Abstract We consider the multi-view data completion problem, i.e., to complete a matrix U = [ U 1 | U 2 ] where the ranks of U, U1, and U2 are given. In particular, we investigate the fundamental conditions on the sampling pattern, i.e., locations of the sampled entries for finite completability of such a multi-view data given the corresponding rank constraints. We provide a geometric analysis on the manifold structure for multi-view data to incorporate more than one rank constraint. We derive a probabilistic condition in terms of the number of samples per column that guarantees finite completability with high probability. Finally, we derive the guarantees for unique completability. Numerical results demonstrate reduced sampling complexity when the multi-view structure is taken into account as compared to when only low-rank structure of individual views is taken into account. Then, we propose an apporach using Newton’s method to almost achieve these information-theoretic bounds for mulit-view data retrieval by taking advantage of the rank decomposition and the analysis in this work.

中文翻译：

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低秩多视图数据完成的基本采样模式

摘要 我们考虑多视图数据完成问题，即完成一个矩阵 U = [ U 1 | U 2 ] 其中给出了 U、U1 和 U2 的等级。特别是，我们研究了采样模式的基本条件，即在给定相应等级约束的情况下，对于这种多视图数据的有限可完成性，采样条目的位置。我们为多视图数据的流形结构提供了几何分析，以包含多个秩约束。我们根据每列的样本数推导出一个概率条件，以高概率保证有限的可完成性。最后，我们推导出唯一可完成性的保证。数值结果表明，与仅考虑单个视图的低秩结构相比，考虑多视图结构时采样复杂度降低。然后，我们提出了一种使用牛顿方法的方法，通过利用秩分解和这项工作中的分析，几乎可以实现多视图数据检索的这些信息理论界限。