tensorder

Tensorder is a theoretical concept used to quantify the complexity of representing a tensor through a structured factorization. It is defined as the smallest integer k for which a given tensor T admits a tensorder decomposition consisting of k components. Each component is a low-arity product of smaller tensors arranged in a hierarchical or networked pattern. In this sense, tensorder generalizes the idea of rank: for a matrix (a second-order tensor), tensorder equals the usual rank; for higher-order tensors, it provides an alternative, often coarser, complexity measure compared to CP or Tucker ranks.

Properties and computation: Tensorder depends on the chosen class of decompositions; different tensorder schemes can yield different k for the same tensor. Computing the exact tensorder is generally NP-hard, and practitioners rely on heuristic algorithms based on tensor networks to obtain practical upper bounds.

Applications and interpretation: The tensorder concept is discussed in the context of data compression, machine learning with structured tensors, and theoretical studies of tensor networks. A low tensorder representation supports efficient storage and fast contractions, while a high tensorder indicates greater representational complexity.

History and relation: The term tensorder has appeared in speculative literature and some conference discussions; it has not achieved broad standardization and remains one of several measures of tensor complexity, alongside tensor rank, CP decomposition, Tucker decomposition, and TT-rank. See also: tensor rank; CP decomposition; Tucker decomposition; Tensor Train; Hierarchical Tucker; tensor networks.