Kernel Vectors Research Articles

Editorial for the special issue “Control‐theoretic approaches for systems in the life sciences”

Editorial for the special issue “Control<scp>‐theoretic</scp> approaches for systems in the life sciences”

Sign‐sensitivity of metabolic networks: Which structures determine the sign of the responses

AbstractPerturbations are ubiquitous in metabolism. A central tool to understand and control their influence on metabolic networks is sensitivity analysis, which investigates how the network responds to external perturbations. We follow here a structural approach: the analysis is based on the network stoichiometry only and it does not require any quantitative knowledge of the reaction rates. We consider perturbations of reaction rates and metabolite concentrations, at equilibrium, and we investigate the responses in the network. For general metabolic systems, this paper focuses on the sign of the responses, that is, whether a response is positive, negative or whether its sign depends on the parameters of the system. In particular, we identify and describe the subnetworks that are the main players in the sign description. These subnetworks are associated to certain kernel vectors of the stoichiometric matrix and are thus independent from the chosen kinetics.

Inverse numerical range and Abel-Jacobi map of Hermitian determinantal representation

Let A be an n×n matrix. The Hermitian parts of A are denoted by ℜ(A)=(A+A⁎)/2 and ℑ(A)=(A−A⁎)/(2i). The kernel vectors of the linear pencil xℜ(A)+yℑ(A)+zIn play a role for the inverse numerical range of A. This kernel vector technique was applied to perform the inverse numerical range of 3×3 symmetric matrices. In this paper, we follow the kernel vector method and apply the Abel theorem for 3×3 Hermitian matrices. We present the elliptic curve group structure of the cubic curve associated to the ternary form of the matrix, and characterize the Abel type additive structure of the divisors of the cubic curve. A numerical example is given to illustrate the characterization related to the Riemann theta representation.

An algebraic multigrid method for elasticity based on an auxiliary topology with edge matrices

AbstractThis article introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second‐order partial differential equations. Based on a discrete energy made up of edge and vertex contributions, we are able to develop coarsening criteria that guarantee two‐level convergence even for systems of equations such as linear elasticity . This energy also allows us to construct prolongations with prescribed sparsity pattern that still preserve kernel vectors exactly. These allow for a straightforward optimization that simplifies parallelization and reduces communication on coarse levels. Numerical experiments demonstrate efficiency and robustness of the method and scalability of the implementation.

Analytical Harmonic Method for Modeling High-Frequency Oscillation With Applications to Aircraft Piston Pump Vibration Analysis

Undesired oscillations are often encountered in controlled systems at steady state due to high-frequency (HF) periodic disturbance and/or feedback noises. This article presents a computationally efficient alternative to derive an analytical harmonic model (AHM) that expresses the oscillatory variable as a series of harmonic kernel vectors and a position-independent harmonic amplitude vector for identifying the dominant harmonic components of the undesired effects on the manipulated variables. Illustrated in the context of an aircraft pressure-controlled piston pump (PC-PP) where the pressure feedback pulsations and the torque disturbances from the piston/slipper assemblies lead to HF swash-plate (SP) oscillation at steady state, the AHM and its significance were investigated experimentally on a PC-PP test-rig capable of simultaneously measuring the HF feedback pressure and SP-angle. Good agreements between the experimental and numerical results validate the AHM, and reveal that the HF SP-oscillation is dominated by its fundamental harmonic component, and primarily contributed by the disturbance torque.

A characterization of bases of tropical kernels in terms of Cramer's rule

The tropical kernel of a Min-Plus matrix is a linear space. It has a finite basis that is uniquely determined up to scalar multiplication. In this paper, we give a characterization of vectors in a basis of the kernel using the tropical analogue of Cramer's rule. We first show that each finite vector in a basis can be obtained by applying Cramer's rule to a submatrix consisting of certain rows of the original matrix. We prove the main results by presenting an algorithm for choosing such rows. Further, we extend the result for kernel vectors that contain infinity in its entries.

Reservoir-based surrogate modeling of dynamic user equilibrium

This paper proposes a new dynamic user equilibrium (DUE) traffic assignment model using reservoir-based network reduction techniques and surrogate dynamic network loading models. A traffic network is decomposed into a reservoir structure, and the DUE problem is formulated as a variational inequality, with an embedded surrogate model for the path delay operator to describe traffic dynamics at the reservoir level. The surrogate model is further enhanced by the reproducing kernel Hilbert space and adaptive sampling to reduce approximation error and improve computational efficiency. To solve the proposed surrogate-based DUE problem on reduced networks, we develop a customized algorithm that integrates the kernel trick with the generalized projection framework. A pre-computation scheme is proposed, which calculates and stores the of kernel matrices and vectors, could further reduce the computational burden. Numerical experiments of the proposed methods show significant reduction of the computational times, by up to 90%, while maintaining low approximation errors (MAPE below 6%), when compared to the exact models and solution methods.

Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations

The paper aims to present the inverse scattering transforms and soliton solutions for nonlocal reverse-time nonlinear Schrödinger equations. The inverse scattering problems are formulated via Riemann–Hilbert problems, and their solutions are determined by the Sokhotski–Plemelj formula, which close the systems for the Jost solutions. Soliton solutions, corresponding to the reflectionless transforms, are generated from zeros and kernel vectors of the Riemann–Hilbert problems with the identity jump matrix.

Reservoir-based surrogate modeling of dynamic user equilibrium

Abstract This paper proposes a new dynamic user equilibrium (DUE) traffic assignment model using reservoir-based network reduction techniques and surrogate dynamic network loading models. A traffic network is decomposed into a reservoir structure, and the DUE problem is formulated as a variational inequality, with an embedded surrogate model for the path delay operator to describe traffic dynamics at the reservoir level. The surrogate model is further enhanced by the reproducing kernel Hilbert space and adaptive sampling to reduce approximation error and improve computational efficiency. To solve the proposed surrogate-based DUE problem on reduced networks, we develop a customized algorithm that integrates the kernel trick with the generalized projection framework. A pre-computation scheme is proposed, which calculates and stores the of kernel matrices and vectors, could further reduce the computational burden. Numerical experiments of the proposed methods show significant reduction of the computational times, by up to 90 % , while maintaining low approximation errors (MAPE below 6 % ), when compared to the exact models and solution methods.

Kernel Cross-Correlator

Cross-correlator plays a significant role in many visual perception tasks, such as object detection and tracking. Beyond the linear cross-correlator, this paper proposes a kernel cross-correlator (KCC) that breaks traditional limitations. First, by introducing the kernel trick, the KCC extends the linear cross-correlation to non-linear space, which is more robust to signal noises and distortions. Second, the connection to the existing works shows that KCC provides a unified solution for correlation filters. Third, KCC is applicable to any kernel function and is not limited to circulant structure on training data, thus it is able to predict affine transformations with customized properties. Last, by leveraging the fast Fourier transform (FFT), KCC eliminates direct calculation of kernel vectors, thus achieves better performance yet still with a reasonable computational cost. Comprehensive experiments on visual tracking and human activity recognition using wearable devices demonstrate its robustness, flexibility, and efficiency. The source codes of both experiments are released at https://github.com/wang-chen/KCC.

Commutation Relations and Discrete Garnier Systems

We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the factors. We also reparameterize these systems in terms of the image and kernel vectors at singular points to obtain a separate birational form. A distinguishing feature of this study is the presence of a symmetry condition on the associated linear problems that only appears as a necessary feature of the Lax pairs for the least degenerate discrete Painlev\'e equations.

On-line quality prediction of batch processes using a new kernel multiway partial least squares method

The conventional data-driven soft sensor methods such as multiway partial least squares have been encountering nonlinear problems in predictions of batch processes, and kernel methods have been used to deal with these problems. In this work, a new data-driven soft sensor method is proposed by developing a Reduced Dual Kernel multiway partial least squares algorithm. First, the number of kernel vectors is reduced by the feature vector selection method. Then, by projecting both input data and the output data into two reduced kernel spaces, dual kernel matrices are established. These two matrices can be used to build PLS models. Finally, the predicted data in the kernel space can be reversely projected onto its original space during online prediction. Comparisons were made among the proposed method and some pervious algorithms through a numerical example and an Escherichia coli fermentation batch process.

Damage localization in a residential-sized wind turbine blade by use of the SDDLV method

The stochastic dynamic damage location vector (SDDLV) method has previously proved to facilitate effective damage localization in truss- and plate-like structures. The method is based on interrogating damage-induced changes in transfer function matrices in cases where these matrices cannot be derived explicitly due to unknown input. Instead, vectors from the kernel of the transfer function matrix change are utilized; vectors which are derived on the basis of the system and state-to-output mapping matrices from output-only state-space realizations. The idea is then to convert the kernel vectors associated with the lowest singular values into static pseudo-loads and apply these alternately to an undamaged reference model with known stiffness matrix. By doing so, the stresses in the potentially damaged elements will, theoretically, approach zero. The present paper demonstrates an application of the SDDLV method for localization of structural damages in a cantilevered residential-sized wind turbine blade. The blade was excited by an unmeasured multi-impulse load and the resulting dynamic response was captured through accelerometers mounted along the blade. The static pseudo-loads were applied to a finite element (FE) blade model, which was tuned against the modal parameters of the actual blade. In the experiments, an undamaged blade configuration was analysed along with different damage scenarios, hereby testing the applicability of the SDDLV method.

Statistical evaluation of characteristic SDDLV-induced stress resultants to discriminate between undamaged and damaged elements

The stochastic dynamic damage location vector (SDDLV) method utilizes the vectors from the kernel of a damaged-induced transfer function matrix change to localize damages in a structure. The kernel vectors associated with the lowest singular values are converted into static pseudo-loads and applied alternately to an undamaged reference model with known stiffness matrix, hereby, theoretically, yielding characteristic stress resultants approaching zero in the damaged elements. At present, the discrimination between potentially damaged elements and undamaged ones is typically conducted on the basis of modified characteristic stress resultants, which are compared to a pre-defined tolerance value, without any thorough statistical evaluation. In the present paper, it is tested whether three widely-used statistical pattern-recognition-based damage-detection methods can provide an effective statistical evaluation of the characteristic stress resultants, hence facilitating general discrimination between damaged and undamaged elements. The three detection methods in question enable outlier analysis on the basis of, respectively, Euclidian distance, Hotelling's T2 statistics, and Mahalanobis distance. The study of the applicability of these methods is based on experimentally obtained accelerations of a cantilevered residential-sized wind turbine blade subjected to an unmeasured multi-impulse load. The characteristic stress resultants are derived by applying the static pseudo-loads to a representative finite element (FE) model of the actual blade.

Computing nullity and kernel vectors using NF-package: Counterexamples

A computational technique for calculating nullity vectors and kernel vectors, using the new Finsler package, is introduced. As an application, three interesting counterexamples are given. The first counterexample shows that the two distributions KerR and NR do not coincide. The second shows that the nullity distribution NP∘ is not completely integrable. The third shows that the nullity distribution NR is not a sub-distribution of the nullity distribution NR∘.

Projective interpolation of polynomial vectors and improved key recovery attack on SFLASH

SFLASH is an instance of the famous C* $$^{-}$$ - multivariate public key cryptographic schemes and it was chosen by the NESSIE cryptographic project of the European Consortium in 2003 as a candidate signature algorithm used for digital signatures on limited-resource devices. Recently, a successful private key recovery attack on SFLASH was proposed by Bouillaguet, Fouque and Macario-Rat by uncovering the kernel properties of quadratic forms of the central map. The most expensive step in the attack is the calculation of kernel vectors of skew-symmetric matrices over a bivariate polynomial ring. Bouillaguet et al. proposed two methods to accomplish this computation. Both methods involve symbolic computation on bivariate polynomials. The first method computes characteristic polynomials of matrices of polynomials and is very expensive. The second method involves a Grobner basis computation and so its complexity is difficult to estimate. In this paper, we show this critical step of calculating kernel vectors can be done by numerical computation on field elements instead of symbolic computation. Our method uses a nondeterministic interpolation of polynomial vectors called projective interpolation, and its complexity can be explicitly evaluated. Experiments show that it is much faster, making the total attack on SFLASH about 30 times faster (the critical step is about 100 times faster) than the first method of Bouillaguet et al. The new method is also slighter faster than their second method.

An efficient KPCA algorithm based on feature correlation evaluation

Classic kernel principal component analysis (KPCA) is less computationally efficient when extracting features from large data sets. In this paper, we propose an algorithm, that is, efficient KPCA (EKPCA), that enhances the computational efficiency of KPCA by using a linear combination of a small portion of training samples, referred to as basic patterns, to approximately express the KPCA feature extractor, that is, the eigenvector of the covariance matrix in the feature extraction. We show that the feature correlation (i.e., the correlation between different feature components) can be evaluated by the cosine distance between the kernel vectors, which are the column vectors in the kernel matrix. The proposed algorithm can be easily implemented. It first uses feature correlation evaluation to determine the basic patterns and then uses these to reconstruct the KPCA model, perform feature extraction, and classify the test samples. Since there are usually many fewer basic patterns than training samples, EKPCA feature extraction is much more computationally efficient than that of KPCA. Experimental results on several benchmark data sets show that EKPCA is much faster than KPCA while achieving similar classification performance.

A new method for constructing kernel vectors in morphological associative memories of binary patterns

Kernel vectors represent an elegant representation for the retrieval of pattern associations, where the input patterns are corrupted by both erosive and dilative noise. However, their action completely fails when a particular kind of erosive noise, even of very low percentage, corrupts the input pattern. In this paper, a theoretical justification of this fact is given and a new method is proposed for the construction of kernel vectors for binary patterns associations. The new kernels are not binary but ?gray?, because they contain elements with values in the interval [0, 1]. It is shown, both theoretically and experimentally that the new kernel vectors carry the good properties of conventional kernel vectors and, at the same time, they can be easily computed. Moreover, they do not suffer from the particular noise deficiency of the conventional kernel vectors. The recalling result is in general a gray pattern, which in the sequel undergoes a simple thresholding action and passes through a simple Hamming network to produce high recall rates, even in heavily corrupted patterns. Retrieval of pattern associations is very significant for a variety of scientific disciplines including data analysis, signal and image understanding and intelligent control.

Nonlinear System Identification Using Deterministic Multilevel Sequences

A new exact method of measuring the Volterra kernels of finite-order discrete nonlinear systems is presented. The kernels are rearranged in terms of multivariate crossproducts in vector form. The one-, two-, . . . , and l-dimensional kernel vectors are determined using a deterministic multilevel sequence with l distinct levels at the input of the system. It is shown that the defined multilevel sequence with l distinct levels is persistently exciting for a truncated Volterra filter with nonlinearities of polynomial degree l. Examples demonstrating the rearrangement of the Volterra kernels and a novel method for estimation of the kernels are presented. Simulation results are given to illustrate the effectiveness of the proposed method.

Observations on morphological associative memories and the kernel method

The ability of human beings to retrieve information on the basis of associated cues continues to elicit great interest among researchers. Investigations of how the brain is capable to make such associations from partial information have led to a variety of theoretical neural network models that act as associative memories. Several researchers have had significant success in retrieving complete stored patterns from noisy or incomplete input pattern keys by using morphological associative memories. Thus far morphological associative memories have been employed in two different ways: a direct approach which is suitable for input patterns containing either dilative or erosive noise and an indirect one for arbitrarily corrupted input patterns which is based on kernel vectors. In a recent paper (P. Sussner, in: Proceedings of the International ICSA/IFAC Symposium on Neural Computation, Vienna, September 1998), we suggested how to select these kernel vectors and we deduced exact statements on the amount of noise which is permissible for perfect recall. In this paper, we establish the proofs for all our claims made about the choice of kernel vectors and perfect recall in kernel method applications. Moreover, we provide arguments for the success of both approaches beyond the experimental results presented up to this point.