Title: Graph-guided Sparse Learning via Boolean Relaxation
Primary area: learning on graphs and other geometries & topologies
Keywords: Sparse learning, Boolean convex program, Boolean relaxation, Generalized fused lasso

Abstract: We propose a novel graph-guided sparse learning model using $\ell_0$ norm. The proposed model addresses a key limitation of existing methods, which enforce neighboring variables to have similar coefficients. We introduce a novel relaxation for the proposed problem. Our approach is based on reformulating the original model exactly as a Boolean convex program. We analyze the first-order relaxation and derive the necessary and sufficient conditions for exactness. We further show that these conditions are satisfied with high probability on random ensembles. Unlike existing methods, our relaxations provide lower bounds on the objective and can be verified whether the relaxation is exact. When the relaxation is not exact, we show that a rounding scheme based on the relaxed solutions leads to provably good feasible solutions. We numerically illustrate the outperformance of our novel relaxation in both simulation data and the real-world gene regulation inference task, demonstrating significant improvement of the proposed model.