A Spectral Framework for Multi-Scale Nonlinear Dimensionality Reduction

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.