Training Neural Networks to Approach the Optimum Bayes Estimator in Dense Multi-Emitter Localization

We study the estimation of a $K$-dimensional simplex from $N$ i.i.d.\ points sampled uniformly from its interior; the observations are convex combinations of $K+1$ unknown prototypes. Existing polynomial-time estimators need cubic per-sample work or $O(NK)$ storage and are impractical at $N\sim 10^6$--$10^8$. We propose DeepMVSA, which re-expresses the minimum-volume principle in neural implicit form: a lightweight coordinate network generates the mixing weights and a triangular LU-type parameterization the dual simplex matrix, reducing the trainable-state memory to $O(K^2)$, independent of $N$, and the cost per data pass to $O(NK^2)$. We prove a non-asymptotic sample-complexity bound of the polynomial-time benchmark order for a localized surrogate estimator; an oracle inequality for every global minimizer of the neural objective, with volume-inflation control and an explicit shrinkage bias; a conditional end-to-end error budget separating statistical, approximation, optimization, and enclosure-residual terms on an explicit envelope event; and two-point lower bounds: at any noise level $σ>0$ fixed independently of $N$, the $N^{-1/2}$ scaling is unimprovable in its $N$-exponent. Experiments with up to $N=10^8$ synthetic observations are consistent with the predicted accuracy and scaling, and feasibility on real scenes of $\sim 10^7$ pixels is demonstrated.

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.

This letter presents LiteLoc, a novel and efficient localizer built on 3D Gaussian Splatting (3DGS). The previous state-of-the-art (SoTA) sparse-to-dense localizer, STDLoc, has shown remarkable localization capability but suffers from severe storage redundancy and computational latency. By revisiting its design decisions, we derive two simple yet highly effective improvements that cumulatively make LiteLoc much more efficient in both memory and computation, while also being easier to train. One key observation is that the color field, inherited directly from Feature 3DGS, is functionally useless for localization. Yet, its reconstruction of high-frequency photometric details necessitates excessive Gaussian primitives, resulting in a tightly coupled color-feature representation with significant memory overhead and sub-optimal feature field optimization. To resolve this, we propose a color-free decoupled feature field that constructs a compact Gaussian scene representation by retaining only task-essential feature attributes, thereby eliminating approximately 94% of redundant storage with no loss of localization-relevant information. We further find that the primary computational bottleneck lies in the dense Perspective-n-Point (PnP) solver, where most matches contribute saturated geometric constraints with diminishing accuracy gains. Accordingly, we propose a condensing strategy that distills dense matches into a subset of 5% representative matches, enabling a nearly 19-fold speedup in robust estimation with negligible performance drop. Extensive experiments show that LiteLoc surpasses STDLoc in multiple scenes with considerable efficiency benefits, opening up exciting prospects for latency-sensitive visual localization.