  Tensor data with rich structural information becomes increasingly important
in process modeling, monitoring, and diagnosis. Here structural information is
referred to structural properties such as sparsity, smoothness, low-rank, and
piecewise constancy. To reveal useful information from tensor data, we propose
to decompose the tensor into the summation of multiple components based on
different structural information of them. In this paper, we provide a new
definition of structural information in tensor data. Based on it, we propose an
additive tensor decomposition (ATD) framework to extract useful information
from tensor data. This framework specifies a high dimensional optimization
problem to obtain the components with distinct structural information. An
alternating direction method of multipliers (ADMM) algorithm is proposed to
solve it, which is highly parallelable and thus suitable for the proposed
optimization problem. Two simulation examples and a real case study in medical
image analysis illustrate the versatility and effectiveness of the ATD
framework.
