text: Many tensor-based data completion methods aim to solve image and video in-painting problems. But, all methods were only developed for a single dataset. In most of real applications, we can usually obtain more than one dataset to reflect one phenomenon, and all the datasets are mutually related in some sense. Thus one question raised whether such the relationship can improve the performance of data completion or not? In the paper, we proposed a novel and efficient method by exploiting the relationship among datasets for multi-video data completion. Numerical results show that the proposed method significantly improve the performance of video in-painting, particularly in the case of very high missing percentage.
  @cite_4: A method operates on observed relationship data between pairs of entities of a set of entities including entities of at least two (and optionally at least three) different entity types. An observed collective symmetric matrix is constructed in which element (n,m)=element (m,n) stores the observed relationship between entities indexed n and m when the observed relationship data includes this observed relationship. A prediction collective symmetric matrix is optimized in order to minimize a loss function comparing the observed collective symmetric matrix and the prediction collective symmetric matrix. A relationship between two entities of the set of entities is predicted using the optimized prediction collective symmetric matrix. Entities of the same entity type may be indexed using a contiguous set of indices such that the entity type maps to a contiguous set of rows and corresponding contiguous set of columns in the observed collective symmetric matrix.
  @cite_8: 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.
  @cite_0: Real-world data are often acquired as a collection of matrices rather than as a single matrix. Such multiblock data are naturally linked and typically share some common features while at the same time exhibiting their own individual features, reflecting the underlying data generation mechanisms. To exploit the linked nature of data, we propose a new framework for common and individual feature extraction (CIFE) which identifies and separates the common and individual features from the multiblock data. Two efficient algorithms termed common orthogonal basis extraction (COBE) are proposed to extract common basis is shared by all data, independent on whether the number of common components is known beforehand. Feature extraction is then performed on the common and individual subspaces separately, by incorporating dimensionality reduction and blind source separation techniques. Comprehensive experimental results on both the synthetic and real-world data demonstrate significant advantages of the proposed CIFE method in comparison with the state-of-the-art.
  @cite_2: CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of user-item, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrices, which enables transferring information between them. The existing solutions, however, break down when the individual matrices have low-rank structure not shared with others. In this work we present a novel CMF solution that allows each of the matrices to have a separate low-rank structure that is independent of the other matrices, as well as structures that are shared only by a subset of them. We compare MAP and variational Bayesian solutions based on alternating optimization algorithms and show that the model automatically infers the nature of each factor using group-wise sparsity. Our approach supports in a principled way continuous, binary and count observations and is efficient for sparse matrices involving missing data. We illustrate the solution on a number of examples, focusing in particular on an interesting use-case of augmented multi-view learning.
  @cite_5: Abstract The problem of incomplete data – i.e., data with missing or unknown values – in multi-way arrays is ubiquitous in biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, communication networks, etc. We consider the problem of how to factorize data sets with missing values with the goal of capturing the underlying latent structure of the data and possibly reconstructing missing values (i.e., tensor completion). We focus on one of the most well-known tensor factorizations that captures multi-linear structure, CANDECOMP PARAFAC (CP). In the presence of missing data, CP can be formulated as a weighted least squares problem that models only the known entries. We develop an algorithm called CP-WOPT (CP Weighted OPTimization) that uses a first-order optimization approach to solve the weighted least squares problem. Based on extensive numerical experiments, our algorithm is shown to successfully factorize tensors with noise and up to 99 missing data. A unique aspect of our approach is that it scales to sparse large-scale data, e.g., 1000 × 1000 × 1000 with five million known entries (0.5 dense). We further demonstrate the usefulness of CP-WOPT on two real-world applications: a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes and the problem of modeling computer network traffic where data may be absent due to the expense of the data collection process.
summary: In many different field, multi-data analysis or group analysis was also deeply discussed @cite_0 @cite_2 @cite_4 . But the majority of methods focus on common and distinctive component extraction. In this paper, we do not care latent features in each dataset. Thus we can reduce the complexity of algorithms compared to tensor decomposition based method like @cite_5 @cite_8 . A multi-matrix completion method Convex Collective Matrix Factorization (cCMF) was proposed in @cite_4 . Compare to cCMF, The proposed method in this paper can be considered as an extension of cCMF into tensor problem.