Show that the subgradient of the nuclear norm is given by $$\partial|X| = UV^T + W$$ where $X = U \Sigma V^T $ is the compact SVD of $X$, $W$ is a matrix such that $U^T W = 0$ and $WV = 0$, and $\|W\|_2 \le 1$.

This is exact that question. But I don't know how to get $W$. Can some explain that paper explicitly?


  • $\begingroup$ Please provide some more context behind this question: Is there something in particular you are struggling with? What have you tried? Why are you interested in this? $\endgroup$ – Kitter Catter Mar 3 '17 at 2:04
  • $\begingroup$ I could get the $UV^T $ , but I don't know how to get W $\endgroup$ – Kevin Z Mar 3 '17 at 2:25
  • $\begingroup$ When asking for an explanation of "that paper explicitly", it surely makes sense to give a citation to the paper with title, author, journal, and date. $\endgroup$ – hardmath Mar 3 '17 at 21:24
  • $\begingroup$ This is exact that question, but I don't know how to get W. And I hope some one could explain that paper concerning this . Characterization of the Subdifferential of Some Matrix Norms $\endgroup$ – Kevin Z Mar 3 '17 at 22:11