Geometric Estimation of Multivariate Dependency

Area of Science:

• Statistics
• Machine Learning
• Information Theory

Background:

• Measuring dependency between multivariate random variables is crucial in various scientific fields.
• Traditional methods like mutual information often require density estimation, limiting scalability to large datasets.
• Existing dependency measures may struggle with complex, high-dimensional data structures.

Purpose of the Study:

• To propose a novel geometric estimator for quantifying dependency between multivariate random variables.
• To introduce Geometric Mutual Information (GMI) as a scalable and effective dependency measure.
• To demonstrate the practical advantages of the proposed GMI estimator through empirical evaluation.

Main Methods:

• Utilizing a randomly permuted geometric graph, specifically the minimal spanning tree (MST), on paired multivariate samples.
• Developing an empirical estimator for GMI that avoids density estimation by constructing an MST over original and permuted data.
• Establishing asymptotic convergence and analyzing bias/variance convergence rates for smooth multivariate density functions.

Main Results:

• The proposed estimator converges to Geometric Mutual Information (GMI), equivalent to the Henze-Penrose divergence.
• GMI exhibits desirable properties similar to standard Mutual Information (MI) but is computationally scalable.
• Empirical experiments demonstrate the advantages of the geometric dependency estimator over existing methods.

Conclusions:

• The geometric dependency estimator based on MST provides a scalable and effective approach for measuring multivariate variable relationships.
• GMI offers a practical alternative to traditional dependency measures, particularly for large and complex datasets.
• The proposed method shows promise for applications in statistical analysis, machine learning, and data mining.