The definition of Affine Invariant Riemannian Metric (AIRM)
For any two symmetric positive definite (SPD) matrices $A$ and $B$, the Affine Invariant Riemannian Metric (AIRM) between them is defined as [1], [2]:
$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F$,
where $log(A)$ is the matrix logrithm of $A$, and $||A||_F$ is the Frobenius norm of $A$.
On the other hand, it is also shown that (e.g., see [3])
$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F = ||log (A^{-1}B) ||_F$.
However, when I used Matlab function logm, I found that $||log (A^{-1/2}B A^{-1/2}) ||_F ne ||log (A^{-1}B) ||_F$ at all (but their eigenvalues are the same).
I was wondering if there is anything wrong with my understanding the definition of AIRM?
Thanks very much!
[1] R. Bhatia, Positive Definite Matrices. Princeton University Press, 2009.
[2] X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” International Journal of Computer Vision, vol. 66, no. 1, pp. 41–66, 2006.
[3] M. Moakher, “A differential geometric approach to the geometric mean of symmetric Positive-Definite matrices,” SIAM J. Matrix Anal. Appl., vol. 26, no. 3, pp. 735–747, 2005.
[4] I. Horev, F. Yger, and M. Sugiyama, “Geometry-aware principal component analysis for symmetric positive definite matrices,” Machine Learning, 2017.
matrices riemannian-geometry
asked Nov 16 '18 at 7:10
user3138073
1708
add a comment |
For any two symmetric positive definite (SPD) matrices $A$ and $B$, the Affine Invariant Riemannian Metric (AIRM) between them is defined as [1], [2]:
$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F$,
where $log(A)$ is the matrix logrithm of $A$, and $||A||_F$ is the Frobenius norm of $A$.
On the other hand, it is also shown that (e.g., see [3])
$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F = ||log (A^{-1}B) ||_F$.
However, when I used Matlab function logm, I found that $||log (A^{-1/2}B A^{-1/2}) ||_F ne ||log (A^{-1}B) ||_F$ at all (but their eigenvalues are the same).
I was wondering if there is anything wrong with my understanding the definition of AIRM?
Thanks very much!
[1] R. Bhatia, Positive Definite Matrices. Princeton University Press, 2009.
[2] X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” International Journal of Computer Vision, vol. 66, no. 1, pp. 41–66, 2006.
[3] M. Moakher, “A differential geometric approach to the geometric mean of symmetric Positive-Definite matrices,” SIAM J. Matrix Anal. Appl., vol. 26, no. 3, pp. 735–747, 2005.
[4] I. Horev, F. Yger, and M. Sugiyama, “Geometry-aware principal component analysis for symmetric positive definite matrices,” Machine Learning, 2017.
matrices riemannian-geometry
asked Nov 16 '18 at 7:10
user3138073
1708
add a comment |
For any two symmetric positive definite (SPD) matrices $A$ and $B$, the Affine Invariant Riemannian Metric (AIRM) between them is defined as [1], [2]:
$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F$,
where $log(A)$ is the matrix logrithm of $A$, and $||A||_F$ is the Frobenius norm of $A$.
On the other hand, it is also shown that (e.g., see [3])
$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F = ||log (A^{-1}B) ||_F$.
However, when I used Matlab function logm, I found that $||log (A^{-1/2}B A^{-1/2}) ||_F ne ||log (A^{-1}B) ||_F$ at all (but their eigenvalues are the same).
I was wondering if there is anything wrong with my understanding the definition of AIRM?
Thanks very much!
[1] R. Bhatia, Positive Definite Matrices. Princeton University Press, 2009.
[2] X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” International Journal of Computer Vision, vol. 66, no. 1, pp. 41–66, 2006.
[3] M. Moakher, “A differential geometric approach to the geometric mean of symmetric Positive-Definite matrices,” SIAM J. Matrix Anal. Appl., vol. 26, no. 3, pp. 735–747, 2005.
[4] I. Horev, F. Yger, and M. Sugiyama, “Geometry-aware principal component analysis for symmetric positive definite matrices,” Machine Learning, 2017.
matrices riemannian-geometry
asked Nov 16 '18 at 7:10
user3138073
1708
For any two symmetric positive definite (SPD) matrices $A$ and $B$, the Affine Invariant Riemannian Metric (AIRM) between them is defined as [1], [2]:
$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F$,
where $log(A)$ is the matrix logrithm of $A$, and $||A||_F$ is the Frobenius norm of $A$.
On the other hand, it is also shown that (e.g., see [3])
$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F = ||log (A^{-1}B) ||_F$.
However, when I used Matlab function logm, I found that $||log (A^{-1/2}B A^{-1/2}) ||_F ne ||log (A^{-1}B) ||_F$ at all (but their eigenvalues are the same).
I was wondering if there is anything wrong with my understanding the definition of AIRM?
Thanks very much!
[1] R. Bhatia, Positive Definite Matrices. Princeton University Press, 2009.
[2] X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” International Journal of Computer Vision, vol. 66, no. 1, pp. 41–66, 2006.
[3] M. Moakher, “A differential geometric approach to the geometric mean of symmetric Positive-Definite matrices,” SIAM J. Matrix Anal. Appl., vol. 26, no. 3, pp. 735–747, 2005.
[4] I. Horev, F. Yger, and M. Sugiyama, “Geometry-aware principal component analysis for symmetric positive definite matrices,” Machine Learning, 2017.
matrices riemannian-geometry
matrices riemannian-geometry
asked Nov 16 '18 at 7:10
user3138073
1708
asked Nov 16 '18 at 7:10
user3138073
1708
asked Nov 16 '18 at 7:10
user3138073
1708
asked Nov 16 '18 at 7:10
user3138073
1708
asked Nov 16 '18 at 7:10
user3138073
1708
1708
add a comment |
add a comment |
1 Answer
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The reason for this is that $log(A^{-1}B)$ isn't even a symmetric matrix. Notice that while $A$ and $B$ are symmetric, $A^{-1}B$ isn't necessarily so, even though $A^{-1/2}BA^{-1/2}$ is symmetric positive-definite. The logarithm likely obscures the way that Matlab computes the Frobenius norm of a matrix.
answered Dec 26 '18 at 23:24
Mnifldz
6,81311534
Thanks for the feedback! I agree with you that $A^{-1}B$ is not symmetric, but still, in [3], the author used $||log(A^{-1}B)||_F$ as AIRM, which is confusing to me.
– user3138073
Dec 27 '18 at 3:39
add a comment |
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1 Answer
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1 Answer
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The reason for this is that $log(A^{-1}B)$ isn't even a symmetric matrix. Notice that while $A$ and $B$ are symmetric, $A^{-1}B$ isn't necessarily so, even though $A^{-1/2}BA^{-1/2}$ is symmetric positive-definite. The logarithm likely obscures the way that Matlab computes the Frobenius norm of a matrix.
answered Dec 26 '18 at 23:24
Mnifldz
6,81311534
Thanks for the feedback! I agree with you that $A^{-1}B$ is not symmetric, but still, in [3], the author used $||log(A^{-1}B)||_F$ as AIRM, which is confusing to me.
– user3138073
Dec 27 '18 at 3:39
add a comment |
The reason for this is that $log(A^{-1}B)$ isn't even a symmetric matrix. Notice that while $A$ and $B$ are symmetric, $A^{-1}B$ isn't necessarily so, even though $A^{-1/2}BA^{-1/2}$ is symmetric positive-definite. The logarithm likely obscures the way that Matlab computes the Frobenius norm of a matrix.
answered Dec 26 '18 at 23:24
Mnifldz
6,81311534
Thanks for the feedback! I agree with you that $A^{-1}B$ is not symmetric, but still, in [3], the author used $||log(A^{-1}B)||_F$ as AIRM, which is confusing to me.
– user3138073
Dec 27 '18 at 3:39
add a comment |
The reason for this is that $log(A^{-1}B)$ isn't even a symmetric matrix. Notice that while $A$ and $B$ are symmetric, $A^{-1}B$ isn't necessarily so, even though $A^{-1/2}BA^{-1/2}$ is symmetric positive-definite. The logarithm likely obscures the way that Matlab computes the Frobenius norm of a matrix.
answered Dec 26 '18 at 23:24
Mnifldz
6,81311534
The reason for this is that $log(A^{-1}B)$ isn't even a symmetric matrix. Notice that while $A$ and $B$ are symmetric, $A^{-1}B$ isn't necessarily so, even though $A^{-1/2}BA^{-1/2}$ is symmetric positive-definite. The logarithm likely obscures the way that Matlab computes the Frobenius norm of a matrix.
answered Dec 26 '18 at 23:24
Mnifldz
6,81311534
answered Dec 26 '18 at 23:24
Mnifldz
6,81311534
answered Dec 26 '18 at 23:24
Mnifldz
6,81311534
answered Dec 26 '18 at 23:24
Mnifldz
6,81311534
6,81311534
Thanks for the feedback! I agree with you that $A^{-1}B$ is not symmetric, but still, in [3], the author used $||log(A^{-1}B)||_F$ as AIRM, which is confusing to me.
– user3138073
Dec 27 '18 at 3:39
add a comment |
Thanks for the feedback! I agree with you that $A^{-1}B$ is not symmetric, but still, in [3], the author used $||log(A^{-1}B)||_F$ as AIRM, which is confusing to me.
– user3138073
Dec 27 '18 at 3:39
Thanks for the feedback! I agree with you that $A^{-1}B$ is not symmetric, but still, in [3], the author used $||log(A^{-1}B)||_F$ as AIRM, which is confusing to me.
– user3138073
Dec 27 '18 at 3:39
Thanks for the feedback! I agree with you that $A^{-1}B$ is not symmetric, but still, in [3], the author used $||log(A^{-1}B)||_F$ as AIRM, which is confusing to me.
– user3138073
Dec 27 '18 at 3:39
add a comment |
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