Cholesky space

The Cholesky space is a Riemannian manifold on the lower triangular matrices. Its metric is based on the cholesky decomposition. The CholeskySpace is used to define the LogCholeskyMetric on the manifold of SymmetricPositiveDefinite matrices.

Manifolds.CholeskySpace — Type

CholeskySpace{N} <: Manifold{ℝ}

The manifold of lower triangular matrices with positive diagonal and a metric based on the cholesky decomposition. The formulae for this manifold are for example summarized in Table 1 of ^[Lin2019].

Constructor

CholeskySpace(n)

Generate the manifold of $n× n$ lower triangular matrices with positive diagonal.

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Base.exp — Method

exp(M::CholeskySpace, p, X)

Compute the exponential map on the CholeskySpace M emanating from the lower triangular matrix with positive diagonal p towards the lower triangular matrix X The formula reads

\[\exp_p X = ⌊ p ⌋ + ⌊ X ⌋ + \operatorname{diag}(p) \operatorname{diag}(p)\exp\bigl( \operatorname{diag}(X)\operatorname{diag}(p)^{-1}\bigr),\]

where $⌊\cdot⌋$ denotes the strictly lower triangular matrix, and $\operatorname{diag}$ extracts the diagonal matrix.

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Base.log — Method

log(M::CholeskySpace, X, p, q)

Compute the logarithmic map on the CholeskySpace M for the geodesic emanating from the lower triangular matrix with positive diagonal p towards q. The formula reads

\[\log_p q = ⌊ p ⌋ - ⌊ q ⌋ + \operatorname{diag}(p)\log\bigl(\operatorname{diag}(q)\operatorname{diag}(p)^{-1}\bigr),\]

where $⌊\cdot⌋$ denotes the strictly lower triangular matrix, and $\operatorname{diag}$ extracts the diagonal matrix.

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ManifoldsBase.check_manifold_point — Method

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

Check whether the matrix p lies on the CholeskySpace M, i.e. it's size fits the manifold, it is a lower triangular matrix and has positive entries on the diagonal. The tolerance for the tests can be set using the kwargs....

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ManifoldsBase.check_tangent_vector — Method

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

Check whether v is a tangent vector to p on the CholeskySpace M, i.e. after check_manifold_point(M,p), X has to have the same dimension as x and a symmetric matrix. The optional parameter check_base_point indicates whether to call check_manifold_point for p. The tolerance for the tests can be set using the kwargs....

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ManifoldsBase.distance — Method

distance(M::CholeskySpace, p, q)

Compute the Riemannian distance on the CholeskySpace M between two matrices p, q that are lower triangular with positive diagonal. The formula reads

\[d_{\mathcal M}(p,q) = \sqrt{\sum_{i>j} (p_{ij}-q_{ij})^2 + \sum_{j=1}^m (\log p_{jj} - \log q_{jj})^2 }\]

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ManifoldsBase.inner — Method

inner(M::CholeskySpace, p, X, Y)

Compute the inner product on the CholeskySpace M at the lower triangular matric with positive diagonal p and the two tangent vectors X,Y, i.e they are both lower triangular matrices with arbitrary diagonal. The formula reads

\[g_p(X,Y) = \sum_{i>j} X_{ij}Y_{ij} + \sum_{j=1}^m X_{ii}Y_{ii}p_{ii}^{-2}\]

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ManifoldsBase.manifold_dimension — Method

manifold_dimension(M::CholeskySpace)

Return the manifold dimension for the CholeskySpace M, i.e.

\[ \dim(\mathcal M) = \frac{N(N+1)}{2}.\]

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ManifoldsBase.representation_size — Method

reporesentation_size(M::CholeskySpace)

Return the representation size for the CholeskySpace{N} M, i.e. (N,N).

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ManifoldsBase.vector_transport_to — Method

vector_transport_to(M::CholeskySpace, p, X, q, ::ParallelTransport)

Parallely transport the tangent vector X at p along the geodesic to q on the CholeskySpace manifold M. The formula reads

\[\mathcal P_{q←p}(X) = ⌊ X ⌋ + \operatorname{diag}(q)\operatorname{diag}(p)^{-1}\operatorname{diag}(X),\]

where $⌊\cdot⌋$ denotes the strictly lower triangular matrix, and $\operatorname{diag}$ extracts the diagonal matrix.

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ManifoldsBase.zero_tangent_vector — Method

zero_tangent_vector(M::CholeskySpace, p)

Return the zero tangent vector on the CholeskySpace M at p.

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Literature

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