Multidimensional Scaling With Heaviside Weighting: Extensions to Curvilinear Component Analysis and Curvilinear Distance Analysis

ABSTRACT In this article, I propose extensions to multidimensional scaling (MDS) variants that use Heaviside weighting for manifold learning, namely, curvilinear component analysis and curvilinear distance analysis. These methods weight the approximation error with a Heaviside function to ignore larger fitted distances in the configuration, thus effectively providing a localized version of MDS. I extend these methods with (i) optimal scaling that allows for linear, affine or spline transformations of the proximities and (ii) fitting power transformations of proximities and distances. I suggest a quasi‐MM algorithm for optimization and present the methods either for a fixed neighbourhood parameter or as self‐organizing. I employ both geodesic distances and arbitrary dissimilarities as proximities. This yields new MDS variants that I call extended curvilinear component analysis (eCLCA), extended curvilinear power component analysis (eCLPCA), extended curvilinear distance analysis (eCLDA) and extended curvilinear power distance analysis (eCLPDA). With three data examples, I show how the extensions can improve learning a manifold.