Iterative Algorithm Research Articles

Generalized Derivatives and Optimality Conditions in Nonconvex Optimization

In this paper, we study the radial epiderivative notion for nonconvex functions, which extends the (classical) directional derivative concept. The paper presents new definition and new properties for this notion and establishes relationships between the radial epiderivative, the Clarke’s directional derivative, the Rockafellar’s subderivative and the directional derivative. The radial epiderivative notion is used to establish new regularity conditions without convexity conditions. The paper presents explicit formulations for computing the radial epiderivatives in terms of weak subgradients and vice versa. We also present an iterative algorithm for approximate computing of radial epiderivatives and show that the algorithm terminates in a finite number of iterations. The paper analyzes necessary and sufficient conditions for global optimums in nonconvex optimization via the radial epiderivatives. We formulate a necessary and sufficient condition for a global descent direction for radially epidifferentiable nonconvex functions. All the properties and theorems presented in this paper are illustrated and interpreted on examples.

Effective Causal Discovery under Identifiable Heteroscedastic Noise Model

Capturing the underlying structural causal relations represented by Directed Acyclic Graphs (DAGs) has been a fundamental task in various AI disciplines. Causal DAG learning via the continuous optimization framework has recently achieved promising performance in terms of accuracy and efficiency. However, most methods make strong assumptions of homoscedastic noise, i.e., exogenous noises have equal variances across variables, observations, or even both. The noises in real data usually violate both assumptions due to the biases introduced by different data collection processes. To address the heteroscedastic noise issue, we introduce relaxed implementable sufficient conditions and prove the identifiability of a general class of SEM subject to those conditions. Based on the identifiable general SEM, we propose a novel formulation for DAG learning which accounts for the noise variance variation across variables and observations. We then propose an effective two-phase iterative DAG learning algorithm to address the increasing optimization difficulties and learn a causal DAG from data with heteroscedastic variables noise under varying variance. We show significant empirical gains of the proposed approaches over state-of-the-art methods on both synthetic data and real data.

Iterative Regularization with k-support Norm: An Important Complement to Sparse Recovery

Sparse recovery is ubiquitous in machine learning and signal processing. Due to the NP-hard nature of sparse recovery, existing methods are known to suffer either from restrictive (or even unknown) applicability conditions, or high computational cost. Recently, iterative regularization methods have emerged as a promising fast approach because they can achieve sparse recovery in one pass through early stopping, rather than the tedious grid-search used in the traditional methods. However, most of those iterative methods are based on the l1 norm which requires restrictive applicability conditions and could fail in many cases. Therefore, achieving sparse recovery with iterative regularization methods under a wider range of conditions has yet to be further explored. To address this issue, we propose a novel iterative regularization algorithm, IRKSN, based on the k-support norm regularizer rather than the l1 norm. We provide conditions for sparse recovery with IRKSN, and compare them with traditional conditions for recovery with l1 norm regularizers. Additionally, we give an early stopping bound on the model error of IRKSN with explicit constants, achieving the standard linear rate for sparse recovery. Finally, we illustrate the applicability of our algorithm on several experiments, including a support recovery experiment with a correlated design matrix.

Deep Unfolded Network with Intrinsic Supervision for Pan-Sharpening

Existing deep pan-sharpening methods lack the learning of complementary information between PAN and MS modalities in the intermediate layers, and exhibit low interpretability due to their black-box designs. To this end, an interpretable deep unfolded network with intrinsic supervision for pan-sharpening is proposed. Building upon the observation degradation process, it formulates the pan-sharpening task as a variational model minimization with spatial consistency prior and spectral projection prior. The former prior requires a joint component decomposition of PAN and MS images to extract intrinsic features. By being supervised in the intermediate layers, it can selectively provide high-frequency information for spatial enhancement. The latter prior constrains the intensity correlation between MS and PAN images derived from physical observations, so as to improve spectral fidelity. To further enhance the transparency of network design, we develop an iterative solution algorithm following the half-quadratic splitting to unfold the deep model. It rigorously adheres to the variational model, significantly enhancing the interpretability behind network design and efficiently alternating the optimization of the network. Extensive experiments demonstrate the advantages of our method compared to state-of-the-arts, showcasing its remarkable generalization capability to real-world scenes. Our code is publicly available at https://github.com/Baixuzx7/DISPNet.

Point2Real: Bridging the Gap between Point Cloud and Realistic Image for Open-World 3D Recognition

Recognition in open-world scenarios is an important and challenging field, where Vision-Language Pre-training paradigms have greatly impacted the 2D domain. This inspires a growing interest in introducing 2D pre-trained models, such as CLIP, into the 3D domain to enhance the ability of point cloud understanding. Considering the difference between discrete 3D point clouds and real-world 2D images, reducing the domain gap is crucial. Some recent works project point clouds onto a 2D plane to enable 3D zero-shot capabilities without training. However, this simplistic approach leads to an unclear or even distorted geometric structure, limiting the potential of 2D pre-trained models in 3D. To address the domain gap, we propose Point2Real, a training-free framework based on the realistic rendering technique to automate the transformation of the 3D point cloud domain into the Vision-Language domain. Specifically, Point2Real leverages a shape recovery module that devises an iterative ball-pivoting algorithm to convert point clouds into meshes, narrowing the gap in shape at first. To simulate photo-realistic images, a set of refined textures as candidates is applied for rendering, where the CLIP confidence is utilized to select the suitable one. Moreover, to tackle the viewpoint challenge, a heuristic multi-view adapter is implemented for feature aggregation, which exploits the depth surface as an effective indicator of view-specific discriminability for recognition. We conduct experiments on ModelNet10, ModelNet40, and ScanObjectNN datasets, and the results demonstrate that Point2Real outperforms other approaches in zero-shot and few-shot tasks by a large margin.

Sketched Newton Value Iteration for Large-Scale Markov Decision Processes

Value Iteration (VI) is one of the most classic algorithms for solving Markov Decision Processes (MDPs), which lays the foundations for various more advanced reinforcement learning algorithms, such as Q-learning. VI may take a large number of iterations to converge as it is a first-order method. In this paper, we introduce the Newton Value Iteration (NVI) algorithm, which eliminates the impact of action space dimension compared to some previous second-order methods. Consequently, NVI can efficiently handle MDPs with large action spaces. Building upon NVI, we propose a novel approach called Sketched Newton Value Iteration (SNVI) to tackle MDPs with both large state and action spaces. SNVI not only inherits the stability and fast convergence advantages of second-order algorithms, but also significantly reduces computational complexity, making it highly scalable. Extensive experiments demonstrate the superiority of our algorithms over traditional VI and previously proposed second-order VI algorithms.

Physical‐layer security enhancement in wireless sensor networks through artificial noise optimization

AbstractEnsuring network security is of utmost importance, especially in wireless sensor networks (WSNs), where the confidentiality of data is at risk due to eavesdropping. This paper introduces an artificial‐noise‐aided secure communication scheme for WSNs. This scheme comprises numerous sensor nodes, the sink, a friendly jammer, and an eavesdropper. The primary aim is to enhance the secrecy rate of the network by optimizing the jamming power. The friendly jammer plays a crucial role in degrading the wiretap channel between the sensor nodes and the eavesdropper. The optimization process revolves around maximizing the secrecy rate, while carefully managing the jamming power to prevent any negative impact on the main transmission link between the sink node and the sensor nodes. Due to the nonconvexity of the secrecy rate problem, the convex approximation‐based algorithm is recommened to get a safe convex approximation of the original solution. To address this optimization problem, an iterative algorithm based on quadratic transformation is proposed. Afterwards, the optimum jamming power to achieve high security is obtained. The simulation outcomes underscore the dual impact of the jammer in WSNs. The jammer diminishes the eavesdropper channel. Unfortunately, it introduces interference that consequently affects the security enhancement. In response, we concentrate on the optimization of jamming power. The proposed scheme consistently elevates secrecy rates across a spectrum of eavesdropper positions, adeptly addressing worst‐case scenarios. Notably, the algorithm showcases resilience across diverse node distributions.

Variational-like inclusion problems involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p. and fixed point problems

This paper is devoted to the introduction of a new class of accretive mappings in the framework of q-uniformly smooth Banach spaces called A-maximal m-relaxed η-accretive mappings in the sense of semi-inner product of type (p) (for short, s.i.p.(p)). The resolvent operator associated with such a mapping is defined; its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed under some suitable conditions. At the same time, the problem of finding a common element of the set of fixed points of a given nearly asymptotically nonexpansive mapping and the set of solutions of a variational-like inclusion problem involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is investigated. To achieve this goal, a new iterative algorithm is constructed. Using the notions of graph convergence and resolvent operator associated with an A-maximal m-relaxed η-accretive mapping in the sense of s.i.p.(p), a new equivalence relationship between the graph convergence and the resolvent operator convergence of a sequence of A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is established. As an application of this equivalence, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is demonstrated under some suitable assumptions imposed on the parameters. The results presented in this paper unify, improve and generalize some recent works in this field.

Simultaneous optimal control of directional missed discovery rates in data stream diagnosis

High-dimensional data streams are ubiquitous in modern manufacturing, due to their ability to provide valuable information about the industrial system’s performance on a real-time basis. If a shift occurs in a production process, fault diagnosis based on the data streams is of critical importance for identifying the root cause. Existing methods have largely focused on controlling the total missed discovery rate without distinguishing missed signals for positive versus negative components of the shift vector. In practice, however, losses incurred from the two directional shifts can differ substantially, so it is desirable to constrain the proportions of missed signals for positive and negative components at two distinctive levels. In this article, we propose a fault classification procedure that controls the two proportions separately. By formulating the problem as Lagrangian multiplier optimization, we show that the proposed procedure is optimal in the sense that it minimizes the expected number of false discoveries. We also suggest an iterative adjustment algorithm that converges to the optimal Lagrangian parameters. The asymptotic optimality for the data-driven version of our procedure is also established. Theoretical justification and numerical comparison with state-of-the-art methods show that the proposed procedure works well in applications.

Local regularization assisted split augmented Lagrangian shrinkage algorithm for feature selection in condition monitoring

Feature selection plays a vital role in improving the efficiency and accuracy of condition monitoring by constructing sparse but effective models. In this study, an advanced feature selection algorithm named the local regularization assisted split augmented Lagrangian shrinkage algorithm (LR-SALSA) is proposed. The feature selection is realized by solving a l1-norm optimization problem, which usually selects more sparse and representative features at less computational costs. The proposed algorithm operates in two stages, namely variable selection and coefficient estimation. In the stage of variable selection, the primal problem is converted into three subproblems which can be solved separately. Then individual penalty parameters are applied to every coefficient of the model when dealing with the first subproblem. Under the Bayesian evidence framework, an iterative algorithm is derived to optimize these hyperparameters. During the optimization process, redundant variables will be pruned to guarantee model sparsity and improve computational efficiency at the same time. In the second stage, the coefficients for the selected model terms are determined using the least squares technique. The superior performance and efficiency of the proposed LR-SALSA method are validated through two numerical examples and a real-world cutting tool wear prediction case study. Compared with the existing methods, the proposed method can generate a sparse model and ensure a good trade-off between estimation accuracy and computational efficiency.

Deep-unrolling architecture for image-domain least-squares migration

Deep-image prior (DIP) is a novel approach to solving ill-posed inverse problems whose solution is parameterized with an untrained deep neural network and cascaded with the forward modeling operator. A key component to the success of such a method is represented by the choice of the network architecture, which must act as a natural prior to the inverse problem at hand and provide a strong inductive bias toward the desired solution. Inspired by the close link between neural networks and iterative algorithms in classical optimization, we apply an unrolled version of the gradient descent (GD) algorithm as our DIP network architecture, denoted as the deep-unrolling (DU) architecture. Each layer of the unrolled network is comprised of two parts: the first part corresponds to the GD step of the data-fidelity term, whereas the second part, formed by a six-layer convolutional neural network (CNN), plays the role of a regularizer in the associated objective function. The developed DU architecture is applied to the problem of image-domain least-squares migration (IDLSM) to invert migrated seismic images for their underlying reflectivity and is denoted as DU-IDLSM. As such, the DU architecture parameterizes the reflectivity, and the input of each layer of the unrolled network is the reflectivity at the previous layer. Similar to the classical DIP approach, the parameters of the DU architecture are optimized in an unsupervised fashion by minimizing the data misfit function itself. Through experiments with a part of the Sigsbee2A model and a marine field data set, we test the effectiveness of the DU-IDLSM approach and highlight two key benefits. First, the DU architecture can effectively regularize the inversion process, resulting in reflectivity estimates with fewer artifacts and higher image resolution than those produced by conventional IDLSM approaches. Second, we indicate that by including dropout layers in the CNN architecture, DU-IDLSM can produce a qualitative measure of the uncertainty associated with the least-squares migration process.

On fixed point convergence results for class of nonexpansivemappings in hyperbolic spaces via PJ iteration process

In this paper, we provide certain fixed point results for a mean nonexpansive mapping, as well as a new iterative algorithm called PJ-iterationfor approximating the fixed point of this class of mappings in the setting of hyperbolic spaces. Furthermore, we establish strong and∆-convergence theorem for mean nonexpansive mapping in hyperbolic space. Finally, we present a numerical example to illustrate ourmain result and then display the efficiency of the proposed algorithm compared to different iterative algorithms in the literature. Our resultsobtained in this paper improve, extend and unify some related results in the literature.

Network sum-rate maximization for network-coded clustered uplink NOMA networks with SWIPT-enabled relays

This paper considers network sum-rate maximization for network-coded clustered uplink non-orthogonal multiple-access (NOMA) networks with simultaneous wireless information and power transfer (SWIPT)-enabled relays. Specifically, the aim is to perform joint power allocation, power-splitting, and relay selection (J-PA-PS-RS), subject to quality-of-service (QoS) requirements. The formulated problem is excessively computationally-intensive, as it is a non-convex optimization problem. To efficiently solve it, it is split into two sub-problems: (1) joint power allocation and power-splitting per relay, and (2) relay selection. Particularly, a solution procedure is proposed, where the joint user and relay power allocation, and power-splitting per relay sub-problem is solved via a low-complexity iterative three-layer algorithm, and then followed by optimal relay selection. Simulation results have revealed that the proposed solution procedure can efficiently be used to maximize the network sum-rate, achieve near-optimal solutions, and outperform other benchmark schemes, while satisfying QoS constraints.

Optimal control for continuous-time Markov jump singularly perturbed systems : A hybrid reinforcement learning scheme

This article discusses the adaptive optimal control problem for continuous-time Markov jump singularly perturbed systems with unknown system dynamics. First, the subsystems transformation technique is introduced to reconstruct the system with stochastic jump characteristics, yielding a set of parallel subsystems. Next, under the framework of reinforcement learning, an offline model-based hybrid iteration algorithm is developed to approximate the solution of the full-order coupled algebraic Riccati equations. Following this, to escape from the system model constraints, an online model-free hybrid iteration algorithm is introduced and relevant convergence proof is given. Compared to traditional value iteration and policy iteration algorithms, the hybrid iteration algorithm has ideal convergence rate and eliminates the requirement of initial stabilizing control policy. At the end, the availability of the online algorithm is demonstrated by means of an operational amplifier circuit model as a simulation example.

A Single-Shot Scattering Medium Imaging Method via Bispectrum Truncation.

Imaging using scattering media is a very important yet challenging technology. As one of the most widely used scattering imaging methods, speckle autocorrelation technology has important applications in several fields. However, traditional speckle autocorrelation imaging methods usually use iterative phase recovery algorithms to obtain the Fourier phase of hidden objects, posing issues such as large data calculation volumes and uncertain reconstruction results. Here, we propose a single-shot scattering imaging method based on the bispectrum truncation method. The bispectrum analysis is utilized for hidden object phase recovery, the truncation method is used to avoid the computation of redundant data when calculating the bispectrum data, and the method is experimentally verified. The experimental results show that our method does not require uncertain iterative calculations and can reduce the bispectrum data computation by more than 80% by adjusting the truncation factor without damaging the imaging quality, which greatly improves imaging efficiency. This method paves the way for rapid imaging through scattering media and brings benefits for imaging in dynamic situations.

Surrogate optimisation strategies for intraocular lens formula constant optimisation.

To investigate surrogate optimisation (SO) as a modern, purely data-driven, nonlinear adaptive iterative strategy for lens formula constant optimisation in intraocular lens power calculation. A SO algorithm was implemented for optimising the root mean squared formula prediction error (rmsPE, defined as predicted refraction minus achieved refraction) for the SRKT, Hoffer Q, Holladay, Haigis and Castrop formulae in a dataset of N = 888 cataractous eyes with implantation of the Hoya Vivinex hydrophobic acrylic aspheric lens. A Gaussian Process estimator was used as the model, and the SO was initialised with equidistant datapoints within box constraints, and the number of iterations restricted to either 200 (SRKT, Hoffer Q, Holladay) or 700 (Haigis, Castrop). The performance of the algorithm was compared to the classical gradient-based Levenberg-Marquardt algorithm. The SO algorithm showed stable convergence after fewer than 50/150 iterations (SRKT, HofferQ, Holladay, Haigis, Castrop). The rmsPE was reduced systematically to 0.4407/0.4288/0.4265/0.3711/0.3449 dioptres. The final constants were A = 119.2709, pACD = 5.7359, SF = 1.9688, -a0 = 0.5914/a1 = 0.3570/a2 = 0.1970, C = 0.3171/H = 0.2053/R = 0.0947 for the SRKT, Hoffer Q, Holladay, Haigis and Castrop formula and matched the respective constants optimised in previous studies. The SO proves to be a powerful adaptive nonlinear iteration algorithm for formula constant optimisation, even in formulae with one or more constants. It acts independently of a gradient and is in general able to search within a (box) constrained parameter space for the best solution, even where there are multiple local minima of the target function.

Enhancing the spatial resolution of time-of-flight based non-line-of-sight imaging via instrument response function deconvolution.

Non-line-of-sight (NLOS) imaging retrieves the hidden scenes by utilizing the signals indirectly reflected by the relay wall. Benefiting from the picosecond-level timing accuracy, time-correlated single photon counting (TCSPC) based NLOS imaging can achieve theoretical spatial resolutions up to millimeter level. However, in practical applications, the total temporal resolution (also known as total time jitter, TTJ) of most current TCSPC systems exceeds hundreds of picoseconds due to the combined effects of multiple electronic devices, which restricts the underlying spatial resolution of NLOS imaging. In this paper, an instrument response function deconvolution (IRF-DC) method is proposed to overcome the constraints of a TCSPC system's TTJ on the spatial resolution of NLOS imaging. Specifically, we model the transient measurements as Poisson convolution process with the normalized IRF as convolution kernel, and solve the inverse problem with iterative deconvolution algorithm, which significantly improves the spatial resolution of NLOS imaging after reconstruction. Numerical simulations show that the IRF-DC facilitates light-cone transform and frequency-wavenumber migration solver to achieve successful reconstruction even when the system's TTJ reaches 1200 ps, which is equivalent to what was previously possible when TTJ was about 200ps. In addition, the IRF-DC produces satisfactory reconstruction outcomes when the signal-to-noise ratio (SNR) is low. Furthermore, the effectiveness of the proposed method has also been experimentally verified. The proposed IRF-DC method is highly applicable and efficient, which may promote the development of high-resolution NLOS imaging.

Artifact removal and motor imagery classification in EEG using advanced algorithms and modified DNN

This paper presents an advanced approach for EEG artifact removal and motor imagery classification using a combination of Four Class Iterative Filtering and Filter Bank Common Spatial Pattern Algorithm with a Modified Deep Neural Network (DNN) classifier. The research aims to enhance the accuracy and reliability of BCI systems by addressing the challenges posed by EEG artifacts and complex motor imagery tasks.The methodology begins by introducing FCIF, a novel technique for ocular artifact removal, utilizing iterative filtering and filter banks. FCIF's mathematical formulation allows for effective artifact mitigation, thereby improving the quality of EEG data. In tandem, the FC-FBCSP algorithm is introduced, extending the Filter Bank Common Spatial Pattern approach to handle four-class motor imagery classification. The Modified DNN classifier enhances the discriminatory power of the FC-FBCSP features, optimizing the classification process.The paper showcases a comprehensive experimental setup, featuring the utilization of BCI Competition IV Dataset 2a & 2b. Detailed preprocessing steps, including filtering and feature extraction, are presented with mathematical rigor. Results demonstrate the remarkable artifact removal capabilities of FCIF and the classification prowess of FC-FBCSP combined with the Modified DNN classifier. Comparative analysis highlights the superiority of the proposed approach over baseline methods and the method achieves the mean accuracy of 98.575%.

Dual-drive coordinated operation strategy for internet data centers by exploiting spatiotemporal flexibility

With the rapid expansion of computational infrastructure, the electricity consumption of geographically dispersed Internet Data Centers (IDCs) has imposed a substantial influence on the power grid. Geographically distributed IDCs, which present a versatile electricity demand with significant operational flexibility, hold potential for mitigating cost issues and balancing regional supply and load demand. To exploit this flexibility, this paper has developed a dual-drive coordinated operational strategy, which integrates locational electricity prices and unit carbon emission costs based on the dynamic carbon emission intensity (DCEI) of various regions as interaction signals to improve coordination effectiveness. The upper-level determines the consumption behaviour of IDCs, the lower-level provides feedback on dual-drive signals factoring in IDCs' involvement in grid operation, whilst an iterative algorithm is suggested to solve the bi-level model. Case studies based on real-world datasets demonstrate that the proposed operation strategy can achieve arbitrage through multi-regional carbon-electricity markets, significantly reduce operating costs, and support the rapid development of IDCs. Additionally, this strategy facilitates optimal matching of power load in IDCs with renewable energy sources, thereby mitigating imbalances in regional renewable energy supply and demand.

A Modified Iterative Approach for Fixed Point Problem in Hadamard Spaces

The role of iterative algorithms is vital in exploring the diverse domains of science and has proven to be a powerful tool for solving complex computational problems in the most trending branches of computer science. Taking motivation from this fact, we develop and apply a modified four-step iterative algorithm to solve the fixed point problem in the Hadamard spaces using a total asymptotic nonexpansive mapping. MATLAB R2018b is used for numerical experiments to ensure a better convergence rate of the proposed iterative algorithm with existing results.