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.

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