How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{X}} \frac{1}{2} \| \boldsymbol{X - Y} \|_F^2 + \lambda \| \boldsymbol{X} \|_{*} \end{equation} where $F$ denotes the Frobenius norm and $*$ denotes the nuclear norm. $\boldsymbol{Y}$ and $\lambda$ are known. $\boldsymbol{X},\boldsymbol{Y} \in C^{N \times M}$.

  • $\begingroup$ are you sure that you need to write it $||x-y||^2_F$ instead of $||x-y||^2_2$. I have seen that people use L2-norm instead of Frobenius norm. $\endgroup$ – user2806363 Jan 5 '17 at 11:15
  • $\begingroup$ Are you sure you need it in the Complex Domain? $\endgroup$ – Royi Mar 10 '18 at 14:18

Are you familiar with proximal algorithms? You are asking how to evaluate the prox operator of the nuclear norm. The answer is given in slide 3-41 here.

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    $\begingroup$ I think the nuance is that question deals with Complex Matrices. Does it hold? As the whole course there is for Real Matrices. $\endgroup$ – Royi Mar 12 '18 at 6:41

Theorem 2.1 in the paper A Singular Value Thresholding Algorithm for Matrix Completion

  • $\begingroup$ But, it's still true for complex matrices? $\endgroup$ – Yesid Fonseca V. Mar 11 '18 at 1:47

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