Title: QDOT: An Efficient Quantile-weighted Distance Metric for Geometric Comparison via Optimal Transport
Primary area: learning on graphs and other geometries & topologies
Keywords: Optimal Transport; Isometry; Distribution comparison; Geometric matching

Abstract: Measuring the discrepancy between data distributions in heterogeneous metric spaces is a fundamental challenge. 
Existing methods, typically based on geometric structures, address this by embedding distributions into a shared space. 
However, these approaches face fundamental limitations, including the loss of geometric information, computationally intractable representations, and inability to preserve essential structural features. 
In this work, we introduce the Quantile-weighted Distance Optimal Transport (QDOT), a novel and efficient metric for geometric comparison. 
QDOT constructs a family of isometry-invariant distance representations by leveraging distance quantiles as structural weights in Euclidean space, thereby preserving essential geometric characteristics and enabling optimal transport coupling within a common space. 
We prove that, under mild conditions, QDOT is a well-defined metric with a convergence rate no slower than the classical Wasserstein distance. Moreover, we present an integral version that computes the loss in complexity of $\mathcal{O}(n\log n)$. 
Extensive experiments demonstrate that our methods achieves strong performance across diverse applications, including cross-space comparison, transfer learning, and molecule generation, while also achieving state-of-the-art results on several key metrics.

TL;DR: We introduce QDOT, an efficient metric that makes cross-space distribution comparison practical with theoretical guarantees in metric properity and empirical convergence rate. QDOT achieves strong performance across a diverse range of experiments.