2024/10/22 by Koshi Watanabe, Watanabe, Koshi, Keisuke Maeda +4
Computer Science · #FOS: Computer and information sciences #Face and Expression Recognition #Machine Learning (cs.LG) #Neural Networks and Applications #Time Series Analysis and Forecasting
paper · pdf · doi:10.48550/arxiv.2410.16698
openalex publication_date 2024/10/22 · openalex created_date 2024/11/13 · openalex updated_date 2026/07/28
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.