Journal
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
Volume 23, Issue 3, Pages 519-526Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TNNLS.2011.2178037
Keywords
Flexible semi-supervised discriminant analysis; semi-supervised dimension reduction; trace ratio
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Funding
- Singapore National Research Foundation
- Interactive Digital Media Research and Development Program Office, MDA [NRF2008IDM-IDM004-018]
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In this brief, we address the trace ratio (TR) problem for semi-supervised dimension reduction. We first reformulate the objective function of the recent work semi-supervised discriminant analysis (SDA) in a TR form. We also observe that in SDA the low-dimensional data representation F is constrained to be in the linear subspace spanned by the training data matrix X (i.e., F = (XW)-W-T). In order to relax this hard constraint, we introduce a flexible regularizer parallel to F -(XW)-W-T parallel to(2) which models the regression residual into the reformulated objective function. With such relaxation, our method referred to as TR based flexible SDA (TR-FSDA) can better cope with data sampled from a certain type of nonlinear manifold that is somewhat close to a linear subspace. In order to address the non-trivial optimization problem in TR-FSDA, we further develop an iterative algorithm to simultaneously solve for the low-dimensional data representation F and the projection matrix W. Moreover, we theoretically prove that our iterative algorithm converges to the optimum based on the Newton-Raphson method. The experiments on two face databases, one shape image database and one webpage database demonstrate that TR-FSDA outperforms the existing semi-supervised dimension reduction methods.
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