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\begin{enumerate}  \item When $\mathbf{A}_0$ is orthogonal, $f(\mathbf{q}, \mathbf{Y})$ has no spurious local minima. In fact, each local minimizer $\hat{\mathbf{q}}$ corresponds to a row of $\mathbf{X}_0$ by $\hat{\mathbf{q}}^T \mathbf{Y} = \alpha \mathbf{e}_i^T \mathbf{X}_0$.   \item When $\mathbf{A}_0$ is not orthogonal, it can be reduced to the  orthogonal case by preconditioning $\hat{\mathbf{Y}} = \big(\frac{1}{p \theta} \mathbf{Y} \mathbf{Y}^T \big)^{-1/2} \mathbf{Y}$. \item All orthogonal $\mathbf{A}_0$ are equivalent since $f(\mathbf{q}; \mathbf{A}_0 \mathbf{X}_0) = f(\mathbf{A_0}^T \mathbf{q}; \mathbf{X}_0)$, corresponding to a rotation on $\mathbf{q}$.  \item The feasible region $\mathbf{q} \in \mathbb{S}^{n-1}$ is a sphere with $2n$ symmetric sections, corresponding to those centered around $\pm \mathbf{e}_i$ for $i=1,\cdots,n$.