Consider the strictly convex optimization problem \begin{equation} (1)\quad\quad\min_{f} J[f] \end{equation} where $f$ is a differentiable mapping $\mathbb{S}^N\times \mathbb{S}^N \to\mathbb{R}$ and $J$ is differentiable. Without further constraints, given the strong convexity of $J$, there is a unique minimizer $f^{*}$. Now suppose that $f$ needs to have the form $$ f(x, y) = h(x)^T h(y) $$ where $x, y\in\mathbb{S}^N$ and $h: \mathbb{S}^N \to\mathbb{S}^N$. This turns (1) into an optimization problem over $h$, \begin{equation} \label{eq:hopt} \min_{h} J[h]. \end{equation}
Question: Is it true that, if $h^{*}$ is a local minimizer of $J[h]$, then any other minimum $h'$ must have the form $h' = Rh^{*}$ for some rotation $R$?