  This paper presents a deep nonlinear metric learning framework for data
visualization on an image dataset. We propose the Triangular Similarity and
prove its equivalence to the Cosine Similarity in measuring a data pair. Based
on this novel similarity, a geometrically motivated loss function - the
triangular loss - is then developed for optimizing a metric learning system
comprising two identical CNNs. It is shown that this deep nonlinear system can
be efficiently trained by a hybrid algorithm based on the conventional
backpropagation algorithm. More interestingly, benefiting from classical
manifold learning theories, the proposed system offers two different views to
visualize the outputs, the second of which provides better classification
results than the state-of-the-art methods in the visualizable spaces.
