Magnitude of Gaussian Random Projection
up vote
0
down vote
favorite
Let $X= [X_1 dots X_n]$ be a $ptimes n$ random matrix, where $X_1, dots, X_n$ are i.i.d and follow $N(0,Sigma)$, and let $R$ be a random unit vector uniformly distributed on $S^{p-1}$ independently with $X$. I am interested in the distribution of $|X^top R|^2 = sum_{i=1}^n |X_i^top R|^2$.
An easy case to start with would be $Sigma = I_p$. Given any unit vector $r$, $X_i^top rsim N(0,1)$, so $|X^top R|^2sim chi^2_n$. However, this observation does not apply to general $Sigma$.
Here is another simple observation. Without loss of generality, we can always assume $Sigma$ is a diagonal matrix, since we can rotate $X_i$ and $R$ without changing the distribution of $|X^top R|^2$.
Any help would be greatly appreciated.
probability probability-distributions
asked Nov 14 at 19:51
Mayu
63
add a comment |
up vote
0
down vote
favorite
Let $X= [X_1 dots X_n]$ be a $ptimes n$ random matrix, where $X_1, dots, X_n$ are i.i.d and follow $N(0,Sigma)$, and let $R$ be a random unit vector uniformly distributed on $S^{p-1}$ independently with $X$. I am interested in the distribution of $|X^top R|^2 = sum_{i=1}^n |X_i^top R|^2$.
An easy case to start with would be $Sigma = I_p$. Given any unit vector $r$, $X_i^top rsim N(0,1)$, so $|X^top R|^2sim chi^2_n$. However, this observation does not apply to general $Sigma$.
Here is another simple observation. Without loss of generality, we can always assume $Sigma$ is a diagonal matrix, since we can rotate $X_i$ and $R$ without changing the distribution of $|X^top R|^2$.
Any help would be greatly appreciated.
probability probability-distributions
asked Nov 14 at 19:51
Mayu
63
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X= [X_1 dots X_n]$ be a $ptimes n$ random matrix, where $X_1, dots, X_n$ are i.i.d and follow $N(0,Sigma)$, and let $R$ be a random unit vector uniformly distributed on $S^{p-1}$ independently with $X$. I am interested in the distribution of $|X^top R|^2 = sum_{i=1}^n |X_i^top R|^2$.
An easy case to start with would be $Sigma = I_p$. Given any unit vector $r$, $X_i^top rsim N(0,1)$, so $|X^top R|^2sim chi^2_n$. However, this observation does not apply to general $Sigma$.
Here is another simple observation. Without loss of generality, we can always assume $Sigma$ is a diagonal matrix, since we can rotate $X_i$ and $R$ without changing the distribution of $|X^top R|^2$.
Any help would be greatly appreciated.
probability probability-distributions
asked Nov 14 at 19:51
Mayu
63
Let $X= [X_1 dots X_n]$ be a $ptimes n$ random matrix, where $X_1, dots, X_n$ are i.i.d and follow $N(0,Sigma)$, and let $R$ be a random unit vector uniformly distributed on $S^{p-1}$ independently with $X$. I am interested in the distribution of $|X^top R|^2 = sum_{i=1}^n |X_i^top R|^2$.
An easy case to start with would be $Sigma = I_p$. Given any unit vector $r$, $X_i^top rsim N(0,1)$, so $|X^top R|^2sim chi^2_n$. However, this observation does not apply to general $Sigma$.
Here is another simple observation. Without loss of generality, we can always assume $Sigma$ is a diagonal matrix, since we can rotate $X_i$ and $R$ without changing the distribution of $|X^top R|^2$.
Any help would be greatly appreciated.
probability probability-distributions
probability probability-distributions
asked Nov 14 at 19:51
Mayu
63
asked Nov 14 at 19:51
Mayu
63
asked Nov 14 at 19:51
Mayu
63
asked Nov 14 at 19:51
Mayu
63
asked Nov 14 at 19:51
Mayu
63
63
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2998748%2fmagnitude-of-gaussian-random-projection%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
