Preference-based Centrality and Ranking in General Metric Spaces

TL;DR

This paper tackles centrality and ranking for observations residing in general metric spaces where no natural order exists. It defines a population preference functional $p_F$ based on a dissimilarity $\delta$ and derives an induced centrality $r_F$, then uses a Bradley--Terry--Luce (BTL) projection to obtain a one-dimensional calibrated score $\theta^*$ with strength $s^*$ that preserves the $r_F$ ordering. Two practical estimators are developed: CORE-GD, a convex M-estimation-based method, and CORE-Spectral, a fast spectral approximation; both yield scores that satisfy canonical depth properties under mild conditions. Theoretical results establish existence, uniqueness, and a monotone link between $\theta^*$ and $r_F$, while simulations demonstrate stable, high-resolution rankings in high-dimensional and non-Euclidean settings, outperforming classical depth methods in non-elliptical regimes. The framework is scalable and broadly applicable to non-Euclidean data, enabling robust centrality-based ordering and out-of-sample scoring via kernel smoothing of the learned scores.

Abstract

Assessing centrality or ranking observations in multivariate or non-Euclidean spaces is challenging because such data lack an intrinsic order and many classical depth notions lose resolution in high-dimensional or structured settings. We propose a preference-based framework that defines centrality through population pairwise proximity comparisons: a point is central if a typical draw from the underlying distribution tends to lie closer to it than to another. This perspective yields a well-defined statistical functional that generalizes data depth to arbitrary metric spaces. To obtain a coherent one-dimensional representation, we study a Bradley-Terry-Luce projection of the induced preferences and develop two finite-sample estimators based on convex M-estimation and spectral aggregation. The resulting procedures are consistent, scalable, and applicable to high-dimensional and non-Euclidean data, and across a range of examples they exhibit stable ranking behavior and improved resolution relative to classical depth-based methods.

Figures (3)

• Figure 1: One-dimensional comparison of four CORE estimators. Each panel corresponds to a distribution (Normal or skewed Laplace) and a sample size $n\in\{50,200,500,2000\}$. Estimated scores $\hat{\theta}_i$ are plotted against the sample points $x_i$. The vertical dashed line marks the population maximizer $\mu^\star$ of $r(\cdot)$.
• Figure 2: Comparison of the four estimators. Each panel fixes a distribution (Normal or skewed Laplace), and overlays the estimated score curves obtained at sample sizes $n\in\{50,200,500,2000\}$. In each panel, the scores $\hat{\theta}_i$ are plotted against the ordered sample points $x_{(i)}$. The vertical dashed line marks the population maximizer $\mu^\star=0$ for the Normal case and $\mu^\star=-0.462$ for the skewed Laplace case.
• Figure 3: Mean Pearson correlation between the estimated score vectors and the Population-GD benchmark over $R=20$ independent replications. Points indicate the mean correlation and vertical error bars denote $\pm 1$ standard deviation.

Theorems & Definitions (8)

• Remark 1: Discrete scores versus parametric scoring
• Remark 2: Win-rate scores and their relation to CORE
• Theorem 1: Existence and uniqueness
• Remark 3: Correct specification as a special case
• Theorem 2: Monotone calibration link between $\theta^\star$ and $r$
• Theorem 3: Depth properties
• Proposition 1
• Proposition 2