The variational autoencoder (VAE) is a powerful generative model that can estimate the probability of a data point by using latent variables. In the VAE, the posterior of the latent variable given the data point is regularized by the prior of the latent variable using Kullback Leibler (KL) divergence. Although the standard Gaussian distribution is usually used for the prior, this simple prior incurs over-regularization. As a sophisticated prior, the aggregated posterior has been introduced, which is the expectation of the posterior over the data distribution. This prior is optimal for the VAE in terms of maximizing the training objective function. However, KL divergence with the aggregated posterior cannot be calculated in a closed form, which prevents us from using this optimal prior. With the proposed method, we introduce the density ratio trick to estimate this KL divergence without modeling the aggregated posterior explicitly. Since the density ratio trick does not work well in high dimensions, we rewrite this KL divergence that contains the high-dimensional density ratio into the sum of the analytically calculable term and the low-dimensional density ratio term, to which the density ratio trick is applied. Experiments on various datasets show that the VAE with this implicit optimal prior achieves high density estimation performance.
First, we focus on the works about the prior. Although the optimal prior for the VAE is the aggregated posterior, the KL divergence with the aggregated posterior cannot be calculated in a closed form. As described in Section , the VampPrior @cite_12 has been presented to solve this problem. However, it has sensitive hyperparameters such as the number of mixtures @math . Since the VampPrior requires a heavy computational cost, these hyperparameters are difficult to tune. In contrast to this, our approach can estimate the KL divergence more easily and robustly than the VampPrior since it does not need to model the aggregated posterior explicitly. In addition, since the computational cost of our approach is much more lightweight than that of VampPrior, the hyperparameters of our approach are easier to tune than those of VampPrior.
@cite_12
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Many different methods to train deep generative models have been introduced in the past. In this paper, we propose to extend the variational auto-encoder (VAE) framework with a new type of prior which we call "Variational Mixture of Posteriors" prior, or VampPrior for short. The VampPrior consists of a mixture distribution (e.g., a mixture of Gaussians) with components given by variational posteriors conditioned on learnable pseudo-inputs. We further extend this prior to a two layer hierarchical model and show that this architecture with a coupled prior and posterior, learns significantly better models. The model also avoids the usual local optima issues related to useless latent dimensions that plague VAEs. We provide empirical studies on six datasets, namely, static and binary MNIST, OMNIGLOT, Caltech 101 Silhouettes, Frey Faces and Histopathology patches, and show that applying the hierarchical VampPrior delivers state-of-the-art results on all datasets in the unsupervised permutation invariant setting and the best results or comparable to SOTA methods for the approach with convolutional networks. Three Bayesian related approaches, namely, variational Bayesian (VB), minimum message length (MML) and Bayesian Ying-Yang (BYY) harmony learning, have been applied to automatically determining an appropriate number of components during learning Gaussian mixture model (GMM). This paper aims to provide a comparative investigation on these approaches with not only a Jeffreys prior but also a conjugate Dirichlet-Normal-Wishart (DNW) prior on GMM. In addition to adopting the existing algorithms either directly or with some modifications, the algorithm for VB with Jeffreys prior and the algorithm for BYY with DNW prior are developed in this paper to fill the missing gap. The performances of automatic model selection are evaluated through extensive experiments, with several empirical findings: 1) Considering priors merely on the mixing weights, each of three approaches makes biased mistakes, while considering priors on all the parameters of GMM makes each approach reduce its bias and also improve its performance. 2) As Jeffreys prior is replaced by the DNW prior, all the three approaches improve their performances. Moreover, Jeffreys prior makes MML slightly better than VB, while the DNW prior makes VB better than MML. 3) As the hyperparameters of DNW prior are further optimized by each of its own learning principle, BYY improves its performances while VB and MML deteriorate their performances when there are too many free hyper-parameters. Actually, VB and MML lack a good guide for optimizing the hyper-parameters of DNW prior. 4) BYY considerably outperforms both VB and MML for any type of priors and whether hyper-parameters are optimized. Being different from VB and MML that rely on appropriate priors to perform model selection, BYY does not highly depend on the type of priors. It has model selection ability even without priors and performs already very well with Jeffreys prior, and incrementally improves as Jeffreys prior is replaced by the DNW prior. Finally, all algorithms are applied on the Berkeley segmentation database of real world images. Again, BYY considerably outperforms both VB and MML, especially in detecting the objects of interest from a confusing background. An encompassing prior (EP) approach to facilitate Bayesian model selection for nested models with inequality constraints has been previously proposed. In this approach, samples are drawn from the prior and posterior distributions of an encompassing model that contains an inequality restricted version as a special case. The Bayes factor in favor of the inequality restriction then simplifies to the ratio of the proportions of posterior and prior samples consistent with the inequality restriction. This formalism has been applied almost exclusively to models with inequality or ''about equality'' constraints. It is shown that the EP approach naturally extends to exact equality constraints by considering the ratio of the heights for the posterior and prior distributions at the point that is subject to test (i.e., the Savage-Dickey density ratio). The EP approach generalizes the Savage-Dickey ratio method, and can accommodate both inequality and exact equality constraints. The general EP approach is found to be a computationally efficient procedure to calculate Bayes factors for nested models. However, the EP approach to exact equality constraints is vulnerable to the Borel-Kolmogorov paradox, the consequences of which warrant careful consideration. We study a variant of the variational autoencoder model (VAE) with a Gaussian mixture as a prior distribution, with the goal of performing unsupervised clustering through deep generative models. We observe that the known problem of over-regularisation that has been shown to arise in regular VAEs also manifests itself in our model and leads to cluster degeneracy. We show that a heuristic called minimum information constraint that has been shown to mitigate this effect in VAEs can also be applied to improve unsupervised clustering performance with our model. Furthermore we analyse the effect of this heuristic and provide an intuition of the various processes with the help of visualizations. Finally, we demonstrate the performance of our model on synthetic data, MNIST and SVHN, showing that the obtained clusters are distinct, interpretable and result in achieving competitive performance on unsupervised clustering to the state-of-the-art results.