A Spectral Framework for Multi-Scale Nonlinear Dimensionality Reduction

2026-04-02

AI Summary

This work addresses the challenges in nonlinear dimensionality reduction of simultaneously preserving global and local structures and providing interpretability during embedding. We propose a novel embedding method that integrates spectral decomposition with multiscale cross-entropy optimization. By analyzing the influence of spectral modes on embeddings from a graph frequency-domain perspective, our approach uniquely combines spectral basis representations with multiscale nonlinear dimensionality reduction. This integration enables a balanced preservation of both global and local manifold structures while maintaining continuity. To enhance interpretability, we introduce glyph-augmented scatterplots for visual exploration of the embedding process. Quantitative evaluations and case studies demonstrate the effectiveness of our method in achieving structurally faithful and interpretable low-dimensional representations.

Abstract

Dimensionality reduction (DR) is characterized by two longstanding trade-offs. First, there is a global-local preservation tension: methods such as t-SNE and UMAP prioritize local neighborhood preservation, yet may distort global manifold structure, while methods such as Laplacian Eigenmaps preserve global geometry but often yield limited local separation. Second, there is a gap between expressiveness and analytical transparency: many nonlinear DR methods produce embeddings without an explicit connection to the underlying high-dimensional structure, limiting insight into the embedding process. In this paper, we introduce a spectral framework for nonlinear DR that addresses these challenges. Our approach embeds high-dimensional data using a spectral basis combined with cross-entropy optimization, enabling multi-scale representations that bridge global and local structure. Leveraging linear spectral decomposition, the framework further supports analysis of embeddings through a graph-frequency perspective, enabling examination of how spectral modes influence the resulting embedding. We complement this analysis with glyph-based scatterplot augmentations for visual exploration. Quantitative evaluations and case studies demonstrate that our framework improves manifold continuity while enabling deeper analysis of embedding structure through spectral mode contributions.

Problem

• dimensionality reduction
• global-local preservation
• spectral analysis
• manifold structure
• embedding interpretability

Innovation

• spectral framework
• multi-scale representation
• nonlinear dimensionality reduction
• graph frequency analysis
• manifold preservation