A Triangle Inequality for Cosine Similarity

Similarity search is a fundamental problem for many data analysis techniques. Many efficient techniques rely on the triangle inequality of metrics, which allows pruning parts space based transitive bounds distances. Recently, cosine similarity has become popular alternative choice to standard Euclidean metric, in particular context textual and neural network embeddings. Unfortunately, not metric does satisfy inequality. Instead, approximation such as locality sensitive hashing. In this paper, we derive that suitable with structures (such VP-tree, Cover-tree, M-tree); show bound tight discuss fast approximations it. We hope spurs new research accelerating exact similarity, possible other measures beyond existing work distance metrics.