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Semester 1, 2023

Tutorial 6

COMP4528: Computer Vision

Question 1 Matrix Algebra

1. Let A = [2(2)   0(6)] and B =  [2(0)   8(1)], compute AB.

2. Let x =  [1(5)] and y =  [8(0)], compute ∥x − y∥2 .

3. Let w =  [w(w)2(1)], x =  [6(8)] and L = 2(1) (w⊤x − 4)2 , compute w(L) 1 .

Question 2 Back Propagation

1.  Back propagation through the computational graph. The current values are w0  = 0.2, w1  = 0.2, w2  = 0.3, x0  = 2, x1  = 3. p and q define the intermediate variables that are calculated during training, at the specified points in the computation graph. L is the output of the computational graph. Please provide the gradient and based on the back-propagated gradient calculation. w0

2.  Given the linear regression model ˆ(y) = w⊤x + b  and the loss function is defined as L(y,ˆ(y)) =  2(y − yˆ) 2 . The initial model weights are w =  − 6 4  and b = −8. What is the new model weights after performing one gradient descent step with learning rate 0.01 and training data x =  [1(8)], y = 1.

3. !! Given a logistic regression model Softmax(Wx+b) where W =  l 2(2)     2(0)   6(4) and b =

−1    1    4 −2 .

The Softmax function is defined as Softmax(xi) = . For the training data, you have

x = −5   and the ground truth label y =    0   . Calculate the gradient of the cross-entropy loss (L = −ΣcM=1yo,clog (po,c))2 with respect to the bias vector b.