Optimal Riemannian quantization with an application to air traffic analysis

TL;DR

This paper introduces a Riemannian quantization algorithm for summarizing complex data on manifolds, with applications to air traffic analysis, demonstrating its convergence and practical utility.

Contribution

The paper presents CLRQ, an online Riemannian quantization algorithm with proven convergence, tailored for data on manifolds, and applies it to real-world air traffic image data.

Findings

CLRQ converges on sphere and hyperbolic plane examples.

Effective summarization of air traffic images using Riemannian quantization.

Clusterings reflect airspace homogeneity and can be used for further analysis.

Abstract

The goal of optimal quantization is to find the best approximation of a probability distribution by a discrete measure with finite support. When dealing with empirical distributions, this boils down to finding the best summary of the data by a smaller number of points, and automatically yields a K-means-type clustering. In this paper, we introduce Competitive Learning Riemannian Quantization (CLRQ), an online algorithm that computes the optimal summary when the data does not belong to a vector space, but rather a Riemannian manifold. We prove its convergence and show simulated examples on the sphere and the hyperbolic plane. We also provide an application to real data by using CLRQ to create summaries of images of covariance matrices estimated from air traffic images. These summaries are representative of the air traffic complexity and yield clusterings of the airspaces into zones that are homogeneous with respect to that criterion. They can then be compared using discrete optimal transport and be further used as inputs of a machine learning algorithm or as indexes in a traffic database.