Flexible Adaptive Graph Embedding for Semi-supervised Dimension Reduction

Graph-based semi-supervised dimension reduction can use the inherent graph structure of samples to propagate label information, and has become a hot research field in machine learning. However, most current methods have strict linear constraints and cannot handle data sampled from nonlinear manifolds; and rely on predefined graphs, which cannot capture the local structure information of data and cannot handle complex non-Gaussian data. To address these issues, this paper proposes a new locality-preserved flexible dimension reduction framework, called Semi-supervised Flexible Adaptive Graph Embedding (SFAG), which learns the embedding space that can preserve the local neighborhood structure by constructing a k1-nearest neighbor graph over labeled samples. Then, another k2-nearest neighbor graph is constructed on all samples to adaptively construct the optimal graph, clustering labeled and unlabeled embedding sample points with neighborhood relations into the same sub-manifold sharing the same label information. Last but not least, the hard linear projection constraint is relaxed by adding residual terms to obtain not only the nonlinear embedding of the training samples, but also the linear projection matrix applied directly to the out-of-sample. In addition, two different semi-supervised dimension reduction methods for adaptive construction of optimal graphs are proposed based on the SFAG framework. Several evaluation experiments validate the effectiveness of our method in exploring manifold structures and classification tasks.