td><td>46.4</td><td>40.2</td><td>35.8</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>89.6</td><td>87.0</td><td>99.1*</td><td>83.9</td><td>74.0</td><td>67.9</td><td>63.7</td></tr><tr><td rowspan=\"2\">Res-101</td><td>TI-DIM</td><td>74.9</td><td>69.8</td><td>70.5</td><td>98.7*</td><td>52.6</td><td>49.1</td><td>37.8</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>86.4</td><td>82.6</td><td>84.6</td><td>99.0*</td><td>72.6</td><td>66.8</td><td>56.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/7a65a35f0b7050470d7ed9a3641828f46273de1a19a0c1846280493362c005d4.jpg",
+        "table_caption": [
+            "Table 2: Attack success rates $( \\% )$ of adversarial attacks against seven models under multimodel setting. \\* indicates the white-box models being attacked. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>Inc-v3*</td><td>Inc-v4*</td><td>IncRes-v2*</td><td>Res-101*</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td></tr><tr><td>TIM</td><td>99.9</td><td>99.3</td><td>99.3</td><td>99.8</td><td>71.6</td><td>67.0</td><td>53.2</td></tr><tr><td>SI-NI-TIM(Ours)</td><td>100.0</td><td>100.0</td><td>100.0</td><td>100.0</td><td>93.2</td><td>90.1</td><td>84.5</td></tr><tr><td>DIM</td><td>99.7</td><td>99.2</td><td>98.9</td><td>98.9</td><td>66.4</td><td>60.9</td><td>41.6</td></tr><tr><td>SI-NI-DIM (Ours)</td><td>100.0</td><td>100.0</td><td>100.0</td><td>99.9</td><td>88.2</td><td>85.1</td><td>69.7</td></tr><tr><td>TI-DIM</td><td>99.6</td><td>98.8</td><td>98.8</td><td>98.9</td><td>85.2</td><td>80.2</td><td>73.3</td></tr><tr><td>SI-NI-TI-DIM(Ours)</td><td>99.9</td><td>99.9</td><td>99.9</td><td>99.9</td><td>96.0</td><td>94.3</td><td>90.3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ferability of adversarial examples and attacking on the normally trained models, SI-NI-TI-DIM can fool the adversarially trained models and other advanced defense mechanism, raising a new security issue for the development of more robust deep learning models. Some adversarial examples generated by SI-NI-TI-DIM are shown in Appendix B. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/937d47b806dda8a411643928f1e88eddd0b160fb493cd96388fc327dcb9601e5.jpg",
+        "table_caption": [
+            "Table 3: Attack success rates $( \\% )$ of adversarial attacks against the advanced defense methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>HGD</td><td>R&amp;P</td><td>NIPS-r3</td><td>FD</td><td>ComDefend</td><td>RS</td><td>Average</td></tr><tr><td>MI-FGSM</td><td>36.9</td><td>29.3</td><td>40.8</td><td>51.6</td><td>47.5</td><td>27.1</td><td>38.9</td></tr><tr><td>TI-DIM</td><td>84.8</td><td>75.3</td><td>80.7</td><td>83.5</td><td>79.1</td><td>50.3</td><td>75.6</td></tr><tr><td>SI-NI-TI-DIM (Ours)</td><td>96.1</td><td>91.3</td><td>94.4</td><td>95.4</td><td>93.3</td><td>71.4</td><td>90.3</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.6 FURTHER ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "NI-FGSM vs. MI-FGSM. We perform additional analysis for the difference between NI-FGSM with MI-FGSM (Dong et al., 2018). The adversarial examples are crafted on Inc-v3 with various number of iterations ranging from 4 to 16, and then transfer to attack Inc-v4 and IncRes-v2. As shown in Figure 2, NI-FGSM yields higher attack success rates than MI-FGSM with the same number of iterations. In another view, NI-FGSM needs fewer number of iterations to gain the same attack success rate of MI-FGSM. The results not only indicate that NI-FGSM has a better transferability, but also demonstrate that with the property of looking ahead, NI-FGSM can accelerate the generation of adversarial examples. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/beff38286a98867ed12e1c3c6e0befc7110d9eebcfcd4d74e1818e65be7e46bc.jpg",
+        "image_caption": [
+            "Figure 2: Attack success rates $( \\% )$ of NI-FGSM and MI-FGSM on various number of iterations. The adversarial examples are crafted on Inc-v3 model against (a) Inc-v3 model, (b) Inc-v4 model and (c) IncRes-v2 model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            256,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Comparison with classic attacks. We consider to make addition comparison with classic attacks, including FGSM (Goodfellow et al., 2014), I-FGSM (Kurakin et al., 2016), PGD (Madry et al., 2018) and C&W (Carlini & Wagner, 2017). As shown in Table 4, our methods achieve $100 \\%$ attack success rate which is the same as C&W under the white-box setting, and significantly outperform other methods under the black-box setting. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/0ed3d52723b66945cd611b0f7ce407409c4e189666e9b26e9abf6026a3b8eaee.jpg",
+        "table_caption": [
+            "Table 4: Attack success rates $( \\% )$ of adversarial attacks against the models. The adversarial examples are crafted on Inc-v3 using FGSM, I-FGSM, PGD, C&W, NI-FGSM, and SI-NI-FGSM. \\* indicates the white-box model being attacked. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>Inc-v3*</td><td>Inc-v4</td><td>IncRes-v2</td><td>Res-101</td><td>Inc-v3ens3</td><td>Inc-v3ens4</td><td>IncRes-v2ens</td><td>Average</td></tr><tr><td>FGSM</td><td>67.1</td><td>26.7</td><td>25.0</td><td>24.4</td><td>10.5</td><td>10.0</td><td>4.5</td><td>24.0</td></tr><tr><td>I-FGSM</td><td>99.9</td><td>20.7</td><td>18.5</td><td>15.3</td><td>3.6</td><td>5.8</td><td>2.9</td><td>23.8</td></tr><tr><td>PGD</td><td>99.5</td><td>17.3</td><td>15.1</td><td>13.1</td><td>6.1</td><td>5.6</td><td>3.1</td><td>20.9</td></tr><tr><td>C&amp;W</td><td>100.0</td><td>18.4</td><td>16.2</td><td>14.3</td><td>3.8</td><td>4.7</td><td>2.7</td><td>22.9</td></tr><tr><td>NI-FGSM(Ours)</td><td>100.0</td><td>52.6</td><td>51.4</td><td>41.0</td><td>12.9</td><td>12.8</td><td>6.4</td><td>39.6</td></tr><tr><td>SI-NI-FGSM(Ours)</td><td>100.0</td><td>76.0</td><td>73.3</td><td>67.6</td><td>31.6</td><td>30.0</td><td>17.4</td><td>56.6</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            766,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose two new attack methods, namely Nesterov Iterative Fast Gradient Sign Method (NI-FGSM) and Scale-Invariant attack Method (SIM), to improve the transferability of adversarial examples. NI-FGSM aims to adopt Nesterov accelerated gradient method into the gradientbased attack, and SIM aims to achieve model augmentation by leveraging the scale-invariant property of models. NI-FGSM and SIM can be naturally combined to build a robust attack, namely SINI-FGSM. Moreover, by integrating SI-NI-FGSM with the baseline attacks, we can further improve the transferability of adversarial examples. Extensive experiments demonstrate that our methods not only yield higher success rates on adversarially trained models but also break other strong defense mechanism. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our work of NI-FGSM suggests that other momentum methods (e.g. Adam) may also be helpful to build a strong attack, which will be our future work, and the key is how to migrate the optimization method to the gradient-based iterative attack. Our work also shows that deep neural networks have the scale-invariant property, which we utilized to design the SIM to improve the attack transferability. However, it is not clear why the scale-invariant property holds. Possibly it is due to the batch normalization at each convolutional layer, that may mitigate the impact of the scale change. We will also explore the reason more thoroughly in our future work. ",
+        "bbox": [
+            174,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This work is supported by the Fundamental Research Funds for the Central Universities (2019kfyXKJC021) and Microsoft Research Asia. ",
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+    {
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+        "text": "Cihang Xie, Zhishuai Zhang, Yuyin Zhou, Song Bai, Jianyu Wang, Zhou Ren, and Alan L Yuille. Improving transferability of adversarial examples with input diversity. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2730–2739, 2019. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Runtian Zhai, Tianle Cai, Di He, Chen Dan, Kun He, John E. Hopcroft, and Liwei Wang. Adversarially robust generalization just requires more unlabeled data. CoRR, abs/1906.00555, 2019. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A DETAILS OF THE ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            467,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The algorithm of SI-NI-TI-DIM attack is summarized in Algorithm 2. We can get the SI-NI-DIM attack algorithm by removing Step 10 of Algorithm 2, and get the SI-NI-TIM attack algorithm by removing $T ( \\cdot ; p )$ in Step 7 of Algorithm 2. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            176
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 SI-NI-TI-DIM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            193,
+            361,
+            208
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Input: A clean example $x$ with ground-truth label $y ^ { t r u e }$ ; a classifier $f$ with loss function $J$ ; Input: Perturbation size $\\epsilon$ ; maximum iterations $T$ ; number of scale copies $m$ and decay factor $\\mu$ . Output: An adversarial example $x ^ { a d v }$ ",
+        "bbox": [
+            173,
+            217,
+            812,
+            260
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "1: $\\alpha = \\epsilon / T$   \n2 : g0 = 0; xadv0 = x   \n3: for $t = 0$ to $T - 1$ do   \n4: $g = 0$   \n5: Get $\\boldsymbol { x } _ { t } ^ { n e s }$ by Eq.(6) $\\triangleright$ make a jump in the direction of previous accumulated gradients   \n6: for $i = 0$ to $m - 1$ do $\\triangleright$ sum the gradients over the scale copies of the input image   \n7: Get the gradients by $\\nabla _ { \\boldsymbol { x } } J ( T ( S _ { i } ( x _ { t } ^ { n e s } ) ; p ) , y ^ { t r u e } ) \\ \\circ$ apply random resizing and padding   \nto the inputs with the probability $p$   \n8: Sum the gradients as $g = \\bar { g } + \\nabla _ { x } J ( T ( S _ { i } ( x _ { t } ^ { n e s } ) ; p ) , y ^ { t r u e } )$   \n9: Get average gradients as $\\begin{array} { r } { g = \\frac { 1 } { m } \\cdot g } \\end{array}$   \n10: Convolve the gradients by $g = W * g \\ \\triangleright$ convolve gradient with the pre-defined kernel W   \n11: Update gt+1 by gt+1 = µ · gt + gkgk   \n12: Update $x _ { t + 1 } ^ { a d v }$ by Eq.(8) ",
+        "bbox": [
+            179,
+            266,
+            826,
+            455
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "13: return $x ^ { a d v } = x _ { T } ^ { a d v }$ ",
+        "bbox": [
+            176,
+            457,
+            336,
+            473
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B VISUALIZATION OF ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            505,
+            596,
+            521
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We visualize 12 randomly selected benign images and their corresponding adversarial images in Figure 3. The adversarial images are crafted on the ensemble models, including Inc-v3, Inc-v4, IncRes-v2 and Res-101, using the proposed SI-NI-TI-DIM. We see that these generated adversarial perturbations are human imperceptible. ",
+        "bbox": [
+            173,
+            536,
+            825,
+            593
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/0263936b72439ee188ead0f4050d98be89136e2374cb18d3b6e6cf2bddbe4389.jpg",
+        "image_caption": [
+            "Figure 3: Visualization of randomly picked benign images and their corresponding adversarial images, crafted on the ensemble models using SI-NI-TI-DIM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            608,
+            754,
+            875
+        ],
+        "page_idx": 11
+    }
+]

parse/train/Skeh1krtvH/Skeh1krtvH.md

@@ -0,0 +1,250 @@
+# WAVEFLOW: A COMPACT FLOW-BASED MODEL FOR RAW AUDIO
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In this work, we present WaveFlow, a small-footprint generative flow for raw audio, which is trained with maximum likelihood without density distillation and auxiliary losses as used in Parallel WaveNet. It provides a unified view of flowbased models for raw audio, including autoregressive flow (e.g., WaveNet) and bipartite flow (e.g., WaveGlow) as special cases. We systematically study these likelihood-based generative models for raw waveforms in terms of test likelihood and speech fidelity. We demonstrate that WaveFlow can synthesize high-fidelity speech and obtain comparable likelihood as WaveNet, while only requiring a few sequential steps to generate very long waveforms. In particular, our small-footprint WaveFlow has 5.91M parameters and can generate $2 2 . 0 5 \mathrm { k H z }$ high-fidelity speech $4 2 . 6 \times$ faster than real-time on a GPU without engineered inference kernels.
+
+# 1 INTRODUCTION
+
+Deep generative models have obtained noticeable successes for modeling raw audio in high-fidelity speech synthesis and music generation (e.g., van den Oord et al., 2016; Dieleman et al., 2018). Autoregressive models are among the best performing generative models for raw audio waveforms, providing the highest likelihood scores and generating high quality samples (e.g., van den Oord et al., 2016; Kalchbrenner et al., 2018). One of the most successful examples is WaveNet (van den Oord et al., 2016), an autoregressive model for waveform synthesis. It operates at the high temporal resolution of raw audio (e.g., 24kHz) and sequentially generates waveform samples at inference. As a result, WaveNet is prohibitively slow for speech synthesis and one has to develop highly engineered kernels for real-time inference (Arık et al., 2017a; Pharris, 2018). 2
+
+Flow-based models (Dinh et al., 2014; Rezende and Mohamed, 2015) are a family of generative models, in which a simple initial density is transformed into a complex one by applying a series of invertible transformations. One group of models are based on autoregressive transformation, including autoregressive flow (AF) and inverse autoregressive flow (IAF) as the “dual” of each other (Kingma et al., 2016; Papamakarios et al., 2017; Huang et al., 2018). AF is analogous to autoregressive models, which performs parallel density evaluation and sequential synthesis. In contrast, IAF performs parallel synthesis but sequential density evaluation, making likelihood-based training very slow. Parallel WaveNet (van den Oord et al., 2018) distills an IAF from a pretrained autoregressive WaveNet, which gets the best of both worlds. However, it requires the density distillation with Monte Carlo approximation and a set of auxiliary losses for good performance, which complicates the training pipeline and increases the cost of development. Instead, ClariNet (Ping et al., 2019) simplifies the density distillation by computing a regularized KL divergence in closed-form.
+
+Another group of flow-based models are based on bipartite transformation (Dinh et al., 2017; Kingma and Dhariwal, 2018), which provide parallel density evaluation and parallel synthesis. Most recently, WaveGlow (Prenger et al., 2019) and FloWaveNet (Kim et al., 2019) successfully applies Glow (Kingma and Dhariwal, 2018) and RealNVP (Dinh et al., 2017) for waveform synthesis, respectively. However, the bipartite transformations are less expressive than the autoregressive transformations (see Section 2.3 for detailed discussion). In general, these bipartite flows require deeper layers, larger hidden size, and huge number of parameters to reach comparable capacities as autoregressive models. For example, WaveGlow and FloWaveNet have 87.88M and 182.64M parameters with 96 layers and 256 residual channels, respectively. In contrast, a 30-layer WaveNet has only 4.57M parameters with 128 residual channels.
+
+In this work, we present WaveFlow, a compact flow-based model for raw audio. Specifically, we make the following contributions:
+
+1. WaveFlow is trained with maximum likelihood without density distillation and auxiliary losses used in Parallel WaveNet (van den Oord et al., 2018) and ClariNet (Ping et al., 2019), which simplifies the training pipeline and reduces the cost of development.   
+2. WaveFlow squeezes the 1-D raw waveforms into a 2-D matrix and produces the whole audio within a fixed sequential steps. It also provides a unified view of flow-based models for raw audio and allows us to explicitly trade inference efficiency for model capacity. We implement WaveFlow with a dilated 2-D convolutional architecture (Yu and Koltun, 2015), and it includes both Gaussian WaveNet (Ping et al., 2019) and WaveGlow (Prenger et al.,   
+2019) as special cases.   
+3. We systematically study the likelihood-based generative models for raw audios in terms of test likelihood and speech quality. We demonstrate that WaveFlow can obtain comparable likelihood and synthesize high-fidelity speech as WaveNet (van den Oord et al., 2016), while only requiring a few sequential steps to generate very long waveforms.   
+4. Our small-footprint WaveFlow has only 5.91M parameters and synthesizes $2 2 . 0 5 \mathrm { k H z }$ highfidelity speech (MOS: 4.32) more than $4 0 \times$ faster than real-time on a Nvidia V100 GPU. In contrast, WaveGlow (Prenger et al., 2019) requires 87.8M parameters for generating high-fidelity speech. The small memory footprint is preferred in production TTS systems, especially for on-device deployment.
+
+We organize the rest of the paper as follows. Section 2 reviews the flow-based models with autoregressive and bipartite transformations. We present WaveFlow in Section 3 and discuss related work in Section 4. We report experimental results in Section 5 and conclude the paper in Section 6.
+
+# 2 FLOW-BASED GENERATIVE MODELS
+
+Flow-based models (Dinh et al., 2014; 2017; Rezende and Mohamed, 2015) transform a simple density of latent variables $p ( z )$ (e.g., isotropic Gaussian) into a complex data distribution $p ( { \pmb x } )$ by applying a bijection $\begin{array} { r } { \mathbf { \boldsymbol { x } } = f ( \boldsymbol { z } ) } \end{array}$ , where $_ { \textbf { \em x } }$ and $_ z$ are both $n$ -dimensional. The probability density of $_ { \textbf { \em x } }$ can be obtained through the change of variables formula:
+
+$$
+p ( \pmb { x } ) = p ( z ) \left| \operatorname* { d e t } \left( \frac { \partial f ^ { - 1 } ( \pmb { x } ) } { \partial \pmb { x } } \right) \right| ,
+$$
+
+where $z = f ^ { - 1 } ( { \pmb x } )$ is the inverse transformation, and det $\big ( \frac { \partial { f ^ { - 1 } ( { \pmb x } ) } } { \partial { \pmb x } } \big )$ is the determinant of its Jacobian. In general, it takes $O ( n ^ { 3 } )$ to compute the determinant, which is not scalable to high-dimensional data. There are two notable groups of flow-based models with triangular Jacobians and tractable determinants. They are based on autoregressive and bipartite transformations, respectively.
+
+# 2.1 AUTOREGRESSIVE TRANSFORMATION
+
+The autoregressive flow (AF) and inverse autoregressive flow (IAF) (Kingma et al., 2016; Papamakarios et al., 2017) use autoregressive transformations. Specifically, AF defines the inverse transformation ${ \pmb z } = f ^ { - 1 } ( { \pmb x } ; \vartheta )$ as:
+
+$$
+z _ { t } = x _ { t } \cdot \sigma _ { t } ( x _ { < t } ; \pmb { \vartheta } ) + \mu _ { t } ( x _ { < t } ; \pmb { \vartheta } ) ,
+$$
+
+where the shifting variables $\mu _ { t } ( \boldsymbol { x } _ { < t } ; \boldsymbol { \vartheta } )$ and scaling variables $\sigma _ { t } ( x _ { < t } ; \vartheta )$ are modeled by an autoregressive architecture parameterized by $\vartheta$ (e.g., WaveNet). Note that, the $t$ -th variable $z _ { t }$ only depends on , thus the Jacobian is a triangular matrix as illustrated in Figure 1(a) and its determinant is the product of the diagonal entries: $\begin{array} { r } { \operatorname* { d e t } \left( \frac { \partial f ^ { - 1 } ( \pmb { x } ) } { \partial \pmb { x } } \right) = \prod _ { t } \sigma _ { t } ( \pmb { x } _ { < t } ; \pmb { \vartheta } ) } \end{array}$ . The density $p ( { \pmb x } )$ can be easily evaluated by change of variables formula, because $z = f ^ { - 1 } ( { \pmb x } )$ can be computed in parallel from Eq. (2) (i.e., the required $O ( n )$ operations can be done in $O ( 1 )$ time on modern GPU hardware). However, AF has to do sequential synthesis, because the forward transformation $\begin{array} { r } { { \bf x } = f ( z ) } \end{array}$ is autoregressive: xt = zt−µt(x<t;ϑ)σ (x ;ϑ) . In contrast, IAF uses an autoregressive transformation for $z = f ^ { - 1 } ( { \pmb x } )$ :
+
+![](images/daf5a61a74ad30d096f8cce589f021f56c0f47157d2b6026cff69270c14f49ca.jpg)  
+Figure 1: The Jacobian ∂f−1(x)∂x of (a) an autoregressive transformation, and (b) a bipartite transformation. The blank cells are 0s and represent the independent relations between $z _ { i }$ and $x _ { j }$ . The light-blue cells are scaling variables and represent the linear dependencies between $z _ { i }$ and $x _ { i }$ . The dark-blue cells represent complex non-linear dependencies defined by neural networks.
+
+$$
+z _ { t } = \frac { x _ { t } - \mu _ { t } ( z _ { < t } ; \vartheta ) } { \sigma _ { t } ( z _ { < t } ; \vartheta ) } ,
+$$
+
+making density evaluation impractically slow for training, but it can do parallel synthesis by $x _ { t } =$ $z _ { t } \cdot \sigma _ { t } ( z _ { < t } ; \vartheta ) + \mu _ { t } ( z _ { < t } ; \vartheta )$ . Parallel WaveNet (van den Oord et al., 2018) and ClariNet (Ping et al., 2019) are based on IAF, which lacks efficient density evaluation and relies on distillation from a pretrained autoregressive WaveNet.
+
+# 2.2 BIPARTITE TRANSFORMATION
+
+RealNVP (Dinh et al., 2017) and Glow (Kingma and Dhariwal, 2018) use bipartite transformation by partitioning the data $_ { \textbf { \em x } }$ into two groups $\scriptstyle { \mathbf { \mathcal { x } } } _ { a }$ and $\scriptstyle { \mathbf { { \mathit { x } } } } _ { b }$ , where the indices sets $a \cup b = \{ 1 , \cdots , n \}$ and $a \cap b = \phi$ . Then, the inverse transformation $z = f ^ { - 1 } ( x , \theta )$ is defined as:
+
+$$
+z _ { a } = x _ { a } , \quad z _ { b } = x _ { b } \cdot \sigma _ { b } ( x _ { a } ; \pmb \theta ) + \mu _ { b } ( x _ { a } ; \pmb \theta ) .
+$$
+
+where the shifting variables $\mu _ { b } ( x _ { a } ; \pmb \theta )$ and scaling variables $\sigma _ { b } ( x _ { a } ; \theta )$ are modeled by a feed-forward neural network. The Jacobian ∂f−1(x)∂x is a special triangular matrix as illustrated in Figure 1 (b). By definition, the forward transformation $\pmb { x } = f ( \pmb { z } , \pmb { \theta } )$ is,
+
+$$
+x _ { a } = z _ { a } , \quad x _ { b } = \frac { z _ { b } - \mu _ { b } ( x _ { a } ; \pmb \theta ) } { \sigma _ { b } ( x _ { a } ; \pmb \theta ) } ,
+$$
+
+and can also be done in parallel. As a result, the bipartite transformation provides both parallel density evaluation and parallel synthesis. In previous work, WaveGlow (Prenger et al., 2019) and FloWaveNet (Kim et al., 2019) both squeeze the adjacent audio samples on the channel dimension, and apply the bipartite transformation on the partitioned channel dimension.
+
+# 2.3 CONNECTIONS
+
+It is worthwhile to mention that the autoregressive transformation is more expressive than bipartite transformation in general. As illustrated in Figure 1(a) and (b), the autoregressive transformation introduces $\frac { n \times ( n - 1 ) } { 2 }$ complex non-linear dependencies (dark-blue cells) and $n$ linear dependencies between data $_ { \textbf { \em x } }$ and latents $_ z$ . In contrast, bipartite transformation introduces only $\frac { n ^ { 2 } } { 4 }$ non-linear dependencies and $\frac { n } { 2 }$ linear dependencies. Indeed, one can reduce an autoregressive transformation ${ \pmb z } = f ^ { - 1 } ( { \pmb x } ; \vartheta )$ to a bipartite transformation ${ z = f ^ { - 1 } ( x ; \theta ) }$ by: (i) picking an autoregressive order $^ o$ such that all of the indices in set $a$ rank early than the indices in $b$ , and (ii) setting the shifting and scaling variables as,
+
+$$
+\mu _ { t } ( \boldsymbol { x } _ { < t } ; \vartheta ) = \left\{ \begin{array} { l l } { 0 } & { \mathrm { f o r ~ } t \in a } \\ { \mu _ { t } ( \boldsymbol { x } _ { a } ; \theta ) } & { \mathrm { f o r ~ } t \in b } \end{array} \right. , \quad \sigma _ { t } ( \boldsymbol { x } _ { < t } ; \vartheta ) = \left\{ \begin{array} { l l } { 1 } & { \mathrm { f o r ~ } t \in a } \\ { \sigma _ { t } ( \boldsymbol { x } _ { a } ; \theta ) } & { \mathrm { f o r ~ } t \in b } \end{array} \right. .
+$$
+
+Given the less expressive building block, the bipartite transformation-based flows generally require many more layers and larger hidden size to match the capacity of a compact autoregressive models (e.g., as measured by test likelihood) (Kingma and Dhariwal, 2018; Prenger et al., 2019).
+
+# 3 WAVEFLOW
+
+In this section, we present WaveFlow and its implementation with dilated 2-D convolutions.
+
+# 3.1 DEFINITION
+
+We denote the high dimensional 1-D waveform as $\pmb { x } = \{ x _ { 1 } , \cdots , x _ { n } \}$ . We first squeeze $_ { \textbf { \em x } }$ into a $h$ -row 2-D matrix $\mathbf { \bar { \boldsymbol { X } } } \in \mathbb { R } ^ { h \times w }$ by column-major order, where $\begin{array} { r } { w = { \frac { n } { h } } } \end{array}$ and adjacent samples are in the same column. We assume $\boldsymbol { Z } \in \mathbb { R } ^ { h \times w }$ are sampled from an isotropic Gaussian, and define the inverse transformation $Z = f ^ { - 1 } ( X ; \Theta )$ as,
+
+$$
+Z _ { i , j } = \sigma _ { i , j } ( X _ { < i , \bullet } ; \Theta ) \cdot X _ { i , j } + \mu _ { i , j } ( X _ { < i , \bullet } ; \Theta ) ,
+$$
+
+where $X _ { < i , \bullet }$ represents all elements above $i$ -th row (see Figure 2 for an illustration). Note that, i) the receptive fields over the squeezed inputs $X$ for computing $Z _ { i , j }$ in WaveFlow is strictly larger than that of WaveGlow when $h > 2$ . ii) WaveNet is equivalent to an autoregressive flow with column-major order on the squeezed inputs $X$ . iii) Both WaveFlow and WaveGlow look at future waveform samples in original $_ { \textbf { \em x } }$ for computing $Z _ { i , j }$ , whereas WaveNet can not. iv) The autoregressive flow with row-major order has larger receptive fields than WaveFlow and WaveGlow.
+
+The shifting variables $\mu _ { i , j } ( X _ { < i , \bullet } ; \Theta )$ and scaling variables $\sigma _ { i , j } ( X _ { < i , \bullet } ; \Theta )$ in Eq. (6) are modeled by a 2-D convolutional neural network detailed in Section 3.2. By definition, the variable $Z _ { i , j }$ only depends on the current $X _ { i , j }$ and previous $X _ { < i , \bullet }$ in raw-major order, thus the Jacobian is a triangular matrix and its determinant is:
+
+$$
+\operatorname* { d e t } \left( \frac { \partial f ^ { - 1 } ( X ) } { \partial X } \right) = \prod _ { i = 1 } ^ { h } \prod _ { j = 1 } ^ { w } \sigma _ { i , j } ( X _ { < i , \bullet } ; \Theta ) .
+$$
+
+As a result, the log-likelihood can be calculated in parallel by change of variable formula in Eq. (1),
+
+$$
+\log p ( \boldsymbol { X } ) = - \sum _ { i = 1 } ^ { h } \sum _ { j = 1 } ^ { w } \left( Z _ { i , j } ^ { 2 } + \frac { 1 } { 2 } \log ( 2 \pi ) \right) + \sum _ { i = 1 } ^ { h } \sum _ { j = 1 } ^ { w } \log \sigma _ { i , j } ( X _ { < i , \bullet } ; \Theta ) ,
+$$
+
+and one can do maximum likelihood training efficiently. At synthesis, one may first sample $Z$ from the isotropic Gaussian and apply the forward transformation $X = f ( Z ; \Theta )$ :
+
+$$
+X _ { i , j } = \frac { Z _ { i , j } - \mu _ { i , j } ( X _ { < i , \bullet } ; \Theta ) } { \sigma _ { i , j } ( X _ { < i , \bullet } ; \Theta ) } ,
+$$
+
+which is only autoregressive on height dimension. Thus, it requires $h$ sequential steps to generate the whole waveform $X$ . In practice, a small $h$ (e.g., 8 or 16) works well, thus we can generate very long waveforms within a few sequential steps.
+
+# 3.2 IMPLEMENTATION WITH DILATED 2-D CONVOLUTIONS
+
+In this work, we implement WaveFlow with a dilated 2-D convolutional architecture. Specifically, we use a stack of 2-D convolution layers (e.g., 8 layers in all experiments) to model the shifting variables $\mu _ { i , j } ( X _ { < i , \bullet } ; \Theta )$ and scaling variables $\sigma _ { i , j } ( X _ { < i , \bullet } ; \Theta )$ in Eq. (6). We use the similar architecture as
+
+![](images/451fb9d0bedeea61af9e4fc3a8784df0fc7a453e7d6966601dfcf60555ea7947.jpg)  
+Figure 2: The receptive fields over the squeezed inputs $X$ for computing $Z _ { i , j }$ in (a) WaveFlow, (b) WaveGlow, (c) autoregressive flow with column-major order (e.g., WaveNet), and (d) autoregressive flow with row-major order.
+
+Table 1: The test log-likelihoods (LLs) of WaveFlow with different dilation cycles on the height dimension. Both models are stacked with 8 flows and each flow has 8 convolutional layers.   
+
+<table><tr><td>Model</td><td>res.channels</td><td>dilations d</td><td>receptive field r</td><td>test LLs</td></tr><tr><td>WaveFlow (h = 32)</td><td>128</td><td>1,1,1,1,1,1,1,1</td><td>17</td><td>4.960</td></tr><tr><td>WaveFlow (h = 32)</td><td>128</td><td>1,2,4,1,2,4,1,2</td><td>35</td><td>5.055</td></tr></table>
+
+Table 2: The heights and corresponding dilations used in our experiments. Note that, the receptive fields are only slightly larger than height $h$ .   
+
+<table><tr><td>Height h</td><td>filter size k</td><td>dilations d</td><td>receptive field r</td></tr><tr><td>8</td><td>3</td><td>1,1,1,1,1,1,1,1</td><td>17</td></tr><tr><td>16</td><td>3</td><td>1,1,1,1,1,1,1,1</td><td>17</td></tr><tr><td>32</td><td>3</td><td>1,2,4,1,2,4,1,2</td><td>35</td></tr><tr><td>64</td><td>3</td><td>1,2,4,8,16,1,2,4</td><td>77</td></tr></table>
+
+WaveNet (van den Oord et al., 2016) by replacing the dilated 1-D convolution to 2-D convolution (Yu and Koltun, 2015), while still keeping the gated-tanh nonlinearities, residual connections and skip connections.
+
+We set the filter sizes as 3 for both height and width dimensions. We use non-causal convolutions on width dimension and set the dilation cycle as $[ 1 , 2 , 4 , \cdots , 2 ^ { 7 } ]$ . The convolutions on height dimension are causal with an autoregressive constraint, and their dilation cycle needs to be designed carefully. In practice, we find the following rules of thumb are important to obtain good results:
+
+• As motivated by the dilation cycle of WaveNet (van den Oord et al., 2016), the dilations of 8 layers should be set as $\pmb { d } = [ 1 , 2 , \cdots , 2 ^ { s } , 1 , 2 , \cdots , 2 ^ { s } , \cdots ] ,$ , where $s \leq 7$ . 3 • The receptive field $r$ over the height dimension should be larger than the squeezed height $h$ Otherwise, it explicitly introduces unnecessary conditional independence and leads to lower likelihood (see Table 1 for an example). Note that, the receptive field of a stack of dilated convolutional layers is: $r = ( k { - } 1 ) { \times } \textstyle \sum _ { i } d _ { i } { + } 1$ , where $k$ is the filter size and $d _ { i }$ is the dilation at $i$ -th layer. Thus, the sum of dilations should satisfy: $\textstyle \sum _ { i } d _ { i } \geq { \frac { h - 1 } { k - 1 } }$ . However, when $h$ is larger than or equal to $2 ^ { 8 } = 5 1 2$ , we simply set the dilation cycle as $[ 1 , 2 , 4 , \cdots , 2 ^ { 7 } ]$ . When the receptive field $r$ has already been larger than $h$ , we find that convolutions with smaller dilation and fewer holes provide larger likelihood.
+
+We summarize the heights and preferred dilations in our experiments in Table 2. Note that, WaveFlow becomes fully autoregressive when we squeeze $_ { \textbf { \em x } }$ by its length (i.e. $h = n$ ) and set its filter size as 1 over the width dimension, which is equivalent to a Gaussian WaveNet learned by MLE (Ping et al., 2019). If we squeeze $_ { \textbf { \em x } }$ by $h = 2$ and set the filter size as 1 on the height dimension, WaveFlow becomes a bipartite flow and is equivalent to WaveGlow with squeezed channels 2.
+
+# 3.3 CONDITIONAL GENERATION
+
+In neural speech synthesis, a neural vocoder (e.g., WaveNet) synthesizes the time-domain waveforms. It can be conditioned on linguistic features (van den Oord et al., 2016; Arık et al., 2017a), the mel-spectrograms from a text-to-spectrogram model (Ping et al., 2018; Shen et al., 2018), or the learned hidden representation within a text-to-wave architecture (Ping et al., 2019). In this work, we test WaveFlow by conditioning it on ground truth mel-spectrograms as in previous work (Prenger et al., 2019; Kim et al., 2019). The mel-spectrogram is upsampled to the same resolution as waveform samples by transposed 2-D convolutions (Ping et al., 2019). To aligned with the squeezed waveform, they are squeezed to the shape $c \times h \times w$ , where $c$ is the feature dimension (e.g, bands of the spectrogram). After a $1 \times 1$ convolution mapping the features to residual channels, they are added as the bias term at each layer (van den Oord et al., 2016).
+
+Table 3: The test LLs of WaveFlow with different permutation strategies. All models consist of 8 flows and each flow has 8 convolutional layers with filter sizes 3.   
+
+<table><tr><td>Model</td><td>res.channels</td><td>permutation strategy</td><td>test LLs</td></tr><tr><td>WaveFlow (h = 16)</td><td>64</td><td>none</td><td>4.551</td></tr><tr><td>WaveFlow (h=1 16)</td><td>64</td><td>(i) 8 reverse</td><td>4.954</td></tr><tr><td>WaveFlow (h = 16)</td><td>64</td><td>(ii) 4 reverse +4 split &amp; reverse</td><td>4.971</td></tr></table>
+
+# 3.4 STACKING MULTIPLE FLOWS WITH PERMUTATIONS OVER HEIGHT DIMENSION
+
+Flow-based models require a series of transformations until the distribution $p ( X )$ reaches a desired level of complexity (e.g., Rezende and Mohamed, 2015). We let $X = Z ^ { ( n ) }$ and repeatedly apply the transformation $Z ^ { ( i - 1 ) } = f ^ { - 1 } ( Z ^ { ( i ) } ; \Theta ^ { ( i ) } )$ defined in Eq. (6) from $Z ^ { ( n ) } \to \dots Z ^ { ( \bar { i } ) } \to \dots \bar { Z } ^ { ( \bar { 0 } ) }$ . We assume $Z ^ { ( 0 ) }$ is from the isotropic Gaussian distribution. The likelihood $p ( X )$ can be evaluated by iteratively applying the chain rule:
+
+$$
+p ( X ) = p ( Z ^ { ( 0 ) } ) \prod _ { i = 1 } ^ { n } \left. \operatorname* { d e t } \left( \frac { \partial f ^ { - 1 } ( Z ^ { ( i ) } ; \Theta ^ { ( i ) } ) } { \partial Z ^ { ( i ) } } \right) \right. .
+$$
+
+We find that permuting each $Z ^ { ( i ) }$ over the height dimension after each transformation can significantly improve the likelihood scores. In particular, we test two permutation strategies for WaveFlow models stacked with 8 flows (i.e., $X = Z ^ { ( 8 ) }$ ) in Table 3: (i) we reverse each $Z ^ { ( i ) }$ over the height dimension after each transformation, and (ii) we reverse $Z ^ { ( 7 ) } , Z ^ { ( 6 ) } , Z ^ { ( 5 ) } , Z ^ { ( 4 ) }$ over the height dimension as before, but split $Z ^ { ( 3 ) } , Z ^ { ( 2 ) } , Z ^ { ( 1 ) } , Z ^ { ( 0 ) }$ in the middle of the height dimension then reverse each part respectively. 4 Note that, one also needs to permute the conditioner on the height dimension accordingly, which is aligned with $Z ^ { ( i ) }$ . From Table 3, both (i) and (ii) significantly outperform the model without permutations mainly because of bidirectional modeling. Strategy (ii) outperforms (i) because of its diverse autoregressive orders.
+
+# 4 RELATED WORK
+
+Deep neural networks for speech synthesis (a.k.a. text-to-speech) have received a lot of attention. Over the past few years, several neural text-to-speech (TTS) systems have been introduced, including WaveNet (van den Oord et al., 2016), Deep Voice (Arık et al., 2017a), Deep Voice 2 (Arık et al., 2017b), Deep Voice 3 (Ping et al., 2018), Tacotron (Wang et al., 2017), Tacotron 2 (Shen et al., 2018), Char2Wav (Sotelo et al., 2017), VoiceLoop (Taigman et al., 2018), WaveRNN (Kalchbrenner et al., 2018), ClariNet (Ping et al., 2019), Transformer TTS (Li et al., 2019), ParaNet (Peng et al., 2019) and FastSpeech (Ren et al., 2019).
+
+Neural vocoders, such as WaveNet, play the most important role in recent advances of speech synthesis. In previous work, the state-of-the-art neural vocoders are autoregressive models (van den Oord et al., 2016; Mehri et al., 2017; Kalchbrenner et al., 2018). Several engineering endeavors have been advocated for speeding up their sequential generation process (Arık et al., 2017a; Kalchbrenner et al., 2018). In particular, Subscale WaveRNN (Kalchbrenner et al., 2018) folds a long waveform sequence ${ \pmb x } _ { 1 : n }$ into a batch of shorter sequences and can produces up to 16 samples per step, thus it requires at least $\frac { n } { 1 6 }$ steps to generate the whole audio. Note that, this is different from the proposed WaveFlow, which can generate ${ \pmb x } _ { 1 : n }$ within a fixed number of steps (e.g., 16). Most recently, flowbased models have been successfully applied for parallel waveform synthesis with comparable fidelity as autoregressive models (van den Oord et al., 2018; Ping et al., 2019; Prenger et al., 2019; Kim et al., 2019; Yamamoto et al., 2019; Serrà et al., 2019). Among these models, WaveGlow (Prenger et al., 2019) and FloWaveNet (Kim et al., 2019) have a simple training pipeline as they solely use the maximum likelihood objective. However, both of them are less expressive than autoregressive models as indicated by their lower likelihood scores.
+
+Flow-based models can either represent the approximate posteriors for variational inference (Rezende and Mohamed, 2015; Kingma et al., 2016; Berg et al., 2018), or can be trained directly on data using the change of variables formula (Dinh et al., 2014; 2017; Kingma and Dhariwal, 2018; Grathwohl et al., 2018). In previous work, Glow (Kingma and Dhariwal, 2018) extends RealNVP (Dinh et al., 2017) with invertible $1 \times 1$ convolution, and can generate high quality images. Later on, Hoogeboom et al. (2019) generalizes the $1 \times 1$ convolution to invertible $d \times d$ convolutions which operate both channel and spatial axes.
+
+# 5 EXPERIMENT
+
+In this section, we compare likelihood-based generative models for raw audio in term of test likelihood, speech quality and synthesis speed.
+
+Data: We use the LJ speech dataset (Ito, 2017) containing about 24 hours of audio with a sampling rate of $2 2 . 0 5 \mathrm { k H z }$ recorded on a MacBook Pro in a home enviroment. It consists of 13, 100 audio clips of a single female speaker reading passages from 7 non-fiction books.
+
+Models: We evaluate several likelihood-based generative models, including Gaussian WaveNet, WaveGlow, WaveFlow and autoregressive flow (AF). As in Section 3.2, we implement autoregressive flow from WaveFlow by squeezing the waveforms by its length and setting the filter size as 1 for width dimension. Both WaveNet and AF have 30 layers with dilation cycle $[ 1 , 2 , \cdots , 5 1 2 ]$ and filter size 3. For WaveGlow and WaveFlow, we investigate different setups, including the number of flows, size of residual channels, and squeezed height $h$ .
+
+Conditioner: We use the 80-band mel-spectrogram of the original audio as the conditioner for WaveNet, WaveGlow, and WaveFlow. We use FFT size 1024, hop size 256, and window size 1024. For WaveNet and WaveFlow, we upsample the mel conditioner 256 times by applying two layers of transposed 2-D convolution (in time and frequency) interleaved with leaky ReLU $( \alpha = 0 . 4 )$ ). The upsampling strides in time are 16 and the 2-D convolution filter sizes are [32, 3] for both layers. For WaveGlow, we directly use the open source implementation. 5
+
+Training: We train all models on 8 Nvidia 1080Ti GPUs using randomly chosen short clips of 16, 000 samples from each utterance. For WaveFlow and WaveNet, we use the Adam optimizer (Kingma and Ba, 2015) with a batch size of 8 and a constant learning rate of $2 \times 1 0 ^ { - 4 }$ . For WaveGlow, we use the Adam optimizer with a batch size of 16 and a learning rate of $1 \times 1 0 ^ { - 4 }$ . We applied weight normalization (Salimans and Kingma, 2016) whenever possible.
+
+# 5.1 LIKELIHOOD
+
+The test log-likelihoods (LLs) of all models are evaluate at 1M training steps. Note that, i) all of the LLs decrease slowly after 1M steps and ii) it took one month to train the largest WaveGlow (residual channels $= 5 1 2$ ) for 1M steps. Thus, we chose 1M as the cut-off to compare these models. We summarize the results in Table 4 with models from row (a) to (t). We draw the following observations:
+
+• Stacking a large number of flows improves LLs for WaveFlow, autoregressive flow, and WaveGlow. For example, (m) WaveFlow with 8 flows provide larger LL than (l) WaveFlow with 6 flows. The $( b )$ autoregressive flow obtains the highest likelihood and even outperforms (a) WaveNet with the same amount of parameters. Indeed, AF provides bidirectional modeling by stacking 3 flows interleaved with reverse operations. WaveFlow has much larger likelihood than WaveGlow with comparable number of parameters. In particular, a small-footprint $( k )$ WaveFlow has only 5.91M parameters but can provide comparable likelihood (5.023 vs. 5.026) as the largest (g) WaveGlow with 268.29M parameters.
+
+Table 4: The test log-likelihoods (LLs) of all models conditioned on mel-spectrograms. For $a \times b = c$ in the flows $\times$ layers column, $a$ is number of flows, $b$ is number of layers in each flow, and $c$ is the total number of layers. In WaveFlow, $h$ is the squeezed height. Models with bolded test LLs are mentioned in the text.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>flows×layers</td><td rowspan=1 colspan=1>res.channels</td><td rowspan=1 colspan=1># param</td><td rowspan=1 colspan=1>test LLs</td></tr><tr><td rowspan=1 colspan=1>(a)</td><td rowspan=1 colspan=1>WaveNet</td><td rowspan=1 colspan=1>1×30=30</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>4.57 M</td><td rowspan=1 colspan=1>5.059</td></tr><tr><td rowspan=2 colspan=1>(b)C</td><td rowspan=2 colspan=1>Autoregressive flowWaveGlow</td><td rowspan=1 colspan=1>3×10=30</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>4.54M</td><td rowspan=1 colspan=1>5.161</td></tr><tr><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>17.59 M</td><td rowspan=1 colspan=1>4.804</td></tr><tr><td rowspan=5 colspan=1>de</td><td rowspan=2 colspan=1>WaveGlowWaveGlow</td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>34.83M</td><td rowspan=1 colspan=1>4.927</td></tr><tr><td rowspan=1 colspan=1>6×8=48</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>47.22M</td><td rowspan=1 colspan=1>4.922</td></tr><tr><td rowspan=2 colspan=1>WaveGlowWaveGlow</td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>87.88 M</td><td rowspan=1 colspan=1>5.018</td></tr><tr><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>268.29 M</td><td rowspan=1 colspan=1>5.026</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>4.935</td></tr><tr><td rowspan=1 colspan=1>（i）</td><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91M</td><td rowspan=1 colspan=1>4.954</td></tr><tr><td rowspan=11 colspan=1>09r(s(t）</td><td rowspan=1 colspan=1>WaveFlow (h = 32)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>5.002</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 64)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>5.023</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>6×8=48</td><td rowspan=1 colspan=1>96</td><td rowspan=1 colspan=1>9.58 M</td><td rowspan=1 colspan=1>4.946</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>96</td><td rowspan=1 colspan=1>12.78 M</td><td rowspan=1 colspan=1>4.977</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>96</td><td rowspan=1 colspan=1>12.78M</td><td rowspan=1 colspan=1>5.007</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>6×8=48</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>16.69 M</td><td rowspan=1 colspan=1>4.990</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>22.25M</td><td rowspan=1 colspan=1>5.009</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>22.25 M</td><td rowspan=2 colspan=1>5.0285.055</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 32)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>22.25 M</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>6×8=48</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>64.64M</td><td rowspan=2 colspan=1>5.0645.101</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>86.18 M</td></tr></table>
+
+• As we increase $h$ , the likelihood of WaveFlow steadily increases (can be seen from $( h ) – ( k ) )$ , and its inference is getting slower with more sequential steps. In the limit, it is equivalent to an autoregressive flow. It illustrates the trade-off between model capacity and inference efficiency.
+
+• $( r )$ WaveFlow with 128 residual channels can obtain comparable likelihood (5.055 vs 5.059) as $( a )$ WaveNet with 128 residual channels. A larger (t) WaveFlow with 256 residual channels can obtain even larger likelihood than WaveNet (5.101 vs 5.059).
+
+# 5.2 SPEECH FIDELITY AND SYNTHESIS SPEED
+
+We train WaveNet for 1M steps. We train WaveGlow and WaveFlow for 2M steps with small residual channels (64, 96 and 128). We train larger models (res. channels 256 and 512) for 1M steps due to the practical time constraint. At synthesis, we sampled $Z$ from an isotropic Gaussian with standard deviation 1.0 and 0.6 (default) for WaveFlow and WaveGlow, respectively. For WaveFlow and WaveGlow, we run synthesis under NVIDIA Apex with 16-bit floating point (FP16) arithmetic, which does not introduce any degradation of audio fidelity and brings about $2 \times$ speedup. We use the crowdMOS tookit (Ribeiro et al., 2011) for naturalness evaluation, where test utterances from these models were presented to workers on Mechanical Turk. We also test the synthesis speed on a Nvidia V100 GPU without using any customized inference kernels. We only implement convolution queues (Paine et al., 2016) in Python to cache the intermediate hidden states within WaveFlow for autoregressive inference over the height dimension, which brings about $4 \times$ speedup. We use the permutation strategy (ii) described in Section 3.4 for WaveFlow.
+
+We report the 5-scale Mean Opinion Score (MOS), synthesis speed and model footprint in Table 5. We draw the following observations:
+
+• The small WaveFlow (res. channels 64) has 5.91M parameters and can synthesize 22.05 kHz high-fidelity speech (MOS: 4.32) $4 2 . 6 0 \times$ faster than real-time. In contrast, the speech quality of small WaveGlow (res. channels 64) is significantly worse (MOS: 2.17). Indeed, WaveGlow (res. channels 256) requires $8 7 . 8 8 \mathrm { M }$ parameters for generating high-fidelity speech. • The large WaveFlow (res. channels 256) outperforms the same size WaveGlow in terms of speech fidelity (MOS: 4.43 vs. 4.34). It also matches the state-of-the-art WaveNet while generating speech $8 . 4 2 \times$ faster than real-time, because it only requires 128 sequential steps (number of flows $\times$ height $h$ ) to synthesize very long waveforms.
+
+Table 5: The synthesis speed over real-time and the 5-scale Mean Opinion Score (MOS) ratings with $9 5 \%$ confidence intervals. We use 30-layer WaveNet, 96-layer WaveGlow, and 64-layer WaveFlow. Models with bolded numbers are mentioned in the text.   
+
+<table><tr><td rowspan=1 colspan=2>Model</td><td rowspan=1 colspan=1>flows×layers</td><td rowspan=1 colspan=1>res.channels</td><td rowspan=1 colspan=1># param</td><td rowspan=1 colspan=1>syn.speed</td><td rowspan=1 colspan=1>MOS</td></tr><tr><td rowspan=2 colspan=2>WaveNetWaveGlow</td><td rowspan=1 colspan=1>1×30=30</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>4.57 M</td><td rowspan=1 colspan=1>0.002×</td><td rowspan=1 colspan=1>4.43 ± 0.14</td></tr><tr><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>17.59 M</td><td rowspan=1 colspan=1>93.53×</td><td rowspan=1 colspan=1>2.17 ± 0.13</td></tr><tr><td rowspan=1 colspan=2>WaveGlow</td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>34.83 M</td><td rowspan=1 colspan=1>69.88×</td><td rowspan=1 colspan=1>2.97 ± 0.15</td></tr><tr><td rowspan=1 colspan=2>WaveGlow</td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>87.88 M</td><td rowspan=1 colspan=1>34.69×</td><td rowspan=5 colspan=1>4.34 ± 0.114.32 ± 0.124.26 ± 0.124.32 ± 0.084.34 ± 0.13</td></tr><tr><td rowspan=1 colspan=1>WaveGlow</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>268.29 M</td><td rowspan=1 colspan=1>8.08×</td></tr><tr><td rowspan=1 colspan=2>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>47.61×</td></tr><tr><td rowspan=2 colspan=2>WaveFlow (h = 16)WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>42.60×</td></tr><tr><td rowspan=1 colspan=2>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>96</td><td rowspan=1 colspan=1>12.78 M</td><td rowspan=1 colspan=1>26.23×</td></tr><tr><td rowspan=1 colspan=2>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>22.25M</td><td rowspan=1 colspan=1>21.32×</td><td rowspan=2 colspan=1>4.38 ± 0.094.43 ± 0.10</td></tr><tr><td rowspan=1 colspan=2>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>86.18 M</td><td rowspan=1 colspan=1>8.42×</td></tr><tr><td rowspan=1 colspan=2>Ground-truth</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>4.56± 0.09</td></tr></table>
+
+![](images/f333ae51a57dae0719d4616b59533acfac3cba891ab3bead8cbef71e027b1312.jpg)  
+Figure 3: The test log-likelihoods (LLs) vs. MOS scores for all likelihood-based models in Table 5.
+
+• We find a positive correlation between the test likelihoods and MOS scores for these likelihoodbased generative models (see Figure 3 for an illustration). One can see that larger LLs correspond to higher MOS scores even when we compare all models in Figure 3 (a). The correlations become more evident, as we consider WaveFlow and WaveGlow in Figure 3 (b).
+
+# 6 CONCLUSION
+
+We propose WaveFlow, a compact flow-based model for raw audio, which can be directly trained with maximum likelihood estimation. It provides a unified view of flow-based models for time-domain waveforms, and includes WaveNet and WaveGlow as special cases. WaveFlow requires a small number of sequential steps to generate high-fidelity speech and obtains likelihood comparable to WaveNet. In the end, our small-footprint WaveFlow can generate $2 2 . 0 5 \mathrm { k H z }$ high-fidelity speech more than $4 0 \times$ faster than real-time on a GPU without engineered inference kernels.
+
+# REFERENCES
+
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+        "text": "WAVEFLOW: A COMPACT FLOW-BASED MODEL FOR RAW AUDIO ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
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+        "text": "ABSTRACT ",
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+        "text": "In this work, we present WaveFlow, a small-footprint generative flow for raw audio, which is trained with maximum likelihood without density distillation and auxiliary losses as used in Parallel WaveNet. It provides a unified view of flowbased models for raw audio, including autoregressive flow (e.g., WaveNet) and bipartite flow (e.g., WaveGlow) as special cases. We systematically study these likelihood-based generative models for raw waveforms in terms of test likelihood and speech fidelity. We demonstrate that WaveFlow can synthesize high-fidelity speech and obtain comparable likelihood as WaveNet, while only requiring a few sequential steps to generate very long waveforms. In particular, our small-footprint WaveFlow has 5.91M parameters and can generate $2 2 . 0 5 \\mathrm { k H z }$ high-fidelity speech $4 2 . 6 \\times$ faster than real-time on a GPU without engineered inference kernels. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
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+        "text": "Deep generative models have obtained noticeable successes for modeling raw audio in high-fidelity speech synthesis and music generation (e.g., van den Oord et al., 2016; Dieleman et al., 2018). Autoregressive models are among the best performing generative models for raw audio waveforms, providing the highest likelihood scores and generating high quality samples (e.g., van den Oord et al., 2016; Kalchbrenner et al., 2018). One of the most successful examples is WaveNet (van den Oord et al., 2016), an autoregressive model for waveform synthesis. It operates at the high temporal resolution of raw audio (e.g., 24kHz) and sequentially generates waveform samples at inference. As a result, WaveNet is prohibitively slow for speech synthesis and one has to develop highly engineered kernels for real-time inference (Arık et al., 2017a; Pharris, 2018). 2 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Flow-based models (Dinh et al., 2014; Rezende and Mohamed, 2015) are a family of generative models, in which a simple initial density is transformed into a complex one by applying a series of invertible transformations. One group of models are based on autoregressive transformation, including autoregressive flow (AF) and inverse autoregressive flow (IAF) as the “dual” of each other (Kingma et al., 2016; Papamakarios et al., 2017; Huang et al., 2018). AF is analogous to autoregressive models, which performs parallel density evaluation and sequential synthesis. In contrast, IAF performs parallel synthesis but sequential density evaluation, making likelihood-based training very slow. Parallel WaveNet (van den Oord et al., 2018) distills an IAF from a pretrained autoregressive WaveNet, which gets the best of both worlds. However, it requires the density distillation with Monte Carlo approximation and a set of auxiliary losses for good performance, which complicates the training pipeline and increases the cost of development. Instead, ClariNet (Ping et al., 2019) simplifies the density distillation by computing a regularized KL divergence in closed-form. ",
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+    {
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+        "text": "Another group of flow-based models are based on bipartite transformation (Dinh et al., 2017; Kingma and Dhariwal, 2018), which provide parallel density evaluation and parallel synthesis. Most recently, WaveGlow (Prenger et al., 2019) and FloWaveNet (Kim et al., 2019) successfully applies Glow (Kingma and Dhariwal, 2018) and RealNVP (Dinh et al., 2017) for waveform synthesis, respectively. However, the bipartite transformations are less expressive than the autoregressive transformations (see Section 2.3 for detailed discussion). In general, these bipartite flows require deeper layers, larger hidden size, and huge number of parameters to reach comparable capacities as autoregressive models. For example, WaveGlow and FloWaveNet have 87.88M and 182.64M parameters with 96 layers and 256 residual channels, respectively. In contrast, a 30-layer WaveNet has only 4.57M parameters with 128 residual channels. ",
+        "bbox": [
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+        "text": "In this work, we present WaveFlow, a compact flow-based model for raw audio. Specifically, we make the following contributions: ",
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+    {
+        "type": "text",
+        "text": "1. WaveFlow is trained with maximum likelihood without density distillation and auxiliary losses used in Parallel WaveNet (van den Oord et al., 2018) and ClariNet (Ping et al., 2019), which simplifies the training pipeline and reduces the cost of development.   \n2. WaveFlow squeezes the 1-D raw waveforms into a 2-D matrix and produces the whole audio within a fixed sequential steps. It also provides a unified view of flow-based models for raw audio and allows us to explicitly trade inference efficiency for model capacity. We implement WaveFlow with a dilated 2-D convolutional architecture (Yu and Koltun, 2015), and it includes both Gaussian WaveNet (Ping et al., 2019) and WaveGlow (Prenger et al.,   \n2019) as special cases.   \n3. We systematically study the likelihood-based generative models for raw audios in terms of test likelihood and speech quality. We demonstrate that WaveFlow can obtain comparable likelihood and synthesize high-fidelity speech as WaveNet (van den Oord et al., 2016), while only requiring a few sequential steps to generate very long waveforms.   \n4. Our small-footprint WaveFlow has only 5.91M parameters and synthesizes $2 2 . 0 5 \\mathrm { k H z }$ highfidelity speech (MOS: 4.32) more than $4 0 \\times$ faster than real-time on a Nvidia V100 GPU. In contrast, WaveGlow (Prenger et al., 2019) requires 87.8M parameters for generating high-fidelity speech. The small memory footprint is preferred in production TTS systems, especially for on-device deployment. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We organize the rest of the paper as follows. Section 2 reviews the flow-based models with autoregressive and bipartite transformations. We present WaveFlow in Section 3 and discuss related work in Section 4. We report experimental results in Section 5 and conclude the paper in Section 6. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 FLOW-BASED GENERATIVE MODELS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Flow-based models (Dinh et al., 2014; 2017; Rezende and Mohamed, 2015) transform a simple density of latent variables $p ( z )$ (e.g., isotropic Gaussian) into a complex data distribution $p ( { \\pmb x } )$ by applying a bijection $\\begin{array} { r } { \\mathbf { \\boldsymbol { x } } = f ( \\boldsymbol { z } ) } \\end{array}$ , where $_ { \\textbf { \\em x } }$ and $_ z$ are both $n$ -dimensional. The probability density of $_ { \\textbf { \\em x } }$ can be obtained through the change of variables formula: ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/573ea02803b13aa58500df4a52f7f6509de60c3e1ac0b53773ba78f121be26a3.jpg",
+        "text": "$$\np ( \\pmb { x } ) = p ( z ) \\left| \\operatorname* { d e t } \\left( \\frac { \\partial f ^ { - 1 } ( \\pmb { x } ) } { \\partial \\pmb { x } } \\right) \\right| ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            637,
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+            672
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $z = f ^ { - 1 } ( { \\pmb x } )$ is the inverse transformation, and det $\\big ( \\frac { \\partial { f ^ { - 1 } ( { \\pmb x } ) } } { \\partial { \\pmb x } } \\big )$ is the determinant of its Jacobian. In general, it takes $O ( n ^ { 3 } )$ to compute the determinant, which is not scalable to high-dimensional data. There are two notable groups of flow-based models with triangular Jacobians and tractable determinants. They are based on autoregressive and bipartite transformations, respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 AUTOREGRESSIVE TRANSFORMATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The autoregressive flow (AF) and inverse autoregressive flow (IAF) (Kingma et al., 2016; Papamakarios et al., 2017) use autoregressive transformations. Specifically, AF defines the inverse transformation ${ \\pmb z } = f ^ { - 1 } ( { \\pmb x } ; \\vartheta )$ as: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27ea8aabce78cbd83559e1b9eb75e679e7a990d3e49c3bc34b3fd9d590dc42ee.jpg",
+        "text": "$$\nz _ { t } = x _ { t } \\cdot \\sigma _ { t } ( x _ { < t } ; \\pmb { \\vartheta } ) + \\mu _ { t } ( x _ { < t } ; \\pmb { \\vartheta } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where the shifting variables $\\mu _ { t } ( \\boldsymbol { x } _ { < t } ; \\boldsymbol { \\vartheta } )$ and scaling variables $\\sigma _ { t } ( x _ { < t } ; \\vartheta )$ are modeled by an autoregressive architecture parameterized by $\\vartheta$ (e.g., WaveNet). Note that, the $t$ -th variable $z _ { t }$ only depends on , thus the Jacobian is a triangular matrix as illustrated in Figure 1(a) and its determinant is the product of the diagonal entries: $\\begin{array} { r } { \\operatorname* { d e t } \\left( \\frac { \\partial f ^ { - 1 } ( \\pmb { x } ) } { \\partial \\pmb { x } } \\right) = \\prod _ { t } \\sigma _ { t } ( \\pmb { x } _ { < t } ; \\pmb { \\vartheta } ) } \\end{array}$ . The density $p ( { \\pmb x } )$ can be easily evaluated by change of variables formula, because $z = f ^ { - 1 } ( { \\pmb x } )$ can be computed in parallel from Eq. (2) (i.e., the required $O ( n )$ operations can be done in $O ( 1 )$ time on modern GPU hardware). However, AF has to do sequential synthesis, because the forward transformation $\\begin{array} { r } { { \\bf x } = f ( z ) } \\end{array}$ is autoregressive: xt = zt−µt(x<t;ϑ)σ (x ;ϑ) . In contrast, IAF uses an autoregressive transformation for $z = f ^ { - 1 } ( { \\pmb x } )$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/daf5a61a74ad30d096f8cce589f021f56c0f47157d2b6026cff69270c14f49ca.jpg",
+        "image_caption": [
+            "Figure 1: The Jacobian ∂f−1(x)∂x of (a) an autoregressive transformation, and (b) a bipartite transformation. The blank cells are 0s and represent the independent relations between $z _ { i }$ and $x _ { j }$ . The light-blue cells are scaling variables and represent the linear dependencies between $z _ { i }$ and $x _ { i }$ . The dark-blue cells represent complex non-linear dependencies defined by neural networks. "
+        ],
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+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            366,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5918018b7b09a573ecca341c3edd728e0cbf3bf38c8535343c16262e7c1d5db4.jpg",
+        "text": "$$\nz _ { t } = \\frac { x _ { t } - \\mu _ { t } ( z _ { < t } ; \\vartheta ) } { \\sigma _ { t } ( z _ { < t } ; \\vartheta ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "making density evaluation impractically slow for training, but it can do parallel synthesis by $x _ { t } =$ $z _ { t } \\cdot \\sigma _ { t } ( z _ { < t } ; \\vartheta ) + \\mu _ { t } ( z _ { < t } ; \\vartheta )$ . Parallel WaveNet (van den Oord et al., 2018) and ClariNet (Ping et al., 2019) are based on IAF, which lacks efficient density evaluation and relies on distillation from a pretrained autoregressive WaveNet. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 BIPARTITE TRANSFORMATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "RealNVP (Dinh et al., 2017) and Glow (Kingma and Dhariwal, 2018) use bipartite transformation by partitioning the data $_ { \\textbf { \\em x } }$ into two groups $\\scriptstyle { \\mathbf { \\mathcal { x } } } _ { a }$ and $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { b }$ , where the indices sets $a \\cup b = \\{ 1 , \\cdots , n \\}$ and $a \\cap b = \\phi$ . Then, the inverse transformation $z = f ^ { - 1 } ( x , \\theta )$ is defined as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3bc7488c84842d49d67e7771831200aefbb655844f4031f938eedc24c6518cc.jpg",
+        "text": "$$\nz _ { a } = x _ { a } , \\quad z _ { b } = x _ { b } \\cdot \\sigma _ { b } ( x _ { a } ; \\pmb \\theta ) + \\mu _ { b } ( x _ { a } ; \\pmb \\theta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            642,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the shifting variables $\\mu _ { b } ( x _ { a } ; \\pmb \\theta )$ and scaling variables $\\sigma _ { b } ( x _ { a } ; \\theta )$ are modeled by a feed-forward neural network. The Jacobian ∂f−1(x)∂x is a special triangular matrix as illustrated in Figure 1 (b). By definition, the forward transformation $\\pmb { x } = f ( \\pmb { z } , \\pmb { \\theta } )$ is, ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3a287749460930e0d3e5f8ab63a1f06419df672eb155a4fdb3c33ffc3a877c25.jpg",
+        "text": "$$\nx _ { a } = z _ { a } , \\quad x _ { b } = \\frac { z _ { b } - \\mu _ { b } ( x _ { a } ; \\pmb \\theta ) } { \\sigma _ { b } ( x _ { a } ; \\pmb \\theta ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            719,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and can also be done in parallel. As a result, the bipartite transformation provides both parallel density evaluation and parallel synthesis. In previous work, WaveGlow (Prenger et al., 2019) and FloWaveNet (Kim et al., 2019) both squeeze the adjacent audio samples on the channel dimension, and apply the bipartite transformation on the partitioned channel dimension. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 CONNECTIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            833,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It is worthwhile to mention that the autoregressive transformation is more expressive than bipartite transformation in general. As illustrated in Figure 1(a) and (b), the autoregressive transformation introduces $\\frac { n \\times ( n - 1 ) } { 2 }$ complex non-linear dependencies (dark-blue cells) and $n$ linear dependencies between data $_ { \\textbf { \\em x } }$ and latents $_ z$ . In contrast, bipartite transformation introduces only $\\frac { n ^ { 2 } } { 4 }$ non-linear dependencies and $\\frac { n } { 2 }$ linear dependencies. Indeed, one can reduce an autoregressive transformation ${ \\pmb z } = f ^ { - 1 } ( { \\pmb x } ; \\vartheta )$ to a bipartite transformation ${ z = f ^ { - 1 } ( x ; \\theta ) }$ by: (i) picking an autoregressive order $^ o$ such that all of the indices in set $a$ rank early than the indices in $b$ , and (ii) setting the shifting and scaling variables as, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37609e6eaea45dc0666c848f03734cfe9a2abd2e0592c74cce9a86b397db64da.jpg",
+        "text": "$$\n\\mu _ { t } ( \\boldsymbol { x } _ { < t } ; \\vartheta ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { f o r ~ } t \\in a } \\\\ { \\mu _ { t } ( \\boldsymbol { x } _ { a } ; \\theta ) } & { \\mathrm { f o r ~ } t \\in b } \\end{array} \\right. , \\quad \\sigma _ { t } ( \\boldsymbol { x } _ { < t } ; \\vartheta ) = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathrm { f o r ~ } t \\in a } \\\\ { \\sigma _ { t } ( \\boldsymbol { x } _ { a } ; \\theta ) } & { \\mathrm { f o r ~ } t \\in b } \\end{array} \\right. .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given the less expressive building block, the bipartite transformation-based flows generally require many more layers and larger hidden size to match the capacity of a compact autoregressive models (e.g., as measured by test likelihood) (Kingma and Dhariwal, 2018; Prenger et al., 2019). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 WAVEFLOW ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present WaveFlow and its implementation with dilated 2-D convolutions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 DEFINITION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We denote the high dimensional 1-D waveform as $\\pmb { x } = \\{ x _ { 1 } , \\cdots , x _ { n } \\}$ . We first squeeze $_ { \\textbf { \\em x } }$ into a $h$ -row 2-D matrix $\\mathbf { \\bar { \\boldsymbol { X } } } \\in \\mathbb { R } ^ { h \\times w }$ by column-major order, where $\\begin{array} { r } { w = { \\frac { n } { h } } } \\end{array}$ and adjacent samples are in the same column. We assume $\\boldsymbol { Z } \\in \\mathbb { R } ^ { h \\times w }$ are sampled from an isotropic Gaussian, and define the inverse transformation $Z = f ^ { - 1 } ( X ; \\Theta )$ as, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43bb81c4ea546f605bdc762df0b66f91df8824c278beb94966e67039fafaa88f.jpg",
+        "text": "$$\nZ _ { i , j } = \\sigma _ { i , j } ( X _ { < i , \\bullet } ; \\Theta ) \\cdot X _ { i , j } + \\mu _ { i , j } ( X _ { < i , \\bullet } ; \\Theta ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $X _ { < i , \\bullet }$ represents all elements above $i$ -th row (see Figure 2 for an illustration). Note that, i) the receptive fields over the squeezed inputs $X$ for computing $Z _ { i , j }$ in WaveFlow is strictly larger than that of WaveGlow when $h > 2$ . ii) WaveNet is equivalent to an autoregressive flow with column-major order on the squeezed inputs $X$ . iii) Both WaveFlow and WaveGlow look at future waveform samples in original $_ { \\textbf { \\em x } }$ for computing $Z _ { i , j }$ , whereas WaveNet can not. iv) The autoregressive flow with row-major order has larger receptive fields than WaveFlow and WaveGlow. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The shifting variables $\\mu _ { i , j } ( X _ { < i , \\bullet } ; \\Theta )$ and scaling variables $\\sigma _ { i , j } ( X _ { < i , \\bullet } ; \\Theta )$ in Eq. (6) are modeled by a 2-D convolutional neural network detailed in Section 3.2. By definition, the variable $Z _ { i , j }$ only depends on the current $X _ { i , j }$ and previous $X _ { < i , \\bullet }$ in raw-major order, thus the Jacobian is a triangular matrix and its determinant is: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a75d98b1d4d946cbf7fbd6d298be0731d09805273bd1ae739d7d613feda5bbba.jpg",
+        "text": "$$\n\\operatorname* { d e t } \\left( \\frac { \\partial f ^ { - 1 } ( X ) } { \\partial X } \\right) = \\prod _ { i = 1 } ^ { h } \\prod _ { j = 1 } ^ { w } \\sigma _ { i , j } ( X _ { < i , \\bullet } ; \\Theta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As a result, the log-likelihood can be calculated in parallel by change of variable formula in Eq. (1), ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/176ccc0b344954f3cd268a6c1d65ea935833e89a28c4d1e8b8e55fd005bacad5.jpg",
+        "text": "$$\n\\log p ( \\boldsymbol { X } ) = - \\sum _ { i = 1 } ^ { h } \\sum _ { j = 1 } ^ { w } \\left( Z _ { i , j } ^ { 2 } + \\frac { 1 } { 2 } \\log ( 2 \\pi ) \\right) + \\sum _ { i = 1 } ^ { h } \\sum _ { j = 1 } ^ { w } \\log \\sigma _ { i , j } ( X _ { < i , \\bullet } ; \\Theta ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and one can do maximum likelihood training efficiently. At synthesis, one may first sample $Z$ from the isotropic Gaussian and apply the forward transformation $X = f ( Z ; \\Theta )$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fab170fc04705e7f3c31c710a11c2a5519e657561d8b7c54f3f536058e77721b.jpg",
+        "text": "$$\nX _ { i , j } = \\frac { Z _ { i , j } - \\mu _ { i , j } ( X _ { < i , \\bullet } ; \\Theta ) } { \\sigma _ { i , j } ( X _ { < i , \\bullet } ; \\Theta ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which is only autoregressive on height dimension. Thus, it requires $h$ sequential steps to generate the whole waveform $X$ . In practice, a small $h$ (e.g., 8 or 16) works well, thus we can generate very long waveforms within a few sequential steps. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 IMPLEMENTATION WITH DILATED 2-D CONVOLUTIONS ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this work, we implement WaveFlow with a dilated 2-D convolutional architecture. Specifically, we use a stack of 2-D convolution layers (e.g., 8 layers in all experiments) to model the shifting variables $\\mu _ { i , j } ( X _ { < i , \\bullet } ; \\Theta )$ and scaling variables $\\sigma _ { i , j } ( X _ { < i , \\bullet } ; \\Theta )$ in Eq. (6). We use the similar architecture as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/451fb9d0bedeea61af9e4fc3a8784df0fc7a453e7d6966601dfcf60555ea7947.jpg",
+        "image_caption": [
+            "Figure 2: The receptive fields over the squeezed inputs $X$ for computing $Z _ { i , j }$ in (a) WaveFlow, (b) WaveGlow, (c) autoregressive flow with column-major order (e.g., WaveNet), and (d) autoregressive flow with row-major order. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/326ad10f7afd2dae1cfeb8369def469d9b29618fa58cb4566fe04b9ae62c4d27.jpg",
+        "table_caption": [
+            "Table 1: The test log-likelihoods (LLs) of WaveFlow with different dilation cycles on the height dimension. Both models are stacked with 8 flows and each flow has 8 convolutional layers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>res.channels</td><td>dilations d</td><td>receptive field r</td><td>test LLs</td></tr><tr><td>WaveFlow (h = 32)</td><td>128</td><td>1,1,1,1,1,1,1,1</td><td>17</td><td>4.960</td></tr><tr><td>WaveFlow (h = 32)</td><td>128</td><td>1,2,4,1,2,4,1,2</td><td>35</td><td>5.055</td></tr></table>",
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+        "type": "table",
+        "img_path": "images/e3086372c3784a697ea3d2b625e1d426df62e36c008f4c9fc56f6ca825ef05ba.jpg",
+        "table_caption": [
+            "Table 2: The heights and corresponding dilations used in our experiments. Note that, the receptive fields are only slightly larger than height $h$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Height h</td><td>filter size k</td><td>dilations d</td><td>receptive field r</td></tr><tr><td>8</td><td>3</td><td>1,1,1,1,1,1,1,1</td><td>17</td></tr><tr><td>16</td><td>3</td><td>1,1,1,1,1,1,1,1</td><td>17</td></tr><tr><td>32</td><td>3</td><td>1,2,4,1,2,4,1,2</td><td>35</td></tr><tr><td>64</td><td>3</td><td>1,2,4,8,16,1,2,4</td><td>77</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "WaveNet (van den Oord et al., 2016) by replacing the dilated 1-D convolution to 2-D convolution (Yu and Koltun, 2015), while still keeping the gated-tanh nonlinearities, residual connections and skip connections. ",
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+        "type": "text",
+        "text": "We set the filter sizes as 3 for both height and width dimensions. We use non-causal convolutions on width dimension and set the dilation cycle as $[ 1 , 2 , 4 , \\cdots , 2 ^ { 7 } ]$ . The convolutions on height dimension are causal with an autoregressive constraint, and their dilation cycle needs to be designed carefully. In practice, we find the following rules of thumb are important to obtain good results: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• As motivated by the dilation cycle of WaveNet (van den Oord et al., 2016), the dilations of 8 layers should be set as $\\pmb { d } = [ 1 , 2 , \\cdots , 2 ^ { s } , 1 , 2 , \\cdots , 2 ^ { s } , \\cdots ] ,$ , where $s \\leq 7$ . 3 • The receptive field $r$ over the height dimension should be larger than the squeezed height $h$ Otherwise, it explicitly introduces unnecessary conditional independence and leads to lower likelihood (see Table 1 for an example). Note that, the receptive field of a stack of dilated convolutional layers is: $r = ( k { - } 1 ) { \\times } \\textstyle \\sum _ { i } d _ { i } { + } 1$ , where $k$ is the filter size and $d _ { i }$ is the dilation at $i$ -th layer. Thus, the sum of dilations should satisfy: $\\textstyle \\sum _ { i } d _ { i } \\geq { \\frac { h - 1 } { k - 1 } }$ . However, when $h$ is larger than or equal to $2 ^ { 8 } = 5 1 2$ , we simply set the dilation cycle as $[ 1 , 2 , 4 , \\cdots , 2 ^ { 7 } ]$ . When the receptive field $r$ has already been larger than $h$ , we find that convolutions with smaller dilation and fewer holes provide larger likelihood. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We summarize the heights and preferred dilations in our experiments in Table 2. Note that, WaveFlow becomes fully autoregressive when we squeeze $_ { \\textbf { \\em x } }$ by its length (i.e. $h = n$ ) and set its filter size as 1 over the width dimension, which is equivalent to a Gaussian WaveNet learned by MLE (Ping et al., 2019). If we squeeze $_ { \\textbf { \\em x } }$ by $h = 2$ and set the filter size as 1 on the height dimension, WaveFlow becomes a bipartite flow and is equivalent to WaveGlow with squeezed channels 2. ",
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+    {
+        "type": "text",
+        "text": "3.3 CONDITIONAL GENERATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In neural speech synthesis, a neural vocoder (e.g., WaveNet) synthesizes the time-domain waveforms. It can be conditioned on linguistic features (van den Oord et al., 2016; Arık et al., 2017a), the mel-spectrograms from a text-to-spectrogram model (Ping et al., 2018; Shen et al., 2018), or the learned hidden representation within a text-to-wave architecture (Ping et al., 2019). In this work, we test WaveFlow by conditioning it on ground truth mel-spectrograms as in previous work (Prenger et al., 2019; Kim et al., 2019). The mel-spectrogram is upsampled to the same resolution as waveform samples by transposed 2-D convolutions (Ping et al., 2019). To aligned with the squeezed waveform, they are squeezed to the shape $c \\times h \\times w$ , where $c$ is the feature dimension (e.g, bands of the spectrogram). After a $1 \\times 1$ convolution mapping the features to residual channels, they are added as the bias term at each layer (van den Oord et al., 2016). ",
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+    {
+        "type": "table",
+        "img_path": "images/996226454d25fc499974bb8dc77abdb614a3c7ad3c681b6a4bcb5a80bb60feb7.jpg",
+        "table_caption": [
+            "Table 3: The test LLs of WaveFlow with different permutation strategies. All models consist of 8 flows and each flow has 8 convolutional layers with filter sizes 3. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>res.channels</td><td>permutation strategy</td><td>test LLs</td></tr><tr><td>WaveFlow (h = 16)</td><td>64</td><td>none</td><td>4.551</td></tr><tr><td>WaveFlow (h=1 16)</td><td>64</td><td>(i) 8 reverse</td><td>4.954</td></tr><tr><td>WaveFlow (h = 16)</td><td>64</td><td>(ii) 4 reverse +4 split &amp; reverse</td><td>4.971</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.4 STACKING MULTIPLE FLOWS WITH PERMUTATIONS OVER HEIGHT DIMENSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Flow-based models require a series of transformations until the distribution $p ( X )$ reaches a desired level of complexity (e.g., Rezende and Mohamed, 2015). We let $X = Z ^ { ( n ) }$ and repeatedly apply the transformation $Z ^ { ( i - 1 ) } = f ^ { - 1 } ( Z ^ { ( i ) } ; \\Theta ^ { ( i ) } )$ defined in Eq. (6) from $Z ^ { ( n ) } \\to \\dots Z ^ { ( \\bar { i } ) } \\to \\dots \\bar { Z } ^ { ( \\bar { 0 } ) }$ . We assume $Z ^ { ( 0 ) }$ is from the isotropic Gaussian distribution. The likelihood $p ( X )$ can be evaluated by iteratively applying the chain rule: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a393dd99ffcedbe40d4739dfdceb8e833bc37f0f8df57355fa70498ccf546eb.jpg",
+        "text": "$$\np ( X ) = p ( Z ^ { ( 0 ) } ) \\prod _ { i = 1 } ^ { n } \\left. \\operatorname* { d e t } \\left( \\frac { \\partial f ^ { - 1 } ( Z ^ { ( i ) } ; \\Theta ^ { ( i ) } ) } { \\partial Z ^ { ( i ) } } \\right) \\right. .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We find that permuting each $Z ^ { ( i ) }$ over the height dimension after each transformation can significantly improve the likelihood scores. In particular, we test two permutation strategies for WaveFlow models stacked with 8 flows (i.e., $X = Z ^ { ( 8 ) }$ ) in Table 3: (i) we reverse each $Z ^ { ( i ) }$ over the height dimension after each transformation, and (ii) we reverse $Z ^ { ( 7 ) } , Z ^ { ( 6 ) } , Z ^ { ( 5 ) } , Z ^ { ( 4 ) }$ over the height dimension as before, but split $Z ^ { ( 3 ) } , Z ^ { ( 2 ) } , Z ^ { ( 1 ) } , Z ^ { ( 0 ) }$ in the middle of the height dimension then reverse each part respectively. 4 Note that, one also needs to permute the conditioner on the height dimension accordingly, which is aligned with $Z ^ { ( i ) }$ . From Table 3, both (i) and (ii) significantly outperform the model without permutations mainly because of bidirectional modeling. Strategy (ii) outperforms (i) because of its diverse autoregressive orders. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks for speech synthesis (a.k.a. text-to-speech) have received a lot of attention. Over the past few years, several neural text-to-speech (TTS) systems have been introduced, including WaveNet (van den Oord et al., 2016), Deep Voice (Arık et al., 2017a), Deep Voice 2 (Arık et al., 2017b), Deep Voice 3 (Ping et al., 2018), Tacotron (Wang et al., 2017), Tacotron 2 (Shen et al., 2018), Char2Wav (Sotelo et al., 2017), VoiceLoop (Taigman et al., 2018), WaveRNN (Kalchbrenner et al., 2018), ClariNet (Ping et al., 2019), Transformer TTS (Li et al., 2019), ParaNet (Peng et al., 2019) and FastSpeech (Ren et al., 2019). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Neural vocoders, such as WaveNet, play the most important role in recent advances of speech synthesis. In previous work, the state-of-the-art neural vocoders are autoregressive models (van den Oord et al., 2016; Mehri et al., 2017; Kalchbrenner et al., 2018). Several engineering endeavors have been advocated for speeding up their sequential generation process (Arık et al., 2017a; Kalchbrenner et al., 2018). In particular, Subscale WaveRNN (Kalchbrenner et al., 2018) folds a long waveform sequence ${ \\pmb x } _ { 1 : n }$ into a batch of shorter sequences and can produces up to 16 samples per step, thus it requires at least $\\frac { n } { 1 6 }$ steps to generate the whole audio. Note that, this is different from the proposed WaveFlow, which can generate ${ \\pmb x } _ { 1 : n }$ within a fixed number of steps (e.g., 16). Most recently, flowbased models have been successfully applied for parallel waveform synthesis with comparable fidelity as autoregressive models (van den Oord et al., 2018; Ping et al., 2019; Prenger et al., 2019; Kim et al., 2019; Yamamoto et al., 2019; Serrà et al., 2019). Among these models, WaveGlow (Prenger et al., 2019) and FloWaveNet (Kim et al., 2019) have a simple training pipeline as they solely use the maximum likelihood objective. However, both of them are less expressive than autoregressive models as indicated by their lower likelihood scores. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Flow-based models can either represent the approximate posteriors for variational inference (Rezende and Mohamed, 2015; Kingma et al., 2016; Berg et al., 2018), or can be trained directly on data using the change of variables formula (Dinh et al., 2014; 2017; Kingma and Dhariwal, 2018; Grathwohl et al., 2018). In previous work, Glow (Kingma and Dhariwal, 2018) extends RealNVP (Dinh et al., 2017) with invertible $1 \\times 1$ convolution, and can generate high quality images. Later on, Hoogeboom et al. (2019) generalizes the $1 \\times 1$ convolution to invertible $d \\times d$ convolutions which operate both channel and spatial axes. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare likelihood-based generative models for raw audio in term of test likelihood, speech quality and synthesis speed. ",
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+    },
+    {
+        "type": "text",
+        "text": "Data: We use the LJ speech dataset (Ito, 2017) containing about 24 hours of audio with a sampling rate of $2 2 . 0 5 \\mathrm { k H z }$ recorded on a MacBook Pro in a home enviroment. It consists of 13, 100 audio clips of a single female speaker reading passages from 7 non-fiction books. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Models: We evaluate several likelihood-based generative models, including Gaussian WaveNet, WaveGlow, WaveFlow and autoregressive flow (AF). As in Section 3.2, we implement autoregressive flow from WaveFlow by squeezing the waveforms by its length and setting the filter size as 1 for width dimension. Both WaveNet and AF have 30 layers with dilation cycle $[ 1 , 2 , \\cdots , 5 1 2 ]$ and filter size 3. For WaveGlow and WaveFlow, we investigate different setups, including the number of flows, size of residual channels, and squeezed height $h$ . ",
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+    {
+        "type": "text",
+        "text": "Conditioner: We use the 80-band mel-spectrogram of the original audio as the conditioner for WaveNet, WaveGlow, and WaveFlow. We use FFT size 1024, hop size 256, and window size 1024. For WaveNet and WaveFlow, we upsample the mel conditioner 256 times by applying two layers of transposed 2-D convolution (in time and frequency) interleaved with leaky ReLU $( \\alpha = 0 . 4 )$ ). The upsampling strides in time are 16 and the 2-D convolution filter sizes are [32, 3] for both layers. For WaveGlow, we directly use the open source implementation. 5 ",
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+    {
+        "type": "text",
+        "text": "Training: We train all models on 8 Nvidia 1080Ti GPUs using randomly chosen short clips of 16, 000 samples from each utterance. For WaveFlow and WaveNet, we use the Adam optimizer (Kingma and Ba, 2015) with a batch size of 8 and a constant learning rate of $2 \\times 1 0 ^ { - 4 }$ . For WaveGlow, we use the Adam optimizer with a batch size of 16 and a learning rate of $1 \\times 1 0 ^ { - 4 }$ . We applied weight normalization (Salimans and Kingma, 2016) whenever possible. ",
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+    {
+        "type": "text",
+        "text": "5.1 LIKELIHOOD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The test log-likelihoods (LLs) of all models are evaluate at 1M training steps. Note that, i) all of the LLs decrease slowly after 1M steps and ii) it took one month to train the largest WaveGlow (residual channels $= 5 1 2$ ) for 1M steps. Thus, we chose 1M as the cut-off to compare these models. We summarize the results in Table 4 with models from row (a) to (t). We draw the following observations: ",
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+    {
+        "type": "text",
+        "text": "• Stacking a large number of flows improves LLs for WaveFlow, autoregressive flow, and WaveGlow. For example, (m) WaveFlow with 8 flows provide larger LL than (l) WaveFlow with 6 flows. The $( b )$ autoregressive flow obtains the highest likelihood and even outperforms (a) WaveNet with the same amount of parameters. Indeed, AF provides bidirectional modeling by stacking 3 flows interleaved with reverse operations. WaveFlow has much larger likelihood than WaveGlow with comparable number of parameters. In particular, a small-footprint $( k )$ WaveFlow has only 5.91M parameters but can provide comparable likelihood (5.023 vs. 5.026) as the largest (g) WaveGlow with 268.29M parameters. ",
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+    {
+        "type": "table",
+        "img_path": "images/8a8aac3bf44940e8e6670f231731843ee7138a60ebc2326a4a17f3154c66f042.jpg",
+        "table_caption": [
+            "Table 4: The test log-likelihoods (LLs) of all models conditioned on mel-spectrograms. For $a \\times b = c$ in the flows $\\times$ layers column, $a$ is number of flows, $b$ is number of layers in each flow, and $c$ is the total number of layers. In WaveFlow, $h$ is the squeezed height. Models with bolded test LLs are mentioned in the text. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>flows×layers</td><td rowspan=1 colspan=1>res.channels</td><td rowspan=1 colspan=1># param</td><td rowspan=1 colspan=1>test LLs</td></tr><tr><td rowspan=1 colspan=1>(a)</td><td rowspan=1 colspan=1>WaveNet</td><td rowspan=1 colspan=1>1×30=30</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>4.57 M</td><td rowspan=1 colspan=1>5.059</td></tr><tr><td rowspan=2 colspan=1>(b)C</td><td rowspan=2 colspan=1>Autoregressive flowWaveGlow</td><td rowspan=1 colspan=1>3×10=30</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>4.54M</td><td rowspan=1 colspan=1>5.161</td></tr><tr><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>17.59 M</td><td rowspan=1 colspan=1>4.804</td></tr><tr><td rowspan=5 colspan=1>de</td><td rowspan=2 colspan=1>WaveGlowWaveGlow</td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>34.83M</td><td rowspan=1 colspan=1>4.927</td></tr><tr><td rowspan=1 colspan=1>6×8=48</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>47.22M</td><td rowspan=1 colspan=1>4.922</td></tr><tr><td rowspan=2 colspan=1>WaveGlowWaveGlow</td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>87.88 M</td><td rowspan=1 colspan=1>5.018</td></tr><tr><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>268.29 M</td><td rowspan=1 colspan=1>5.026</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>4.935</td></tr><tr><td rowspan=1 colspan=1>（i）</td><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91M</td><td rowspan=1 colspan=1>4.954</td></tr><tr><td rowspan=11 colspan=1>09r(s(t）</td><td rowspan=1 colspan=1>WaveFlow (h = 32)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>5.002</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 64)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>5.023</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>6×8=48</td><td rowspan=1 colspan=1>96</td><td rowspan=1 colspan=1>9.58 M</td><td rowspan=1 colspan=1>4.946</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>96</td><td rowspan=1 colspan=1>12.78 M</td><td rowspan=1 colspan=1>4.977</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>96</td><td rowspan=1 colspan=1>12.78M</td><td rowspan=1 colspan=1>5.007</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>6×8=48</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>16.69 M</td><td rowspan=1 colspan=1>4.990</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>22.25M</td><td rowspan=1 colspan=1>5.009</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>22.25 M</td><td rowspan=2 colspan=1>5.0285.055</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 32)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>22.25 M</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>6×8=48</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>64.64M</td><td rowspan=2 colspan=1>5.0645.101</td></tr><tr><td rowspan=1 colspan=1>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>86.18 M</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• As we increase $h$ , the likelihood of WaveFlow steadily increases (can be seen from $( h ) – ( k ) )$ , and its inference is getting slower with more sequential steps. In the limit, it is equivalent to an autoregressive flow. It illustrates the trade-off between model capacity and inference efficiency. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• $( r )$ WaveFlow with 128 residual channels can obtain comparable likelihood (5.055 vs 5.059) as $( a )$ WaveNet with 128 residual channels. A larger (t) WaveFlow with 256 residual channels can obtain even larger likelihood than WaveNet (5.101 vs 5.059). ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 SPEECH FIDELITY AND SYNTHESIS SPEED ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We train WaveNet for 1M steps. We train WaveGlow and WaveFlow for 2M steps with small residual channels (64, 96 and 128). We train larger models (res. channels 256 and 512) for 1M steps due to the practical time constraint. At synthesis, we sampled $Z$ from an isotropic Gaussian with standard deviation 1.0 and 0.6 (default) for WaveFlow and WaveGlow, respectively. For WaveFlow and WaveGlow, we run synthesis under NVIDIA Apex with 16-bit floating point (FP16) arithmetic, which does not introduce any degradation of audio fidelity and brings about $2 \\times$ speedup. We use the crowdMOS tookit (Ribeiro et al., 2011) for naturalness evaluation, where test utterances from these models were presented to workers on Mechanical Turk. We also test the synthesis speed on a Nvidia V100 GPU without using any customized inference kernels. We only implement convolution queues (Paine et al., 2016) in Python to cache the intermediate hidden states within WaveFlow for autoregressive inference over the height dimension, which brings about $4 \\times$ speedup. We use the permutation strategy (ii) described in Section 3.4 for WaveFlow. ",
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+    {
+        "type": "text",
+        "text": "We report the 5-scale Mean Opinion Score (MOS), synthesis speed and model footprint in Table 5. We draw the following observations: ",
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+    {
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+        "text": "• The small WaveFlow (res. channels 64) has 5.91M parameters and can synthesize 22.05 kHz high-fidelity speech (MOS: 4.32) $4 2 . 6 0 \\times$ faster than real-time. In contrast, the speech quality of small WaveGlow (res. channels 64) is significantly worse (MOS: 2.17). Indeed, WaveGlow (res. channels 256) requires $8 7 . 8 8 \\mathrm { M }$ parameters for generating high-fidelity speech. • The large WaveFlow (res. channels 256) outperforms the same size WaveGlow in terms of speech fidelity (MOS: 4.43 vs. 4.34). It also matches the state-of-the-art WaveNet while generating speech $8 . 4 2 \\times$ faster than real-time, because it only requires 128 sequential steps (number of flows $\\times$ height $h$ ) to synthesize very long waveforms. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5f684be8d3f93414f3a0f3da23e9f25baf09b6fd303e04f4c98d3bc968efb7cd.jpg",
+        "table_caption": [
+            "Table 5: The synthesis speed over real-time and the 5-scale Mean Opinion Score (MOS) ratings with $9 5 \\%$ confidence intervals. We use 30-layer WaveNet, 96-layer WaveGlow, and 64-layer WaveFlow. Models with bolded numbers are mentioned in the text. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>Model</td><td rowspan=1 colspan=1>flows×layers</td><td rowspan=1 colspan=1>res.channels</td><td rowspan=1 colspan=1># param</td><td rowspan=1 colspan=1>syn.speed</td><td rowspan=1 colspan=1>MOS</td></tr><tr><td rowspan=2 colspan=2>WaveNetWaveGlow</td><td rowspan=1 colspan=1>1×30=30</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>4.57 M</td><td rowspan=1 colspan=1>0.002×</td><td rowspan=1 colspan=1>4.43 ± 0.14</td></tr><tr><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>17.59 M</td><td rowspan=1 colspan=1>93.53×</td><td rowspan=1 colspan=1>2.17 ± 0.13</td></tr><tr><td rowspan=1 colspan=2>WaveGlow</td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>34.83 M</td><td rowspan=1 colspan=1>69.88×</td><td rowspan=1 colspan=1>2.97 ± 0.15</td></tr><tr><td rowspan=1 colspan=2>WaveGlow</td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>87.88 M</td><td rowspan=1 colspan=1>34.69×</td><td rowspan=5 colspan=1>4.34 ± 0.114.32 ± 0.124.26 ± 0.124.32 ± 0.084.34 ± 0.13</td></tr><tr><td rowspan=1 colspan=1>WaveGlow</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>12×8=96</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>268.29 M</td><td rowspan=1 colspan=1>8.08×</td></tr><tr><td rowspan=1 colspan=2>WaveFlow (h = 8)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>47.61×</td></tr><tr><td rowspan=2 colspan=2>WaveFlow (h = 16)WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>5.91 M</td><td rowspan=1 colspan=1>42.60×</td></tr><tr><td rowspan=1 colspan=2>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>96</td><td rowspan=1 colspan=1>12.78 M</td><td rowspan=1 colspan=1>26.23×</td></tr><tr><td rowspan=1 colspan=2>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>22.25M</td><td rowspan=1 colspan=1>21.32×</td><td rowspan=2 colspan=1>4.38 ± 0.094.43 ± 0.10</td></tr><tr><td rowspan=1 colspan=2>WaveFlow (h = 16)</td><td rowspan=1 colspan=1>8×8=64</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>86.18 M</td><td rowspan=1 colspan=1>8.42×</td></tr><tr><td rowspan=1 colspan=2>Ground-truth</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>4.56± 0.09</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/f333ae51a57dae0719d4616b59533acfac3cba891ab3bead8cbef71e027b1312.jpg",
+        "image_caption": [
+            "Figure 3: The test log-likelihoods (LLs) vs. MOS scores for all likelihood-based models in Table 5. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "text": "• We find a positive correlation between the test likelihoods and MOS scores for these likelihoodbased generative models (see Figure 3 for an illustration). One can see that larger LLs correspond to higher MOS scores even when we compare all models in Figure 3 (a). The correlations become more evident, as we consider WaveFlow and WaveGlow in Figure 3 (b). ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            631,
+            318,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We propose WaveFlow, a compact flow-based model for raw audio, which can be directly trained with maximum likelihood estimation. It provides a unified view of flow-based models for time-domain waveforms, and includes WaveNet and WaveGlow as special cases. WaveFlow requires a small number of sequential steps to generate high-fidelity speech and obtains likelihood comparable to WaveNet. In the end, our small-footprint WaveFlow can generate $2 2 . 0 5 \\mathrm { k H z }$ high-fidelity speech more than $4 0 \\times$ faster than real-time on a GPU without engineered inference kernels. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
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Kavukcuoglu. WaveNet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016.   \nA. van den Oord, Y. Li, I. Babuschkin, K. Simonyan, O. Vinyals, K. Kavukcuoglu, G. v. d. Driessche, E. Lockhart, L. C. Cobo, F. Stimberg, et al. Parallel WaveNet: Fast high-fidelity speech synthesis. In ICML, 2018.   \nY. Wang, R. Skerry-Ryan, D. Stanton, Y. Wu, R. J. Weiss, N. Jaitly, Z. Yang, Y. Xiao, Z. Chen, S. Bengio, Q. Le, Y. Agiomyrgiannakis, R. Clark, and R. A. Saurous. Tacotron: Towards end-to-end speech synthesis. In Interspeech, 2017.   \nR. Yamamoto, E. Song, and J.-M. Kim. Probability density distillation with generative adversarial networks for high-quality parallel waveform generation. arXiv preprint arXiv:1904.04472, 2019.   \nF. Yu and V. Koltun. Multi-scale context aggregation by dilated convolutions. arXiv preprint arXiv:1511.07122, 2015. ",
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parse/train/wS0UFjsNYjn/wS0UFjsNYjn.md

@@ -0,0 +1,336 @@
+# META-GMVAE: MIXTURE OF GAUSSIAN VAES FOR UNSUPERVISED META-LEARNING
+
+Dong Bok Lee1, Dongchan ${ { \bf { M } } { \bf { i n } } ^ { 1 } }$ , Seanie Lee1, and Sung Ju Hwang1,2 KAIST1, AITRICS2, South Korea {markhi,alsehdcks95,lsnfamily02,sjhwang82}@kaist.ac.kr
+
+# ABSTRACT
+
+Unsupervised learning aims to learn meaningful representations from unlabeled data which can capture its intrinsic structure, that can be transferred to downstream tasks. Meta-learning, whose objective is to learn to generalize across tasks such that the learned model can rapidly adapt to a novel task, shares the spirit of unsupervised learning in that the both seek to learn more effective and efficient learning procedure than learning from scratch. The fundamental difference of the two is that the most meta-learning approaches are supervised, assuming full access to the labels. However, acquiring labeled dataset for meta-training not only is costly as it requires human efforts in labeling but also limits its applications to pre-defined task distributions. In this paper, we propose a principled unsupervised meta-learning model, namely Meta-GMVAE, based on Variational Autoencoder (VAE) and set-level variational inference. Moreover, we introduce a mixture of Gaussian (GMM) prior, assuming that each modality represents each class-concept in a randomly sampled episode, which we optimize with Expectation-Maximization (EM). Then, the learned model can be used for downstream few-shot classification tasks, where we obtain task-specific parameters by performing semi-supervised EM on the latent representations of the support and query set, and predict labels of the query set by computing aggregated posteriors. We validate our model on Omniglot and Mini-ImageNet datasets by evaluating its performance on downstream few-shot classification tasks. The results show that our model obtains impressive performance gains over existing unsupervised metalearning baselines, even outperforming supervised MAML on a certain setting.
+
+# 1 INTRODUCTION
+
+Unsupervised learning is one of the most fundamental and challenging problems in machine learning, due to the absence of target labels to guide the learning process. Thanks to the enormous research efforts, there now exist many unsupervised learning methods that have shown promising results on real-world domains, including image recognition (Le, 2013) and natural language understanding (Ramachandran et al., 2017). The essential goal of unsupervised learning is obtaining meaningful feature representations that best characterize the data, which can be later utilized to improve the performance of the downstream tasks, by training a supervised task-specific model on the top of the learned representations (Reed et al., 2014; Cheung et al., 2015; Chen et al., 2016) or fine-tuning the entire pre-trained models (Erhan et al., 2010).
+
+Meta-learning, whose objective is to learn general knowledge across diverse tasks, such that the learned model can rapidly adapt to novel tasks, shares the spirit of unsupervised learning in that both seek more efficient and effective learning procedure over learning from scratch. However, the essential difference between the two is that most meta-learning approaches have been built on the supervised learning scheme, and require human-crafted task distributions to be applied in fewshot classification. Acquiring labeled dataset for meta-training may require a massive amount of human efforts, and more importantly, meta-learning limits its applications to the pre-defined task distributions (e.g. classification of specific set of classes).
+
+Two recent works have proposed unsupervised meta-learning that can bridge the gap between unsupervised learning and meta-learning by focusing on constructing supervised tasks with pseudo-labels from the unlabeled data. To do so, CACTUs (Hsu et al., 2019) clusters data in the embedding space learned with several unsupervised learning methods, while UMTRA (Khodadadeh et al., 2019) assumed that each randomly drawn sample represents a different class and augmented each pseudoclass with data augmentation (Cubuk et al., 2018). After constructing the meta-training dataset with such heuristics, they simply apply supervised meta-learning algorithms as usual. Despite the success of the existing unsupervised meta-learning methods, they are fundamentally limited, since 1) they only consider unsupervised learning for heuristic pseudo-labeling of unlabeled data, and 2) the two-stage approach makes it impossible to recover from incorrect pseudo-class assignment when learning the unsupervised representation space.
+
+![](images/563eefd04e52ad314f1bdd51a97274257cf051ecf8262431bf4a369965c3c0b1.jpg)  
+Figure 1: During meta-training, Meta-GMVAE learns multi-modal latent space that can best explain the unlabeled data using EM algorithm. At meta-test time, we use semi-supervised EM to map both the support (labeled data) and queries (unlabeled data) to each mode learned during meta-training.
+
+In this paper, we propose a principled unsupervised meta-learning model based on Variational Autoencoder (VAE) (Kingma & Welling, 2014) and set-level variational inference using self-attention (Vaswani et al., 2017). Moreover, we introduce multi-modal prior distributions, a mixture of Gaussians (GMM), assuming that each modality represents each class-concept in any given tasks. Then the parameter of GMM is optimized by running Expectation-Maximization (EM) on the observations sampled from the set-dependent variational posterior. In this framework, however, there is no guarantee that each modality obtained from EM algorithm corresponds to a label. To realize modality as label, we deploy semi-supervised EM at meta-test time, considering the support set and query set as labeled and unlabeled observations, respectively. We refer to our method as Meta-Gaussian Mixture Variational Autoencoders (Meta-GMVAE) (See Figure 1 for high-level concept). While our method can be used as a full generative model for generating the samples (images), the ability to generalize to generate samples may not be necessary for capturing the meta-knowledge for non-generative downstream tasks. Thus, we propose another version of Meta-GMVAE that reconstructs high-level features learned by unsupervised representation learning approaches (e.g. Chen et al. (2020)).
+
+To investigate the effectiveness of our framework, we run experiments on two benchmark fewshot image classification datasets, namely Omiglot (Lake et al., 2011) and Mini-Imagenet (Ravi & Larochelle, 2017). The experimental results show that our Meta-GMVAE obtains impressive performance gains over the relevant unsupervised meta-learning baselines on both datasets, obtaining even better accuracy than fully supervised MAML (Finn et al., 2017) while utilizing as small as $0 . 1 \%$ of the labeled data on one-shot settings in Omniglot dataset. Moreover, our model can generalize to classification tasks with different number of ways (classes) without loss of accuracy. Our contribution is threefold:
+
+• We propose a novel unsupervised meta-learning model, namely Meta-GMVAE, which metalearns the set-conditioned prior and posterior network for a VAE. Our Meta-GMVAE is a principled unsupervised meta-learning method, unlike existing methods on unsupervised meta-learning that combines heuristic pseudo-labeling with supervised meta-learning.   
+• We propose to learn the multi-modal structure of a given dataset with the Gaussian mixture prior, such that it can adapt to a novel dataset via the EM algorithm. This flexible adaptation to a new task, is not possible with existing methods that propose VAEs with Gaussian mixture priors for single task learning.   
+We show that Meta-GMVAE largely outperforms relevant unsupervised meta-learning baselines on two benchmark datasets, while obtaining even better performance than a supervised metalearning model under a specific setting. We further show that Meta-GMVAE can generalize to classification tasks with different number of ways (classes).
+
+# 2 RELATED WORK
+
+Unsupervised learning Many prior unsupervised learning methods have developed proxy objectives which is either based on reconstruction (Vincent et al., 2010; Higgins et al., 2017), adversarially obtained image fidelity (Radford et al., 2016; Salimans et al., 2016; Donahue et al., 2017; Dumoulin et al., 2017), disentanglement (Bengio et al., 2013; Reed et al., 2014; Cheung et al., 2015; Chen et al., 2016; Mathieu et al., 2016; Denton & Birodkar, 2017; Kim & Mnih, 2018; Ding et al., 2020), clustering (Coates & Ng, 2012; Krahenb ¨ uhl et al., 2016; Bojanowski & Joulin, 2017; Caron ¨ et al., 2018), or contrastive learning (Chen et al., 2020). In the unsupervised learning literature, the most relevant work to ours are methods that use Gaussian Mixture priors for variational autoencoders. Dilokthanakul et al. (2016); Jiang et al. (2017) consider single task learning and therefore, the learned prior parameter is fixed after training, and thus cannot adapt to new tasks. CURL (Rao et al., 2019) learns a network that outputs Gaussian mixture priors over a sequence of tasks for unsupervised continual learning. However CURL cannot adapt to a new task without training on it, while our framework can generalize to a new task without any training, via amortized inference with a dataset (task) encoder. Also, our model does not learn Gaussian mixture priors but rather obtain them on the fly using the expectation-maximization algorithm.
+
+Meta-learning Meta-learning (Thrun & Pratt, 1998) shares the intuition of unsupervised learning in that it aims to improve the model performance on an unseen task by leveraging prior knowledge, rather than learning from scratch. While the literature on meta-learning is vast, we only discuss relevant existing works for few-shot image classification. Metric-based meta-learning (Koch et al., 2015; Vinyals et al., 2016; Snell et al., 2017; Oreshkin et al., 2018; Mishra et al., 2018) is one of the most popular approaches, where it learns to embed the data instances of the same class to be closer in the shared embedding space. One can measure the distance in the embedding space by cosine similarity (Vinyals et al., 2016), or Euclidean distance (Snell et al., 2017). On the other hand, gradient-based meta-learning (Finn et al., 2017; 2018; Li et al., 2017; Lee & Choi, 2018; Ravi & Beatson, 2019; Flennerhag et al., 2020) aims at learning a global initialization of parameters, which can rapidly adapt to a novel task with only a few gradient steps. Moreover, some previous works (Hewitt et al., 2018; Edwards & Storkey, 2017; Garnelo et al., 2018) tackle meta-learning by modeling the set-dependent variational posterior with a single global latent variable, however, we model the variational posterior conditioned on each data instances. Moreover, while all of these works assume supervised learning scenarios where one has access to full labels in meta-training stage, we focus on unsupervised setting in this paper.
+
+Unsupervised meta-learning One of the main limitations of conventional meta-learning methods is that their application is strictly limited to the tasks from a pre-defined task distribution. A few works (Hsu et al., 2019; Khodadadeh et al., 2019) have been proposed to resolve this issue by combining unsupervised learning with meta-learning. The main idea is to construct meta-training dataset in an unsupervised manner by leveraging existing supervised meta-learning models. CACTUs (Hsu et al., 2019) deploy several deep metric learning (Berthelot et al., 2019; Donahue et al., 2017; Caron et al., 2018; Chen et al., 2016) to episodically cluster the unlabeled dataset, and then train MAML (Finn et al., 2017) and Prototypical Networks (Snell et al., 2017) on the constructed data. UMTRA (Khodadadeh et al., 2019) assumes that each randomly drawn sample is from a different class from others, and use data augmentation (Cubuk et al., 2018) to construct synthetic task distribution for meta-training. Instead of only deploying unsupervised learning for constructing meta-training task distributions, we propose an unsupervised meta-learning model that meta-learns set-level variational posterior by matching the multi-modal prior distribution representing latent classes.
+
+# 3 UNSUPERVISED META-LEARNING WITH META-GMVAES
+
+In this section, we describe our problem setting with respect to unsupervised meta-learning, and demonstrate our approach. The graphical illustration of our model for unsupervised meta-training and supervised meta-test is depicted in Figure 2.
+
+# 3.1 PROBLEM STATEMENT
+
+Our goal is to learn unsupervised feature representations which can be transferred to wide range of downstream few-shot classification tasks. As suggested by Hsu et al. (2019); Khodadadeh et al.
+
+![](images/896c87ed6809fff031fc3b44b087bf0883056c94c91be9c37ba269a21e3cdb9c.jpg)  
+Figure 2: The graphical illustration of Meta-GMVAE. The dotted lines denote either variational inference or Expectation Maximization. (a): We introduce the multimodal distribution $p _ { \psi } ( \mathbf { z } )$ into prior distribution, and its optimal task-specific parameter $\psi _ { i } ^ { * }$ is obtained by EM in an episodic manner. (b): For meta-test, we obtain task-specific parameter $\psi _ { i } ^ { * }$ by semi-supervised EM using $\mathbf { x } _ { s } , \mathbf { y } _ { s }$ , and $\mathbf { x } _ { q }$ .
+
+(2019), we only assume an unlabeled dataset toward applying the knowledge learned durin $\mathcal { D } _ { u } = \{ \mathbf { x } _ { u } \} _ { u = 1 } ^ { U }$ in the meta-training stage. We aimmeta-training stage to novel tasks in meta-test stage, which comes with a modest amount of labeled data (or as few as a single example per class) for each task. As with most meta-learning methods, we further assume that the labeled data are drawn from the same distribution as that of the unlabeled data, with a different set of classes. Specifically, the goal of a $K$ -way $S$ -shot classification task $\tau$ is to correctly predict the labels of query data pclass, where nts is r $\mathcal { Q } = \{ \mathbf { x } _ { q } \} _ { q = 1 } ^ { Q }$ , using  (i.e. bet $S$ support data points and labels een 1 and 50). $\mathbf { \mathcal { S } } = \{ ( \mathbf { x } _ { s } , \mathbf { y } _ { s } ) \} _ { s = 1 } ^ { S }$ per $S$
+
+# 3.2 META-LEVEL GAUSSIAN MIXTURE VAE
+
+Unsupervised meta-training We now describe the meta-learning framework for learning unsupervised latent representations that can be transferred to human-designed few-shot image-classification tasks. In particular, we aim toward learning multi-modal latent spaces for Variational Autoencoder (VAE) in an episodic manner. We use the Gaussian mixture for the prior distribution $\begin{array} { r } { p _ { \psi } ( \mathbf { z } ) = \sum _ { k = 1 } ^ { K } p _ { \psi } ( \mathbf { \bar { y } } ^ { \mathbf { \bar { \psi } } } = k ) p _ { \psi } ( \mathbf { z } | \mathbf { y } = k ) } \end{array}$ , where $\psi$ is the parameter of the prior network. Then the generative process can be described as follows:
+
+• $\mathbf { y } \sim p _ { \psi } ( \mathbf { y } )$ , where y corresponds to the categorical L.V. for a single mode.   
+• $\mathbf { z } \sim p _ { \psi } ( \mathbf { z } | \mathbf { y } )$ , where $\mathbf { z }$ corresponds to the Gaussian L.V. responsible for data generation.   
+• $\mathbf { x } \sim p _ { \theta } ( \mathbf { x } | \mathbf { z } )$ , where $\theta$ is the parameter of the generative model.
+
+The above generative process is similar to those from the previous works (Dilokthanakul et al., 2016; Jiang et al., 2017) on modeling the VAE prior with Gaussian mixtures. However, they target single-task learning and the parameter of the prior network is fixed after training such as equation 1c in Dilokthanakul et al. (2016) and equation 5 in Jiang et al. (2017), which is suboptimal since a meta-learning model should be able to adapt and generalize to a novel task.
+
+To learn the set-dependent multi-modalities, we further assume that there exists a parameter $\psi _ { i }$ for each episodic dataset we derive the variatio $\mathcal { D } _ { i } = \{ \mathbf { x } _ { j } \} _ { j = 1 } ^ { M }$ , which is randomly drawn from the d for the marginal log-likelihood of abeled dataset as follows: $\mathcal { D } _ { u }$ . Then $\mathcal { D } _ { i }$
+
+$$
+\begin{array} { l } { { \displaystyle \log p _ { \theta } ( \boldsymbol { D } _ { i } ) = \sum _ { j = 1 } ^ { M } \log p _ { \theta } ( { \bf x } _ { j } ) = \sum _ { j = 1 } ^ { M } \log \int p _ { \theta } ( { \bf x } _ { j } | { \bf z } _ { j } ) p _ { \psi _ { i } } ( { \bf z } _ { j } ) \frac { q _ { \phi } ( { \bf z } _ { j } | { \bf x } _ { j } , \mathcal { D } _ { i } ) } { q _ { \phi } ( { \bf z } _ { j } | { \bf x } _ { j } , \mathcal { D } _ { i } ) } d { \bf z } _ { j } } \ ~ } \\ { { \displaystyle \ ~ \geq \sum _ { j = 1 } ^ { M } \left[ \mathbb { E } _ { { \bf z } _ { j } \sim q _ { \phi } ( { \bf z } _ { j } | { \bf x } _ { j } , \mathcal { D } _ { i } ) } \left[ \log p _ { \theta } ( { \bf x } _ { j } | { \bf z } _ { j } ) + \log p _ { \psi _ { i } } ( { \bf z } _ { j } ) - \log q _ { \phi } ( { \bf z } _ { j } | { \bf x } _ { j } , \mathcal { D } _ { i } ) ) \right] \right] } } \\ { { \displaystyle ~ \approx \sum _ { j = 1 } ^ { M } \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \left[ \log p _ { \theta } ( { \bf x } _ { j } | { \bf z } _ { j } ^ { ( n ) } ) + \log p _ { \psi _ { i } } ( { \bf z } _ { j } ^ { ( n ) } ) - \log q _ { \phi } ( { \bf z } _ { j } ^ { ( n ) } | { \bf x } _ { j } , \mathcal { D } _ { i } ) \right] } } \\ { { \displaystyle ~ = : \mathcal { L } ( \theta , \phi , \psi _ { i } , \mathcal { D } _ { i } ) , ~ { \bf z } _ { j } ^ { ( n ) } \triangleq q _ { \phi } ( { \bf z } _ { j } | { \bf x } _ { j } , \mathcal { D } _ { i } ) } . } \end{array}
+$$
+
+Here the lower bound for each datapoint is approximated by Monte Carlo estimation with the sample size $N$ . Following the convention of the VAE literature, we assume that the variational posterior $q _ { \phi } ( \mathbf { z } _ { j } | \mathbf { x } _ { j } , \mathcal { D } _ { i } )$ follows an isotropic Gaussian distribution.
+
+<table><tr><td colspan="2">Algorithm1Meta-training</td><td>Algorithm2 Meta-test for an episode</td></tr><tr><td colspan="2">Require: An unlabeled dataset Du</td><td>Require: A test task T= SU Q</td></tr><tr><td colspan="2">1: Initialize parameters 0,Φ</td><td></td></tr><tr><td colspan="2">2: while not done do Sample B episode datasets {Di}=1 from Du</td><td>2:Draw n MC samples from q(zj|xj,D)</td></tr><tr><td colspan="2">3: 4: for all i∈[1,B] do</td><td>1 v(n) 3:Initialize μk = ys k and σ²=I</td></tr><tr><td colspan="2">5: Draw n MC samples from q(zj|xj,Di) Initialize πk as1/K and randomly choose K</td><td>n=1 11(n)=k ys</td></tr><tr><td colspan="2">6: different points for μk·</td><td>4: Compute optimal parameter &amp;* using Eq 10 5: Compute p(yq|xq,D) using Eq 11</td></tr></table>
+
+Set-dependent variational posterior Our derivation of the evidence lower bound in $\mathrm { E q } 4$ is similar to that of the hierarchical VAE framework, such as equation 3 in Edwards & Storkey (2017) and equation 4 in Hewitt et al. (2018), in that we use the i.i.d assumption that the log likelihood of a dataset equals the sum over the log-likelihoods of each individual data point. Yet, previous works assume that each input set consists of data instances from a single concept (e.g. a class), therefore, they encode the dataset into a single global latent variable (e.g. $q _ { \phi } ( \mathbf { z } | \mathcal { D } ) )$ . This is not appropriate for unsupervised meta-learning where labels are unavailable. Thus we learn a set-conditioned variational posterior $q _ { \phi } ( \mathbf { z } _ { j } | \mathbf { x } _ { j } , \mathcal { D } _ { i } )$ , which models a latent variable to encode each data $\mathbf { x } _ { j }$ within the given dataset $\mathcal { D } _ { i }$ into the latent space. Specifically, we model the variational posterior $\bar { q _ { \phi } } ( \mathbf { z } _ { j } | \mathbf { x } _ { j } , \mathcal { D } _ { i } )$ using the self-attention mechanism (Vaswani et al., 2017) as follows:
+
+$$
+\begin{array} { c } { { H = \mathrm { T r a n s f o r m e r E n c o d e r } ( f ( \mathcal { D } _ { i } ) ) } } \\ { { { \pmb \mu } _ { j } = W _ { \pmb \mu } H _ { j } + { \bf b } _ { \pmb \mu } , \quad { \pmb \sigma } _ { j } ^ { 2 } = e x p ( W _ { \pmb \sigma ^ { 2 } } H _ { j } + { \bf b } _ { \pmb \sigma ^ { 2 } } ) } } \\ { { q _ { \phi } ( { \bf z } _ { j } | { \bf x } _ { j } , \mathcal { D } _ { i } ) = \mathcal { N } ( { \bf z } _ { j } ; { \pmb \mu } _ { j } , { \pmb \sigma } _ { j } ^ { 2 } ) } } \end{array}
+$$
+
+Here we deploy TransformerEncoder $\cdot ( \cdot )$ , a neural network based on the multi-head self-attention mechanism proposed by Vaswani et al. (2017), to model the dependency between data instances, and $f$ is a convolutional neural network (or an identity function for the Mini-ImageNet) which takes each data in $\mathcal { D } _ { i }$ as an input. Moreover, we use the reparameterization trick (Kingma & Welling, 2014) to train the model with backpropagation since the stochastic sampling process z(nj ${ \mathbf z } _ { j } ^ { ( n ) } \overset { i . i . d } { \sim } q _ { \phi } ( { \mathbf z } _ { j } | { \mathbf x } _ { j } , { D } _ { i } )$ is non-differentiable.
+
+Expectation Maximization As discussed before, we assume that the parameter $\psi _ { i }$ of the prior Gaussian Mixture is task-specific and characterizes the given dataset $\mathcal { D } _ { i }$ . To obtain the task-specific parameter that optimally explain the given dataset, we propose to locally maximize the lower bound in Eq 4 with respect to the prior parameter $\psi _ { i }$ . We can obtain the optimal parameter $\psi _ { i } ^ { * }$ by solving the following optimization problem:
+
+$$
+\psi _ { i } ^ { * } = \underset { \psi _ { i } } { \arg \operatorname* { m a x } } \mathcal { L } ( \theta , \phi , \psi _ { i } , \mathcal { D } _ { i } ) = \underset { \psi _ { i } } { \arg \operatorname* { m a x } } \sum _ { j , n = 1 } ^ { M , N } \log p _ { \psi } ( \mathbf { z } _ { j } ^ { ( n ) } ) , \quad \mathbf { z } _ { j } ^ { ( n ) } \overset { i , i . i . d } { \sim } q _ { \phi } ( \mathbf { z } _ { j } | \mathbf { x } _ { j } , \mathcal { D } _ { i } ) ,
+$$
+
+where we only consider the term related to the task-specific parameter $\psi _ { i }$ , and eliminate the normalization term $\textstyle { \frac { 1 } { N } }$ since it does not change the solution of the optimization problem. The above formula implies that the optimal parameter maximizes the log-likelihood of observations which can be drawn from the variational posterior distribution. However, we do not have an analytic solution for Maximum Likelihood Estimation (MLE) of a GMM.
+
+The most prevalent approach for estimating the parameters for the mixture of Gaussian is solving it with Expectation Maximization (EM) algorithm. To this end, we propose to optimize the taskspecific parameter of GMM prior distribution using EM algorithm as follows:
+
+$$
+\begin{array} { r l } & { \mathrm { ( E \mathrm { - } s t e p ) } \quad Q _ { j , n } ( k ) : = p ( \mathbf { y } _ { j } ^ { ( n ) } = k | \mathbf { z } _ { j } ^ { ( n ) } ) = \frac { \pi _ { k } \mathcal { N } ( \mathbf { z } _ { j } ^ { ( n ) } ; \mu _ { k } , I ) } { \sum _ { k } \pi _ { k } \mathcal { N } ( \mathbf { z } _ { j } ^ { ( n ) } ; \mu _ { k } , I ) } } \\ & { \mathrm { ( M \mathrm { - } s t e p ) } \quad \mu _ { k } : = \frac { \sum _ { j , n = 1 } ^ { M , N } Q _ { j , n } ( k ) \mathbf { z } _ { j } ^ { ( n ) } } { \sum _ { j , n = 1 } ^ { M , N } Q _ { j , n } ( k ) } , \quad \pi _ { k } : = \frac { \sum _ { j , n = 1 } ^ { M , N } Q _ { j , n } ( k ) } { \sum _ { k = 1 } ^ { K } \sum _ { j , n = 1 } ^ { M , N } Q _ { j , n } ( k ) } } \\ & { \qquad \psi _ { i } : = \{ ( \mu _ { k } , I , \pi _ { k } ) \} _ { k = 1 } ^ { K } , } \end{array}
+$$
+
+where $\pi _ { k } , \mu _ { k }$ , and $\mathcal { N } ( \cdot )$ denote the mixing probability of $k$ -th component, mean parameter, and normal distribution, respectively. We assume that the covariance matrix of Gaussian distribution is fixed with the identity matrix $\pmb { I }$ , following the assumption of original VAE on the prior distribution. We initialize $\{ \pi _ { k } \} _ { k = 1 } ^ { K }$ and $\{ \mu _ { k } \} _ { k = 1 } ^ { K }$ as $\begin{array} { r } { \frac { 1 } { K } } \end{array}$ and randomly drawn $K$ different points, respectively. We can obtain MLE solution for the parameters of GMM, by iteratively performing E-step and M-step until the log-likelihood converges. We found that using a fixed number of iterations for the EM algorithm does not degrade the performance, and consider it as a hyperparameter of our framework.
+
+Training objective Note that we want to maximize the variational lower bound of the marginal loglikelihood over all the episode datasets $\mathcal { D } _ { i }$ that can be sampled from $\mathcal { D } _ { u }$ . We use stochastic gradient ascent with respect to the variational parameter $\phi$ and the generative parameter $\theta$ , to maximize the following objective:
+
+$$
+\begin{array} { r l r } {  { \mathcal { L } ( \theta , \phi , \{ \mathcal { D } _ { i } \} _ { i = 1 } ^ { B } ) : = \frac { 1 } { B } \sum _ { i = 1 } ^ { B } [ \operatorname* { m a x } _ { \psi _ { i } } \mathcal { L } ( \theta , \phi , \psi _ { i } , \mathcal { D } _ { i } ) ] } } \\ & { } & { = \frac { 1 } { B } \sum _ { i = 1 } ^ { B } \sum _ { j = 1 } ^ { M } \frac { 1 } { N } \sum _ { n = 1 } ^ { N } [ \log p _ { \theta } ( \mathbf { x } _ { j } | \mathbf { z } _ { j } ^ { ( n ) } ) + \log p _ { \psi _ { i } ^ { * } } ( \mathbf { z } _ { j } ^ { ( n ) } ) - \log q _ { \phi } ( \mathbf { z } _ { j } ^ { ( n ) } | \mathbf { x } _ { j } , \mathcal { D } _ { i } ) ] . } \end{array}
+$$
+
+Here we use $B$ mini-batch of episode datasets, where each dataset consists of $M$ datapoints. The task-specific parameter $\psi _ { i } ^ { * }$ for each episode dataset $\mathcal { D } _ { i }$ is obtained by EM algorithm in Eq 7.
+
+Supervised meta-test By introducing the multi-modal prior distribution into a generative learning framework, our model learns pseudo-class concepts by clustering latent features with EM algorithm. However, there is no guarantee that each modality obtained by EM algorithm corresponds to the label we are interested in at the meta-test stage. To realize modality as label in downstream fewshot image classification tasks, we deploy semi-supervised EM algorithm instead. Given a task $\tau$ consisting of support set $\mathcal { S } = \{ ( \mathbf { x } _ { s } , \mathbf { y } _ { s } ) \} _ { s = 1 } ^ { S }$ and query set $\mathcal { Q } = \{ \bar { \mathbf { x } } _ { q } \} _ { q = 1 } ^ { Q }$ , we use both the support set and query set as an episode dataset $\mathcal { D } = \{ \mathbf { x } _ { s } \} _ { s = 1 } ^ { S } \cup \{ \mathbf { x } _ { q } \} _ { q = 1 } ^ { Q }$ and draw latent variables from the variational posterior $q _ { \phi } ( \mathbf { z } _ { j } | \mathbf { x } _ { j } , \mathcal { D } )$ . Note that we abbreviate the index $i$ since we consider a single task for now. We then perform semi-supervised EM algorithm as follows:
+
+$$
+\begin{array} { r l } & { ( \mathrm { E } { \mathrm { - } } \ s \mathrm { t e p } ) \quad Q _ { q , n } ( k ) : = p ( \mathsf { y } _ { q } ^ { ( n ) } = k | \mathbf { z } _ { q } ^ { ( n ) } ) = \frac { \mathcal { N } ( { \mathbf { z } } _ { q } ^ { ( n ) } ; \mu _ { k } , \sigma _ { k } ^ { 2 } ) } { \sum _ { k } \mathcal { N } ( { \mathbf { z } } _ { q } ^ { ( n ) } ; \mu _ { k } , \sigma _ { k } ^ { 2 } ) } } \\ & { \mu _ { k } : = \frac { \sum _ { s , n = 1 } ^ { S , N } { \mathbf { 1 } } _ { \mathbf { y } _ { \mathrm { s } } ^ { ( \mathrm { a } ) } = \mathbf { k } } ^ { \mathbf { z } _ { s } ^ { ( n ) } } + \sum _ { q , n = 1 } ^ { Q , N } Q _ { q , n } ( k ) \mathbf { z } _ { q } ^ { ( n ) } } { \sum _ { s , n = 1 } ^ { S , N } { \mathbf { 1 } } _ { \mathbf { y } _ { \mathrm { s } } ^ { ( \mathrm { a } ) } = \mathbf { k } } + \sum _ { q , n = 1 } ^ { Q , N } Q _ { q , n } ( k ) } , } \\ & { \sigma _ { k } ^ { 2 } : = \frac { \sum _ { s , n = 1 } ^ { S , N } { \mathbf { 1 } } _ { \mathbf { y } _ { \mathrm { a } } ^ { ( \mathrm { a } ) } = \mathbf { k } } ( { \mathbf { z } } _ { n } ^ { ( \mathrm { s } ) } - \mu _ { k } ) ^ { 2 } + \sum _ { q , n = 1 } ^ { Q , N } Q _ { q , n } ( k ) ( \mathbf { z } _ { q } ^ { ( n ) } - \mu _ { k } ) ^ { 2 } } { \sum _ { s , n = 1 } ^ { S , N } { \mathbf { 1 } } _ { \mathbf { y } _ { \mathrm { s } } ^ { ( \mathrm { a } ) } = \mathbf { k } } + \sum _ { q , n = 1 } ^ { Q , N } Q _ { q , n } ( k ) } } \\ & { \quad \quad \quad \psi : = \{ ( \mu _ { k } , \sigma _ { k } ^ { 2 } , \frac { 1 } { K } ) { \mathbb { I } } _ { k = 1 } ^ { K } , } \end{array}
+$$
+
+where 1 denotes an indicator function. We fix the mixing probability as $\textstyle { \frac { 1 } { K } }$ since the labels in each task $\tau$ are uniformly distributed. Moreover, we utilize diagonal covariance $\sigma _ { k } ^ { 2 }$ to obtain more accurate statistics for the inference. We initialize $\mu _ { k }$ and $\sigma _ { k } ^ { 2 }$ as the average value of support latent representations and the identity matrix $\pmb { I }$ , respectively. Similar to the meta-training stage, we obtain the MLE solution for the parameters of GMM, by performing E-step and M-step for a fixed number of iterations. Finally, we compute the conditional probability of $p ( \mathbf { y } _ { q } | \mathbf { x } _ { q } , \mathcal { D } )$ using the obtained parameters $\psi ^ { * }$ as follows:
+
+$$
+p ( \mathbf { y } _ { q } | \mathbf { x } _ { q } , \mathcal { D } ) = \mathbb { E } _ { q _ { \phi } ( \mathbf { z } _ { q } | \mathbf { x } _ { q } , \mathcal { D } ) } \left[ p _ { \psi ^ { * } } ( \mathbf { y } _ { q } | \mathbf { z } _ { q } ) \right] \approx \frac { 1 } { N } \sum _ { n = 1 } ^ { N } p _ { \psi ^ { * } } ( \mathbf { y } _ { q } | \mathbf { z } _ { q } ^ { ( n ) } ) , \quad \mathbf { z } _ { q } ^ { ( n ) } \overset { i , i . d } { \sim } q _ { \phi } ( \mathbf { z } _ { q } | \mathbf { x } _ { q } , \mathcal { D } ) .
+$$
+
+Here we compute $p _ { \psi ^ { * } } ( \mathbf { y } _ { q } | \mathbf { z } _ { q } ^ { ( n ) } )$ with Bayes rule, and we reuse $N$ different Monte Carlo samples that is drawn for $\mathrm { E q ~ } 1 0 $ , where the prediction of query $\hat { \mathbf { y } _ { q } } = \mathop { \arg \operatorname* { m a x } } _ { k } p ( \mathbf { y } _ { q } = k | \mathbf { x } _ { q } , \mathcal { D } )$ . We present the pseudo-code of the algorithm for training and inference of Meta-GMVAE in the Algorithm 1 and 2.
+
+Visual feature reconstruction While our method is a generative model that can generate samples from output distribution, the ability to generate samples may not be necessary for discriminative downstream tasks (Chen et al., 2020). Moreover, we found that VAEs almost fail to learn in MiniImageNet dataset with the architecturally limited constraints of the meta-learning literature. Thus, we propose a high-level feature reconstruction objective instead for Mini-ImageNet dataset. We experimentally find that the recently proposed constrastive learning framework, namely SimCLR (Chen et al., 2020), is the most effective for our settings. Specifically, SimCLR learns high-level representation by performing a constrastive prediction task on pairs of augmented examples derived from a minibatch. We train SimCLR on the unsupervised dataset $\mathcal { D } _ { u } = \{ \mathbf { \bar { x } } _ { u } \} _ { u = 1 } ^ { U }$ , and use high-level features extracted by SimCLR as an input for our framework.
+
+# 4 EXPERIMENT
+
+In this section, we now validate the effectiveness of our Meta-GMVAE on several downstream fewshot classification tasks. The source codes are available at https://github.com/db-Lee/ Meta-GMVAE.
+
+# 4.1 EXPERIMENTAL SETUPS
+
+Baselines and ours We now describe two supervised meta-learning approaches which we consider as “oracles”, unsupervised meta-learning baselines, and the proposed Meta-GMVAE. 1) MAML (oracle): Model Agnostic Meta Learning by Finn et al. (2017). We compare against its performance reported in Hsu et al. (2019). 2) ProtoNets (oracle): Euclidean distance-based meta-learning approach by Snell et al. (2017). We also compare against it using its performance reported in Hsu et al. (2019). 3) CACTUs: Clustering to Automatically Construct Tasks for Unsupervised meta-learning by Hsu et al. (2019). It automatically constructs tasks by clustering the unsupervised dataset in embedding space learned by ACAI (Berthelot et al., 2019), BiGAN (Donahue et al., 2017), and DeepCluster (Caron et al., 2018). Then they train either MAML or ProtoNets using the cluster indices as pseudo-labels. 4) UMTRA: Unsupervised Meta-learning with Tasks constructed by Random sampling and Augmentation by Khodadadeh et al. (2019). For constructing a K-way 1-shot task, it randomly samples K-way datapoints from unsupervised dataset and augments each datapoint. Then MAML is trained on the constructed tasks. 5) Meta-GMVAE: Our proposed Meta-level Gaussian Mixture VAE. It learns a latent representation by matching set-level amortized variational posterior and task-specific multimodal prior optimized by EM algorithm.
+
+Datasets We validate all the models on two benchmark datasets for few-shot classification. 1) Omniglot: This is a collection of $2 8 \times 2 8$ gray-scale hand-written characters that describe 1623 different alphabets, each of which contains 20 instances. Following the experimental setup of Hsu et al. (2019), we use 1200 classes for unsupervised meta-training, 100 classes for meta-validation and the remaining 323 classes for meta-test. We further augment each class by rotating the images 90, 180, and 270 degrees, such that the total number of classes is $1 6 2 3 \times 4$ , following the convention. 2) Mini-ImageNet: This is a subset of ILSVRC-2012 (Deng et al., 2009) introduced by Ravi & Larochelle (2017), consisting of 100 classes that comes with 600 images of size $8 4 \times 8 4$ that describe different instances. We use 64 classes for unsupervised meta-training, 16 classes for meta-validation, and the remaining 20 classes for meta-test, following the standard protocol.
+
+Implementation details We now introduce the specific implementation details of Meta-GMVAE on the two benchmark datasets. 1) Variational posterior network $q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathcal { D } _ { i } )$ : we use the standard Conv4 architecture on Omniglot dataset for a fair comparison against relevant baselines. On top of the Conv4 architecture, we stack two TransformerEncoder layers and an affine transformation layer to predict the mean and log-variance of Gaussian distribution. For Mini-ImageNet dataset, we only utilize two TransformerEncoder layers and an affine transformation layer since the input used for Mini-ImageNet is already a high-level visual representation extracted from the Conv5 architecture trained with SimCLR. For both datasets, we set the dimensionality of the latent variable to 64. 2) Generative network $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ : For Omniglot dataset, the architecture of generative network is symmetric to the Conv4 architecture of variational posterior network. The last layer outputs the parameter of output Bernoulli distribution. For Mini-ImageNet dataset, we use 3-layer MLP with ReLU activation to predict the mean of output Gaussian distribution. 3) Other details: we utilize Adam optimizer (Kingma & Ba, 2015) with a constant learning rate of 0.001 and 0.0001 for Omniglot and MiniImageNet experiments, respectively. We set the number of iterations for EM algorithm as 10 for all the experiments. For the more details, please see the Appendix.
+
+<table><tr><td></td><td colspan="5">Omniglot (way, shot)</td><td colspan="4">Mini-ImagNet (way, shot)</td></tr><tr><td>Method</td><td>Clustering</td><td>(5,1)</td><td>(5,5)</td><td>(20,1)</td><td>(20,5)</td><td>(5,1)</td><td>(5,5)</td><td>(5,20)</td><td>(5,50)</td></tr><tr><td>Training from Scratch</td><td>N/A</td><td>52.50</td><td>74.78</td><td>24.91</td><td>47.62</td><td>27.59</td><td>38.48</td><td>51.53</td><td>59.63</td></tr><tr><td>CACTUs-MAML</td><td>BiGAN</td><td>58.18</td><td>78.66</td><td>35.56</td><td>58.62</td><td>36.24</td><td>51.28</td><td>61.33</td><td>66.91</td></tr><tr><td>CACTUs-ProtoNets</td><td>BiGAN</td><td>54.74</td><td>71.69</td><td>33.40</td><td>50.62</td><td>36.62</td><td>50.16</td><td>59.56</td><td>63.27</td></tr><tr><td>CACTUs-MAML</td><td>ACAI/DC</td><td>68.84</td><td>87.78</td><td>48.09</td><td>73.36</td><td>39.90</td><td>53.97</td><td>63.84</td><td>69.64</td></tr><tr><td>CACTUs-ProtoNets</td><td>ACAI/DC</td><td>68.12</td><td>83.58</td><td>47.75</td><td>66.27</td><td>39.18</td><td>53.36</td><td>61.54</td><td>63.55</td></tr><tr><td>UMTRA</td><td>N/A</td><td>83.80</td><td>95.43</td><td>74.25</td><td>92.12</td><td>39.93</td><td>50.73</td><td>61.11</td><td>67.15</td></tr><tr><td>Meta-GMVAE (ours)</td><td>N/A</td><td>94.92</td><td>97.09</td><td>82.21</td><td>90.61</td><td>42.82</td><td>55.73</td><td>63.14</td><td>68.26</td></tr><tr><td>MAML (oracle)</td><td>N/A</td><td>94.46</td><td>98.83</td><td>84.60</td><td>96.29</td><td>46.81</td><td>62.13</td><td>71.03</td><td>75.54</td></tr><tr><td>ProtoNets (oracle)</td><td>N/A</td><td>98.35</td><td>99.58</td><td>95.31</td><td>98.81</td><td>46.56</td><td>62.29</td><td>70.05</td><td>72.04</td></tr></table>
+
+Table 1: The few-shot classification results (way, shot) on the Omniglot and Mini-ImageNet datasets. DC denotes DeepCluster. We report the average of accuracies evaluated over 1000 episodes. All the values are based on the reported performance in Hsu et al. (2019) and Khodadadeh et al. (2019), except for ours.
+
+![](images/fe23846753dba483d5ca21495ad5cf596b52e1447752f68b584a8500ed917ca4.jpg)  
+Figure 3: The samples obtained and generated for each mode at unsupervised meta-training and supervised meta-test step of Meta-GMVAE. Samples in each row are in the same modality obtained by EM.
+
+# 4.2 EXPERIMENTAL RESULTS
+
+Few-shot classification Table 1 shows the few-shot classification results obtained by supervised meta-learning baselines (oracle), the two unsupervised meta-learning baselines, and our MetaGMVAE. For the Omniglot dataset, the Meta-GMVAE outperforms all the baselines that only utilize unsupervised-learning for constructing meta-training tasks, except for the UMTRA on the 20-shot 5-shot classification. Meta-GMVAE also outperforms baselines on Mini-ImageNet 1-shot, and 5- shot settings which are the most widely used settings, while it matches the performance of baselines in 20-shot, and 50-shot settings. This shows that meta-learning the posterior network can capture the multi-modal distribution of any given tasks with Meta-GMVAE, is indeed more effective over unsupervised meta-learning baselines which simply trains supervised meta-learning models with pseudo-labels obtained from unlabeled data. Moreover, our Meta-GMVAE obtains better performance than supervised MAML on Omniglot 5-way 1-shot classification, while utilizing as small as $0 . 1 \%$ of the labeled data. This matches the observation in Chen et al. (2020) that well-calibrated unsupervised learning approaches with a modest amount of labels can obtain a performance comparable to or even better than supervised approaches.
+
+Visualization To better understand how our Meta-GMVAE learns and realizes class-concepts in fewshot classification tasks, we visualize the actual samples in an episode classified by Meta-GMVAE and ones generated by generative network $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ , during unsupervised meta-training and supervised meta-testing. We visualize the actual samples and generated ones that have a same modality in a same row. In Figure 3-a, b, we can observe that our Meta-GMVAE captures the similar visual structure in each modality during meta-training, but the modalities are not the class-concepts. However, as shown Figure 3-c, d, our Meta-GMVAE easily realizes each modality as each class-concept at meta-test time.
+
+Ablation study Furthermore, we compare the performance of our model variants by eliminating each of the most important components for our model. We describe the each variant as follows: 1) LR (SimCLR): This performs the logistic regression using support set on top of features pretrained by SimCLR. 2) Vanilla VAE: We train Vanilla VAE on $\mathcal { D } _ { u }$ and predict labels using semi-supervised EM with fixed identity covariance I. 3) Vanilla VAE (SimCLR): This is same as 2) Vanilla VAE except that it is trained on features pretrained by SimCLR. 4) Ep: Meta-GMVAE with an episodic training with task specific parameter $\psi _ { i } ^ { * }$ obtained by EM. 5) Set: Meta-GMM whether having setlevel variational posterior (i.e. $q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathcal { D } _ { i } ) )$ or not (i.e. $q _ { \phi } ( { \bf z } | { \bf x } ) )$ . 6) $\sigma ^ { 2 }$ : Meta-GMM performs semi-supervised EM algorithm whether using diagonal covariance matrix or fixing it with identity matrix $\pmb { I }$ . Table 2-Left shows that all the components we consider are critical for the performance on the few-shot classification tasks as expected. The best performance gain comes from Ep, which supports our proposal on meta-learning the set-level variational posterior by matching it with the multi-modal prior, where the task-specific parameter is obtained with EM.
+
+<table><tr><td>Method</td><td>Ep Set</td><td>9²</td><td>0</td><td>M</td></tr><tr><td>Training From Scratch</td><td></td><td></td><td>24.91</td><td>27.59</td></tr><tr><td>LR (SimCLR)</td><td></td><td></td><td>N/A</td><td>40.11</td></tr><tr><td>Vanilla VAE</td><td></td><td></td><td>69.68</td><td>N/A</td></tr><tr><td>Vanilla VAE (SimCLR)</td><td></td><td></td><td>N/A</td><td>38.40</td></tr><tr><td rowspan="4">Meta-GMVAE</td><td>&gt;&gt;&gt;</td><td></td><td>78.64</td><td>40.51</td></tr><tr><td></td><td>√</td><td>81.65</td><td>41.13</td></tr><tr><td></td><td>√</td><td>80.94</td><td>40.92</td></tr><tr><td>&lt;</td><td>√ √</td><td>82.21</td><td>42.82</td></tr><tr><td>MAML (oracle)</td><td>√</td><td></td><td>84.60</td><td>46.81</td></tr><tr><td>ProtoNets (oracle)</td><td>√</td><td></td><td>95.31</td><td>46.56</td></tr></table>
+
+Table 2: Left: The results of the ablation study on Meta-GMVAE (O: 20-way 1-shot classification on Omniglot, M: 5-way 1-shot classification on Mini-ImageNet). Right: The results of cross-way 1-shot experiments on Omniglot. The values in the parenthesis indicate that a model is trained based on the (way, shot) setting.
+
+<table><tr><td></td><td>Training ( g(way,shot)</td></tr><tr><td>Test way</td><td>(5,1) (5,5) (20,1) (20,5)</td></tr><tr><td>2-way</td><td>98.26 98.00 98.36 98.23</td></tr><tr><td>5-way</td><td>94.92 94.57 93.93 94.01</td></tr><tr><td>10-way</td><td>89.87 89.99 89.10 89.30</td></tr><tr><td>15-way</td><td>85.11 85.12 85.36 85.33</td></tr><tr><td>20-way 81.38</td><td>81.11 82.21 81.98</td></tr><tr><td>30-way 77.80</td><td>77.42 78.40 77.24</td></tr><tr><td>40-way 73.76</td><td>73.15 74.03 73.56</td></tr><tr><td>50-way 70.92</td><td>70.85 69.86 70.02</td></tr></table>
+
+Cross-way classification We then experiment our Meta-GMVAE by varying the number of way (between 2 and 50) and fixing the number of shot as 1. In particular, we set the number of component $k$ as the Test way for the meta-test and perform semi-supervised EM algorithm in Eq 10. Table 2-Right shows that the difference in the number of way used for training and test does not significantly affect the performance, which demonstrates the robustness of Meta-GMVAE on varying number of way. We
+
+![](images/65edb0d1502a941e5c1fccb35eb2a0add38414cfb28b77a75fafe9539fb30dc5.jpg)  
+Figure 4: The visualization of the latent space for the cross-shot generalization experiment.
+
+also visualize the latent space for the cross-shot experiment using t-SNE (Rauber et al., 2016), in Figure 4, which shows that Meta-GMVAE trained with 20-way can cluster 5-way meta-test task.
+
+# 5 CONCLUSION
+
+We proposed a novel unsupervised meta-learning model, namely Meta-GMVAE, which can generate a task-dependent posterior for a given unseen task with multi-modal Gaussian Mixture priors. Given a random episode that consists of samples from diverse classes, we optimize the task-specific parameter of the mixture of Gaussian prior with Expectation-Maximization algorithm, such that each mode can capture intrinsic groupings in the given data. We meta-train the variational posterior network over such data-driven prior obtained over large number of episodes. Then, at the metatest step, we realize each modality with a label by deploying semi-supervised EM algorithm with both the support and the query set. We validate our method on two few-shot image classification benchmark datasets, and show that Meta-GMVAE largely outperforms the relevant unsupervised meta-learning baselines, even achieving better performance than supervised MAML on Omniglot 5-way 1-shot experiments.
+
+Acknowledgements This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019- 0-00075, Artificial Intelligence Graduate School Program (KAIST)), Samsung Research Funding Center of Samsung Electronics (No. SRFC-IT1502-51), Samsung Electronics (IO201214-08145- 01), and the Engineering Research Center Program through the National Research Foundation of Korea (NRF) funded by the Korean Government MSIT (NRF-2018R1A5A1059921). We sincerely thank the anonymous reviewers for their constructive comments which helped us significantly improve our paper during the rebuttal period. We also appreciate D. Khue Lˆ e-Huu for the valuable ˆ discussion on Rao et al. (2019).
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+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December 2017, Long Beach, CA, USA, pp. 5998–6008, 2017.
+
+Pascal Vincent, Hugo Larochelle, Isabelle Lajoie, Yoshua Bengio, and Pierre-Antoine Manzagol. Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion. J. Mach. Learn. Res., 11:3371–3408, 2010.
+
+Oriol Vinyals, Charles Blundell, Tim Lillicrap, Koray Kavukcuoglu, and Daan Wierstra. Matching networks for one shot learning. In Daniel D. Lee, Masashi Sugiyama, Ulrike von Luxburg, Isabelle Guyon, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain, pp. 3630–3638, 2016.
+
+# A OMNIGLOT EXPERIMENTS
+
+# A.1 TRAINING PROCEDURE
+
+Omniglot is a collection of $2 8 \times 2 8$ gray-scale hand-written characters that describe 1623 different alphabets, each of which contains 20 instances. Following the experimental setup of Hsu et al. (2019), we use 1200 classes for unsupervised meta-training, 100 classes for meta-validation and the remaining 323 classes for meta-test. We further augment each class by rotating the images 90, 180, and 270 degrees, such that the total number of classes is $1 6 2 3 \times 4$ , following the convention. We evaluate the trained model using 1000 randomly selected tasks from test set. During evaluation, $K \times S$ data instances are used as support inputs and $K \times 1 5$ data instances are used as query inputs. We use the Adam (Kingma & Ba, 2015) optimizer with a constant learning rate of 0.001 to train all models. All models are trained for 60,000 iterations. For the 5-way experiments (i.e. $K = 5$ ), we set the mini-batch size, the number of datapoints, and Monte Carlo sample size as 4, 200, and 32, respectively (i.e. $B = 4 , M = 2 0 0 .$ , and $N = 3 2$ ). For the 20-way experiments (i.e. $K = 2 0$ ), we set them as 4, 300, and 32 (i.e. $B = 4 , M = 3 0 0$ , and $N = 3 2$ ). We set the number of EM iterations as 10.
+
+# A.2 NETWORK ARCHITECTURE
+
+We summarize the network architecture in the following Table 3, and 4. We assume that the output follows Bernoulli distribution, therefore, the output of generative network $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ is the mean parameter.
+
+Set-level variational posterior network $q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathcal { D } _ { i } )$   
+
+<table><tr><td>Output Size</td><td>Layers</td></tr><tr><td>1× 28× 28</td><td>Input Images</td></tr><tr><td>64 × 14 ×14</td><td>conv2d(3 × 3, stride 1, padding 1), BatchNorm2D,ReLU,Maxpool(2 × 2, stride 2)</td></tr><tr><td>64×7×7</td><td>conv2d(3 × 3,stride 1, padding 1),BatchNorm2D,ReLU,Maxpool(2 × 2, stride 2)</td></tr><tr><td>64×4×4</td><td>conv2d(3 × 3,stride 1, padding 1),BatchNorm2D,ReLU,Maxpool(2 × 2, stride 2)</td></tr><tr><td>64×2×2</td><td>conv2d(3 × 3, stride 1,padding 1),BatchNorm2D,ReLU,Maxpool(2 × 2, stride 2)</td></tr><tr><td>256</td><td>Flatten</td></tr><tr><td>256</td><td>TransformerEncoder(dmodel = 256,dff = 256,h = 4,ELU,LayerNorm=False)</td></tr><tr><td>256</td><td>TransformerEncoder(dmodel = 256,dff = 256,h = 4,ELU,LayerNorm=False)</td></tr><tr><td>64×2</td><td>Linear(256,64 × 2)</td></tr></table>
+
+Table 3: Set-level variational posterior network used for Omniglot dataset. We refer the hyperparameter notation of TransformerEncoder to Vaswani et al. (2017).   
+Generative network $p _ { \theta } ( \mathbf { x } | \mathbf { z } )$   
+Table 4: Generative Network for $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ for Omniglot dataset.   
+
+<table><tr><td>Output Size</td><td>Layers</td></tr><tr><td>64</td><td>Latent code</td></tr><tr><td>256</td><td>Linear(64,256),ELU</td></tr><tr><td>256</td><td>Linear(256,256),ELU</td></tr><tr><td>256</td><td>Linear(256,256),ELU</td></tr><tr><td>64×2×2</td><td>Unflatten</td></tr><tr><td>64×4×4</td><td>deconv2d(4 × 4, stride 2,padding1),BatchNorm2D,ReLU</td></tr><tr><td>64×7×7</td><td>deconv2d(3 × 3, stride 2,padding1),BatchNorm2D,ReLU</td></tr><tr><td>64×14 × 14</td><td>deconv2d(4 × 4, stride 2, padding 1), BatchNorm2D,ReLU</td></tr><tr><td>1×28×28</td><td>deconv2d(4 × 4,stride 2,padding 1), Sigmoid</td></tr></table>
+
+# A.3 $9 5 \%$ CONFIDENCE INTERVAL
+
+We provide the standard errors of our model’s performance at $9 5 \%$ confidence interval over 1000 episodes on the Omniglot dataset in Table 5.
+
+<table><tr><td>Omniglot</td><td>(5,1)</td><td>(5,5)</td><td>(20,1)</td><td>(20,5)</td></tr><tr><td>Meta-GMVAE</td><td>94.92 ± 0.42</td><td>97.09±0.20</td><td>82.21± 0.44</td><td>90.61± 0.19</td></tr></table>
+
+Table 5: The few-shot classification results (way, shot) with $9 5 \%$ confidence interval on the Omniglot.
+
+# B MINI-IMAGENET EXPERIMENTS
+
+# B.1 TRAINING PROCEDURE
+
+Mini-ImageNet is a subset of ILSVRC-2012 (Deng et al., 2009) introduced by Ravi & Larochelle (2017), consisting of 100 classes that comes with 600 images of size $8 4 \times 8 4$ that describe different instances. We first train Conv5 feature extractor using SimCLR objective with temperature term $\tau = 0 . 5$ , on Mini-ImageNet unsupervised meta-training dataset. We train the feature extractor using Adam optimizer with learning rate of 0.0001 for 400 epochs. We use 64 classes for unsupervised meta-training, 16 classes for meta-validation, and the remaining 20 classes for meta-test, following the standard protocol. We evaluate the trained model using 1000 randomly selected tasks from test set. During evaluation, $5 \times S$ data instances are used as support inputs and $5 \times 1 5$ data instances are used as query inputs. For all the experiments, we use the Adam (Kingma & Ba, 2015) optimizer with a constant learning rate of 0.0001, and set the mini-batch size, the number of datapoints, and Monte Carlo sample size as 16, 5, and 256, respectively (i.e. $B = 1 6 , M = 5$ , and $N = 2 5 6$ ) for the 1, 5, and 20-shot experiments. For the 50-shot experiment, we set them 4, 200, and 256, respectively (i.e. $B = , M = 2 0 0$ , and $N = 2 5 6$ ). We train the models for 5K, 10K, 15k, 25K, and 30K for 1, 5, 20, and 50-shot experiments, respectively. We set the number of EM iterations as 10.
+
+# B.2 NETWORK ARCHITECTURE
+
+We summarize the network architecture in the following Table 6, 7, and 8. We assume that the output follows Gaussian distribution, therefore, the output of generative network $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ is the mean parameter. Moreover, the variance of output Gaussian distribution is obtained as suggested in Rybkin et al. (2020).
+
+Feature Extractor for SimCLR   
+Table 6: Feature Extractor trained on Mini-ImageNet dataset using SimCLR objective.   
+
+<table><tr><td>Output Size</td><td>Layers</td></tr><tr><td>3×84×84</td><td>Input Images</td></tr><tr><td>64×42× 42</td><td>conv2d(3 × 3, stride 1, padding1), BatchNorm2D,ReLU,Maxpool(2 × 2, stride 2)</td></tr><tr><td>64× 21× 21</td><td>conv2d(3 × 3,stride 1,padding 1),BatchNorm2D,ReLU,Maxpool(2 × 2,stride 2)</td></tr><tr><td>64×10×10</td><td>conv2d(3 × 3, stride 1,padding1),BatchNorm2D,ReLU,Maxpool(2 × 2, stride 2)</td></tr><tr><td>64×5×5</td><td>conv2d(3 × 3, stride1,padding1), BatchNorm2D,ReLU,Maxpool(2 × 2,stride 2)</td></tr><tr><td>64×2×2</td><td>conv2d(3 × 3, stride 1,padding 1),BatchNorm2D,ReLU,Maxpool(2 × 2, stride 2)</td></tr><tr><td>256</td><td>Flatten</td></tr></table>
+
+Set-level variational posterior network $q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathcal { D } _ { i } )$   
+Table 7: Set-level variational posterior network used for Mini-ImageNet dataset. We refer the hyperparameter notation of TransformerEncoder to Vaswani et al. (2017).   
+
+<table><tr><td>Output Size</td><td>Layers</td></tr><tr><td>256</td><td>Input Features</td></tr><tr><td>256</td><td>TransformerEncoder(dmodel = 256,dff = 256,h = 4,ReLU,LayerNorm=False)</td></tr><tr><td>256</td><td>TransformerEncoder(dmodel = 256,dff = 256,h = 4,ReLU,LayerNorm=False)</td></tr><tr><td>64×2</td><td>Linear(256,64 × 2)</td></tr></table>
+
+Generative network $p _ { \theta } ( \mathbf { x } | \mathbf { z } )$   
+Table 8: Generative Network for $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ for Mini-ImageNet dataset.   
+
+<table><tr><td>Output Size</td><td>Layers</td></tr><tr><td>64</td><td>Latent code</td></tr><tr><td>512</td><td>Linear(64, 512), ReLU</td></tr><tr><td>512</td><td>Linear(512,512),ReLU</td></tr><tr><td>256</td><td>Linear(512,256), ReLU</td></tr></table>
+
+# B.3 $9 5 \%$ CONFIDENCE INTERVAL
+
+We provide the standard errors at $9 5 \%$ confidence interval over 1000 episodes on the Mini-ImageNet dataset in Table 9.   
+
+<table><tr><td>Mini-ImageNet</td><td>(5,1)</td><td>(5,5)</td><td>(20,1)</td><td>(20,5)</td></tr><tr><td>Meta-GMVAE</td><td>42.82 ± 0.56</td><td>55.73 ±0.48</td><td>63.14 ± 0.47</td><td>68.26 ± 0.42</td></tr></table>
+
+Table 9: The few-shot classification results (way, shot) with $9 5 \%$ confidence interval on the Mini-ImageNet.
+
+# B.4 ADDITIONAL COMPARISON USING SIMCLR
+
+To further understand where the improvement of Meta-GMVAE on the Mini-ImageNet dataset, we ran experiment on baselines with SimCLR pretrained features. For CACTUs, we cluster in the embedding space pretrained by SimCLR. For UMTRA, we follow the exact same procedure to generate training episode, which is proposed by the authors. Moreover, we fix the pretrained SimCLR features as the setting of Meta-GMVAE for the both of baselines. Table 10 shows that Meta-GMVAE outperforms the baselines with SimCLR, which supports the effectiveness of Meta-GMVAE combined with SimCLR pretrained features.
+
+<table><tr><td>Mini-ImageNet</td><td>(5,1) (5,5) (20,1) (20,5)</td></tr><tr><td>CACTUs-MAML (SimCLR) 40.39</td><td>52.35 61.09 64.89</td></tr><tr><td>UMTRA (SimCLR) 40.85</td><td>51.47 61.03 67.30</td></tr><tr><td>Meta-GMVAE 42.82</td><td>55.73 63.14 68.26</td></tr></table>
+
+Table 10: The comparison on the few-shot classification results (way, shot) using SimCLR.