Symmetric Positive Semidefinite Matrices of Fixed Rank

SymmetricPositiveSemidefiniteFixedRank{T,𝔽} <: AbstractDecoratorManifold{𝔽}

The AbstractManifold $\operatorname{SPS}_k(n)$ consisting of the real- or complex-valued symmetric positive semidefinite matrices of size $n×n$ and rank $k$, i.e. the set

\[\operatorname{SPS}_k(n) = \bigl\{ p ∈ 𝔽^{n×n}\ \big|\ p^{\mathrm{H}} = p, apa^{\mathrm{H}} \geq 0 \text{ for all } a ∈ 𝔽 \text{ and } \operatorname{rank}(p) = k\bigr\},\]

where $⋅^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transpose, and the field $𝔽 ∈ \{ ℝ, ℂ\}$. We sometimes $\operatorname{SPS}_{k,𝔽}(n)$, when distinguishing the real- and complex-valued manifold is important.

An element is represented by $q ∈ 𝔽^{n×k}$ from the factorization $p = qq^{\mathrm{H}}$. Note that since for any unitary (orthogonal) $A ∈ 𝔽^{n×n}$ we have $(Aq)(Aq)^{\mathrm{H}} = qq^{\mathrm{H}} = p$, the representation is not unique, or in other words, the manifold is a quotient manifold of $𝔽^{n×k}$.

The tangent space at $p$, $T_p\operatorname{SPS}_k(n)$, is also represented by matrices $Y ∈ 𝔽^{n×k}$ and reads as

\[T_p\operatorname{SPS}_k(n) = \bigl\{ X ∈ 𝔽^{n×n}\,|\,X = qY^{\mathrm{H}} + Yq^{\mathrm{H}} \text{ i.e. } X = X^{\mathrm{H}} \bigr\}.\]

Note that the metric used yields a non-complete manifold. The metric was used in [JBAS10][MA20].

Constructor

SymmetricPositiveSemidefiniteFixedRank(n::Int, k::Int, field::AbstractNumbers=ℝ; parameter::Symbol=:type)

Generate the manifold of $n×n$ symmetric positive semidefinite matrices of rank $k$ over the field of real numbers ℝ or complex numbers ℂ.

exp(M::SymmetricPositiveSemidefiniteFixedRank, q, Y)

Compute the exponential map on the SymmetricPositiveSemidefiniteFixedRank, which just reads

\[ \exp_q Y = q+Y.\]

log(M::SymmetricPositiveSemidefiniteFixedRank, q, p)

Compute the logarithmic map on the SymmetricPositiveSemidefiniteFixedRank manifold by minimizing $\lVert p - qY\rVert$ with respect to $Y$.

isapprox(M::SymmetricPositiveSemidefiniteFixedRank, p, q; kwargs...)

test, whether two points p, q are (approximately) nearly the same. Since this is a quotient manifold in the embedding, the test is performed by checking their distance, if they are not the same, i.e. that $d_{\mathcal M}(p,q) \approx 0$, where the comparison is performed with the classical isapprox. The kwargs... are passed on to this accordingly.

check_point(M::SymmetricPositiveSemidefiniteFixedRank, q; kwargs...)

Check whether q is a valid manifold point on the SymmetricPositiveSemidefiniteFixedRank M, i.e. whether p=q*q' is a symmetric matrix of size (n,n) with values from the corresponding AbstractNumbers 𝔽. The symmetry of p is not explicitly checked since by using q p is symmetric by construction. The tolerance for the symmetry of p can and the rank of q*q' be set using kwargs....

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

Check whether X is a tangent vector to manifold point p on the SymmetricPositiveSemidefiniteFixedRank M. This requires X to be a symmetric matrix of size (n,n) with values from the correct AbstractNumbers.

Due to the reduced representation this is fulfilled as soon as the matrix is of correct size.

distance(M::SymmetricPositiveSemidefiniteFixedRank, p, q)

Compute the distance between two points p, q on the SymmetricPositiveSemidefiniteFixedRank, which is the Frobenius norm of $Y$ which minimizes $\lVert p - qY\rVert$ with respect to $Y$.

is_flat(::SymmetricPositiveSemidefiniteFixedRank)

Return false. SymmetricPositiveSemidefiniteFixedRank is not a flat manifold. See Theorem A.18 in [MA20].

manifold_dimension(M::SymmetricPositiveSemidefiniteFixedRank)

Return the dimension of the SymmetricPositiveSemidefiniteFixedRank matrix M over the number system 𝔽, i.e.

\[\begin{aligned} \dim \operatorname{SPS}_{k,ℝ}(n) &= kn - \frac{k(k-1)}{2},\\ \dim \operatorname{SPS}_{k,ℂ}(n) &= 2kn - k^2, \end{aligned}\]

where the last $k^2$ is due to the zero imaginary part for Hermitian matrices diagonal

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

Transport the tangent vector X at p to q by projecting it onto the tangent space at q.

 zero_vector(M::SymmetricPositiveSemidefiniteFixedRank, p)

returns the zero tangent vector in the tangent space of the symmetric positive definite matrix p on the SymmetricPositiveSemidefiniteFixedRank manifold M.