Target localization, identification and sensing using latent symmetries

Primary motivation in blind inverse problems is to recover signals of interest from corrupted observations without knowing the obfuscating mechanism. Blind deconvolution is a prominent approach when the corruption is convolutional, but it is not applicable when general linear transformations obfuscate the domain structure. In this work, we propose an unsupervised framework for recovering latent domains and signals by discovering symmetries of the data distribution. Our framework models observations as linear measurements of signals sampled from a latent random field, and optimizes a shallow group-convolutional network by imposing stationarity and locality regularization at the model output. The model learns a latent symmetry action and an appropriate filter, thereby mapping unstructured observations to a symmetry-based representation that reveals latent signals. Experiments on stochastic processes, Ising models, shuffled and bit-scrambled images, and neural recordings show that the method recovers latent domains and signals from unstructured observations, suggesting symmetry discovery as a new direction for unsupervised structure learning and blind inverse problems.

Wireless localization of permanent magnets enables occlusion-free guidance for medical interventions, yet its practical accuracy is fundamentally limited by two coupled challenges: the poor observability of conventional planar sensor arrays and the simulation-to-reality (Sim-to-Real) gap of learning-based estimators. To address these issues, this article presents a unified framework that combines information-theoretic sensor geometry optimization with physics-aware deep learning. First, a rigorous Fisher Information Matrix (FIM)-based evaluation framework is established to quantify geometry-induced observability limitations. The results show that a staggered split-array topology provides a substantially stronger observability foundation for localization while remaining compatible with practical external deployment. Second, building on this optimized sensing configuration, we propose Phy-GAANet, a calibration-free estimator trained entirely on hardware-aware synthetic data. By incorporating Physics-Informed Features (PIF) for saturation modeling and Geometry-Aware Attention (GAA) for preserving cross-layer vector structure, the network effectively bridges the Sim-to-Real gap. Extensive real-world experiments demonstrate state-of-the-art performance, achieving a position error of 1.84 mm and an orientation error of 3.18 degrees at a refresh rate exceeding 270 Hz. The proposed method consistently outperforms classical Levenberg--Marquardt solvers and generic convolutional baselines, particularly in suppressing catastrophic outliers and maintaining robustness in challenging near-field boundary regions. Beyond the proposed network, the FIM-guided analysis also provides a framework for sensor geometry design in magnetic localization systems under practical deployment constraints.

Future integrated sensing and communication (ISAC) architectures separate the sensing entity (SE) that acquires measurements from the sensing function (SF) that performs inference, creating a need for compact, task-oriented feedback on the SE-SF interface. Forwarding the raw channel frequency response or full per-link delay-Doppler-azimuth-elevation (DDAE) tensor is prohibitively expensive, while peak-only reporting discards target-discriminative structure under clutter. We propose a learning-based coarse-to-fine sensing pipeline with candidate-latent feedback for single-target estimation. At the SE, a lightweight convolutional scorer produces a dense delay-Doppler proposal map from pilot-based OFDM channel estimates, and a learned encoder constructs K compact C-dimensional candidate tokens by fusing per-candidate azimuth-elevation patches, normalized position, and confidence cues. The latents are uniformly quantized post-training to b bits and transmitted under a finite budget B_fb = bKC + 18K + 16 bits to the SF, which performs cross-candidate refinement, reranking, and joint four-parameter estimation. On a ray-traced urban scene with static and dynamic clutter, three operating points in the (K, C, b) design space achieve 96.33-98.88% detection at 107-806 bytes per coherent processing interval, compression ratios of 1.2-9.2 x 10^4 over the 8-bit DDAE magnitude tensor, reducing the SE-SF interface from multi-Gbit/s to sub-Mbit/s rates. Cross-scene evaluation on an independent campus-scale environment achieves 98.79-99.50% detection and at-or-better angular accuracy without retraining, indicating that the learned representation captures target-relevant structure that transports across scenes of comparable or lower clutter density.