Semi-Supervised Hyperbolic Hierarchical Clustering with Set-Level Structural Priors

Semi-supervised hierarchical clustering aims to learn a tree structure consistent with data patterns and user-provided supervision. Supervision is usually given as leaf-level relations, such as pairwise must-link/cannot-link constraints or triplet-wise must-link-before constraints. Although useful for regulating local sample relations, such supervision does not directly indicate which samples should form coherent subtrees. Consequently, the non-leaf structure of the learned tree may deviate from the hierarchical organization preferred by ground-truth labels. To address this limitation, we propose a semi-supervised hyperbolic hierarchical clustering method with set-level structural priors. The main contribution is to introduce sets as basic modeling units for hierarchy learning. Each set denotes samples expected to cohere within a subtree and is induced from leaf-level supervision together with a learned constraint-consistent similarity structure. These sets act as soft structural priors for subtree-level supervision, allowing supervision to guide non-leaf hierarchy formation beyond local leaf-level relations. Specifically, we first learn constraint-consistent embeddings to obtain a reliable set partition, then construct constraint-induced sets and estimate inter-set similarities to form set-level structural priors. Finally, these priors are incorporated into a hyperbolic hierarchy objective for continuous tree optimization. Experiments on eleven benchmark datasets and ablation studies show that the proposed method consistently improves label consistency over representative hierarchical clustering baselines while also enhancing similarity-based tree quality.

Worst-Case Riemannian Optimization with Uncertain Target Steering Vector for Slow-Time Transmit Sequence of Cognitive Radar

Optimization of slow-time transmit sequence endows cognitive radar with the ability to suppress strong clutter in the range-Doppler domain. However, in practice, inaccurate target velocity information or random phase error would induce uncertainty about the actual target steering vector, which would in turn severely deteriorate the the performance of the slow-time matched filter. In order to solve this problem, we propose a new optimization method for slow-time transmit sequence design. The proposed method transforms the original non-convex optimization with an uncertain target steering vector into a two-step worst-case optimization problem. For each sub-problem, we develop a corresponding trust-region Riemannian optimization algorithm. By iteratively solving the two sub-problems, a sub-optimal solution can be reached without accurate information about the target steering vector. Furthermore, the convergence property of the proposed algorithms has been analyzed and detailed proof of the convergence is given. Unlike the traditional waveform optimization method, the proposed method is designed to work with an uncertain target steering vector and therefore, is more robust in practical radar systems. Numerical simulation results in different scenarios verify the effectiveness of the proposed method in suppressing the clutter and show its advantages in terms of the output signal-to-clutter plus noise ratio (SCNR) over traditional methods.