title: A proper vector similarity index
body: <p>I'm trying to adjust cosine similarity to determine how similar two vectors are, with respect to entries. Since the obtained measure is invariant under vector scale {(0, 1, 2) and (0, 2, 4) have cosine similarity of 1}, what would be the way to extend the similarity measure to account for the initial vector scale? I thought of multiplying by min{|v1|, |v2|}/max{|v1|, |v2|}, with |v| denoting a vector v norm, to preserve the bounds of -1 and 1. Any suggestions are highly appreciated.</p>