Graph theoretic methods for matrix completion problems

Given an undirected graph G=(V,E) with node set V=[1,n], a subset $S\subseteq V$, and a rational vector $a\in {\rm\bf Q}^{S\cup E}$, the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n × n positive semidefinite matrix X=(xij) satisfying xii=ai ($i\in S$) and xij=aij ($ij\in E$). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fill-in, the minimum fill-in of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits us to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph K4.

This paper studies the problem of the restoration of images corrupted by mixed Gaussian-impulse noise. In recent years, low-rank matrix reconstruction has become a research hotspot in many scientific and engineering domains such as machine learning, image processing, computer vision and bioinformatics, which mainly involves the problem of matrix completion and robust principal component analysis, namely recovering a low-rank matrix from an incomplete but accurate sampling subset of its entries and from an observed data matrix with an unknown fraction of its entries being arbitrarily corrupted, respectively. Inspired by these ideas, we consider the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors, which is defined as the problem of matrix completion from corrupted samplings and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the -norm in this paper. Meanwhile, we put forward a novel and effective algorithm called augmented Lagrange multipliers to exactly solve the problem. For mixed Gaussian-impulse noise removal, we regard it as the problem of matrix completion from corrupted samplings, and restore the noisy image following an impulse-detecting procedure. Compared with some existing methods for mixed noise removal, the recovery quality performance of our method is dominant if images possess low-rank features such as geometrically regular textures and similar structured contents; especially when the density of impulse noise is relatively high and the variance of Gaussian noise is small, our method can outperform the traditional methods significantly not only in the simultaneous removal of Gaussian noise and impulse noise, and the restoration ability for a low-rank image matrix, but also in the preservation of textures and details in the image.

formula omitted] completions on partial matrices

The matrix completion problem can be stated as follows: Suppose $X\in \mathbb{R}^{n_{1}\times n_{2}}$ is unknown except for an upper bound $r$ on its rank. By measuring a small number $m\ll n_{1}n_{2}$ of elements of $X$ , is it possible to recover $X$ exactly, or at least, to construct a reasonable approximation of $X$ ? There are two existing approaches to choosing the elements to be measured, namely: choosing them at random or in some deterministic fashion. Practically all of the existing literature focuses on the case where the sampled locations are chosen at random. In contrast, the focus in the present paper is on deterministic methods for choosing the elements to be sampled. Specifically, we choose the measurement matrix to be the adjacency or biadjacency matrix of a so-called Ramanujan graph. Existing explicit techniques for constructing Ramanujan graphs lead to graphs whose degree $d$ is bounded by $n^{1/3}$ where $n$ is the number of vertices. However, for the matrix completion problem, this degree is too small. We point out that the well-known Lubotzky-Phillips-Sarnak construction of Ramanujan graphs can be used to generate graphs of very high degree, and use these for the matrix construction problem. Then we carry out numerical studies of “phase transition” in the matrix completion problem, by comparing the behavior of sampling at random with choosing the sampled locations using a Ramanujan graph. Specifically, we show that for a fixed sampling pattern, there is a maximum rank $\bar{r}$ for which randomly generated matrices of rank $r\leq\overline{r}$ can be recovered with probability one, and the recovery probability drops very sharply to zero if $r$ is increased by just two or three beyond $\bar{r}$ . The technique proposed here is applicable only to the recovery of square matrices. The recovery of rectangular matrices requires so-called asymmetric Ramanujan graphs, and is studied in a separate paper.

This paper considers the matrix completion problem. We show that it is not necessary to assume joint incoherence, which is a standard but unintuitive and restrictive condition that is imposed by previous studies. This leads to a sample complexity bound that is order-wise optimal with respect to the incoherence parameter (as well as to the rank $r$ and the matrix dimension $n$ up to a log factor). As a consequence, we improve the sample complexity of recovering a semidefinite matrix from $O(nr^{2}\log^{2}n)$ to $O(nr\log^{2}n)$, and the highest allowable rank from $\Theta(\sqrt{n}/\log n)$ to $\Theta(n/\log^{2}n)$. The key step in proof is to obtain new bounds on the $\ell_{\infty,2}$-norm, defined as the maximum of the row and column norms of a matrix. To illustrate the applicability of our techniques, we discuss extensions to SVD projection, structured matrix completion and semi-supervised clustering, for which we provide order-wise improvements over existing results. Finally, we turn to the closely-related problem of low-rank-plus-sparse matrix decomposition. We show that the joint incoherence condition is unavoidable here for polynomial-time algorithms conditioned on the Planted Clique conjecture. This means it is intractable in general to separate a rank-$\omega(\sqrt{n})$ positive semidefinite matrix and a sparse matrix. Interestingly, our results show that the standard and joint incoherence conditions are associated respectively with the information (statistical) and computational aspects of the matrix decomposition problem.

In the present paper, we consider the problem of matrix completion with noise. Unlike previous works, we consider quite general sampling distribution and we do not need to know or to estimate the variance of the noise. Two new nuclear-norm penalized estimators are proposed, one of them of "square-root" type. We analyse their performance under high-dimensional scaling and provide non-asymptotic bounds on the Frobenius norm error. Up to a logarithmic factor, these performance guarantees are minimax optimal in a number of circumstances.

In this paper, the P0,1+-matrix completion is introduced. A real n × n matrix is a P0,1+-matrix if it is a P0+-matrix with all positive diagonal entries. Firstly some necessary conditions for P0,1+-matrix completion are obtained and later on some sufficient conditions are also shown. Also the relationship between the P0,1+-matrix completion problem and other completion problems of p-matrix classes are discussed. Finally most of the digraphs of order at most 4 having P0,1+-completion are classified.

We derive square-root based algorithms for structured matrices and discuss potential applications to interpolation and matrix completion problems. The mathematical machinery used here is based on a standard Gaussian elimination technique and on simple results from matrix and linear system theory. We show how to exploit the inherent displacement structure in order to construct a convenient transmission-line cascade that makes evident the required interpolation and completion conditions. We also introduce the concept of time-variant structured matrices and discuss its applications to matrix completions and time-variant interpolation problems.

Detect potential relations by link prediction in multi-relational social networks

The essence of distantly supervised relation extraction is that it is an incomplete multi-label classification problem with sparse and noisy features. To tackle the sparsity and noise challenges, we propose solving the classification problem using matrix completion on factorized matrix of minimized rank. We formulate relation classification as completing the unknown labels of testing items (entity pairs) in a sparse matrix that concatenates training and testing textual features with training labels. Our algorithmic framework is based on the assumption that the rank of item-byfeature and item-by-label joint matrix is low. We apply two optimization models to recover the underlying low-rank matrix leveraging the sparsity of feature-label matrix. The matrix completion problem is then solved by the fixed point continuation (FPC) algorithm, which can find the global optimum. Experiments on two widely used datasets with different dimensions of textual features demonstrate that our low-rank matrix completion approach significantly outperforms the baseline and the state-of-the-art methods.

Since lots of miRNA-disease associations have been verified, it is meaningful to discover more miRNA-disease associations for serving disease diagnosis and prevention of human complex diseases. However, it is not practical to identify potential associations using traditional biological experimental methods since the process is expensive and time consuming. Therefore, it is necessary to develop efficient computational methods to accomplish this task. In this work, we introduced a matrix completion model with dual Laplacian regularization (DLRMC) to infer unknown miRNA-disease associations in heterogeneous omics data. Specifically, DLRMC transformed the task of miRNA-disease association prediction into a matrix completion problem, in which the potential missing entries of the miRNA-disease association matrix were calculated, the missing association can be obtained based on the prediction scores after the completion procedure. Meanwhile, the miRNA functional similarity and the disease semantic similarity were fully exploited to serve the miRNA-disease association matrix completion by using a dual Laplacian regularization term. In the experiments, we conducted global and local Leave-One-Out Cross Validation (LOOCV) and case studies to evaluate the efficacy of DLRMC on the Human miRNA-disease associations dataset obtained from the HMDDv2.0 database. As a result, the AUCs of DLRMC is 0.9174 and 0.8289 in global LOOCV and local LOOCV, respectively, which significantly outperform a variety of previous methods. In addition, in the case studies on four significant diseases related to human health including Colon Neoplasms, Kidney neoplasms, Lymphoma and Prostate neoplasms, 90%, 92%, 92% and 94% out of the top 50 predicted miRNAs has been confirmed, respectively.

To reconstruct the ocean sound speed field (SSF), a matrix in horizontal could be formed by gridding the sea area, where some sparse entries were filled by the observation sound speed profiles (SSPs). The reconstruction of the SSF can be modeled as a matrix completion problem, using limited and noisy observation values to reconstruct the entire matrix. Furthermore, the spatial correlation of the SSF could be modeled by graph space model, utilizing with the low-rank property of the SSF matrix, the problem of graph regularization low-rank matrix completion was developed. The observed average SSPs in six cross section were estimated from the four underwater acoustic tomography moorings, and nine local SSPs were obtained from pressure inverted echo sounders (PIESs) inversion. By combining the method of deep matrix factorization and the graph regularization, the Graph Regularized Deep Matrix Factorization (GRDMF) method was proposed. The simulation results showed that the GRDMF performs better than the spatial interpolation algorithm in SSF reconstruction. Considering the application scenarios of deep-sea acoustic tomography, an average constraint (AC) item was designed additionally, and then a customized algorithm named GRDMF-AC is proposed. Finally, data of the experiment at the South China Sea in 2021 is processed to verify GRDMF-AC, the results showed that the reconstructed SSF captures mesoscale eddy in the experiment.

The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of its entries. Here, we investigate the case of highly quantized observations when the measurements can take only a small number of values. These quantized outputs are generated according to a probability distribution parametrized by the unknown matrix of interest. This model corresponds, for example, to ratings in recommender systems or labels in multi-class classification. We consider a general, non-uniform, sampling scheme and give theoretical guarantees on the performance of a constrained, nuclear norm penalized maximum likelihood estimator. One important advantage of this estimator is that it does not require knowledge of the rank or an upper bound on the nuclear norm of the unknown matrix and, thus, it is adaptive. We provide lower bounds showing that our estimator is minimax optimal. An efficient algorithm based on lifted coordinate gradient descent is proposed to compute the estimator. A limited Monte-Carlo experiment, using both simulated and real data is provided to support our claims.

Recovering a low-rank matrix from some of its linear measurements is a popular problem in many areas of science and engineering. One special case of it is the matrix completion problem, where we need to reconstruct a low-rank matrix from incomplete samples of its entries. A lot of efficient algorithms have been proposed to solve this problem and they perform well when Gaussian noise with a small variance is added to the given data. But they can not deal with the sparse random-valued noise in the measurements. In this paper, we propose a robust method for recovering the low-rank matrix with adaptive outlier pursuit when part of the measurements are damaged by outliers. This method will detect the positions where the data is completely ruined and recover the matrix using correct measurements. Numerical experiments show the accuracy of noise detection and high performance of matrix completion for our algorithms compared with other algorithms.

State statistics of a linear system obey certain structural constraints that arise from the underlying dynamics and the directionality of input disturbances. Herein, we formulate completion problems of partially known state statistics with the added freedom of identifying disturbance dynamics. The goal of the proposed completion problem is to obtain information about input excitations that explain observed sample statistics. Our formulation aims at low-complexity models for admissible disturbances. The complexity represents the dimensionality of the subspace of the state-dynamics that is directly affected by disturbances. An example is provided to illustrate that colored-in-time stochastic processes can be effectively used to explain available data.