🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.001383s)
  1. The Geometry of Synchronization Problems and Learning Group Actions (2019)

    Tingran Gao, Jacek Brodzki, Sayan Mukherjee
    Abstract We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group G on connected graph ΓΓ\Gamma with a flat principal G-bundle over ΓΓ\Gamma , thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of ΓΓ\Gamma into G. We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal G-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham–Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions—partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations—and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.
  2. Lung Topology Characteristics in Patients With Chronic Obstructive Pulmonary Disease (2018)

    Francisco Belchi, Mariam Pirashvili, Joy Conway, Michael Bennett, Ratko Djukanovic, Jacek Brodzki
    Abstract Quantitative features that can currently be obtained from medical imaging do not provide a complete picture of Chronic Obstructive Pulmonary Disease (COPD). In this paper, we introduce a novel analytical tool based on persistent homology that extracts quantitative features from chest CT scans to describe the geometric structure of the airways inside the lungs. We show that these new radiomic features stratify COPD patients in agreement with the GOLD guidelines for COPD and can distinguish between inspiratory and expiratory scans. These CT measurements are very different to those currently in use and we demonstrate that they convey significant medical information. The results of this study are a proof of concept that topological methods can enhance the standard methodology to create a finer classification of COPD and increase the possibilities of more personalized treatment.
  3. Improved Understanding of Aqueous Solubility Modeling Through Topological Data Analysis (2018)

    Mariam Pirashvili, Lee Steinberg, Francisco Belchi Guillamon, Mahesan Niranjan, Jeremy G. Frey, Jacek Brodzki
    Abstract Topological data analysis is a family of recent mathematical techniques seeking to understand the ‘shape’ of data, and has been used to understand the structure of the descriptor space produced from a standard chemical informatics software from the point of view of solubility. We have used the mapper algorithm, a TDA method that creates low-dimensional representations of data, to create a network visualization of the solubility space. While descriptors with clear chemical implications are prominent features in this space, reflecting their importance to the chemical properties, an unexpected and interesting correlation between chlorine content and rings and their implication for solubility prediction is revealed. A parallel representation of the chemical space was generated using persistent homology applied to molecular graphs. Links between this chemical space and the descriptor space were shown to be in agreement with chemical heuristics. The use of persistent homology on molecular graphs, extended by the use of norms on the associated persistence landscapes allow the conversion of discrete shape descriptors to continuous ones, and a perspective of the application of these descriptors to quantitative structure property relations is presented.