Search Results (73)

This paper studies $(p,q)$-harmonic maps by unified geometric analytic methods. First, we deduce variation formulas of the $(p,q)$-energy functional. Second, we analyze weakly conformal and horizontally conformal $(p,q)$-harmonic maps and prove Liouville results for $(p,q)$-harmonic maps under Hessian and asymptotic conditions on complete Riemannian manifolds. Finally, we define the $(p,q)$-$SSU$ manifold and prove that non-constant stable $(p,q)$-harmonic maps do not exist. Full article

This paper investigates the distributed time-varying (TV) resource management problem (RMP) for microgrids (MGs) within a multi-agent system (MAS) framework. A novel fixed-time (FXT) distributed optimization algorithm is proposed, capable of operating over switching communication graphs and handling both local inequality and global equality constraints. By incorporating a time-decaying penalty function, the algorithm achieves an FXT consensus on marginal costs and ensures asymptotic convergence to the optimal TV solution of the original RMP. Unlike the prior methods with centralized coordination, the proposed algorithm is fully distributed, scalable, and privacy-preserving, making it suitable for real-time deployment in dynamic MG environments. Rigorous theoretical analysis establishes FXT convergence under both identical and nonidentical Hessian conditions. Simulations on the IEEE 14-bus system validate the algorithm’s superior performance in convergence speed, plug-and-play adaptability, and robustness to switching topologies. Full article

Federated learning represents a newly emerging methodology in the field of machine learning that enables distributed agents to collaboratively learn a centralized model without sharing their raw data. Some scholars have already proposed many first-order algorithms and second-order algorithms for federated learning to reduce communication costs and speed up convergence. However, these algorithms generally rely on gradient or Hessian information, and we find it difficult to solve such federated optimization problems when the analytical expression of the loss function is not available, that is, when gradient information is not available. Therefore, we employed derivative-free federated zero-order optimization in this paper, which does not rely on specific gradient information, but instead utilizes the changes in function values or model outputs to estimate the optimization direction. Furthermore, to enhance the performance of derivative-free zero-order optimization, we propose an effective adaptive algorithm that can dynamically adjust the learning rate and other hyperparameters based on the performance during the optimization process, aiming to accelerate convergence. We rigorously analyze the convergence of our approach, and the experimental findings demonstrate our method can indeed achieve faster convergence speed on the MNIST, CIFAR-10 and Fashion-MNIST datasets in cases where gradient information is not available. Full article

In this paper, we utilize advanced optimization techniques on Riemannian submanifolds to establish two distinct inequalities concerning the generalized normalized $\delta$-Casorati curvatures of warped product pointwise semi-slant (WPPSS) submanifolds within complex space forms. We further identify the precise conditions under which these inequalities attain equality, providing valuable insights into their geometric and structural significance. Additionally, we also present results involving harmonic and Hessian functions, revealing a broader connection between curvature properties and analytic functions. Full article

In this paper, we propose a new finite-difference method for nonconvex absolute value equations. The nonsmooth unconstrained optimization problem equivalent to the absolute value equations is considered. The finite-difference technique is considered to compose the linear programming subproblems for obtaining the search direction. The algorithm avoids the computation of gradients and Hessian matrices of problems. The new finite-difference parameter correction technique is considered to ensure the monotonic descent of the objective function. The convergence of the algorithm is analyzed, and numerical experiments are reported, indicating the effectiveness by comparison against a state-of-the-art absolute value equations. Full article

Deep learning models are often perceived as black boxes, making it challenging to analyze the causal relationships between inputs and outputs. For this reason, the explainability of model learning has garnered increasing attention in recent years. Some previous studies proposed influence functions, which evaluate how the weighting of data impacts the model by mathematical analysis, thereby explaining how it realizes the data. This inspires us to suggest that when data in an upstream task is affected by varying levels of noise interference, it is practical to set up a downstream model to apply Taylor expansion in conjunction with the Hessian matrix to estimate perturbations that each data point cause in the model. Additionally, utilizing Integrated Gradients to compute the loss difference between the original data instances and a baseline instance which does not affect the model is powerful to yield a memorization matrix that allows researchers to observe the changes in model reasoning before and after noise interference, helping to analyze the causes of erroneous inference. Full article

Accurate liver segmentation from computed tomography (CT) scans is essential for liver cancer diagnosis and liver surgery planning. Convolutional neural network (CNN)-based models have limited segmentation performance due to their localized receptive fields. Hybrid models incorporating CNNs and transformers that can capture long-range dependencies have shown promising performance in liver segmentation with the cost of high model complexity. Therefore, a new network architecture named G-UNETR++ is proposed to improve accuracy in liver segmentation with moderate model complexity. Two gradient-based encoders that take the second-order partial derivatives (the first two elements from the last column of the Hessian matrix of a CT scan) as inputs are proposed to learn the 3D geometric features such as the boundaries between different organs and tissues. In addition, a hybrid loss function that combines dice loss, cross-entropy loss, and Hausdorff distance loss is designed to address class imbalance and improve segmentation performance in challenging cases. The proposed method was evaluated on three public datasets, the Liver Tumor Segmentation (LiTS) dataset, the 3D Image Reconstruction for Comparison of Algorithms Database (3D-IRCADb), and the Segmentation of the Liver Competition 2007 (Sliver07) dataset, and achieved 97.38%, 97.50%, and 97.32% in terms of the dice similarity coefficient for liver segmentation on the three datasets, respectively. The proposed method outperformed the other state-of-the-art models on the three datasets, which demonstrated the strong effectiveness, robustness, and generalizability of the proposed method in liver segmentation. Full article

Seismic coherence attributes are valuable for identifying structural features, but they often face challenges due to significant background noise and non-feature-related stratigraphic discontinuities. To address this, it is necessary to apply attribute conditioning to the coherence to enhance the visibility of these structures. The primary challenge of attribute conditioning lies in finding a concise structural representation that isolates only the true interpretive features while effectively removing noise and stratigraphic interference. In this study, we choose sheet-like structures as this concise structural representation, as faults are typically characterized by their thin and narrow profiles. Inspired by multiscale Hessian-based filtering (MHF) and its application on vascular structure detection, we propose a method called anisotropic multiscale Hessian-based sheet-enhancing filtering (AMHSF). This method is specifically designed to extract and magnify sheet-like structures from noisy coherence images, with a novel enhancement function distinct from those traditionally used in vascular enhancement. The effectiveness of our AMHSF is demonstrated through experiments on both synthetic and real datasets, showcasing its potential to improve the identification of structural features in coherence images. Full article

The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented. Full article

This paper explores the properties of projective vector fields on semi-Riemannian manifolds. The main result establishes that if a projective vector field P on such a manifold is also a conformal vector field with potential function $\psi$ and the vector field $\zeta$ dual to $d\psi$ does not change its causal character, then P is homothetic, or $\zeta$ is a light-like vector field. Additionally, it is shown that a complete Riemannian manifold admits a projective vector field that is also conformal and non-Killing if and only if it is locally Euclidean. The paper also presents other results related to the characterization of Killing and parallel vector fields using the Ricci curvature and the Hessian of the function given by the inner product of the vector field. Full article

In this study, we investigate the tangent bundle $TM$ of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection $\tilde{\nabla }$. Our primary goal is to establish the necessary and sufficient conditions for $TM$ to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for $TM$ to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to $TM$, alongside specific constraints on the conformal factor $\lambda$ and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve $\lambda$ and the Hessian of the potential function. Similarly, for $TM$ to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of $\lambda$, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry. Full article

In addition to the filter coefficients, the location of the microphone array is a crucial factor in improving the overall performance of a beamformer. The optimal microphone array placement can considerably enhance speech quality. However, the optimization problem with microphone configuration variables is non-convex and highly non-linear. Heuristic algorithms that are frequently employed take a long time and have a chance of missing the optimal microphone array placement design. We extend the Bayesian optimization method to solve the microphone array configuration design problem. The proposed Bayesian optimization method does not depend on gradient and Hessian approximations and makes use of all the information available from prior evaluations. Furthermore, Gaussian process regression and acquisition functions make up the Bayesian optimization method. The objective function is given a prior probabilistic model through Gaussian process regression, which exploits this model while integrating out uncertainty. The acquisition function is adopted to decide the next placement point based upon the incumbent optimum with the posterior distribution. Numerical experiments have demonstrated that the Bayesian optimization method could find a similar or better microphone array placement compared with the hybrid descent method and computational time is significantly reduced. Our proposed method is at least four times faster than the hybrid descent method to find the optimal microphone array configuration from the numerical results. Full article

Artificial intelligence (AI) technology has advanced significantly, now capable of performing tasks previously believed to be exclusive to skilled humans. However, AI models, in contrast to humans who can develop skills with relatively less data, often require substantial amounts of data to emulate human cognitive abilities in specific areas. In situations where adequate pre-training data is not available, meta-learning becomes a crucial method for enhancing generalization. The Model Agnostic Meta-Learning (MAML) algorithm, which employs second-order derivative calculations to fine-tune initial parameters for better starting points, plays a pivotal role in this area. However, the computational demand of this method can be challenging for modern models with a large number of parameters. The concept of the Approximate Hessian Effect is introduced in this context, examining the effectiveness of second-order derivatives in identifying initial parameters conducive to high generalization performance. The study suggests the use of cosine similarity and squared error (L2 loss) as a loss function within the Approximate Hessian Effect framework to modify gradient weights, aiming for more generalizable model parameters. Additionally, an algorithm that relies on first-order calculations is presented, designed to achieve performance levels comparable to MAML. This approach was tested and compared with traditional MAML methods using both the MiniImagenet dataset and a modified MNIST dataset. The results were analyzed to evaluate its efficiency. Compared to previous studies that achieved good performance using only the first derivative, this approach is more efficient because it does not require iterative loops to converge on additional loss functions. Additionally, there is potential for further performance enhancement through hyperparameter tuning. Full article

The T-WING project, a CS2-CPW (Clean Sky 2 call for core partner waves) research initiative within FRC IADP (Fast Rotor-Craft Innovative Aircraft Demonstrator Platform), focuses on developing, qualifying and testing the new wing of the Next-Generation Civil Tilt-Rotor (NGCTR). This paper introduces a case study about a methodology for refining the stick-beam model for the NGCTR wing, aligning it with the GFEM (Global Finite Element Model) wing’s dynamic characteristics in terms of modal frequencies and mode shapes. The initial stick-beam model was generated through the static condensation of the GFEM wing. The tuning process was formulated as an optimization problem, adjusting beam properties to minimize the sum of weighted quadratic errors in modal frequencies and Modal Assurance Criterion (MAC) values. Throughout the optimization, the MAC analysis ensured that the target modes were tracked, and, at each iteration, a new set of variable estimates were determined based on the gradient vector and Hessian matrix of the objective function. This methodology effectively fine-tunes the stick-beam model for various mass cases, such as maximum take-off weight (MTOW) and maximum zero-fuel weight (MZFW). Full article

Medical imaging-based biomarkers derived from small objects (e.g., cell nuclei) play a crucial role in medical applications. However, detecting and segmenting small objects (a.k.a. blobs) remains a challenging task. In this research, we propose a novel 3D small blob detector called BlobCUT. BlobCUT is an unpaired image-to-image (I2I) translation model that falls under the Contrastive Unpaired Translation paradigm. It employs a blob synthesis module to generate synthetic 3D blobs with corresponding masks. This is incorporated into the iterative model training as the ground truth. The I2I translation process is designed with two constraints: (1) a convexity consistency constraint that relies on Hessian analysis to preserve the geometric properties and (2) an intensity distribution consistency constraint based on Kullback-Leibler divergence to preserve the intensity distribution of blobs. BlobCUT learns the inherent noise distribution from the target noisy blob images and performs image translation from the noisy domain to the clean domain, effectively functioning as a denoising process to support blob identification. To validate the performance of BlobCUT, we evaluate it on a 3D simulated dataset of blobs and a 3D MRI dataset of mouse kidneys. We conduct a comparative analysis involving six state-of-the-art methods. Our findings reveal that BlobCUT exhibits superior performance and training efficiency, utilizing only 56.6% of the training time required by the state-of-the-art BlobDetGAN. This underscores the effectiveness of BlobCUT in accurately segmenting small blobs while achieving notable gains in training efficiency. Full article