isotropic gaussian

Isotropic gaussian

A isotropic gaussian is a multivariate distribution where the covariance matrix $\Sigma$ is diagonal and has the same variance across all dimensions:

$$\Sigma ={\sigma }^{2}I$$

Here, $I$ is the identity matrix and ${\sigma }^2$ is the variance.
The distribution is circular / spherical.
Dimensions / variables are independent.

The isotropic gaussian is a factored gaussian with fixed varriance

$$\mathcal{N}(\mu ,\Sigma )=\prod _{i=1}^n\mathcal{N}({\mu }_i,{\sigma }^2)$$

Isotropic gaussians are spherical distributions

Isotropic distribution $⟹$ $\text{Cov}(X)=cI$ (constrains only second moments, )
Spherical distribution (centered) $⟹$ $QX\sim X$ (constrains all moments, stronger)
So in general, isotropic $\text{centernot}⟹$ spherical, because the isotropic property only constrains the second moments, while the spherical property constrains all moments.

But since gaussians are fully determined by their first two moments, isotropic gaussians are also spherical: $Q\Sigma Q^T=Q⟹\Sigma ={\sigma }^{2}I$, for all orthogonal matrices $Q$.
See also: isotropic distribution

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Isotropic gaussians are rotation invariant around their mean. E.g. standard normal distribution: If $X\sim \mathcal{N}(0,I)$ and $Q$ is an orthogonal matrix, then $QX\sim \mathcal{N}(0,I)$.

Standard normal distribution is an isotropic gaussian

Let $f_X(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$ and $f_Y(y)=\frac{1}{\sqrt{2\pi }}{e}^{-y^2/2}$ be two independent standard normal distributions.
Then the joint distribution $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ is an isotropic gaussian:

$$f_{X,Y}(x,y)=\frac{1}{2\pi }{e}^{-(x^2+y^2)/2}$$

$(x^2+y^2)$ is the squared distance from the origin, so the density is constant on circles around the origin.