M-estimators for robust multidimensional scaling employing ℓ2,1 norm regularization

Abstract

Multidimensional Scaling (MDS) has been exploited to visualise the hidden structures among a set of entities in a reduced dimensional metric space. Here, we are interested in cases whenever the initial dissimilarity matrix is contaminated by outliers. It is well-known that the state-of-the-art algorithms for solving the MDS problem generate erroneous embeddings due to the distortion introduced by such outliers. To remedy this vulnerability, a unified framework for the solution of MDS problem is proposed, which resorts to half-quadratic optimization and employs potential functions of M-estimators in combination with ℓ2,1 norm regularization. Two novel algorithms are derived. Their performance is assessed for various M-estimators against state-of-the-art MDS algorithms on four benchmark data sets. The numerical tests demonstrate that the proposed algorithms perform better than the competing alternatives.