Further developments of Multidimensional Scaling via Euclidean Distance Matrix Optimization

Abstract

Multidimensional scaling (MDS) is a method that maps a set of observations into low dimensional space using the pairwise dissimilarity data. Regarding strategies for solving MDS problems, compared with the semi-definite programming (SDP) approach and majorization method SMACOF, Euclidean distance matrix (EDM) optimization is proved to be numerically outstanding. This thesis aims to develop the MDS in different directions via EDM optimization. Firstly, we proposed to improve slacked supervised maximum variance (SMVU) by MDS with raw stress via EDM optimization. When EDM reformulates SMVU, the dissimilarity preservation term can be described by raw stress. The usability of raw stress combined with the efficiency of the penalized majorization-projection algorithm for solving the EDM model is validated by the performance of SMVU theoretically and numerically. An energy-based out-of-sample embedding model derived from SMVU is proposed; the testing on time-series datasets verifies its effectiveness. Back to the MDS model, we balance the robust MDS and
MDS with raw stress, improving the robustness of MDS when Gaussian noise and outliers exist at the same time. The efficiency of the new balancing model is testified on test problems such as sensor network localization and molecular conformation. In the last part of this thesis, the application of EDM optimization is extended to non-metric MDS. As a variant of MDS, non-metric MDS construct low-dimensional representation by order relation between observations. Numerical testing on reality data verifies the effectiveness of this extension.

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Identifiers

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

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