@inproceedings{
deng2024neucmds,
title={Neuc-{MDS}: Non-Euclidean Multidimensional Scaling Through Bilinear Forms},
author={Chengyuan Deng and Jie Gao and Kevin Lu and Feng Luo and Hongbin Sun and Cheng Xin},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=8W5ADJOKcv}
}
We introduce \textbf{N}on-\textbf{Euc}lidean-\textbf{MDS} (Neuc-MDS), which extends Multidimensional Scaling (MDS) to generate outputs that can be non-Euclidean and non-metric. The main idea is to generalize the inner product to other symmetric bilinear forms to utilize the negative eigenvalues of dissimiliarity 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.
Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms
Chengyuan Deng Jie Gao Kevin Lu Feng Luo Hongbin Sun Cheng Xin