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通常考虑的低秩结构，适用于具有低描述复杂度的一般矩阵，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。支撑该理论的数学概念是“可整流性”——几何测度论中的一个基本概念。该理论所涵盖的矩阵集合的复杂度通过豪斯多夫维数和闵可夫斯基维数来衡量。我们的目标是刻画线性测量的数量（重点强调秩-1测量），这些测量是算法存在所必需的，该算法能根据设置实现完美重建（概率为1）或任意小的误差概率。具体来说，我们证明：取自集合$\mathcal{U}$（满足$\mathcal{U}-\mathcal{U}$的豪斯多夫维数为$s$，或为可数$s$-可整流）的矩阵，可以从$k>s$次测量中恢复；支撑在豪斯多夫维数为$s$的集合$\mathcal{U}$（或可数$s$-可整流集合）上的随机矩阵，可以从$k>s$次测量中以概率1恢复。此外，我们还建立了$\beta$-赫尔德连续解码器的存在性：对于取自上闵可夫斯基维数为$s$的集合中的矩阵，从$k>2s/(1-\beta)$次测量中恢复；对于支撑在上闵可夫斯基维数为$s$的集合上的随机矩阵，从$k>s/(1-\beta)$次测量中恢复，且误差概率任意小。