Semi-supervised local multi-manifold Isomap by linear embedding for feature extraction

In this paper, we mainly propose a semi-supervised local multi-manifold Isomap learning framework by linear embedding, termed SSMM-Isomap, that can apply the labeled and unlabeled training samples to perform the joint learning of neighborhood preserving local nonlinear manifold features and a linear feature extractor. The formulation of SSMM-Isomap aims at minimizing pairwise distances of intra-class points in the same manifold and maximizing the distances over different manifolds. To enhance the performance of nonlinear manifold feature learning, we also incorporate the neighborhood reconstruction error to preserve local topology structures between both labeled and unlabeled samples. To enable our SSMM-Isomap to extract local manifold features from the outside new data, we also add a feature approximation error that correlates manifold features with embedded features by the jointly learnt feature extractor. Thus, the learnt linear extractor can extract the local manifold features from the new data efficiently by direct embedding. To optimize the proposed objective function, two effective schemes are presented, i.e., Scaling by MAjorizing a Complicated Function and Eigen-decomposition. Notice that the comparison of the proposed two solvers is also described. We mainly evaluate SSMM-Isomap for manifold feature learning, data clustering and classification. Extensive simulation results verify the effectiveness of our SSMM-Isomap algorithm, compared with other related feature learning techniques.