A multiscale environment for learning by diffusion

•We introduce the MELD data model: a diffusion framework for multiscale clustering.•We show how cluster coherence and separation interact with diffusion in MELD clusterings.•We introduce the M-LUND multiscale clustering algorithm and guarantee its performance.•We guarantee that M-LUND recovers the M...

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Bibliographic Details
Published in:Applied and computational harmonic analysis Vol. 57; pp. 58 - 100
Main Authors: Murphy, James M., Polk, Sam L.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.03.2022
Subjects:
Clustering
Diffusion geometry
Hierarchical clustering
Machine learning
Spectral graph theory
Clustering
Hierarchical clustering
Spectral graph theory
Diffusion geometry
Machine learning
ISSN:1063-5203
Online Access:Get full text
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Summary:•We introduce the MELD data model: a diffusion framework for multiscale clustering.•We show how cluster coherence and separation interact with diffusion in MELD clusterings.•We introduce the M-LUND multiscale clustering algorithm and guarantee its performance.•We guarantee that M-LUND recovers the MELD data model from many datasets.•We show M-LUND extracts latent multiscale structure in synthetic and real datasets. 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.
ISSN:1063-5203
DOI:10.1016/j.acha.2021.11.004