Differentiation

Documentation for Manifolds.jl's methods and types for finite differences and automatic differentiation.

Differentiation backends

Manifolds.AbstractDiffBackend — Type

AbstractDiffBackend

An abstract type for diff backends. See FiniteDifferencesBackend for an example.

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Manifolds.CurrentDiffBackend — Type

CurrentDiffBackend(backend::AbstractDiffBackend)

A mutable struct for storing the current differentiation backend in a global constant _current_diff_backend.

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Manifolds.FiniteDifferencesBackend — Type

FiniteDifferencesBackend(method::FiniteDifferenceMethod = central_fdm(5, 1))

Differentiation backend based on the FiniteDifferences package.

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Manifolds._derivative — Function

_derivative(f, t[, backend::AbstractDiffBackend])

Compute the derivative of a callable f at time t computed using the given backend, an object of type Manifolds.AbstractDiffBackend. If the backend is not explicitly specified, it is obtained using the function Manifolds.diff_backend.

This function calculates plain Euclidean derivatives, for Riemannian differentiation see for example differential.

Note

Not specifying the backend explicitly will usually result in a type instability and decreased performance.

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Manifolds._gradient — Function

_gradient(f, p[, backend::AbstractDiffBackend])

Compute the gradient of a callable f at point p computed using the given backend, an object of type AbstractDiffBackend. If the backend is not explicitly specified, it is obtained using the function diff_backend.

This function calculates plain Euclidean gradients, for Riemannian gradient calculation see for example gradient.

Note

Not specifying the backend explicitly will usually result in a type instability and decreased performance.

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Manifolds.diff_backend! — Method

diff_backend!(backend::AbstractDiffBackend)

Set current backend for differentiation to backend.

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Manifolds.diff_backend — Method

diff_backend() -> AbstractDiffBackend

Get the current differentiation backend.

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Manifolds.diff_backends — Method

diff_backends() -> Vector{AbstractDiffBackend}

Get vector of currently valid differentiation backends.

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Manifolds._current_diff_backend — Constant

_current_diff_backend

The instance of Manifolds.CurrentDiffBackend that stores the globally default differentiation backend.

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Manifolds._diff_backends — Constant

_diff_backends

A vector of valid Manifolds.AbstractDiffBackend.

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Riemannian differentiation backends

Manifolds.AbstractRiemannianDiffBackend — Type

AbstractRiemannianDiffBackend

An abstract type for diff backends. See RiemannianONBDiffBackend for an example.

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Manifolds.CurrentRiemannianDiffBackend — Type

CurrentRiemannianDiffBackend(backend::AbstractRiemannianDiffBackend)

A mutable struct for storing the current Riemannian differentiation backend in global constants Manifolds._current_rgradient_backend and Manifolds._current_rdifferential_backend.

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Manifolds.RiemannianONBDiffBackend — Type

RiemannianONBDiffBackend(
    diff_backend::AbstractDiffBackend
    retraction::AbstractRetractionMethod
    inverse_retraction::AbstractInverseRetractionMethod
    basis::Union{AbstractOrthonormalBasis,CachedBasis{<:AbstractOrthonormalBasis}},
) <: AbstractRiemannianDiffBackend

Riemannian differentiation based on differentiation in an AbstractOrthonormalBasis basis using specified retraction, inverse_retraction and using backend diff_backend.

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Manifolds.RiemannianProjectionGradientBackend — Type

RiemannianProjectionGradientBackend(
    diff_backend::AbstractDiffBackend
) <: AbstractRiemannianDiffBackend

Riemannian differentiation based on differentiation in the ambient space and projection to the given manifold. Differentiation in the ambient space is performed using the backend diff_backend.

Only valid for manifolds that are embedded in a special way in the Euclidean space. See ^[Absil2008], Section 3.6.1 for details.

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Manifolds.differential — Method

differential(M::Manifold, f, t::Real, backend::AbstractDiffBackend = rdifferential_backend())

Compute the Riemannian differential of a curve $f: ℝ\to M$ on a manifold M represented by function f at time t using the given backend. It is calculated as the tangent vector equal to $\mathrm{d}f_t(t)[1]$.

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Manifolds.gradient — Method

gradient(M::Manifold, f, p, backend::AbstractRiemannianDiffBackend = rgradient_backend())

Compute the Riemannian gradient $∇f(p)$ of a real field on manifold M represented by function f at point p using the given backend.

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Manifolds.rdifferential_backend! — Method

rdifferential_backend!(backend::AbstractRiemannianDiffBackend)

Set current Riemannian differential backend for differentiation to backend.

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Manifolds.rdifferential_backend — Method

rdifferential_backend() -> AbstractRiemannianDiffBackend

Get the current differentiation backend for Riemannian differentials.

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Manifolds.rgradient_backend! — Method

rgradient_backend!(backend::AbstractRiemannianDiffBackend)

Set current Riemannian gradient backend for differentiation to backend.

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Manifolds.rgradient_backend — Method

rgradient_backend() -> AbstractRiemannianDiffBackend

Get the current differentiation backend for Riemannian gradients.

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Manifolds._current_rdifferential_backend — Constant

_current_rdifferential_backend

The instance of Manifolds.CurrentRiemannianDiffBackend that stores the globally default differentiation backend for calculating differentials.

See also

Manifolds.differential

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Manifolds._current_rgradient_backend — Constant

_current_rgradient_backend

The instance of Manifolds.CurrentRiemannianDiffBackend that stores the globally default differentiation backend for calculating gradients.

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Absil2008
Absil, P. A., et al. Optimization Algorithms on Matrix Manifolds. 2008.