Picking a random vector from Spherical Gaussian Distribution

Question

I was recently reading a research paper on Probabilistic Matrix Factorization and the authors were picking a random vector from a spherical gaussian distribution

ui ∼N (0,λ−1IK).

Where lambda is a regularization parameter and IK is Kth dimensional identity matrix. They provided no details on how this is actually done.

Can any one point me in the right direction for achieving that?

Answer 1

Any mean zero Gaussian random vector $\newcommand{\bR}{\mathbb{R}}$ on $ X=(X_1,\dotsc, X_n)\in\bR^n$ is uniquely determined by its covariance matrix $C$. This is a symmetric $n\times n$ matrix with entries $\newcommand{\bE}{\mathbb{E}}$

$$ C_{ij}=\bE[X_iX_j],\;\;1\leq i,j\leq n, $$

$\bE=$ expectation. The matrix $C$ is positive semidefinite, i.e., $\newcommand{\bx}{\boldsymbol{x}}$ $\newcommand{\by}{\boldsymbol{y}}$

$$ (C\bx,\bx)\geq 0,\;\;\forall \bx\in\bR^n. $$

To simulate (sample) such a random vector proceed as follows.

1. Compute the square root of $C$.This is the unique symmetric positive definite $n\times n$ matrix $A$ such that $A^2=C$.
2. Generate (simulate) $n$ independent standard normal random variables $Y_1,\dotsc, Y_n$. Denote by $Y$ the random vector $(Y_1,\dotsc, Y_n)$.
3. The random vector $AY$ is Gaussian with covariance matrix $C$.

Answer 2

Since this is a "spherical" Gaussian Distribution there is no interaction between the dimensions. This can also be seen from the fact that the second parameter of N contains the identity matrix, so all the off-diagonal values (corresponding to correlations between dimensions) are 0. (I do not understand the meaning of parameter lambda here though) In that case, if the Gaussian distribution is really spherical, you can just sample each dimension separately for each vector.