THE LOGARITHMIC SOBOLEV INEQUALITY FOR A SUBMANIFOLD IN MANIFOLDS WITH ASYMPTOTICALLY NONNEGATIVE SECTIONAL CURVATURE

In this note,we prove a logarithmic Sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional curvature.Like the Michale-Simon Sobolev inequality,this inequality contains a term involving the mean curvature.

Proximal Alternating-Direction-Method-of-Multipliers-Incorporated Nonnegative Latent Factor Analysis

High-dimensional and incomplete(HDI)data subject to the nonnegativity constraints are commonly encountered in a big data-related application concerning the interactions among numerous nodes.A nonnegative latent factor analysis(NLFA)model can perform representation learning to HDI data efficiently.However,existing NLFA models suffer from either slow convergence rate or representation accuracy loss.To address this issue,this paper proposes a proximal alternating-directionmethod-of-multipliers-based nonnegative latent factor analysis(PAN)model with two-fold ideas:(1)adopting the principle of alternating-direction-method-of-multipliers to implement an efficient learning scheme for fast convergence and high computational efficiency;and(2)incorporating the proximal regularization into the learning scheme to suppress the optimization fluctuation for high representation learning accuracy to HDI data.Theoretical studies verify that PAN converges to a Karush-KuhnTucker(KKT)stationary point of its nonnegativity-constrained learning objective with its learning scheme.Experimental results on eight HDI matrices from real applications demonstrate that the proposed PAN model outperforms several state-of-the-art models in both estimation accuracy for missing data of an HDI matrix and computational efficiency.

Vertex centrality of complex networks based on joint nonnegative matrix factorization and graph embedding

Finding crucial vertices is a key problem for improving the reliability and ensuring the effective operation of networks,solved by approaches based on multiple attribute decision that suffer from ignoring the correlation among each attribute or the heterogeneity between attribute and structure. To overcome these problems, a novel vertex centrality approach, called VCJG, is proposed based on joint nonnegative matrix factorization and graph embedding. The potential attributes with linearly independent and the structure information are captured automatically in light of nonnegative matrix factorization for factorizing the weighted adjacent matrix and the structure matrix, which is generated by graph embedding. And the smoothness strategy is applied to eliminate the heterogeneity between attributes and structure by joint nonnegative matrix factorization. Then VCJG integrates the above steps to formulate an overall objective function, and obtain the ultimately potential attributes fused the structure information of network through optimizing the objective function. Finally, the attributes are combined with neighborhood rules to evaluate vertex's importance. Through comparative analyses with experiments on nine real-world networks, we demonstrate that the proposed approach outperforms nine state-of-the-art algorithms for identification of vital vertices with respect to correlation, monotonicity and accuracy of top-10 vertices ranking.

基于Nonnegative Garrote的ARX和ARMA模型定阶方法

针对传统ARX和ARMA模型定阶方法计算量大和稳定性不强的缺点,提出采用非负绞杀法对ARX和ARMA模型进行定阶。通过分析ARX和ARMA模型的特点,对常规非负绞杀方法进行了改进,使其更适用于动态系统辨识问题,进而给出了相应的求解算法。仿真实验的结果表明了该方法的有效性,且在稳定性上优于传统的信息量准则法。

NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF INHOMOGENEOUS p-LAPLACE EQUATION

Let Ω be a smooth bounded domain in R^n. In this article, we consider the homogeneous boundary Dirichlet problem of inhomogeneous p-Laplace equation --△pu = ｜u｜^q-1 u ＋ λf（x） on Ω, and identify necessary and sufficient conditions on Ω and f（x） which ensure the existence, or multiplicities of nonnegative solutions for the problem under consideration.

Bispectrum Feature Extraction of Gearbox Faults Based on Nonnegative Tucker3 Decomposition with 3D Calculations

Nonnegative Tucker3 decomposition（NTD） has attracted lots of attentions for its good performance in 3D data array analysis. However, further research is still necessary to solve the problems of overfitting and slow convergence under the anharmonic vibration circumstance occurred in the field of mechanical fault diagnosis. To decompose a large-scale tensor and extract available bispectrum feature, a method of conjugating Choi-Williams kernel function with Gauss-Newton Cartesian product based on nonnegative Tucker3 decomposition（NTD_EDF） is investigated. The complexity of the proposed method is reduced from o（nNlgn） in 3D spaces to o（RiR2nlgn） in 1D vectors due to its low rank form of the Tucker-product convolution. Meanwhile, a simultaneously updating algorithm is given to overcome the overfitting, slow convergence and low efficiency existing in the conventional one-by-one updating algorithm. Furthermore, the technique of spectral phase analysis for quadratic coupling estimation is used to explain the feature spectrum extracted from the gearbox fault data by the proposed method in detail. The simulated and experimental results show that the sparser and more inerratic feature distribution of basis images can be obtained with core tensor by the NTD EDF method compared with the one by the other methods in bispectrum feature extraction, and a legible fault expression can also be performed by power spectral density（PSD） function. Besides, the deviations of successive relative error（DSRE） of NTD_EDF achieves 81.66 dB against 15.17 dB by beta-divergences based on NTD（NTD_Beta） and the time-cost of NTD EDF is only 129.3 s, which is far less than 1 747.9 s by hierarchical alternative least square based on NTD （NTD_HALS）. The NTD_EDF method proposed not only avoids the data overfitting and improves the computation efficiency but also can be used to extract more inerratic and sparser bispectrum features of the gearbox fault.

STRUCTURE OF NONNEGATIVE NONTRIVIAL AND POSITIVE SOLUTIONS OF SINGULARLY PERTURBED p-LAPLACE EQUATIONS

Structure of nonnegative nontrivial and positive solutions was precisely studied for some singularly perturbed p-Laplace equations. By virtue of sub- and supersolution method, it is shown that there are many nonnegative nontrivial spike-layer solutions and positive intermediate spike-layer solutions. Moreover, the upper and lower bound on the measure of each spike-layer were estimated when the parameter is sufficiently small.

Estimation for Nonnegative First-Order Autoregressive Processes with an Unknown Location Parameter

Consider a first-order autoregressive processes , where the innovations are nonnegative random variables with regular variation at both the right endpoint infinity and the unknown left endpoint θ. We propose estimates for the autocorrelation parameter f and the unknown location parameter θ by taking the ratio of two sample values chosen with respect to an extreme value criteria for f and by taking the minimum of over the observed series, where represents our estimate for f. The joint limit distribution of the proposed estimators is derived using point process techniques. A simulation study is provided to examine the small sample size behavior of these estimates.

Open Manifolds with Nonnegative Ricci Curvature and Large Volume Growth

In this paper, we prove that if M is an open manifold with nonnegativeRicci curvature and large volume growth, positive critical radius, then sup Cp = ∞.As an application, we give a theorem which supports strongly Petersen's conjecture.

Nonnegative Solutions for a Riemann-Liouville Fractional Boundary Value Problem

We investigate the existence of nonnegative solutions for a Riemann-Liouville fractional differential equation with integral terms, subject to boundary conditions which contain fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main results, we use the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators.

Area-Correlated Spectral Unmixing Based on Bayesian Nonnegative Matrix Factorization

To solve the problem of the spatial correlation for adjacent areas in traditional spectral unmixing methods, we propose an area-correlated spectral unmixing method based on Bayesian nonnegative matrix factorization. In the proposed me-thod, the spatial correlation property between two adjacent areas is expressed by a priori probability density function, and the endmembers extracted from one of the adjacent areas are used to estimate the priori probability density func-tions of the endmembers in the current area, which works as a type of constraint in the iterative spectral unmixing process. Experimental results demonstrate the effectivity and efficiency of the proposed method both for synthetic and real hyperspectral images, and it can provide a useful tool for spatial correlation and comparation analysis between ad-jacent or similar areas.

Orthogonal nonnegative matrix factorization based local hidden Markov model for multimode process monitoring

Traditional data driven fault detection methods assume that the process operates in a single mode so that they cannot perform well in processes with multiple operating modes. To monitor multimode processes effectively,this paper proposes a novel process monitoring scheme based on orthogonal nonnegative matrix factorization(ONMF) and hidden Markov model(HMM). The new clustering technique ONMF is employed to separate data from different process modes. The multiple HMMs for various operating modes lead to higher modeling accuracy.The proposed approach does not presume the distribution of data in each mode because the process uncertainty and dynamics can be well interpreted through the hidden Markov estimation. The HMM-based monitoring indication named negative log likelihood probability is utilized for fault detection. In order to assess the proposed monitoring strategy, a numerical example and the Tennessee Eastman process are used. The results demonstrate that this method provides efficient fault detection performance.

Unsupervised hyperspectral unmixing based on robust nonnegative dictionary learning

Considering the sparsity of hyperspectral images(HSIs),dictionary learning frameworks have been widely used in the field of unsupervised spectral unmixing.However,it is worth mentioning here that existing dictionary learning method-based unmixing methods are found to be short of robustness in noisy contexts.To improve the performance,this study specifically puts forward a new unsupervised spectral unmixing solution.For the reason that the solution only functions in a condition that both endmembers and the abundances meet non-negative con-straints,a model is built to solve the unsupervised spectral un-mixing problem on the account of the dictionary learning me-thod.To raise the screening accuracy of final members,a new form of the target function is introduced into dictionary learning practice,which is conducive to the growing robustness of noisy HSI statistics.Then,by introducing the total variation(TV)terms into the proposed spectral unmixing based on robust nonnega-tive dictionary learning(RNDLSU),the context information under HSI space is to be cited as prior knowledge to compute the abundances when performing sparse unmixing operations.Ac-cording to the final results of the experiment,this method makes favorable performance under varying noise conditions,which is especially true under low signal to noise conditions.

Extracting Sub-Networks from Brain Functional Network Using Graph Regularized Nonnegative Matrix Factorization

Currently,functional connectomes constructed from neuroimaging data have emerged as a powerful tool in identifying brain disorders.If one brain disease just manifests as some cognitive dysfunction,it means that the disease may affect some local connectivity in the brain functional network.That is,there are functional abnormalities in the sub-network.Therefore,it is crucial to accurately identify them in pathological diagnosis.To solve these problems,we proposed a sub-network extraction method based on graph regularization nonnegative matrix factorization(GNMF).The dynamic functional networks of normal subjects and early mild cognitive impairment(eMCI)subjects were vectorized and the functional connection vectors(FCV)were assembled to aggregation matrices.Then GNMF was applied to factorize the aggregation matrix to get the base matrix,in which the column vectors were restored to a common sub-network and a distinctive sub-network,and visualization and statistical analysis were conducted on the two sub-networks,respectively.Experimental results demonstrated that,compared with other matrix factorization methods,the proposed method can more obviously reflect the similarity between the common subnetwork of eMCI subjects and normal subjects,as well as the difference between the distinctive sub-network of eMCI subjects and normal subjects,Therefore,the high-dimensional features in brain functional networks can be best represented locally in the lowdimensional space,which provides a new idea for studying brain functional connectomes.

Image Fusion Based on Complex Contourlet Transform and Nonnegative Matrix Factorization

An image fusion method combining complex contourlet transform(CCT) with nonnegative matrix factorization(NMF) is proposed in this paper.After two images are decomposed by CCT,NMF is applied to their highand low-frequency components,respectively,and finally an image is synthesized.Subjective-visual-quality of the image fusion result is compared with those of the image fusion methods based on NMF and the combination of wavelet /contourlet /nonsubsampled contourlet with NMF.The experimental results are evaluated quantitatively,and the running time is also contrasted.It is shown that the proposed image fusion method can gain larger information entropy,standard deviation and mean gradient,which means that it can better integrate featured information from all source images,avoid background noise and promote space clearness in the fusion image effectively.

On Open Problems of Nonnegative Inverse Eigenvalues Problem

In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.

A Note on Gap Phenomena of K＆#168;ahler Manifolds with Nonnegative Curvature

In this paper, we study the complex structure and curvature decay of Khler manifolds with nonnegative curvature. Using a recent result obtained by Ni-Shi-Tam, we get a gap theorem of Ricci curvature on Khler manifold.

Complete Open Manifolds with Nonnegative Ricci Curvature

In this paper, we study complete open manifolds with nonnegative Ricci curvature and injectivity radius bounded from below. We find that this kind of manifolds are diffeomorphic to a Euclidean space when certain distance functions satisfy a reasonable condition.

Nonnegative Matrix Factorization with Zellner Penalty

Nonnegative matrix factorization (NMF) is a relatively new unsupervised learning algorithm that decomposes a nonnegative data matrix into a parts-based, lower dimensional, linear representation of the data. NMF has applications in image processing, text mining, recommendation systems and a variety of other fields. Since its inception, the NMF algorithm has been modified and explored by numerous authors. One such modification involves the addition of auxiliary constraints to the objective function of the factorization. The purpose of these auxiliary constraints is to impose task-specific penalties or restrictions on the objective function. Though many auxiliary constraints have been studied, none have made use of data-dependent penalties. In this paper, we propose Zellner nonnegative matrix factorization (ZNMF), which uses data-dependent auxiliary constraints. We assess the facial recognition performance of the ZNMF algorithm and several other well-known constrained NMF algorithms using the Cambridge ORL database.