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+# Disentangling Identifiable Features from Noisy Data with Structured Nonlinear ICA
+
+Hermanni Hälvä1 ∗ Sylvain Le Corff2 Luc Lehéricy3
+
+Jonathan So4 Yongjie Zhu1 Elisabeth Gassiat5 † Aapo Hyvärinen1 †
+
+1Department of Computer Science, University of Helsinki, Finland 2 Samovar, Télécom SudParis, département CITI, Institut Polytechnique de Paris, Palaiseau, France 3Laboratoire J. A. Dieudonné, Université Côte d’Azur, CNRS, 06100, Nice, France 4Department of Engineering, University of Cambridge, UK 5Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France
+
+# Abstract
+
+We introduce a new general identifiable framework for principled disentanglement referred to as Structured Nonlinear Independent Component Analysis (SNICA). Our contribution is to extend the identifiability theory of deep generative models for a very broad class of structured models. While previous works have shown identifiability for specific classes of time-series models, our theorems extend this to more general temporal structures as well as to models with more complex structures such as spatial dependencies. In particular, we establish the major result that identifiability for this framework holds even in the presence of noise of unknown distribution. Finally, as an example of our framework’s flexibility, we introduce the first nonlinear ICA model for time-series that combines the following very useful properties: it accounts for both nonstationarity and autocorrelation in a fully unsupervised setting; performs dimensionality reduction; models hidden states; and enables principled estimation and inference by variational maximum-likelihood.
+
+# 1 Introduction
+
+A central tenet of unsupervised deep learning is that noisy and high dimensional real world data is generated by a nonlinear transformation of lower dimensional latent factors. Learning such lower dimensional features is valuable as they may allow us to understand complex scientific observations in terms of much simpler, semantically meaningful, representations (Morioka et al., 2020; Zhou and Wei, 2020). Access to a ground truth generative model and its latent features would also greatly enhance several other downstream tasks such as classification (Klindt et al., 2021; Banville et al., 2021), transfer learning (Khemakhem et al., 2020b), as well as causal inference (Monti et al., 2019; Wu and Fukumizu, 2020).
+
+A recently popular approach to deep representation learning has been to learn disentangled features. Whilst not rigorously defined, the general methodology has been to use deep generative models such as VAEs (Kingma and Welling, 2014; Higgins et al., 2017) to estimate semantically distinct factors of variation that generate and encode the data. A substantial problem with the vast majority of work on disentanglement learning is that the models used are not identifiable – that is, they do not learn the true generative features, even in the limit of infinite data – in fact, this task has been proven impossible without inductive biases on the generative model (Hyvärinen and Pajunen, 1999; Locatello et al., 2019). Lack of identifiability plagues deep learning models broadly and has been implicated as one of the reasons for unexpectedly poor behaviour when these models are deployed in real world applications (D’Amour et al., 2020). Fortunately, in many applications the data have dependency structures, such as temporal dependencies which introduce inductive biases. Recent advances in both identifiability theory and practical algorithms for nonlinear ICA (Hyvärinen and Morioka, 2016, 2017; Hälvä and Hyvärinen, 2020; Morioka et al., 2021; Klindt et al., 2021; Oberhauser and Schell, 2021) exploit this and offer a principled approach to disentanglement for such data. Learning statistically independent nonlinear features in such models is well-defined, i.e. those models are identifiable.
+
+However, the existing nonlinear ICA models suffer from numerous limitations. First, they only exploit specific types of temporal structures, such as either temporal dependencies or nonstationarity. Second, they often work under the assumption that some ’auxiliary’ data about a latent process is observed, such as knowledge of the switching points of a nonstationary process as in Hyvärinen and Morioka (2016); Khemakhem et al. (2020a) . Furthermore, all the nonlinear ICA models cited above, with the exception of Khemakhem et al. (2020a), assume that the data are fully observed and noise-free, even though observation noise is very common in practice, and even Khemakhem et al. (2020a) assumes the noise distribution to be exactly known. This approach of modelling observation noise explicitly is in stark contrast to the approach taken in papers, such as Locatello et al. (2020), who instead consider general stochasticity of their model to be captured by latent variables – this approach would be ill-suited to the type of denoising one would often need in practice. Lastly, the identifiability theorems in previous nonlinear ICA works usually restrict the latent components to a specific class of models such as exponential families (but see Hyvärinen and Morioka (2017)).
+
+In this paper we introduce a new framework for identifiable disentanglement, Structured Nonlinear ICA (SNICA), which removes each of the aforementioned limitations in a single unifying framework. Furthermore, the framework guarantees identifiability of a rich class of nonlinear ICA models that is able to exploit dependency structures of any arbitrary order and thus, for instance, extends to spatially structured data. This is the first major theoretical contribution of our paper.
+
+The second important theoretical contribution of our paper proves that models within the SNICA framework are identifiable even in the presence of additive output noise of arbitrary, unknown distribution. We achieve this by extending the theorems by Gassiat et al. (2020b,a). The subsequent practical implication is that SNICA models can perform dimensionality reduction to identifiable latent components and de-noise observed data. We note that noisy-observation part of the identifiability theory is not even limited to nonlinear ICA but applies to any system observed under noise.
+
+Third, we give mild sufficient conditions, relating to the strength and the non-Gaussian nature of the temporal or spatial dependencies, enabling identifiability of nonlinear independent components in this general framework. An important implication is that our theorems can be used, for example, to develop models for disentangling identifiable features from spatial or spatio-temporal data.
+
+As an example of the flexibility of the SNICA framework, we present a new nonlinear ICA model called $\Delta$ -SNICA . It achieves the following very practical properties which have previously been unattainable in the context of nonlinear ICA: the ability to account for both nonstationarity and autocorrelation in a fully unsupervised setting; ability perform dimensionality reduction; model latent states; and to enable principled estimation and inference by variational maximum-likelihood methods. We demonstrate the practical utility of the model in an application to noisy neuroimaging data that is hypothesized to contain meaningful lower dimensional latent components and complex temporal dynamics.
+
+# 2 Background
+
+We start by giving some brief background on Nonlinear ICA and identifiability. Consider a model where the distribution of observed data $\mathbf { x }$ is given by $p _ { X } ( \mathbf { x } ; \pmb \theta )$ for some parameter vector $\pmb \theta$ . This model is called identifiable if the following condition is fulfilled:
+
+$$
+\forall ( \pmb \theta , \pmb \theta ^ { \prime } ) \qquad p _ { X } ( \mathbf x ; \pmb \theta ) = p _ { X } ( \mathbf x ; \pmb \theta ^ { \prime } ) \Rightarrow \pmb \theta = \pmb \theta ^ { \prime } .
+$$
+
+In other words, based on the observed data distribution alone, we can uniquely infer the parameters that generated the data. For models parameterized with some nonparametric function estimator $\mathbf { f }$ , such as a deep neural network, we can replace $\pmb { \theta }$ with $\mathbf { f }$ in the equation above. In practice, identifiability
+
+might hold for some parameters, not all; and parameters might be identifiable up to some more or less trivial indeterminacies, such as scaling.
+
+In a typical nonlinear ICA setting we observe some $\mathbf { x } \in \mathbb { R } ^ { N }$ which has been generated by an invertible nonlinear mixing function f from latent independent components $\mathbf { s } \in \mathbb { R } ^ { N }$ , with $\begin{array} { r } { p ( \mathbf { s } ) = \mathbf { \dot { \prod } } _ { i = 1 } ^ { N } p ( s ^ { ( i ) } ) } \end{array}$ , as per:
+
+$$
+\begin{array} { r } { { \bf x } = { \bf f } ( { \bf s } ) , } \end{array}
+$$
+
+Identifiability of f would then mean that we can in theory find the true $\mathbf { f }$ , and subsequently the true data generating components. Unfortunately, without some additional structure this model is unidentifiable, as shown by Hyvärinen and Pajunen (1999): there is an infinite number of possible solutions and these have no trivial relation with each other. To solve this problem, previous work (Sprekeler et al., 2014; Hyvärinen and Morioka, 2016, 2017) developed models with temporal structure. Such time series models were generalized and expressed in a succinct way by Hyvärinen et al. (2019); Khemakhem et al. (2020a) by assuming the independent components are conditionally independent upon some observed auxiliary variable $\begin{array} { r } { \bar { u } _ { t } \colon p ( \mathbf { s } _ { t } | \bar { u _ { t } } ) = \prod _ { i = 1 } ^ { N } \bar { p ( s _ { t } ^ { ( i ) } | u _ { t } ) } . } \end{array}$ . In a time series context, the auxiliary variable might be history, e.g. $u _ { t } = \mathbf { x } _ { t - 1 }$ , or the index of a time segment to model nonstationarity (or piece-wise stationarity). (It could also be data from another modality, such as audio data used to condition video data (Arandjelovic and Zisserman, 2017).)
+
+Notice that the mixing function f in (2) is assumed bijective and thus identifiable dimension reduction is not possible in most of the models discussed above. The only exceptions, we are aware of, are Khemakhem et al. (2020a); Klindt et al. (2021) who choose f as injective rather than bijective. Further, Khemakhem et al. (2020a) assume additive noise on the observations $\mathbf { x } = \mathbf { f } ( \mathbf { s } ) + \boldsymbol \varepsilon$ , which allows to estimate posterior of s by an identifiable VAE (iVAE). We will take a similar strategy in what follows.
+
+# 3 Definition of Structured Nonlinear ICA
+
+In this section, we first present the new framework of Structured Nonlinear ICA (SNICA) – a broad class of models for identifiable disentanglement and learning of independent components when data has structural dependencies. Next, we give an example of a particularly useful specific model that fits within our framework, called $\Delta$ -SNICA , by using switching linear dynamical latent processes.
+
+# 3.1 Structured Nonlinear ICA framework
+
+Consider observations $( \mathbf { x } _ { t } ) _ { t \in \mathbb { T } } = ( ( x _ { t } ^ { ( 1 ) } , \dots , x _ { t } ^ { ( M ) } ) ) _ { t \in \mathbb { T } }$ where $\mathbb { T }$ is a discrete indexing set of arbitrary dimension. For discrete time-series models, like previous works, $\mathbb { T }$ would be a subset of $\mathbb { N }$ . Crucially, however, we allow it to be any arbitrary indexing variable that describes a desired structure. For instance, $\mathbb { T }$ could be a subset of $\mathbb { N } ^ { 2 }$ for spatial data.
+
+We assume the data is generated according the following nonlinear ICA model. First, there exist latent components $\mathbf { s } ^ { ( i ) } \bar { = } ( s _ { t } ^ { ( i ) } ) _ { t \in \mathbb { T } }$ for $i \in \{ 1 , \ldots , N \}$ where for any $t , t ^ { \prime } \in \mathbb { T }$ , the distributions of $( \mathbf { s } _ { t } ^ { ( i ) } ) _ { 1 \leqslant i \leqslant N }$ and $( \mathbf { s } _ { t ^ { \prime } } ^ { ( i ) } ) _ { 1 \leqslant i \leqslant N }$ are the same, which is a weak form of stationarity. Second, we assume t that for any $m \in \mathbb { N } ^ { * }$ and $( t _ { 1 } , \ldots , t _ { m } ) \in \mathbb { T } ^ { m }$ , $\begin{array} { r } { \underline { { p } } ( \mathbf { s } _ { t _ { 1 } } , \ldots , \mathbf { s } _ { t _ { m } } ) = \prod _ { i = 1 } ^ { N } p ( s _ { t _ { 1 } } ^ { ( i ) } , \ldots , s _ { t _ { m } } ^ { ( i ) } ) } \end{array}$ : that is, the $\mathbf { f } : \dot { \mathbb { R } ^ { N } } \xrightarrow { } \mathbb { R } ^ { M }$ with $M \geqslant N$ is injective, so there may be more observed variables than components. Finally, denote observational noise by $\boldsymbol { \varepsilon } _ { t } \in \mathbb { R } ^ { M }$ and assume that they are i.i.d. for all $t \in \mathbb { T }$ and independent of the signals $\mathbf { s } ^ { ( i ) }$ . Putting these together, we assume the mixing model where for each $t \in \mathbb { T }$ ,
+
+$$
+\mathbf { x } _ { t } = \mathbf { f } ( \mathbf { s } _ { t } ) + \pmb { \varepsilon } _ { t } ,
+$$
+
+where $\mathbf { s } _ { t } = ( s _ { t } ^ { ( 1 ) } , \dots , s _ { t } ^ { ( N ) } )$ . Importantly, $\varepsilon _ { t }$ can have any arbitrary unknown distribution, even with dependent entries; in fact, it may even not have finite moments.
+
+The main appeal of this framework is that, under the conditions given in next section, we can now guarantee identifiability for a very broad and rich class of models.
+
+First, notice that all previous Nonlinear ICA time-series models can be reformulated and often improved upon when viewed through this new unifying framework. In other words, we can create models that are very much like those previous works, and capture their dependency profiles, but with the changes that by assuming unconditional independence and output noise we now allow them to perform dimension reduction (this does also require some additional assumptions needed in our identifiability theorems below). To see this, consider the model in Hälvä and Hyvärinen (2020) which captures nonstationarity in the independent components through a global hidden Markov chain. We can transform this model into the SNICA framework if we instead model each independent component as its own HMM (Figure 1a), with the added benefit that we now have marginally independent components and are able to perform dimensionality reduction into low dimensional latent components. Nonlinear ICA with time-dependencies, such as in an autoregressive model, proposed by Hyvärinen and Morioka (2017) is also a special case of our framework (Figure 1b), but again with the extension of dimensionality reduction. Furthermore, this framework allows for a plethora of new Nonlinear ICA models to be developed. As described above, these do not have to be limited to time-series but could for instance be a process on a two-dimensional graph with appropriate (in)dependencies (see Figure 1c). However, we now proceed to introduce a particularly useful time-series model using our framework.
+
+![](images/d5b846d3be1fcd8b8bd5707832745a6e06c4941953b67729900625560231feac.jpg)  
+Figure 1: Graphical models for the SNICA framework
+
+# 3.2 $\Delta$ -SNICA $:$ Nonlinear ICA with switching linear dynamical systems
+
+While the above framework has great generality, any practical application will need a specific model. Next we propose one which combines the following properties of previous nonlinear ICA models into a single model: ability to account for both nonstationarity and autocorrelation in a fully unsupervised setting, to perform dimensionality reduction and model hidden states. Real world processes, such as video/audio data, financial time-series, and brain signals, exhibit these properties – disentangling latent features in such data would hence be very useful.
+
+Our new model is depicted in Figure 1d. The independent components are generated by a Switching Linear Dynamical System (SLDS) (Ackerson and Fu, 1968; Chang and Athans, 1978; Hamilton, 1990; Ghahramani and Hinton, 2000) with additional latent variables to express rich dynamics. Formally, for each independent component $i \in \{ 1 , \ldots , N \}$ , consider the following SLDS over some latent vector $\mathbf { y } _ { t } ^ { ( i ) }$ :
+
+$$
+\mathbf { y } _ { t } ^ { ( i ) } = \mathbf { B } _ { u _ { t } } ^ { ( i ) } \mathbf { y } _ { t - 1 } ^ { ( i ) } + \mathbf { b } _ { u _ { t } } ^ { ( i ) } + \varepsilon _ { u _ { t } } ^ { ( i ) } ,
+$$
+
+where $u _ { t } : = u _ { t } ^ { ( i ) }$ is a state of a first-order hidden Markov chain $( u _ { t } ^ { ( i ) } ) _ { t = 1 : T }$ . Crucially, we assume that t tthe independent components at each time-point are the first elements $y _ { t , 1 } ^ { ( i ) }$ of $\mathbf { y } _ { t } ^ { ( i ) } = ( y _ { t , 1 } ^ { ( i ) } , \ldots , y _ { t , d } ^ { ( i ) } ) ^ { T }$ i.e. $s _ { t } ^ { ( i ) } = y _ { t , 1 } ^ { ( i ) }$ . The rest of the elements in $\mathbf { y } _ { t } ^ { ( i ) }$ are latent variables modelling hidden dynamics. The great utility of using such a higher-dimensional latent variable is that this model allows us, for example, as a special case, to consider higher-order ARMA processes, thus modelling each $s _ { t } ^ { ( i ) }$ as switching between ARMA processes of an order determined by the dimensionality of $\mathbf { y } _ { t }$ . We call the ensuing model $\Delta$ -SNICA ("Delta-SNICA", with delta as in "dynamic").
+
+# 4 Identifiability
+
+In this section, we present two very general identifiability theorems for SNICA. We basically decouple the problem into two parts. First, we consider identifying the noise-free distribution of $\mathbf { f } \left( \mathbf { s } _ { t } \right)$ from noisy data. Theorem 1 states conditions—on tail behaviour, non-degeneracy, and non-Gaussianity— under which it is possible to recover the distribution of a process based on noisy data with unknown noise distribution. Second, we consider demixing of the nonlinearly mixed data. Theorem 2 provides general conditions—on temporal or spatial dependencies, and non-Gaussianity—that allow recovery of the mixing function f when there is no more noise. We then consider application of these theorems to SNICA.
+
+# 4.1 Identifiability with unknown noise distribution
+
+Consider the model
+
+$$
+{ \bf x } _ { t } = { \bf z } _ { t } + \varepsilon _ { t } ,
+$$
+
+where $( \mathbf { z } _ { t } ) _ { t \in \mathbb { T } }$ is a family of random variables in $\mathbb { R } ^ { M }$ such that all $\mathbf { z } _ { t }$ , $t \in \mathbb { T }$ , have the same marginal distribution, and $\textstyle ( \varepsilon _ { t } ) _ { t \in \mathbb { T } }$ is a family of independent (over $t$ ) and identically distributed random variables, independent of $( \mathbf { z } _ { t } ) _ { t \in \mathbb { T } }$ . Let $P$ be the common distribution of each $\varepsilon _ { t }$ , for $t \in \mathbb { T }$ . Let $t _ { 1 }$ and $t _ { 2 }$ in $\mathbb { T }$ , and consider the following assumptions.
+
+• (A1) [Tail behaviour] For some $\rho < 3$ , there exist $A$ and $B$ such that for all $\boldsymbol { \lambda } \in \mathbb { R } ^ { N }$ , $\mathbb { E } [ \exp ( \langle \lambda , \mathbf { z } _ { t _ { 1 } } \rangle ) ] \leqslant A \exp ( B \| \lambda \| ^ { \rho } ) .$   
+• (A2) [Non-degeneracy] For any $\ b { \eta } \in \mathbb { C } ^ { M }$ , $\mathbb { E } [ \exp \{ \langle \eta , \mathbf { z } _ { t _ { 2 } } \rangle \} | \ \mathbf { z } _ { t _ { 1 } } ]$ is not the null random variable.   
+• (A3) [Non-Gaussianity] The following assertion is false: there exist a vector $\boldsymbol { \eta } \in \mathbb { R } ^ { M }$ and independent random variables $\tilde { z }$ and $u$ , such that $u$ is a non dirac Gaussian random variable and $\langle \eta , \mathbf { z } _ { t _ { 1 } } \rangle$ has the same distribution as $\tilde { z } + u$ .
+
+We defer the detailed discussion on the practical meaning of the assumptions (A1-A3) in the context of SNICA to Section 4.3. We next present Theorem 1 which establishes identifiability under unknown noise (its proof is postponed to Section A.1 in the Supplementary Material):
+
+Theorem 1 Assume that assumptions (A1), (A2) and (A3) hold for some $( t _ { 1 } , t _ { 2 } ) \in  { \mathbb { T } } ^ { 2 }$ . Then, up to translation, for all $m \geqslant 2$ , for all $( t _ { 3 } , \ldots , t _ { m } ) \in  { \mathbb { T } } ^ { m - 2 }$ , the application that associates the distribution of $\left( \mathbf { z } _ { t _ { 1 } } , \ldots , \mathbf { z } _ { t _ { m } } \right)$ and $P$ to the distribution of $\left( \mathbf { x } _ { t _ { 1 } } , \ldots , \mathbf { x } _ { t _ { m } } \right)$ is one-to-one.
+
+Here, up to translation means that adding a constant vector to all $\varepsilon _ { t }$ , and substracting this constant to all $\mathbf { z } _ { t }$ , $t \in \{ t _ { 1 } , \ldots , t _ { m } \}$ , does not change the distribution of $( \mathbf { x } _ { t _ { 1 } } , \ldots , \mathbf { x } _ { t _ { m } } )$ . The proof of Theorem 1 extends that of Theorem 1 in (Gassiat et al., 2020b), see also (Gassiat et al., 2020a), which assumed sub-Gaussian noise-free data. Our extension allows the noise-free data to have heavier tails, which is important since (noise-free) data in many real-world applications is super-Gaussian, i.e. heavy-tailed, as is well-known in work on linear ICA (Hyvärinen et al., 2001).
+
+Importantly, there is no assumption on the unknown noise distribution in Theorem 1. In fact, it does not even assume a mixing as in ICA, and thus extends greatly outside of the framework of this paper.
+
+# 4.2 Identifiability of the mixing function
+
+Based on Theorem 1, it is possible to recover the distribution of the noise-free data in SNICA in (3) by setting $\mathbf { z } _ { t } = \mathbf { f } ( \mathbf { s } _ { t } )$ . Next, we consider under which conditions the mixing function f is identifiable. Denote by $S = S ^ { ( 1 ) } \times \cdots \times S ^ { ( N ) }$ the support of the distribution of all $\mathbf { s } _ { t }$ . We consider the situation where each $S ^ { ( i ) } \subset \mathbb { R }$ , $1 \leqslant i \leqslant N$ , is connected, so that each $S ^ { ( i ) }$ is an interval. We assume moreover that the injective mixing function f is a $\mathcal { C } ^ { 2 }$ diffeomorphism between $S$ and a $\mathcal { C } ^ { 2 }$ differentiable manifold $\mathcal { M } \subset \mathbb { R } ^ { M }$ . Formally, this means that there exists an atlas $\{ \varphi _ { \vartheta } : U _ { \vartheta } \to \mathbb { R } ^ { N } \} _ { \vartheta \in \Theta }$ of $\mathcal { M }$ such that for all $\vartheta , \vartheta ^ { \prime } \in \Theta$ , the map $\varphi _ { \vartheta } \circ \varphi _ { \vartheta ^ { \prime } } ^ { - 1 }$ is a $\mathcal { C } ^ { 2 }$ map, and f is a bijection $\mathbb { R } ^ { N } \to \mathcal { M }$ such that for all $\vartheta \in \Theta$ , $\varphi _ { \vartheta } \circ \mathbf { f }$ and $\mathbf { f } ^ { - 1 } \circ \varphi _ { \vartheta } ^ { - 1 }$ have continuous second derivatives. The sets $U _ { \vartheta }$ , $\vartheta \in \Theta$ , cover $\mathcal { M }$ and are open in $\mathcal { M }$ . The proof of Theorem 2 is postponed to Section A.2 in the Supplementary Material.
+
+ssume that thehas a density xist wh $\qquad m \ \geqslant \ 2$ anon $( t _ { 1 } , \ldots , t _ { m } ) \in \mathbb { T } ^ { m }$ such that the vectorreover that there exist $( s _ { t _ { 1 } } ^ { ( i ) } , \ldots , s _ { t _ { m } } ^ { ( i ) } )$ $p _ { m } ^ { ( i ) }$ $\mathcal { C } ^ { 2 }$ $( S ^ { ( i ) } ) ^ { m }$ $( k , l ) \in \{ 1 , \ldots , m \} ^ { 2 }$ with $k \neq l$ such that the following assumptions hold with .
+
+• $( B l )$ (Uniform $( k , l )$ -dependency). For all $i \in \{ 1 , \ldots , N \}$ , the set of zeros of $\frac { \partial ^ { 2 } } { \partial s _ { t _ { k } } ^ { ( i ) } \partial s _ { t _ { l } } ^ { ( i ) } } Q _ { m } ^ { ( i ) }$ is a meagre subset of $( S ^ { ( i ) } ) ^ { m }$ , i.e. it contains no open subset.
+
+• $( B 2 )$ (Local $( k , l )$ -non quasi Gaussianity). For any open subset $A \subset S ^ { m }$ , there exists at most one $i \in \{ 1 , \ldots , N \}$ such that there exists a function $\alpha : \mathbb { R } ^ { m - 1 }  \mathbb { R }$ and a constant $c \in \mathbb { R }$ such that for all $s \in A$ ,
+
+$$
+\frac { \partial ^ { 2 } } { \partial s _ { t _ { k } } ^ { ( i ) } \partial s _ { t _ { l } } ^ { ( i ) } } Q _ { m } ^ { ( i ) } = c \alpha ( s _ { t _ { k } } ^ { ( i ) } , \mathbf { s } _ { ( - t _ { k } , - t _ { l } ) } ^ { ( i ) } ) \alpha ( s _ { t _ { l } } ^ { ( i ) } , \mathbf { s } _ { ( - t _ { k } , - t _ { l } ) } ^ { ( i ) } ) ,
+$$
+
+where $\mathbf { s } _ { ( - t _ { k } , - t _ { l } ) } ^ { ( i ) } \ i s \ ( s _ { t _ { 1 } } ^ { ( i ) } , \ldots , s _ { t _ { m } } ^ { ( i ) } )$ without the coordinates $t _ { k }$ and $t _ { l }$ .
+
+Then, $\mathbf { f } ^ { - 1 }$ can be recovered up to permutation and coordinate-wise transformations from the distribution of $( \mathbf { f } ( \mathbf { s } _ { t _ { 1 } } ) , \dots , \mathbf { f } ( \mathbf { s } _ { t _ { m } } ) )$ .
+
+# 4.3 Applications to SNICA
+
+In this section, we provide additional comments on the assumptions (A1-A3) and (B1-B2) and their verification in the context of SNICA.
+
+Assumption (A1) is a condition on the tails of the noise-free data: it allows tails that are somewhat heavier than Gaussian tails. It is in fact equivalent to assuming that for some $\tilde { \rho } > 3 / 2$ , there exists $A ^ { \prime } , B ^ { \prime } > 0$ such that for all $t > 0$ , $\| \mathbf { z } _ { t _ { 1 } } \| \geqslant t ) \leqslant A ^ { \prime } \exp ( - \bar { B ^ { \prime } } t ^ { \tilde { \rho } } )$ .
+
+Assumption (A2) is a non-degeneracy condition likely to be fulfilled for any randomly chosen SNICA model parameters. As an example, consider a model such as Fig. 1c, where there exist hidden variables $( u _ { t } ) _ { t \in \mathbb { T } }$ taking values in a finite set $\{ 1 , \ldots , K \}$ such that the pairs of variables $\left( \mathbf { z } _ { t } , u _ { t } \right)$ have the same distribution for all $t \in \mathbb { T }$ , and such that conditioned on $( u _ { t } ) _ { t \in \mathbb { T } }$ , the variables $( \mathbf { z } _ { t } ) _ { t \in \mathbb { T } }$ are independent and the distribution of $\mathbf { z } _ { t }$ only depends on $u _ { t }$ . (As a special case, this model includes the temporal HMM setting described in Fig. 1a.) Let $( t _ { 1 } , t _ { 2 } ) \in \mathbb { T } ^ { 2 }$ . For all $u , v \in \{ 1 , \ldots , K \}$ , let $\pi ( u ) = \overline { { p _ { u _ { t _ { 1 } } } ( u ) } }$ be the mass function of $\boldsymbol { u } _ { t _ { 1 } }$ , $Q ( u , v ) = p _ { u _ { t _ { 2 } } | u _ { t _ { 1 } } } ( v | u )$ be the transition matrix from $u _ { t _ { 1 } }$ to $u _ { t _ { 2 } }$ , and $\gamma _ { u } ( \mathbf { z } ) = p _ { \mathbf { z } _ { t _ { 1 } } | u _ { t _ { 1 } } } ( \mathbf { z } | u )$ be the density of $\mathbf { z } _ { t _ { 1 } }$ conditionally to $u _ { t _ { 1 } } = u$ . By assumption, it is also the density of $\mathbf { z } _ { t _ { 2 } }$ conditionally to $u _ { t _ { 2 } } = u$ . Theorem 3 provides sufficient conditions for assumption (A2) to hold:
+
+Theorem 3 Assume that $Q$ has full rank, $\operatorname* { m i n } _ { u } \pi ( u ) > 0$ and the $( \gamma _ { u } ) _ { 1 \leqslant u \leqslant K }$ are linearly independent, then (A2) is satisfied as soon as the functions $\begin{array} { r } { \langle \eta \mapsto \int \exp ( \langle \eta , \mathbf { z } \rangle ) \gamma _ { v } ( \mathbf { z } ) d \mathbf { z } ) _ { 1 \leqslant v \leqslant K } } \end{array}$ do not have simultaneous zeros.
+
+Besides the non-simultaneous zeros assumption, the assumptions of Theorem 3 are reminiscent of those used for the identifiability of non-parametric hidden Markov models, see for instance Gassiat et al. (2016); Lehéricy (2019). The key element is that $\mathbf { z } _ { t _ { 1 } }$ and $\mathbf { z } _ { t _ { 2 } }$ are not independent. Thus, we see that (A2) holds if the $\pi$ and the $\gamma$ are not degenerate (in the precise sense given by Theorem 3), for the latent state models in Figs. 1a,1c.Another situation where (A2) holds is when $\mathbf { z } _ { t _ { 2 } }$ is a complete statistic (Lehmann and Casella, 2006) in the statistical model $\{ \mathbb { P } _ { \mathbf { z } _ { t _ { 2 } } | \mathbf { z } _ { t _ { 1 } } } \left( \cdot | \mathbf { z } _ { t _ { 1 } } \right) \} _ { \mathbf { z } _ { t _ { 1 } } }$ , where $\mathbb { P } _ { \mathbf { z } _ { t _ { 2 } } \mid \mathbf { z } _ { t _ { 1 } } } \left( \cdot | \mathbf { z } _ { t _ { 1 } } \right)$ is the distribution of $\mathbf { z } _ { t _ { 2 } }$ conditionally to $\mathbf { z } _ { t _ { 1 } }$ . Consider the two following examples where this holds: 1) When the model $\{ \mathbb { P } _ { \mathbf { z } _ { t _ { 2 } } | \mathbf { z } _ { t _ { 1 } } } ^ { - } \left( \cdot | \mathbf { z } _ { t _ { 1 } } \right) \} _ { \mathbf { z } _ { t _ { 1 } } }$ is an exponential family. In this situation, complete statistics are known. 2) Autoregressive models with additive innovation of the form ${ \bf z } _ { t _ { 2 } } = { \bf h } ( { \bf z } _ { t _ { 1 } } ) + { \bf v } _ { t _ { 2 } }$ for some bijective function $\mathbf { h }$ when the additive noise $\mathbf { v } _ { t _ { 2 } }$ is a complete statistic in the statistical model $\{ \mathbb { P } _ { \mathbf { v } _ { t _ { 2 } } | \mathbf { z } _ { t _ { 1 } } } \left( \cdot | \mathbf { z } _ { t _ { 1 } } \right) \} _ { \mathbf { z } _ { t _ { 1 } } }$ (note that $\mathbf { v } _ { t _ { 2 } }$ cannot be independent of $\mathbf { z } _ { t _ { 1 } }$ here). The case in Fig. 1b is typically covered by this example.
+
+Assumption (A3) states that no direction of the noise free data has a non Dirac Gaussian variable component. It holds as soon as $\mathbf { z } _ { t } = \mathbf { f } ( \mathbf { s } _ { t } )$ and the range of $\mathbf { f }$ is such that its orthogonal projection on any line is not the full line. This assumption holds for instance in the following cases: 1) The range of f is compact, or 2) the range of f is contained in a half-cylinder, that is, there exists a hyperplane such that the range of f is only on one side of this hyperplane and the projection of the range of f on this hyperplane is bounded.
+
+Assumption (B1) and Assumption (B2) are similar to those in (Hyvärinen and Morioka, 2017; Oberhauser and Schell, 2021) in the special case of time-series, i.e. $\mathbb { T } = \mathbb { N }$ . (B1) then entails that there must be sufficiently strong statistical dependence between nearby time points. (B2) is a condition which excludes Gaussian processes and processes which can be trivially transformed to be Gaussian. (For treatment of the Gaussian case, see Appendix B in Supplementary Material.) We can further provide a simple and equivalent formulation when the independent components $\mathbf { s } ^ { ( i ) }$ follow independent and stationary HMMs with two hidden states, which is a special case of SNICA. Denote by $\gamma _ { 0 } ^ { ( i ) }$ and $\gamma _ { 1 } ^ { ( i ) }$ the densities of $s _ { t } ^ { ( i ) }$ conditionally to $\{ u _ { t } ^ { ( i ) } = 0 \}$ and $\bar { \{ u _ { t } ^ { ( i ) } = 1 \} }$ respectively.
+
+Theorem 4 Assume that the stationary distribution $\pi$ of the hidden chain is such that $0 < \pi ( 0 ) < 1$ and that its transition matrix is invertible. Then $( B l )$ and $( B 2 )$ are satisfied with $m = 2$ if and only $i f$ on any open interval, γ(i)0 a nd $\gamma _ { 1 } ^ { ( i ) }$ are not proportional.
+
+Thus, a very simple HMM leads to these conditions being verified. Hyvärinen and Morioka (2017) already showed that the conditions (B1) and (B2) also hold in the case of non-Gaussian autoregressive models. Thus, we see that our identifiability theory applies both in the case HMM’s (Fig 1a) and autoregressive models (Fig 1b), the two principal kinds of temporal structure proposed in previous work, while extending them to further cases and combinations such as in Fig 1c,1d.
+
+A simplification of (B1,B2) It is also possible to combine the assumptions (B1) and (B2) in one, while slightly weakening the generality. The key is to notice that (6) in (B2) implies the derivative in (B1) is zero, by setting $c = 0$ . But there is still the difference that (B2) considers all but one index while (B1) considers all indices $i$ . If we simply assume (6) does not hold for any $i$ , we can replace (B1) and (B2) by the new condition:
+
+• $\mathbf { ( B ^ { \prime } ) }$ For any open subset $A \subset S ^ { m }$ and for any $i \in \{ 1 , \ldots , N \}$ , a function $\alpha : \mathbb { R } ^ { m - 1 }  \mathbb { R }$ and a constant $c \in \mathbb { R }$ do not exist such that (6) would hold for all $s \in A$ .
+
+Note that Hyvärinen and Morioka (2017) defined uniform dependency and (non-)quasi-Gaussianity as two separate properties, but in fact their assumption of non-quasi-Gaussianity was weaker than ours: it did not consider all open subsets separately, which is why this simplification was not possible for them. We believe their definition of non-quasi-Gaussianity was in fact not quite sufficient to prove their theorem, and our stronger version may be needed, in line with Oberhauser and Schell (2021).
+
+# 5 Experiments
+
+Estimation method One challenge is that it is not practically possible to learn $\Delta$ -SNICA by exact maximum-likelihood methods. Instead, we perform learning and inference using Structured VAEs (Johnson et al., 2016) – the current state-of-art in variational inference for structured models. Specifically, this consists of assuming that the latent posterior factorizes as per $q ( \mathbf { y } _ { 1 : T } ^ { ( 1 : N ) } , u _ { 1 : T } ^ { ( 1 : N ) } ) =$ $\begin{array} { r } { \prod _ { i = 1 } ^ { N } q ( \mathbf { y } _ { 1 : T } ^ { ( i ) } ) q ( u _ { 1 : T } ^ { ( i ) } ) } \end{array}$ , which allows us to optimize the resulting evidence lower bound (ELBO):
+
+$$
+\begin{array} { r l } & { \log \widehat { \mathcal { L } } = \mathbb { E } _ { q } \left[ \displaystyle \sum _ { t = 1 } ^ { T } \log p ( \mathbf { x } _ { t } \mid \mathbf { s } _ { t } ^ { ( 1 ) } , . . . , \mathbf { s } _ { t } ^ { ( N ) } ) \right] + \displaystyle \sum _ { i = 1 } ^ { N } \left( - \mathrm { K L } \Big [ q ( u _ { 1 : T } ^ { ( i ) } ) \Big | p ( u _ { 1 : T } ^ { ( i ) } ) \Big ] + \mathrm { H } \Big [ q ( \mathbf { s } _ { 1 : T } ^ { ( i ) } ) \Big ] \right. } \\ & { \qquad \left. + \mathbb { E } _ { q } \left[ \log p ( \mathbf { s } _ { 1 } ^ { ( i ) } \mid u _ { 1 } ^ { ( i ) } ) \right] + \displaystyle \sum _ { t = 2 } ^ { T } \mathbb { E } _ { q } \left[ \log p ( \mathbf { s } _ { t } ^ { ( i ) } \mid \mathbf { s } _ { t - 1 } ^ { ( i ) } , u _ { t } ^ { ( i ) } ) \right] \right) . } \end{array}
+$$
+
+Since all the distributions are in conjugate exponential families (encoder neural network is used to approximate the natural parameters of the nonlinear likelihood term) efficient message passing can be used for inference, and the mixing function is learned as decoder neural network. Even though this method lacks consistency guarantees (but see Wang and Blei (2018)), we find that our model performs very well. A more detailed treatment of estimation and inference of $\Delta$ -SNICA is given in Appendix C. Our code will be openly available at https://github.com/HHalva/snica.
+
+![](images/5fd030a3253ec2319636d147c7d753a552ca81974f6635d2ea2a6a146c22d51d.jpg)  
+Figure 2: (a) Mean absolute correlation coefficients between ground-truth independent components and their estimates by $\Delta$ -SNICA , IIA-HMM, LGSSM and $\mathrm { i V A E ^ { * } }$ , with different orders of complexity (number of layers) and two different dimensions of observed (12, 24) and latent (6, 12) data. (b) Mean absolute correlation coefficient between estimated noise free data and ground-truth noise free data for same set of models except IIA-HMM. Please note the difference in y-axis scales.
+
+# 5.1 Experiments on simulated data
+
+The identifiability theorems stated above hold in the limit of infinite data. Additionally, a consistent estimator would be required to learn the ground-truth components. In the real world, we are limited by data and estimation methods and hence it is unclear as to what extent we are actually able to estimate identifiable components – and whether identifiability reflects in better performance in real world tasks. To explore this, we first performed experiments on simulated data. We compared the performance of our model to the current state-of-the-art, IIA-HMM (Morioka et al., 2021), as well as identifiable VAE (iVAE) (Khemakhem et al., 2020a) and standard linear Gaussian state-space model (LGSSM). Since iVAE is not able to handle latent auxiliary variables, we allow it to "cheat" by giving it access to the true data generating latent-state, thereby creating a presumably challenging baseline (denoted $\mathrm { i V A E ^ { * } }$ in our figures). LGSSM was included as a naive baseline which is only able to estimate linear mixing function.
+
+Investigating identifiability and consistency We simulated 100K long time-sequences from the $\Delta$ - SNICA model and computed the mean absolute correlation coefficient (MCC) between the estimated latent components and ground truth independent components (see Supplementary material for further implementation details). More precisely, to illustrate the dimensionality reduction capabilities we considered two settings where the observed data dimension $M$ , was either 12 or 24 and the number of independent components, $N$ was 3 and 6, respectively. Since IIA-HMM is unable to do dimensionality reduction, we used PCA to get the data dimension to match that of the latent states. We considered four levels of mixing of increasing complexity by randomly initialized MLPs of the following number of layers: 1 (linear ICA), 2, 3, and 5. The results in Figure 2a) illustrate the clearly superior performance of our model. The especially poor performance of IIA-HMM maybe explained by lack of noise model, much simpler latent dynamics, and lost information due to PCA pre-processing. See Appendix D for further discussion and training details.
+
+Application to denoising $\Delta$ -SNICA is able to denoise time-series signals by learning the generative model and then performing inference on latent variables. Specifically, SVAE learns the encoder network which is used to perform inference on the posterior of the independent components. We illustrate this using the same settings as above, with the exception that we now use our learned encoder and inference to get the posterior means of the independent components and input these in to the estimated decoder to get predicted noise-free observations, denoted as $\widehat { \mathbf { f } } \left( \mathbf { s } _ { t } \right)$ – we measured the correlation between $\widehat { \mathbf { f } } \left( \mathbf { s } _ { t } \right)$ and the ground-truth $\mathbf { f } \left( \mathbf { s } _ { t } \right)$ . Note that IIA-HMM, is not able to perform this task. The results in Figure 2b) show that the other models, designed to handle denoising, perform well at this task, as would be expected – identifiability of the latent state is not necessary for good denoising performance. For LGSSM, denoising is done with the Kalman Smoother algorithm.
+
+# 5.2 Experiments on real MEG data
+
+To demonstrate real-data applicability, $\Delta$ -SNICA was applied to multivariate time series of electrical activity in the human brain, measured by magnetoencephalography (MEG). Recently, many studies have demonstrated the existence of fast transient networks measured by MEG in the resting state and the dynamic switching between different brain networks (Baker et al., 2014; Vidaurre et al., 2017). Additionally, such MEG data is high-dimensional and very noisy. Thus this data provides an excellent target for $\Delta$ -SNICA to disentangle the underlying low-dimensional components.
+
+Data and Preprocessing We considered a resting state MEG sessions from the Cam-CAN dataset. During the resting state recording, subjects sat still with their eyes closed. In the task-session data, the subjects carried out a (passive) audio–visual task including visual stimuli and auditory stimuli. We exclusively used the resting-session data for the training of the network, and task-session data was only used in the evaluation. The modality of the sensory stimulation provided a class label that we used in the evaluation, giving in total two classes. We band-pass filtered the data between $4 \mathrm { H z }$ and $3 0 \mathrm { H z }$ (see Supplementary Material for the details of data and settings).
+
+Methods The resting-state data from all subjects were temporally concatenated and used for training. The number of layers of the decoder and encoder were equal and took values 2, 3, 4. We fixed the number of independent components to 5 so that our result can be fairly compared to those in Morioka et al. (2021). To evaluate the obtained features, we performed classification of the sensory stimulation categories by applying feature extractors trained with (unlabeled) resting-state data to (labeled) task-session data. Classification was performed using a linear support vector machine (SVM) classifier trained on the stimulation modality labels and sliding-window-averaged features obtained for each trial. The performance was evaluated by the generalizability of a classifier across subjects. i.e., one-subject-out cross-validation. For comparison, we evaluated the baseline methods: IIA-HMM and IIA-TCL (Morioka et al., 2021). We also visualized the spatial activity patterns obtained by $\Delta$ -SNICA , using the weight vectors from encoder neural network across each layer.
+
+Results Figure 3 a) shows the classification accuracies of the stimulus categories, across different methods and the number of layers for each model. The performances by $\Delta$ -SNICA were consistently higher than those by the other (baseline) methods, which indicates the importance of the modeling of the MEG signals by $\Delta$ -SNICA . Figure $^ { 3 \mathrm { ~ b ~ } }$ ) shows an example of spatial patterns from the encoder network learned by the $\Delta$ -SNICA . We used the visualization method presented in (Hyvärinen and Morioka, 2016). We manually picked one out of the hidden nodes from the third layer in encoder network, and plotted its weighted-averaged sensor signals, We also visualized the most strongly contributing second- and first-layer nodes. We see progressive pooling of L1 units to form left lateral frontal, right lateral frontal and parietal patterns in L2 which are then all pooled together in L3 resulting in a lateral frontoparietal pattern. Most of the spatial patterns in the third layer (not shown) are actually similar to those previously reported using MEG (Brookes et al., 2011). Appendix E provides more detail to the interpretation of the $\Delta$ -SNICA results.
+
+# 6 Related work
+
+Previous works on nonlinear ICA have exploited autocorrelations (Hyvärinen and Morioka, 2017; Oberhauser and Schell, 2021) and nonstationarities (Hyvärinen and Morioka, 2016; Hälvä and Hyvärinen, 2020) for identifiability. The SNICA setting provides a unifying framework which allows for both types of temporal dependencies, and further, extends identifiability to other temporal structures as well as any arbitrary higher order data structures which has not previously been considered in the context of nonlinear ICA. Another major theoretical contribution here is to show that identifiability with noise of unknown, arbitrary distribution, while previous work on noisy nonlinear ICA assumed noise of known distribution and known variance (Khemakhem et al., 2020a).
+
+Importantly, the SNICA framework is fully probabilistic and thus accommodates higher order latent variables, leading to "purely unsupervised" learning. This is in large contrast to previous research which have been developed for the case where we are able to observe some additional auxiliary variable, such as audio signals accompanying video (Hyvärinen et al., 2019; Khemakhem et al., 2020a,b), or heuristically define the auxiliary variable based on time structure (Hyvärinen and Morioka, 2016). In practice this means that we are able to estimate our models using (variational)
+
+![](images/a5ff5727e16244565822b63bcda491f3e6db8ebd27f9bbe3c0993da73fbece75.jpg)  
+Figure 3: $\Delta$ -SNICA on MEG data. (a) Classification accuracies of linear SVMs trained with auditoryvisual data to predict stimulus category, with feature extractors trained by $\Delta$ -SNICA in advance with resting-state data. Each point represents a testing accuracy on a target subject (chance level: $50 \%$ ). Horizontal dotted line is PCA-only baseline. (b) Example of spatial patterns of the components learned by $\Delta$ -SNICA $\left( \mathrm { L } { = } 3 \right)$ . Each topography corresponds to one spatial pattern. L3: approximate total spatial pattern of one third-layer unit. L2: the patterns of the three second-layer units maximally contributing to this L3 unit. L1: for each L2 unit, the two most strongly contributing first-layer units.
+
+MLE, which is more principled than the heuristic self-supervised methods in most earlier papers. The only existing frameworks allowing MLE (Hälvä and Hyvärinen, 2020; Khemakhem et al., 2020a) used model restricted to exponential families, and had either no HMM or a very simple one.
+
+The switching linear dynamical model, $\Delta$ -SNICA in Section 3.2, shows the above benefits in the form of a single model. That is, unlike previous nonlinear ICA models, it combines: 1) temporal dependencies and "non-stationarity" (or HMM) in a single model 2) dimensionality reduction within a rigorous maximum likelihood learning and inference framework, and 3) a separate observation equation with general observational noise. This results in a very rich, realistic, and principled model for time series.
+
+Very recently, Morioka et al. (2021) proposed a related model by considering innovations of time series to be nonstationary. However, their model is noise-free, restricted to exponential families of at least order two, and not applicable to the spatial case, thus making our identifiability results significantly stronger. From a more practical viewpoint, their model suffers from the fact that it either does not allow for dimensionality reduction (if an HMM is used) or requires a manual segmentation (if HMM is not used). Nor does it have a clear distinction into a state dynamics equation and a measurement equation which allows for cleaning or denoising of the data.
+
+Limitations Our identifiability theory makes some restrictive assumptions, and it remains to be seen if they could be lifted in future work. In particular, the data is not allowed to have too heavy tails; the noise must be additive, and independent of the signal; and the practical interpretation of some of the assumptions, such as (A3) is difficult. It is also difficult to say whether our assumption of unconditionally independent components is realistic in practice. Regarding practical applications, our specific model only scratches the surface of what is possible in this framework. In particular, we did not develop a model with spatial distributions, nor did we model non-Gaussian observational noise – our main aim was to lay the foundations for the relevant identification theory. Future work should aim to make the estimation more efficient computationally; this is a ubiquitous problem in deep learning, but specific solutions for this concrete problem may be achievable (Gresele et al., 2020).
+
+# 7 Conclusion
+
+We proposed a new general framework for identifiable disentanglement, based on nonlinear ICA with very general temporal dynamics or spatial structure. Observational noise of arbitrary unknown distribution is further included. We prove identifiability of the models in this framework with high generality and mathematical rigour. For real data analysis, we propose a special case which subsumes the properties of all existing time series models in nonlinear ICA, while generalizing them in many ways (see Section 6 for details). We hope this work will contribute to wide-spread application of identifiable methods for disentanglement in a highly principled, probabilistic framework.

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+    {
+        "type": "text",
+        "text": "Disentangling Identifiable Features from Noisy Data with Structured Nonlinear ICA ",
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+    {
+        "type": "text",
+        "text": "Hermanni Hälvä1 ∗ Sylvain Le Corff2 Luc Lehéricy3 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Jonathan So4 Yongjie Zhu1 Elisabeth Gassiat5 † Aapo Hyvärinen1 † ",
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+    {
+        "type": "text",
+        "text": "1Department of Computer Science, University of Helsinki, Finland 2 Samovar, Télécom SudParis, département CITI, Institut Polytechnique de Paris, Palaiseau, France 3Laboratoire J. A. Dieudonné, Université Côte d’Azur, CNRS, 06100, Nice, France 4Department of Engineering, University of Cambridge, UK 5Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France ",
+        "bbox": [
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
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+    {
+        "type": "text",
+        "text": "We introduce a new general identifiable framework for principled disentanglement referred to as Structured Nonlinear Independent Component Analysis (SNICA). Our contribution is to extend the identifiability theory of deep generative models for a very broad class of structured models. While previous works have shown identifiability for specific classes of time-series models, our theorems extend this to more general temporal structures as well as to models with more complex structures such as spatial dependencies. In particular, we establish the major result that identifiability for this framework holds even in the presence of noise of unknown distribution. Finally, as an example of our framework’s flexibility, we introduce the first nonlinear ICA model for time-series that combines the following very useful properties: it accounts for both nonstationarity and autocorrelation in a fully unsupervised setting; performs dimensionality reduction; models hidden states; and enables principled estimation and inference by variational maximum-likelihood. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "A central tenet of unsupervised deep learning is that noisy and high dimensional real world data is generated by a nonlinear transformation of lower dimensional latent factors. Learning such lower dimensional features is valuable as they may allow us to understand complex scientific observations in terms of much simpler, semantically meaningful, representations (Morioka et al., 2020; Zhou and Wei, 2020). Access to a ground truth generative model and its latent features would also greatly enhance several other downstream tasks such as classification (Klindt et al., 2021; Banville et al., 2021), transfer learning (Khemakhem et al., 2020b), as well as causal inference (Monti et al., 2019; Wu and Fukumizu, 2020). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A recently popular approach to deep representation learning has been to learn disentangled features. Whilst not rigorously defined, the general methodology has been to use deep generative models such as VAEs (Kingma and Welling, 2014; Higgins et al., 2017) to estimate semantically distinct factors of variation that generate and encode the data. A substantial problem with the vast majority of work on disentanglement learning is that the models used are not identifiable – that is, they do not learn the true generative features, even in the limit of infinite data – in fact, this task has been proven impossible without inductive biases on the generative model (Hyvärinen and Pajunen, 1999; Locatello et al., 2019). Lack of identifiability plagues deep learning models broadly and has been implicated as one of the reasons for unexpectedly poor behaviour when these models are deployed in real world applications (D’Amour et al., 2020). Fortunately, in many applications the data have dependency structures, such as temporal dependencies which introduce inductive biases. Recent advances in both identifiability theory and practical algorithms for nonlinear ICA (Hyvärinen and Morioka, 2016, 2017; Hälvä and Hyvärinen, 2020; Morioka et al., 2021; Klindt et al., 2021; Oberhauser and Schell, 2021) exploit this and offer a principled approach to disentanglement for such data. Learning statistically independent nonlinear features in such models is well-defined, i.e. those models are identifiable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, the existing nonlinear ICA models suffer from numerous limitations. First, they only exploit specific types of temporal structures, such as either temporal dependencies or nonstationarity. Second, they often work under the assumption that some ’auxiliary’ data about a latent process is observed, such as knowledge of the switching points of a nonstationary process as in Hyvärinen and Morioka (2016); Khemakhem et al. (2020a) . Furthermore, all the nonlinear ICA models cited above, with the exception of Khemakhem et al. (2020a), assume that the data are fully observed and noise-free, even though observation noise is very common in practice, and even Khemakhem et al. (2020a) assumes the noise distribution to be exactly known. This approach of modelling observation noise explicitly is in stark contrast to the approach taken in papers, such as Locatello et al. (2020), who instead consider general stochasticity of their model to be captured by latent variables – this approach would be ill-suited to the type of denoising one would often need in practice. Lastly, the identifiability theorems in previous nonlinear ICA works usually restrict the latent components to a specific class of models such as exponential families (but see Hyvärinen and Morioka (2017)). ",
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+    {
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+        "text": "In this paper we introduce a new framework for identifiable disentanglement, Structured Nonlinear ICA (SNICA), which removes each of the aforementioned limitations in a single unifying framework. Furthermore, the framework guarantees identifiability of a rich class of nonlinear ICA models that is able to exploit dependency structures of any arbitrary order and thus, for instance, extends to spatially structured data. This is the first major theoretical contribution of our paper. ",
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+    {
+        "type": "text",
+        "text": "The second important theoretical contribution of our paper proves that models within the SNICA framework are identifiable even in the presence of additive output noise of arbitrary, unknown distribution. We achieve this by extending the theorems by Gassiat et al. (2020b,a). The subsequent practical implication is that SNICA models can perform dimensionality reduction to identifiable latent components and de-noise observed data. We note that noisy-observation part of the identifiability theory is not even limited to nonlinear ICA but applies to any system observed under noise. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Third, we give mild sufficient conditions, relating to the strength and the non-Gaussian nature of the temporal or spatial dependencies, enabling identifiability of nonlinear independent components in this general framework. An important implication is that our theorems can be used, for example, to develop models for disentangling identifiable features from spatial or spatio-temporal data. ",
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+        "text": "As an example of the flexibility of the SNICA framework, we present a new nonlinear ICA model called $\\Delta$ -SNICA . It achieves the following very practical properties which have previously been unattainable in the context of nonlinear ICA: the ability to account for both nonstationarity and autocorrelation in a fully unsupervised setting; ability perform dimensionality reduction; model latent states; and to enable principled estimation and inference by variational maximum-likelihood methods. We demonstrate the practical utility of the model in an application to noisy neuroimaging data that is hypothesized to contain meaningful lower dimensional latent components and complex temporal dynamics. ",
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+        "text": "2 Background ",
+        "text_level": 1,
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+        "text": "We start by giving some brief background on Nonlinear ICA and identifiability. Consider a model where the distribution of observed data $\\mathbf { x }$ is given by $p _ { X } ( \\mathbf { x } ; \\pmb \\theta )$ for some parameter vector $\\pmb \\theta$ . This model is called identifiable if the following condition is fulfilled: ",
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+        "img_path": "images/49617abbb89ab39cc2c3ecdbff66248028105eee7dd768aac0a017ddd6bf74aa.jpg",
+        "text": "$$\n\\forall ( \\pmb \\theta , \\pmb \\theta ^ { \\prime } ) \\qquad p _ { X } ( \\mathbf x ; \\pmb \\theta ) = p _ { X } ( \\mathbf x ; \\pmb \\theta ^ { \\prime } ) \\Rightarrow \\pmb \\theta = \\pmb \\theta ^ { \\prime } .\n$$",
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+        "type": "text",
+        "text": "In other words, based on the observed data distribution alone, we can uniquely infer the parameters that generated the data. For models parameterized with some nonparametric function estimator $\\mathbf { f }$ , such as a deep neural network, we can replace $\\pmb { \\theta }$ with $\\mathbf { f }$ in the equation above. In practice, identifiability ",
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+    {
+        "type": "text",
+        "text": "might hold for some parameters, not all; and parameters might be identifiable up to some more or less trivial indeterminacies, such as scaling. ",
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+        "type": "text",
+        "text": "In a typical nonlinear ICA setting we observe some $\\mathbf { x } \\in \\mathbb { R } ^ { N }$ which has been generated by an invertible nonlinear mixing function f from latent independent components $\\mathbf { s } \\in \\mathbb { R } ^ { N }$ , with $\\begin{array} { r } { p ( \\mathbf { s } ) = \\mathbf { \\dot { \\prod } } _ { i = 1 } ^ { N } p ( s ^ { ( i ) } ) } \\end{array}$ , as per: ",
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+        "type": "equation",
+        "img_path": "images/f0678148a07288a6581a7d3b9745f0dd6050863aceeba0a86ae6e0bce8b7047a.jpg",
+        "text": "$$\n\\begin{array} { r } { { \\bf x } = { \\bf f } ( { \\bf s } ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Identifiability of f would then mean that we can in theory find the true $\\mathbf { f }$ , and subsequently the true data generating components. Unfortunately, without some additional structure this model is unidentifiable, as shown by Hyvärinen and Pajunen (1999): there is an infinite number of possible solutions and these have no trivial relation with each other. To solve this problem, previous work (Sprekeler et al., 2014; Hyvärinen and Morioka, 2016, 2017) developed models with temporal structure. Such time series models were generalized and expressed in a succinct way by Hyvärinen et al. (2019); Khemakhem et al. (2020a) by assuming the independent components are conditionally independent upon some observed auxiliary variable $\\begin{array} { r } { \\bar { u } _ { t } \\colon p ( \\mathbf { s } _ { t } | \\bar { u _ { t } } ) = \\prod _ { i = 1 } ^ { N } \\bar { p ( s _ { t } ^ { ( i ) } | u _ { t } ) } . } \\end{array}$ . In a time series context, the auxiliary variable might be history, e.g. $u _ { t } = \\mathbf { x } _ { t - 1 }$ , or the index of a time segment to model nonstationarity (or piece-wise stationarity). (It could also be data from another modality, such as audio data used to condition video data (Arandjelovic and Zisserman, 2017).) ",
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+    {
+        "type": "text",
+        "text": "Notice that the mixing function f in (2) is assumed bijective and thus identifiable dimension reduction is not possible in most of the models discussed above. The only exceptions, we are aware of, are Khemakhem et al. (2020a); Klindt et al. (2021) who choose f as injective rather than bijective. Further, Khemakhem et al. (2020a) assume additive noise on the observations $\\mathbf { x } = \\mathbf { f } ( \\mathbf { s } ) + \\boldsymbol \\varepsilon$ , which allows to estimate posterior of s by an identifiable VAE (iVAE). We will take a similar strategy in what follows. ",
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+        "type": "text",
+        "text": "3 Definition of Structured Nonlinear ICA ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we first present the new framework of Structured Nonlinear ICA (SNICA) – a broad class of models for identifiable disentanglement and learning of independent components when data has structural dependencies. Next, we give an example of a particularly useful specific model that fits within our framework, called $\\Delta$ -SNICA , by using switching linear dynamical latent processes. ",
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+        "type": "text",
+        "text": "3.1 Structured Nonlinear ICA framework ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Consider observations $( \\mathbf { x } _ { t } ) _ { t \\in \\mathbb { T } } = ( ( x _ { t } ^ { ( 1 ) } , \\dots , x _ { t } ^ { ( M ) } ) ) _ { t \\in \\mathbb { T } }$ where $\\mathbb { T }$ is a discrete indexing set of arbitrary dimension. For discrete time-series models, like previous works, $\\mathbb { T }$ would be a subset of $\\mathbb { N }$ . Crucially, however, we allow it to be any arbitrary indexing variable that describes a desired structure. For instance, $\\mathbb { T }$ could be a subset of $\\mathbb { N } ^ { 2 }$ for spatial data. ",
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+        "type": "text",
+        "text": "We assume the data is generated according the following nonlinear ICA model. First, there exist latent components $\\mathbf { s } ^ { ( i ) } \\bar { = } ( s _ { t } ^ { ( i ) } ) _ { t \\in \\mathbb { T } }$ for $i \\in \\{ 1 , \\ldots , N \\}$ where for any $t , t ^ { \\prime } \\in \\mathbb { T }$ , the distributions of $( \\mathbf { s } _ { t } ^ { ( i ) } ) _ { 1 \\leqslant i \\leqslant N }$ and $( \\mathbf { s } _ { t ^ { \\prime } } ^ { ( i ) } ) _ { 1 \\leqslant i \\leqslant N }$ are the same, which is a weak form of stationarity. Second, we assume t that for any $m \\in \\mathbb { N } ^ { * }$ and $( t _ { 1 } , \\ldots , t _ { m } ) \\in \\mathbb { T } ^ { m }$ , $\\begin{array} { r } { \\underline { { p } } ( \\mathbf { s } _ { t _ { 1 } } , \\ldots , \\mathbf { s } _ { t _ { m } } ) = \\prod _ { i = 1 } ^ { N } p ( s _ { t _ { 1 } } ^ { ( i ) } , \\ldots , s _ { t _ { m } } ^ { ( i ) } ) } \\end{array}$ : that is, the $\\mathbf { f } : \\dot { \\mathbb { R } ^ { N } } \\xrightarrow { } \\mathbb { R } ^ { M }$ with $M \\geqslant N$ is injective, so there may be more observed variables than components. Finally, denote observational noise by $\\boldsymbol { \\varepsilon } _ { t } \\in \\mathbb { R } ^ { M }$ and assume that they are i.i.d. for all $t \\in \\mathbb { T }$ and independent of the signals $\\mathbf { s } ^ { ( i ) }$ . Putting these together, we assume the mixing model where for each $t \\in \\mathbb { T }$ , ",
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+    {
+        "type": "equation",
+        "img_path": "images/d2558c4b2a69dda20acb8f60770fa18be879d57076fb8e3cc4087280a0cb7795.jpg",
+        "text": "$$\n\\mathbf { x } _ { t } = \\mathbf { f } ( \\mathbf { s } _ { t } ) + \\pmb { \\varepsilon } _ { t } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { s } _ { t } = ( s _ { t } ^ { ( 1 ) } , \\dots , s _ { t } ^ { ( N ) } )$ . Importantly, $\\varepsilon _ { t }$ can have any arbitrary unknown distribution, even with dependent entries; in fact, it may even not have finite moments. ",
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+        "text": "The main appeal of this framework is that, under the conditions given in next section, we can now guarantee identifiability for a very broad and rich class of models. ",
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+        "type": "text",
+        "text": "First, notice that all previous Nonlinear ICA time-series models can be reformulated and often improved upon when viewed through this new unifying framework. In other words, we can create models that are very much like those previous works, and capture their dependency profiles, but with the changes that by assuming unconditional independence and output noise we now allow them to perform dimension reduction (this does also require some additional assumptions needed in our identifiability theorems below). To see this, consider the model in Hälvä and Hyvärinen (2020) which captures nonstationarity in the independent components through a global hidden Markov chain. We can transform this model into the SNICA framework if we instead model each independent component as its own HMM (Figure 1a), with the added benefit that we now have marginally independent components and are able to perform dimensionality reduction into low dimensional latent components. Nonlinear ICA with time-dependencies, such as in an autoregressive model, proposed by Hyvärinen and Morioka (2017) is also a special case of our framework (Figure 1b), but again with the extension of dimensionality reduction. Furthermore, this framework allows for a plethora of new Nonlinear ICA models to be developed. As described above, these do not have to be limited to time-series but could for instance be a process on a two-dimensional graph with appropriate (in)dependencies (see Figure 1c). However, we now proceed to introduce a particularly useful time-series model using our framework. ",
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+        "img_path": "images/d5b846d3be1fcd8b8bd5707832745a6e06c4941953b67729900625560231feac.jpg",
+        "image_caption": [
+            "Figure 1: Graphical models for the SNICA framework "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "3.2 $\\Delta$ -SNICA $:$ Nonlinear ICA with switching linear dynamical systems ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "While the above framework has great generality, any practical application will need a specific model. Next we propose one which combines the following properties of previous nonlinear ICA models into a single model: ability to account for both nonstationarity and autocorrelation in a fully unsupervised setting, to perform dimensionality reduction and model hidden states. Real world processes, such as video/audio data, financial time-series, and brain signals, exhibit these properties – disentangling latent features in such data would hence be very useful. ",
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+        "type": "text",
+        "text": "Our new model is depicted in Figure 1d. The independent components are generated by a Switching Linear Dynamical System (SLDS) (Ackerson and Fu, 1968; Chang and Athans, 1978; Hamilton, 1990; Ghahramani and Hinton, 2000) with additional latent variables to express rich dynamics. Formally, for each independent component $i \\in \\{ 1 , \\ldots , N \\}$ , consider the following SLDS over some latent vector $\\mathbf { y } _ { t } ^ { ( i ) }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/f4f24ac61862b008dc62602729f8b4e9a543e2f570d0059281ed12b53efc6dfd.jpg",
+        "text": "$$\n\\mathbf { y } _ { t } ^ { ( i ) } = \\mathbf { B } _ { u _ { t } } ^ { ( i ) } \\mathbf { y } _ { t - 1 } ^ { ( i ) } + \\mathbf { b } _ { u _ { t } } ^ { ( i ) } + \\varepsilon _ { u _ { t } } ^ { ( i ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $u _ { t } : = u _ { t } ^ { ( i ) }$ is a state of a first-order hidden Markov chain $( u _ { t } ^ { ( i ) } ) _ { t = 1 : T }$ . Crucially, we assume that t tthe independent components at each time-point are the first elements $y _ { t , 1 } ^ { ( i ) }$ of $\\mathbf { y } _ { t } ^ { ( i ) } = ( y _ { t , 1 } ^ { ( i ) } , \\ldots , y _ { t , d } ^ { ( i ) } ) ^ { T }$ i.e. $s _ { t } ^ { ( i ) } = y _ { t , 1 } ^ { ( i ) }$ . The rest of the elements in $\\mathbf { y } _ { t } ^ { ( i ) }$ are latent variables modelling hidden dynamics. The great utility of using such a higher-dimensional latent variable is that this model allows us, for example, as a special case, to consider higher-order ARMA processes, thus modelling each $s _ { t } ^ { ( i ) }$ as switching between ARMA processes of an order determined by the dimensionality of $\\mathbf { y } _ { t }$ . We call the ensuing model $\\Delta$ -SNICA (\"Delta-SNICA\", with delta as in \"dynamic\"). ",
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+        "type": "text",
+        "text": "4 Identifiability ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we present two very general identifiability theorems for SNICA. We basically decouple the problem into two parts. First, we consider identifying the noise-free distribution of $\\mathbf { f } \\left( \\mathbf { s } _ { t } \\right)$ from noisy data. Theorem 1 states conditions—on tail behaviour, non-degeneracy, and non-Gaussianity— under which it is possible to recover the distribution of a process based on noisy data with unknown noise distribution. Second, we consider demixing of the nonlinearly mixed data. Theorem 2 provides general conditions—on temporal or spatial dependencies, and non-Gaussianity—that allow recovery of the mixing function f when there is no more noise. We then consider application of these theorems to SNICA. ",
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+        "type": "text",
+        "text": "4.1 Identifiability with unknown noise distribution ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Consider the model ",
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+    {
+        "type": "equation",
+        "img_path": "images/ed421181254fdfbb0d092d0653b0da5d055e876b5b84eb4437cddae0f5d08e27.jpg",
+        "text": "$$\n{ \\bf x } _ { t } = { \\bf z } _ { t } + \\varepsilon _ { t } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $( \\mathbf { z } _ { t } ) _ { t \\in \\mathbb { T } }$ is a family of random variables in $\\mathbb { R } ^ { M }$ such that all $\\mathbf { z } _ { t }$ , $t \\in \\mathbb { T }$ , have the same marginal distribution, and $\\textstyle ( \\varepsilon _ { t } ) _ { t \\in \\mathbb { T } }$ is a family of independent (over $t$ ) and identically distributed random variables, independent of $( \\mathbf { z } _ { t } ) _ { t \\in \\mathbb { T } }$ . Let $P$ be the common distribution of each $\\varepsilon _ { t }$ , for $t \\in \\mathbb { T }$ . Let $t _ { 1 }$ and $t _ { 2 }$ in $\\mathbb { T }$ , and consider the following assumptions. ",
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+        "type": "text",
+        "text": "• (A1) [Tail behaviour] For some $\\rho < 3$ , there exist $A$ and $B$ such that for all $\\boldsymbol { \\lambda } \\in \\mathbb { R } ^ { N }$ , $\\mathbb { E } [ \\exp ( \\langle \\lambda , \\mathbf { z } _ { t _ { 1 } } \\rangle ) ] \\leqslant A \\exp ( B \\| \\lambda \\| ^ { \\rho } ) .$   \n• (A2) [Non-degeneracy] For any $\\ b { \\eta } \\in \\mathbb { C } ^ { M }$ , $\\mathbb { E } [ \\exp \\{ \\langle \\eta , \\mathbf { z } _ { t _ { 2 } } \\rangle \\} | \\ \\mathbf { z } _ { t _ { 1 } } ]$ is not the null random variable.   \n• (A3) [Non-Gaussianity] The following assertion is false: there exist a vector $\\boldsymbol { \\eta } \\in \\mathbb { R } ^ { M }$ and independent random variables $\\tilde { z }$ and $u$ , such that $u$ is a non dirac Gaussian random variable and $\\langle \\eta , \\mathbf { z } _ { t _ { 1 } } \\rangle$ has the same distribution as $\\tilde { z } + u$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "We defer the detailed discussion on the practical meaning of the assumptions (A1-A3) in the context of SNICA to Section 4.3. We next present Theorem 1 which establishes identifiability under unknown noise (its proof is postponed to Section A.1 in the Supplementary Material): ",
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+        "type": "text",
+        "text": "Theorem 1 Assume that assumptions (A1), (A2) and (A3) hold for some $( t _ { 1 } , t _ { 2 } ) \\in  { \\mathbb { T } } ^ { 2 }$ . Then, up to translation, for all $m \\geqslant 2$ , for all $( t _ { 3 } , \\ldots , t _ { m } ) \\in  { \\mathbb { T } } ^ { m - 2 }$ , the application that associates the distribution of $\\left( \\mathbf { z } _ { t _ { 1 } } , \\ldots , \\mathbf { z } _ { t _ { m } } \\right)$ and $P$ to the distribution of $\\left( \\mathbf { x } _ { t _ { 1 } } , \\ldots , \\mathbf { x } _ { t _ { m } } \\right)$ is one-to-one. ",
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+    {
+        "type": "text",
+        "text": "Here, up to translation means that adding a constant vector to all $\\varepsilon _ { t }$ , and substracting this constant to all $\\mathbf { z } _ { t }$ , $t \\in \\{ t _ { 1 } , \\ldots , t _ { m } \\}$ , does not change the distribution of $( \\mathbf { x } _ { t _ { 1 } } , \\ldots , \\mathbf { x } _ { t _ { m } } )$ . The proof of Theorem 1 extends that of Theorem 1 in (Gassiat et al., 2020b), see also (Gassiat et al., 2020a), which assumed sub-Gaussian noise-free data. Our extension allows the noise-free data to have heavier tails, which is important since (noise-free) data in many real-world applications is super-Gaussian, i.e. heavy-tailed, as is well-known in work on linear ICA (Hyvärinen et al., 2001). ",
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+    {
+        "type": "text",
+        "text": "Importantly, there is no assumption on the unknown noise distribution in Theorem 1. In fact, it does not even assume a mixing as in ICA, and thus extends greatly outside of the framework of this paper. ",
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+    {
+        "type": "text",
+        "text": "4.2 Identifiability of the mixing function ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Based on Theorem 1, it is possible to recover the distribution of the noise-free data in SNICA in (3) by setting $\\mathbf { z } _ { t } = \\mathbf { f } ( \\mathbf { s } _ { t } )$ . Next, we consider under which conditions the mixing function f is identifiable. Denote by $S = S ^ { ( 1 ) } \\times \\cdots \\times S ^ { ( N ) }$ the support of the distribution of all $\\mathbf { s } _ { t }$ . We consider the situation where each $S ^ { ( i ) } \\subset \\mathbb { R }$ , $1 \\leqslant i \\leqslant N$ , is connected, so that each $S ^ { ( i ) }$ is an interval. We assume moreover that the injective mixing function f is a $\\mathcal { C } ^ { 2 }$ diffeomorphism between $S$ and a $\\mathcal { C } ^ { 2 }$ differentiable manifold $\\mathcal { M } \\subset \\mathbb { R } ^ { M }$ . Formally, this means that there exists an atlas $\\{ \\varphi _ { \\vartheta } : U _ { \\vartheta } \\to \\mathbb { R } ^ { N } \\} _ { \\vartheta \\in \\Theta }$ of $\\mathcal { M }$ such that for all $\\vartheta , \\vartheta ^ { \\prime } \\in \\Theta$ , the map $\\varphi _ { \\vartheta } \\circ \\varphi _ { \\vartheta ^ { \\prime } } ^ { - 1 }$ is a $\\mathcal { C } ^ { 2 }$ map, and f is a bijection $\\mathbb { R } ^ { N } \\to \\mathcal { M }$ such that for all $\\vartheta \\in \\Theta$ , $\\varphi _ { \\vartheta } \\circ \\mathbf { f }$ and $\\mathbf { f } ^ { - 1 } \\circ \\varphi _ { \\vartheta } ^ { - 1 }$ have continuous second derivatives. The sets $U _ { \\vartheta }$ , $\\vartheta \\in \\Theta$ , cover $\\mathcal { M }$ and are open in $\\mathcal { M }$ . The proof of Theorem 2 is postponed to Section A.2 in the Supplementary Material. ",
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+        "type": "text",
+        "text": "ssume that thehas a density xist wh $\\qquad m \\ \\geqslant \\ 2$ anon $( t _ { 1 } , \\ldots , t _ { m } ) \\in \\mathbb { T } ^ { m }$ such that the vectorreover that there exist $( s _ { t _ { 1 } } ^ { ( i ) } , \\ldots , s _ { t _ { m } } ^ { ( i ) } )$ $p _ { m } ^ { ( i ) }$ $\\mathcal { C } ^ { 2 }$ $( S ^ { ( i ) } ) ^ { m }$ $( k , l ) \\in \\{ 1 , \\ldots , m \\} ^ { 2 }$ with $k \\neq l$ such that the following assumptions hold with . ",
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+        "type": "text",
+        "text": "• $( B l )$ (Uniform $( k , l )$ -dependency). For all $i \\in \\{ 1 , \\ldots , N \\}$ , the set of zeros of $\\frac { \\partial ^ { 2 } } { \\partial s _ { t _ { k } } ^ { ( i ) } \\partial s _ { t _ { l } } ^ { ( i ) } } Q _ { m } ^ { ( i ) }$ is a meagre subset of $( S ^ { ( i ) } ) ^ { m }$ , i.e. it contains no open subset. ",
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+    {
+        "type": "text",
+        "text": "• $( B 2 )$ (Local $( k , l )$ -non quasi Gaussianity). For any open subset $A \\subset S ^ { m }$ , there exists at most one $i \\in \\{ 1 , \\ldots , N \\}$ such that there exists a function $\\alpha : \\mathbb { R } ^ { m - 1 }  \\mathbb { R }$ and a constant $c \\in \\mathbb { R }$ such that for all $s \\in A$ , ",
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+        "type": "equation",
+        "img_path": "images/f6015e3b7a0f99a18206b8e3b1d9c748c479b76487a74fb6bcdaf96865088154.jpg",
+        "text": "$$\n\\frac { \\partial ^ { 2 } } { \\partial s _ { t _ { k } } ^ { ( i ) } \\partial s _ { t _ { l } } ^ { ( i ) } } Q _ { m } ^ { ( i ) } = c \\alpha ( s _ { t _ { k } } ^ { ( i ) } , \\mathbf { s } _ { ( - t _ { k } , - t _ { l } ) } ^ { ( i ) } ) \\alpha ( s _ { t _ { l } } ^ { ( i ) } , \\mathbf { s } _ { ( - t _ { k } , - t _ { l } ) } ^ { ( i ) } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { s } _ { ( - t _ { k } , - t _ { l } ) } ^ { ( i ) } \\ i s \\ ( s _ { t _ { 1 } } ^ { ( i ) } , \\ldots , s _ { t _ { m } } ^ { ( i ) } )$ without the coordinates $t _ { k }$ and $t _ { l }$ . ",
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+    {
+        "type": "text",
+        "text": "Then, $\\mathbf { f } ^ { - 1 }$ can be recovered up to permutation and coordinate-wise transformations from the distribution of $( \\mathbf { f } ( \\mathbf { s } _ { t _ { 1 } } ) , \\dots , \\mathbf { f } ( \\mathbf { s } _ { t _ { m } } ) )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 Applications to SNICA ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we provide additional comments on the assumptions (A1-A3) and (B1-B2) and their verification in the context of SNICA. ",
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+        "type": "text",
+        "text": "Assumption (A1) is a condition on the tails of the noise-free data: it allows tails that are somewhat heavier than Gaussian tails. It is in fact equivalent to assuming that for some $\\tilde { \\rho } > 3 / 2$ , there exists $A ^ { \\prime } , B ^ { \\prime } > 0$ such that for all $t > 0$ , $\\| \\mathbf { z } _ { t _ { 1 } } \\| \\geqslant t ) \\leqslant A ^ { \\prime } \\exp ( - \\bar { B ^ { \\prime } } t ^ { \\tilde { \\rho } } )$ . ",
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+        "type": "text",
+        "text": "Assumption (A2) is a non-degeneracy condition likely to be fulfilled for any randomly chosen SNICA model parameters. As an example, consider a model such as Fig. 1c, where there exist hidden variables $( u _ { t } ) _ { t \\in \\mathbb { T } }$ taking values in a finite set $\\{ 1 , \\ldots , K \\}$ such that the pairs of variables $\\left( \\mathbf { z } _ { t } , u _ { t } \\right)$ have the same distribution for all $t \\in \\mathbb { T }$ , and such that conditioned on $( u _ { t } ) _ { t \\in \\mathbb { T } }$ , the variables $( \\mathbf { z } _ { t } ) _ { t \\in \\mathbb { T } }$ are independent and the distribution of $\\mathbf { z } _ { t }$ only depends on $u _ { t }$ . (As a special case, this model includes the temporal HMM setting described in Fig. 1a.) Let $( t _ { 1 } , t _ { 2 } ) \\in \\mathbb { T } ^ { 2 }$ . For all $u , v \\in \\{ 1 , \\ldots , K \\}$ , let $\\pi ( u ) = \\overline { { p _ { u _ { t _ { 1 } } } ( u ) } }$ be the mass function of $\\boldsymbol { u } _ { t _ { 1 } }$ , $Q ( u , v ) = p _ { u _ { t _ { 2 } } | u _ { t _ { 1 } } } ( v | u )$ be the transition matrix from $u _ { t _ { 1 } }$ to $u _ { t _ { 2 } }$ , and $\\gamma _ { u } ( \\mathbf { z } ) = p _ { \\mathbf { z } _ { t _ { 1 } } | u _ { t _ { 1 } } } ( \\mathbf { z } | u )$ be the density of $\\mathbf { z } _ { t _ { 1 } }$ conditionally to $u _ { t _ { 1 } } = u$ . By assumption, it is also the density of $\\mathbf { z } _ { t _ { 2 } }$ conditionally to $u _ { t _ { 2 } } = u$ . Theorem 3 provides sufficient conditions for assumption (A2) to hold: ",
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+        "type": "text",
+        "text": "Theorem 3 Assume that $Q$ has full rank, $\\operatorname* { m i n } _ { u } \\pi ( u ) > 0$ and the $( \\gamma _ { u } ) _ { 1 \\leqslant u \\leqslant K }$ are linearly independent, then (A2) is satisfied as soon as the functions $\\begin{array} { r } { \\langle \\eta \\mapsto \\int \\exp ( \\langle \\eta , \\mathbf { z } \\rangle ) \\gamma _ { v } ( \\mathbf { z } ) d \\mathbf { z } ) _ { 1 \\leqslant v \\leqslant K } } \\end{array}$ do not have simultaneous zeros. ",
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+    {
+        "type": "text",
+        "text": "Besides the non-simultaneous zeros assumption, the assumptions of Theorem 3 are reminiscent of those used for the identifiability of non-parametric hidden Markov models, see for instance Gassiat et al. (2016); Lehéricy (2019). The key element is that $\\mathbf { z } _ { t _ { 1 } }$ and $\\mathbf { z } _ { t _ { 2 } }$ are not independent. Thus, we see that (A2) holds if the $\\pi$ and the $\\gamma$ are not degenerate (in the precise sense given by Theorem 3), for the latent state models in Figs. 1a,1c.Another situation where (A2) holds is when $\\mathbf { z } _ { t _ { 2 } }$ is a complete statistic (Lehmann and Casella, 2006) in the statistical model $\\{ \\mathbb { P } _ { \\mathbf { z } _ { t _ { 2 } } | \\mathbf { z } _ { t _ { 1 } } } \\left( \\cdot | \\mathbf { z } _ { t _ { 1 } } \\right) \\} _ { \\mathbf { z } _ { t _ { 1 } } }$ , where $\\mathbb { P } _ { \\mathbf { z } _ { t _ { 2 } } \\mid \\mathbf { z } _ { t _ { 1 } } } \\left( \\cdot | \\mathbf { z } _ { t _ { 1 } } \\right)$ is the distribution of $\\mathbf { z } _ { t _ { 2 } }$ conditionally to $\\mathbf { z } _ { t _ { 1 } }$ . Consider the two following examples where this holds: 1) When the model $\\{ \\mathbb { P } _ { \\mathbf { z } _ { t _ { 2 } } | \\mathbf { z } _ { t _ { 1 } } } ^ { - } \\left( \\cdot | \\mathbf { z } _ { t _ { 1 } } \\right) \\} _ { \\mathbf { z } _ { t _ { 1 } } }$ is an exponential family. In this situation, complete statistics are known. 2) Autoregressive models with additive innovation of the form ${ \\bf z } _ { t _ { 2 } } = { \\bf h } ( { \\bf z } _ { t _ { 1 } } ) + { \\bf v } _ { t _ { 2 } }$ for some bijective function $\\mathbf { h }$ when the additive noise $\\mathbf { v } _ { t _ { 2 } }$ is a complete statistic in the statistical model $\\{ \\mathbb { P } _ { \\mathbf { v } _ { t _ { 2 } } | \\mathbf { z } _ { t _ { 1 } } } \\left( \\cdot | \\mathbf { z } _ { t _ { 1 } } \\right) \\} _ { \\mathbf { z } _ { t _ { 1 } } }$ (note that $\\mathbf { v } _ { t _ { 2 } }$ cannot be independent of $\\mathbf { z } _ { t _ { 1 } }$ here). The case in Fig. 1b is typically covered by this example. ",
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+    {
+        "type": "text",
+        "text": "Assumption (A3) states that no direction of the noise free data has a non Dirac Gaussian variable component. It holds as soon as $\\mathbf { z } _ { t } = \\mathbf { f } ( \\mathbf { s } _ { t } )$ and the range of $\\mathbf { f }$ is such that its orthogonal projection on any line is not the full line. This assumption holds for instance in the following cases: 1) The range of f is compact, or 2) the range of f is contained in a half-cylinder, that is, there exists a hyperplane such that the range of f is only on one side of this hyperplane and the projection of the range of f on this hyperplane is bounded. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Assumption (B1) and Assumption (B2) are similar to those in (Hyvärinen and Morioka, 2017; Oberhauser and Schell, 2021) in the special case of time-series, i.e. $\\mathbb { T } = \\mathbb { N }$ . (B1) then entails that there must be sufficiently strong statistical dependence between nearby time points. (B2) is a condition which excludes Gaussian processes and processes which can be trivially transformed to be Gaussian. (For treatment of the Gaussian case, see Appendix B in Supplementary Material.) We can further provide a simple and equivalent formulation when the independent components $\\mathbf { s } ^ { ( i ) }$ follow independent and stationary HMMs with two hidden states, which is a special case of SNICA. Denote by $\\gamma _ { 0 } ^ { ( i ) }$ and $\\gamma _ { 1 } ^ { ( i ) }$ the densities of $s _ { t } ^ { ( i ) }$ conditionally to $\\{ u _ { t } ^ { ( i ) } = 0 \\}$ and $\\bar { \\{ u _ { t } ^ { ( i ) } = 1 \\} }$ respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 4 Assume that the stationary distribution $\\pi$ of the hidden chain is such that $0 < \\pi ( 0 ) < 1$ and that its transition matrix is invertible. Then $( B l )$ and $( B 2 )$ are satisfied with $m = 2$ if and only $i f$ on any open interval, γ(i)0 a nd $\\gamma _ { 1 } ^ { ( i ) }$ are not proportional. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Thus, a very simple HMM leads to these conditions being verified. Hyvärinen and Morioka (2017) already showed that the conditions (B1) and (B2) also hold in the case of non-Gaussian autoregressive models. Thus, we see that our identifiability theory applies both in the case HMM’s (Fig 1a) and autoregressive models (Fig 1b), the two principal kinds of temporal structure proposed in previous work, while extending them to further cases and combinations such as in Fig 1c,1d. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A simplification of (B1,B2) It is also possible to combine the assumptions (B1) and (B2) in one, while slightly weakening the generality. The key is to notice that (6) in (B2) implies the derivative in (B1) is zero, by setting $c = 0$ . But there is still the difference that (B2) considers all but one index while (B1) considers all indices $i$ . If we simply assume (6) does not hold for any $i$ , we can replace (B1) and (B2) by the new condition: ",
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+    {
+        "type": "text",
+        "text": "• $\\mathbf { ( B ^ { \\prime } ) }$ For any open subset $A \\subset S ^ { m }$ and for any $i \\in \\{ 1 , \\ldots , N \\}$ , a function $\\alpha : \\mathbb { R } ^ { m - 1 }  \\mathbb { R }$ and a constant $c \\in \\mathbb { R }$ do not exist such that (6) would hold for all $s \\in A$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that Hyvärinen and Morioka (2017) defined uniform dependency and (non-)quasi-Gaussianity as two separate properties, but in fact their assumption of non-quasi-Gaussianity was weaker than ours: it did not consider all open subsets separately, which is why this simplification was not possible for them. We believe their definition of non-quasi-Gaussianity was in fact not quite sufficient to prove their theorem, and our stronger version may be needed, in line with Oberhauser and Schell (2021). ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Estimation method One challenge is that it is not practically possible to learn $\\Delta$ -SNICA by exact maximum-likelihood methods. Instead, we perform learning and inference using Structured VAEs (Johnson et al., 2016) – the current state-of-art in variational inference for structured models. Specifically, this consists of assuming that the latent posterior factorizes as per $q ( \\mathbf { y } _ { 1 : T } ^ { ( 1 : N ) } , u _ { 1 : T } ^ { ( 1 : N ) } ) =$ $\\begin{array} { r } { \\prod _ { i = 1 } ^ { N } q ( \\mathbf { y } _ { 1 : T } ^ { ( i ) } ) q ( u _ { 1 : T } ^ { ( i ) } ) } \\end{array}$ , which allows us to optimize the resulting evidence lower bound (ELBO): ",
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+        "img_path": "images/92af25847025f25bd24e56fe05994dad77027e69998f5e5fe4b96e9b9cc97f22.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log \\widehat { \\mathcal { L } } = \\mathbb { E } _ { q } \\left[ \\displaystyle \\sum _ { t = 1 } ^ { T } \\log p ( \\mathbf { x } _ { t } \\mid \\mathbf { s } _ { t } ^ { ( 1 ) } , . . . , \\mathbf { s } _ { t } ^ { ( N ) } ) \\right] + \\displaystyle \\sum _ { i = 1 } ^ { N } \\left( - \\mathrm { K L } \\Big [ q ( u _ { 1 : T } ^ { ( i ) } ) \\Big | p ( u _ { 1 : T } ^ { ( i ) } ) \\Big ] + \\mathrm { H } \\Big [ q ( \\mathbf { s } _ { 1 : T } ^ { ( i ) } ) \\Big ] \\right. } \\\\ & { \\qquad \\left. + \\mathbb { E } _ { q } \\left[ \\log p ( \\mathbf { s } _ { 1 } ^ { ( i ) } \\mid u _ { 1 } ^ { ( i ) } ) \\right] + \\displaystyle \\sum _ { t = 2 } ^ { T } \\mathbb { E } _ { q } \\left[ \\log p ( \\mathbf { s } _ { t } ^ { ( i ) } \\mid \\mathbf { s } _ { t - 1 } ^ { ( i ) } , u _ { t } ^ { ( i ) } ) \\right] \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Since all the distributions are in conjugate exponential families (encoder neural network is used to approximate the natural parameters of the nonlinear likelihood term) efficient message passing can be used for inference, and the mixing function is learned as decoder neural network. Even though this method lacks consistency guarantees (but see Wang and Blei (2018)), we find that our model performs very well. A more detailed treatment of estimation and inference of $\\Delta$ -SNICA is given in Appendix C. Our code will be openly available at https://github.com/HHalva/snica. ",
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+    {
+        "type": "image",
+        "img_path": "images/5fd030a3253ec2319636d147c7d753a552ca81974f6635d2ea2a6a146c22d51d.jpg",
+        "image_caption": [
+            "Figure 2: (a) Mean absolute correlation coefficients between ground-truth independent components and their estimates by $\\Delta$ -SNICA , IIA-HMM, LGSSM and $\\mathrm { i V A E ^ { * } }$ , with different orders of complexity (number of layers) and two different dimensions of observed (12, 24) and latent (6, 12) data. (b) Mean absolute correlation coefficient between estimated noise free data and ground-truth noise free data for same set of models except IIA-HMM. Please note the difference in y-axis scales. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.1 Experiments on simulated data ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The identifiability theorems stated above hold in the limit of infinite data. Additionally, a consistent estimator would be required to learn the ground-truth components. In the real world, we are limited by data and estimation methods and hence it is unclear as to what extent we are actually able to estimate identifiable components – and whether identifiability reflects in better performance in real world tasks. To explore this, we first performed experiments on simulated data. We compared the performance of our model to the current state-of-the-art, IIA-HMM (Morioka et al., 2021), as well as identifiable VAE (iVAE) (Khemakhem et al., 2020a) and standard linear Gaussian state-space model (LGSSM). Since iVAE is not able to handle latent auxiliary variables, we allow it to \"cheat\" by giving it access to the true data generating latent-state, thereby creating a presumably challenging baseline (denoted $\\mathrm { i V A E ^ { * } }$ in our figures). LGSSM was included as a naive baseline which is only able to estimate linear mixing function. ",
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+        "type": "text",
+        "text": "Investigating identifiability and consistency We simulated 100K long time-sequences from the $\\Delta$ - SNICA model and computed the mean absolute correlation coefficient (MCC) between the estimated latent components and ground truth independent components (see Supplementary material for further implementation details). More precisely, to illustrate the dimensionality reduction capabilities we considered two settings where the observed data dimension $M$ , was either 12 or 24 and the number of independent components, $N$ was 3 and 6, respectively. Since IIA-HMM is unable to do dimensionality reduction, we used PCA to get the data dimension to match that of the latent states. We considered four levels of mixing of increasing complexity by randomly initialized MLPs of the following number of layers: 1 (linear ICA), 2, 3, and 5. The results in Figure 2a) illustrate the clearly superior performance of our model. The especially poor performance of IIA-HMM maybe explained by lack of noise model, much simpler latent dynamics, and lost information due to PCA pre-processing. See Appendix D for further discussion and training details. ",
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+    {
+        "type": "text",
+        "text": "Application to denoising $\\Delta$ -SNICA is able to denoise time-series signals by learning the generative model and then performing inference on latent variables. Specifically, SVAE learns the encoder network which is used to perform inference on the posterior of the independent components. We illustrate this using the same settings as above, with the exception that we now use our learned encoder and inference to get the posterior means of the independent components and input these in to the estimated decoder to get predicted noise-free observations, denoted as $\\widehat { \\mathbf { f } } \\left( \\mathbf { s } _ { t } \\right)$ – we measured the correlation between $\\widehat { \\mathbf { f } } \\left( \\mathbf { s } _ { t } \\right)$ and the ground-truth $\\mathbf { f } \\left( \\mathbf { s } _ { t } \\right)$ . Note that IIA-HMM, is not able to perform this task. The results in Figure 2b) show that the other models, designed to handle denoising, perform well at this task, as would be expected – identifiability of the latent state is not necessary for good denoising performance. For LGSSM, denoising is done with the Kalman Smoother algorithm. ",
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+    {
+        "type": "text",
+        "text": "5.2 Experiments on real MEG data ",
+        "text_level": 1,
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "To demonstrate real-data applicability, $\\Delta$ -SNICA was applied to multivariate time series of electrical activity in the human brain, measured by magnetoencephalography (MEG). Recently, many studies have demonstrated the existence of fast transient networks measured by MEG in the resting state and the dynamic switching between different brain networks (Baker et al., 2014; Vidaurre et al., 2017). Additionally, such MEG data is high-dimensional and very noisy. Thus this data provides an excellent target for $\\Delta$ -SNICA to disentangle the underlying low-dimensional components. ",
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+    {
+        "type": "text",
+        "text": "Data and Preprocessing We considered a resting state MEG sessions from the Cam-CAN dataset. During the resting state recording, subjects sat still with their eyes closed. In the task-session data, the subjects carried out a (passive) audio–visual task including visual stimuli and auditory stimuli. We exclusively used the resting-session data for the training of the network, and task-session data was only used in the evaluation. The modality of the sensory stimulation provided a class label that we used in the evaluation, giving in total two classes. We band-pass filtered the data between $4 \\mathrm { H z }$ and $3 0 \\mathrm { H z }$ (see Supplementary Material for the details of data and settings). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Methods The resting-state data from all subjects were temporally concatenated and used for training. The number of layers of the decoder and encoder were equal and took values 2, 3, 4. We fixed the number of independent components to 5 so that our result can be fairly compared to those in Morioka et al. (2021). To evaluate the obtained features, we performed classification of the sensory stimulation categories by applying feature extractors trained with (unlabeled) resting-state data to (labeled) task-session data. Classification was performed using a linear support vector machine (SVM) classifier trained on the stimulation modality labels and sliding-window-averaged features obtained for each trial. The performance was evaluated by the generalizability of a classifier across subjects. i.e., one-subject-out cross-validation. For comparison, we evaluated the baseline methods: IIA-HMM and IIA-TCL (Morioka et al., 2021). We also visualized the spatial activity patterns obtained by $\\Delta$ -SNICA , using the weight vectors from encoder neural network across each layer. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Results Figure 3 a) shows the classification accuracies of the stimulus categories, across different methods and the number of layers for each model. The performances by $\\Delta$ -SNICA were consistently higher than those by the other (baseline) methods, which indicates the importance of the modeling of the MEG signals by $\\Delta$ -SNICA . Figure $^ { 3 \\mathrm { ~ b ~ } }$ ) shows an example of spatial patterns from the encoder network learned by the $\\Delta$ -SNICA . We used the visualization method presented in (Hyvärinen and Morioka, 2016). We manually picked one out of the hidden nodes from the third layer in encoder network, and plotted its weighted-averaged sensor signals, We also visualized the most strongly contributing second- and first-layer nodes. We see progressive pooling of L1 units to form left lateral frontal, right lateral frontal and parietal patterns in L2 which are then all pooled together in L3 resulting in a lateral frontoparietal pattern. Most of the spatial patterns in the third layer (not shown) are actually similar to those previously reported using MEG (Brookes et al., 2011). Appendix E provides more detail to the interpretation of the $\\Delta$ -SNICA results. ",
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+    {
+        "type": "text",
+        "text": "6 Related work ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Previous works on nonlinear ICA have exploited autocorrelations (Hyvärinen and Morioka, 2017; Oberhauser and Schell, 2021) and nonstationarities (Hyvärinen and Morioka, 2016; Hälvä and Hyvärinen, 2020) for identifiability. The SNICA setting provides a unifying framework which allows for both types of temporal dependencies, and further, extends identifiability to other temporal structures as well as any arbitrary higher order data structures which has not previously been considered in the context of nonlinear ICA. Another major theoretical contribution here is to show that identifiability with noise of unknown, arbitrary distribution, while previous work on noisy nonlinear ICA assumed noise of known distribution and known variance (Khemakhem et al., 2020a). ",
+        "bbox": [
+            174,
+            710,
+            825,
+            821
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Importantly, the SNICA framework is fully probabilistic and thus accommodates higher order latent variables, leading to \"purely unsupervised\" learning. This is in large contrast to previous research which have been developed for the case where we are able to observe some additional auxiliary variable, such as audio signals accompanying video (Hyvärinen et al., 2019; Khemakhem et al., 2020a,b), or heuristically define the auxiliary variable based on time structure (Hyvärinen and Morioka, 2016). In practice this means that we are able to estimate our models using (variational) ",
+        "bbox": [
+            174,
+            827,
+            823,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/a5ff5727e16244565822b63bcda491f3e6db8ebd27f9bbe3c0993da73fbece75.jpg",
+        "image_caption": [
+            "Figure 3: $\\Delta$ -SNICA on MEG data. (a) Classification accuracies of linear SVMs trained with auditoryvisual data to predict stimulus category, with feature extractors trained by $\\Delta$ -SNICA in advance with resting-state data. Each point represents a testing accuracy on a target subject (chance level: $50 \\%$ ). Horizontal dotted line is PCA-only baseline. (b) Example of spatial patterns of the components learned by $\\Delta$ -SNICA $\\left( \\mathrm { L } { = } 3 \\right)$ . Each topography corresponds to one spatial pattern. L3: approximate total spatial pattern of one third-layer unit. L2: the patterns of the three second-layer units maximally contributing to this L3 unit. L1: for each L2 unit, the two most strongly contributing first-layer units. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
+            87,
+            728,
+            219
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "MLE, which is more principled than the heuristic self-supervised methods in most earlier papers. The only existing frameworks allowing MLE (Hälvä and Hyvärinen, 2020; Khemakhem et al., 2020a) used model restricted to exponential families, and had either no HMM or a very simple one. ",
+        "bbox": [
+            174,
+            351,
+            826,
+            392
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The switching linear dynamical model, $\\Delta$ -SNICA in Section 3.2, shows the above benefits in the form of a single model. That is, unlike previous nonlinear ICA models, it combines: 1) temporal dependencies and \"non-stationarity\" (or HMM) in a single model 2) dimensionality reduction within a rigorous maximum likelihood learning and inference framework, and 3) a separate observation equation with general observational noise. This results in a very rich, realistic, and principled model for time series. ",
+        "bbox": [
+            173,
+            398,
+            825,
+            482
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Very recently, Morioka et al. (2021) proposed a related model by considering innovations of time series to be nonstationary. However, their model is noise-free, restricted to exponential families of at least order two, and not applicable to the spatial case, thus making our identifiability results significantly stronger. From a more practical viewpoint, their model suffers from the fact that it either does not allow for dimensionality reduction (if an HMM is used) or requires a manual segmentation (if HMM is not used). Nor does it have a clear distinction into a state dynamics equation and a measurement equation which allows for cleaning or denoising of the data. ",
+        "bbox": [
+            174,
+            488,
+            825,
+            585
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Limitations Our identifiability theory makes some restrictive assumptions, and it remains to be seen if they could be lifted in future work. In particular, the data is not allowed to have too heavy tails; the noise must be additive, and independent of the signal; and the practical interpretation of some of the assumptions, such as (A3) is difficult. It is also difficult to say whether our assumption of unconditionally independent components is realistic in practice. Regarding practical applications, our specific model only scratches the surface of what is possible in this framework. In particular, we did not develop a model with spatial distributions, nor did we model non-Gaussian observational noise – our main aim was to lay the foundations for the relevant identification theory. Future work should aim to make the estimation more efficient computationally; this is a ubiquitous problem in deep learning, but specific solutions for this concrete problem may be achievable (Gresele et al., 2020). ",
+        "bbox": [
+            174,
+            599,
+            826,
+            739
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            758,
+            299,
+            775
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We proposed a new general framework for identifiable disentanglement, based on nonlinear ICA with very general temporal dynamics or spatial structure. Observational noise of arbitrary unknown distribution is further included. We prove identifiability of the models in this framework with high generality and mathematical rigour. For real data analysis, we propose a special case which subsumes the properties of all existing time series models in nonlinear ICA, while generalizing them in many ways (see Section 6 for details). We hope this work will contribute to wide-spread application of identifiable methods for disentanglement in a highly principled, probabilistic framework. ",
+        "bbox": [
+            174,
+            789,
+            825,
+            887
+        ],
+        "page_idx": 9
+    }
+]

parse/train/BylaUTNtPS/BylaUTNtPS.md

@@ -0,0 +1,582 @@
+# RECURRENT INDEPENDENT MECHANISMS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Learning modular structures which reflect the dynamics of the environment can lead to better generalization and robustness to changes which only affect a few of the underlying causes. We propose Recurrent Independent Mechanisms (RIMs), a new recurrent architecture in which multiple groups of recurrent cells operate with nearly independent transition dynamics, communicate only sparingly through the bottleneck of attention, and are only updated at time steps where they are most relevant. We show that this leads to specialization amongst the RIMs, which in turn allows for dramatically improved generalization on tasks where some factors of variation differ systematically between training and evaluation.
+
+# 1 INTRODUCTION
+
+Physical processes in the world often have a modular structure, with complexity emerging through combinations of simpler subsystems. Machine learning seeks to uncover and use regularities in the physical world. Although these regularities manifest themselves as statistical dependencies, they are ultimately due to dynamic processes governed by physics. These processes are often independent and only interact sparsely. For instance, we can model the motion of two balls as separate independent mechanisms even though they are both gravitationally coupled to Earth as well as (weakly) to each other. They may, however, occasionally strongly interact via collisions.
+
+The notion of independent or autonomous mechanisms has been influential in the field of causal inference, where it is applied not only to dynamic processes but also to time independent datasets. For instance, it has been argued that the conditional distribution of the average annual temperature given the altitude of a place is an abstraction of a causal mechanism (subsuming complex physical processes involving air pressure, etc.) that is independent of the distribution of the altitudes of settlements (Peters et al., 2017), and will thus apply invariantly for, say, different countries in the same climate zone with different altitude distributions.
+
+A complex generative model, temporal or not, can be thought of as the composition of independent mechanisms or “causal” modules. In the causality community, this is often considered a prerequisite of being able to perform localized interventions upon variables determined by such models (Pearl, 2009). It has been argued that the individual modules tend to remain robust or invariant even as other modules change, e.g., in the case of distribution shift (Schölkopf et al., 2012; Peters et al., 2017). One may hypothesize that if a brain is able to solve multiple problems beyond a single i.i.d. (independent and identically distributed) task, it would be economical to learn structures aligned with this, by learning independent mechanisms that can flexibly be reused, composed and re-purposed.
+
+In the dynamic setting, we think of an overall system being assayed as composed of a number of fairly independent subsystems that evolve over time, responding to forces and interventions. A learning agent then need not devote equal attention to all subsystems at all times: only those aspects that significantly interact need to be considered jointly when taking a decision or forming a plan (Bengio, 2017). Such sparse interactions can reduce the difficulty of learning since few interactions need to be considered at a time, reducing unnecessary interference when a subsystem is adapted. Models learned this way may be more likely to capture the compositional generative (or causal) structure of the world, and thus better generalize across tasks where a (small) subset of mechanisms change while most of them remain invariant (Simon, 1991; Peters et al., 2017; Parascandolo et al., 2018). The central question motivating our work is how a machine learning approach can learn independent but sparsely interacting recurrent mechanisms in order to benefit from such modularity.
+
+![](images/954aa4362fa00b84dc9401779be8747757d91477403fdb1eecd279c8b24803c1.jpg)  
+Figure 1: Illustration of Recurrent Independent Mechanisms (RIMs). A single step under the proposed model occurs in four stages (left figure shows two steps). In the first stage, individual RIMs produce a query which is used to read from the current input. In the second stage, an attention based competition mechanism is used to select which RIMs to activate (right figure) based on encoded visual input (blue RIMs are active, based on attention score, white RIMs remain inactive). In the third stage, individual activated RIMs follow their own default transition dynamics while non-activated RIMs remain unchanged. In the fourth stage, the RIMs sparsely communicate information between themselves, also using attention.
+
+# 2 RECURRENT INDEPENDENT MECHANISMS WITH SPARSE INTERACTIONS
+
+Our approach to modelling a dynamical system of interest divides the overall model into $k$ small subsystems (or modules), each of which is recurrent in order to be able to capture dynamics. We refer to these subsystems as Recurrent Independent Mechanisms $( R I M s )$ , where each RIM has distinct functions that are learned automatically from data1. We refer to RIM $k$ at time step $t$ as having state $h _ { t , k }$ , where $t = 1 , \dots , T$ . Each RIM has parameters $\theta _ { k }$ , which are shared across all time steps.
+
+At a high level (see. Fig. 1), we want each RIM to have its own independent dynamics operating by default, and occasionally to interact with other relevant RIMs and with selected elements of the encoded input. The total number of parameters can be kept small since RIMs can specialize on simple sub-problems, similar to Parascandolo et al. (2018). This specialization and modularization not only has computational and statistical advantages (Baum & Haussler, 1989; Bengio et al., 2019), but also prevents individual RIMs from dominating and modelling complex, composite mechanisms. We expect this to lead to more robust systems than training one big homogeneous neural network (Schmidhuber, 2018). Moreover, modularity also has the desirable implication that a RIM should maintain its own independent functionality even as other RIMs are changed. A more detailed account of the desiderata for the model is given in Appendix A.
+
+# 2.1 INDEPENDENT RIM DYNAMICS
+
+Now, consider the default transition dynamics which we apply for each RIM independently and during which no information passes between RIMs. We use $\tilde { h }$ for the hidden state after the independent dynamics are applied (and before attention is applied). First, for the RIMs which are not activated (we refer to the activated set as $S _ { t }$ ), the hidden state remains unchanged:
+
+$$
+\tilde { h } _ { t + 1 , k } = h _ { t , k } \qquad \forall k \notin S _ { t } .
+$$
+
+Note that the gradient still flows through a RIM on a step where it is not activated. For the RIMs that are activated, we run a per-RIM independent transition dynamics. The form of this is somewhat flexible, but in this work we opted to use either a GRU (Chung et al., 2015) or an LSTM (Hochreiter & Schmidhuber, 1997). We generically refer to these independent transition dynamics as $D _ { k }$ , and we emphasize that each RIM has its own separate parameters. Aside from being RIM-specific, the internal operation of the LSTM and GRU remain unchanged, and the active RIMs are updated by
+
+$$
+\tilde { h } _ { t + 1 , k } = D _ { k } ( h _ { t , k } ) = L S T M ( h _ { t , k } , A _ { k } ^ { ( i n ) } ; \theta _ { k } ^ { ( D ) } ) \qquad \forall k \in S _ { t }
+$$
+
+as a function of the attention mechanism A(in)k applied on the current input, described in the next two subsections below, after explaining the key-value mechanism used to select arguments for this update.
+
+2.2 KEY-VALUE ATTENTION TO PROCESS SETS OF NAMED INTERCHANGEABLE VARIABLES
+
+Each RIM should be activated and updated when the input is relevant to it. We thus utilize competition to allocate representational and computational resources. As argued by Parascandolo et al. (2018), this tends to produce independence among learned mechanisms, provided the training data has been generated by a set of independent physical mechanisms. In contrast to Parascandolo et al. (2018), we use an attention mechanism for this purpose. In doing so, we are inspired by findings from experimental psychology in the study of the interplay of top-down attention and bottom-up information flow, conceptualized in the biased competition theory of selective attention (Desimone & Duncan, 1995): A brain’s capacity for parallel processing of complex entities is limited, and many brain systems representing visual information use competition (operating in parallel across the visual field) to allocate resources, often biased by feedback from higher brain areas.
+
+The introduction of content-based soft-attention mechanisms (Bahdanau et al., 2014) has opened the door to neural networks which operate on sets of typed interchangeable objects. This idea has been remarkably successful and widely applied to most recent Transformer-style multi-head dot product self attention models (Vaswani et al., 2017; Santoro et al., 2018), achieving new state-of-the-art results in many tasks. Soft-attention uses the product of a query (or read key) $Q$ of dimensionality $N _ { r } \times d$ matrix $Q$ , and $d$ dimension of each key) to a set of $N _ { o }$ objects each associated with a key (or write-key) matrix $K ^ { T }$ $( N _ { o } \times d )$ , and after normalization with a softmax yields outputs in the convex hull of the values (or write-values) $V _ { i }$ (row $i$ of matrix $V$ ). Its result is computed as
+
+$$
+{ \mathrm { A t t e n t i o n } } ( Q , K , V ) = { \mathrm { s o f t m a x } } \left( { \frac { Q K ^ { T } } { \sqrt { d } } } \right) V ,
+$$
+
+where the softmax is applied to each row of its argument matrix, yielding a set of convex weights. As a result, one obtains a convex combination of the values $V$ . If the attention is focused on one element for a particular row (i.e., the softmax is saturated), this simply selects one of the objects and copies its value to row $j$ of the result. Note that the $d$ dimensions in the key can be split into heads which then have their attention matrix and write values computed separately.
+
+When the inputs and outputs of each RIM are a set of objects or entities (each associated with a key and value vector), the RIM processing becomes a generic object-processing machine which can operate on “variables” in a sense analogous to variables in a programming language: as interchangeable arguments of functions. Because each object has a key embedding (which one can understand both as a name and as a type), the same RIM processing can be applied to any variable which fits an expected "distributed type" (specified by a query vector). Each attention head then corresponds to a typed argument of the function computed by the RIM. When the key of an object matches the query, it can be used as input for the RIM. Whereas in regular neural networks (without attention) neurons operate on fixed variables (the neurons which are feeding them from the previous layer), the key-value attention mechanisms make it possible to select on the fly which variable instance (i.e. which entity or object) is going to be used as input for each of the arguments of the RIM dynamics, with a different set of query embeddings for each RIM. These inputs can come from the external input or from the output of other RIMs. So, if the individual RIMs can represent these “functions with typed arguments,” then they can “bind” to whatever input is currently available and best suited according to its attention score: the “input attention” mechanism would look at the candidate input object’s key and evaluate if its “type” matches with what this RIM expects (specified in the query).
+
+# 2.3 SELECTIVE ACTIVATION OF RIMS AS A FORM OF TOP-DOWN MODULATION
+
+The proposed model learns to dynamically select those RIMs for which the current input is relevant. We give each RIM the choice between attending to the actual input instances or a special null input. The null input consists entirely of zeros and thus contains no information. At each step, we select the top- $k _ { A }$ (out of $k _ { T }$ ) RIMs in terms of their value of the softmax for the real input. Intuitively, the RIMs must compete on each step to read from the input, and only the RIMs that win this competition will be able to read from the input and have their state updated.
+
+In our use of key-value attention, the queries come from the RIMs, while the keys and values come from the current input. The mechanics of this attention mechanism follow from the Transformer (Vaswani et al., 2017) and the RMC (Santoro et al., 2018), with the modification that the parameters of the attention mechanism itself are separate for each RIM. The input attention for a particular RIM is described as follows. The input $x _ { t }$ at time $t$ is seen as a set of elements, structured as rows of a matrix (for image data, it can be the output of the CNN). We first concatenate a row full of zeros, to obtain
+
+$$
+X = \varnothing \oplus x _ { t } .
+$$
+
+$\oplus$ refers to the row-level concatenation operator. Then, linear transformations are used to construct keys $K = X W ^ { k }$ , one per input element and for the null element), values $\mathbf { \nabla } V = X W ^ { v }$ , again one per element), and queries $\mathbf { \bar { \rho } } Q = R W _ { k } ^ { q }$ , one per RIM attention head) where $R$ is a matrix with each row $r _ { i }$ corresponding to the hidden state of an individual RIM (i.e $h _ { t , k }$ ). $W ^ { v }$ is a simple matrix mapping from an input element to the corresponding value vector for the weighted attention and $W ^ { k }$ is similarly a weight matrix which maps the input to the keys. $\boldsymbol { W } _ { k } ^ { q }$ is a per-RIM weight matrix which maps from the RIM’s hidden state to its queries. The attention thus is
+
+$$
+A _ { k } ^ { ( i n ) } = \mathrm { s o f t m a x } \left( \frac { R W _ { k } ^ { q } ( X W ^ { k } ) ^ { T } } { \sqrt { d _ { e } } } \right) X W ^ { v } , \mathrm { w h e r e } \theta _ { k } ^ { ( i n ) } = ( W _ { k } ^ { q } , W ^ { e } , W ^ { v } ) .
+$$
+
+Based on the softmax values in (4), we select the top $k _ { A }$ RIMs (out of the total $K$ RIMs) to be activated for each step, which have the least attention on the null input (and thus put the highest attention on the input), and we call this set $S _ { t }$ . Since the queries depend on the state of the RIMs, this enables individual RIMs to attend only to the part of the input that is relevant for that particular RIM, thus enabling selective attention based on a top-down attention process (see. Fig 1). In practice, we use multiheaded attention, and multi-headed attention doesn’t change the essential computation, but when we do use it for input-attention we compute RIM activation by averaging the attention scores over the heads.
+
+# 2.4 COMMUNICATION BETWEEN RIMS
+
+Although the RIMs operate independently by default, the attention mechanism allows sharing of information among the RIMs. Specifically, we allow the activated RIMs to read from all other RIMs (activated or not). The intuition behind this is that non-activated RIMs are not related to the current input, so their value should not change. However they may still store contextual information that is relevant for activated RIMs. For this communication between RIMs, we use a residual connection as in (Santoro et al., 2018) to prevent vanishing or exploding gradients over long sequences.
+
+$$
+\begin{array} { r l } & { Q _ { t , k } = \tilde { W } _ { k } ^ { q } \tilde { h } _ { t , k } , \quad \forall k \in \mathcal { S } _ { t } } \\ & { K _ { t , k } = \tilde { W } _ { k } ^ { e } \tilde { h } _ { t , k } , \quad \forall k } \\ & { V _ { t , k } = \tilde { W } _ { k } ^ { v } \tilde { h } _ { t , k } , \quad \forall k } \\ & { h _ { t + 1 , k } = \mathrm { s o f t m a x } \left( \frac { Q _ { t , k } \left( K _ { t , : } \right) ^ { T } } { \sqrt { d _ { e } } } \right) V _ { t , : } + \tilde { h } _ { t , k } \quad \forall k \in \mathcal { S } _ { t } , \mathrm { ~ w h e r e ~ } \theta _ { k } ^ { ( c ) } = ( \tilde { W } _ { k } ^ { q } , \tilde { W } _ { k } ^ { e } , \tilde { W } _ { k } ^ { v } ) . } \end{array}
+$$
+
+As in the Transformer and RMC Vaswani et al. (2017); Santoro et al. (2018), we use multiple heads (as well as input attention (as in Sec 2.3) by producing different sets of queries, keys, and values to compute a linear transformation for each head (different heads have different parameters).
+
+# 2.5 VARIATIONS ON THE RIMS ARCHITECTURE
+
+The RIMs architecture that we study is highly homogeneous and generally the only hyperparameters are the number of RIMs $K$ and how many RIMs are activated on each time step $K _ { A }$ . All of the datasets that we consider are temporal, yet there is a distinction between datasets where the input on each time step is highly structured (such as a video, where each time step is an image) and where this is not the case (such as language modeling, where each step is a word or character). In the former case, we can get further improvements by making the activation of RIMs not just sparse across time but also sparse across the (spatial) structure.
+
+# 3 RELATED WORK
+
+Neural Turing Machine (NTM) and Relational Memory Core (RMC): the NTM (Graves et al., 2014a) consists of a sequence of independent memory cells, and uses an attention mechanism while performing targeted read and write operations. This shares a key idea with RIMs: that input information should only impact a sparse subset of the memory by default, while keeping most of the memory unaltered. RMC (Santoro et al., 2018) uses a multi-head attention mechanism to share information between multiple memory elements. We encourage the RIMs to remain separate as much as possible, whereas Santoro et al. (2018) allow information between elements to flow on each step in an unsconstrained way. Instead, each RIM has its own default dynamics, while in RMC, all the processes interact with each other.
+
+Separate Recurrent Models: EnTNet (Henaff et al., 2016) and IndRNN (Li et al., 2018) can be viewed as a set of separate recurrent models. In IndRNN, each recurrent unit has completely independent dynamics, whereas EntNet uses an independent gate for writing to each memory slot. RIMs use different recurrent models (with separate parameters), but we allow the RIMs to communicate with each other sparingly using an attention mechanism.
+
+Modularity and Neural Networks: A neural network is composed of several neural modules, where each module is meant to perform a distinct function, and hence can be seen as a combination of experts (Jacobs et al., 1991; Bottou & Gallinari, 1991; Ronco et al., 1997; Reed & De Freitas, 2015; Andreas et al., 2016; Parascandolo et al., 2018; Rosenbaum et al., 2017; Fernando et al., 2017; Shazeer et al., 2017; Kirsch et al., 2018; Rosenbaum et al., 2019) routing information through a gated activation of layers. These works generally assume that only a single expert is active at a particular time step. In the proposed method, multiple RIMs can be active, interact and share information.
+
+Computation on demand: There are various architectures (El Hihi & Bengio, 1996; Koutnik et al., 2014; Chung et al., 2016; Neil et al., 2016; Jernite et al., 2016; Krueger et al., 2016) where parts of the LSTM’s hidden state are kept dormant at times. The major differences as compared to the proposed architecture are that (a) we modularize the dynamics of recurrent cells (using RIMs), and (b) we also control the inputs of each module (using transformer style attention), while many previous gating methods did not control the inputs of each module, but only whether they should be executed or not.
+
+# 4 EXPERIMENTS
+
+The main goal of our experiments is to show that the use of RIMs improves generalization across changing environments and/or in modular tasks, and to explore how it does so. Our goal is not to outperform highly optimized baselines; rather, we want to show the versatility of our approach by applying it to a range of diverse tasks, focusing on tasks that involve a changing environment. We organize our results by the capabilities they illustrate: we address generalization based on temporal patterns, based on objects, and finally consider settings where both of these occur together.
+
+# 4.1 RIMS IMPROVE GENERALIZATION BY SPECIALIZING OVER TEMPORAL PATTERNS
+
+We first show that when RIMs are presented with sequences containing distinct temporal patterns, they are able to specialize so that different RIMs are activated on different patterns. As a result, RIMs are able to generalize well when we modify a subset of the patterns (especially those unrelated to the class label) while most recurrent models fail to generalize well to these variations.
+
+# 4.1.1 COPYING TASK
+
+First we turn our attention to the task of receiving a short sequence of characters, then receiving blank inputs for a large number of steps, and then being asked to reproduce the original sequence. We can think of this as consisting of two temporal patterns which are independent: one where the sequence is received and another “dormant” pattern where no input is provided.
+
+![](images/518f0a836a3b408cfdde94d535a861d0a1b0836b71b60aed880d0d403dfbe64f.jpg)  
+Figure 2: Copying Task RIM Activation Pattern for a model with $K = 6$ RIMs and $K _ { A } = 3$ active RIMs per step (the activated RIMs are in black, non-activated in white). We can see that the RIM activation pattern is distinct during the dormant part of the sequence.
+
+Table 1: Performance on the copying task (left) and sequential MNIST resolution task right). Error (CE on the last 10 time steps) on the copying task. Note that while all of the methods are able to learn to copy for the length seen during training, the RIMs model generalizes to sequences longer than those seen during training whereas the LSTM, RMC, and NTM degrade. Sequential MNIST resolution: Test Accuracy $\%$ on the Sequential MNIST resolution generalization task (see text) after 100 epochs. Both the proposed and the Baseline model (LSTM) were trained on $1 4 \mathrm { x } 1 4$ resolution but evaluated at different resolutions; results averaged over 3 different trials.   
+
+<table><tr><td colspan="2">Copying k</td><td>kA</td><td>hsize</td><td>Train(50) CE</td><td>Test(200) CE</td></tr><tr><td rowspan="5">RIMs</td><td>6</td><td></td><td>600</td><td>0.01</td><td>3.5</td></tr><tr><td>6</td><td>5 4</td><td>600</td><td>0.00</td><td>0.00</td></tr><tr><td>6</td><td>3</td><td>600</td><td>0.00</td><td>0.00</td></tr><tr><td>6</td><td>2</td><td>600</td><td>0.00</td><td>0.00</td></tr><tr><td>5</td><td>3</td><td>500</td><td>0.00</td><td>0.00</td></tr><tr><td rowspan="2">LSTM</td><td></td><td>=</td><td>300</td><td>0.00</td><td>2.28</td></tr><tr><td></td><td>=</td><td>600</td><td>0.00</td><td>3.56</td></tr><tr><td>NTM</td><td></td><td>=</td><td>-</td><td>0.00</td><td>2.54</td></tr><tr><td>RMC</td><td></td><td></td><td>-</td><td>0.00</td><td>0.13</td></tr><tr><td>Transformers -</td><td></td><td></td><td>-</td><td>0.00</td><td>0.54</td></tr></table>
+
+<table><tr><td colspan="4">Sequential MNIST</td><td rowspan="2">16 x 16 19 x 19 Accuracy</td><td rowspan="2"></td><td rowspan="2">24 x 24 Accuracy</td></tr><tr><td>kT</td><td>kA</td><td>hsize</td><td>Accuracy</td></tr><tr><td rowspan="3">RIMs</td><td>6</td><td>6</td><td>600</td><td>85.5</td><td>56.2</td><td>30.9</td></tr><tr><td>6</td><td>5</td><td>600</td><td>88.3</td><td>43.1</td><td>22.1</td></tr><tr><td>6</td><td>4</td><td>600</td><td>90.0</td><td>73.4</td><td>38.1</td></tr><tr><td rowspan="2">LSTM</td><td></td><td>-</td><td>300</td><td>86.8</td><td>42.3</td><td>25.2</td></tr><tr><td></td><td>=</td><td>600</td><td>84.5</td><td>52.2</td><td>21.9</td></tr><tr><td>EntNet -</td><td></td><td>=</td><td>: 一</td><td>89.2</td><td>52.4</td><td>23.5</td></tr><tr><td>RMC</td><td></td><td></td><td>-|</td><td>89.58</td><td>54.23</td><td>27.75</td></tr><tr><td rowspan="2">DNC Transformers-</td><td></td><td></td><td></td><td>87.2</td><td>44.1</td><td>19.8</td></tr><tr><td></td><td></td><td></td><td>91.2</td><td>51.6</td><td>22.9</td></tr></table>
+
+As an example of out-of-distribution generalization, we find that using RIMs, we can extend the length of this dormant phase from 50 during training to 200 during testing and retain perfect performance (Table 1), whereas baseline methods including LSTM, NTM, and RMC substantially degrade. In addition, we find that this result is robust to the number of RIMs used as well as to the number of RIMs activated per-step. Our results (Appendix C.5) show that communication between different RIMs as well as input attention is necessary to achieve good generalization. We consider this preliminary evidence that RIMs can specialize over distinct patterns in the data and improve generalization to settings where these patterns change.
+
+# 4.1.2 SEQUENTIAL MNIST RESOLUTION TASK
+
+RIMs are motivated by the hypothesis that generalization performance can be improved by having modules which only activate on relevant parts of the sequence. For further evidence that RIMs can achieve this out-of-distribuution, we consider the task of classifying MNIST digits as sequences of pixels (Krueger et al., 2016) and assay generalization to images of resolutions different from those seen during training. Our intuition is that the RIMs model should have distinct subsets of the RIMs activated for pixels with the digit and empty pixels. As a result, RIMs should generalize better to greater resolutions by keeping the RIMs which store pixel information dormant over the empty regions of the image.
+
+Results: Table 1 shows the result of the proposed model on the Sequential MNIST Resolution Task. If the train and test sequence lengths agree, both models achieve comparable test set performance. However, the RIMs model was relatively robust to changing the sequence length (by changing the image resolution), whereas the LSTM performance degraded more severely. This can be seen as a more involved analogue of the copying task, as MNIST digits contain large empty regions. It is essential that the model be able to store information and pass gradients through these regions. The RIMs outperform strong baselines such as Transformers, EntNet, RMC, as well as the Differentiable Neural Computer (DNC) (Graves et al., 2016).
+
+# 4.2 RIMS LEARN TO SPECIALIZE OVER OBJECTS AND GENERALIZE BETWEEN THEM
+
+We have presented evidence that RIMs can specialize over temporal patterns. We now turn our attention to showing that RIMs can specialize to objects, and show improved generalization to settings where we add or remove objects at test time.
+
+# 4.2.1 BOUNCING BALL ENVIRONMENT
+
+We consider a synthetic “bouncing balls” task in which multiple balls (of different masses and sizes) move using basic Newtonian physics (Van Steenkiste et al., 2018). What makes this task particularly suited to RIMs is that the balls move independently most of the time, except when they collide. During training, we predict the next frame at each time step using teacher forcing (Williams & Zipser, 1989). We can then use this model to generate multi-step rollouts.
+
+As a preliminary experiment, we train on sequences of length 51 (the previous standard), using a binary cross entropy loss when predicting the next frame. We consider LSTM as baseline. We then produce rollouts, finding that RIMs are better able to predict future motion (examples in Figure 3, Figure 10 in Appendix and quantitative comparisons in Figure 4).
+
+![](images/df76c63b8385c9c428d2b3b306ec95aaaae2d4a82055de303c7856790ad7b75c.jpg)  
+Figure 3: Predicting Movement of Bouncing Balls. The first 15 frames of ground truth are given (last 6 of those shown) and then the system is rolled out for the next 15 time steps. We find that RIMs perform better than the LSTMs (predictions are in black, ground truth in blue). Notice the blurring of LSTM predictions.
+
+![](images/9b012418bbb493449dadc87b5479cde1a7a43c8ece186051cf9f84f5d77ab619.jpg)  
+Figure 4: Handling Novel Out-of-Distribution Variations. Here, we study the performance of our proposed model compared to an LSTM baseline. The first 15 frames of ground truth are fed in and then the system is rolled out for the next 10 time steps. During the rollout phase, RIMs perform better than the LSTMs in accurately predicting the dynamics of the balls as reflected by the lower Cross Entropy (CE) [see blue for RIMs, purple for LSTM]. Notice the substantially better out-of-distribution generalization of RIMs when testing on a different number of objects than during training.
+
+We take this further by evaluating RIMs on environments where the setup is different from the training setup. First we consider training with 4 balls and evaluating on an environment with 6-8 balls. Second, we consider training with 6-8 balls and evaluating with just 4 balls. Robustness in these settings requires a degree of invariance w.r.t. the number of balls.
+
+In addition, we consider a task where we train on 4 balls and then evaluate on sequences where part of the visual space is occluded by a “curtain”. This allows us to assess the ability of balls to be tracked (or remembered) through the occluding region. Our experimental results on these generalization tasks (Figure 4) show that RIMs substantially improve over an LSTM baseline. We found that increasing the capacity of the LSTM from 256 to 512 units did not substantially change the performance gap, suggesting that the improvement from RIMs is not primarily a result of increased capacity.
+
+# 4.2.2 ENVIRONMENT WITH NOVEL DISTRACTORS
+
+We next consider an object-picking reinforcement learning task from BabyAI (Chevalier-Boisvert et al., 2018) in which an agent must retrieve a specific object in the presence of distractors. We use a partially observed formulation of the task, where the agent only sees a small number of squares ahead of it. These tasks are difficult to solve (Chevalier-Boisvert et al., 2018) with standard RL algorithms, due to (1) the partial observability of the environment and (2) the sparsity of the reward, given that the agent receives a reward only after reaching the goal. During evaluation, we introduce new distractors to the environment which were not observed during training.
+
+![](images/81730a9871233a0360965fadf9a9c547130335e291305c39c693bfd35dead9a2.jpg)  
+Figure 5 shows that RIMs outperform LSTMs on this task (details in appendix). When evaluating with known distractors, the RIM model achieves perfect performance while the LSTM struggles. When evaluating in an environment with novel unseen distractors the RIM doesn’t achieve perfect performance but still outperforms the LSTM. An LSTM with a single memory flow may struggle to keep the distracting elements separate from elements which are necessary for the task, while the RIMs model uses attention to control which RIMs receive infor  
+Figure 5: Robustness to Novel Distractors:. Left: performance of the proposed method compared to an LSTM baseline in solving the object picking task in the presence of distractors. Right: performance of proposed method and the baseline when novel distractors are added.
+
+mation at each step as well as what information they receive (as a function of their hidden state). This "top-down" bias results in a diminished representation of the distractor, not only enhancing the target visual information, but also suppressing irrelevant information. The notion that enhancement of the relevant information necessarily results in suppression of irrelevant information is fundamental to biased competition theory (Desimone & Duncan, 1995).
+
+# 4.3 RIMS IMPROVE GENERALIZATION IN COMPLEX ENVIRONMENTS
+
+We have investigated how RIMs use specialization to improve generalization to changing important factors of variation in the data. While these improvements have often been striking, it raises a question: what factors of variation should be changed between training and evaluation? One setting where factors of variation change naturally is in reinforcement learning, as the data received from an environment changes as the agent learns and improves. We conjecture that when applied to reinforcement learning, an agent using RIMs may be able to learn faster as its specialization leads to improved generalization to previously unseen aspects of the environment.
+
+To investigate this we use an RL agent trained using Proximal Policy Optimization (PPO) (Schulman et al., 2017) with a recurrent network producing the policy. We employ an LSTM as a baseline, and compare results to the RIMs architecture. This was a simple drop-in replacement and did not require changing any of the hyperparameters for PPO. We experiment on the whole suite of Atari games and find that simply replacing the LSTM with RIMs greatly improves performance (Figure 6).
+
+There is also an intriguing connection between the selective activation in RIMs and the concept of affordances from cognitive psychology (Gibson, 1977; Cisek & Kalaska, 2010). To perform well in environments with a dynamic combination of risks and opportunities, an agent should be ready to adapt immediately, releasing into execution actions which are at least partially prepared. This suggests agents should process sensory information in a contextual manner, building representations of potential actions that the environment currently affords. For instance, in Demon Attack, one of the games where RIMs exhibit strong performance gains, the agent must quickly choose between targeting distant aliens to maximize points and avoiding fire from close-by aliens to avoid destruction (indeed both types of aliens are always present, but which is relevant depends on the player’s position). We hypothesize that in cases like this, selective activation of RIMs allows the agent to rapidly adapt its information processing to the types of actions relevant to the current context.
+
+# 4.4 ABLATIONS
+
+Role of Top-Down Modulation: Removing Input Attention We study the scenario where we remove the input attention process (Section 2.3) but still allow communication between RIMs (Section 2.4). We train this agent on 30 ATARI games for 30M time-steps each and compare the performance of this agent with the normal RIMs-PPO agent. We find that the RIMs agent still outperform this agent on 11 out of 30 games, while on 1 game (Frostbite) we see the proposed baseline agent substantially improves the performance. For more details regarding the training curves, refer to Fig. 25 (in Appendix).
+
+Importance of communication between RIMs: For copying, we performed an ablation where we remove the communication between RIMs. We also varied the number of RIMs as well as the number of activated RIMs (Table 5). We found that the communication between RIMs is essential for good performance. We found similar results for the sequential MNIST resolution task.
+
+![](images/754484a8563a8e92fd2d0f0ed253edcf977342cd79304b5cf8ba1bbf74a2e7c2.jpg)  
+Figure 6: RIMs-PPO relative score improvement over LSTM-PPO baseline (Schulman et al., 2017) across all Atari games averaged over 3 trials per game. In both cases, PPO was used with the exact same settings, and the only change is the choice of recurrent architecture. More detailed experiments with learning curves are in Appendix C.
+
+Importance of sparsity of activation of the RIMs For the copying task, as well as for the sequential MNIST changed resolution task, we performed an ablation where we kept all RIMs active for all time steps (Table 5). We found that we were not able to achieve strong generalization as compared to the best performing RIMs model. On Atari we found that using $k _ { A } = 5$ slightly improved over results compared with $k _ { A } = 4$ , but both had similar performance across the vast majority of games, suggesting that the $k _ { A }$ hyperparameter is reasonably flexible in practice.
+
+Varying the number of attention heads for communication: Here, we study what happens if the output of RIMs only has one ’object’ rather than multiple ones (Section 2.2). The intuition is that RIM processing can be applied to any “head” which matches the query by an individual RIM. So, having more heads should help, as different heads could be used by different RIMs, rather than every RIM competing for the same head. We study this in the context of bouncing balls. We found that using multiple heads improves the performance, thus validating our hypothesis (Sec. 2.2). See Appendix C.11 for details.
+
+Randomly Dropping Out RIMs: Modular structures are aggregates of mechanisms that can perform functions without affecting the remainder of the system, and interact as needed. To what extent are trained RIMs able to model meaningful phenomena when other RIMs are removed? We performed an experiment on moving MNIST digits where we train normally and “dropout” a random RIM at test time. We found that in the absence of selective activation (i.e. when $k _ { A } = k _ { T }$ Section C.13) the performance degraded very badly, but the performance degrades much less with selective activation. See Appendix C.13 for details.
+
+# 5 CONCLUSION
+
+Many systems of interest comprise multiple dynamical processes that operate relatively independently and only occasionally have meaningful interactions. Despite this, most machine learning models employ the opposite inductive bias, i.e., that all processes interact. This can lead to poor generalization (if data is limited) and lack of robustness to changing task distributions. We have proposed a new architecture, Recurrent Independent Mechanisms (RIMs), in which we learn multiple recurrent modules that are independent by default, but interact sparingly. Our positive experimental results lend support to the consciousness prior (Bengio, 2017), i.e., the importance of computational elements which focus on few mechanisms at a time in order to determine how a high-level state evolves over time, with many aspects of the state not being affected by this attentive dynamics (i.e., following default dynamics). For the purposes of this paper, we note that the notion of RIMs is not limited to the particular architecture employed here. The latter is used as a vehicle to assay and validate our overall hypothesis (cf. Appendix A), but better architectures for the RIMs model can likely be found.
+
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+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998–6008, 2017.
+
+H. L. F. von Helmholtz. Handbuch der physiologischen Optik, volume III. Voss, 1867.
+
+Erich von Holst and Horst Mittelstaedt. Das reafferenzprinzip. Naturwissenschaften, 37(20):464–476, Jan 1950. doi: 10.1007/BF00622503.
+
+Nicholas Watters, Daniel Zoran, Theophane Weber, Peter Battaglia, Razvan Pascanu, and Andrea Tacchetti. Visual interaction networks: Learning a physics simulator from video. In Advances in neural information processing systems, pp. 4539–4547, 2017.
+
+Ronald J Williams and David Zipser. A learning algorithm for continually running fully recurrent neural networks. Neural computation, 1(2):270–280, 1989.
+
+# A DESIDERATA FOR RECURRENT INDEPENDENT MECHANISMS
+
+We have laid out a case for building models composed of modules which by default operate independently and can interact in a limited manner. Accordingly, our approach to modelling the dynamics of the world starts by dividing the overall model into small subsystems (or modules), referred to as Recurrent Independent Mechanisms (RIMs), with distinct functions learned automatically from data.Our model encourages sparse interaction, i.e., we want most RIMs to operate independently and follow their default dynamics most of the time, only rarely sharing information. Below, we lay out desiderata for modules to capture modular dynamics with sparse interactions.
+
+Competitive Mechanisms: Inspired by the observations in the main paper, we propose that RIMs utilize competition to allocate representational and computational resources. As argued by (Parascandolo et al., 2018), this tends to produce independence among learned mechanisms if the training data has been generated by independent physical mechanisms.
+
+Top Down Attention: The points mentioned in Section 2 in principle pertain to synthetic and natural intelligent systems alike. Hence, it is not surprising that they also appear in neuroscience. For instance, suppose we are looking for a particular object in a large scene, using limited processing capacity. The biased competition theory of selective attention conceptualizes basic findings of experimental psychology and neuroscience (Desimone & Duncan, 1995): our capacity of parallel processing of and reasoning with high-level concepts is limited, and many brain systems representing visual information use competition to allocate resources. Competitive interactions among multiple objects occur automatically and operate in parallel across the visual field. Second, the principle of selectivity amounts to the idea that a perceiver has the ability to filter out unwanted information and selectively process the rest of the information. Third, top-down bias originating from higher brain areas enables us to selectively devote resources to input information that may be of particular interest or relevance. This may be accomplished by units matching the internal model of an object or process of interest being pre-activated and thus gaining an advantage during the competition of brain mechanisms.
+
+Sparse Information Flow: Each RIMs’ dynamics should only be affected by RIMs which are deemed relevant. The fundamental challenge is centered around establishing sensible communication between RIMs. In the presence of noisy or distracting information, a large subset of RIMs should stay dormant, and not be affected by the noise. This way, training an ensemble of these RIMs can be more robust to out-of-distribution or distractor observations than training one big homogeneous neural network (Schmidhuber, 2018).
+
+Modular Computation Flow and Modular Parameterization: Each RIM should have its own dynamics operating by default, in the absence of interaction with other RIMs. The total number of parameters (i.e. weights) can be reduced since the RIMs can specialize on simple sub-problems, similar to (Parascandolo et al., 2018). This can speed up computation and improve the generalisation ability of the system (Baum & Haussler, 1989). The individuals RIMs in the ensemble should be simple also to prevent individual RIMs from dominating and modelling complex, composite mechanisms. We refer to a parameterization as modular if most parameters are associated to individuals RIMs only. This has the desirable property that a RIM should maintain its own independent functionality even as other RIMs are changed (due to its behavior being determined by its own self-contained parameters).
+
+# B EXTENDED RELATED WORK
+
+Table 2: A concise comparison of recurrent models with modular memory.   
+
+<table><tr><td rowspan=1 colspan=1>Method /Property</td><td rowspan=1 colspan=1>ModularMemory</td><td rowspan=1 colspan=1>SparseInformation Flow</td><td rowspan=1 colspan=1>ModularComputation Flow</td><td rowspan=1 colspan=1>ModularParameterization</td></tr><tr><td rowspan=1 colspan=1>LSTM/RNN</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>Relational RNN (Santoro et al., 2018)</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></tr><tr><td rowspan=1 colspan=1>NTM (Graves et al.,2014b)</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>SAB(Ke et al., 2018)</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></tr><tr><td rowspan=1 colspan=1>IndRNN(Li et al.,2018)</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>√</td></tr><tr><td rowspan=1 colspan=1>RIMs</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></tr></table>
+
+The present section provides further details on related work, thus extending Section 3.
+
+Neural Turing Machine (NTM). The NTM (Graves et al., 2014a) has a Turing machine inspired memory with a sequence of independent memory cells, and uses an attention mechanism to move heads over the cells while performing targeted read and write operations. This shares a key idea with RIMs: that input information should only impact a sparse subset of the memory by default, while keeping most of the memory unaltered. The RIM model introduces the idea that each RIM has its own independent dynamics, whereas the mechanism for updating memory cells update is shared.
+
+Relational RNN. The Relational Models paper (Santoro et al., 2018) is based on the idea of using a multi-head attention mechanism to share information between multiple parts of memory. It is related to our idea but a key difference is that we encourage the RIMs to remain separate as much as possible, whereas (Santoro et al., 2018) allows information between the parts to flow on each step (in effect making the part distribution only relevant to a particular step). Additionally, RIMs has the notion of each RIM having its own independent transition dynamics which operate by default, whereas the Relational RNN only does computation and updating of the memory using attention.
+
+Sparse Attentive Backtracking (SAB). The SAB architecture (Ke et al., 2018) explores RNNs with selfattention across time steps as well as variants where the attention is sparse in the forward pass and where the gradient is sparse in the backward pass. It shares the motivation of using sparse attention to keep different pieces of information separated, but differs from the RIMs model in that it considers separation between time steps rather than separation between RIMs.
+
+Independently Recurrent Neural Network (IndRNN). The IndRNN (Li et al., 2018) replaces the full transition matrix in a vanilla RNN (between time steps) to a diagonal transition weight matrix. In other words, each recurrent unit has completely independent dynamics. Intriguingly they show that this gives much finer control over the gating of information, and allows for such an RNN to learn long-term dependencies without vanishing or exploding gradients. Analysis of the gradients shows that having smaller recurrent transition matrices mitigates the vanishing and exploding gradient issue. This may provide further explanation for why RIMs perform well on long sequences.
+
+Consciousness Prior (Bengio, 2017): This is based on the assumption of a sparse graphical model describing the interactions between high-level variables, using gating mechanisms to select only a subset of high-level variables to interact at any particular time. This is closely related to our work in the sense high level abstract representation is based on the representations of the RIMs, which are activated sparsely and interact sparsely. Our paper thus helps to validate the consciousness prior idea.
+
+Recurrent Entity Networks: EnTNet (Henaff et al., 2016) can be viewed as a set of separate recurrent models whose hidden states store the memory slots. These hidden states are either fixed by the gates, or modified through a simple RNN-style update. Moreover, EntNet uses an independent gate for writing to each memory slot. Our work is related in the sense that we also have different recurrent models (i.e.,RIMs, though each RIM has different parameters), but we allow the RIMs to communicate with each other sparingly using an attention mechanism.
+
+Capsules and Dynamic Routing: EM Capsules (Hinton et al., 2018) and the preceding Dynamic Capsules (Sabour et al., 2017) use the poses of parts and learned part object relationships to vote for the poses of objects. When multiple parts cast very similar votes, the object is assumed to be present, which is facilitated by an interactive inference (routing) algorithm.
+
+Relational Graph Based Methods: Recent graph-based architectures have studied combinatorial generalization in the context of modeling dynamical systems like physics simulation, multi-object scenes, and motion-capture data, and multiagent systems (Scarselli et al., 2008; Bronstein et al., 2017; Watters et al., 2017; Raposo et al., 2017; Santoro et al., 2017; Gilmer et al., 2017; Van Steenkiste et al., 2018; Kipf et al., 2018; Battaglia et al., 2018; Tacchetti et al., 2018). One can also view our proposed model as a relational graph neural network, where nodes are parameterized as individual RIMs and edges are parameterized by the attention mechanism. Though, its important to emphasize that the topology of the graph induced in the proposed model is dynamic, while in most graph neural networks the topology is fixed.
+
+Default Behaviour: Our work is also related to work in behavioural research that deals with two modes of decision making (Dickinson, 1985; Botvinick & Braver, 2015; Kool & Botvinick, 2018): an automatic systems that relies on habits and a controlled system that uses some privileged information for making decision making. The proposed model also has two modes of input processing, RIMs which activate uses some external sensory information, and hence analogous to controlled system. RIMs which don’t activate, they are synonymous to habit based system. There is some work done trying in Reinforcement learning, trying to learn default policies, which have shown to improve transfer and generalization in multi-task RL (Teh et al., 2017; Goyal et al., 2019a). The proposed method is different in the sense, we are not trying to learn default policies which effect the environment, instead we want to learn mechanisms, which try to understand the environment. State dependent activation of different primitive policies was also studied in (Goyal et al., 2019b), and the authors showed that they can learn different primitives, but they also consider that only a single primitive can be active at a particular time step. Also, note that primitive policies try to effect the environment, whereas mechanism try to understand the enviornment.
+
+# C EXPERIMENTAL DETAILS AND HYPERPARAMETERS
+
+# C.1 RIMS IMPLEMENTATION
+
+The RIMs model consists of three main components: the input attention, the process for selecting activated RIMs, and the communication between RIMs. The input attention closely follows the attention mechanism of (Santoro et al., 2018) but with a significant modification: that all of the weights within the attention mechanism are separate per-block. Thus we remove the normal linear layers and replace them with a batch matrix multiplication over the RIMs (as each block has its own weight matrix). Note that the read-key (or query) is a function of the hidden state of each RIM.
+
+For selecting activated RIMs, we compute the top- $\mathbf { \nabla } \cdot \mathbf { k }$ attention weight on the null input over the RIMs. We then select the activated RIMs by using a mask.
+
+We compute the independent dynamics over all RIMs by using a separate LSTM for each RIM. Following this, we compute the communication between RIMs as a multihead attention (Santoro et al., 2018), with the earlier-discussed modification of having separate weight parameters for each block, and also that we added a skip-connection around the attention mechanism. This attention mechanism used 4 heads and in general used a key size and value size of 32. We computed the updates for all RIMs but used the activated-block mask to selectively update only the activated subset of the RIMs.
+
+The use of RIMs introduces two additional hyperparameters over an LSTM/GRU: the number of RIMs and the number of activated RIMs per step. We also observed that having too few activated RIMs tends to hurt optimization and having too many activated RIMs attenuates the improvements to generalization. For the future it would be interesting to explore dynamic ways of controlling how many RIMs to activate.
+
+# C.2 DETAILED MODEL HYPERPARAMETERS
+
+Table 3 lists the different hyperparameters.
+
+Table 3: Hyperparameters   
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Optimizer learning rate batch size</td><td>Adam(Kingma &amp; Ba, 2014) 7·10-4 64</td></tr><tr><td>Inp keys Inp Values Inp Heads</td><td>64 Size of individual RIM * 4</td></tr><tr><td>Inp Dropout</td><td>4 0.1</td></tr><tr><td>Comm keys Comm Values</td><td>32 32</td></tr><tr><td>Comm heads Comm Dropout</td><td>4</td></tr></table>
+
+# C.3 FUTURE ARCHITECTURAL CHANGES
+
+We have not conducted systematic optimizations of the proposed architecture. We believe that even principled hyperparameter tuning may significantly improve performance for many of the tasks we have considered in the paper. We briefly mention a few architectural changes which we have studied:
+
+• On the output side, we concatenate the representations of the different RIMs, and use the concatenated representation for learning a policy (in RL experiments) or for predicting the input at the next time step (for bouncing balls as well as all other experiments). We empirically found that adding another layer of (multi-headed) key value attention on the output seems to improve the results. We have not included this change In our experiments, we shared the same decoder for all the RIMs, i.e., we concatenate the representations of different RIMS, and feed the concatenated representations to the decoder. In the future it would be interesting to think of ways to allow a more “structured” decoder. The reason for this is that even if the RIMs generalize to new environments, the shared decoder can fail to do so. So changing the structure of decoder could be helpful.
+
+• For the RL experiments, we also tried providing the previous actions, rewards, language instruction as input to decide the activation of RIMs. This is consistent with the idea of efference copies as proposed by von Helmholtz (1867); von Holst & Mittelstaedt (1950), i.e., using copies of motor signals as inputs. Preliminary experiments shows that this improves the performance in Atari games.
+
+# C.4 LANGUAGE MODELING
+
+Table 4: Wikitext-2 results   
+
+<table><tr><td>Approach</td><td>Num.Parameters</td><td>Train PPL</td><td>Valid PPL</td><td>Test PPL</td></tr><tr><td>LSTM (2-layer)</td><td>21.2M</td><td>39.78</td><td>109.25</td><td>102.53</td></tr><tr><td>Relational Memory (Santoro et al., 2018)</td><td>11M</td><td>n/a</td><td>112.77</td><td>107.21</td></tr><tr><td>RIMs (2-layer, kT = 6,k A = 6)</td><td>23.7M</td><td>41.27</td><td>103.60</td><td>98.66</td></tr></table>
+
+We investigate the task of word-based language modeling. We ran experiments on the wikitext-2 dataset (Merity et al., 2016). We ran each experiment for a fixed 100 epochs. These results are in Table 4. Our goal in this experiment is to demonstrate the breadth of the approach by showing that RIMs performs well even on datasets which are noisy and drawn from the real-world.
+
+# C.5 COPYING TASK
+
+We used a learning rate of 0.001 with the Adam Optimizer and trained each model for 150 epochs (unless the model was stuck, we found that this was enough to bring the training error close to zero). For the RIMs model we used 600 units split across 6 RIMs (100 units per block). For the LSTM we used a total of 600 units. We did not explore this extensively but we qualitatively found that the results on copying were not very sensitive to the exact number of units.
+
+The sequences to be copied first have 10 random digits (from 0-8), then a span of zeros of some length, followed by a special indicator “9” in the input which instructs the model to begin outputting the copied sequence.
+
+In our experiments, we trained the models with “zero spans” of length 50 and evaluated on the model with “zero spans” of length 200. We note that all the ablations were run with the default parameters (i.e number of keys, values as for RIMs model) for 100 epochs. Tab. 5 shows the effect of two baselines as compared to the RIMs model (a) When we allow the input attention for activation of different RIMs but we dont allow different RIMs to communicate. (b) No Input attention, but we allow different RIMs to communicate with each other. Tab. 5 shows that the proposed method is better than both of these baselines. For copy task, we used 1 head in input attention, and 4 heads for RIMs communication. We note that even with 1 RIM, its not exactly same as a LSTM, because each RIM can still reference itself.
+
+# C.6 SEQUENTIAL MNIST TASK
+
+In this task we considered classifying binary MNIST digits by feeding the pixels to an RNN (in a fixed order scanning over the image). As the focus of this work is on generalization, we introduced a variant on this task where the training digits are at a resolution of $1 4 \times 1 4$ (sequence length of 196). We then evaluated on MNIST digits of different higher resolutions ( $1 6 \times 1 6$ , $1 9 \times 1 9$ , and $2 4 \times 2 4$ ). When re-scaling the images, we used the nearest-neighbor based down-scaling and performed binarization after re-scaling. We trained with a learning rate of 0.0001 and the Adam optimizer. For RIMs we used a total of 600 hidden units split across 6 RIMs (100 units per block). For the LSTM we used a total of 600 units. We ran proposed model as well as baselines for 100 epochs. For sequential MNIST task, we used 1 head in input attention, and 4 heads for RIMs communication.
+
+C.7 IMITATION LEARNING: ROBUSTNESS TO NOISE IN STATE DISTRIBUTION
+
+Here, we consider imitation learning where we have training trajectories generated from an expert (Table 6). We evaluate our model on continuous control tasks in Mujoco (in our case, Half-Cheetah) (Todorov et al., 2012). We take the rendered images as input and compared the proposed model with recurrent policy (i.e., LSTM). Since, using rendered image of the input does not tell anything about the velocity of the Half-Cheetah, it makes the task partially observable. In order to test how well the proposed model generalizes during test, we add some noise (in the joints of the half-cheetah body). As one can see, after adding noise LSTM baselines performs poorly. On the other hand, for the proposed model, there’s also a drop in performance but not as bad as for the LSTM baseline.
+
+Table 5: Error (CE for last 10 time steps) on the copying task. Note that while all of the methods are able to learn to copy on the length seen during training, the RIMs model generalizes to sequences longer than those seen during training whereas the LSTM fails catastrophically.   
+
+<table><tr><td>Approach Train Length 50 Test Length 200</td></tr><tr><td>RIMs 0.00</td></tr><tr><td>0.00</td></tr><tr><td>With input Attention and No Communication</td></tr><tr><td>RIMs (kt=4,kA=2) 2.3 1.6</td></tr><tr><td>RIMs (kt=4,kA= 3) 1.7 4.3</td></tr><tr><td>RIMs (kT=5,kA=2) 2.5 4,7</td></tr><tr><td>RIMs (kT=5,kA=3) 0.4 4.0</td></tr><tr><td>RIMs (kT=5,kA=4) 0.2 0.7 3.3 2.4</td></tr><tr><td>RIMs(kT=6,kA=2) RIMs (kT=6,kA=3) 1.2 1.0</td></tr><tr><td>RIMs (kT=6,kA=4) 0.7 5.0</td></tr><tr><td>RIMs (kT=6,kA= 5) 0.22 0.56</td></tr><tr><td></td></tr><tr><td>With No input Attention and Full Communication</td></tr><tr><td>RIMs (kT =6,kA = 6, hdim =600) 0.0 0.7</td></tr><tr><td>RIMs (kT =5,kA=5,hdim =500) 0.0 1.7</td></tr><tr><td>RIMs(kT=2,kA=2,hdim= 256) 0.0 2.9</td></tr><tr><td>RIMs (kT =2,kA =2,hdim=512) 0.0 1.8 0.2</td></tr><tr><td>RIMs (kT =1,kA=1, hdim =512) 0.0</td></tr></table>
+
+Table 6: Imitation Learning: Results on the half-cheetah imitation learning task. RIMs outperforms a baseline LSTM when we evaluate with perturbations not observed during training (left). An example of an input image fed to the model (right).   
+
+<table><tr><td>Method / Setting</td><td>Training Observed Reward</td><td>Perturbed States Observed Reward</td></tr><tr><td>LSTM (Recurrent Policy)</td><td>5400 ±100</td><td>2500 ± 300</td></tr><tr><td>RIMs (kT = 6, kA = 3)</td><td>5300 ± 200</td><td>3800± 200</td></tr><tr><td>RIMs (kt = 6, kA = 6)</td><td>5500 ±100</td><td>2700 ± 400</td></tr><tr><td>RIMs (without Input attention)</td><td>5400 ±100</td><td>3200 ± 50</td></tr></table>
+
+![](images/8278f86cb617c0b7d2087c03df7ff71f70f89213ff4c63da1c8ec01c7da0686e.jpg)
+
+We use the convolutional network from (Ha & Schmidhuber, 2018) as our encoder, a GRU (Chung et al., 2015) with 600 units as deterministic path in the dynamics model, and implement all other functions as two fully connected layers of size 256 with ReLU activations. Since, here we are using images as input, which makes the task, partially observable. Hence, we concatenate the past 4 observations, and then feed the concatenated observations input to GRU (or our model). For our model, we use 6 RIMs, each of size 100, and we set $k _ { a } = 3$ . We follow the same setting as in (Hafner et al., 2018; Sodhani et al., 2019). We also compare the proposed method to the baseline where we dont include input attention (or top-down attention). AS 6 shows, there’s a decline in performance if we dont use input attention, hence justifying the importance
+
+# C.8 GENERALIZATION TO DISTRACTORS: ALGORITHM IMPLEMENTATION DETAILS
+
+We evaluate the proposed framework using Adavantage Actor-Critic (A2C) to learn a policy $\pi _ { \boldsymbol { \theta } } ( a | \boldsymbol { s } , \boldsymbol { g } )$ conditioned on the goal. To evaluate the performance of proposed method, we use a range of maze multi-room tasks from the gym-minigrid framework (Chevalier-Boisvert & Willems, 2018) and the A2C implementation from (Chevalier-Boisvert & Willems, 2018). For the maze tasks, we used agent’s relative distance to the absolute goal position as "goal".
+
+For the maze environments, we use A2C with 48 parallel workers. Our actor network and critic networks consist of two and three fully connected layers respectively, each of which have 128 hidden units. The encoder network is also parameterized as a neural network, which consists of 1 fully connected layer. We use RMSProp with an initial learning rate of 0.0007 to train the models. Due to the partially observable nature of the environment, we further use a LSTM to encode the state and summarize the past observations.
+
+![](images/2f2b495d11c7a214e5a9cf3d288c94fdf5886c28c727e11727a6453fe525a48a.jpg)  
+Figure 7: An example of the minigrid task.
+
+# C.9 MINIGRID ENVIRONMENTS FOR OPENAI GYM
+
+The MultiRoom environments used for this research are part of MiniGrid, which is an open source gridworld package2. This package includes a family of reinforcement learning environments compatible with the OpenAI Gym framework. Many of these environments are parameterizable so that the difficulty of tasks can be adjusted (e.g., the size of rooms is often adjustable).
+
+# C.9.1 THE WORLD
+
+In MiniGrid, the world is a grid of size NxN. Each tile in the grid contains exactly zero or one object. The possible object types are wall, door, key, ball, box and goal. Each object has an associated discrete color, which can be one of red, green, blue, purple, yellow and grey. By default, walls are always grey and goal squares are always green.
+
+# C.9.2 REWARD FUNCTION
+
+Rewards are sparse for all MiniGrid environments. In the MultiRoom environment, episodes are terminated with a positive reward when the agent reaches the green goal square. Otherwise, episodes are terminated with zero reward when a time step limit is reached. In the FindObj environment, the agent receives a positive reward if it reaches the object to be found, otherwise zero reward if the time step limit is reached.
+
+The formula for calculating positive sparse rewards is $1 - 0 . 9 * ( s t e p \_ c o u n t / m a x \_ s t e p s )$ . That is, rewards are always between zero and one, and the quicker the agent can successfully complete an episode, the closer to 1 the reward will be. The max_steps parameter is different for each environment, and varies depending on the size of each environment, with larger environments having a higher time step limit.
+
+# C.9.3 ACTION SPACE
+
+There are seven actions in MiniGrid: turn left, turn right, move forward, pick up an object, drop an object, toggle and done. For the purpose of this paper, the pick up, drop and done actions are irrelevant. The agent can use the turn left and turn right action to rotate and face one of 4 possible directions (north, south, east, west). The move forward action makes the agent move from its current tile onto the tile in the direction it is currently facing, provided there is nothing on that tile, or that the tile contains an open door. The agent can open doors if they are right in front of it by using the toggle action.
+
+# C.9.4 OBSERVATION SPACE
+
+Observations in MiniGrid are partial and egocentric. By default, the agent sees a square of $7 { \bf x } 7$ tiles in the direction it is facing. These include the tile the agent is standing on. The agent cannot see through walls or closed doors. The observations are provided as a tensor of shape $7 \mathrm { x } 7 \mathrm { x } 3$ . However, note that these are not RGB images. Each tile is encoded using 3 integer values: one describing the type of object contained in the cell, one describing its color, and a flag indicating whether doors are open or closed. This compact encoding was chosen for space efficiency and to enable faster training. The fully observable RGB image view of the environments shown in this paper is provided for human viewing.
+
+# C.9.5 LEVEL GENERATION
+
+The level generation in this task works as follows: (1) Generate the layout of the map (X number of rooms with different sizes (at most size Y) and green goal) (2) Add the agent to the map at a random location in the first room. (3) Add the goal at a random location in the last room. A neural network parameterized as CNN is used to process the visual observation.
+
+We follow the same architecture as (Chevalier-Boisvert & Willems, 2018) but we replace the LSTM layer with BlockLSTM.
+
+# C.10 BOUNCING BALLS
+
+We use the bouncing-ball dataset from (Van Steenkiste et al., 2018). The dataset consists of 50,000 training examples and 10,000 test examples showing ${ \sim } 5 0$ frames of either 4 solid balls bouncing in a confined square geometry, 6-8 balls bouncing in a confined geometry, or 3 balls bouncing in a confined geometry with a random occluded region. In all cases, the balls bounce off the wall as well as off one another. We train baselines as well as proposed model for about 100 epochs using 0.0007 as learning rate and using Adam as optimizer (Kingma & Ba, 2014). We use the same architecture for encoder as well as decoder as in (Van Steenkiste et al., 2018). We train the proposed model as well as the baselines for 100 epochs. Our goal in this section is to give more thorough experimental results omitted from the main paper for the sake of brevity. Below, we highlight a few different results.
+
+![](images/a0bec43d89bceb9b14fffccdc3799d55e6da8f302dfba866960da333fecd155e.jpg)  
+C.10.1 DIFFERENT RIMS ATTEND TO DIFFERENT BALLS   
+Figure 8: Different RIMs attending to Different Balls. For understanding what each RIM is actually doing, we associate each with a separate encoder, which are spatially masked. Only 4 encoders can be active at any particular instant and there are four different balls. We did this to check if there would be the expected geometric activation of RIMs. 1.) Early in training, RIM activations correlated more strongly with the locations of the four different balls. Later in training, this correlation decreased and the active strips did not correlate as strongly with the location of balls. As the model got better at predicting the location, it needed to attend less to the actual objects. The top row shows every 5th frame when the truth is fed in and the bottom shows the results during rollout. The gray region shows the active block. In the top row, the orange corresponds to the prediction and in the bottom, green corresponds to the prediction.
+
+In order to visualize what each RIM is doing, we associate each RIM with a different encoder. By performing spatial masking on the input, we can control the possible spatial input to each RIM. We use six non-overlapping horizontal strips and allow only 4 RIMs to be active at a time (shown in Fig. 8). The mask is fixed mask of zeros with a band of ones that is multiplied by the input to each encoder. Therefore, each of the 6 encoders gets 1/6th of the input. The goal was to see how the RIM activation patterns changed/correlated with the locations of the balls. We find that early in training, the RIMs’ activations are strongly correlated with the location of the 4 balls. However, after training has proceeded for some time this correlation deteriorates. This is likely because the predictable dynamics of the system do not necessitate constant attention.
+
+# C.10.2 COMPARISON WITH LSTM BASELINES
+
+In Figures 9, 10, 11, and 12 we highlight different baselines and how these compare to the proposed RIMs model.
+
+![](images/31aae17a6489fd1255dc264b1cb1cbddd23587ec9143a4aed15403f7df2d89f9.jpg)  
+Figure 9: Example of the other LSTM baselines. For the 2 other experiments that we consider, here we show example outputs of our LSTM baselines. In each row, the top panel represents the ground truth and the bottom represents the prediction. All shown examples use an LSTM with 250 hidden units, as shown in Fig. 4. Frames are plotted every 3rd time step. The red line marks 10 rollout frames. This is marked because after this we do not find BCE to be a reliable measure of dissimilarity.
+
+![](images/f6ad9188f023a860229b04ca67ed9e3bcd1500ed199126cb88148a3553d37ce8.jpg)  
+Figure 10: Comparison of RIMs to LSTM baseline. For 4 different experiments in the text, we compare RIMs to two different LSTM baselines. In all cases we find that during rollout, RIMs perform better than the LSTMs at accurately capturing the trajectories of the balls through time. Due to the number of hard collisions, accurate modeling is very difficult. In all cases, the first 15 frames of ground truth are fed in (last 6 shown) and then the system is rolled out for the next 15 time steps, computing the binary cross entropy between the prediction and the true balls at each instant, as in Van Steenkiste et al. (2018). See the Appendix for losses over the entire 35 frame rollout trajectory. In the predictions, the transparent blue shows the ground truth, overlaid to help guide the eye.
+
+# C.10.3 OCCLUSION
+
+In Fig. 13, we show the performance of RIMs on the curtain dataset. We find RIMs are able to track balls through the occlusion without difficulty. Note that the LSTM baseline, is also able to track the ball through the “invisible” curtain.
+
+# C.10.4 STUDY OF TRANSFER
+
+It is interesting to ask how models trained on a dataset with 6-8 balls perform on a dataset with 4 balls. In Fig. 14 we show predictions during feed-in and rollout phases.
+
+![](images/f4e6369f54617ee402a0fa84ed83ef4340bf67036200f7898233373e36e42168.jpg)  
+Figure 11: Comparison between RIMs and LSTM baseline. For the 4 ball task and the 6-8 ball extrapolation task, here we show an example output of from our LSTM baseline and from RIMs. All shown examples use an LSTM with 250 hidden units, as shown in Fig. 4. Frames are plotted every 3rd time step. The red line marks 10 rollout frames. This is marked because after this we do not find BCE to be a reliable measure of dissimilarity.
+
+![](images/6265561a3e53a9eeda0891f9644d6250d7d45e9c04344e203ff2681ae8d241fd.jpg)  
+Figure 12: Comparison of RIMs to LSTM baseline. For 4 different experiments in the text, we compare RIMs to two different LSTM baselines. In all cases we find that during rollout, RIMs perform better than the LSTMs at accurately capturing the trajectories of the balls through time. Due to the number of hard collisions, accurate modeling is very difficult.
+
+![](images/ed604b66bf9dea912914a9e4b1c235c383d7d053d7810a569459daf4f9578f9e.jpg)  
+Figure 13: RIMs on dataset with an occlusion. We show two trajectories (top and bottom) of three balls. For the left frames, at each step the true frame is used as input. On the right, outlined in black, the previous output is used as input.
+
+# C.11 ABLATIONS
+
+We present one ablation in addition to the ones in Section 4.4. In this experiment, we study the effect on input attention (i.e top down attention) as well as the use of multi-headed head key-value attention. We compare the proposed model (with input attention as well as multi-headed key value attention) with 2 baselines: (a) In which we remove the input attention (and force all the RIMs to communicate with each other (b) We use 1 head for key value attention as compared to multi-headed key-value attention. Results comparing the proposed model, with these two baselines is shown in Fig. 15.
+
+![](images/e70da60a6a8cfd1f817c3ca4cf9004c1e47c17aac09d5797349c02df53e09245.jpg)  
+Figure 14: RIMs transferred on new data. We train the RIMs model on the 6-8 ball dataset (as shown in the top row). Then, we apply the model to the 4 ball dataset, as shown in the bottom.
+
+![](images/8187dd8303fd62d4a4c58ccbdad59074b7739da6a1b6cc5d5b8e00714a2dfba8.jpg)  
+Figure 15: Ablation loss For the normal, a one-head model, and without input attention, we show the loss during training and the loss for the 4th and 5th frame of rollout. We find that the one-head and without input attention models perform worse than the normal RIMs model during the rollout phase.
+
+In Fig. 16, we show the predictions that result from the model with only one active head.
+
+![](images/3878333431e176704cd58f53e869b12aa2109ec8609abb6b2e42ea90df1c9045.jpg)  
+Figure 16: One head and no attention Using one head and no attention models, we show the rollout predictions in blue. On top we show results on the 4 ball dataset and on the bottom we show results on the curtains dataset.
+
+<table><tr><td>Environment</td><td>LSTM-PPO</td><td>RIMs-PPO</td></tr><tr><td>Alien</td><td>1612 ± 44</td><td>2152 ±81</td></tr><tr><td>Amidar</td><td>1000 ±58</td><td>1800 ± 43</td></tr><tr><td>Assault</td><td>4000 ± 213</td><td>5400 ± 312</td></tr><tr><td>Asterix</td><td>3090 ±420</td><td>21040 ± 548</td></tr><tr><td>Asteroids</td><td>1611.0 ± 200</td><td>3801 ±89</td></tr><tr><td>Atlantis</td><td>3280000 ± 200000</td><td>3500000 ± 120000</td></tr><tr><td>BankHeist</td><td>1153 ± 23</td><td>1195 ±4</td></tr><tr><td>BattleZone</td><td>21000 ± 232.0</td><td>22000 ±324</td></tr><tr><td>BeamRider</td><td>698 ±100</td><td>5320±300</td></tr><tr><td>Bowling</td><td>30±5</td><td>42 ±13</td></tr><tr><td>Boxing</td><td>80±3</td><td>95±10</td></tr><tr><td>Breakout</td><td>593 ± 90</td><td>590 ±10</td></tr><tr><td>Centipede</td><td>4600 ±312</td><td>5534±283</td></tr><tr><td>ChopperCommand</td><td>11000 ± 790</td><td>12303 ± 412</td></tr><tr><td>CrazyClimber</td><td>138000 ± 2412</td><td>132039±1221</td></tr><tr><td>DemonAttack</td><td>26320 ± 3234</td><td>230324±4032</td></tr><tr><td>DoubleDunk</td><td>-3.0 ± 0.5</td><td>-3.8 ± 0.3</td></tr><tr><td>Enduro</td><td>1600 ± 200</td><td>2800 ± 232</td></tr><tr><td>FishingDerby</td><td>20±4</td><td>38±8</td></tr><tr><td>Freeway</td><td>29±2</td><td>33±2</td></tr><tr><td>Gopher</td><td>7000.0 ± 402</td><td>33000 ± 2210</td></tr><tr><td>Gravitar</td><td>500 ±100</td><td>1090 ± 80</td></tr><tr><td>IceHockey</td><td>-5±0.3</td><td>-4±1</td></tr><tr><td>Jamesbond</td><td>425±25</td><td>800 ±100</td></tr><tr><td>Kangaroo</td><td>13000 ± 500</td><td>1800 ±400</td></tr><tr><td>Krull</td><td>10000 ±500</td><td>7900 ± 200</td></tr><tr><td>KungFuMaster</td><td>28000 ± 2000</td><td>51000 ± 800</td></tr><tr><td>NameThisGame</td><td>4200± 400</td><td>6800 ±300</td></tr><tr><td>Pong</td><td>20±1</td><td>20±1</td></tr><tr><td>PrivateEye</td><td>90±3</td><td>100±0</td></tr><tr><td>Qbert</td><td>22000 ± 300</td><td>22500 ± 400</td></tr><tr><td>Riverraid</td><td>7500 ± 300</td><td>12000 ± 100</td></tr><tr><td>RoadRunner</td><td>53000 ±120</td><td>53430 ±300</td></tr><tr><td>Robotank</td><td>3±1</td><td>11±2</td></tr><tr><td>SpaceInvaders</td><td>1600 ± 40</td><td>2800 ± 80</td></tr><tr><td>StarGunner</td><td>35000 ± 800</td><td>70000 ±1200</td></tr><tr><td>TimePilot</td><td>4000 ±100</td><td>10000 ± 689</td></tr><tr><td>UpNDown</td><td>70000 ± 6000</td><td>390000 ± 20000</td></tr><tr><td>VideoPinball</td><td>90000 ± 5000</td><td>220000 ±9000</td></tr><tr><td>WizardOfWor</td><td>3833 ±400</td><td>10800 ± 700</td></tr><tr><td>Zaxxon</td><td>200 ±100</td><td>15000 ± 600</td></tr></table>
+
+Table 7: Scores obtained using PPO with the LSTM architecture and PPO with the RIMs architecture with $k _ { A } = 5$ .
+
+We used open-source implementation of PPO from (Kostrikov, 2018) with default parameters. We ran the proposed algorihtm with 6 RIMs, and kept the number of activated RIMs to 4/5. We have not done any hyper-parameter search for Atari experiments.
+
+![](images/57727618e025c071e282e20dcf5274e882a4381ab7a9acb61ec4951496a9a08c.jpg)  
+Figure 17: A comparison showing relative improvement of RIMs with $k _ { A } = 5$ over a $k _ { A } = 4$ baseline. Using $k _ { A } = 5$ performs slightly worse than $k _ { A } = 4$ but still outperforms PPO, and has similar results across the majority of games.
+
+![](images/42cf0f8ace9b7f79a6db90ffe13490cb2297cd994becddc419bd711e4a2be4b2.jpg)  
+Figure 18: RIMs-PPO relative score improvement over LSTM-PPO baseline (Schulman et al., 2017) across all Atari games averaged over 3 trials per game. In both cases PPO was used with the exact same settings with the only change being the choice of the recurrent architecture (RIMs with $k _ { A } = 5$ ).
+
+# C.12.1 TRANSFER ON ATARI
+
+As a very preliminary result, we investigate feature transfer between randomly selected Atari games. In order to study this question, we follow the experimental protocol of Rusu et al. (2016).
+
+We start by training RIMs on three source games (Pong, River Raid, and Seaquest) and test if the learned features transfer to a different subset of randomly selected target games (Alien, Asterix, Boxing, Centipede, Gopher, Hero, James Bond, Krull, Robotank, Road Runner, Star Gunner, and Wizard of Wor). We observe, that RIMs result in positive transfer in 9 out of 12 target games, with three cases of negative transfer. On the other hand progressive networks (Rusu et al., 2016) result in positive transfer in 8 out of 12 target games, and two cases of negative transfer. We also compare to LSTM baseline, which yields positive transfer in 3 of 12 games.
+
+# C.13 BOUNCING MNIST: DROPPING OFF RIMS
+
+We use the Stochastic Moving MNIST (SM-MNIST) (Denton & Fergus, 2018) dataset which consists of sequences of frames of size $6 4 \times 6 4$ , containing one or two MNIST digits moving and bouncing off the walls. Training sequences are generated on the fly by sampling two different MNIST digits from the training set (60k total digits) and two distinct trajectories.
+
+Here, we show the effect of masking out a particular RIM and study the effect of the masking on the ensemble of RIMs. Ideally, we would want different RIMs not to co-adapt with each other. So, masking out a particular RIM should not really effect the dynamics of the entire model. We show qualitative comparisons in Fig. 19, 20, 21, 22, 23. In each of these figures, the model gets the ground truth image as input for first 5 time steps, and then asked to simulate the dynamics for next 25 time-steps. We find that sparsity is needed otherwise different RIMs co-adapt with each other (for ex. see Fig. 20, 22, 23). We tried similar masking experiments for different models like RMC, Transformers, EntNet (which learns a mixture of experts), LSTMs, but all of them failed to do anything meaningful (after masking). We suspect this is partly due to learning a homogeneous network.
+
+![](images/f971a83cd70dd9062b60fb3f8631f70f96be6d7cb8bbaeae0987672d224778b3.jpg)
+
+![](images/0a3225ccf0b76ce97d7cd0a0daea7e48c4c1c85d038662a11f2fde7d86c98e27.jpg)
+
+![](images/af31640d6753f73793f736bdc835946c36c294a4665ea8a90e20217110370e5c.jpg)
+
+![](images/4452f4916ac7e52efc306c23fc22d1d7bddd4357420ce2321fda768d6babd0a3.jpg)  
+Figure 19: 4 RIMs, (top ${ \mathrm { k } } = 2$ ). Each sub-figure shows the effect of masking a particular RIM and studying the effect of masking on the other RIMs. For example, the top figure shows the effect of masking the first RIM, the second figure shows the effect of masking the second RIM etc.
+
+![](images/a9eb098881486623a43996135b76047a8d2f0e9eb2b751f300569637680de534.jpg)
+
+![](images/f7bf98b8d99f61083779a60ad93760d277735965473ecc608d4908b9759b2b18.jpg)
+
+![](images/f2ea091fa4163e2ded29aeceb61719eecfdb5b4f3c5763a6ccbcc850f0d33fd2.jpg)
+
+![](images/ef3f3462cf305e361263b8e2d19a92cb7d874ca05d39f69e65d2d23e66ab9b64.jpg)  
+Figure 20: 4 RIMs, (top ${ \bf k } = 3$ ). Each sub-figure shows the effect of masking a particular RIM and studying the effect of masking on the other RIMs. For example, the top figure shows the effect of masking the first RIM, the second figure shows the effect of masking the second RIM etc.
+
+![](images/c8c57c3630e25c553a994aaaf21740cf2a5aa4d75eb4de052284dd55ca9d6a0a.jpg)  
+Figure 21: 400dim, 5 RIMs, (top ${ \bf k } = 2$ ). Each sub-figure shows the effect of masking a particular RIM and studying the effect of masking on the other RIMs. For example, the top figure shows the effect of masking the first RIM, the second figure shows the effect of masking the second RIM etc.
+
+![](images/ec9b38a16235c0c5a950cd54777abbc6e4e8aa8fc69f27ad2fdc16b7cf522cf6.jpg)  
+Figure 22: 400dim, 5 blocks, (top ${ \bf k } = 3$ ). Each sub-figure shows the effect of masking a particular RIM and studying the effect of masking on the other RIMs. For examples, the top figure shows the effect of masking the first RIM, the second figure shows the effect of masking the second RIM etc.
+
+![](images/42501d75b99c2917900abba1c9a092072c33e658d0c3ae3efc6dcfa6fdcfc7cd.jpg)  
+Figure 23: 400dim, 5 blocks, (top $\mathrm { k } = 4$ ). Each sub-figure shows the effect of masking a particular RIM and studying the effect of masking on the other RIMs. For example, the top figure shows the effect of masking the first RIM, the second figure shows the effect of masking the second RIM etc.
+
+![](images/39ef30e68f0cd11115f6385496da9f0eff513b6cea1d7de50d8d9c96896d8063.jpg)  
+Figure 24: Comparing RIMs-PPO with LSTM-PPO: Learning curves for $k _ { A } = 4$ , $k _ { A } = 5$ RIMs-PPO models and the LSTM-PPO baseline across all Atari games.
+
+# C.13.2 ATARI RESULTS: NO INPUT ATTENTION
+
+Here we compare the proposed method to the baseline, where we dont use input attention, and we force different RIMs to communicate with each at all the time steps.
+
+![](images/3b2c554deff8b2e6e8ad321d8a192ec97ed6d20f62baebf9eb73f270ddd41c71.jpg)  
+Figure 25: Baseline agent with no input attention mechanism: Here we compare the RIMs to the baseline, where their is no input attention (i.e., top down attention) as well as all the RIMs communicate with each other at all the time steps. Learning curves for RIMs-PPO models, Baseline Agent, the LSTM-PPO baseline across 30 Atari games.

parse/train/zfO1MwBFu-/zfO1MwBFu-.md

@@ -0,0 +1,476 @@
+# INFORMATION THEORETIC REGULARIZATION FOR LEARNING GLOBAL FEATURES BY SEQUENTIAL VAE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Sequential variational autoencoders (VAEs) with global latent variable $z$ have been studied for the purpose of disentangling the global features of data, which is useful in many downstream tasks. To assist the sequential VAEs further in obtaining meaningful $z$ , an auxiliary loss that maximizes the mutual information (MI) between the observation and $z$ is often employed. However, by analyzing the sequential VAEs from the information theoretic perspective, we can claim that simply maximizing the MI encourages the latent variables to have redundant information and prevents the disentanglement of global and local features. Based on this analysis, we derive a novel regularization method that makes $z$ informative while encouraging the disentanglement. Specifically, the proposed method removes redundant information by minimizing the MI between $z$ and the local features by using adversarial training. In the experiments, we trained state-space and autoregressive model variants using speech and image datasets. The results indicate that the proposed method improves the performance of the downstream classification and data generation tasks, thereby supporting our information theoretic perspective in the learning of global representations.
+
+# 1 INTRODUCTION
+
+Uncovering the global factors of variation from high-dimensional data is a significant and relevant problem in representation learning (Bengio et al., 2013). For example, a global representation of images that presents only the identity of the objects and is invariant to the detailed texture would assist in downstream semi-supervised classification (Ma et al., 2019). In addition, the representation is known to be useful in the controlled generation of data. Obtaining the representation allows us to manipulate the voice of the speaker in speeches (Yingzhen & Mandt, 2018), or generate images that share similar global structures (e.g. the structure of objects) but varying details (Razavi et al., 2019).
+
+Sequential variational autoencoders (VAEs) with a global latent variable $z$ have played an important role in the unsupervised learning of the global features. Specifically, we consider the sequential VAEs with a structured data generating process in which an observation $x$ at time $t$ (denoted as $\boldsymbol { x } _ { t }$ ) is generated from a global feature $z$ and local feature $s _ { t }$ . Then, the $z$ of such sequential VAEs can acquire only global information invariant to $t$ . For example, Yingzhen & Mandt (2018) demonstrated that a disentangled sequential autoencoder (DSAE), which combines state-space models (SSMs) with a global latent variable $z$ , can uncover the speaker information from speeches. Furthermore, Chen et al. (2017); Gulrajani et al. (2017) proposed a VAE with a PixelCNN decoder (denoted as PixelCNN-VAE), which combines autoregressive models (ARMs) and $z$ . In both methods, the hidden state of the sequential model (either SSMs or ARMs) is designed to capture local information, while an additional latent variable $z$ captures global information.
+
+Unfortunately, the design of the aforementioned structured data generating process alone is insufficient to uncover the global features in practice. A typical issue is that latent variable $z$ is ignored by a decoder (SSMs or ARMs) and becomes uninformative. This phenomenon occurs as follows: with expressive decoders, such as SSMs or ARMs, the additional latent variable $z$ cannot assist in improving the evidence lower bound (ELBO), which is the objective function of VAEs; therefore, the decoders will not use $z$ (Chen et al., 2017; Alemi et al., 2018). The phenomenon in which the latent variables are ignored is referred to as posterior collapse (PC). To alleviate this issue, several studies have proposed regularizing the mutual information (MI) between $x$ and $z$ to be large, e.g., using $\beta$ -VAE (Alemi et al., 2018). A higher MI $I ( x ; z )$ indicates that $z$ has significant information regarding $x$ ; this regularization prevents $z$ from becoming uninformative.
+
+![](images/0b258f734715cf2986514558f7e9cddd40505e45248ccff47f57367dd7340bab.jpg)  
+Figure 1: Comparison of (a) MI-maximizing regularization and (b) the proposed method, using a Venn diagram of information theoretic measures of $x , z ,$ , and $s$ .
+
+In this paper, we further analyze the MI-maximization and claim that merely maximizing $I ( x ; z )$ is insufficient to uncover the global factors of variation. Figure 1-(a) summarizes the issue of the MI-maximization. As illustrated in the Venn diagram, the MI can be decomposed into $I ( x ; z ) =$ $I ( x ; z | s ) + I ( x ; z ; s )$ . Maximizing the first term $I ( x ; z | s )$ is beneficial, as it measures the informativeness of $z$ about $x$ given a local feature $s$ . However, maximizing the second term $I ( x ; z ; s )$ might cause a negative effect, because it would also increase $I ( z ; s )$ . In other words, maximizing $I ( x ; z )$ would encourage latent variables to have redundant information. For example, when $I ( x ; z )$ becomes so large that $z$ retains all (local and global) information of $x$ , the downstream classification performance would be degrated. Also, when local variables still contain global information due to large $I ( z ; s )$ , it becomes difficult to control speaker information in speeches using a DSAE. See Appendix A for empirical evidence that the MI-maximization increases $I ( z ; s )$ , as discussed above.
+
+Based on the analysis, we propose a new information theoretic regularization method for disentangling the global factors. Specifically, our method minimizes $I ( z ; s )$ , in addition to maximizing $I ( x ; z )$ similar to prior work (Figure 1-(b)). As $I ( z ; s )$ measures the dependence between $z$ and $s$ , our method encourages $z$ and $s$ to have different information, i.e., the disentanglement of global and local factors. We call our method CMI-maximizing regularization, as it is the lower bound of the conditional mutual information (CMI) $I ( x ; z | s )$ . Furthermore, we introduce an adversarial training technique for estimating the CMI. A simple way to estimate it would be considering $I ( x ; z )$ and $I ( z ; s )$ independently, but it might result in compounding approximation errors. Instead, we use the formularization of $\beta$ -VAE and adversarial training (Ganin et al., 2016), which reduce the number of terms to be approximated. Specifically, we approximate the upper bound of $I ( z ; s )$ using a density ratio trick (DRT) (Nguyen et al., 2008), where an adversarial classifier models the density ratio. Once we estimate the bound, $I ( z ; s )$ can be minimized via backpropagation through the classifier.
+
+In our experiments, we used DSAE and PixelCNN-VAE as illustrative examples of the SSM and ARM variants. In addition to evaluate the quality of global latent variable as in previous studies, we also evaluated the ability of controlled generation using a novel evaluation method inspired by Ravuri & Vinyals (2019). In the experiments, the CMI-maximizing regularization consistently outperformed the MI-maximizing one on image and speech datasets. These results support (i) our information theoretic view of learning global features: the sequential VAEs can suffer from obtaining redundant features when merely maximizing the MI. Also, the results support that (ii) regularizing $I ( x ; z )$ and $I ( z ; s )$ is complementary: learning global features can be facilitated by not only making $z$ informative, but also the control for which aspect of $x$ information (global or local) goes into $z$ .
+
+Our contribution can be summarized as follows: (i) through our analysis, we reveal the potential negative side-effect of MI-maximizing regularization, which has been standard in learning global representation with sequential VAEs. Then, the analysis encourages the sequential VAE community to seek for new regularization approach. (ii) In order to learn good global representation, we proposed regularizing $I ( x ; z )$ and $I ( z ; s )$ at the same time. $I ( x ; z )$ and $I ( x ; z )$ are robustly shown to work complementary by our experiments using two models and two domains (speech and image datasets). This finding would help improve various sequential VAEs proposed before.
+
+![](images/152cb765e8fa03b9d56bfc624cf03e8229f97fbaac3e805b4a10db3a792607f0.jpg)  
+Figure 2: Graphical models for (a) DSAE and (b) PixelCNN-VAE.
+
+# 2 PRELIMINARY
+
+# 2.1 SEQUENTIAL VAES FOR LEARNING GLOBAL REPRESENTATIONS
+
+Here we first explain the standard VAE; then, we give overviews of the DSAE and PixelCNN-VAE. Both models are shown to be interpreted as having two types of the latent variables, global $z$ and local $s _ { t }$ ; although it is not explicitly stated for PixelCNN-VAE. Here, $s _ { t }$ is designed to influence particular timesteps or dimensions of $x$ (e.g., a single-frame in a speech or a small area of pixels in an image). On the other hand, $z$ influences all the timesteps of $x$ , although $z$ of DSAE and PixelCNN-VAE are imposed on different architectural constraints (discussed in Appendix C).
+
+Variational autoencoder (VAE) Let $\begin{array} { r } { p ( x ) : = \int p ( z ) p ( x | z ) d z } \end{array}$ be a latent variable model, whose decoder $p ( x | z )$ is parameterized by a deep neural network (DNN). Using an encoder distribution $q ( z | x )$ , which is also parameterized by a DNN, the VAEs maximize ELBO:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { E L B O } } : = \mathbb { E } _ { p _ { d } ( x ) } \big [ \mathbb { E } _ { q ( z | x ) } [ \log p ( x | z ) ] - D _ { \mathrm { K L } } ( q ( z | x ) | | p ( z ) ) \big ] . } \end{array}
+$$
+
+Here, $p _ { d } ( x )$ denotes the data distribution. ELBO contains two terms: the reconstruction error and the Kullback-Leibler (KL) divergence between encoder $q ( z | x )$ and the prior $p ( z )$ .
+
+Disentangled sequential autoencoder DSAE (Yingzhen & Mandt, 2018) is an extension of the SSMs for modeling the global and local features of sequential data as separate latent variables. Using the disentangled variables, DSAE can control the outputs (e.g., perform voice conversion). DSAE has a global latent variable $z$ and a local latent variable $s _ { t }$ , and generates an observation $x _ { t }$ at time $t$ from $z$ and $s _ { t }$ . The ELBO can be expressed as follows:
+
+$$
+\begin{array} { r l r } {  { \mathcal { L } _ { \mathrm { S S M } } : = - \mathrm { R e c o n - K L } ( z ) - \mathrm { K L } ( s ) , } } \\ & { } & \\ & { } & { \quad \mathrm { w h e r e ~ } \mathrm { R e c o n } = - \mathbb { E } _ { q ( x , z , s ) } [ \underset { t = 1 } { \overset { T } { \sum } } \log p ( x _ { t } | s _ { t } , z ) ] , \mathrm { K L } ( z ) = \mathbb { E } _ { q ( x , z , s ) } [ D _ { \mathrm { K L } } ( q ( z | x _ { \leq T } ) | | p ( z ) ) ] , } \\ & { } & \\ & { } & { \quad \mathrm { K L } ( s ) = \mathbb { E } _ { q ( x , z , s ) } [ \underset { t = 1 } { \overset { T } { \sum } } D _ { \mathrm { K L } } ( q ( s _ { t } | x _ { \leq T } , z , s _ { t - 1 } ) | | p ( s _ { t } | s _ { t - 1 } ) ) ] . } \end{array}
+$$
+
+Here, $p ( s _ { t } | s _ { t - 1 } )$ is a prior, $q { \big ( } z | x _ { \leq T } )$ and $q \big ( s _ { t } | x _ { \le T } , z , s _ { t - 1 } \big )$ are encoders, $p ( \boldsymbol { x } _ { t } | \boldsymbol { s } _ { t } , z )$ is a decoder, and $q ( x , z , s ) : = p _ { d } ( x ) q ( z | x ) q ( s | x , z )$ . Furthermore, $x _ { < t }$ denotes all the elements of the sequences up to $t$ , and $x$ denote $x : = x _ { \leq T }$ . Figure 2-(a) illustrates the data generating process.
+
+VAE with PixelCNN decoder PixelCNN-VAE is designed to take advantage of both PixelCNN and VAEs. VAEs are known to fail in terms of capturing the local features of images, such as textures and sharp edges. Conversely, PixelCNN is good at capturing the local features, but often fails to generate globally coherent images and has no latent variables. Then, successfully trained PixelCNN-VAEs would generate high-fidelity data and induce latent variable $z$ , which maintains only the global information by discarding the local information (Gulrajani et al., 2017).
+
+PixelCNN-VAE can be interpreted as a structured VAE in which $x _ { t }$ is generated from the global latent variable $z$ and local variable $s _ { t }$ as follows. First, the autoregressive decoder is expressed as $p ( x _ { \le T } | z ) \ = \ \Pi _ { t = 1 } ^ { T } p ( x _ { t } | z , x _ { < t } )$ . This means that for every timestep $t .$ , $x _ { t }$ is sampled from $p ( x _ { t } | \boldsymbol { z } , \boldsymbol { x } _ { < t } )$ using previous observations $x _ { < t }$ and the latent variable $z$ . Secondly, we assume that the decoder can be decomposed as $p ( x _ { t } | \boldsymbol { z } , \boldsymbol { x } _ { < t } ) = p ( x _ { t } | \boldsymbol { z } , f ( \boldsymbol { x } _ { < t } ) )$ using a neural network $f$ (PixelCNN). Finally, we introduce a random variable $s _ { t }$ and its distribution ${ \overline { { q } } } ( s _ { t } | x _ { < t } ) = p ( s _ { t } | x _ { < t } ) : =$ $\delta ( s _ { t } - f ( x _ { < t } ) )$ , where $\delta$ denotes the Dirac’s delta, and $q ( s _ { t } | x _ { < t } ) \ = \ p ( s _ { t } | x _ { < t } )$ is employed to simplify the notation. With this notation, the decoder can be decomposed as $p ( x _ { t } | \boldsymbol { z } , \boldsymbol { x } _ { < t } ) \ =$ $p ( x _ { t } | \boldsymbol { z } , f ( \boldsymbol { x } _ { < t } ) ) \ = \ p ( x _ { t } | \boldsymbol { z } , s _ { t } ) p ( s _ { t } | \boldsymbol { x } _ { < t } )$ (details on the practical decomposition using PixelCNN have been provided in Appendix B). Thus, $x _ { t }$ can be regarded to be generated from $z$ and $s _ { t }$ , which is sampled from $p ( s _ { t } | x _ { < t } )$ (see, Figure 2-(b)). Furthermore, the ELBO is given as follows:
+
+$$
+{ \mathcal { L } } _ { \mathrm { A R M } } = - \ \mathrm { R e c o n } - \mathrm { K L } ( z ) .
+$$
+
+# 2.2 MUTUAL INFORMATION-MAXIMIZING REGULARIZATION FOR SEQUENTIAL VAES
+
+Despite the intentional data generating process of the sequential VAEs, the global latent variable $z$ often becomes uninformative. To alleviate this issue, MI-maximizing regularization methods are often employed to encourage $z$ to have $x$ information. Note that, here we consider the MI defined by the encoder (which corresponds to the representational MI in Alemi et al. (2018)):
+
+$$
+I ( x ; z ) = \mathbb { E } _ { p _ { d } ( x ) q ( z | x ) } \big [ \log \frac { p _ { d } ( x ) q ( z | x ) } { p _ { d } ( x ) q ( z ) } \big ] ,
+$$
+
+A representative example of the MI-maximizing regularization is $\beta$ -VAE, which was shown to work well in Alemi et al. (2018) and used as a baseline in He et al. (2019) (other methods are presented in Section 4). Because the ELBO (Eq. 1) contains a positive lower bound and a negative upper bound of $I ( x ; z )$ , the MI can be controlled by balancing the two terms using a weighting parameter $\beta$ . The concrete $\beta$ -VAE objectives for DSAE and PixelCNN-VAE are:
+
+$$
+\begin{array} { r l } & { \mathcal { V } _ { \mathrm { S S M } } : = - \mathrm { R e c o n } - \beta \mathrm { K L } ( z ) - \mathrm { K L } ( s ) , } \\ & { \mathcal { V } _ { \mathrm { A R M } } : = - \mathrm { R e c o n } - \beta \mathrm { K L } ( z ) . } \end{array}
+$$
+
+Alemi et al. (2018) use $\beta < 1$ to regularize $I ( x ; z )$ to be large, although $\beta$ -VAE was originally invented to encourage the independence of each dimension of $z$ with $\beta > 1$ by Higgins et al. (2017).
+
+# 3 PROPOSED METHOD
+
+# 3.1 DECOMPOSITION OF MUTUAL INFORMATION
+
+Sequential VAEs with a global latent variable $z$ can in principle uncover global representation of data by exploiting its structured data generating process. Previous studies for the sequential VAEs have further regularized mutual information $I ( x ; z )$ to be large in order to alleviate posterior collapse (PC) (further discussed in Section 4). Unfortunately, the MI maximization is insufficient to uncover the global factor of variations, because it cannot control the type of information going into $z$ . More specifically, as indicated in Section 1, the MI is decomposed as follows:
+
+$$
+I ( x ; z ) = I ( x ; z ; s ) + I ( x ; z | s ) = I ( z ; s ) - I ( z ; s | x ) + I ( x ; z | s ) .
+$$
+
+Simply maximizing $I ( x ; z )$ can increase $I ( z ; s )$ in the right-hand side, which is observed in our preliminary experiment in Appendix A. When $I ( z ; s )$ becomes large, $z$ is likely to have redundant local information, or conversely $s$ becomes to have global information. Which phenomenon occurs could depend on the network architecture, as discussed in Appendix C. In both cases, the performance of downstream tasks, e.g., classification from $z$ to the labels, or controlling the global characteristics of the decoder output using $z$ , is likely to be degraded. Furthermore, as well as the MI $I ( x ; z )$ , this MI $I ( z ; s )$ is defined by the encoder distribution $q ( z , s )$ (Appendix E). Then, although the graphical model of DSAE is designed such that $z$ and $s$ are independent, $I ( z ; s )$ is not necessarily zero, i.e., $p ( z , s ) = p ( z ) p ( s )$ does not necessarily mean $q ( z , s ) \stackrel { - } { = } q ( z ) q ( s )$ .
+
+# 3.2 CONDITIONAL MUTUAL INFORMATION-MAXIMIZING REGULARIZATION
+
+Considering the limitations of MI regularization, we need a method that can encourage both (i) the increasing of $I ( x ; z )$ to prevent $z$ from becoming uninformative, and (ii) the decreasing of $I ( z ; s )$ to prevent $z$ and $s$ from having information that is irrelevant to the global and local structure, respectively. Therefore, we propose maximizing the following objective as a regularization approach:
+
+$$
+I ( x ; z ) - \alpha I ( z ; s ) .
+$$
+
+As $I ( z ; s )$ measures the mutual dependence between $s$ and $z$ , minimizing $I ( z ; s )$ encourages $z$ and $s$ not to have redundant information. Then, the induced global variable $z$ would have more $x$ information, while $z$ and $s$ maintains only the global and local information, respectively. The $\alpha$ is a weighting parameter for balancing the two terms. In practice, we found that $\alpha = 1$ works reasonably; therefore, we used $\alpha = 1$ for the reminder of the study.
+
+Furthermore, it is noteworthy that our method is closely related to CMI estimation. Specifically, when assuming $\alpha \geq 1$ ,
+
+$$
+I ( x ; z | s ) = I ( x ; z ) - I ( z ; s ) + I ( z ; s | x ) \geq I ( x ; z ) - \alpha I ( z ; s ) = : I _ { \mathrm { C M I } ^ { \prime } } .
+$$
+
+It indicates that $I _ { \mathrm { C M I ^ { \prime } } } = I ( x ; z ) - \alpha I ( z ; s )$ is equal to the lower bound of CMI (further discussed in Appendix D). CMI is known to be useful in selecting the features that are both individually informative and two-by-two weakly dependant (Fleuret, 2004). Therefore, maximizing $I ( x ; z | s )$ as regularization would make the features $z$ and $s$ informative but disentangled.
+
+Then, we present one of the tractable instances to estimate ${ \cal I } _ { \mathrm { C M I ^ { \prime } } }$ . A simple way to estimate ${ \cal I } _ { \mathrm { C M I ^ { \prime } } }$ may be to consider $I ( x ; z )$ and $I ( z ; s )$ independently; however, it must approximate both, $I ( x ; z )$ and $I ( z ; s )$ , which may complicate optimization. Fortunately, when we assume $\alpha = 1$ , we can reduce the number of terms to be approximated to only one, utilizing the $\beta$ -VAE formularization. First, we express ${ \cal I } _ { \mathrm { C M I ^ { \prime } } }$ as follows (the derivation is given in Appendix F):
+
+$$
+I _ { \mathrm { C M I ^ { \prime } } } = I ( x ; z ) - I ( z ; s ) = \mathbb { E } _ { p _ { d } ( x ) } \big [ D _ { \mathrm { K L } } ( q ( z | x ) | | p ( z ) ) \big ] - D _ { \mathrm { K L } } ( q ( z , s ) | | p ( z ) q ( s ) ) .
+$$
+
+The first term is the same as $\operatorname { K L } ( z )$ in Eqs. 5 and 6, and is used for the $\beta$ -VAE formularization. The second term is the upper bound of $I ( z ; s )$ because $D _ { \mathrm { K L } } ( q ( \boldsymbol { z } , \boldsymbol { s } ) | | p ( \boldsymbol { z } ) q ( \boldsymbol { s } ) ) = I ( \boldsymbol { z } ; \boldsymbol { s } ) +$ $D _ { \mathrm { K L } } ( q ( \boldsymbol { z } ) | | p ( \boldsymbol { z } ) )$ .
+
+Because the second term is difficult to calculate analytically, we estimate it using the DRT (Nguyen et al., 2008; Sugiyama et al., 2012), as performed in standard generative adversarial networks (Mohamed & Lakshminarayanan, 2017). By introducing the labels $y = 1$ for samples from $q ( z , s )$ and $y = 0$ for those from $p ( z ) q ( s )$ , we re-express these distributions in conditional form, i.e., $q ( z , s ) : = p ( z , s | y = 1 )$ and $p ( \bar { z } ) \dot { q } ( s ) : = p ( \bar { z } , \bar { s } | y = 0 )$ . The density ratio betwwen $q ( z , s )$ and $p ( z ) q ( s )$ can be computed using these conditional distributions as follows:
+
+$$
+\frac { q ( z , s ) } { p ( z ) q ( s ) } = \frac { p ( z , s | y = 1 ) } { p ( z , s | y = 0 ) } = \frac { p ( y = 1 | z , s ) } { p ( y = 0 | z , s ) } ,
+$$
+
+where we used Bayes’ rule and assumed that the marginal class probabilities are equal, i.e. $p ( y =$ $0 ) = p ( y = 1 )$ . Here, $p ( y | z , s )$ can be approximated with a discriminator $D ( z , s )$ , which outputs $D = 1$ when $z , s \sim _ { i . i . d }$ . $q ( z , s )$ , and $D = 0$ when $z , s \sim _ { i . i . d }$ . $q ( s ) p ( z )$ . Then, Eq. 10 can be approximated as follows:
+
+$$
+I _ { \mathrm { C M I ^ { \prime } } } \approx \mathbb { E } _ { p _ { d } ( x ) } [ D _ { \mathrm { K L } } ( q ( { \boldsymbol { z } } | { \boldsymbol { x } } ) | | { \boldsymbol { p } } ( { \boldsymbol { z } } ) ) ] - \mathbb { E } _ { q ( { \boldsymbol { z } } , s ) } \big [ \log \frac { D ( { \boldsymbol { z } } , s ) } { 1 - D ( { \boldsymbol { z } } , s ) } \big ] = : I _ { \mathrm { C M I - D R T } } .
+$$
+
+We parameterize $D ( z , s )$ with a DNN and train it alternately with the VAE objectives. Specifically, we train $D$ to maximize the following objective with Monte Carlo estimates:
+
+$$
+\begin{array} { r } { \mathbb { E } _ { q ( z , s ) } [ \log D ( z , s ) ] + \mathbb { E } _ { p ( z ) q ( s ) } [ \log \bigl ( 1 - D ( z , s ) \bigr ) ] . } \end{array}
+$$
+
+Now, we can introduce the concrete objectives of the DSAEs and PixelCNN-VAEs with a CMI regularization term. Adding $I _ { \mathrm { C M I - D R T } }$ as a regularization term to Eqs. 2 and 3 with a weighting parameter $\gamma$ , we obtain the objective functions of our proposed method that need to be maximized:
+
+$$
+\begin{array} { r l } & { \mathcal { I } _ { \mathrm { S S M } } : = \mathcal { L } _ { \mathrm { S S M } } + \gamma I _ { \mathrm { C M I - D R T } } = - \mathrm { R e c o n - K L } ( s ) - ( 1 - \gamma ) \mathrm { K L } ( z ) - \gamma I ^ { \prime } ( z ; s ) , } \\ & { \mathcal { I } _ { \mathrm { A R M } } : = \mathcal { L } _ { \mathrm { A R M } } + \gamma I _ { \mathrm { C M I - D R T } } = - \mathrm { R e c o n - } ( 1 - \gamma ) \mathrm { K L } ( z ) - \gamma I ^ { \prime } ( z ; s ) , } \\ & { \quad \quad \quad \mathrm { w h e r e } I ^ { \prime } ( z ; s ) = \mathbb { E } _ { q ( z , s ) } [ \log \displaystyle \frac { D ( z , s ) } { 1 - D ( z , s ) } ] . } \end{array}
+$$
+
+Considering $( 1 - \gamma ) \mathrm { K L } ( z )$ is equivalent to the weighting technique in $\beta$ -VAE (note that $1 - \gamma = \beta$ , and see Eqs. 5 and 6), the proposed method consists of the $\beta$ -VAE objective and $- I ^ { \prime } ( z ; s )$ . As noted in Section 2.2, $\beta$ -VAE is effective for alleviating PC. However, because $\beta$ -VAE alone is insufficient for decreasing the redundancy of $z$ and $s$ , minimizing $I ^ { \prime } ( z , s )$ is employed.
+
+Finally, we discuss alternative choices to estimate the second term of Eq. 10. While we chose to approximate the term with a discriminator, it can also be approximated with other distance such as maximum mean discripancy (MMD), or it can be minimized via Stein variational gradient (see, Zhao et al. (2019)). However, a weakness of these methods is that they are difficult to apply efficiently in high dimensions. Unfortunately, because the second term treats the random variable $[ z , s _ { 1 } , . . . , s _ { T } ]$ , the dimension size becomes high when $T$ is large. On the other hand, adversarial training requires only one assumption, i.e., $D$ perfectly approximates the true density ratio. In practice, while this assumption does not always hold true (Moyer et al., 2018; Iwasawa et al., 2020), it is also empirically known that original objectives (in our case, minimizing $D _ { \mathrm { K L } } ( q ( \boldsymbol { z } , \boldsymbol { s } ) | | p ( \boldsymbol { z } ) q ( \boldsymbol { s } ) ) )$ can be reasonably achieved (Ganin et al., 2016). In addition, various studies (e.g., Miyato et al. (2018); Iwasawa et al. (2020)) have proposed techniques to improve the robustness of adversarial training, and it has been shown to scale to high dimensions when carefully designed (Brock et al., 2019).
+
+# 4 RELATED WORKS
+
+This study is closely related to the literature on disentangled representation. Locatello et al. (2019) claimed that pure unsupervised disentangling (Chen et al., 2016; Higgins et al., 2017; Kim & Mnih, 2018) is fundamentally impossible, whereas using rich supervision (Kulkarni et al., 2015) can be costly. Thus, the use of inductive bias or weak supervision (Shu et al., 2020) has been encouraged. The assumption that data are generated from global and local factors is a representative example of the inductive bias. Such data generating process can be probabilistically expressed by the sequential VAEs with a global latent variable. Then, the sequential VAEs have been studied for disentangling styles and topics of texts (Bowman et al., 2016), object identities from the detailed textures of images (Chen et al., 2017), content and motion of movies (Hsieh et al., 2018), and the speaker and linguistic information of speeches (Hsu et al., 2017; Yingzhen & Mandt, 2018). Although this paper focused on DSAE and PixelCNN-VAE as examples, our method could be also combined with them.
+
+Unfortunately, the design of the structured data generating processes alone is often insufficient to learn the global features. To address this issue, Bowman et al. (2016); Chen et al. (2017) initially proposed to weaken the decoder because PC often occurs when using highly expressive decoders. Subsequently, various methods have been proposed to control the MI $I ( x ; z )$ with a regularization term, which does not require problem-specific architectural constraints of Bowman et al. (2016); Chen et al. (2017). Concrete examples of MI-maximizing regularization methods are as follows: InfoVAE: (Zhao et al., 2019) estimates $I ( x ; z )$ using the MMD or adversarial training. $\beta$ -VAE: Alemi et al. (2018) proposed targeting a specific rate (the KL term value) via $\beta$ -VAE , and observed that the objective with $\beta < 1$ produces solutions to alleviate PC. $\beta$ -VAE is a simpler than InfoVAE since it does not require an approximation of $I ( x ; z )$ . Auxiliary loss: (Lucas & Verbeek, 2018) uses the auxiliary tasks of predicting $x$ from $z$ , which approximates the minimization of conditional entropy $H ( x | z )$ . The minimization of $H ( x | z )$ is equivalent to maximizing $I ( x ; z )$ because the data entropy $H ( x )$ is constant. Discriminative objective: (Hsu et al., 2017) predicts a sequence index from $z$ , which also approximates $H ( x | z )$ minimization in the finite sample case.
+
+Various studies have attemted to separate relevant from irrelevant information via informationtheoretic regularization. Namely, the studies in the literature regarding domain-invariant representation proposed to learn the invariant representation using adversarial training (Ganin et al., 2016; Xie et al., 2017; Liu et al., 2018), variational information bottleneck frameworks (Moyer et al., 2018), or Hilbert-Schmidt independence criterion (Jaiswal et al., 2019). Our regularization term of minimizing $I ( z ; s )$ is inspired and similar to these studies; however, it differs in considering PC at the same time (i.e., maximizing $I ( x ; z ) )$ . Also, the separation could be achieved by the design of network architectures, as was performed in VQ-VAE2 (Razavi et al., 2019). Our proposal is the regularization term and orthogonal to such architecture choices.
+
+Also, our analysis is similar to that of Moyer et al. (2018), but differs in two ways. Firstly, while the phenomenon that ”large $I ( x ; z )$ results in large $I ( z ; s ) ^ { \prime }$ was discussed by them, the whole mechanism that ”MI-maximizing regularization for alleviating PC has a negative side-effect to increase $I ( z ; s ) ^ { , }$ ” has been overlooked in the sequential VAE community. Secondly, by explicitly considering the relationship between the two latent variables $z$ and $s$ , our analysis is able to highlight a new problem. For example, Moyer et al. (2018) consider the relationship between the latent variable $z$ and the observed nuisance factor $s$ . Then, their focus is only on removing the redundant information from $z$ . On the other hand, our analysis highlights the need to consider removing the redundant information from $s$ at the same time as removing the redundant information from $z$ . Although the former has been overlooked, it is an important issue in applications such as voice conversion.
+
+Some studies have also proposed methods for alleviating PC, which are complementary to MI maximization. Lucas et al. (2019) argued that the variance of the decoder influences the stability of local stationary points corresponding to PC. He et al. (2019) proposed a method that remedies ill-trainingdynamics. Our study differs in aiming at obtaining informative and disentangled representation with sequential VAEs, although they could, in principle, be combined with our method.
+
+From technical perspective, our work is also related to a feature selection technique based on CMI (Fleuret, 2004). CMI is known to be useful in selecting the features that are both individually informative and two-by-two weakly dependant. Then, the CMI-based technique is different from the MI-based one in considering the independence of the features. Moreover, it is different from previous studies for disentangled representation learning, e.g., Higgins et al. (2017); Kim & Mnih (2018); Liu et al. (2018) control only the independence of latent factors. Also, Mukherjee et al. (2019) first proposed the estimation of CMI using DNNs; however, our method is different in utilizing the encoder distribution of VAEs to improve the estimation (Zhao et al., 2019; Poole et al., 2019).
+
+# 5 EXPERIMENTS
+
+# 5.1 SETTINGS
+
+We performed experiments to confirm the effect of regularizing both $I ( x ; z )$ and $I ( z ; s )$ for learning good representation, using DSAE and PixelCNN-VAE as representative examples of the sequential VAEs. We used the speech corpus TIMIT (Garofolo et al., 1992) for the DSAE, and evaluated representation quality using a speaker verification task, as was performed in previous studies (Hsu et al., 2017; Yingzhen & Mandt, 2018). For PixelCNN-VAE, we trained the VAE with a 13-layer PixelCNN decoder on the statically binarized MNIST and Fashion-MNIST (Xiao et al., 2017) datasets. Using the trained models, we performed linear classification from $z$ to class labels to evaluated representation quality, as was performed in (Razavi et al., 2019), and then evaluated the ability of controlled generation. $z$ , which has a dimensional size of 32, was concatenated with the feature map outputted from the fifth layer of the PixelCNN (which corresponds to $s$ , see Appendix B), and was passed to the sixth layer. Further details are given in Appendix G.
+
+As the proposed method, we employed the objective functions $\mathcal { I } _ { \mathrm { S S M } }$ and $\mathcal { I } _ { \mathrm { A R M } }$ in Eqs. 13 and 14 (denoted as CMI-VAE). We implemented a discriminator $D$ as a CNN that receives $s$ and $z$ as inputs (Appendix I), and trained it alternately with the VAEs. As baseline methods, we employed $\beta$ -VAE (see, Section 2.2). The objectives of $\beta$ -VAE are given in Eqs. 5 and 6, and are equal to CMI-VAE, except for not having the $I ^ { \prime } ( z ; s )$ term. Moreover, we employed the regularization method proposed in Makhzani & Frey (2017); Zhao et al. (2019), which directly estimates and maximizes $I ( x ; z )$ with adversarial training (denoted as MI-VAE). In the method, $I ( x ; z )$ is added to ELBO (Eqs. 2 and 3) as a regularization term, along with a weighting term $\gamma$ (details can be found in Appendix H). The performances of the models were verified by changing the value of $\gamma$ .
+
+# 5.2 SPEAKER VERIFICATION WITH DISENTANGLED SEQUENTIAL AUTOENCODERS
+
+For a quantitative assessment of the global representation of DSAE, we evaluate whether $z$ can uncover the speaker individualities, which are the global features of speech. Specifically, we extract $z$ and $s _ { \leq T }$ from the test utterances using the mean of the encoders of the learned DSAE. Subsequently, we performed speaker verification by measuring the cosine similarity of the variables and evaluated the equal error rate (EER). Here, EER is measured for both $z$ and $s$ (denoted as $\operatorname { E E R } ( z )$ and EER(s), respectively), and $s _ { \leq T }$ is averaged over each utterance prior to its measurement. A lower $\operatorname { E E R } ( z )$ is preferable because it indicates that the model has an improved global representation, containing sufficient information of the speakers in a linear-separable form. Furthermore, a higher EER(s) is preferable because it indicates that $s$ does not have the redundant speaker information. In addition, we report $\operatorname { K L } ( z )$ (see, Eq. 2), which approximates the amount of information in $z$ .
+
+Table 1 presents the values of $\operatorname { K L } ( z )$ and EER for the vanilla DSAE, $\beta$ -VAE, and CMI-VAE. Note that our results for vanilla DSAE differ from those reported in Yingzhen & Mandt (2018) $( \mathrm { D S A E ^ { * } }$ in the table), which may be due to differences in the unreported training settings. The table presents that (1) a lower $\gamma$ (such as 0) provides a lower EER(s), which indicates that $s$ have global information instead of $z$ owing to PC, without regularizing $I ( x ; z )$ . Furthermore, (2) given a fixed $\gamma$ , CMI-VAE consistently achieved a lower $\operatorname { E E R } ( z )$ and a higher EER(s) while having the same level of $\operatorname { K L } ( z )$ compared to $\beta$ -VAE. This indicates that regularizing $I ( z ; s )$ is complementary to MI-maximization ( $\beta$ -VAE), yielding a better $z$ and $s$ that have sufficient global or local information but are well compressed. Note that $\gamma \geq 0 . 8$ yields a higher $\operatorname { E E R } ( z )$ than $\gamma = 0 . 4$ , which may be due to the fact that the independence of each dimension of $z$ is worsened by increasing $\gamma$ , as indicated in Higgins et al. (2017), and the induced non-linear relation cannot be measured by the cosine similarity. In fact, $\gamma \geq 0 . 8$ presented a better performance in the voice conversion experiment in Appendix J, indicating that $z$ with $\gamma \geq 0 . 8$ has more global information, although the $\operatorname { E E R } ( z )$ is lower. Also, note that the $\operatorname { E E R } ( Z )$ reported in Hsu et al. (2017) is lower than the results for CMI-CAE here. However, we believe that our claim, ”regularizing $I ( x ; z )$ and $I ( z ; s )$ is complementary”, is defended even if we could not achieve state-of-the-art results.
+
+Table $1 \colon \ \mathrm { K L }$ term and EER values of DSAE trained using TIMIT. Each model was trained with a weight $\gamma$ . The $\uparrow$ and $\downarrow$ indicate that the purpose was to obtain a high and low score, respectively.   
+
+<table><tr><td>Model</td><td>Y</td><td>KL(z)</td><td>EER(z)↓</td><td>EER(s) ↑</td></tr><tr><td>DSAE*</td><td>0.00</td><td>-</td><td>4.82</td><td>18.89</td></tr><tr><td>DSAE(our implementation)</td><td></td><td>18.00</td><td>11.01</td><td>18.64</td></tr><tr><td>+ β-VAE</td><td>0.40 (β=0.6)</td><td>53.28</td><td>3.88</td><td>29.45</td></tr><tr><td>+ CMI-VAE</td><td></td><td>54.13</td><td>3.43</td><td>30.96</td></tr><tr><td>+ β-VAE</td><td>0.80 (β=0.2)</td><td>145.88</td><td>4.33</td><td>38.84</td></tr><tr><td>+ CMI-VAE</td><td></td><td>145.09</td><td>3.99</td><td>41.30</td></tr><tr><td>+ β-VAE</td><td>0.90(β=1e-) 1)</td><td>202.52</td><td>4.55</td><td>39.42</td></tr><tr><td>+ CMI-VAE</td><td></td><td>199.89</td><td>4.39</td><td>41.25</td></tr><tr><td>+ β-VAE</td><td>0.99 (β= 1e-2)</td><td>364.71</td><td>6.33</td><td>38.63</td></tr><tr><td>+ CMI-VAE</td><td></td><td>361.03</td><td>5.06</td><td>40.08</td></tr></table>
+
+# 5.3 VAES WITH PIXELCNN DECODER
+
+Unsupervised learning for image classification For a quantitative assessment of the representation $z$ of PixelCNN-VAEs, we performed a logistic regression from $z$ to the class labels $y$ on MNIST and Fashion-MNIST. Specifically, first, we extracted $z$ from 1000 training samples using the mean of $q ( z | x )$ , where each of the 10 classes had 100 samples, and trained the classifier with a total of 1000 samples. Then, we evaluated the acccuracy of the logistic regression (AoLR) on the test data. A high AoLR indicates that $z$ succeeds in capturing the label information in a linear-separable form.
+
+Figures 3(a) and 3(b) present AoLR for $\beta$ -VAE, MI-VAE, and CMI-VAE, along with the ELBO and $\operatorname { K L } ( z )$ . In the figures, an upper left curve indicates that the method balance better compression (low $\operatorname { K L } ( z ) )$ and high downstream task performance. As shown in the figures, given a fixed $\gamma$ , the AoLRs for CMI-VAE are consistently better than those for $\beta$ -VAE and MI-VAE, although all the methods have the same level of $\operatorname { K L } ( z )$ . This indicates that CMI-VAE can extract more global information when compressing data to the same size as $\beta$ -VAE does. Note that a small $\gamma$ (such as $\gamma = 0$ ) and very large $\gamma$ degrade the AoLRs, which may be attributed to the same reason as explained in Section 5.2. Furthermore, the AoLRs of MI-VAE are lower than those of $\beta$ -VAE, which may be due to the adversarial training in MI-VAE causing optimization difficulities, as stated in Alemi et al. (2018).
+
+Controlled generation Most previous works (Yingzhen & Mandt, 2018; He et al., 2019) have primarily focused on evaluating the quality of global representation. However, a better representation does not necessarily improve the performance of the controlled generation, as Nie et al. (2020) claimed. Then, to evaluate the ability of the controlled generation, we propose a modified version of the classification accuracy score (CAS) (Ravuri & Vinyals, 2019), called mCAS. CAS trains a classifier which predicts class labels only from the samples generated from conditional generative models, and then evaluates the classification accuracy on real images, thus measuring the sample quality and diversity of the model. CAS is not directly applicable to non-conditional models such as PixelCNN-VAEs. Instead, mCAS measures the ability of the model to produce high quality, diverse, but globally coherent (i.e., belonging to the same class) images for a given $z$ .
+
+![](images/446accb0e66385757e545169fe366d600a90ef082518976659d404bde57a4d4a.jpg)  
+Figure 3: Comparison of CMI-VAE with $\beta$ -VAE and MI-VAE. Each maker for $\beta$ -VAE is annotated with the value of $\gamma$ . In the figures, an upper left curve is desirable because it shows the method balance better compression (low $\operatorname { K L } ( z ) )$ and high downstream task performance (AoLR and mCAS, see explanations in Section 5.3). Also, detailed results can be found in Appendix K.2.
+
+In mCAS, we first prepared 100 real images $\{ x _ { i } \} _ { i = 1 } ^ { 1 0 0 }$ , along with their class labels $\{ y _ { i } \} _ { i = 1 } ^ { 1 0 0 }$ , where each of the 10 classes had 10 samples. Then, using the trained VAEs, we encoded each $x _ { i }$ into $z _ { i }$ , and decoded $z _ { i }$ to obtain 10 images $\{ \hat { x } _ { i , j } \} _ { j = 1 } ^ { 1 0 }$ for every $z _ { i }$ , thereby resulting in 1000 generated images (sample images $\hat { x }$ can be found in Appendix K.3). Finally, we trained the logistic classifier with the pairs $\{ ( \hat { x } _ { i , j } , y _ { i } ) | i \in \{ 1 , . . . , 1 0 0 \} , j \in \{ 1 , . . . , 1 0 \} \}$ and evaluated the performance on real test images. Intuitively, when the decoder ignores $z$ , the generated samples might belong to a class different from the original ones, which produces label errors. Moreover, when $z$ has excessive information regarding $x$ and the VAE resembles an identity mapping, the diversity of the generated samples decreases (recall that 10 samples are generated for every $z _ { i }$ ), which induces overfitting of the classifier. Therefore, to achieve a high mCAS, $z$ should capture only the global (label) information.
+
+Figure 3(c) and 3(d) compares the mCAS along with $\operatorname { K L } ( z )$ on MNIST and Fashion-MNIST. In addition, the black horizontal line indicates the classification accuracy when the classifier is trained on 100 real samples $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { 1 0 0 }$ , and evaluated on real test images, which will be referred to as the baseline score. The following can be observed from the figures: (1) The mCAS of the three methods outperformed the baseline score, despite using only 100 labeled samples, as well as in the baseline score, indicating that properly regularized PixelCNN-VAEs could be used for data augmentation. (2) As expected, a significantly low $\operatorname { K L } ( z )$ gives a low mCAS because the decoder of the VAE does not utilize $z$ . Moreover, a significantly high $\operatorname { K L } ( z )$ also tends to degrade mCAS, because the decoder might resemble a one-to-one mapping from $z$ to $x$ and therefore, degrade the diversity. This phoenomenon can also be observed in the sample images in appendix K.3: there seems to be little diversity in samples drawn from $\beta$ -VAE and CMI-VAE with $\gamma = 0 . 6$ . (3) The curves for CMI-VAE are consistently left to those for $\beta$ -VAE, indicating that regularizing $I ( z ; s )$ is also complementary to regularizing $I ( x ; z )$ at the controlled generation.
+
+# 6 DISCUSSIONS AND FUTURE WORKS
+
+Based on the experimental results, it was confirmed that regularizing $I ( z ; s )$ is complementary to regularizing $I ( x ; z )$ , and leads to an improvement in the learning of global latent variables. Here, we chose to extend $\beta$ -VAE to construct the proposed objective function because we believe $\beta$ -VAE is the simplest MI-maximization method that requires fewer hyperparameters, widely used in (sequential) VAE community (e.g., He et al. (2019); Alemi et al. (2018)). However, other MI estimation methods, such as discriminative objective and MMD-based InfoVAE, can be extended to CMI regularization by the addition of the $I ( z ; s )$ minimization term (see, Section 3.2). Incorporating such MI maximization methods into the estimation of CMI, or stabilizing adversarial training with some technique (Miyato et al., 2018) might improve the performance, and this remains an issue to be addressed in a future work. Also, it would be interesting to approximate $I ( x ; z )$ and $I ( z ; s )$ separately, and tune the strength of them independently. Future studies may also apply the proposed method to encourage the learning of the representation that captures the global factors of the environment such as maps, to support reinforcement learning, as suggested in Gregor et al. (2019).
+
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+![](images/a5ab5dfcfcd1246d64e3ee4400e3f212f55a6a9d7aed2dc419626f7d3d88794f.jpg)  
+Figure 4: $\hat { I } ( z ; s )$ values of DSAE trained with $\beta$ -VAE objective.
+
+APPENDIX
+
+# A EMPIRICAL EVIDENCE OF LIMITATION OF MI-MAXIMIZING REGULARIZATION
+
+In this section, we evaluate the $I ( z ; s )$ values of DSAE, and the MI-maximizing regularization is shown to result in increasing it, contrary to the intention of learning the disentangled global features. Note that the $I ( z ; s )$ of DSAE ideally equals to zero because its graphical model is designed such that $z$ and $s$ are independent. However, $I ( z ; s )$ is not necessarily zero because the representational MI is considered (see, also, Section 3.1). Because $I ( z ; s )$ is intractable, we used the value of $\hat { I } ( z ; s )$ estimated with DRT was used in a similar manner to Section 3.2. Namely, we used the following equation:
+
+$$
+{ \frac { q ( z , s ) } { q ( z ) q ( s ) } } = : { \frac { p ( z , s | y = 1 ) } { p ( z , s | y = 0 ) } } = { \frac { p ( y = 1 | z , s ) } { p ( y = 0 | z , s ) } } , { \mathrm { ~ w h e r e ~ } } p ( y = 0 ) = p ( y = 1 ) = { \frac { 1 } { 2 } } .
+$$
+
+$p ( y | z , s )$ can be approximated with a discriminator $D ( z , s )$ that outputs $D = 1$ when $z , s \sim _ { i . i . d }$ $q ( z , s )$ , and $D = 0$ when $z , s \sim _ { i . i . d . } q ( s ) q ( z )$ . Then, $I ( z ; s )$ can be approximated as follows:
+
+$$
+I ( z ; s ) \approx \mathbb { E } _ { p _ { d } ( x ) q ( z , s | x ) } \left[ \log \frac { D ( z , s ) } { 1 - D ( z , s ) } \right] = : \hat { I } ( z ; s ) .
+$$
+
+$D ( z , s )$ is parameterized with a DNN, and trained alternately with the VAEs’ objectives. The other training settings are the same as those in Section 5.2, and can be found in Appendix G.
+
+Figure 4 presents $\hat { I } ( z ; s )$ values of DSAE trained with the $\beta$ -VAE objective (Eqs. 5 and 6), where $\operatorname { K L } ( z )$ is reweighted with parameter $\beta = 1 - \gamma$ . The figure indicates that if $\beta \geq 0 . 2$ , a smaller $\beta$ results in a larger $\hat { I } ( z ; s )$ . This indicates that when we simply regularize $I ( x ; z )$ to be large using $\beta$ -VAE objective, $z$ and $s$ become to have redundant information. In contrast, when we do not regularize $I ( x ; z )$ (i.e., $\beta = 1$ ), $\hat { I } ( z ; s )$ becomes small; however, $z$ also becomes uninformative regarding $x$ .
+
+One may notice that even if $\beta$ is smaller than 0.2, $\hat { I } ( z ; s )$ will not increase. This is probably due to the following reason. Firstly, in DSAE, it is difficult for the $z$ to have redundant local information due to the architectural constraint (Appendix C). Then, the $\hat { I } ( z ; s )$ might become larger when $s$ , not $z$ , has redundant information. That is, regardless of the value of $\beta$ , $s$ may retain a certain degree of global information due to PC (which is supported by the experiments in Section 5.2). Therefore, when $z$ has global information as $\beta$ decreases, $I ( z ; s )$ increases. However, once $z$ has enough global information, there is no room for $I ( z ; s )$ to increase beyond a certain point.
+
+# B PRACTICAL DECOMPOSITION OF AUTOREGRESSIVE DATA GENERATINGPROCESS
+
+Here we show the practical decomposition of the autoregressive data generating process $\Pi _ { t = 1 } ^ { T } p ( x _ { t } | \boldsymbol { z } , \boldsymbol { x } _ { < t } ) = \dot { \Pi _ { t = 1 } ^ { T } } p ( x _ { t } | \boldsymbol { z } , \boldsymbol { s } _ { t } ) q ( \dot { s } _ { t } | \boldsymbol { x } _ { < t } )$ for PixelCNN. We consider the 13-layer PixelCNN used in He et al. (2019), which has five $( 7 \times 7 )$ -kernel-size layers, followed by, four $( 5 \mathrm { ~ x ~ } 5 )$ layers, and then four $( 3 \mathrm { ~ x ~ } 3 )$ layers. Each layer has 64 feature maps with dimensions $2 8 \times 2 8$ dimensions. The latent variable $z$ is extracted by an encoder, linearly transformed into (28, 28, 4) feature maps, and then concatenated to the each layer of the PixelCNN feature maps after the sixth layer. We denote the output of the $i$ -th $( i \in \{ 1 , . . . , 1 3 \} )$ ) layer as $h _ { i , t }$ , where $t$ denotes the timestep ( $\mathbf { \dot { x } }$ and y coordinates, and $t \in \{ 1 , . . . , 2 8 \times 2 \dot { 8 } = 7 8 4 \}$ ). Then, we can put $s _ { t } : = h _ { 6 , < t }$ and the decomposition $\Pi _ { t = 1 } ^ { T } p ( x _ { t } | \boldsymbol { z } , \boldsymbol { x } _ { < t } ) = \Pi _ { t = 1 } ^ { T } p ( x _ { t } | \boldsymbol { z } , \boldsymbol { s } _ { t } ) q ( \boldsymbol { s } _ { t } | \boldsymbol { x } _ { < t } )$ holds because only $h _ { 6 , < t }$ (not $h _ { 6 , \geq t , }$ ) are used to generate $x _ { t }$ with causal convolution.
+
+One might wonder whether the activations of the PixelCNN, which is the deterministic function of $x$ , can be treated as random variables. However, because we regularize $s$ only via minimizing $I ( z ; s )$ (Section 3.2), $s$ can be determined to be treated as random variables. specifically, $I ( z ; s )$ is defined by the joint distribution $\begin{array} { r } { q ( z ; s ) = \int p _ { d } ( x ) q ( z | x ) q ( s | x , z ) d x } \end{array}$ (see definition in Appendix E), where $q ( s | x , z ) = q ( s | x ) = \Pi _ { t = 1 } ^ { T } \delta ( s _ { t } - f ( x _ { < t } ) )$ would be integrated over a random variable $x$ . Therefore, $z$ and $s$ have no deterministic relation and $s$ can be meaningfully referred to as local latent variables. Furthermore, it is common to treat the activations of hidden layers as random variables and to consider their MI (or conditional entropy) in the literature on domain-invariant representation learning (Xie et al., 2017).
+
+Note that, the definition of $s$ is an important factor for the “control” of what will be learned in $z$ ; however, anything is acceptable as long as $s _ { t }$ has sufficiently large receptive fields. For example, Chen et al. (2017) proposed improving global representation $z$ by using smaller receptive fields for $q ( s _ { t } | \boldsymbol x _ { < t } )$ and constraining $s _ { t }$ to more local information. Although this architectural constraint can make $z$ informative, it requires weakening the expressiveness of PixelCNN and can degrade ELBO (Chen et al., 2017). By contrast, our method can be applied regardless of the size of the receptive fields because it prevents $s$ from having global information with an information theoretic regularization term. Therefore, the architectural change of Chen et al. (2017) was not employed and large receptive fields were used to balance sufficient ELBO and representation quality.
+
+# C DSAE AND PIXELCNN-VAE HAVE DIFFERENT ARCHITECTURAL CONSTRAINTS ON GLOBAL LATENT VARIABLES
+
+$z$ of DSAE and PixelCNN-VAE are imposed on different architectural constraints. In DSAE, the $z$ is constrained to have no local information. On the other hand, the $z$ of PixelCNN-VAEs has no such architectural constraints, although it is designed to capture global features via the structured data generating process. Here, we distinguish the structured data generating process from the architectural constraints: the former is the constraint based on the probabilistic graphical model, while the latter is the constraint based on the neural network structures. Specifically, the $z$ of DSAE is concatenated with $s _ { t }$ and feeded into the fully connected neural network decoder for all timestep $t \in \{ 1 , . . . , T \}$ , so the $z$ may have the same effects on each timestep $t$ . On the other hand, the $z$ of PixelCNN-VAE is linearly transformed into (28, 28, 4) feature maps, and then concatenated to the each layer of PixelCNN feature maps (see, Appendix B). Since the linear transformation creates the feature maps that depend on timesteps (x and y coordinates), it becomes easy for the $z$ to have different effects on each timestep $t$ . Note that, such linear transformation is commonly employed in previous studies of PixelCNN-VAEs (e.g., He et al. (2019)) in order to improve expressiveness of the decoder.
+
+Due to these architectural differences, different phenomena can be observed in DSAE and PixelCNN-VAE when using MI-maximizing regularization. First, both the models have in common that $I ( z ; s )$ would become larger when using the MI regularization (see, Section 3.1). However, in DSAE, it is likely that $s$ has redundant global features, not that $z$ has redundant local features, because it is difficult for the $z$ to have local information due to the architectural constraint. On the other hand, in PixelCNN-VAE, $z$ can have redundant local features. Then, these architectural differences would cause different problems in the controlled generation using DSAEs and PixelCNN-VAEs. DSAE can hopefully change speaker individualities while preserving the linguistic contents (i.e., perform voice conversion), by swapping the $z$ of two utterances and reconstructing them. However, if $s$ still contains speaker information due to the redundancy, the decoder can extract speaker information from either $s$ or $z$ and there is no guarantee that $z$ will be used (see, also, Appendix J). For PixelCNN-VAE, previous studies (Alemi et al., 2018; Razavi et al., 2019) have shown that by stochastically sampling $x$ from PixelCNN-VAE with a given $z$ , one can obtain images with different local patterns but similar global characteristics (e.g. color background, scale, and structure of objects). However, when $z$ has all (local and global) information, the diversity of the generated images would decrease, because the decoder resembles one-to-one mapping from $z$ to $x$ (see, also, Section 5.3).
+
+# D THE LOWER BOUND OF $I ( x ; z | s )$
+
+Here we derive Eq. 9 and discuss the approximation error between $I ( x ; z | s )$ and ${ \cal I } _ { \mathrm { C M I ^ { \prime } } }$ . Firstly, we can take the lower bound:
+
+$$
+I ( x ; z | s ) = I ( x ; z ) - I ( z ; s ) + I ( z ; s | x ) \geq I ( x ; z ) - I ( z ; s ) ,
+$$
+
+since the MI $I ( z ; s | x )$ is positive. Then, the lower bound $I ( x ; z ) - I ( z ; s )$ has approximation error $I ( z ; s | x )$ . Note that the error can be small under a particular condition. Namely, the error can be decomposed as:
+
+$$
+I ( z ; s | x ) = H ( z | x ) - H ( z | x , s ) .
+$$
+
+Here, both $H ( z | x )$ and $H ( z | x , s )$ is thought to be small when $x$ is high-dimensional data such as images, movies, and audios, because observing such $x$ would enable us to predict $z$ accurately. Also, empirically, it has been shown that the performance of inference model did not drop much even if the encoders of DSAE are decomposed into $q ( z , s | x ) = q ( z | x ) q ( s | x )$ (Yingzhen $\&$ Mandt, 2018), which indicates the error is small.
+
+In addition, as long as $\alpha \geq 1$ , the following condition holds:
+
+$$
+I ( x ; z ) - I ( z ; s ) \geq I ( x ; z ) - \alpha I ( z ; s ) ,
+$$
+
+because the MI $I ( z ; s )$ is positive. This approximation error becomes the smallest when $\alpha = 1$
+
+# E DEFINITION OF $I ( z ; s )$
+
+This paper considers the MI $I ( z ; s )$ defined by the encoder (which corresponds to representational MI in Alemi et al. (2018)). Namely,
+
+$$
+I ( z ; s ) = \mathbb { E } _ { q ( z , s ) } [ \log \frac { q ( z ) q ( s | z ) } { q ( z ) q ( s ) } ] ,
+$$
+
+where the joint distribution is $\begin{array} { r } { q ( z ; s ) : = \int p _ { d } ( x ) q ( z | x ) q ( s | x , z ) d x } \end{array}$ .
+
+# F DERIVING EQ. 10
+
+Here we present the deriviation of Eq. 10:
+
+$$
+\begin{array} { r l } { I ( x ; z ) - I ( z ; s ) = \mathbb { E } _ { q ( x , z ) } [ \log \displaystyle \frac { q ( z | x ) } { q ( z ) } ] - \mathbb { E } _ { q ( z , s ) } [ \log \displaystyle \frac { q ( z | s ) } { q ( z ) } ] } & { } \\ { = \mathbb { E } _ { q ( x , z , s ) } [ \log \displaystyle \frac { q ( z | x ) q ( z ) p ( z ) } { q ( z | s ) q ( z ) p ( z ) } ] } & { } \\ { = \mathbb { E } _ { q ( x , z ) } [ \log \displaystyle \frac { q ( z | x ) } { p ( z ) } ] - \mathbb { E } _ { q ( z , s ) } [ \log \displaystyle \frac { q ( z , s ) } { p ( z ) q ( s ) } ] } & { } \\ { = \mathbb { E } _ { p _ { d } ( x ) } \big [ D _ { \mathrm { K L } } ( q ( z | x ) | | p ( z ) ) \big ] - D _ { \mathrm { K L } } ( q ( z , s ) | | p ( z ) q ( s ) ) . } \end{array}
+$$
+
+# G DETAILS OF EXPERIMENTAL SETTINGS
+
+# G.1 DISENTANGLED SEQUENTIAL AUTOENCODER
+
+Data preprocessing We use the TIMIT data (Garofolo et al., 1992), which contains broadband 16kHz recordings of phonetically-balanced read speech. A total of 6300 utterances (5.4 hours) are presented with 10 sentences from each of the 630 speakers $70 \%$ male and $30 \%$ female). Garofolo et al. (1992) have originally split the data into train/test subset, and we further split the train subset into $90 \%$ of train and $10 \%$ of validation subset. We followed Hsu et al. (2017); Yingzhen & Mandt (2018) for data preprocessing: the raw speech waveforms are first split into sub-sequences of $2 0 0 \mathrm { m s }$ , and then preprocessed with sparse fast Fourier transform to obtain a 201 dimensional log-magnitude spectrum, with the window size $2 5 \mathrm { m s }$ and shift size $1 0 \mathrm { m s }$ . This results in $T = 2 0$ for the observation $x _ { 1 : T }$ .
+
+Optimization we follow Yingzhen & Mandt (2018) for model architecture, data preprocessing, and evaluation procedures. The dimensionality of $s _ { t }$ and $z$ were fixed at 64; we set $T = 2 0$ for the observation $x _ { \le T }$ . We used Adam optimizer with learning rate 2e-4 for the VAE and 2e-3 for the discriminator, and trained the models for 6000 epochs to get good convergence on the training set. The VAE architecture followed full model in Yingzhen & Mandt (2018), and the discriminator architecture is described in Appendix I. The discriminator is updated twice while the VAE is updated once. The results are averaged over three random seed trials.
+
+# G.2 PIXELCNN-VAE
+
+Data preprocessing We use the statically binarized version of MNIST and Fashion-MNIST datasets: each pixel value $\in [ 0 , 1 ]$ is binarized with the threshold 0.5. The datasets are originally split into train/test subsets, and we further split the train subsets into $80 \%$ of train and $20 \%$ of validation subsets.
+
+Optimization Regarding the optimization of VAEs, we used the Adam optimizer with a learning rate of 0.0001, trained for 300 epochs. We reported the values for the test data when the objective function for the validation data was maximized. Regarding the discriminators, we used the Adam optimizer with learning rate 0.001. The discriminator architecture is described in Appendix I, and is updated twice while the VAE is updated once. As for the PixelCNN architecture, see Appendix B.
+
+# H DETAILS OF MI-VAE IN OUR EXPERIMENTS
+
+We employ $I ( x ; z )$ maximization method proposed by Makhzani & Frey (2017); Zhao et al. (2019) as a baseline method in our experiment. Briefly, we add $I ( x ; z )$ to the standard VAE objectives as a regularization term with weighting term $\gamma$ .
+
+To estimate $I ( x ; z )$ , Makhzani & Frey (2017) utilize the follwing relation based on the density ratio trick:
+
+$$
+{ \frac { q ( z ) } { p ( z ) } } = : { \frac { p ( z | y = 1 ) } { p ( z | y = 0 ) } } = { \frac { p ( y = 1 | z ) } { p ( y = 0 | z ) } } ,
+$$
+
+where $\begin{array} { r } { p ( y = 0 ) = \frac { 1 } { 2 } } \end{array}$ and $\begin{array} { r } { p ( y = 1 ) = \frac { 1 } { 2 } } \end{array}$ . Then, although conditional probability $p ( y | z )$ cannot be obtained, it can be approximated with a discriminator $D ( z )$ , which outputs $D = 1$ when $z \sim _ { i . i . d }$ . $q ( z )$ and $D = 0$ when $z \sim _ { i . i . d . } p ( z )$ . Then, $I ( x ; z )$ can be approximated as follows:
+
+$$
+\begin{array} { l l } { I ( x ; z ) = \mathbb { E } _ { p _ { d } ( x ) } [ D _ { K L } [ q ( z | x ) | | p ( z ) ] - D _ { K L } ( q ( z ) | | p ( z ) ) ] } \\ { \approx \mathbb { E } [ D _ { K L } [ q ( z | x ) | | p ( z ) ] - \log \displaystyle \frac { D ( z ) } { 1 - D ( z ) } ] } \\ { \hfill } \end{array}
+$$
+
+$D ( z )$ is parameterized with some DNN, and trained alternately with VAEs’ objectives. Namely, $D$ is trained to maximize the following objective with Monte Carlo sampling:
+
+$$
+\begin{array} { r } { \mathbb { E } _ { q ( \boldsymbol { z } ) } [ \log D ( \boldsymbol { z } ) ] + \mathbb { E } _ { p ( \boldsymbol { z } ) } [ \log \left( 1 - D ( \boldsymbol { z } ) \right) ] . } \end{array}
+$$
+
+Finally, we introduce the concrete objective of PixelCNN-VAE with the regularization term $I _ { M I }$ . Adding $I _ { \mathrm { M I - D R T } }$ to the objective of PixelCNN-VAE, we obtain the objective functions of MI-VAE:
+
+$$
+\mathcal { W } _ { \mathrm { A R M } } : = \mathcal { L } _ { \mathrm { A R M } } + \gamma I _ { \mathrm { M I - D R T } } = - \mathrm { R e c o n } - ( 1 - \gamma ) \mathrm { K L } ( z ) - \gamma \mathbb { E } [ \log \frac { D ( z ) } { 1 - D ( z ) } ] .
+$$
+
+In short, the objective only differs from our CMI maximization method in that the discriminator is added on the purpose of minimizing $D _ { K L } ( q ( z ) | | p ( z ) )$ , while our method minimizes $D _ { K L } ( q ( \boldsymbol { z } , \boldsymbol { s } ) | | p ( \boldsymbol { z } ) q ( \boldsymbol { s } ) )$ and encourages disentanglement of $z$ and $s$ .
+
+# I DISCRIMINATOR SETTINGS
+
+We have used discriminators for CMI-VAE and MI-VAE (whose details can be found in Appendix H). For the discriminator of CMI-VAE, we first applied convolutional encoder and took mean pooling, obtaining the embedding of $s _ { 1 : T }$ . Then, in PixelCNN-VAE, we took innner product of the embedding and $z$ and treated it as logit of the discriminator. On the other hand, in DSAE, we took cosine similarity of the embedding and $z$ , multiplied the similarity by a learnable scale parameter, and treated it as logit of the discriminator. The encoder architectures for PixelCNN-VAE and DSAE are summarized as follows, with the format Conv (depth, kernel size, stride, padding):
+
+# PixelCNN-VAE
+
+• Input (28, 28, 1)   
+• Conv2D (256, 4, 2, 1)   
+• BatchNorm   
+• ReLU   
+• Conv2D (256, 4, 2, 1)   
+• BatchNorm   
+• ReLU   
+• Conv2D ( $z$ -dim, 4, 2, 1)
+
+# DSAE
+
+• Input (20, 201)   
+• Conv1D (256, 4, 2, 1)   
+• BatchNorm   
+• ReLU   
+• Conv1D (256, 4, 2, 1)   
+• BatchNorm   
+• ReLU   
+• Conv1D $z$ -dim, 4, 2, 1)
+
+The discriminator architecture for MI-VAE is summarized as follows, with the format Linear (input size, output size):
+
+• Input $z$ -dim)   
+• Linear $z$ -dim, 400)   
+• ReLU   
+• Linear (400, 1)   
+• Softmax
+
+# J VOICE CONVERSION EXPERIMENTS USING DISENTANGLED SEQUENTIALAUTOENCODER
+
+For a quantitative assessment of controlled generation by DSAE, we performed voice conversion and evaluated the models with a score similar to mCAS (see, Section 5.3), which we call VC-mCAS. First, we prepared 500 real speeches (spectrograms with $T = 2 0$ ) $\{ x _ { i } \} _ { i = 1 } ^ { 5 0 0 }$ , along with their gender labels (male or female) $\{ y _ { i } \} _ { i = 1 } ^ { 5 0 0 }$ , where each of the 2 classes had 250 samples. Then, we randomly created 250 pairs $\{ ( x _ { i } , x _ { j } ) | y _ { i } \neq y _ { j } \}$ , i.e., each pair consists of one male and one female speech. Using the trained DSAE, we encoded each $x$ into $z$ and $s$ , created the pairs $\{ ( z _ { i } , s _ { i } , z _ { j } , s _ { j } ) | y _ { i } \bar { \neq } y _ { j } \}$ , and decoded $z _ { i }$ and $s _ { j }$ $( z _ { j }$ and $s _ { i }$ ) to obtain $\hat { x } _ { i , j }$ $( \hat { x } _ { j , i } )$ , which ideally has the speaker characteristics of $x _ { i }$ and the linguistic contents of $x _ { j }$ . Thus, we obtain 500 generated samples, where each $\hat { x } _ { i , j }$ was labeled with $y _ { i }$ assuming that the characteristics that tend to depend on gender (such as pitch) were successfully converted. Finally, we trained a logistic classifier with the 500 pairs $\{ ( \stackrel { - } { x } _ { i , j } , y _ { i } ) \}$ and evaluated the performance on real test speeches. Note that, because the raw $\hat { x } _ { i , j }$ has an excessively high dimension $( 2 0 ( T ) \times 2 0 1$ (features)) for the logistic classifier, $\hat { x }$ was averaged over the timeaxis prior to its measurement. Intuitively, when the decoder ignores $z$ , the generated samples might belong to a class different from the original ones, which produces label errors. Therefore, to achieve a high VC-mCAS, $z$ should capture global information but $s$ should not. Aos, the generated samples should be realistic to reduce the domain gap between train (generated) and test (real) data.
+
+Table 2: VC-mCAS for $\beta$ -VAE and CMI-VAE.   
+
+<table><tr><td>Model</td><td>Y</td><td>VC-mCAS(mean)</td><td>VC-mCAS(max)</td></tr><tr><td>DSAE + β-VAE</td><td>0.4000</td><td>83.73</td><td>85.8</td></tr><tr><td>DSAE + CMI-VAE</td><td>0.4000</td><td>84.33</td><td>86.6</td></tr><tr><td>DSAE + β-VAE</td><td>0.8000</td><td>87.27</td><td>87.6</td></tr><tr><td>DSAE + CMI-VAE</td><td>0.8000</td><td>87.47</td><td>88.0</td></tr><tr><td>DSAE + β-VAE</td><td>0.9000</td><td>87.20</td><td>87.6</td></tr><tr><td>DSAE + CMI-VAE</td><td>0.9000</td><td>87.00</td><td>87.6</td></tr><tr><td>DSAE + β-VAE</td><td>0.9900</td><td>87.33</td><td>87.6</td></tr><tr><td>DSAE + CMI-VAE</td><td>0.9900</td><td>87.60</td><td>88.6</td></tr><tr><td>DSAE+β-VAE</td><td>0.9990</td><td>87.27</td><td>87.4</td></tr><tr><td>DSAE + CMI-VAE</td><td>0.9990</td><td>87.60</td><td>88.0</td></tr><tr><td>DSAE + β-VAE</td><td>0.9999</td><td>86.87</td><td>87.2</td></tr><tr><td>DSAE + CMI-VAE</td><td>0.9999</td><td>87.73</td><td>88.0</td></tr></table>
+
+Table 2 presents the values of VC-mCAS for the objectives of $\beta$ -VAE and CMI-VAE. Note that we report the mean and best scores within three random seed trials for each $\gamma$ . The table illustrates that given a fixed $\gamma$ , CMI-VAE nearly consistently achieved a higher VC-mCAS compared to $\beta$ -VAE, indicating that regularizing $I ^ { \prime } ( z ; s )$ is complementary to $\beta$ -VAE. Furthermore, although $\gamma = 0 . 8$ yields a higher $\operatorname { E E R } ( z )$ than those with $\gamma = 0 . 4$ in Table 1, it yields a higher VC-mCAS. Therefore, in addition to measuring EER, as was done in previous studies (Hsu et al., 2017; Yingzhen & Mandt, 2018), we claim that it is necessary to consider the performance of the controlled generation for evaluating the usefulness of the global representation.
+
+# K DETAILED EXPERIMENTAL RESULTS
+
+# K.1 DETAILED EXPERIMENTAL RESULTS FOR DSAE
+
+Table 3 presents the ELBO, KL, Recon, EER, and $\hat { I } ( z ; s )$ values of DSAE on TIMIT corpus. Regarding the estimation of $\hat { I } ( z ; s )$ , please refer to Appendix A. Also, note that $\operatorname { K L } ( z )$ approximates $I ( x ; z )$ because it upper bounds $I ( x ; z )$ , and has been used for the metric to assess whether a decoder ignores $z$ or not (Bowman et al., 2016; Alemi et al., 2018; He et al., 2019). Here, the ELBO, Recon, and $\mathrm { K L } ( S )$ values are not divided by $T = 2 0$ . Also, the Recon values can be nevative because the variance of our decoder are learnable parameters.
+
+As shown in the table, (i) given a fixed $\gamma$ , the two methods ( $\beta$ -VAE and CMI-VAE) have the same level of $\operatorname { K L } ( z )$ ; therefore, both the methods can be used to alleviate PC. (ii) On the other hand, given a fixed $\gamma$ , CMI-VAE achieved the lower $\hat { I } ( z ; s )$ values in most cases, suggesting that it facilitates the learning of good global representation. (iii) Finally, we have confirmed that even for large $\gamma$ , there is still reasonable reconstruction performance for the both methods.
+
+Table 3: The ELBO, KL, Recon, EER, and $\hat { I } ( z ; s )$ values of DSAE on TIMIT corpus. Each model was trained with a weighting parameter $\gamma$ . ”se” denotes standard error.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">~</td><td rowspan="2">ELBO mean</td><td rowspan="2">se</td><td rowspan="2">KL(z) mean</td><td rowspan="2"></td><td rowspan="2">KL(s) mean</td><td rowspan="2">se</td><td rowspan="2">Recon</td><td rowspan="2">mean</td><td rowspan="2">se mean</td><td rowspan="2">EER(z) se</td><td rowspan="2">EER(s)</td><td rowspan="2">se</td><td rowspan="2">1（z;8） mean</td><td rowspan="2">se</td></tr><tr><td>mean</td></tr><tr><td>DSAE</td><td>0.00</td><td>6299.08</td><td>14.06</td><td>18.00</td><td>0.09</td><td>495.92</td><td>4.45</td><td>-6813.00</td><td>13.37</td><td>11.01</td><td>0.52</td><td>18.64</td><td>1.04</td><td>1.61</td><td>0.04</td></tr><tr><td>DSAE + β-VAE</td><td>0.40</td><td>6289.48</td><td>11.72</td><td>53.28</td><td>0.97</td><td>483.40</td><td>5.73</td><td>-6826.16</td><td>15.46</td><td>3.88</td><td>0.15</td><td>29.45</td><td>0.29</td><td>2.60</td><td>0.01</td></tr><tr><td>DSAE+ CMI-VAE</td><td>0.40</td><td>6288.64</td><td>4.82</td><td>54.13</td><td>2.51</td><td>468.06</td><td>14.03</td><td>-6810.83</td><td>16.04</td><td>3.43</td><td>0.26</td><td>30.96</td><td>0.81</td><td>2.44</td><td>0.03</td></tr><tr><td>DSAE + β-VAE</td><td>0.80</td><td>6228.58</td><td>4.81</td><td>145.88</td><td>0.45</td><td>430.73</td><td>7.19</td><td>-6805.20</td><td>10.76</td><td>4.33</td><td>0.26</td><td>38.84</td><td>0.47</td><td>2.84</td><td>0.05</td></tr><tr><td>DSAE + CMI-VAE</td><td>0.80</td><td>6222.22</td><td>4.60</td><td>145.09</td><td>0.17</td><td>430.59</td><td>5.60</td><td>-6797.90</td><td>1.53</td><td>3.99</td><td>0.29</td><td>41.30</td><td>0.66</td><td>2.83</td><td>0.04</td></tr><tr><td>DSAE+β-VAE</td><td>0.90</td><td>6172.96</td><td>3.83</td><td>202.52</td><td>1.34</td><td>432.20</td><td>3.88</td><td>-6807.67</td><td>2.13</td><td>4.55</td><td>0.15</td><td>39.42</td><td>1.06</td><td>2.89</td><td>0.01</td></tr><tr><td>DSAE+CMI-VAE</td><td>0.90</td><td>6192.05</td><td>6.58</td><td>199.89</td><td>1.10</td><td>429.53</td><td>7.84</td><td>-6821.46</td><td>14.59</td><td>4.39</td><td>0.23</td><td>41.25</td><td>2.10</td><td>2.70</td><td>0.03</td></tr><tr><td>DSAE+ β-VAE</td><td>0.99</td><td>6019.10</td><td>11.52</td><td>364.71</td><td>2.08</td><td>433.59</td><td>9.40</td><td>-6817.40</td><td>19.39</td><td>6.33</td><td>0.34</td><td>38.63</td><td>1.18</td><td>3.27</td><td>0.15</td></tr><tr><td>DSAE + CMI-VAE</td><td>0.99</td><td>6031.71</td><td>5.78</td><td>361.03</td><td>1.85</td><td>434.55</td><td>8.67</td><td>-6827.29</td><td>12.56</td><td>5.06</td><td>0.23</td><td>40.08</td><td>1.09</td><td>2.85</td><td>0.07</td></tr></table>
+
+# K.2 DETAILED EXPERIMENTAL RESULTS FOR PIXELCNN-VAE
+
+Tables 4 and 5 present the ELBO, $\operatorname { K L } ( z )$ , Recon, $\hat { I } ( z ; s )$ , mCAS, and AoLR values of PixelCNNVAEs on MNIST and Fashion-MNIST. Moreover, these tables present mCAS(SVM) and AoSVM, which are the same with $\mathrm { m C A S }$ and AoLR except for using a support vector machine (SVM) with RBF kernel, i.e., a more powerful non-linear classifier, instead of the logistic classifier. Regarding the estimation of $\hat { I } ( z ; s )$ , please refer to Appendix A. Also, note that $\operatorname { K L } ( z )$ approximates $I ( x ; z )$ because it upper bounds $I ( x ; z )$ , and has been used for the metric to assess whether a decoder ignores $z$ or not (Bowman et al., 2016; Alemi et al., 2018; He et al., 2019). Here, the ELBO and Recon values are not divided by $T = 2 8 \times 2 8$ .
+
+As shown in the tables, (i) given a fixed $\gamma$ , the three methods ( $\beta$ -VAE, MI-VAE, and CMI-VAE) have the same level of $\operatorname { K L } ( z )$ ; therefore, all the methods can be used to alleviate PC. (ii) On the other hand, given a fixed $\gamma$ , CMI-VAE achieved the lower $\hat { I } ( z ; s )$ values in most cases, suggesting that it facilitates the learning of good global representation. (iii) Finally, even if we used a non-linear classifier SVM to calculate mCAS(SVM) and AoSVM, CMI-VAE achieved competitive or higher performance than the baselines in most cases. Note that, the exception is that given a $\gamma > 0 . 4$ , there were not much differences in mCAS(SVM) for Fashion-MNIST within the three methods. One possible reason is that using the non-linear classifier increases the number of factors to be considered, such as overfitting, and makes fair comparisons difficult. Also, we note that using a very large $\gamma$ for PixelCNN-VAEs might not be a good idea. It is because when $\gamma$ becomes too large, the decoder of PixelCNN tends to resemble an identity mapping from $z$ to its output, regardless of the regularization method (e.g., see, generated samples for $\gamma = 0 . 6$ in Appendix K.3). To improve performance while avoiding this phenomenon, it could be useful to using a weighting parameter $\alpha > 1$ in Eq. 8 (e.g., using $\gamma = 0 . 3$ and $\alpha > 1$ ), and this remains an issue to be addressed in a future work as noted in Section 6.
+
+# K.3 SAMPLE IMAGES FOR PIXELCNN-VAE
+
+Figures 5 present the generated images with PixelCNN-VAEs.
+
+Table 4: The ELBO, $\operatorname { K L } ( z )$ , Recon, mCAS, AoLR, mCAS(SVM), and AoSVM values of PixelCNNVAEs on MNIST. Each model was trained with a weighting parameter $\gamma$ .   
+
+<table><tr><td colspan="2"></td><td rowspan="2">ELBO</td><td rowspan="2">KL(z)</td><td rowspan="2">Recon</td><td rowspan="2">mCAS</td><td rowspan="2">AoLR</td><td rowspan="2">mCAS(SVM)</td><td rowspan="2">AoSVM</td><td rowspan="2">i（z;s)</td></tr><tr><td>Y</td><td>Model</td></tr><tr><td>0.0</td><td>β-VAE</td><td>56.21</td><td>3.60</td><td>52.61</td><td>0.3966</td><td>0.6094</td><td>0.5722</td><td>0.6447</td><td>1.81</td></tr><tr><td>0.1</td><td>β-VAE</td><td>56.28</td><td>5.33</td><td>50.95</td><td>0.5917</td><td>0.8161</td><td>0.7172</td><td>0.8458</td><td>2.11</td></tr><tr><td rowspan="3"></td><td>CMI-VAE</td><td>56.24</td><td>4.80</td><td>51.43</td><td>0.5421</td><td>0.7243</td><td>0.6499</td><td>0.7809</td><td>1.97</td></tr><tr><td>MI-VAE</td><td>56.23</td><td>5.16</td><td>51.07</td><td>0.5325</td><td>0.7602</td><td>0.6651</td><td>0.8080</td><td>2.05</td></tr><tr><td>β-VAE</td><td>56.68</td><td>9.21</td><td>47.47</td><td>0.6924</td><td>0.8479</td><td>0.7750</td><td>0.8980</td><td>2.45</td></tr><tr><td rowspan="3">0.2</td><td>CMI-VAE</td><td>56.68</td><td>9.32</td><td>47.36</td><td>0.7229</td><td>0.8692</td><td>0.7934</td><td>0.9026</td><td>2.29</td></tr><tr><td>MI-VAE</td><td>56.50</td><td>8.90</td><td>47.60</td><td>0.6734</td><td>0.8208</td><td>0.7606</td><td>0.8762</td><td>2.38</td></tr><tr><td>β-VAE</td><td>58.33</td><td>18.04</td><td>40.29</td><td>0.7448</td><td>0.8354</td><td>0.8109</td><td>0.9027</td><td>2.85</td></tr><tr><td rowspan="3">0.3</td><td>CMI-VAE</td><td>58.03</td><td>17.52</td><td>40.51</td><td>0.7716</td><td>0.8630</td><td>0.8136</td><td>0.9182</td><td>1.80</td></tr><tr><td>MI-VAE</td><td>57.81</td><td>16.37</td><td>41.44</td><td>0.7399</td><td>0.8238</td><td>0.8014</td><td>0.8938</td><td>2.85</td></tr><tr><td>β-VAE</td><td>61.22</td><td>29.00</td><td>32.21</td><td>0.7476</td><td>0.8204</td><td>0.8063</td><td>0.8962</td><td>3.28</td></tr><tr><td rowspan="3"></td><td>CMI-VAE</td><td>61.27</td><td>29.56</td><td>31.71</td><td>0.7664</td><td>0.8401</td><td>0.8122</td><td>0.9114</td><td>2.21</td></tr><tr><td>MI-VAE</td><td>60.52</td><td>27.10</td><td>33.43</td><td>0.7461</td><td>0.7903</td><td>0.7985</td><td>0.8742</td><td>3.11</td></tr><tr><td>β-VAE</td><td>64.55</td><td>37.79</td><td>26.76</td><td>0.7555</td><td>0.8279</td><td>0.8060</td><td>0.9010</td><td>3.16</td></tr><tr><td rowspan="3"></td><td>CMI-VAE</td><td>64.69</td><td>38.40</td><td>26.28</td><td>0.7575</td><td>0.8405</td><td>0.8075</td><td>0.9088</td><td>1.79</td></tr><tr><td>MI-VAE</td><td>63.57</td><td>35.62</td><td>27.95</td><td>0.7454</td><td>0.7719</td><td>0.7963</td><td>0.8582</td><td>3.18</td></tr><tr><td>β-VAE</td><td>68.74</td><td>46.29</td><td>22.45</td><td>0.7642</td><td>0.8306</td><td>0.8120</td><td>0.9056</td><td>3.27</td></tr><tr><td rowspan="3"></td><td>CMI-VAE</td><td>69.00</td><td>47.11</td><td>21.90</td><td>0.7618</td><td>0.8486</td><td>0.8079</td><td>0.9190</td><td>2.57</td></tr><tr><td>MI-VAE</td><td>67.72</td><td>44.16</td><td>23.57</td><td>0.7484</td><td>0.7722</td><td>0.7949</td><td>0.8572</td><td>3.19</td></tr><tr><td>β-VAE</td><td>74.47</td><td>56.04</td><td>18.43</td><td>0.7531</td><td>0.8454</td><td>0.7984</td><td>0.9148</td><td>3.27</td></tr><tr><td rowspan="3">0.7</td><td>CMI-VAE</td><td>74.71</td><td>56.28</td><td>18.43</td><td>0.7553</td><td>0.8486</td><td>0.8019</td><td>0.9179</td><td>1.81</td></tr><tr><td>MI-VAE</td><td>72.92</td><td>52.88</td><td>20.05</td><td>0.7496</td><td>0.7753</td><td>0.7912</td><td>0.8601</td><td>3.19</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 5: The ELBO, $\operatorname { K L } ( z )$ , Recon, mCAS, AoLR, mCAS(SVM), and AoSVM values of PixelCNNVAEs on Fashion-MNIST Each model was trained with a weighting parameter $\gamma$ .   
+
+<table><tr><td></td><td></td><td>ELBO</td><td>KL(z)</td><td>Recon</td><td>mCAS</td><td>AoLR</td><td>mCAS(SVM)</td><td>AoSVM</td><td>i(z;s)</td></tr><tr><td>Y</td><td>Model</td><td></td><td></td><td></td><td>0.4820</td><td></td><td></td><td></td><td></td></tr><tr><td>0.0</td><td>β-VAE β-VAE</td><td>88.60 88.86</td><td>4.02 6.71</td><td>84.58 82.16</td><td>0.5716</td><td>0.6906 0.7172</td><td>0.6342 0.6556</td><td>0.7552 0.7790</td><td>2.00 2.40</td></tr><tr><td rowspan="3">0.1</td><td>CMI-VAE</td><td>88.90</td><td>6.47</td><td>82.43</td><td>0.5837</td><td>0.7201</td><td>0.6618</td><td>0.7757</td><td>2.53</td></tr><tr><td>MI-VAE</td><td>88.84</td><td>6.72</td><td>82.12</td><td>0.5645</td><td>0.7166</td><td>0.6592</td><td>0.7742</td><td>2.37</td></tr><tr><td>β-VAE</td><td>89.94</td><td>10.95</td><td>78.99</td><td>0.6174</td><td>0.7092</td><td>0.6792</td><td>0.7708</td><td>2.78</td></tr><tr><td>0.2</td><td>CMI-VAE</td><td>90.25</td><td>11.49</td><td>78.76</td><td>0.6463</td><td>0.7258</td><td>0.6885</td><td>0.7779</td><td>2.46</td></tr><tr><td rowspan="3">0.3</td><td>MI-VAE</td><td>89.78</td><td>10.55</td><td>79.23</td><td>0.6132</td><td>0.6991</td><td>0.6760</td><td>0.7655</td><td>2.69</td></tr><tr><td>β-VAE</td><td>91.45</td><td>16.11</td><td>75.34</td><td>0.6427</td><td>0.7030</td><td>0.6811</td><td>0.7665</td><td>2.94</td></tr><tr><td>CMI-VAE</td><td>91.55</td><td>16.39</td><td>75.16</td><td>0.6578</td><td>0.7238</td><td>0.6873</td><td>0.7713</td><td>2.03</td></tr><tr><td></td><td>MI-VAE</td><td>91.14</td><td>15.09</td><td>76.06</td><td>0.6241</td><td>0.6817</td><td>0.6897</td><td>0.7507</td><td>2.92</td></tr><tr><td rowspan="3">0.4</td><td>β-VAE</td><td>93.98</td><td>24.33</td><td>69.65</td><td>0.6536</td><td>0.7062</td><td>0.6858</td><td>0.7636</td><td>3.24</td></tr><tr><td>CMI-VAE</td><td>93.81</td><td>23.35</td><td>70.46</td><td>0.6662</td><td>0.7145</td><td>0.6868</td><td>0.7701</td><td>2.11</td></tr><tr><td>MI-VAE</td><td>93.36</td><td>22.38</td><td>70.98</td><td>0.6366</td><td>0.6787</td><td>0.6886</td><td>0.7278</td><td>3.21</td></tr><tr><td rowspan="3">0.5</td><td>β-VAE</td><td>97.36</td><td>33.02</td><td>64.34</td><td>0.6547</td><td>0.6919</td><td>0.6845</td><td>0.7521</td><td>3.36</td></tr><tr><td>CMI-VAE</td><td>97.14</td><td>32.14</td><td>65.00</td><td>0.6651</td><td>0.7193</td><td>0.6867</td><td>0.7688</td><td>2.29</td></tr><tr><td>MI-VAE</td><td>96.47</td><td>31.19</td><td>65.27</td><td>0.6343</td><td>0.6506</td><td>0.6823</td><td>0.7056</td><td>3.31</td></tr><tr><td rowspan="3">0.6</td><td>β-VAE</td><td>101.76</td><td>42.40</td><td>59.36</td><td>0.6633</td><td>0.6899</td><td>0.6853</td><td>0.7524</td><td>3.39</td></tr><tr><td>CMI-VAE</td><td>101.83</td><td>41.97</td><td>59.86</td><td>0.6685</td><td>0.7112</td><td>0.6904</td><td>0.7659</td><td>2.43</td></tr><tr><td>MI-VAE</td><td>100.66</td><td>39.96</td><td>60.70</td><td>0.6535</td><td>0.6420</td><td>0.6854</td><td>0.6961</td><td>3.33</td></tr><tr><td>0.7</td><td>β-VAE</td><td>107.77</td><td>52.57</td><td>55.20</td><td>0.6680</td><td>0.6983</td><td>0.6900</td><td>0.7545</td><td>3.41</td></tr><tr><td rowspan="3"></td><td>CMI-VAE</td><td>107.49</td><td>51.97</td><td>55.52</td><td>0.6729</td><td>0.7103</td><td>0.6910</td><td>0.7681</td><td>2.00</td></tr><tr><td>MI-VAE</td><td>106.05</td><td>49.14</td><td>56.91</td><td>0.6578</td><td>0.6278</td><td>0.6838</td><td>0.6791</td><td>3.34</td></tr><tr><td>β-VAE</td><td>117.00</td><td>65.76</td><td>51.25</td><td>0.6649</td><td>0.7044</td><td>0.6855</td><td>0.7665</td><td>3.37</td></tr><tr><td rowspan="3">0.8</td><td>CMI-VAE</td><td>116.21</td><td>64.68</td><td>51.53</td><td>0.6634</td><td>0.7098</td><td>0.6851</td><td>0.7705</td><td>2.35</td></tr><tr><td>MI-VAE</td><td>113.98</td><td>60.19</td><td>53.79</td><td>0.6599</td><td>0.6108</td><td>0.6890</td><td>0.6709</td><td>3.31</td></tr><tr><td>β-VAE</td><td>132.62</td><td>85.13</td><td>47.49</td><td>0.6695</td><td>0.7097</td><td>0.6844</td><td>0.7787</td><td>3.47</td></tr><tr><td rowspan="3">0.9</td><td>CMI-VAE</td><td>131.01</td><td>82.97</td><td>48.04</td><td>0.6703</td><td>0.7169</td><td>0.6860</td><td>0.7780</td><td>2.74</td></tr><tr><td>MI-VAE</td><td>128.34</td><td>77.72</td><td>50.61</td><td>0.6565</td><td>0.6209</td><td>0.6880</td><td>0.6882</td><td>3.31</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+![](images/af2043da9e1347c598a8bec69769934f82d1bb083963660017155df9556c5689.jpg)  
+e 5: Real images (the first column) and generated images by PixelCNN-VAEs (the other 1 columns). The images in each row are stochastically sampled from the decoder $p ( x | z )$ using the same $z$ , which is extracted from $x$ in the first column. The figures present that the diversity of the images in $\gamma = 0 . 3$ is better than that in $\gamma = 0 . 6$ , which may be because PixelCNN-VAE would resemble an identity mapping with a large $\gamma$ . In contrast, $\gamma = 0 . 3$ apparently produces more label errors than $\gamma = 0 . 6$ because the decoder ignores $z$ with a small $\gamma$ (see, e.g., the rows for 3 and 4). Furthermore, when comparing (a) (CMI-VAE with $\gamma = 0 . 3 $ ) and (b) ( $\beta$ -VAE with $\gamma = 0 . 3$ ), apparently, (a) produces less label errors (see, e.g., the rows for 2 and 3). This result is consistent with the mCAS scores in Figure 3 (Section 5.3), which indicates that CMI-VAE achieved better diversity and less label errors than $\beta$ -VAE.