基于Gershgorin圆盘完全对齐的无投影图分类器学习
Projection-free Graph-based Classifier Learning using Gershgorin Disc Perfect Alignment
摘要：
在基于半监督图的二元分类器学习中，已知标签的子集$\hat{x}_i$用于推断未知标签，假设标签信号$\mathbf{x}$相对于由拉普拉斯矩阵指定的相似性图是平滑的。当将标签$x_i$限制为二进制值时，问题是NP难的。虽然可以使用例如交替方向乘法器（ADMM）在多项式时间内求解常规的半定编程松弛（SDR），但是每次迭代将候选矩阵$\mathbf{M}$投影到半正定（PSD）锥（$\mathbf{M} \succeq 0$）上的复杂度仍然很高。在本文中，利用最近的线性代数理论称为Gershgorin圆盘完美对齐（GDPA），我们提出了一个快速的投影免费的方法，而不是解决一系列的线性规划（LP）。具体来说，我们首先将SDR转换为它的对偶，其中可行解$\mathbf{H} \succeq 0$被解释为对应于平衡符号图减去最后一个节点的拉普拉斯矩阵。为了实现图的平衡，我们将最后一个节点分成两个，每个节点保留原始的正/负边，从而产生一个新的拉普拉斯$\bar{\mathbf{H}}$。我们将SDR对偶用于解$\bar{\mathbf{H}}$，然后将PSD锥约束$\bar{\mathbf{H}} \succeq 0$替换为来自GDPA的线性约束--确保$\bar{\mathbf{H}}$是PSD的充分条件--使得每次迭代的优化成为LP。最后，我们从收敛的解决方案$\bar{\mathbf{H}}$中提取预测标签。实验表明，我们的算法享有$28\times$加速比下一个最快的计划，同时实现可比的标签预测性能。
英文摘要：
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 $\mathbf{x}$ 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 relaxation (SDR) can be solved in polynomial time using, for example, the alternating direction method of multipliers (ADMM), the complexity of projecting a candidate matrix $\mathbf{M}$ onto the positive semi-definite (PSD) cone ($\mathbf{M} \succeq 0$) per iteration 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 SDR to its dual, where a feasible solution $\mathbf{H} \succeq 0$ is interpreted as a Laplacian matrix corresponding to a balanced signed graph minus the last node. To achieve graph balance, we split the last node into two, each retains the original positive / negative edges, resulting in a new Laplacian $\bar{\mathbf{H}}$. We repose the SDR 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 the optimization becomes an LP per iteration. Finally, we extract predicted labels from converged solution $\bar{\mathbf{H}}$. Experiments show that our algorithm enjoyed a $28\times$ speedup over the next fastest scheme while achieving comparable label prediction performance.
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