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In science and industry, data often arise as tensors, or multidimensional arrays, collected along various dimensions such as time, space, or frequency. Examples include video sequences in computer vision, 2D+ images in engineering and biomedical research, audio signals, and text embeddings in natural language processing. Preserving tensor structure in analysis can provide significant statistical and computational advantages over routine vectorization methods. Tensors retain the inherent multidimensional relationships within data, leading to more accurate and interpretable representations of complex phenomena. Additionally, tensor operations enable efficient manipulation of high-dimensional data, resulting in substantial savings in computation time and memory usage.
However, the mathematical theory of tensors remains somewhat elusive and is still under active development. While the maximum rank of a matrix of given dimensions is well understood, determining the maximum rank of a tensor is an open problem. Similarly, while the rank of a matrix can be easily determined using established algorithms like QR or SVD, finding the rank of a tensor is generally NP-hard. There is ample room for theoretical advances in tensor algebra and geometry, as well as in tensor-based optimization and statistics.