🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.001091s)

  1. Branching and Circular Features in High Dimensional Data (2011)

    B. Wang, B. Summa, V. Pascucci, M. Vejdemo-Johansson
    Abstract Large observations and simulations in scientific research give rise to high-dimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of highdimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto low-dimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original high-dimensional data. Our solution is to utilize topological techniques to recover important structures in high-dimensional data that contains non-trivial topology. Specifically, we are interested in high-dimensional branching structures. We construct local circle-valued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Additionally, we study the effects of global circular structures on visualizations. Our results reveal never-before-seen structures on real-world data sets from a variety of applications.

  2. Topology Identifies Emerging Adaptive Mutations in SARS-CoV-2 (2021)

    Michael Bleher, Lukas Hahn, Juan Angel Patino-Galindo, Mathieu Carriere, Ulrich Bauer, Raul Rabadan, Andreas Ott
    Abstract The COVID-19 pandemic has lead to a worldwide effort to characterize its evolution through the mapping of mutations in the genome of the coronavirus SARS-CoV-2. Ideally, one would like to quickly identify new mutations that could confer adaptive advantages (e.g. higher infectivity or immune evasion) by leveraging the large number of genomes. One way of identifying adaptive mutations is by looking at convergent mutations, mutations in the same genomic position that occur independently. However, the large number of currently available genomes precludes the efficient use of phylogeny-based techniques. Here, we establish a fast and scalable Topological Data Analysis approach for the early warning and surveillance of emerging adaptive mutations based on persistent homology. It identifies convergent events merely by their topological footprint and thus overcomes limitations of current phylogenetic inference techniques. This allows for an unbiased and rapid analysis of large viral datasets. We introduce a new topological measure for convergent evolution and apply it to the GISAID dataset as of February 2021, comprising 303,651 high-quality SARS-CoV-2 isolates collected since the beginning of the pandemic. We find that topologically salient mutations on the receptor-binding domain appear in several variants of concern and are linked with an increase in infectivity and immune escape, and for many adaptive mutations the topological signal precedes an increase in prevalence. We show that our method effectively identifies emerging adaptive mutations at an early stage. By localizing topological signals in the dataset, we extract geo-temporal information about the early occurrence of emerging adaptive mutations. The identification of these mutations can help to develop an alert system to monitor mutations of concern and guide experimentalists to focus the study of specific circulating variants.