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

## TL;DR

This paper introduces a novel approach to efficiently project onto a three-view cardinality constrained set, which is useful in various applications like bioinformatics and crowdsourcing, by transforming the problem into a linear program.

## Contribution

It proposes a new method for projection under Three-view Cardinality Structure (TVCS) that simplifies the NP-hard problem into a linear programming solution with polynomial complexity.

## Key findings

- Vertex solution of LP yields the projection in TVCS
- Method reduces complexity to linear in variables and constraints
- Validated with synthetic and real-world bioinformatics and crowdsourcing data

## Abstract

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 cardinality constraints. In this paper, we consider the scenario where the 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 into a linear programming, and show that for TVCS, the vertex solution of this linear programming is the solution for the original projection problem. We further prove that such solution can be found with the complexity proportional to the number of variables and constraints. We finally use synthetic experiments and two interesting applications in bioinformatics and crowdsourcing to validate the proposed TVCS model and method.