bv_cvxbook_extra_exercises

Distinguish two cases m n and n m and give the most

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Unformatted text preview: mation. In approximate total variation de-noising, we use Newton’s method to minimize ψ (x) = x − xcor (The parameters µ > 0 and ǫ > 0 are given.) 76 2 2 + µφatv (x). (a) Find expressions for the gradient and Hessian of ψ . (b) Explain how you would exploit the structure of the Hessian to compute the Newton direction for ψ efficiently. (Your explanation can be brief.) Compare the approximate flop count for your method with the flop count for a generic method that does not exploit any structure in the Hessian of ψ . (c) Implement Newton’s method for approximate total variation de-noising. Get the corrupted signal xcor from the file approx_tv_denoising_data.m, and compute the de-noised signal x⋆ , using parameters ǫ = 0.001, µ = 50 (which are also in the file). Use line search parameters α = 0.01, β = 0.5, initial point x(0) = 0, and stopping criterion λ2 /2 ≤ 10−8 . Plot the Newton decrement versus iteration, to verify asymptotic quadratic convergence. Plot the final smoothed signal x⋆ , along with the corrupted one xcor . 8.7 Derive the Newton equation for the unconstrained minimization problem mini...
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