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.
In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from laborious model selection problem due to high model sensitivity. Especially for tensor ring (TR) decomposition, the number of model possibility grows exponentially with the tensor order, which makes it rather challenging to find the optimal TR decomposition. In this paper, by exploiting the low-rank structure on TR latent space, we propose a novel tensor completion method, which is robust to model selection. In contrast to imposing low-rank constraint on the data space, we introduce nuclear norm regularization on the latent TR factors, resulting in that the optimization step using singular value decomposition (SVD) can be performed at a much smaller scale. By leveraging the alternating direction method of multipliers (ADMM) scheme, the latent TR factors with optimal rank and the recovered tensor can be obtained simultaneously. Our proposed algorithm effectively alleviates the burden of TR-rank selection, therefore the computational cost is greatly reduced. The extensive experimental results on synthetic data and real-world data demonstrate the superior high performance and efficiency of the proposed approach against the state-of-the-art algorithms.
Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in data mining, computer vision, signal processing, and neuroscience, etc. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data analytics characterized by diverse variety, large volume, and high velocity. Towards a better comprehension and comparison of vast existing advances, we summarize and categorize them into four groups including general tensor completion algorithms, tensor completion with auxiliary information (variety), scalable tensor completion algorithms (volume) and dynamic tensor completion algorithms (velocity). Besides, we introduce their applications on real-world data-driven problems and present an open-source package covering several widely used tensor decomposition and completion algorithms. Our goal is to summarize these popular methods and introduce them to researchers for promoting the research process in this field and give an available repository for practitioners. In the end, we also discuss some challenges and promising research directions in this community for future explorations.
Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging problem is related to determination of model complexity (i.e., multilinear rank), especially when noise and missing data are present. In addition, existing methods cannot take into account uncertainty information of latent factors, resulting in low generalization performance. To address these issues, we present a class of probabilistic generative Tucker models for tensor decomposition and completion with structural sparsity over multilinear latent space. To exploit structural sparse modeling, we introduce two group sparsity inducing priors by hierarchial representation of Laplace and Student-t distributions, which facilitates fully posterior inference. For model learning, we derived variational Bayesian inferences over all model (hyper)parameters, and developed efficient and scalable algorithms based on multilinear operations. Our methods can automatically adapt model complexity and infer an optimal multilinear rank by the principle of maximum lower bound of model evidence. Experimental results and comparisons on synthetic, chemometrics and neuroimaging data demonstrate remarkable performance of our models for recovering ground-truth of multilinear rank and missing entries.
In multi-data learning, it is usually assumed that common latent factors exist among multi-datasets, but it may lead to deteriorated performance when datasets are heterogeneous and unbalanced. In this paper, we propose a novel common structure for multi-data learning. Instead of common latent factors, we assume that datasets share Common Adjacency Graph (CAG) structure, which is more robust to heterogeneity and unbalance of datasets. Furthermore, we utilize CAG structure to develop a new method for multi-tensor completion, which exploits the common structure in datasets to improve the completion performance. Numerical results demostrate that the proposed method not only outperforms state-of-the-art methods for video in-painting, but also can recover missing data well even in cases that conventional methods are not applicable.