# Robust Distance Metric Learning via Simultaneous L1-Norm Minimization and Maximization

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Hua Wang,
Feiping Nie,
Heng Huang
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Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):1836-1844, 2014.

#### Abstract

Traditional distance metric learning with side information usually formulates the objectives using the covariance matrices of the data point pairs in the two constraint sets of must-links and cannot-links. Because the covariance matrix computes the sum of the squared L2-norm distances, it is prone to both outlier samples and outlier features. To develop a robust distance metric learning method, in this paper we propose a new objective for distance metric learning using the L1-norm distances. However, the resulted objective is very challenging to solve, because it simultaneously minimizes and maximizes (minmax) a number of non-smooth L1-norm terms. As an important theoretical contribution of this paper, we systematically derive an efficient iterative algorithm to solve the general L1-norm minmax problem, which is rarely studied in literature. We have performed extensive empirical evaluations, where our new distance metric learning method outperforms related state-of-the-art methods in a variety of experimental settings to cluster both noiseless and noisy data.

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