MD is not RSGD, but RSGD also does M from MD

The whole idea of trying to parallel mirror descent with following geodesics as in RSGD has come to naught. And not the way one would expect, because MD still seems "type-correct" and RSGD doesn't yet. Long story short: in RSGD we're pulling back COTANGENTS but updating along a TANGENT.


Update! Before updating, we're raising an index of cotangent by applying inverse metric tensor, thus making it a tangent! Thanks to @ferrine for the idea.

Following \(\mathbb{R}^m\to\mathbb{R}^n\) analogy of previous posts:

\begin{equation*} F:M\to N, \end{equation*}
\begin{equation*} \xi = F^*\eta\in\mathcal{T}^*M,~\text{for}~\eta\in\mathcal{T}^*N, \end{equation*}
\begin{equation*} X = \xi^\sharp = g^{-1}(\xi) = g^{-1} \xi^\top~\text{so it becomes a column}. \end{equation*}

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