arXiv subjects

Hanxin Liu

Publications and source records attributed to Hanxin Liu.

4 recordsLinked to original sources

arXiv2026

$\mathrm M$-Eigenpairs of Partially Symmetric Tensors: Exact Reformulation and Perturbation Bounds

In this paper, we consider the computation of $M$-eigenpairs of fourth order partially symmetric tensors arising from elasticity theory. First, a lifted fourth order tensor is constructed, and the original $M$-eigenvalue problem is reformulated as a parameterized generalized tensor eigenvalue problem under the $\mathbf B_{\alpha,\beta}$-normalization. Then, an exact correspondence between the two eigenvalue problems is established, which provides a procedure for computing all real $M$-eigenpairs through the proposed reformulation. Furthermore, perturbation bounds for the largest $M$-eigenvalue are derived, and the lifted reformulation is shown to preserve these bounds without introducing any additional relaxation. Finally, numerical experiments are reported to show the effectiveness of the proposed method.

math.OC↗

arXiv2026

New Bounds for Limited Zarankiewicz Numbers from $K_{5t}$ Blocks

The restricted augmented Zarankiewicz number $z_L(m,n)$ yields core combinatorial lower bounds for the maximal SOS rank of biquadratic forms. All previously known infinite admissible graph families rely on $K_{4t}$ incidence graphs, attaining an asymptotic relative gap limit of $1/4$. This work develops a new infinite family built from $K_{5t}$ incidence bipartite graphs with $\mathbb{Z}_5$ cyclic labeling for block partitions. We construct valid nondegenerate intra-block and inter-block 2-edges, derive a quadratic closed-form lower bound of $z_L$, and prove its relative gap converges asymptotically to $2/5$. Full enumeration for $t=1$ verifies the exact value $z_L(10,5)=23$. Under nondegenerate and generalized $C_4$-free constraints, the ratio $2/5$ is shown to be the maximal asymptotic ratio attainable under this block framework. Our results expand the library of extremal bipartite graphs and sharpen lower bounds for biquadratic SOS rank, with further open problems for general $K_{kt}$ constructions outlined in closing.

math.OC↗

arXiv2025

An Image Noise Level Estimation Based on Tensor T-Product

Currently, the noise level of color images is estimated by many algorithms through separate selection of each page of the third-order tensor using sliding blocks of size ${M_1} \times {M_1}$. The data structure of the tensor is disrupted by this method, leading to errors in the estimation results. In order not to disrupt the data structure of the tensor, we directly select the tensor using a sliding block of size ${M_1} \times {M_1} \times 3$ and then re-arrange it. The newly obtained tensor is decomposed into a block diagonal matrix form through T-product. It is demonstrated that the eigenvalues of this matrix are related to the noise level of the color image. Then train the relationship coefficients through learning methods, thereby obtaining the estimated noise level. The effectiveness of the algorithm was verified through numerical experiments, and it also achieved high estimation accuracy.

math.OC↗

arXiv2024

Generalized Low-Rank Matrix Completion Model with Overlapping Group Error Representation

The low-rank matrix completion (LRMC) technology has achieved remarkable results in low-level visual tasks. There is an underlying assumption that the real-world matrix data is low-rank in LRMC. However, the real matrix data does not satisfy the strict low-rank property, which undoubtedly present serious challenges for the above-mentioned matrix recovery methods. Fortunately, there are feasible schemes that devise appropriate and effective priori representations for describing the intrinsic information of real data. In this paper, we firstly model the matrix data ${\bf{Y}}$ as the sum of a low-rank approximation component $\bf{X}$ and an approximation error component $\cal{E}$. This finer-grained data decomposition architecture enables each component of information to be portrayed more precisely. Further, we design an overlapping group error representation (OGER) function to characterize the above error structure and propose a generalized low-rank matrix completion model based on OGER. Specifically, the low-rank component describes the global structure information of matrix data, while the OGER component not only compensates for the approximation error between the low-rank component and the real data but also better captures the local block sparsity information of matrix data. Finally, we develop an alternating direction method of multipliers (ADMM) that integrates the majorization-minimization (MM) algorithm, which enables the efficient solution of the proposed model. And we analyze the convergence of the algorithm in detail both theoretically and experimentally. In addition, the results of numerical experiments demonstrate that the proposed model outperforms existing competing models in performance.

cs.CV↗