Autocorrelation Ac-shaped Learning

Dimensional reduction techniques are used for representing the relationship among higher-dimensional data by a lower-dimensional structure. Many algorithms for manifold learning and nonlinear dimensional reduction such as Isomeitric Mapping(Isomap),Local Linear Embedding (LLE) and Local Tangent Space Alignment (LTSA), merge the local linear geometry obtained by Euclidean neighborhood to a global embedding space. However, when the underlying lower-dimensional manifold is self-intersecting at some points, it's difficult to obtain local lower-dimensional structure from the Euclidean neighborhoods. In this case, these dimensional reduction methods can not be used directly. To solve this problem,we present a method of self-intersecting manifold learning. The core of this method is to select neighborhood points based on tangent space clustering. We construct a layered neighbor-graph to storage the trusted information of low dimensional neighborhood. We first define regular points by local low-dimensional approximation, strongly regularize their neighborhood based on clustering tangent spaces, and build the neighbor relationship among these regular points. Based on this relationship, we gradually confirm other points' neighborhood,and build the neighbor relationship among all the points. We illustrate our methods using data from self-intersecting curves, surfaces in 2D/3D and higher-dimensional spaces.