An Ordinal Weighted EDM Model for Nonmetric Multidimensional Scaling

Qing-Na Li (), Chi Zhang and Mengzhi Cao ()
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Qing-Na Li: School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing, Institute of Technology, Beijing 100081, P. R. China
Chi Zhang: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China
Mengzhi Cao: Data Science, Worcester Polytechnic Institute, Worcester 01609, USA

Asia-Pacific Journal of Operational Research (APJOR), 2022, vol. 39, issue 03, 1-22

Abstract: Multidimensional scaling (MDS) is to recover a set of points by making use of noised pairwise Euclidean distances. In some situations, the observed Euclidean distances may contain large errors or even missing values. In such cases, the order of the distances is far more important than their magnitude. Non-metric multidimensional scaling (NMDS) is then to deal with this problem by taking use of the ordinal information. The challenge of NMDS is to tackle the large number of ordinal constraints on distances (for n points, this will be of O(n4)), which will slow down existing numerical algorithms. In this paper, we propose an ordinal weighted Euclidean distance matrix model for NMDS. By designing an ordinal weighted matrix, we get rid of the large number of ordinal constraints and tackle the ordinal constraints in a soft way. We then apply our model to image ranking. The key insight is to view the image ranking problem as NMDS in the kernel space. We conduct extensive numerical test on two state-of-the-art datasets: FG-NET aging dataset and MSRA-MM dataset. The results show the improvement of the proposed approach over the existing methods.

Keywords: Nonmetric multidimensional scaling; image ranking; distance metric learning; optimization methods; Euclidean distance matrix (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0217595921500330

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