We propose a theory for matrix completion that goes beyond the low-rank structure commonly considered in the literature and applies to general matrices of low description complexity. Specifically, complexity of the sets of matrices encompassed by the theory is measured in terms of Hausdorff and upper Minkowski dimensions. Our goal is the characterization of the number of linear measurements, with an emphasis on rank-$1$ measurements, needed for the existence of an algorithm that yields reconstruction, either perfect, with probability 1, or with arbitrarily small probability of error, depending on the setup. Concretely, we show that matrices taken from a set $\mathcal{U}$ such that $\mathcal{U}-\mathcal{U}$ has Hausdorff dimension $s$ can be recovered from $k>s$ measurements, and random matrices supported on a set $\mathcal{U}$ of Hausdorff dimension $s$ can be recovered with probability 1 from $k>s$ measurements. What is more, we establish the existence of recovery mappings that are robust against additive perturbations or noise in the measurements. Concretely, we show that there are $\beta$-H\"older continuous mappings recovering matrices taken from a set of upper Minkowski dimension $s$ from $k>2s/(1-\beta)$ measurements and, with arbitrarily small probability of error, random matrices supported on a set of upper Minkowski dimension $s$ from $k>s/(1-\beta)$ measurements. The numerous concrete examples we consider include low-rank matrices, sparse matrices, QR decompositions with sparse R-components, and matrices of fractal nature.

翻译：我们提出了一种矩阵补全理论，超越了文献中通常考虑的低秩结构，适用于一般低描述复杂度的矩阵。具体而言，该理论所涵盖的矩阵集合的复杂度通过豪斯多夫维数和上闵可夫斯基维数来度量。我们的目标是刻画所需线性测量（特别强调秩-1测量）的数量，以确保存在一种算法能够实现重构——根据具体设定，重构可以是完美的（以概率1实现），或者具有任意小的误差概率。具体来说，我们证明了从集合$\mathcal{U}$中选取的矩阵，若$\mathcal{U}-\mathcal{U}$的豪斯多夫维数为$s$，则可通过$k>s$次测量进行恢复；而支撑在豪斯多夫维数为$s$的集合$\mathcal{U}$上的随机矩阵，可通过$k>s$次测量以概率1恢复。此外，我们建立了存在能够抵抗测量中加性扰动或噪声的恢复映射。具体而言，我们证明了存在$\beta$-赫尔德连续映射，能够从$k>2s/(1-\beta)$次测量中恢复取自上闵可夫斯基维数为$s$的集合的矩阵，并且能够以任意小的误差概率，从$k>s/(1-\beta)$次测量中恢复支撑在上闵可夫斯基维数为$s$的集合上的随机矩阵。我们考虑的众多具体示例包括低秩矩阵、稀疏矩阵、具有稀疏R分量的QR分解，以及具有分形性质的矩阵。