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Richard Guo edited paragraph_Symmetry_and_reduction_begin__.tex
over 8 years ago
Commit id: 5c5be09ab13ab607fcc9baa98d9005ac08cb0310
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diff --git a/paragraph_Symmetry_and_reduction_begin__.tex b/paragraph_Symmetry_and_reduction_begin__.tex
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--- a/paragraph_Symmetry_and_reduction_begin__.tex
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\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 $\Gamma := \{\mathbf{w}: \|\mathbf{w}\|_2 < \sqrt{\frac{4n-1}{4n}} \}$. In this set, $\mathbf{w} = \mathbf{0}$ corresponds to the local
minima. minima ($\mathbf{A}_0 = \mathbf{I}).
\end{enumerate}