Mixing
Derivate of an Inverse of a Matrix
I have the following loss function.
$$||theta - (X^T X)^{-1} X^T y||_2^2$$
$$Xspace text{ is a matrix, } theta text{ and } y text{ are known vectors.}$$
I have another constraint for $X$, which is $X = f(lambda)$ for some function $f$ that I didn't include here.
The idea is that I want to initialize $lambda$ to some random vector, compute an $X$ with $X=f(lambda)$, and then use gradient descent or some iterative method to minimize the loss function given above by updating $lambda$ at each step. However, I am having trouble taking the gradient of this loss function that could be used in this iterative algorithm.
How would I do this?
optimization matrix-calculus gradient-descent
asked Nov 20 '18 at 18:10
codemirel
113
add a comment |
I have the following loss function.
$$||theta - (X^T X)^{-1} X^T y||_2^2$$
$$Xspace text{ is a matrix, } theta text{ and } y text{ are known vectors.}$$
I have another constraint for $X$, which is $X = f(lambda)$ for some function $f$ that I didn't include here.
The idea is that I want to initialize $lambda$ to some random vector, compute an $X$ with $X=f(lambda)$, and then use gradient descent or some iterative method to minimize the loss function given above by updating $lambda$ at each step. However, I am having trouble taking the gradient of this loss function that could be used in this iterative algorithm.
How would I do this?
optimization matrix-calculus gradient-descent
asked Nov 20 '18 at 18:10
codemirel
113
Wasn't that posted like 15 minutes ago ?
– Rebellos
Nov 20 '18 at 18:12
add a comment |
I have the following loss function.
$$||theta - (X^T X)^{-1} X^T y||_2^2$$
$$Xspace text{ is a matrix, } theta text{ and } y text{ are known vectors.}$$
I have another constraint for $X$, which is $X = f(lambda)$ for some function $f$ that I didn't include here.
The idea is that I want to initialize $lambda$ to some random vector, compute an $X$ with $X=f(lambda)$, and then use gradient descent or some iterative method to minimize the loss function given above by updating $lambda$ at each step. However, I am having trouble taking the gradient of this loss function that could be used in this iterative algorithm.
How would I do this?
optimization matrix-calculus gradient-descent
asked Nov 20 '18 at 18:10
codemirel
113
I have the following loss function.
$$||theta - (X^T X)^{-1} X^T y||_2^2$$
$$Xspace text{ is a matrix, } theta text{ and } y text{ are known vectors.}$$
I have another constraint for $X$, which is $X = f(lambda)$ for some function $f$ that I didn't include here.
The idea is that I want to initialize $lambda$ to some random vector, compute an $X$ with $X=f(lambda)$, and then use gradient descent or some iterative method to minimize the loss function given above by updating $lambda$ at each step. However, I am having trouble taking the gradient of this loss function that could be used in this iterative algorithm.
How would I do this?
optimization matrix-calculus gradient-descent
optimization matrix-calculus gradient-descent
asked Nov 20 '18 at 18:10
codemirel
113
asked Nov 20 '18 at 18:10
codemirel
113
asked Nov 20 '18 at 18:10
codemirel
113
asked Nov 20 '18 at 18:10
codemirel
113
asked Nov 20 '18 at 18:10
codemirel
113
113
Wasn't that posted like 15 minutes ago ?
– Rebellos
Nov 20 '18 at 18:12
add a comment |
Wasn't that posted like 15 minutes ago ?
– Rebellos
Nov 20 '18 at 18:12
Wasn't that posted like 15 minutes ago ?
– Rebellos
Nov 20 '18 at 18:12
Wasn't that posted like 15 minutes ago ?
– Rebellos
Nov 20 '18 at 18:12
add a comment |
1 Answer
1
active
oldest
votes
Define some new variables
$$eqalign{
M &= (X^TX)^{-1}X^T cr
p &= My - theta cr
}$$
and their differentials
$$eqalign{
dM &= (X^TX)^{-1},dX^T - (X^TX)^{-1},d(X^TX),(X^TX)^{-1}X^T cr
&= (X^TX)^{-1},dX^T - (X^TX)^{-1},dX^T,XM - M,dX,M cr
dp &= dM,y cr
}$$
Write the cost function in terms of these new variables.
Then find its differential and gradient.
$$eqalign{
phi &= p:p crcr
dphi
&= p:dM,y cr
&= py^T:dM cr
&= py^T:(X^TX)^{-1},dX^T - py^T:(X^TX)^{-1},dX^T,XM - py^T:M,dX,M cr
&= (X^TX)^{-1}py^T:dX^T - (X^TX)^{-1}py^TM^TX^T:dX^T - M^Tpy^TM^T:dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k},dlambda_k crcr
frac{partialphi}{partial lambda_k}
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k} crcr
}$$
The colon is a convenient product notation for the trace, i.e. $,,A:B={rm Tr}(A^TB)$.
Rules for rearranging terms in a colon product follow from the properties of the trace.
answered Nov 21 '18 at 2:52
greg
7,5251821
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Define some new variables
$$eqalign{
M &= (X^TX)^{-1}X^T cr
p &= My - theta cr
}$$
and their differentials
$$eqalign{
dM &= (X^TX)^{-1},dX^T - (X^TX)^{-1},d(X^TX),(X^TX)^{-1}X^T cr
&= (X^TX)^{-1},dX^T - (X^TX)^{-1},dX^T,XM - M,dX,M cr
dp &= dM,y cr
}$$
Write the cost function in terms of these new variables.
Then find its differential and gradient.
$$eqalign{
phi &= p:p crcr
dphi
&= p:dM,y cr
&= py^T:dM cr
&= py^T:(X^TX)^{-1},dX^T - py^T:(X^TX)^{-1},dX^T,XM - py^T:M,dX,M cr
&= (X^TX)^{-1}py^T:dX^T - (X^TX)^{-1}py^TM^TX^T:dX^T - M^Tpy^TM^T:dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k},dlambda_k crcr
frac{partialphi}{partial lambda_k}
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k} crcr
}$$
The colon is a convenient product notation for the trace, i.e. $,,A:B={rm Tr}(A^TB)$.
Rules for rearranging terms in a colon product follow from the properties of the trace.
answered Nov 21 '18 at 2:52
greg
7,5251821
add a comment |
Define some new variables
$$eqalign{
M &= (X^TX)^{-1}X^T cr
p &= My - theta cr
}$$
and their differentials
$$eqalign{
dM &= (X^TX)^{-1},dX^T - (X^TX)^{-1},d(X^TX),(X^TX)^{-1}X^T cr
&= (X^TX)^{-1},dX^T - (X^TX)^{-1},dX^T,XM - M,dX,M cr
dp &= dM,y cr
}$$
Write the cost function in terms of these new variables.
Then find its differential and gradient.
$$eqalign{
phi &= p:p crcr
dphi
&= p:dM,y cr
&= py^T:dM cr
&= py^T:(X^TX)^{-1},dX^T - py^T:(X^TX)^{-1},dX^T,XM - py^T:M,dX,M cr
&= (X^TX)^{-1}py^T:dX^T - (X^TX)^{-1}py^TM^TX^T:dX^T - M^Tpy^TM^T:dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k},dlambda_k crcr
frac{partialphi}{partial lambda_k}
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k} crcr
}$$
The colon is a convenient product notation for the trace, i.e. $,,A:B={rm Tr}(A^TB)$.
Rules for rearranging terms in a colon product follow from the properties of the trace.
answered Nov 21 '18 at 2:52
greg
7,5251821
add a comment |
Define some new variables
$$eqalign{
M &= (X^TX)^{-1}X^T cr
p &= My - theta cr
}$$
and their differentials
$$eqalign{
dM &= (X^TX)^{-1},dX^T - (X^TX)^{-1},d(X^TX),(X^TX)^{-1}X^T cr
&= (X^TX)^{-1},dX^T - (X^TX)^{-1},dX^T,XM - M,dX,M cr
dp &= dM,y cr
}$$
Write the cost function in terms of these new variables.
Then find its differential and gradient.
$$eqalign{
phi &= p:p crcr
dphi
&= p:dM,y cr
&= py^T:dM cr
&= py^T:(X^TX)^{-1},dX^T - py^T:(X^TX)^{-1},dX^T,XM - py^T:M,dX,M cr
&= (X^TX)^{-1}py^T:dX^T - (X^TX)^{-1}py^TM^TX^T:dX^T - M^Tpy^TM^T:dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k},dlambda_k crcr
frac{partialphi}{partial lambda_k}
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k} crcr
}$$
The colon is a convenient product notation for the trace, i.e. $,,A:B={rm Tr}(A^TB)$.
Rules for rearranging terms in a colon product follow from the properties of the trace.
answered Nov 21 '18 at 2:52
greg
7,5251821
Define some new variables
$$eqalign{
M &= (X^TX)^{-1}X^T cr
p &= My - theta cr
}$$
and their differentials
$$eqalign{
dM &= (X^TX)^{-1},dX^T - (X^TX)^{-1},d(X^TX),(X^TX)^{-1}X^T cr
&= (X^TX)^{-1},dX^T - (X^TX)^{-1},dX^T,XM - M,dX,M cr
dp &= dM,y cr
}$$
Write the cost function in terms of these new variables.
Then find its differential and gradient.
$$eqalign{
phi &= p:p crcr
dphi
&= p:dM,y cr
&= py^T:dM cr
&= py^T:(X^TX)^{-1},dX^T - py^T:(X^TX)^{-1},dX^T,XM - py^T:M,dX,M cr
&= (X^TX)^{-1}py^T:dX^T - (X^TX)^{-1}py^TM^TX^T:dX^T - M^Tpy^TM^T:dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):dX cr
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k},dlambda_k crcr
frac{partialphi}{partial lambda_k}
&= Big(yp^T(X^TX)^{-1} - XMyp^T(X^TX)^{-1} - M^Tpy^TM^TBig):frac{partial X}{partiallambda_k} crcr
}$$
The colon is a convenient product notation for the trace, i.e. $,,A:B={rm Tr}(A^TB)$.
Rules for rearranging terms in a colon product follow from the properties of the trace.
answered Nov 21 '18 at 2:52
greg
7,5251821
edited Nov 21 '18 at 4:50
answered Nov 21 '18 at 2:52
greg
7,5251821
answered Nov 21 '18 at 2:52
greg
7,5251821
answered Nov 21 '18 at 2:52
greg
7,5251821
7,5251821
add a comment |
add a comment |
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Wasn't that posted like 15 minutes ago ?
– Rebellos
Nov 20 '18 at 18:12