Research Of Robust Dimensionality Reduction Algorithms Based On TL1 And Its Applications

With the development of science and technology,it has become easier for people to obtain large-scale data.However,the process of data collection is affected by many factors,such as hardware equipment,environmental factors,and abnormal operations.The collected data inevitably contains noise,which is far from the true distribution.When the classical dimensionality reduction algorithms are used to analyze the data with noise,it is difficult to extract robust features.In order to overcome this problem,lots of robust dimensionality reduction algorithms have been proposed by replacing the norms sensitive to noise with various norms insensitive to noise,such as L1-norm-based principal component analysis and L1-norm-based linear discrimination Analysis etc.In order to generalize the L1-norm,the Lp-norm((27)(27)20 p)is proposed as the metric of the dimensionality reduction algorithms.For(27)(27)10 p,the Lp-norm-based dimensionality reduction algorithms seem to be more robust than the L1-norm-based ones.However,Lp-norm does not satisfy Lipschitz continuity and boundedness.In addition,recently,as multi-view data has become more abundant,dimensionality reduction algorithms based on multi-view data can themselves improve robustness to a certain extent.Aiming at the deficiencies of the Lp-norm,this paper proposes a new robust metric and applies it to principal component analysis and linear discriminant analysis;in view of the advantages of multi-view learning,considering that the data of each view may contain noise,we propose a robust multi-view learning algorithm.The above work further extends the robust dimensionality reduction algorithms.The main work of this article can be summarized as follows:(1)In relevant literature,it is found that TL1 and Lp-norm((27)(27)10 p)are similar to a certain extent,and TL1 also has a positive parameter that needs to be adjusted.The study finds that TL1 can be as close as possible to the corresponding Lp norm by adjusting a,indicating that there seems to be a certain corresponding relationship with the corresponding Lp-norm.And TL1 has Lipschitz continuity and boundedness.However,TL1 is usually used as a penalty instead of a metric,so we consider TL1 as a metric in this paper.Obviously,TL1 is a robust metric.Therefore,we propose a TL1-based principal component analysis(TL1PCA),which uses TL1 as a measure to maximize the dispersion.It not only improves the robustness of the model,but also improves the continuity of the objective function.However,the optimization model of TL1 PCA is non-convex and non-smooth,which is difficult to solve.To this end,we design an iterative algorithm satisfying convergence and clarify its convergence.Experimental results show that TL1 PCA is more robust than PCAs based on L2-norm,L1-norm and,Lp-norm.(2)But TL1 PCA is only used for vector data.When TL1 PCA is applied to matrix data(such as image data),the matrix data need to be converted into vector data,which lead to the loss of spatial structure information.However,the classical two-dimensional principal component analysis uses the F norm as a metric,which is sensitive to noise.To overcome this problem,inspired by the success of TL1 in one-dimensional principal component analysis,this paper proposes a TL1-based two-dimensional principal component analysis(2DPCA-TL1),which is essentially a generalization of TL1 PCA on matrix data.Compared with the classical two-dimensional principal component analysis,the robustness of2DPCA-TL1 has been significantly improved;compared with TL1 PCA,2DPCA-TL1 not only uses the spatial structure information in the data,but also improves the performance and computational efficiency of the model.We conduct experiments on some real image datasets with noise,and the results show that 2DPCA-TL1 significantly improves the performance of two-dimensional principal component analysis,thus demonstrating the effectiveness of the algorithm.(3)TL1PCA and 2DPCA-TL1 improve the robustness of the model,but both are unsupervised algorithms and do not use the label information in the data.To overcome this problem,this paper proposes one-dimensional linear discriminant analysis based on TL1(TL1LDA)and two-dimensional linear discriminant analysis(2DLDA-TL1).TL1 LDA and2DLDA-TL1 not only improve the robustness of the model,but also use the label information in the data,so they theoretically have better performance than TL1 PCA and 2DPCA-TL1.Compared with TL1 LDA,2DLDA-TL1 also uses the spatial structure information in the data,so 2DLDA-TL1 theoretically has better performance than TL1 LDA.To solve these two algorithms,we combine the research of TL1 PCA and 2DPCA-TL1,and merge TL1 LDA and2DLDA-TL1 into the same iterative algorithm to solve.The experimental results show that TL1 LDA and 2DLDA-TL1 perform better than TL1 PCA and 2DPCA-TL1,respectively,and2DLDA-TL1 performs better than TL1 LDA and 2DPCA-TL1.And the experimental results are also consistent with the theoretical analysis.(4)The above work has conducted in-depth research on the single-view learning algorithms.However,with the development of data collection technology,multi-view data is becoming more and more abundant,and algorithms based on multi-view data can themselves improve robustness to a certain extent.Consider that the data of each view may contain noise,but most current multi-view learning algorithms essentially use L2-norm or F-norm as metric,which are sensitive to noise.Since L1-norm is a robust metric,this paper proposes a L1-norm-based robust multi-view discriminant analysis(L1-Mv DA-VC),and on the basis of L1-Mv DA-VC,a TL1-based robust multi-view discriminant analysis TL1-Mv DA-VC is further proposed.In addition to using multi-view data,L1-Mv DA-VC and TL1-Mv DA-VC also use label information.In order to further enhance the robustness of the algorithm,we have introduced structural information from the views.Due to the difficulty of solving the optimization problem of TL1-Mv DA-VC,we focuse on constructing a simple and effective iterative algorithm for L1-Mv DA-VC and prove its monotonicity.The experimental results show the effectiveness of L1-Mv DA-VC.