2017/07/21 by Dario Gasbarra, Gasbarra, Dario, Sinisa Pajevic +3
Mathematics · Medicine · #62E20 #62K #62P10 #92C55 #94A08 #Advanced Neuroimaging Techniques and Applications #FOS: Computer and information sciences #Methodology (stat.ME) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1707.06953
openalex publication_date 2017/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Tensor-valued and matrix-valued measurements of different physical properties\nare increasingly available in material sciences and medical imaging\napplications. The eigenvalues and eigenvectors of such multivariate data\nprovide novel and unique information, but at the cost of requiring a more\ncomplex statistical analysis. In this work we derive the distributions of\neigenvalues and eigenvectors in the special but important case of m \× m\nsymmetric random matrices, D, observed with isotropic matrix-variate Gaussian\nnoise. The properties of these distributions depend strongly on the symmetries\nof the mean tensor/matrix, D. When D has repeated eigenvalues,\nthe eigenvalues of D are not asymptotically Gaussian, and repulsion is\nobserved between the eigenvalues corresponding to the same D\neigenspaces. We apply these results to diffusion tensor imaging (DTI), with\nm=3, addressing an important problem of detecting the symmetries of the\ndiffusion tensor, and seeking an experimental design that could potentially\nyield an isotropic Gaussian distribution. In the 3-dimensional case, when the\nmean tensor is spherically symmetric and the noise is Gaussian and isotropic,\nthe asymptotic distribution of the first three eigenvalue central moment\nstatistics is simple and can be used to test for isotropy. In order to apply\nsuch tests, we use quadrature rules of order t \≥ 4 with constant weights on\nthe unit sphere to design a DTI-experiment with the property that isotropy of\nthe underlying true tensor implies isotropy of the Fisher information. We also\nexplain the potential implications of the methods using simulated DTI data with\na Rician noise model.\n