A Manifold Unfolding Method Based on Boundary Constraints

Abstract

In the spectral methods of manifold learning, the manifold unfolding tasks are formulated as optimization problems. The optimal solutions to these problems will embed all samples into one point. To avoid the degenerate solutions, the spectral methods impose a unit covariance constraint to the embedding coordinates. However, this constraint usually causes highly distorted embeddings. A new manifold unfolding method is proposed in this paper, which discards the unit covariance constraint completely. The central idea is to embed the manifold boundary at first, then the inner regions. The embedding positions of inner samples will be pulled out by the embedded boundary to avoid collapsing into one point. The embedding of inner samples is obtained by solving a linear system that reffects local isometry requirement, using the embedding of boundary as a boundary condition. The embedding of boundary is determined by a simplified version of manifold, and a manifold boundary detection algorithm and a manifold graph simplification algorithm are thus also proposed. Experimental results on synthetic and real data sets demonstrate the effectiveness of our method, which results in less mapping distortion than spectral methods.