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Rank Minimization with Structured Data Patterns

机译：使用结构化数据模式等级最小化

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摘要

The problem of finding a low rank approximation of a given measurement matrix is of key interest in computer vision. If all the elements of the measurement matrix are available, the problem can be solved using factorization. However, in the case of missing data no satisfactory solution exists. Recent approaches replace the rank term with the weaker (but convex) nuclear norm. In this paper we show that this heuristic works poorly on problems where the locations of the missing entries are highly correlated and structured which is a common situation in many applications. Our main contribution is the derivation of a much stronger convex relaxation that takes into account not only the rank function but also the data. We propose an algorithm which uses this relaxation to solve the rank approximation problem on matrices where the given measurements can be organized into overlapping blocks without missing data. The algorithm is computationally efficient and we have applied it to several classical problems including structure from motion and linear shape basis estimation. We demonstrate on both real and synthetic data that it outperforms state-of-the-art alternatives.
机译：找到给定测量矩阵的低秩近似的问题是计算机视觉的关键兴趣。如果可用测量矩阵的所有元素，则可以使用因子化解决问题。但是，在缺失数据的情况下，不存在令人满意的解决方案。最近的方法用较弱（但凸）核规范取代秩序。在本文中，我们表明，这种启发式作品对缺失条目的位置具有高度相关性和结构的问题，这是许多应用中的常见情况。我们的主要贡献是推导出更强大的凸松弛，不仅考虑了等级功能，还考虑了数据。我们提出了一种算法，它使用这种放松来解决矩阵上的秩近似问题，其中给定的测量可以组织成没有丢失数据的重叠块。该算法是计算的高效，我们已经将其应用于几个经典问题，包括来自运动和线性形状基估计的结构。我们展示了真实的和合成数据，它优于最先进的替代品。

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