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

2024-11-16

arXiv.org

AI Summary

Traditional multidimensional scaling (MDS) struggles with non-Euclidean and non-metric dissimilarity data due to its reliance on Euclidean embedding and the positive semi-definiteness of the Gram matrix. To address this, we propose a generalized MDS framework based on symmetric bilinear forms. Our method explicitly models both positive and negative eigenvalues and jointly optimizes the corresponding subspaces under the STRESS criterion, achieving the theoretical optimal lower bound. This is the first systematic integration of symmetric bilinear forms into the MDS paradigm, accompanied by rigorous proofs of convergence and optimality. Experiments on synthetic and real-world datasets demonstrate that our approach significantly outperforms classical MDS, t-SNE, and UMAP—accurately recovering non-Euclidean geometric structures while markedly improving embedding fidelity and robustness.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

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.

Problem

• Multidimensional Scaling
• Complex Data Features
• Data Characterization

Innovation

• Neuc-MDS
• Complex Data Handling
• Error Minimization