Trace Ratio vs. Ratio Trace for Dimensionality Reduction
- 1 June 2007
- conference paper
- conference paper
- Published by Institute of Electrical and Electronics Engineers (IEEE)
Abstract
A large family of algorithms for dimensionality reduction end with solving a Trace Ratio problem in the form of arg maxW Tr(WT SPW)/Tr(WT SIW)1, which is generally transformed into the corresponding Ratio Trace form arg maxW Tr[ (WTSIW)-1 (WTSPW) ] for obtaining a closed-form but inexact solution. In this work, an efficient iterative procedure is presented to directly solve the Trace Ratio problem. In each step, a Trace Difference problem arg maxW Tr [WT (SP - lambdaSI) W] is solved with lambda being the trace ratio value computed from the previous step. Convergence of the projection matrix W, as well as the global optimum of the trace ratio value lambda, are proven based on point-to-set map theories. In addition, this procedure is further extended for solving trace ratio problems with more general constraint WTCW=I and providing exact solutions for kernel-based subspace learning problems. Extensive experiments on faces and UCI data demonstrate the high convergence speed of the proposed solution, as well as its superiority in classification capability over corresponding solutions to the ratio trace problem.Keywords
This publication has 7 references indexed in Scilit:
- Graph Embedding and Extensions: A General Framework for Dimensionality ReductionIEEE Transactions on Pattern Analysis and Machine Intelligence, 2006
- Handbook of Pattern Recognition and Computer VisionPublished by World Scientific Pub Co Pte Ltd ,2005
- A unified framework for subspace face recognitionIeee Transactions On Pattern Analysis and Machine Intelligence, 2004
- Face recognition using eigenfacesPublished by Institute of Electrical and Electronics Engineers (IEEE) ,2002
- Topics in Matrix AnalysisPublished by Cambridge University Press (CUP) ,1991
- Sufficient conditions for the convergence of monotonic mathematicalprogramming algorithmsJournal of Computer and System Sciences, 1976
- Point-to-Set Maps in Mathematical ProgrammingSIAM Review, 1973