Decomposability of Tensors

Abstract

Tensor decomposition has recently become a relevant topic, both for theoretical and applied mathematics, due to its interdisciplinary nature, which ranges from multilinear algebra and algebraic geometry to numerical analysis, algebraic statistics, quantum physics, signal processing, artificial intelligence, etc. The starting point behind the study of a decomposition relies on the idea that knowledge of elementary components of a tensor is fundamental to implement procedures able to understand and efficiently handle the information that a tensor encodes. Recent advances started with a systematic application of classical methods (some of them of geometric nature) to determine effective results on tensor decompositions. The methods range from the applications of the geometry of secant varieties in tensor spaces, to the study of symmetries in the decomposition of a specific tensor, to the determination of the sensitivity of a decomposition to small variations (deformations) of the data. Thanks to new applications of theoretic results, criteria for understanding when a given decomposition is minimal or unique, both for generic or specific tensors, have been recently introduced or significantly improved. New types of decompositions, of which elementary blocks can be chosen in a range of different possible models (e.g., Chow decompositions or mixed decompositions) are now systematically studied, and produce a deeper insight on the topic, with fruitful consequences on applications. The aim of this Special Issue is to collect papers that illustrate some directions in which recent research moves, as well as to provide a wide overview on several new approaches to the problem of tensor decomposition.