Nonlinear Dimensionality Reduction Research Based On Differential Manifold

In the modern information age, almost all the message carrier in our daily life can be stored and processed as unstructured data, such as article, voice and image. However, the high-dimensional data could not be processed by existing machine learning, data mining and data analysis methods efficiently. How to discover the meaningful and valuable underlying factors in the massive high-dimensional data is a basic question in information science. Data dimensionality reduction is the main approach to deal with this problem. Data dimensionality reduction can be divided into linear dimensionality reduction methods and nonlinear dimensionality reduction methods. Since the distribution of data often has characteristic of nonlinearity, linear dimensionality reduction methods have many limitations. Therefore, nonlinear dimensionality reduction methods play an important role and had been widely researched in machine learning, computer vision, data mining, image analysis and many other fields.By now, the nonlinear dimensionality reduction methods mainly include Neural Network based methods, Kernel function based methods and Manifold Learning methods. Among them, manifold learning methods attract an extensive attention by its explicit geometric interpretation and biological foundation. At present, manifold learning is still in theoretical research and there are many problems hinder its application in real problems. This paper focuses on several key problems in application and theory of manifold learning.Firstly, this paper analyzed the existing manifold learning methods in theoretical, such as Isomap, LLE, LTSA, and applications in face recognition, pose estimation and 3D human gait capture problem. Since the recognition problem and pose estimation problem are interrelated and interact on each other, this paper proposed a framework to deal with these two problems simultaneously and verified it by simulation experiments on FacePix database. Since the manifold could reflect the motion of target object, this paper utilized Isomap to find low-dimensional manifold in pixel space and 3D skeleton point space and adopted generalized regression neural network to build mapping between 2D gait image and 3D gait skeleton points and verified this method by simulation experiments on Weizamann and CMU database.Secondly, this paper analyzed the assessment indicators, the shortcomings of existing neighborhood selection methods and characteristics of a reasonable neighborhood in theoretical and the experiments. Two adaptive maximal linear neighborhood methods are proposed which based on the included angle between normal hyper plane and neighborhood expansion respectively. These two methods could overcome “short edges” and many other shortcomings in tradition neighborhood selection methods. To overcome the problem that dimensionality reduction results are sensitive to neighborhood parameter selection, this paper improved Isomap and LLE by substituting the KNN neighborhood with maximal linear neighborhood and added weights into objective function. The experiments show that these methods improved the performance and robustness of manifold learning.Thirdly, this paper proposed a novel manifold learning method based on atlas compatibility transformation. By analyzing the physical meaning of atlas and compatibility in scatter point database, this paper constructed the objective function to minimize the sum of distances of the coordinates of intersection points. The object function could be turned into an extremum problem with constraints and sovled by generalized eigenvalue decomposition. The methods of incremental learning and reconstruction were also given. The validity of method was illustrated by scatter point cloud dataset and image dataset experiments.Finally, to reduce the dimensionality reduction error that caused by noise and non-uniform of data points, this paper analyzed the dimensionality reduction results at different intensity noise and different sparse degree and proposed the de-noise method and data interpolation method. These methods utilized linear projection, coordinate transform and fast convex hull algorithm and improved the performance of manifold learning. The validity of proposed methods is illustrated by experiments.