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While taking the arithmetic, geometric, or harmonic mean of a sample data set, equal importance is assigned to all the data points. However, all the values may not always be equally important in some data sets. An intrinsic bias might make it more important to give more weightage to specific values over others.
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Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases
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Tensor Recovery With Weighted Tensor Average Rank.

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    Summary
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    This study introduces the weighted tensor average rank (WTAR) to address information loss in tensor recovery algorithms. WTAR analyzes how different transpositions of observation tensors affect results, improving tensor robust principal component analysis (TRPCA).

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    Area of Science:

    • Data Science
    • Numerical Analysis
    • Linear Algebra

    Background:

    • Tensor recovery algorithms can be sensitive to the transposition of observation tensors.
    • Information loss may occur when observation tensors are transposed differently.
    • Existing methods lack a robust way to quantify the impact of tensor transpositions.

    Purpose of the Study:

    • To investigate the phenomenon of varying results in tensor recovery based on tensor transposition.
    • To propose a novel tensor rank, weighted tensor average rank (WTAR), to capture relationships between tensors under different transpositions.
    • To develop and evaluate a generalized tensor robust principal component analysis (TRPCA) framework using WTAR.

    Main Methods:

    • Introduction of the weighted tensor average rank (WTAR) metric.
    • Application of WTAR to tensor robust principal component analysis (TRPCA).
    • Development of a generalized TRPCA model with convex and nonconvex surrogates.
    • Proposal of a generalized tensor singular value thresholding (GTSVT) method and optimization algorithm.

    Main Results:

    • WTAR effectively learns relationships between tensors resulting from various transpose operators.
    • The generalized TRPCA model demonstrates effectiveness in tensor recovery.
    • Worst-case error bounds for the recovered tensor are established.
    • The proposed GTSVT method and optimization algorithm efficiently solve the generalized model.

    Conclusions:

    • The proposed weighted tensor average rank (WTAR) provides a robust measure for tensor recovery.
    • The generalized TRPCA framework with WTAR offers improved effectiveness and solvability.
    • The developed GTSVT algorithm is efficient for solving the generalized tensor recovery model.