Browsing by Author "Micka, Samuel"

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Reconstructing embedded graphs from persistence diagrams
(2020-10) Belton, Robin Lynne; Fasy, Brittany T.; Mertz, Rostik; Micka, Samuel; Millman, David L.; Salinas, Daniel; Schenfisch, Anna; Schupbach, Jordan; Williams, Lucia
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes using a finite set of descriptors. A related inverse problem is the following: given a set of PDs (or an oracle we can query for persistence diagrams), what is underlying geometric simplicial complex? In this paper, we present an algorithm for reconstructing embedded graphs in Rd (plane graphs in R2) with n vertices from n2 −n+d+1 directional (augmented) PDs. Additionally, we empirically validate the correctness and time-complexity of our algorithm in R2 on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.
Sampling Bounds for Topological Descriptors
(Undergraduate Scholars Program, 2024-04) Fasy, Brittany; Millman, David; Micka, Samuel; Padula, Luke; Makarchuk, Maksym
Increasingly, topological descriptors like the Euler characteristic curve and persistence diagrams are utilized to represent complex data. Recent studies suggest that a meticulously selected set of these descriptors can encode geometric and topological information about shapes in d-dimensional space. In practical applications, epsilon-nets are employed to sample data, presenting two extremes: oversampling, where epsilon is small enough to ensure a comprehensive representation but may lead to computational inefficiencies, and undersampling, where epsilon lacks a grounded rationale, offering faster computations but risking an incomplete shape description without theoretical guarantees. This research investigates phenomena of oversampling and undersampling, delving into their prevalence across synthetic and real-world datasets. It experimentally verifies excessive oversampling in theory-guided approaches and examines the implications of undersampling, shedding light on the behavior and consequences of both extremes. We establish lower bounds on the number of descriptors required for exact encodings and explore the trade-offs associated with undersampling, contributing insights into the potential information loss and the resulting impact on the overall shape representation.