You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
A standard Variational Autoencoder, with a Euclidean latent space, is structurally incapable of capturing topological properties of certain datasets. To remove topological obstructions, we introduce Diffusion Variational Autoencoders with arbitrary manifolds as a latent space. A Diffusion Variational Autoencoder uses transition kernels of Brownian motion on the manifold. In particular, it uses properties of the Brownian motion to implement the reparametrization trick and fast approximations to the KL divergence. We show that the Diffusion Variational Autoencoder is capable of capturing topological properties of synthetic datasets. Additionally, we train MNIST on spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a natural dataset like MNIST does not have latent variables with a clear-cut topological structure, training it on a manifold can still highlight topological and geometrical properties.

Cited Abstract(s):
[1]: 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.

[2]: We define and address the problem of unsupervised learning of disentangled representations on data generated from independent factors of variation. We propose FactorVAE, a method that disentangles by encouraging the distribution of representations to be factorial and hence independent across the dimensions. We show that it improves upon @math -VAE by providing a better trade-off between disentanglement and reconstruction quality. Moreover, we highlight the problems of a commonly used disentanglement metric and introduce a new metric that does not suffer from them.

[3]: Learning an interpretable factorised representation of the independent data generative factors of the world without supervision is an important precursor for the development of artificial intelligence that is able to learn and reason in the same way that humans do. We introduce beta-VAE, a new state-of-the-art framework for automated discovery of interpretable factorised latent representations from raw image data in a completely unsupervised manner. Our approach is a modification of the variational autoencoder (VAE) framework. We introduce an adjustable hyperparameter beta that balances latent channel capacity and independence constraints with reconstruction accuracy. We demonstrate that beta-VAE with appropriately tuned beta > 1 qualitatively outperforms VAE (beta = 1), as well as state of the art unsupervised (InfoGAN) and semi-supervised (DC-IGN) approaches to disentangled factor learning on a variety of datasets (celebA, faces and chairs). Furthermore, we devise a protocol to quantitatively compare the degree of disentanglement learnt by different models, and show that our approach also significantly outperforms all baselines quantitatively. Unlike InfoGAN, beta-VAE is stable to train, makes few assumptions about the data and relies on tuning a single hyperparameter, which can be directly optimised through a hyper parameter search using weakly labelled data or through heuristic visual inspection for purely unsupervised data.

[4]: We decompose the evidence lower bound to show the existence of a term measuring the total correlation between latent variables. We use this to motivate our @math -TCVAE (Total Correlation Variational Autoencoder), a refinement of the state-of-the-art @math -VAE objective for learning disentangled representations, requiring no additional hyperparameters during training. We further propose a principled classifier-free measure of disentanglement called the mutual information gap (MIG). We perform extensive quantitative and qualitative experiments, in both restricted and non-restricted settings, and show a strong relation between total correlation and disentanglement, when the latent variables model is trained using our framework.

[5]: 

[6]: How can intelligent agents solve a diverse set of tasks in a data-efficient manner? The disentangled representation learning approach posits that such an agent would benefit from separating out (disentangling) the underlying structure of the world into disjoint parts of its representation. However, there is no generally agreed-upon definition of disentangling, not least because it is unclear how to formalise the notion of world structure beyond toy datasets with a known ground truth generative process. Here we propose that a principled solution to characterising disentangled representations can be found by focusing on the transformation properties of the world. In particular, we suggest that those transformations that change only some properties of the underlying world state, while leaving all other properties invariant, are what gives exploitable structure to any kind of data. Similar ideas have already been successfully applied in physics, where the study of symmetry transformations has revolutionised the understanding of the world structure. By connecting symmetry transformations to vector representations using the formalism of group and representation theory we arrive at the first formal definition of disentangled representations. Our new definition is in agreement with many of the current intuitions about disentangling, while also providing principled resolutions to a number of previous points of contention. While this work focuses on formally defining disentangling - as opposed to solving the learning problem - we believe that the shift in perspective to studying data transformations can stimulate the development of better representation learning algorithms.
Our work originated out of the search for algorithms that find semantically meaningful latent factors of data. The use of VAEs and their extensions to this end has mostly taken place in the context of [3] [6] [1] . Examples of extensions that aim at disentangling latent factors are the @math -VAE [3] , the factor-VAE [2] , the @math -TCVAE [4] and the DIP-VAE [5] .