Fast methods for denoising matrix completion formulations, with applications to robust seismic data interpolation

TL;DR

This paper introduces LR-BPDN, a fast SVD-free matrix completion algorithm tailored for large-scale applications like seismic data interpolation, incorporating extensions for known subspace information and robustness to measurement errors.

Contribution

The paper presents LR-BPDN, a novel, efficient matrix completion method that targets specific error levels and includes extensions for subspace weighting and robustness to data contamination.

Findings

LR-BPDN outperforms existing methods on collaborative filtering datasets.

The method achieves high-quality seismic data reconstructions with real contaminated data.

Extensions improve results when prior subspace knowledge or measurement errors are present.

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 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.