Vector-valued Inputs Research Articles

Chaos-guided neural key coordination for improving security of critical energy infrastructures

In this paper, chaos-guided artificial neural learning-based session key coordination for industrial internet-of-things (IIoT) to enhance the security of critical energy infrastructures (CEI) is proposed. An intruder might pose several security problems since the data are transferred across a public network. Although there have been substantial efforts to solve security problems in the IIoT, the majority of them have relied on traditional methods. A wide range of privacy issues (secrecy, authenticity, and access control) must be addressed to protect IIoT systems against attack. Owing to the unique characteristics of IIoT nodes, existing solutions do not properly address the entire security range of IIoT networks. To deal with this, a chaos-based triple layer vector-valued neural network (TLVVNN) is proposed in this paper. A chaos-based exchange of common seed value for the generation of the identical input vector at both transmitter and receiver is also proposed. This technique has several advantages, including (1) it protects IIoT devices by utilizing TLVVNN synchronization to improve CEI security. (2) Here, artificial neural coordination is utilized for the exchange of neural keys between two IIoT nodes. (3) Using this suggested methodology, chaotic synchronization can be achieved, enabling the chaos-based PRNG seed exchange. (4) Vector-valued inputs and weights are taken into consideration for TLVVNN networks. (5) The deep internal architecture is made up of three hidden layers of the neural network and a vector value as input. As a result, the attacker would have great difficulty interpreting the internal structure. Experiments to verify the performance of the proposed technique are conducted, and the findings demonstrate that the proposed technique has greater performance benefits than the existing related techniques.

Periodic Signal Compressors

We propose and study a compression scheme for lossless causal compression of periodic signals. Our compressors produce scalar-valued signals from periodic vector-valued input signals by taking the inner product with another periodic signal. In the simplest case, this amounts to switching through the different components of the input signal with a periodic switch. We refer to such compressors as periodic compressors. In both continuous and discrete time, we investigate conditions under which the original signal can be reconstructed from the compressed signal, i.e., we characterize lossless periodic compressors. Our conditions amount to certain non-resonances between the periods of our two signals. Finally, we insert our periodic compressor into a feedback loop and investigate how this affects the stability properties of a control system.

Piecewise Linear Approximation of Vector-Valued Images and Curves via Second-Order Variational Model.

Variational models are known to work well for addressing image restoration/regularization problems. However, most of the methods proposed in the literature are defined for scalar inputs and are used on multiband images (such as RGB or multispectral imagery) by the composition of a simple band-wise processing. This involves suboptimal results and may introduce artifacts. Only in a few cases, variational models are extended to the case of vector-valued inputs. However, the known implementations are restricted to the first-order models, while the second-order models are never considered. Thus, typical problems of the first-order models, such as the staircasing effect cannot be overtaken. This paper considers a second-order functional model to function approximation with free discontinuities given by Blake-Zisserman (BZ) and proposes an efficient minimization algorithm in the case of vector-valued inputs. In the BZ model, the Hessian of the solution is penalized outside a set of finite length, therefore the solution is forced to be piecewise linear. Moreover, the model allows the formation of free discontinuities and free gradient discontinuities. The proposed algorithm is applied to difficult color image restoration/regularization problems and to piecewise linear approximation of curves in space.

SkData: data sets and algorithm evaluation protocols in Python

Machine learning benchmark data sets come in all shapes and sizes, whereas classification algorithms assume sanitized input, such as (x, y) pairs with vector-valued input x and integer class label y. Researchers and practitioners know all too well how tedious it can be to get from the URL of a new data set to a NumPy ndarray suitable for e.g. pandas or sklearn. The SkData library handles that work for a growing number of benchmark data sets (small and large) so that one-off in-house scripts for downloading and parsing data sets can be replaced with library code that is reliable, community-tested, and documented. The SkData library also introduces an open-ended formalization of training and testing protocols that facilitates direct comparison with published research. This paper describes the usage and architecture of the SkData library.

Reduced dimensional Gaussian process emulators of parametrized partial differential equations based on Isomap

In this paper, Isomap and kernel Isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a Gaussian process emulator of parametrized partial differential equations. The output space consists of spatial or spatio-temporal fields that are functions of multiple input variables. For such problems, standard multi-output Gaussian process emulation strategies are computationally impractical and/or make restrictive assumptions regarding the correlation structure. The method we develop can be applied without modification to any problem involving vector-valued targets and vector-valued inputs. It also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space. Comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction.

A representation result for hysteresis operators with vector valued inputs and its application to models for magnetic materials

In this work, hysteresis operators mapping continuous vector-valued input functions being piecewise monotaffine, i.e. being piecewise the composition of a monotone with an affine function, to vector-valued output functions are considered.It is shown that the operator can be generated by a unique defined function on the convexity triple free strings. A formulation of a congruence property for periodic inputs is presented and reformulated as a condition for the generating string function.

Representation of hysteresis operators for vector-valued inputs by functions on strings

In 1996, Brokate and Sprekels have shown that scalar-valued hysteresis operators for scalar-valued continuous input functions being piecewise monotone can be uniquely represented by functionals defined on the set of all finite alternating strings of real numbers.In this work, it is shown that a similar result can also be derived for hysteresis operators dealing with inputs in a general normed vector space. Considering hysteresis operators defined for continuous inputs that are piecewise monotaffine, it will be shown that these operators can be uniquely represented by functionals acting on an appropriate set of finite strings of elements of this space.

$\Bbb L^2$ Sufficient Conditions for End-Constrained Optimal Control Problems With Inputs in a Polyhedron

An $\hbox{{\bbb L}}^2$-local optimality sufficiency theorem is proved for a class of structured infinite-dimensional nonconvex programs with constraints of the form $u\in\Omega$ and h(u)=0, where $\Omega$ is a set of Lebesgue measurable essentially bounded vector-valued functions $u(\cdot ): [0,1]\rightarrow \hbox{{\bbb R}}^m$ with range in a polyhedron U, and h is a smooth map of the space of essentially bounded functions $u(\cdot )$ into ${\bbb R}^k$. The sufficiency theorem is based on formal counterparts of the finite-dimensional Karush--Kuhn--Tucker sufficient conditions in a Cartesian product of polyhedra, a strengthened variant of Pontryagin's necessary condition, and structure and continuity conditions on the first and second differentials of the objective function and equality constraint functions. The new sufficient conditions are directly applicable to nonconvex continuous-time Bolza optimal control problems with control-quadratic Hamiltonians, unqualified affine inequality constraints on vector-valued control inputs, and equality constraints on the terminal state vector or equivalent isoperimetric constraints on integrals of functions depending on the state and control variables.

Approximation of Myopic Systems Whose Inputs Need Not BeContinuous

Our main result is a theorem that gives, in a certain setting, a necessary and sufficient condition under which multidimensional shift-invariant maps with vector-valued inputs drawn from a certain large set can be uniformly approximated arbitrarily well using a structure consisting of a linear preprocessing stage followed by a memoryless nonlinear network. The inputs considered need not be continuous, and noncausal as well as causal maps are addressed. Approximations for noncausal maps for which inputs and outputs are functions of more than one variable are of current interest in connection with, for example, image processing.

Uniform approximation of multidimensional myopic maps

Our main result is a theorem that gives, in a certain setting, a necessary and sufficient condition under which multidimensional shift-invariant input-output maps with vector-valued inputs drawn from a certain large set can be uniformly approximated arbitrarily well using a structure consisting of a linear preprocessing stage followed by a memoryless nonlinear network. Noncausal as well as causal maps are considered. Approximations for noncausal maps for which inputs and outputs are functions of more than one variable are of current interest in connection with, for example, image processing.

Uniform Approximation of Discrete-Space Multidimensional Myopic Maps

Our main result is a theorem that gives, in a certain setting, a necessary and sufficient condition under which discrete-space multidimensional shift-invariant input-output maps with vector-valued inputs drawn from a certain large set can be uniformly approximated arbitrarily well using a structure consisting of a linear preprocessing stage followed by a memoryless nonlinear network. Noncausal as well as causal maps are considered. Approximations for noncausal maps for which inputs and outputs are functions of more than one variable are of current interest in connection with, for example, image processing.

On ${\text{L}}^2 $ Sufficient Conditions and the Gradient Projection Method for Optimal Control Problems

${\text{L}}^2 $-local convergence and active constraint identification theorems are proved for gradient projection iterates in sets of ${\text{L}}^\infty $ functions on $[0,1]$ with range in a polyhedron $U \subset \mathbb{R}^m $. These theorems extend earlier results for $U = [0,\infty ) \subset \mathbb{R}^1 $ and are based on an infinite-dimensional variant of the Karush–Kuhn–Tucker second-order sufficient conditions in polyhedral subsets of $\mathbb{R}^n $. The new sufficient conditions and convergence results proved here are directly applicable to continuous-time optimal control problems with smooth nonconvex nonquadratic objective functions and Hamiltonians that are quadratic in the control input vector u. In particular, these theorems apply to nonconvex nonquadratic regulator problems with control-linear state equations and vector-valued inputs $u(t)$ satisfying unqualified affine inequality constraints at almost all t in $[0,1]$.

G-Representations and differential equations

Earlier results concerning g- and h-representations for nonlinear input-output maps are extended so that vector-valued inputs and outputs can be treated. It is established that the range of applicability of the extended results is very wide, and it is illustrated how they can be used, within the contexts of systems governed by differential equations. In particular, it is shown that the nonlinear variation of constants formula is a special case of the main representation theorem. >

Vector median filters

Two nonlinear algorithms for processing vector-valued signals are introduced. The algorithms, called vector median operations, are derived from two multidimensional probability density functions using the maximum-likelihood-estimate approach. The underlying probability densities are exponential, and the resulting operations have properties very similar to those of the median filter. In the vector median approach, the samples of the vector-valued input signal are processed as vectors. The operation inherently utilizes the correlation between the signal components, giving the filters some desirable properties. General properties as well as the root signals of the vector median filters are studied. The vector median operation is combined with linear filtering, resulting in filters with improved noise attenuation and filters with very good edge response. An efficient algorithm for implementing long vector median filters is presented. The noise attenuation of the filters is discussed, and an application to velocity filtering is shown.

Axon shape as a basis for multinode functional units in a hierarchical neural model

The ability of animals to perform fixed action patterns and to access information by categories suggests that there are several types of hierarchical organization in the nervous system. This paper employs data about axon shape and neurotransmitter effect to demonstrate the emergence of hierarchical structure in a neural model. Two dimensions of neural classification, axon shape and neurotransmitter effect, are used to generate a five-node-type neural model. Neurons are classified as interneurons, relay cells, and monoamine transmitters on the basis of axon shape; the transmitter classifications include excitatory, inhibitory, and parameter-changing. The five types of nodes in the model correspond to all the biologically observed combinations: excitatory and inhibitory short-range, excitatory and inhibitory long-range-directional, and long-lasting long-range-diffuse nodes. The emergence of multinode functional units (MFUs) from the five-node-type model is mathematically demonstrated. These units correspond to cortical columns anatomically defined by the axon fields of relay cells, and are called columnar multinode functional units (CMFUs). CMFUs may, in turn, be part of larger functional groups designated coherent populations, which consist of widely distributed CMFUs in retinotopically equivalent locations. The existence of coherent populations imposes a three-level hierarchical structure on the model. To represent this hierarchical structure, a new type of CMFU node, which has a set of vector-valued inputs and outputs, is introduced. Each CMFU node contains a system of short-range nodes which supplies it with vector-valued inputs. Sets of long-range-diffuse nodes are also treated as vector-valued nodes whose outputs control the size and number of coherent populations. The role of coherent populations and hierarchical organization in the nervous system is discussed for such cognitive tasks as visual perception, attention and learning. Physiological and behavioral evidence are cited which support the existence of a similar three-level hierarchy in vertebrate brains.