  In recent years, there has been a growing trend of incorporating hyperbolic
geometry methods into computer vision. While these methods have achieved
state-of-the-art performance on various metric learning tasks using hyperbolic
distance measurements, the underlying theoretical analysis supporting this
superior performance remains under-exploited. In this study, we investigate the
effects of integrating hyperbolic space into metric learning, particularly when
training with contrastive loss. We identify a need for a comprehensive
comparison between Euclidean and hyperbolic spaces regarding the temperature
effect in the contrastive loss within the existing literature. To address this
gap, we conduct an extensive investigation to benchmark the results of Vision
Transformers (ViTs) using a hybrid objective function that combines loss from
Euclidean and hyperbolic spaces. Additionally, we provide a theoretical
analysis of the observed performance improvement. We also reveal that
hyperbolic metric learning is highly related to hard negative sampling,
providing insights for future work. This work will provide valuable data points
and experience in understanding hyperbolic image embeddings. To shed more light
on problem-solving and encourage further investigation into our approach, our
code is available online (https://github.com/YunYunY/HypMix).
