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@ARTICLE{Albers:1041480,
      author       = {Albers, Jasper and Kurth, Anno and Gutzen, Robin and
                      Morales-Gregorio, Aitor and Grün, Sonja and Diesmann,
                      Markus},
      title        = {{A}ssessing the similarity of real matrices with arbitrary
                      shape},
      publisher    = {arXiv},
      reportid     = {FZJ-2025-02266},
      year         = {2024},
      abstract     = {Assessing the similarity of matrices is valuable for
                      analyzing the extent to which data sets exhibit common
                      features in tasks such as data clustering, dimensionality
                      reduction, pattern recognition, group comparison, and graph
                      analysis. Methods proposed for comparing vectors, such as
                      cosine similarity, can be readily generalized to matrices.
                      However, this approach usually neglects the inherent
                      two-dimensional structure of matrices. Here, we propose
                      singular angle similarity (SAS), a measure for evaluating
                      the structural similarity between two arbitrary, real
                      matrices of the same shape based on singular value
                      decomposition. After introducing the measure, we compare SAS
                      with standard measures for matrix comparison and show that
                      only SAS captures the two-dimensional structure of matrices.
                      Further, we characterize the behavior of SAS in the presence
                      of noise and as a function of matrix dimensionality.
                      Finally, we apply SAS to two use cases: square non-symmetric
                      matrices of probabilistic network connectivity, and
                      non-square matrices representing neural brain activity. For
                      synthetic data of network connectivity, SAS matches
                      intuitive expectations and allows for a robust assessment of
                      similarities and differences. For experimental data of brain
                      activity, SAS captures differences in the structure of
                      high-dimensional responses to different stimuli. We conclude
                      that SAS is a suitable measure for quantifying the shared
                      structure of matrices with arbitrary shape.},
      keywords     = {Neurons and Cognition (q-bio.NC) (Other) / Data Analysis,
                      Statistics and Probability (physics.data-an) (Other) /
                      Quantitative Methods (q-bio.QM) (Other) / FOS: Biological
                      sciences (Other) / FOS: Physical sciences (Other)},
      cin          = {IAS-6 / INM-10},
      cid          = {I:(DE-Juel1)IAS-6-20130828 / I:(DE-Juel1)INM-10-20170113},
      pnm          = {5231 - Neuroscientific Foundations (POF4-523) / BMBF
                      03ZU1106CB - NeuroSys: Algorithm-Hardware Co-Design (Projekt
                      C) - B (BMBF-03ZU1106CB) / DFG project G:(GEPRIS)313856816 -
                      SPP 2041: Computational Connectomics (313856816) / HBP SGA3
                      - Human Brain Project Specific Grant Agreement 3 (945539) /
                      EBRAINS 2.0 - EBRAINS 2.0: A Research Infrastructure to
                      Advance Neuroscience and Brain Health (101147319) / JL SMHB
                      - Joint Lab Supercomputing and Modeling for the Human Brain
                      (JL SMHB-2021-2027)},
      pid          = {G:(DE-HGF)POF4-5231 / G:(DE-Juel1)BMBF-03ZU1106CB /
                      G:(GEPRIS)313856816 / G:(EU-Grant)945539 /
                      G:(EU-Grant)101147319 / G:(DE-Juel1)JL SMHB-2021-2027},
      typ          = {PUB:(DE-HGF)25},
      doi          = {10.48550/arXiv.2403.17687},
      url          = {https://juser.fz-juelich.de/record/1041480},
}