Tensor-Based Least-Squares Solutions for Multirelational Signals and Applications

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.