With the emergence of deep learning, metric learning has gained significant popularity in numerous machine learning tasks dealing with complex and large-scale datasets, such as information retrieval, object recognition and recommendation systems. Metric learning aims to maximize and minimize inter- and intra-class similarities. However, existing models mainly rely on distance measures to obtain a separable embedding space and implicitly maximize the intra-class similarity while neglecting the inter-class relationship. We argue that to enable metric learning as a service for high-performance deep learning applications, we should also wisely deal with inter-class relationships to obtain a more advanced and meaningful embedding space representation. In this paper, a novel metric learning is presented as a service methodology that incorporates covariance to signify the direction of the linear relationship between data points in an embedding space. Unlike conventional metric learning, our covariance-embedding-enhanced approach enables metric learning as a service to be more expressive for computing similar or dissimilar measures and can capture positive, negative, or neutral relationships. Extensive experiments conducted using various benchmark datasets, including natural, biomedical, and facial images, demonstrate that the proposed model as a service with covariance-embedding optimizations can obtain higher-quality, more separable, and more expressive embedding representations than existing models.