This assignment does not count towards the admission. Nonetheless you should try to upload your solution as PDF to Ilias. This will help you and us in testing the platform and identifying potential problems.
The rst two tasks were taken from \Exercise 1" of the previous semester held by Prof.
Dr. Marc Toussaint.
• Matrix equations
1. Let X, YA be arbitrary matrices, A invertible. Solve for X: XA+A> =I
2. Let X, A, B be arbitrary matrices, (C 2A>) invertible. Solve for X:
X>C = [2A(X + B)]>
3. Let x 2 Rn, y 2 Rd, A 2 Rd n, A>A invertible. Solve for x:
(Ax y)>A = 0>n
4. As above, additionally B 2 Rn n, B positive-de nite. Solve for x:
(Ax y)>A + x>B = 0>n
2 Vector derivatives
Let x 2 Rn, y 2 Rd, A 2 Rd n.
1. What is @x@ x? (Of what type/dimension is this thing?)
2. What is @x@ [x>x]?
3. Let B be symmetric and positive de nite. What is the minimum of (Ax y)>(Ax y) + x>Bx w.r.t. x?
1
3 Error Measures
Let y; y^ 2 Rn be n true and predicted values of a regression problem.
1. Formally de ne the error measures Mean Squared Error (MSE) and Mean Absolute Error between y and y^.
2. How would choosing MAE or MSE as objective function for a regression problem impact the resulting prediction model? How do MSE and MAE di er?
3. Let y; y^ 2 f0; 1gn be n true and predicted labels of a binary classi cation problem. What would the MSE and MAE calculate in this case?
2
