Multivariate Extension of Matrix-based Renyi's \alpha-order Entropy Functional

TL;DR

This paper extends matrix-based Renyi's -order entropy to multivariate cases, enabling estimation of joint and interactive information among multiple variables, with applications in feature selection and hyperspectral image analysis.

Contribution

It introduces a novel multivariate matrix-based Renyi's -order entropy functional, facilitating the estimation of complex multivariate information quantities.

Findings

Defined multivariate joint entropy using matrix-based Renyi's -order entropy.

Demonstrated the estimation of interactive information and total correlation.

Applied the method to feature selection in hyperspectral images.

Abstract

The matrix-based Renyi's \alpha-order entropy functional was recently introduced using the normalized eigenspectrum of a Hermitian matrix of the projected data in a reproducing kernel Hilbert space (RKHS). However, the current theory in the matrix-based Renyi's \alpha-order entropy functional only defines the entropy of a single variable or mutual information between two random variables. In information theory and machine learning communities, one is also frequently interested in multivariate information quantities, such as the multivariate joint entropy and different interactive quantities among multiple variables. In this paper, we first define the matrix-based Renyi's \alpha-order joint entropy among multiple variables. We then show how this definition can ease the estimation of various information quantities that measure the interactions among multiple variables, such as interactive information and total correlation. We finally present an application to feature selection to show how our definition provides a simple yet powerful way to estimate a widely-acknowledged intractable quantity from data. A real example on hyperspectral image (HSI) band selection is also provided.