Qibin Zhao

Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination

CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance.

Sparse Bayesian Classification of EEG for Brain–Computer Interface

Regularization has been one of the most popular approaches to prevent overfitting in electroencephalogram (EEG) classification of brain-computer interfaces (BCIs). The effectiveness of regularization is often highly dependent on the selection of regularization parameters that are typically determined by cross-validation (CV). However, the CV imposes two main limitations on BCIs: 1) a large amount of training data is required from the user and 2) it takes a relatively long time to calibrate the classifier. These limitations substantially deteriorate the systems practicability and may cause a user to be reluctant to use BCIs. In this paper, we introduce a sparse Bayesian method by exploiting Laplace priors, namely, SBLaplace, for EEG classification. A sparse discriminant vector is learned with a Laplace prior in a hierarchical fashion under a Bayesian evidence framework. All required model parameters are automatically estimated from training data without the need of CV. Extensive comparisons are carried out between the SBLaplace algorithm and several other competing methods based on two EEG data sets. The experimental results demonstrate that the SBLaplace algorithm achieves better overall performance than the competing algorithms for EEG classification.

Linked Component Analysis From Matrices to High-Order Tensors: Applications to Biomedical Data

With the increasing availability of various sensor technologies, we now have access to large amounts of multiblock (also called multiset, multirelational, or multiview) 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 article, 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 multiblock multiway (tensor) data. We show how constrained multiblock 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 multiblock 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.

Spatial-Temporal Discriminant Analysis for ERP-Based Brain-Computer Interface

Linear discriminant analysis (LDA) has been widely adopted to classify event-related potential (ERP) in brain-computer interface (BCI). Good classification performance of the ERP-based BCI usually requires sufficient data recordings for effective training of the LDA classifier, and hence a long system calibration time which however may depress the system practicability and cause the users resistance to the BCI system. In this study, we introduce a spatial-temporal discriminant analysis (STDA) to ERP classification. As a multiway extension of the LDA, the STDA method tries to maximize the discriminant information between target and nontarget classes through finding two projection matrices from spatial and temporal dimensions collaboratively, which reduces effectively the feature dimensionality in the discriminant analysis, and hence decreases significantly the number of required training samples. The proposed STDA method was validated with dataset II of the BCI Competition III and dataset recorded from our own experiments, and compared to the state-of-the-art algorithms for ERP classification. Online experiments were additionally implemented for the validation. The superior classification performance in using few training samples shows that the STDA is effective to reduce the system calibration time and improve the classification accuracy, thereby enhancing the practicability of ERP-based BCI.

Multiway canonical correlation analysis for frequency components recognition in SSVEP-Based BCIs

Steady-state visual evoked potential (SSVEP)-based brain computer-interface (BCI) is one of the most popular BCI systems. An efficient SSVEP-based BCI system in shorter time with higher accuracy in recognizing SSVEP has been pursued by many studies. This paper introduces a novel multiway canonical correlation analysis (Multiway CCA) approach to recognize SSVEP. This approach is based on tensor CCA and focuses on multiway data arrays. Multiple CCAs are used to find appropriate reference signals for SSVEP recognition from different data arrays. SSVEP is then recognized by implementing multiple linear regression (MLR) between EEG and optimized reference signals. The proposed Multiway CCA is verified by comparing to the standard CCA and power spectral density analysis (PSDA). Results showed that the Multiway CCA achieved higher recognition accuracy within shorter time than that of the CCA and PSDA.

Bayesian Robust Tensor Factorization for Incomplete Multiway Data

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-CANDECOMP/PARAFAC (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 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 the 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 data sets demonstrate the superiorities of our method from several perspectives.

Smooth PARAFAC Decomposition for Tensor Completion

In recent years, low-rank based tensor completion, which is a higher order extension of matrix completion, has received considerable attention. However, the low-rank assumption is not sufficient for the recovery of visual data, such as color and 3D images, when the ratio of missing data is extremely high. In this paper, we consider “smoothness” constraints as well as low-rank approximations and propose an efficient algorithm for performing tensor completion that is particularly powerful regarding visual data. The proposed method admits significant advantages, owing to the integration of smooth PARAFAC decomposition for incomplete tensors and the efficient selection of models in order to minimize the tensor rank. Thus, our proposed method is termed as “smooth PARAFAC tensor completion (SPC).” In order to impose the smoothness constraints, we employ two strategies, total variation (SPC-TV) and quadratic variation (SPC-QV), and invoke the corresponding algorithms for model learning. Extensive experimental evaluations on both synthetic and real-world visual data illustrate the significant improvements of our method, in terms of both prediction performance and efficiency, compared with many state-of-the-art tensor completion methods.

Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness

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, the 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.

Uncorrelated Multiway Discriminant Analysis for Motor Imagery EEG Classification

Motor imagery-based brain-computer interfaces (BCIs) training has been proved to be an effective communication system between human brain and external devices. A practical problem in BCI-based systems is how to correctly and efficiently identify and extract subject-specific features from the blurred scalp electroencephalography (EEG) and translate those features into device commands in order to control external devices. In real BCI-based applications, we usually define frequency bands and channels configuration that related to brain activities beforehand. However, a steady configuration usually loses effects due to individual variability among different subjects in practical applications. In this study, a robust tensor-based method is proposed for a multiway discriminative subspace extraction from tensor-represented EEG data, which performs well in motor imagery EEG classification without the prior neurophysiologic knowledge like channels configuration and active frequency bands. Motor imagery EEG patterns in spatial-spectral-temporal domain are detected directly from the multidimensional EEG, which may provide insights to the underlying cortical activity patterns. Extensive experiment comparisons have been performed on a benchmark dataset from the famous BCI competition III as well as self-acquired data from healthy subjects and stroke patients. The experimental results demonstrate the superior performance of the proposed method over the contemporary methods.

Multi-domain feature of event-related potential extracted by nonnegative tensor factorization: 5 vs. 14 electrodes EEG data

As nonnegative tensor factorization (NTF) is particularly useful for the problem of underdetermined linear transform model, we performed NTF on the EEG data recorded from 14 electrodes to extract the multi-domain feature of N170 which is a visual event-related potential (ERP), as well as 5 typical electrodes in occipital-temporal sites for N170 and in frontal-central sites for vertex positive potential (VPP) which is the counterpart of N170, respectively. We found that the multi-domain feature of N170 from 5 electrodes was very similar to that from 14 electrodes and more discriminative for different groups of participants than that of VPP from 5 electrodes. Hence, we conclude that when the data of typical electrodes for an ERP are decomposed by NTF, the estimated multi-domain feature of this ERP keeps identical to its counterpart extracted from the data of all electrodes used in one ERP experiment.