A universal expectation bound on empirical projections of deformed random matrices

Let $C$ be a real-valued $M\times M$ matrix with singular values ${\lambda }_1\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge {\lambda }_M$ and $E$ a random matrix of centered i.i.d. entries with finite fourth moment. In this paper we give a universal upper bound on the expectation of $\mathrm{\mid }\mathrm{\mid }{\widehat{\pi }}_{r}X\mathrm{\mid }{\mathrm{\mid }}_{S_2}^2-\mathrm{\mid }\mathrm{\mid }{\pi }_{r}X\mathrm{\mid }{\mathrm{\mid }}_{S_2}^2$, where $X:=C+E$ and ${\widehat{\pi }}_r$ (resp. ${\pi }_r$) is a rank-$r$ projection maximizing the Hilbert-Schmidt norm $\mathrm{\mid }\mathrm{\mid }{\tilde{\pi }}_{r}X\mathrm{\mid }{\mathrm{\mid }}_{S_2}$ (resp. $\mathrm{\mid }\mathrm{\mid }{\tilde{\pi }}_{r}C\mathrm{\mid }{\mathrm{\mid }}_{S_2}$) over the set ${\mathrm{\S }}_{M,r}$ of all orthogonal rank-$r$ projections. This result is a generalization of a theorem for Gaussian matrices due to Rohde (2012). Our approach differs substantially from the techniques of the mentioned article. We analyze $\mathrm{\mid }\mathrm{\mid }{\widehat{\pi }}_{r}X\mathrm{\mid }{\mathrm{\mid }}_{S_2}^2-\mathrm{\mid }\mathrm{\mid }{\pi }_{r}X\mathrm{\mid }{\mathrm{\mid }}_{S_2}^2$ from a rather deterministic point of view by an upper bound on $\mathrm{\mid }\mathrm{\mid }{\widehat{\pi }}_{r}X\mathrm{\mid }{\mathrm{\mid }}_{S_2}^2-\mathrm{\mid }\mathrm{\mid }{\pi }_{r}X\mathrm{\mid }{\mathrm{\mid }}_{S_2}^2$, whose randomness is totally determined by the largest singular value of $E$.