通过对低秩之外的额外结构建模来推进矩阵完备化
Advancing Matrix Completion by Modeling Extra Structures beyond Low-Rankness
摘要：
Cand{è}s和Recht建立了一种基于凸优化的著名的低秩矩阵完备化方法。虽然理论上是完备的，但该方法可能无法完全解决低秩矩阵完备化问题。这是因为该方法仅捕获低秩属性，其仅给出数据点位于某个低维子空间上的粗略约束，但通常忽略更详细地指定数据点如何位于子空间上的额外结构。当数据点的几何分布不均匀时，数据的相干参数可能很大，因此，即使我们想要恢复的潜在矩阵是相当低的秩，该方法也可能失败。为了更好地处理非均匀数据，在本文中，我们提出了一种称为低秩因子分解（LRFD）的方法，该方法施加了额外的限制，即数据点必须表示为预先构建或学习的字典中的基的线性组合。我们表明，LRFD可以很好地处理非均匀数据，只要字典配置正确：我们从数学上证明，如果字典本身是低秩的，那么LRFD是免疫的相干参数，这可能是大的非均匀数据。这为学习LRFD中的字典提供了一个基本原则，并且自然地导致了推进矩阵完成的实用算法。在随机生成的矩阵和运动数据集上进行的大量实验显示了令人鼓舞的结果。
英文摘要：
A well-known method for completing low-rank matrices based on convex optimization has been established by Cand{è}s and Recht. Although theoretically complete, the method may not entirely solve the low-rank matrix completion problem. This is because the method captures only the low-rankness property which gives merely a rough constraint that the data points locate on some low-dimensional subspace, but generally ignores the extra structures which specify in more detail how the data points locate on the subspace. Whenever the geometric distribution of the data points is not uniform, the coherence parameters of data might be large and, accordingly, the method might fail even if the latent matrix we want to recover is fairly low-rank. To better handle non-uniform data, in this paper we propose a method termed Low-Rank Factor Decomposition (LRFD), which imposes an additional restriction that the data points must be represented as linear combinations of the bases in a dictionary constructed or learnt in advance. We show that LRFD can well handle non-uniform data, provided that the dictionary is configured properly: We mathematically prove that if the dictionary itself is low-rank then LRFD is immune to the coherence parameters which might be large on non-uniform data. This provides an elementary principle for learning the dictionary in LRFD and, naturally, leads to a practical algorithm for advancing matrix completion. Extensive experiments on randomly generated matrices and motion datasets show encouraging results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
