In linear distance metric learning, we are given data in one Euclidean metric
space and the goal is to find an appropriate linear map to another Euclidean
metric space which respects certain distance conditions as much as possible. In
this paper, we formalize a simple and elegant method which reduces to a general
continuous convex loss optimization problem, and for different noise models we
derive the corresponding loss functions. We show that even if the data is
noisy, the ground truth linear metric can be learned with any precision
provided access to enough samples, and we provide a corresponding sample
complexity bound. Moreover, we present an effective way to truncate the learned
model to a low-rank model that can provably maintain the accuracy in loss
function and in parameters -- the first such results of this type. Several
experimental observations on synthetic and real data sets support and inform
our theoretical results.

本文提出了一种基于连续凸损失优化的简单优雅方法，用于线性距离度量学习，并针对不同的噪声模型推导了相应的损失函数。研究结果表明，即使数据存在噪声，只要样本充足就可以学习到地面事实线性度量，并提供相应的样本复杂度限制。此外，我们还提出了一种有效地将学习模型截断为低秩模型的方法，该方法可证明在损失函数和参数方面都能保持准确性，这是该领域首次出现这种结果。实验结果表明了理论结果的正确性。