Low rank tensor ring model is powerful for image completion which recovers missing entries in data acquisition and transformation. The recently proposed tensor ring (TR) based completion algorithms generally solve the low rank optimization problem by alternating least squares method with predefined ranks, which may easily lead to overfitting when the unknown ranks are set too large and only a few measurements are available. In this paper, we present a Bayesian low rank tensor ring model for image completion by automatically learning the low rank structure of data. A multiplicative interaction model is developed for the low-rank tensor ring decomposition, where core factors are enforced to be sparse by assuming their entries obey Student-T distribution. Compared with most of the existing methods, the proposed one is free of parameter-tuning, and the TR ranks can be obtained by Bayesian inference. Numerical Experiments, including synthetic data, color images with different sizes and YaleFace dataset B with respect to one pose, show that the proposed approach outperforms state-of-the-art ones, especially in terms of recovery accuracy.
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…
A novel regularizer of the PARAFAC decomposition factors capturing the tensor's rank is proposed in this paper, as the key enabler for completion of three-way data arrays with missing entries. Set in a Bayesian framework, the tensor completion method incorporates prior information to enhance its smoothing and prediction capabilities. This probabilistic approach can naturally accommodate general models for the data distribution, lending itself to various fitting criteria that yield optimum estimates in the maximum-a-posteriori sense. In particular, two algorithms are devised for Gaussian- and P…
Department of Mathematics, UC Santa Barbara（加州大学圣巴巴拉分校数学系）；Department of ECE, UC Santa Barbara（加州大学圣巴巴拉分校电气工程与计算机科学系）
Streaming tensor factorization is a powerful tool for processing high-volume and multi-way temporal data in Internet networks, recommender systems and image/video data analysis. Existing streaming tensor factorization algorithms rely on least-squares data fitting and they do not possess a mechanism for tensor rank determination. This leaves them susceptible to outliers and vulnerable to over-fitting. This paper presents a Bayesian robust streaming tensor factorization model to identify sparse outliers, automatically determine the underlying tensor rank and accurately fit low-rank structure. We…
This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly unknown but small compared to the number of considered data points. The focus is on high-dimensional problems. We recast the considered problem into an optimization problem over the set of low-rank positive semidefinite matrices and propose two efficient algorithms for low-rank distance matrix completion. In addition, we propose a strategy to determine the dimension of the embedding space. Th…
Wisconsin Institutes for Discovery（威斯康星发现研究院）；UW-Madison（威斯康星大学麦迪逊分校）；University of Michigan Ann Arbor（密歇根大学安娜堡分校）
Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is distorted by a (typically unknown) nonlinear transformation. This paper addresses the challenge of matrix completion in the face of such nonlinearities. Given a few observations of a matrix that are obtained by applying a Lipschitz, monotonic function to a low rank matrix, our task is to estimate the remaining unobserved entries. We propose a novel matrix complet…
In this paper we study the problem of bilinear regression and we further address the case when the response matrix contains missing data that referred as the problem of inductive matrix completion. We propose a quasi-Bayesian approach first to the problem of bilinear regression where a quasi-likelihood is employed. Then, we adapt this approach to the context of inductive matrix completion. Under a low-rankness assumption and leveraging PAC-Bayes bound technique, we provide statistical properties for our proposed estimators and for the quasi-posteriors. We propose a Langevin Monte Carlo method …
Carnegie Mellon University（卡内基梅隆大学）；University of Pittsburgh（匹兹堡大学）；Microsoft Research（微软研究院）
Knowledge Graph (KG) completion research usually focuses on densely connected benchmark datasets that are not representative of real KGs. We curate two KG datasets that include biomedical and encyclopedic knowledge and use an existing commonsense KG dataset to explore KG completion in the more realistic setting where dense connectivity is not guaranteed. We develop a deep convolutional network that utilizes textual entity representations and demonstrate that our model outperforms recent KG completion methods in this challenging setting. We find that our model's performance improvements stem pr…
Matrix and tensor completion are frameworks for a wide range of problems, including collaborative filtering, missing data, and image reconstruction. Missing entries are estimated by leveraging an assumption that the matrix or tensor is low-rank. Most existing Bayesian techniques encourage rank-sparsity by modelling factorized matrices and tensors with Normal-Gamma priors. However, the Horseshoe prior and other "global-local" formulations provide tuning-parameter-free solutions which may better achieve simultaneous rank-sparsity and missing-value recovery. We find these global-local priors outp…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
This paper presents a multi-dimensional computational method to predict the spatial variation data inside and across multiple dies of a wafer. This technique is based on tensor computation. A tensor is a high-dimensional generalization of a matrix or a vector. By exploiting the hidden low-rank property of a high-dimensional data array, the large amount of unknown variation testing data may be predicted from a few random measurement samples. The tensor rank, which decides the complexity of a tensor representation, is decided by an available variational Bayesian approach. Our approach is validat…
We consider the problem of uncertainty quantification for an unknown low-rank matrix $\mathbf{X}$, given a partial and noisy observation of its entries. This quantification of uncertainty is essential for many real-world problems, including image processing, satellite imaging, and seismology, providing a principled framework for validating scientific conclusions and guiding decision-making. However, existing literature has mainly focused on the completion (i.e., point estimation) of the matrix $\mathbf{X}$, with little work on investigating its uncertainty. To this end, we propose in this work…
University of Potsdam（波茨坦大学）；École Nationale de la Statistique et de l’administration économique（国家统计与经济管理学院）；CNRS（法国国家科学研究中心）；Université Paris-Dauphine（巴黎九大）；PSL Research University（巴黎文理研究大学）
We present a case-study demonstrating the usefulness of Bayesian hierarchical mixture modelling for investigating cognitive processes. In sentence comprehension, it is widely assumed that the distance between linguistic co-dependents affects the latency of dependency resolution: the longer the distance, the longer the retrieval time (the distance-based account). An alternative theory, direct-access, assumes that retrieval times are a mixture of two distributions: one distribution represents successful retrievals (these are independent of dependency distance) and the other represents an initial…
University of Electronic Science and Technology of China（电子科技大学）；Stevens Institute of Technology（斯蒂文斯理工学院）
The problem of low rank matrix completion is considered in this paper. To exploit the underlying low-rank structure of the data matrix, we propose a hierarchical Gaussian prior model, where columns of the low-rank matrix are assumed to follow a Gaussian distribution with zero mean and a common precision matrix, and a Wishart distribution is specified as a hyperprior over the precision matrix. We show that such a hierarchical Gaussian prior has the potential to encourage a low-rank solution. Based on the proposed hierarchical prior model, a variational Bayesian method is developed for matrix co…
Norwegian University of Science and Technology（挪威科技大学）
We study the problem of matrix completion in this paper. A spectral scaled Student prior is exploited to favour the underlying low-rank structure of the data matrix. We provide a thorough theoretical investigation for our approach through PAC-Bayesian bounds. More precisely, our PAC-Bayesian approach enjoys a minimax-optimal oracle inequality which guarantees that our method works well under model misspecification and under general sampling distribution. Interestingly, we also provide efficient gradient-based sampling implementations for our approach by using Langevin Monte Carlo. More specifi…
Columbia University（哥伦比亚大学）；Purdue University（普渡大学）
Recently, fundamental conditions on the sampling patterns have been obtained for finite completability of low-rank matrices or tensors given the corresponding ranks. In this paper, we consider the scenario where the rank is not given and we aim to approximate the unknown rank based on the location of sampled entries and some given completion. We consider a number of data models, including single-view matrix, multi-view matrix, CP tensor, tensor-train tensor and Tucker tensor. For each of these data models, we provide an upper bound on the rank when an arbitrary low-rank completion is given. We…
Matrix completion is a problem that arises in many data-analysis settings where the input consists of a partially-observed matrix (e.g., recommender systems, traffic matrix analysis etc.). Classical approaches to matrix completion assume that the input partially-observed matrix is low rank. The success of these methods depends on the number of observed entries and the rank of the matrix; the larger the rank, the more entries need to be observed in order to accurately complete the matrix. In this paper, we deal with matrices that are not necessarily low rank themselves, but rather they contain …
The goal of tensor completion is to recover a tensor from a subset of its entries, often by exploiting its low-rank property. Among several useful definitions of tensor rank, the low-tubal-rank was shown to give a valuable characterization of the inherent low-rank structure of a tensor. While some low-tubal-rank tensor completion algorithms with favorable performance have been recently proposed, these algorithms utilize second-order statistics to measure the error residual, which may not work well when the observed entries contain large outliers. In this paper, we propose a new objective funct…
University of Electronic Science and Technology of China（电子科技大学）
Tensor completion estimates missing components by exploiting the low-rank structure of multi-way data. The recently proposed methods based on tensor train (TT) and tensor ring (TR) show better performance in image recovery than classical ones. Compared with TT and TR, the projected entangled pair state (PEPS), which is also called tensor grid (TG), allows more interactions between different dimensions, and may lead to more compact representation. In this paper, we propose to perform image completion based on low-rank tensor grid. A two-stage density matrix renormalization group algorithm is us…
Weifang University（潍坊学院）；The University of Hong Kong（香港大学）；Central China Normal University（华中师范大学）
One of the key problems in tensor completion is the number of uniformly random sample entries required for recovery guarantee. The main aim of this paper is to study $n_1 \times n_2 \times n_3$ third-order tensor completion based on transformed tensor singular value decomposition, and provide a bound on the number of required sample entries. Our approach is to make use of the multi-rank of the underlying tensor instead of its tubal rank in the bound. In numerical experiments on synthetic and imaging data sets, we demonstrate the effectiveness of our proposed bound for the number of sample entr…
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…
University of Electronic Science and Technology of China（电子科技大学）
Low-rank tensor completion recovers missing entries based on different tensor decompositions. Due to its outstanding performance in exploiting some higher-order data structure, low rank tensor ring has been applied in tensor completion. To further deal with its sensitivity to sparse component as it does in tensor principle component analysis, we propose robust tensor ring completion (RTRC), which separates latent low-rank tensor component from sparse component with limited number of measurements. The low rank tensor component is constrained by the weighted sum of nuclear norms of its balanced …
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…
Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学机械化重点实验室）
Despite the popularity of low-rank matrix completion, the majority of its theory has been developed under the assumption of random observation patterns, whereas very little is known about the practically relevant case of non-random patterns. Specifically, a fundamental yet largely open question is to describe patterns that allow for unique or finitely many completions. This paper provides two such families of patterns for any rank. A key to achieving this is a novel formulation of low-rank matrix completion in terms of Plucker coordinates, the latter a traditional tool in computer vision. This…
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 …
Massachusetts Institute of Technology（麻省理工学院）；Carnegie Mellon University（卡内基梅隆大学）
The problem of low-rank matrix completion with heterogeneous and sub-exponential (as opposed to homogeneous and Gaussian) noise is particularly relevant to a number of applications in modern commerce. Examples include panel sales data and data collected from web-commerce systems such as recommendation engines. An important unresolved question for this problem is characterizing the distribution of estimated matrix entries under common low-rank estimators. Such a characterization is essential to any application that requires quantification of uncertainty in these estimates and has heretofore onl…
Baidu Research（百度研究院）；Paradigm Inc.（第四范式公司）；Tsinghua University（清华大学）；Hong Kong University of Science and Technology（香港科技大学）
Matrix learning is at the core of many machine learning problems. A number of real-world applications such as collaborative filtering and text mining can be formulated as a low-rank matrix completion problem, which recovers incomplete matrix using low-rank assumptions. To ensure that the matrix solution has a low rank, a recent trend is to use nonconvex regularizers that adaptively penalize singular values. They offer good recovery performance and have nice theoretical properties, but are computationally expensive due to repeated access to individual singular values. In this paper, based on th…
We study alternating minimization for matrix completion in the simplest possible setting: completing a rank-one matrix from a revealed subset of the entries. We bound the asymptotic convergence rate by the variational characterization of eigenvalues of a reversible consensus problem. This leads to a polynomial upper bound on the asymptotic rate in terms of number of nodes as well as the largest degree of the graph of revealed entries.
Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion
National Key Laboratory for Novel Software Technology（计算机软件新技术国家重点实验室）；Nanjing University（南京大学）；the University of Iowa（爱荷华大学）；Michigan State University（密歇根州立大学）
This paper presents novel algorithms which exploit the intrinsic algebraic and combinatorial structure of the matrix completion task for estimating missing en- tries in the general low rank setting. For positive data, we achieve results out- performing the state of the art nuclear norm, both in accuracy and computational efficiency, in simulations and in the task of predicting athletic performance from partially observed data.
In this paper, a new definition of tensor p-shrinkage nuclear norm (p-TNN) is proposed based on tensor singular value decomposition (t-SVD). In particular, it can be proved that p-TNN is a better approximation of the tensor average rank than the tensor nuclear norm when p < 1. Therefore, by employing the p-shrinkage nuclear norm, a novel low-rank tensor completion (LRTC) model is proposed to estimate a tensor from its partial observations. Statistically, the upper bound of recovery error is provided for the LRTC model. Furthermore, an efficient algorithm, accelerated by the adaptive momentum s…
Due to challenging applications such as collaborative filtering, the matrix completion problem has been widely studied in the past few years. Different approaches rely on different structure assumptions on the matrix in hand. Here, we focus on the completion of a (possibly) low-rank matrix with binary entries, the so-called 1-bit matrix completion problem. Our approach relies on tools from machine learning theory: empirical risk minimization and its convex relaxations. We propose an algorithm to compute a variational approximation of the pseudo-posterior. Thanks to the convex relaxation, the c…
We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. In addition, varieties can be used to model a richer class of nonlinear quadratic and higher degree curves and surfaces. We…
Robust low-rank matrix completion (RMC), or robust principal component analysis with partially observed data, has been studied extensively for computer vision, signal processing and machine learning applications. This problem aims to decompose a partially observed matrix into the superposition of a low-rank matrix and a sparse matrix, where the sparse matrix captures the grossly corrupted entries of the matrix. A widely used approach to tackle RMC is to consider a convex formulation, which minimizes the nuclear norm of the low-rank matrix (to promote low-rankness) and the l1 norm of the sparse…
Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study matrix completion from corrupted data under a novel low-rank missing mechanism. The probability matrix of observation is estimated via a high dimensional low-rank matrix estimation procedure, and further used to complete the target matrix via inverse probabilities weighting. Due to…
Matrix completion constantly receives tremendous attention from many research fields. It is commonly applied for recommender systems such as movie ratings, computer vision such as image reconstruction or completion, multi-task learning such as collaboratively modeling time-series trends of multiple sensors, and many other applications. Matrix completion techniques are usually computationally exhaustive and/or fail to capture the heterogeneity in the data. For example, images usually contain a heterogeneous set of objects, and thus it is a challenging task to reconstruct images with high levels…
Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete $d \times N$ matrix is finitely rank-$r$ completable if there are at most finitely many rank-$r$ matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a determinis…
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…
Boston University（波士顿大学）；University of Wisconsin - Madison（威斯康星大学麦迪逊分校）
This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion problem to situations in which the matrix rank can be quite high or even full rank. Since the columns belong to a union of subspaces, this problem may also be viewed as a missing-data version of the subspace clustering problem. Let X be an n x N matrix whose (complete) columns lie in a union of at most k subspaces, each of rank <= r < n, and assume N >> kn.…
This work studies low-rank approximation of a positive semidefinite matrix from partial entries via nonconvex optimization. We characterized how well local-minimum based low-rank factorization approximates a fixed positive semidefinite matrix without any assumptions on the rank-matching, the condition number or eigenspace incoherence parameter. Furthermore, under certain assumptions on rank-matching and well-boundedness of condition numbers and eigenspace incoherence parameters, a corollary of our main theorem improves the state-of-the-art sampling rate results for nonconvex matrix completion …
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…
In this paper we offer a review and bibliography of work on Hankel low-rank approximation and completion, with particular emphasis on how this methodology can be used for time series analysis and forecasting. We begin by describing possible formulations of the problem and offer commentary on related topics and challenges in obtaining globally optimal solutions. Key theorems are provided, and the paper closes with some expository examples.
Low-rank matrix estimation from incomplete measurements recently received increased attention due to the emergence of several challenging applications, such as recommender systems; see in particular the famous Netflix challenge. While the behaviour of algorithms based on nuclear norm minimization is now well understood, an as yet unexplored avenue of research is the behaviour of Bayesian algorithms in this context. In this paper, we briefly review the priors used in the Bayesian literature for matrix completion. A standard approach is to assign an inverse gamma prior to the singular values of …
We study low rank matrix and tensor completion and propose novel algorithms that employ adaptive sampling schemes to obtain strong performance guarantees. Our algorithms exploit adaptivity to identify entries that are highly informative for learning the column space of the matrix (tensor) and consequently, our results hold even when the row space is highly coherent, in contrast with previous analyses. In the absence of noise, we show that one can exactly recover a $n \times n$ matrix of rank $r$ from merely $Ω(n r^{3/2}\log(r))$ matrix entries. We also show that one can recover an order $T$ te…
The completion of tensors, or high-order arrays, attracts significant attention in recent research. Current literature on tensor completion primarily focuses on recovery from a set of uniformly randomly measured entries, and the required number of measurements to achieve recovery is not guaranteed to be optimal. In addition, the implementation of some previous methods is NP-hard. In this article, we propose a framework for low-rank tensor completion via a novel tensor measurement scheme we name Cross. The proposed procedure is efficient and easy to implement. In particular, we show that a thir…
The matrix-completion problem has attracted a lot of attention, largely as a result of the celebrated Netflix competition. Two popular approaches for solving the problem are nuclear-norm-regularized matrix approximation (Candes and Tao, 2009, Mazumder, Hastie and Tibshirani, 2010), and maximum-margin matrix factorization (Srebro, Rennie and Jaakkola, 2005). These two procedures are in some cases solving equivalent problems, but with quite different algorithms. In this article we bring the two approaches together, leading to an efficient algorithm for large matrix factorization and completion t…
Factorization-based gradient descent is a scalable and efficient algorithm for solving low-rank matrix completion. Recent progress in structured non-convex optimization has offered global convergence guarantees for gradient descent under certain statistical assumptions on the low-rank matrix and the sampling set. However, while the theory suggests gradient descent enjoys fast linear convergence to a global solution of the problem, the universal nature of the bounding technique prevents it from obtaining an accurate estimate of the rate of convergence. In this paper, we perform a local analysis…
