Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the nuclear norm for low rankness has an excellent analogy with the $\ell_1$ norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the nuclear norm. We provide a tensor-norm-constrained estimator for the recovery of approximately low-rank matrices from local measurements corrupted with noise. A tensor-norm regularizer is designed to adapt to the local structure. We derive statistical analysis of the estimator over matrix completion and decentralized sketching by applying Maurey's empirical method to tensor products of Banach spaces. The estimator provides a near-optimal error bound in a minimax sense and admits a polynomial-time algorithm for these applications.

翻译：低秩矩阵模型在众多应用中具有广泛用途，从经典系统辨识到信号处理与统计学中更现代的矩阵补全。核范数因其能诱导逆问题的低秩解，被用作低秩性的凸替代。尽管通过奇异值分解，低秩性的核范数与稀疏性的ℓ1范数存在极好的类比关系，但其他矩阵范数同样能诱导低秩性。特别地，当将矩阵解释为巴拿赫空间之间的线性算子时，各种张量积范数推广了核范数的角色。我们提出一种张量范数约束估计器，用于从含噪局部测量中恢复近似低秩矩阵。该估计器设计的张量范数正则化项可自适应局部结构。通过将Maurey经验方法应用于巴拿赫空间的张量积，我们推导了矩阵补全与分布式草图化场景下该估计器的统计分析。该估计器在极小极大意义下达到了近最优误差界，并为这些应用提供了多项式时间算法。