1 consider a linear mapping y wx with y e r2x r2 and w r22 with 21 w 12 1 then the singular valu

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Consider a linear mapping y = Wx with y âˆˆ R^2, x âˆˆ R^2 and W âˆˆ R^2x2 with W = [1 2] Then the singular values of W are Ïƒ1 = 3, Ïƒ2 = 1. What is the condition number of W? Why does the condition number matter? (3 marks) 2. Corrected_text: Consider a linear mapping y = Wx with y âˆˆ R^2, x âˆˆ R^2 and W âˆˆ R^2x2 with W = [0.1 0.2; 0.1 0.3] Perform one gradient descent update step to W with a learning rate of Î± = 0.1 with the data point (x,yT) = ([1,1]T, [1,2]T) (one data point means we are doing "online learning"). Use the loss function L(W) = ||y - W||^2 (6 marks). 3. Corrected_text: Describe the purpose of using momentum as opposed to vanilla gradient descent. Explain the relevant problem with gradient descent and how momentum fixes this problem. (4 marks)

          Consider a linear mapping y = Wx with y âˆˆ R^2, x âˆˆ R^2 and W âˆˆ R^2x2 with
W = [1 2]
Then the singular values of W are Ïƒ1 = 3, Ïƒ2 = 1. What is the condition number of W? Why does the condition number matter? (3 marks)

2. Corrected_text: Consider a linear mapping y = Wx with y âˆˆ R^2, x âˆˆ R^2 and W âˆˆ R^2x2 with W = [0.1 0.2; 0.1 0.3]
Perform one gradient descent update step to W with a learning rate of Î± = 0.1 with the data point (x,yT) = ([1,1]T, [1,2]T) (one data point means we are doing "online learning"). Use the loss function L(W) = ||y - W||^2 (6 marks).

3. Corrected_text: Describe the purpose of using momentum as opposed to vanilla gradient descent. Explain the relevant problem with gradient descent and how momentum fixes this problem. (4 marks)

1 consider a linear mapping y wx with y e r2x r2 and w r22 with 21 w 12 1 then the singular values of w are 1 3 2 1 what is the condition number of w why does the condition number matter 3 m 76199

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