Fast Graph Metric Learning via Gershgorin Disc Alignment
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We propose a fast general projection-free metric learning framework, where the minimization objective min_∈ Q() is a convex differentiable function of the metric matrix , and resides in the set of generalized graph Laplacian matrices for connected graphs with positive edge weights and node degrees. Unlike low-rank metric matrices common in the literature, 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 as signal-adaptive linear constraints via Gershgorin disc alignment, so that the alternating optimization of the diagonal and off-diagonal terms in 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 $̌ of, 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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