Low-Rank Matrix Factorization With Adaptive Graph Regularizer.

Multi-view low-rank matrix factorization using multiple manifold regularization

Data missing remains a difficult and important problem in the transportation information system, which seriously restricts the application of the intelligent transportation system (ITS), dominatingly on traffic monitoring, e.g., traffic data collection, traffic state estimation, and traffic control. Numerous traffic data imputation methods had been proposed in the last decade. However, lacking of sufficient temporal variation characteristic analysis as well as spatial correlation measurements leads to limited completion precision, and poses a major challenge for an ITS. Leveraging the low-rank nature and the spatial-temporal correlation of traffic network data, this paper proposes a novel approach to reconstruct the missing traffic data based on low-rank matrix factorization, which elaborates the potential implications of the traffic matrix by decomposed factor matrices. To further exploit the temporal evolvement characteristics and the spatial similarity of road links, we design a time-series constraint and an adaptive Laplacian regularization spatial constraint to explore the local relationship with road links. The experimental results on six real-world traffic data sets show that our approach outperforms the other methods and can successfully reconstruct the road traffic data precisely for various structural loss modes.

Low-rank matrix factorizations such as Principal Component Analysis (PCA), Singular Value Decomposition (SVD) and Non-negative Matrix Factorization (NMF) are a large class of methods for pursuing the low-rank approximation of a given data matrix. The conventional factorization models are based on the assumption that the data matrices are contaminated stochastically by some type of noise. Thus the point estimations of low-rank components can be obtained by Maximum Likelihood (ML) estimation or Maximum a posteriori (MAP). In the past decade, a variety of probabilistic models of low-rank matrix factorizations have emerged. The most significant difference between low-rank matrix factorizations and their corresponding probabilistic models is that the latter treat the low-rank components as random variables. This paper makes a survey of the probabilistic models of low-rank matrix factorizations. Firstly, we review some probability distributions commonly-used in probabilistic models of low-rank matrix factorizations and introduce the conjugate priors of some probability distributions to simplify the Bayesian inference. Then we provide two main inference methods for probabilistic low-rank matrix factorizations, i.e., Gibbs sampling and variational Bayesian inference. Next, we classify roughly the important probabilistic models of low-rank matrix factorizations into several categories and review them respectively. The categories are performed via different matrix factorizations formulations, which mainly include PCA, matrix factorizations, robust PCA, NMF and tensor factorizations. Finally, we discuss the research issues needed to be studied in the future.

With the rapid growth of scientific publications, it is hard for researchers to acquire appropriate papers that meet their expectations. Recommendation system for scientific articles is an essential technology to overcome this problem. In this paper, we propose a novel low-rank and sparse matrix factorization-based paper recommendation (LSMFPRec) method for authors. The proposed method seamlessly combines low-rank and sparse matrix factorization method with fine-grained paper and author affinity matrixes that are extracted from heterogeneous scientific network. Thus, it can effectively alleviate the sparsity and cold start problems that exist in traditional matrix factorization based collaborative filtering methods. Moreover, LSMFPRec can significantly reduce the error propagated from intermediate outputs. In addition, the proposed method essentially captures the low-rank and sparse characteristics that exist in scientific rating activities; therefore, it can generate more reasonable predicted ratings for influential and uninfluential papers. The effectiveness of the proposed LSMFPRec is demonstrated by the recommendation evaluation conducted on the AAN and CiteULike data sets.

Many computer vision problems can be posed as learning a low-dimensional subspace from high dimensional data. The low rank matrix factorization (LRMF) represents a commonly utilized subspace learning strategy. Most of the current LRMF techniques are constructed on the optimization problem using L_1 norm and L_2 norm, which mainly deal with Laplacian and Gaussian noise, respectively. To make LRMF capable of adapting more complex noise, this paper proposes a new LRMF model by assuming noise as Mixture of Exponential Power (MoEP) distributions and proposes a penalized MoEP model by combining the penalized likelihood method with MoEP distributions. Such setting facilitates the learned LRMF model capable of automatically fitting the real noise through MoEP distributions. Each component in this mixture is adapted from a series of preliminary super-or sub-Gaussian candidates. An Expectation Maximization (EM) algorithm is also designed to infer the parameters involved in the proposed PMoEP model. The advantage of our method is demonstrated by extensive experiments on synthetic data, face modeling and hyperspectral image restoration.

Matrix completion refers to recovering a matrix from a small subset of its entries. It is an important topic because numerous real-world data can be modeled as low-rank matrices. One popular approach for matrix completion is based on low-rank matrix factorization, but it requires knowing the matrix rank, which is difficult to accurately determine in many practical scenarios. We propose a novel algorithm based on rank-one approximation that a matrix can be decomposed as a sum of outer products. The key idea is to find the basis vectors of the underlying matrix according to the observed entries, and gradually increase the vector number until an appropriate rank estimate is reached. In contrast to the conventional rank-one schemes that employ unchanging rank-one basis matrices, our algorithm performs completion from the vector viewpoint and is able to generate continuously updated rank-one basis matrices. Besides, we theoretically show that the developed method has a linear convergence rate and a smaller recovery error than existing rank-one based algorithms. Experimental results using both synthetic data and real-world images demonstrate that our solution has the best recovery performance among the competing algorithms when the observations are contaminated by Gaussian noise.

The explosive growth of data has caused users to spend considerable time and effort finding the items they need. Various recommender systems have been created to provide convenience for users. This paper proposes a low-rank and sparse matrix factorization with prior relations (LSMF-PR) recommendation model, which predicts users’ ratings for items through a sum of the learned low-rank matrix and sparse matrix. Thus, unlike traditional matrix factorization approaches, our method can alleviate the error propagation produced by intermediate outputs. The LSMF-PR integrates user relationships and item relationships as prior information. User relationships in different recommendation scenarios are extracted by the corresponding social relations of the users, and item relationships are obtained from the similarity of the item content. Therefore, the sparsity and cold start problems can be effectively reduced with prior information. Furthermore, our model has better interpretability since it reveals the low-rank and sparse features of the ratings. Experiments are conducted on four real-world datasets to validate the performance of our proposed method.

<abstract><p>Trace norm regularization is a widely used approach for learning low rank matrices. A standard optimization strategy is based on formulating the problem as one of low rank matrix factorization which, however, leads to a non-convex problem. In practice this approach works well, and it is often computationally faster than standard convex solvers such as proximal gradient methods. Nevertheless, it is not guaranteed to converge to a global optimum, and the optimization can be trapped at poor stationary points. In this paper we show that it is possible to characterize all critical points of the non-convex problem. This allows us to provide an efficient criterion to determine whether a critical point is also a global minimizer. Our analysis suggests an iterative meta-algorithm that dynamically expands the parameter space and allows the optimization to escape any non-global critical point, thereby converging to a global minimizer. The algorithm can be applied to problems such as matrix completion or multitask learning, and our analysis holds for any random initialization of the factor matrices. Finally, we confirm the good performance of the algorithm on synthetic and real datasets.</p></abstract>

The total variation (TV) regularization method has been used to solve the ill-posed inverse problem of electrical resistance tomography (ERT), owing to its good ability to preserve edges. However, the quality of the reconstructed images, especially in the flat region, is often degraded by noise. To optimize the regularization term and the regularization factor according to the spatial feature and to improve the resolution of reconstructed images, a spatially adaptive total variation (SATV) regularization method is proposed. A kind of effective spatial feature indicator named difference curvature is used to identify which region is a flat or edge region. According to different spatial features, the SATV regularization method can automatically adjust both the regularization term and regularization factor. At edge regions, the regularization term is approximate to the TV functional to preserve the edges; in flat regions, it is approximate to the first-order Tikhonov (FOT) functional to make the solution stable. Meanwhile, the adaptive regularization factor determined by the spatial feature is used to constrain the regularization strength of the SATV regularization method for different regions. Besides, a numerical scheme is adopted for the implementation of the second derivatives of difference curvature to improve the numerical stability. Several reconstruction image metrics are used to quantitatively evaluate the performance of the reconstructed results. Both simulation and experimental results indicate that, compared with the TV (mean relative error 0.288, mean correlation coefficient 0.627) and FOT (mean relative error 0.295, mean correlation coefficient 0.638) regularization methods, the proposed SATV (mean relative error 0.259, mean correlation coefficient 0.738) regularization method can endure a relatively high level of noise and improve the resolution of reconstructed images.

We here adapt an extended version of the adaptive cubic regularization method with dynamic inexact Hessian information for nonconvex optimization in Bellavia et al. [Adaptive cubic regularization methods with dynamic inexact hessian information and applications to finite-sum minimization. IMA Journal of Numerical Analysis. 2021;41(1):764–799] to the stochastic optimization setting. While exact function evaluations are still considered, this novel variant inherits the innovative use of adaptive accuracy requirements for Hessian approximations introduced in the just quoted paper and additionally employs inexact computations of the gradient. Without restrictions on the variance of the errors, we assume that these approximations are available within a sufficiently large, but fixed, probability and we extend, in the spirit of Cartis and Scheinberg [Global convergence rate analysis of unconstrained optimization methods based on probabilistic models. Math Program Ser A. 2018;159(2):337–375], the deterministic analysis of the framework to its stochastic counterpart, showing that the expected number of iterations to reach a first-order stationary point matches the well-known worst-case optimal complexity. This is, in fact, still given by , with respect to the first-order ϵ tolerance. Finally, numerical tests on nonconvex finite-sum minimization confirm that using inexact first- and second-order derivatives can be beneficial in terms of the computational savings.

In this paper, a novel multiple kernel learning (MKL) algorithm is proposed for the classification of hyperspectral images. The proposed MKL algorithm adopts a two-step strategy to learn a multiple kernel machine. In the first step, unsupervised learning is carried out to learn a combined kernel from the predefined base kernels. In our algorithms, low-rank nonnegative matrix factorization (NMF) is used to carry out the unsupervised learning and learn an optimal combined kernel. Furthermore, the kernel NMF (KNMF) is introduced to substitute NMF for enhancing the ability of the unsupervised learning with the predefined base kernels. In the second step, the optimal kernel is embedded into the standard optimization routine of support vector machine (SVM). In addition, we address a major challenge in hyperspectral data classification, i.e., using very few labeled samples in a high-dimensional space. Experiments are conducted on three real hyperspectral datasets, and the experimental results show that the proposed algorithms, especially for KNMF-based MKL, achieve the outstanding performance for hyperspectral image classification with few labeled samples when compared with several state-of-the-art algorithms.

Adaptive regularizations in an iterative regularized pseudoinverse method

In many domains such as digital photography, remote sensing, and medical imaging, improving image clarity is essential. It is frequently difficult for traditional methods to strike a balance between preserving significant image details and reducing noise. This work presents a novel method for improving images by utilizing adaptive regularization techniques. In order to effectively reduce noise and preserve fine details, the suggested method dynamically modifies the regularization parameters based on local image characteristics. This study offers a novel method for improving image clarity through adaptive regularization. Our approach produces better noise reduction while maintaining important details, especially edges, by dynamically modifying the regularization parameter based on image properties. By contrasting our method with conventional regularization techniques, experimental results demonstrate effectiveness of adaptive regularization method.

We propose an adaptive regularization scheme in a variational framework where a convex composite energy functional is optimized. We consider a number of imaging problems including denoising, segmentation and motion estimation, which are considered as optimal solutions of the energy functionals that mainly consist of data fidelity, regularization and a control parameter for their trade-off. We presents an algorithm to determine the relative weight between data fidelity and regularization based on the residual that measures how well the observation fits the model. Our adaptive regularization scheme is designed to locally control the regularization at each pixel based on the assumption that the diversity of the residual of a given imaging model spatially varies. The energy optimization is presented in the alternating direction method of multipliers (ADMM) framework where the adaptive regularization is iteratively applied along with mathematical analysis of the proposed algorithm. We demonstrate the robustness and effectiveness of our adaptive regularization through experimental results presenting that the qualitative and quantitative evaluation results of each imaging task are superior to the results with a constant regularization scheme. The desired properties, robustness and effectiveness, of the regularization parameter selection in a variational framework for imaging problems are achieved by merely replacing the static regularization parameter with our adaptive one.

This article is an extended version of previous work of Lesieur et al (2015 IEEE Int. Symp. on Information Theory Proc. pp 1635–9 and 2015 53rd Annual Allerton Conf. on Communication, Control and Computing (IEEE) pp 680–7) on low-rank matrix estimation in the presence of constraints on the factors into which the matrix is factorized. Low-rank matrix factorization is one of the basic methods used in data analysis for unsupervised learning of relevant features and other types of dimensionality reduction. We present a framework to study the constrained low-rank matrix estimation for a general prior on the factors, and a general output channel through which the matrix is observed. We draw a parallel with the study of vector-spin glass models—presenting a unifying way to study a number of problems considered previously in separate statistical physics works. We present a number of applications for the problem in data analysis. We derive in detail a general form of the low-rank approximate message passing (Low-RAMP) algorithm, that is known in statistical physics as the TAP equations. We thus unify the derivation of the TAP equations for models as different as the Sherrington–Kirkpatrick model, the restricted Boltzmann machine, the Hopfield model or vector (xy, Heisenberg and other) spin glasses. The state evolution of the Low-RAMP algorithm is also derived, and is equivalent to the replica symmetric solution for the large class of vector-spin glass models. In the section devoted to result we study in detail phase diagrams and phase transitions for the Bayes-optimal inference in low-rank matrix estimation. We present a typology of phase transitions and their relation to performance of algorithms such as the Low-RAMP or commonly used spectral methods.