Explanation of figure 8

[UFO-101]

I am quite confused about figure 8 in your paper. Assuming the shared vocabulary size is significantly larger than the embedding size in all models (which it appears to be from figure 2), I would expect matrices A and B to span the whole embedding space. I would also expect a randomly initialized matrix to span the full embedding space. In which case the maximum canonical angle would always be 0. What am I missing?

[mimno]

The rows of an embedding matrix would almost certainly span the D-dimensional space, we're looking at the columns, which are in V-dimensional space. We're calculating the angles of the DxD rotation matrix that best maps one embedding to another.