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, including sparse matrices, matrices with sparse factorizations such as, e.g., sparse R-factors in their QR-decomposition, and algebraic combinations of matrices of low description complexity. The mathematical concept underlying this theory is that of rectifiability, a basic notion in geometric measure theory. Complexity of the sets of matrices encompassed by the theory is measured in terms of Hausdorff and 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. Specifically, we show that matrices taken from a set $\mathcal{U}$ such that $\mathcal{U}-\mathcal{U}$ has Hausdorff dimension $s$ %(or is countably $s$-rectifiable) can be recovered from $k>s$ measurements, and random matrices supported on a set $\mathcal{U}$ of Hausdorff dimension $s$ %(or a countably $s$-rectifiable set) can be recovered with probability 1 from $k>s$ measurements. What is more, we establish the existence of $\beta$-H\"older continuous decoders 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.

翻译：我们提出了一种矩阵补全理论，该理论超越了文献中常讨论的低秩结构，适用于低描述复杂度的广义矩阵，包括稀疏矩阵、具有稀疏分解（例如QR分解中的稀疏R因子）的矩阵，以及低描述复杂度矩阵的代数组合。该理论的数学基础是可求长性（rectifiability）——几何测度论中的一个基本概念。理论所涵盖的矩阵集合的复杂度通过Hausdorff维数和Minkowski维数进行度量。我们的目标是刻画线性测量数量（尤其关注秩-1测量）的特征，以确定是否存在一种算法能够实现矩阵重构——根据具体设定，可以是完美重构、以概率1成功重构，或任意小误差概率重构。具体而言，我们证明：取自集合$\mathcal{U}$（满足$\mathcal{U}-\mathcal{U}$的Hausdorff维数为$s$）的矩阵，可通过$k>s$次测量恢复；而支撑在Hausdorff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$\beta$-Hölder连续解码器的存在性：对于取自上Minkowski维数为$s$的集合的矩阵，可通过$k>2s/(1-\beta)$次测量恢复；对于支撑在上Minkowski维数为$s$的集合上的随机矩阵，可通过$k>s/(1-\beta)$次测量实现任意小误差概率的重构。