去噪矩阵完备公式的快速方法及其在稳健地震数据插值中的应用
Fast methods for denoising matrix completion formulations, with applications to robust seismic data interpolation
摘要：
最近的无SVD矩阵分解公式已经为具有数百万行和列的系统实现了秩最小化，为超大规模应用（如地震数据插值）中的矩阵完成铺平了道路。在本文中，我们考虑矩阵完成配方设计，以达到目标数据拟合误差水平的用户提供的，并提出了一种算法称为LR-BPDN，能够利用因子分解配方来解决相应的优化问题。由于从业者通常对目标误差水平有很强的先验知识，因此这种创新使得在实践中应用该算法变得容易，只留下因子秩待确定。在既定的框架内，我们提出了两个扩展，是高度相关的解决数据插值的实际挑战。首先，我们提出了一个加权的扩展，允许已知的子空间信息，以改善矩阵完成配方的结果。我们展示了如何使用这种加权的情况下，频率的延续，地震数据插值的一个重要方面。第二，我们提出了矩阵完成配方，是强大的大测量误差，在现有的数据。我们使用MovieLens 1 M，10 M和Netflix 100 M数据集说明了LR-BPDN在协同过滤问题上的优势。然后，我们使用新的方法，连同其强大的和子空间重新加权扩展，获得高质量的重建大规模地震插值问题与真实数据，即使在数据污染的存在。
英文摘要：
Recent SVD-free matrix factorization formulations have enabled rank minimization for systems with millions of rows and columns, paving the way for matrix completion in extremely large-scale applications, such as seismic data interpolation. In this paper, we consider matrix completion formulations designed to hit a target data-fitting error level provided by the user, and propose an algorithm called LR-BPDN that is able to exploit factorized formulations to solve the corresponding optimization problem. Since practitioners typically have strong prior knowledge about target error level, this innovation makes it easy to apply the algorithm in practice, leaving only the factor rank to be determined. Within the established framework, we propose two extensions that are highly relevant to solving practical challenges of data interpolation. First, we propose a weighted extension that allows known subspace information to improve the results of matrix completion formulations. We show how this weighting can be used in the context of frequency continuation, an essential aspect to seismic data interpolation. Second, we propose matrix completion formulations that are robust to large measurement errors in the available data. We illustrate the advantages of LR-BPDN on the collaborative filtering problem using the MovieLens 1M, 10M, and Netflix 100M datasets. Then, we use the new method, along with its robust and subspace re-weighted extensions, to obtain high-quality reconstructions for large scale seismic interpolation problems with real data, even in the presence of data contamination.
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