Gaussian distribution is isotropic?

I was in a seminar today and the lecturer said that the gaussian distribution is isotropic. What does it mean for a distribution to be isotropic? It seems like he is using this property for the pseudo-independence of vectors where each entry is sampled from the normal distribution.

Answer

TLDR: An isotropic gaussian is one where the covariance matrix is represented by the simplified matrix Σ=σ2I.

Some motivations:

Consider the traditional gaussian distribution:

N(μ,Σ)

where μ is the mean and Σ is the covariance matrix.

Consider how the number of free parameters in this Gaussian grows as the number of dimensions grows.

μ will have a linear growth.
Σ will have a quadratic growth!

This quadratic growth can be very computationally expensive, so Σ is often restricted as Σ=σ2I where σ2I is a scalar variance multiplied by an identity matrix.

Note that this results in Σ where all dimensions are independent and where the variance of each dimension is the same. So the gaussian will be circular/spherical.

Disclaimer: Not a mathematician, and I only just learned about this so may be missing some things 🙂

Hope that helps!

Attribution
Source : Link , Question Author : Astaboom , Answer Author : suharshs

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