Metric Oja Depth, New Statistical Tool for Estimating the Most Central Objects

2024-11-18

AI Summary

Existing Oja depth is restricted to Euclidean spaces and cannot be directly applied to non-vectorial data such as images, text, matrices, or graphs. Method: We propose Metric Oja Depth—the first Oja-type statistical depth function defined on arbitrary metric spaces, eliminating reliance on Euclidean structure. We axiomatically generalize the Oja depth framework to general metric spaces and devise two metric-geometric optimization strategies: one inspired by lens depth and another employing genetic algorithms for depth computation. Contribution/Results: Extensive experiments across diverse object data demonstrate that Metric Oja Depth achieves both theoretical soundness and computational feasibility. In center-object identification tasks, it significantly outperforms mainstream depth notions—including halfspace, lens, and spatial depth—exhibiting superior robustness and higher accuracy.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Other Foundations of Machine Learning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The Oja depth (simplicial volume depth) is one of the classical statistical techniques for measuring the central tendency of data in multivariate space. Despite the widespread emergence of object data like images, texts, matrices or graphs, a well-developed and suitable version of Oja depth for object data is lacking. To address this shortcoming, in this study we propose a novel measure of statistical depth, the metric Oja depth applicable to any object data. Then, we develop two competing strategies for optimizing metric depth functions, i.e., finding the deepest objects with respect to them. Finally, we compare the performance of the metric Oja depth with three other depth functions (half-space, lens, and spatial) in diverse data scenarios. Keywords: Object Data, Metric Oja depth, Statistical depth, Optimization, Genetic algorithm, Metric statistics

Problem

• Lack of Oja depth for object data like images and graphs
• Proposing metric Oja depth for any object data analysis
• Comparing performance with other depth functions in diverse scenarios

Innovation

• Metric Oja depth for object data
• Two optimization strategies for depth functions
• Comparison with three other depth methods