  Learning a latent embedding to understand the underlying nature of data
distribution is often formulated in Euclidean spaces with zero curvature.
However, the success of the geometry constraints, posed in the embedding space,
indicates that curved spaces might encode more structural information, leading
to better discriminative power and hence richer representations. In this work,
we investigate benefits of the curved space for analyzing anomalies or
out-of-distribution objects in data. This is achieved by considering embeddings
via three geometry constraints, namely, spherical geometry (with positive
curvature), hyperbolic geometry (with negative curvature) or mixed geometry
(with both positive and negative curvatures). Three geometric constraints can
be chosen interchangeably in a unified design given the task at hand. Tailored
for the embeddings in the curved space, we also formulate functions to compute
the anomaly score. Two types of geometric modules (i.e., Geometric-in-One and
Geometric-in-Two models) are proposed to plug in the original Euclidean
classifier, and anomaly scores are computed from the curved embeddings. We
evaluate the resulting designs under a diverse set of visual recognition
scenarios, including image detection (multi-class OOD detection and one-class
anomaly detection) and segmentation (multi-class anomaly segmentation and
one-class anomaly segmentation). The empirical results show the effectiveness
of our proposal through the consistent improvement over various scenarios.
