Bayesian Hyperbolic Multidimensional Scaling

TL;DR

This paper introduces a Bayesian hyperbolic multidimensional scaling method that effectively models hierarchical data structures, reduces computational complexity, and provides uncertainty quantification, demonstrated through various real-world datasets.

Contribution

It presents a novel Bayesian hyperbolic MDS approach with a case-control likelihood approximation for efficient large-scale inference.

Findings

Outperforms existing methods in simulations and real data.

Reduces computational complexity from O(n^2) to O(n).

Provides uncertainty estimates for the low-dimensional embeddings.

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.