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Adjoint Differentials

Manopt.adjoint_differential_bezier_control — Method
adjoint_differential_bezier_control(
    M::MAnifold,
    B::AbstractVector{<:BezierSegment},
    t::Float64,
    X
)

evaluate the adjoint of the differential of a composite Bézier curve on the manifold M with respect to its control points b based on a points T$=(t_i)_{i=1}^n$ that are pointwise in $t_i\in[0,1]$ on the curve and given corresponding tangential vectors $X = (\eta_i)_{i=1}^n$, $\eta_i\in T_{\beta(t_i)}\mathcal M$

See de_casteljau for more details on the curve.

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Manopt.adjoint_differential_bezier_control — Method
adjoint_differential_bezier_control(
    M::Manifold,
    b::BezierSegment,
    t::Array{Float64,1},
    X::Array{Q,1}
)

evaluate the adjoint of the differential of a Bézier curve on the manifold M with respect to its control points b based on a points T$=(t_i)_{i=1}^n that are pointwise in $ t_i\in[0,1]$ on the curve and given corresponding tangential vectors $X = (\eta_i)_{i=1}^n$, $\eta_i\in T_{\beta(t_i)}\mathcal M$

See de_casteljau for more details on the curve and[BergmannGousenbourger2018].

source
Manopt.adjoint_differential_bezier_control — Method
adjoint_differential_bezier_control(
    M::Manifold,
    b::BezierSegment,
    t::Float64,
    η::Q)

evaluate the adjoint of the differential of a Bézier curve on the manifold M with respect to its control points b based on a point t $\in[0,1]$ on the curve and a tangent vector $\eta\in T_{\beta(t)}\mathcal M$.

See de_casteljau for more details on the curve.

source
Manopt.adjoint_differential_forward_logs — Method
Y = adjoint_differential_forward_logs(M, p, X)

Compute the adjoint differential of forward_logs $F$ orrucirng, in the power manifold array p, the differential of the function

\[F_i(p) = \sum_{j ∈ \mathcal I_i} \log_{p_i} p_j\]

where $i$ runs over all indices of the PowerManifold manifold M and $\mathcal I_i$ denotes the forward neighbors of $i$ Let $n$ be the number dimensions of the PowerManifold manifold (i.e. length(size(x))). Then the input tangent vector lies on the manifold $\mathcal M' = \mathcal M^n$.

Input

  • M – a PowerManifold manifold
  • p – an array of points on a manifold
  • X – a tangent vector to from the n-fold power of p, where n is the ndims of p

Ouput

Y – resulting tangent vector in $T_p\mathcal M$ representing the adjoint differentials of the logs.

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