Math 309 HW4 - Adam Oberman Math

(extending HW1). Let $b \in R^n$ be an arbitrary vector, and let $\vec 1 \in R^n$ be the vector of all ones.

What is the solution of the norm approximation problem with one scalar variable $x\in R$

$\min \| x\vec 1 - b\|_p$

for (a) $p = 2$ , (b) $p = \infty$ , (c) $p = 1$

For a monotone increasing function $f (x)$ defined on $[0,1]$ , prove that $\| f'(x) \|_1 = \int |f'(x)| dx$ is simply the total increase in height of the function, $f (1) - f (0)$ .
Over all differentiable functions with $f (0) = 0, f (1) = H$ , find the one which minimizes $\| f'(x) \|_2 = \sqrt{\int (f'(x))^2 dx}$ . What is the minimal value?
Prove, on the other hand, the minimum over all differentiable functions with $f (0) = 0, f (1) = H$ of $\| f'(x) \|_1$ is simply H. Notice that there is more than one function for which the minimum is attained. Describe all functions which are minimizers in this case. (Hint: use the earlier problem).

Modify the code in Matlab Code for Week 7 to denoise the sounds signal using the multi-objective formulation (instead of the constrained formulation in the code). Change the problem formulation to the code below (as in the second code example) plot the solution for several different values of the parameter. Which is the (approximate) best value of the parameter?
Run the second code sample in Matlab Code for Week 7 with different values of the parameter delta. Describe what happens for the larger and smaller values of delta in the cases p =1,2 with the smooth and non-smooth signals. What are the best values of p, delta for each of the signals? Include plots of the best denoised solutions in each case.
Use the third code sample in Matlab Code for Week 7 to allow for
1. A different matrix M, which is [1 -2 1] along the diagonal. (the curvature matrix).
2. Put the one norm, instead of the default two norm in the fidelity term. Do these changes make a difference for the denoising of the signals? Describe your results and draw conclusions (if any).
3. Use the version of the signal with a straight line. Describe what happens using the Dx matrix versus the Dxx matrix. Which combination gives the best denoising?

Code for 1:

cvx_begin
variable x(n)
minimize ( norm(x - Scorrupt) + delta*norm(D*x,p) )
cvx_end