🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.000901s)
  1. Geometric Anomaly Detection in Data (2020)

    Bernadette J. Stolz, Jared Tanner, Heather A. Harrington, Vidit Nanda
    Abstract The quest for low-dimensional models which approximate high-dimensional data is pervasive across the physical, natural, and social sciences. The dominant paradigm underlying most standard modeling techniques assumes that the data are concentrated near a single unknown manifold of relatively small intrinsic dimension. Here, we present a systematic framework for detecting interfaces and related anomalies in data which may fail to satisfy the manifold hypothesis. By computing the local topology of small regions around each data point, we are able to partition a given dataset into disjoint classes, each of which can be individually approximated by a single manifold. Since these manifolds may have different intrinsic dimensions, local topology discovers singular regions in data even when none of the points have been sampled precisely from the singularities. We showcase this method by identifying the intersection of two surfaces in the 24-dimensional space of cyclo-octane conformations and by locating all of the self-intersections of a Henneberg minimal surface immersed in 3-dimensional space. Due to the local nature of the topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.
  2. A Topological Measurement of Protein Compressibility (2015)

    Marcio Gameiro, Yasuaki Hiraoka, Shunsuke Izumi, Miroslav Kramar, Konstantin Mischaikow, Vidit Nanda
    Abstract In this paper we partially clarify the relation between the compressibility of a protein and its molecular geometric structure. To identify and understand the relevant topological features within a given protein, we model its molecule as an alpha filtration and hence obtain multi-scale insight into the structure of its tunnels and cavities. The persistence diagrams of this alpha filtration capture the sizes and robustness of such tunnels and cavities in a compact and meaningful manner. From these persistence diagrams, we extract a measure of compressibility derived from those topological features whose relevance is suggested by physical and chemical properties. Due to recent advances in combinatorial topology, this measure is efficiently and directly computable from information found in the Protein Data Bank (PDB). Our main result establishes a clear linear correlation between the topological measure and the experimentally-determined compressibility of most proteins for which both PDB information and experimental compressibility data are available. Finally, we establish that both the topological measurement and the linear correlation are stable with respect to small perturbations in the input data, such as those arising from experimental errors in compressibility and X-ray crystallography experiments.