Complex Grassmann manifold Documentation

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Stuart Rogers

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Jul 24, 2021, 8:15:50 PMJul 24
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In the Manifolds section of the Tutorial (https://www.manopt.org/tutorial.html#manifolds), should the complex Grassmann manifold constraint be X^* X = I_p instead of  X^T X = I_p? X^* denotes the Hermitian (conjugate) transpose of X, while X^T denotes the unconjugated transpose of X.

Ronny Bergmann

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Jul 25, 2021, 7:36:18 AMJul 25
to Manopt
Hej,
I think that is correct, it should either read \bar X°T or better X°* (sometimes I also see X°H for the Hermitian), compare for example https://juliamanifolds.github.io/Manifolds.jl/stable/manifolds/grassmann.html#Manifolds.Grassmann

Nicolas Boumal

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Jul 26, 2021, 5:32:14 AMJul 26
to Ronny Bergmann, Manopt
Thanks for pointing this out! I made the change.

Best,
Nicolas


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Stuart Rogers

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Jul 26, 2021, 1:25:34 PMJul 26
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Why are the Grassmann manifold tangent space formulas different in the Manopt.jl documentation (https://juliamanifolds.github.io/Manifolds.jl/stable/manifolds/grassmann.html)? The tangent space formula under "Manifolds.Grassmann - Type" lacks a complex conjugate overbar on the second summand, whereas the tangent space formula under "ManifoldsBase.check_vector—Method" has  a complex conjugate overbar on the second summand.

Ronny Bergmann

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Jul 26, 2021, 1:40:39 PMJul 26
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Manopt.jl documentation? Do you mean the Manifolds.jl documentation? Those are two separate packages in Julia, that is you can use the manifolds for anything not just for optimisation :)

Well – besides that small detail, that is also a typo. We at some point switched from \bar{\cdot}^T to ^H (that is Hermitian is complex conjugate transpose); we might have missed to adapt the documentation correctly there. I just adapted that in https://github.com/JuliaManifolds/Manifolds.jl/commit/fd189ff1f951d0a013dee9b47ef7ac9722f3fd23 - deleting once the mention of the overbear, since we have the notation of the Hermitian already and removing the overline in the check_vektor documentation. Thanks for noticing.

Stuart Rogers

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Jul 26, 2021, 4:27:22 PMJul 26
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Yes, I was referring to Manifolds.jl. You may also want to remove the text "\overline{\cdot}⋅ the (elementwise) complex conjugate," from the documentation of  "ManifoldsBase.check_vector—Method".
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