# A Multiscale Environment for Learning by Diffusion

arXiv: https://arxiv.org/abs/2102.00500 · HTML: https://mlsanity.com/paper/2102.00500/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 68 references that are in the map.
- First posted on arXiv 31 Jan 2021.

## Builds on (most cited first, 15 of 15)

- [Hierarchical Block Structures and High-resolution Model Selection in Large Networks](https://mlsanity.com/paper/1310.4377/) (2013)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks](https://mlsanity.com/paper/1502.04381/) (2015)
- [Community Detection and Classification in Hierarchical Stochastic Blockmodels](https://mlsanity.com/paper/1503.02115/) (2015)
- [A simple SVD algorithm for finding hidden partitions](https://mlsanity.com/paper/1404.3918/) (2014)
- [Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms](https://mlsanity.com/paper/1712.06206/) (2017)
- [Spectral Clustering Based on Local PCA](https://mlsanity.com/paper/1301.2007/) (2013)
- [The geometry of kernelized spectral clustering](https://mlsanity.com/paper/1404.7552/) (2014)
- [Learning by Unsupervised Nonlinear Diffusion](https://mlsanity.com/paper/1810.06702/) (2018)
- [Geometric structure of graph Laplacian embeddings](https://mlsanity.com/paper/1901.10651/) (2019)
- [Spectral-Spatial Diffusion Geometry for Hyperspectral Image Clustering](https://mlsanity.com/paper/1902.05402/) (2019)
- [Graph-based data clustering via multiscale community detection](https://mlsanity.com/paper/1909.04491/) (2019)
- [Statistical Inference for Cluster Trees](https://mlsanity.com/paper/1605.06416/) (2016)
- [Hyperspectral Image Clustering with Spatially-Regularized Ultrametrics](https://mlsanity.com/paper/2004.05048/) (2020)
- [Diffusion State Distances: Multitemporal Analysis, Fast Algorithms, and Applications to Biological Networks](https://mlsanity.com/paper/2003.03616/) (2020)

## Built on by (most cited first, 8 of 8)

- [Superpixel-based and Spatially-regularized Diffusion Learning for Unsupervised Hyperspectral Image Clustering](https://mlsanity.com/paper/2312.15447/) (2023)
- [Multiscale Clustering of Hyperspectral Images Through Spectral-Spatial Diffusion Geometry](https://mlsanity.com/paper/2103.15783/) (2021)
- [Active Diffusion and VCA-Assisted Image Segmentation of Hyperspectral Images](https://mlsanity.com/paper/2204.06298/) (2022)
- [Unsupervised Diffusion and Volume Maximization-Based Clustering of Hyperspectral Images](https://mlsanity.com/paper/2203.09992/) (2022)
- [Classification of Hyperspectral Images Using SVM with Shape-adaptive Reconstruction and Smoothed Total Variation](https://mlsanity.com/paper/2203.15619/) (2022)
- [Time-inhomogeneous diffusion geometry and topology](https://mlsanity.com/paper/2203.14860/) (2022)
- [Unsupervised detection of ash dieback disease (Hymenoscyphus fraxineus) using diffusion-based hyperspectral image clustering](https://mlsanity.com/paper/2204.09041/) (2022)
- [Unsupervised Spatial-spectral Hyperspectral Image Reconstruction and Clustering with Diffusion Geometry](https://mlsanity.com/paper/2204.13497/) (2022)

## Abstract (arXiv)

Clustering algorithms partition a dataset into groups of similar points. The clustering problem is very general, and different partitions of the same dataset could be considered correct and useful. To fully understand such data, it must be considered at a variety of scales, ranging from coarse to fine. We introduce the Multiscale Environment for Learning by Diffusion (MELD) data model, which is a family of clusterings parameterized by nonlinear diffusion on the dataset. We show that the MELD data model precisely captures latent multiscale structure in data and facilitates its analysis. To efficiently learn the multiscale structure observed in many real datasets, we introduce the Multiscale Learning by Unsupervised Nonlinear Diffusion (M-LUND) clustering algorithm, which is derived from a diffusion process at a range of temporal scales. We provide theoretical guarantees for the algorithm's performance and establish its computational efficiency. Finally, we show that the M-LUND clustering algorithm detects the latent structure in a range of synthetic and real datasets.