Yuanyuan Liu

Zhijin Ge, Hongying Liu, Xiaosen Wang et al.

Transfer-based attack adopts the adversarial examples generated on the surrogate model to attack various models, making it applicable in the physical world and attracting increasing interest. Recently, various adversarial attacks have emerged to boost adversarial transferability from different perspectives. In this work, inspired by the observation that flat local minima are correlated with good generalization, we assume and empirically validate that adversarial examples at a flat local region tend to have good transferability by introducing a penalized gradient norm to the original loss function. Since directly optimizing the gradient regularization norm is computationally expensive and intractable for generating adversarial examples, we propose an approximation optimization method to simplify the gradient update of the objective function. Specifically, we randomly sample an example and adopt a first-order procedure to approximate the curvature of Hessian/vector product, which makes computing more efficient by interpolating two neighboring gradients. Meanwhile, in order to obtain a more stable gradient direction, we randomly sample multiple examples and average the gradients of these examples to reduce the variance due to random sampling during the iterative process. Extensive experimental results on the ImageNet-compatible dataset show that the proposed method can generate adversarial examples at flat local regions, and significantly improve the adversarial transferability on either normally trained models or adversarially trained models than the state-of-the-art attacks. Our codes are available at: https://github.com/Trustworthy-AI-Group/PGN.

Zhijin Ge, Fanhua Shang, Hongying Liu et al.

Deep neural networks are vulnerable to adversarial examples crafted by applying human-imperceptible perturbations on clean inputs. Although many attack methods can achieve high success rates in the white-box setting, they also exhibit weak transferability in the black-box setting. Recently, various methods have been proposed to improve adversarial transferability, in which the input transformation is one of the most effective methods. In this work, we notice that existing input transformation-based works mainly adopt the transformed data in the same domain for augmentation. Inspired by domain generalization, we aim to further improve the transferability using the data augmented from different domains. Specifically, a style transfer network can alter the distribution of low-level visual features in an image while preserving semantic content for humans. Hence, we propose a novel attack method named Style Transfer Method (STM) that utilizes a proposed arbitrary style transfer network to transform the images into different domains. To avoid inconsistent semantic information of stylized images for the classification network, we fine-tune the style transfer network and mix up the generated images added by random noise with the original images to maintain semantic consistency and boost input diversity. Extensive experimental results on the ImageNet-compatible dataset show that our proposed method can significantly improve the adversarial transferability on either normally trained models or adversarially trained models than state-of-the-art input transformation-based attacks. Code is available at: https://github.com/Zhijin-Ge/STM.

Junkang Liu, Yuxuan Tian, Fanhua Shang et al.

To prevent inference attacks in Federated Learning (FL) and reduce the leakage of sensitive information, Client-level Differentially Private Federated Learning (CL-DPFL) is widely used. However, current CL-DPFL methods usually result in sharper loss landscapes, which leads to a decrease in model generalization after differential privacy protection. By using Sharpness Aware Minimization (SAM), the current popular federated learning methods are to find a local flat minimum value to alleviate this problem. However, the local flatness may not reflect the global flatness in CL-DPFL. Therefore, to address this issue and seek global flat minima of models, we propose a new CL-DPFL algorithm, DP-FedPGN, in which we introduce a global gradient norm penalty to the local loss to find the global flat minimum. Moreover, by using our global gradient norm penalty, we not only find a flatter global minimum but also reduce the locally updated norm, which means that we further reduce the error of gradient clipping. From a theoretical perspective, we analyze how DP-FedPGN mitigates the performance degradation caused by DP. Meanwhile, the proposed DP-FedPGN algorithm eliminates the impact of data heterogeneity and achieves fast convergence. We also use Rényi DP to provide strict privacy guarantees and provide sensitivity analysis for local updates. Finally, we conduct effectiveness tests on both ResNet and Transformer models, and achieve significant improvements in six visual and natural language processing tasks compared to existing state-of-the-art algorithms. The code is available at https://github.com/junkangLiu0/DP-FedPGN

Junkang Liu, Fanhua Shang, Kewen Zhu et al.

AdamW has become one of the most effective optimizers for training large-scale models. We have also observed its effectiveness in the context of federated learning (FL). However, directly applying AdamW in federated learning settings poses significant challenges: (1) due to data heterogeneity, AdamW often yields high variance in the second-moment estimate $\boldsymbol{v}$; (2) the local overfitting of AdamW may cause client drift; and (3) Reinitializing moment estimates ($\boldsymbol{v}$, $\boldsymbol{m}$) at each round slows down convergence. To address these challenges, we propose the first \underline{Fed}erated \underline{AdamW} algorithm, called \texttt{FedAdamW}, for training and fine-tuning various large models. \texttt{FedAdamW} aligns local updates with the global update using both a \textbf{local correction mechanism} and decoupled weight decay to mitigate local overfitting. \texttt{FedAdamW} efficiently aggregates the \texttt{mean} of the second-moment estimates to reduce their variance and reinitialize them. Theoretically, we prove that \texttt{FedAdamW} achieves a linear speedup convergence rate of $\mathcal{O}(\sqrt{(L Δσ_l^2)/(S K R ε^2)}+(L Δ)/R)$ without \textbf{heterogeneity assumption}, where $S$ is the number of participating clients per round, $K$ is the number of local iterations, and $R$ is the total number of communication rounds. We also employ PAC-Bayesian generalization analysis to explain the effectiveness of decoupled weight decay in local training. Empirically, we validate the effectiveness of \texttt{FedAdamW} on language and vision Transformer models. Compared to several baselines, \texttt{FedAdamW} significantly reduces communication rounds and improves test accuracy. The code is available in https://github.com/junkangLiu0/FedAdamW.

Junkang Liu, Fanhua Shang, Junchao Zhou et al.

The core bottleneck of Federated Learning (FL) lies in the communication rounds. That is, how to achieve more effective local updates is crucial for reducing communication rounds. Existing FL methods still primarily use element-wise local optimizers (Adam/SGD), neglecting the geometric structure of the weight matrices. This often leads to the amplification of pathological directions in the weights during local updates, leading deterioration in the condition number and slow convergence. Therefore, we introduce the Muon optimizer in local, which has matrix orthogonalization to optimize matrix-structured parameters. Experimental results show that, in IID setting, Local Muon significantly accelerates the convergence of FL and reduces communication rounds compared to Local SGD and Local AdamW. However, in non-IID setting, independent matrix orthogonalization based on the local distributions of each client induces strong client drift. Applying Muon in non-IID FL poses significant challenges: (1) client preconditioner leading to client drift; (2) moment reinitialization. To address these challenges, we propose a novel Federated Muon optimizer (FedMuon), which incorporates two key techniques: (1) momentum aggregation, where clients use the aggregated momentum for local initialization; (2) local-global alignment, where the local gradients are aligned with the global update direction to significantly reduce client drift. Theoretically, we prove that \texttt{FedMuon} achieves a linear speedup convergence rate without the heterogeneity assumption, where $S$ is the number of participating clients per round, $K$ is the number of local iterations, and $R$ is the total number of communication rounds. Empirically, we validate the effectiveness of FedMuon on language and vision models. Compared to several baselines, FedMuon significantly reduces communication rounds and improves test accuracy.

Yuanyuan Liu, Haiyang Mei, Dongyang Zhan et al.

3D visual grounding (3DVG) identifies objects in 3D scenes from language descriptions. Existing zero-shot approaches leverage 2D vision-language models (VLMs) by converting 3D spatial information (SI) into forms amenable to VLM processing, typically as composite inputs such as specified view renderings or video sequences with overlaid object markers. However, this VLM + SI paradigm yields entangled visual representations that compel the VLM to process entire cluttered cues, making it hard to exploit spatial semantic relationships effectively. In this work, we propose a new VLM x SI paradigm that externalizes the 3D SI into a form enabling the VLM to incrementally retrieve only what it needs during reasoning. We instantiate this paradigm with a novel View-on-Graph (VoG) method, which organizes the scene into a multi-modal, multi-layer scene graph and allows the VLM to operate as an active agent that selectively accesses necessary cues as it traverses the scene. This design offers two intrinsic advantages: (i) by structuring 3D context into a spatially and semantically coherent scene graph rather than confounding the VLM with densely entangled visual inputs, it lowers the VLM's reasoning difficulty; and (ii) by actively exploring and reasoning over the scene graph, it naturally produces transparent, step-by-step traces for interpretable 3DVG. Extensive experiments show that VoG achieves state-of-the-art zero-shot performance, establishing structured scene exploration as a promising strategy for advancing zero-shot 3DVG.

Yuanyuan Liu, Wenbin Wang, Chuanxu Feng et al.

The recent success of Transformer has provided a new direction to various visual understanding tasks, including video-based facial expression recognition (FER). By modeling visual relations effectively, Transformer has shown its power for describing complicated patterns. However, Transformer still performs unsatisfactorily to notice subtle facial expression movements, because the expression movements of many videos can be too small to extract meaningful spatial-temporal relations and achieve robust performance. To this end, we propose to decompose each video into a series of expression snippets, each of which contains a small number of facial movements, and attempt to augment the Transformer's ability for modeling intra-snippet and inter-snippet visual relations, respectively, obtaining the Expression snippet Transformer (EST). In particular, for intra-snippet modeling, we devise an attention-augmented snippet feature extractor (AA-SFE) to enhance the encoding of subtle facial movements of each snippet by gradually attending to more salient information. In addition, for inter-snippet modeling, we introduce a shuffled snippet order prediction (SSOP) head and a corresponding loss to improve the modeling of subtle motion changes across subsequent snippets by training the Transformer to identify shuffled snippet orders. Extensive experiments on four challenging datasets (i.e., BU-3DFE, MMI, AFEW, and DFEW) demonstrate that our EST is superior to other CNN-based methods, obtaining state-of-the-art performance.

Hua Huang, Fanhua Shang, Yuanyuan Liu et al.

Federated Learning (FL) has become an active and promising distributed machine learning paradigm. As a result of statistical heterogeneity, recent studies clearly show that the performance of popular FL methods (e.g., FedAvg) deteriorates dramatically due to the client drift caused by local updates. This paper proposes a novel Federated Learning algorithm (called IGFL), which leverages both Individual and Group behaviors to mimic distribution, thereby improving the ability to deal with heterogeneity. Unlike existing FL methods, our IGFL can be applied to both client and server optimization. As a by-product, we propose a new attention-based federated learning in the server optimization of IGFL. To the best of our knowledge, this is the first time to incorporate attention mechanisms into federated optimization. We conduct extensive experiments and show that IGFL can significantly improve the performance of existing federated learning methods. Especially when the distributions of data among individuals are diverse, IGFL can improve the classification accuracy by about 13% compared with prior baselines.

Lin Kong, Wei Sun, Fanhua Shang et al.

Recently, the study on learned iterative shrinkage thresholding algorithm (LISTA) has attracted increasing attentions. A large number of experiments as well as some theories have proved the high efficiency of LISTA for solving sparse coding problems. However, existing LISTA methods are all serial connection. To address this issue, we propose a novel extragradient based LISTA (ELISTA), which has a residual structure and theoretical guarantees. In particular, our algorithm can also provide the interpretability for Res-Net to a certain extent. From a theoretical perspective, we prove that our method attains linear convergence. In practice, extensive empirical results verify the advantages of our method.

Tao Xu, Fanhua Shang, Yuanyuan Liu et al.

In this paper, we study efficient differentially private alternating direction methods of multipliers (ADMM) via gradient perturbation for many machine learning problems. For smooth convex loss functions with (non)-smooth regularization, we propose the first differentially private ADMM (DP-ADMM) algorithm with performance guarantee of $(ε,δ)$-differential privacy ($(ε,δ)$-DP). From the viewpoint of theoretical analysis, we use the Gaussian mechanism and the conversion relationship between Rényi Differential Privacy (RDP) and DP to perform a comprehensive privacy analysis for our algorithm. Then we establish a new criterion to prove the convergence of the proposed algorithms including DP-ADMM. We also give the utility analysis of our DP-ADMM. Moreover, we propose an accelerated DP-ADMM (DP-AccADMM) with the Nesterov's acceleration technique. Finally, we conduct numerical experiments on many real-world datasets to show the privacy-utility tradeoff of the two proposed algorithms, and all the comparative analysis shows that DP-AccADMM converges faster and has a better utility than DP-ADMM, when the privacy budget $ε$ is larger than a threshold.

Hongying Liu, Zhenyu Zhou, Fanhua Shang et al.

Deep neural networks (DNNs) are playing key roles in various artificial intelligence applications such as image classification and object recognition. However, a growing number of studies have shown that there exist adversarial examples in DNNs, which are almost imperceptibly different from original samples, but can greatly change the network output. Existing white-box attack algorithms can generate powerful adversarial examples. Nevertheless, most of the algorithms concentrate on how to iteratively make the best use of gradients to improve adversarial performance. In contrast, in this paper, we focus on the properties of the widely-used ReLU activation function, and discover that there exist two phenomena (i.e., wrong blocking and over transmission) misleading the calculation of gradients in ReLU during the backpropagation. Both issues enlarge the difference between the predicted changes of the loss function from gradient and corresponding actual changes, and mislead the gradients which results in larger perturbations. Therefore, we propose a universal adversarial example generation method, called ADV-ReLU, to enhance the performance of gradient based white-box attack algorithms. During the backpropagation of the network, our approach calculates the gradient of the loss function versus network input, maps the values to scores, and selects a part of them to update the misleading gradients. Comprehensive experimental results on \emph{ImageNet} demonstrate that our ADV-ReLU can be easily integrated into many state-of-the-art gradient-based white-box attack algorithms, as well as transferred to black-box attack attackers, to further decrease perturbations in the ${\ell _2}$-norm.

Fanhua Shang, Bingkun Wei, Hongying Liu et al.

Large-scale non-convex sparsity-constrained problems have recently gained extensive attention. Most existing deterministic optimization methods (e.g., GraSP) are not suitable for large-scale and high-dimensional problems, and thus stochastic optimization methods with hard thresholding (e.g., SVRGHT) become more attractive. Inspired by GraSP, this paper proposes a new general relaxed gradient support pursuit (RGraSP) framework, in which the sub-algorithm only requires to satisfy a slack descent condition. We also design two specific semi-stochastic gradient hard thresholding algorithms. In particular, our algorithms have much less hard thresholding operations than SVRGHT, and their average per-iteration cost is much lower (i.e., O(d) vs. O(d log(d)) for SVRGHT), which leads to faster convergence. Our experimental results on both synthetic and real-world datasets show that our algorithms are superior to the state-of-the-art gradient hard thresholding methods.

Rui Luo, Qiang Zhang, Yuanyuan Liu

We propose a new sampler that integrates the protocol of parallel tempering with the Nosé-Hoover (NH) dynamics. The proposed method can efficiently draw representative samples from complex posterior distributions with multiple isolated modes in the presence of noise arising from stochastic gradient. It potentially facilitates deep Bayesian learning on large datasets where complex multimodal posteriors and mini-batch gradient are encountered.

Fanhua Shang, James Cheng, Yuanyuan Liu et al.

The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions generally leads to challenging non-convex, non-smooth and non-Lipschitz problems, and makes existing algorithms very slow for large-scale applications. Together with the analytic solutions to lp-norm minimization with two specific values of p, i.e., p=1/2 and p=2/3, we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis. We first define the double nuclear norm and Frobenius/nuclear hybrid norm penalties, and then prove that they are in essence the Schatten-1/2 and 2/3 quasi-norms, respectively, which lead to much more tractable and scalable Lipschitz optimization problems. Our experimental analysis shows that both our methods yield more accurate solutions than original Schatten quasi-norm minimization, even when the number of observations is very limited. Finally, we apply our penalties to various low-level vision problems, e.g., text removal, moving object detection, image alignment and inpainting, and show that our methods usually outperform the state-of-the-art methods.

Fanhua Shang, Yuanyuan Liu, James Cheng

The Schatten quasi-norm was introduced to bridge the gap between the trace norm and rank function. However, existing algorithms are too slow or even impractical for large-scale problems. Motivated by the equivalence relation between the trace norm and its bilinear spectral penalty, we define two tractable Schatten norms, i.e.\ the bi-trace and tri-trace norms, and prove that they are in essence the Schatten-$1/2$ and $1/3$ quasi-norms, respectively. By applying the two defined Schatten quasi-norms to various rank minimization problems such as MC and RPCA, we only need to solve much smaller factor matrices. We design two efficient linearized alternating minimization algorithms to solve our problems and establish that each bounded sequence generated by our algorithms converges to a critical point. We also provide the restricted strong convexity (RSC) based and MC error bounds for our algorithms. Our experimental results verified both the efficiency and effectiveness of our algorithms compared with the state-of-the-art methods.

Fanhua Shang, Yuanyuan Liu, Kaiwen Zhou et al.

In this paper, we propose a novel sufficient decrease technique for stochastic variance reduced gradient descent methods such as SVRG and SAGA. In order to make sufficient decrease for stochastic optimization, we design a new sufficient decrease criterion, which yields sufficient decrease versions of stochastic variance reduction algorithms such as SVRG-SD and SAGA-SD as a byproduct. We introduce a coefficient to scale current iterate and to satisfy the sufficient decrease property, which takes the decisions to shrink, expand or even move in the opposite direction, and then give two specific update rules of the coefficient for Lasso and ridge regression. Moreover, we analyze the convergence properties of our algorithms for strongly convex problems, which show that our algorithms attain linear convergence rates. We also provide the convergence guarantees of our algorithms for non-strongly convex problems. Our experimental results further verify that our algorithms achieve significantly better performance than their counterparts.

Yuanyuan Liu, Fanhua Shang, James Cheng

Recently, many variance reduced stochastic alternating direction method of multipliers (ADMM) methods (e.g.\ SAG-ADMM, SDCA-ADMM and SVRG-ADMM) have made exciting progress such as linear convergence rates for strongly convex problems. However, the best known convergence rate for general convex problems is O(1/T) as opposed to O(1/T^2) of accelerated batch algorithms, where $T$ is the number of iterations. Thus, there still remains a gap in convergence rates between existing stochastic ADMM and batch algorithms. To bridge this gap, we introduce the momentum acceleration trick for batch optimization into the stochastic variance reduced gradient based ADMM (SVRG-ADMM), which leads to an accelerated (ASVRG-ADMM) method. Then we design two different momentum term update rules for strongly convex and general convex cases. We prove that ASVRG-ADMM converges linearly for strongly convex problems. Besides having a low per-iteration complexity as existing stochastic ADMM methods, ASVRG-ADMM improves the convergence rate on general convex problems from O(1/T) to O(1/T^2). Our experimental results show the effectiveness of ASVRG-ADMM.

Yaodong Yang, Rui Luo, Yuanyuan Liu

The Tweedie Compound Poisson-Gamma model is routinely used for modeling non-negative continuous data with a discrete probability mass at zero. Mixed models with random effects account for the covariance structure related to the grouping hierarchy in the data. An important application of Tweedie mixed models is pricing the insurance policies, e.g. car insurance. However, the intractable likelihood function, the unknown variance function, and the hierarchical structure of mixed effects have presented considerable challenges for drawing inferences on Tweedie. In this study, we tackle the Bayesian Tweedie mixed-effects models via variational inference approaches. In particular, we empower the posterior approximation by implicit models trained in an adversarial setting. To reduce the variance of gradients, we reparameterize random effects, and integrate out one local latent variable of Tweedie. We also employ a flexible hyper prior to ensure the richness of the approximation. Our method is evaluated on both simulated and real-world data. Results show that the proposed method has smaller estimation bias on the random effects compared to traditional inference methods including MCMC; it also achieves a state-of-the-art predictive performance, meanwhile offering a richer estimation of the variance function.

Fanhua Shang, Yuanyuan Liu, James Cheng et al.

Recently, research on accelerated stochastic gradient descent methods (e.g., SVRG) has made exciting progress (e.g., linear convergence for strongly convex problems). However, the best-known methods (e.g., Katyusha) requires at least two auxiliary variables and two momentum parameters. In this paper, we propose a fast stochastic variance reduction gradient (FSVRG) method, in which we design a novel update rule with the Nesterov's momentum and incorporate the technique of growing epoch size. FSVRG has only one auxiliary variable and one momentum weight, and thus it is much simpler and has much lower per-iteration complexity. We prove that FSVRG achieves linear convergence for strongly convex problems and the optimal $\mathcal{O}(1/T^2)$ convergence rate for non-strongly convex problems, where $T$ is the number of outer-iterations. We also extend FSVRG to directly solve the problems with non-smooth component functions, such as SVM. Finally, we empirically study the performance of FSVRG for solving various machine learning problems such as logistic regression, ridge regression, Lasso and SVM. Our results show that FSVRG outperforms the state-of-the-art stochastic methods, including Katyusha.

Fanhua Shang, Yuanyuan Liu, James Cheng et al.

In this paper, we propose a novel sufficient decrease technique for variance reduced stochastic gradient descent methods such as SAG, SVRG and SAGA. In order to make sufficient decrease for stochastic optimization, we design a new sufficient decrease criterion, which yields sufficient decrease versions of variance reduction algorithms such as SVRG-SD and SAGA-SD as a byproduct. We introduce a coefficient to scale current iterate and satisfy the sufficient decrease property, which takes the decisions to shrink, expand or move in the opposite direction, and then give two specific update rules of the coefficient for Lasso and ridge regression. Moreover, we analyze the convergence properties of our algorithms for strongly convex problems, which show that both of our algorithms attain linear convergence rates. We also provide the convergence guarantees of our algorithms for non-strongly convex problems. Our experimental results further verify that our algorithms achieve significantly better performance than their counterparts.

Fanhua Shang, Yuanyuan Liu, James Cheng

The Schatten-p quasi-norm $(0<p<1)$ is usually used to replace the standard nuclear norm in order to approximate the rank function more accurately. However, existing Schatten-p quasi-norm minimization algorithms involve singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each iteration, and thus may become very slow and impractical for large-scale problems. In this paper, we first define two tractable Schatten quasi-norms, i.e., the Frobenius/nuclear hybrid and bi-nuclear quasi-norms, and then prove that they are in essence the Schatten-2/3 and 1/2 quasi-norms, respectively, which lead to the design of very efficient algorithms that only need to update two much smaller factor matrices. We also design two efficient proximal alternating linearized minimization algorithms for solving representative matrix completion problems. Finally, we provide the global convergence and performance guarantees for our algorithms, which have better convergence properties than existing algorithms. Experimental results on synthetic and real-world data show that our algorithms are more accurate than the state-of-the-art methods, and are orders of magnitude faster.

Fanhua Shang, Yuanyuan Liu, James Cheng

The Schatten quasi-norm can be used to bridge the gap between the nuclear norm and rank function, and is the tighter approximation to matrix rank. However, most existing Schatten quasi-norm minimization (SQNM) algorithms, as well as for nuclear norm minimization, are too slow or even impractical for large-scale problems, due to the SVD or EVD of the whole matrix in each iteration. In this paper, we rigorously prove that for any p, p1, p2>0 satisfying 1/p=1/p1+1/p2, the Schatten-p quasi-norm of any matrix is equivalent to minimizing the product of the Schatten-p1 norm (or quasi-norm) and Schatten-p2 norm (or quasi-norm) of its two factor matrices. Then we present and prove the equivalence relationship between the product formula of the Schatten quasi-norm and its weighted sum formula for the two cases of p1 and p2: p1=p2 and p1\neq p2. In particular, when p>1/2, there is an equivalence between the Schatten-p quasi-norm of any matrix and the Schatten-2p norms of its two factor matrices, where the widely used equivalent formulation of the nuclear norm can be viewed as a special case. That is, various SQNM problems with p>1/2 can be transformed into the one only involving smooth, convex norms of two factor matrices, which can lead to simpler and more efficient algorithms than conventional methods. We further extend the theoretical results of two factor matrices to the cases of three and more factor matrices, from which we can see that for any 0<p<1, the Schatten-p quasi-norm of any matrix is the minimization of the mean of the Schatten-(p3+1)p norms of all factor matrices, where p3 denotes the largest integer not exceeding 1/p. In other words, for any 0<p<1, the SQNM problem can be transformed into an optimization problem only involving the smooth, convex norms of multiple factor matrices.

Fanhua Shang, Yuanyuan Liu, James Cheng

Higher-order tensors are becoming prevalent in many scientific areas such as computer vision, social network analysis, data mining and neuroscience. Traditional tensor decomposition approaches face three major challenges: model selecting, gross corruptions and computational efficiency. To address these problems, we first propose a parallel trace norm regularized tensor decomposition method, and formulate it as a convex optimization problem. This method does not require the rank of each mode to be specified beforehand, and can automatically determine the number of factors in each mode through our optimization scheme. By considering the low-rank structure of the observed tensor, we analyze the equivalent relationship of the trace norm between a low-rank tensor and its core tensor. Then, we cast a non-convex tensor decomposition model into a weighted combination of multiple much smaller-scale matrix trace norm minimization. Finally, we develop two parallel alternating direction methods of multipliers (ADMM) to solve our problems. Experimental results verify that our regularized formulation is effective, and our methods are robust to noise or outliers.