mlsanityData to 14 Oct 2026

Hyperboloid GPLVM for Discovering Continuous Hierarchies via Nonparametric Estimation

arXiv 2410.16698 ↗First posted 22 Oct 2024cs.LGcited by 0

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Abstract (arXiv)

Dimensionality reduction (DR) offers a useful representation of complex high-dimensional data. Recent DR methods focus on hyperbolic geometry to derive a faithful low-dimensional representation of hierarchical data. However, existing methods are based on neighbor embedding, frequently ruining the continual relation of the hierarchies. This paper presents hyperboloid Gaussian process (GP) latent variable models (hGP-LVMs) to embed high-dimensional hierarchical data with implicit continuity via nonparametric estimation. We adopt generative modeling using the GP, which brings effective hierarchical embedding and executes ill-posed hyperparameter tuning. This paper presents three variants that employ original point, sparse point, and Bayesian estimations. We establish their learning algorithms by incorporating the Riemannian optimization and active approximation scheme of GP-LVM. For Bayesian inference, we further introduce the reparameterization trick to realize Bayesian latent variable learning. In the last part of this paper, we apply hGP-LVMs to several datasets and show their ability to represent high-dimensional hierarchies in low-dimensional spaces.

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References · 24

  1. 2018UMAP: Uniform Manifold Approximation and Projection for Dimension Reductioncited by 2,898
  2. 2017Poincar'e Embeddings for Learning Hierarchical Representationscited by 684
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  4. 2018Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometrycited by 265
  5. 2018Representation Tradeoffs for Hyperbolic Embeddingscited by 206
  6. 2020Low-Dimensional Hyperbolic Knowledge Graph Embeddingscited by 195
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  9. 2021Hyperbolic Deep Neural Networks: A Surveycited by 139
  10. 2023Hyperbolic Image-Text Representationscited by 136
  11. 2019Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoderscited by 112
  12. 2022Hyperbolic Image Segmentationcited by 102
12 more
  1. 2020Mat'ern Gaussian processes on Riemannian manifoldscited by 82
  2. 2016Understanding Probabilistic Sparse Gaussian Process Approximationscited by 68
  3. 2019TriMap: Large-scale Dimensionality Reduction Using Tripletscited by 60
  4. 2014Geodesic Exponential Kernels: When Curvature and Linearity Conflictcited by 47
  5. 2022Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact casecited by 25
  6. 2023Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spacescited by 15
  7. 2020Manifold GPLVMs for discovering non-Euclidean latent structure in neural datacited by 14
  8. 2023Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Spacecited by 11
  9. 2022A Rotated Hyperbolic Wrapped Normal Distribution for Hierarchical Representation Learningcited by 10
  10. 2022Bringing motion taxonomies to continuous domains via GPLVM on hyperbolic manifoldscited by 10
  11. 2023Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metricscited by 6
  12. 2022Revisiting Active Sets for Gaussian Process Decoderscited by 5

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