On The Manifold Learning Algorithm Based On Homotopy Mapping

The issue of manifold learning,which is a branch of dimension reduction of high-dimensional data,is concerned in this thesis.Faced with the problem of "curse of dimensionality",many data analysis algorithms will be powerless.So,before the analysis of high-dimensional data,we usually use the method of data dimensionality reduction to process high-dimensional data.In this way,on the basis of maintaining the most effective information,we can effectively improve the efficiency of data analysis.Among many dimensionality reduction methods of high-dimensional data,the nonlinear dimensionality reduction method has become a new research hotspot,and the development of manifold learning has attracted more and more attention.In this thesis,the problem of dimensionality reduction of high-dimensional manifold data is studied based on homotopy mapping and the Isomap algorithm is restudied based on correlation entropy.The main contributions are as follows:1.With the help of homotopy mapping,we study the dimensionality reduction of nonlinear manifold data.The main idea here is to map the data points generated by the nonlinear manifold to a linear manifold in the same space by homotopy mapping in order to remove the non-linear structure of the original data points.With the " unfolding" of the high dimensional manifold,the dimensionality reduction of high-dimensional data will be achieved.Due to the analytical expression of the nonlinear manifold which generates the high dimensional data is unknown,in this thesis,an adaptive neural fuzzy inference system(Anfis)is adopted to approximate the high dimensional data point set.By iterations,the Anfis is mapped onto a linear manifold.Because the mapping process is realized in the same space,the corresponding relationship between the original nonlinear data point set and the reduced dimension data point set is obvious.Therefore,compared with the classical manifold algorithms,such as LLE and Isomap,the proposed algorithm in this thesis has great advantages.Furthermore,because of the characteristics of homotopy mapping,the local linear structure and local neighborhood of the original data are well preserved.This makes the dimension reduced data reflect the structural characteristics of the original data well,which will provide a good support for the subsequent data analysis process.The simulation results show that the data change process is continuous.The dimensionality reduction process is very intuitive.The effect of data dimensionality reduction is very good.2.The Isomap algorithm is restudied based on the correlation entropy.In the proposed algorithm,we replace the Euclidean distance measure with the correlation entropy measure,and reset the shortest distance as the geodesic distance.When using the traditional Isomap algorithm to reduce the dimension of high-dimensional data,if the neighborhood of the data set is large,there will be a short circuit phenomenon.The improved Isomap dimension reduction algorithm can still achieve the purpose of dimension reduction when the data set neighborhood is large.The innovation of this thesis is using homotopy mapping method to reduce the dimension of data.Different from the previous dimensionality reduction algorithms,such as LLE and Isomap,which needs to calculate the eigenvalues and eigenvectors of large matrices,which is more complex,the proposed homotopy mapping dimension reduction algorithm is to map the high-dimensional nonlinear manifold to a low-dimensional linear manifold,which greatly reduces the computational complexity of data dimension reduction,and the dimension reduction process is reversible.