Proof Of Convergence Research Articles

On the splitting iteration method for Pareto eigenvalue complementarity problems of H +-matrices

We present a class of inexact splitting-modulus iteration methods for solving the Pareto Eigenvalue Complementarity Problem (EiCP) when the system matrix A is an H + -matrix. Our method first employs the matrix splitting technique to transform the original eigenvalue complementarity problem (EiCP) into an easily solvable Linear Complementarity Problem (LCP), and the inner iterative process corresponds to the inexact split iterative process when the modulus iteration method is used, where we employed an effective and fast inner iterative stopping criterion. We then provide a convergence proof for inexact matrix splitting iteration methods. Numerical experiments demonstrate that our proposed algorithms exhibit significant superiority over the classical matrix splitting iteration method in terms of computational time, especially for solving large-scale symmetric and asymmetric eigenvalue complementarity problems of H + -matrices.

Broad learning system based on maximum multi-kernel correntropy criterion

The broad learning system (BLS) is an effective machine learning model that exhibits excellent feature extraction ability and fast training speed. However, the traditional BLS is derived from the minimum mean square error (MMSE) criterion, which is highly sensitive to non-Gaussian noise. In order to enhance the robustness of BLS, this paper reconstructs the objective function of BLS based on the maximum multi-kernel correntropy criterion (MMKCC), and obtains a new robust variant of BLS (MKC-BLS). For the multitude of parameters involved in MMKCC, an effective parameter optimization method is presented. The fixed-point iteration method is employed to further optimize the model, and a reliable convergence proof is provided. In comparison to the existing robust variants of BLS, MKC-BLS exhibits superior performance in the non-Gaussian noise environment, particularly in the multi-modal noise environment. Experiments on multiple public datasets and real application validate the efficacy of the proposed method.

Building forecasting model for time series based on improvements in establishing fuzzy relationships and clustering algorithm

This research proposes a new forecasting model for time series with important improvements. The first improvement is the use of the variation between two consecutive times as a universal set and dividing it into intervals with an appropriate number using an automatic clustering technique. The second improvement is the establishment of fuzzy relationships between the built intervals and between each element in the series and these intervals. Finally, using the established relationships, a new forecasting rule is created. The model is presented step by step and detailed with numerical examples, and the proofs for algorithm convergence are provided. It outperforms existing models on well-known datasets including the M3 Competition with 3003 series and M4 Competition datasets with 100,000 series. Another important contribution of this study is the establishment of the R procedure to effectively apply the proposed model to real series.

Elementary proof of QAOA convergence

The quantum alternating operator ansatz (QAOA) and its predecessor, the quantum approximate optimization algorithm, are one of the most widely used quantum algorithms for solving combinatorial optimization problems. However, as there is yet no rigorous proof of convergence for the QAOA, we provide one in this paper. The proof involves retracing the connection between the quantum adiabatic algorithm and the QAOA, and naturally suggests a refined definition of the ‘phase separator’ and ‘mixer’ keywords.

Dual hypergraphs with feature weighted and latent space learning for the diagnosis of Alzheimer’s disease

In recent years of research on the diagnosis of Alzheimer’s disease, capturing data relationships can help improve model performance. However, the simple graph structure can only capture pairwise relationships between data, which cannot model the complex data relationships in real situations. In addition, the redundant features and noise from the original data space also harm the model performance. Dual hypergraphs with feature-weighted and latent space learning (DHFWLSL) method for Alzheimer’s disease diagnosis are proposed to solve the abovementioned problems. Hypergraphs are used to capture higher-order data relationships and construct hypergraphs for both sample and feature relationships to fully utilize the auxiliary ability of data relationships for model classification. In order to remove noise and redundant features from the original data space, latent space learning is utilized to project the original data into the latent space, and then data is projected into the label space for diagnosis. Finally, a feature-weighted strategy helps to improve the model performance further. An alternating optimization algorithm with the proof of convergence is used to solve the proposed model. Sufficient experiments support the validity and advantages of the proposed model.

Distributed optimal coordination algorithm for nonlinear second-order multi-agent systems and its application to vehicle platoon

This paper addresses the problem of distributed optimal coordination over second-order multi-agent systems, where the dynamics are nonlinear and unknown. To solve this problem, a state-based observer is designed to handle the unknown matched and unmatched nonlinear terms. Using this observer and the proportional–integral control concept, a distributed optimal coordination algorithm with fixed control parameters is constructed through the consensus scheme and gradient flow method. With the help of the Routh–Hurwitz stability criterion, the exponential convergence proof is guaranteed under the condition that the nonlinear term and the convex local cost function are all locally Lipschitz. Finally, we present four simulation examples, including the vehicle platoon’s cooperative control, to demonstrate the efficiency of our proposed algorithms.

Nonexpansive maps in nonlinear smooth spaces

We introduce the notion of a nonlinear smooth space generalizing both CAT ⁡ ( 0 ) \operatorname {CAT}(0) spaces as well as smooth Banach spaces. We show that this notion allows for a unified treatment of several results in functional analysis. Namely, we substantiate the usefulness of this setting by establishing a nonlinear generalization of an important result due to Reich in Banach spaces. On par with the linear context, this nonlinear version entails a convergence proof of several other methods. Here, we establish the convergence of a general form of the Halpern-type schema for resolvent-like families of functions. We furthermore prove the convergence of the viscosity generalization of Halpern’s iteration (even for families of maps) generalizing a result due to Chang. This work is set in the context of the ‘proof mining’ program, and the results are complemented with quantitative information like rates of convergence and of metastability (in the sense of T. Tao).

Efficient learning in spiking neural networks

Spiking neural networks (SNNs) are a large class of neural model distinct from ‘classical’ continuous-valued networks such as multilayer perceptrons (MLPs). With event-driven dynamics and a continuous-time model in contrast to the discrete-time model of their classical counterparts, they offer interesting advantages in representational capacity and energy consumption. Spiking networks may also be more biologically plausible, offering more insights into neuroscience. However, developing models of learning for SNNs has historically proven challenging: as discrete-time systems, their dynamics are much more complex and they cannot benefit from the strong theoretical developments in MLPs such as convergence proofs and optimal gradient descent. Nor do they gain automatically from algorithmic improvements that have produced efficient matrix inversion and batch training methods. Most of the existing research has focused on the most well-studied learning mechanism in SNNs, spike-timing-dependent plasticity (STDP), and although there has been progress, there are also notable pathologies that have often been solved with a variety of ad-hoc techniques. While efforts have been made to map SNNs to classical convolutional neural networks (CNNs), these have not yet shown any decisive efficiency advantage over conventional CNNs. More promising research directions lie in the realm of pure spiking learning models that exploit the inherent temporal dynamics (and often leverage recurrency). Metrics are needed; one possibility would be a measure of total energy cost per unit reduction in error. This tutorial overview looks at existing techniques for learning in SNNs and offers some thoughts for future directions.

Stick–Slip Suppression in Drill String Systems Using a Novel Adaptive Sliding Mode Control Approach

A novel control technique is presented in this paper, which is based on a first-order adaptive sliding mode that ensures convergence in a finite time without any prior information on the upper limits of the parametric uncertainties and/or external disturbances. Based on an exponent reaching law, this controller uses two dynamically adaptive control gains. Once the sliding mode is reached, the dynamic gains decrease in order to loosen the system’s constraints, which guarantees minimal control effort. The proof of convergence was demonstrated according to Lyapunov’s criterion. The proposed algorithm was applied to a drill string system to evaluate its performance because such systems present variable operating conditions caused by, for example, the type of rock. The effectiveness of the proposed controller was evaluated by conducting a comparative study that involved comparing it against a commonly used sliding mode controller, as well as other recent adaptive sliding mode control techniques. The different mathematical performance measures included energy consumption. The proposed algorithm had the best performance measures with the lowest energy consumption and it was able to significantly improve the functioning of the drill string system. The results indicated that the proposed controller had 20% less chattering than the classic SM controller. Finally, the proposed controller was the most robust to uncertainties in system parameters and external disturbances, thus demonstrating the auto-adjustable features of the controller.

Quadratic Tracking Control of Linear Stochastic Systems with Unknown Dynamics Using Average Off-Policy Q-Learning Method

This article investigates the optimal tracking control problem for data-based stochastic discrete-time linear systems. An average off-policy Q-learning algorithm is proposed to solve the optimal control problem with random disturbances. Compared with the existing off-policy reinforcement learning (RL) algorithm, the proposed average off-policy Q-learning algorithm avoids the assumption of an initial stability control. First, a pole placement strategy is used to design an initial stable control for systems with unknown dynamics. Second, the initial stable control is used to design a data-based average off-policy Q-learning algorithm. Then, this algorithm is used to solve the stochastic linear quadratic tracking (LQT) problem, and a convergence proof of the algorithm is provided. Finally, numerical examples show that this algorithm outperforms other algorithms in a simulation.

Fusion of low-rankness and smoothness under learnable nonlinear transformation for tensor completion

The recently proposed tensor correlated total variation (t-CTV) has achieved success in tensor completion. It utilizes the low-rank structure of the gradient tensor under a unified linear transform to jointly encode low-rankness and smoothness priors. However, fixed linear transforms have inherent limitations in fully characterizing gradient tensors in different directions and adapting them to tensors from diverse categories. In this work, we propose the nonlinear tensor correlated total variation (NTCTV) regularization term that leverages the low-rank correlations of the gradient tensor under the learnable nonlinear transformation, providing a more natural approach to fuse the low-rankness and smoothness priors. Specifically, our approach learns the optimal nonlinear implicit low-rank structure of the gradient tensor along different modes separately, and then achieves the expression of fused prior information in a coupled manner. Furthermore, we propose the NTCTV-based tensor completion model and design the proximal alternating minimization (PAM) algorithm to efficiently solve the optimization model. Moreover, we provide a theoretical proof of the global convergence of the algorithm to a critical point. Comprehensive experimental results for hyperspectral images, medical images, multispectral images, and videos demonstrate that the proposed method achieves substantial quantitative and qualitative improvements over many state-of-the-art tensor completion techniques.

Another modified version of RMIL conjugate gradient method

Due to their simplicity and global convergence properties, the conjugate gradient (CG) methods are widely used for solving unconstrained optimization problems, especially those of large scale. To establish the global convergence and to obtain better numerical performance in practice, much effort has been devoted to develop new CG methods or even to modify well- known methods. In 2012, Rivaie et al., have proposed a new CG method, called RMIL which has good numerical results and globally convergent under the exact line search. However, in 2016, Dai has pointed out a mistake in the steps of the proof of global convergence of RMIL and hence to guarantee the global convergence he suggested a modified version of RMIL, called RMIL+. In this paper, we present another modified version of RMIL, which is globally convergent via the exact line search. Furthermore, to support the theoretical proof of the global convergence of the modified version in practical computation, a numerical experiment based on comparing it with RMIL, RMIL+, and CG-DESCENT was done.

TPMS2STEP: Error-Controlled and [formula omitted] Continuity-Preserving Translation of TPMS Models to STEP Files Based on Constrained-PIA

Triply periodic minimal surface (TPMS) is emerging as an important way of designing microstructures. However, there has been limited use of commercial CAD/CAM/CAE software packages for TPMS design and manufacturing. This is mainly because TPMS is consistently described in the functional representation (F-rep) format, while modern CAD/CAM/CAE tools are built upon the boundary representation (B-rep) format. One possible solution to this gap is translating TPMS to STEP, which is the standard data exchange format of CAD/CAM/CAE. Following this direction, this paper proposes a new translation method with error-controlling and C2 continuity-preserving features. It is based on an approximation error-driven TPMS sampling algorithm and a constrained-PIA algorithm. The sampling algorithm controls the deviation between the original and translated models. With it, an error bound of 2ϵ on the deviation can be ensured if two conditions called ϵ-density and ϵ-approximation are satisfied. The constrained-PIA algorithm enforces C2 continuity constraints during TPMS approximation, and meanwhile attaining high efficiency. A theoretical convergence proof of this algorithm is also given. The effectiveness of the translation method has been demonstrated by a series of examples and comparisons. The code will be open-sourced upon publication.

Newton Geometric Iterative Method for B-Spline Curve and Surface Approximation

We introduce a progressive and iterative method for B-spline curve and surface approximation, incorporating parameter correction based on the Newton iterative method. While parameter corrections have been used in existing Geometric Approximation (GA) methods to enhance approximation quality, they suffer from low computational efficiency. Our approach unifies control point updates and parameter corrections in a progressive and iterative procedure, employing a one-step strategy for parameter correction. We provide a theoretical proof of convergence for the algorithm, demonstrating its superior computational efficiency compared to current GA methods. Furthermore, the provided convergence proof offers a methodology for proving the convergence of existing GA methods with location parameter correction.

Functional central limit theorems for rough volatility

The non-Markovian nature of rough volatility makes Monte Carlo methods challenging, and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker’s approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent H∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H \\in (0,1)$\\end{document}. Some of the most relevant consequences of this ‘rough Donsker (rDonsker) theorem’ are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme and find remarkable agreement (for a large range of values of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}). Our rDonsker theorem further provides a weak convergence proof for the hybrid scheme itself and allows constructing binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options.

Semi-explicit integration of second order for weakly coupled poroelasticity

We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes.

Enforced Block Diagonal Graph Learning for Multikernel Clustering

The existing multikernel graph clustering (MKGC) methods have emerged with notable success on nonlinear clustering tasks since the graph learning can effectively capture graph structure similarity between sample pair. Ideally, a high-quality graph should enjoy the good block diagonal property, i.e. the intercluster similarities correspond to zeros, while intracluster similarities represent nonzeros. Meanwhile, the number of diagonal blocks for a graph equals to the number of clusters on the dataset. However, most of the existing MKGC methods design a corpulent three-parts graph leaning process that poses challenges for hyperparameter tuning, time cost, and clustering performance. To overcome these challenging issues, we propose an enforced block diagonal graph learning for multikernel clustering (EBDGL-MKC) method, where we pursue a high-quality block diagonal graph via well-designed one-part graph leaning scheme rather than three parts. Inspired by symmetric matrix factorization (SMF), we first design a one-part block diagonal graph learning scheme to learn multiple block diagonal graphs, by exploring an explicit theoretical connection between the clustering partition of kernel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula> -means and the excellent block diagonal graph. Then, these block diagonal graphs are stacked into a low-rank tensor for exploiting the high-order structure information hidden in the nonlinear data. After that, an effective alternate algorithm with convergence proof is performed on extensive experiments to demonstrate the superiority of EBDGL compared with the state-of-the-art multikernel clustering (MKC) methods.

On the reconstruction of unknown driving forces from low-mode observations in the 2D Navier–Stokes equations

This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which makes use of a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier–Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decreases to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned.

Connection probabilities of multiple FK-Ising interfaces

We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight q∈[1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${q \\in [1,4)}$$\\end{document}. Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of q than the FK-Ising model (q=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=2$$\\end{document}). Given the convergence of interfaces, the conjectural formulas for other values of q could be verified similarly with relatively minor technical work. The limit interfaces are variants of SLEκ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SLE}_\\kappa $$\\end{document} curves (with κ=16/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa = 16/3$$\\end{document} for q=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=2$$\\end{document}). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all q∈[1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q \\in [1,4)$$\\end{document}, thus providing further evidence of the expected CFT description of these models.

Convergence proof for the GenCol algorithm in the case of two-marginal optimal transport

The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport (OT) plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor β \beta . Here we prove that for β ≥ 2 \beta \geq 2 and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from O ( ℓ 2 ) O(\ell ^2) to O ( ℓ ) O(\ell ) without any loss in accuracy, where ℓ \ell is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.