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Metric learning aims at finding a suitable distance metric over the inputspace, to improve the performance of distance-based learning algorithms. Inhigh-dimensional settings, metric learning can also play the role ofdimensionality reduction, by imposing a low-rank restriction to the learntmetric. In this paper, instead of training a low-rank metric onhigh-dimensional data, we consider a randomly compressed version of the data,and train a full-rank metric there. We give theoretical guarantees on the errorof distance-based metric learning, with respect to the random compression,which do not depend on the ambient dimension. Our bounds do not make anyexplicit assumptions, aside from i.i.d. data from a bounded support, andautomatically tighten when benign geometrical structures are present.Experimental results on both synthetic and real data sets support ourtheoretical findings in high-dimensional settings.

The Effect of Intrinsic Dimension on Metric Learning under Compression