  To achieve a low computational cost when performing online metric learning
for large-scale data, we present a one-pass closed-form solution namely OPML in
this paper. Typically, the proposed OPML first adopts a one-pass triplet
construction strategy, which aims to use only a very small number of triplets
to approximate the representation ability of whole original triplets obtained
by batch-manner methods. Then, OPML employs a closed-form solution to update
the metric for new coming samples, which leads to a low space (i.e., $O(d)$)
and time (i.e., $O(d^2)$) complexity, where $d$ is the feature dimensionality.
In addition, an extension of OPML (namely COPML) is further proposed to enhance
the robustness when in real case the first several samples come from the same
class (i.e., cold start problem). In the experiments, we have systematically
evaluated our methods (OPML and COPML) on three typical tasks, including UCI
data classification, face verification, and abnormal event detection in videos,
which aims to fully evaluate the proposed methods on different sample number,
different feature dimensionalities and different feature extraction ways (i.e.,
hand-crafted and deeply-learned). The results show that OPML and COPML can
obtain the promising performance with a very low computational cost. Also, the
effectiveness of COPML under the cold start setting is experimentally verified.
