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I am trying to derive the proximal operator $\operatorname{prox}_{\lambda}f(v), v \in \mathbb R^n$ of the addition between a quadratic term and a lasso penalty
$f(x) = \lVert b - Ax \rVert_2^2 + \tau \|x\|_1.$

This is different from the question asked here.

The soft-thresholding operator is the proximal of the lasso regularizer $\|x\|_1$. I also know how to compute the proximal of the squared error loss $\lVert b - Ax \rVert_2^2$ as it is a quadratic function of $x.$

I may use the pre-composition property (Section 2.2, Proximal Algorithms, Parikh & Boyd) for deriving the proximity operator of $f$, but I am unsure how to define it.

In the same book, they propose this derivation.

If $f(x) = \phi(x) + (\rho/2)\lvert x - a \rvert_2^2$, then $$\boldsymbol{\operatorname{prox}}_{\lambda f}(\nu) = \boldsymbol{\operatorname{prox}}_{\tilde \lambda \phi} \left( (\tilde \lambda / \lambda ) \nu + (\rho \tilde \lambda) a \right).$$

The function $\phi$ can be taken as Lasso. However, in my case $x$ is multiplied by a matrix $A.$

user418560
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  • Has been asked here before. – ViktorStein Oct 26 '22 at 08:07
  • E.g. here. – ViktorStein Oct 26 '22 at 09:05
  • @Ramanujan It's not the same question. I'm interested in the proximal of the sum between the l1-norm and a quadratic term. – user418560 Oct 26 '22 at 09:19
  • I'm not sure if this is explicitly written out anywhere. Under some conditions on $A$, you can often determine $\textrm{prox}_{f\circ A}$ in terms of $\textrm{prox}_f$, $A$, and $A^\top$ (The book by Bauschke & Combettes, vol. 2 has these sort of identities). However, since a linear operator only appears in one of the terms, these results don't immediately apply. So you may need to derive this result yourself, but those proofs should provide candidates for potential assumptions on $A$. Also see http://proximity-operator.net/multivariatefunctions.html for a good reference – Zim Jan 12 '23 at 13:23
  • Thank you for the answer @Zim – user418560 Jun 13 '23 at 15:09

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