Tensor network decomposition for data recovery: Recent advancements and future prospects

Tensor network (TN) decomposition stands as a pivotal technique for characterizing the essential features of high-dimensional data, attracting significant interest and achieving notable success in high-dimensional data recovery. In recent years, there has been a steady stream of scholarly articles on TN decomposition, reflecting its growing significance. However, a comprehensive review that encapsulates the recent advancements and future prospects of TN decomposition remains conspicuously absent. To bridge this gap, in this article, we present a detailed review and insightful discussion on TN decomposition. Our goal is to furnish a comprehensive resource that equips researchers to navigate the complexities of TN decomposition and catalyze future innovative breakthroughs. We start by explaining concepts, operational rules, and computational properties on tensors. We then review various TN decompositions with different topologies, discussing their benefits, limitations, and common numerical algorithms. In particular, we emphasize the relationship between different TN ranks and unfolding matrix ranks, offering deeper insights for practical applications. Next, we focus on the critical TN structure search (TN-SS) problem, reviewing different TN-SS methods and discussing their advantages and disadvantages. Subsequently, we present a range of TN decomposition-based high-dimensional data recovery methods and evaluate their performance through numerical experiments. Finally, we delve into the realm of challenging problems and analyze potential future solutions.