It has been shown that using geometric spaces with non-zero curvature instead of plain Euclidean spaces with zero curvature improves performance on a range of Machine Learning tasks for learning representations. Recent work has leveraged these geometries to learn latent variable models like Variational Autoencoders (VAEs) in spherical and hyperbolic spaces with constant curvature. While these approaches work well on particular kinds of data that they were designed for e.g. tree-like data for a hyperbolic VAE, there exists no generic approach unifying all three models. We develop a Mixed-curvature Variational Autoencoder, an efficient way to train a VAE whose latent space is a product of constant curvature Riemannian manifolds, where the per-component curvature can be learned. This generalizes the Euclidean VAE to curved latent spaces, as the model essentially reduces to the Euclidean VAE if curvatures of all latent space components go to 0.
Learning graph representations via lowdimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-like structures better than Euclidean geometry. Second, we view hierarchical relations as partial orders defined using a family of nested geodesically convex cones. We prove that these entailment cones admit an optimal shape with a closed form expression both in the Euclidean and hyperbolic spaces, and they canonically define the embedding learning process. Experiments show significant improvements of our method over strong recent baselines both in terms of representational capacity and generalization.
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases (e.g., hierarchical structures benefit from hyperbolic geometry). However, going beyond embedding spaces of constant sectional curvature, while potentially more representationally powerful, proves to be challenging as one can easily lose the appeal of computationally tractable tools such as geodesic distances or Riemannian gradients. Here, we explore two computationally efficient matrix manifolds, showcasing how to learn and optimize graph embeddings in these Riemannian spaces. Empirically, we demonstrate consistent improvements over Euclidean geometry while often outperforming hyperbolic and elliptical embeddings based on various metrics that capture different graph properties. Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.
Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network layers. This makes it hard to use hyperbolic embeddings in downstream tasks. Here, we bridge this gap in a principled manner by combining the formalism of Möbius gyrovector spaces with the Riemannian geometry of the Poincaré model of hyperbolic spaces. As a result, we derive hyperbolic versions of important deep learning tools: multinomial logistic regression, feed-forward and recurrent neural networks such as gated recurrent units. This allows to embed sequential data and perform classification in the hyperbolic space. Empirically, we show that, even if hyperbolic optimization tools are limited, hyperbolic sentence embeddings either outperform or are on par with their Euclidean variants on textual entailment and noisy-prefix recognition tasks.
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