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L-One Norm of Derivative Objective

Another way to add smoothness constraint is to add $ \ensuremath {L_1}$ -norm of the derivative to the objective:

$\displaystyle \mathrm{minimize}\quad \delta +\eta \left\Vert \Delta h\right\Vert _1$ (4.82)

Note that the $ \ensuremath {L_1}$ norm is sensitive to all the derivatives, not just the largest.

We can formulate an LP problem by adding a vector of optimization parameters $ \tau$ which bound derivatives:

$\displaystyle -\tau _{i}\leq \Delta h_{i}\leq \tau _{i}\qquad i=1,\ldots ,L-1.$ (4.83)

In matrix form,

$\displaystyle \left[\begin{array}{r} -\mathbf{D}\\ \mathbf{D}\end{array} \right]h-\left[\begin{array}{c} -\tau \\ -\tau \end{array} \right]\le 0.$ (4.84)

The objective function becomes

$\displaystyle \mathrm{minimize}\quad \delta +\eta \mathbf1^{T}\tau .$ (4.85)

See Fig.3.41 and Fig.3.42 for example results.

Figure: $ \ensuremath {L_1}$ norm of diff(h) added to the objective function ($ \eta =1$ )
\includegraphics[width=\twidth,height=6.5in]{eps/print_lone_chebwin_1}

Figure: Six times the $ \ensuremath {L_1}$ norm of diff(h) added to the objective function ($ \eta =6$ )
\includegraphics[width=\twidth,height=6.5in]{eps/print_lone_chebwin_2}


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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