32bit (1/4 reduction) case. **Right**: behavior of PRHs across some dimensions. The meaning of PRH(m, n, $\lambda$) is shown in Figure \[experiment:gauss\]. \[experiment:sift1m\_reduct\]](./eps/graph_sift1m_64.pdf "fig:") (a) 64bit (1/2 dimension reduction) results.

![Top-10 NN retrieval results of SIFT1M data with PCA dimension reduction. **Left**:64bit (1/2 reduction) case. **Middle**: 32bit (1/4 reduction) case. **Right**: behavior of PRHs across some dimensions. The meaning of PRH(m, n, $\lambda$) is shown in Figure \[experiment:gauss\]. \[experiment:sift1m\_reduct\]](./eps/graph_sift1m_32.pdf "fig:") (b) 32bit (1/4 dimension reduction) results.

![Top-10 NN retrieval results of SIFT1M data with PCA dimension reduction. **Left**:64bit (1/2 reduction) case. **Middle**: 32bit (1/4 reduction) case. **Right**: behavior of PRHs across some dimensions. The meaning of PRH(m, n, $\lambda$) is shown in Figure \[experiment:gauss\]. \[experiment:sift1m\_reduct\]](./eps/graph_sift1m_reduct.pdf "fig:") (c) PRHs for different degree of reductions.

The SIFT1M case is considered next. Fig. \[experiment:sift1m\] is the retrieval results. In common with the gaussian case, the completely isotropic PRH (PRH(7, 0, 0.0)) leads unfavorable accuracy. PCAT and RSPCA attain good performance. Especially, RSPCA achieves remarkably better retrieval result compared to other methods in figure (d).

### Relation to dimension reduction

Although a sparse dimension reduction scheme is not devised in this study, we examine the effect of dimension reduction on the performance of the 