The Variational Auto-Encoder (VAE) is one of the most used unsupervised machine learning models. But although the default choice of a Gaussian distribution for both the prior and posterior represents a mathematically convenient distribution often leading to competitive results, we show that this parameterization fails to model data with a latent hyperspherical structure. To address this issue we propose using a von Mises-Fisher (vMF) distribution instead, leading to a hyperspherical latent space. Through a series of experiments we show how such a hyperspherical VAE, or @math -VAE, is more suitable for capturing data with a hyperspherical latent structure, while outperforming a normal, @math -VAE, in low dimensions on other data types.
Another approach by @cite_8 , capitalizes on the hierarchical structure present in some data types. By learning the embeddings for a graph in a non-euclidean negative curvature hyperbolical space, they show this topology has clear advantages over embedding these objects in a Euclidean space. Although they did not use a VAE-based approach, that is, they did not build a probabilistic generative model of the data interpreting the embeddings as latent variables, this approach shows the merit of explicitly adjusting the choice of latent topology to the data used.
@cite_8
1522301498
We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.