Special Transformations And Their Application In Doubling Algorithm And Tensor Feature Selection

Algebraic Riccati equation is a special kind of matrix equations,which plays an important role in scientific computation and engineering application.Many problems which we model in the real life,are related to algebraic Riccati equations,including optimal control,queue model,transport theory,applications related to particle beam transport and Markov process.The doubling algorithm is often used to solve the algebraic Riccati matrix equation and its related equations.By using the special structure of matrix and matrix pencil,invariant subspaces or deflating subspaces provide the solutions of the matrix equations.Feature selection is to select a desirable feature subset from the original feature set,which is the main method and effective means of dimensionality reduction of high-dimensional data.In order to build a simpler,more understandable model,we often need to select the feature of the data first in the field of machine learning and data mining.Generally speaking,feature selection problem is mathematical combinatorial optimization problem.We mainly study two mathematical problems from practical application.The first problem is to solve the minimum non-negative solution for nonsymmetric algebraic Riccati equation.The second is feature selection for tensor data.Because special transformations can play an important role in both problems,we study special transformations and their application in doubling algorithm and tensor feature selection.Since the doubling algorithm can effectively obtain the minimum non-negative solution of nonsymmetric algebraic Riccati equation,we study the theory of the doubling algorithm and compare several common doubling algorithms firstly.Af-ter analyzing and comparing the performance of several doubling algorithms,we improve one of them.Numerical results also show that the improved algorithm is better than the original one.Meanwhile,a new generalized transformation is proposed after analyzing the transformation used in several doubling algorithms.It inherits some nice proper-ties of the shift-and-shrink transform and the generalized Cayley transform.The doubling algorithm based on the proposed generalized transformation is presented and shown to be well-defined.Moreover,the convergence result and the compar-ison theorem on convergent rate are established.We prove theoretically that the new doubling algorithm converges faster than other doubling algorithms.Subse-quently,we use numerical experiments to verify the above theory.Finally,we consider a kind of tensor feature selection problem with practical application background.It mainly aims at the problem of feature selection and classification of high order tensor data represented by face data.Firstly,face data is processed by special transformations to solve the influence of external inter-ference such as illumination change.Since the support tensor machine does not have the ability to select features directly,it needs to be improved before feature selection.According to the condition of tensor rank-one decomposition,combined with the linear support higher-order tensor machine model and the threshold con-trol function,we propose a new algorithm for tensor feature selection.Numerical results show that the new algorithm is better than the original one.