Nuclear Norm Based Matrix Regression Research Articles

Enhanced nuclear norm based matrix regression for occluded face recognition

An effective approach for the task of face recognition is proposed in this paper, which formulates the problem as an enhanced nuclear norm based matrix regression model and explores the low-rank property of the reconstructed image. Previous works have already leveraged the nuclear norm to obtain a low-rank representation of the error image and get a promising recognition rate. Motivated by the low-rank property of the reconstructed image through theoretical observation, our model imposes the nuclear norm constraints not only on the representation residual but also on the reconstructed image. The proposed method preserves the 2D structural information of the error images and reconstructs images, which is significant for the face recognition tasks. To further improve the performance of the proposed model, we explore the impact of different regularization terms under various scenarios. Extensive experiments on several benchmark datasets show the efficacy of the proposed model especially in terms of robustness against contiguous occlusion and illumination changes, which achieves superior performance over the most competitive methods.

Nonconvex regularizer and latent pattern based robust regression for face recognition

Nuclear norm based matrix regression (NMR) approaches have yielded encouraging recognition results in the presence of image-level noise. However, NMR methods have two shortcomings. One is that test samples are directly used in the reconstruction process, and this may degrade recognition performance, especially when the samples are severely corrupted by noise. The other is that these methods exploit the nuclear norm to characterize low-rank structural information of residual images, which may lead to suboptimal solutions. To overcome these two deficiencies, we present a robust regression model that coalesces a nonconvex regularizer with a latent pattern (NRLPR). The latent pattern is essentially obtained by removing noise from a test sample, and therefore it can be closer to the reconstructed sample than the test sample. Furthermore, the nonconvex regularizer penalizes larger values less while penalizing smaller values more, accordingly, it can efficiently approximate the rank function. Additionally, we integrate the iteratively reweighted least squares method and the alternating direction method of multipliers to devise an efficient iterative algorithm (IR-ADMM) for NRLPR. Meanwhile, a nonconvex function is used for the design of the classifier. Finally, numerous experiments demonstrate the superiority of the proposed methods over state-of-the-art approaches for handling structural and mixed noise.

Robust mixed-norm constrained regression with application to face recognitions

Most existing regression-based classification methods cope with pixelwise noise via $$\ell _1$$ -norm or $$\ell _2$$ -norm, but neglect the structural information between pixels. To the best of our knowledge, nuclear norm-based matrix regression approaches have achieved great success for addressing imagewise noise, but may result in unreasonable regression and incorrect classification, especially when test images are extremely corrupted by larger occlusions and severe illumination variations, since they apply the corrupted test images to reconstruction process directly, and the influence of noise will be unavoidable. To overcome this limitation, this paper presents a robust mixed-norm constrained regression model to deal with the structural noise corruption. To be more specific, nuclear norm of the error between corrupted test image and its corresponding recovered image is exploited as a regular term for characterizing the low rank noise structure, and Frobenius norm is utilized to depict the difference between the recovered image and restructured image on account of the less noise of recovered image. Then, we adopt the alternating direction method of multipliers to settle our proposed approaches efficiently. Furthermore, the theoretical convergence proof and detailed analysis of computational complexity are provided to assess our algorithms. Eventually, extensive experiments on five well-known face databases have manifested that the proposed methods outperform some state-of-the-art regression-based approaches for primarily addressing noise caused by occlusion and illumination changes.

Ensemble learning-based computational imaging method for electrical capacitance tomography

Electrical capacitance tomography (ECT) is a potential measurement technology for industrial process monitoring, but its applicability is generally limited by low-quality tomographic images. Boosting the performance of inverse computing imaging algorithms is the key to improving the reconstruction quality (RQ). Common regularization iteration imaging methods with analytical prior regularizers are less flexible in dealing with actual reconstruction tasks, leading to large reconstruction errors. To address the challenge, this study proposes a new imaging method, including reconstruction model and optimizer. The data-driven regularizer from a new ensemble learning model and the analytical prior regularizer with the focus on the sparsity of imaging objects are combined into a new optimization model for imaging. In the proposed ensemble learning model, the generalized low rank approximations of matrices (GLRAM) method is used to carry out the dimensionality reduction for decreasing the redundancy of the input data and improving the diversity, the extreme learning machine (ELM) serves as a base learner and the nuclear norm based matrix regression (NNMR) method is developed to aggregate the ensemble of solutions. The singular value thresholding method (SVTM) and the fast iterative shrinkage-thresholding algorithm (FISTA) are inserted into the split Bregman method (SBM) to generate a powerful optimizer for the built computational model. Its comparison to other competing methods through numerical experiments on typical imaging targets demonstrates that the developed algorithm reduces reconstruction error and achieves much more improvement in imaging quality and robustness.

Image decomposition based matrix regression with applications to robust face recognition

The previous matrix regression based methods mainly focus on designing a robust error term to characterize the occlusion and illumination changes. In actually, it is very challenging to give a strong model for solving the original images directly since the images contains rich and complex structure information. To address this problem, we aim to simplify the complex images and propose a simple and robust matrix regression based classification model. In our method, we firstly employ the local gradient distribution to decompose the image into a series of gradient images (LID for short). Each gradient image reveals the local structure information in different gradient orientations. Subsequently, we consider each gradient image as the diagonal block element and construct the diagonal block matrix for image representation. Nuclear norm based matrix regression model (NMR) is then applied to complete the classification tasks. The proposed model can be called ID-NMR for short. We further design a fast ADMM optimization algorithm to solve the proposed ID-NMR due to the fact that the big diagonal block matrix will increase the computational load. Experimental results show that the proposed method performs favorably compared with state-of-the-art regression based classification methods.

A sparse regularized nuclear norm based matrix regression for face recognition with contiguous occlusion

Nuclear norm based matrix regression (NMR) method has been proposed to alleviate the influence of contiguous occlusion on face recognition problems. NMR considers that the error image of a test sample has low-rank structure due to the contiguous nature of occlusion. Based on the observation that l1-norm can uncover more natural sparsity of representations than l2-norm, we propose a sparse regularized NMR (SR_NMR) algorithm by imposing the l1-norm constraint rather than l2-norm on the representations of NMR framework. SR_NMR seamlessly integrates the nuclear norm based error matrix regression and l1-norm based sparse representation into one joint framework. Finally, we use the training samples to learn a linear classifier to implement efficient classification. Extensive experiments on three face databases show the proposed SR_NMR can achieve better recognition performance compared with the traditional NMR and other regression methods which indicates that sparse representations are very helpful to recover low-rank error images in the presence of severe occlusion and illumination changes.

Nuclear norm-based matrix regression preserving embedding for face recognition

Abstract Recently, using linear reconstruction technique to construct intrinsic graph for projection-based dimensionality reduction (DR) has aroused broad interest in face recognition. However, current methods either lack robustness to corruptions or require to perform vectorization which causes loss of local geometrical information of images. To this end, a novel nuclear norm-based matrix regression preserving embedding (NN-MRPE) method is proposed in this paper. First, NN-MRPE constructs an intrinsic graph by using the nuclear norm to evaluate the residual errors to resist data corruptions. Second, a matrix-based embedding cost function is formulated to seek two transformation matrices which can preserve the geometrical structure reflected by the intrinsic graph exactly. Finally, based on the linear regression theory, we summarize a general DR framework called linear regression preserving embedding that preserves the intrinsic structure of data by recovering the reconstruction relationship in the original space. Specifically, many existing approaches are the special cases of the linear regression preserving embedding. Experiments on five public face databases with different types of corruptions are conducted to demonstrate the efficiency of the proposed NN-MRPE method.

An adaptive line search scheme for approximated nuclear norm based matrix regression

Abstract Face recognition with partial occlusion is common in the real world and has become a hot topic in the pattern recognition research. Recently, nuclear norm based matrix regression model (NMR) has been shown a great potential in dealing with the structural noise. And yet, it uses Alternating Direction Method of Multipliers (ADMM) to achieve the optimal regression coefficients, which needs to bring into an auxiliary variable and only exploits the convexity of NMR. Comparing with ADMM, the gradient based methods are simpler. To take advantage of these methods, this paper considers the Approximated NMR model with Elastic-net regularization (ANMRE). By virtue of the soft-thresholding (shrinkage) operator, singular value shrinkage operator and strong convexity of ANMRE, the dual problem of ANMRE (DANMRE) is derived and a crucial result is obtained: the primal optimal solution of ANMRE can be converted as the matrix function associated with the dual optimal solution. Leveraging the differentiability of DANMRE, an adaptive line search scheme is developed to solve it. This approach makes full use of the advantages of the accelerated gradient technique and adaptive parameters updating strategy. Meanwhile, a convergence rate of O(1/N2) can be guaranteed under mild conditions. Experimental results on five standard databases show the superiority of the proposed algorithm over some existing methods.

A Nuclear Norm Based Matrix Regression Based Projections Method for Feature Extraction

In the traditional graph embedding framework, the graph is usually built by k-NN or r-ball. Since it is difficult to manually set the parameters k and r in the high-dimensional space, sparse representation-based methods are usually introduced to automatically build the graphs. In recent years, nuclear norm-based matrix regression (NMR) has been proposed for face recognition using the low rank structural information (i.e., the image matrix-based error model). Inspired by NMR, we give a NMR-based projections (NMRP) method for feature extraction and recognition. The experiments on FERET and extended Yale B face databases show that NMR can be used to build the graph while NMRP is an effective feature extraction method.

Nonconvex relaxation based matrix regression for face recognition with structural noise and mixed noise

For face recognition, sparse coding based classification methods have demonstrated the inspiring attraction in dealing with the pixel-level based sparse or gaussian noise, and nuclear norm based matrix regression (NMR) methods have shown the robustness in handling with the image-level based structural noise (e.g., occlusions, illuminations). Such regression methods have two limitations: one is that both ignore the label information of training samples and the similarity relationship between the training samples and the testing ones, the other is that nuclear norm as the rank relaxation can make the obtained solution deviate from the original matrix since its over-relaxations to the larger singular values. To overcome both limitations, this paper presents nonconvex relaxation based matrix regression (NRMR) methods: one is called locality-constrained group sparsity regularized nonconvex matrix gamma norm regression model for the structural noise (NRMRS), and the other extends it to the locality-constrained group sparsity regularized matrix gamma and minimax concave plus (MCP) regression model for the mixed noise (NRMRM), i.e., structural noise plus sparse noise. Two variants of inexact augmented Lagrange multiplier (IALM) algorithms including nonconvex IALM (NC-IALM) and majorization–minization IALM (MM-IALM) are devised to solve the proposed models, respectively. Finally, experiments on Extended Yale B, AR and CUHKFS databases can show the superiority of our methods to face recognition with structural noise and mixed noise.

Robust Nuclear Norm-Based Matrix Regression With Applications to Robust Face Recognition.

Face recognition (FR) via regression analysis-based classification has been widely studied in the past several years. Most existing regression analysis methods characterize the pixelwise representation error via l1 -norm or l2 -norm, which overlook the 2D structure of the error image. Recently, the nuclear norm-based matrix regression model is proposed to characterize low-rank structure of the error image. However, the nuclear norm cannot accurately describe the low-rank structural noise when the incoherence assumptions on the singular values does not hold, since it overpenalizes several much larger singular values. To address this problem, this paper presents the robust nuclear norm to characterize the structural error image and then extends it to deal with the mixed noise. The majorization-minimization (MM) method is applied to derive a iterative scheme for minimization of the robust nuclear norm optimization problem. Then, an efficiently alternating direction method of multipliers (ADMM) method is used to solve the proposed models. We use weighted nuclear norm as classification criterion to obtain the final recognition results. Experiments on several public face databases demonstrate the effectiveness of our models in handling with variations of structural noise (occlusion, illumination, and so on) and mixed noise.

Tree-Structured Nuclear Norm Approximation with Applications to Robust Face Recognition.

Structured sparsity, as an extension of standard sparsity, has shown the outstanding performance when dealing with some highly correlated variables in computer vision and pattern recognition. However, the traditional mixed (L1, L2) or (L1, L∞) group norm becomes weak in characterizing the internal structure of each group since they cannot alleviate the correla-tions between variables. Recently, nuclear norm has been vali-dated to be useful for depicting a spatially structured matrix variable. It considers the global structure of the matrix variable but overlooks the local structure. To combine the advantages of structured sparsity and nuclear norm, this paper presents a tree-structured nuclear norm approximation (TSNA) model as-suming that the representation residual with tree-structured prior is a random matrix variable and follows a dependent matrix dis-tribution. The Extended Alternating Direction Method of Multi-pliers (EADMM) is utilized to solve the proposed model. An effi-cient bound condition based on the extended restricted isometry constants is provided to show the exact recovery of the proposed model under the given noisy case. In addition, TSNA is connected with some newest methods such as sparse representation based classifier (SRC), nuclear-L1 norm joint regression (NL1R) and nuclear norm based matrix regression (NMR), which can be re-garded as the special cases of TSNA. Experiments with face re-construction and recognition demonstrate the benefits of TSNA over other approaches.

Robust Face Recognition via Multi-Scale Patch-Based Matrix Regression.

In many real-world applications such as smart card solutions, law enforcement, surveillance and access control, the limited training sample size is the most fundamental problem. By making use of the low-rank structural information of the reconstructed error image, the so-called nuclear norm-based matrix regression has been demonstrated to be effective for robust face recognition with continuous occlusions. However, the recognition performance of nuclear norm-based matrix regression degrades greatly in the face of the small sample size problem. An alternative solution to tackle this problem is performing matrix regression on each patch and then integrating the outputs from all patches. However, it is difficult to set an optimal patch size across different databases. To fully utilize the complementary information from different patch scales for the final decision, we propose a multi-scale patch-based matrix regression scheme based on which the ensemble of multi-scale outputs can be achieved optimally. Extensive experiments on benchmark face databases validate the effectiveness and robustness of our method, which outperforms several state-of-the-art patch-based face recognition algorithms.

Nuclear Norm Based Matrix Regression with Applications to Face Recognition with Occlusion and Illumination Changes.

Recently, regression analysis has become a popular tool for face recognition. Most existing regression methods use the one-dimensional, pixel-based error model, which characterizes the representation error individually, pixel by pixel, and thus neglects the two-dimensional structure of the error image. We observe that occlusion and illumination changes generally lead, approximately, to a low-rank error image. In order to make use of this low-rank structural information, this paper presents a two-dimensional image-matrix-based error model, namely, nuclear norm based matrix regression (NMR), for face representation and classification. NMR uses the minimal nuclear norm of representation error image as a criterion, and the alternating direction method of multipliers (ADMM) to calculate the regression coefficients. We further develop a fast ADMM algorithm to solve the approximate NMR model and show it has a quadratic rate of convergence. We experiment using five popular face image databases: the Extended Yale B, AR, EURECOM, Multi-PIE and FRGC. Experimental results demonstrate the performance advantage of NMR over the state-of-the-art regression-based methods for face recognition in the presence of occlusion and illumination variations.

Nuclear-L1 norm joint regression for face reconstruction and recognition with mixed noise

Occlusion, real disguise and illumination are still the common difficulties encountered in face recognition. The sparse representation based classifier (SRC) has shown a great potential in handling pixel-level sparse noise, while the nuclear norm based matrix regression (NMR) model has been demonstrated to be powerful for dealing with the image-wise structural noise. Both methods, however, might be not very effective for handling the mixed noise: the structural noise plus the sparse noise. In this paper, we present two nuclear-L1 norm joint matrix regression (NL1R) models for face recognition with mixed noise, which are derived by using MAP (maximum a posteriori probability estimation). The first model considers the mixed noise as a whole, while the second model assumes the mixed noise is an additive combination of two independent componenral nts: sparse noise and structuoise. The proposed models can be solved by the alternating direction method of multipliers (ADMM). We validate the effectiveness of the proposed models through a series of experiments on face reconstruction and recognition.