TY - JOUR

T1 - A note on the trace quotient problem

AU - Zhang, Lei Hong

AU - Yang, Wei Hong

AU - LIAO, Lizhi

N1 - Funding Information:
Acknowledgments We would like to thank the Editor and the anonymous referees for their helpful suggestions that have improved the presentation of this paper. The starting point strategy presented in Sect. 4 was motivated by one referee’s comments. The work of L.-H. Zhang was supported in part by the National Natural Science Foundation of China NSFC-11101257 and the Basic Academic Discipline Program, the 11th 5 year plan of 211 Project for Shanghai University of Finance and Economics. The work of L.-Z. Liao was supported in part by FRG grants from Hong Kong Baptist University and GRF grants from the Research Grant Council of Hong Kong.

PY - 2014/1

Y1 - 2014/1

N2 - The trace quotient problem or the trace ratio problem (TRP) is to find an orthogonal matrix (Formula presented.) that maximizes the quotient (Formula presented.) for a given symmetric matrix B∈ℝm×m and a symmetric positive definite matrix W∈ℝm×m. It has a crucial role in linear discriminant analysis and has many other applications in computer vision and machine learning as well. In this short note, we first establish the classical first and second order optimality conditions for TRP. As a straightforward application of these optimality conditions, we contribute a simple proof for the property that TRP does not admit local non-global maximizer, which is first proved by Shen et al. (A geometric revisit to the trace quotient problem, proceedings of the 19th International Symposium of Mathematical Theory of Networks and Systems, 2010) based on Grassmann manifold. Without involving much knowledge of the underlying differential geometry, our proof primarily uses basic properties in linear algebra, which also leads to an effective starting pointing strategy for any monotonically convergent iteration to find the global maximizer of TRP.

AB - The trace quotient problem or the trace ratio problem (TRP) is to find an orthogonal matrix (Formula presented.) that maximizes the quotient (Formula presented.) for a given symmetric matrix B∈ℝm×m and a symmetric positive definite matrix W∈ℝm×m. It has a crucial role in linear discriminant analysis and has many other applications in computer vision and machine learning as well. In this short note, we first establish the classical first and second order optimality conditions for TRP. As a straightforward application of these optimality conditions, we contribute a simple proof for the property that TRP does not admit local non-global maximizer, which is first proved by Shen et al. (A geometric revisit to the trace quotient problem, proceedings of the 19th International Symposium of Mathematical Theory of Networks and Systems, 2010) based on Grassmann manifold. Without involving much knowledge of the underlying differential geometry, our proof primarily uses basic properties in linear algebra, which also leads to an effective starting pointing strategy for any monotonically convergent iteration to find the global maximizer of TRP.

KW - Global maximizer

KW - Grassmann manifold

KW - Karush-Kuhn-Tucker conditions

KW - Linear discriminant analysis

KW - Trace ratio (quotient) problem

UR - http://www.scopus.com/inward/record.url?scp=84901615819&partnerID=8YFLogxK

U2 - 10.1007/s11590-013-0680-z

DO - 10.1007/s11590-013-0680-z

M3 - Article

AN - SCOPUS:84901615819

VL - 8

SP - 1637

EP - 1645

JO - Optimization Letters

JF - Optimization Letters

SN - 1862-4472

IS - 5

ER -