Understanding the proof of Johnson-Lindenstrauss (JL) lemma I have some questions about proof of Johnson-Lindenstrauss (JL) lemma. I appreciate any responses in advance.


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*It is stated in the following paper: An elementary proof of JL lemma
we argued: "Hence the aim is to estimate the length of a unit vector in $\mathbb{R}^{d}$ when it is projected onto a random k-dimensional subspace. However, this length has the same distribution as the length of a random unit vector projected down onto a fixed k-dimensional subspace." I have no clue why this is the case.

*In the JL context by projection of a vector we mean $u=\mathbf{A}v$ where $\mathbf{A} \in \mathbb{R}^{k \times d}$, $v \in \mathbb{R}^{d}$ and $u \in \mathbb{R}^{k}$, right? If the answer is yes, how about the case that instead of $\mathbf{A}$ we have access to the projection operator (matrix) $\mathbf{P} \in \mathbb{R}^{d \times d}$ to the k-dimensional subspace. What can we say about distribution of $||\mathbf{P}v||_{2}$, that is, distribution of norm after projection? Is the result equivalent to the previous (common) case? If yes, why?
Again, any comment, answer, or reply is appreciated.
 A: For the first question, I received the following answer from a friend:
"This is how I understand it. This is not rigorous, but it gives the intuition.
Consider the case of projecting a unit vector $u$ in $\mathbb{R}^{d}$ onto a random $k$-dimensional subspace. One way of selecting a random $k$−dimensional subspace is through randomly selecting $k$ orthonormal vectors that form the basis of the subspace. Lets assume we randomly selected these $k$ vectors. Now lets rotate the coordinate axis such that our $k$ vectors become the first $k$ standard basis. This operation doesn't change the length of projection of $u$ onto the subspace.
Now consider the point $u$ in this rotated space. Since we randomly selected our $k$ vectors (independent of $u$), we expect $u$ to be uniformly distributed in the rotated space. So intuitively, the length of $u$ projected onto a random subspace has the same distribution as the length of a random unit vector projected onto a fixed $k$-dimensional subspace."
For the second question, I was able to verify by means of simulation that expected value of $||A^{T}u||_2$ is approximately equal to expected value of $||Pu||_2$ where $A$ is a basis (not necessarily orthogonal) for the $k$-dimensional subspace and $P=AA^{\dagger}$ is the orthogonal projection operator onto the $k$-dimensional subspace. However I cannot provide a proof for this argument yet.
