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As far as you use the cosine as similarity measure, the matrix is a correlation matrix. For this situation in statistics there is the concept of "canonical correlation", and this might be then the most appropriate for your case: it gives an index how much "variance of one set of variables is explained by the other". The two set of variables are the two sets of vectors $\small v_i$ here.
As far as you use the cosine as similarity measure, the matrix is a correlation matrix. For this situation in statistics there is the concept of "canonical correlation", and this might be then the most appropriate for your case: it gives an index how much "variance of one set of variables is explained by the other". The two set of variables are the two sets of vectors $\small v_i$ here.