Top-$k$ feature selection in sparse learning is a fundamental problem in machine learning. It is difficult to conquer due to the rigid $\ell _{2,0}$-norm constraint. Existing literature mostly relaxes the constraint and seeks the approximation of the selection matrix, degenerating primitive models and missing the genuine solutions. This research tackles the primitive top-$k$ feature selection model in sparse learning. From the perspective of universality, we investigate both supervised and semi-supervised models of top-$k$ feature selection in sparse learning. By disassembling the feature selection matrix, it is revealed that two different objectives could be unified into one general ratio-trace problem, which is a non-convex optimization problem. The accelerated coordinate descent method is raised to efficiently solve the non-convex objective, through which the local optimal solution of top-$k$ feature indices is obtained with a competitive time cost. To verify the proposed algorithm, we design toy experiments that could visualize the advantages of the selected features. Meanwhile, experimental results on nine normal datasets and the large-scale ImageNet dataset comprehensively show the superiority of our methods compared to representative and state-of-the-art supervised and semi-supervised algorithms.

We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its $\overline{\partial}$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative.

A trust region method for uncertain multiobjective optimization: comparative analysis with existing descent methods

Abstract Quantum physics is used for communications, computing, and sensor applications beyond classical capabilities. Uniform Circular Array (UCA) antennas are one of the means of generating Orbital Angular Momentum waves (OAM), which are known as quantum waves that travel at microwave frequencies. The research focuses on the mathematical analysis of OAM wave generation using both classical and quantum optics. These quantum waves are simulated by developing cross-phased omnidirectional antennas and UCAs with 10-element patches using MATLAB. Antennas generating OAM waves are tested at frequencies 1.8 GHz, 40 GHz, 50 GHz, and 90 GHz. The antenna radiation range, elevation angle, and azimuth angles are analyzed. One UCA Antenna at 6 GHz is designed, fabricated, and tested for its radiation, circular polarization, and gain. UCAs are used in applications such as estimating radar cross-section (RCS) and locating the direction of targets. Simulations are carried out to detect targets in 3D (R, θ , Φ) and to calculate the RCS. Since deep learning (DL) techniques help solve large databases of RCS calculations, we utilise gradient descent and backpropagation methods for target detection. These techniques can estimate the position and surface structure of the target using RCS. DL models prove their greater robustness in direction of arrival estimation (DoA) in low SNR conditions. It shows high accuracy and less computational complexity, even in nonlinearity, as compared to traditional optimization methods such as the MUSIC algorithm. Hence, the design of Quantum antennas in DL is proposed for future wireless communication, frequency-aware designs, and high-resolution sensing.

In this paper, we present a rapid gradient descent method for solving low-rank matrix recovery problems. Our method extends the conventional gradient descent framework by exploiting the problem’s unique features to develop an innovative fast gradient computation technique that lowers the computational cost of gradient evaluation. The introduced adaptive step size selection strategy not only eliminates the need for the heavy calculations usually involved in finding the descent direction but also guarantees a consistent decrease in the objective function at every iteration. Additionally, we offer a proof confirming the algorithm’s convergence. Numerical experiments are provided to show the efficiency of the proposed algorithm.

Multi-view spectral clustering (MVSC) has garnered growing interest across various real-world applications, owing to its flexibility in managing diverse data space structures. Nevertheless, the fusion of multiple $n\times n$ similarity matrices and the separate post- discretization process hinder the utilization of MVSC in large-scale tasks, where $n$ denotes the number of samples. Moreover, noise in different similarity matrices, along with the two-stage mismatch caused by the post- discretization, results in a reduction in clustering effectiveness. To overcome these challenges, we establish a novel fast multi-view discrete clustering (FMVDC) model via spectral embedding fusion, which integrates spectral embedding matrices ($n\times c$, $c\ll n$) to directly obtain discrete sample categories, where $c$ indicates the number of clusters, bypassing the need for both similarity matrix fusion and post- discretization. To further enhance clustering efficiency, we employ an anchor-based spectral embedding strategy to decrease the computational complexity of spectral analysis from cubic to linear. Since gradient descent methods are incapable of discrete models, we propose a fast optimization strategy based on the coordinate descent method to solve the FMVDC model efficiently. Extensive studies demonstrate that FMVDC significantly improves clustering performance compared to existing state-of-the-art methods, particularly in large-scale clustering tasks.

Tempered fractional gradient descent: Theory, algorithms, and robust learning applications.

This chapter describes the Gradient Descent method to predict the inhibition of protein targets. Protein systems are well-suited to study with artificial intelligence techniques, including machine learning methods. Here, we employ two variants of the Gradient Descent method: Batch Gradient Descent and Stochastic Gradient Descent. The last one is available in the Scikit-Learn library (SGDRegressor class). We can integrate Scikit-Learn methods into pipelines to build regression models addressing protein targets employed for drug discovery. In this work, we adopt a hands-on approach and show how to make a regression model to predict the inhibition of cyclin-dependent kinase 2, a protein target for anticancer drugs. We combine pair interaction data determined using the docking program AutoDock Vina and the SGDRegressor class implemented in the program SAnDReS 2.0 to create models to determine enzyme inhibition. All Jupyter Notebooks and datasets examined in this work are at GitHub: https://github.com/azevedolab/docking#readme . We made the program SAnDReS 2.0 available at https://github.com/azevedolab/sandres .

The subject of this work is the problem of optimizing the configuration of cuts for skin grafting in order to improve the efficiency of the procedure. We consider the optimization problem in the framework of a linear elasticity model. We choose three mechanical measures that define optimality via related objective functionals: the compliance, the L^p-norm of the von Mises stress, and the area covered by the stretched skin. We provide a proof of the existence of the solution for each problem, but we cannot claim uniqueness. We compute the gradient of the objectives with respect to the cut configuration using concepts from shape calculus. To solve the problem numerically, we apply the gradient descent method, which performs well under uniaxial stretching. However, in more complex cases, such as multidirectional stretching, its effectiveness is limited due to the low sensitivity of the functionals under consideration.To avoid this difficulty, we use a combination of the genetic algorithm and the gradient descent method, which leads to a significant improvement in the results.

A stochastic column-block gradient descent method for solving nonlinear systems of equations

Abstract We study a discrete variant of the Airy equation, formulated as an advance-delay equation, to reveal that discretization induces the higher-order Stokes phenomenon, which is not present in the continuous Airy function and is typically only encountered in solutions to third-order or higher linear homogeneous, or nonlinear, differential equations. Using steepest descent and direct series methods, we derive asymptotic solutions and the Stokes structure. Our analysis shows that discretization produces a more intricate Stokes structure, containing higher-order Stokes phenomena and infinite accumulations of Stokes and anti-Stokes curves. The latter feature is a strictly nonlinear effect in continuous differential equations. We show that this unusual behavior can be generated in a discrete equation from a linear discretization. Numerical simulations confirm the predictions, and a direct comparison with the continuous Airy equation explains how the discretization alters the Stokes structure.

The Cekirge Global σ-Regularized Deterministic Method introduces a non-iterative learning framework in which model parameters are obtained through a single closed-form computation rather than through gradient-based optimization. For more than half a century, supervised learning has relied on gradient descent, stochastic gradient descent, and conjugate gradient descent—methods requiring learning rates, batching rules, random initialization, and stopping heuristics, whose outcomes vary with floating-point resolution, operating-system effects, and hardware drift. As dimensions increase or matrices become ill-conditioned, these iterative processes frequently diverge or yield inconsistent results. The σ-Regularized Deterministic Method replaces this instability with a σ-regularized quadratic formulation whose stationary point is analytically unique; even very small σ values eliminate ill-conditioning and ensure machine-independent reproducibility. Learning is reframed not as a search, but as the direct computation of an equilibrium determined by the structural geometry of the data matrix. To address the common reviewer concern that stability must be demonstrated across progressive system sizes, the method is validated sequentially—from small 5×5 and 8×8 matrices, whose full algebra is explicitly inspectable, through 20×20, 100×100, and ultimately 1000×1000. Across all scales, the deterministic σ-solution remains stable and identical across platforms, whereas gradient-based algorithms begin to degrade even at moderate sizes. In practice, the σ-Regularized Deterministic Method requires only a single algebraic evaluation, eliminating the repeated matrix passes and energy expenditure inherent to iterative algorithms. Its runtime scales linearly with the number of partitions rather than the number of iterations, yielding substantial time and energy savings even in very large systems.

Abstract We propose a gradient-based online optimization framework for solving stochastic programming problems that frequently arise in the context of cyber-physical and robotic systems. Our problem formulation accommodates constraints that model the evolution of a cyber-physical system, which has, in general, a continuous state and action space, is nonlinear, and where the state is only partially observed. We also incorporate an approximate model of the dynamics as prior knowledge into the learning process and show that even rough estimates of the dynamics can significantly improve the convergence of our algorithms. Our online optimization framework encompasses both gradient descent and quasi-Newton methods, and we provide a unified convergence analysis of our algorithms in a non-convex setting. We also characterize the impact of modeling errors in the system dynamics on the convergence rate of the algorithms. Finally, we evaluate our algorithms in simulations of a flexible beam, a four-legged walking robot, and in real-world experiments with a ping-pong playing robot.

The inverse problem of reconstructing the vertical profiles of CO\(_2\) in the atmosphere by IR spectra of the solar light transmission is investigated. To solve this problem, we propose a two-stage method. At the first stage, we use the modified Tikhonov method. At the second stage, to approximate a solution of the regularized equation, we apply a nonlinear analogue of the modified steepest descent method. The convergence theorem is formulated and the results of numerical experiments for retrieving the concentration of carbon dioxide in the atmosphere from measured spectra are discussed.

Insulator detection is an important task for safe and reliable operation of smart grid. Due to various background interferences in insulator images, most traditional image processing methods cannot achieve good performance. In this paper, a new method based on Log AdaBoost is proposed for insulator detection. Firstly, our boosting algorithm optimizes Polylog loss function rather than Exponential function in classical AdaBoost. We use gradient descent to optimize our loss function while the coordinate descent method is used in classical AdaBoost. Secondly, a new weight updating strategy is taken to find the weak classifier relevant to the label under the current weight distribution. In other word, the weight is updated towards the negative gradient of loss function to find the optimal weak classifier. Thirdly, a neighborhood feature is proposed in this paper, and this Haar-like feature can make the pixel difference between the insulator and the background obvious. Experimental results on two databases (UCI and ACDC) show that the proposed algorithm achieves the lowest test error on 11 of the 20 UCI datasets (second-lowest on the other nine), and on ACDC it yields lower testing error with the fewest weak classifiers and the smallest margin variance across the four labels, indicating better generalization than other AdaBoost variants. Finally, on the CPLID insulator detection dataset, the proposed method achieves an AUC of 0.82 with only 21k parameters.

In computer graphics, photorealistic lighting simulation and efficient rendering technology have always faced the dual challenges of computational complexity and visual fidelity. Traditional global illumination algorithms rely on many ray sampling and iterative calculations. For example, path tracing needs to emit thousands of rays per pixel to converge. However, joint optimization problems of hundreds of dimensions, such as light source parameters and material reflectivity in dynamic scenes, often cause traditional gradient descent methods to fall into local optimization. The L-BFGS algorithm stores historical gradient information through a limited memory strategy and constructs an iterative model approximating the inverse of the Hessian matrix. While maintaining the fast convergence characteristics of second-order optimization, the memory consumption is reduced to the order of O (mn) (m is the number of memory steps), which provides a new idea for large-scale lighting parameter optimization. Experimental results demonstrate that the L-BFGS optimization achieves convergence of the energy function to 10-6 within 500 iterations in scenes with dynamic light sources and complex materials, reducing computation time by 38% compared to traditional BFGS. When integrated into NeRF training, the hybrid L-BFGS strategy reduces geometric reconstruction error to 0.12 mm, improving accuracy by 52% over pure stochastic gradient descent. In real-time rendering, GPU-accelerated L-BFGS optimizes shadow mapping parameters for 256 virtual point lights per frame, maintaining 60 FPS at 4K resolution with 1.2 GB VRAM usage. For mobile AR, a quantized L-BFGS variant achieves material reflection calibration in 8.3 ms with ±0.5% azimuth accuracy, while the Monte Carlo-L-BFGS framework reduces indirect illumination precomputation from 14.6 hours to 2.3 hours with 98.7% visual fidelity. These technological advances provide a new paradigm for integrating movie-level offline and real-time rasterized rendering pipelines and promote the development of efficient visualization in emerging fields such as digital twins and the metaverse.

Abstract The equilibrium selection problem in the variational Generalized Nash Equilibrium Problem (v-GNEP) has been reported as an optimization problem defined over the solution set of v-GNEP, called in this paper the lower-level v-GNEP. However, to make such a selection fair for every player, we have to rely on an unrealistic assumption, that is, the availability of a trusted center that does not induce any bias among players. In this paper, to ensure fairness for every player even in the process of equilibrium selection, we propose a new equilibrium selection problem, named the upper-level v-GNEP. The proposed upper-level v-GNEP is formulated as a v-GNEP defined over the solution set of the lower-level v-GNEP. We also present an iterative algorithm, of guaranteed convergence to a solution of the upper-level v-GNEP, as an application of the hybrid steepest descent method to a fixed point set characterization of the solution of the lower-level v-GNEP. Numerical experiments illustrate the proposed equilibrium selection and algorithm.

ABSTRACT Space‐time‐coding metasurfaces (STCMs) enable simultaneous controls of electromagnetic wave across multiple harmonics, but designing high‐performance coding sequences in real time remains challenging. Here, we propose an unsupervised physics‐informed deep learning framework that can generate optimal spatiotemporal coding patterns for arbitrary single‐ and dual‐beam requirements at each harmonic frequency. The proposed method features three key innovations: physics‐informed mechanisms to enable unsupervised learning without requiring paired training data, a dedicated strategy for multi‐bit metasurface configurations, and the Conflict Averse Gradient descent (CAGrad) method to coordinate the parameter optimization across harmonics in multi‐task learning. Experiments on a 2‐bit STCM demonstrate robust beamforming capabilities over five harmonics, achieving an average radiation difference of 1.55 dB and real‐time design <0.1s. This is a 4‐order‐of‐magnitude improvement in computational efficiency compared with the particle swarm optimization methods. This work establishes a real‐time and physics‐aware design paradigm for intelligent metasurfaces in the next‐generation wireless systems.

The article examines the processes of implementing transparency and ensuring transparent accountability in the work of both public authorities and local governments within the country. The purpose of the article is to substantiate the managerial principles of ensuring transparency and accountability in the activities of public authorities and local self-government bodies, as well as to develop a model for their integration into the public administration system. The study is based on a systematic methodological approach that combines dialectical, logical methods and the method of descent from the abstract to the concrete to analyze transparency and accountability in public administration. It is established that transparency and accountability are the basis of effective governance, providing feedback to society and preventing dysfunctions, such as serving narrow group interests, through the introduction of open procedures and public monitoring. A model of strategic governance at the regional level is proposed that integrates transparency through public approval of decisions, digital tools, and public control, ensuring that local and national priorities in the field of public administration and the work of public authorities in the country are in line with each other. Further research can be aimed at developing the component support for the level of effectiveness of the proposed model in different countries through the analysis of open data. It is also advisable to study the impact of digital technologies on accountability in the context of regional peculiarities of public administration.

The metasurface has emerged as a powerful platform for achieving versatile and precise light control due to its extraordinary optical properties. However, multifunctional metasurface-based devices, usually employing shared apertures or cascading with different metasurfaces, suffer from inherent interference and complex configurations. Here, we propose a single-layer transmissive multifunctional metasurface performing distinct manipulations of light at specific wavelengths, free from inter-wavelength interference, benefiting from the strong dispersion of the silicon dielectric nanostructure. The metasurface is optimized using an adjoint-based inverse design with a gradient descent method. The simulated and experimental results confirm the successful integration of three functionalities within a single metasurface: beam stopper, linear polarizer, and half waveplate. The proposed multifunctional metasurface features an ultra-compact design with a simple configuration, which has the potential for applications in various fields, including photonic data transmission, hyperspectral imaging, and advanced optical sensing systems.