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Entropy 2017, 19(3), 89; doi:10.3390/e19030089

Optimization of Alpha-Beta Log-Det Divergences and their Application in the Spatial Filtering of Two Class Motor Imagery Movements

1
Department of Sensor and Biomedical Technology, School of Electronics Engineering, VIT University, Vellore, Tamil Nadu 632014, India
2
Departamento de Teoría de la Señal y Comunicaciones, Universidad de Sevilla, Camino de los Descubrimientos s/n, Seville 41092, Spain
3
Laboratory for Advanced Brain Signal Processing, Brain Science Institute, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
4
Systems Research Institute, Polish Academy of Sciences, Warsaw 01-447, Poland
5
Skolkovo Institute of Science and Technology (Skoltech), Moscow 143026, Russia
*
Author to whom correspondence should be addressed.
Academic Editor: Kevin H. Knuth
Received: 13 December 2016 / Revised: 7 February 2017 / Accepted: 22 February 2017 / Published: 25 February 2017
(This article belongs to the Section Information Theory)
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Abstract

The Alpha-Beta Log-Det divergences for positive definite matrices are flexible divergences that are parameterized by two real constants and are able to specialize several relevant classical cases like the squared Riemannian metric, the Steins loss, the S-divergence, etc. A novel classification criterion based on these divergences is optimized to address the problem of classification of the motor imagery movements. This research paper is divided into three main sections in order to address the above mentioned problem: (1) Firstly, it is proven that a suitable scaling of the class conditional covariance matrices can be used to link the Common Spatial Pattern (CSP) solution with a predefined number of spatial filters for each class and its representation as a divergence optimization problem by making their different filter selection policies compatible; (2) A closed form formula for the gradient of the Alpha-Beta Log-Det divergences is derived that allows to perform optimization as well as easily use it in many practical applications; (3) Finally, in similarity with the work of Samek et al. 2014, which proposed the robust spatial filtering of the motor imagery movements based on the beta-divergence, the optimization of the Alpha-Beta Log-Det divergences is applied to this problem. The resulting subspace algorithm provides a unified framework for testing the performance and robustness of the several divergences in different scenarios. View Full-Text
Keywords: similarity measures; Common Spatial Pattern (CSP); generalized divergences for symmetric positive definite matrices; Brain Computer Interface (BCI); Alpha-Beta Log-Det divergence similarity measures; Common Spatial Pattern (CSP); generalized divergences for symmetric positive definite matrices; Brain Computer Interface (BCI); Alpha-Beta Log-Det divergence
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Thiyam, D.B.; Cruces, S.; Olias, J.; Cichocki, A. Optimization of Alpha-Beta Log-Det Divergences and their Application in the Spatial Filtering of Two Class Motor Imagery Movements. Entropy 2017, 19, 89.

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