Why are the non-diagonals in a Jacobian zeros?
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From the Matrix Calculus for Deep Learning, in the "Derivatives of vector element-wise binary operators" section, it says
Any time the general function is a vector, we know that $f_i(w)$ reduces to $f_i(w_i) = w_i$.
Why is it that $f_i(w)$ reduces to $f_i(w_i) = w_i$ for $y=w+x$ ?
Can someone show an example of why when i != j, the partial derivatives are zero?
Is this answer satisfactory? Something like this?
$frac{d}{dw_i}f_i(w_j)$ when i != j, the scalar derivative is 0
matrix-calculus jacobian
asked Jan 15 at 13:37
alvasalvas
679
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migrated from stats.stackexchange.com Jan 15 at 14:13
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
add a comment |
$begingroup$
From the Matrix Calculus for Deep Learning, in the "Derivatives of vector element-wise binary operators" section, it says
Any time the general function is a vector, we know that $f_i(w)$ reduces to $f_i(w_i) = w_i$.
Why is it that $f_i(w)$ reduces to $f_i(w_i) = w_i$ for $y=w+x$ ?
Can someone show an example of why when i != j, the partial derivatives are zero?
Is this answer satisfactory? Something like this?
$frac{d}{dw_i}f_i(w_j)$ when i != j, the scalar derivative is 0
matrix-calculus jacobian
asked Jan 15 at 13:37
alvasalvas
679
$endgroup$
migrated from stats.stackexchange.com Jan 15 at 14:13
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
$begingroup$
In the quoted passage I don't see anything about $ineq j.$ It's $w_i$ and $x_i.$ Also, the passage assumes at the start that $f(mathbf w) = mathbf w,$ so that simplifies a lot of things right away.
$endgroup$
– David K
Jan 15 at 14:18
add a comment |
$begingroup$
From the Matrix Calculus for Deep Learning, in the "Derivatives of vector element-wise binary operators" section, it says
Any time the general function is a vector, we know that $f_i(w)$ reduces to $f_i(w_i) = w_i$.
Why is it that $f_i(w)$ reduces to $f_i(w_i) = w_i$ for $y=w+x$ ?
Can someone show an example of why when i != j, the partial derivatives are zero?
Is this answer satisfactory? Something like this?
$frac{d}{dw_i}f_i(w_j)$ when i != j, the scalar derivative is 0
matrix-calculus jacobian
asked Jan 15 at 13:37
alvasalvas
679
$endgroup$
From the Matrix Calculus for Deep Learning, in the "Derivatives of vector element-wise binary operators" section, it says
Any time the general function is a vector, we know that $f_i(w)$ reduces to $f_i(w_i) = w_i$.
Why is it that $f_i(w)$ reduces to $f_i(w_i) = w_i$ for $y=w+x$ ?
Can someone show an example of why when i != j, the partial derivatives are zero?
Is this answer satisfactory? Something like this?
$frac{d}{dw_i}f_i(w_j)$ when i != j, the scalar derivative is 0
matrix-calculus jacobian
matrix-calculus jacobian
asked Jan 15 at 13:37
alvasalvas
679
asked Jan 15 at 13:37
alvasalvas
679
asked Jan 15 at 13:37
alvasalvas
679
asked Jan 15 at 13:37
alvasalvas
679
asked Jan 15 at 13:37
alvasalvas
679
679
migrated from stats.stackexchange.com Jan 15 at 14:13
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
migrated from stats.stackexchange.com Jan 15 at 14:13
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
$begingroup$
In the quoted passage I don't see anything about $ineq j.$ It's $w_i$ and $x_i.$ Also, the passage assumes at the start that $f(mathbf w) = mathbf w,$ so that simplifies a lot of things right away.
$endgroup$
– David K
Jan 15 at 14:18
add a comment |
$begingroup$
In the quoted passage I don't see anything about $ineq j.$ It's $w_i$ and $x_i.$ Also, the passage assumes at the start that $f(mathbf w) = mathbf w,$ so that simplifies a lot of things right away.
$endgroup$
– David K
Jan 15 at 14:18
$begingroup$
In the quoted passage I don't see anything about $ineq j.$ It's $w_i$ and $x_i.$ Also, the passage assumes at the start that $f(mathbf w) = mathbf w,$ so that simplifies a lot of things right away.
$endgroup$
– David K
Jan 15 at 14:18
$begingroup$
In the quoted passage I don't see anything about $ineq j.$ It's $w_i$ and $x_i.$ Also, the passage assumes at the start that $f(mathbf w) = mathbf w,$ so that simplifies a lot of things right away.
$endgroup$
– David K
Jan 15 at 14:18
add a comment |
1 Answer
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$begingroup$
The reason that the off-diagonal elements are 0 is because the derivative of a constant is 0. The key bit is that $w_i$ looks like a constant if we take the derivative with respect to $w_j$ for $j ne i$.
Here is more of the matrix calculus article I co-authored:
answered Jan 15 at 17:56
Terence ParrTerence Parr
1093
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The reason that the off-diagonal elements are 0 is because the derivative of a constant is 0. The key bit is that $w_i$ looks like a constant if we take the derivative with respect to $w_j$ for $j ne i$.
Here is more of the matrix calculus article I co-authored:
answered Jan 15 at 17:56
Terence ParrTerence Parr
1093
$endgroup$
add a comment |
$begingroup$
The reason that the off-diagonal elements are 0 is because the derivative of a constant is 0. The key bit is that $w_i$ looks like a constant if we take the derivative with respect to $w_j$ for $j ne i$.
Here is more of the matrix calculus article I co-authored:
answered Jan 15 at 17:56
Terence ParrTerence Parr
1093
$endgroup$
add a comment |
$begingroup$
The reason that the off-diagonal elements are 0 is because the derivative of a constant is 0. The key bit is that $w_i$ looks like a constant if we take the derivative with respect to $w_j$ for $j ne i$.
Here is more of the matrix calculus article I co-authored:
answered Jan 15 at 17:56
Terence ParrTerence Parr
1093
$endgroup$
The reason that the off-diagonal elements are 0 is because the derivative of a constant is 0. The key bit is that $w_i$ looks like a constant if we take the derivative with respect to $w_j$ for $j ne i$.
Here is more of the matrix calculus article I co-authored:
answered Jan 15 at 17:56
Terence ParrTerence Parr
1093
answered Jan 15 at 17:56
Terence ParrTerence Parr
1093
answered Jan 15 at 17:56
Terence ParrTerence Parr
1093
answered Jan 15 at 17:56
Terence ParrTerence Parr
1093
1093
add a comment |
add a comment |
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$begingroup$
In the quoted passage I don't see anything about $ineq j.$ It's $w_i$ and $x_i.$ Also, the passage assumes at the start that $f(mathbf w) = mathbf w,$ so that simplifies a lot of things right away.
$endgroup$
– David K
Jan 15 at 14:18