Projected gradient descent method for cardinality-constrained portfolio optimization

We study the best low-rank Tucker decomposition of symmetric tensors. The motivating application is decomposing higher-order multivariate moments. Moment tensors have special structure and are important to various data science problems. We advocate for projected gradient descent (PGD) method and higher-order eigenvalue decomposition (HOEVD) approximation as computation schemes. Most importantly, we develop scalable adaptations of the basic PGD and HOEVD methods to decompose sample moment tensors. With the help of implicit and streaming techniques, we evade the overhead cost of building and storing the moment tensor. Such reductions make computing the Tucker decomposition realizable for large data instances in high dimensions. Numerical experiments demonstrate the efficiency of the algorithms and the applicability of moment tensor decompositions to real-world datasets. Finally we study the convergence on the Grassmannian manifold, and prove that the update sequence derived by the PGD solver achieves first-and second-order criticality.

Application of gradient descent method to the sedimentary grain-size distribution fitting

Gradient descent (GD) method is used to fit the the laser grain-size distribution of the sediments with a sum of four weighted lognormal functions. The numerical results indicate that the GD method is not only easy to operate but also could effectively optimize the parameters of the fitting function with the error decreasing steadily. Meanwhile the overall fitting results are satisfactory. As a new way of data fitting, the GD method could also be used to solve other optimization problems.

ZeroGrad: Costless conscious remedies for catastrophic overfitting in the FGSM adversarial training

We investigate multiuser uplink communications from multiple single-antenna users to a base station (BS), which is equipped with multiple fluid antennas (FAs) and adopts zero-forcing receivers to decode multiple signals. We aim to optimize antennas’ positions at the BS, to minimize the total transmit power of all users subject to the minimum rate requirement. After applying transformations, we show that the problem is equivalent to minimizing the sum of each eigenvalue’s reciprocal of a matrix, which is a function of all antennas’ positions at the BS. Subsequently, the projected gradient descent (PGD) method is utilized to find a locally optimal solution. In particular, different from the latest related work, we exploit the eigenvalue decomposition to successfully derive a closed-form gradient for the PGD, which facilitates the practical implementation greatly. We demonstrate by simulations that via careful optimization for all antennas’ positions in our proposed design, the total transmit power of all users can be decreased significantly as compared to competitive benchmarks.

In this paper, the projected-gradient-descent (PGD) -based detector for massive MIMO system, which consists of two basic operations — projection and gradient descent (GD), is studied to achieve the performance improvement. Since the projection and GD step have different loss functions, necessary compromise has to be made to balance them during iterations. For this reason, the generalized PGD (GPGD) method is proposed with flexible choices of projection and GD. Different from performing projection and GD alternatively, we show that implementing projection after every multiple GD steps is a better solution. Meanwhile, the step-size of GD is also investigated for convergence efficiency. After that, by unfolding this proposed GPGD method with deep neural networks (DNN), the self-corrected auto-detector (SAD) is established to achieve better decoding performance, where enhancement by attention mechanism and extension by another iterative method are also given for performance improvement and efficiency.

BackgroundVision language models (VLMs) are increasingly integrated into medical workflows for diagnostic support and clinical decision-making. While recent studies have demonstrated susceptibility of proprietary VLMs to prompt injection attacks in medical contexts [1], the security landscape of domain-specific medical VLMs remains largely unexplored. This study comprehensively evaluates the vulnerability of multiple VLMs to both prompt injection and adversarial perturbation attacks [2], investigating white-box attacks on MedGemma and black-box transfer attacks across medical-domain and proprietary models.MethodsWe conducted systematic vulnerability assessment using medical images with histologically confirmed malignant lesions spanning multiple modalities: CT, MRI, ultrasound, pathology, endoscopy, and dermatology (n = 18 cases, 3 per modality). For prompt injection, we embedded malicious instructions within text prompts and visual elements. For adversarial perturbations, we used Projected Gradient Descent and optimization-based methods. White-box attacks utilized full model access to MedGemma, while black-box attacks employed transfer-based methods using surrogate models (OpenCLIP, BiomedCLIP, BLIP).ResultsMedGemma achieved the lowest prompt injection vulnerability (38% ASR), followed by Claude 4 Sonnet (48%), GPT-5 (57%), and Claude 4.1 Opus (69%), suggesting domain-specific medical training enhances resistance. For adversarial perturbations, white-box attacks on MedGemma exceeded 80% ASR. Black-box transfer attacks showed varying vulnerability: GPT-5 (44%), MedGemma (37%), Claude 4.1 Opus (17%), and Claude 4 Sonnet (6%). Vulnerability rankings differed notably between attack modalities.ConclusionsThis study provides the first comparative security assessment across medical-domain and proprietary VLMs. Results reveal complex vulnerability patterns with no single model providing universal robustness across different attack vectors. These findings emphasize that robust medical AI security requires comprehensive, multi-layered defenses targeting both text-based and image-based attack vectors, with model-specific threat considerations for medical applications.

Time series anomaly detection is extensively studied in statistics, economics, and computer science. Over the years, numerous methods have been proposed for time series anomaly detection using deep learning-based methods. Many of these methods demonstrate state-of-the-art performance on benchmark datasets, giving the false impression that these systems are robust and deployable in many practical and industrial real-world scenarios. In this paper, we demonstrate that the performance of state-of-the-art anomaly detection methods is degraded substantially by adding only small adversarial perturbations to the sensor data. We use different scoring metrics such as prediction errors, anomaly, and classification scores over several public and private datasets ranging from aerospace applications, server machines, to cyber-physical systems in power plants. Under well-known adversarial attacks from Fast Gradient Sign Method (FGSM) and Projected Gradient Descent (PGD) methods, we demonstrate that state-of-the-art deep neural networks (DNNs) and graph neural networks (GNNs) methods, which claim to be robust against anomalies and have been possibly integrated in real-life systems, have their performance drop to as low as 0%. To the best of our understanding, we demonstrate, for the first time, the vulnerabilities of anomaly detection systems against adversarial attacks. The overarching goal of this research is to raise awareness towards the adversarial vulnerabilities of time series anomaly detectors.

With the widespread use of deep learning models in various applications. People are gradually realizing the vulnerability of these models to adversarial attacks. Adversarial training is an effective strategy to defend against adversarial attacks. Based on the advantages and disadvantages of the current mainstream Fast Gradient Sign Method (FGSM) adversarial training and Projected Gradient Descent (PGD) adversarial training, this paper proposes a hybrid adversarial training that integrates FGSM and PGD methods and uses the ResNet-18 model and SVHN dataset for testing. Experimental results show that hybrid adversarial training can effectively reduce training time. Its accuracy on the original data set is higher than that of PGD adversarial training, which is improved by about 2%. The performance when facing FGSM attacks is almost the same as that of single FGSM adversarial training. The performance when facing PGD attacks decreases more significantly, which is about 2% to 3% lower than that of PGD adversarial training. This study not only helps to understand the robustness of hybrid adversarial training to models facing adversarial attacks but also helps in studying new adversarial training strategies.

Fuzzy logic systems have been extensively applied for solving many real world application problems because they are found to be universal approximators and many methods, particularly, gradient descent (GD) methods have been widely adopted for the optimization of fuzzy membership functions. Despite its popularity, GD still suffers some drawbacks in terms of its slow learning and convergence. In this study, the use of decoupled extended Kalman filter (DEKF) to optimize the parameters of an interval type-2 intuitionistic fuzzy logic system of Tagagi-Sugeno-Kang (IT2IFLS-TSK) fuzzy inference is proposed and results compared with IT2IFLS gradient descent learning. The resulting systems are evaluated on a real world dataset from Australia's electricity market. The IT2IFLS-DEKF is also compared with its type-1 variant and interval type-2 fuzzy logic system (IT2FLS). Analysis of results reveal performance superiority of IT2IFLS trained with DEKF (IT2IFLS-DEKF) over IT2IFLS trained with gradient descent (IT2IFLS-GD). The proposed IT2IFLS-DEKF also outperforms its type-1 variant and IT2FLS on the same learning platform.

Malware threatens cybersecurity by enabling data theft, unauthorized access, and extortion. Traditional malware detection systems (MDS) struggle with the increasing volume and complexity of malware. While machine learning (ML) and deep learning (DL) offer promising solutions, they remain vulnerable to adversarial attacks that evade detection. Recent research focuses on developing adversarial datasets to retrain ML/DL-based malware detection systems, enhancing their robustness against adversarial attacks. While these methods improve detection of adversarial samples, they also cause more misclassification of non-adversarial data due to overfitting. These methods lack scalability when ML/DL-based MDS are retrained in isolation, without utilizing knowledge from other MDS with retrained models, leading to inefficiency and waste. To tackle these issues, we introduce ProDef-MDS, a proactive defense system that integrates an Adversarial Restoration (AR) module to mitigate adversarial perturbations and recover inputs to a correctly classifiable form before passing them into the malware classification model. We focus on portable executable (PE) malware within Windows OS to evaluate our approach’s effectiveness across various scenarios, including those with adversarial data generated from five white-box attacks, including Fast Gradient Sign Method (FGSM), Projected Gradient Descent (PGD)10, PGD100, DeepFool, Carlini and Wagner (CW) and one black-box attack Auxiliary Classifier Generative Adversarial Networks (ACGAN). Additionally, we assess our approach with non-adversarial data to demonstrate its effectiveness in adversarial detection without compromising non-adversarial performance. The results obtained from the real-world dataset indicate enhanced robustness and minimal overhead, offering a proactive solution to adversarial threats in MDS. This approach outperforms retraining defense method in five white-box attacks and also indicates better performance in non-adversarial scenarios.

Based on an internal tidal model, the practical performances of the limited-memory BFGS (L-BFGS) method and two gradient descent (GD) methods (the normal one with Wolfe’s line search and the simplified one) are investigated computationally through a series of ideal experiments in which the open boundary conditions (OBCs) are inverted by assimilating the interior observations with the adjoint method. In the case that the observations closer to the unknown boundary are included for assimilation, the L-BFGS method performs the best. As compared with the simplified GD method, the normal one really uses less iteration to reach a satisfactory solution, but its advantage over the simplified one is much smaller than expected. In the case that only the observations that are further from the unknown boundary are assimilated, the simplified GD method performs the best instead, whereas the performances of the other two methods are not satisfactory. The advanced L-BFGS algorithm and Wolfe’s line search still need to be improved when applied to the practical cases. The simplified GD method, which is controllable and easy to implement, should be regarded seriously as a choice, especially when the classical advanced optimization techniques fail or perform poorly.

Nowadays, the influence of machine learning algorithms on our lives is booming gradually, and gradient descent (GD) method is a typical big data analysis method that can be applied to various machine learning algorithms. Traditional GD method has its advantages in dealing with the optimal parameter finding issue for partial loss functions, but the optimal parameter finding process may be trapped into local minimum sometimes if the loss function with certain convexity; moreover, traditional GD cannot meet the requirement of high accuracy. Fractional order GD, which is the core of fixed memory step GD, has better performance by comparing with traditional GD, but still have some problems in the aspect of convergence. On the other hand, t-distributed machine learning algorithm for dimensionality reduction, can reduce the dimensionality of data in high-dimensional space to two-dimensional or three-dimensional space, and can convert the original data into more comprehensible images. In order to solve the problem mentioned above reasonably, in this paper, a fixed-memory step GD method combined with t-distributed stochastic neighbor embedding (t-SNE) is employed to replace the traditional GD method, so as to keep the consistency of low-dimensional data being mapped from high-dimensional data, and to achieve efficient data visualization. The effectiveness of proposed t-SNE algorithm based on fixed memory step gradient descent method has been verified finally via numerical experiment.

One of the earliest and the best method to minimize a function is the classical steepest descent (SD) method. In this paper, a new modification of SD method is suggested using a new search direction, d k. The numerical results are presented based on number of iterations and CPU time. It shows that the new d k are efficient when compared to the classical SD.

The importance of robust proton treatment planning to mitigate the impact of uncertainty is well understood. However, its computational cost grows with the number of uncertainty scenarios, prolonging the treatment planning process. We developed a fast and scalable distributed optimization platform that parallelizes the robust proton treatment plan computation over the uncertaintyscenarios. We modeled the robust proton treatment planning problem as a weighted least-squares problem. To solve it, we employed an optimization technique called the alternating direction method of multipliers with Barzilai-Borwein step size (ADMM-BB). We reformulated the problem in such a way as to split the main problem into smaller subproblems, one for each proton therapy uncertainty scenario. The subproblems can be solved in parallel, allowing the computational load to be distributed across multiple processors (e.g., CPU threads/cores). We evaluated ADMM-BB on four head-and-neck proton therapy patients, each with 13 scenarios accounting for 3 mm setup and 3.5% range uncertainties. We then compared the performance of ADMM-BB with projected gradient descent (PGD) applied to the sameproblem. For each patient, ADMM-BB generated a robust proton treatment plan that satisfied all clinical criteria with comparable or better dosimetric quality than the plan generated by PGD. However, ADMM-BB's total runtime averaged about 6 to 7 times faster. This speedup increased with the number ofscenarios. ADMM-BB is a powerful distributed optimization method that leverages parallel processing platforms, such as multicore CPUs, GPUs, and cloud servers, to accelerate the computationally intensive work of robust proton treatment planning. This results in (1) a shorter treatment planning process and (2) the ability to consider more uncertainty scenarios, which improves planquality.