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Richard Guo added paragraph_Symmetry_and_reduction_begin__.tex
over 8 years ago
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\paragraph{Symmetry and reduction}
\begin{enumerate}
\item When $\mathbf{A}_0$ is not orthogonal, it can be reduced to the orthogonal case by preconditioning $\hat{\mathbf{Y}} = \big(\frac{1}{p \theta} \mathbf{Y} \mathbf{Y}^T \big)^{-1/2} \mathbf{Y}$ for some $i \in [n]$ and $\alpha \neq 0$.
\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 $\mathbf{q}$. Therefore, it suffices to study $\mathbf{A}_0 = \mathbf{I}$ only.
\item The feasible region $\mathbf{q} \in \mathbb{S}^{n-1}$ is a sphere with $2n$ symmetric hemispheres, corresponding to those centered around $\pm \mathbf{e}_i$ for $i=1,\cdots,n$. It suffices to study only the one centered around $\mathbf{e}_n$.
\item The hemisphere can be bijectively mapped to the equatorial plane $\mathbf{e}_n^{\bot}$ with the coordinates $\mathbf{w} \in \mathbb{B}^{n-1}$ in the unit ball, by
\[ \mathbf{q}(\mathbf{w}) = (\mathbf{w}, \sqrt{1-\|\mathbf{w}\|_2^2}). \]
\item It suffices to study $g(\mathbf{w}, \mathbf{X}_0) := f(\mathbf{q}(\mathbf{w}); \mathbf{X_0})$ over the set $\{\mathbf{w}: \|\mathbf{w}\|_2 < \sqrt{\frac{4n-1}{4n}} \}$.
\end{enumerate}
