A $p$-spherical section property for matrix Schatten-$p$ quasi-norm minimization

Abstract

<p style='text-indent:20px;'>Low-rank matrix recovery has become a popular research topic with various applications in recent years. One of the most popular methods to dual with this problem for overcoming its NP-hardness is to relax it into some tractable optimization problems. In this paper, we consider a nonconvex relaxation, the Schatten-<inline-formula><tex-math id="M1">\begin{document}$p$\end{document}</tex-math></inline-formula> quasi-norm minimization (<inline-formula><tex-math id="M2">\begin{document}$0&lt;p&lt;1$\end{document}</tex-math></inline-formula>), and discuss conditions for the equivalence between the original problem and this nonconvex relaxation. Specifically, based on null space analysis, we propose a <inline-formula><tex-math id="M3">\begin{document}$p$\end{document}</tex-math></inline-formula>-spherical section property for the exact and approximate recovery via the Schatten-<inline-formula><tex-math id="M4">\begin{document}$p$\end{document}</tex-math></inline-formula> quasi-norm minimization (<inline-formula><tex-math id="M5">\begin{document}$0&lt;p&lt;1$\end{document}</tex-math></inline-formula>).