Showing papers on "Function (mathematics) published in 2021"
TL;DR: A new deep neural network called DeepONet can lean various mathematical operators with small generalization error and can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations.
Abstract: It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. We study different formulations of the input function space and its effect on the generalization error for 16 different diverse applications. Neural networks are known as universal approximators of continuous functions, but they can also approximate any mathematical operator (mapping a function to another function), which is an important capability for complex systems such as robotics control. A new deep neural network called DeepONet can lean various mathematical operators with small generalization error.
1,667 citations
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TL;DR: Wang et al. as discussed by the authors proposed an Efficient Intersection over Union (EIOU) loss, which explicitly measures the discrepancies of three geometric factors in BBR, i.e., the overlap area, the central point and the side length.
Abstract: In object detection, bounding box regression (BBR) is a crucial step that determines the object localization performance. However, we find that most previous loss functions for BBR have two main drawbacks: (i) Both $\ell_n$-norm and IOU-based loss functions are inefficient to depict the objective of BBR, which leads to slow convergence and inaccurate regression results. (ii) Most of the loss functions ignore the imbalance problem in BBR that the large number of anchor boxes which have small overlaps with the target boxes contribute most to the optimization of BBR. To mitigate the adverse effects caused thereby, we perform thorough studies to exploit the potential of BBR losses in this paper. Firstly, an Efficient Intersection over Union (EIOU) loss is proposed, which explicitly measures the discrepancies of three geometric factors in BBR, i.e., the overlap area, the central point and the side length. After that, we state the Effective Example Mining (EEM) problem and propose a regression version of focal loss to make the regression process focus on high-quality anchor boxes. Finally, the above two parts are combined to obtain a new loss function, namely Focal-EIOU loss. Extensive experiments on both synthetic and real datasets are performed. Notable superiorities on both the convergence speed and the localization accuracy can be achieved over other BBR losses.
917 citations
TL;DR: In this article, the problem of tracking control for a class of nonlinear time-varying full state constrained systems is investigated, and the intelligent controller and adaptive law are developed.
Abstract: In this article, the problem of tracking control for a class of nonlinear time-varying full state constrained systems is investigated. By constructing the time-varying asymmetric barrier Lyapunov function (BLF) and combining it with the backstepping algorithm, the intelligent controller and adaptive law are developed. Neural networks (NNs) are utilized to approximate the uncertain function. It is well known that in the past research of nonlinear systems with state constraints, the state constraint boundary is either a constant or a time-varying function. In this article, the constraint boundaries both related to state and time are investigated, which makes the design of control algorithm more complex and difficult. Furthermore, by employing the Lyapunov stability analysis, it is proven that all signals in the closed-loop system are bounded and the time-varying full state constraints are not violated. In the end, the effectiveness of the control algorithm is verified by numerical simulation.
246 citations
TL;DR: A time-varying function-based preset-time approach is proposed to realize the convergence in predetermined time to achieve bipartite consensus tracking for second-order multiagent systems with signed directed graphs.
Abstract: This article is concerned with bipartite consensus tracking for second-order multiagent systems with signed directed graphs. A time-varying function-based preset-time approach is proposed to realize the convergence in predetermined time. First, a class of time-varying functions with generalized properties are presented. Second, two time-varying function-based auxiliaries and a corresponding manifold are constructed. Under a structurally balanced and strongly connected graph, a time-varying function-based controller considering the neighboring state is proposed to guarantee that the system trajectory is constrained on the manifold such that bipartite consensus tracking is achieved in preset-time. Third, for first-order multiagent systems, a preset-time controller is further developed with simplified design. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed controllers.
201 citations
TL;DR: A hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions is described, and how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space is shown.
Abstract: We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.
181 citations
13 Sep 2021
TL;DR: In this article, the path loss of a channel consisting of a passive reflectarray-type RIS is characterized as a function of RIS size, link geometry, and the method used to set the element states.
Abstract: A reconfigurable intelligent surface (RIS) employs an array of individually-controllable elements to scatter incident signals in a desirable way; for example, to facilitate links between base stations and mobile stations that would otherwise be blocked. A principal consideration in the study of RIS-enabled propagation channels is path loss. This paper presents a simple yet broadly-applicable method for calculating the path loss of a channel consisting of a passive reflectarray-type RIS. This model is then used to characterize path loss as a function of RIS size, link geometry, and the method used to set the element states. Whereas previous work presumes either (1) an array of parameterizable element patterns and spacings (most useful for analysis of specific designs) or (2) a continuous electromagnetic surface (most useful for determining scaling laws and theoretical limits), this work begins with (1) and is then shown to be consistent with (2), making it possible to identify specific practical designs and scenarios that exhibit the performance predicted using (2). This model is used to further elucidate the matter of path loss of the RIS-enabled channel relative to that of the free space direct and specular reflection channels, which is an important consideration in the design of networks employing RIS technology.
173 citations
TL;DR: This paper proposes a meta-optimization problem to find the least conservative predictors and prescriptors subject to constraints on their out-of-sample disappointment and proves that the best predictor-prescriptor-pair is obtained by solving a distributionally robust optimization problem over all distributions within a given relative entropy distance from the empirical distribution of the data.
Abstract: We study stochastic programs where the decision-maker cannot observe the distribution of the exogenous uncertainties but has access to a finite set of independent samples from this distribution. In this setting, the goal is to find a procedure that transforms the data to an estimate of the expected cost function under the unknown data-generating distribution, i.e., a predictor, and an optimizer of the estimated cost function that serves as a near-optimal candidate decision, i.e., a prescriptor. As functions of the data, predictors and prescriptors constitute statistical estimators. We propose a meta-optimization problem to find the least conservative predictors and prescriptors subject to constraints on their out-of-sample disappointment. The out-of-sample disappointment quantifies the probability that the actual expected cost of the candidate decision under the unknown true distribution exceeds its predicted cost. Leveraging tools from large deviations theory, we prove that this meta-optimization problem admits a unique solution: The best predictor-prescriptor pair is obtained by solving a distributionally robust optimization problem over all distributions within a given relative entropy distance from the empirical distribution of the data.
135 citations
TL;DR: In this paper, the authors derived a lower bound on the variance of the gradient, which depends mainly on the width of the circuit causal cone of each term in the Pauli decomposition of the cost function.
Abstract: Variational quantum algorithms rely on gradient based optimization to iteratively minimize a cost function evaluated by measuring output(s) of a quantum processor. A barren plateau is the phenomenon of exponentially vanishing gradients in sufficiently expressive parametrized quantum circuits. It has been established that the onset of a barren plateau regime depends on the cost function, although the particular behavior has been demonstrated only for certain classes of cost functions. Here we derive a lower bound on the variance of the gradient, which depends mainly on the width of the circuit causal cone of each term in the Pauli decomposition of the cost function. Our result further clarifies the conditions under which barren plateaus can occur.
124 citations
TL;DR: In this paper, it was shown that deep networks are Kolmogorov-optimal approximants for unit balls in Besov spaces and modulation spaces, and that for sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.
Abstract: This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracy—i.e., the approximation error decays exponentially in the number of nonzero weights in the network—of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function—a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.
120 citations
TL;DR: In this article, the data-based two-player zero-sum game problem is considered for linear discrete-time systems and it is proved that the PIQL algorithm is equivalent to the Newton iteration method in the Banach space by using the Fréchet derivative.
Abstract: In this article, the data-based two-player zero-sum game problem is considered for linear discrete-time systems. This problem theoretically depends on solving the discrete-time game algebraic Riccati equation (DTGARE), while it requires complete system dynamics. To avoid solving the DTGARE, the $Q$ -function is introduced and a data-based policy iteration $Q$ -learning (PIQL) algorithm is developed to learn the optimal $Q$ -function by using data collected from the real system. Writing the $Q$ -function in a quadratic form, it is proved that the PIQL algorithm is equivalent to the Newton iteration method in the Banach space by using the Frechet derivative. Then, the convergence of the PIQL algorithm can be guaranteed by Kantorovich’s theorem. For the realization of the PIQL algorithm, the off-policy learning scheme is proposed using real data rather than the system model. Finally, the efficiency of the developed data-based PIQL method is validated through simulation studies.
120 citations
TL;DR: A three-hidden-layer neural network with super approximation power is introduced, which overcomes the curse of dimensionality in approximation power when the variation of ωf(r) as r→0 is moderate and is extended to general bounded continuous functions on a bounded set E⊆Rd.
TL;DR: The Fourier series expansion and radial basis function neural network are incorporated into a function approximator to model time-varying-disturbed function with a known period in nonlinear systems to deal with the problems of the dead zone output and unknown control direction.
Abstract: This article considers the Nussbaum gain adaptive control issue for a type of nonlinear systems, in which some sophisticated and challenging problems, such as periodic disturbances, dead zone output, and unknown control direction are addressed. The Fourier series expansion and radial basis function neural network are incorporated into a function approximator to model time-varying-disturbed function with a known period in nonlinear systems. To deal with the problems of the dead zone output and unknown control direction, the Nussbaum-type function is recommended in the design of the control algorithm. Applying the Lyapunov stability theory and backstepping technique, the proposed control strategy ensures that the tracking error is pulled back to a small neighborhood of origin and all closed-loop signals are bounded. Finally, simulation results are presented to show the availability and validity of the analysis approach.
TL;DR: A novel normalization based on the logarithmic hyperbolic cosine function is proposed to achieve the stabilization for the case of large initial weight errors, which generates a logarathmic HCAF (LHCAF) and a variable scaling factor and step-size LHCAF and VSS-LH CAF are proposed to improve the filtering accuracy and stability.
Abstract: The hyperbolic cosine function with high-order errors can be utilized to improve the accuracy of adaptive filters. However, when initial weight errors are large, the hyperbolic cosine-based adaptive filter (HCAF) may be unstable. In this paper, a novel normalization based on the logarithmic hyperbolic cosine function is proposed to achieve the stabilization for the case of large initial weight errors, which generates a logarithmic HCAF (LHCAF). Actually, the cost function of LHCAF is the logarithmic hyperbolic cosine function that is robust to large errors and smooth to small errors. The transient and steady-state analyses of LHCAF in terms of the mean-square deviation (MSD) are performed for a stationary white input with an even probability density function in a stationary zero-mean white noise. The convergence and stability of LHCAF can be therefore guaranteed as long as the filtering parameters satisfy certain conditions. The theoretical results based on the MSD are supported by the simulations. In addition, a variable scaling factor and step-size LHCAF (VSS-LHCAF) is proposed to improve the filtering accuracy of LHCAF further. The proposed LHCAF and VSS-LHCAF are superior to HCAF and other robust adaptive filters in terms of filtering accuracy and stability.
TL;DR: DeepMoD as discussed by the authors discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives, using a neural network as function approximator and its output to construct the function library, allowing to perform the sparse regression within the neural network.
TL;DR: A global adaptive neural network tracking control method for uncertain strict-feedback nonlinear system with output constraint and dead zone to achieve predefined-time convergence of the tracking error to predefined accuracy within predefined time is proposed.
Abstract: This article proposes a global adaptive neural network tracking control method for uncertain strict-feedback nonlinear system with output constraint and dead zone to achieve predefined-time convergence of the tracking error to predefined accuracy. First, an integral-type Barrier Lyapunov function (BLF) is constructed to handle output constraint. Next, radial basis function neural network (RBFNN) control and robust control are used to tackle unknown nonlinear function. The continuous switching function is designed to switch RBFNN control to robust control when the arguments of the unknown function exceed the active region of neural network. Utilizing the property of radial basis function, we derive the upper bound of the term containing the unknown nonlinear function and design adaptive laws to determine the derived upper bound and robust control gain. Then, the predefined time virtual control inputs are obtained and their derivatives are estimated by finite time differentiator. Finally, we use the dead zone inverse technique to obtain the actual control input. Stability analysis shows the presented control scheme achieves global convergence of the errors to predefined accuracy within predefined time. The simulation results verify the effectiveness of the presented control scheme.
TL;DR: In this paper, the authors characterize the set of extreme points of monotonic functions that are either majorized by a given function f or themselves majorize f and show that these extreme points play a crucial role in many economic design problems.
Abstract: We characterize the set of extreme points of monotonic functions that are either majorized by a given function f or themselves majorize f and show that these extreme points play a crucial role in many economic design problems. Our main results show that each extreme point is uniquely characterized by a countable collection of intervals. Outside these intervals the extreme point equals the original function f and inside the function is constant. Further consistency conditions need to be satisfied pinning down the value of an extreme point in each interval where it is constant. We apply these insights to a varied set of economic problems: equivalence and optimality of mechanisms for auctions and (matching) contests, Bayesian persuasion, optimal delegation, and decision making under uncertainty.
TL;DR: The robust stability and stabilization of Boolean networks with stochastic function perturbations is studied and it is proved that the finite-time stability is reduced to stability in distribution when the intersection of perturbed set and complement set of parameterized set is nonempty.
Abstract: In genetic regulatory networks (GRNs), gene mutations often occur in a stochastic manner. As an important model of GRNs, gene mutations of Boolean networks are always described as function perturbations. This article studies the robust stability and stabilization of Boolean networks with stochastic function perturbations. A kind of parameterized set is constructed, and it is revealed that under the stochastic function perturbations, the property of finite-time stability remains unchanged when the perturbed set and the parameterized set are disjoint. In addition, it is proved that the finite-time stability is reduced to stability in distribution when the intersection of perturbed set and complement set of parameterized set is nonempty. As an application, the robust stabilization problem of Boolean control networks with stochastic function perturbations is discussed, and several necessary and sufficient conditions are presented for the robustness of feedback stabilizers. Finally, the obtained results are used to study the Drosophila melanogaster segmentation polarity gene network and the lac operon in the bacterium Escherichia coil.
TL;DR: In this article, the authors derive modifications of the Kolmogorov-Arnold representation that transfer smoothness properties of the represented function to the outer function and can be well approximated by ReLU networks.
TL;DR: In this article, the best linear predictor of a structural function is estimated using machine learning techniques, such as conditional average structural and treatment effects, and structural derivatives, based on modern machine learning (ML) tools.
Abstract: This paper provides estimation and inference methods for the best linear predictor (approximation) of a structural function, such as conditional average structural and treatment effects, and structural derivatives, based on modern machine learning (ML) tools. We represent this structural function as a conditional expectation of an unbiased signal that depends on a nuisance parameter, which we estimate by modern machine learning techniques. We first adjust the signal to make it insensitive (Neyman-orthogonal) with respect to the first-stage regularization bias. We then project the signal onto a set of basis functions, growing with sample size, which gives us the best linear predictor of the structural function. We derive a complete set of results for estimation and simultaneous inference on all parameters of the best linear predictor, conducting inference by Gaussian bootstrap. When the structural function is smooth and the basis is sufficiently rich, our estimation and inference result automatically targets this function. When basis functions are group indicators, the best linear predictor reduces to group average treatment/structural effect, and our inference automatically targets these parameters. We demonstrate our method by estimating uniform confidence bands for the average price elasticity of gasoline demand conditional on income.
TL;DR: Considering the uncertain nonstrict nonlinear system with dead-zone input, an adaptive neural network (NN)-based finite-time online optimal tracking control algorithm is proposed, using the tracking errors and the Lipschitz linearized desired tracking function as the new state vector.
Abstract: Considering the uncertain nonstrict nonlinear system with dead-zone input, an adaptive neural network (NN)-based finite-time online optimal tracking control algorithm is proposed. By using the tracking errors and the Lipschitz linearized desired tracking function as the new state vector, an extended system is present. Then, a novel Hamilton–Jacobi–Bellman (HJB) function is defined to associate with the nonquadratic performance function. Further, the upper limit of integration is selected as the finite-time convergence time, in which the dead-zone input is considered. In addition, the Bellman error function can be obtained from the Hamiltonian function. Then, the adaptations of the critic and action NN are updated by using the gradient descent method on the Bellman error function. The semiglobal practical finite-time stability (SGPFS) is guaranteed, and the tracking errors convergence to a compact set by zero in a finite time.
1 Aug 2021
TL;DR: PairRE as mentioned in this paper proposes PairRE, a model with paired vectors for each relation representation, which enable an adaptive adjustment of the margin in loss function to fit for different complex relations.
Abstract: Distance based knowledge graph embedding methods show promising results on link prediction task, on which two topics have been widely studied: one is the ability to handle complex relations, such as N-to-1, 1-to-N and N-to-N, the other is to encode various relation patterns, such as symmetry/antisymmetry. However, the existing methods fail to solve these two problems at the same time, which leads to unsatisfactory results. To mitigate this problem, we propose PairRE, a model with paired vectors for each relation representation. The paired vectors enable an adaptive adjustment of the margin in loss function to fit for different complex relations. Besides, PairRE is capable of encoding three important relation patterns, symmetry/antisymmetry, inverse and composition. Given simple constraints on relation representations, PairRE can encode subrelation further. Experiments on link prediction benchmarks demonstrate the proposed key capabilities of PairRE. Moreover, We set a new state-of-the-art on two knowledge graph datasets of the challenging Open Graph Benchmark.
TL;DR: It is shown that explicitly respecting the analytic "Nevanlinna" structure of the Green's function leads to intrinsically positive and normalized spectral functions, and a continued fraction expansion that yields all possible functions consistent with the analytic structure is presented.
Abstract: Simulations of finite temperature quantum systems provide imaginary frequency Green's functions that correspond one to one to experimentally measurable real-frequency spectral functions. However, due to the bad conditioning of the continuation transform from imaginary to real frequencies, established methods tend to either wash out spectral features at high frequencies or produce spectral functions with unphysical negative parts. Here, we show that explicitly respecting the analytic "Nevanlinna" structure of the Green's function leads to intrinsically positive and normalized spectral functions, and we present a continued fraction expansion that yields all possible functions consistent with the analytic structure. Application to synthetic trial data shows that sharp, smooth, and multipeak data is resolved accurately. Application to the band structure of silicon demonstrates that high energy features are resolved precisely. Continuations in a realistic correlated setup reveal additional features that were previously unresolved. By substantially increasing the resolution of real frequency calculations our work overcomes one of the main limitations of finite-temperature quantum simulations.
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TL;DR: In this article, the authors investigate the connection between three different landscape features that have been observed for PQCs: (1) exponentially vanishing gradients (called barren plateaus), (2) exponential cost concentration about the mean, and (3) the exponential narrowness of minina.
Abstract: Optimizing parameterized quantum circuits (PQCs) is the leading approach to make use of near-term quantum computers. However, very little is known about the cost function landscape for PQCs, which hinders progress towards quantum-aware optimizers. In this work, we investigate the connection between three different landscape features that have been observed for PQCs: (1) exponentially vanishing gradients (called barren plateaus), (2) exponential cost concentration about the mean, and (3) the exponential narrowness of minina (called narrow gorges). We analytically prove that these three phenomena occur together, i.e., when one occurs then so do the other two. A key implication of this result is that one can numerically diagnose barren plateaus via cost differences rather than via the computationally more expensive gradients. More broadly, our work shows that quantum mechanics rules out certain cost landscapes (which otherwise would be mathematically possible), and hence our results are interesting from a quantum foundations perspective.
TL;DR: It is concluded, based on the analysis and numerical experiments, that the shifted formulas are of similar accuracy to the unshifted ones, so can safely be used, but that a division-free variant of softmax can suffer from loss of accuracy.
Abstract: Evaluating the log-sum-exp function or the softmax function is a key step in many modern data science algorithms, notably in inference and classification. Because of the exponentials that these functions contain, the evaluation is prone to overflow and underflow, especially in low precision arithmetic. Software implementations commonly use alternative formulas that avoid overflow and reduce the chance of harmful underflow, employing a shift or another rewriting. Although mathematically equivalent, these variants behave differently in floating-point arithmetic
ew{and shifting can introduce subtractive cancellation}. We give rounding error analyses of different evaluation algorithms and interpret the error bounds using condition numbers for the functions. We conclude, based on the analysis and numerical experiments, that the shifted formulas are of similar accuracy to the unshifted ones, so can safely be used, but that a division-free variant of softmax can suffer from loss of accuracy.
TL;DR: In this article, an additive bias correction (ABC) model based on intensity inhomogeneity is proposed, which divides the observed image into three parts: additive bias function, reflection edge structure function and Gaussian noise.
Abstract: Intensity inhomogeneity brings great difficulties to image segmentation. This problem is partly solved by the multiplicative bias field correction model. However, some other problems still exist, such as slow segmentation speed and narrow application field. In this paper, an additive bias correction (ABC) model based on intensity inhomogeneity is proposed. The model divides the observed image into three parts: additive bias function, reflection edge structure function and Gaussian noise. Firstly, the local area and local clustering criterion of intensity inhomogeneity are defined. Secondly, by introducing the level set function, the local clustering criterion is transformed into an energy function based on the level set model. Finally, the structure of the estimated bias field and the reflection edge is computed through the process of minimizing the energy function while the image is segmented. In order to improve the stability of the system, a de-parameterized regularization function and an adaptive data-driven term function are designed. Compared with the traditional multiplicative model, the addition model has faster calculation speed. The proposed model can obtain ideal segmentation effect for images with intensity inhomogeneity. Experiment results show that the proposed method is more robust, faster and more accurate than traditional piecewise and multiplicative models.
TL;DR: A novel ordered SIMP-like interpolation function is proposed to realize the relaxed and scaled stress interpolation, so that only a unique set of density variables are required for the SMMTO problem.
TL;DR: A by-product of this analysis is a tuning recommendation for several existing (non-accelerated) distributed algorithms yielding provably faster (worst-case) convergence rate for the class of problems under consideration.
Abstract: We study distributed composite optimization over networks: agents minimize a sum of smooth (strongly) convex functions–the agents’ sum-utility–plus a nonsmooth (extended-valued) convex one. We propose a general unified algorithmic framework for such a class of problems and provide a convergence analysis leveraging the theory of operator splitting. Distinguishing features of our scheme are: (i) When each of the agent’s functions is strongly convex, the algorithm converges at a linear rate, whose dependence on the agents’ functions and network topology is decoupled ; (ii) When the objective function is convex (but not strongly convex), similar decoupling as in (i) is established for the coefficient of the proved sublinear rate. This also reveals the role of function heterogeneity on the convergence rate. (iii) The algorithm can adjust the ratio between the number of communications and computations to achieve a rate (in terms of computations) independent on the network connectivity; and (iv) A by-product of our analysis is a tuning recommendation for several existing (non-accelerated) distributed algorithms yielding provably faster (worst-case) convergence rate for the class of problems under consideration.
TL;DR: A new comparison rule is obtained, whereby two different IVPFNs may be distinguished, and the proposed operators in MAGDM problems can eliminate bad influences of extreme evaluation values from biased decision makers and capture the interaction between attributes.
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TL;DR: A dimension reduction technique for Bayesian inverse problems with nonlinear forward operators, non-Gaussian priors, and non- Gaussian observation noise is proposed and an analysis that enables control of the posterior approximation error due to this sampling is provided.
Abstract: We propose a dimension reduction technique for Bayesian inverse problems with nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation noise. The likelihood function is approximated by a ridge function, i.e., a map which depends non-trivially only on a few linear combinations of the parameters. We build this ridge approximation by minimizing an upper bound on the Kullback-Leibler divergence between the posterior distribution and its approximation. This bound, obtained via logarithmic Sobolev inequalities, allows one to certify the error of the posterior approximation. Computing the bound requires computing the second moment matrix of the gradient of the log-likelihood function. In practice, a sample-based approximation of the upper bound is then required. We provide an analysis that enables control of the posterior approximation error due to this sampling. Numerical and theoretical comparisons with existing methods illustrate the benefits of the proposed methodology.
TL;DR: A novel concept of dependency is proposed: inner product dependency to describe the classification error, and a criterion function to evaluate the importance of candidate features is proposed to overcome this weakness.
Abstract: Classical fuzzy rough set often uses fuzzy rough dependency as an evaluation function of feature selection. However, this function only retains the maximum member- ship degree of a sample to one decision class, it can not describe the classification error. Therefore, in this work, a novel criterion function for feature selection is proposed to overcome this weakness. To characterize the classification error rate, we first introduce a class of irreflexive and symmetric fuzzy binary relations to redefine the concepts of fuzzy rough approximations. Then, we propose a novel concept of dependency: inner product dependency to describe the classification error, and construct a criterion function to evaluate the importance of candidate features. The proposed criterion function not only can maintain a maximum dependency function, but also guarantees the minimum classification error. The experimental analysis shows that the proposed criterion function is effective for data sets with a large overlap between different categories.