Online and Batch Learning of Generalized Cosine Similarities

In this paper, we define an online algorithm to learn the generalized cosine similarity measures for k-NN classification and hence a similarity matrix A corresponding to a bilinear form. In contrary to the standard cosine measure, the normalization is itself dependent on the similarity matrix which makes it impossible to use directly the algorithms developed for learning Mahanalobis distances, based on positive, semi-definite (PSD) matrices. We follow the approach where we first find an appropriate matrix and then project it onto the cone of PSD matrices, which we have adapted to the particular form of generalized cosine similarities, and more particularly to the fact that such measures are normalized. The resulting online algorithm as well as its batch version is fast and has got better accuracy as compared with state-of-the-art methods on standard data sets.