\sigma_{22}$. It is proved that the Isotropic Hashing [@Kong2012] is quantization-error-minimizing hashing for gaussian distribution. We can see the isotropy as a measure of quantization error.

Entropy and Quantization Error {#sec:ent_qerr}
------------------------------

![Schematic illustration of two-dimensional gaussian distribution. An ellipse describes the shape of distribution. **Left:** eigenvalue and angle parameterization(Eq. (\[eq:r-theta\_representation\_of\_2d\_gauss\])). $\lambda_1$ and $\lambda_2$ are the eigenvalues of variance-covariance matrix. **Right:** rotation angle for Isotropic or PCA transformation. $\theta_{\text{iso}}$ and $\theta_{\text{pca}}$ are given in (\[eq:iso\_rotation\]), (\[eq:pca\_rotation\]). \[fig:ellipse\]](./eps/gauss_0.pdf "fig:") ![Schematic illustration of two-dimensional gaussian distribution. An ellipse describes the shape of distribution. **Left:** eigenvalue and angle parameterization(Eq. (\[eq:r-theta\_representation\_of\_2d\_gauss\])). $\lambda_1$ and $\lambda_2$ are the eigenvalues of variance-covariance matrix. **Right:** rotation angle for Isotropic or PCA transformation. $\theta_{\text{iso}}$ and $\theta_{\text{pca}}$ are given in (\[eq:iso\_rotation\]), (\[eq:pca\_rotation\]). \[fig:ellipse\]](./eps/gauss_1.pdf "fig:")

![Quantization error and entropy of binary code for $\lambda_1 = 2, \lambda_2 = 1$. The x-axis is angle $\theta$ given in equation (\[eq:r-theta\_representation\_of\_2d\_gauss\]). From the symmetry of gaussian distribution, it is enough to consider range $\theta \in [0, \pi / 2]$ \[fig:qerr\_entropy\]](./eps/qerr.pdf "fig:") ![Quantization error and entropy of binary code for $\lambda_1 = 2, \lambda_2 = 1$. The x-axis is angle $\theta$ given in equation (\[eq:r-theta\_representation\_of\_2d\_gauss\]). From the symmetry of gaussian distribution, it is enough to 