mlsanityData to 14 Oct 2026

Mixed-curvature Variational Autoencoders

arXiv 1911.08411 ↗First posted 19 Nov 2019cs.LG stat.MLcited by 86

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Abstract (arXiv)

Euclidean geometry has historically been the typical "workhorse" for machine learning applications due to its power and simplicity. However, it has recently been shown that geometric spaces with constant non-zero curvature improve representations and performance on a variety of data types and downstream tasks. Consequently, generative models like Variational Autoencoders (VAEs) have been successfully generalized to elliptical and hyperbolic latent spaces. While these approaches work well on data with particular kinds of biases e.g. tree-like data for a hyperbolic VAE, there exists no generic approach unifying and leveraging 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 is fixed or learnable. This generalizes the Euclidean VAE to curved latent spaces and recovers it when curvatures of all latent space components go to 0.

Key facts. Cited by 86 papers in the map (data through 14 Oct 2026). First posted on arXiv 19 Nov 2019.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 15; 2022: 13; 2023: 17; 2024: 19; 2025: 8; 2026: 14.

86 papers.

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  1. 2021Hyperbolic Deep Neural Networks: A Surveycited by 139
  2. 2022Riemannian Diffusion Modelscited by 122
  3. 2021A Self-supervised Mixed-curvature Graph Neural Networkcited by 33
  4. 2022Self-Supervised Continual Graph Learning in Adaptive Riemannian Spacescited by 32
  5. 2021Pseudo-Riemannian Graph Convolutional Networkscited by 30
  6. 2024Hyperbolic Fine-Tuning for Large Language Modelscited by 27
  7. 2021Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approachcited by 23
  8. 2022Heterogeneous manifolds for curvature-aware graph embeddingcited by 23
  9. 2022Bayesian Learning for Neural Networks: an algorithmic surveycited by 23
  10. 2024Manifold Integrated Gradients: Riemannian Geometry for Feature Attributioncited by 21
  11. 2021Equivariant Manifold Flowscited by 20
  12. 2023Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approachcited by 18
38 more
  1. 2023Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learningcited by 18
  2. 2023Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimizationcited by 18
  3. 2021Neural Distance Embeddings for Biological Sequencescited by 17
  4. 2021Unsupervised Hyperbolic Representation Learning via Message Passing Auto-Encoderscited by 15
  5. 2022Curved Geometric Networks for Visual Anomaly Recognitioncited by 15
  6. 2023Exploring Data Geometry for Continual Learningcited by 15
  7. 2023Contrastive Graph Clustering in Curvature Spacescited by 15
  8. 2022Hyperbolic VAE via Latent Gaussian Distributionscited by 14
  9. 2023A Survey on Temporal Knowledge Graph Completion: Taxonomy, Progress, and Prospectscited by 14
  10. 2024HyperVQ: MLR-based Vector Quantization in Hyperbolic Spacecited by 13
  11. 2024Matrix Manifold Neural Networks++cited by 13
  12. 2021Linear Classifiers in Product Space Formscited by 11
  13. 2021On Incorporating Inductive Biases into VAEscited by 11
  14. 2022Bringing motion taxonomies to continuous domains via GPLVM on hyperbolic manifoldscited by 10
  15. 2022Curvature Graph Generative Adversarial Networkscited by 9
  16. 2022Wrapped Distributions on homogeneous Riemannian manifoldscited by 9
  17. 2023Projections of Model Spaces for Latent Graph Inferencecited by 9
  18. 2021Highly Scalable and Provably Accurate Classification in Poincare Ballscited by 8
  19. 2024Pulling back symmetric Riemannian geometry for data analysiscited by 8
  20. 2024RMLR: Extending Multinomial Logistic Regression into General Geometriescited by 8
  21. 2024Generalizing Knowledge Graph Embedding with Universal Orthogonal Parameterizationcited by 7
  22. 2021Switch Spaces: Learning Product Spaces with Sparse Gatingcited by 6
  23. 2023Hyperbolic vs Euclidean Embeddings in Few-Shot Learning: Two Sides of the Same Coincited by 6
  24. 2022Neural Networks as Paths through the Space of Representationscited by 5
  25. 2026Hyperbolic Busemann Neural Networkscited by 5
  26. 2022AMCAD: Adaptive Mixed-Curvature Representation based Advertisement Retrieval Systemcited by 4
  27. 2023Hyperbolic Geometry in Computer Vision: A Surveycited by 4
  28. 2023SINCERE: Sequential Interaction Networks representation learning on Co-Evolving RiEmannian manifoldscited by 4
  29. 2023Improving Heterogeneous Graph Learning with Weighted Mixed-Curvature Product Manifoldcited by 4
  30. 2023Capacity Bounds for Hyperbolic Neural Network Representations of Latent Tree Structurescited by 4
  31. 2024Robust Hyperbolic Learning with Curvature-Aware Optimizationcited by 4
  32. 2024Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometrycited by 4
  33. 2025Curvature Learning for Generalization of Hyperbolic Neural Networkscited by 4
  34. 2024Contrastive Sequential Interaction Network Learning on Co-Evolving Riemannian Spacescited by 3
  35. 2025Hyperbolic Genome Embeddingscited by 3
  36. 2025Modality Alignment across Trees on Heterogeneous Hyperbolic Manifoldscited by 3
  37. 2026Intrinsic Lorentz Neural Networkcited by 3
  38. 2026The Post-GCN Decade Revisited: Curvature-Stratified Evaluation of Relational Learningcited by 3

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