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I am struggling to work out how to calculate the proximity norm of $ \lambda {\left\| x \right\|}_{2} $.

I believe this should be done using Moreau decomposition $ v = \operatorname{Prox} \left( x \right) + \operatorname{Prox}^{\ast} \left( v \right) $ but I am unsure how this works. Can anyone show me some steps as to what to do?

Royi
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CEB12
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2 Answers2

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Given $ f \left( x \right) = \left\| x \right\| $ is a norm function its Prox is given by (For any Norm):

$$ \operatorname{Prox}_{\lambda f \left( \cdot \right)} \left( v \right) = v - \lambda \operatorname{Proj}_{ \mathcal{B}_{ \left\| \cdot \right\|_{\ast} } } \left( \frac{v}{\lambda} \right) $$

Where $ \operatorname{Proj}_{ \mathcal{B}_{ \left\| \cdot \right\|_{\ast} } } \left( \cdot \right) $ is the Orthogonal Projection Operator and $ \mathcal{B}_{ \left\| \cdot \right\|_{\ast} } $ is the Norm Unit Ball (Of the Dual Norm).

In your case we're dealing with the $ {L}_{2} $ Norm which is self dual.
Moreover, the Projection onto the $ {L}_{2} $ Unit Ball is given by:

$$ \operatorname{Proj}_{ \mathcal{B}_{ \left\| \cdot \right\|_{2} } } \left( x \right) = \begin{cases} \frac{x}{ \left\| x \right\|_{2} } & \text{ if } \left\| x \right\|_{2} > 1 \\ x & \text{ if } \left\| x \right\|_{2} \leq 1 \end{cases} $$

In summary:

$$ \operatorname{Prox}_{\lambda \left\| \cdot \right\|_{2}} \left( v \right) = v - \lambda \operatorname{Proj}_{ \mathcal{B}_{ \left\| \cdot \right\|_{2} } } \left( \frac{v}{\lambda} \right) = \begin{cases} v - \lambda \frac{ \frac{v}{\lambda} }{ \left\| \frac{v}{\lambda} \right\|_{2} } & \text{ if } \left\| \frac{v}{\lambda} \right\|_{2} > 1 \\ v - \lambda \frac{v}{\lambda} & \text{ if } \left\| \frac{v}{\lambda} \right\|_{2} \leq 1 \end{cases} = \left(1 - \frac{\lambda}{ \left\| v \right\|_{2} } \right)_{+} v $$

Where $ \left( x \right)_{+} = \max \left\{ 0, x \right\} $.

Royi
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I answered exactly the same question here, using exactly the same Moreau decomposition idea you're thinking about :)

dohmatob
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  • Thanks, I've had a look at it but still don't fully understand where you go from the steps you have done. I haven't done any questions involving Moreau decomposition before so I am unsure as to what I need to do next. Are you able to provide any more help with this? – CEB12 Mar 19 '17 at 15:14