Symmetric positive definite matrices

Manifolds.SymmetricPositiveDefinite — Type

SymmetricPositiveDefinite{N} <: AbstractEmbeddedManifold{ℝ,DefaultEmbeddingType}

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

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

Constructor

SymmetricPositiveDefinite(n)

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

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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

ManifoldsBase.check_manifold_point — Method

check_manifold_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_tangent_vector — Method

check_tangent_vector(M::SymmetricPositiveDefinite, p, X; check_base_point = true, kwargs... )

Check whether X is a tangent vector to p on the SymmetricPositiveDefinite M, i.e. atfer check_manifold_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 optional parameter check_base_point indicates, whether to call check_manifold_point for p. The tolerance for the last test can be set using the kwargs....

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ManifoldsBase.injectivity_radius — Method

injectivity_radius(M::SymmetricPositiveDefinite[, p])
injectivity_radius(M::MetricManifold{SymmetricPositiveDefinite,LinearAffineMetric}[, p])
injectivity_radius(M::MetricManifold{SymmetricPositiveDefinite,LogCholeskyMetric}[, p])

Return the injectivity radius of the SymmetricPositiveDefinite. Since M is a Hadamard manifold with respect to the LinearAffineMetric and the LogCholeskyMetric, the injectivity radius is globally $∞$.

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ManifoldsBase.manifold_dimension — Method

manifold_dimension(M::SymmetricPositiveDefinite)

returns the dimension of SymmetricPositiveDefinite M$=\mathcal P(n), n ∈ ℕ$, i.e.

\[\dim \mathcal P(n) = \frac{n(n+1)}{2}.\]

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ManifoldsBase.representation_size — Method

representation_size(M::SymmetricPositiveDefinite)

Return the size of an array representing an element on the SymmetricPositiveDefinite manifold M, i.e. $n × n$, the size of such a symmetric positive definite matrix on $\mathcal M = \mathcal P(n)$.

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ManifoldsBase.zero_tangent_vector — Method

zero_tangent_vector(M::SymmetricPositiveDefinite,x)

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

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Default metric: the linear affine metric

Manifolds.LinearAffineMetric — Type

LinearAffineMetric <: Metric

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 — 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 — 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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ManifoldsBase.distance — 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 — Method

[Ξ,κ] = get_basis(M::SymmetricPositiveDefinite, p, B::DiagonalizingOrthonormalBasis)
[Ξ,κ] = get_basis(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 has curvature 0.

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ManifoldsBase.inner — Method

inner(M::SymmetricPositiveDefinite, p, X, Y)
inner(M::MetricManifold{SymmetricPositiveDefinite,LinearAffineMetric}, p, X, Y)

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

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

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ManifoldsBase.vector_transport_to — Method

vector_transport_to(M::SymmetricPositiveDefinite, p, X, q, ::ParallelTransport)
vector_transport_to(M::MetricManifold{SymmetricPositiveDefinite,LinearAffineMetric}, p, X, y, ::ParallelTransport)

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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The log Euclidean metric

Manifolds.LogEuclideanMetric — Type

LogEuclideanMetric <: Metric

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 — 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

Manifolds.LogCholeskyMetric — Type

LogCholeskyMetric <: Metric

The Log-Cholesky metric imposes a metric based on the Cholesky decomposition as introduced by ^[Lin2019].

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Base.exp — 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 — 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 — 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

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

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

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ManifoldsBase.inner — 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.vector_transport_to — 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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Statistics

Statistics.mean — Method

mean(
    M::SymmetricPositiveDefinite,
    x::AbstractVector,
    [w::AbstractWeights,]
    method = GeodesicInterpolation();
    kwargs...,
)

Compute the Riemannian mean of x using GeodesicInterpolation.

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Literature

Lin2019
Lin, Zenhua: "Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition", arXiv: 1908.09326.