Adjoint Differentials

Adjoint Differentials

Manopt.AdjDforwardLogs — Method.
ξ = AdjDforwardLogs(M,x,ν)

compute the adjoibnt differential of forwardLogs $F$ orrucirng, in the power manifold array, the differential of the function

\[F_i(x) = \sum_{j\in\mathcal I_i} \log_{x_i} x_j\]

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

Input

  • M – a Power manifold
  • x – a PowPoint.
  • ν – a PowTVector from $T_X\mathcal M'$, where $X = (x,...,x)\in\mathcal M'$ is an $n$-fold copy of $x$ where \mathcal N (x,...,x)N.

Ouput

  • ξ – resulting tangent vector in $T_x\mathcal M$ representing the adjoint differentials of the logs.
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Manopt.AdjDxExp — Method.
AdjDxExp(M,x,ξ,η)

computes the adjoint of $D_x\exp_x\xi[\eta]$.

See also

DxExp, adjointJacobiField

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Manopt.AdjDxGeo — Method.
AdjDxGeo(M,x,y,t,η)

computes the adjoint of $D_xg(t;x,y)[\eta]$.

See also

DxGeo, adjointJacobiField

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Manopt.AdjDxLog — Method.
AdjDxLog(M,x,y,η)

computes the adjoint of $D_xlog_xy[\eta]$.

See also

DxLog, adjointJacobiField

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Manopt.AdjDyGeo — Method.
AdjDyGeo(M,x,y,t,η)

computes the adjoint of $D_yg(t;x,y)[\eta]$.

See also

DyGeo, adjointJacobiField

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Manopt.AdjDyLog — Method.
AdjDyLog(M,x,y,η)

computes the adjoint of $D_ylog_xy[\eta]$.

See also

DyLog, adjointJacobiField

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Manopt.AdjDξExp — Method.
AdjDξExp(M,x,ξ,η)

computes the adjoint of $D_\xi\exp_x\xi[\eta]$. Note that $\xi\in T_\xi(T_x\mathcal M) = T_x\mathcal M$ is still a tangent vector.

See also

DξExp, adjointJacobiField

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