Optimal Schatten-q and Ky-Fan-k Norm Rate of Low Rank Matrix Estimation
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In this paper, we consider low rank matrix estimation using either matrix-version Dantzig Selector Â_λ^d or matrix-version LASSO estimator Â_λ^L. We consider sub-Gaussian measurements, i.e., the measurements X_1,...,X_n∈R^m× m have i.i.d. sub-Gaussian entries. Suppose rank(A_0)=r. We proved that, when n≥ Cm[r^2∨ r(m)(n)] for some C>0, both Â_λ^d and Â_λ^L 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.
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