基于Gershgorin圆盘对齐的图度量学习
Graph Metric Learning via Gershgorin Disc Alignment
摘要：
我们提出了一个快速通用的无投影度量学习框架，其中最小化目标$\min_{\textbf{M} \in \mathcal{S}} Q(\textbf{M})$是度量矩阵$\textbf{M}$的凸可微函数，并且$\textbf{M}$驻留在具有正边权重和节点度的连通图的广义图拉普拉斯矩阵集合$\mathcal{S}$中。与文献中常见的低秩度量矩阵不同，$\mathcal{S}$将重要的仅正对角矩阵作为极限的特殊情况。快速优化的关键思想是通过Gershgorin圆盘对齐将$\mathcal{S}$中的正定锥约束重写为信号自适应线性约束，从而通过Frank-Wolfe迭代将$\textbf{M}$中对角项和非对角项的交替优化有效地求解为线性规划。我们证明，Gershgorin光盘可以完全对齐使用的第一特征向量$\textbf{v}$的$\textbf{M}$，我们迭代更新使用局部最优块预处理共轭梯度（LOBPCG）与热启动作为对角/非对角项进行优化。实验表明，我们有效计算的图度量矩阵优于使用竞争方法学习的度量分类任务。
英文摘要：
We propose a fast general projection-free metric learning framework, where the minimization objective $\min_{\textbf{M} \in \mathcal{S}} Q(\textbf{M})$ is a convex differentiable function of the metric matrix $\textbf{M}$, and $\textbf{M}$ resides in the set $\mathcal{S}$ of generalized graph Laplacian matrices for connected graphs with positive edge weights and node degrees. Unlike low-rank metric matrices common in the literature, $\mathcal{S}$ includes the important positive-diagonal-only matrices as a special case in the limit. The key idea for fast optimization is to rewrite the positive definite cone constraint in $\mathcal{S}$ as signal-adaptive linear constraints via Gershgorin disc alignment, so that the alternating optimization of the diagonal and off-diagonal terms in $\textbf{M}$ can be solved efficiently as linear programs via Frank-Wolfe iterations. We prove that the Gershgorin discs can be aligned perfectly using the first eigenvector $\textbf{v}$ of $\textbf{M}$, which we update iteratively using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as diagonal / off-diagonal terms are optimized. Experiments show that our efficiently computed graph metric matrices outperform metrics learned using competing methods in terms of classification tasks.
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