Mixing
Upper Bound to an expected value
$begingroup$
I have a vector $x in mathbb{R}^d$ with norm one. I also have random noise vector $eta in mathbb{R}^d$ with norm $1/k$ taken at random from the sphere $S^{d-1}$ of radius 1/k, where k is a large natural number.
Let's now define the softmax function $sigma: mathbb{R}^d rightarrow mathbb{R}^d$ defined in the following way:
$sigma(x)_i = frac{e^{x_i}}{sum_j e^{x_j}}$
I need to compute an upper bound to the following expected value
$$ mathbb{E}[leftlVert sigma(x) - sigma(x+eta) rightrVert]$$
Any suggestions?
Thank you!
probability statistics norm
asked Jan 22 at 16:00
AlfredAlfred
357
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add a comment |
$begingroup$
I have a vector $x in mathbb{R}^d$ with norm one. I also have random noise vector $eta in mathbb{R}^d$ with norm $1/k$ taken at random from the sphere $S^{d-1}$ of radius 1/k, where k is a large natural number.
Let's now define the softmax function $sigma: mathbb{R}^d rightarrow mathbb{R}^d$ defined in the following way:
$sigma(x)_i = frac{e^{x_i}}{sum_j e^{x_j}}$
I need to compute an upper bound to the following expected value
$$ mathbb{E}[leftlVert sigma(x) - sigma(x+eta) rightrVert]$$
Any suggestions?
Thank you!
probability statistics norm
asked Jan 22 at 16:00
AlfredAlfred
357
$endgroup$
$begingroup$
edited, thanks!!
$endgroup$
– Alfred
Jan 22 at 16:26
add a comment |
$begingroup$
I have a vector $x in mathbb{R}^d$ with norm one. I also have random noise vector $eta in mathbb{R}^d$ with norm $1/k$ taken at random from the sphere $S^{d-1}$ of radius 1/k, where k is a large natural number.
Let's now define the softmax function $sigma: mathbb{R}^d rightarrow mathbb{R}^d$ defined in the following way:
$sigma(x)_i = frac{e^{x_i}}{sum_j e^{x_j}}$
I need to compute an upper bound to the following expected value
$$ mathbb{E}[leftlVert sigma(x) - sigma(x+eta) rightrVert]$$
Any suggestions?
Thank you!
probability statistics norm
asked Jan 22 at 16:00
AlfredAlfred
357
$endgroup$
I have a vector $x in mathbb{R}^d$ with norm one. I also have random noise vector $eta in mathbb{R}^d$ with norm $1/k$ taken at random from the sphere $S^{d-1}$ of radius 1/k, where k is a large natural number.
Let's now define the softmax function $sigma: mathbb{R}^d rightarrow mathbb{R}^d$ defined in the following way:
$sigma(x)_i = frac{e^{x_i}}{sum_j e^{x_j}}$
I need to compute an upper bound to the following expected value
$$ mathbb{E}[leftlVert sigma(x) - sigma(x+eta) rightrVert]$$
Any suggestions?
Thank you!
probability statistics norm
probability statistics norm
asked Jan 22 at 16:00
AlfredAlfred
357
asked Jan 22 at 16:00
AlfredAlfred
357
edited Jan 22 at 16:26
Alfred
asked Jan 22 at 16:00
AlfredAlfred
357
asked Jan 22 at 16:00
AlfredAlfred
357
asked Jan 22 at 16:00
AlfredAlfred
357
357
$begingroup$
edited, thanks!!
$endgroup$
– Alfred
Jan 22 at 16:26
add a comment |
$begingroup$
edited, thanks!!
$endgroup$
– Alfred
Jan 22 at 16:26
$begingroup$
edited, thanks!!
$endgroup$
– Alfred
Jan 22 at 16:26
$begingroup$
edited, thanks!!
$endgroup$
– Alfred
Jan 22 at 16:26
add a comment |
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$begingroup$
edited, thanks!!
$endgroup$
– Alfred
Jan 22 at 16:26