Robust nonlinear dimensionality reduction by topologically constrained semi-isometric embedding

Abstract

Real world data is usually high dimensional, and dimensionality reduction can significantly improve the efficiency of data processing and analysis. Existing approaches relying on distances between neighboring features typically suffer from the unreliable estimation of the true distance on a feature manifold due to its non-convexity. An approach is proposed to solve the problem by discarding long geodesics poisoned by boundary points indiscriminately. However, despite the improved performance, there are two major shortcomings with the approach. First, many long geodesics poisoned by few boundary points, which contribute little to the distortion of a manifold, are thrown away, as may decrease the robustness without improving the distortion of the manifold. Second, since short geodesics are sensitive to noise, retaining the whole effect of them may result in the bad robustness. This paper presents a regularization framework for nonlinear dimensionality reduction that incorporates long geodesics poisoned by few boundary points and reduces the effect of short geodesics, to realize isometry largely. In addition, the approach is sensitive to non-uniform sampling. To cope with the issue, we describe an improved robust boundary detection method. Experimental results are presented to illustrate the better performance of the proposed algorithm on two standard data sets.