Timeline for How to compare two similarity matrices?
Current License: CC BY-SA 3.0
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| when toggle format | what | by | license | comment | |
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| Mar 12, 2012 at 22:41 | answer | added | Gottfried Helms | timeline score: 0 | |
| Mar 12, 2012 at 19:41 | comment | added | Ahmet | @Tom: The similarity matrix specify the distances between the items. I'm not sure whether I'm using the correct terminology but we can think that each similarity matrix specifies the relative positions of the items in a space. What I'm wondering is this: how can we make sense of these two different relative positions of items specified by the two similarity matrices? Currently my only requirement of the measure of distance is to give me some sense of how different are the relative positions of items. | |
| Mar 12, 2012 at 19:26 | history | edited | Ahmet | CC BY-SA 3.0 |
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| Mar 12, 2012 at 19:19 | comment | added | Tom Leinster | @Ahmet: you should edit your question to include your clarification - that way, everyone will see it immediately. (There's a button marked "edit".) Also, you should explain what "cosine(v_i, v_j)" means. Do you mean the cosine of the angle between v_i and v_j? Finally, I'd like to see your response to Federico's question: what properties do you require of this measure of distance between matrices? | |
| Mar 12, 2012 at 19:12 | comment | added | Steve Huntsman | @Ahmet- Your cosine similarity matrix is the Gram matrix of a set of normalized Euclidean vectors. If I recall correctly, given a Gram matrix G, you can form a distance matrix as D(i,j) = G(i,i) + G(j,j) - G(i,j) - G(j,i). Then you can apply Gromov-Hausdorff. | |
| Mar 12, 2012 at 19:00 | comment | added | Ahmet | Let me clarify what I mean by a similarity matrix. Suppose that we have n items. And suppose that each item i is represented with a vector of numbers. Then each element of the similarity matrix S(i,j) = cosine(v_i, v_j). Where v_i and v_j are the ith and jth item vectors. Cosine is the cosine similarity metric (other metrics can also be used). | |
| Mar 12, 2012 at 18:32 | comment | added | Federico Poloni | This is a modelling problem. We cannot say anything unless the original poster clarifies what he/she means. Some questions that might help: Why is Frobenius distance not ok? Can you give an example in which it over/underestimates the quantity that you need? How are your matrices normalized/constructed? | |
| Mar 12, 2012 at 18:06 | comment | added | Steve Huntsman | Given a symmetric dissimilarity matrix, you can always consider (e.g.) the maximal subdominant ultrametric if it's not already a metric, and I should have been explicit here as well in my answer. | |
| Mar 12, 2012 at 17:56 | comment | added | Tom Leinster | Thanks, Steve, but I'm still not up to speed. What's the definition of dissimilarity matrix? I ask because I'm familiar with these terms being used in the literature on quantification of biodiversity, but they're not used very precisely there, and I guess the same terms are also used differently by other people in other contexts. In particular, your answer made me think you were using some definition of (dis)similarity matrix that had a kind of triangle inequality built in. | |
| Mar 12, 2012 at 17:51 | comment | added | Steve Huntsman | @Tom: I realize that I conflated similarity and dissimilarity matrices. The latter is generally regarded as a metric. I'm not sure if there is a generally used transformation from one to the other, but taking inverses and setting the diagonal as a special case should do the trick. | |
| Mar 12, 2012 at 17:43 | comment | added | Tom Leinster | What definition of similarity matrix are you (and Steve H) using? | |
| Mar 12, 2012 at 17:23 | answer | added | Steve Huntsman | timeline score: 2 | |
| Mar 12, 2012 at 17:18 | history | asked | Ahmet | CC BY-SA 3.0 |