Maximum Variance Unfolding

Abstract

Unlike Isomap method that preserves geodesic distances, MVU method learns the data from the similarities, preserving both local distances and angles between the pairs of all neighbors of each point in the data set. Since the method keeps the local maximum variance in dimensionality reduction processing, it is called maximum variance unfolding (MVU). Like multidimensional scaling (MDS), MVU can be applied to the cases that only the local similarities of objects in a set are given. In these cases, MVU tries to find a configuration that preserves the given similarities. Technically, MVU adopts semidefinite programming (SDP) to solve the DR problems. Hence, it is also called semidefinite embedding (SDE) or SDP. Solving a DR problem using MVU is expensive regarding both memory cost and time cost. To overcome the shortage of high computational cost, landmark MVU (LMVU) is introduced. In Section 9.1, we describe the MVU method and the corresponding maximization model. In Section 9.2, we give a brief review of SDP and introduce several popular SDP software packages. The experiments and applications of MVU are included in Section 9.3. The LMVU is discussed in Section 9.4.