Optimal Schatten-q and Ky-Fan-k Norm Rate of Low Rank Matrix Estimation
This provides theoretical guarantees for matrix estimation in high-dimensional settings, but it is incremental as it extends existing norm analyses to more general cases.
The paper tackles low-rank matrix estimation using Dantzig Selector and LASSO variants with sub-Gaussian measurements, proving that these estimators achieve optimal upper bounds (up to logarithmic terms) under spectral, Schatten-q, and Ky-Fan-k norms when the sample size is sufficiently large, with matching minimax lower bounds established for spectral and Schatten-q norms.
In this paper, we consider low rank matrix estimation using either matrix-version Dantzig Selector $\hat{A}_λ^d$ or matrix-version LASSO estimator $\hat{A}_λ^L$. We consider sub-Gaussian measurements, $i.e.$, the measurements $X_1,\ldots,X_n\in\mathbb{R}^{m\times m}$ have $i.i.d.$ sub-Gaussian entries. Suppose $\textrm{rank}(A_0)=r$. We proved that, when $n\geq Cm[r^2\vee r\log(m)\log(n)]$ for some $C>0$, both $\hat{A}_λ^d$ and $\hat{A}_λ^L$ 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 $1\leq q\leq\infty$. As a direct corollary, we show both upper bounds and minimax lower bounds of estimation accuracy under Ky-Fan-k norms for every $1\leq k\leq m$.