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Need to check the gradient of data fitting function with respective the two vector variables
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Derivative of a composed function with several variablesGradient vector function using sum and scalarCheck if the following gradient is correctWhy isn't the gradient vector of a parametric curve parallel to the tangent vector?Gradient of a function (check if I did it well)How to apply gradient with respect to a vectorCan you use the chain rule in vector calculus to compute the gradient of a matrix?Deriving the gradient vector of a Probit modelderivative of a modulus of a gradient of a function with respect to the same functionWhy is the first derivative of a scalar-valued function with respect to a vector its gradient?
$begingroup$
Suppose $mathbfa_1$, $mathbfa_2$ and $mathbfc$ are given vectors, $lin mathbbR_+$ and $0<p<1$. Let $mathbfb_1,mathbfb_2inmathbbR^M$ be two vector variables and define a function as follows
$$
J(mathbfb_1,mathbfb_2)=sum_n=1^99Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2^p \+
Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2^p
$$
What is the gradient of $J$ with respect to $mathbfb_1$ and $mathbfb_2$?
My attempt:
Let
$$
u_n(mathbfb_1,mathbfb_2) = Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2, n = 1,dots,99,
$$
and
$$
u_100(mathbfb_1,mathbfb_2) = Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2.
$$
The partial derivative are derived as following chain rule
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1triangleq fracsum_n=1^100partial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial mathbfb_1 =sum_n=1^100fracpartial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial u_n(mathbfb_1,mathbfb_2)fracpartial u_n(mathbfb_1,mathbfb_2)partial mathbfb_1.
$$
Then, the partial derivative of this problem can be obtained as follows:
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1 \=sum_n=1^99frac-np100big(u_n(mathbfb_1,mathbfb_2)big)^p-2Bigg(fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg) \+ (-0.5p)big(u_100(mathbfb_1,mathbfb_2)big)^p-2Bigg(0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg).
$$
calculus linear-algebra vector-analysis
asked Apr 2 at 9:25
Roy HsuRoy Hsu
335
$endgroup$
add a comment |
$begingroup$
Suppose $mathbfa_1$, $mathbfa_2$ and $mathbfc$ are given vectors, $lin mathbbR_+$ and $0<p<1$. Let $mathbfb_1,mathbfb_2inmathbbR^M$ be two vector variables and define a function as follows
$$
J(mathbfb_1,mathbfb_2)=sum_n=1^99Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2^p \+
Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2^p
$$
What is the gradient of $J$ with respect to $mathbfb_1$ and $mathbfb_2$?
My attempt:
Let
$$
u_n(mathbfb_1,mathbfb_2) = Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2, n = 1,dots,99,
$$
and
$$
u_100(mathbfb_1,mathbfb_2) = Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2.
$$
The partial derivative are derived as following chain rule
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1triangleq fracsum_n=1^100partial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial mathbfb_1 =sum_n=1^100fracpartial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial u_n(mathbfb_1,mathbfb_2)fracpartial u_n(mathbfb_1,mathbfb_2)partial mathbfb_1.
$$
Then, the partial derivative of this problem can be obtained as follows:
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1 \=sum_n=1^99frac-np100big(u_n(mathbfb_1,mathbfb_2)big)^p-2Bigg(fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg) \+ (-0.5p)big(u_100(mathbfb_1,mathbfb_2)big)^p-2Bigg(0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg).
$$
calculus linear-algebra vector-analysis
asked Apr 2 at 9:25
Roy HsuRoy Hsu
335
$endgroup$
add a comment |
$begingroup$
Suppose $mathbfa_1$, $mathbfa_2$ and $mathbfc$ are given vectors, $lin mathbbR_+$ and $0<p<1$. Let $mathbfb_1,mathbfb_2inmathbbR^M$ be two vector variables and define a function as follows
$$
J(mathbfb_1,mathbfb_2)=sum_n=1^99Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2^p \+
Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2^p
$$
What is the gradient of $J$ with respect to $mathbfb_1$ and $mathbfb_2$?
My attempt:
Let
$$
u_n(mathbfb_1,mathbfb_2) = Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2, n = 1,dots,99,
$$
and
$$
u_100(mathbfb_1,mathbfb_2) = Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2.
$$
The partial derivative are derived as following chain rule
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1triangleq fracsum_n=1^100partial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial mathbfb_1 =sum_n=1^100fracpartial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial u_n(mathbfb_1,mathbfb_2)fracpartial u_n(mathbfb_1,mathbfb_2)partial mathbfb_1.
$$
Then, the partial derivative of this problem can be obtained as follows:
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1 \=sum_n=1^99frac-np100big(u_n(mathbfb_1,mathbfb_2)big)^p-2Bigg(fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg) \+ (-0.5p)big(u_100(mathbfb_1,mathbfb_2)big)^p-2Bigg(0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg).
$$
calculus linear-algebra vector-analysis
asked Apr 2 at 9:25
Roy HsuRoy Hsu
335
$endgroup$
Suppose $mathbfa_1$, $mathbfa_2$ and $mathbfc$ are given vectors, $lin mathbbR_+$ and $0<p<1$. Let $mathbfb_1,mathbfb_2inmathbbR^M$ be two vector variables and define a function as follows
$$
J(mathbfb_1,mathbfb_2)=sum_n=1^99Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2^p \+
Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2^p
$$
What is the gradient of $J$ with respect to $mathbfb_1$ and $mathbfb_2$?
My attempt:
Let
$$
u_n(mathbfb_1,mathbfb_2) = Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2, n = 1,dots,99,
$$
and
$$
u_100(mathbfb_1,mathbfb_2) = Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2.
$$
The partial derivative are derived as following chain rule
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1triangleq fracsum_n=1^100partial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial mathbfb_1 =sum_n=1^100fracpartial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial u_n(mathbfb_1,mathbfb_2)fracpartial u_n(mathbfb_1,mathbfb_2)partial mathbfb_1.
$$
Then, the partial derivative of this problem can be obtained as follows:
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1 \=sum_n=1^99frac-np100big(u_n(mathbfb_1,mathbfb_2)big)^p-2Bigg(fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg) \+ (-0.5p)big(u_100(mathbfb_1,mathbfb_2)big)^p-2Bigg(0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg).
$$
calculus linear-algebra vector-analysis
calculus linear-algebra vector-analysis
asked Apr 2 at 9:25
Roy HsuRoy Hsu
335
asked Apr 2 at 9:25
Roy HsuRoy Hsu
335
edited Apr 2 at 12:42
Roy Hsu
asked Apr 2 at 9:25
Roy HsuRoy Hsu
335
asked Apr 2 at 9:25
Roy HsuRoy Hsu
335
asked Apr 2 at 9:25
Roy HsuRoy Hsu
335
335
add a comment |
add a comment |
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