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Add one further reference.
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kellertuer committed Aug 21, 2023
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15 changes: 14 additions & 1 deletion docs/src/references.bib
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Expand Up @@ -3,7 +3,7 @@ @incollection{AbsilMahonyTrumpf:2013
YEAR = {2013},
DOI = {10.1007/978-3-642-40020-9_39},
EDITOR = {Nielsen, Frank
and Barbaresco, Fr{\'e}d{\'e}ric},
and Barbaresco, Frédéric},
ISBN = {978-3-642-40020-9},
PUBLISHER = {Springer Berlin Heidelberg},
PAGES = {361--368},
Expand Down Expand Up @@ -35,4 +35,17 @@ @article{Vandereycken:2013
PAGES = {1214--1236},
TITLE = {Low-rank matrix completion by Riemannian optimization},
VOLUME = {23}
}

@article{ZimmermannHueper:2022,
AUTHOR = {Zimmermann, Ralf and Hüper, Knut},
TITLE = {Computing the Riemannian Logarithm on the Stiefel Manifold: Metrics, Methods, and Performance},
JOURNAL = {SIAM Journal on Matrix Analysis and Applications},
VOLUME = {43},
NUMBER = {2},
PAGES = {953-980},
YEAR = {2022},
DOI = {10.1137/21M1425426},
EPRINT = {2103.12046},
EPRINTTYPE = {arXiv},
}
14 changes: 5 additions & 9 deletions src/manifolds/StiefelSubmersionMetric.jl
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Expand Up @@ -8,17 +8,13 @@ The family, with a single real parameter ``α>-1``, has two special cases:
- ``α = 0``: [`CanonicalMetric`](@ref)
The family was described in [^HüperMarkinaLeite2021]. This implementation follows the
description in [^ZimmermanHüper2022].
description in [ZimmermannHueper:2022](@cite).
[^HüperMarkinaLeite2021]:
> Hüper, M., Markina, A., Leite, R. T. (2021)
> "A Lagrangian approach to extremal curves on Stiefel manifolds"
> Journal of Geometric Mechanics, 13(1): 55-72.
> doi: [10.3934/jgm.2020031](http://dx.doi.org/10.3934/jgm.2020031)
[^ZimmermanHüper2022]:
> Ralf Zimmerman and Knut Hüper. (2022).
> "Computing the Riemannian logarithm on the Stiefel manifold: metrics, methods and performance."
> arXiv: [2103.12046](https://arxiv.org/abs/2103.12046)
# Constructor
Expand All @@ -45,7 +41,7 @@ The exponential map is given by
X p^\mathrm{T} - p X^\mathrm{T}
\bigr) p \operatorname{Exp}\bigl(\frac{\alpha}{\alpha+1} p^\mathrm{T} X\bigr)
````
This implementation is based on [^ZimmermanHüper2022].
This implementation is based on [ZimmermannHueper:2022](@cite).
For ``k < \frac{n}{2}`` the exponential is computed more efficiently using
[`StiefelFactorization`](@ref).
Expand Down Expand Up @@ -207,8 +203,8 @@ end
Compute the logarithmic map on the [`Stiefel(n,k)`](@ref) manifold with respect to the [`StiefelSubmersionMetric`](@ref).
The logarithmic map is computed using [`ShootingInverseRetraction`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/retractions.html#ManifoldsBase.ShootingInverseRetraction). For
``k ≤ \lfloor\frac{n}{2}\rfloor``, this is sped up using the ``k``-shooting method of
[^ZimmermanHüper2022]. Keyword arguments are forwarded to `ShootingInverseRetraction`; see
``k ≤ \lfloor\frac{n}{2}\rfloor``, this is sped up using the ``k``-shooting method of [ZimmermannHueper:2022](@cite).
Keyword arguments are forwarded to `ShootingInverseRetraction`; see
that documentation for details. Their defaults are:
- `num_transport_points=4`
- `tolerance=sqrt(eps())`
Expand Down Expand Up @@ -313,7 +309,7 @@ end
@doc raw"""
StiefelFactorization{UT,XT} <: AbstractManifoldPoint
Represent points (and vectors) on `Stiefel(n, k)` with ``2k × k`` factors.[^ZimmermanHüper2022]
Represent points (and vectors) on `Stiefel(n, k)` with ``2k × k`` factors [ZimmermannHueper:2022](@cite).
Given a point ``p ∈ \mathrm{St}(n, k)`` and another matrix ``B ∈ ℝ^{n × k}`` for
``k ≤ \lfloor\frac{n}{2}\rfloor`` the factorization is
Expand Down

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