Open AccessAn adaptation for iterative structured matrix completion
Author(s) -
Henry Adams,
Lara Kassab,
Deaneedell
Publication year - 2021
Publication title -
foundations of data science
Language(s) - English
Resource type - Journals
ISSN - 2639-8001
DOI - 10.3934/fods.2021028
Subject(s) - matrix completion , missing data , matrix (chemical analysis) , algorithm , minification , computer science , matrix norm , low rank approximation , data matrix , mathematics , mathematical optimization , machine learning , gaussian , pure mathematics , clade , biochemistry , eigenvalues and eigenvectors , physics , materials science , chemistry , quantum mechanics , tensor (intrinsic definition) , gene , composite material , phylogenetic tree
The task of predicting missing entries of a matrix, from a subset of known entries, is known as matrix completion . 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 sparsity-based 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 \begin{document}$ 1000 \times 1000 $\end{document} .