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Most modern learning problems are highly overparameterized, i.e., have many more model parameters than the number of training data points. As a result, the training loss may have infinitely many global minima (parameter vectors that perfectly ``interpolate'' the training data). It is therefore imperative to understand which interpolating solutions we converge to, how they depend on the initialization and learning algorithm, and whether they yield different test errors. In this article, we study these questions for the family of stochastic mirror descent (SMD) algorithms, of which stochastic gradient descent (SGD) is a special case. Recently, it has been shown that for overparameterized linear models, SMD converges to the closest global minimum to the initialization point, where closeness is in terms of the Bregman divergence corresponding to the potential function of the mirror descent. With appropriate initialization, this yields convergence to the minimum-potential interpolating solution, a phenomenon referred to as implicit regularization. On the theory side, we show that for sufficiently- overparameterized nonlinear models, SMD with a (small enough) fixed step size converges to a global minimum that is ``very close'' (in Bregman divergence) to the minimum-potential interpolating solution, thus attaining approximate implicit regularization. On the empirical side, our experiments on the MNIST and CIFAR-10 datasets consistently confirm that the above phenomenon occurs in practical scenarios. They further indicate a clear difference in the generalization performances of different SMD algorithms: experiments on the CIFAR-10 dataset with different regularizers, l₁ to encourage sparsity, l₂ (SGD) to encourage small Euclidean norm, and l to discourage large components, surprisingly show that the l norm consistently yields better generalization performance than SGD, which in turn generalizes better than the l₁ norm.

Low-rank plus sparse matrix decomposition (LSD) is an important problem in computer vision and machine learning. It has been solved using convex relaxations of the matrix rank and l0-pseudo-norm, which are the nuclear norm and l1-norm, respectively. Convex approximations are known to result in biased estimates, to overcome which, nonconvex regularizers such as weighted nuclear-norm minimization and weighted Schatten p-norm minimization have been proposed. However, works employing these regularizers have used heuristic weight-selection strategies. We propose weighted minimax-concave penalty (WMCP) as the nonconvex regularizer and show that it admits an equivalent representation that enables weight adaptation. Similarly, an equivalent representation to the weighted matrix gamma norm (WMGN) enables weight adaptation for the low-rank part. The optimization algorithms are based on the alternating direction method of multipliers technique. We show that the optimization frameworks relying on the two penalties, WMCP and WMGN, coupled with a novel iterative weight update strategy, result in accurate low-rank plus sparse matrix decomposition. The algorithms are also shown to satisfy descent properties and convergence guarantees. On the applications front, we consider the problem of foreground-background separation in video sequences. Simulation experiments and validations on standard datasets, namely, I2R, CDnet 2012, and BMC 2012 show that the proposed techniques outperform the benchmark techniques.

We extend Quine's bound on the number of self-intersection of curves with polynomial parameterization to the case of Laurent polynomials. As an application, we show that circle embeddings are dense among all maps from a circle to a plane with respect to an integral norm., Comment: 10 pages

This article identifies a new upper bound norm for the intervalized interconnection matrices pertaining to delayed dynamical neural networks under the parameter uncertainties. By formulating the appropriate Lyapunov functional and slope-bounded activation functions, the derived new upper bound norms provide new sufficient conditions corresponding to the equilibrium point of the globally asymptotic robust stability with respect to the delayed neural networks. The new upper bound norm also yields the optimized minimum results as compared with some existing methods. Numerical examples are given to demonstrate the effectiveness of the proposed results obtained through the new upper bound norm method.

A recommender system (RS) is highly efficient in filtering people's desired information from high-dimensional and sparse (HiDS) data. To date, a latent factor (LF)-based approach becomes highly popular when implementing a RS. However, current LF models mostly adopt single distance-oriented Loss like an L₂ norm-oriented one, which ignores target data's characteristics described by other metrics like an L₁ norm-oriented one. To investigate this issue, this article proposes an L₁-and-L₂-norm-oriented LF (L³F) model. It adopts twofold ideas: 1) aggregating L₁ norm's robustness and L₂ norm's stability to form its Loss and 2) adaptively adjusting weights of L₁ and L₂ norms in its Loss. By doing so, it achieves fine aggregation effects with L₁ norm-oriented Loss's robustness and L₂ norm-oriented Loss's stability to precisely describe HiDS data with outliers. Experimental results on nine HiDS datasets generated by real systems show that an L³F model significantly outperforms state-of-the-art models in prediction accuracy for missing data of an HiDS dataset. Its computational efficiency is also comparable with the most efficient LF models. Hence, it has good potential for addressing HiDS data from real applications.

A dilemma faced by classification is that the data is not collected at the same frequency in some applications. We investigate the mixed frequency data in a new way and recognize them as a special style of multi-view data, in which each view data is collected at a different sampling frequency. This paper proposes a discriminative dictionary pair learning method constrained by ordinal locality for mixed frequency data classification (shorted by DPLOL-MF). This method integrates synthesis dictionary and analysis dictionary into a dictionary pair, which not only improves computational cost caused by the ${\ell_0}$ or ${\ell_1}$ -norm constraint, but also can deal with the sampling frequency inconsistency. The DPLOL-MF utilizes a synthesis dictionary to learn class-specified reconstruction information and employs an analysis dictionary to generate coding coefficients by analyzing samples. Particularly, the ordinal locality preserving term is leveraged to constrain the atoms of dictionaries pair to further facilitate the learned dictionary pair to be more discriminative. Besides, we design a specific classification scheme for the inconsistent sample size of mixed frequency data. This paper illustrates a novel idea to solve the classification task of mixed frequency data and the experimental results demonstrate the effectiveness of the proposed method.

Inspired by the mean calculation of RPCA_OM and inductiveness of IRPCA, we first propose an inductive robust principal component analysis method with removing the optimal mean automatically, which is shorted as IRPCA_OM. Furthermore, IRPCA_OM is extended to Schatten-p norm and a more general framework (i.e., EIRPCA_OM) is presented. The objective function of EIRPCA_OM includes two terms, the first term is a robust reconstruction error term constrained by an l2,1-norm and the second term is a regularization term constrained by a Schatten-p norm. The proposed EIRPCA_OM method is robust, inductive and accurate. However, on the high-dimensional data, it would spend a large computation cost in training stage. To this end, a fast version of EIRPCA_OM called as FEIRPCA_OM is proposed, and its basic idea is to eliminate the zero eigenvalues of data matrix. More importantly, an effective theoretical proof is presented to ensure that FEIRPCA_OM has faster processing speed than EIRPCA_OM when processing high-dimensional data, but without any performance loss. Based on it, we also can exchange the less performance loss for the higher computation efficiency by removing the small eigenvalues of data matrix. Experimental results on the public datasets demonstrate that FEIRPCA_OM works efficiently on the high-dimensional data.

lsub1/sub-norm quantile regression is a common choice if there exists outlier or heavy-tailed error in high-dimensional data sets. However, it is computationally expensive to solve this problem when the feature size of data is ultra high. As far as we know, existing screening rules can not speed up the computation of the lsub1/sub-norm quantile regression, which dues to the non-differentiability of the quantile function/pinball loss. In this paper, we introduce the dual circumscribed sphere technique and propose a novel lsub1/sub-norm quantile regression screening rule. Our rule is expressed as the closed-form function of given data and eliminates inactive features with a low computational cost. Numerical experiments on some simulation and real data sets show that this screening rule can be used to eliminate almost all inactive features. Moreover, this rule can help to reduce up to 23 times of computational time, compared with the computation without our screening rule.

Non-destructive testing & evaluation techniques play an essential role in ensuring safety of materials in operation at various industry sectors. Pulse compressed favourable thermal wave imaging is one of the widely used non-destructive testing techniques due to its excellent noise rejection capabilities. However, the high dimensional thermal imaging data needs to be encoded into lossless compressed form to highlight the hidden defects inside the materials. This paper proposes a novel constrained and regularized autoencoder based thermography approach for sub-surface defect detection in a mild steel specimen. Certain properties such as non-correlation of encoded data, weight orthogonality, and weights with unit norm length have been highlighted which are non-existent in linear autoencoders but are responsible for better defect detection inside the materials inspected by frequency modulated thermal wave imaging. Novel constraints are formulated for autoencoder cost function to incorporate these significant properties. The proposed approach is able to provide better defect detection, in terms of signal to noise ratio of defects, than linear autoencoder as well as traditional principal component thermography approach. Also, non-correlation of encoded data is found to be the most significant factor in achieving better defect detection followed by properties ensuring weight orthogonality and weights with unit norm length.

It is well known that metric spaces are an instance of categorical enrichment in a particular quantale. We show that in a categorically natural way a notion of Lipschitz norm arises in the context of an arbitrary diagram of quantales, instead of just one particular quantale. The generalised Lipschitz norm we present depends functorially on the diagram and is itself a functor to the indexing category of the diagram. The entire process is, in a way we make precise, an instance of a concrete Grothendieck construction.

The law of importation between fuzzy implications and conjunctions is an important property in both the theory and the application of fuzzy logic. Although many interesting results have been reported in the literature, the related open problem of finding all possible pairs of implications and conjunctions satisfying the law of importation has not been fully solved. As a continuation of two articles S. Massanet et al. (2018) [23] ; S. Massanet et al. (2018) [24] , this article aims to characterize fuzzy implication solutions with a continuous natural negation to the law of importation with respect to a fixed conjunctive uninorm having continuous underlying operators in several cases. Such solutions for the cases where the continuous underlying triangular norm and triangular conorm of a prefixed uninorm are idempotent or Archimedean are completely characterized, and the solutions for the cases where the continuous underlying operators are given by ordinal sums are characterized under some additional assumptions. Finally, some comments on relations to recent independent work W.-H. Li and F. Qin (2021) [25] are given. These two independent articles will go a further step towards answering the open problem regarding the law of importation.

In big data time, selecting informative features has become an urgent need. However, due to the huge cost of obtaining enough labeled data for supervised tasks, researchers have turned their attention to semisupervised learning, which exploits both labeled and unlabeled data. In this article, we propose a sparse discriminative semisupervised feature selection (SDSSFS) method. In this method, the e-dragging technique for the supervised task is extended to the semisupervised task, which is used to enlarge the distance between classes in order to obtain a discriminative solution. The flexible l2,p norm is implicitly used as regularization in the new model. Therefore, we can obtain a more sparse solution by setting smaller p. An iterative method is proposed to simultaneously learn the regression coefficients and e-dragging matrix and predicting the unknown class labels. Experimental results on ten real-world datasets show the superiority of our proposed method.

A fuzzy first order and second moment method (FFOSM) is proposed to efficiently estimate time-independent and time-dependent failure credibility in the presence of fuzzy uncertainty. For the time-independent linear performance function with independently, regularly and identically distributed fuzzy inputs, the analytical relationship is firstly derived between the failure credibility and fuzzy safety index, and the relation form is established between fuzzy safety index and fuzzy first two moments of the performance function. Secondly, in the standard space of fuzzy inputs, the minimum infinite norm distance from the coordinate origin to the linear limit state surface is analytically derived, and the point with the smallest infinite norm distance located on the limit state surface is defined as the fuzzy design point. Based on this definition, the geometric and physical meanings are explained for the fuzzy safety index and fuzzy design point, and the fuzzy first order and second moment method for time-independent failure credibility analysis is established for both linear and nonlinear performance function. Thirdly, for time-dependent performance functions, the equivalence is proved between time-dependent fuzzy safety index and the minimum value of transient fuzzy safety indices, on which the iteration steps of fuzzy first order and second moment method is designed for time-dependent fuzzy safety index and failure credibility estimation. Since the proposed fuzzy first order and second moment method is an approximately analytical approach, it has higher efficiency than the fuzzy simulation method and the cut level optimization method. Finally, the effectiveness and applicability of the proposed method are demonstrated by different examples.

We present a complete representation theorem for nullnorms on bounded lattices. Explicitly, any nullnorm on a bounded lattice can be represented in terms of two order-preserving maps, a triangular conorm, a triangular norm and a conditionally associative function. As a particular case, we retrieve the known representation of nullnorms only taking values that are comparable with the annihilator (in terms of two order-preserving maps, a triangular conorm and a triangular norm). As an illustration of the representation theorem, we generate construction methods for nullnorms, especially those taking at least one value that is incomparable with the annihilator.

We provide several characterizations of integral input-to-state stability (iISS) of time-delay systems. These characterizations differ in the way the considered Lyapunov-Krasovskii functionals (LKF) dissipate along the solutions of the system. This dissipation can involve the LKF itself, as in existing iISS characterizations, but can alternatively involve the instantaneous value of the solution's norm (point-wise dissipation), the supremum norm of the state history (history-wise dissipation) or a mix of the two ( $\mathcal{KL}$ dissipation). We show that all of them guarantee iISS. By relying on a recent converse result by Y. Lin \& Y. Wang, we show that most of them are also necessary for iISS. These relaxed dissipation rates simplify the iISS analysis of time-delay systems and contribute to uniforming iISS theory with that of input-free systems. Proofs rely on several results for time-delay systems that may be of interest on their own, including a novel characterization of global asymptotic stability and the fact that iISS is equivalent to global asymptotic stability of the input-free system plus a uniform bounded energy-bounded state property.

With the increase in number of VSC based HVDC systems, it is inevitable that there would be interconnections with weak ac grids. The paper presents the analytical basis for identifying coupling in the power loops which is dependent on grid inductance and operating point by utilizing the impedance-based modeling technique. The formulation is used to define an inverse-based decoupling outer-loop controller for VSC-HVDC system. The concept is extended to HVDC grids and it is shown that the gains obtained using simple inverse may lead to instability. Robust controller tuning with pole-placement and minimization of $H_\infty$ norm is presented for HVDC grids, which form a restricted feedback system. The results show that the proposed method of decoupling results in effective decoupling of the power loops over a large range of SCR values as compared to that with traditional vector control. The system stability is also maintained over a large range of SCR values and PLL gains when a sudden variation of parameters is considered. The stability margins can be enhanced by pole-placement. The observations from the small-signal models are confirmed using detailed time-domain simulations. The proposal is expected to be useful in controller design for HVDC grids.

This paper presents a simple control method for the fully automatic parking of an unmanned vehicle. This method is based on the switching control algorithm and backstepping theory. The fully automatic parking is hard to accomplish since it cannot converge to a specified norm, guarantee the convergence rate, and large uncertainties. Global exponential convergence to any prescribed norm bound (e-convergence) is guaranteed, and the convergence rate is explicitly given. For the design of steering laws, a method based on backstepping is proposed and analyzed in detail. The backstepping-based design of a multichain system is obtained. Moreover, an exponentially e-convergent control algorithm is adopted to guarantee both converge norm and convergence rate. Real road experiment results are presented and the results show the effectiveness of the control strategies.

In this present paper, we propose and analyze a C 1 -conforming virtual element method to solve the so-called one-layer stationary quasi-geostrophic equations (QGE) with applications in the large scale wind-driven ocean circulation, formulated in terms of the stream-function. This problem corresponds to a nonlinear fourth order partial differential equation. The C 1 virtual space and the discrete scheme are built in a straightforward way due to the flexibility of the virtual approach. Under the assumption of small data, we prove well-posedness of the discrete problem by using a fixed-point strategy and under standard assumptions on the computational domain, we establish error estimates in H 2 -norm for the stream-function. Finally, we report four numerical experiments that illustrate the behavior of the proposed scheme and confirm our theoretical results on different families of polygonal meshes.

This paper investigates the finite-time delay-free stability criteria for fractional-order gene regulatory networks (FOGRNs) with feedback regulation time delays. The existence of unique equilibrium points of considered systems is established based on the Banach fixed point theorem and Cauchy–Schwarz inequality. Further, when the order γ satisfies 0 γ 1 and 1 γ 2 , some novel delay-free sufficient conditions are derived to ensure the finite-time stability for addressing fractional order systems by using the ideas of generalized Gronwall–Bellman inequality, equivalent norm techniques, and Laplace transform. In the end, the article comes up with two numerical cases to manifest the applicability and superiority of the obtained theoretical results.

This paper considers the problem of bilinearization and optimal control of a control-affine nonlinear system by projecting the system dynamics onto the Koopman eigenspace. Although there are linearization techniques like Carleman linearization for embedding a finite-dimensional nonlinear system into an infinite-dimensional space, they depend on the analytic property of the vector fields and work only on polynomial space. The proposed method utilizes the Koopman Canonical Transform, specifically the Koopman eigenfunctions of the drift vector field, to transform the dynamics into a bilinear system under certain assumptions. While the bilinearization is exact if there exists a Koopman-invariant finite-dimensional subspace for the drift vector field, sometimes this condition is too conservative. An approximate approach is to minimize an $\mc{L}^2$ norm on the truncated state space. The approximation can be carried out from time-series data without explicit knowledge of the drift vector field. Controllability of the bilinear system is analyzed using the Myhill semigroup method and Lie algebraic structures. Pontryagin's Principle is applied to the bilinear system to yield a two-point bou

This paper investigates the state estimation problem for a class of complex networks, in which the dynamics of each node is subject to Gaussian noise, system uncertainties and nonlinearities. Based on a regularized least-squares approach, the estimation problem is reformulated as an optimization problem, solving for a solution in a distributed way by utilizing a decoupling technique. Then, based on this solution, a class of estimators is designed to handle the system dynamics and constraints. A novel feature of this design lies in the unified modeling of uncertainties and nonlinearities, the decoupling of nodes, and the construction of recursive approximate covariance matrices for the optimization problem. Furthermore, the feasibility of the proposed estimators and the boundedness of the mean-square errors are ensured under a developed criterion, which is easier to check than some typical estimation strategies including the linear matrix inequalities-based and the variance-constrained ones. Finally, the effectiveness of the theoretical results is verified by a numerical simulation.

In this article, we propose a novel feature selection approach, named unsupervised feature selection with constrained l2,0-norm (row-sparsity constrained) and optimized graph (RSOGFS), which unifies feature selection and similarity matrix construction into a general framework instead of independently performing the two-stage process; thus, the similarity matrix preserving the local manifold structure of data can be determined adaptively. Unlike those sparse learning-based feature selection methods that can only solve the relaxation or approximation problems by introducing sparsity regularization term into the objective function, the proposed method directly tackles the original l2,0-norm constrained problem to achieve group feature selection. Two optimization strategies are provided to solve the original sparse constrained problem. The convergence and approximation guarantees for the new algorithms are rigorously proved, and the computational complexity and parameter determination are theoretically analyzed. Experimental results on real-world data sets show that the proposed method for solving a nonconvex problem is superior to the state of the arts for solving the relaxed or approximate convex problems.

Motion planning in unstructured and low-speed environments is a fundamental and difficult task for all mobile robotics. If we view motion planning as an optimization problem, the non-convex collision avoidance constraints and the nonlinear vehicle dynamic constraints make motion planning challenging and time-consuming. In this paper, we propose a Successive Linearization in Feasible Set (SLiFS) algorithm to address these two difficulties. SLiFS consists of two steps. The first step is to iteratively construct convex feasible sets around the current trajectory to approximate non-convex collision avoidance constraints. The second step is to successively linearize the nonlinear dynamic constraints along the current trajectory and further penalize them into the objective function to avoid infeasible linearized constraints, so that the current trajectory can be reshaped within the obtained convex feasible sets by iteratively solving the linearized optimization problem. The main innovation of SLiFS algorithm is that we consider the L₁ norm type penalty function in the second step. We find that the sparsity of the L₁ norm might help to satisfy the robotic dynamic constraints by numerical experiments. Numerical testing results show that our proposed SLiFS algorithm has a high success rate to find feasible trajectories and costs much less time than the classical interior-point method.

Locality-preserving projection (LPP) has been widely used in feature extraction. However, LPP does not use data category information and uses the 2 L -norm for distance measurement, which is highly sensitive to outliers. In this paper, we consider the LPP weight matrix from a supervised perspective and combine the low-rank regression method to propose a new model to discover and extract features. By using the L2,1-norm to constrain the loss function and the regression matrix, not only is the sensitivity to outliers reduced but the low-rank condition of the regression matrix is also restricted. Then, we propose a solution to the optimization problem. Finally, we apply the method to a series of face databases, handwriting digital datasets and palmprint datasets to test the performance, and the experimental results show that this method is effective compared with some existing methods.

In order to improve the effect of the ExpSign activation function (ESAF) and Sinh-Sign activation function (SSAF) on the convergence and robustness of the ZNN model, two fuzzy adaptive activation functions (AFs), named FAESAF and FASSAF, are constructed by using a Mamdani fuzzy logic controller (MFLC) in this paper. Thus, a novel zeroing neural network (ZNN) with FAESAF and FASSAF is proposed to solve the time-varying linear matrix equation (TVLME). Different from the ESAF and SSAF whose parameters are fixed, the newly constructed FAESAF and FASSAF have adaptive property, which come from the fact that their parameters are intelligently generated by the MFLC according to the error norm of the ZNN model. In order to highlight the superior predefined time convergence and robustness of the corresponding ZNN model with the FAESAF and FASSAF, several theorems are provided and the corresponding proof is given in detail. Furthermore, the ESAF and SSAF with different values of parameters are used as a comparison in numerical experiments to verify the superior performance of the FAESAF and FASSAF. From theoretical analysis and numerical results, we can conclude that the ZNN model with the FAESAF and FASSAF has better predefined time convergence and robustness compared to the ZNN model with the ESAF and SSAF under the same conditions.

With intrinsic non-uniform illumination variations in laparoscopic images, such images often suffer from insufficient lighting and low visibility. This in turn may cause incorrect targeting, surgical risk, and extended operating time during laparoscopic surgery. Although various methods for nature image enhancement have been proposed in past decades, surgical laparoscopic image enhancement still needs improving due to issues including naturalness, blurred texture, color cast, and extensive computational time. To address these problems, this paper proposes a distributed Retinex optimization method for laparoscopic image enhancement using refined information fusion. The proposed method minimizes a distributed optimization model with l 2 -norm regularization terms, resulting in a low computational complexity. By integrating the refined image information into the optimization process, our method significantly enhances dark regions while preserving naturalness and texture structures. Experimental results show that our optimization algorithm is more effective than conventional enhancement algorithms in terms of vision augmentation, performance index, and computation time.

This article proposes a second-order conic programming (SOCP) approach to solve distributionally robust two-stage linear programs over 1-Wasserstein balls. We start from the case with distribution uncertainty only in the objective function and then explore the case with distribution uncertainty only in constraints. The former program is exactly reformulated as a tractable SOCP problem, whereas the latter one is proved to be generally NP-hard as it involves a norm maximization problem over a polyhedron. However, it reduces to an SOCP problem if the extreme points of the polyhedron are given as a prior. This motivates the design of a constraint generation algorithm with provable convergence to approximately solve the NP-hard problem. Moreover, the least favorable distribution achieving the worst case cost is given as an ``empirical'' distribution by simply perturbing each original sample for both cases. Finally, experiments illustrate the advantages of the proposed model in terms of the out-of-sample performance and computational complexity.

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists for all positive times and converges to flat Euclidean space. In particular our result implies that the initial manifold must have been diffeomorphic to $\mathbb{R}^n$., Comment: 33 pages, comments welcome

Estimating endmembers and corresponding abundances from mixed pixels are essential steps for hyperspectral unmixing. In hyperspectral unmixing, obtaining accurate unmixing results is difficult since less prior knowledge is available. Besides, the unmixing results are influenced by noise and highly correlated endmembers, so that the obtained abundance maps exist small values which are not present in the image. In this paper, we separate each abundance map into a feature layer and a sparse layer to protect the obtained abundance maps from the above-mentioned factors. The feature layer represents the main information of the abundance map. And the sparse layer contains outliers dominated by the above-mentioned factors. In particular, we design a feature extraction regularization to describe the feature layer and use a weighted l 1 norm to describe the sparse layer. Then, under the framework of non-negative tensor factorization (NTF), we propose a novel unmixing algorithm named spatial feature extraction NTF (SFE-NTF) for hyperspectral unmixing. The proposed SFE-NTF is based on an augmented multiplicative algorithm. Experimental results on both synthetic and real hyperspectral data demonstrate that the proposed algorithm outperforms other state-of-the-art algorithms.

In order to improve the detection effect and simplify the number of sensors of electronic nose system for wound infection detection, the sensor arrays are optimized based on Hilbert-Schmidt independence criterion (HSIC) in this paper. Specifically, the HSIC optimization algorithm of Gaussian kernel function is exploited. The criterion of Hilbert-Schmidt independence is obtained by empirical estimation of norm of Hilbert-Schmidt cross-covariance operator. The main idea of its application in sensor array optimization is that the larger the empirical norm estimates of Hilbert-Schmidt cross-covariance operator, the stronger the correlation between sensor features and labels, the greater the role of sensor features in classification and recognition. The experimental results show that the HSIC optimization method (Gaussian kernel function) performs better than the linear discriminant analysis (LDA) method and the principal component analysis (PCA).

We show that a constant factor approximation of the shortest and closest lattice vector problem in any l p -norm can be computed in time 2 ( 0.802 + e ) n . This matches the currently fastest constant factor approximation algorithm for the shortest vector problem in the l 2 norm. To obtain our result, we combine the latter algorithm for l 2 with geometric insights related to coverings.

One-dimensional bin-packing is a well-known combinatorial optimization problem which is strongly NP-hard. It consists of allocating a given set of items of different sizes into bins of the same capacity to minimize the number of bins used. The capacity of each bin cannot be exceeded. This paper deals with some variants of this problem to take into account the cases when there are items with uncertain sizes. The goal is to obtain robust solutions taking into account possible variations of item sizes around their nominal values. First, two robust approaches are considered which are based on a stability radius calculation, to ensure that the stability radius, measured either with the Manhattan or Chebyshev norm, is not below a given threshold. Then, a complementary robust approach is applied which is based on a relative resiliency calculation. To solve to optimality these robust variants of the bin-packing problem, a compact 0-1 linear programming formulation, which is also valid for the standard bin-packing problem, is introduced. Then, a Dantzig-Wolfe decomposition is suggested in order to provide a set-cover reformulation with a stronger linear relaxation, but an exponential number of columns. Finally, to obtain integer optimal solutions, a branch-and-price algorithm is developed, whose linear relaxation of the set-cover formulation is solved by a dynamic column generation. Numerical experiments are conducted on adapted benchmark sets from the literature. The performance of the branch-and-price algorithm allows us to investigate what protection against uncertainty is offered by each approach, and at which cost of robustness.

In order to resist periodic interfere in robot hardware or environment, a Runge-Kutta type discrete-time circadian rhythms neural network (RK-DCRNN) model is proposed, and investigated to plan the motion of redundant robot manipulators. To achieve the optimal control, a quadratic programming-based acceleration-level hybrid tri-criteria (ALHT) scheme is first designed, which simultaneously minimize the acceleration norm, torque norm, and joint-angle shift-free indices. Second, according to the neural dynamic design method, a continuous-time circadian rhythms neural network model is exploited, and then based on the Runge-Kutta numerical differential method, a discrete-time circadian rhythms neural network model is obtained. Third, the convergence of the proposed RK-DCRNN model is proved by detailed mathematical derivation. Fourth, comparative simulations and physical experiments verify that the proposed RK-DCRNN model can suppress the accumulation of position error in the motion planning of manipulators.

This paper investigates the event-triggered observer-based security consensus and fault detection problem for nonlinear multi-agent systems (MASs) under external disturbances and stochastic false data injection attacks (FDIAs) over a directed communication network. The randomly occurring FDIAs are modeled by random variables that follow the Bernoulli distribution. An observer-based event-triggered control strategy using only local measurements and information from neighboring agents is developed, where the Zeno behavior of event-triggered mechanism (ETM) is excluded. Interestingly, the observer errors are first regarded as disturbance and thus not only the observer errors but also the external disturbances can be attenuated by the $\mathcal{H}_{\infty}$ norm bound. Meanwhile, it is worth highlighting here that the same information used by the state observers is also adopted to construct residuals with adaptive thresholds, whose aim is to detect the faults occurring on any agents. In addition, the accuracy of the observer and the performance of the fault detection mechanism are improved by introducing the disturbance compensation mechanism. Finally, simulation results are provided to illustrate the usefulness and advantages of the proposed strategy.

A nonlinear robust fault detection scheme for synchronous generators in power systems is presented. The higher-order model of a synchronous generator having nonlinearity in dynamics is considered. A nonlinear observer-based fault detection (FD) scheme is proposed so that the designed FD is robust against disturbances, uncertainties, and sensitive to faults. To this end, a mixed L /L induced norm-based framework is proposed, and various faults that are encountered in synchronous generators are investigated. The success of the proposed scheme is demonstrated through an application to IEEE 9-bus and 3-machine systems.

Compared with a single fault, the occurrence, evolution, and composition of coupling faults have more uncertainties and diversities, which make coupling fault classification a challenging topic in academic research and industrial application areas. This brief addresses the classification problems of coupling faults from a new perspective. Specifically, the main innovations are: 1) a classification framework for coupling faults is first proposed, which integrates multiple kernel learning and multilabel dimensionality reduction; 2) label correlations and nonlinear characteristics among coupling faults are fully explored aiming at improving classification performance; and 3) a trace ratio form of l_1 norm-based objective function is designed for improving the robustness of multilabel classifier. Extensive experiments on the hot rolling process (HRP) are finally given to validate the effectiveness of the proposed scheme.

This paper is devoted to the study of high-order finite difference scheme for the nonlinear Schrodinger equation with wave operator. The difference scheme is three level and a five point stencil is used for spatial variable. Existence of solutions is shown using a variant of Brouwer fixed point theorem. The unconditional stability as well as uniqueness of the difference scheme are also discussed in detail. The convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete maximum norm without any restrictions on the mesh sizes. Finally, some numerical experiments and comparisons with other existing methods in the literature are presented. The numerical results show that the high-order difference scheme of this article improves the accuracy of the space and time direction.

This paper investigates the performance of cubic observers in state estimation of linear systems. In particular, the proposed observer yields a smaller estimation error norm in comparison with a linear one. It is then shown that cubic observers can be designed to perform similar to linear observers in presence of disturbances and delays. It also compares a cubic observer with a nonlinear extended observer in a simulation example.

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

In this paper, we introduce a new concept of fuzzy measure and fuzzy integration which has a dynamic position and is different from previous approaches. Our definition of a new type of fuzzy measure deals with distance functions (special L-fuzzy numbers) and is based on continuous triangular norms. By this concept, we construct a new version of measure theory and integration which is more flexible since the measure of the set both depends on the set itself and on the other parameter named by time. Our approach is related to the idea of fuzzy metric spaces. We study some fuzzy measures induced by classical measures. An integral based on the introduced measures is proposed and studied, too. To complete our paper, we prove some limits and convergence theorems.

In this paper, the linearized Backward-Euler (BE) and backward-differential-formula (BDF) schemes are developed for the nonlinear and coupled prey-predator system with finite element method (FEM). The stability of the discrete solution in H 1 -norm is proved and used to deal with the quadratic term appeared in the system. Thanks to the boundness of the discrete solution obtained from the stability, the superclose results without time-step restriction are derived, in which a novel splitting technique is adopted to cope with the estimations related to the discrete time derivatives. Then, based on the high accuracy analysis, the unconditional superconvergence results of order O ( h 2 + τ ) and O ( h 2 + τ 2 ) in H 1 -norm are derived for BE and BDF schemes through post-processing approach, respectively. Here, h is the subdivision parameter and τ, the time step.

The paper aims to develop the stable numerical schemes for generalized time-fractional diffusion equations (GTFDEs) with smooth and non-smooth solutions on the non-uniform grid. In time, the generalized Caputo derivative is discretized by a difference scheme of order ( 2 − α ) on a non-uniform grid where 0 α 1 . Choosing the non-uniform meshes in the case of the smooth and non-smooth solution is also essential, so we graded the mesh in both cases separately. Stability and convergence for smooth as well as non-smooth solutions are obtained in L 2 -norm and L ∞ -norm respectively. Several numerical results are presented to show how the grading of meshes is essential. Also, numerical results validate the efficiency and effectiveness of proposed schemes and show how a non-uniform grid produces better results.

Since the speed that the pth moment of an output signal norm tends to 0 can be used to measure the quality of the output signal, the stability of the pth moment of complex networks has been widely concerned. This article studies fixed-time synchronization in the pth moment for time-varying delay stochastic multilayer networks, where fixed-time synchronization of multilayer networks is realized by using nonlinear feedback or delay feedback, and the concrete expression of fixed settling time is evaluated. Compared with some existing fixed-time control methods, the optimal combination relationship between the exponential power of fixed-time controllers and the pth moment is given, and the designed controllers are continuous function. Finally, the effectiveness of the method is verified by numerical simulations.

Redundant manipulators have been studied and applied in many fields. The trajectory tracking of redundant manipulators is an important topic to explore for applications. This article aims to develop a planning scheme for achieving the trajectory tracking of redundant manipulators, from the receding horizon control (RHC) perspective. For the nonlinear model of manipulators, the linearization operation is conducted to obtain predictive outputs through the forward kinematic equation. Subsequently, an RHC scheme, which minimizes tracking error, velocity norm, and acceleration norm, and directly considers joint limits at three levels as well as the terminal equality constraint, is constructed and further simplified as a convex quadratic programming problem. Furthermore, a recurrent neural network (RNN) model is designed for the constructed RHC scheme, with the help of the technique of converting inequality constraints into equality constraints. The proposed RHC scheme solved by the RNN model is compared with other existing planning schemes and solvers through computer simulations and experiments, without and with the sudden external interference. Simulation and experiment results show that the proposed RHC scheme solved by the RNN model is able to make the redundant manipulator track the given trajectory excellently, and is superior to other existing schemes and solvers in terms of high efficiency, quick-response capacity, and strong robustness.

This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t = 0 . A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L 2 -norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method.

In this paper, we propose two new exponential integrator Fourier pseudo-spectral schemes for nonlinear Dirac (NLD) equation. The proposed schemes are time symmetric, unconditionally stable and preserve the total energy in the discrete level. We give rigorously error analysis and establish error bounds in the general H m -norm for the numerical solutions of the new schemes applied to the NLD equation. In more details, the proposed schemes have the second-order temporal accuracy and spectral spatial accuracy, respectively, without any CFL-type condition constraint. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity to bound the numerical approximate solutions or the mathematical induction. Extensive numerical results are reported to confirm our error bounds and theoretical analysis.

In this paper we formulate a general problem of extreme functional interpolation of real-valued functions of one variable (for finite differences, this is the Yanenko–Stechkin–Subbotin problem) in terms of divided differences. The least value of the -th derivative in , , needs to be calculated over the class of functions interpolating any given infinite sequence of real numbers on an arbitrary grid of nodes, infinite in both directions, on the number axis for the class of interpolated sequences for which the sequence of -th order divided differences belongs to . In the present paper this problem is solved in the case when . The indicated value is estimated from above and below using the greatest and the least step of the grid of nodes.

Let $\mathit{\Pi}$ be a cohomological irreducible cuspidal automorphic representation of ${\rm GL}_2(\mathbb{A}_{\mathbb F})$ with central character $\omega_{\mathit{\Pi}}$ over a totally real number field ${\mathbb F}$. In this paper, we prove the algebraicity of the near central non-critical value of the symmetric fourth $L$-function of $\mathit{\Pi}$ twisted by $\omega_{\mathit{\Pi}}^{-2}$. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of $\mathit{\Pi}$ and the top degree Whittaker period of the Gelbart-Jacquet lift ${\rm Sym}^2\mathit{\Pi}$ of $\mathit{\Pi}$.