Nonlinear Dimensionality Reduction by Local Orthogonality Preserving Alignment

We present a new manifold learning algorithm called Local Orthogonality Preserving Alignment (LOPA). Our algorithm is inspired by the Local Tangent Space Alignment (LTSA) method that aims to align multiple local neighborhoods into a global coordinate system using affine transformations. However, LTSA often fails to preserve original geometric quantities such as distances and angles. Although an iterative alignment procedure for preserving orthogonality was suggested by the authors of LTSA, neither the corresponding initialization nor the experiments were given. Procrustes Subspaces Alignment (PSA) implements the orthogonality preserving idea by estimating each rotation transformation separately with simulated annealing. However, the optimization in PSA is complicated and multiple separated local rotations may produce globally contradictive results. To address these difficulties, we first use the pseudo-inverse trick of LTSA to represent each local orthogonal transformation with the unified global coordinates. Second the orthogonality constraints are relaxed to be an instance of semi-definite programming (SDP). Finally a two-step iterative procedure is employed to further reduce the errors in orthogonal constraints. Extensive experiments show that LOPA can faithfully preserve distances, angles, inner products, and neighborhoods of the original datasets. In comparison, the embedding performance of LOPA is better than that of PSA and comparable to that of state-of-the-art algorithms like MVU and MVE, while the runtime of LOPA is significantly faster than that of PSA, MVU and MVE.