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.
The Ohio State University（俄亥俄州立大学）；The University of Hong Kong（香港大学）；Carnegie Mellon University（卡内基梅隆大学）
Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent variables can be exploited by pairwise learning using the graph regularized matrix factorization (GRMF) method. However, existing GRMF approaches often use the squared loss to measure the pairwise differences, which may be overly influenced by dissimilar pairs and lead to inferior prediction. To fully empower pairwise learning for matrix completion, we prop…
Latent Structural Categorical Matrix Completion with Application to Quasispecies Analysis
Engineering Systems and Design, Singapore University of Technology and Design（新加坡科技设计大学工程系统与设计系）；Institute of Statistics and Big Data, Renmin University of China（中国人民大学统计与大数据研究院）
University of Missouri - Columbia（密苏里大学哥伦比亚分校）；Stanford University（斯坦福大学）
In this paper we provide a latent-variable formulation and solution to the recommender system (RS) problem in terms of a fundamental property that any reasonable solution should be expected to satisfy. Specifically, we examine a novel tensor completion method to efficiently and accurately learn parameters of a model for the unobservable personal preferences that underly user ratings. By regularizing the tensor decomposition with a single latent invariant, we achieve three properties for a reliable recommender system: (1) uniqueness of the tensor completion result with minimal assumptions, (2) …
London School of Economics and Political Science（伦敦政治经济学院）；University of Minnesota（明尼苏达大学）
Matrix completion is a class of machine learning methods that concerns the prediction of missing entries in a partially observed matrix. This paper studies matrix completion for mixed data, i.e., data involving mixed types of variables (e.g., continuous, binary, ordinal). We formulate it as a low-rank matrix estimation problem under a general family of non-linear factor models and then propose entrywise consistent estimators for estimating the low-rank matrix. Tight probabilistic error bounds are derived for the proposed estimators. The proposed methods are evaluated by simulation studies and …
Saitama Institute of Technology（埼玉工业大学）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能项目中心）；Imperial College London（帝国理工学院）；Hangzhou Dianzi University（杭州电子科技大学）；Guangdong University of Technology（广东工业大学）
In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model possibilities 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 of the TR latent space, we propose a novel tensor completion method which is robust to model selection. In contrast to imposing the low-rank constraint on the data space, we introd…
Abstract reasoning ability is fundamental to human intelligence. It enables humans to uncover relations among abstract concepts and further deduce implicit rules from the relations. As a well-known abstract visual reasoning task, Raven's Progressive Matrices (RPM) are widely used in human IQ tests. Although extensive research has been conducted on RPM solvers with machine intelligence, few studies have considered further advancing the standard answer-selection (classification) problem to a more challenging answer-painting (generating) problem, which can verify whether the model has indeed unde…
Hong Kong Baptist University（香港浸会大学）；The Chinese University of Hong Kong（香港中文大学）
Regular medical records are useful for medical practitioners to analyze and monitor patient health status especially for those with chronic disease, but such records are usually incomplete due to unpunctuality and absence of patients. In order to resolve the missing data problem over time, tensor-based model is suggested for missing data imputation in recent papers because this approach makes use of low rank tensor assumption for highly correlated data. However, when the time intervals between records are long, the data correlation is not high along temporal direction and such assumption is no…
Factoring a matrix into two low rank matrices is at the heart of many problems. The problem of matrix completion especially uses it to decompose a sparse matrix into two non sparse, low rank matrices which can then be used to predict unknown entries of the original matrix. We present a scalable and decentralized approach in which instead of learning two factors for the original input matrix, we decompose the original matrix into a grid blocks, each of whose factors can be individually learned just by communicating (gossiping) with neighboring blocks. This eliminates any need for a central serv…
Matrix completion has received vast amount of attention and research due to its wide applications in various study fields. Existing methods of matrix completion consider only nonlinear (or linear) relations among entries in a data matrix and ignore linear (or nonlinear) relationships latent. This paper introduces a new latent variables model for data matrix which is a combination of linear and nonlinear models and designs a novel deep-neural-network-based matrix completion algorithm to address both linear and nonlinear relations among entries of data matrix. The proposed method consists of two…
BRIDGE: Predicting Human Task Completion Time From Model Performance
Mila - Quebec AI Institute（魁北克人工智能研究所）；McGill University（麦吉尔大学）；Polytechnique Montréal（蒙特利尔理工学院）；Periodic Labs（Periodic实验室）；Canada CIFAR AI Chair（加拿大CIFAR人工智能主席）；ServiceNow Research（ServiceNow研究）
We address the collective matrix completion problem of jointly recovering a collection of matrices with shared structure from partial (and potentially noisy) observations. To ensure well--posedness of the problem, we impose a joint low rank structure, wherein each component matrix is low rank and the latent space of the low rank factors corresponding to each entity is shared across the entire collection. We first develop a rigorous algebra for representing and manipulating collective--matrix structure, and identify sufficient conditions for consistent estimation of collective matrices. We then…
Weifang University（潍坊学院）；Hong Kong Baptist University（香港浸会大学）；Central China Normal University（华中师范大学）
In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform matrix. This would be more effective for robust tensor completion. Experimental results for hyperspectral, video and face datasets have shown that the recovery performance for the robust tensor compl…
Recovering Wasserstein Distance Matrices from Few Measurements
Department of Mathematics, University of Texas at Arlington（德克萨斯理工大学数学系）；Department of Mathematics, Tufts University（塔夫茨大学数学系）；Department of Statistics and Data Science and Department of Computer Science University of Central Florida（中央佛罗里达大学统计与数据科学系和计算机科学系）；Division of Data Science, University of Texas at Arlington（德克萨斯理工大学数据科学 division）
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
In this paper we study the matrix completion problem: Suppose $X \in {\mathbb R}^{n_r \times n_c}$ is unknown except for a known upper bound $r$ on its rank. By measuring a small number $m \ll n_r n_c$ of elements of $X$, is it possible to recover $X$ exactly with noise-free measurements, or to construct a good approximation of $X$ with noisy measurements? Existing solutions to these problems involve sampling the elements uniformly and at random, and can guarantee exact recovery of the unknown matrix only with high probability. In this paper, we present a \textit{deterministic} sampling method…
University of California, Davis（加利福尼亚大学戴维斯分校）；Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）
Matrix completion has attracted attention in many fields, including statistics, applied mathematics, and electrical engineering. Most of the works focus on the independent sampling models under which the observed entries are sampled independently. Motivated by applications in the integration of knowledge graphs derived from multi-source biomedical data such as those from Electronic Health Records (EHR) and biomedical text, we propose the {\bf B}lock-wise {\bf O}verlapping {\bf N}oisy {\bf M}atrix {\bf I}ntegration (BONMI) to treat blockwise missingness of symmetric matrices representing relate…
One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian …
Low-rank matrix completion is an important problem with extensive real-world applications. When observations are uniformly sampled from the underlying matrix entries, existing methods all require the matrix to be incoherent. This paper provides the first working method for coherent matrix completion under the standard uniform sampling model. Our approach is based on the weighted nuclear norm minimization idea proposed in several recent work, and our key contribution is a practical method to compute the weighting matrices so that the leverage scores become more uniform after weighting. Under su…
Recent advances in matrix completion enable data imputation in full-rank matrices by exploiting low dimensional (nonlinear) latent structure. In this paper, we develop a new model for high rank matrix completion (HRMC), together with batch and online methods to fit the model and out-of-sample extension to complete new data. The method works by (implicitly) mapping the data into a high dimensional polynomial feature space using the kernel trick; importantly, the data occupies a low dimensional subspace in this feature space, even when the original data matrix is of full-rank. We introduce an ex…
University of Technology Sydney（悉尼科技大学）；Alibaba DAMO Academy（阿里巴巴达摩院）
Contemporary data-driven methods are typically fed with full supervision on large-scale datasets which limits their applicability. However, in the actual systems with limitations such as measurement error and data acquisition problems, people usually obtain incomplete data. Although data completion has attracted wide attention, the underlying data pattern and relativity are still under-developed. Currently, the family of latent variable models allows learning deep latent variables over observed variables by fitting the marginal distribution. As far as we know, current methods fail to perceive …
In this paper, we propose a simple algorithm to cluster nonnegative data lying in disjoint subspaces. We analyze its performance in relation to a certain measure of correlation between said subspaces. We use our clustering algorithm to develop a matrix completion algorithm which can outperform standard matrix completion algorithms on data matrices satisfying certain natural conditions.
Beihang University（北京航空航天大学）；Guangzhou University（广州大学）；Chongqing University of Posts and Telecommunications（重庆邮电大学）
High-dimensional and sparse (HiDS) matrices are frequently adopted to describe the complex relationships in various big data-related systems and applications. A Position-transitional Latent Factor Analysis (PLFA) model can accurately and efficiently represent an HiDS matrix. However, its involved latent factors are optimized by stochastic gradient descent with the specific gradient direction step-by-step, which may cause a suboptimal solution. To address this issue, this paper proposes a Sequential-Group-Differential- Evolution (SGDE) algorithm to refine the latent factors optimized by a PLFA …
We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming that the underlying matrix has low rank, or is well-approximated by a low rank matrix. In this paper, we propose a richer model based on what we term the "permutation-rank" of a matrix. We first describe how the classical non-negative rank model enforces restrictions that may be undesirable in practice, and how and these restrictions can be avoided by using the…
University of Electronic Science and Technology of China（中国电子科技大学）
Coupled tensor decomposition reveals the joint data structure by incorporating priori knowledge that come from the latent coupled factors. The tensor ring (TR) decomposition is invariant under the permutation of tensors with different mode properties, which ensures the uniformity of decomposed factors and mode attributes. The TR has powerful expression ability and achieves success in some multi-dimensional data processing applications. To let coupled tensors help each other for missing component estimation, in this paper we utilize TR for coupled completion by sharing parts of the latent facto…
Completing a data matrix X has become an ubiquitous problem in modern data science, with applications in recommender systems, computer vision, and networks inference, to name a few. One typical assumption is that X is low-rank. A more general model assumes that each column of X corresponds to one of several low-rank matrices. This paper generalizes these models to what we call mixture matrix completion (MMC): the case where each entry of X corresponds to one of several low-rank matrices. MMC is a more accurate model for recommender systems, and brings more flexibility to other completion and c…
New Jersey Institute of Technology（新泽西理工学院）；Central Michigan University（中密歇根大学）
Firms earning prediction plays a vital role in investment decisions, dividends expectation, and share price. It often involves multiple tensor-compatible datasets with non-linear multi-way relationships, spatiotemporal structures, and different levels of sparsity. Current non-linear tensor completion algorithms tend to learn noisy embedding and incur overfitting. This paper focuses on the embedding learning aspect of the tensor completion problem and proposes a new multi-layer neural network architecture for tensor factorization and completion (MLCTR). The network architecture entails multiple…
Extracting the latent information in high-dimensional and incomplete matrices is an important and challenging issue. The Latent Factor Analysis (LFA) model can well handle the high-dimensional matrices analysis. Recently, Particle Swarm Optimization (PSO)-incorporated LFA models have been proposed to tune the hyper-parameters adaptively with high efficiency. However, the incorporation of PSO causes the premature problem. To address this issue, we propose a sequential Adam-adjusting-antennae BAS (A2BAS) optimization algorithm, which refines the latent factors obtained by the PSO-incorporated LF…
University of Salerno（萨勒诺大学）；IMT School for Advanced Studies（IMT高等研究院）
World Input-Output (I/O) matrices provide the networks of within- and cross-country economic relations. In the context of I/O analysis, the methodology adopted by national statistical offices in data collection raises the issue of obtaining reliable data in a timely fashion and it makes the reconstruction of (part of) the I/O matrices of particular interest. In this work, we propose a method combining hierarchical clustering and Matrix Completion (MC) with a LASSO-like nuclear norm penalty, to impute missing entries of a partially unknown I/O matrix. Through simulations based on synthetic matr…
Let M be a random (alpha n) x n matrix of rank r<<n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(rn) observed entries with relative root mean square error RMSE <= C(rn/|E|)^0.5 . Further, if r=O(1), M can be reconstructed exactly from |E| = O(n log(n)) entries. These results apply beyond random matrices to general low-rank incoherent matrices. This settles (in the case of bounded rank) a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The c…
Technische Universität München（慕尼黑工业大学）；Google Inc.（谷歌公司）
We propose a novel model for 3D semantic completion from a single depth image, based on a single encoder and three separate generators used to reconstruct different geometric and semantic representations of the original and completed scene, all sharing the same latent space. To transfer information between the geometric and semantic branches of the network, we introduce paths between them concatenating features at corresponding network layers. Motivated by the limited amount of training samples from real scenes, an interesting attribute of our architecture is the capacity to supplement the exi…
Univ Paris Nanterre（巴黎楠泰尔大学）；ESSEC Business School（ESSEC商学院）；CREST（法国经济与统计研究所）
Matrix completion aims to reconstruct a data matrix based on observations of a small number of its entries. Usually in matrix completion a single matrix is considered, which can be, for example, a rating matrix in recommendation system. However, in practical situations, data is often obtained from multiple sources which results in a collection of matrices rather than a single one. In this work, we consider the problem of collective matrix completion with multiple and heterogeneous matrices, which can be count, binary, continuous, etc. We first investigate the setting where, for each source, th…
Institute for Molecular Medicine Finland (FIMM), University of Helsinki（赫尔辛基大学芬兰分子医学研究所（FIMM））；Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；Aalto University（阿尔托大学）
Matrix completion aims to predict missing elements in a partially observed data matrix which in typical applications, such as collaborative filtering, is large and extremely sparsely observed. A standard solution is matrix factorization, which predicts unobserved entries as linear combinations of latent variables. We generalize to non-linear combinations in massive-scale matrices. Bayesian approaches have been proven beneficial in linear matrix completion, but not applied in the more general non-linear case, due to limited scalability. We introduce a Bayesian non-linear matrix completion algor…
University of Oxford（牛津大学）；e-Therapeutics plc（e-Therapeutics公司）
We propose an algorithm for low rank matrix completion for matrices with binary entries which obtains explicit binary factors. Our algorithm, which we call TBMC (\emph{Tiling for Binary Matrix Completion}), gives interpretable output in the form of binary factors which represent a decomposition of the matrix into tiles. Our approach is inspired by a popular algorithm from the data mining community called PROXIMUS: it adopts the same recursive partitioning approach while extending to missing data. The algorithm relies upon rank-one approximations of incomplete binary matrices, and we propose a …
We study the problem of exact completion for $m \times n$ sized matrix of rank $r$ with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces (which we call \textit{sparsity-number} here). Using this relation, we propose matrix completion algorithms that exactly recovers the target matrix. These algorithms are superior to previous works in two important ways. First, our algorithms exactly recovers $μ_0$-coherent column space matrices by probability at least $1 - ε$ using much smaller observations complexity than …
University of Liège（列日大学）；University of Cambridge（剑桥大学）
We exploit the versatile framework of Riemannian optimization on quotient manifolds to develop R3MC, a nonlinear conjugate-gradient method for low-rank matrix completion. The underlying search space of fixed-rank matrices is endowed with a novel Riemannian metric that is tailored to the least-squares cost. Numerical comparisons suggest that R3MC robustly outperforms state-of-the-art algorithms across different problem instances, especially those that combine scarcely sampled and ill-conditioned data.
Chongqing University of Posts and Telecommunications（重庆邮电大学）；Southwest University（西南大学）；University of Chinese Academy of Sciences（中国科学院大学）
Missing data is an inevitable and common problem in data-driven intelligent transportation systems (ITS). In the past decade, scholars have done many research on the recovery of missing traffic data, however how to make full use of spatio-temporal traffic patterns to improve the recovery performance is still an open problem. Aiming at the spatio-temporal characteristics of traffic speed data, this paper regards the recovery of missing data as a matrix completion problem, and proposes a spatio-temporal traffic data completion method based on hidden feature analysis, which discovers spatio-tempo…
We aim to provably complete a sparse and highly-missing tensor in the presence of covariate information along tensor modes. Our motivation comes from online advertising where users click-through-rates (CTR) on ads over various devices form a CTR tensor that has about 96% missing entries and has many zeros on non-missing entries, which makes the standalone tensor completion method unsatisfactory. Beside the CTR tensor, additional ad features or user characteristics are often available. In this paper, we propose Covariate-assisted Sparse Tensor Completion (COSTCO) to incorporate covariate inform…
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；NAVER LABS Europe（NAVER LABS 欧洲实验室）；University College London（伦敦大学学院）；Bloomsbury AI
In statistical relational learning, knowledge graph completion deals with automatically understanding the structure of large knowledge graphs---labeled directed graphs---and predicting missing relationships---labeled edges. State-of-the-art embedding models propose different trade-offs between modeling expressiveness, and time and space complexity. We reconcile both expressiveness and complexity through the use of complex-valued embeddings and explore the link between such complex-valued embeddings and unitary diagonalization. We corroborate our approach theoretically and show that all real sq…
Columbia University（哥伦比亚大学）；Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）
Graph data completion is a fundamentally important issue as data generally has a graph structure, e.g., social networks, recommendation systems, and the Internet of Things. We consider a graph where each node has a data matrix, represented as a \textit{graph-tensor} by stacking the data matrices in the third dimension. In this paper, we propose a \textit{Convolutional Graph-Tensor Net} (\textit{Conv GT-Net}) for the graph data completion problem, which uses deep neural networks to learn the general transform of graph-tensors. The experimental results on the ego-Facebook data sets show that the…
We consider the problem of exact recovery of any $m\times n$ matrix of rank $\varrho$ from a small number of observed entries via the standard nuclear norm minimization framework. Such low-rank matrices have degrees of freedom $(m+n)\varrho - \varrho^2$. We show that any arbitrary low-rank matrices can be recovered exactly from a $Θ\left(((m+n)\varrho - \varrho^2)\log^2(m+n)\right)$ randomly sampled entries, thus matching the lower bound on the required number of entries (in terms of degrees of freedom), with an additional factor of $O(\log^2(m+n))$. To achieve this bound on sample size we obs…
Autoencoders are popular among neural-network-based matrix completion models due to their ability to retrieve potential latent factors from the partially observed matrices. Nevertheless, when training data is scarce their performance is significantly degraded due to overfitting. In this paper, we mit- igate overfitting with a data-dependent regularization technique that relies on the principles of multi-task learning. Specifically, we propose an autoencoder-based matrix completion model that performs prediction of the unknown matrix values as a main task, and manifold learning as an auxiliary …
Efficient Tensor Completion Algorithms for Highly Oscillatory Operators
Department of Computer Science, University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校计算机科学系）；Lawrence Berkeley National Laboratory（伯克利国家实验室）；Applied Mathematics and Computational Research Division（应用数学与计算研究部）
