Fixed-rank matrices

FixedRankMatrices{𝔽, T} <: AbstractDecoratorManifold{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 $p = U_p S V_p^\mathrm{H}$, where $⋅^{\mathrm{H}}$ 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 $p=U_p S V_p^\mathrm{H}$ is given by

\[T_p\mathcal M = \bigl\{ U_p M V_p^\mathrm{H} + U_X V_p^\mathrm{H} + U_p V_X^\mathrm{H} : M ∈ 𝔽^{k×k}, U_X ∈ 𝔽^{m×k}, V_X ∈ 𝔽^{n×k} \text{ s.t. } U_p^\mathrm{H}U_X = 0_k, V_p^\mathrm{H}V_X = 0_k \bigr\},\]

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

The (default) metric of this manifold is obtained by restricting the metric on $ℝ^{m×n}$ to the tangent bundle [Van13].

Constructor

FixedRankMatrices(m, n, k[, field=ℝ])

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

OrthographicInverseRetraction <: AbstractInverseRetractionMethod

Retractions that are related to orthographic projections, which was first used in [AM12].

OrthographicRetraction <: AbstractRetractionMethod

Retractions that are related to orthographic projections, which was first used in [AM12].

SVDMPoint <: AbstractManifoldPoint

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 $V^\mathrm{H}$ (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. implicitlyk=\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

UMVTangentVector <: AbstractTangentVector

A tangent vector that can be described as a product $U_p M V_p^\mathrm{H} + U_X V_p^\mathrm{H} + U_p V_X^\mathrm{H}$, where $X = U_X S V_X^\mathrm{H}$ is its base point, see for example FixedRankMatrices.

The base point $p$ is required for example embedding this point, but it is not stored. The fields of the tangent vector are U for $U_X$, M and Vt to store $V_X^\mathrm{H}$

Constructors

• UMVTangentVector(U,M,Vt) store umv factors to initialize the UMVTangentVector
• UMVTangentVector(U,M,Vt,k) store the umv factors after shortening them down to inner dimensions k.

Random.rand(M::FixedRankMatrices; vector_at=nothing, kwargs...)

If vector_at is nothing, return a random point on the FixedRankMatrices manifold. The orthogonal matrices are sampled from the Stiefel manifold and the singular values are sampled uniformly at random.

If vector_at is not nothing, generate a random tangent vector in the tangent space of the point vector_at on the FixedRankMatrices manifold M.

Y = riemannian_Hessian(M::FixedRankMatrices, p, G, H, X)
riemannian_Hessian!(M::FixedRankMatrices, Y, p, G, H, X)

Compute the Riemannian Hessian $\operatorname{Hess} f(p)[X]$ given the Euclidean gradient $∇ f(\tilde p)$ in G and the Euclidean Hessian $∇^2 f(\tilde p)[\tilde X]$ in H, where $\tilde p, \tilde X$ are the representations of $p,X$ in the embedding,.

The Riemannian Hessian can be computed as stated in Remark 4.1 [Ngu23] or Section 2.3 [Van13], that B. Vandereycken adopted for Manopt (Matlab).

inverse_retract_orthographic!(M::AbstractManifold, X, p, q)

Compute the in-place variant of the OrthographicInverseRetraction.

retract_orthographic!(M::AbstractManifold, q, p, X)

Compute the in-place variant of the OrthographicRetraction.

check_point(M::FixedRankMatrices, p; kwargs...)

Check whether the matrix or SVDMPoint x ids a valid point on the FixedRankMatrices 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.

check_vector(M::FixedRankMatrices, p, X; kwargs...)

Check whether the tangent UMVTangentVector X is from the tangent space of the SVDMPoint p on the FixedRankMatrices M. This means that X.U and X.Vt are (columnwise) orthogonal to p.U and p.Vt, respectively, and that its dimensions are consistent with p and X.M, i.e. correspond to m-by-n matrices of rank k.

default_inverse_retraction_method(M::FixedRankMatrices)

Return PolarInverseRetraction as the default inverse retraction for the FixedRankMatrices manifold.

default_retraction_method(M::FixedRankMatrices)

Return PolarRetraction as the default retraction for the FixedRankMatrices manifold.

default_vector_transport_method(M::FixedRankMatrices)

Return the ProjectionTransport as the default vector transport method for the FixedRankMatrices manifold.

embed(M::FixedRankMatrices, p, X)

Embed the tangent vector X at point p in M from its UMVTangentVector representation into the set of $m×n$ matrices.

The formula reads

\[U_pMV_p^{\mathrm{H}} + U_XV_p^{\mathrm{H}} + U_pV_X^{\mathrm{H}}\]

embed(::FixedRankMatrices, p::SVDMPoint)

Embed the point p from its SVDMPoint representation into the set of $m×n$ matrices by computing $USV^{\mathrm{H}}$.

injectivity_radius(::FixedRankMatrices)

Return the incjectivity radius of the manifold of FixedRankMatrices, i.e. 0. See [HU17].

inner(M::FixedRankMatrices, p::SVDMPoint, X::UMVTangentVector, Y::UMVTangentVector)

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.

inverse_retract(M, p, q, ::OrthographicInverseRetraction)

Compute the orthographic inverse retraction FixedRankMatrices M by computing

\[ X = P_{T_{p}M}(q - p) = qVV^\mathrm{T} + UU^{\mathrm{T}}q - UU^{\mathrm{T}}qVV^{\mathrm{T}} - p,\]

where $p$ is a SVDMPoint(U,S,Vt) and $P_{T_{p}M}$ is the projection onto the tangent space at $p$.

For more details, see [AO14].

is_flat(::FixedRankMatrices)

Return false. FixedRankMatrices is not a flat manifold.

manifold_dimension(M::FixedRankMatrices)

Return the manifold dimension for the 𝔽-valued FixedRankMatrices M of dimension mxn of rank k, namely

\[\dim(\mathcal M) = k(m + n - k) \dim_ℝ 𝔽,\]

where $\dim_ℝ 𝔽$ is the real_dimension of 𝔽.

project(M, p, A)

Project the matrix $A ∈ ℝ^{m,n}$ or from the embedding the tangent space at $p$ on the FixedRankMatrices M, further decomposing the result into $X=UMV^\mathrm{H}$, i.e. a UMVTangentVector.

representation_size(M::FixedRankMatrices)

Return the element size of a point on the FixedRankMatrices M, i.e. the size of matrices on this manifold $(m,n)$.

retract(M::FixedRankMatrices, p, X, ::OrthographicRetraction)

Compute the OrthographicRetraction on the FixedRankMatrices M by finding the nearest point to $p + X$ in

\[ p + X + N_{p}\mathcal M \cap \mathcal M\]

where $N_{p}\mathcal M$ is the Normal Space of $T_{p}\mathcal M$.

If $X$ is sufficiently small, then the nearest such point is unique and can be expressed by

\[ q = (U(S + M) + U_{p})(S + M)^{-1}((S + M)V^{\mathrm{T}} + V^{\mathrm{T}}_{p}),\]

where $p$ is a SVDMPoint(U,S,Vt) and $X$ is an UMVTangentVector(Up,M,Vtp).

For more details, see [AO14].

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 $USV^\mathrm{H}=p+X$, 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.

vector_transport_to(M::FixedRankMatrices, p, X, q, ::ProjectionTransport)

Compute the vector transport of the tangent vector X at p to q, using the project of X to q.

zero_vector(M::FixedRankMatrices, p::SVDMPoint)

Return a UMVTangentVector representing the zero tangent vector in the tangent space of p on the FixedRankMatrices M, for example all three elements of the resulting structure are zero matrices.