Convert SPDs!

docs/src/references.bib

@@ -9,4 +9,18 @@ @incollection{AbsilMahonyTrumpf:2013
     PAGES     = {361--368},
     TITLE     = {An Extrinsic Look at the Riemannian Hessian},
     BOOKTITLE = {Geometric Science of Information},
+}
+@article{Nguyen:2023,
+    DOI       = {10.1007/s10957-023-02242-z},
+    EPRINT    = {2009.10159},
+    EPRINTTYPE = {arXiv},
+    YEAR      = {2023},
+    MONTH     = jun,
+    PUBLISHER = {Springer Science and Business Media {LLC}},
+    VOLUME    = {198},
+    NUMBER    = {1},
+    PAGES     = {135--164},
+    AUTHOR    = {Du Nguyen},
+    TITLE     = {Operator-Valued Formulas for Riemannian Gradient and Hessian and Families of Tractable Metrics in Riemannian Optimization},
+    JOURNAL   = {Journal of Optimization Theory and Applications}
 }

src/manifolds/SymmetricPositiveDefiniteAffineInvariant.jl

@@ -412,6 +412,31 @@ function parallel_transport_to!(M::SymmetricPositiveDefinite{N}, Y, p, X, q) whe
     return copyto!(Y, vtp)
 end
 
+@doc raw"""
+    riemannian_Hessian(M::SymmetricPositiveDefinite, p, eG, eH, X)
+
+The Riemannian Hessian can be computed as stated in Eq. (7.3) [Nguyen:2023](@cite).
+Let ``\nabla f(p)`` denote the Euclidean gradient `eG`,
+``\nabla^2 f(p)[X]`` the Euclidean Hessian `eH`, and
+``\operatorname{sym}(X) = \frac{1}{2}\bigl(X^{\mathrm{T}+X\bigr)```
+the symmetrization operator. Then the formula reads
+
+```math
+    \operatorname{Hess}f(p)[X]
+    =
+    p\operatorname{sym}(\nabla^2 f(p)[X])p
+    + \operatorname{sym}\Bigl( X\operatorname{sym}\bigl(\nabla f(p)\bigr)p)
+```
+"""
+riemannian_Hessian(M::SymmetricPositiveDefinite, p, eG, eH, X)
+function riemannian_Hessian(::SymmetricPositiveDefinite, Y, p, eG, eH, X)
+    # The following formulae even work for complex, H-symmetric positive definite matrices.
+    # the term within the seconds summands outer \sym:
+    Y .= X * 0.5 * (eG' + eG) * p
+    Y .= p * 1 / 2 * (eH' + eH) * p + 1 / 2 * (Y' + Y)
+    return Y
+end
+
 @doc raw"""
     riemann_tensor(::SymmetricPositiveDefinite, p, X, Y, Z)