论文标题
在新参数化下完成的最佳精确矩阵完成
Optimal Exact Matrix Completion Under new Parametrization
论文作者
论文摘要
我们研究了自适应采样方法的$ m \ times n $ suble级矩阵的精确完成问题。 我们介绍了确切完成问题的关系与列和行空间的最稀少向量（我们称之为\ textit {sparsity-number}此处）。使用此关系，我们提出了矩阵完成算法，以恢复目标矩阵。这些算法以两种重要方式优于以前的作品。首先，我们的算法精确地恢复了$μ_0$ - coherent列列矩阵，概率至少$1-ε$，使用比$ \ MATHCAL {O}较小的复杂性（μ_0rn \ mathrm {logRm {log} \ frac} \ frac {r} + frac {r}ε）。具体而言，许多先前的自适应采样方法都需要在柱空间高度连贯时观察整个矩阵。但是，我们表明我们的方法仍然能够通过在许多情况下观察一小部分条目来恢复这种类型的矩阵。其次，我们提出了一种精确的完成算法，该算法需要最小的预报，因为行或列空间没有高度相干。在本文的最后，我们提供了实验结果，以说明此处提出的算法的强度。
We study the problem of exact completion for $m \times n$ sized matrix of rank $r$ with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces (which we call \textit{sparsity-number} here). Using this relation, we propose matrix completion algorithms that exactly recovers the target matrix. These algorithms are superior to previous works in two important ways. First, our algorithms exactly recovers $μ_0$-coherent column space matrices by probability at least $1 - ε$ using much smaller observations complexity than $\mathcal{O}(μ_0 rn \mathrm{log}\frac{r}ε)$ the state of art. Specifically, many of the previous adaptive sampling methods require to observe the entire matrix when the column space is highly coherent. However, we show that our method is still able to recover this type of matrices by observing a small fraction of entries under many scenarios. Second, we propose an exact completion algorithm, which requires minimal pre-information as either row or column space is not being highly coherent. At the end of the paper, we provide experimental results that illustrate the strength of the algorithms proposed here.
