  Deep-learning architectures for classification problems involve the
cross-entropy loss sometimes assisted with auxiliary loss functions like center
loss, contrastive loss and triplet loss. These auxiliary loss functions
facilitate better discrimination between the different classes of interest.
However, recent studies hint at the fact that these loss functions do not take
into account the intrinsic angular distribution exhibited by the low-level and
high-level feature representations. This results in less compactness between
samples from the same class and unclear boundary separations between data
clusters of different classes. In this paper, we address this issue by
proposing the use of geometric constraints, rooted in Riemannian geometry.
Specifically, we propose Angular Margin Contrastive Loss (AMC-Loss), a new loss
function to be used along with the traditional cross-entropy loss. The AMC-Loss
employs the discriminative angular distance metric that is equivalent to
geodesic distance on a hypersphere manifold such that it can serve a clear
geometric interpretation. We demonstrate the effectiveness of AMC-Loss by
providing quantitative and qualitative results. We find that although the
proposed geometrically constrained loss-function improves quantitative results
modestly, it has a qualitatively surprisingly beneficial effect on increasing
the interpretability of deep-net decisions as seen by the visual explanations
generated by techniques such as the Grad-CAM. Our code is available at
https://github.com/hchoi71/AMC-Loss.
