Global Optimization Methods for Extended Fisher Discriminant Analysis

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Satoru Iwata, Yuji Nakatsukasa, Akiko Takeda ;
Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, PMLR 33:411-419, 2014.

Abstract

The Fisher discriminant analysis (FDA) is a common technique for binary classification. A parametrized extension, which we call the extended FDA, has been introduced from the viewpoint of robust optimization. In this work, we first give a new probabilistic interpretation of the extended FDA. We then develop algorithms for solving an optimization problem that arises from the extended FDA: computing the distance between a point and the surface of an ellipsoid. We solve this problem via the KKT points, which we show are obtained by solving a generalized eigenvalue problem. We speed up the algorithm by taking advantage of the matrix structure and proving that a globally optimal solution is a KKT point with the smallest Lagrange multiplier, which can be computed efficiently as the leftmost eigenvalue. Numerical experiments illustrate the efficiency and effectiveness of the extended FDA model combined with our algorithm.

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