Bayesian Hyperbolic Multidimensional Scaling
This method addresses the need for uncertainty assessment and regularization in hierarchical data analysis, such as in text or genetics, but is incremental as it builds on existing MDS techniques.
The authors tackled the problem of representing high-dimensional data with tree-like structures by proposing a Bayesian approach to multidimensional scaling on hyperbolic manifolds, achieving computational efficiency with a reduction from O(n^2) to O(n) through a case-control likelihood approximation.
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.