The escalating threat posed by low-altitude swarm targets underscores the critical need for precise tracking and situation awareness to secure key areas. While existing tracking methods based on random matrix theory offer promising opportunities, they face significant challenges. The high similarity among swarm targets, combined with radar resolution limitations, often leads to instabilities in target counts and measurements due to occlusion, environmental factors, and other disturbances, significantly increasing tracking complexity. To address these challenges, we design a digital staring radar system integrated with an adaptive random matrix method for efficient tracking of low-altitude swarm targets. The system achieves full spatiotemporal coverage without beam scanning or complex resource scheduling, enabling simultaneous detection and tracking of multiple targets. Algorithmically, the random matrix model is enhanced by introducing extension parameters to accurately capture the dynamic changes in swarm shape. Leveraging an adaptive Rao-Blackwellized Particle Filter (RBPF), the presented method jointly estimates the motion and extension states of swarm targets. Extensive simulation experiments and real-data validation demonstrate that the proposed method significantly improves the estimation accuracy for swarm extension states under complex shape variations while maintaining high precision in motion state estimation. This work provides a practical and effective solution for countering low-altitude swarm threats, with strong potential for real-world security applications.

Covariance matrices, known as symmetric positive definite (SPD) matrices, are usually regarded as points lying on Riemannian manifolds. We describe a new covariance descriptor, which could improve the discriminative learning ability of region covariance descriptor by taking into account the mean of feature vectors. Due to the specific geometry of Riemannian manifolds, classical learning methods cannot be directly used on it. In this paper, we propose a subspace projection framework for the classification task on Riemannian manifolds and give the mathematical derivation for it. It is different from the common technique used for Riemannian manifolds, which is to explicitly project the points from a Riemannian manifold onto Euclidean space based upon a linear hypothesis. Under the proposed framework, we define a Gaussian Radial Basis Function‐ (RBF‐) based kernel with a Log‐Euclidean Riemannian Metric (LERM) to embed a Riemannian manifold into a high‐dimensional Reproducing Kernel Hilbert Space (RKHS) and then project it onto a subspace of the RKHS. Finally, a variant of Linear Discriminative Analyze (LDA) is recast onto the subspace. Experiments demonstrate the considerable effectiveness of the mixed region covariance descriptor and the proposed method.

In manifold learning, alignment is performed with the objective of deriving the global low‐dimensional coordinates of input data from their local coordinates. In virtually all alignment processes, the relation between the local and global coordinates is designed intuitively, without mathematical deduction and detailed analysis. In this study, the authors propose a local non‐linear alignment manifold learning algorithm (LNA) for non‐linear dimensionality reduction, based on the concept of local pullback and the mathematical characteristics of a manifold. According to mathematical manifold theory, a function defined on a manifold cannot be differentiated directly on the manifold directly. Instead, it has to be pulled back to Euclidean space with the help of local homeomorphism between the manifold and Euclidean space, where it is then differentiated. In the authors’ proposed algorithm, the component functions of global homeomorphism are regarded as the functions defined on the manifold and pulled back to the Euclidean space. Then, Taylor expansion is utilised up to the second order to establish the relation between the global and local coordinates. The objective function in LNA is based on the alignment error and can be solved with an eigenvalue problem. The experimental results conducted on various datasets verify the validity of the authors’ method.

This paper presents a novel semi-supervised learning algorithm with local adaptive kernels named local feature adaptive learning (LFAL) for data classification. We employ multiple kernels to explore the local feature of data distribution and learn a novel Hilbert space from the kernel matrix of data derived from multiple kernels to extract more data information. Next, we construct a new semi-supervised learning problem with the learned Hilbert space under the general framework of semi-supervised learning. The advantage of the LFAL is that it learns more local data feature from data samples, and provides an adaptive way to learn the solution space with multiple kernels for learning problem. Experimental results on four real world data sets show the effectiveness of our algorithm.