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 $\mathcal{O}(μ_0 rn \mathrm{log}\frac{r}ε)$ the state of art. Specifically, many of the previous adaptive sampling methods require to observe the entire matrix when the column space is highly coherent. However, we show that our method is still able to recover this type of matrices by observing a small fraction of entries under many scenarios. Second, we propose an exact completion algorithm, which requires minimal pre-information as either row or column space is not being highly coherent. At the end of the paper, we provide experimental results that illustrate the strength of the algorithms proposed here.
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…
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…
Sloan School of Management（斯隆管理学院）；Operations Research Center（运筹学研究中心）；Massachusetts Institute of Technology（麻省理工学院）
We formulate the problem of matrix completion with and without side information as a non-convex optimization problem. We design fastImpute based on non-convex gradient descent and show it converges to a global minimum that is guaranteed to recover closely the underlying matrix while it scales to matrices of sizes beyond $10^5 \times 10^5$. We report experiments on both synthetic and real-world datasets that show fastImpute is competitive in both the accuracy of the matrix recovered and the time needed across all cases. Furthermore, when a high number of entries are missing, fastImpute is over …
Institute for Advanced Study（高等研究院）；Cornell University（康奈尔大学）
We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in $\mathbb R^n$ from $r\cdot \tilde O(n^{1.5})$ randomly observed entries of the tensor. This bound improves over the previous best one of $r\cdot \tilde O(n^{2})$ by reduction to exact matrix completion. Our bound also matches the best known results for the easier problem of approximate tensor completion (Barak & Moitra, 2015). Our algorithm and analysis extends s…
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…
In this paper we show that simple semidefinite programs inspired by degree $4$ SOS can exactly solve the tensor nuclear norm, tensor decomposition, and tensor completion problems on tensors with random asymmetric components. More precisely, for tensor nuclear norm and tensor decomposition, we show that w.h.p. these semidefinite programs can exactly find the nuclear norm and components of an $(n\times n\times n)$-tensor $\mathcal{T}$ with $m\leq n^{3/2}/polylog(n)$ random asymmetric components. Unlike most of the previous algorithms, our algorithm provides a certificate for the decomposition, d…
We consider the problem of noisy 1-bit matrix completion under an exact rank constraint on the true underlying matrix $M^*$. Instead of observing a subset of the noisy continuous-valued entries of a matrix $M^*$, we observe a subset of noisy 1-bit (or binary) measurements generated according to a probabilistic model. We consider constrained maximum likelihood estimation of $M^*$, under a constraint on the entry-wise infinity-norm of $M^*$ and an exact rank constraint. This is in contrast to previous work which has used convex relaxations for the rank. We provide an upper bound on the matrix es…
This paper conducts a rigorous analysis for provable estimation of multidimensional arrays, in particular third-order tensors, from a random subset of its corrupted entries. Our study rests heavily on a recently proposed tensor algebraic framework in which we can obtain tensor singular value decomposition (t-SVD) that is similar to the SVD for matrices, and define a new notion of tensor rank referred to as the tubal rank. We prove that by simply solving a convex program, which minimizes a weighted combination of tubal nuclear norm, a convex surrogate for the tubal rank, and the $\ell_1$-norm, …
We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures. For exact recovery of a low-rank matrix, our algorithm judiciously selects a few columns to observe in full and, with few additional measurements, projects the remaining columns onto their span. This algorithm exactly recovers an $n \times n$ rank $r$ matrix using $O(nrμ_0 \log^2(r…
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…
Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear norm minimization will exactly recover the low rank matrix. In this article, we analyze the scenario in which the non-uniform sampling distribution may or may not not align with the underlying matrix's leverage scores. Here we explore learning the parameters for weighted nuclear norm minimization in terms of the empirical sampling distribution. We provide a…
We study tensor completion in the agnostic setting. In the classical tensor completion problem, we receive $n$ entries of an unknown rank-$r$ tensor and wish to exactly complete the remaining entries. In agnostic tensor completion, we make no assumption on the rank of the unknown tensor, but attempt to predict unknown entries as well as the best rank-$r$ tensor. For agnostic learning of third-order tensors with the square loss, we give the first polynomial time algorithm that obtains a "fast" (i.e., $O(1/n)$-type) rate improving over the rate obtained by reduction to matrix completion. Our pre…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Microsoft Research, India（微软研究院（印度））
The problem of low-rank matrix completion has recently generated a lot of interest leading to several results that offer exact solutions to the problem. However, in order to do so, these methods make assumptions that can be quite restrictive in practice. More specifically, the methods assume that: a) the observed indices are sampled uniformly at random, and b) for every new matrix, the observed indices are sampled afresh. In this work, we address these issues by providing a universal recovery guarantee for matrix completion that works for a variety of sampling schemes. In particular, we show t…
Microsoft Research India（微软印度研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the problem of inductive matrix completion in the exact recovery setting. That is, we assume that the ratings matrix is generated by applying feature vectors to a low-rank matrix and the goal is to recover back the underlying matrix. Furthermore, we generalize the problem to that of low-rank matrix estimatio…
We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends on the values of the known entries. If the entries of the matrix are complex numbers, then for a fixed pattern of locations of specified and unspecified entries there is a unique completion rank which occurs with positive probability. We call this rank the generic completion rank. Over the real numbers there can be multiple ranks that occur with positive prob…
This paper is about iteratively reweighted basis-pursuit algorithms for compressed sensing and matrix completion problems. In a first part, we give a theoretical explanation of the fact that reweighted basis pursuit can improve a lot upon basis pursuit for exact recovery in compressed sensing. We exhibit a condition that links the accuracy of the weights to the RIP and incoherency constants, which ensures exact recovery. In a second part, we introduce a new algorithm for matrix completion, based on the idea of iterative reweighting. Since a weighted nuclear "norm" is typically non-convex, it c…
We improve existing results in the field of compressed sensing and matrix completion when sampled data may be grossly corrupted. We introduce three new theorems. 1) In compressed sensing, we show that if the m \times n sensing matrix has independent Gaussian entries, then one can recover a sparse signal x exactly by tractable \ell1 minimimization even if a positive fraction of the measurements are arbitrarily corrupted, provided the number of nonzero entries in x is O(m/(log(n/m) + 1)). 2) In the very general sensing model introduced in "A probabilistic and RIPless theory of compressed sensing…
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…
We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative thresholding methods. In spite of their empirical success, the theoretical guarantees of such iterative thresholding methods are poorly understood. The goal of this paper is to provide strong theo-retical guarantees, similar to those obtained for nuclear-norm penalization methods and one step thresholding methods, for an iterative thresholding algorithm which is…
We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless case our algorithm is exact. For rank-one matrices, the new algorithm is fast, admits a highly-parallel implementation, and produces an error minimizing estimate that is qualitatively close to our theoretical and the state-of-the-are Nuclear Norm and OptSpace methods.
We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called typical ranks. We give a combinatorial description of the patterns of specified entires of $n\times n$ symmetric matrices that have $n$ as a typical rank. Moreover, we describe exactly when such a generic partial matrix is…
Exact matrix completion and low rank matrix estimation problems has been studied in different underlying conditions. In this work we study exact low-rank completion under non-degenerate noise model. Non-degenerate random noise model has been previously studied by many researchers under given condition that the noise is sparse and existing in some of the columns. In this paper, we assume that each row can receive random noise instead of columns and propose an interactive algorithm that is robust to this noise. We show that we use a parametrization technique to give a condition when the underlyi…
Singapore Management University (SMU)（新加坡管理大学）；Czech Technical University in Prague (CTU)（捷克理工大学）；Hong Kong Baptist University (HKBU)（香港浸会大学）；Technische Universität Kaiserslautern (TUK)（凯泽斯劳滕工业大学）
We study inductive matrix completion (matrix completion with side information) under an i.i.d. subgaussian noise assumption at a low noise regime, with uniform sampling of the entries. We obtain for the first time generalization bounds with the following three properties: (1) they scale like the standard deviation of the noise and in particular approach zero in the exact recovery case; (2) even in the presence of noise, they converge to zero when the sample size approaches infinity; and (3) for a fixed dimension of the side information, they only have a logarithmic dependence on the size of th…
National University of Defense Technology（国防科技大学）；University of Iowa（爱荷华大学）
The common task in matrix completion (MC) and robust principle component analysis (RPCA) is to recover a low-rank matrix from a given data matrix. These problems gained great attention from various areas in applied sciences recently, especially after the publication of the pioneering works of Cand`es et al.. One fundamental result in MC and RPCA is that nuclear norm based convex optimizations lead to the exact low-rank matrix recovery under suitable conditions. In this paper, we extend this result by showing that strongly convex optimizations can guarantee the exact low-rank matrix recovery as…
Morgridge Institute for Research（莫格里奇研究所）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
In this paper, we investigate the sample size requirement for exact recovery of a high order tensor of low rank from a subset of its entries. We show that a gradient descent algorithm with initial value obtained from a spectral method can, in particular, reconstruct a ${d\times d\times d}$ tensor of multilinear ranks $(r,r,r)$ with high probability from as few as $O(r^{7/2}d^{3/2}\log^{7/2}d+r^7d\log^6d)$ entries. In the case when the ranks $r=O(1)$, our sample size requirement matches those for nuclear norm minimization (Yuan and Zhang, 2016a), or alternating least squares assuming orthogonal…
University of Electronic Science and Technology of China (UESTC)（电子科技大学）；School of Information and Communication Engineering, University of Electronic Science and Technology of China (UESTC)（电子科技大学信息与通信工程学院）
Tensor completion recovers a multi-dimensional array from a limited number of measurements. Using the recently proposed tensor ring (TR) decomposition, in this paper we show that a d-order tensor of dimensional size n and TR rank r can be exactly recovered with high probability by solving a convex optimization program, given n^{d/2} r^2 ln^7(n^{d/2})samples. The proposed TR incoherence condition under which the result holds is similar to the matrix incoherence condition. The experiments on synthetic data verify the recovery guarantee for TR completion. Moreover, the experiments on real-world d…
Low-rank matrix completion has been studied extensively under various type of categories. The problem could be categorized as noisy completion or exact completion, also active or passive completion algorithms. In this paper we focus on adaptive matrix completion with bounded type of noise. We assume that the matrix $\mathbf{M}$ we target to recover is composed as low-rank matrix with addition of bounded small noise. The problem has been previously studied by \cite{nina}, in a fixed sampling model. Here, we study this problem in adaptive setting that, we continuously estimate an upper bound for…
Matrix completion problem has been previously studied under various adaptive and passive settings. Previously, researchers have proposed passive, two-phase and single-phase algorithms using coherence parameter, and multi phase algorithm using sparsity-number. It has been shown that the method using sparsity-number reaching to theoretical lower bounds in many conditions. However, the aforementioned method is running in many phases through the matrix completion process, therefore it makes much more informative decision at each stage. Hence, it is natural that the method outperforms previous algo…
