Differentiation of a vector
$begingroup$
I have a matrix $A$ which is $n$ by $m$ and a vector $b$ which is $m$ by 1. I have the following expression:
$frac{delta}{delta b} mathbf{1}^{T}f(Ab)$
Where $mathbf{1}$ is a vector of ones which is $n$ by $1$, $f$ is a function which is applied elementwise and $^{T}$ represents transpose.
How can I express this just in terms of $f'$, $A$ and $b$? And does anyone have any resources on vector differentiation that could help me.
matrices derivatives
asked Jan 11 at 23:36
JDoe2JDoe2
896
$endgroup$
add a comment |
$begingroup$
I have a matrix $A$ which is $n$ by $m$ and a vector $b$ which is $m$ by 1. I have the following expression:
$frac{delta}{delta b} mathbf{1}^{T}f(Ab)$
Where $mathbf{1}$ is a vector of ones which is $n$ by $1$, $f$ is a function which is applied elementwise and $^{T}$ represents transpose.
How can I express this just in terms of $f'$, $A$ and $b$? And does anyone have any resources on vector differentiation that could help me.
matrices derivatives
asked Jan 11 at 23:36
JDoe2JDoe2
896
$endgroup$
add a comment |
$begingroup$
I have a matrix $A$ which is $n$ by $m$ and a vector $b$ which is $m$ by 1. I have the following expression:
$frac{delta}{delta b} mathbf{1}^{T}f(Ab)$
Where $mathbf{1}$ is a vector of ones which is $n$ by $1$, $f$ is a function which is applied elementwise and $^{T}$ represents transpose.
How can I express this just in terms of $f'$, $A$ and $b$? And does anyone have any resources on vector differentiation that could help me.
matrices derivatives
asked Jan 11 at 23:36
JDoe2JDoe2
896
$endgroup$
I have a matrix $A$ which is $n$ by $m$ and a vector $b$ which is $m$ by 1. I have the following expression:
$frac{delta}{delta b} mathbf{1}^{T}f(Ab)$
Where $mathbf{1}$ is a vector of ones which is $n$ by $1$, $f$ is a function which is applied elementwise and $^{T}$ represents transpose.
How can I express this just in terms of $f'$, $A$ and $b$? And does anyone have any resources on vector differentiation that could help me.
matrices derivatives
matrices derivatives
asked Jan 11 at 23:36
JDoe2JDoe2
896
asked Jan 11 at 23:36
JDoe2JDoe2
896
asked Jan 11 at 23:36
JDoe2JDoe2
896
asked Jan 11 at 23:36
JDoe2JDoe2
896
asked Jan 11 at 23:36
JDoe2JDoe2
896
896
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
Let $y=Ab,,$ then the differential of the elementwise function $f(y)$ is given by
$$eqalign{
df &= f'odot dy cr
}$$ where $odot$ is the elementwise/Hadamard product.
Now find the differential and gradient of the cost function.
$$eqalign{
phi &= 1:f cr
dphi &= 1:df = 1:(f'odot dy) = f':dy = f':A,db = A^Tf':db cr
frac{partialphi}{partial b} &= A^Tf' cr
}$$ where : is the trace/Frobenius product, i.e. $,A:B={rm Tr}(A^TB)$
answered Jan 12 at 3:06
greggreg
8,7751824
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add a comment |
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1 Answer
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oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Let $y=Ab,,$ then the differential of the elementwise function $f(y)$ is given by
$$eqalign{
df &= f'odot dy cr
}$$ where $odot$ is the elementwise/Hadamard product.
Now find the differential and gradient of the cost function.
$$eqalign{
phi &= 1:f cr
dphi &= 1:df = 1:(f'odot dy) = f':dy = f':A,db = A^Tf':db cr
frac{partialphi}{partial b} &= A^Tf' cr
}$$ where : is the trace/Frobenius product, i.e. $,A:B={rm Tr}(A^TB)$
answered Jan 12 at 3:06
greggreg
8,7751824
$endgroup$
add a comment |
$begingroup$
Let $y=Ab,,$ then the differential of the elementwise function $f(y)$ is given by
$$eqalign{
df &= f'odot dy cr
}$$ where $odot$ is the elementwise/Hadamard product.
Now find the differential and gradient of the cost function.
$$eqalign{
phi &= 1:f cr
dphi &= 1:df = 1:(f'odot dy) = f':dy = f':A,db = A^Tf':db cr
frac{partialphi}{partial b} &= A^Tf' cr
}$$ where : is the trace/Frobenius product, i.e. $,A:B={rm Tr}(A^TB)$
answered Jan 12 at 3:06
greggreg
8,7751824
$endgroup$
add a comment |
$begingroup$
Let $y=Ab,,$ then the differential of the elementwise function $f(y)$ is given by
$$eqalign{
df &= f'odot dy cr
}$$ where $odot$ is the elementwise/Hadamard product.
Now find the differential and gradient of the cost function.
$$eqalign{
phi &= 1:f cr
dphi &= 1:df = 1:(f'odot dy) = f':dy = f':A,db = A^Tf':db cr
frac{partialphi}{partial b} &= A^Tf' cr
}$$ where : is the trace/Frobenius product, i.e. $,A:B={rm Tr}(A^TB)$
answered Jan 12 at 3:06
greggreg
8,7751824
$endgroup$
Let $y=Ab,,$ then the differential of the elementwise function $f(y)$ is given by
$$eqalign{
df &= f'odot dy cr
}$$ where $odot$ is the elementwise/Hadamard product.
Now find the differential and gradient of the cost function.
$$eqalign{
phi &= 1:f cr
dphi &= 1:df = 1:(f'odot dy) = f':dy = f':A,db = A^Tf':db cr
frac{partialphi}{partial b} &= A^Tf' cr
}$$ where : is the trace/Frobenius product, i.e. $,A:B={rm Tr}(A^TB)$
answered Jan 12 at 3:06
greggreg
8,7751824
answered Jan 12 at 3:06
greggreg
8,7751824
answered Jan 12 at 3:06
greggreg
8,7751824
answered Jan 12 at 3:06
greggreg
8,7751824
8,7751824
add a comment |
add a comment |
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