Qingsong Wang

Qingsong Wang, Chunfeng Cui, Deren Han

Recently, there has been a growing interest in the exploration of Nonlinear Matrix Decomposition (NMD) due to its close ties with neural networks. NMD aims to find a low-rank matrix from a sparse nonnegative matrix with a per-element nonlinear function. A typical choice is the Rectified Linear Unit (ReLU) activation function. To address over-fitting in the existing ReLU-based NMD model (ReLU-NMD), we propose a Tikhonov regularized ReLU-NMD model, referred to as ReLU-NMD-T. Subsequently, we introduce a momentum accelerated algorithm for handling the ReLU-NMD-T model. A distinctive feature, setting our work apart from most existing studies, is the incorporation of both positive and negative momentum parameters in our algorithm. Our numerical experiments on real-world datasets show the effectiveness of the proposed model and algorithm. Moreover, the code is available at https://github.com/nothing2wang/NMD-TM.

Qingsong Wang, Yunfei Qu, Chunfeng Cui et al.

Despite the remarkable success of low-rank estimation in data mining, its effectiveness diminishes when applied to data that inherently lacks low-rank structure. To address this limitation, in this paper, we focus on non-negative sparse matrices and aim to investigate the intrinsic low-rank characteristics of the rectified linear unit (ReLU) activation function. We first propose a novel nonlinear matrix decomposition framework incorporating a comprehensive regularization term designed to simultaneously promote useful structures in clustering and compression tasks, such as low-rankness, sparsity, and non-negativity in the resulting factors. This formulation presents significant computational challenges due to its multi-block structure, non-convexity, non-smoothness, and the absence of global gradient Lipschitz continuity. To address these challenges, we develop an accelerated alternating partial Bregman proximal gradient method (AAPB), whose distinctive feature lies in its capability to enable simultaneous updates of multiple variables. Under mild and theoretically justified assumptions, we establish both sublinear and global convergence properties of the proposed algorithm. Through careful selection of kernel generating distances tailored to various regularization terms, we derive corresponding closed-form solutions while maintaining the $L$-smooth adaptable property always holds for any $L\ge 1$. Numerical experiments, on graph regularized clustering and sparse NMF basis compression confirm the effectiveness of our model and algorithm.

Chengcheng Yan, Jiawei Xu, Qingsong Wang et al.

The stochastic gradient descent (SGD) algorithm has achieved remarkable success in training deep learning models. However, it has several limitations, including susceptibility to vanishing gradients, sensitivity to input data, and a lack of robust theoretical guarantees. In recent years, alternating minimization (AM) methods have emerged as a promising alternative for model training by employing gradient-free approaches to iteratively update model parameters. Despite their potential, these methods often exhibit slow convergence rates. To address this challenge, we propose a novel Triple-Inertial Accelerated Alternating Minimization (TIAM) framework for neural network training. The TIAM approach incorporates a triple-inertial acceleration strategy with a specialized approximation method, facilitating targeted acceleration of different terms in each sub-problem optimization. This integration improves the efficiency of convergence, achieving superior performance with fewer iterations. Additionally, we provide a convergence analysis of the TIAM algorithm, including its global convergence properties and convergence rate. Extensive experiments validate the effectiveness of the TIAM method, showing significant improvements in generalization capability and computational efficiency compared to existing approaches, particularly when applied to the rectified linear unit (ReLU) and its variants.

Qingsong Wang, Jiaxing He, Bingzhe Hou et al.

Building practical filtrations on objects to detect topological and geometric features is an important task in the field of Topological Data Analysis (TDA). In this paper, leveraging the ability of the Laplacian of Gaussian operator to enhance the boundaries of medical images, we define the G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration to generate the features more suitable for multi-parameter persistence module. By modeling volumetric images as bounded functions, then we prove the interleaving distance on the persistence modules obtained from our bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions. Finally, we conduct experiments on the MedMNIST dataset, comparing our bi-filtration against single-parameter filtration and the established deep learning baselines, including Google AutoML Vision, ResNet, AutoKeras and auto-sklearn. Experiments results demonstrate that our bi-filtration significantly outperforms single-parameter filtration. Notably, a simple Multi-Layer Perceptron (MLP) trained on the topological features generated by our bi-filtration achieves performance comparable to complex deep learning models trained on the original dataset.

Qingsong Wang, Mikhail Belkin, Yusu Wang

Guiding unconditional diffusion models typically requires either retraining with conditional inputs or per-step gradient computations (e.g., classifier-based guidance), both of which incur substantial computational overhead. We present a general recipe for efficiently steering unconditional diffusion {without gradient guidance during inference}, enabling fast controllable generation. Our approach is built on two observations about diffusion model structure: Noise Alignment: even in early, highly corrupted stages, coarse semantic steering is possible using a lightweight, offline-computed guidance signal, avoiding any per-step or per-sample gradients. Transferable concept vectors: a concept direction in activation space once learned transfers across both {timesteps} and {samples}; the same fixed steering vector learned near low noise level remains effective when injected at intermediate noise levels for every generation trajectory, providing refined conditional control with efficiency. Such concept directions can be efficiently and reliably identified via Recursive Feature Machine (RFM), a light-weight backpropagation-free feature learning method. Experiments on CIFAR-10, ImageNet, and CelebA demonstrate improved accuracy/quality over gradient-based guidance, while achieving significant inference speedups.

Qingsong Wang, Tao Wu, Wang Lin et al.

Large Language Models (LLMs) have demonstrated strong performance in open-ended generation tasks. However, they often struggle to adapt content to users with differing cognitive capacities, leading to a phenomenon we term cognitive misalignment. This issue arises in two forms: knowledge-level misalignment, where content is too complex or too simplistic relative to user understanding, and presentation-style misalignment, where the structure or tone hinders effective comprehension. To address these challenges, we propose the Cognitive-Level Alignment Framework (CLAF), a general-purpose generation framework that aligns both knowledge complexity and presentation style with user cognition. CLAF integrates a capability-aware retrieval module based on a hierarchical knowledge graph and a style optimization module guided by Bloom's taxonomy and preference learning. Additionally, a knowledge-controllable generation component ensures consistency and relevance throughout the output. To support training and evaluation, we construct SCALE, a cognitively annotated dataset containing responses at multiple comprehension levels per query. Empirical results show that CLAF enhances the adaptability and informativeness of LLM outputs across a range of user profiles, offering a robust solution to cognitive-level alignment in real-world applications.

Chengcheng Yan, Jiawei Xu, Zheng Peng et al.

The training of deep neural networks is inherently a nonconvex optimization problem, yet standard approaches such as stochastic gradient descent (SGD) require simultaneous updates to all parameters, often leading to unstable convergence and high computational cost. To address these issues, we propose a novel method, Stochastic Alternating Minimization with Trainable Step Sizes (SAMT), which updates network parameters in an alternating manner by treating the weights of each layer as a block. By decomposing the overall optimization into sub-problems corresponding to different blocks, this block-wise alternating strategy reduces per-step computational overhead and enhances training stability in nonconvex settings. To fully leverage these benefits, inspired by meta-learning, we proposed a novel adaptive step size strategy to incorporate into the sub-problem solving steps of alternating updates. It supports different types of trainable step sizes, including but not limited to scalar, element-wise, row-wise, and column-wise, enabling adaptive step size selection tailored to each block via meta-learning. We further provide a theoretical convergence guarantee for the proposed algorithm, establishing its optimization soundness. Extensive experiments for multiple benchmarks demonstrate that SAMT achieves better generalization performance with fewer parameter updates compared to state-of-the-art methods, highlighting its effectiveness and potential in neural network optimization.

Qingsong Wang

Symmetric matrix decomposition is an active research area in machine learning. This paper focuses on exploiting the low-rank structure of non-negative and sparse symmetric matrices via the rectified linear unit (ReLU) activation function. We propose the ReLU-based nonlinear symmetric matrix decomposition (ReLU-NSMD) model, introduce an accelerated alternating partial Bregman (AAPB) method for its solution, and present the algorithm's convergence results. Our algorithm leverages the Bregman proximal gradient framework to overcome the challenge of estimating the global $L$-smooth constant in the classic proximal gradient algorithm. Numerical experiments on synthetic and real datasets validate the effectiveness of our model and algorithm.

Qingsong Wang, Shengze Xu, Xiaojiao Tong et al.

Image deblurring remains a central research area within image processing, critical for its role in enhancing image quality and facilitating clearer visual representations across diverse applications. This paper tackles the optimization problem of image deblurring, assuming a known blurring kernel. We introduce an improved optimal proximal gradient algorithm (IOptISTA), which builds upon the optimal gradient method and a weighting matrix, to efficiently address the non-blind image deblurring problem. Based on two regularization cases, namely the $l_1$ norm and total variation norm, we perform numerical experiments to assess the performance of our proposed algorithm. The results indicate that our algorithm yields enhanced PSNR and SSIM values, as well as a reduced tolerance, compared to existing methods.

Zhengchao Wan, Qingsong Wang, Gal Mishne et al.

Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigorous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applicability. In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories. Central to our theory is the discovery that the denoiser, a key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data geometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and local clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submanifold-cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These findings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.

Leon Bungert, Nicolás García Trillos, Matt Jacobs et al.

Although deep neural networks have achieved super-human performance on many classification tasks, they often exhibit a worrying lack of robustness towards adversarially generated examples. Thus, considerable effort has been invested into reformulating standard Risk Minimization (RM) into an adversarially robust framework. Recently, attention has shifted towards approaches which interpolate between the robustness offered by adversarial training and the higher clean accuracy and faster training times of RM. In this paper, we take a fresh and geometric view on one such method -- Probabilistically Robust Learning (PRL). We propose a mathematical framework for understanding PRL, which allows us to identify geometric pathologies in its original formulation and to introduce a family of probabilistic nonlocal perimeter functionals to rectify them. We prove existence of solutions to the original and modified problems using novel relaxation methods and also study properties, as well as local limits, of the introduced perimeters. We also clarify, through a suitable $Γ$-convergence analysis, the way in which the original and modified PRL models interpolate between risk minimization and adversarial training.

Dunbiao Niu, Chengjing Wang, Peipei Tang et al.

Support vector machines (SVMs) are successful modeling and prediction tools with a variety of applications. Previous work has demonstrated the superiority of the SVMs in dealing with the high dimensional, low sample size problems. However, the numerical difficulties of the SVMs will become severe with the increase of the sample size. Although there exist many solvers for the SVMs, only few of them are designed by exploiting the special structures of the SVMs. In this paper, we propose a highly efficient sparse semismooth Newton based augmented Lagrangian method for solving a large-scale convex quadratic programming problem with a linear equality constraint and a simple box constraint, which is generated from the dual problems of the SVMs. By leveraging the primal-dual error bound result, the fast local convergence rate of the augmented Lagrangian method can be guaranteed. Furthermore, by exploiting the second-order sparsity of the problem when using the semismooth Newton method,the algorithm can efficiently solve the aforementioned difficult problems. Finally, numerical comparisons demonstrate that the proposed algorithm outperforms the current state-of-the-art solvers for the large-scale SVMs.