Isotropic vs. Anisotropic

Posted on Oct 24, 2019

Tags: Covariance Function

I came across isotropic and anisotropic covariance functions in kjytay’s blog, and then I found more materials, chapter 4 from the book Gaussian Processes for Machine Learning, via the reference in StackExchange: What is an isotropic (spherical) covariance matrix?.

Firstly of all, a covariance function must be symmetric and positive semi-definite.

A stationary covariance function is a function of $\x - \x’$. Thus it is invariant to translations in the input space.

For a stochastic process,

• weakly stationary: a process has constant mean and whose covariance function is invariant to translations
• strictly stationary: all of its finite dimensional distributions are invariant to translations.

If further the covariance function is a function only of $\vert \x -\x’\vert$ then it is called isotropic. These are also known as radial basis functions (RBFs).

A covariance is said to be anisotropic if it is not isotropic, such as setting

\[r^2(\x,\x') = (\x - \x')^TM(\x - \x')\]

for some positive semidefinite $M$.