Low Rank Tensor Completion via Adaptive ADMM

arXiv stat.ML / 5/6/2026

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Key Points

The paper introduces a new low-rank tensor completion (TC) algorithm for partially observed tensors, extending matrix completion to the tensor setting.
It reformulates nuclear-norm minimization into multiple subproblems and solves them iteratively using an ADMM optimization framework with closed-form proximal operators.
The method incorporates over-relaxation and an adaptive penalty-parameter update strategy to accelerate convergence and improve performance.
Experiments show the proposed approach achieves lower normalized mean square error (NMSE) than existing state-of-the-art methods, including standard nuclear-norm approaches and hybrids with matrix factorization.
Convergence is further improved by initializing ADMM with the solution from a current state-of-the-art method, indicating practical ways to enhance runtime/accuracy.

Abstract

We consider a novel algorithm, for the completion of partially observed low-rank tensors, as a generalization of matrix completion. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, by leveraging the alternating direction method of multipliers (ADMM) optimization framework. To that extend the original NN minimization problem is reformulated into multiple subproblems, which are then solved iteratively via closed-form proximal operators, making use of over-relaxation and an adaptive penalty parameter update scheme, to further speed up convergence and improve the overall performance of the method. Simulation results demonstrate the superior performance of the new method in terms of normalized mean square error (NMSE), compared to the conventional state-of-the-art (SotA) techniques, including NN minimization approaches, as well as a mixture of the latter with a matrix factorization approach, while its convergence can be significantly improved by initializing the algorithm with the solution of the SotA.