  Self-supervised metric learning has been a successful approach for learning a
distance from an unlabeled dataset. The resulting distance is broadly useful
for improving various distance-based downstream tasks, even when no information
from downstream tasks is utilized in the metric learning stage. To gain
insights into this approach, we develop a statistical framework to
theoretically study how self-supervised metric learning can benefit downstream
tasks in the context of multi-view data. Under this framework, we show that the
target distance of metric learning satisfies several desired properties for the
downstream tasks. On the other hand, our investigation suggests the target
distance can be further improved by moderating each direction's weights. In
addition, our analysis precisely characterizes the improvement by
self-supervised metric learning on four commonly used downstream tasks: sample
identification, two-sample testing, $k$-means clustering, and $k$-nearest
neighbor classification. When the distance is estimated from an unlabeled
dataset, we establish the upper bound on distance estimation's accuracy and the
number of samples sufficient for downstream task improvement. Finally,
numerical experiments are presented to support the theoretical results in the
paper.
