this method only determines the partition of the entire space to the set of small subspaces. Rotations in each subspaces are not under discussion. By contrast, our analysis can consider rotational optimality in the two-dimensional subspaces because we investigate the entropy of binary codes directly.

Another example is [@Jin2013]. The authors proposed two criteria, that is the “crossing sparse region” and the “balanced buckets”. The first criterion can be interpreted as quantization error minimization, and the second means quantizer entropy maximization. We think that it is possible to interpret many existing methods as such trade-off problem of quantization error minimization and entropy maximization.

Methods
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A binary hashing algorithm based on the above-discussed theory is developed as follows. We are going to have very sparse transformation matrices, which substantially decrease encoding cost.

Problem Statement
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The problem is to yield a linear transformation matrix $A$ in equation (\[eq:binhash\]). Most existing methods split the transformation into dimension reduction projection $W \in \mathbb{R}^{m \times n}$ and transformation in reduced space $Q \in \mathbb{R}^{m \times m}$. PCA is commonly used for reducing the number of dimension. However, a PCA transformation matrix is dense, it is difficult to get transformation and efficient encoding calculation 