Generalized Multimodal ELBO

1 INTRODUCTION . The availability of multiple data types provides a rich source of information and holds promise for learning representations that generalize well across multiple modalities ( Baltrušaitis et al. , 2018 ) . Multimodal data naturally grants additional self-supervision in the form of shared information connecting the different data types . Further , the understanding of different modalities and the interplay between data types are non-trivial research questions and long-standing goals in machine learning research . While fully-supervised approaches have been applied successfully ( Karpathy & Fei-Fei , 2015 ; Tsai et al. , 2019 ; Pham et al. , 2019 ; Schoenauer-Sebag et al. , 2019 ) , the labeling of multiple data types remains time consuming and expensive . Therefore , it requires models that efficiently learn from multiple data types in a self-supervised fashion . Self-supervised , generative models are suitable for learning the joint distribution of multiple data types without supervision . We focus on VAEs ( Kingma & Welling , 2014 ; Rezende et al. , 2014 ) which are able to jointly infer representations and generate new observations . Despite their success on unimodal datasets , there are additional challenges associated with multimodal data ( Suzuki et al. , 2016 ; Vedantam et al. , 2018 ) . In particular , multimodal generative models need to represent both modality-specific and shared factors and generate semantically coherent samples across modalities . Semantically coherent samples are connected by the information which is shared between data types ( Shi et al. , 2019 ) . These requirements are not inherent to the objective—the evidence lower bound ( ELBO ) —of unimodal VAEs . Hence , adaptions to the original formulation are required to cater to and benefit from multiple data types . Furthermore , to handle missing modalities , there is a scalability issue in terms of the number of modalities : naively , it requires 2M different encoders to handle all combinations for M data types . Thus , we restrict our search for an improved multimodal ELBO to the class of scalable multimodal VAEs . Among the class of scalable multimodal VAEs , there are two dominant strains of models , based on either the multimodal variational autoencoder ( MVAE , Wu & Goodman , 2018 ) or the Mixture-ofExperts multimodal variational autoencoder ( MMVAE , Shi et al. , 2019 ) . However , we show that these approaches differ merely in their choice of joint posterior approximation functions . We draw a theoretical connection between these models , showing that they can be subsumed under the class ∗Equal contribution . of abstract mean functions for modeling the joint posterior . This insight has practical implications , because the choice of mean function directly influences the properties of a model ( Nielsen , 2019 ) . The MVAE uses a geometric mean , which enables learning a sharp posterior , resulting in a good approximation of the joint distribution . On the other hand , the MMVAE applies an arithmetic mean which allows better learning of the unimodal and pairwise conditional distributions . We generalize these approaches and introduce the Mixture-of-Products-of-Experts-VAE that combines the benefits of both methods without considerable trade-offs . In summary , we derive a generalized multimodal ELBO formulation that connects and generalizes two previous approaches . The proposed method , termed MoPoE-VAE , models the joint posterior approximation as a Mixture-of-Products-of-Experts , which encompasses the MVAE ( Product-ofExperts ) and MMVAE ( Mixture-of-Experts ) as special cases ( Section 3 ) . In contrast to previous models , the proposed model approximates the joint posterior for all subsets of modalities , an advantage that we validate empirically in Section 4 , where our model achieves state-of-the-art results . 2 RELATED WORK . This work extends and generalizes existing work in self-supervised multimodal generative models that are scalable in the number of modalities . Scalable in the sense that a single model approximates the joint distribution over all modalities ( including all marginal and conditional distributions ) instead of requiring individual models for every subset of modalities ( e.g. , Huang et al. , 2018 ; Tian & Engel , 2019 ; Hsu & Glass , 2018 ) . The latter approach requires a prohibitive number of models , exponential in number of modalities . Multimodal VAEs Among multimodal generative models , multimodal VAEs ( Suzuki et al. , 2016 ; Vedantam et al. , 2018 ; Kurle et al. , 2019 ; Tsai et al. , 2019 ; Wu & Goodman , 2018 ; Shi et al. , 2019 ; 2020 ; Sutter et al. , 2020 ) have recently been the dominant approach . Multimodal VAEs are not only suitable to learn a joint distribution over multiple modalities , but also enable joint inference given a subset of modalities . However , to approximate the joint posterior for all subsets of modalities efficiently , it is required to introduce additional assumptions on the form of the joint posterior . To overcome the issue of scalability , previous work relies on either the product ( Kurle et al. , 2019 ; Wu & Goodman , 2018 ) or the mixture ( Shi et al. , 2019 ; 2020 ) of unimodal posteriors . While both approaches have their merits , there are also disadvantages associated with them . We unite these approaches in a generalized formulation—a mixture of products joint posterior—that encapsulates both approaches and combines their benefits without significant trade-offs . Multimodal posteriors The MVAE ( Wu & Goodman , 2018 ) assumes that the joint posterior is a product of unimodal posteriors—a Product-of-Experts ( PoE , Hinton , 2002 ) . The PoE has the benefit of aggregating information across any subset of unimodal posteriors and therefore provides an efficient way of dealing with missing modalities for specific types of unimodal posteriors ( e.g. , Gaussians ) . However , to handle missing modalities the MVAE relies on an additional sub-sampling of unimodal log-likelihoods , which no longer guarantees a valid lower bound on the joint log-likelihood ( Wu & Goodman , 2019 ) . Previous work provides empirical results that exhibit the shortcomings of the MVAE , attributing them to a precision miscalibration of experts ( Shi et al. , 2019 ) or to the averaging over inseparable individual beliefs ( Kurle et al. , 2019 ) . Our results suggest that the PoE works well in practice , if it is also applied on all subsets of modalities , which naturally leads to the proposed Mixture-of-Products-of-Experts ( MoPoE ) generalization , which yields a valid lower bound on the joint log-likelihood . On the other hand , the MMVAE ( Shi et al. , 2019 ) assumes that the joint posterior is a mixture of unimodal posteriors—a Mixture-of-Experts ( MoE ) . The MMVAE is suitable for the approximation of unimodal posteriors and for translation between pairs of modalities , however , it can not take advantage of multiple modalities being present , because it only takes the unimodal posteriors into account during training . In contrast , the proposed MoPoE-VAE computes the joint posterior for all subsets of modalities and therefore enables efficient many-to-many translations . Extensions of the MVAE and MMVAE ( Kurle et al. , 2019 ; Daunhawer et al. , 2020 ; Shi et al. , 2020 ; Sutter et al. , 2020 ) have introduced additional loss terms , however , these are also applicable to and can be added on top of the proposed model . Table 1 summarizes the properties of previous multimodal VAEs and highlights the benefits of the proposed model : the ability to aggregate multiple modalities , to learn a multi-modal posterior ( in the statistical sense ) , and to efficiently handle missing modalities at test time . 3 METHOD . 3.1 PRELIMINARIES . We consider a dataset { X ( i ) } Ni=1 of N i.i.d . samples , each of which is a set of M modalities X ( i ) = { x ( i ) j } Mj=1 . We assume that the data is generated by some random process involving a joint hidden random variable z such that inter-modality dependencies are unknown . The marginal log-likelihood can be decomposed into a sum over marginal log-likelihoods of individual sets log pθ ( { X ( i ) } Ni=1 ) = ∑N i=1 log pθ ( X ( i ) ) , which can be written as : log pθ ( X ( i ) ) = DKL ( qφ ( z|X ( i ) ) ||pθ ( z|X ( i ) ) ) + L ( θ , φ ; X ( i ) ) , ( 1 ) with L ( θ , φ ; X ( i ) ) : = Eqφ ( z|X ( i ) ) [ log pθ ( X ( i ) |z ) ] −DKL ( qφ ( z|X ( i ) ) ||pθ ( z ) ) . ( 2 ) L ( θ , φ ; X ( i ) ) is called evidence lower bound ( ELBO ) on the marginal log-likelihood of the i-th set . It forms a tractable objective to approximate the joint data distribution log pθ ( X ( i ) ) . qφ ( z|X ( i ) ) is the posterior approximation distribution with learnable parameters φ . From the non-negativity of the KL divergence , it follows that log pθ ( X ( i ) ) ≥ L ( θ , φ ; X ( i ) ) . If the posterior approximation qφ ( z|X ( i ) ) is identical to the true posterior distribution pθ ( z|X ( i ) ) , the bound holds with equality . Hence , maximizing the ELBO in Equation ( 2 ) minimizes the otherwise intractable KL-divergence between approximate and true posterior distribution : argmin φ DKL ( qφ ( z|X ( i ) ) ||pθ ( z|X ( i ) ) ) . ( 3 ) Adaptations to the ELBO formulation in Equation ( 2 ) include an additional hyperparameter β which weights the KL-divergence relative to the log-likelihood ( Higgins et al. , 2017 ) . To improve readability , we will omit the superscript ( i ) in the remaining part of this work . 3.2 APPROXIMATING pθ ( z|X ) IN CASE OF MISSING DATA TYPES For a dataset of M modalities , there are 2M different subsets contained in the powerset P ( X ) . If , for a particular observation , we only have access to a subset of data types Xk ∈ P ( X ) , the approximation of pθ ( Xk ) would result in a different ELBO formulation L ( θ , φk ; Xk ) where the true posterior pθ ( z|Xk ) of subset Xk is approximated . Instead , we are interested in the true posterior pθ ( z|X ) of all data types X , even when only a subset Xk , i.e . q̃φk ( z|Xk ) , is available . The desired ELBO for the available subset Xk is given by Lk ( θ , φk ; X ) : = Eq̃φk ( z|Xk ) [ log ( pθ ( X|z ) ] −DKL ( q̃φk ( z|Xk ) ||pθ ( z ) ) . ( 4 ) The subtle but important difference between Lk ( θ , φk ; X ) and L ( θ , φk ; Xk ) is that the former still yields a valid lower bound on pθ ( X ) , whereas the latter forms a lower bound on log pθ ( Xk ) , which is no longer a valid bound on the desired log pθ ( X ) . Different from previous work , we argue for an optimization of the powerset P ( X ) , i.e. , the joint optimization of all ELBOs Lk ( θ , φk ; X ) defined by the posterior subset approximation q̃φk ( z|Xk ) . Since maximizing the ELBO in Equation ( 2 ) is equivalent to minimizing the KL-divergence in Equation ( 3 ) , the joint optimization of the powerset P ( X ) is equal to the minimization of the following convex combination of KL-divergences of the power set P ( X ) .1 argmin φ ∑ Xk∈P ( X ) DKL ( q̃φ ( z|Xk ) ||pθ ( z|X ) ) ( 5 ) Hence , we propose to optimize Equation ( 4 ) for all subsets Xk . Lemma 1 . The sum of KL-divergences in Equation ( 5 ) describes the joint probability log pθ ( X ) as follows : log pθ ( X ) = 1 2M ∑ Xk∈P ( X ) DKL ( q̃φ ( z|Xk ) ||pθ ( z|X ) ) + 1 2M ∑ Xk∈P ( X ) Eq̃φ ( z|Xk ) [ log pθ ( X|z ) pθ ( z ) q̃φ ( z|Xk ) ] Following Lemma 1 ( see Appendix A.1 for the proof ) and the non-negativity of the KL-divergence , we see that the convex combination of expectations over the powerset P ( X ) is an ELBO on the joint probability log pθ ( X ) . Since this would require 2M different inference networks in a naive implementation , we use a more efficient approach utilizing abstract mean functions .

Summary: This paper focuses on providing a more generalize multimodal ELBO to encompass previous PoE and MoE as special cases and combines their benefits. To this end, the authors first define the new ELBO L_{MoPoE} which is an interesting extension of PoE and MoE. Different from PoE (product of experts) and MoE (mixture of experts), MoPoE (mixture of product of experts) explores a more general way by mixing more experts where each expert is the product of a subset of all modalities’ posterior. In this way, as illustrated by the author, PoE and MoE can be seen as specific cases of MoPoE easily. The proposed model achieves competitive results compared with PoE and MoE.