A note on the trace quotient problem

Abstract

The trace quotient problem or the trace ratio problem (TRP) is to find an orthogonal matrix $V\in \mathbb{R }^{m\times \ell } ~(m\ge \ell )$ that maximizes the quotient ${{\mathrm{tr}}}(V^\top BV)/{{\mathrm{tr}}}(V^\top WV)$ for a given symmetric matrix $B\in \mathbb R ^{m\times m}$ and a symmetric positive definite matrix $W\in \mathbb R ^{m\times m}.$ 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.