Topological Learning under Sparsity through Huberized Distance Functionals
Abstract
Learning topological structure from sparsely contaminated data requires robust statistical methods, as even a small fraction of atypical observations can substantially distort the inferred topology. This article introduces a robust framework for topological learning through the Huberized Distance-to-Measure, which provides a robust filtration-generating functional for persistent homology. We establish large-sample properties of the proposed functional, and show that the associated filtration induces stable and consistent persistence diagrams. We further derive a maximum bias bound for persistence diagrams in bottleneck distance, thereby providing a quantitative measure of the sparseness associated with topological inference. The practical utility of the methodology thorough testing of statistical hypothesis problem is illustrated for a seven-circle packing configuration under sparse contamination, where it recovers the underlying topological structure. Its applicability to real data is further demonstrated through an analysis of the M-ferritin protein structure, where the method identifies a persistent H2 feature corresponding to the iron-storage cavity.
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BibTeX
@misc{mahato2026topological,
title = {{Topological Learning under Sparsity through Huberized Distance Functionals}},
author = {Tuhin Subhra Mahato and Subhra Sankar Dhar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24170},
eprint = {2610.24170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24170},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Topological Learning under Sparsity through Huberized Distance Functionals
AU - Tuhin Subhra Mahato
AU - Subhra Sankar Dhar
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24170
UR - https://arxiv.org/abs/2610.24170
AB - Learning topological structure from sparsely contaminated data requires robust statistical methods, as even a small fraction of atypical observations can substantially distort the inferred topology. This article introduces a robust framework for topological learning through the Huberized Distance-to-Measure, which provides a robust filtration-generating functional for persistent homology. We establish large-sample properties of the proposed functional, and show that the associated filtration induces stable and consistent persistence diagrams. We further derive a maximum bias bound for persistence diagrams in bottleneck distance, thereby providing a quantitative measure of the sparseness associated with topological inference. The practical utility of the methodology thorough testing of statistical hypothesis problem is illustrated for a seven-circle packing configuration under sparse contamination, where it recovers the underlying topological structure. Its applicability to real data is further demonstrated through an analysis of the M-ferritin protein structure, where the method identifies a persistent H_2 feature corresponding to the iron-storage cavity.
ER -