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Find the closed-form solution for the denoised signal ?f for both the weight matrix–based and the Laplacian-based frameworks. Note that only the expression for S 2 ( f) is different in each framework. Hint: Take the derivative with respect to f and equate to zero.

This problem is about graph signal denoising via regularization. Usually the measurements

are corrupted by noise, and the task of graph signal denoising is to recover the true graph

signal from noisy measurements. Consider a noisy graph signal measurement



where

f is the true graph signal and



n is the noise signal. The goal is to recover original signal



f from noisy measurements



y. In the classical signal processing, discrete-time signals and

digital images are denoised via regularization, where the regularization term usually enforces

smoothness. Similarly, a graph signal denoising task can be formulated as the following

optimization problem:



where S2(f) is 2-Dirichlet form as given by Equation (10.5.16) and ? is

Redraw the diagram created in number 3 above, after deleting the node containing Chicago.

What will be an ideal response?