Despite the fact that PRH is extremely simple and sparse, it sometimes achieves entropically inferior isotropic states. However, this inferiority is reasonably overcome by PCAT or RSPCA without loss of sparsity.

The lower middle plot of Fig. \[experiment:gauss\] indicates that almost-PCA angle ($\lambda \sim 1$) at each pairwise rotation leads to good performance in the sharp gaussian distribution. It is important to distinguish our sequential pairwise almost-PCA rotation and PCA hashing rotation. To obtain exact PCA basis, it is necessary to account for all $n(n-1)/2$ possible pairs. PCAT, however, only deals with $\mathrm{O}(n \log_2 n)$ pairs.

Real Datasets
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### Low Dimensional Case

[c]{}

![Top-10 NN retrieval results of 128-dimensional SIFT1M data. The meaning of PRH(m, n, $\lambda$) is shown in Figure \[experiment:gauss\].[]{data-label="experiment:sift1m"}](./eps/graph_sift1m.pdf "fig:") (a) comparison with existing methods.

![Top-10 NN retrieval results of 128-dimensional SIFT1M data. The meaning of PRH(m, n, $\lambda$) is shown in Figure \[experiment:gauss\].[]{data-label="experiment:sift1m"}](./eps/graph_sift1m_lambdas.pdf "fig:") (b) effect of PCAT. $\qquad \qquad \qquad$

![Top-10 NN retrieval results of 128-dimensional SIFT1M data. The meaning of PRH(m, n, $\lambda$) is shown in Figure \[experiment:gauss\].[]{data-label="experiment:sift1m"}](./eps/graph_sift1m_rpcas.pdf "fig:") (c) effect of RSPCA. $\qquad \qquad \qquad$

[c]{}

![Top-10 NN retrieval results of SIFT1M data with PCA dimension reduction. **Left**:64bit (1/2 reduction) case. **Middle**: 