scholarly journals Local low-rank approach to nonlinear-matrix completion

2021 ◽  
Author(s):  
Ryohei Sasaki ◽  
Katsumi Konishi ◽  
Tomohiro Takahashi ◽  
Toshihiro Furukawa
Keyword(s):  
Matrix Completion ◽  
Full Rank ◽  
Differentiable Manifold ◽  
Column Vector ◽  
Low Rank ◽  
Estimation Accuracy ◽  
Dimensional Linear Subspace ◽  
Target Matrix ◽  
The Matrix ◽  
Low Dimensional

Abstract This paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a p-th order polynomial, and that the rank of a matrix whose column vectors are d-th monomial features of target column vectors is defficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.

scholarly journals Local low-rank approach to nonlinear-matrix completion

2020 ◽  
Author(s):  
Ryohei Sasaki ◽  
Katsumi Konishi ◽  
Tomohiro Takahashi ◽  
Toshihiro Furukawa
Keyword(s):  
Matrix Completion ◽  
Full Rank ◽  
Differentiable Manifold ◽  
Column Vector ◽  
Low Rank ◽  
Estimation Accuracy ◽  
Dimensional Linear Subspace ◽  
Target Matrix ◽  
The Matrix ◽  
Low Dimensional

Abstract This paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a p-th order polynomial, and that the rank of a matrix whose column vectors are d-th monomial features of target column vectors is deficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.

scholarly journals Local low-rank approach to nonlinear matrix completion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ryohei Sasaki ◽  
Katsumi Konishi ◽  
Tomohiro Takahashi ◽  
Toshihiro Furukawa
Keyword(s):  
Matrix Completion ◽  
Full Rank ◽  
Differentiable Manifold ◽  
Column Vector ◽  
Low Rank ◽  
Estimation Accuracy ◽  
Dimensional Linear Subspace ◽  
Target Matrix ◽  
The Matrix ◽  
Low Dimensional

AbstractThis paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a pth order polynomial and that the rank of a matrix whose column vectors are dth monomial features of target column vectors is deficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.

scholarly journals Recovering spectral images from compressive measurements using designed coded apertures and Matrix Completion Theory

2017 ◽  
Vol 21 (2) ◽  
Author(s):  
Tatiana Gelvez ◽  
Hoover Rueda ◽  
Henry Arguello
Keyword(s):  
Imaging Techniques ◽  
Matrix Completion ◽  
Spectral Imaging ◽  
Spatial Location ◽  
Low Rank ◽  
Coded Aperture ◽  
Reconstruction Algorithms ◽  
Spectral Image ◽  
The Matrix ◽  
Spectral Imager

<p>Spectral imaging aims to capture and process a 3-dimensional spectral image with a large amount of spectral information for each spatial location. Compressive spectral imaging techniques (CSI) increases the sensing speed and reduces the amount of collected data compared to traditional spectral imaging methods. The coded aperture snapshot spectral imager (CASSI) is an optical architecture to sense a spectral image in a single 2D coded projection by applying CSI. Typically, the 3D scene is recovered by solving an L1-based optimization problem that assumes the scene is sparse in some known orthonormal basis. In contrast, the matrix completion technique (MC) allows to recover the scene without such prior knowledge. The MC reconstruction algorithms rely on a low-rank structure of the scene. Moreover, the CASSI system uses coded aperture patterns that determine the quality of the estimated scene. Therefore, this paper proposes the design of an optimal coded aperture set for the MC methodology. The designed set is attained by maximizing the distance between the translucent elements in the coded aperture. Visualization of the recovered spectral signals and simulations over different databases show average improvement when the designed coded set is used between 1-3 dBs compared to the complementary coded aperture set, and between 3-9 dBs compared to the conventional random coded aperture set.</p>

scholarly journals Low-rank matrix completion and denoising under Poisson noise

2020 ◽  
Author(s):  
Andrew D McRae ◽  
Mark A Davenport
Keyword(s):  
Matrix Completion ◽  
Universal Constant ◽  
Poisson Noise ◽  
Nuclear Norm ◽  
Low Rank ◽  
Frobenius Norm ◽  
Rank Matrix ◽  
The Matrix ◽  
Low Rank Matrix Completion ◽  
Low Rank Matrix

Abstract This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyse several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares and a non-convex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed and the maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low-rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.

scholarly journals On the Parameterized Complexity of Clustering Incomplete Data into Subspaces of Small Rank

2020 ◽  
Vol 34 (04) ◽  
pp. 3906-3913
Author(s):  
Robert Ganian ◽  
Iyad Kanj ◽  
Sebastian Ordyniak ◽  
Stefan Szeider
Keyword(s):  
Machine Learning ◽  
Parameterized Complexity ◽  
Linear Codes ◽  
Matrix Completion ◽  
Low Rank ◽  
Input Constraints ◽  
Binary Linear Codes ◽  
Completion Problem ◽  
The Matrix ◽  
Complete Matrix

We consider a fundamental matrix completion problem where we are given an incomplete matrix and a set of constraints modeled as a CSP instance. The goal is to complete the matrix subject to the input constraints and in such a way that the complete matrix can be clustered into few subspaces with low rank. This problem generalizes several problems in data mining and machine learning, including the problem of completing a matrix into one with minimum rank. In addition to its ubiquitous applications in machine learning, the problem has strong connections to information theory, related to binary linear codes, and variants of it have been extensively studied from that perspective. We formalize the problem mentioned above and study its classical and parameterized complexity. We draw a detailed landscape of the complexity and parameterized complexity of the problem with respect to several natural parameters that are desirably small and with respect to several well-studied CSP fragments.

scholarly journals Numerical Comparisons Between Bayesian and Frequentist Low-Rank Matrix Completion: Estimation Accuracy and Uncertainty Quantification

2021 ◽  
Author(s):  
The Tien Mai
Keyword(s):  
Confidence Intervals ◽  
Estimation Error ◽  
Matrix Completion ◽  
Low Rank ◽  
Small Samples ◽  
Estimation Accuracy ◽  
Bayesian Approaches ◽  
Rank Matrix ◽  
Low Rank Matrix Completion ◽  
Low Rank Matrix

In this paper we perform numerous numerical studies for the problem of low-rank matrix completion. We compare the Bayesian approaches and a recently introduced de-biased estimator which provides a useful way to build confidence intervals of interest. From a theoretical viewpoint, the de-biased estimator comes with a sharp minimax-optimal rate of estimation error whereas the Bayesian approach reaches this rate with an additional logarithmic factor. Our simulation studies show originally interesting results that the de-biased estimator is just as good as the Bayesian estimators. Moreover, Bayesian approaches are much more stable and can outperform the de-biased estimator in the case of small samples. However, we also find that the length of the confidence intervals revealed by the de-biased estimator for an entry is absolutely shorter than the length of the considered credible interval. These suggest further theoretical studies on the estimation error and the concentration for Bayesian methods as they are being quite limited up to present.

scholarly journals Finding metastabilities in reversible Markov chains based on incomplete sampling

2017 ◽  
Vol 5 (1) ◽  
pp. 73-81
Author(s):  
Konstantin Fackeldey ◽  
Amir Niknejad ◽  
Marcus Weber
Keyword(s):  
Markov Chains ◽  
Invariant Subspace ◽  
Stationary Distribution ◽  
Transition Matrix ◽  
Matrix Completion ◽  
Sampling Strategy ◽  
Low Rank ◽  
Sampling Effort ◽  
The Matrix ◽  
Partial Sampling

Abstract In order to fully characterize the state-transition behaviour of finite Markov chains one needs to provide the corresponding transition matrix P. In many applications such as molecular simulation and drug design, the entries of the transition matrix P are estimated by generating realizations of the Markov chain and determining the one-step conditional probability Pij for a transition from one state i to state j. This sampling can be computational very demanding. Therefore, it is a good idea to reduce the sampling effort. The main purpose of this paper is to design a sampling strategy, which provides a partial sampling of only a subset of the rows of such a matrix P. Our proposed approach fits very well to stochastic processes stemming from simulation of molecular systems or random walks on graphs and it is different from the matrix completion approaches which try to approximate the transition matrix by using a low-rank-assumption. It will be shown how Markov chains can be analyzed on the basis of a partial sampling. More precisely. First, we will estimate the stationary distribution from a partially given matrix P. Second, we will estimate the infinitesimal generator Q of P on the basis of this stationary distribution. Third, from the generator we will compute the leading invariant subspace, which should be identical to the leading invariant subspace of P. Forth, we will apply Robust Perron Cluster Analysis (PCCA+) in order to identify metastabilities using this subspace.

scholarly journals Simultaneous completion and spatiotemporal grouping of corrupted motion tracks

2021 ◽  
Author(s):  
Antonio Agudo ◽  
Vincent Lepetit ◽  
Francesc Moreno-Noguer
Keyword(s):  
Matrix Completion ◽  
Low Rank ◽  
Partial Observations ◽  
Rank Matrix ◽  
The Matrix ◽  
Temporal And Spatial ◽  
Point Trajectories ◽  
Low Rank Matrix Completion ◽  
Noisy Measurements ◽  
Low Rank Matrix

AbstractGiven an unordered list of 2D or 3D point trajectories corrupted by noise and partial observations, in this paper we introduce a framework to simultaneously recover the incomplete motion tracks and group the points into spatially and temporally coherent clusters. This advances existing work, which only addresses partial problems and without considering a unified and unsupervised solution. We cast this problem as a matrix completion one, in which point tracks are arranged into a matrix with the missing entries set as zeros. In order to perform the double clustering, the measurement matrix is assumed to be drawn from a dual union of spatiotemporal subspaces. The bases and the dimensionality for these subspaces, the affinity matrices used to encode the temporal and spatial clusters to which each point belongs, and the non-visible tracks, are then jointly estimated via augmented Lagrange multipliers in polynomial time. A thorough evaluation on incomplete motion tracks for multiple-object typologies shows that the accuracy of the matrix we recover compares favorably to that obtained with existing low-rank matrix completion methods, specially under noisy measurements. In addition, besides recovering the incomplete tracks, the point trajectories are directly grouped into different object instances, and a number of semantically meaningful temporal primitive actions are automatically discovered.

scholarly journals Robust Matrix Completion By Exploiting Dynamic Low-Dimensional Structures

2021 ◽  
Author(s):  
Ren Wang ◽  
Pengzhi Gao ◽  
Meng Wang
Keyword(s):  
Matrix Completion ◽  
Synthetic Data ◽  
Original Data ◽  
Low Rank ◽  
Temporal Correlations ◽  
Completion Problem ◽  
Reconstruction Performance ◽  
Matrix Completion Problem ◽  
Recovery Error ◽  
Low Dimensional

Abstract This paper studies the robust matrix completion problem for time-varying models. Leveraging the low-rank property and the temporal information of the data, we develop novel methods to recover the original data from partially observed and corrupted measurements. We show that the reconstruction performance can be improved if one further leverages the information of the sparse corruptions in addition to the temporal correlations among a sequence of matrices. The dynamic robust matrix completion problem is formulated as a nonconvex optimization problem, and the recovery error is quantified analytically and proved to decay in the same order as that of the state-of-the-art method when there is no corruption. A fast iterative algorithm with convergence guarantee to the stationary point is proposed to solve the nonconvex problem. Experiments on synthetic data and real video dataset demonstrate the effectiveness of our method.

A Recovery Method of Data Lost in Network Communication

2019 ◽  
Vol 23 (2) ◽  
pp. 248-252
Author(s):  
Bo Zhang ◽  
Keyword(s):  
Matrix Completion ◽  
Low Rank ◽  
Network Communication ◽  
Data Loss ◽  
Recovery Method ◽  
Random Loss ◽  
The Matrix ◽  
Rank Decomposition ◽  
Low Rank Decomposition ◽  
Recovery Problem

At present, ScanDisk is used to recover the data lost in network communication. But this method is limited in scope, and once the lost data is covered, it’s difficult or impossible to recover it, which results in low recovery degree. Accordingly, a recovery method for lost data in network communication based on RAID6 is proposed. Firstly, according to the mechanism of data loss in network communication, the missing data is divided into three categories: random loss, completely random loss and nonrandom loss, and then according to the results of classification, the recovery problem of the data loss in network communication is converted into the problem of matrix completion, finally, a low-rank decomposition model is proposed, according to the low rank characteristics of the matrix, the lost data in the matrix is recovered, thus the recovery of the lost data in network communication is finished. Experimental results show that the proposed method can easily recover the lost data in network communication with a simple operation, low computing complexity and strong applicability, and can be used as a universal recovery method for data lost in network communication.

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