T-ARC: Topology-Aware Randomized Clustering via Distributionally Robust Stochastic Block Models
Abstract
In this work, we introduce a new clustering method, namely T-ARC (Topology-Aware Randomized Clustering), that corrects the geometric bias of K-means by embedding topological information directly into the optimization objective. Building on the assumption that the data admits an underlying hidden structure modeled via a latent graph, the idea is to uncover this information through the interplay between the standard K-means data-fidelity term and a graph-cut penalty, which discourages cluster assignments inconsistent with the connectivity structure of the data. To render this coupling tractable, the latent graph is modeled as a random realization from a Stochastic Block Model (SBM), whose scalar parameter is optimized within a Distributionally Robust Optimization (DRO) framework, yielding a closed-form proximal update. Both SBM and DRO are informed by a persistence-based similarity matrix derived from zero-dimensional persistent homology (H0), which translates the multiscale connectivity structure of the data into a pairwise topological prior. The overall optimization proceeds via Block Coordinate Descent; convergence is established through a global Lyapunov functional: the deterministic blocks satisfy monotonic descent, while the stochastic graph update satisfies descent in expectation, so that the expected energy converges. Experiments on synthetic datasets with non-convex geometries and on random subsets of Fashion-MNIST show that T-ARC recovers latent topological structures where K-means fails, achieving the highest accuracy on curved and interleaved clusters while remaining competitive, and markedly more stable than K-means, on real data.
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BibTeX
@misc{benedictis2026tarc,
title = {{T-ARC: Topology-Aware Randomized Clustering via Distributionally Robust Stochastic Block Models}},
author = {Serena Grazia De Benedictis and Andersen Ang and Nicoletta Del Buono and Flavia Esposito and Laura Selicato},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39466},
eprint = {2609.39466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39466},
}
RIS
TY - UNPB
TI - T-ARC: Topology-Aware Randomized Clustering via Distributionally Robust Stochastic Block Models
AU - Serena Grazia De Benedictis
AU - Andersen Ang
AU - Nicoletta Del Buono
AU - Flavia Esposito
AU - Laura Selicato
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.39466
UR - https://arxiv.org/abs/2609.39466
AB - In this work, we introduce a new clustering method, namely T-ARC (Topology-Aware Randomized Clustering), that corrects the geometric bias of K-means by embedding topological information directly into the optimization objective. Building on the assumption that the data admits an underlying hidden structure modeled via a latent graph, the idea is to uncover this information through the interplay between the standard K-means data-fidelity term and a graph-cut penalty, which discourages cluster assignments inconsistent with the connectivity structure of the data. To render this coupling tractable, the latent graph is modeled as a random realization from a Stochastic Block Model (SBM), whose scalar parameter is optimized within a Distributionally Robust Optimization (DRO) framework, yielding a closed-form proximal update. Both SBM and DRO are informed by a persistence-based similarity matrix derived from zero-dimensional persistent homology (H_0), which translates the multiscale connectivity structure of the data into a pairwise topological prior. The overall optimization proceeds via Block Coordinate Descent; convergence is established through a global Lyapunov functional: the deterministic blocks satisfy monotonic descent, while the stochastic graph update satisfies descent in expectation, so that the expected energy converges. Experiments on synthetic datasets with non-convex geometries and on random subsets of Fashion-MNIST show that T-ARC recovers latent topological structures where K-means fails, achieving the highest accuracy on curved and interleaved clusters while remaining competitive, and markedly more stable than K-means, on real data.
ER -