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Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases
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On Koopman mode decomposition and tensor component analysis.

William T Redman1

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Chaos (Woodbury, N.Y.)
|July 9, 2021
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This summary is machine-generated.

Koopman mode decomposition and tensor component analysis (CANDECOMP/PARAFAC) are unified, showing they yield identical results under specific data conditions. This bridges two scientific communities and enables new algorithmic approaches.

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Area of Science:

  • Data analysis
  • Dynamical systems theory
  • Optimization theory

Background:

  • Koopman mode decomposition and tensor component analysis (CANDECOMP/PARAFAC) are popular methods for decomposing high-dimensional data.
  • These methods aim to capture relevant features and dynamics but are used by different scientific communities with distinct mathematical formulations.

Purpose of the Study:

  • To examine the relationship between Koopman mode decomposition and tensor component analysis.
  • To establish conditions under which these two decomposition methods yield the same results.
  • To foster communication between scientific communities and explore new algorithmic possibilities.

Main Methods:

  • Comparative analysis of Koopman mode decomposition and tensor component analysis.
  • Theoretical examination of decomposition under specific data conditions.

Main Results:

  • Demonstration that tensor component analysis provides the same theoretical decomposition as Koopman mode decomposition under certain data conditions.
  • Establishment of a theoretical bridge between the two methods.

Conclusions:

  • The findings provide a unified perspective on Koopman mode decomposition and tensor component analysis.
  • This unification facilitates interdisciplinary communication and opens avenues for novel algorithmic development.
  • The study highlights the utility of Koopman operator theory in problems traditionally addressed by optimization theory.