Search

This paper studies consensus problems of high-order heterogeneous continuous-time networks with bounded communication delays and dynamically changing digraphs. The heterogeneous continuous-time networks consist of different order agents from first-order to lth-order agents, where different order networks correspond to different linear consensus protocols. In order to use the properties of Metzler matrix with zero row sums, we make model transformations of system matrix. Giving some restrictive conditions, we obtain the sufficient condition for consensus problems of heterogeneous networks with varying bounded communication delays. Although each communication graph may have no spanning trees, all agents of high-order heterogeneous networks can reach stationary consensus with varying bounded communication delays and dynamically changing digraphs. Finally, we give an example to validate the correctness of our obtained results.

It is known that the stability of a Metzler matrix can be characterized in a Routh–Hurwitz-like fashion based on a recursive application of scalar Schur complements [1] . Our objective in this brief note is to show that recently obtained stability conditions are equivalent statements of this result and can be deduced directly therefrom using only elementary results from linear algebra. Implications of this equivalence are also discussed and several examples are given to illustrate potentially interesting system-theoretic applications of this observation.

Applications of nonnegative matrices have been of immense interest to both social and physical scientists, particularly to economists and statisticians. This paper considers the question as to when a nonnegative singular matrix can be decomposed as a product of nonnegative idempotents analogous to the well-known result for any arbitrary matrix. It is shown that (i) all singular nonnegative matrices of rank , (ii) all singular nonnegative matrices that have a nonnegative von Neumann inverse, (iii) (0-1) nonnegative definite matrices and (iv) periodic matrices have the property that they decompose into a product of nonnegative idempotents. An example is given, in general, showing that this need not be true for singular matrices of rank three or higher, including stochastic and symmetric matrices. Besides computational techniques, a recent result that a singular nonnegative quasi-permutation matrix is a product of nonnegative idempotents plays a key role in the proofs of the results.

International audience; This paper is concerned with the analysis and synthesis of interconnected systems constructed from heterogeneous positive subsystems and a nonnegative interconnection matrix. We first show that admissibility, to be defined in this paper, is an essential requirement in constructing such interconnected systems. Then, we clarify that the interconnected system is admissible and stable if and only if a Metzler matrix, which is built from the coefficient matrices of positive subsystems and the nonnegative interconnection matrix, is Hurwitz stable. By means of this key result, we further provide several results that characterize the admissibility and stability of the interconnected system in terms of the Frobenius eigenvalue of the interconnection matrix and the weighted L1- induced norm of the positive subsystems again to be defined in this paper. Moreover, in the case where every subsystem is SISO, we provide explicit conditions under which the interconnected system has the property of persistence, i.e., its state converges to a unique strictly positive vector (that is known in advance up to a strictly positive constant multiplicative factor) for any nonnegative and nonzero initial state. As an important consequence of this property, we show that the output of the interconnected system converges to a scalar multiple of the right eigenvector of a nonnegative matrix associated with its Frobenius eigenvalue, where the nonnegative matrix is nothing but the interconnection matrix scaled by the steady-stage gains of the positive subsystems. This result is then naturally and effectively applied to formation control of multiagent systems with positive dynamics. This result can be seen as a generalization of a well-known consensus algorithm that has been basically applied to interconnected systems constructed from integrators.

A real matrix is called totally nonnegative if all of its minors are nonnegative. In this paper, the minors are determined from which the maximum allowable entry perturbation of a totally nonnegative matrix can be found, such that the perturbed matrix remains totally nonnegative. Also, the total nonnegativity of the first and second subdirect sum of two totally nonnegative matrices is considered.

In this paper, fault estimation for high-speed railway traction motor with sensor fault and disturbances is investigated based on the interval observer and unknown input observer (IO-UIO). First, the proposed method, which can completely eliminate the external disturbance, is studied based on the disturbance isolation characteristic of the unknown input observer. Then, an interval observer is constructed to deal with the nonlinear part, which sandwiched the actual system between the upper and lower bounds. Moreover, the Metzler matrix is constructed using an equivalent transformation and through the unified design based on the concept of the augmented state to form a global fault augmented model. Finally, simulation results are presented to illustrate the effectiveness and advantages of the proposed IO-UIO.

In order to deal with the complexity of system, an interval observer-based fault diagnosis method was proposed recently. The design approach of interval observer may be classified as linear method and nonlinear method according to the type of fault diagnosis system. Besides, it is divided into direct method and coordinate transformation approach for the linear systems. Finally, the problems to be solved and future research directions of interval observer-based fault diagnosis are addressed.

Decomposition of the controllability and observability matrices into symmetrical and asymmetrical parts of electrical circuits composed of resistors, coils, capacitors and voltage sources are addressed. It is shown that: 1) the Metzler matrix is Hurwitz if and only if its symmetrical part is Hurwitz; 2) if the linear electrical circuit is controllable (observable) then at least its symmetrical or asymmetrical part is controllable (observable).

This paper investigates the cluster synchronization problem of linearly coupled complex networks via aperiodically intermittent control (AIC). The dynamical behaviors of nodes in different clusters are assumed to be governed by different dynamical functions, while the dynamical behaviors of nodes in the same cluster are the same. Moreover, the original functions of nodes are defined by continuous-time ordinary differential equations with time-varying delays. As for the coupling matrix, we assume it is a Metzler matrix with zero row sums. The main contribution is that we pin some simple AIC to realize the cluster synchronization. Furthermore, as for the pinning control gains, both static and adaptive control cases are considered and some criteria are obtained. We also present some numerical simulations to verify the theoretical results.

Arising from the classification of the matrix-roots of a nonnegative imprimitive irreducible matrix, we present results concerning the Jordan chains of an $h$-cyclic matrix. We also present ancillary results applicable to nonnegative imprimitive irreducible matrices and demonstrate these results via examples., Comment: To appear in the special issue "The Legacy of Hans Schneider" in Linear Algebra and its Applications

The max-times algebra is the set R + of nonnegative reals with operations ⊕ : ( a , b ) → max ⁡ { a , b } and ⊙ : ( a , b ) → a b . We discuss the property of matrices to be squares of max-times or conventional nonnegative matrices. We prove that there exists a matrix having a conventional nonnegative square root but no max-times square root. Also, we present a set S of cardinality three for which there is a nonnegative matrix M with spectrum S, and every such M has both conventional and max-times square roots. These results answer two questions from the recent paper by Tam and Huang.

An extended Kalman-Yakubovich-Popov (KYP) Lemma for positive systems is derived. The main difference compared to earlier versions is that non-strict inequalities are treated. Matrix assumptions are also less restrictive. Moreover, a new equivalence is introduced in terms of linear programming rather than semi-definite programming. As a complement to the KYP lemma, it is also proved that a symmetric Metzler matrix with $m$ nonzero entries above the diagonal is negative semi-definite if and only if it can be written as a sum of $m$ negative semi-definite matrices, each of which has only four nonzero entries. This is useful in the context large-scale optimization.

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) asks when is a list \[ \sigma=(\lambda_1, \lambda_2,\ldots,\lambda_n)\] consisting of real numbers the spectrum of an $n \times n$ nonnegative matrix $A$. In that case, $\sigma$ is said to be realizable and $A$ is a realizing matrix. In a recent paper dealing with RNIEP, P.~Paparella considered cases of realizable spectra where a realizing matrix can be taken to have a special form, more precisely such that the entries of each row are obtained by permuting the entries of the first row. A matrix of this form is called permutative. Paparella raised the question whether any realizable list $\sigma$ can be realized by a permutative matrix or a direct sum of permutative matrices. In this paper, it is shown that in general the answer is no.

A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition and cooperation is presented and analysed. The state of each agent is represented by its payoff, which the agent aims at maximising. The role of control variables is played by the model parameters, which account for the agents' decisions to either cooperate with or boycott the other agents and can vary in time within assigned intervals. Alliances and enmities can be established at any time, according to either a greedy or a longsighted criterion. The general nonlinear case is first considered. It is proved that, under realistic assumptions, the system evolution is bounded positive (no extinction) and there is a unique globally-stable equilibrium point. As is somehow expected, the optimal decision for all agents corresponds to full cooperation (decision parameters kept at their positive maximum value) in the case of both shortsighted and farsighted criteria. This is not true if some parameters have negative upper bounds (meaning that some agents systematically boycott some others). Then, in the linear case, it is shown that the system is stable for arbitrarily-varying decision parameters, provided that a Metzler matrix associated with full cooperation is Hurwitz. A characterisation of the long-term behaviour of the linear system is also provided. In particular, it is proved that, under stability conditions, a Nash equilibrium exists if a steady strategy is adopted.

This work presents a mathematical model for the spread of lymphatic filariasis in a population. We use the Metzler Matrix Theory and the Kamgang–Sallet [2] (Math Biosci 213, 1–12, 2008) algorithm to compute the threshold conditions and global stability of the disease-free equilibrium. We showed that if \(R_0 1\), the disease-free equilibrium is unstable.

In this paper, we investigate the consensus problem for multiple autonomous vehicles with nonconvex constraints. We propose a distributed control algorithm and prove that consensus can be reached if the communication network jointly has a spanning tree. The analysis approach is based on the model transformation and the properties of the Metzler matrix and the stochastic matrix.

This paper studies the edge consensus problems of undirected networked systems with positivity constraint and input saturation. Based on a given nodal network, the definition of edge network is introduced, and the edge dynamics is described by positive systems subject to input saturation. With the connections between the edge network and the nodal network, the lower and upper bounds of the nonzero eigenvalues of the edge Laplacian matrix are presented. Rigorous convergence analysis is carried out. Based on the special properties of positive systems, sufficient conditions of positive edge consensus with input saturation are given without using global topology information, that is, the topology information in the given results only relates to the edge and vertex number of the nodal network. Furthermore, the nonnegative edge consensus problem with uncertain parameters is also studied based on the properties of Metzler matrix. Different from many consensus problems with input saturation which achieve semi-global consensus, the results in this paper are global consensus. The feedback matrix can be computed by solving linear matrix inequality. Some illustrative numerical and practical simulation examples are finally introduced to verify the proposed methods in this paper.

Bubonic plague is a serious bacterial disease, mainly transmitted to human beings and rodents through flea bite. However, the disease may also be transmitted upon the interaction with the infected materials or surfaces in the environment. In this study, a deterministic model for bubonic plague disease with Yersinia pestis in the environment is developed and analyzed. Conditions for existence and stability of the equilibrium points are established. Using Jacobian method disease free equilibrium (DFE) point, E0 was proved to be locally asymptotically stable. The Metzler matrix method was used to prove that the DFE was globally asymptotically stable when R0 R0 > 1. Numerical simulations are done to verify the analytical predictions. The results show that bubonic plague can effectively be controlled or even be eradicated if efforts are made to ensure that there are effective and timely control strategies.

We study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. We show that nonnegative, real, and complex ranks are all equal for a general nonnegative tensor of nonnegative rank strictly less than the complex generic rank. In addition, such nonnegative tensors always have unique nonnegative rank-$r$ decompositions if the real tensor space is $r$-identifiable. We determine conditions under which a best nonnegative rank-$r$ approximation has a unique nonnegative rank-$r$ decomposition: for $r \le 3$, this is always the case; for general $r$, this is the case when the best nonnegative rank-$r$ approximation does not lie on the boundary of $D_r$. Many of our general identifiability results also apply to real tensors and real symmetric tensors., 25 pages, to appear in SIMAX

In this paper, we investigate the cluster synchronization problem of complex networks via pinning control. Nodes in the same cluster are governed by the same dynamical function, while the functions for different clusters are different. For the coupling scheme, the effects of the coupling are assumed to be only cooperative, i.e., the coupling matrix is a Metzler matrix with zero row sums, and the connections between the same cluster can be few. For this type of coupling, the main difference of this paper with previous works is that the cluster synchronization is realized by pinning control technique. A simple pinning control strategy is proposed and some sufficient criteria to realize cluster synchronization with both static and adaptive control strength are given. Finally, numerical simulations are presented to verify our theoretical results.

The purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property (∗) that each right (left) singular element is a product of idempotents, and (2) to consider the question: “when is a singular nonnegative square matrix a product of nonnegative idempotent matrices?” The importance of the class of quasi- Euclidean rings in connection with the property (∗) is given by the first three authors and T.Y. Lam [Journal of Algebra, 406:154–170, 2014], where it is shown that every singular matrix over a right and left quasi-Euclidean domain is a product of idempotents, generalizing the results of J. A Erdos [Glasgow Mathematical Journal, 8: 118–122, 1967] for matrices over fields and that of T. J. Laffey [Linear and Multilinear Algebra, 14:309–314, 1983] for matrices over commutative Euclidean domains. We have shown in this paper that quasi-Euclidean rings appear among many interesting classes of rings and hence they are in abundance. We analyze the properties of triangular matrix rings and upper triangular matrices with respect to the decomposition into product of idempotents and show, in particular, that nonnegative nilpotent matrices are products of nonnegative idempotent matrices. We study as to when each singular matrix is a product of idempotents in special classes of rings. Regarding the second question for nonnegative matrices, bounds are obtained for a rank one nonnegative matrix to be a product of two idempotent matrices. It is shown that every nonnegative matrix of rank one is a product of three nonnegative idempotent matrices. For matrices of higher orders, we show that some power of a group monotone matrix is a product of idempotent matrices.

This paper considers the non-convex problem of finding the nearest Metzler matrix to a given possibly unstable matrix. Linear systems whose state vector evolves according to a Metzler matrix have many desirable properties in analysis and control with regard to scalability. This motivates the question, how close (in the Frobenius norm of coefficients) to the nearest Metzler matrix are we? Dropping the Metzler constraint, this problem has recently been studied using the theory of dissipative Hamiltonian (DH) systems, which provide a helpful characterization of the feasible set of stable matrices. This work uses the DH theory to provide a block coordinate descent algorithm consisting of a quadratic program with favourable structural properties and a semidefinite program for which recent diagonal dominance results can be used to improve tractability., To Appear in Proc. of 56th IEEE CDC

Metzler matrices are algebraically characterized by having nonnegative off-diagonal entries and, from the dynamical point of view, define continuous-time positive systems. Our work studies polytopes of Metzler matrices and linear dynamics generated by such sets. The first part of the paper explores the algebraic properties of matrix polytopes, by focusing on the estimation of a right outer bound for all eigenvalues. The second part analyzes the dynamical properties of positive polytopic systems, by revealing connections between the estimated right outer bound and the evolution of trajectories (related to copositive Lyapunov functions and invariant sets).

A new approach to design interval observers for a class of nonlinear systems is proposed in this paper. Based on the matrix theory, a simple elementary operation is given to find a positive realization for the systems with bounded uncertainties. Compared with previous works, the main contribution of this work is to provide a method to guarantee the existence of an observer gain by means of such a positive representation. Finally, a numerical example is provided to demonstrate efficiency of the proposed approach.

This paper considers the stabilization problem of linear dynamical systems with combined structural properties of symmetry and positivity. Such systems arise in various applications including electromechanical systems, aerodynamics, structural vibration, robotics and compartmental systems, in which the stabilization and performance improvement are the main objectives. Although the goal of the paper is to design controller for such systems, we broaden the scope by designing controller for general systems such that the closed-loop systems admits the structural constraints of symmetry and positivity. We concentrate on two different classes of symmetric positive systems. The first class has the state-space symmetric representation A = AT, C = BT with A being a Metzler matrix and the positive pair of matrices (B, C). The stability conditions of this class are used to formulate and solve symmetric positive stabilization by means of state feedback for systems with arbitrary state-space parameters. The second class is defined through the block controllable canonical form in which the block sub-matrices associated with the system matrix A are symmetric Metzlerian. Assuming that such systems are unstable, we design state feedback control law such that the closed-loop system becomes stable and maintains its structure. A generalized symmetric Metzlerian stabilization algorithm is provided through a set of chain equations to achieve this goal. Numerical examples are provided for stabilization of both classes.

The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for . If the matrix A is required to be normal, the problem will be called the normal nonnegative inverse eigenvalue problem (NNIEP). Sufficient conditions for a list of complex numbers to be the spectrum of a normal nonnegative matrix were obtained by Xu [Linear Multilinear Algebra. 1993;34:353–364]. In this paper, we give a normal version of a rank-r perturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305–311], which allow us to obtain new sufficient conditions for the NNIEP to have a solution. These new conditions significantly improve Xu’s conditions. We also apply our results to construct nonnegative matrices with arbitrarily prescribed elementary divisors.

In this note we consider a supervisory control scheme that achieves either asymptotic stability or consensus for a group of homogenous agents described by a positive state-space model. Each agent is modeled by means of the same SISO positive state-space model, and the supervisory controller, representing the information exchange among the agents, is implemented via a static output feedback. Necessary and sufficient conditions for the asymptotic stability, or the consensus of all agents, are derived under the positivity constraint.

The goal of this paper is to study the switched stochastic control problem of discrete-time linear systems with multiplicative noises. We consider both the quadratic and the H∞ criteria for the performance evaluation. Initially we present a sufficient condition based on some Lyapunov–Metzler inequalities to guarantee the stochastic stability of the switching system. Moreover, we derive a sufficient condition for obtaining a Metzler matrix that will satisfy the Lyapunov–Metzler inequalities by directly solving a set of linear matrix inequalities, and not bilinear matrix inequalities as usual in the literature of switched systems. We believe that this result is an interesting contribution on its own. In the sequel we present sufficient conditions, again based on Lyapunov–Metzler inequalities, to obtain the state feedback gains and the switching rule so that the closed loop system is stochastically stable and the quadratic and H∞ performance costs are bounded above by a constant value. These results are illu...

Tridiagonal matrices are considered which are totally nonnegative, i. e., all their mi- nors are nonnegative. The largest amount is given by which the single entries of such a matrix can be perturbed without losing the property of total nonnegativity.

This paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities (LMIs). We first show that the celebrated Perron–Frobenius theorem can be proved concisely by a duality-based argument. Again by duality, we next clarify a necessary and sufficient condition under which a Hurwitz stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries as the solution of the Lyapunov inequality. This new result leads to an alternative proof of the recent result by Tanaka and Langbort on the existence of a diagonal Lyapunov matrix for the LMI characterizing the H ∞ performance of continuous-time positive systems. In addition, we further derive a new LMI for the H ∞ performance analysis where the variable corresponding to the Lyapunov matrix is allowed to be non-symmetric. We readily extend these results to discrete-time positive systems and derive new LMIs for the H ∞ performance analysis and synthesis. We finally illustrate their effectiveness by numerical examples on robust state-feedback H ∞ controller synthesis for discrete-time positive systems affected by parametric uncertainties.

In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an att...

This paper studies the construction problems of five order nonnegative matrices from spectrum data. Let be a list of complex numbers with . Necessary and sufficient conditions for the existence of an entry-wise nonnegative 5×5 matrix with spectrum are presented.

The basin of attraction to the origin in the space is obtained for the m dimensional system of difference equations of the formwhen coefficient is a Metzler matrix. Some information about the dynamics of the solutions outside of the basin is also given.

Given any nonnegative matrix $A \in \mathbb{R}^{m \times n}$ , it is always possible to express A as the sum of a series of nonnegative rank-one matrices. Among the many possible representations of A, the number of terms that contributes the shortest nonnegative rank-one series representation is called the nonnegative rank of A. Computing the exact nonnegative rank and the corresponding factorization are known to be NP-hard. Even if the nonnegative rank is known a priori, no simple procedure exists presently that is able to perform the nonnegative factorization. Based on the Wedderburn rank reduction formula, this paper proposes a heuristic approach to tackle this difficult problem numerically. Starting with A, the idea is to recurrently extrat, whenever possible, a rank-one nonnegative portion from the previous matrix while keeping the residual nonnegative and lowering its rank by one. With a slight modification for symmetry, the method can equally be applied to another important class of completely positive matrices. No convergence can be guaranteed, but repeated restart might help alleviate the difficulty. Extensive numerical testing seems to suggest that the proposed algorithm might serve as a first-step numerical means for exploring the intriguing problem of nonnegative rank factorization.

This article presents a novel MM-MHE algorithm for online prediction of the component survival functions based on their usage profiles. The framework employs Cox PHM based on offline and online data for the RUL prediction. The proposed approach has been validated by way of application to data derived from an automotive ETC system simulator. The MM-MHE algorithm shows excellent performance (R2 and MSE) in the presence of significant measurement noise over all windows and converges to the correct cluster number. The future work includes application of this approach to continuous PID and to account for the uncertainty in RUL estimation. In the near future, by simple transformations, the authors plan on implementing the MM-MHE algorithm for measurements and states between (-∞,∞) and considering the effects of process noise, hence, modifying the cost function accordingly. A potential extension of the Cox PHM framework for prognosis of coupled systems will be to model the coupled survival dynamics as monotone positive linear systems or monotone Markov processes in which the state matrix is a Metzler matrix (i.e., has nonnegative off-diagonal elements).

To simplify the analysis of complex dynamical networks, we have recently proposed an approach that decomposes the overall system into the sign-definite interconnection of subsystems with a Positive Impulse Response (PIR). PIR systems include and significantly generalise input-output monotone systems, and the PIR property (or equivalently, for linear systems, the Monotonic Step Response property) can be evinced from experimental data, without an explicit model of the system. An aggregate of PIR subsystems can be associated with a signed matrix of interaction weights, hence with a signed graph where the nodes represent the subsystems and the arcs represent the interactions among them. In this paper, we prove that stability is structurally ensured (for any choice of the PIR subsystems) if a Metzler matrix depending on the interaction weights is Hurwitz; this condition is non-conservative. We also show how to compute an influence matrix that represents the steady-state effects of the interactions among PIR subsystems.

Real data are usually complex and contain various components. For example, face images have expressions and genders. Each component mainly reflects one aspect of data and provides information others do not have. Therefore, exploring the semantic information of multiple components as well as the diversity among them is of great benefit to understand data comprehensively and in-depth. However, this cannot be achieved by current nonnegative matrix factorization (NMF)-based methods, despite that NMF has shown remarkable competitiveness in learning parts-based representation of data. To overcome this limitation, we propose a novel multi-component nonnegative matrix factorization (MCNMF). Instead of seeking for only one representation of data, MCNMF learns multiple representations simultaneously, with the help of the Hilbert Schmidt Independence Criterion (HSIC) as a diversity term. HSIC explores the diverse information among the representations, where each representation corresponds to a component. By integrating the multiple representations, a more comprehensive representation is then established. A new iterative updating optimization scheme is derived to solve the objective function of MCNMF, along with its correctness and convergence guarantees. Extensive experimental results on real-world datasets have shown that MCNMF not only achieves more accurate performance over the state-of-the-arts using the aggregated representation, but also interprets data from different aspects with the multiple representations, which is beyond what current NMFs can offer.

Many real-world applications involve multi-label classification where each sample is usually associated with a set of labels. Although many methods have been proposed, most of them are just applicable to single-view data neglecting the complementary information among multiple views. Besides, most existing methods are supervised, hence they cannot handle the case where only a few labeled data are available. To address these issues, we propose a novel semi-supervised multi-view multi-label classification method based on nonnegative matrix factorization (NMF). Specifically, it explores the complementary information by adopting multi-view NMF, regularizes the learned labels of each view towards a common consensus labeling, and obtains the labels of the unlabeled data guided by supervised information. Experimental results on real-world benchmark datasets demonstrate the superior performance of our method over the state-of-the-art methods.

Nonnegative matrix factorization/approximation (NMF/NMA) is a widely used method for data analysis. So far, many multiplicative update algorithms have been developed for NMF. In this paper, we propose a nonnegative projection based NMF algorithm, which is different from the conventional multiplicative update NMF algorithms and decreases the objective function by performing Procrustes rotation and nonnegative projection alternately. The experiment results demonstrate that the new algorithm converges much faster than traditional ones.

In this paper, we theoretically clarify the phase transition structure of the variational Bayesian nonnegative matrix factorization (VBNMF). By asymptotic analysis of the objective functional in variational inference, we find that the variational posterior distribution of the VBNMF is drastically changed by hyperparameters; we call this phenomenon phase transition of the VBNMF. We also discuss a numerical experiment demonstrating our theoretical results.