Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion

• URL: http://arxiv.org/abs/2504.21468v1
• Date: Wed, 30 Apr 2025 09:44:09 GMT
• Title: Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion
• Authors: Yu Guo, Guoqing Chen, Tieyong Zeng, Qiyu Jin, Michael Kwok-Po Ng,
• Abstract summary: This paper introduces the quaternion model normart state the Frobenius framework for this problem.<n>We prove that the QNOF can be simplified to solving the $L1/L$ problem.<n>We also extend the QNOF to robust completion of the quaternion matrix.
• Score: 20.11953064373745
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recovering hidden structures from incomplete or noisy data remains a pervasive challenge across many fields, particularly where multi-dimensional data representation is essential. Quaternion matrices, with their ability to naturally model multi-dimensional data, offer a promising framework for this problem. This paper introduces the quaternion nuclear norm over the Frobenius norm (QNOF) as a novel nonconvex approximation for the rank of quaternion matrices. QNOF is parameter-free and scale-invariant. Utilizing quaternion singular value decomposition, we prove that solving the QNOF can be simplified to solving the singular value $L_1/L_2$ problem. Additionally, we extend the QNOF to robust quaternion matrix completion, employing the alternating direction multiplier method to derive solutions that guarantee weak convergence under mild conditions. Extensive numerical experiments validate the proposed model's superiority, consistently outperforming state-of-the-art quaternion methods.

Related papers

• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution [6.635903943457569]
  We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from non-commutativity of quaternion multiplication.
  arXiv  Detail & Related papers  (2023-07-04T17:28:58Z)
• Symmetric Tensor Networks for Generative Modeling and Constrained Combinatorial Optimization [72.41480594026815]
  Constrained optimization problems abound in industry, from portfolio optimization to logistics. One of the major roadblocks in solving these problems is the presence of non-trivial hard constraints which limit the valid search space. In this work, we encode arbitrary integer-valued equality constraints of the form Ax=b, directly into U(1) symmetric networks (TNs) and leverage their applicability as quantum-inspired generative models.
  arXiv  Detail & Related papers  (2022-11-16T18:59:54Z)
• Orthogonal Nonnegative Matrix Factorization with Sparsity Constraints [0.0]
  This article presents a novel approach to solving the sparsity-constrained Orthogonal Nonnegative Matrix Factorization (SCONMF) problem.<n>By reformulating SCONMF as a capacity-constrained facility-location problem, the proposed method naturally integrates non-negativity, orthogonality, and sparsity constraints.<n>Specifically, our approach integrates control-barrier function (CBF) based framework used for dynamic optimal control design problems with maximum-entropy-principle-based framework used for facility location problems to enforce these constraints while ensuring robust factorization.
  arXiv  Detail & Related papers  (2022-10-06T04:30:59Z)
• Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness [9.649070872824957]
  Matrix functions are utilized to rewrite smooth spectral constrained matrix optimization problems. A new convergence analysis is provided for cubic-regularized Newton for matrix vector spaces.
  arXiv  Detail & Related papers  (2022-09-02T18:11:05Z)
• Log-based Sparse Nonnegative Matrix Factorization for Data Representation [55.72494900138061]
  Nonnegative matrix factorization (NMF) has been widely studied in recent years due to its effectiveness in representing nonnegative data with parts-based representations. We propose a new NMF method with log-norm imposed on the factor matrices to enhance the sparseness. A novel column-wisely sparse norm, named $ell_2,log$-(pseudo) norm, is proposed to enhance the robustness of the proposed method.
  arXiv  Detail & Related papers  (2022-04-22T11:38:10Z)
• Relative Pose from SIFT Features [50.81749304115036]
  We derive a new linear constraint relating the unknown elements of the fundamental matrix and the orientation and scale. The proposed constraint is tested on a number of problems in a synthetic environment and on publicly available real-world datasets on more than 80000 image pairs.
  arXiv  Detail & Related papers  (2022-03-15T14:16:39Z)
• Robust Implicit Networks via Non-Euclidean Contractions [63.91638306025768]
  Implicit neural networks show improved accuracy and significant reduction in memory consumption. They can suffer from ill-posedness and convergence instability. This paper provides a new framework to design well-posed and robust implicit neural networks.
  arXiv  Detail & Related papers  (2021-06-06T18:05:02Z)
• Quaternion Factorization Machines: A Lightweight Solution to Intricate Feature Interaction Modelling [76.89779231460193]
  factorization machine (FM) is capable of automatically learning high-order interactions among features to make predictions without the need for manual feature engineering. We propose the quaternion factorization machine (QFM) and quaternion neural factorization machine (QNFM) for sparse predictive analytics.
  arXiv  Detail & Related papers  (2021-04-05T00:02:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.