A Subspace Dimension Reduction Algorithm Based On Trace Ratio

In many practical applications,such as information retrieval and data mining in ar-tificial intelligence,one often encounters high-dimensional data processing.In order to make the data more compact and reduce the computational complexity of the calculation,people consider the dimensionality reduction methods which project the data reasonably onto the low-dimensional space and meet the aforementioned two requirements well,and the key to the problem becomes how to find a satisfactory projection.Many dimensionality reduction problems will eventually be attributed to the trace ratio optimization problem which maximizes the ratio of two matrix traces,??V?=Tr?V^TAV?/Tr?V^TBV?,where V is a n×p column orthonormal matrix,A is a symmetric matrix,B is a symmetric positive definite matrix.It is difficult to solve the original prob-lem directly,so the problem is often transformed into arg maxTr[?V^TBV?^-1?V^TAV?]as the approximate solution of the trace ratio problem.The ^VI^TT^VR^=I?Iterative Trace Ratio?al-gorithm which is a viable method for this problem has been proposed.It needs to solve _Vm_TVa₌x_ITr[V^T?A-?_kB?V]at each step,where?_kis the result of previous iteration.This is a general symmetric eigenvalue problem.In this paper,we first introduce the Trace Ratio problem and the ITR algorithm.Then we make several improvements on the ITR algorithm,including the selection of the stopping criterion,the refined projection method which is used to compute the eigen-vector in the inner iteration,and the initial value estimated by solving the generalized eigenvalue problem rather than a random selection.These improvements have some ad-vantages in terms of the number of iterations and time.Finally,the convergence of the algorithm is also carried out in detail.The previous convergence analysis is a little,but we proved that{?_k}and{V_k}are both globally convergent and asymptotic quadratic.