I'm currently implementing coordinate descent for the LASSO with complex-valued data. For this, one needs a complex version of the soft-thresholding operator, which seems hardly available on the net.

For real-valued data, the definition of the soft-thresholding operator is given in the famous paper of Friedmann, Hastie und Tibshirani, Regularization Paths for Generalized Linear Models via Coordinate Descent, as \begin{aligned} S(z, \lambda) \ = \ \text{sign}(z)\, (|z|-\gamma)_+ \qquad \text{for } z, \, \gamma\in \mathbb R. \end{aligned}

From this, I think it is straightforward to extend it to the complex domain as \begin{aligned} S(z, \lambda) \ = \ e^{i\,\text{arg}(z)} \,(|z|-\gamma)_+ \qquad \text{for } z\in \mathbb C, \, \gamma \in \mathbb R.\\ \end{aligned}

Is this complex-version of the soft-thresholding operator correct?

From a geometric view, this extension seems obvious: all one does is to rotate the real soft-thresholding curve into the complex domain, as visualized in the following picture, which shows the penalization of the radial part for $S(z,5)$:

enter image description here

  • $\begingroup$ Check this paper for details... Single-snapshot DoA estimation using adaptive elastic net in the complex domain $\endgroup$
    – MNTabassum
    Nov 1 '18 at 5:39
  • $\begingroup$ Thanks for your answer. From your nickname you seem to be one of the authors of the paper, so could you please elaborate your answer a bit? Link-only answers are discouraged here (even more because your link does not work), and this is probably the reason why some other guy voted -1 ... which I compensated by my +1 :-) $\endgroup$
    – davidhigh
    Nov 1 '18 at 6:37
  • $\begingroup$ For completeness: The approach in the paper is basically the same than in my question (with the only difference that the case $z=0$ is further explicitly set to zero). $\endgroup$
    – davidhigh
    Nov 1 '18 at 22:59
  • $\begingroup$ Wouldn't it be a pairwise L2 norm regularization? $\endgroup$
    – csjch3cook
    Sep 1 '20 at 7:45

Yes, you're absolutely correct. In Foucart's book ("A mathematical introduction to compressive sensing") you can find this exact same expression on page 72 when they define the soft thresholding operator in the complex case.


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