Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment
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 S of generalized gra...
| Publié dans: | IEEE transactions on pattern analysis and machine intelligence. - 1979. - 44(2022), 10 vom: 23. Okt., Seite 7219-7234 |
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| Auteur principal: | |
| Autres auteurs: | , |
| Format: | Article en ligne |
| Langue: | English |
| Publié: |
2022
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| Accès à la collection: | IEEE transactions on pattern analysis and machine intelligence |
| Sujets: | Journal Article |
| Résumé: | 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 S of generalized graph Laplacian matrices corresponding to balanced signed graphs. M ∈ S that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, 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 M ∈ S and diagonal matrix S, where Sii = 1/vi and v is the first eigenvector of M, we prove that Gershgorin disc left-ends of similarity transform B = SMS-1 are perfectly aligned at the smallest eigenvalue λmin. 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 |
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| Description: | Date Revised 16.09.2022 published: Print-Electronic Citation Status PubMed-not-MEDLINE |
| ISSN: | 1939-3539 |
| DOI: | 10.1109/TPAMI.2021.3091682 |