Projection-free Graph-based Classifier Learning using Gershgorin Disc Perfect Alignment

In semi-supervised graph-based binary classifier learning, a subset of known labels x̂i are used to infer unknown labels, assuming that the label signal x is smooth with respect to a similarity graph specified by a Laplacian matrix. When restricting labels xi to binary values, the problem is NP-hard. While a conventional semidefinite programming (SDP) relaxation can be solved in polynomial time using, for example, the alternating direction method of multipliers (ADMM), the complexity of iteratively projecting a candidate matrix M onto the positive semi-definite (PSD) cone (M 0) remains high. In this paper, leveraging a recent linear algebraic theory called Gershgorin disc perfect alignment (GDPA), we propose a fast projection-free method by solving a sequence of linear programs (LP) instead. Specifically, we first recast the SDP relaxation to its SDP dual, where a feasible solution H 0 can be interpreted as a Laplacian matrix corresponding to a balanced signed graph sans the last node. To achieve graph balance, we split the last node into two that respectively contain the original positive and negative edges, resulting in a new Laplacian H̄. We repose the SDP dual for solution H̄, then replace the PSD cone constraint H̄ 0 with linear constraints derived from GDPA— sufficient conditions to ensure H̄ is PSD—so that the optimization becomes an LP per iteration. Finally, we extract predicted labels from our converged LP solution H̄. Experiments show that our algorithm enjoyed a 40× speedup on average over the next fastest scheme while retaining comparable label prediction performance.