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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.)
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
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- Fall '13
- The Aeneid