Although distance metric learning has been successfully applied to many real-world applications, learning a distance metric from large-scale and high-dimensional data remains a challenging problem. Due to the PSD constraint, the computational complexity of previous algorithms per iteration is at least O(d~2) where d is the dimensionality of the data. In this paper, we develop an efficient stochastic algorithm for a class of distance metric learning problems with nuclear norm regularization, referred to as low-rank DML. By utilizing the low-rank structure of the intermediate solutions and stochastic gradients, the complexity of our algorithm has a linear dependence on the dimensionality d. The key idea is to maintain all the iterates in factorized representations and construct stochastic gradients that are low-rank. In this way, the projection onto the PSD cone can be implemented efficiently by incremental SVD. Experimental results on several data sets validate the effectiveness and efficiency of our method.
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