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English is not my first language, so please excuse any mistakes.

I'm working with two similarity matrices on the same data set: Suppose I have $n$ items, and I calculated the similarity of each item based on different properties of the items. I have two different vector representations of an item, and I used cosine similarity to compute the pairwise similarity between all items.

As a result, I have two $n \times n$ matrices with different similarity scores. I defined the similarity of an item to itself always as $0$. Now I want to compare these two matrices: I have the hypothesis that if two items are similar in matrix A, they are also similar in matrix B, but I'm missing a mathematical way of comparing the two matrices.

In particular, I have the problem that in similarity matrix A, the values range between $0$ and $1$, but are often $0$. In similarity matrix B, most values are around $0.9$, but still vary - just not so much as in matrix A. I'm looking for a way to somehow normalize these values and then compare them.

Do you have an advice which metric might be helpful?

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    $\begingroup$ could you illustrate(give a small example) your procedure. it is not clear what you mean by similarity matrix $\endgroup$
    – vidyarthi
    Commented Dec 10, 2019 at 10:53
  • $\begingroup$ In principle, you could transform the similarity matrices into metrics, possibly with dissimilarity matrices as intermediate representations (this can be as simple as entrywise inversion). Once you have metrics, the Gromov-Hausdorff distance on metrics could be used. Unfortunately, GH distance is computationally intractable. $\endgroup$
    – Steve Huntsman
    Commented Dec 10, 2019 at 14:08

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