From 50ef3eac4ac860bbc07c4c3c201ed2c003726865 Mon Sep 17 00:00:00 2001 From: Ronny Bergmann Date: Mon, 21 Aug 2023 18:26:13 +0200 Subject: [PATCH] Add one further reference. --- docs/src/references.bib | 15 ++++++++++++++- src/manifolds/StiefelSubmersionMetric.jl | 14 +++++--------- 2 files changed, 19 insertions(+), 10 deletions(-) diff --git a/docs/src/references.bib b/docs/src/references.bib index dc3f8b43ac..cf4d1f5713 100644 --- a/docs/src/references.bib +++ b/docs/src/references.bib @@ -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}, @@ -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}, } \ No newline at end of file diff --git a/src/manifolds/StiefelSubmersionMetric.jl b/src/manifolds/StiefelSubmersionMetric.jl index 2a1897b5eb..7b1b0b8198 100644 --- a/src/manifolds/StiefelSubmersionMetric.jl +++ b/src/manifolds/StiefelSubmersionMetric.jl @@ -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 @@ -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). @@ -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())` @@ -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