Bayesian Hyperbolic Multidimensional Scaling

• URL: http://arxiv.org/abs/2210.15081v3
• Date: Tue, 15 Aug 2023 11:20:59 GMT
• Title: Bayesian Hyperbolic Multidimensional Scaling
• Authors: Bolun Liu, Shane Lubold, Adrian E. Raftery, Tyler H. McCormick
• Abstract summary: We propose a Bayesian approach to multidimensional scaling when the low-dimensional manifold is hyperbolic. A case-control likelihood approximation allows for efficient sampling from the posterior distribution in larger data settings. We evaluate the proposed method against state-of-the-art alternatives using simulations, canonical reference datasets, Indian village network data, and human gene expression data.
• Score: 2.5944208050492183
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Multidimensional scaling (MDS) is a widely used approach to representing high-dimensional, dependent data. MDS works by assigning each observation a location on a low-dimensional geometric manifold, with distance on the manifold representing similarity. We propose a Bayesian approach to multidimensional scaling when the low-dimensional manifold is hyperbolic. Using hyperbolic space facilitates representing tree-like structures common in many settings (e.g. text or genetic data with hierarchical structure). A Bayesian approach provides regularization that minimizes the impact of measurement error in the observed data and assesses uncertainty. We also propose a case-control likelihood approximation that allows for efficient sampling from the posterior distribution in larger data settings, reducing computational complexity from approximately $O(n^2)$ to $O(n)$. We evaluate the proposed method against state-of-the-art alternatives using simulations, canonical reference datasets, Indian village network data, and human gene expression data.

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