Encoding sparse and competitive structures among tasks in multi-task learning.

•we develop a new formulation for MTL based on the decomposition of the coeffcient matrix into a Hadamard (element-wise) product of two matrices. Comparing with conventional MTL methods and EL, the advantages of the proposed approach can be summarized as follows: (1) It is capable of capturing the competitive structure among tasks. (2) Unimportant features which are common across the tasks can be removed from the final model. Moreover, we propose to employ an alternating optimization method to iteratively estimate the coeffcients of the two components in the SpEL objective function.•We also provide an analysis of the proposed model based on the element- wise product decomposition framework to highlight its advantage.•We conduct experimental studies on both synthetic and real data in different application domains which include handwritten digit data and gene expression analysis. The experimental results demonstrate the effectiveness of the proposed model, and suggest potential applications of the proposed method.