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Huguet, G.* ; Tong, A.Z.* ; Rieck, B. ; Huang, J.* ; Kuchroo, M.* ; Hirn, M.* ; Wolf, G.* ; Krishnaswamy, S.*

Time-inhomogeneous diffusion geometry and topology.

SIAM J. Math. Data Sci. 5, 346-372 (2023)
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Diffusion condensation is a dynamic process that yields a sequence of multiscale data representations that aim to encode meaningful abstractions. It has proven effective for manifold learning, denoising, clustering, and visualization of high-dimensional data. Diffusion condensation is constructed as a time-inhomogeneous process where each step first computes a diffusion operator and then applies it to the data. We theoretically analyze the convergence and evolution of this process from geometric, spectral, and topological perspectives. From a geometric perspective, we obtain convergence bounds based on the smallest transition probability and the radius of the data, whereas from a spectral perspective, our bounds are based on the eigenspectrum of the diffusion kernel. Our spectral results are of particular interest since most of the literature on data diffusion is focused on homogeneous processes. From a topological perspective, we show that diffusion condensation generalizes centroid-based hierarchical clustering. We use this perspective to obtain a bound based on the number of data points, independent of their location. To understand the evolution of the data geometry beyond convergence, we use topological data analysis. We show that the condensation process itself defines an intrinsic condensation homology. We use this intrinsic topology, as well as the ambient persistent homology, of the condensation process to study how the data changes over diffusion time. We demonstrate both types of topological information in well-understood toy examples. Our work gives theoretical insight into the convergence of diffusion condensation and shows that it provides a link between topological and geometric data analysis.
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Publication type Article: Journal article
Document type Scientific Article
Corresponding Author
Keywords diffusion; time-inhomogeneous process; topological data analysis; persistent homology; hierarchical clustering; Persistence
ISSN (print) / ISBN 2577-0187
e-ISSN 2577-0187
Journal SIAM Journal on Mathematics of Data Science
Quellenangaben Volume: 5, Issue: 2, Pages: 346-372 Article Number: , Supplement: ,
Publisher SIAM Publications
Publishing Place 3600 Univ City Science Center, Philadelphia, Pa 19104-2688 Usa
Non-patent literature Publications
Reviewing status Peer reviewed
Institute(s) Institute of AI for Health (AIH)
Grants CIFAR AI Chair
IVADO Professor funds