$$\begin{aligned}
\label{eq:qerr_marginal}
&E_{q}^{N\to \infty} = \int_{-\infty}^{\infty}\!\!\! dx_1 \int_{-\infty}^{\infty}\!\!\! dx_2 \left[ x_1^2 + x_2^2 + 2\right] p(\bm{x}) \nonumber \\
&-2 \bigg[ \int_{0}^{\infty}\!\!\! dx_1 \, x_1 p_1(x_1) - \int_{-\infty}^{0}\!\!\! dx_1\, x_1 p_1(x_1) + \int_{0}^{\infty}\!\!\! dx_2 \, x_2 p_2(x_2) - \int_{-\infty}^{0}\!\!\! dx_2\, x_2 p_2(x_2) \bigg].\end{aligned}$$ where $p_1(\cdot)$ and $p_2(\cdot)$ are marginal distributions with respect to $x_1$ and $x_2$.

If it is assumed that the distribution is gaussian with mean centering and variance-covariance matrix $\Sigma$, (\[eq:qerr\_marginal\]) is calculated as follows, $$\begin{aligned}
 \label{eq:qerr_2d_gauss}
E_q^{N\to \infty} = 2 + \text{Tr}(\Sigma) - 2\sqrt{\frac{2}{\pi}} \left( \sqrt{\sigma_{11}} + \sqrt{\sigma_{22}} \right), \ \
\Sigma =
\begin{pmatrix}
\sigma_{11}& \sigma_{12} \\
\sigma_{12}& \sigma_{22}
\end{pmatrix},\end{aligned}$$ where $\sqrt{\sigma_{11}}$ and $\sqrt{\sigma_{22}}$ are standard deviations of each dimension. These results are straightforwardly extended to higher dimensions case.

Quantization Error Minimization in Gaussian Distribution with Rotational Transformation
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Some binary hashing algorithms use orthogonal transformation [@Gong2013b; @Gong2013]. This means that their purpose is to find cost minimizing point in rotational group $SO(n)$. We also consider rotational group in this paper.

With rotational group transformation, minimizing (\[eq:qerr\_2d\_gauss\]) is equivalent to maximizing $\sqrt{\sigma_{11}} + \sqrt{\sigma_{22}}$ subject to $\text{Tr}(\Sigma) = \text{const}$. Since constancy of $\text{Tr}(\Sigma)$ constrains a two dimensional vector $(\sqrt{\sigma_{11}},  \sqrt{\sigma_{22}})$ onto a circle, the solution is $\sigma_{11} = 