State covariances and the matrix completion problem

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.

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.

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.

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.

Graph theoretic methods for matrix completion problems

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.

formula omitted] completions on partial matrices

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.

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

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.

Using noncommutative Gröbner bases in solving partially prescribed matrix inverse completion problems

This paper investigates an iterative approach to solve the Rank Minimization Problems (RMPs) constrained in a convex set. The matrix rank function is discontinuous and nonconvex and the general RMP is classified as NP-hard. A continuous function is firstly introduced to approximately represent the matrix rank function with prescribed accuracy by selecting appropriate parameters. The RMPs are then converted to rank constrained optimization problems. An Iterative Rank Minimization (IRM) method is proposed to gradually approach the constrained rank. Convergence proof of the IRM method using the duality theory and Karush-Kuhn-Tucker conditions is provided. Two representative applications of RMP, matrix completion and output feedback stabilization problems, are presented to verify the feasibility and improved performance of the proposed IRM method.

The problem of recovering a low-rank matrix from partial entries, known as low-rank matrix completion, has been extensively investigated in recent years. It can be viewed as a special case of the affine constrained rank minimization problem which is NP-hard in general and is computationally hard to solve in practice. One widely studied approach is to replace the matrix rank function by its nuclear-norm, which leads to the convex nuclear-norm minimization problem solved efficiently by many popular convex optimization algorithms. In this paper, we propose a new nonconvex approach to better approximate the rank function. The new approximation function is actually the Moreau envelope of the rank function (MER) which has an explicit expression. The new approximation problem of low-rank matrix completion based on MER can be converted to an optimization problem with two variables. We then adapt the proximal alternating minimization algorithm to solve it. The convergence (rate) of the proposed algorithm is proved and its accelerated version is also developed. Numerical experiments on completion of low-rank random matrices and standard image inpainting problems have shown that our algorithms have better performance than some state-of-art methods.

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 first identify how the classical non-negative rank model enforces restrictions that may be undesirable in practice. We propose a richer model based on what we term the “permutation-rank” of a matrix and show how the restrictions due to classical low rank assumptions can be avoided by using the richer permutation-rank model. We establish information-theoretic lower bounds on the rates of estimation, and design an estimator which we prove is simultaneously optimal (up to logarithmic factors) for both the permutation-rank and the low-rank models. Our results thus show that the proposed permutation-rank model and estimator enjoy a surprising win-win in terms of the statistical bias-variance tradeoff as compared to the classical low-rank models. An extended version of this paper is available on arXiv [1].

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.