**PCA Hashing (PCA)**: PCA hashing, as its name suggests, uses linear transformation to PCA basis. As discussed in section \[sec:ent\_qerr\], PCA basis is the opposite extreme of the isotropic basis with regard to the trade-off relationship between quantization error and entropy. **K-means Hashing (KMH)**[@He2013]: This is a recently proposed state-of-the-art method. It uses k-means Vector Quantization and binary code assignment optimization for each cluster center. It is thus a kind of nonlinear method. It is selected for evaluating binary hashing performance compared to nonlinear methods. We use algorithm parameter $b=4, M=\text{ndim}/b$ and 50 iteration number defined in [@He2013]. **Bilinear Projection-based Binary Codes (BPBC)**[@Gong2013]: This is the state-of-the-art high-dimensional hashing method using bilinear transformation. It is considered the baseline method. We use algorithm parameter $d_1 = 128, d_2 = \text{ndim} / d_1$, and 50 iteration number.

Toy-data Experiment {#sec:toydata_experiment}
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![Top-10 NN retrieval results for 128-dimensional gaussian data. PRH(m, n, $\lambda$) indicates the proposed method with $m$-times basic isotropic rotation, $n$-times basic PCA rotation, and PCAT parameter $\lambda$ (Eq. (\[eq:pca\_rotation\])). Upper row is for the data with log-variance value of one. Lower row is for the data with log-variance of three (described in Section \[sec:toydata\_experiment\]). Abbreviated legends of plot (d) 