Dynamical Clustering and its Limiting Differential Equations
Abstract
A novel, non-parametric clustering algorithm is developed, based on flows in the space of soft assignments Pki, representing the probability that point xi belongs to class k. These flows have two drivers: a nonlinear reaction component that relaxes P to an evolving Bayesian-like posterior, and a diffusive component that relaxes P to its local average, with a diffusivity ν that adapts dynamically so as to balance the two components. The methodology extends to semi-supervised classification, where a subset of the labels are known. In the limit of infinitely many observations, it yields a set of non-standard reaction-diffusion equations, which produces sharp boundaries between species that separate spontaneously into well-balanced domains. Including more than one diffusive network opens the way to a broader class of applications, including the detection of regime changes in time series and a clustering procedure based on numerous features that defeats the curse of dimensionality. In the continuous limit, this extension results in a novel, non-local class of diffusive operators.
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BibTeX
@misc{milewski2026dynamical,
title = {{Dynamical Clustering and its Limiting Differential Equations}},
author = {Paul A. Milewski and Esteban G. Tabak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11608},
eprint = {2609.11608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11608},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Dynamical Clustering and its Limiting Differential Equations
AU - Paul A. Milewski
AU - Esteban G. Tabak
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11608
UR - https://arxiv.org/abs/2609.11608
AB - A novel, non-parametric clustering algorithm is developed, based on flows in the space of soft assignments P_k^i, representing the probability that point x^i belongs to class k. These flows have two drivers: a nonlinear reaction component that relaxes P to an evolving Bayesian-like posterior, and a diffusive component that relaxes P to its local average, with a diffusivity ν that adapts dynamically so as to balance the two components. The methodology extends to semi-supervised classification, where a subset of the labels are known. In the limit of infinitely many observations, it yields a set of non-standard reaction-diffusion equations, which produces sharp boundaries between species that separate spontaneously into well-balanced domains. Including more than one diffusive network opens the way to a broader class of applications, including the detection of regime changes in time series and a clustering procedure based on numerous features that defeats the curse of dimensionality. In the continuous limit, this extension results in a novel, non-local class of diffusive operators.
ER -