Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms

We introduce Non-Euclidean-MDS (Neuc-MDS), an extension of classical Multidimensional Scaling (MDS) that accommodates non-Euclidean and non-metric inputs. The main idea is to generalize the standard inner product to symmetric bilinear forms to utilize the negative eigenvalues of dissimilarity Gram m...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Deng, Chengyuan, Gao, Jie, Lu, Kevin, Luo, Feng, Sun, Hongbin, Cheng, Xin
Format: Article
Language:English
Subjects:
Eigenvalues
Error analysis
Error reduction
Lower bounds
Online Access:Get full text
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Summary:We introduce Non-Euclidean-MDS (Neuc-MDS), an extension of classical Multidimensional Scaling (MDS) that accommodates non-Euclidean and non-metric inputs. The main idea is to generalize the standard inner product to symmetric bilinear forms to utilize the negative eigenvalues of dissimilarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods.
ISSN:2331-8422