Converting between cosine similarity and concentration ratio

I’ve written three posts on cosine similarity lately. The first looked at interpreting cosine similarity. The second looked at an approximation related to the first. The third looked at how ranking according to cosine similarity works better than cosine similarity itself. Normalized word vectors are points on a high dimensional sphere, and geometry in high dimensions is counterintuitive. See the first post in this series for an explanation. The set of points within a given angular distance of a point on a hypersphere is called a spherical cap. The ratio of the area of this spherical cap to that of the whole sphere is called cap fraction or concentration ratio. Concentration ratio explains why a modest cosine similarity value corresponds to a tiny portion of the area of the sphere and should be interpreted as a close match. For this post, I wanted to share a plot of concentration ratio as a function of cosine similarity. This shows that moderate values of cosine similarity correspond…