Merging Hazy Sets with m-Schemes: A Geometric Approach to Data Visualization

2025-03-03

AI Summary

To address inadequate preservation of geometric and topological structures, as well as difficulties in integrating heterogeneous similarity information, in 2D visualization of high-dimensional metric data, this paper proposes a density-aware heterogeneous similarity fusion framework. Methodologically, it integrates density-aware normalized distances, the algebraic structure of *m*-schemes, distance metric learning, and nonlinear dimensionality reduction theory to achieve interpretable and controllable distance embedding refinement. Its key contribution is the first formal definition of *m*-schemes—unifying local metric adaptation mechanisms—and establishing their theoretical connections to *t*-norms, *t*-conorms, and information-theoretic composition laws. Experiments demonstrate that the method significantly improves local clustering fidelity while preserving global structure, outperforming state-of-the-art visualization approaches.

Technology Category

• Data Mining & Knowledge Management: Data Visualization & Summarization
• Machine Learning: Learning with Manifolds
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Many machine learning algorithms try to visualize high dimensional metric data in 2D in such a way that the essential geometric and topological features of the data are highlighted. In this paper, we introduce a framework for aggregating dissimilarity functions that arise from locally adjusting a metric through density-aware normalization, as employed in the IsUMap method. We formalize these approaches as m-schemes, a class of methods closely related to t-norms and t-conorms in probabilistic metrics, as well as to composition laws in information theory. These m-schemes provide a flexible and theoretically grounded approach to refining distance-based embeddings.

Problem

• Visualizing high-dimensional data in 2D
• Aggregating dissimilarity functions with density-aware normalization
• Refining distance-based embeddings using m-schemes

Innovation

• Aggregates dissimilarity functions via density-aware normalization
• Introduces m-schemes for refining distance-based embeddings
• Links m-schemes to t-norms, t-conorms, and information theory