scholarly journals An adaptation for iterative structured matrix completion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Henry Adams ◽  
Lara Kassab ◽  
Deanna Needell
Keyword(s):  
Large Scale ◽  
Matrix Completion ◽  
Low Rank ◽  
Norm Minimization ◽  
Structured Matrix ◽  
Rank Matrix ◽  
Processing Step ◽  
Data Completion ◽  
Low Rank Matrix Completion ◽  
Low Rank Matrix

<p style='text-indent:20px;'>The task of predicting missing entries of a matrix, from a subset of known entries, is known as <i>matrix completion</i>. 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 <i>sparsity-based</i> 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 <inline-formula><tex-math id="M1">\begin{document}$ 1000 \times 1000 $\end{document}</tex-math></inline-formula>.</p>

Three-dimensional irregular seismic data reconstruction via low-rank matrix completion

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. V181-V192 ◽  
Author(s):  
Jianwei Ma
Keyword(s):  
Seismic Data ◽  
Matrix Completion ◽  
Low Rank ◽  
Nuclear Norm Minimization ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Low Rank Matrix Completion ◽  
Value Decomposition ◽  
Low Rank Matrix ◽  
Seismic Data Reconstruction

We have developed a new algorithm for the reconstruction of seismic traces randomly missing from a uniform grid of a 3D seismic volume. Several algorithms have been developed for such reconstructions, based on properties of the seismic wavefields and on signal processing concepts, such as sparse signal representation in a transform domain. We have investigated a novel approach, originally introduced for noise removal, which is based on the premise that for suitable representation of the seismic data as matrices or tensors, the rank of the seismic data (computed by singular value decomposition) increases with noise or missing traces. Thus, we apply low-rank matrix completion (MC) with a designed texture-patch transformation to 3D seismic data reconstruction. Low-rank components capture geometrically meaningful structures in seismic data that encompass conventional local features such as events and dips. The low-rank MC is based on nuclear-norm minimization. An efficient [Formula: see text]-norm minimizing algorithm, named approximate message passing, is extended to use for a general nonconvex nuclear-norm minimization problem. A fast MC algorithm named low-rank matrix fitting (LMaFit), which avoids the computation of singular value decomposition, was also considered for the 3D reconstruction. Empirical studies on synthetic and real data have shown promising performance of the method, in comparison with traditional projection onto convex sets.

scholarly journals Clipped Matrix Completion: A Remedy for Ceiling Effects

2019 ◽  
Vol 33 ◽  
pp. 5151-5158
Author(s):  
Takeshi Teshima ◽  
Miao Xu ◽  
Issei Sato ◽  
Masashi Sugiyama
Keyword(s):  
Matrix Completion ◽  
Low Rank ◽  
Trace Norm ◽  
Minimization Algorithm ◽  
Benchmark Data ◽  
Hinge Loss ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Recovery Error ◽  
Low Rank Matrix

We consider the problem of recovering a low-rank matrix from its clipped observations. Clipping is conceivable in many scientific areas that obstructs statistical analyses. On the other hand, matrix completion (MC) methods can recover a low-rank matrix from various information deficits by using the principle of low-rank completion. However, the current theoretical guarantees for low-rank MC do not apply to clipped matrices, as the deficit depends on the underlying values. Therefore, the feasibility of clipped matrix completion (CMC) is not trivial. In this paper, we first provide a theoretical guarantee for the exact recovery of CMC by using a trace-norm minimization algorithm. Furthermore, we propose practical CMC algorithms by extending ordinary MC methods. Our extension is to use the squared hinge loss in place of the squared loss for reducing the penalty of overestimation on clipped entries. We also propose a novel regularization term tailored for CMC. It is a combination of two trace-norm terms, and we theoretically bound the recovery error under the regularization. We demonstrate the effectiveness of the proposed methods through experiments using both synthetic and benchmark data for recommendation systems.

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