On the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization

arXiv:2607.04875 · cs.CC, cs.IR, math.CO, stat.ML · Submitted 2026-07-06 · Read on arXiv

Nicolas Gillis, Subhayan Saha, Stefano Sicilia, Arnaud Vandaele

cs.CC, cs.IR, math.CO, stat.ML

Submitted: 2026-07-06

Comments: 28 pages, new title and we refined some parts of the paper. code available from https://gitlab.com/ngillis/rank-r_signing/

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• NP-hardness — 18× in this paper · explained in 3 episodes
• Low-Rank Matrix Signing — 10× in this paper
• Matrix Completion — 3× in this paper · explained in 4 episodes
• Sparsity — 3× in this paper · explained in 5 episodes
• Nuclear Norm — 2× in this paper · explained in 2 episodes
• Workshop on Machine Learning for Signal Processing — 2× in this paper
• Clarity — 1× in this paper · explained in 3 episodes
• Composition — 1× in this paper · explained in 2 episodes
• Consistency — 1× in this paper · explained in 3 episodes
• Core — 1× in this paper · explained in 2 episodes
• Federated Learning — 1× in this paper · explained in 19 episodes
• Generalization — 1× in this paper · explained in 14 episodes
• Interpretability — 1× in this paper · explained in 15 episodes
• Loss Function — 1× in this paper · explained in 2 episodes
• Objective Function — 1× in this paper · explained in 2 episodes
• Parameterization — 1× in this paper · explained in 2 episodes
• Singular Value Decomposition (SVD) — 1× in this paper · explained in 3 episodes

Sources

• Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions
• The complexity of computing the minimum rank of a sign pattern matrix
• An extrapolated and provably convergent algorithm for nonlinear matrix decomposition with the ReLU function
• Manifold-based Algorithms for the Hadamard Decomposition
• Binary component decomposition Part II: The asymmetric case
• The square root rank of the correlation polytope is exponential

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