Mixing
Existence of a sub-matrix capturing most of the power of the whole matrix
$begingroup$
I would like to verify the following argument:
Consider an $M times N$ complex matrix $mathbf{X}$ with $M le N$. There exists a sub-matrix $tilde{mathbf{X}}$ containing $M$ columns of $mathbf{X}$ such that
begin{align}
logdet(mathbf{I}_M+mathbf{X}mathbf{X}^H) - logdet(mathbf{I}_M + tilde{mathbf{X}} tilde{mathbf{X}}^H) = O(1),
end{align}
where $O(1)$ denotes a function whose value is bounded by constants as $|mathbf{X}|_F^2 to infty$.
A sufficient condition for the argument to be true is that $$tilde{mathbf{X}} tilde{mathbf{X}}^H succeq alpha mathbf{X} mathbf{X}^H$$ for some sub-matrix $tilde{mathbf{X}}$ and some possitive constant $alpha$. I think this is correct but do not know how to show it.
Does anyone have an idea how to prove it or beware of any known result stating the argument?
inequality determinant positive-definite
asked Jan 8 at 17:51
Khac-Hoang NgoKhac-Hoang Ngo
113
$endgroup$
add a comment |
$begingroup$
I would like to verify the following argument:
Consider an $M times N$ complex matrix $mathbf{X}$ with $M le N$. There exists a sub-matrix $tilde{mathbf{X}}$ containing $M$ columns of $mathbf{X}$ such that
begin{align}
logdet(mathbf{I}_M+mathbf{X}mathbf{X}^H) - logdet(mathbf{I}_M + tilde{mathbf{X}} tilde{mathbf{X}}^H) = O(1),
end{align}
where $O(1)$ denotes a function whose value is bounded by constants as $|mathbf{X}|_F^2 to infty$.
A sufficient condition for the argument to be true is that $$tilde{mathbf{X}} tilde{mathbf{X}}^H succeq alpha mathbf{X} mathbf{X}^H$$ for some sub-matrix $tilde{mathbf{X}}$ and some possitive constant $alpha$. I think this is correct but do not know how to show it.
Does anyone have an idea how to prove it or beware of any known result stating the argument?
inequality determinant positive-definite
asked Jan 8 at 17:51
Khac-Hoang NgoKhac-Hoang Ngo
113
$endgroup$
add a comment |
$begingroup$
I would like to verify the following argument:
Consider an $M times N$ complex matrix $mathbf{X}$ with $M le N$. There exists a sub-matrix $tilde{mathbf{X}}$ containing $M$ columns of $mathbf{X}$ such that
begin{align}
logdet(mathbf{I}_M+mathbf{X}mathbf{X}^H) - logdet(mathbf{I}_M + tilde{mathbf{X}} tilde{mathbf{X}}^H) = O(1),
end{align}
where $O(1)$ denotes a function whose value is bounded by constants as $|mathbf{X}|_F^2 to infty$.
A sufficient condition for the argument to be true is that $$tilde{mathbf{X}} tilde{mathbf{X}}^H succeq alpha mathbf{X} mathbf{X}^H$$ for some sub-matrix $tilde{mathbf{X}}$ and some possitive constant $alpha$. I think this is correct but do not know how to show it.
Does anyone have an idea how to prove it or beware of any known result stating the argument?
inequality determinant positive-definite
asked Jan 8 at 17:51
Khac-Hoang NgoKhac-Hoang Ngo
113
$endgroup$
I would like to verify the following argument:
Consider an $M times N$ complex matrix $mathbf{X}$ with $M le N$. There exists a sub-matrix $tilde{mathbf{X}}$ containing $M$ columns of $mathbf{X}$ such that
begin{align}
logdet(mathbf{I}_M+mathbf{X}mathbf{X}^H) - logdet(mathbf{I}_M + tilde{mathbf{X}} tilde{mathbf{X}}^H) = O(1),
end{align}
where $O(1)$ denotes a function whose value is bounded by constants as $|mathbf{X}|_F^2 to infty$.
A sufficient condition for the argument to be true is that $$tilde{mathbf{X}} tilde{mathbf{X}}^H succeq alpha mathbf{X} mathbf{X}^H$$ for some sub-matrix $tilde{mathbf{X}}$ and some possitive constant $alpha$. I think this is correct but do not know how to show it.
Does anyone have an idea how to prove it or beware of any known result stating the argument?
inequality determinant positive-definite
inequality determinant positive-definite
asked Jan 8 at 17:51
Khac-Hoang NgoKhac-Hoang Ngo
113
asked Jan 8 at 17:51
Khac-Hoang NgoKhac-Hoang Ngo
113
asked Jan 8 at 17:51
Khac-Hoang NgoKhac-Hoang Ngo
113
asked Jan 8 at 17:51
Khac-Hoang NgoKhac-Hoang Ngo
113
asked Jan 8 at 17:51
Khac-Hoang NgoKhac-Hoang Ngo
113
113
add a comment |
add a comment |
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