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New Hardness Results for Low-Rank Matrix Completion

Authors Dror Chawin, Ishay Haviv

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ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Machine learning theory
  • Theory of computation → Problems, reductions and completeness
Keywords
  • hardness of approximation
  • low-rank matrix completion
  • graph coloring

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Abstract

The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy additional structural constraints, such as positive semi-definiteness or a bounded infinity norm. The problem arises in various research fields, including machine learning, statistics, and theoretical computer science, and has broad real-world applications.
This paper presents new NP-hardness results for low-rank matrix completion problems. We show that for every sufficiently large integer d and any real number ε ∈ [2^{-O(d)},1/7], given a partial matrix A with exposed values of magnitude at most 1 that admits a positive semi-definite completion of rank d, it is NP-hard to find a positive semi-definite matrix that agrees with each given value of A up to an additive error of at most ε, even when the rank is allowed to exceed d by a multiplicative factor of O (1/(ε²⋅log(1/ε))). This strengthens a result of Hardt, Meka, Raghavendra, and Weitz (COLT, 2014), which applies to multiplicative factors smaller than 2 and to ε that decays polynomially in d. We establish similar NP-hardness results for the case where the completed matrix is constrained to have a bounded infinity norm (rather than be positive semi-definite), for which all previous hardness results rely on complexity assumptions related to the Unique Games Conjecture. Our proofs involve a novel notion of nearly orthonormal representations of graphs, the concept of line digraphs, and bounds on the rank of perturbed identity matrices.

Cite As Get BibTex

Dror Chawin and Ishay Haviv. New Hardness Results for Low-Rank Matrix Completion. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.37

Author Details

Dror Chawin
  • The Academic College of Tel Aviv-Yaffo, Tel Aviv, Israel
Ishay Haviv
  • The Academic College of Tel Aviv-Yaffo, Tel Aviv, Israel

Funding

  • Chawin, Dror: Research supported by the Israel Science Foundation (grant No. 1218/20).
  • Haviv, Ishay: Research supported by the Israel Science Foundation (grant No. 1218/20).

Acknowledgements

We thank the anonymous reviewers for their insightful comments and suggestions that improved the presentation of this paper.

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