Approximately harmonic projection: Theoretical analysis and an algorithm

Manifold learning have attracted considerable attention over the last decade. The most frequently used functional is the l2-norm of the gradient of the function. In this paper, we consider the linear manifold learning problem by minimizing this functional with appropriate constraint. We provide theoretical analysis on both the functional and the constraint, which shows the affine hulls of the manifold and the connected components are essential to linear manifold learning problem. Based on the theoretical analysis, we introduce a novel linear manifold learning algorithm called approximately harmonic projection (AHP). Unlike canonical linear methods such as principal component analysis, our method is sensitive to the connected components. This makes our method especially applicable to data clustering. We conduct several experimental results on three real data sets to demonstrate the effectiveness of our proposed method.