Flow Manifold Learning Algorithm

The development of scientific technology has moved us into an information era. People often work with huge amounts of various data, such as biology data, image data and so on. Although the increasing of the dimensionality of the data provides us more rich information, it has brought us great challenge to deal with these data. In order to discover the intrinsic low dimensional structures hidden in the high dimensional observations, various dimensionality reduction methods have been proposed. Recently, manifold learning method for dimensionality reduction has aroused broad attentions, such as isometric mapping algorithm (Isomap), locally linear embedding algorithm (LLE), Laplacian eigenmap algorithm, Hessian eigenmap algorithm (HLLE), Local tangent space alignment (LTSA), and Diffusion Map algorithm. The problems with which these algorithms often confront are how to select the neighborhood parameters, how to improve the robustness to noise, and how to deal with the new query samples efficiently. In order to select the neighborhood parameters properly, we propose a novel method for constructing an optimal neighborhood graph from a data set. In the proposed method, two key neighborhood parameters are selected based on the relationship of the average shortest distance of the graph with the neighborhood size, two neighborhood graphs are constructed respectively, and then the optimal neighborhood graph is determined, by integrating the two graphs after deleting the "short circuit edges" in them. The optimal neighborhood graph has few short-circuit edges, and better approximates the geodesic distances between the data points. Furthermore, different point has adaptive variant number of neighbor points with the local characteristics of the data point in the optimal neighborhood graph. Experimental results show that the proposed method yields better performances in symmetrically sampling data points free of noise than the recent methods. It is also shown that its topologically stability and degree of noise tolerance can be significantly improved.Since the mapping functions are not explicitly given by the manifold learning algorithms based on graphic theory, when we need to know a query sample's low dimensional representation, we have to put the query sample into the original sample sets, and then reconstruct neighborhood graph and redo the process of determining the low dimensional embedding. In order to deal with the new query samples efficiently, we propose a method by combining a nonlinear interpolation technique i.e. generalized regression network (GRNN) with manifold learning methods. In the proposed method the GRNN is trained using the low dimensional embedding of the training samples, the trained GRNN can approximate the mapping function of the manifold learning methods, and the low dimensional representation of the query sample can be determined directly by using the trained GRNN. Experimental results show that the proposed method can perform as accurately as manifold learning method with a lower computation cost.