Spherization Layer: Representation Using Only Angles

In neural network literature, angular similarity between feature vectors is frequently used for interpreting or re-using learned representations. However, the inner product in neural networks partially disperses information over the scales and angles of the involved input vectors and weight vectors. Therefore, when using only angular similarity on representations trained with the inner product, information loss occurs in downstream methods, which limits their performance. In this paper, we proposed the \textit{spherization layer} to represent all information on angular similarity. The layer 1) maps the pre-activations of input vectors into the specific range of angles, 2) converts the angular coordinates of the vectors to Cartesian coordinates with an additional dimension, and 3) trains decision boundaries from hyperplanes, without bias parameters, passing through the origin.