MODEL QUESTION PAPERS Mahatma Gandhi University, Kottayam First
Question 1amcollins/cs4705-spring2020/... · Question 1a The leaf variables are W1, b1, W2, b2, V,...
Transcript of Question 1amcollins/cs4705-spring2020/... · Question 1a The leaf variables are W1, b1, W2, b2, V,...
Question 1a
Question 1a
The leaf variables are W 1, b1, W 2, b2, V , γ, xi, yi
Each leaf in the graph has a single directed path leading to theoutput o. To calculate the partial derivative, take the product ofJacobians along the path
∂o
∂V
∣∣∣∣f̄o
=∂o
∂q
∣∣∣∣fo
× ∂q
∂l
∣∣∣∣fq
× ∂l
∂V
∣∣∣∣f l
Question 1b
∂o
∂W 1
∣∣∣∣f̄o
=∂o
∂q
∣∣∣∣fo
× ∂q∂l
∣∣∣∣fq
× ∂l
∂h2
∣∣∣∣f l
× ∂h2
∂z2
∣∣∣∣fh2
× ∂z2
∂h1
∣∣∣∣fz2
× ∂h1
∂z1
∣∣∣∣fh1
× ∂z1
∂W 1
∣∣∣∣fz1
Question 1b
∂o
∂W 2
∣∣∣∣f̄o
=∂o
∂q
∣∣∣∣fo
× ∂q
∂l
∣∣∣∣fq
× ∂l
∂h2
∣∣∣∣f l
× ∂h2
∂z2
∣∣∣∣fh2
× ∂z2
∂W 2
∣∣∣∣fz2
Question 2a
Question 2a
We get valuesu3 = u1 × u2 = 12
u4 = u1 + u2 = 7
u5 = 2× u3 × u4 = 168
Question 2b
un = 2×u3×u4 = 2×(u1×u2)×(u1+u2) = 2×(u1)2×u2+2×u1×(u2)2
hence
f̄n(u1, u2) = 2× (u1)2 × u2 + 2× u1 × (u2)2
hence∂un
∂u1= 4× u1 × u2 + 2× (u2)2 = 80
∂un
∂u2= 2× (u1)2 + 4× u1 × u2 = 66
J1→3(A3) = u2 = 4
J2→3(A3) =∂
∂u2
(u1 × u2
)= u1 = 3
J1→4(A4) =∂
∂u1
(u1 + u2
)= 1
J2→4(A4) =∂
∂u2
(u1 + u2
)= 1
J3→5(A5) =∂
∂u3
(2× u3 × u4
)= 2× u4 = 14
J4→5(A5) =∂
∂u4
(2× u3 × u4
)= 2× u3 = 24
p5 = 1
p4 = p5 × J4→5(A5) = 1× 24 = 24
p3 = p5 × J3→5(A5) = 1× 14 = 14
p2 = p3 × J2→3(A3) + p4 × J2→4(A4) = 14× 3 + 24× 1 = 66
p1 = p3 × J1→3(A3) + p4 × J1→4(A4) = 14× 4 + 24× 1 = 80
Question 3a
Equations:
z1 ∈ Rm = Wxi,1 + b
h1 ∈ Rm = g(z1)
l1 ∈ RK = V h1 + γ
q1 ∈ RK = LS(l1)
z2 ∈ Rm = Wxi,2 + b
h2 ∈ Rm = g(z2)
l2 ∈ RK = V h2 + γ
q2 ∈ RK = LS(l2)
o ∈ R = −q1yi,1 − q2
yi,2
Question 3a
Question 3b
There are two directed paths from V to o in the computationalgraph. To calculate the partial derivative, take a sum of productsof Jacobians on these paths
∂o
∂V
∣∣∣∣f̄o
=∂o
∂q1
∣∣∣∣fo
× ∂q1
∂l1
∣∣∣∣fq1
× ∂l1
∂V
∣∣∣∣f l1
+∂o
∂q2
∣∣∣∣fo
× ∂q2
∂l2
∣∣∣∣fq2
× ∂l2
∂V
∣∣∣∣f l2
Question 3c
f z(W,xi,1, xi,2, b) = [Wxi,1 + b;Wxi,2 + b]
fh([z1; z2]) = [g(z1); g(z2)]
f l([h1;h2]) = [V h1 + γ;V h2 + γ]
f q([l1; l2]) = [LS(l1); LS(l2)]
f o([q1; q2], yi,1, yi,2) = −q1yi,1 − q2
yi,2