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In semi-supervised graph-based binary classifier learning, a subset of knownlabels $\hat{x}_i$ are used to infer unknown labels, assuming that the labelsignal $\mathbf{x}$ is smooth with respect to a similarity graph specified by aLaplacian matrix. When restricting labels $x_i$ to binary values, the problemis NP-hard. While a conventional semi-definite programming relaxation (SDR) canbe solved in polynomial time using, for example, the alternating directionmethod of multipliers (ADMM), the complexity of projecting a candidate matrix$\mathbf{M}$ onto the positive semi-definite (PSD) cone ($\mathbf{M} \succeq0$) per iteration remains high. In this paper, leveraging a recent linearalgebraic theory called Gershgorin disc perfect alignment (GDPA), we propose afast projection-free method by solving a sequence of linear programs (LP)instead. Specifically, we first recast the SDR to its dual, where a feasiblesolution $\mathbf{H} \succeq 0$ is interpreted as a Laplacian matrixcorresponding to a balanced signed graph minus the last node. To achieve graphbalance, we split the last node into two, each retains the original positive /negative edges, resulting in a new Laplacian $\bar{\mathbf{H}}$. We repose theSDR dual for solution $\bar{\mathbf{H}}$, then replace the PSD cone constraint$\bar{\mathbf{H}} \succeq 0$ with linear constraints derived from GDPA --sufficient conditions to ensure $\bar{\mathbf{H}}$ is PSD -- so that theoptimization becomes an LP per iteration. Finally, we extract predicted labelsfrom converged solution $\bar{\mathbf{H}}$. Experiments show that our algorithmenjoyed a $28\times$ speedup over the next fastest scheme while achievingcomparable label prediction performance.

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