Random projection of a fixed point
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 $zin mathbb{R}^n$ 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 $zsim Unif(S^{n-1})$. 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 $z$.
linear-algebra probability probability-theory rotations projection
asked Jan 3 at 22:14
S_Alex
1159
add a comment |
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 $zin mathbb{R}^n$ 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 $zsim Unif(S^{n-1})$. 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 $z$.
linear-algebra probability probability-theory rotations projection
asked Jan 3 at 22:14
S_Alex
1159
add a comment |
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 $zin mathbb{R}^n$ 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 $zsim Unif(S^{n-1})$. 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 $z$.
linear-algebra probability probability-theory rotations projection
asked Jan 3 at 22:14
S_Alex
1159
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 $zin mathbb{R}^n$ 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 $zsim Unif(S^{n-1})$. 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 $z$.
linear-algebra probability probability-theory rotations projection
linear-algebra probability probability-theory rotations projection
asked Jan 3 at 22:14
S_Alex
1159
asked Jan 3 at 22:14
S_Alex
1159
edited Jan 3 at 22:34
asked Jan 3 at 22:14
S_Alex
1159
asked Jan 3 at 22:14
S_Alex
1159
asked Jan 3 at 22:14
S_Alex
1159
1159
add a comment |
add a comment |
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