迭代结构矩阵完备化的一种改进
An Adaptation for Iterative Structured Matrix Completion
摘要：
从已知条目的子集预测矩阵的缺失条目的任务被称为\textit{matrix completion}。在当今数据驱动的世界中，数据完成无论是主要目标还是预处理步骤都至关重要。结构化矩阵补全包括数据不是均匀随机丢失的任何设置。在最近的工作中，修改的标准核范数最小化（NNM）矩阵完成已被开发，以考虑到\emph{sparsity-based}结构的缺失条目。这种结构的概念在包括推荐系统在内的许多设置中都是有动机的，其中条目被观察到的概率取决于条目的值。我们建议调整迭代重加权最小二乘（IRLS）算法的低秩矩阵完成，以考虑到稀疏结构的缺失条目。我们还提出了一个迭代的梯度投影为基础的实现算法，可以处理大规模矩阵。最后，我们提出了一个强大的阵列的数值实验矩阵的大小，秩和结构的水平。我们表明，我们提出的方法是与调整后的NNM小尺寸的矩阵，往往优于IRLS算法在结构化设置的矩阵大小$1000 \times 1000$。
英文摘要：
The task of predicting missing entries of a matrix, from a subset of known entries, is known as \textit{matrix completion}. In today's data-driven world, data completion is essential whether it is the main goal or a pre-processing step. Structured matrix completion includes any setting in which data is not missing uniformly at random. In recent work, a modification to the standard nuclear norm minimization (NNM) for matrix completion has been developed to take into account \emph{sparsity-based} structure in the missing entries. This notion of structure is motivated in many settings including recommender systems, where the probability that an entry is observed depends on the value of the entry. We propose adjusting an Iteratively Reweighted Least Squares (IRLS) algorithm for low-rank matrix completion to take into account sparsity-based structure in the missing entries. We also present an iterative gradient-projection-based implementation of the algorithm that can handle large-scale matrices. Finally, we present a robust array of numerical experiments on matrices of varying sizes, ranks, and level of structure. We show that our proposed method is comparable with the adjusted NNM on small-sized matrices, and often outperforms the IRLS algorithm in structured settings on matrices up to size $1000 \times 1000$.
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