Definition and Computation of Tensor‐Based Generalized Function Composition

ABSTRACT Functions are fundamental to mathematics as they offer a structured and analytical framework to express relations between variables. While scalar and matrix‐based functions are well‐established, higher‐order tensor‐based functions have not been as extensively explored. Tensors, which generalize matrices to multiple dimensions, are vital in science and engineering, especially in data analysis, machine learning, and computational mathematics. Recently, the concept of “t‐function” emphasizing the tubal view of a 3‐order tensor (i.e., a cube) has been introduced within the framework of the t‐algebra; however, definition and computation for functions applied to multilinear arrays based on Tucker or canonical polyadic (CP) models are still lacking. This work introduces a definition of tensor‐based functions that is tailored to the complex and rich multilinear structure of a ‐order tensor. The proposed tensor‐based function is defined using the composition of generalized functions, combining unfolding and folding operations. We show that the tensor‐based function based on the higher‐order singular‐value decomposition (HOSVD) is “propagated” onto the core of the decomposition as in the matrix‐based function case with respect to the SVD. In addition, lower and upper error bounds on a fidelity criterion are obtained by invoking the bi‐Lipschitz condition. The computational aspects related to the generalized function composition for the dense core tensor of the HOSVD, as well as a diagonal core that arises in the CP model, are discussed. The proposed formalism is illustrated through three standard yet challenging applications: tensor‐based entropy for uniform hypergraphs, interpolation of covariance tensors within the log‐Euclidean (LE) framework, and denoising and edge detection for volumetric imaging.