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    This study introduces a new mathematical framework for exact tensor-based least squares (TLS) solutions, overcoming limitations of current methods for high-dimensional tensor data. This enables precise linear regression analysis in machine learning and signal processing.

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    Area of Science:

    • * Mathematics
    • * Computer Science
    • * Signal Processing

    Background:

    • * Least squares (LSs) is a standard method for linear regression, applicable to various systems.
    • * Current LS methods are limited to matrix data and cannot handle high-dimensional tensor data directly.
    • * Existing tensor-based LS approximations lack exactness due to the absence of a suitable mathematical framework.

    Purpose of the Study:

    • * To present a novel mathematical framework for exact tensor-based least squares (TLS) solutions.
    • * To address the limitations of existing methods in handling high-dimensional tensor data.

    Main Methods:

    • * Development of a new mathematical framework for exact TLS solutions.
    • * Application of the framework to tensor data in linear regression problems.
    • * Numerical experiments to validate the proposed scheme.

    Main Results:

    • * Successful demonstration of a new mathematical framework for exact TLS solutions.
    • * Validation of the framework's applicability through experiments in machine learning and speech recognition.
    • * Analysis of memory and computational complexities associated with the new scheme.

    Conclusions:

    • * The proposed mathematical framework enables exact TLS solutions for tensor data.
    • * This advancement overcomes the limitations of previous approximation techniques.
    • * The new scheme shows promise for applications in machine learning and robust signal processing.