In the book "High-dimensional probability by Vershynin", page 111, in the proof of Johnson-Lindenstrauss Lemma, let $E$ be a random $m$-dimensional subspace in $\mathbb{R}^n$ uniformly distributed in the Grassmannian $G_{n,m}$, i.e,
$$E \sim Unif(G_{n,m}))$$.
Denote the orthogonal projection onto $E$ by $P$. Let $z\in \mathbb{R}^n$ be a fixed point such that $||z||_2=1$.
The book intuitively claim that instead of random projection $P$ acting on a fixed vector $z$, we consider a fixed projection $P$ acting on a random vector $z\sim Unif(S^{n-1})$. Then, the distribution of $||Pz||_2$ is unchanged.
I was wondering if there is a rigorous way to show the invariance of distribution. I thin we may use the rotation invariance property of $z$.
Best Answer
You can think of the Grassmannian as being some canonical $m$ dimensional subspace with a uniformly distributed random orthogonal transformation (i.e. it is a homogeneous space). To get this uniform distribution on the orthogonal group we use the Haar measure $\lambda$, which for any group $G$ acts like a uniform distribution on $G$. In particular, for compact groups, this measure is both left and right invariant. By this I mean that for any $g \in G$ and any measurable subset $A$ of $G$ we have $\lambda(gA) = \lambda(Ag) = \lambda(A)$. Moreover this means that if $M \sim \lambda$ then $gM \sim \lambda$. Using this on the two norm and with $G$ as the orthogonal group gives:
$||Pz||_2^2 = ||R D R^\top z ||_2^2 = ||DR^\top z||_2^2 = ||D x||_2^2$
where $R \sim \lambda$, $D$ is a fixed diagonal projection matrix and $x = R^\top z$ is uniform on the sphere. (There is a property called unimodularity that means that $R^\top \sim R$.)