Submanifold Decomposition

a novel submanifold decomposition algorithm, which simultaneously considers two manif

Extracting low-dimensional structures from high-dimensional space through spectral analysis has been prevalent in the fields of machine learning and computer vision. Many existing manifold learning methods assume that there is a dominant low-dimensional manifold, while other variations are usually considered as noise or even ignored. This paper proposes a novel submanifold decomposition (SMD) algorithm, which simultaneously considers two manifolds intertwined in the same high-dimensional space for decomposition.
Three contributions are made by this paper:
1) a submanifold framework is proposed to model the high-dimension 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 existing manifold learning methods that only extract one manifold, the proposed SMD and its linear extension are capable of extracting two submanifolds discriminatively and effectively.
Moreover, the two extracted manifolds can complement each other to elevate the performance.
Comparative experiments on both artificial data and real data indicate that the proposed method outperforms state-of-the-art manifold learning algorithms in recognition tasks.
It is remarkable that SMD is applicable not only to the two-manifold situation, but also to cases where three or more manifolds are fused.