(Fig. \[fig:ellipse\]). It corresponds to $\theta = 0$ in equation (\[eq:r-theta\_representation\_of\_2d\_gauss\]). Entropy is increased with this tilting, since the PCA angle is the entropy-maximizing angle. Rotation matrix is derived as $$\begin{aligned}
 \label{eq:pca_rotation}
&R = 
\begin{pmatrix}
\cos \theta(\lambda) & -\sin \theta(\lambda) \\
\sin \theta(\lambda) & \cos \theta(\lambda)
\end{pmatrix}, \nonumber \\
\theta(\lambda) = \theta_{\text{iso}} + \lambda (&\theta_{\text{pca}} - \theta_{\text{iso}}), \quad \theta_{\text{pca}} = \tan^{-1} \left( \frac{1}{2} \frac{\sigma_{12}}{\sigma_{11} - \sigma_{22}}\right),\end{aligned}$$ where $\theta_{\text{iso}}$ is given in Eq. (\[eq:iso\_rotation\]).

$\lambda$ is a tuning parameter ranging from zero (completely isotropic) to one (completely PCA). We can control a degree of balance between isotropy and entropy by tuning $\lambda$. Since PCA tilting does not lead to completely isotropic variances, there is no definite reason to stop applying basic rotations at $\log_2 n$ times. However, it is not necessary increase the number of basic rotations because it practically leads enough accuracy with $\log_2 n$ times application. The number of fill-ins of the transformation matrix therefore does not need to change.

### Random Sparse PCA Rotation (RSPCA)

The second method applies additional sparse rotations after having a completely isotropic transformation. The additional matrices have the same form as basic isotropic rotation, but $n/2$ rotational pairs are randomly chosen and PCA 