gaussian random matrix transforms vectors into gaussian random vectors

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Proposition

Let $G \sim N (0, 1)^{\otimes d \times m}$ be a random matrix and $y \in R^{m}$ . Then

\text{Law}(Gy) = {\cal N}(0, \lvert \lvert y \rvert \rvert {#2} I_{d})

^statement

Proof

$G y$ is linear in $G$ . The entries of $G$ form a gaussian random vector, and so $G y$ must be Gaussian as well. Thus, it suffices to calculate the mean and covariance to find its law.

Since each entry of $G$ , call them $G_{i α}$ , is iid $N (0, 1)$ , we get

\begin{aligned} E (G y)_{i} & = E [\sum_{α = 1}^{m} G_{i α} y_{α}] \\ = 0 \end{aligned}

And for the variance

\begin{align} \text{Cov}(\,(Gy)_{i}, (Gy)_{j}\,) &= \mathbb{E}[\,(Gy)_{i}(Gy)_{j}\,] \\ &= \mathbb{E}\left( \left( \sum_{\alpha=1}^m G_{i\alpha} y_{\alpha} \right) \left( \sum_{\beta=1}^m G_{j\beta}y_{\beta} \right) \right) \\ &= \sum_{\alpha,\beta=1}^m y_{\alpha}y_{\beta} \cdot \mathbb{E} [G_{i\alpha} G_{j\beta}] \\ &= \begin{cases} 0 & i \neq j \\ \lvert \lvert y \rvert \rvert {#2} & i=j \end{cases} \\ \implies \text{Cov}(Gy,Gy) &= \lvert \lvert y \rvert \rvert {#2} I_{d} \end{align}

\begin{matrix} ◼ \end{matrix}

^proof-0

Above, we found the expectation and covariance in terms of the entry-wise products and sums. We can also do the calculation in terms of matrices.

Proof (matrices)

Expectation
If there is only one random matrix involved, we can write

E [G y] = E [G] y (= 0)^{(*)}

$_^{(*)}$ In terms of our same setting above. This is a simple result of linearity of expectation.

Covariance
If $g_{1}, \dots, g_{m} \in R^{d}$ are the columns of $G$ so that each $g_{α} \sim N (0, I_{d})$ iid, then we see that

\begin{align} \text{Cov}(Gy) &= \mathbb{E}\left[ (Gy)(Gy)^{\intercal} \right] \\ &= \mathbb{E}\left[ \left( \sum_{\alpha=1}^m y_{\alpha} g_{\alpha} \right)\left( \sum_{\beta=1}^m y_{\beta} g_{\beta}^{\intercal} \right) \right] \\ &= \sum_{\alpha,\beta=1}^m y_{\alpha} y_{\beta} \cdot \mathbb{E}\left[ g_{\alpha}g_{\beta}^{\intercal} \right] \\ (*) &= \sum_{\alpha=1}^m y_{\alpha}^2 \cdot I_{d} \\ &= \lvert \lvert y \rvert \rvert {#2} I_{d} \end{align}

Where again $(*)$ is in terms of the setting above. In this line we use the law of the $g_{α}$ .

\begin{matrix} ◼ \end{matrix}

References

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gaussian random matrix transforms vectors into gaussian random vectors

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