{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/signed-graph-metric-learning-via-gershgorin","title":"Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment","arxiv_id":"2006.08816","date":"2020-06-15","proceeding":null,"authors":["Cheng Yang","Gene Cheung","Wei Hu"],"abstract":"Given a convex and differentiable objective $Q(\\M)$ for a real symmetric matrix $\\M$ 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 $\\M$ resides in a space $\\cS$ of generalized graph Laplacian matrices corresponding to balanced signed graphs. $\\M \\in \\cS$ that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, $\\cS$ 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 $\\M \\in \\cS$ and diagonal matrix $\\S$, where $S_{ii} = 1/v_i$ and $\\v$ is $\\M$'s first eigenvector, we prove that Gershgorin disc left-ends of similarity transform $\\B = \\S \\M \\S^{-1}$ 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 $\\M$ can be solved efficiently as linear programs via the Frank-Wolfe method. We update $\\v$ using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in $\\M$ are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes, and produces competitive binary classification performance.","url_abs":"https://arxiv.org/abs/2006.08816v6","url_pdf":"https://arxiv.org/pdf/2006.08816v6.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"signed-graph-metric-learning-via-gershgorin","repo_url":"https://github.com/bobchengyang/SGML","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"binary-classification","task_name":"Binary Classification"},{"task_slug":"metric-learning","task_name":"Metric Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}