Bayesian tensor network structure search and its application to tensor completion.

Tensor network (TN) has demonstrated remarkable efficacy in the compact representation of high-order data. In contrast to the TN methods with pre-determined structures, the recently introduced tensor network structure search (TNSS) methods automatically learn a compact TN structure from the data, gaining increasing attention. Nonetheless, TNSS requires time-consuming manual adjustments of the penalty parameters that control the model complexity to achieve better performance, especially in the presence of missing or noisy data. To provide an effective solution to this problem, in this paper, we propose a parameters tuning-free TNSS algorithm based on Bayesian modeling, aiming at conducting TNSS in a fully data-driven manner. Specifically, the uncertainty in the data corruption is well-incorporated in the prior setting of the probabilistic model. For TN structure determination, we reframe it as a rank learning problem of the fully-connected tensor network (FCTN), integrating the generalized inverse Gaussian (GIG) distribution for low-rank promotion. To eliminate the need for hyperparameter tuning, we adopt a fully Bayesian approach and propose an efficient Markov chain Monte Carlo (MCMC) algorithm for posterior distribution sampling. Compared with the previous TNSS method, experiment results demonstrate the proposed algorithm can effectively and efficiently find the latent TN structures of the data under various missing and noise conditions and achieves the best recovery results. Furthermore, our method exhibits superior performance in tensor completion with real-world data compared to other state-of-the-art tensor-decomposition-based completion methods.

A Novel Tensor Ring Sparsity Measurement for Image Completion.

As a promising data analysis technique, sparse modeling has gained widespread traction in the field of image processing, particularly for image recovery. The matrix rank, served as a measure of data sparsity, quantifies the sparsity within the Kronecker basis representation of a given piece of data in the matrix format. Nevertheless, in practical scenarios, much of the data are intrinsically multi-dimensional, and thus, using a matrix format for data representation will inevitably yield sub-optimal outcomes. Tensor decomposition (TD), as a high-order generalization of matrix decomposition, has been widely used to analyze multi-dimensional data. In a direct generalization to the matrix rank, low-rank tensor modeling has been developed for multi-dimensional data analysis and achieved great success. Despite its efficacy, the connection between TD rank and the sparsity of the tensor data is not direct. In this work, we introduce a novel tensor ring sparsity measurement (TRSM) for measuring the sparsity of the tensor. This metric relies on the tensor ring (TR) Kronecker basis representation of the tensor, providing a unified interpretation akin to matrix sparsity measurements, wherein the Kronecker basis serves as the foundational representation component. Moreover, TRSM can be efficiently computed by the product of the ranks of the mode-2 unfolded TR-cores. To enhance the practical performance of TRSM, the folded-concave penalty of the minimax concave penalty is introduced as a nonconvex relaxation. Lastly, we extend the TRSM to the tensor completion problem and use the alternating direction method of the multipliers scheme to solve it. Experiments on image and video data completion demonstrate the effectiveness of the proposed method.

Imbalanced low-rank tensor completion via latent matrix factorization.

Tensor completion has been widely used in computer vision and machine learning. Most existing tensor completion methods empirically assume the intrinsic tensor is simultaneous low-rank in all over modes. However, tensor data recorded from real-world applications may conflict with these assumptions, e.g., face images taken from different subjects often lie in a union of low-rank subspaces, which may result in a quite high rank or even full rank structure in its sample mode. To this aim, in this paper, we propose an imbalanced low-rank tensor completion method, which can flexibly estimate the low-rank incomplete tensor via decomposing it into a mixture of multiple latent tensor ring (TR) rank components. Specifically, each latent component is approximated using low-rank matrix factorization based on TR unfolding matrix. In addition, an effective proximal alternating minimization algorithm is developed and theoretically proved to maintain the global convergence property, that is, the whole sequence of iterates is convergent and converges to a critical point. Extensive experiments on both synthetic and real-world tensor data demonstrate that the proposed method achieves more favorable completion results with less computational cost when compared to the state-of-the-art tensor completion methods.

SNP Data Science for Classification of Bipolar Disorder I and Bipolar Disorder II.

Bipolar disorder I (BD-I) and bipolar disorder II (BD-II) have specific characteristics and clear diagnostic criteria, but quite different treatment guidelines. In clinical practice, BD-II is commonly mistaken as a mild form of BD-I. This study uses data science technique to identify the important Single Nucleotide Polymorphisms (SNPs) significantly affecting the classifications of BD-I and BD-II, and develops a set of complementary diagnostic classifiers to enhance the diagnostic process. Screening assessments and SNP genotypes of 316 Han Chinese were performed with the Affymetrix Axiom Genome-Wide TWB Array Plate. The results show that the classifier constructed by 23 SNPs reached the area under curve of ROC (AUC) level of 0.939, while the classifier constructed by 42 SNPs reached the AUC level of 0.9574, which is a mere addition of 1.84 percent. The accuracy rate of classification increased by 3.46 percent. This study also uses Gene Ontology (GO) and Pathway to conduct a functional analysis and identify significant items, including calcium ion binding, GABA-A receptor activity, Rap1 signaling pathway, ECM proteoglycans, IL12-mediated signaling events, Nicotine addiction), and PI3K-Akt signaling pathway. The study can address time-consuming SNPs identification and also quantify the effect of SNP-SNP interactions.