Multivariate Extension of Matrix-based Rényi's α-order Entropy Functional. Yu, S., Member, S., Gonzalo Sánchez Giraldo, L., Jenssen, R., Príncipe, J., C., Yu, S., Príncipe, J., C., & Sánchez Giraldo, L., G. 2018.
abstract   bibtex   
The matrix-based Rényi's α-order entropy functional was recently introduced using the normalized eigenspectrum of an Hermitian matrix of the projected data in the reproducing kernel Hilbert space (RKHS). However, the current theory in the matrix-based Rényi's α-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 Rényi's α-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. Index Terms Rényi's α-order entropy functional, Multivariate information quantities, Feature selection.
@article{
 title = {Multivariate Extension of Matrix-based Rényi's α-order Entropy Functional},
 type = {article},
 year = {2018},
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 pages = {1-24},
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 created = {2019-04-01T11:22:40.574Z},
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 abstract = {The matrix-based Rényi's α-order entropy functional was recently introduced using the normalized eigenspectrum of an Hermitian matrix of the projected data in the reproducing kernel Hilbert space (RKHS). However, the current theory in the matrix-based Rényi's α-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 Rényi's α-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. Index Terms Rényi's α-order entropy functional, Multivariate information quantities, Feature selection.},
 bibtype = {article},
 author = {Yu, Shujian and Member, Student and Gonzalo Sánchez Giraldo, Luis and Jenssen, Robert and Príncipe, José C and Yu, Shujian and Príncipe, José C and Sánchez Giraldo, L G}
}
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