Research On Several Issues In Manifold Alignment

With the rapid development of information technology, the size and dimensions of data is growing more and more quickly. Obviously, it is important to discover the low-dimensional structures from these data. So nonlinear dimension reduction have widely application. However, manifold learning are designed to discover the low-dimensional features of the single points lying on a single manifold. In many real-world applications, it is required to discover the latent features of two or more disparate input data sets. As a new spot direction in manifold learning algorithms, manifold alignment algorithms provide a good platform form single manifold to multi-manifold. Manifold alignment not only attempt to identify the low dimensional manifold structure of that data set and preserve that structure in a low dimensional embedding of the data set, but also encourage corresponding instances across data sets to have similar locations in the embedding.However,there are some drawback in manifold alignment. At first, in manifold alignment, A key issue that determine the effectiveness of the manifold alignment approaches is how to accurately build connections between the data sets. But in the real world, it is very difficult to obtain the correspondence information. Since the points from different data sets are represented by different features, it is difficult to compare them directly. Though there are some efforts on solving this problem, they have a high time complexity, and they can only used in certain situations. Second, in many situation,we may know the labels in each data set rather than correspondence. It is an other point we study that how to align two manifold when we only know labels.So, this paper mainly focuses on the manifold alignment with few correspondence and manifold alignment base on labels information. The mainly contributions of this paper are as follows:1. This paper gives a new semi-supervised manifold alignment with few correspondences. The key step that determines the effectiveness of the manifold alignment approaches is how to accurately build connections between the input data sets.Assume that very limited correspondence information such as few pairwise correspondences is given. We use the geodesic distances along the manifold between the point and the correspondences to characterize the point’s manifold structure in order to build the connection of each manifold. In addition, we verify the theories that we could find the accurate correspondence form different manifold by d+1 points in d-dimension manifold. At last, we give some experiments to show the effectiveness of the proposed manifold alignment algorithm.2. Manifold alignment based on classification. This article give different classification methods based on two assumes. The first case is both the correspondence and the labels are know, we rearrange the Laplace graph matrix and utilize the semi-supervised model to classify the test set. The second case is that we only know the labels of some points. In this case we employ a two step alignment method. At first, we embedding the data sets to low-dimensional space. Then, alignment of two manifolds can be achieved by removing some components(like rotational, scaling components and translating) from one manifold leaving another untouched. At last, we give some experiments to show the effectiveness of the proposed manifold alignment algorithm.