Yimin Wei

Maolin Che, Liqun Qi, Yimin Wei

The main purpose of this note is to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as, symmetric positive definite tensors and copositive tensors, we derive the existence theorems on a solution of these kinds of nonlinear complementarity problems. We prove that a unique solution of the NCP exists under the condition of diagonalizable tensors.

Huai-An Diao, Yimin Wei, Sanzheng Qiao

Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured normwise, mixed and componentwise condition numbers for the Tikhonov regularization are introduced and their explicit expressions are derived. For the general linear structure, we prove the structured condition numbers are smaller than their corresponding unstructured counterparts based on the derived expressions. By means of the power method and small sample condition estimation, the fast condition estimation algorithms are proposed. Our estimation methods can be integrated into Tikhonov regularization algorithms that use the generalized singular value decomposition (GSVD). The structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts. Our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones, and the proposed condition estimation algorithms are reliable.

Yuchao Wang, Maolin Che, Yimin Wei

The tensor-train (TT) decomposition is widely used to compress large tensors into a more compact form by exploiting their inherent data structures. A fundamental approach for constructing the TT format is the well-known TT-SVD method, which performs singular value decompositions (SVDs) on the successive matrices sequentially. But in practical applications, it is often unnecessary to compute full SVDs. In this article, we propose a new method called the TT-UTV. It utilizes the virtues of rank-revealing UTV decomposition to compute the TT format for a large-scale tensor, resulting in lower computational cost. We analyze the error bounds on the accuracy of these algorithms in both the URV and ULV cases and then recommend different sweep patterns for these two cases. Based on the theoretical analysis, we also formulate the rank-adaptive algorithms with prescribed accuracy. Numerical experiments on various applications, including magnetic resonance imaging data completion, are performed to illustrate their good performance in practice.

Liping Zhang, Yimin Wei

In this paper, we apply randomized algorithms to approximate the total least squares (TLS) solution of the problem $Ax\approx b$ in the large-scale discrete ill-posed problems. A regularization technique, based on the multiplicative randomization and the subspace iteration, is proposed to obtain the approximate core problem.In the error analysis, we provide upper bounds %in terms of the $(k\!\!+\!\!1)$-th singular value of $A$ for the errors of the solution and the residual of the randomized core reduction. Illustrative numerical examples and comparisons are presented.

Ting-ting Feng, Gang Wu, Yimin Wei

We revisit the shift-and-invert Arnoldi method proposed in [S. Lee, H. Pang, and H. Sun. {\it Shift-invert Arnoldi approximation to the Toeplitz matrix exponential}, SIAM J. Sci. Comput., 32: 774--792, 2010] for numerical approximation to the product of Toeplitz matrix exponential with a vector. In this approach, one has to solve two large scale Toeplitz linear systems in advance. However, if the desired accuracy is high, the cost will be prohibitive. Therefore, it is interesting to investigate how to solve the Toeplitz systems inexactly in this method. The contribution of this paper is in three regards. First, we give a new stability analysis on the Gohberg-Semencul formula (GSF) and define the GSF condition number of a Toeplitz matrix. It is shown that, when the size of the Toeplitz matrix is large, our result is sharper than the one given in [M. Gutknecht and M. Hochbruck. {\it The stability of inversion formulas for Toeplitz matrices}, Linear Algebra Appl., 223/224: 307--324, 1995]. Second, we establish a relation between the error of Toeplitz systems and the residual of Toeplitz matrix exponential. We show that if the GSF condition number of the Toeplitz matrix is medium sized, then the Toeplitz systems can be solved in a low accuracy. Third, based on this relationship, we present a practical stopping criterion for relaxing the accuracy of the Toeplitz systems, and propose an inexact shift-and-invert Arnoldi algorithm for the Toeplitz matrix exponential problem. Numerical experiments illustrate the numerical behavior of the new algorithm, and show the effectiveness of our theoretical results.

Feiyang Han, Yun Miao, Zhaoyi Sun et al.

Image classification is one of the most fundamental tasks in Computer Vision. In practical applications, the datasets are usually not as abundant as those in the laboratory and simulation, which is always called as Data Hungry. How to extract the information of data more completely and effectively is very important. Therefore, an Adaptive Data Augmentation Framework based on the tensor T-product Operator is proposed in this paper, to triple one image data to be trained and gain the result from all these three images together with only less than 0.1% increase in the number of parameters. At the same time, this framework serves the functions of column image embedding and global feature intersection, enabling the model to obtain information in not only spatial but frequency domain, and thus improving the prediction accuracy of the model. Numerical experiments have been designed for several models, and the results demonstrate the effectiveness of this adaptive framework. Numerical experiments show that our data augmentation framework can improve the performance of original neural network model by 2%, which provides competitive results to state-of-the-art methods.

Feiyang Han, Yimin Wei, Zhaofeng Liu et al.

Reinforcement Learning from Human Feedback (RLHF) has played a crucial role in the success of large models such as ChatGPT. RLHF is a reinforcement learning framework which combines human feedback to improve learning effectiveness and performance. However, obtaining preferences feedback manually is quite expensive in commercial applications. Some statistical commercial indicators are usually more valuable and always ignored in RLHF. There exists a gap between commercial target and model training. In our research, we will attempt to fill this gap with statistical business feedback instead of human feedback, using AB testing which is a well-established statistical method. Reinforcement Learning from Statistical Feedback (RLSF) based on AB testing is proposed. Statistical inference methods are used to obtain preferences for training the reward network, which fine-tunes the pre-trained model in reinforcement learning framework, achieving greater business value. Furthermore, we extend AB testing with double selections at a single time-point to ANT testing with multiple selections at different feedback time points. Moreover, we design numerical experiences to validate the effectiveness of our algorithm framework.

Juefei Chen, Longxiu Huang, Yimin Wei

Nonnegative Matrix Factorization (NMF) is an important unsupervised learning method to extract meaningful features from data. To address the NMF problem within a polynomial time framework, researchers have introduced a separability assumption, which has recently evolved into the concept of coseparability. This advancement offers a more efficient core representation for the original data. However, in the real world, the data is more natural to be represented as a multi-dimensional array, such as images or videos. The NMF's application to high-dimensional data involves vectorization, which risks losing essential multi-dimensional correlations. To retain these inherent correlations in the data, we turn to tensors (multidimensional arrays) and leverage the tensor t-product. This approach extends the coseparable NMF to the tensor setting, creating what we term coseparable Nonnegative Tensor Factorization (NTF). In this work, we provide an alternating index selection method to select the coseparable core. Furthermore, we validate the t-CUR sampling theory and integrate it with the tensor Discrete Empirical Interpolation Method (t-DEIM) to introduce an alternative, randomized index selection process. These methods have been tested on both synthetic and facial analysis datasets. The results demonstrate the efficiency of coseparable NTF when compared to coseparable NMF.

Renjie Xu, Chong Wu, Maolin Che et al.

We propose sparseGeoHOPCA, a novel framework for sparse higher-order principal component analysis (SHOPCA) that introduces a geometric perspective to high-dimensional tensor decomposition. By unfolding the input tensor along each mode and reformulating the resulting subproblems as structured binary linear optimization problems, our method transforms the original nonconvex sparse objective into a tractable geometric form. This eliminates the need for explicit covariance estimation and iterative deflation, enabling significant gains in both computational efficiency and interpretability, particularly in high-dimensional and unbalanced data scenarios. We theoretically establish the equivalence between the geometric subproblems and the original SHOPCA formulation, and derive worst-case approximation error bounds based on classical PCA residuals, providing data-dependent performance guarantees. The proposed algorithm achieves a total computational complexity of $O\left(\sum_{n=1}^{N} (k_n^3 + J_n k_n^2)\right)$, which scales linearly with tensor size. Extensive experiments demonstrate that sparseGeoHOPCA accurately recovers sparse supports in synthetic settings, preserves classification performance under 10$\times$ compression, and achieves high-quality image reconstruction on ImageNet, highlighting its robustness and versatility.

Yimin Wei, Hao Liu, Tingting Xie et al.

Effective learning of spatial-temporal information within a point cloud sequence is highly important for many down-stream tasks such as 4D semantic segmentation and 3D action recognition. In this paper, we propose a novel framework named Point Spatial-Temporal Transformer (PST2) to learn spatial-temporal representations from dynamic 3D point cloud sequences. Our PST2 consists of two major modules: a Spatio-Temporal Self-Attention (STSA) module and a Resolution Embedding (RE) module. Our STSA module is introduced to capture the spatial-temporal context information across adjacent frames, while the RE module is proposed to aggregate features across neighbors to enhance the resolution of feature maps. We test the effectiveness our PST2 with two different tasks on point cloud sequences, i.e., 4D semantic segmentation and 3D action recognition. Extensive experiments on three benchmarks show that our PST2 outperforms existing methods on all datasets. The effectiveness of our STSA and RE modules have also been justified with ablation experiments.

Xuezhong Wang, Maolin Che, Liqun Qi et al.

Nonlinear gradient dynamic approach for solving the tensor complementarity problem (TCP) is presented. Theoretical analysis shows that each of the defined dynamical system models ensures the convergence performance. The computer simulation results further substantiate that the considered dynamical system can solve the tensor complementarity problem (TCP).

Lizhu Sun, Baodong Zheng, Changjiang Bu et al.

Let $\mathcal{A}$ be an order $t$ dimension $m\times n\times \cdots \times n$ tensor over complex field. In this paper, we study some {generalized inverses} of $\mathcal{A}$, the {$k$-T-idempotent tensors} and the idempotent tensors based on the general tensor product. Using the tensor generalized inverse, some solutions of the equation $\mathcal{A}\cdot x^{t-1}=b$ are given, where $x$ and $b$ are dimension $n$ and $m$ vectors, respectively. The {generalized inverses} of some block tensors, the eigenvalues of {$k$-T-idempotent tensors} and idempotent tensors are given. And the relation between the generalized inverses of tensors and the $k$-T-idempotent tensors is also showed.

Pengpeng Xie, Yimin Wei, Hua Xiang

In this paper, we present perturbation analysis and randomized algorithms for the total least squares (TLS) problems. We derive the perturbation bound and check its sharpness by numerical experiments. Motivated by the recently popular probabilistic algorithms for low-rank approximations, we develop randomized algorithms for the TLS and the truncated total least squares (TTLS) solutions of large-scale discrete ill-posed problems, which can greatly reduce the computational time and still keep good accuracy.