In semi-supervised graph-based binary classifier learning, a subset of known labels $\hat{x}_i$ 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 $x_i$ to binary values, the problem is NP-hard. While a conventional semi-definite 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 ($M \succeq 0$) 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 $H \succeq 0$ 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 $\bar{H}$. We repose the SDP dual for solution $\bar{H}$, then replace the PSD cone constraint $\bar{H} \succeq 0$ with linear constraints derived from GDPA -- sufficient conditions to ensure $\bar{H}$ is PSD -- so that the optimization becomes an LP per iteration. Finally, we extract predicted labels from our converged LP solution $\bar{H}$. Experiments show that our algorithm enjoyed a $40\times$ speedup on average over the next fastest scheme while retaining comparable label prediction performance.
翻译:在半监督的基于图形的二进制分类器学习中,使用一组已知的标签 $\ h{x{x} 来推断未知的标签, 假设标签信号$x$对于一个拉普拉西亚矩阵指定的类似图形来说是平滑的。 当限制标签$x_ 美元对二进制值时, 问题在于 NP- 硬。 虽然常规的半确定性程序(SDP) 松绑可以在多盘调时解决, 例如, 使用乘数交替方向法( ADMM ), 反复将候选人的基数$M$反复投放到正的半确定性( PSD) 。 在本文中, 将最近的直线性平面理论称为 Gershgorin 完全对齐( GDP ), 我们建议快速投影化方法, 而不是用线性程序序列(LP) 。 我们首先将SDP 放松到 SDP 双向, 可行的解决方案从 $succeqion $ 0\ deal deal deal deal deal deal max the the the tral deal pal deal deal deal deal deal deal deal deal deal deal deal mas the slate sil deal deal slate slate slate slate sl ex the sla deal deal deal deal deal deal deal s pral deal deal deal pral pral deal pral de ex pral pral pral pral pril.