Classical multidimensional scaling: A subspace perspective, over-denoising and outlier detection

Abstract

The classical Multi-Dimensional Scaling (cMDS) has become a cornerstone for analyzing metric dissimilarity data due to its simplicity in derivation, low computational complexity and its nice interpretation via the principle component analysis. This paper focuses on its capability of denoising and outlier detection.Our new interpretation shows that \cMDS\ always overly denoises a sparsely perturbed data by subtracting a fully dense denoising matrix in a subspace from the given data matrix.This leads us to consider two types of sparsity-driven models: Subspace sparse MDS and Full-space sparse MDS, which respectively uses the $\ell_1$ and $\ell_{1-2}$ regularization to induce sparsity. We then develop fast majorization algorithms for both models and establish their convergence. In particular, we are able to control the sparsity level at every iterate provided that the sparsity control parameter is above a computable threshold. This is a desirable property that has not been enjoyed by any of existing sparse MDS methods.Our numerical experiments on both artificial and real data demonstrates that \cMDS\ with appropriate regularization can perform the tasks of denoising and outlier detection, and inherits the efficiency of \cMDS\ in comparisonwith several state-of-the-art sparsity-driven MDS methods.

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ORCID for Hou-Duo Qi: orcid.org/0000-0003-3481-4814

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