Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the Čech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers. This lack of robustness has been frequently addressed in the literature. In this paper, we take a novel perspective by asking the following question: When can we guarantee that an observed persistent feature (a bar) is inherent to the underlying data in the presence of a limited number of unknown, arbitrary outliers. We formalize this question by introducing the notion of adversarial robustness, and study the problem of deciding whether a given bar in the barcode of a filtered simplicial complex is adversarially robust. We show that this problem is essentially equivalent to a homological variant of the minimum cut problem in simplicial complexes, which we believe to be of independent interest. As our main technical contribution, we provide the first computational complexity results for this problem, consisting of an efficient algorithm in 0-dimensional homology, NP-hardness for the general problem, and an efficient algorithm for codimension-1 in n-dimensional complexes embedded in ℝⁿ. We also analyze its natural linear programming relaxation, whose dual defines a homological analog of the max-flow problem in graphs. We show that a max-flow/min-cut theorem does not hold in our setting, implying that the LP relaxation is not tight in general. Finally, in the special case of the Rips filtration, we provide a global heuristic based on the Hausdorff distance that guarantees adversarial robustness of sufficiently long bars. This connects adversarial robustness to standard stability theorems in persistent homology.

Persistent homology has been widely used to study the topology of point clouds in Rn. Standard approaches are very sensitive to outliers, and their computational complexity depends badly on the number of data points. In this paper we introduce a novel persistence module for a point cloud using the theory of Christoffel-Darboux kernels. This module is robust to (statistical) outliers in the data, and can be computed in time linear in the number of data points. We illustrate the benefits and limitations of our new module with various numerical examples in Rn, for n = 1, 2, 3. Our work expands upon recent applications of Christoffel-Darboux kernels in the context of statistical data analysis and geometric inference [13]. There, these kernels are used to construct a polynomial whose level sets capture the geometry of a point cloud in a precise sense. We show that the persistent homology associated to the sublevel set filtration of this polynomial is stable with respect to the Wasserstein distance. Moreover, we show that the persistent homology of this filtration can be computed in singly exponential time in the ambient dimension n, using a recent algorithm of Basu & Karisani [1].