On The Projection Operator to A Three-view Cardinality Constrained Set

The cardinality constraint is an intrinsic way to restrict the solution structure in many domains, for example, sparse learning, feature selection, and compressed sensing. To solve a cardinality constrained problem, the key challenge is to solve the projection onto the cardinality constraint set, which is NP-hard in general when there exist multiple overlapped cardiaiality constraints. In this paper, we consider the scenario where overlapped cardinality constraints satisfy a Three-view Cardinality Structure (TVCS), which reflects the natural restriction in many applications, such as identification of gene regulatory networks and task-worker assignment problem. We cast the projection onto the TVCS set into a linear programming, and prove that its solution can be obtained by finding an integer solution to such linear programming. We further prove that such integer solution can be found with the complexity proportional to the problem scale. We finally use synthetic experiments and two interesting applications in bioinformatics and crowdsourcing to validate the proposed TVCS model and method.