Multidimensional Scaling in the Poincare Disk

Multidimensional scaling (MDS) is a class of projective algorithms traditionally used in Euclidean space to produce two- or three-dimensional visualizations of datasets of multidimensional points or point distances. More recently however, several authors have pointed out that for certain datasets, hyperbolic target space may provide a better fit than Euclidean space. In this paper we develop PD-MDS, a metric MDS algorithm designed specifically for the Poincare disk (PD) model of the hyperbolic plane. Emphasizing the importance of proceeding from first principles in spite of the availability of various black box optimizers, our construction is based on an elementary hyperbolic line search and reveals numerous particulars that need to be carefully addressed when implementing this as well as more sophisticated iterative optimization methods in a hyperbolic space model.

💡 Research Summary

The paper introduces PD‑MDS, a metric multidimensional scaling algorithm that operates directly in the Poincaré disk (PD) model of the hyperbolic plane. Traditional MDS techniques embed data in Euclidean space, which can be inadequate for datasets that exhibit hierarchical or tree‑like relationships because Euclidean distances cannot faithfully represent the exponential growth of distances inherent in such structures. Hyperbolic geometry, and specifically the PD model, offers a natural setting for these cases due to its constant negative curvature and conformal properties.

The authors begin by formalizing the PD as the unit disk in the complex plane, D = {z ∈ ℂ | |z| < 1}, and present the closed‑form hyperbolic distance between two points z_j and z_k:
d_D(z_j, z_k) = 2 atanh (|z_j − z_k| / |1 − z_j z̅_k|).
Using this distance, they define a least‑squares stress function E(z) that measures the discrepancy between the target dissimilarities Δ = {δ_jk} and the embedded hyperbolic distances, optionally weighted by a matrix W and masked by an indicator matrix I to handle missing entries.

A key technical contribution is the adaptation of gradient‑based optimization to the hyperbolic setting. Instead of moving points along Euclidean straight lines, the algorithm moves each point along a hyperbolic geodesic (distance‑realizing path) that preserves the conformal structure of the disk. Lemma 1 shows that, given a current point z, a unit‑complex direction γ, and a hyperbolic step size s, the new point can be computed analytically as
z′ = (γ tanh(s/2) + z) / (γ z tanh(s/2) + 1),
which is a Möbius transformation that keeps the point inside the disk.

The step size is parameterized by a scalar r related to the hyperbolic distance by r = tanh(s/2)/‖g‖∞, where g is the complex gradient of the stress function. To avoid crossing the disk boundary, the algorithm enforces r < 1/‖g‖∞, and defines a maximum admissible step r_M based on a user‑specified maximal hyperbolic travel distance s_M (e.g., s_M = 10).

For line search, the authors argue that an exact minimization of the stress along the geodesic is unnecessary and computationally expensive. Instead, they employ an inexact (approximate) line search that satisfies a sufficient‑decrease condition of Armijo type: λ(r) = q(0) + p q′(0) r, with 0 < p < 1, where q(r) = E(z(r)) is the stress as a function of the step size. An efficient binary search finds a suitable r within