Boundary Constrained Manifold Unfolding

A new manifold learning algorithm is proposed in this paper. Our method is motivated by the unit covariance constraint problem of spectral embedding methods, where a unit covariance constraint is imposed to avoid degenerate solutions that map all manifold samples to one point. This constraint distorts the aspect ratio and introduces unwanted correlation between different components of embedding coordinates. Instead, our method uses boundary conditions to pull apart mapped points, and obtains the embedding by solving linear systems under boundary conditions. The mapping of boundary samples is decided by that of a coarse version of manifold, obtained by a graph simplification algorithm designed by us. Comparisons between our method and several other representative manifold learning methods are made, and the results demonstrate the effectiveness of the proposed method.