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：我们提出了一种超越现有文献中常考虑的低秩结构、适用于一般低描述复杂度矩阵的矩阵补全理论。具体而言，该理论所涵盖的矩阵集合的复杂度将通过豪斯多夫维数和上闵可夫斯基维数来衡量。我们的目标是刻画线性测量（尤其关注秩-1测量）的数量，以确保存在某种算法能够实现完美重建（概率为1），或根据设定以任意小的误差概率完成重建。具体地，我们证明了：若取自集合 $\mathcal{U}$ 的矩阵满足 $\mathcal{U}-\mathcal{U}$ 的豪斯多夫维数为 $s$，则可通过 $k>s$ 次测量恢复该矩阵；而支撑在豪斯多夫维数 $s$ 的集合 $\mathcal{U}$ 上的随机矩阵，则能以概率1通过 $k>s$ 次测量恢复。此外，我们证明了存在对测量中的加性扰动或噪声具有鲁棒性的恢复映射。具体地，我们展示了：存在 $\beta$-赫尔德连续映射可从 $k>2s/(1-\beta)$ 次测量中恢复取自上闵可夫斯基维数 $s$ 的集合中的矩阵；而支撑在上闵可夫斯基维数 $s$ 的集合上的随机矩阵，则能以任意小的误差概率通过 $k>s/(1-\beta)$ 次测量恢复。我们考虑的具体实例包括低秩矩阵、稀疏矩阵、具有稀疏R分量的QR分解以及分形矩阵。