scholarly journals Linear Total Variation Approximate Regularized Nuclear Norm Optimization for Matrix Completion

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xu Han ◽  
Jiasong Wu ◽  
Lu Wang ◽  
Yang Chen ◽  
Lotfi Senhadji ◽  
...  
Keyword(s):  
Total Variation ◽  
Missing Values ◽  
Matrix Completion ◽  
Nuclear Norm ◽  
Low Rank ◽  
Visual Data ◽  
Total Variation Norm ◽  
Rank Matrix ◽  
Variation Norm ◽  
Low Rank Matrix

Matrix completion that estimates missing values in visual data is an important topic in computer vision. Most of the recent studies focused on the low rank matrix approximation via the nuclear norm. However, the visual data, such as images, is rich in texture which may not be well approximated by low rank constraint. In this paper, we propose a novel matrix completion method, which combines the nuclear norm with the local geometric regularizer to solve the problem of matrix completion for redundant texture images. And in this paper we mainly consider one of the most commonly graph regularized parameters: the total variation norm which is a widely used measure for enforcing intensity continuity and recovering a piecewise smooth image. The experimental results show that the encouraging results can be obtained by the proposed method on real texture images compared to the state-of-the-art methods.

Low rank matrix completion using truncated nuclear norm and sparse regularizer

2018 ◽  
Vol 68 ◽  
pp. 76-87 ◽  
Author(s):  
Jing Dong ◽  
Zhichao Xue ◽  
Jian Guan ◽  
Zi-Fa Han ◽  
Wenwu Wang
Keyword(s):  
Matrix Completion ◽  
Nuclear Norm ◽  
Low Rank ◽  
Rank Matrix ◽  
Low Rank Matrix Completion ◽  
Low Rank Matrix ◽  
Truncated Nuclear Norm

DOA Estimation for Antenna Array with Partially-Well Sensors via Low-Rank Matrix Completion

2013 ◽  
Vol 756-759 ◽  
pp. 3977-3981 ◽  
Author(s):  
Hua Xing Yu ◽  
Xiao Fei Zhang ◽  
Jian Feng Li ◽  
De Ben
Keyword(s):  
Missing Values ◽  
Linear Array ◽  
Matrix Completion ◽  
Doa Estimation ◽  
Low Rank ◽  
Uniform Linear Array ◽  
Completion Problem ◽  
Rank Matrix ◽  
Low Rank Matrix Completion ◽  
Low Rank Matrix

In this paper, we address the angle estimation problem in linear array with some ill sensors (partially-well sensors), which only work well randomly. The output of the array will miss some values, and this can be regarded as a low-rank matrix completion problem due to the property that the number of sources is smaller than the number of the total sensors. The output of the array, which is corrupted by the missing values and the noise, can be complete via the Optspace method, and then the angles can be estimated according to the complete output. The proposed algorithm works well for the array with some ill sensors; moreover, it is suitable for non-uniform linear array. Simulation results illustrate performance of the algorithm.

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 Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion

2011 ◽  
Vol 39 (5) ◽  
pp. 2302-2329 ◽  
Author(s):  
Vladimir Koltchinskii ◽  
Karim Lounici ◽  
Alexandre B. Tsybakov
Keyword(s):  
Matrix Completion ◽  
Nuclear Norm ◽  
Low Rank ◽  
Rank Matrix ◽  
Low Rank Matrix Completion ◽  
Low Rank Matrix

scholarly journals Clipped Matrix Completion: A Remedy for Ceiling Effects

2019 ◽  
Vol 33 ◽  
pp. 5151-5158
Author(s):  
Takeshi Teshima ◽  
Miao Xu ◽  
Issei Sato ◽  
Masashi Sugiyama
Keyword(s):  
Matrix Completion ◽  
Low Rank ◽  
Trace Norm ◽  
Minimization Algorithm ◽  
Benchmark Data ◽  
Hinge Loss ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Recovery Error ◽  
Low Rank Matrix

We consider the problem of recovering a low-rank matrix from its clipped observations. Clipping is conceivable in many scientific areas that obstructs statistical analyses. On the other hand, matrix completion (MC) methods can recover a low-rank matrix from various information deficits by using the principle of low-rank completion. However, the current theoretical guarantees for low-rank MC do not apply to clipped matrices, as the deficit depends on the underlying values. Therefore, the feasibility of clipped matrix completion (CMC) is not trivial. In this paper, we first provide a theoretical guarantee for the exact recovery of CMC by using a trace-norm minimization algorithm. Furthermore, we propose practical CMC algorithms by extending ordinary MC methods. Our extension is to use the squared hinge loss in place of the squared loss for reducing the penalty of overestimation on clipped entries. We also propose a novel regularization term tailored for CMC. It is a combination of two trace-norm terms, and we theoretically bound the recovery error under the regularization. We demonstrate the effectiveness of the proposed methods through experiments using both synthetic and benchmark data for recommendation systems.

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