Fast Methods for Denoising Matrix Completion Formulations, with Applications to Robust Seismic Data Interpolation

Abstract

Abstract. 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 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.1. Introduction. Sparsity-and rank-regularization have had significant impact in many areas over the last several decades. Sparsity in certain transform domains has been exploited to solve underdetermined linear systems with applications to compressed sensing [11,9], natural image denoising/inpainting [41,28,30], and seismic image processing [20,33,18,31]. Analogously, low-rank structure has been used to efficiently solve matrix completion problems, such as the Netflix Prize problem, along with many other applications, including control, system identification, signal processing, and combinatorial optimization [12,35,7], and seismic data interpolation and denoising [34].Regularization formulations for both types of problems introduce a regularization functional of the decision variable, either by adding an explicit penalty to the data-fitting term

Inversion of the Multidimensional Marchenko Equation

“…th Note that for the inversion of equation 4, we do not necessarily require complete data, as we do for iterative substitution (Van der Neut et al, 2015). Even if a system of equations is underdetermined, irregularly sampled and corrupted with noise, satisfying solutions can sometimes still be obtained by posing additional constraints on the solution (Aravkin et al, 2014). With an inversion-based Marchenko methodology, we might benefit from similar advantages.…”

A joint matrix minimization approach for seismic wavefield recovery

“…This approach employs matrix to expresses multiple signals simultaneously. The measurement of matrix row coefficients are expected to exhibit the compact priori of multiple observations which is different from the existed methods based on matrix nuclear-norm minimization 31–34 .…”

“…Recently, matrix-minimization methods with nuclear norm have been developed for seismic wavefield recovery 31–34 which mainly considers the rank reduction as the sparse pattern in 2D cases. To avoid the expensive computations in solving the involved matrix completion optimization problems, a matrix factorization strategy was developed in 31,32 .…”

“…To avoid the expensive computations in solving the involved matrix completion optimization problems, a matrix factorization strategy was developed in 31,32 . This paper proposes a different matrix minimization approach based on l 2, q − l 2, p norm which naturally generalizes the representative vector to matrix in joint distribution sense.…”

“…This approach employs matrix to expresses multiple signals simultaneously. The measurement of matrix row coefficients are expected to exhibit the compact priori of multiple observations which is different from the existed methods based on matrix nuclear-norm minimization 31–34 .A unified algorithm is developed to solve the mixed matrix optimization problem (7) for any p   (0, 1]. This algorithm needs only matrix-vector operations but not matrix factorization which can be easily adapted to large-scale cases.…”