Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment

Given a convex and differentiable objective $Q({\mathbf M})$ for a real symmetric matrix ${\mathbf M}$ in the positive definite (PD) cone—used to compute Mahalanobis distances—we propose a fast general metric learning framework that is entirely projection-free. We first assume that ${\mathbf M}$ resides in a space ${\mathcal S}$ of generalized graph Laplacian matrices corresponding to balanced signed graphs. ${\mathbf M}\in {\mathcal S}$ that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, ${\mathcal S}$ includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc perfect alignment (GDPA): given ${\mathbf M}\in {\mathcal S}$ and diagonal matrix ${\mathbf S}$ , where $S_{ii} = 1/v_i$ and ${\mathbf v}$ is the first eigenvector of ${\mathbf M}$ , we prove that Gershgorin disc left-ends of similarity transform ${\mathbf B}= {\mathbf S}{\mathbf M}{\mathbf S}^{-1}$ are perfectly aligned at the smallest eigenvalue $\lambda _{\min }$ . Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in ${\mathbf M}$ can be solved efficiently as linear programs via the Frank-Wolfe method. We update ${\mathbf v}$ using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in ${\mathbf M}$ are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes, and produces competitive binary classification performance.