This paper presents a new paradigm of explicitly modeling and harnessing the data structure to address the inverse problems in structural dynamics, identification, and data-driven health monitoring. In particular, it is shown that the structural dynamic features and damage information, intrinsic within the structural vibration response measurement data, possesses sparse and low-rank structure, which can be effectively modeled and processed by emerging mathematical tools such as sparse representation and compressed sensing, low-rank matrix decomposition and completion, as well as the unsupervised multivariate blind source separation. It is also discussed that explicitly modeling and harnessing the sparse and low-rank data structure could benefit future work in developing data-driven approaches toward rapid, unsupervised, and effective system identification, damage detection, as well as massive SHM data sensing and management. Copyright (C) 2016 John Wiley & Sons, Ltd.