Guoxu Zhou

Andong Wang, Chao Li, Mingyuan Bai et al. · tsinghua

Achieving efficient and robust multi-channel data learning is a challenging task in data science. By exploiting low-rankness in the transformed domain, i.e., transformed low-rankness, tensor Singular Value Decomposition (t-SVD) has achieved extensive success in multi-channel data representation and has recently been extended to function representation such as Neural Networks with t-product layers (t-NNs). However, it still remains unclear how t-SVD theoretically affects the learning behavior of t-NNs. This paper is the first to answer this question by deriving the upper bounds of the generalization error of both standard and adversarially trained t-NNs. It reveals that the t-NNs compressed by exact transformed low-rank parameterization can achieve a sharper adversarial generalization bound. In practice, although t-NNs rarely have exactly transformed low-rank weights, our analysis further shows that by adversarial training with gradient flow (GF), the over-parameterized t-NNs with ReLU activations are trained with implicit regularization towards transformed low-rank parameterization under certain conditions. We also establish adversarial generalization bounds for t-NNs with approximately transformed low-rank weights. Our analysis indicates that the transformed low-rank parameterization can promisingly enhance robust generalization for t-NNs.

Yichun Qiu, Weijun Sun, Guoxu Zhou et al. · tsinghua

Efficient and accurate low-rank approximation (LRA) methods are of great significance for large-scale data analysis. Randomized tensor decompositions have emerged as powerful tools to meet this need, but most existing methods perform poorly in the presence of noise interference. Inspired by the remarkable performance of randomized block Krylov iteration (rBKI) in reducing the effect of tail singular values, this work designs an rBKI-based Tucker decomposition (rBKI-TK) for accurate approximation, together with a hierarchical tensor ring decomposition based on rBKI-TK for efficient compression of large-scale data. Besides, the error bound between the deterministic LRA and the randomized LRA is studied. Numerical experiences demonstrate the efficiency, accuracy and scalability of the proposed methods in both data compression and denoising.

Qi Jiang, Guoxu Zhou, Qibin Zhao · tsinghua

Concept Factorization (CF), as a novel paradigm of representation learning, has demonstrated superior performance in multi-view clustering tasks. It overcomes limitations such as the non-negativity constraint imposed by traditional matrix factorization methods and leverages kernel methods to learn latent representations that capture the underlying structure of the data, thereby improving data representation. However, existing multi-view concept factorization methods fail to consider the limited labeled information inherent in real-world multi-view data. This often leads to significant performance loss. To overcome these limitations, we propose a novel semi-supervised multi-view concept factorization model, named SMVCF. In the SMVCF model, we first extend the conventional single-view CF to a multi-view version, enabling more effective exploration of complementary information across multiple views. We then integrate multi-view CF, label propagation, and manifold learning into a unified framework to leverage and incorporate valuable information present in the data. Additionally, an adaptive weight vector is introduced to balance the importance of different views in the clustering process. We further develop targeted optimization methods specifically tailored for the SMVCF model. Finally, we conduct extensive experiments on four diverse datasets with varying label ratios to evaluate the performance of SMVCF. The experimental results demonstrate the effectiveness and superiority of our proposed approach in multi-view clustering tasks.

Yuning Qiu, Guoxu Zhou, Qibin Zhao et al.

Tensor completion is a fundamental tool for incomplete data analysis, where the goal is to predict missing entries from partial observations. However, existing methods often make the explicit or implicit assumption that the observed entries are noise-free to provide a theoretical guarantee of exact recovery of missing entries, which is quite restrictive in practice. To remedy such drawbacks, this paper proposes a novel noisy tensor completion model, which complements the incompetence of existing works in handling the degeneration of high-order and noisy observations. Specifically, the tensor ring nuclear norm (TRNN) and least-squares estimator are adopted to regularize the underlying tensor and the observed entries, respectively. In addition, a non-asymptotic upper bound of estimation error is provided to depict the statistical performance of the proposed estimator. Two efficient algorithms are developed to solve the optimization problem with convergence guarantee, one of which is specially tailored to handle large-scale tensors by replacing the minimization of TRNN of the original tensor equivalently with that of a much smaller one in a heterogeneous tensor decomposition framework. Experimental results on both synthetic and real-world data demonstrate the effectiveness and efficiency of the proposed model in recovering noisy incomplete tensor data compared with state-of-the-art tensor completion models.

Peilin Yang, Yonghui Huang, Yuning Qiu et al.

Tensor completion aimes at recovering missing data, and it is one of the popular concerns in deep learning and signal processing. Among the higher-order tensor decomposition algorithms, the recently proposed fully-connected tensor network decomposition (FCTN) algorithm is the most advanced. In this paper, by leveraging the superior expression of the fully-connected tensor network (FCTN) decomposition, we propose a new tensor completion method named the fully connected tensor network weighted optization(FCTN-WOPT). The algorithm performs a composition of the completed tensor by initialising the factors from the FCTN decomposition. We build a loss function with the weight tensor, the completed tensor and the incomplete tensor together, and then update the completed tensor using the lbfgs gradient descent algorithm to reduce the spatial memory occupation and speed up iterations. Finally we test the completion with synthetic data and real data (both image data and video data) and the results show the advanced performance of our FCTN-WOPT when it is applied to higher-order tensor completion.

Peilin Yang, Weijun Sun, Qibin Zhao et al.

The prevalent fully-connected tensor network (FCTN) has achieved excellent success to compress data. However, the FCTN decomposition suffers from slow computational speed when facing higher-order and large-scale data. Naturally, there arises an interesting question: can a new model be proposed that decomposes the tensor into smaller ones and speeds up the computation of the algorithm? This work gives a positive answer by formulating a novel higher-order tensor decomposition model that utilizes latent matrices based on the tensor network structure, which can decompose a tensor into smaller-scale data than the FCTN decomposition, hence we named it Latent Matrices for Tensor Network Decomposition (LMTN). Furthermore, three optimization algorithms, LMTN-PAM, LMTN-SVD and LMTN-AR, have been developed and applied to the tensor-completion task. In addition, we provide proofs of theoretical convergence and complexity analysis for these algorithms. Experimental results show that our algorithm has the effectiveness in both deep learning dataset compression and higher-order tensor completion, and that our LMTN-SVD algorithm is 3-6 times faster than the FCTN-PAM algorithm and only a 1.8 points accuracy drop.

Tianle Hu, Weijun Lv, Na Han et al.

Domain adaptive retrieval aims to transfer knowledge from a labeled source domain to an unlabeled target domain, enabling effective retrieval while mitigating domain discrepancies. However, existing methods encounter several fundamental limitations: 1) neglecting class-level semantic alignment and excessively pursuing pair-wise sample alignment; 2) lacking either pseudo-label reliability consideration or geometric guidance for assessing label correctness; 3) directly quantizing original features affected by domain shift, undermining the quality of learned hash codes. In view of these limitations, we propose Prototype-Based Semantic Consistency Alignment (PSCA), a two-stage framework for effective domain adaptive retrieval. In the first stage, a set of orthogonal prototypes directly establishes class-level semantic connections, maximizing inter-class separability while gathering intra-class samples. During the prototype learning, geometric proximity provides a reliability indicator for semantic consistency alignment through adaptive weighting of pseudo-label confidences. The resulting membership matrix and prototypes facilitate feature reconstruction, ensuring quantization on reconstructed rather than original features, thereby improving subsequent hash coding quality and seamlessly connecting both stages. In the second stage, domain-specific quantization functions process the reconstructed features under mutual approximation constraints, generating unified binary hash codes across domains. Extensive experiments validate PSCA's superior performance across multiple datasets.

Tenghui Li, Guoxu Zhou, Xuyang Zhao et al.

As the length of input text grows, the key-value (KV) cache in LLMs imposes prohibitive GPU memory costs and limits long-context inference on resource constrained devices. Existing approaches, such as KV quantization and pruning, reduce memory usage but suffer from numerical precision loss or suboptimal retention of key-value pairs. We introduce Low Rank Query and Key attention (LRQK), a two-stage framework that jointly decomposes the full-precision query and key matrices into compact rank-\(r\) factors during the prefill stage, and then uses these low-dimensional projections to compute proxy attention scores in \(\mathcal{O}(lr)\) time at each decode step. By selecting only the top-\(k\) tokens and a small fixed set of recent tokens, LRQK employs a mixed GPU-CPU cache with a hit-and-miss mechanism that transfers only missing full-precision KV pairs, thereby preserving exact attention outputs while reducing CPU-GPU data movement. Extensive experiments on the RULER and LongBench benchmarks with LLaMA-3-8B and Qwen2.5-7B demonstrate that LRQK matches or surpasses leading sparse-attention methods in long context settings, while delivering significant memory savings with minimal loss in accuracy. Our code is available at https://github.com/tenghuilee/LRQK.

Ganxi Xu, Zhao-Rong Lai, Yuting Tang et al.

Brain encoding models not only serve to decipher how visual stimuli are transformed into neural responses, but also represent a critical step toward visual prostheses that restore vision for patients with severe vision disorders. Brain encoding involves two fundamental steps: achieving faithful reconstruction of neural responses and establishing cross-modal alignment between visual stimuli and neural responses. To this end, we propose ViBE, a novel brain encoding framework for generating magnetoencephalography (MEG) and electroencephalography (EEG) signals from visual stimuli. Specifically, we first design a spatio-temporal convolutional variational autoencoder (TSC-VAE) that captures the spatio-temporal characteristics of M/EEG signals for effective neural response reconstruction. To bridge the modality gap between visual features and neural representations, we employ Q-Former to map CLIP image embeddings to the TSC-VAE latent space, producing neural proxy embeddings. For comprehensive cross-modal alignment, we combine mean squared error (MSE) loss for point-wise feature matching with sliced Wasserstein distance (SWD) for probability distribution alignment between the neural proxy embeddings and TSC-VAE latent embeddings. We conduct extensive experiments on the THINGS-EEG2 and THINGS-MEG datasets, demonstrating the effectiveness of our approach in generating high-quality M/EEG signals from visual stimuli.

Junhua Zeng, Chao Li, Zhun Sun et al.

Tensor networks are efficient for extremely high-dimensional representation, but their model selection, known as tensor network structure search (TN-SS), is a challenging problem. Although several works have targeted TN-SS, most existing algorithms are manually crafted heuristics with poor performance, suffering from the curse of dimensionality and local convergence. In this work, we jump out of the box, studying how to harness large language models (LLMs) to automatically discover new TN-SS algorithms, replacing the involvement of human experts. By observing how human experts innovate in research, we model their common workflow and propose an automatic algorithm discovery framework called tnGPS. The proposed framework is an elaborate prompting pipeline that instruct LLMs to generate new TN-SS algorithms through iterative refinement and enhancement. The experimental results demonstrate that the algorithms discovered by tnGPS exhibit superior performance in benchmarks compared to the current state-of-the-art methods.

Yue Huang, Zi'ang Li, Tianle Hu et al.

Although dual-stream architectures have achieved remarkable success in single image reflection removal, they fail to fully exploit inter-layer complementarity in their physical modeling and network design, which limits the quality of image separation. To address this fundamental limitation, we propose two targeted improvements to enhance dual-stream architectures: First, we introduce a novel inter-layer complementarity model where low-frequency components extracted from the residual layer interact with the transmission layer through dual-stream architecture to enhance inter-layer complementarity. Meanwhile, high-frequency components from the residual layer provide inverse modulation to both streams, improving the detail quality of the transmission layer. Second, we propose an efficient inter-layer complementarity attention mechanism which first cross-reorganizes dual streams at the channel level to obtain reorganized streams with inter-layer complementary structures, then performs attention computation on the reorganized streams to achieve better inter-layer separation, and finally restores the original stream structure for output. Experimental results demonstrate that our method achieves state-of-the-art separation quality on multiple public datasets while significantly reducing both computational cost and model complexity.

Tenghui Li, Guoxu Zhou, Xuyang Zhao et al.

Large language models have demonstrated predictable scaling behaviors with respect to model parameters and training data. This study investigates whether a similar scaling relationship exist for vision-language models with respect to the number of vision tokens. A mathematical framework is developed to characterize a relationship between vision token number and the expected divergence of distance between vision-referencing sequences. The theoretical analysis reveals two distinct scaling regimes: sublinear scaling for less vision tokens and linear scaling for more vision tokens. This aligns with model performance relationships of the form \(S(n) \approx c / n^{α(n)}\), where the scaling exponent relates to the correlation structure between vision token representations. Empirical validations across multiple vision-language benchmarks show that model performance matches the prediction from scaling relationship. The findings contribute to understanding vision token scaling in transformers through a theoretical framework that complements empirical observations.

Zhenhao Huang, Yuning Qiu, Xinqi Chen et al.

Robust tensor completion (RTC) aims to recover a low-rank tensor from its incomplete observation with outlier corruption. The recently proposed tensor ring (TR) model has demonstrated superiority in solving the RTC problem. However, the existing methods either require a pre-assigned TR rank or aggressively pursue the minimum TR rank, thereby often leading to biased solutions in the presence of noise. In this paper, a Bayesian robust tensor ring decomposition (BRTR) method is proposed to give more accurate solutions to the RTC problem, which can avoid exquisite selection of the TR rank and penalty parameters. A variational Bayesian (VB) algorithm is developed to infer the probability distribution of posteriors. During the learning process, BRTR can prune off slices of core tensor with marginal components, resulting in automatic TR rank detection. Extensive experiments show that BRTR can achieve significantly improved performance than other state-of-the-art methods.

Yuyuan Yu, Guoxu Zhou, Haonan Huang et al.

Distinguishing the importance of views has proven to be quite helpful for semi-supervised multi-view learning models. However, existing strategies cannot take advantage of semi-supervised information, only distinguishing the importance of views from a data feature perspective, which is often influenced by low-quality views then leading to poor performance. In this paper, by establishing a link between labeled data and the importance of different views, we propose an auto-weighted strategy to evaluate the importance of views from a label perspective to avoid the negative impact of unimportant or low-quality views. Based on this strategy, we propose a transductive semi-supervised auto-weighted multi-view classification model. The initialization of the proposed model can be effectively determined by labeled data, which is practical. The model is decoupled into three small-scale sub-problems that can efficiently be optimized with a local convergence guarantee. The experimental results on classification tasks show that the proposed method achieves optimal or sub-optimal classification accuracy at the lowest computational cost compared to other related methods, and the weight change experiments show that our proposed strategy can distinguish view importance more accurately than other related strategies on multi-view datasets with low-quality views.

Yuning Qiu, Guoxu Zhou, Zhenhao Huang et al.

Tensor robust principal component analysis (TRPCA) is a fundamental model in machine learning and computer vision. Recently, tensor train (TT) decomposition has been verified effective to capture the global low-rank correlation for tensor recovery tasks. However, due to the large-scale tensor data in real-world applications, previous TRPCA models often suffer from high computational complexity. In this letter, we propose an efficient TRPCA under hybrid model of Tucker and TT. Specifically, in theory we reveal that TT nuclear norm (TTNN) of the original big tensor can be equivalently converted to that of a much smaller tensor via a Tucker compression format, thereby significantly reducing the computational cost of singular value decomposition (SVD). Numerical experiments on both synthetic and real-world tensor data verify the superiority of the proposed model.

Tenghui Li, Guoxu Zhou, Yuning Qiu et al.

We make an attempt to understanding convolutional neural network by exploring the relationship between (deep) convolutional neural networks and Volterra convolutions. We propose a novel approach to explain and study the overall characteristics of neural networks without being disturbed by the horribly complex architectures. Specifically, we attempt to convert the basic structures of a convolutional neural network (CNN) and their combinations to the form of Volterra convolutions. The results show that most of convolutional neural networks can be approximated in the form of Volterra convolution, where the approximated proxy kernels preserve the characteristics of the original network. Analyzing these proxy kernels may give valuable insight about the original network. Base on this setup, we presented methods to approximating the order-zero and order-one proxy kernels, and verified the correctness and effectiveness of our results.

Xinhai Zhao, Yuyuan Yu, Guoxu Zhou et al.

For the high dimensional data representation, nonnegative tensor ring (NTR) decomposition equipped with manifold learning has become a promising model to exploit the multi-dimensional structure and extract the feature from tensor data. However, the existing methods such as graph regularized tensor ring decomposition (GNTR) only models the pair-wise similarities of objects. For tensor data with complex manifold structure, the graph can not exactly construct similarity relationships. In this paper, in order to effectively utilize the higher-dimensional and complicated similarities among objects, we introduce hypergraph to the framework of NTR to further enhance the feature extraction, upon which a hypergraph regularized nonnegative tensor ring decomposition (HGNTR) method is developed. To reduce the computational complexity and suppress the noise, we apply the low-rank approximation trick to accelerate HGNTR (called LraHGNTR). Our experimental results show that compared with other state-of-the-art algorithms, the proposed HGNTR and LraHGNTR can achieve higher performance in clustering tasks, in addition, LraHGNTR can greatly reduce running time without decreasing accuracy.

Yu Zhang, Tao Zhou, Wei Wu et al.

Accurate electroencephalogram (EEG) pattern decoding for specific mental tasks is one of the key steps for the development of brain-computer interface (BCI), which is quite challenging due to the considerably low signal-to-noise ratio of EEG collected at the brain scalp. Machine learning provides a promising technique to optimize EEG patterns toward better decoding accuracy. However, existing algorithms do not effectively explore the underlying data structure capturing the true EEG sample distribution, and hence can only yield a suboptimal decoding accuracy. To uncover the intrinsic distribution structure of EEG data, we propose a clustering-based multi-task feature learning algorithm for improved EEG pattern decoding. Specifically, we perform affinity propagation-based clustering to explore the subclasses (i.e., clusters) in each of the original classes, and then assign each subclass a unique label based on a one-versus-all encoding strategy. With the encoded label matrix, we devise a novel multi-task learning algorithm by exploiting the subclass relationship to jointly optimize the EEG pattern features from the uncovered subclasses. We then train a linear support vector machine with the optimized features for EEG pattern decoding. Extensive experimental studies are conducted on three EEG datasets to validate the effectiveness of our algorithm in comparison with other state-of-the-art approaches. The improved experimental results demonstrate the outstanding superiority of our algorithm, suggesting its prominent performance for EEG pattern decoding in BCI applications.

Yuyuan Yu, Guoxu Zhou, Ning Zheng et al.

Tensor ring (TR) decomposition is a powerful tool for exploiting the low-rank nature of multiway data and has demonstrated great potential in a variety of important applications. In this paper, nonnegative tensor ring (NTR) decomposition and graph regularized NTR (GNTR) decomposition are proposed, where the former equips TR decomposition with local feature extraction by imposing nonnegativity on the core tensors and the latter is additionally able to capture manifold geometry information of tensor data, both significantly extend the applications of TR decomposition for nonnegative multiway representation learning. Accelerated proximal gradient based methods are derived for NTR and GNTR. The experimental result demonstrate that the proposed algorithms can extract parts-based basis with rich colors and rich lines from tensor objects that provide more interpretable and meaningful representation, and hence yield better performance than the state-of-the-art tensor based methods in clustering and classification tasks.

Jinshi Yu, Chao Li, Qibin Zhao et al.

Tensor ring (TR) decomposition has been successfully used to obtain the state-of-the-art performance in the visual data completion problem. However, the existing TR-based completion methods are severely non-convex and computationally demanding. In addition, the determination of the optimal TR rank is a tough work in practice. To overcome these drawbacks, we first introduce a class of new tensor nuclear norms by using tensor circular unfolding. Then we theoretically establish connection between the rank of the circularly-unfolded matrices and the TR ranks. We also develop an efficient tensor completion algorithm by minimizing the proposed tensor nuclear norm. Extensive experimental results demonstrate that our proposed tensor completion method outperforms the conventional tensor completion methods in the image/video in-painting problem with striped missing values.

Jinshi Yu, Guoxu Zhou, Andrzej Cichocki et al.

Nonsmooth Nonnegative Matrix Factorization (nsNMF) is capable of producing more localized, less overlapped feature representations than other variants of NMF while keeping satisfactory fit to data. However, nsNMF as well as other existing NMF methods is incompetent to learn hierarchical features of complex data due to its shallow structure. To fill this gap, we propose a deep nsNMF method coined by the fact that it possesses a deeper architecture compared with standard nsNMF. The deep nsNMF not only gives parts-based features due to the nonnegativity constraints, but also creates higher-level, more abstract features by combing lower-level ones. The in-depth description of how deep architecture can help to efficiently discover abstract features in dnsNMF is presented. And we also show that the deep nsNMF has close relationship with the deep autoencoder, suggesting that the proposed model inherits the major advantages from both deep learning and NMF. Extensive experiments demonstrate the standout performance of the proposed method in clustering analysis.

Yuning Qiu, Guoxu Zhou, Kan Xie

Nonnegative Matrix Factorization (NMF) is a widely used technique for data representation. Inspired by the expressive power of deep learning, several NMF variants equipped with deep architectures have been proposed. However, these methods mostly use the only nonnegativity while ignoring task-specific features of data. In this paper, we propose a novel deep approximately orthogonal nonnegative matrix factorization method where both nonnegativity and orthogonality are imposed with the aim to perform a hierarchical clustering by using different level of abstractions of data. Experiment on two face image datasets showed that the proposed method achieved better clustering performance than other deep matrix factorization methods and state-of-the-art single layer NMF variants.

Qibin Zhao, Guoxu Zhou, Shengli Xie et al.

Tensor networks have in recent years emerged as the powerful tools for solving the large-scale optimization problems. One of the most popular tensor network is tensor train (TT) decomposition that acts as the building blocks for the complicated tensor networks. However, the TT decomposition highly depends on permutations of tensor dimensions, due to its strictly sequential multilinear products over latent cores, which leads to difficulties in finding the optimal TT representation. In this paper, we introduce a fundamental tensor decomposition model to represent a large dimensional tensor by a circular multilinear products over a sequence of low dimensional cores, which can be graphically interpreted as a cyclic interconnection of 3rd-order tensors, and thus termed as tensor ring (TR) decomposition. The key advantage of TR model is the circular dimensional permutation invariance which is gained by employing the trace operation and treating the latent cores equivalently. TR model can be viewed as a linear combination of TT decompositions, thus obtaining the powerful and generalized representation abilities. For optimization of latent cores, we present four different algorithms based on the sequential SVDs, ALS scheme, and block-wise ALS techniques. Furthermore, the mathematical properties of TR model are investigated, which shows that the basic multilinear algebra can be performed efficiently by using TR representaions and the classical tensor decompositions can be conveniently transformed into the TR representation. Finally, the experiments on both synthetic signals and real-world datasets were conducted to evaluate the performance of different algorithms.

Guoxu Zhou, Qibin Zhao, Yu Zhang et al.

With the increasing availability of various sensor technologies, we now have access to large amounts of multi-block (also called multi-set, multi-relational, or multi-view) data that need to be jointly analyzed to explore their latent connections. Various component analysis methods have played an increasingly important role for the analysis of such coupled data. In this paper, we first provide a brief review of existing matrix-based (two-way) component analysis methods for the joint analysis of such data with a focus on biomedical applications. Then, we discuss their important extensions and generalization to multi-block multiway (tensor) data. We show how constrained multi-block tensor decomposition methods are able to extract similar or statistically dependent common features that are shared by all blocks, by incorporating the multiway nature of data. Special emphasis is given to the flexible common and individual feature analysis of multi-block data with the aim to simultaneously extract common and individual latent components with desired properties and types of diversity. Illustrative examples are given to demonstrate their effectiveness for biomedical data analysis.

Qibin Zhao, Guoxu Zhou, Liqing Zhang et al.

We propose a generative model for robust tensor factorization in the presence of both missing data and outliers. The objective is to explicitly infer the underlying low-CP-rank tensor capturing the global information and a sparse tensor capturing the local information (also considered as outliers), thus providing the robust predictive distribution over missing entries. The low-CP-rank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior, while the sparse tensor is modeled by a hierarchical view of Student-$t$ distribution that associates an individual hyperparameter with each element independently. For model learning, we develop an efficient closed-form variational inference under a fully Bayesian treatment, which can effectively prevent the overfitting problem and scales linearly with data size. In contrast to existing related works, our method can perform model selection automatically and implicitly without need of tuning parameters. More specifically, it can discover the groundtruth of CP rank and automatically adapt the sparsity inducing priors to various types of outliers. In addition, the tradeoff between the low-rank approximation and the sparse representation can be optimized in the sense of maximum model evidence. The extensive experiments and comparisons with many state-of-the-art algorithms on both synthetic and real-world datasets demonstrate the superiorities of our method from several perspectives.

Guoxu Zhou, Andrzej Cichocki, Shengli Xie

Tensor decompositions are promising tools for big data analytics as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most existing approaches are not designed to meet the major challenges posed by big data analytics. This paper attempts to improve the scalability of tensor decompositions and provides two contributions: A flexible and fast algorithm for the CP decomposition (FFCP) of tensors based on their Tucker compression; A distributed randomized Tucker decomposition approach for arbitrarily big tensors but with relatively low multilinear rank. These two algorithms can deal with huge tensors, even if they are dense. Extensive simulations provide empirical evidence of the validity and efficiency of the proposed algorithms.

Guoxu Zhou, Andrzej Cichocki, Qibin Zhao et al.

Nonnegative Tucker decomposition (NTD) is a powerful tool for the extraction of nonnegative parts-based and physically meaningful latent components from high-dimensional tensor data while preserving the natural multilinear structure of data. However, as the data tensor often has multiple modes and is large-scale, existing NTD algorithms suffer from a very high computational complexity in terms of both storage and computation time, which has been one major obstacle for practical applications of NTD. To overcome these disadvantages, we show how low (multilinear) rank approximation (LRA) of tensors is able to significantly simplify the computation of the gradients of the cost function, upon which a family of efficient first-order NTD algorithms are developed. Besides dramatically reducing the storage complexity and running time, the new algorithms are quite flexible and robust to noise because any well-established LRA approaches can be applied. We also show how nonnegativity incorporating sparsity substantially improves the uniqueness property and partially alleviates the curse of dimensionality of the Tucker decompositions. Simulation results on synthetic and real-world data justify the validity and high efficiency of the proposed NTD algorithms.

Yu Zhang, Guoxu Zhou, Jing Jin et al.

Canonical correlation analysis (CCA) has been one of the most popular methods for frequency recognition in steady-state visual evoked potential (SSVEP)-based brain-computer interfaces (BCIs). Despite its efficiency, a potential problem is that using pre-constructed sine-cosine waves as the required reference signals in the CCA method often does not result in the optimal recognition accuracy due to their lack of features from the real EEG data. To address this problem, this study proposes a novel method based on multiset canonical correlation analysis (MsetCCA) to optimize the reference signals used in the CCA method for SSVEP frequency recognition. The MsetCCA method learns multiple linear transforms that implement joint spatial filtering to maximize the overall correlation among canonical variates, and hence extracts SSVEP common features from multiple sets of EEG data recorded at the same stimulus frequency. The optimized reference signals are formed by combination of the common features and completely based on training data. Experimental study with EEG data from ten healthy subjects demonstrates that the MsetCCA method improves the recognition accuracy of SSVEP frequency in comparison with the CCA method and other two competing methods (multiway CCA (MwayCCA) and phase constrained CCA (PCCA)), especially for a small number of channels and a short time window length. The superiority indicates that the proposed MsetCCA method is a new promising candidate for frequency recognition in SSVEP-based BCIs.

Guoxu Zhou, Andrzej Cichocki, Shengli Xie

Canonical Polyadic (or CANDECOMP/PARAFAC, CP) decompositions (CPD) are widely applied to analyze high order tensors. Existing CPD methods use alternating least square (ALS) iterations and hence need to unfold tensors to each of the $N$ modes frequently, which is one major bottleneck of efficiency for large-scale data and especially when $N$ is large. To overcome this problem, in this paper we proposed a new CPD method which converts the original $N$th ($N>3$) order tensor to a 3rd-order tensor first. Then the full CPD is realized by decomposing this mode reduced tensor followed by a Khatri-Rao product projection procedure. This way is quite efficient as unfolding to each of the $N$ modes are avoided, and dimensionality reduction can also be easily incorporated to further improve the efficiency. We show that, under mild conditions, any $N$th-order CPD can be converted into a 3rd-order case but without destroying the essential uniqueness, and theoretically gives the same results as direct $N$-way CPD methods. Simulations show that, compared with state-of-the-art CPD methods, the proposed method is more efficient and escape from local solutions more easily.