On the sum of the largest eigenvalues of a symmetric matrix
SIAM Journal on Matrix Analysis and Applications
Mathematical Programming: Series A and B - Special issue: Festschrift in Honor of Philip Wolfe part II: studies in nonlinear programming
Efficient Pattern Recognition Using a New Transformation Distance
Advances in Neural Information Processing Systems 5, [NIPS Conference]
Null space versus orthogonal linear discriminant analysis
ICML '06 Proceedings of the 23rd international conference on Machine learning
Trace quotient problems revisited
ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part II
Normalized Cuts Revisited: A Reformulation for Segmentation with Linear Grouping Constraints
Journal of Mathematical Imaging and Vision
The Trace Ratio Optimization Problem for Dimensionality Reduction
SIAM Journal on Matrix Analysis and Applications
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The trace quotient problem arises in many applications in pattern classification and computer vision, e.g., manifold learning, low-dimension embedding, etc. The task is to solve a optimization problem involving maximizing the ratio of two traces, i.e., maxW Tr(f(W))/Tr(h(W)). This optimization problem itself is non-convex in general, hence it is hard to solve it directly. Conventionally, the trace quotient objective function is replaced by a much simpler quotient trace formula, i.e., maxW Tr (h(W)-1f(W)), which accommodates a much simpler solution. However, the result is no longer optimal for the original problem setting, and some desirable properties of the original problem are lost. In this paper we proposed a new formulation for solving the trace quotient problem directly. We reformulate the original non-convex problem such that it can be solved by efficiently solving a sequence of semidefinite feasibility problems. The solution is therefore globally optimal. Besides global optimality, our algorithm naturally generates orthonormal projection matrix. Moreover it relaxes the restriction of linear discriminant analysis that the projection matrix's rank can only be at most c - 1, where c is the number of classes. Our approach is more flexible. Experiments show the advantages of the proposed algorithm.