Submanifold Decomposition

Abstract

Low-dimensional structures embedded in high-dimensional data space can be extracted by spectral analysis and manifold learning. Standard approaches to manifold learning are usually based on the assumption that there is a dominant low-dimensional manifold, while other variations are considered with minor priority. We instead consider the scenario that a pair of distinct manifolds intertwined in the same high-dimensional space, which can be decomposed for analysis. The core of this new method is a novel submanifold decomposition (SMD) algorithm. This paper has three contributions: 1) a submanifold framework is proposed to model the high-dimensional dataset, which is dominated by more than one factor; 2) a nonlinear manifold decomposition method, SMD, is presented to extract two intertwined manifolds from a dataset in a discriminative manner; and 3) in order to solve the out-of-sample problem of nonlinear SMD, a linear extension of SMD is developed, which is effective to extract two linear submanifolds. We demonstrate that comparing with the existing manifold learning methods that only extract one dominant manifold, the proposed SMD and its linear extension are capable of extracting a pair of submanifolds discriminatively and effectively. Moreover, the two extracted manifolds can complement each other to enhance the representation performance. Extensive experiments on both artificial data and real data demonstrate that the proposed method outperforms the state-of-the-art manifold learning algorithms in visual recognition tasks.