three times makes then octuple isotropic (Fig. \[illustration:basic\_rotation\]), and so on. Finally, applying the transformation $\lceil \log_2 n \rceil$ times, we have completely isotropic variances. [^1] Finally developed transformation is a product of sparse matrices. It has $2n\lceil \log_2 n \rceil$ fill-ins in total. This factorized form is highly sparse especially for high dimensions. It is possible to use standard sparse matrix data structure. Memory usage and computational cost can be substantially decreased.

Trade-off between Quantization Error and Entropy
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A factorized sparse transformation that makes variances completely isotropic was obtained as described in the preceding section. However, in Section \[sec:theoretical\_background\], a trade-off relationship between isotropy (quantization error minimization) and entropy maximization was revealed. Since entropy reduction degrades retrieval accuracy, a balance between isotropy and entropy should be kept. Accordingly, two methods for such balancing are proposed hereafter. The first one is the simpler one and does not increase the number of fill-ins. The second one is using additional sparse rotation matrices. It increases the number of fill-ins, but has better accuracy than the first one in some cases.

### PCA tilting (PCAT)

In the first method, each pairwise rotation is “tilted” from the isotropic angle to the PCA angle 