Symmetric positive definite matrices

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 $\mathcal P(n) \subset ℝ^{n × n}$

check_manifold_point(M::SymmetricPositiveDefinite, p; kwargs...)

checks, whether p is a valid point on the SymmetricPositiveDefiniteM, 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....

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

Check whether X is a tangent vector to p on the SymmetricPositiveDefiniteM, 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....

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 $∞$.

manifold_dimension(M::SymmetricPositiveDefinite)

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

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

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 $\mathcal M = \mathcal P(n)$.

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.

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.

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 SymmetricPositiveDefiniteM 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.

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.

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.

[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.

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

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

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).

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.

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.

LogCholeskyMetric <: Metric

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

exp(M::MetricManifold{SymmetricPositiveDefinite,LogCholeskyMetric}, p, X)

Compute the exponential map on the SymmetricPositiveDefiniteM 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 $\frac{1}{2}$.

log(M::MetricManifold{SymmetricPositiveDefinite,LogCholeskyMetric}, p, q)

Compute the logarithmic map on SymmetricPositiveDefiniteM 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.

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.

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 $\frac{1}{2}$

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

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

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

Compute the Riemannian mean of x using GeodesicInterpolation.