  In neural networks, the loss function represents the core of the learning
process that leads the optimizer to an approximation of the optimal convergence
error. Convolutional neural networks (CNN) use the loss function as a
supervisory signal to train a deep model and contribute significantly to
achieving the state of the art in some fields of artificial vision.
Cross-entropy and Center loss functions are commonly used to increase the
discriminating power of learned functions and increase the generalization
performance of the model. Center loss minimizes the class intra-class variance
and at the same time penalizes the long distance between the deep features
inside each class. However, the total error of the center loss will be heavily
influenced by the majority of the instances and can lead to a freezing state in
terms of intra-class variance. To address this, we introduce a new loss
function called sigma squared reduction loss ($\sigma^2$R loss), which is
regulated by a sigmoid function to inflate/deflate the error per instance and
then continue to reduce the intra-class variance. Our loss has clear intuition
and geometric interpretation, furthermore, we demonstrate by experiments the
effectiveness of our proposal on several benchmark datasets showing the
intra-class variance reduction and overcoming the results obtained with center
loss and soft nearest neighbour functions.
