Using geometry of the set of symmetric positive semidefinite matrices to classify structural brain networks

Mikhail Belyaev, Yulia Dodonova, Daria Belyaeva, Egor Krivov, Boris Gutman, Joshua Faskowitz, Neda Jahanshad, Paul Thompson

    Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


    This paper presents a method of symmetric positive semidefinite (SPSD) matrices classification and its application to the analysis of structural brain networks (connectomes). Structural connectomes are modeled as weighted graphs in which edge weights are proportional to the number of streamline connections between brain regions detected by a tractography algorithm. The construction of structural brain networks does not typically guarantee that their adjacency matrices lie in some topological space with known properties. This makes them differ from functional connectomes-correlation matrices representing co-activation of brain regions, which are usually symmetric positive definite (SPD). Here, we propose to transform structural connectomes by taking their normalized Laplacians prior to any analysis, to put them into a space of symmetric positive semidefinite (SPSD) matrices, and apply methods developed for manifold-valued data. The geometry of the SPD matrix manifold is well known and used in many classification algorithms. Here, we expand existing SPD matrix-based algorithms to the SPSD geometry and develop classification pipelines on SPSD normalized Laplacians of structural connectomes. We demonstrate the performance of the proposed pipeline on structural brain networks reconstructed from the Alzheimer‘s Disease Neuroimaging Initiative (ADNI) data.

    Original languageEnglish
    Title of host publicationComputational Aspects and Applications in Large-Scale Networks - NET 2017
    EditorsPanos M. Pardalos, Oleg Prokopyev, Irina Utkina, Valery A. Kalyagin
    PublisherSpringer New York LLC
    Number of pages11
    ISBN (Print)9783319962467
    Publication statusPublished - 2018
    Event7th International Conference on Network Analysis, NET 2017 - Nizhny Novgorod, Russian Federation
    Duration: 22 Jun 201724 Jun 2017

    Publication series

    NameSpringer Proceedings in Mathematics and Statistics
    ISSN (Print)2194-1009
    ISSN (Electronic)2194-1017


    Conference7th International Conference on Network Analysis, NET 2017
    Country/TerritoryRussian Federation
    CityNizhny Novgorod


    • Brain networks
    • Riemannian geometry


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