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 โ TypeCholeskySpace{T} <: AbstractManifold{โ}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 [Lin19].
Constructor
CholeskySpace(n; parameter::Symbol=:type)Generate the manifold of $nรn$ lower triangular matrices with positive diagonal.
Base.exp โ Methodexp(M::CholeskySpace, p, X)Compute the exponential map on the CholeskySpaceM emanating from the lower triangular matrix with positive diagonal p towards the lower triangular matrix X The formula reads
\[\exp_p X = โ p โ + โ X โ + \operatorname{diag}(p) \operatorname{diag}(p)\exp\bigl( \operatorname{diag}(X)\operatorname{diag}(p)^{-1}\bigr),\]
where $โโ โ$ denotes the strictly lower triangular matrix, and $\operatorname{diag}$ extracts the diagonal matrix.
Base.log โ Methodlog(M::CholeskySpace, X, p, q)Compute the logarithmic map on the CholeskySpaceM for the geodesic emanating from the lower triangular matrix with positive diagonal p towards q. The formula reads
\[\log_p q = โ p โ - โ q โ + \operatorname{diag}(p)\log\bigl(\operatorname{diag}(q)\operatorname{diag}(p)^{-1}\bigr),\]
where $โโ โ$ denotes the strictly lower triangular matrix, and $\operatorname{diag}$ extracts the diagonal matrix.
ManifoldsBase.check_point โ Methodcheck_point(M::CholeskySpace, p; kwargs...)Check whether the matrix p lies on the CholeskySpaceM, 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....
ManifoldsBase.check_vector โ Methodcheck_vector(M::CholeskySpace, p, X; kwargs... )Check whether v is a tangent vector to p on the CholeskySpaceM, i.e. after check_point(M,p), X has to have the same dimension as p and a symmetric matrix. The tolerance for the tests can be set using the kwargs....
ManifoldsBase.distance โ Methoddistance(M::CholeskySpace, p, q)Compute the Riemannian distance on the CholeskySpaceM between two matrices p, q that are lower triangular with positive diagonal. The formula reads
\[d_{\mathcal M}(p,q) = \sqrt{\sum_{i>j} (p_{ij}-q_{ij})^2 + \sum_{j=1}^m (\log p_{jj} - \log q_{jj})^2 }\]
ManifoldsBase.inner โ Methodinner(M::CholeskySpace, p, X, Y)Compute the inner product on the CholeskySpaceM 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
\[g_p(X,Y) = \sum_{i>j} X_{ij}Y_{ij} + \sum_{j=1}^m X_{ii}Y_{ii}p_{ii}^{-2}\]
ManifoldsBase.is_flat โ Methodis_flat(::CholeskySpace)Return true. CholeskySpace is a flat manifold. See Proposition 8 of [Lin19].
ManifoldsBase.manifold_dimension โ Methodmanifold_dimension(M::CholeskySpace)Return the manifold dimension for the CholeskySpaceM, i.e.
\[ \dim(\mathcal M) = \frac{N(N+1)}{2}.\]
ManifoldsBase.parallel_transport_to โ Methodparallel_transport_to(M::CholeskySpace, p, X, q)Parallely transport the tangent vector X at p along the geodesic to q on the CholeskySpace manifold M. The formula reads
\[\mathcal P_{qโp}(X) = โ X โ + \operatorname{diag}(q)\operatorname{diag}(p)^{-1}\operatorname{diag}(X),\]
where $โโ โ$ denotes the strictly lower triangular matrix, and $\operatorname{diag}$ extracts the diagonal matrix.
ManifoldsBase.representation_size โ Methodrepresentation_size(M::CholeskySpace)Return the representation size for the CholeskySpace{N}M, i.e. (N,N).
ManifoldsBase.zero_vector โ Methodzero_vector(M::CholeskySpace, p)Return the zero tangent vector on the CholeskySpaceM at p.