Flexible Manifold Embedding: A Framework for Semi-Supervised and Unsupervised Dimension Reduction

摘要

We propose a unified manifold learning framework for semi-supervised and unsupervised dimension reduction by employing a simple but effective linear regression function to map the new data points. For semi-supervised dimension reduction, we aim to find the optimal prediction labels $F$ for all the training samples $X$, the linear regression function $h(X)$ and the regression residue $F_0=F-h(X)$ simultaneously. Our new objective function integrates two terms related to label fitness and manifold smoothness as well as a flexible penalty term defined on the residue $F_0$ . Our Semi-Supervised learning framework, referred to as flexible manifold embedding (FME), can effectively utilize label information from labeled data as well as a manifold structure from both labeled and unlabeled data. By modeling the mismatch between $h(X)$ and $F$ , we show that FME relaxes the hard linear constraint $F=h(X)$ in manifold regularization (MR), making it better cope with the data sampled from a nonlinear manifold. In addition, we propose a simplified version (referred to as FME/U) for unsupervised dimension reduction. We also show that our proposed framework provides a unified view to explain and understand many semi-supervised, supervised and unsupervised dimension reduction techniques. Comprehensive experiments on several benchmark databases demonstrate the significant improvement over existing dimension reduction algorithms.