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Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data
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Tensor sparse coding for positive definite matrices.

Ravishankar Sivalingam1, Daniel Boley1, Vassilios Morellas1

  • 1University of Minnesota, Twin Cities, Minneapolis.

IEEE Transactions on Pattern Analysis and Machine Intelligence
|January 25, 2014
PubMed
Summary
This summary is machine-generated.

This study introduces a new sparse coding method for positive definite matrices, preserving their mathematical structure. This approach avoids data vectorization, maintaining eigenvalue properties for improved signal processing.

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Area of Science:

  • Signal Processing
  • Machine Learning
  • Computer Vision

Background:

  • Sparse representation is crucial for vector-valued signals.
  • Vectorizing matrices for sparse coding can destroy inherent data structures, like those in symmetric positive definite (SPD) matrices.
  • Existing methods fail to preserve the unique properties of SPD matrices.

Purpose of the Study:

  • To develop a novel sparse coding technique for positive definite matrices.
  • To respect the Riemannian manifold structure of these matrices.
  • To preserve the positive eigenvalue property without vectorization.

Main Methods:

  • A new sparse coding model designed for the geometry of positive definite matrices.
  • The method operates directly on matrices, avoiding vectorization.
  • Utilizes properties of the Riemannian manifold for SPD matrices.

Main Results:

  • The proposed method successfully preserves the positive eigenvalue structure of SPD matrices.
  • Demonstrated applicability and necessity through synthetic and real-world experiments.
  • Effective use with region covariance descriptors in computer vision.

Conclusions:

  • The novel sparse coding technique effectively models positive definite matrices.
  • This work bridges sparse modeling and the geometry of positive definite matrices.
  • Offers a more suitable approach for structured matrix data compared to vectorization.