  The classification loss functions used in deep neural network classifiers can
be grouped into two categories based on maximizing the margin in either
Euclidean or angular spaces. Euclidean distances between sample vectors are
used during classification for the methods maximizing the margin in Euclidean
spaces whereas the Cosine similarity distance is used during the testing stage
for the methods maximizing margin in the angular spaces. This paper introduces
a novel classification loss that maximizes the margin in both the Euclidean and
angular spaces at the same time. This way, the Euclidean and Cosine distances
will produce similar and consistent results and complement each other, which
will in turn improve the accuracies. The proposed loss function enforces the
samples of classes to cluster around the centers that represent them. The
centers approximating classes are chosen from the boundary of a hypersphere,
and the pairwise distances between class centers are always equivalent. This
restriction corresponds to choosing centers from the vertices of a regular
simplex. There is not any hyperparameter that must be set by the user in the
proposed loss function, therefore the use of the proposed method is extremely
easy for classical classification problems. Moreover, since the class samples
are compactly clustered around their corresponding means, the proposed
classifier is also very suitable for open set recognition problems where test
samples can come from the unknown classes that are not seen in the training
phase. Experimental studies show that the proposed method achieves the
state-of-the-art accuracies on open set recognition despite its simplicity.
