Nonnegativity Property Research Articles

Computational aspects of simultaneous Gaussian quadrature

AbstractIn this paper, we derive a new method to compute the nodes and weights of simultaneous n-point Gaussian quadrature rules. The method is based on the eigendecomposition of the banded lower Hessenberg matrix that contains the coefficients of the recurrence relations for the corresponding multiple orthogonal polynomials. The novelty of the approach is that it uses the property of total nonnegativity of this matrix associated with the particular considered multiple orthogonal polynomials, in order to compute its eigenvalues and eigenvectors in a numerically stable manner. The overall complexity of the computation of all the nodes and weights is $$\mathcal {O}(n^2)$$ O ( n 2 ) .

Second-order accurate and unconditionally stable algorithm with unique solvability for a phase-field model of 3D volume reconstruction

Three-dimensional (3D) volume reconstruction is an important technique in the fields of 3D printing, artificial limb, and medical diagnostic. This work aims to develop an unconditionally stable algorithm with desired accuracy and unique solvability for a phase-field model of 3D volume reconstruction. Based on scattered points of a target object, a smooth narrow volume is reconstructed by solving a Allen–Cahn-type equation with control function. Since the non-negative property of control function, the evolution of governing equation can associate with an energy dissipation law (i.e., energy stability). By modifying the nonlinear term in governing equation into an appropriate truncated form and utilizing a stabilized Crank–Nicolson-type discretization in time, we develop a linear, second-order accurate, and unconditionally energy-stable time-marching scheme to update the solution. The space is discretized with the finite difference method and the fully discrete energy stability is analytically estimated. We also prove the unique solvability of our proposed time-marching scheme. The accuracy, stability, and capability of reconstructing various 3D volumes are verified via extensive numerical simulations. To facilitate the interested readers, we provide the open source codes of 3D volume reconstruction of a teapot on corresponding author's personal website: https://cfdyang521.github.io.

Randomized low rank approximation for nonnegative pure quaternion matrices

Pure quaternion matrix has been widely employed to represent data in real-world applications, such as colour images. Additionally, the imaginary part of quaternion matrix is usually nonnegative due to the natural nonnegativity of real-world data. In this paper, we propose an alternating projection-based algorithm for low rank nonnegative pure quaternion matrix approximation, which can exactly calculate the optimal fixed rank approximation while preserve the pure and nonnegative properties from the given data. More concretely, the proposed algorithm alternatively projects the given quaternion matrix onto the fixed rank quaternion matrix set and nonnegative pure quaternion matrix set in an iterative fashion. Moreover, we establish the theoretical convergence guarantee of the proposed algorithm. To extend the proposed algorithm to large-scaled data, we further propose a randomized algorithm with significant lower computational complexity and comparable accuracy. Numerical experiments on colour images show that our algorithms outperform the other state-of-the-art algorithms.

Graph-Based Soft-Balanced Fuzzy Clustering

Spectral clustering have attracted more and more attention due to their well-defined mathematical frameworks and superior performance. However, there still exist two limitations to be solved: 1) most spectral clustering methods consist of two independent stages, which may cause unpredictable deviation of obtained clustering results from the genuine ones and lead to severe information loss and performance degradation; 2) spectral clustering methods employ the hard clustering mode, which lacks interpretability for data points in the boundary area belonging to multiple clusters. To simultaneously address these challenging issues for spectral clustering, we propose a Graph based soft-Balanced Fuzzy Clustering (GBFC) model. Specifically, we explicitly preserve the nonnegative property of the clustering indicator matrix to enhance the interpretability of clustering results. Moreover, row normalization is imposed on the cluster indicator matrix to show the membership of each data point to different clusters. Additionally, a novel balanced constraint is designed to regularize the clustering results and constrain the size of clusters. We can directly obtain the clustering assignments without any post-processing, and the limitations of the previous two-stage clustering framework can be effectively addressed. Extensive experiments performed on both synthetic datasets and real world datasets demonstrate the superiority and effectiveness of the proposed algorithm compared with several state-of-the-art methods.

A new complex belief entropy of [formula omitted] divergence with its application in cardiac interbeat interval time series analysis

With outstanding capability to express uncertain information, complex evidence theory can be widely applied in a multiplicity of fields of representation and fusion of information. However, in complex evidence theory, conflict measurement is still an open issue. To address this problem, we propose a new complex belief χ2 divergence measure (CBχ2), which has the ability to measure the discrepancy between the complex basic belief assignments (CBBAs). In addition, the proposed CBχ2 divergence measure satisfies the properties of nonnegativity, nondegeneracy and symmetry. Particularly, when the CBBA degenerates to classical basic belief assignments (BBA), the proposed divergence can also measure the conflict well. On this basis, a novel BEoCBχ2D is proposed to measure the complexity of cardiac interbeat interval time series in biological system. An application in cardiac interbeat interval time series analysis demonstrates that BEoCBχ2D has better accuracy.

Hellinger Information Matrix and Hellinger Priors

Hellinger information as a local characteristic of parametric distribution families was first introduced in 2011. It is related to the much older concept of the Hellinger distance between two points in a parametric set. Under certain regularity conditions, the local behavior of the Hellinger distance is closely connected to Fisher information and the geometry of Riemann manifolds. Nonregular distributions (non-differentiable distribution densities, undefined Fisher information or denisities with support depending on the parameter), including uniform, require using analogues or extensions of Fisher information. Hellinger information may serve to construct information inequalities of the Cramer-Rao type, extending the lower bounds of the Bayes risk to the nonregular case. A construction of non-informative priors based on Hellinger information was also suggested by the author in 2011. Hellinger priors extend the Jeffreys rule to nonregular cases. For many examples, they are identical or close to the reference priors or probability matching priors. Most of the paper was dedicated to the one-dimensional case, but the matrix definition of Hellinger information was also introduced for higher dimensions. Conditions of existence and the nonnegative definite property of Hellinger information matrix were not discussed. Hellinger information for the vector parameter was applied by Yin et al. to problems of optimal experimental design. A special class of parametric problems was considered, requiring the directional definition of Hellinger information, but not a full construction of Hellinger information matrix. In the present paper, a general definition, the existence and nonnegative definite property of Hellinger information matrix is considered for nonregular settings.

On the Support Recovery of Jointly Sparse Gaussian Sources via Sparse Bayesian Learning

In this work, we provide non-asymptotic, probabilistic guarantees for successful recovery of the common nonzero support of jointly sparse Gaussian sources in the multiple measurement vector (MMV) problem. The support recovery problem is formulated as the marginalized maximum likelihood (or type-II ML) estimation of the variance hyperparameters of a joint sparsity inducing Gaussian prior on the source signals. We derive conditions under which the resulting nonconvex constrained optimization perfectly recovers the nonzero support of a joint-sparse Gaussian source ensemble with arbitrarily high probability. The support error probability decays exponentially with the number of MMVs at a rate that depends on the smallest restricted singular value and the nonnegative null space property of the self Khatri-Rao product of the sensing matrix. Our analysis confirms that nonzero supports of size as high as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(m^{2})$ </tex-math></inline-formula> are recoverable from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> measurements per sparse vector. Our derived sufficient conditions for support consistency of the proposed constrained type-II ML solution also guarantee the support consistency of any global solution of the multiple sparse Bayesian learning (M-SBL) optimization whose nonzero coefficients lie inside a bounded interval. For the case of noiseless measurements, we further show that a single MMV is sufficient for perfect recovery of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -sparse support by M-SBL, provided all subsets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k + 1$ </tex-math></inline-formula> columns of the sensing matrix are linearly independent.

A novel entropy-based sparsity measure for prognosis of bearing defects and development of a sparsogram to select sensitive filtering band of an axial piston pump

This study aims to establish a novel entropy-based sparsity measure for two main purposes: first is for the prognosis of bearing defects, secondly it is employed to construct sparsogram for identifying sensitive filtering band to carry out envelope analysis for identifying defects in the complex hydraulic machinery (axial piston pump). The newly developed symmetrized directed divergence measure is first authenticated after satisfying the necessary conditions such as convexity, non-negativity and symmetric properties and then reformulated to generate another novel probabilistic entropy measure. The entropy measure so obtained has been mathematically proven to satisfy all essential conditions as laid down by Shannon such as permutationally symmetric, degeneracy, separability, continuity, concavity, maximum value, and non-negativity. Furthermore, the proclaimed probabilistic entropy measure is modified to become a novel sparsity measure that can satisfy all of the six intuitive sparse attributes. The sparsity measure so obtained is applied in two ways. First application is of defect degradation monitoring of the rolling element bearing. Here, LSTM network is used for the carrying out defect prognosis. The degradation monitoring has been carried out on the life time accelerating data of IMS and PRONOSTIA. Second application of the proposed sparsity measure is for the development of a sparsogram for selecting the suitable filtering band to carry out envelop analysis, needed to identify bearing defects in the axial piston pump. To authenticate the validity of the proposed sparsogram, a comparative study has also been done with the existing tool such as fast Kurtogram, spectral kurtosis and Protugram.

Geometric Nature of Relations on Plabic Graphs and Totally Non-negative Grassmannians

Abstract The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms of discrete path integration on planar bicoloured (plabic) graphs in the disc. An alternative parametrization was proposed by T. Lam [38] introducing systems of relations at the vertices of such graphs, depending on some signatures defined on their edges. The problem of characterizing the signatures corresponding to the totally non-negative cells was left open in [38]. In our paper we provide an explicit construction of such signatures, satisfying both the full rank condition and the total non-negativity property on the full positroid cell. If each edge in a graph $\mathcal G$ belongs to some oriented path from the boundary to the boundary, then such signature is unique up to a vertex gauge transformation. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation $\mathcal O$ and the gauge ray direction $\mathfrak l$ on $\mathcal G$. Moreover, we provide a combinatorial representation of the geometric signatures by showing that the total signature of every finite face just depends on the number of white vertices on it. The latter characterization is a Kasteleyn-type property in the case of bipartite graphs [1, 7], and has a different statistical mechanical interpretation otherwise [6]. An explicit connection between the solution of Lam’s system of relations and the value of Postnikov’s boundary measurement map is established using the generalization of Talaska’s formula [51] obtained in [6]. In particular, the components of the edge vectors are rational in the edge weights with subtraction-free denominators. Finally, we provide explicit formulas for the transformations of the signatures under Postnikov’s moves and reductions and amalgamations of networks.

Shrinking-shifting and amplifying-shifting device using transformation optics

Based on transformation optics (TO), this paper uses geometric divisions and linear coordinate transformations to design “shrinking-shifting—and reshaping”, and “amplifying-shifting—and reshaping” devices. The proposed devices can reshape the sizes and locations of the wrapped-objects inside the core-region. The shrinking-shifting device shrinks the larger object into a smaller one and shifts it to different location, whereas the shrinking-reshaping device can generate a smaller-size image with different shape located at different location. In contrast to previously designed shrinking devices, the real object wrapped inside the proposed core-region and the transformed object contains the same material properties, and the location-shifting is another feature. Here, the shifting-region is located inside the physical-space boundaries to achieve the non-negative, homogeneous, and anisotropic material properties of the proposed device, which are easier for real implementations. Thus, we further verified this concept with the amplifying-shifting and -reshaping devices for visually transformation of smaller object into bigger one placed at different location and position. We also applied active scatterer to further validate the working functionality of proposed devices. In addition, the proposed devices behave like the concentrator and (or) rotator effect in the absence of any scatterer. Our findings highlight the role of TO, suggesting directions for future research on bi-functional devices that will be useful for shrinking and amplifying devices, illusion optics, camouflage, and object protection etc.

Co-sparse Non-negative Matrix Factorization.

Non-negative matrix factorization, which decomposes the input non-negative matrix into product of two non-negative matrices, has been widely used in the neuroimaging field due to its flexible interpretability with non-negativity property. Nowadays, especially in the neuroimaging field, it is common to have at least thousands of voxels while the sample size is only hundreds. The non-negative matrix factorization encounters both computational and theoretical challenge with such high-dimensional data, i.e., there is no guarantee for a sparse and part-based representation of data. To this end, we introduce a co-sparse non-negative matrix factorization method to high-dimensional data by simultaneously imposing sparsity in both two decomposed matrices. Instead of adding some sparsity induced penalty such as l1 norm, the proposed method directly controls the number of non-zero elements, which can avoid the bias issues and thus yield more accurate results. We developed an alternative primal-dual active set algorithm to derive the co-sparse estimator in a computationally efficient way. The simulation studies showed that our method achieved better performance than the state-of-art methods in detecting the basis matrix and recovering signals, especially under the high-dimensional scenario. In empirical experiments with two neuroimaging data, the proposed method successfully detected difference between Alzheimer's patients and normal person in several brain regions, which suggests that our method may be a valuable toolbox for neuroimaging studies.

Spatiotemporal Dynamics of a Delayed Viral Infection Model with Two Modes of Transmission and Immune Impairment

In this paper, the dynamical behaviors for a diffusive virus infection model with two general nonlinear incidence functions, CTL immune impairment and discrete delays, are investigated. Cell‐to‐cell transmission plays a more important role in virus spreading in addition to virus‐to‐cell infection. Nonnegativity and boundcedness properties of the solutions are proven to show the well‐posedness of the model. Five reproduction numbers are calculated and verified to be thresholds determining the global properties of the virus model. A significant impact of cell‐to‐cell transmission is that they increase the basic reproduction number. If one neglects either the cell‐to‐cell transmission or virus‐to‐cell infection, the basic reproduction number of the model is underevaluated. Numerical simulations are carried out to illustrate the dynamical behavior of the model.

Structural approach to the control of a reaction system using thermodynamics-based extent model

This work proposes a novel control strategy to stabilize the decoupled dynamics of a homogeneous reaction system, described by vessel extents, using irreversible thermodynamics. Specifically, the non-negative property of the irreversible entropy production due to reaction allows expressing the reaction rate in terms of a thermodynamic quantity, called chemical affinity, that can then be defined in the space of vessel extents using a linear time-invariant transformation. This structural link in turn allows constructing a Lypunov function candidate to synthesize a nonlinear controller for the stabilization of the reaction rate, and thus of the vessel extents. Numerical simulations are carried out for a reversible reaction to illustrate the proposed approach.

AdaNFF: A new method for adaptive nonnegative multi-feature fusion to scene classification

Scene classification is an important basis for many modern intelligent applications, however the performance of pattern recognition or deep learning-based methods are still not sufficient since complicated structure and context of scene images. In this paper, we propose a novel fusion framework of adaptive nonnegative feature fusion (AdaNFF) for scene classification. The AdaNFF integrates nonnegative matrix factorization, adaptive feature fusion and feature fusion boosting into an end-to-end process. Firstly, feature fusion is known as a general strategy to strengthen weak features, and we observe that pixel values and most hand-craft features of the scene image are naturally nonnegative. Therefore we are motivated to build a fusion method based on nonnegative matrix factorization, which can preserve features nonnegative properties and improve their representation performance. Secondly, with the results of fused single or multiple features fusion, we develop an adaptive feature fusion and boosting algorithm to improve the efficiency of image features. Finally, a normalized l2-norm classifier and a deep-learning like multilayer perceptron (MLP) classifier are trained to predict label of scene image. Under this framework, there are two versions of the proposed feature fusion method for nonnegative single-feature fusion and multi-feature fusion. All methods were validated on scene classification benchmarks. Experiment results suggest that the proposed methods can deal with multi-class scene problems and achieve remarkable classification performance.

Integration of process knowledge and statistical learning for the Dow data challenge problem

In this paper, we propose a statistical learning procedure that integrates process knowledge for the Dow data challenge problem presented in Braun et al. (2020). The task is to build an accurate inferential sensor model to predict the impurity in the product stream with apparent drifts. The proposed method consists of i) process data exploratory analysis, ii) a method for variable selection, iii) a method to deal with non-negative physical property modeling using a softplus function; and iv) a method for online bias updating based on known data. We make use of process operation knowledge in all steps of data analytics, including exploratory analysis and feature selection. We report the detection of equipment-switching operations in the data and interpolations found in the impurity data. Partial least squares (PLS) and least angle regression solution (LARS) are adopted to model the data with strong collinearity. Pros and cons of LARS and PLS are given with practical implications.

A novel complex evidential distance with its application in pattern recognition

Complex evidence theory is an effective information fusion method, which can be used to fuse complex basic belief assignments (CBBAs) and obtain a reasonable result. In complex evidence theory, it is still an open issue for conflict management. In order to address this problem, this paper focuses on putting forward a new distance to measure conflict between CBBAs. The newly defined complex evidential distance takes into account not only the singletons, but also the subsets as well as their power sets. Therefore, it has a better performance to measure the conflict between the CBBAs. In addition, the proposed distance satisfies distance properties of nonnegativity, nondegeneracy, symmetry and the triangle inequality. In particular, when the CBBAs degenerate to classical BBAs, the proposed distance can also measure conflict well. Furthermore, a number of numerical examples are given to illustrate the above mentioned properties. Based on the newly devised distance measure, a basic pattern recognition algorithm is proposed. Subsequently, it is extended to a weighted scheme of pattern recognition algorithm. Finally, those two algorithms are applied to solve medical diagnosis to demonstrate their effectiveness.

Autoencoder-like semi-NMF multiple clustering

Clustering is performed to partition samples into disjoint groups for facilitating the discovery of hidden patterns in the data. Many real-world applications involve various clustering methods , most of which only produce a single clustering. As a response to this issue, multiple clustering that aims to generate diverse and high-quality clustering, has emerged recently. This study proposes a novel autoencoder-like semi-nonnegative matrix factorization (NMF) multiple clustering (ASNMFMC) model that generates multiple non-redundant, high-quality clustering. The nonnegative property of the semi-NMF is utilized by the algorithm to enforce non-redundancy. Extensive experimental results demonstrate that the ASNMFMC is superior to the existing multiple clustering methods and can explore diverse high-quality clustering.

Augmenting geophysical interpretation of data-driven operational water supply forecast modeling for a western US river using a hybrid machine learning approach

In the largely dry and increasingly heavily populated western US, operational modeling systems for seasonal river runoff volume forecasting are key elements of the practical water and hydropower management infrastructure. Explainability of model results in terms of known hydroclimatic processes and conditions is a core requirement for these systems. To improve geophysical interpretability of a standard statistical modeling approach to operational water supply forecasting (WSF), we introduce a hybrid statistical-artificial intelligence method. The procedure involves using a recently developed unsupervised machine learning algorithm designed for improved explainability (non-negative matrix factorization with k-means clustering, NMFk) to extract a compact basin-scale hydroclimatic index from available precipitation and snowpack data; that index is then used as the predictor variate in a largely conventional probabilistic regression on seasonal water supply. The resulting method, dominant-signal NMFk regression, is applied to a challenging forecast site, the Owyhee River, drawn from the US Department of Agriculture Natural Resources Conservation Service WSF system. Outcomes demonstrate that improved interpretability and plausibility relative to conventional statistical methods are achieved through physical consistency of NMFk results with nonnegativity of the environmental data being analyzed. In particular, the nonnegativity property facilitates identifying potential geophysical relationships to input variable type (snow water equivalent vs. accumulated precipitation), location, and underlying hydrologic processes; and it encourages nonnegative river runoff predictions, improving physical realism of WSFs over conventional statistical approaches in certain cases. The method also offers straightforward interpretation of relationships to known forms of climate variability. However, testing suggests that with these capabilities come limitations. Its primary anticipated role, at present, is to augment geophysical interpretation when needed, by serving as a complement alongside other methods in a next-generation US West-wide operational forecasting system.

CED: A Distance for Complex Mass Functions.

Evidence theory is an effective methodology for modeling and processing uncertainty that has been widely applied in various fields. In evidence theory, a number of distance measures have been presented, which play an important role in representing the degree of difference between pieces of evidence. However, the existing evidential distances focus on traditional basic belief assignments (BBAs) modeled in terms of real numbers and are not compatible with complex BBAs (CBBAs) extended to the complex plane. Therefore, in this article, a generalized evidential distance measure called the complex evidential distance (CED) is proposed, which can measure the difference or dissimilarity between CBBAs in complex evidence theory. This is the first work to consider distance measures for CBBAs, and it provides a promising way to measure the differences between pieces of evidence in a more general framework of complex plane space. Furthermore, the CED is a strict distance metric with the properties of nonnegativity, nondegeneracy, symmetry, and triangle inequality that satisfies the axioms of a distance. In particular, when the CBBAs degenerate into classical BBAs, the CED will degenerate into Jousselme et al.'s distance. Therefore, the proposed CED is a generalization of the traditional evidential distance, but it has a greater ability to measure the difference or dissimilarity between pieces of evidence. Finally, a decision-making algorithm for pattern recognition is devised based on the CED and is applied to a medical diagnosis problem to illustrate its practicability.

Turán’s inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by L1-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated L1-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties hold (which is very different to L1-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Turán’s inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Turán type inequalities and transformations into more convenient orthogonal polynomial systems.