The reconfigurable intelligent surfaces (RIS)-assisted multi-user communication system has appeared as a promising technology for enhancing capacity and extending coverage, which requires accurate channel state information. However, the associated channel estimation problem is challenging due to the high dimensionality of channels. To realize accurate channel estimation with light training overhead, the key is to exploit the structure of channels to the largest extent. Previous works have exploited the sparsity and tensor decomposition structure separately, while there is still no work jointly leveraging these two channel structures for further enhancing channel estimation performance. Consequently, this letter manages to formulate a novel sparsity-structured tensor decomposition-based channel estimation problem, and derive an efficient algorithm under the alternating optimization framework. The proposed method can reduce the training overhead for RIS-assisted multiple-input multiple-output (MIMO) communications. Simulation results verify the effectiveness of the proposed algorithm.