Introduction to statistical pattern recognition (2nd ed.)
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Matrix computations (3rd ed.)
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Pattern Classification (2nd Edition)
Pattern Classification (2nd Edition)
Local Discriminant Embedding and Its Variants
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 2 - Volume 02
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ICDM '05 Proceedings of the Fifth IEEE International Conference on Data Mining
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AAAI'08 Proceedings of the 23rd national conference on Artificial intelligence - Volume 2
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ACCV'07 Proceedings of the 8th Asian conference on Computer vision - Volume Part II
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ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part II
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This paper considers the problem of optimizing the ratio $\mathrm{Tr}[V^{T}AV]/\mathrm{Tr}[V^{T}BV]$ over all unitary matrices $V$ with $p$ columns, where $A,B$ are two positive definite matrices. This problem is common in supervised learning techniques. However, because its numerical solution is typically expensive it is often replaced by the simpler optimization problem which consists of optimizing $\mathrm{Tr}[V^{T}AV]$ under the constraint that $V^{T}BV=I$, the identity matrix. The goal of this paper is to examine this trace ratio optimization problem in detail, to consider different algorithms for solving it, and to illustrate the use of these algorithms for dimensionality reduction.