bound the expectation of max of sets of functions
0
$begingroup$
Let $Xinmathbb{R}^{ntimes d}$ be a random matrix, and ${m_k}_{k=1}^K, m_k in mathbb{R}^{d}$ be a fixed set of matrices. I was wondering, what is the expectation (or upper bound of the expectation) of
$mathbb{E}max_{k}{||Xm_k||_2^2}$
Any comment or hint will be appreciated!
probability probability-theory expected-value
asked Jan 16 at 23:13
user3138073user3138073
1858
$endgroup$
add a comment |
0
$begingroup$
Let $Xinmathbb{R}^{ntimes d}$ be a random matrix, and ${m_k}_{k=1}^K, m_k in mathbb{R}^{d}$ be a fixed set of matrices. I was wondering, what is the expectation (or upper bound of the expectation) of
$mathbb{E}max_{k}{||Xm_k||_2^2}$
Any comment or hint will be appreciated!
probability probability-theory expected-value
asked Jan 16 at 23:13
user3138073user3138073
1858
$endgroup$
$begingroup$
For an upper bound, notice that $|M_kX|_2^2 leq |M_k|^2|X|_2^2$ (where our first norm is the matrix norm), so $max_k{|M_kX|^2_2} leq max_k{|M_k|^2}|X|_2^2$, and by the linearity of expectation, $mathbb{E}max_k{|M_kX|_2^2leq max_k{|M_k|^2}mathbb{E}|X|_2^2$.
$endgroup$
– user3482749
Jan 16 at 23:18
$begingroup$
Thanks for the comments! I wrote my question in a wrong way (now corrected), but I think your method is still applicable. Thanks a lot!
$endgroup$
– user3138073
Jan 16 at 23:31
add a comment |
0
0
0
$begingroup$
Let $Xinmathbb{R}^{ntimes d}$ be a random matrix, and ${m_k}_{k=1}^K, m_k in mathbb{R}^{d}$ be a fixed set of matrices. I was wondering, what is the expectation (or upper bound of the expectation) of
$mathbb{E}max_{k}{||Xm_k||_2^2}$
Any comment or hint will be appreciated!
probability probability-theory expected-value
asked Jan 16 at 23:13
user3138073user3138073
1858
$endgroup$
Let $Xinmathbb{R}^{ntimes d}$ be a random matrix, and ${m_k}_{k=1}^K, m_k in mathbb{R}^{d}$ be a fixed set of matrices. I was wondering, what is the expectation (or upper bound of the expectation) of
$mathbb{E}max_{k}{||Xm_k||_2^2}$
Any comment or hint will be appreciated!
probability probability-theory expected-value
probability probability-theory expected-value
asked Jan 16 at 23:13
user3138073user3138073
1858
asked Jan 16 at 23:13
user3138073user3138073
1858
edited Jan 16 at 23:30
user3138073
asked Jan 16 at 23:13
user3138073user3138073
1858
asked Jan 16 at 23:13
user3138073user3138073
1858
asked Jan 16 at 23:13
user3138073user3138073
1858
1858
$begingroup$
For an upper bound, notice that $|M_kX|_2^2 leq |M_k|^2|X|_2^2$ (where our first norm is the matrix norm), so $max_k{|M_kX|^2_2} leq max_k{|M_k|^2}|X|_2^2$, and by the linearity of expectation, $mathbb{E}max_k{|M_kX|_2^2leq max_k{|M_k|^2}mathbb{E}|X|_2^2$.
$endgroup$
– user3482749
Jan 16 at 23:18
$begingroup$
Thanks for the comments! I wrote my question in a wrong way (now corrected), but I think your method is still applicable. Thanks a lot!
$endgroup$
– user3138073
Jan 16 at 23:31
add a comment |
$begingroup$
For an upper bound, notice that $|M_kX|_2^2 leq |M_k|^2|X|_2^2$ (where our first norm is the matrix norm), so $max_k{|M_kX|^2_2} leq max_k{|M_k|^2}|X|_2^2$, and by the linearity of expectation, $mathbb{E}max_k{|M_kX|_2^2leq max_k{|M_k|^2}mathbb{E}|X|_2^2$.
$endgroup$
– user3482749
Jan 16 at 23:18
$begingroup$
Thanks for the comments! I wrote my question in a wrong way (now corrected), but I think your method is still applicable. Thanks a lot!
$endgroup$
– user3138073
Jan 16 at 23:31
$begingroup$
For an upper bound, notice that $|M_kX|_2^2 leq |M_k|^2|X|_2^2$ (where our first norm is the matrix norm), so $max_k{|M_kX|^2_2} leq max_k{|M_k|^2}|X|_2^2$, and by the linearity of expectation, $mathbb{E}max_k{|M_kX|_2^2leq max_k{|M_k|^2}mathbb{E}|X|_2^2$.
$endgroup$
– user3482749
Jan 16 at 23:18
$begingroup$
For an upper bound, notice that $|M_kX|_2^2 leq |M_k|^2|X|_2^2$ (where our first norm is the matrix norm), so $max_k{|M_kX|^2_2} leq max_k{|M_k|^2}|X|_2^2$, and by the linearity of expectation, $mathbb{E}max_k{|M_kX|_2^2leq max_k{|M_k|^2}mathbb{E}|X|_2^2$.
$endgroup$
– user3482749
Jan 16 at 23:18
$begingroup$
Thanks for the comments! I wrote my question in a wrong way (now corrected), but I think your method is still applicable. Thanks a lot!
$endgroup$
– user3138073
Jan 16 at 23:31
$begingroup$
Thanks for the comments! I wrote my question in a wrong way (now corrected), but I think your method is still applicable. Thanks a lot!
$endgroup$
– user3138073
Jan 16 at 23:31
add a comment |
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$begingroup$
For an upper bound, notice that $|M_kX|_2^2 leq |M_k|^2|X|_2^2$ (where our first norm is the matrix norm), so $max_k{|M_kX|^2_2} leq max_k{|M_k|^2}|X|_2^2$, and by the linearity of expectation, $mathbb{E}max_k{|M_kX|_2^2leq max_k{|M_k|^2}mathbb{E}|X|_2^2$.
$endgroup$
– user3482749
Jan 16 at 23:18
$begingroup$
Thanks for the comments! I wrote my question in a wrong way (now corrected), but I think your method is still applicable. Thanks a lot!
$endgroup$
– user3138073
Jan 16 at 23:31