Optimal Schatten-q and Ky-Fan-k Norm Rate of Low Rank Matrix Estimation

In this paper, we consider low rank matrix estimation using either matrix-version Dantzig Selector hatAλd or matrix-version LASSO estimator hatAλL. We consider sub-Gaussian measurements, i.e., the measurements X1,ldots,Xn∈ℝm× m have i.i.d. sub-Gaussian entries. Suppose textrmrank(A0)=r. We proved that, when n≥ Cm[r2vee rlog(m)log(n)] for some C>0, both hatAλd and hatAλL can obtain optimal upper bounds(except some logarithmic terms) for estimation accuracy under spectral norm. By applying metric entropy of Grassmann manifolds, we construct (near) matching minimax lower bound for estimation accuracy under spectral norm. We also give upper bounds and matching minimax lower bound(except some logarithmic terms) for estimation accuracy under Schatten-q norm for every 1≤ q≤∞. As a direct corollary, we show both upper bounds and minimax lower bounds of estimation accuracy under Ky-Fan-k norms for every 1≤ k≤ m.