# Fixed-rank matrices

Manifolds.FixedRankMatricesType
FixedRankMatrices{m,n,k,𝔽} <: Manifold{𝔽}

The manifold of $m × n$ real-valued or complex-valued matrices of fixed rank $k$, i.e.

$\bigl\{ p ∈ 𝔽^{m × n}\ \big|\ \operatorname{rank}(p) = k \bigr\},$

where $𝔽 ∈ \{ℝ,ℂ\}$ and the rank is the number of linearly independent columns of a matrix.

Representation with 3 matrix factors

A point $p ∈ \mathcal M$ can be stored using unitary matrices $U ∈ 𝔽^{m × k}$, $V ∈ 𝔽^{n × k}$ as well as the $k$ singular values of $p = USV^\mathrm{H}$, where $\cdot^{\mathrm{H}}$ denotes the complex conjugate transpose or Hermitian. In other words, $U$ and $V$ are from the manifolds Stiefel(m,k,𝔽) and Stiefel(n,k,𝔽), respectively; see SVDMPoint for details.

The tangent space $T_p \mathcal M$ at a point $p ∈ \mathcal M$ with $p=USV^\mathrm{H}$ is given by

$T_p\mathcal M = \bigl\{ UMV^\mathrm{T} + U_pV^\mathrm{H} + UV_p^\mathrm{H} : M ∈ 𝔽^{k × k}, U_p ∈ 𝔽^{m × k}, V_p ∈ 𝔽^{n × k} \text{ s.t. } U_p^\mathrm{H}U = 0_k, V_p^\mathrm{H}V = 0_k \bigr\},$

where $0_k$ is the $k × k$ zero matrix. See UMVTVector for details.

The (default) metric of this manifold is obtained by restricting the metric on $ℝ^{m × n}$ to the tangent bundle[Vandereycken2013].

Constructor

FixedRankMatrics(m, n, k[, field=ℝ])

Generate the manifold of m-by-n (field-valued) matrices of rank k.

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Manifolds.SVDMPointType
SVDMPoint <: MPoint

A point on a certain manifold, where the data is stored in a svd like fashion, i.e. in the form $USV^\mathrm{H}$, where this structure stores $U$, $S$ and $V^\mathrm{H}$. The storage might also be shortened to just $k$ singular values and accordingly shortened $U$ (columns) and $V^\mathrm{T}$ (rows).

Constructors

• SVDMPoint(A) for a matrix A, stores its svd factors (i.e. implicitly $k=\min\{m,n\}$)
• SVDMPoint(S) for an SVD object, stores its svd factors (i.e. implicitly $k=\min\{m,n\}$)
• SVDMPoint(U,S,Vt) for the svd factors to initialize the SVDMPoint (i.e. implicitly $k=\min\{m,n\}$)
• SVDMPoint(A,k) for a matrix A, stores its svd factors shortened to the best rank $k$ approximation
• SVDMPoint(S,k) for an SVD object, stores its svd factors shortened to the best rank $k$ approximation
• SVDMPoint(U,S,Vt,k) for the svd factors to initialize the SVDMPoint, stores its svd factors shortened to the best rank $k$ approximation
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Manifolds.UMVTVectorType
UMVTVector <: TVector

A tangent vector that can be described as a product $UMV^\mathrm{H}$, at least together with its base point, see for example FixedRankMatrices. This vector structure stores the additionally (to the point) required fields.

Constructors

• UMVTVector(U,M,Vt) store umv factors to initialize the UMVTVector
• UMVTVector(U,M,Vt,k) store the umv factors after shortening them down to inner dimensions $k$, i.e. in $UMV^\mathrm{H}$, where $M$ is a $k × k$ matrix.
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ManifoldsBase.check_manifold_pointMethod
check_manifold_point(M::FixedRankMatrices{m,n,k}, p; kwargs...)

Check whether the matrix or SVDMPoint x ids a valid point on the FixedRankMatrices{m,n,k,𝔽} M, i.e. is an m-byn matrix of rank k. For the SVDMPoint the internal representation also has to have the right shape, i.e. p.U and p.Vt have to be unitary. The keyword arguments are passed to the rank function that verifies the rank of p.

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ManifoldsBase.check_tangent_vectorMethod
check_tangent_vector(M:FixedRankMatrices{m,n,k}, p, X; check_base_point = true, kwargs...)

Check whether the tangent UMVTVector X is from the tangent space of the SVDMPoint p on the FixedRankMatrices M, i.e. that v.U and v.Vt are (columnwise) orthogonal to x.U and x.Vt, respectively, and its dimensions are consistent with p and X.M, i.e. correspond to m-by-n matrices of rank k. The optional parameter check_base_point indicates, whether to call check_manifold_point for p.

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ManifoldsBase.innerMethod
inner(M::FixedRankMatrices, p::SVDMPoint, X::UMVTVector, Y::UMVTVector)

Compute the inner product of X and Y in the tangent space of p on the FixedRankMatrices M, which is inherited from the embedding, i.e. can be computed using dot on the elements (U, Vt, M) of X and Y.

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ManifoldsBase.retractMethod
retract(M, p, X, ::PolarRetraction)

Compute an SVD-based retraction on the FixedRankMatrices M` by computing

$q = U_kS_kV_k^\mathrm{H},$

where $U_k S_k V_k^\mathrm{H}$ is the shortened singular value decomposition $USV=p+X$, in the sense that $S_k$ is the diagonal matrix of size $k × k$ with the $k$ largest singular values and $U$ and $V$ are shortened accordingly.

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