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positive semi-definite Procrustes problem #98
Description
This positive semi-definite Procrustes problem is trying to
More details can be found at Woodgate, K. G. (1993, December). A new algorithm for the positive semi-definite Procrustes problem. In Proceedings of 32nd IEEE Conference on Decision and Control (pp. 3596-3601). IEEE..
This will be on the to-do list of our package, but not for this stage.
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PaulWAyers commentedon Apr 12, 2021
This would be really cool to have. It's also nice because it explains (finally!) why the symmetric Procrustes problem is useful!!!
FanwangM commentedon Apr 12, 2021
some more papers related to positive semi-definite Procrustes problem
This list is built for future implementation.
FanwangM commentedon Feb 22, 2022
I would propose these algorithms be prioritized.
The proposed algorithms either come with numerical tests example or source code in Matlab. This makes the testing of the code easy.
PaulWAyers commentedon Feb 23, 2022
Probably just implementing one algorithm here may suffice, at least at first. A related problem with an explicit solution is to find the closest semidefinite matrix to a given matrix (one matrix = I) or the closest semidefinite matrix with a given trace to a given matrix. There are analytic solutions here, and they can be used as testing. If it were possible, it would be nice to be able to solve the positive semi-definite Procrustes problem with a trace constraint, since often positive semidefinite matrices with specified (usually unit) trace show in up quantum-mechanical examples (as (reduced) density matrices).
FanwangM commentedon Feb 23, 2022
Thank you for the suggestions. I have changed the GSoC documentation accordingly.
For the closest semidefinite matrix problem, do you mean something like https://nhigham.com/2021/01/26/what-is-the-nearest-positive-semidefinite-matrix/?
Also, an analog problem is the nearest correlation matrix problem, https://nhigham.com/2013/02/13/the-nearest-correlation-matrix/. I am not sure how this can be used for chemistry, but this is interesting, at least from a math perspective. And it's application examples are listed at the end of the post.
PaulWAyers commentedon Feb 23, 2022
Yep, these references are good leads. The nearest correlation matrix is obviously interesting for machine learning.
The trace-constraint is yet another linear constraint (so similar to the correlation matrix) but has an explicit solution, see the appendix of https://doi.org/10.1063/1.4994618.
In general, adding any set of linear constraint(s) on the solution is helpful (i.e., Tr[Q_k*P] = q_k) where P is positive semidefinite, Q_k is a linear operator (Hermitian in all the cases I can think of), and q_k is a float.