# A Spectral Decomposition Framework for Multiscale Nonlinear Dimensionality Reduction

arXiv: https://arxiv.org/abs/2604.02535 · HTML: https://mlsanity.com/paper/2604.02535/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 89 references that are in the map.
- First posted on arXiv 2 Apr 2026.

## Builds on (most cited first, 30 of 36)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [How Powerful are Spectral Graph Neural Networks](https://mlsanity.com/paper/2205.11172/) (2022)
- [Contrastive and Non-Contrastive Self-Supervised Learning Recover Global and Local Spectral Embedding Methods](https://mlsanity.com/paper/2205.11508/) (2022)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [TriMap: Large-scale Dimensionality Reduction Using Triplets](https://mlsanity.com/paper/1910.00204/) (2019)
- [Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data](https://mlsanity.com/paper/2105.07536/) (2021)
- [NetLSD: Hearing the Shape of a Graph](https://mlsanity.com/paper/1805.10712/) (2018)
- [Attraction-Repulsion Spectrum in Neighbor Embeddings](https://mlsanity.com/paper/2007.08902/) (2020)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [From t-SNE to UMAP with contrastive learning](https://mlsanity.com/paper/2206.01816/) (2022)
- [Revisiting Dimensionality Reduction Techniques for Visual Cluster Analysis: An Empirical Study](https://mlsanity.com/paper/2110.02894/) (2021)
- [Supporting Analysis of Dimensionality Reduction Results with Contrastive Learning](https://mlsanity.com/paper/1905.03911/) (2019)
- [DimReader: Axis lines that explain non-linear projections](https://mlsanity.com/paper/1710.00992/) (2017)
- [Unveiling High-dimensional Backstage: A Survey for Reliable Visual Analytics with Dimensionality Reduction](https://mlsanity.com/paper/2501.10168/) (2025)
- [ZADU: A Python Library for Evaluating the Reliability of Dimensionality Reduction Embeddings](https://mlsanity.com/paper/2308.00282/) (2023)
- [Assessing Neural Network Representations During Training Using Noise-Resilient Diffusion Spectral Entropy](https://mlsanity.com/paper/2312.04823/) (2023)
- [Interactive Dimensionality Reduction for Comparative Analysis](https://mlsanity.com/paper/2106.15481/) (2021)
- [Classes are not Clusters: Improving Label-based Evaluation of Dimensionality Reduction](https://mlsanity.com/paper/2308.00278/) (2023)
- [Dimension Reduction with Locally Adjusted Graphs](https://mlsanity.com/paper/2412.15426/) (2024)
- [Low-dimensional embeddings of high-dimensional data](https://mlsanity.com/paper/2508.15929/) (2025)
- [A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction](https://mlsanity.com/paper/2305.19043/) (2023)
- [Recovering Manifold Structure Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2410.01149/) (2024)
- [Homology-Preserving Multi-Scale Graph Skeletonization Using Mapper on Graphs](https://mlsanity.com/paper/1804.11242/) (2018)
- [Balancing between the Local and Global Structures (LGS) in Graph Embedding](https://mlsanity.com/paper/2308.16403/) (2023)
- [HUMAP: Hierarchical Uniform Manifold Approximation and Projection](https://mlsanity.com/paper/2106.07718/) (2021)
- [DREAMS: Preserving both Local and Global Structure in Dimensionality Reduction](https://mlsanity.com/paper/2508.13747/) (2025)

## Abstract (arXiv)

Dimensionality reduction (DR) involves two longstanding trade-offs. First, preserving local neighborhoods can come at the cost of global structure. Neighbor embedding methods such as t-SNE and UMAP prioritize local similarity preservation but do not explicitly constrain global organization, whereas standard spectral methods such as Laplacian Eigenmaps capture smooth, coarse-scale graph structure but offer limited flexibility to depict finer local structure. Second, the flexibility of nonlinear DR methods often comes at the cost of analytical transparency. Many methods do not explicitly reveal how high-dimensional structure produces patterns in the embedding. We introduce SDMP (Spectral Decomposition for Multiscale Projection), a nonlinear DR framework built on an explicit spectral decomposition. In this formulation, each embedding dimension is expressed as a weighted combination of Laplacian eigenvectors derived from a neighborhood graph, with the weights learned via a UMAP-style cross-entropy objective. By progressively expanding the spectral subspace to capture increasingly fine graph structure, SDMP produces a sequence of embeddings, making the evolving balance between global organization and local detail explicit, controllable, and inspectable. The explicit decomposition also reveals which spectral scales shape the overall embedding and how individual eigenvectors influence point positions. Quantitative evaluations on synthetic, image, and single-cell data show competitive local and global structure preservation, while case studies illustrate how the decomposition supports interpretation of clusters and developmental trajectories across spectral scales.