Theoretical Foundations and Practical Applications in Signal Processing and Machine Learning

Tensor decomposition has emerged as a powerful mathematical framework for analyzing multi-dimensional data, extending classical matrix decomposition techniques to higher-order representations. As modern applications generate increasingly complex datasets with multi-way relationships, tensor methods provide a principled approach to uncovering latent structures, reducing dimensionality, and improving computational efficiency. This paper presents a comprehensive review of tensor decomposition techniques, their theoretical foundations, and their applications in signal processing and machine learning.We begin by introducing the fundamental concepts of tensor algebra, discussing key tensor operations, norms, and properties that form the basis of tensor factorization methods. The two most widely used decompositions—Canonical Polyadic (CP) and Tucker decomposition—are examined in detail, along with alternative factorization techniques such as Tensor Train (TT), Tensor Ring (TR), and Block Term Decomposition (BTD). We explore the computational complexity of these methods and discuss numerical optimization techniques, including Alternating Least Squares (ALS), gradient-based approaches, and probabilistic tensor models.The paper then delves into the applications of tensor decomposition in signal processing, where tensors have been successfully applied to source separation, multi-sensor data fusion, image processing, and compressed sensing. In machine learning, tensor-based models have enhanced feature extraction, deep learning efficiency, and representation learning. We highlight the role of tensor decomposition in reducing the parameter space of deep neural networks, improving generalization, and accelerating training through low-rank approximations.Despite its numerous advantages, tensor decomposition faces several challenges, including the difficulty of determining tensor rank, the computational cost of large-scale tensor factorization, and robustness to noise and missing data. We discuss recent theoretical advancements addressing uniqueness conditions, rank estimation strategies, and adaptive tensor factorization techniques that improve performance in real-world applications. Furthermore, we explore emerging trends in tensor methods, including their integration with quantum computing, neuroscience, personalized medicine, and geospatial analytics.Finally, we provide a detailed discussion of open research questions, such as the need for more scalable decomposition algorithms, automated rank selection mechanisms, and robust tensor models that can handle high-dimensional, noisy, and adversarial data. As data-driven applications continue to evolve, tensor decomposition is poised to become an indispensable tool for uncovering hidden patterns in complex datasets, advancing both theoretical research and practical implementations across multiple scientific domains.