Incoherence-Optimal Matrix Completion

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.

Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property

Sparse matrix decompositions and graph characterizations

We propose a low rank structured matrix completion algorithm for image inpainting problems originated from scanning microscopy. The proposed method exploits the annihilation property observed in Gaussian Markov Random Field (GMRF) or partial differential equation (PDE)-based inpainting approaches. By utilizing the commutative property of the convolution, the annihilation property is embodied into rank-deficient block Hankel structure data matrices and the image inpainting problem is converted into low-rank structured matrix completion problem. To solve the structured low-rank matrix completion problem, an alternating direction method of multiplier (ADMM) method is used with factorization matrix initialization using the low rank matrix fitting (LMaFit) algorithm. Experimental results showed that the proposed method outperforms the existing state-of-the-art image inpainting methods.

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.

formula omitted] completions on partial matrices

Graph theoretic methods for matrix completion problems

In this paper the concept of positive definite bilinear matrix moment functional, acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b] is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Riemann—Lebesgue, matrix property and the bessel—parseval matrix inequality are proved. The concept of total set with respect to a positive definite matrix functional is introduced, and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional is established.

Given a monochrome image and some manually labeled pixels, the colorization problem is a computer-assisted process of adding color to the monochrome image. This paper proposes a novel approach to the colorization problem by formulating it as a matrix completion problem. In particular, taking a monochrome image and parts of the color pixels (labels) as inputs, we develop a robust colorization model and resort to an augmented Lagrange multiplier algorithm for solving the model. Our approach is based on the fact that a matrix can be represented as a low-rank matrix plus a sparse matrix. Our approach is effective because it is able to handle the potential noises in the monochrome image and outliers in the labels. To improve the performance of our method, we further incorporate a so-called local-color-consistency idea into our method. Empirical results on real data sets are encouraging.

Matrix completion is the task of predicting missing entries of matrix from a subset of known entries. Notions of structured matrix completion include any setting in which whether an entry is observed does not occur uniformly at random. In recent work, a modification to the standard nuclear norm minimization for matrix completion has been made to take into account sparsity-based structure in the missing entries, which is motivated e.g. in recommender systems. In this work, we propose adjusting an Iteratively Reweighted Least Squares (IRLS) algorithm for low-rank matrix completion to take into account sparsity-based structure. We also outline an iterative gradient-projection-based implementation of the algorithm that can handle large-scale matrices. Lastly, we present preliminary numerical experiments showcasing the performance of the proposed method compared to the standard IRLS algorithm in structured settings.

ABSTRACTMatrix completion has attracted significant recent attention in many fields including statistics, applied mathematics, and electrical engineering. Current literature on matrix completion focuses primarily on independent sampling models under which the individual observed entries are sampled independently. Motivated by applications in genomic data integration, we propose a new framework of structured matrix completion (SMC) to treat structured missingness by design. Specifically, our proposed method aims at efficient matrix recovery when a subset of the rows and columns of an approximately low-rank matrix are observed. We provide theoretical justification for the proposed SMC method and derive lower bound for the estimation errors, which together establish the optimal rate of recovery over certain classes of approximately low-rank matrices. Simulation studies show that the method performs well in finite sample under a variety of configurations. The method is applied to integrate several ovarian cancer genomic studies with different extent of genomic measurements, which enables us to construct more accurate prediction rules for ovarian cancer survival. Supplementary materials for this article are available online.

In this paper, we propose an analytical solution for recovering a low-rank positive semi-definite (PSD) matrix from its rank-one measurements. We show that by utilizing a set of structured measurement vectors, we can analytically determine the null space of this low-rank PSD matrix. Based on the result, the PSD matrix can be efficiently recovered. Our analysis shows that the proposed method only requires <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(N-K)(2K+1) + K^{2}$</tex-math></inline-formula> measurements to guarantee exact recovery of the PSD matrix, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> respectively denote the dimension and the rank of the PSD matrix. Numerical results show that the proposed method achieves a considerable improvement over existing state-of-the-art methods in terms of both sample complexity and computational efficiency. Specifically, the proposed method helps improve the computational efficiency by an order of magnitude as compared with existing methods.

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.

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.

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.