Trace Ratio Optimization-Based Semi-Supervised Nonlinear Dimensionality Reduction for Marginal Manifold Visualization

Visualizing similarity data of different objects by exhibiting more separate organizations with local and multimodal characteristics preserved is important in multivariate data analysis. Laplacian Eigenmaps (LAE) and Locally Linear Embedding (LLE) aim at preserving the embeddings of all similarity pairs in the close vicinity of the reduced output space, but they are unable to identify and separate interclass neighbors. This paper considers the semi-supervised manifold learning problems. We apply the pairwise Cannot-Link and Must-Link constraints induced by the neighborhood graph to specify the types of neighboring pairs. More flexible regulation on supervised information is provided. Two novel multimodal nonlinear techniques, which we call trace ratio (TR) criterion-based semi-supervised LAE ($({rm S}^2{rm LAE})$) and LLE ($({rm S}^2{rm LLE})$), are then proposed for marginal manifold visualization. We also present the kernelized $({rm S}^2{rm LAE})$ and $({rm S}^2{rm LLE})$. We verify the feasibility of $({rm S}^2{rm LAE})$ and $({rm S}^2{rm LLE})$ through extensive simulations over benchmark real-world MIT CBCL, CMU PIE, MNIST, and USPS data sets. Manifold visualizations show that $({rm S}^2{rm LAE})$ and $({rm S}^2{rm LLE})$ are able to deliver large margins between different clusters or classes with multimodal distributions preserved. Clustering evaluations show they can achieve comparable to or even better results than some widely used methods.