Problem 4. (3pts) Back-propagation. In this problem, you will have to run the back-propagation
algorithm on the following formula.
f=(1)/(2)||Ax||^(2)
Here we have A=[[A_(11),A_(12)],[A_(21),A_(22)]],x=[[x_(1)],[x_(2)]],b=Ax=[[A_(11)x_(1)+A_(12)x_(2)],[A_(21)x_(1)+A_(22)x_(2)]]=[[b_(1)],[b_(2)]],
and f=(1)/(2)||b||^(2)=(1)/(2)(b_(1)^(2)+b_(2)^(2)) is a scalar.
a) (0.5pt) Calculate the partial derivatives of f:(delf)/(delb)=[[(delf)/(delb_(1))],[(delf)/(delb_(2))]]
Answer: ...
b) (0.75pt) Calculate the partial derivatives of b_(1) :
((delb_(1))/(delA_(11)),(delb_(1))/(delA_(12)))
Answer: ...
((delb_(1))/(delA_(21)),(delb_(1))/(delA_(22)))
Answer: ...
((delb_(1))/(delx_(1)),(delb_(1))/(delx_(2)))
Answer: ...
c) (0.75pt) Calculate the partial derivatives of f :
((delf)/(delA_(11)),(delf)/(delA_(12)))
Answer: ...
((delf)/(delA_(11)),(delf)/(delA_(12)))
Answer: ...
((delf)/(delx_(1)),(delf)/(delx_(2)))
Answer: ...
d) (1pt) Assuming that A is an n imes d matrix, and x is a d imes 1 vector.
Find (delf)/(delA) in terms of A and x.
Answer: ...
Find (delf)/(delx) in terms of A and x.
Answer: ...
Problem 4. (3pts) Back-propagation. In this problem, you will have to run the back-propagation algorithm on the following formula.
A11A12 Here we have A = [Az1Azz]
[A11x+A12x2]=[b1 b=Ax= Ax+AzX] Ib-
3 II2
of
Answer: .-
b) (0.75pt) Calculate the partial derivatives of b, :
Answer:
dbi dbi
Answer: .
0b0b xax
Answer: . c) (0.75pt) Calculate the partial derivatives of f :
Answer: .
Answer:
faf axax Answer: . d) (1pt) Assuming that A is an n d matrix, and x is a d 1 vector.
Answer:..
Find &f in terms of A and x. Answer: ..-