Relaxed maximum-variance unfolding

Abstract

Nonlinear dimensionality reduction (recently called manifold learning) is crucial in many research fields. Maximum-variance unfolding (MVU) is one of the most important approaches to it. Unfortunately, due to too strict local constraints, MVU cannot unfold the manifold when short-circuit edges appear or the embedded mapping is conformal but not isometric. A relaxed version, relaxed MVU (RMVU), is proposed. Neighbors are adaptively assigned when short-circuit edges appear, and local distance is rescaled when the manifold is assumed to be angle-preserving. RMVU can effectively solve the preceding problems with MVU. More importantly, RMVU also performs better than MVU in general cases, and hence it has huge potential in many fields. Additionally, the proposed two strategies can also be used in other manifold learning algorithms. Experiments, accompanied with numerical comparisons, were performed on both synthetic and real data sets to demonstrate the effectiveness of RMVU.