Adjoint Differentials

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 $\mathcal M' = \mathcal M^n$.


  • 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


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