in highly dimensional cases. In this paper, dimension reduction is not treated. It is thus assumed that the number of dimensions of the original feature vector and the encoded binary vector are the same, and $A = Q \in \mathbb{R}^{n \times n}$ only is treated. Dimension reduction can be done in the similar way as we are going to discuss below, but detailed study is a future work.

![Schematic illustration of basic isotropic rotations (in 8-dimensions). **Upper**: Structure of the transformation matrix. Non-zero matrix elements are filled with gray color. Sorting matrix is a permutation matrix, which sort variances in descending order. Rotation is done for pair of largest with smallest variance dimensions. Basic rotation is continuously applied $\log_2 n$ times for n dimensions. **Lower**: Behavior of variances under continuous multiplication of basic isotropic rotation. The graphs are sorted variances under sequential multiplication. The rightmost graph is the (sorted) initial state. One basic isotropic rotation makes variances isotropic by pairs. And it is exponentially transformed to globally isotropic state with continuous application of basic rotation. \[illustration:basic\_rotation\]](./eps/mat_var.pdf)

Sequential Pairwise Isotropic Rotation {#sec:completely_isotropic}
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First, we derive transformation that makes variances completely isotropic. We can get very sparse isotropic transformation matrix with 