The number of flows is chosen from { 1, 2, 4, 8} .

We are very grateful to all reviewers for their detailed, insightful, and constructive comments and questions. But we believe that they are very important, and we will pursue them in our ongoing study. The column "FC" is excerpted from Our responses (blue) to reviewers' comments/questions ( black/bold/italic) are as follows. We will refine our claims, and also refer to these SA VI methods. It turns out that it was our faulty claim.

Variational autoencoders employ an amortized inference model to approximate the posterior of latent variables. However, such amortized variational inference faces two challenges: (1) the limited posterior expressiveness of fully-factorized Gaussian assumption and (2) the amortization error of the inference model. We present a novel approach that addresses both challenges. First, we focus on ReLU networks with Gaussian output and illustrate their connection to probabilistic PCA. Building on this observation, we derive an iterative algorithm that finds the mode of the posterior and apply full-covariance Gaussian posterior approximation centered on the mode. Subsequently, we present a general framework named Variational Laplace Autoencoders (VLAEs) for training deep generative models. Based on the Laplace approximation of the latent variable posterior, VLAEs enhance the expressiveness of the posterior while reducing the amortization error. Empirical results on MNIST, Omniglot, Fashion-MNIST, SVHN and CIFAR10 show that the proposed approach significantly outperforms other recent amortized or iterative methods on the ReLU networks.

Inference networks of traditional Variational Autoencoders (VAEs) are typically amortized, resulting in relatively inaccurate posterior approximation compared to instance-wise variational optimization. Recent semi-amortized approaches were proposed to address this drawback; however, their iterative gradient update procedures can be computationally demanding. To address these issues, in this paper we introduce an accurate amortized inference algorithm. We propose a novel recursive mixture estimation algorithm for VAEs that iteratively augments the current mixture with new components so as to maximally reduce the divergence between the variational and the true posteriors. Using the functional gradient approach, we devise an intuitive learning criteria for selecting a new mixture component: the new component has to improve the data likelihood (lower bound) and, at the same time, be as divergent from the current mixture distribution as possible, thus increasing representational diversity. Compared to recently proposed boosted variational inference (BVI), our method relies on amortized inference in contrast to BVI's non-amortized single optimization instance. A crucial benefit of our approach is that the inference at test time requires a single feed-forward pass through the mixture inference network, making it significantly faster than the semi-amortized approaches. We show that our approach yields higher test data likelihood than the state-of-the-art on several benchmark datasets.

In clustering we normally output one cluster variable for each datapoint. However it is not necessarily the case that there is only one way to partition a given dataset into cluster components. For example, one could cluster objects by their colour, or by their type. Different attributes form a hierarchy, and we could wish to cluster in any of them. By disentangling the learnt latent representations of some dataset into different layers for different attributes we can then cluster in those latent spaces. We call this "disentangled clustering". Extending Variational Ladder Autoencoders (Zhao et al., 2017), we propose a clustering algorithm, VLAC, that outperforms a Gaussian Mixture DGM in cluster accuracy over digit identity on the test set of SVHN. We also demonstrate learning clusters jointly over numerous layers of the hierarchy of latent variables for the data, and show component-wise generation from this hierarchical model.

Variational Autoencoder (VAE), a simple and effective deep generative model, has led to a number of impressive empirical successes and spawned many advanced variants and theoretical investigations. However, recent studies demonstrate that, when equipped with expressive generative distributions (aka. decoders), VAE suffers from learning uninformative latent representations with the observation called KL Varnishing, in which case VAE collapses into an unconditional generative model. In this work, we introduce mutual posterior-divergence regularization, a novel regularization that is able to control the geometry of the latent space to accomplish meaningful representation learning, while achieving comparable or superior capability of density estimation. Experiments on three image benchmark datasets demonstrate that, when equipped with powerful decoders, our model performs well both on density estimation and representation learning.

Deep neural networks have been shown to be very successful at learning feature hierarchies in supervised learning tasks. Generative models, on the other hand, have benefited less from hierarchical models with multiple layers of latent variables. In this paper, we prove that hierarchical latent variable models do not take advantage of the hierarchical structure when trained with existing variational methods, and provide some limitations on the kind of features existing models can learn. Finally we propose an alternative architecture that do not suffer from these limitations. Our model is able to learn highly interpretable and disentangled hierarchical features on several natural image datasets with no task specific regularization or prior knowledge.

Representation learning seeks to expose certain aspects of observed data in a learned representation that's amenable to downstream tasks like classification. For instance, a good representation for 2D images might be one that describes only global structure and discards information about detailed texture. In this paper, we present a simple but principled method to learn such global representations by combining Variational Autoencoder (VAE) with neural autoregressive models such as RNN, MADE and PixelRNN/CNN. Our proposed VAE model allows us to have control over what the global latent code can learn and , by designing the architecture accordingly, we can force the global latent code to discard irrelevant information such as texture in 2D images, and hence the VAE only "autoencodes" data in a lossy fashion. In addition, by leveraging autoregressive models as both prior distribution $p(z)$ and decoding distribution $p(x|z)$, we can greatly improve generative modeling performance of VAEs, achieving new state-of-the-art results on MNIST, OMNIGLOT and Caltech-101 Silhouettes density estimation tasks.