Communications, Electronics and Sensing Systems logo

Articles

Vol. 1 No. 4 (2026)

Nonconvex Spectral Regularization for Low-Rank Tensor Recovery from Incomplete Observations

Submitted
July 31, 2026
Published
August 7, 2026

Abstract

The recovery of high-dimensional data from incomplete or corrupted observations is a foundational challenge in signal processing, computer vision, and machine learning. Traditional approaches often rely on matrix unfolding and the convex nuclear norm to approximate low-rank structures, which inherently neglects the multidimensional nature of the data and over-penalizes significant singular values. This paper introduces a comprehensive framework based on nonconvex spectral regularization for low-rank tensor recovery. By operating directly on the tensor structure and applying a nonconvex penalty to the singular values of the unfolded matrices, the proposed methodology achieves tighter approximations of the true tensor rank compared to conventional convex relaxations. The optimization process is driven by a custom adaptation of the alternating direction method of multipliers, designed to handle the nonconvexity of the objective function while ensuring stable convergence properties. Through rigorous theoretical analysis and extensive empirical evaluations on both synthetic and real-world multidimensional datasets, the proposed algorithm demonstrates superior performance in recovering missing entries. The results indicate significant improvements in recovery accuracy and robustness against varying degrees of data loss, thereby providing a robust solution for multidimensional data imputation tasks across various scientific and engineering domains.

References

  1. 1. Gu, C., Zhang, W., Huang, Z., Kou, J., Liu, Z., Zhao, C., Liu, C., Zhang, L., Lin, W., Wang, Z., Deng, J., Xie, Y., Huang, G., Zhang, C., Lu, X., Wang, C., Zhang, Z., Yuan, H., Duan, X., & Fang, Y. (2024). LENS: Layers of evaluation of hallucination in GenAI systems. In 2024 7th International Conference on Universal Village (UV) (pp. 1–85). IEEE. https://doi.org/10.1109/UV63228.2024.11189150
  2. 2. Wang, J., Chen, L., & Al Faruque, M. A. (2023, October). Domino: Domain-invariant hyperdimensional classification for multi-sensor time series data. In2023 IEEE/ACM International Conference on Computer Aided Design (ICCAD)(pp. 1-9). IEEE.
  3. 3. Luo H, Li Q, Zhou H, et al. Weak single phase-to-ground fault time detection with learnable-parameter-driven DSP-enhanced transformer[J]. IEEE Transactions on Smart Grid, 2025.
  4. 4. Liang, R., Xu, S., Xie, C., Chen, J., Ren, F., Yang, S., & Yabe, T. (2026). Abstain Mask Retain Core: Time Series Prediction by Adaptive Masking Loss with Representation Consistency. Advances in Neural Information Processing Systems, 38, 99445-99471.
  5. 5. Sang, Y. (2025, October). AutoCrit: A Meta-Reasoning Framework for Self-Critique and Iterative Error Correction in LLM Chains-of-Thought. In 2025 6th International Conference on Machine Learning and Computer Application (ICMLCA) (pp. 1177-1180). IEEE.
  6. 6. Han, Z., Chen, W., Han, Y., Mao, R., & Qin, J. (2026). Fast Diversified Top-k Rule Discovery via User-Guided Embeddings.IEEE Transactions on Knowledge and Data Engineering.
  7. 7. Yin, F., Xue, X., Zhang, C., Zhang, K., Han, J., Liu, B., ... & Yao, J. (2021). Multifidelity genetic transfer: an efficient framework for production optimization.Spe Journal,26(04), 1614-1635.
  8. 8. Wei, W. D., Fang, H. B., Liu, Y. D., Shi, J. X., Kwok, J., & Li, Y. F. (2026). Activation Compression in LLMs: Theoretical Analysis and Efficient Algorithm. arXiv preprint arXiv:2605.01255.
  9. 9. Cao, C., Xue, X., Zhang, K., Song, L., Zhang, L., Yan, X.,... & Liu, C. (2024). Competitive knowledge transfer–enhanced surrogate-assisted search for production optimization. SPE Journal, 29 (06), 3277-3292.
  10. 10. Xu, Y., Shao, K., Ishikawa, I., Hashimoto, Y., Logothetis, N., & Shen, Z. (2025). A data-driven framework for Koopman semigroup estimation in stochastic dynamical systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(10).
  11. 11. Hu, H., Yang, Y., Zhao, Q., & Zhang, C. (2022, June). On the role of discount factor in offline reinforcement learning. In International conference on machine learning (pp. 9072-9098). PMLR.
  12. 12. Wang, J., & Faruque, M. A. A. (2023). Robust and scalable hyperdimensional computing with brain-like neural adaptations.arXiv preprint arXiv:2311.07705.
  13. 13. Zhang, J., Shi, Y., Ma, Y., Xu, L., Yu, J., & Wang, J. (2023, June). Ikol: Inverse kinematics optimization layer for 3d human pose and shape estimation via gauss-newton differentiation. In Proceedings of the AAAI conference on artificial intelligence (Vol. 37, No. 3, pp. 3454-3462).
  14. 14. Qu, D., Ma, Y., Yan, J., & Pyrozhenko, M. (2026). TriMeta-BFNet: A Tri-Meta Stacked Atypical-Frequency Bayesian Fourier Neural Network for Hallucination-Resistant Community Detection. Mathematics, 14 (13), 2283.
  15. 15. Ma, Y., Qu, D., & Wang, Y. (2026). HALO-GNN: hallucination-resistant temporal graph neural networks for dynamic community detection.Scientific Reports.
  16. 16. Ma, Y., & Qu, D. (2026, February). IMFM-GNN: Interpretable Multimodel GNN Via Fourier-Markov Dynamics for Community Detection in Sustainable Energy Networks. In 2026 International Conference on Artificial Intelligence, Computer, Data Sciences and Applications (ACDSA) (pp. 1-6). IEEE.
  17. 17. Breiman, L. (2001). Random forests. Machine Learning, 45(1), 5–32.
  18. 18. Han, Z., Chen, W., & Han, Y. (2024, August). Data Quality-Driven Top-k Rule Discovery. In International Conference on Machine Learning, Cloud Computing and Intelligent Mining (pp. 24-38). Singapore: Springer Nature Singapore.
  19. 19. Yang Y, Hu H, Mao Y, et al. OPRIDE: Offline Preference-based Reinforcement Learning via In-Dataset Exploration[J]. arXiv preprint arXiv:2604.02349, 2026.
  20. 20. Shor, P. W. (1997). Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5), 1484–1509.
  21. 21. Pei, Zhengqi, Qingming Huang, and Shuhui Wang. "When LLMs develop languages: Symbolic communication for efficient multi-agent reasoning." arXiv preprint arXiv:2606.29354 (2026).
  22. 22. Wei, W. D., Yang, X. W., Shao, J. J., & Guo, L. Z. (2025, August). Curriculum abductive learning for mitigating reasoning shortcuts. In Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence, IJCAI-25 (pp. 6533-6541).
  23. 23. Wang, J., & Al Faruque, M. A. (2025, April). Transformer-based contrastive meta-learning for low-resource generalizable activity recognition. In ICASSP 2025-2025 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)(pp. 1-5). IEEE.
  24. 24. Xue, X., Hu, Y., Feng, L., Zhang, K., Song, L., & Tan, K. C. (2024). Surrogate-assisted search with competitive knowledge transfer for expensive optimization. IEEE Transactions on Evolutionary Computation.
  25. 25. Zhang, Y., Zhang, Y., Zhao, M., Feng, S., & Cheung, Y. M. (2026). How to Ask the AI: A User Perspective Survey for Large Language Model Prompting.IEEE Transactions on Artificial Intelligence.
  26. 26. Xiaoteng Ma, Yiqin Yang, Hao Hu, Qihan Liu, Jun Yang, Chongjie Zhang, Qianchuan Zhao, and
  27. Bin Liang. Offline reinforcement learning with value-based episodic memory. arXiv preprint
  28. arXiv:2110.09796, 2021b.
  29. 27. Qu, D., Ma, Y., & Wang, Y. (2026). STR-GNN: stability-regularized graph neural networks for suppressing spurious temporal variations in dynamic community detection. Scientific Reports.