Junxiang Wang

Junxiang Wang, Junji Jiang, Liang Zhao

Localizing the source of graph diffusion phenomena, such as misinformation propagation, is an important yet extremely challenging task. Existing source localization models typically are heavily dependent on the hand-crafted rules. Unfortunately, a large portion of the graph diffusion process for many applications is still unknown to human beings so it is important to have expressive models for learning such underlying rules automatically. This paper aims to establish a generic framework of invertible graph diffusion models for source localization on graphs, namely Invertible Validity-aware Graph Diffusion (IVGD), to handle major challenges including 1) Difficulty to leverage knowledge in graph diffusion models for modeling their inverse processes in an end-to-end fashion, 2) Difficulty to ensure the validity of the inferred sources, and 3) Efficiency and scalability in source inference. Specifically, first, to inversely infer sources of graph diffusion, we propose a graph residual scenario to make existing graph diffusion models invertible with theoretical guarantees; second, we develop a novel error compensation mechanism that learns to offset the errors of the inferred sources. Finally, to ensure the validity of the inferred sources, a new set of validity-aware layers have been devised to project inferred sources to feasible regions by flexibly encoding constraints with unrolled optimization techniques. A linearization technique is proposed to strengthen the efficiency of our proposed layers. The convergence of the proposed IVGD is proven theoretically. Extensive experiments on nine real-world datasets demonstrate that our proposed IVGD outperforms state-of-the-art comparison methods significantly. We have released our code at https://github.com/xianggebenben/IVGD.

Zheng Zhang, Junxiang Wang, Liang Zhao

Graph Neural Networks (GNNs) have achieved great success in representing data with dependencies by recursively propagating and aggregating messages along the edges. However, edges in real-world graphs often have varying degrees of difficulty, and some edges may even be noisy to the downstream tasks. Therefore, existing GNNs may lead to suboptimal learned representations because they usually treat every edge in the graph equally. On the other hand, Curriculum Learning (CL), which mimics the human learning principle of learning data samples in a meaningful order, has been shown to be effective in improving the generalization ability and robustness of representation learners by gradually proceeding from easy to more difficult samples during training. Unfortunately, existing CL strategies are designed for independent data samples and cannot trivially generalize to handle data dependencies. To address these issues, we propose a novel CL strategy to gradually incorporate more edges into training according to their difficulty from easy to hard, where the degree of difficulty is measured by how well the edges are expected given the model training status. We demonstrate the strength of our proposed method in improving the generalization ability and robustness of learned representations through extensive experiments on nine synthetic datasets and nine real-world datasets. The code for our proposed method is available at https://github.com/rollingstonezz/Curriculum_learning_for_GNNs.

Chen Ling, Junji Jiang, Junxiang Wang et al.

Graph diffusion problems such as the propagation of rumors, computer viruses, or smart grid failures are ubiquitous and societal. Hence it is usually crucial to identify diffusion sources according to the current graph diffusion observations. Despite its tremendous necessity and significance in practice, source localization, as the inverse problem of graph diffusion, is extremely challenging as it is ill-posed: different sources may lead to the same graph diffusion patterns. Different from most traditional source localization methods, this paper focuses on a probabilistic manner to account for the uncertainty of different candidate sources. Such endeavors require overcoming challenges including 1) the uncertainty in graph diffusion source localization is hard to be quantified; 2) the complex patterns of the graph diffusion sources are difficult to be probabilistically characterized; 3) the generalization under any underlying diffusion patterns is hard to be imposed. To solve the above challenges, this paper presents a generic framework: Source Localization Variational AutoEncoder (SL-VAE) for locating the diffusion sources under arbitrary diffusion patterns. Particularly, we propose a probabilistic model that leverages the forward diffusion estimation model along with deep generative models to approximate the diffusion source distribution for quantifying the uncertainty. SL-VAE further utilizes prior knowledge of the source-observation pairs to characterize the complex patterns of diffusion sources by a learned generative prior. Lastly, a unified objective that integrates the forward diffusion estimation model is derived to enforce the model to generalize under arbitrary diffusion patterns. Extensive experiments are conducted on 7 real-world datasets to demonstrate the superiority of SL-VAE in reconstructing the diffusion sources by excelling other methods on average 20% in AUC score.

Chen Ling, Tanmoy Chowdhury, Junji Jiang et al.

Analogical reasoning is the process of discovering and mapping correspondences from a target subject to a base subject. As the most well-known computational method of analogical reasoning, Structure-Mapping Theory (SMT) abstracts both target and base subjects into relational graphs and forms the cognitive process of analogical reasoning by finding a corresponding subgraph (i.e., correspondence) in the target graph that is aligned with the base graph. However, incorporating deep learning for SMT is still under-explored due to several obstacles: 1) the combinatorial complexity of searching for the correspondence in the target graph; 2) the correspondence mining is restricted by various cognitive theory-driven constraints. To address both challenges, we propose a novel framework for Analogical Reasoning (DeepGAR) that identifies the correspondence between source and target domains by assuring cognitive theory-driven constraints. Specifically, we design a geometric constraint embedding space to induce subgraph relation from node embeddings for efficient subgraph search. Furthermore, we develop novel learning and optimization strategies that could end-to-end identify correspondences that are strictly consistent with constraints driven by the cognitive theory. Extensive experiments are conducted on synthetic and real-world datasets to demonstrate the effectiveness of the proposed DeepGAR over existing methods.

Chengyuan Deng, Zhengzhang Chen, Xujiang Zhao et al.

Change point detection aims to identify abrupt shifts occurring at multiple points within a data sequence. This task becomes particularly challenging in the online setting, where different types of changes can occur, including shifts in both the marginal and joint distributions of the data. In this paper, we address these challenges by tracking the Riemannian geometry of correlation matrices, allowing Riemannian metrics to compute the geodesic distance as an accurate measure of correlation dynamics. We introduce Rio-CPD, a non-parametric, correlation-aware online change point detection framework that integrates the Riemannian geometry of the manifold of symmetric positive definite matrices with the cumulative sum (CUSUM) statistic for detecting change points. Rio-CPD employs a novel CUSUM design by computing the geodesic distance between current observations and the Fréchet mean of prior observations. With appropriate choices of Riemannian metrics, Rio-CPD offers a simple yet effective and computationally efficient algorithm. Experimental results on both synthetic and real-world datasets demonstrate that Rio-CPD outperforms existing methods on detection accuracy, average detection delay and efficiency.

Hongyi Li, Junxiang Wang, Yongchao Wang

Massive Multiple-Input Multiple-Out (MIMO) detection is an important problem in modern wireless communication systems. While traditional Belief Propagation (BP) detectors perform poorly on loopy graphs, the recent Graph Neural Networks (GNNs)-based method can overcome the drawbacks of BP and achieve superior performance. Nevertheless, direct use of GNN ignores the importance of edge attributes and suffers from high computation overhead using a fully connected graph structure. In this paper, we propose an efficient GNN-inspired algorithm, called the Edge Graph Neural Network (EGNN), to detect MIMO signals. We first compute graph edge weights through channel correlation and then leverage the obtained weights as a metric to evaluate the importance of neighbors of each node. Moreover, we design an adaptive Edge Drop (ED) scheme to sparsify the graph such that computational cost can be significantly reduced. Experimental results demonstrate that our proposed EGNN achieves better or comparable performance to popular MIMO detection methods for different modulation schemes and costs the least detection time compared to GNN-based approaches.

Junxiang Wang, Liang Zhao

We introduce GraphSL, a new library for studying the graph source localization problem. graph diffusion and graph source localization are inverse problems in nature: graph diffusion predicts information diffusions from information sources, while graph source localization predicts information sources from information diffusions. GraphSL facilitates the exploration of various graph diffusion models for simulating information diffusions and enables the evaluation of cutting-edge source localization approaches on established benchmark datasets. The source code of GraphSL is made available at Github Repository (https://github.com/xianggebenben/GraphSL). Bug reports and feedback can be directed to the Github issues page (https://github.com/xianggebenben/GraphSL/issues).

Chen Ling, Junji Jiang, Junxiang Wang et al.

Influence maximization (IM) is formulated as selecting a set of initial users from a social network to maximize the expected number of influenced users. Researchers have made great progress in designing various traditional methods, and their theoretical design and performance gain are close to a limit. In the past few years, learning-based IM methods have emerged to achieve stronger generalization ability to unknown graphs than traditional ones. However, the development of learning-based IM methods is still limited by fundamental obstacles, including 1) the difficulty of effectively solving the objective function; 2) the difficulty of characterizing the diversified underlying diffusion patterns; and 3) the difficulty of adapting the solution under various node-centrality-constrained IM variants. To cope with the above challenges, we design a novel framework DeepIM to generatively characterize the latent representation of seed sets, and we propose to learn the diversified information diffusion pattern in a data-driven and end-to-end manner. Finally, we design a novel objective function to infer optimal seed sets under flexible node-centrality-based budget constraints. Extensive analyses are conducted over both synthetic and real-world datasets to demonstrate the overall performance of DeepIM. The code and data are available at: https://github.com/triplej0079/DeepIM.

Junxiang Wang, Hongyi Li, Zheng Chai et al.

While Graph Neural Networks (GNNs) are popular in the deep learning community, they suffer from several challenges including over-smoothing, over-squashing, and gradient vanishing. Recently, a series of models have attempted to relieve these issues by first augmenting the node features and then imposing node-wise functions based on Multi-Layer Perceptron (MLP), which are widely referred to as GA-MLP models. However, while GA-MLP models enjoy deeper architectures for better accuracy, their efficiency largely deteriorates. Moreover, popular acceleration techniques such as stochastic-version or data-parallelism cannot be effectively applied due to the dependency among samples (i.e., nodes) in graphs. To address these issues, in this paper, instead of data parallelism, we propose a parallel graph deep learning Alternating Direction Method of Multipliers (pdADMM-G) framework to achieve model parallelism: parameters in each layer of GA-MLP models can be updated in parallel. The extended pdADMM-G-Q algorithm reduces communication costs by introducing the quantization technique. Theoretical convergence to a (quantized) stationary point of the pdADMM-G algorithm and the pdADMM-G-Q algorithm is provided with a sublinear convergence rate $o(1/k)$, where $k$ is the number of iterations. Extensive experiments demonstrate the convergence of two proposed algorithms. Moreover, they lead to a more massive speedup and better performance than all state-of-the-art comparison methods on nine benchmark datasets. Last but not least, the proposed pdADMM-G-Q algorithm reduces communication overheads by up to $45\%$ without loss of performance. Our code is available at \url{https://github.com/xianggebenben/pdADMM-G}.

Junxiang Wang, Zheng Chai, Yue Cheng et al.

Alternating minimization methods have recently been proposed as alternatives to the gradient descent for deep neural network optimization. Alternating minimization methods can typically decompose a deep neural network into layerwise subproblems, which can then be optimized in parallel. Despite the significant parallelism, alternating minimization methods are rarely explored in training deep neural networks because of the severe accuracy degradation. In this paper, we analyze the reason and propose to achieve a compelling trade-off between parallelism and accuracy by a reformulation called Tunable Subnetwork Splitting Method (TSSM), which can tune the decomposition granularity of deep neural networks. Two methods gradient splitting Alternating Direction Method of Multipliers (gsADMM) and gradient splitting Alternating Minimization (gsAM) are proposed to solve the TSSM formulation. Experiments on five benchmark datasets show that our proposed TSSM can achieve significant speedup without observable loss of training accuracy. The code has been released at https://github.com/xianggebenben/TSSM.

Junxiang Wang, Xinwen Xu, Tiancheng Wu et al.

Developing autonomous physical human-robot interaction (pHRI) systems is limited by the scarcity of large-scale training data to learn robust robot behaviors for real-world applications. In this paper, we introduce a zero-shot "text2sim2real" generative simulation framework that automatically synthesizes diverse pHRI scenarios from high-level natural-language prompts. Leveraging Large Language Models (LLMs) and Vision-Language Models (VLMs), our pipeline procedurally generates soft-body human models, scene layouts, and robot motion trajectories for assistive tasks. We utilize this framework to autonomously collect large-scale synthetic demonstration datasets and then train vision-based imitation learning policies operating on segmented point clouds. We evaluate our approach through a user study on two physically assistive tasks: scratching and bathing. Our learned policies successfully achieve zero-shot sim-to-real transfer, attaining success rates exceeding 80% and demonstrating resilience to unscripted human motion. Overall, we introduce the first generative simulation pipeline for pHRI applications, automating simulation environment synthesis, data collection, and policy learning. Additional information may be found on our project website: https://rchi-lab.github.io/gen_phri/

Minhua Lin, Zhengzhang Chen, Yanchi Liu et al.

Time series data is ubiquitous across various domains, including manufacturing, finance, and healthcare. High-quality annotations are essential for effectively understanding time series and facilitating downstream tasks; however, obtaining such annotations is challenging, particularly in mission-critical domains. In this paper, we propose TESSA, a multi-agent system designed to automatically generate both general and domain-specific annotations for time series data. TESSA introduces two agents: a general annotation agent and a domain-specific annotation agent. The general agent captures common patterns and knowledge across multiple source domains, leveraging both time-series-wise and text-wise features to generate general annotations. Meanwhile, the domain-specific agent utilizes limited annotations from the target domain to learn domain-specific terminology and generate targeted annotations. Extensive experiments on multiple synthetic and real-world datasets demonstrate that TESSA effectively generates high-quality annotations, outperforming existing methods.

Junxiang Wang, Guangji Bai, Wei Cheng et al.

Time series domain adaptation stands as a pivotal and intricate challenge with diverse applications, including but not limited to human activity recognition, sleep stage classification, and machine fault diagnosis. Despite the numerous domain adaptation techniques proposed to tackle this complex problem, they primarily focus on domain adaptation from a single source domain. Yet, it is more crucial to investigate domain adaptation from multiple domains due to the potential for greater improvements. To address this, three important challenges need to be overcome: 1). The lack of exploration to utilize domain-specific information for domain adaptation, 2). The difficulty to learn domain-specific information that changes over time, and 3). The difficulty to evaluate learned domain-specific information. In order to tackle these challenges simultaneously, in this paper, we introduce PrOmpt-based domaiN Discrimination (POND), the first framework to utilize prompts for time series domain adaptation. Specifically, to address Challenge 1, we extend the idea of prompt tuning to time series analysis and learn prompts to capture common and domain-specific information from all source domains. To handle Challenge 2, we introduce a conditional module for each source domain to generate prompts from time series input data. For Challenge 3, we propose two criteria to select good prompts, which are used to choose the most suitable source domain for domain adaptation. The efficacy and robustness of our proposed POND model are extensively validated through experiments across 50 scenarios encompassing four datasets. Experimental results demonstrate that our proposed POND model outperforms all state-of-the-art comparison methods by up to $66\%$ on the F1-score.

Yuntong Hu, Junxiang Wang, Liang Zhao

Network tomography is a crucial problem in network monitoring, where the observable path performance metric values are used to infer the unobserved ones, making it essential for tasks such as route selection, fault diagnosis, and traffic control. However, most existing methods either assume complete knowledge of network topology and metric formulas-an unrealistic expectation in many real-world scenarios with limited observability-or rely entirely on black-box end-to-end models. To tackle this, in this paper, we argue that a good network tomography requires synergizing the knowledge from both data and appropriate inductive bias from (partial) prior knowledge. To see this, we propose Deep Network Tomography (DeepNT), a novel framework that leverages a path-centric graph neural network to predict path performance metrics without relying on predefined hand-crafted metrics, assumptions, or the real network topology. The path-centric graph neural network learns the path embedding by inferring and aggregating the embeddings of the sequence of nodes that compose this path. Training path-centric graph neural networks requires learning the neural netowrk parameters and network topology under discrete constraints induced by the observed path performance metrics, which motivates us to design a learning objective that imposes connectivity and sparsity constraints on topology and path performance triangle inequality on path performance. Extensive experiments on real-world and synthetic datasets demonstrate the superiority of DeepNT in predicting performance metrics and inferring graph topology compared to state-of-the-art methods.

Zheng Zhang, Sirui Li, Jingcheng Zhou et al.

Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore, merely combining individual spatial and network representations cannot reveal the underlying interaction mechanism of spatial networks. Besides, existing spatial network representation learning methods can only consider networks embedded in Euclidean space, and can not well exploit the rich geometric information carried by irregular and non-uniform non-Euclidean space. In order to address this issue, in this paper we propose a novel generic framework to learn the representation of spatial networks that are embedded in non-Euclidean manifold space. Specifically, a novel message-passing-based neural network is proposed to combine graph topology and spatial geometry, where spatial geometry is extracted as messages on the edges. We theoretically guarantee that the learned representations are provably invariant to important symmetries such as rotation or translation, and simultaneously maintain sufficient ability in distinguishing different geometric structures. The strength of our proposed method is demonstrated through extensive experiments on both synthetic and real-world datasets.

Yingzhi Mao, Chunkang Zhang, Junxiang Wang et al.

Large Reasoning Models (LRMs) demonstrate remarkable capabilities on complex reasoning tasks but remain vulnerable to severe safety risks, including harmful content generation and jailbreak attacks. Existing mitigation strategies rely on injecting heuristic safety signals during training, which often suppress reasoning ability and fail to resolve the safety-reasoning trade-off. To systematically investigate this issue, we analyze the reasoning trajectories of diverse LRMs and uncover a phenomenon we term Self-Jailbreak, where models override their own risk assessments and justify responding to unsafe prompts. This finding reveals that LRMs inherently possess the ability to reject unsafe queries, but this ability is compromised, resulting in harmful outputs. Building on these insights, we propose the Chain-of-Guardrail (CoG), a training framework that recomposes or backtracks unsafe reasoning steps, steering the model back onto safe trajectories while preserving valid reasoning chains. Extensive experiments across multiple reasoning and safety benchmarks demonstrate that CoG substantially improves the safety of current LRMs while preserving comparable reasoning ability, significantly outperforming prior methods that suffer from severe safety-reasoning trade-offs.

Te Zhang, Yuheng Li, Junxiang Wang et al.

Large multimodal models (LMMs) have advanced significantly by integrating visual encoders with extensive language models, enabling robust reasoning capabilities. However, compressing LMMs for deployment on edge devices remains a critical challenge. In this work, we propose an adaptive search algorithm that optimizes sparsity and KV cache compression to enhance LMM efficiency. Utilizing the Tree-structured Parzen Estimator, our method dynamically adjusts pruning ratios and KV cache quantization bandwidth across different LMM layers, using model performance as the optimization objective. This approach uniquely combines pruning with key-value cache quantization and incorporates a fast pruning technique that eliminates the need for additional fine-tuning or weight adjustments, achieving efficient compression without compromising accuracy. Comprehensive evaluations on benchmark datasets, including LLaVA-1.5 7B and 13B, demonstrate our method superiority over state-of-the-art techniques such as SparseGPT and Wanda across various compression levels. Notably, our framework automatic allocation of KV cache compression resources sets a new standard in LMM optimization, delivering memory efficiency without sacrificing much performance.

Chen Ling, Xujiang Zhao, Jiaying Lu et al.

Large language models (LLMs) have significantly advanced the field of natural language processing (NLP), providing a highly useful, task-agnostic foundation for a wide range of applications. However, directly applying LLMs to solve sophisticated problems in specific domains meets many hurdles, caused by the heterogeneity of domain data, the sophistication of domain knowledge, the uniqueness of domain objectives, and the diversity of the constraints (e.g., various social norms, cultural conformity, religious beliefs, and ethical standards in the domain applications). Domain specification techniques are key to make large language models disruptive in many applications. Specifically, to solve these hurdles, there has been a notable increase in research and practices conducted in recent years on the domain specialization of LLMs. This emerging field of study, with its substantial potential for impact, necessitates a comprehensive and systematic review to better summarize and guide ongoing work in this area. In this article, we present a comprehensive survey on domain specification techniques for large language models, an emerging direction critical for large language model applications. First, we propose a systematic taxonomy that categorizes the LLM domain-specialization techniques based on the accessibility to LLMs and summarizes the framework for all the subcategories as well as their relations and differences to each other. Second, we present an extensive taxonomy of critical application domains that can benefit dramatically from specialized LLMs, discussing their practical significance and open challenges. Last, we offer our insights into the current research status and future trends in this area.

Junxiang Wang, Xuchao Zhang, Bo Zong et al.

During the past several years, a surge of multi-lingual Pre-trained Language Models (PLMs) has been proposed to achieve state-of-the-art performance in many cross-lingual downstream tasks. However, the understanding of why multi-lingual PLMs perform well is still an open domain. For example, it is unclear whether multi-Lingual PLMs reveal consistent token attributions in different languages. To address this, in this paper, we propose a Cross-lingual Consistency of Token Attributions (CCTA) evaluation framework. Extensive experiments in three downstream tasks demonstrate that multi-lingual PLMs assign significantly different attributions to multi-lingual synonyms. Moreover, we have the following observations: 1) the Spanish achieves the most consistent token attributions in different languages when it is used for training PLMs; 2) the consistency of token attributions strongly correlates with performance in downstream tasks.

Junxiang Wang, Hongyi Li, Liang Zhao

As a well-known optimization framework, the Alternating Direction Method of Multipliers (ADMM) has achieved tremendous success in many classification and regression applications. Recently, it has attracted the attention of deep learning researchers and is considered to be a potential substitute to Gradient Descent (GD). However, as an emerging domain, several challenges remain unsolved, including 1) The lack of global convergence guarantees, 2) Slow convergence towards solutions, and 3) Cubic time complexity with regard to feature dimensions. In this paper, we propose a novel optimization framework to solve a general neural network training problem via ADMM (dlADMM) to address these challenges simultaneously. Specifically, the parameters in each layer are updated backward and then forward so that parameter information in each layer is exchanged efficiently. When the dlADMM is applied to specific architectures, the time complexity of subproblems is reduced from cubic to quadratic via a dedicated algorithm design utilizing quadratic approximations and backtracking techniques. Last but not least, we provide the first proof of convergence to a critical point sublinearly for an ADMM-type method (dlADMM) under mild conditions. Experiments on seven benchmark datasets demonstrate the convergence, efficiency, and effectiveness of our proposed dlADMM algorithm.

Hongyi Li, Junxiang Wang, Yongchao Wang et al.

The Graph Convolutional Network (GCN) has been successfully applied to many graph-based applications. Training a large-scale GCN model, however, is still challenging: Due to the node dependency and layer dependency of the GCN architecture, a huge amount of computational time and memory is required in the training process. In this paper, we propose a parallel and distributed GCN training algorithm based on the Alternating Direction Method of Multipliers (ADMM) to tackle the two challenges simultaneously. We first split GCN layers into independent blocks to achieve layer parallelism. Furthermore, we reduce node dependency by dividing the graph into several dense communities such that each of them can be trained with an agent in parallel. Finally, we provide solutions for all subproblems in the community-based ADMM algorithm. Preliminary results demonstrate that our proposed community-based ADMM training algorithm can lead to more than triple speedup while achieving the best performance compared with state-of-the-art methods.

Johnny Torres, Guangji Bai, Junxiang Wang et al.

Multi-task learning is a framework that enforces different learning tasks to share their knowledge to improve their generalization performance. It is a hot and active domain that strives to handle several core issues; particularly, which tasks are correlated and similar, and how to share the knowledge among correlated tasks. Existing works usually do not distinguish the polarity and magnitude of feature weights and commonly rely on linear correlation, due to three major technical challenges in: 1) optimizing the models that regularize feature weight polarity, 2) deciding whether to regularize sign or magnitude, 3) identifying which tasks should share their sign and/or magnitude patterns. To address them, this paper proposes a new multi-task learning framework that can regularize feature weight signs across tasks. We innovatively formulate it as a biconvex inequality constrained optimization with slacks and propose a new efficient algorithm for the optimization with theoretical guarantees on generalization performance and convergence. Extensive experiments on multiple datasets demonstrate the proposed methods' effectiveness, efficiency, and reasonableness of the regularized feature weighted patterns.

Junxiang Wang, Fuxun Yu, Xiang Chen et al.

Alternating Direction Method of Multipliers (ADMM) has been used successfully in many conventional machine learning applications and is considered to be a useful alternative to Stochastic Gradient Descent (SGD) as a deep learning optimizer. However, as an emerging domain, several challenges remain, including 1) The lack of global convergence guarantees, 2) Slow convergence towards solutions, and 3) Cubic time complexity with regard to feature dimensions. In this paper, we propose a novel optimization framework for deep learning via ADMM (dlADMM) to address these challenges simultaneously. The parameters in each layer are updated backward and then forward so that the parameter information in each layer is exchanged efficiently. The time complexity is reduced from cubic to quadratic in (latent) feature dimensions via a dedicated algorithm design for subproblems that enhances them utilizing iterative quadratic approximations and backtracking. Finally, we provide the first proof of global convergence for an ADMM-based method (dlADMM) in a deep neural network problem under mild conditions. Experiments on benchmark datasets demonstrated that our proposed dlADMM algorithm outperforms most of the comparison methods.