  Distance metric learning (DML), which learns a distance metric from labeled
"similar" and "dissimilar" data pairs, is widely utilized. Recently, several
works investigate orthogonality-promoting regularization (OPR), which
encourages the projection vectors in DML to be close to being orthogonal, to
achieve three effects: (1) high balancedness -- achieving comparable
performance on both frequent and infrequent classes; (2) high compactness --
using a small number of projection vectors to achieve a "good" metric; (3) good
generalizability -- alleviating overfitting to training data. While showing
promising results, these approaches suffer three problems. First, they involve
solving non-convex optimization problems where achieving the global optimal is
NP-hard. Second, it lacks a theoretical understanding why OPR can lead to
balancedness. Third, the current generalization error analysis of OPR is not
directly on the regularizer. In this paper, we address these three issues by
(1) seeking convex relaxations of the original nonconvex problems so that the
global optimal is guaranteed to be achievable; (2) providing a formal analysis
on OPR's capability of promoting balancedness; (3) providing a theoretical
analysis that directly reveals the relationship between OPR and generalization
performance. Experiments on various datasets demonstrate that our convex
methods are more effective in promoting balancedness, compactness, and
generalization, and are computationally more efficient, compared with the
nonconvex methods.
