Symmetric positive definite matrices

Manifolds.SymmetricPositiveDefinite β€” Type
SymmetricPositiveDefinite{N} <: AbstractDecoratorManifold{𝔽}

The manifold of symmetric positive definite matrices, i.e.

\[\mathcal P(n) = \bigl\{ p ∈ ℝ^{n Γ— n}\ \big|\ a^\mathrm{T}pa > 0 \text{ for all } a ∈ ℝ^{n}\backslash\{0\} \bigr\}\]

The tangent space at $T_p\mathcal P(n)$ reads

\[ T_p\mathcal P(n) = \bigl\{ X \in \mathbb R^{nΓ—n} \big|\ X=X^\mathrm{T} \bigr\},\]

i.e. the set of symmetric matrices,

Constructor

SymmetricPositiveDefinite(n)

generates the manifold $\mathcal P(n) \subset ℝ^{n Γ— n}$

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This manifold can – for example – be illustrated as ellipsoids: since the eigenvalues are all positive they can be taken as lengths of the axes of an ellipsoids while the directions are given by the eigenvectors.

An example set of data

The manifold can be equipped with different metrics

Common and metric independent functions

Base.rand β€” Method
rand(M::SymmetricPositiveDefinite; Οƒ::Real=1)

Generate a random symmetric positive definite matrix on the SymmetricPositiveDefinite manifold M.

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ManifoldsBase.check_point β€” Method
check_point(M::SymmetricPositiveDefinite, p; kwargs...)

checks, whether p is a valid point on the SymmetricPositiveDefinite M, i.e. is a matrix of size (N,N), symmetric and positive definite. The tolerance for the second to last test can be set using the kwargs....

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ManifoldsBase.check_vector β€” Method
check_vector(M::SymmetricPositiveDefinite, p, X; kwargs... )

Check whether X is a tangent vector to p on the SymmetricPositiveDefinite M, i.e. atfer check_point(M,p), X has to be of same dimension as p and a symmetric matrix, i.e. this stores tangent vetors as elements of the corresponding Lie group. The tolerance for the last test can be set using the kwargs....

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Default metric: the linear affine metric

Manifolds.LinearAffineMetric β€” Type
LinearAffineMetric <: AbstractMetric

The linear affine metric is the metric for symmetric positive definite matrices, that employs matrix logarithms and exponentials, which yields a linear and affine metric.

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This metric is also the default metric, i.e. any call of the following functions with P=SymmetricPositiveDefinite(3) will result in MetricManifold(P,LinearAffineMetric())and hence yield the formulae described in this seciton.

Base.exp β€” Method
exp(M::SymmetricPositiveDefinite, p, X)
exp(M::MetricManifold{SymmetricPositiveDefinite{N},LinearAffineMetric}, p, X)

Compute the exponential map from p with tangent vector X on the SymmetricPositiveDefinite M with its default MetricManifold having the LinearAffineMetric. The formula reads

\[\exp_p X = p^{\frac{1}{2}}\operatorname{Exp}(p^{-\frac{1}{2}} X p^{-\frac{1}{2}})p^{\frac{1}{2}},\]

where $\operatorname{Exp}$ denotes to the matrix exponential.

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Base.log β€” Method
log(M::SymmetricPositiveDefinite, p, q)
log(M::MetricManifold{SymmetricPositiveDefinite,LinearAffineMetric}, p, q)

Compute the logarithmic map from p to q on the SymmetricPositiveDefinite as a MetricManifold with LinearAffineMetric. The formula reads

\[\log_p q = p^{\frac{1}{2}}\operatorname{Log}(p^{-\frac{1}{2}}qp^{-\frac{1}{2}})p^{\frac{1}{2}},\]

where $\operatorname{Log}$ denotes to the matrix logarithm.

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Manifolds.change_metric β€” Method
change_metric(M::SymmetricPositiveDefinite{n}, E::EuclideanMetric, p, X)

Given a tangent vector $X ∈ T_p\mathcal P(n)$ with respect to the EuclideanMetric g_E, this function changes into the LinearAffineMetric (default) metric on the SymmetricPositiveDefinite M.

To be precise we are looking for $c\colon T_p\mathcal P(n) \to T_p\mathcal P(n)$ such that for all $Y,Z ∈ T_p\mathcal P(n)$` it holds

\[⟨Y,Z⟩ = \operatorname{tr}(YZ) = \operatorname{tr}(p^{-1}c(Y)p^{-1}c(Z)) = g_p(c(Z),c(Y))\]

and hence $c(X) = pX$ is computed.

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Manifolds.change_representer β€” Method
change_representer(M::SymmetricPositiveDefinite, E::EuclideanMetric, p, X)

Given a tangent vector $X ∈ T_p\mathcal M$ representing a linear function on the tangent space at p with respect to the EuclideanMetric g_E, this is turned into the representer with respect to the (default) metric, the LinearAffineMetric on the SymmetricPositiveDefinite M.

To be precise we are looking for $Z∈T_p\mathcal P(n)$ such that for all $Y∈T_p\mathcal P(n)$` it holds

\[⟨X,Y⟩ = \operatorname{tr}(XY) = \operatorname{tr}(p^{-1}Zp^{-1}Y) = g_p(Z,Y)\]

and hence $Z = pXp$.

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ManifoldsBase.distance β€” Method
distance(M::SymmetricPositiveDefinite, p, q)
distance(M::MetricManifold{SymmetricPositiveDefinite,LinearAffineMetric}, p, q)

Compute the distance on the SymmetricPositiveDefinite manifold between p and q, as a MetricManifold with LinearAffineMetric. The formula reads

\[d_{\mathcal P(n)}(p,q) = \lVert \operatorname{Log}(p^{-\frac{1}{2}}qp^{-\frac{1}{2}})\rVert_{\mathrm{F}}.,\]

where $\operatorname{Log}$ denotes the matrix logarithm and $\lVert\cdot\rVert_{\mathrm{F}}$ denotes the matrix Frobenius norm.

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ManifoldsBase.get_basis β€” Method
[Ξ,κ] = get_basis(M::SymmetricPositiveDefinite, p, B::DefaultOrthonormalBasis)
[Ξ,κ] = get_basis(M::MetricManifold{SymmetricPositiveDefinite{N},LinearAffineMetric}, p, B::DefaultOrthonormalBasis)

Return a default ONB for the tangent space $T_p\mathcal P(n)$ of the SymmetricPositiveDefinite with respect to the LinearAffineMetric.

\[ g_p(X,Y) = \operatorname{tr}(p^{-1} X p^{-1} Y),\]

The basis constructed here is based on the ONB for symmetric matrices constructed as follows. Let

\[\Delta_{i,j} = (a_{k,l})_{k,l=1}^n \quad \text{ with } a_{k,l} = \begin{cases} 1 & \mbox{ for } k=l \text{ if } i=j\\ \frac{1}{\sqrt{2}} & \mbox{ for } k=i, l=j \text{ or } k=j, l=i\\ 0 & \text{ else.} \end{cases}\]

which forms an ONB for the space of symmetric matrices.

We then form the ONB by

\[ \Xi_{i,j} = p^{\frac{1}{2}}\Delta_{i,j}p^{\frac{1}{2}},\qquad i=1,\ldots,n, j=i,\ldots,n.\]

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ManifoldsBase.get_basis_diagonalizing β€” Method
[Ξ,κ] = get_basis_diagonalizing(M::SymmetricPositiveDefinite, p, B::DiagonalizingOrthonormalBasis)
[Ξ,κ] = get_basis_diagonalizing(M::MetricManifold{SymmetricPositiveDefinite{N},LinearAffineMetric}, p, B::DiagonalizingOrthonormalBasis)

Return a orthonormal basis Ξ as a vector of tangent vectors (of length manifold_dimension of M) in the tangent space of p on the MetricManifold of SymmetricPositiveDefinite manifold M with LinearAffineMetric that diagonalizes the curvature tensor $R(u,v)w$ with eigenvalues κ and where the direction B.frame_direction $V$ has curvature 0.

The construction is based on an ONB for the symmetric matrices similar to get_basis(::SymmetricPositiveDefinite, p, ::DefaultOrthonormalBasis just that the ONB here is build from the eigen vectors of $p^{\frac{1}{2}}Vp^{\frac{1}{2}}$.

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ManifoldsBase.get_coordinates β€” Method
get_coordinates(::SymmetricPositiveDefinite, p, X, ::DefaultOrthonormalBasis)

Using the basis from get_basis the coordinates with respect to this ONB can be simplified to

\[ c_k = \mathrm{tr}(p^{-\frac{1}{2}}\Delta_{i,j} X)\]

where $k$ is trhe linearized index of the $i=1,\ldots,n, j=i,\ldots,n$.

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ManifoldsBase.get_vector β€” Method
get_vector(::SymmetricPositiveDefinite, p, c, ::DefaultOrthonormalBasis)

Using the basis from get_basis the vector reconstruction with respect to this ONB can be simplified to

\[ X = p^{\frac{1}{2}} \Biggl( \sum_{i=1,j=i}^n c_k \Delta_{i,j} \Biggr) p^{\frac{1}{2}}\]

where $k$ is the linearized index of the $i=1,\ldots,n, j=i,\ldots,n$.

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ManifoldsBase.parallel_transport_to β€” Method
parallel_transport_to(M::SymmetricPositiveDefinite, p, X, q)
parallel_transport_to(M::MetricManifold{SymmetricPositiveDefinite,LinearAffineMetric}, p, X, y)

Compute the parallel transport of X from the tangent space at p to the tangent space at q on the SymmetricPositiveDefinite as a MetricManifold with the LinearAffineMetric. The formula reads

\[\mathcal P_{q←p}X = p^{\frac{1}{2}} \operatorname{Exp}\bigl( \frac{1}{2}p^{-\frac{1}{2}}\log_p(q)p^{-\frac{1}{2}} \bigr) p^{-\frac{1}{2}}X p^{-\frac{1}{2}} \operatorname{Exp}\bigl( \frac{1}{2}p^{-\frac{1}{2}}\log_p(q)p^{-\frac{1}{2}} \bigr) p^{\frac{1}{2}},\]

where $\operatorname{Exp}$ denotes the matrix exponential and log the logarithmic map on SymmetricPositiveDefinite (again with respect to the LinearAffineMetric).

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Bures-Wasserstein metric

Base.exp β€” Method
exp(::MatricManifold{ℝ,SymmetricPositiveDefinite,BuresWassersteinMetric}, p, X)

Compute the exponential map on SymmetricPositiveDefinite with respect to the BuresWassersteinMetric given by

\[ \exp_p(X) = p+X+L_p(X)pL_p(X)\]

where $q=L_p(X)$ denotes the Lyapunov operator, i.e. it solves $pq + qp = X$.

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Base.log β€” Method
log(::MatricManifold{SymmetricPositiveDefinite,BuresWassersteinMetric}, p, q)

Compute the logarithmic map on SymmetricPositiveDefinite with respect to the BuresWassersteinMetric given by

\[ \log_p(q) = (pq)^{\frac{1}{2}} + (qp)^{\frac{1}{2}} - 2 p\]

where $q=L_p(X)$ denotes the Lyapunov operator, i.e. it solves $pq + qp = X$.

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Manifolds.change_representer β€” Method
change_representer(M::MetricManifold{ℝ,SymmetricPositiveDefinite,BuresWassersteinMetric}, E::EuclideanMetric, p, X)

Given a tangent vector $X ∈ T_p\mathcal M$ representing a linear function on the tangent space at p with respect to the EuclideanMetric g_E, this is turned into the representer with respect to the (default) metric, the BuresWassersteinMetric on the SymmetricPositiveDefinite M.

To be precise we are looking for $Z∈T_p\mathcal P(n)$ such that for all $Y∈T_p\mathcal P(n)$` it holds

\[⟨X,Y⟩ = \operatorname{tr}(XY) = ⟨Z,Y⟩_{\mathrm{BW}}\]

for all $Y$ and hence we get $Z$= 2(A+A^{\mathrm{T}})$with$A=Xp``.

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ManifoldsBase.distance β€” Method
distance(::MatricManifold{SymmetricPositiveDefinite,BuresWassersteinMetric}, p, q)

Compute the distance with respect to the BuresWassersteinMetric on SymmetricPositiveDefinite matrices, i.e.

\[d(p,q) = \operatorname{tr}(p) + \operatorname{tr}(q) - 2\operatorname{tr}\Bigl( (p^{\frac{1}{2}}qp^{\frac{1}{2}} \bigr)^\frac{1}{2} \Bigr),\]

where the last trace can be simplified (by rotating the matrix products in the trace) to $\operatorname{tr}(pq)$.

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ManifoldsBase.inner β€” Method
inner(::MetricManifold{ℝ,SymmetricPositiveDefinite,BuresWassersteinMetric}, p, X, Y)

Compute the inner product SymmetricPositiveDefinite with respect to the BuresWassersteinMetric given by

\[ ⟨X,Y⟩ = \frac{1}{2}\operatorname{tr}(L_p(X)Y)\]

where $q=L_p(X)$ denotes the Lyapunov operator, i.e. it solves $pq + qp = X$.

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Generalized Bures-Wasserstein metric

Manifolds.GeneralizedBuresWassersteinMetric β€” Type
GeneralizedBurresWassertseinMetric{T<:AbstractMatrix} <: AbstractMetric

The generalized Bures Wasserstein metric for symmetric positive definite matrices, see[HanMishraJawanpuriaGao2021].

This metric internally stores the symmetric positive definite matrix $M$ to generalise the metric, where the name also follows the mentioned preprint.

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Base.exp β€” Method
exp(::MatricManifold{ℝ,SymmetricPositiveDefinite,GeneralizedBuresWassersteinMetric}, p, X)

Compute the exponential map on SymmetricPositiveDefinite with respect to the GeneralizedBuresWassersteinMetric given by

\[ \exp_p(X) = p+X+\mathcal ML_{p,M}(X)pML_{p,M}(X)\]

where $q=L_{M,p}(X)$ denotes the generalized Lyapunov operator, i.e. it solves $pqM + Mqp = X$.

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Base.log β€” Method
log(::MatricManifold{SymmetricPositiveDefinite,GeneralizedBuresWassersteinMetric}, p, q)

Compute the logarithmic map on SymmetricPositiveDefinite with respect to the BuresWassersteinMetric given by

\[ \log_p(q) = M(M^{-1}pM^{-1}q)^{\frac{1}{2}} + (qM^{-1}pM^{-1})^{\frac{1}{2}}M - 2 p.\]

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Manifolds.change_representer β€” Method
change_representer(M::MetricManifold{ℝ,SymmetricPositiveDefinite,GeneralizedBuresWassersteinMetric}, E::EuclideanMetric, p, X)

Given a tangent vector $X ∈ T_p\mathcal M$ representing a linear function on the tangent space at p with respect to the EuclideanMetric g_E, this is turned into the representer with respect to the (default) metric, the GeneralizedBuresWassersteinMetric on the SymmetricPositiveDefinite M.

To be precise we are looking for $Z∈T_p\mathcal P(n)$ such that for all $Y∈T_p\mathcal P(n)$ it holds

\[⟨X,Y⟩ = \operatorname{tr}(XY) = ⟨Z,Y⟩_{\mathrm{BW}}\]

for all $Y$ and hence we get $Z = 2pXM + 2MXp$.

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ManifoldsBase.distance β€” Method
distance(::MatricManifold{SymmetricPositiveDefinite,GeneralizedBuresWassersteinMetric}, p, q)

Compute the distance with respect to the BuresWassersteinMetric on SymmetricPositiveDefinite matrices, i.e.

\[d(p,q) = \operatorname{tr}(M^{-1}p) + \operatorname{tr}(M^{-1}q) - 2\operatorname{tr}\bigl( (p^{\frac{1}{2}}M^{-1}qM^{-1}p^{\frac{1}{2}} \bigr)^{\frac{1}{2}},\]

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Log-Euclidean metric

Manifolds.LogEuclideanMetric β€” Type
LogEuclideanMetric <: RiemannianMetric

The LogEuclidean Metric consists of the Euclidean metric applied to all elements after mapping them into the Lie Algebra, i.e. performing a matrix logarithm beforehand.

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ManifoldsBase.distance β€” Method
distance(M::MetricManifold{SymmetricPositiveDefinite{N},LogEuclideanMetric}, p, q)

Compute the distance on the SymmetricPositiveDefinite manifold between p and q as a MetricManifold with LogEuclideanMetric. The formula reads

\[ d_{\mathcal P(n)}(p,q) = \lVert \operatorname{Log} p - \operatorname{Log} q \rVert_{\mathrm{F}}\]

where $\operatorname{Log}$ denotes the matrix logarithm and $\lVert\cdot\rVert_{\mathrm{F}}$ denotes the matrix Frobenius norm.

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Log-Cholesky metric

Base.exp β€” Method
exp(M::MetricManifold{SymmetricPositiveDefinite,LogCholeskyMetric}, p, X)

Compute the exponential map on the SymmetricPositiveDefinite M with LogCholeskyMetric from p into direction X. The formula reads

\[\exp_p X = (\exp_y W)(\exp_y W)^\mathrm{T}\]

where $\exp_xW$ is the exponential map on CholeskySpace, $y$ is the cholesky decomposition of $p$, $W = y(y^{-1}Xy^{-\mathrm{T}})_\frac{1}{2}$, and $(\cdot)_\frac{1}{2}$ denotes the lower triangular matrix with the diagonal multiplied by $\frac{1}{2}$.

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Base.log β€” Method
log(M::MetricManifold{SymmetricPositiveDefinite,LogCholeskyMetric}, p, q)

Compute the logarithmic map on SymmetricPositiveDefinite M with respect to the LogCholeskyMetric emanating from p to q. The formula can be adapted from the CholeskySpace as

\[\log_p q = xW^{\mathrm{T}} + Wx^{\mathrm{T}},\]

where $x$ is the cholesky factor of $p$ and $W=\log_x y$ for $y$ the cholesky factor of $q$ and the just mentioned logarithmic map is the one on CholeskySpace.

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ManifoldsBase.distance β€” Method
distance(M::MetricManifold{SymmetricPositiveDefinite,LogCholeskyMetric}, p, q)

Compute the distance on the manifold of SymmetricPositiveDefinite nmatrices, i.e. between two symmetric positive definite matrices p and q with respect to the LogCholeskyMetric. The formula reads

\[d_{\mathcal P(n)}(p,q) = \sqrt{ \lVert ⌊ x βŒ‹ - ⌊ y βŒ‹ \rVert_{\mathrm{F}}^2 + \lVert \log(\operatorname{diag}(x)) - \log(\operatorname{diag}(y))\rVert_{\mathrm{F}}^2 }\ \ ,\]

where $x$ and $y$ are the cholesky factors of $p$ and $q$, respectively, $⌊\cdotβŒ‹$ denbotes the strictly lower triangular matrix of its argument, and $\lVert\cdot\rVert_{\mathrm{F}}$ the Frobenius norm.

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ManifoldsBase.inner β€” Method
inner(M::MetricManifold{LogCholeskyMetric,ℝ,SymmetricPositiveDefinite}, p, X, Y)

Compute the inner product of two matrices X, Y in the tangent space of p on the SymmetricPositiveDefinite manifold M, as a MetricManifold with LogCholeskyMetric. The formula reads

\[ g_p(X,Y) = ⟨a_z(X),a_z(Y)⟩_z,\]

where $⟨\cdot,\cdot⟩_x$ denotes inner product on the CholeskySpace, $z$ is the cholesky factor of $p$, $a_z(W) = z (z^{-1}Wz^{-\mathrm{T}})_{\frac{1}{2}}$, and $(\cdot)_\frac{1}{2}$ denotes the lower triangular matrix with the diagonal multiplied by $\frac{1}{2}$

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ManifoldsBase.parallel_transport_to β€” Method
vector_transport_to(
    M::MetricManifold{SymmetricPositiveDefinite,LogCholeskyMetric},
    p,
    X,
    q,
    ::ParallelTransport,
)

Parallel transport the tangent vector X at p along the geodesic to q with respect to the SymmetricPositiveDefinite manifold M and LogCholeskyMetric. The parallel transport is based on the parallel transport on CholeskySpace: Let $x$ and $y$ denote the cholesky factors of p and q, respectively and $W = x(x^{-1}Xx^{-\mathrm{T}})_\frac{1}{2}$, where $(\cdot)_\frac{1}{2}$ denotes the lower triangular matrix with the diagonal multiplied by $\frac{1}{2}$. With $V$ the parallel transport on CholeskySpace from $x$ to $y$. The formula hear reads

\[\mathcal P_{q←p}X = yV^{\mathrm{T}} + Vy^{\mathrm{T}}.\]

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Literature