Timeline for Prove or disprove a matrix logarithm equation
Current License: CC BY-SA 3.0
16 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jul 11, 2015 at 0:16 | review | First posts | |||
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| Jun 25, 2015 at 22:55 | comment | added | nullgeppetto | That's what I was studying right now. It's straight to the point, really good work I think. I believe that killing the curvature is not a problem in my case. Apropos, would you mind suggesting me a good introductory book for beginning studying Riemannian Geometry? Thank you very much again! | |
| Jun 25, 2015 at 22:40 | comment | added | Suvrit | Here is a link to a paper that answers your "kernel" question in complete detail (the log-Euclidean transform indeed leads to a kernel but at the expense of "killing" the curvature): www2.compute.dtu.dk/~sohau/papers/cvpr2015/feragen_cvpr2015.pdf | |
| Jun 25, 2015 at 21:51 | comment | added | nullgeppetto | Well, Mr. @Suvrit, apparently you are familiarized with all this theory. In the last few day, I have read many of your papers. By the way, congratulations for this magnificent work! I hope I'll have the opportunity to cite some of your papers in the foreseeable future! | |
| Jun 25, 2015 at 21:42 | comment | added | nullgeppetto | Thanks for your reply @Suvrit. This is what I have come up with too. I have found that instead of this affine-invariant geodesic distance, I should use the Log-Euclidean distance ($\lVert \log(A)-\log(B)\rVert_{F}$) such that the constructed kernel to be positive-definite. Are you familiarized with this one? | |
| Jun 25, 2015 at 21:38 | comment | added | Suvrit | The Riemannian distance is not negative definite; that is, $\exp(-\gamma d^2(X,Y))$ is not a positive definite function. | |
| Jun 23, 2015 at 8:55 | vote | accept | nullgeppetto | ||
| Jun 23, 2015 at 8:40 | answer | added | user35593 | timeline score: 4 | |
| Jun 23, 2015 at 8:39 | comment | added | nullgeppetto | Well, what I would like to find is a relation between $P$, $Q$ and $A$, $B$, respectively. I am not interested in a numerical solution. So, concerning my question on MathSE, you believe that the squared distance $d^2$ I give is not negative-definite? | |
| Jun 23, 2015 at 8:35 | comment | added | Federico Poloni | I am not a differential geometer, but after reading the formulation in your other question, I am more convinced that it can't be done, at least with a reasonably smooth function. What you ask for is a distance-preserving isomorphism between $\mathbb{S}^n_+$ and $\mathbb{R}^{n^2}$, and this shouldn't be possible because they have different curvature. | |
| Jun 23, 2015 at 8:33 | comment | added | Federico Poloni | What do you mean by "treated"? Matlab, Mathematica and Python (sympy) all have library functions to compute matrix logarithms without trouble. | |
| Jun 23, 2015 at 8:33 | history | edited | nullgeppetto | CC BY-SA 3.0 |
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| Jun 23, 2015 at 8:29 | comment | added | nullgeppetto | Thanks for your reply @FedericoPoloni. No, the truth is that I did not. But how could do that. I cannot figure out how the logarithm should be treated. | |
| Jun 23, 2015 at 8:26 | comment | added | Federico Poloni | Have you done some numerical experiments? In particular, testing longer chains such as $(P-Q)+(Q-R)+(R-S)=(P-S)$. | |
| Jun 23, 2015 at 7:55 | history | asked | nullgeppetto | CC BY-SA 3.0 |