Weighted lens depth: Some applications to supervised classification

TL;DR

This paper extends the concept of lens depth to metric spaces and Riemannian manifolds, demonstrating its theoretical properties and practical applications in supervised classification, including real-world datasets.

Contribution

It introduces a novel extension of lens depth to metric and manifold data, with theoretical proofs and applications in pattern recognition.

Findings

Extended lens depth to Riemannian manifolds.

Proved main properties of the new depth measure.

Applied to real datasets for pattern recognition.

Abstract

Starting with Tukey's pioneering work in the 1970's, the notion of depth in statistics has been widely extended especially in the last decade. These extensions include high dimensional data, functional data, and manifold-valued data. In particular, in the learning paradigm, the depth-depth method has become a useful technique. In this paper we extend the notion of lens depth to the case of data in metric spaces, and prove its main properties, with particular emphasis on the case of Riemannian manifolds, where we extend the concept of lens depth in such a way that it takes into account non-convex structures on the data distribution. Next we illustrate our results with some simulation results and also in some interesting real datasets, including pattern recognition in phylogenetic trees using the depth--depth approach.