TensorKitManifolds

What is new in v0.8

• TensorKitManifolds.jl now provides VectorInterface.jl compatibility for the tangent vectors on the manifolds.
• TensorKitManifolds.jl no longer exports any methods, and only exports the three manifold types. All methods now need to be quantified as e.g. Stiefel.project! , Grassmann.inner etc, or explicitly imported.
• There is a difference between VectorInterface.inner, which computes the complex inner product between two tangent vectors as vectors in a complex Euclidean space, and the inner method in TensorKitManifolds.jl or its submodules, which computes the Riemannian inner product (metric) on the manifold and is always real-valued.

Overview

There are three manifolds: Grassmann , Stiefel and Unitary , corresponding to submodules of TensorKitManifolds, whose names are exported.

These modules have a number of public but non-exported methods, namely:

• Δ = Grassmann.project(!)(X, W; metric = :euclidean): project an arbitrary tensor X onto the tangent space of the manifold at the point W , returned as a tangent vector Δ of a specific type, that also stores the base point W . The exclamation mark denotes that X is destroyed in the process.
• W = Grassmann.base(Δ): return the base point W of the tangent vector Δ.
• s = Grassmann.inner(W, Δ₁, Δ₂; metric = :euclidean): Riemannian inner product (a.k.a. metric) between tangent vectors Δ₁ and Δ₂ at the point W , which is a real-valued scalar s . For the default :euclidean metric, this is equal to the real part of the complex inner product between Δ₁ and Δ₂ as vectors in a complex Euclidean space, but other metrics might also be available.
• W′, Δ′ = Grassmann.retract(W, Δ, α): retract the point W in the direction of tangent vector Δ with step length α , ending up in the point W′ , and return also the local directional derivative along the path Δ′ , which is a tangent vector at W′ .
• Θ′ = Grassmann.transport(!)(Θ, W, Δ, α, W′): transport tangent vector Θ along the retraction of W in the direction of Δ with step length α , which ends at W′ . The resulting transported vector Θ′ is a tangent vector with base point W′ . The method with exclamation mark destroys Θ in the process.

The same methods exist for the Stiefel and Unitary manifolds. When multiple implementations or metrics are avaible, they are specified using a keyword argument to the above methods, or explicitly as Stiefel.inner_euclidean , Stiefel.inner_canonical , Stiefel.project_euclidean(!) , Stiefel.project_canonical(!) , Stiefel.retract_exp , Stiefel.transport_exp(!) , Stiefel.retract_cayley , Stiefel.transport_cayley(!) , Unitary.transport_parallel(!) , Unitary.transport_stiefel(!) .

The tangent vectors Δ returned by project(!) are of specific types defined in the submodules, e.g. GrassmannTangent , StiefelTangent , UnitaryTangent . The corresponding TensorMap object can be obtained via an argumentless getindex , i.e. Δ[] returns the corresponding TensorMap . However, you typically don't need those. The base point W is also stored in Δ and can be returned using W = base(Δ) . Hence, Δ should be assumed to be a point (W, Δ[]) on the tangent bundle of the manifold. Furthermore, these tangent vector types satisfy the behavior of vectors, and support basic Base arithmic, as well as the full set of methods from VectorInterface.jl