Andy Jones

Relationship between the multivariate normal, SVD, and Cholesky decomposition

Consider an $n\times p$ matrix $X$. Its singular value decomposition is $X = UDV^\top$.

Let’s reconstruct this a different way. Let $U = [u_1, \dots, u_n]^\top$ with

\[u_1, \dots, u_n \sim \mathcal{N}\left(0, \frac{1}{n} I_p\right).\]

Further, let $D = \text{diag}(d_1, \dots, d_p)$. Then

\[u_iD \sim \mathcal{N}\left(0, \frac{1}{n} D^2 \right).\]

Consider an orthogonal matrix $V = [v_1, \dots, v_p]$. Then

\[u_iDV^\top \sim \mathcal{N}\left(0, \frac{1}{n} V D^2 V^\top \right).\]

We can immediately notice that this is the $i$th sample of $X$, where $x_i = u_iDV^\top$. Furthermore, we have a decomposition of its covariance matrix

\[\Sigma = V D^2 V^\top.\]

Notice the relationship to the Cholesky decomposition:

\[V D^2 V^\top = V D D^\top V^\top = LL^\top\]

where $L = VD$. This coincides with a popular way to generate multivariate normal samples with covariance $\Sigma$, namely

\[x = LzL^\top=VDzD^\top V, ~~~ z\sim \mathcal{N}(0, I).\]

Furthermore, notice that given an observed data matrix $X \in \mathbb{R}^{n \times p}$, its covariance matrix can be completely described without the rotation matrix U,

\[X^\top X = VDU^\top UDV^\top = VD^2V = LL^\top.\]

In the context of the multivariate normal, this makes sense because the rows of $U$ have spherical covariance. Thus, we can arbitrarily rotate these samples about the origin and still yield the same covariance matrix. Concretely, define $\widetilde{U} = WU$ such that $W^\top W = I_p$ (i.e., let’s rotate the samples of $U$). Then

\[VD\widetilde{U}^\top \widetilde{U}DV^\top = VDU^\top W^\top WUDV^\top = VD^2V^\top,\]

which is the same as the covariance of $X$.