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GLocFM: A Geometry-Aware Foundation Model for 3D Indoor Wireless Localization
Chenghong Bian, Chaozheng Wen, Hongze Chen, Jun Zhang
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Abstract
Abstract:Learning-based wireless localizers often fail to utilize geometric information about the propagation environment, limiting their ability to exploit non-line-of-sight (NLoS) propagation and generalize across scenes. To bridge this gap, we propose GLocFM, a Geometry-aware Localization Foundation Model, which jointly exploits WiFi measurements and scene geometry represented as a 3D point cloud. We formulate localization as a maximum-likelihood (ML) estimation problem, where the goal is to find a transmitter position that maximizes the likelihood of the wireless observations conditioned on the scene geometry. The likelihood of a candidate transmitter position is calculated by a learned scoring function that matches the observed delay--angle-of-arrival (AoA) spectrum against the spectrum predicted for that candidate. A hierarchical scene encoder extracts propagation-relevant features to produce geometric priors for LoS and one-bounce reflection paths. For scenarios with imperfect synchronization, we further introduce a time-of-flight (ToF)-robust GLocFM model to handle unknown ToF offsets. GLocFM is trained on a multi-modal synthetic indoor localization dataset comprising 221 diverse scenes whose associated wireless signals are generated using Sionna RT. On both synthetic and the NeRF$^{2}$ dataset based on real measurements, GLocFM reduces mean 3D localization error relative to one of the state-of-the-art localization baselines by 49.5\% and 48.8\%, respectively. Ablations across different number of receiver, bandwidths, and array sizes further demonstrate the effectiveness and robustness of the proposed framework.
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- Source: https://arxiv.org/abs/2608.09285v1
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iComAILab INTEGRATED COMMUNICATIONSAND AILAB GLocFM: A Geometry-Aware Foundation Model for 3D Indoor Wireless Localization Chenghong Bian †1 , Chaozheng Wen †1 , Hongze Chen 1 , Jun Zhang ∗1 1 Hong Kong University of Science and Technology † Equal Contributions * Corresponding Author Learning-based wireless localizers often fail to utilize geometric information about the propagation environment, limiting their ability to exploit non-line-of-sight (NLoS) propagation and generalize across scenes. To bridge this gap, we propose GLocFM, aGeometry-awareLocalizationFoundation Model, which jointly exploits WiFi measurements and scene geometry represented as a 3D point cloud. We formulate localization as a maximum-likelihood (ML) estimation problem, where the goal is to find a transmitter position that maximizes the likelihood of the wireless observations conditioned on the scene geometry. The likelihood of a candidate transmitter position is calculated by a learned scoring function that matches the observed delay–angle-of-arrival (AoA) spectrum against the spectrum predicted for that candidate. A hierarchical scene encoder extracts propagation-relevant features to produce geometric priors for LoS and one-bounce reflection paths. For scenarios with imperfect synchronization, we further introduce a time-of-flight (ToF)-robust GLocFM model to handle unknown ToF offsets. GLocFM is trained on a multi-modal synthetic indoor localization dataset comprising 221 diverse scenes whose associated wireless signals are generated using Sionna RT. On both synthetic and the NeRF 2 dataset based on real measurements, GLocFM reduces mean 3D localization error relative to one of the state-of-the-art localization baselines by 49.5% and 48.8%, respectively. Ablations across different number of receiver, bandwidths, and array sizes further demonstrate the effectiveness and robustness of the proposed framework. Correspondence to: Jun Zhang (eejzhang@ust.hk) 1 Introduction Accurate indoor localization is a key enabler for autonomous robot navigation, real-time tracking of equipment and inventory, and services in environments where global positioning system (GPS) signals are unavailable or severely attenuated Zafari et al. (2019). Literatures Liu et al. (2007); Zafari et al. (2019) identify NLoS blockage, multipath propagation, material-dependent attenuation as persistent challenges for practical indoor localization, as they make the mapping from wireless observations to transmitter positions ambiguous and scene-dependent. By properly utilizing geometric information, we can identify potential multipath components and thereby improve our understanding of the wireless observations. Moreover, by exploiting geometric priors, it is possible to determine whether a position could contain a transmitter, motivating efficient localization algorithms that use this information to improve performance. Learning-based approaches, with their strong ability to capture important statistics and propagation features, have become mainstream methods for the complicated indoor localization problem. Existing learning-based wireless localization methods can be divided into two categories. The first maps channel state information (CSI) directly to coordinates without explicitly exploiting spatial information Zhao et al. (2024); An et al. (2020); Ayyalasomayajula et al. (2020b); Kotaru et al. (2015). Such models offer efficient inference, but can only learn correlations between wireless observations and Tx positions Studer et al. (2018) by exploring scene geometry implicitly, leading to limited generalization ability. The second group explicitly incorporates spatial information, including 2D floor plans, 3D maps, and meshes, into wireless localizers Vuckovic et al. (2021); 1 arXiv:2608.09285v1 [eess.SP] 10 Aug 2026 GLocFM •Generalizable •Single forward pass •ToF-robust Wireless Geometry Wireless Observations Geometry Prior Figure 1 The overview of the proposed GLocFM which takes both geometry and wireless observation as input to produce 3D position of the transmitter. Kanhere and Rappaport (2025); Han et al. (2025); Hehn et al. (2025); Chu et al. (2026). Spatial information can improve localization accuracy, but it also increases data-acquisition and modeling complexities: spatial representations of scenes must be accurately constructed Ayyalasomayajula et al. (2020a); Suga et al. (2023); more advanced neural models capable of processing images and 3D point clouds must be employed; and, in some methods, ray tracing Han et al. (2025) and inference-time optimization Hehn et al. (2025) must be performed for individual samples, leading to high latency. To this end, we aim to advance prior work by proposing a geometry-aware foundation model that can achieve superior localization performance and is capable of generalizing to unseen scenes using only one feed-forward pass. A critical requirement for obtaining such a foundation model is high-quality training data. Unlike large language models (LLMs), which have access to vast amounts of data on the internet, wireless localization foundation models have far too little real-measurement data available for training. Public available datasets are given in Torres-Sospedra et al. (2014); Alkhateeb (2019), yet the former does not include 3D geometry, and the latter contains too few indoor scenes to meet the generalization objective. We therefore construct a multimodal dataset of 221 diverse geometric layouts, using Infinigen Indoors Raistrick et al. (2024) to produce furnished 3D indoor environments represented as 3D meshes, and use the widely adopted Sionna RT Hoydis et al. (2023) to generate wireless signals within each indoor scenario. Given both wireless observations and scene geometry, we formulate indoor localization as an ML problem in which we first partition the space uniformly into a grid and evaluate which grid center maximizes the likelihood of the wireless observations conditioned on the geometry. Given that the exact likelihood function is intractable, we instead adopt a learned scoring function to model it. In particular, we first calculate the cosine similarity between the observed delay–angular spectrum and the spectrum predicted by considering the LOS and one-bounce reflection paths between each grid center and receiver. This cosine value serves as a prior for the subsequent learning-based scoring function, which directly predicts the probability that the Tx lies within each grid cell. A off-grid component prediction network ensures that the model outputs a continuous localization position rather than a discrete one. We also consider the scenario where an unknown ToF offset exists between the transmitter and receivers. To tackle this, we introduce arbitary ToF offsets during training such that the proposed GLocFM is capable of extracting features that remain informative under a non-zero ToF offset. Numerical experiments are performed to confirm the superiority of the proposed GLocFM over the 2 baselines. We further provide ablation studies to validate its effectiveness and robustness. Our contributions are: •We propose GLocFM, a geometry-aware foundation model grounded in the maximum-likelihood principle, enabling robust, cross-scene generalizable, 3D localization. •We construct a multimodal wireless localization dataset comprising of 221 scenes with 3D geometry and wireless signals simulated via Sionna RT. •We utilize the geometry prior to generate the delay-AoA spectrum of each grid center, and employ learned scoring function to calculate the likelihood of each grid. A ToF-robust GLocFM model is developped to handle unknown ToF offsets between the transmitter and receiver. •We demonstrate mean-error reductions of 49.5–56% on unseen synthetic scenes and 48.8% on NeRF 2 dataset with real measurement. Ablation studies are performed to justify its effectiveness and robustness. 2 Related Work 2.1 Localization without Geometry Most wireless localization systems infer position directly from radio measurements Xiong and Jamieson (2013); Vasisht et al. (2016); Xiong et al. (2015); Xie et al. (2019) without explicitly modeling the environment. SpotFi Kotaru et al. (2015) jointly estimates AoA and ToF and selects the path most likely to be LoS, but this decision is made purely in the signal domain. IARK An et al. (2020) improves multipath-robust AoA estimation across IoT protocols, but introduces neither ranging nor geometric constraints. The authors of LocGPT Zhao et al. (2024) pretrain Transformers for direction estimation and triangulation, while DLoc Ayyalasomayajula et al. (2020b) fuses the AoA–ToF maps from multiple receivers into a location heatmap for localization that is also robust to ToF offsets. Although efficient, these methods learn the radio-to-location mapping Wang et al. (2015, 2016); Chen et al. (2017); Zhang et al. (2022) without checking how the observed paths could propagate through the scene. 2.2 Wireless Localization with Geometry Recently, researchers have started to incorporate scene geometry into localization. MAP-CSI Vuckovic et al. (2021) backtracks dominant AoD–ToA components through a map, while MAP-AT Kanhere and Rappaport (2025) traces resolved angle–delay paths in measured mmWave/THz environments. These approaches make the propagation geometry explicit Gentner et al. (2016), but rely on resolving a small number of individual paths, which is difficult with an 80-MHz frequency band, and are mainly demonstrated in 2D or single-site settings. Another line of work utilize learning-based approach. LocUNet Yapar et al. (2022) combines city and path-loss maps for 2D urban localization, while GEAL Keum et al. (2026) identifies reflection points corresponding to NLoS paths using 2D geometric information. Neither is designed for 3D indoor localization. RayLoc Han et al. (2025) and Wi-GATr Hehn et al. (2025) employ richer physical models, but localize by inverting a forward simulator through gradient optimization, requiring calibrated geometry, material information, or iterative inference. SigMap Chu et al. (2026) is more closely related to the proposed scheme because it also uses complex CFRs and treats a 3D map as a prompt to its foundation model. However, it focuses on outdoor scenarios evaluated on the DeepMIMO dataset, leaving the more challenging indoor localization problem with more NLoS paths unresolved. 3 GLocFM Dataset Generation The proposed multimodal GLocFM dataset is comprised of 221 indoor scenes, whose geometric part is generated using Infinigen Indoors Raistrick et al. (2024), and the wireless part generated using Sionna RT. The scenes cover diverse furnished layouts including offices, bedrooms, bathrooms and kitchens. These scenes occupy areas from 3.25 m 2 to 63.51 m 2 , with a mean of 22.37 m 2 . Each scene consists of 3D object meshes and before simulating the wireless channel in each scene, we assign physically plausible EM properties to each mesh object according to its semantic type. Sionna RT then generates the channel response at each receiver 3 Grid Score 휆 푚 ∈ℝ � 풙=� 풎 푠(흉흀 풎 )(풄 풎 +휹 풎 ) 1 Geometry Branch Scene Encoder 풑 푖 =(풙 푖 ,풏 푖 ,풎 푖 ) Position Normal Materials 2 Wireless Branch RF Encoder 푻 푟,1 푻 푟,2 푻 푟,퐿 ⋮ 푤 푚푟 1 푤 푚푟 2 푤 푚푟퐿 Σ 3 Candidate Scoring Predicted ̂ 퐶 푚,푟 (푣,ℓ) Observed 퐶 푟 (푣,ℓ) Learned Scoring Function 풖 푚푟 =퐸 푝 (흃 푚푟 )+풒 푚푟 +퐸 푟 (풑 푟 )+퐸 표 (푸 푟 ) Transformer 4 Grid Score (λₘ) 4 Final Estimate Off-grid Component 휹 푚 ∈ℝ 3 Geometric prior, candidate 풄 푚 , RX location 풑 LOS Path Reflection Paths 흅 풎 푳 , � ℓ 풎 ퟎ ,풖 풎 ퟎ 흅 풎 풎풓 , � ℓ 풎 ퟏ ,풖 풎 ퟏ Predicted Spectrum � 푪 푚,푟 (푣,ℓ) 풙 푖 ,풓 푖 Cosine Similarity 흃 푚푟 풖 푚,1 풖 푚,2 ⋮ 풖 푚,푵 푹 풓 푚 푞 푚푟 Predicted 퐴푠퐴 퐷 푠퐷 Wireless Observations 풑 푟 ,푸 푟 Path Synthesis 풑 푟 ,푸 푟 풓 푚 풄 풎 � 푪 푚,푟 Grid Score Figure 2 The flowchart of the proposed GLocFM model. antenna array. Although Sionna RT operates on complete 3D meshes, such geometry is seldom available in practical deployments. Thus, we construct the GLocFM dataset by representing the scene geometry using a point cloud, obtained by sampling each mesh surface in proportion to its area. The transmitter and receiver positions used in the dataset are sampled from the free space, denoted by Ω free , which is the volume not occupied by any object mesh. To be precise, letSdenote the scene and let Ω⊂ R 3 be its admissible interior volume. For object meshM j , letO j ⊂Ω,j ∈[1,J] denote the closed region occupied by that object, we define Ω occ = S J j=1 O j and Ω free = Ω\Ω occ . To model ceiling-mounted infrastructure receivers and mobile transmitters below them, receiver candidates lie within 0.15 m of the ceiling, while transmitter heights span 10–70% of the scene height. We split the 221 scenes into 177 training, 22 validation, and 22 test scenes. The channel simulation uses a carrier frequency of 3.5 GHz, 80 MHz bandwidth, and 1,024 OFDM subcarriers (∆f= 78.125 kHz). We include specular and diffuse reflection with a maximum interaction depth of three, and model the transmit antenna pattern of the transmitter as isotropic. Each receiver is a half-wavelength-spaced 4× 4 UPA adopting the 3GPP TR 38.901 antenna radiation pattern. Let Q r ∈ R 3×3 denote the orientation of ther-th receiver withr ∈[1,N R ], which maps UPA-local coordinates to the world coordinates, and Q ⊤ r performs the inverse mapping. For every propagation pathpbetween the transmitter and receiver element a, Sionna RT returns a complex path gainα p,a , delayτ p , and incidence angleθ p . The receiver CFR can be expressed as: H r,a,k = X p∈P r α p,a exp(−j2πf k τ p ),(1) wherea∈[1,A] andk ∈[1,K] denote the antenna and subcarrier indices, respectively, andP r is the set of paths connecting the transmitter to the r-th receiver. 4 GLocFM Model The localization problem can be formulated as an ML estimation problem for the transmitter position given the scene geometry,S, and the observed CFRs atN R receivers. Given that the transmitter location is sampled from the free space, x∈ Ω free , its ML estimate can be expressed as: ˆ x ML = arg max x∈Ω free logp(Y r r∈[1,N R ] | x,S),(2) where N R denotes the number of receivers, and Y r is the CFR at receiver r. Because this continuous search is impractical, we instead adopt a grid search together with a learned scoring function for each grid cell, as described below. 4 4.1 Utilizing Geometric Priors 4.1.1 Generation of the Candidate Grid and Propagation Paths We first illustrate how we partition Ω free into a discrete set of candidate transmitter (Tx) positions and generate the corresponding propagation paths for each candidate Tx position. We denote the position and orientation of ther-th receiver by p r and Q r , respectively. We partition the space uniformly, creatingN x =N y =N horizontal cells and N z vertical cells, giving M≜ N 2 N z candidates c m M m=1 . For each candidate c m , we generate the corresponding propagation paths to each receiverr. Given that the LOS and the one-bounce reflection paths convey the majority of the transmit power, we simplify the propagation modeling of the proposed GLocFM framework by ignoring higher-order interactions. To start with, the LOS path connecting c m and p r has lengthd (0) mr =∥c m − p r ∥ 2 , delayτ (0) mr =d (0) mr /c , and a direction u (0) mr = Q T r c m −p r d (0) mr , expressed in the receiver’s local coordinate system. We then consider one-bounce reflection paths. Let a j be a potential reflection point, which lies on the surface of an object in the scene. It defines a one-bounce propagation path from candidate c m to receiver p r : d (1) mrj =∥c m − a j ∥ 2 +∥a j − p r ∥ 2 ,(3) u (1) mrj = Q T r a j − p r ∥a j − p r ∥ 2 ,(4) where the superscript indicates a single interaction. These quantities are used to calculate key parameters such as amplitude, delay, and AoA, which contribute to the likelihood of each candidate Tx position, as detailed later. 4.1.2 Scene Encoding We then illustrate how we extract essential wireless propagation features from the scene geometry. As shown in Fig. 1, we represent the scene as a 3D point cloud with associated features, which are fed into a scene encoder comprising multiple downsampling and self-attention layers that hierarchically reduce the number of points and aggregate features. In particular, the scene encoder first maps the coordinate x i , normal n i , and EM attributes m i = [ε r,i ,σ i ,S i ,K x,i ], comprising relative permittivity, electrical conductivity, diffuse-scattering coefficient, and cross-polarization coefficient, to a high-dimensional feature: f (0) i =φ 0 ([x i ,n i ,m i ]).Then, at hierarchy levelq,q ∈[1,Q], farthest-point sampling (FPS) selects the subsetI q , and, for each pointi∈I q , a kNN network aggregates features from its k nearest neighbors N k (i): f (q) i = max j∈N k (i) φ q x (q−1) j − x (q−1) i , f (q−1) j ,(5) whereφ q is an MLP layer and max denotes max-pooling operation. At hierarchy levelQ, we apply a self- attention module to the features f (Q) i of the surviving pointsp i , f (Q) i i∈I Q to obtainf i i∈I Q , which are used to compute the geometric priors for each candidate Tx–receiver pair. 4.1.3 Modeling Attenuation of Propagation Paths We first model the attenuation of a LoS path caused by obstruction and free-space path loss. To represent obstruction, we construct the opacity field following the principle of convolutional occupancy networks Peng et al. (2020) using the surviving scene pointsp i ,f i i∈I Q . The resulting opacity field can be queried at an arbitrary 3D location by trilinear interpolation. For a segment with endpoints (x,y), we useTuniformly spaced interior query points, s t = x + t T +1 (y−x),t∈[1,T].Letρ t ≥0 denote the opacity queried at s t , the accumulated opacity along the segment can be expressed as: O(x, y) = ∥y− x∥ 2 T T X t=1 ρ t .(6) We introduce soft visibility, V (x, y), to quantify the obstruction level, which is defined as: logV (x, y) =− softplus(γ)O(x, y), (7) 5 where softplus(γ) > 0 is a learned attenuation factor. Consequently, V (x, y)∈ (0, 1]: it approaches one for a clear segment and decreases along an obstructed segment. The use of a learned soft visibility, rather than a binary ray-tracing label, also provides smoother gradients during training. The LOS geometric prior is then written as: π LOS mr = − β L log(d (0) mr /d 0 ) + logV (c m , p r ) + g L d (0) mr , logV (c m , p r ) , (8) whereβ L is a learned scale factor initialized to the free-space path-loss exponent,d 0 is a positive constant to scale the distance, and g L (·) is a learned function parameterized by an MLP for calibration. The geometric prior for a one-bounce path is evaluated along two segments joined at a reflection point. Specifically,(7)models the obstruction from candidate c m to the reflection point a j ∈I Q , and from a j to receiver p r , while two log-distance terms model distance-dependent propagation loss. The two directions pointing from the reflection point toward the candidate and receiver are denoted by v mj = c m −a j d mj and v rj = p r −a j d rj , respectively. At reflection pointj, an MLP models the reflection-induced attenuation, denoted as α refl mrj , which takes these two directions, the surface normal n j , and the feature f j associated with the reflection point as input: z mrj = concat(v mj , v rj , n j , f j ), α refl mrj = g r (z mrj ).(9) The overall geometric prior for the reflection path is then π refl mrj = − β L [log(d (1) mj /d 0 ) + log(d (1) rj /d 0 )] + logV (c m , a j ) + logV (a j , p r ) + α refl mrj .(10) 4.2 Candidate Scoring Function We then illustrate how we design the scoring function given the observed wireless signals and the predicted ones. Instead of directly matching the CFRs, which are high-dimensional and sensitive to the estimated path parameters, we transform them into delay–angular spectrum which is calculated below: For the observed wireless signals, we first transform its CFR into the delay–array spectrum via IFFT: h r,a,ℓ = 1 K K−1 X k=0 H r,a,k e j2πkℓ/K .(11) Letv g G g=1 denote the set of steering vectors at the receiver, and letℓ∈0,...,L−1denote the delay-bin index. 1 For the r-th receiver, its delay–angular spectrum is: C r (v,ℓ) = log(| X a h r,a,ℓ e −j2πe ⊤ a v | 2 + ε).(12) Here, v∈Vis expressed in the receiver’s local coordinate system. The denominator normalizes the directional response by the received energy at the same delay bin. A propagation path can be represented by its delay, AoA, and attenuation,π LOS mr andπ refl mrj , as defined in(8) and(10), respectively. The delayd mr /cmay correspond to a fractional number of delay bins, denoted by b ℓ =d/∆ rng , where ∆ rng is the range resolution determined by the signal bandwidth; interpolation is therefore required. In principle, the ideal interpolation function for a bandlimited signal is the sinc function. However, its sidelobes produce unstable gradients during training, leading to suboptimal performance, so we instead adopt the Gaussian surrogate: κ ℓ ( b ℓ) = exp[−(ℓ− b ℓ) 2 /(2σ 2 ℓ )] P L−1 v=0 exp[−(v− b ℓ) 2 /(2σ 2 ℓ )] .(13) 1 We setL≪K as only the top delay bins are occupied. 6 The AoA parameter follows standard processing: for array elementaat position e a , the array response for an EM wave arriving from direction u ise j2πe ⊤ a u . Then, the predicted delay–array spectrum can be expressed as: b h m,r,a,ℓ = e π LOS mr e j2πe ⊤ a u (0) mr κ ℓ ( b ℓ (0) mr ) + X j e π refl mrj e j2πe ⊤ a u (1) mrj κ ℓ ( b ℓ (1) mrj ). (14) For candidatem, the predicted spectrum b C m,r (v,ℓ) is obtained from (12) by replacingh r,a,ℓ with b h m,r,a,ℓ . Stacking the sampled values over direction and delay gives C r = [C r (v g ,ℓ)] g,ℓ ∈ R G×L and b C m,r . We then measure the similarity between the observed and predicted one as s mr = ⟨C r , b C m,r ⟩ F ∥C r ∥ F ∥ b C m,r ∥ F + ε ,(15) 4.2.1 Learned Scoring Function for Enhancement The candidate scoring function based on cosine similarity in(15)may not be capable of fully exploiting the rich information contained in the scene geometry and the observed signals. Thus, we introduce a learned scoring function for enhancement. To start with, the learned scoring function maps the similarity scores mr along with the absolute received power P r at ther-th receiver to a feature vectorξ mr containing geometric information:ξ mr =MLP( s mr ,P r ), whereP r can be understood as a ‘confidence’ level of the similarity score. Then, the RF encoder shown in Fig. 2, which comprises Transformer layers, takes the CFR as input and produces tokensT r,ℓ ℓ∈[1,L] for the Ldelay taps. We obtain the feature vector for them-th candidate by performing weighted pooling on T r,ℓ along the delay axis: q mr = L−1 X ℓ=0 w mrℓ T r,ℓ , (16) wherew mrℓ are learnable parameters. By aggregating the aforementioned information which are essential for localization, we represent the candidate–receiver feature as: u mr = E p (ξ mr ) + q mr + E r (p r ) + E o (Q r ), (17) whereE p ,E r , andE o are MLPs that map the path features, receiver position, and orientation to a high- dimensional space. A self-attention block takesu mr r∈[1,N R ] as input and outputs f m for each candidate, which is then used to predict the final score, λ m , for the m-th gird and its off-grid component, δ m : λ m = h score (f m ),r m = h off (f m ), δ m = 1 2 (∆)⊗ tanh(r m ). (18) where ∆ = (∆ x ,∆ y ,∆ z ) contains the spacings between neighboring candidate centers, and⊗denotes the element-wise product, which ensures that c m +δ m still lies within the cell of them-th grid. Theh score and h off are parameterized by MLP. With temperature τ > 0, the estimate is ˆ x = M X m=1 softmax m (τλ m )(c m + δ m ).(19) We note that the softmax function and the off-grid componentδ m are essential to achieve a resolution finer than 1 2 q ∆ 2 x + ∆ 2 y + ∆ 2 z , otherwise GLocFM would simply produce the grid center, c m with the highestλ m value. We train the model by minimizing the ℓ 2 -distance between the predicted position and the ground truth: L train =∥ b x i − x i ∥ 2 2 .(20) 7 N R = 3N R = 4 MethodMean↓ Median↓ RMSE↓ b F e (0.2)↑ Mean↓ Median↓ RMSE↓ b F e (0.2)↑ GLocFM0.17460.11810.253374.8%0.14290.10320.192482.2% LocGPT delay-AoA 0.39720.27010.508536.2%0.28270.21970.405745.2% LocGPT AoA-only0.41430.27810.545332.6%0.30720.22250.429244.5% iARK0.62410.44090.823220.6%0.50790.35790.671828.0% DLoc0.60750.50720.72148.9%0.47460.36790.587516.7% Wi-GATr (RSRP)1.04800.88691.22150.9%0.86820.72810.98275.0% Table 1 3D localization error (m) on the synthetic GLocFM dataset. 4.3 ToF-robust GLocFM The GLocFM model described above assumes perfect synchronization between the transmitter and receivers. We further consider a TDoA setting in which the receivers are mutually synchronized but share an unknown ToF offset relative to the transmitter. For training samplei, the common offsetb i produces the same frequency- domain phase ramp at every receiver: e H i,r,a,k =H i,r,a,k e −j2πν k b i ,r= 1,...,N R ,whereν k is the frequency of thek-th subcarrier. Following Koivisto et al. (2017); Yang et al. (2022), we draw one offset per training sample asb i ∼N(0,σ 2 b ). This bias augmentation exposes the learned scoring function to different ToF offsets whose loss function during training can be expressed as: L train = E b∼N (0,σ 2 b ) ∥ b x i (b)− x i ∥ 2 2 .(21) This differs from the original model as the ToF-robust model optimized using(21)is encouraged to exploit features that remain informative under a non-zero ToF offset, which are essential to retain robustness against unknown ToF offset in real deployment. The network architecture and the corresponding training methodology remain unchanged. During inference, the ToF-robust GLocFM uses a single forward pass without the need to estimate the exact ToF offset. 5 Simulation Results 5.1 Experimental Settings 5.1.1 Datasets We first evaluate the proposed GLocFM model on the synthetic GLocFM dataset detailed above, which consists of 221 generated indoor scenes with corresponding wireless signals. We also evaluate on the NeRF 2 dataset Zhao et al. (2023), which provides 6,123 angular spectrum, of which 4,898 and 1,225 samples are used for training and testing, respectively. In particular, the authors perform real-world wireless experiments in which a single-antenna Tx at different locations communicates with a fixed receiver equipped with a 4×4 UPA. 5.1.2 Baselines Four learning-based localization baselines are considered, including localizers with and without geometry information. We first implement the method in Hehn et al. (2025), which takes the received signal power (RSRP) and the 3D map as inputs and performs optimization during inference. We also implement localizers without geometry information. In particular, LocGPT AoA-only follows the original Zhao et al. (2024), in which only the received AoA spectrum is fed to the Transformer-based model. We furtherimplement LocGPT delay-AoA, in which delay information is also considered. IARK An et al. (2020) is a multipath-resistant CNN-based AoA neural localizer, while DLoc Ayyalasomayajula et al. (2020b) utilizes both delay and AoA information, organizing the wireless signals as multiple AoA–ToF images. All these baselines are trained and tested using the same setup as the proposed GLocFM. 8 0.00.30.60.91.21.5 Localization Error (m) 0.00 0.25 0.50 0.75 1.00 CDF 0.2 m GLocFM LocGPT delay-AoA LocGPT AoA iARK DLoc Array sizeBandwidth (MHz) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Mean localization error (m) 4×4 3×3 2×2 80 40 20 4×4 3×3 2×2 80 40 20 80 MHz fixed4×4 fixed80 MHz fixed4×4 fixed N R = 3N R = 4 ToF offset (ns) 00.511.522.533.54 Mean localization error (m) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Original GLocFM ToF-robust GLocFM No-offset baseline (a)(b)(c) Figure 3 Evaluation of GLocFM. (a) CDFs achieved by different localization schemes. (b) Mean localization error with different system configurations. (c) Mean localization error under different ToF offsets for the original and ToF-robust GLocFM models. 5.1.3 Metrics and Implementation Details The proposed scheme is trained using Adam with a learning rate of 10 −4 for 200 epochs and a cosine scheduler on a device with 8 NVIDIA A6000 GPUs. The default GLocFM model adopts a 12×12×4 candidate grid, a scene-encoder output cardinality of|I p |= 256,L= 20 retained delay taps, and an angular resolution of 5 ◦ . We report mean, median, root mean-square error (RMSE), and the CDF of 3D localization error which are obtained by averaging over multiple trials with different seeds. For sample errore i and thresholdτ, the CDF is defined as: b F e (τ ) = 1 n n X i=1 1[e i ≤ τ ].(22) 5.2 Comparison with Baselines 5.2.1 Comparison on the Synthetic GLocFM Dataset We first evaluate GLocFM on the synthetic GLocFM dataset against the baselines described above. As shown in Table 1, GLocFM achieves the lowest mean, median, and RMSE for differentN R values. In particular, it achieves a mean localization error of 0.1429 m 0.1746 m, which are 49.5% and 56.0% lower than that achieved by the strongest LocGPT delay-AoA baseline forN R = 4 andN R = 3, respectively. We also note that the Wi-GATr Hehn et al. (2025) baseline which also adopt geometric information achieves the worst localization performance. This is due to the fact that it is designed by only adopting the RSRP as input while ignoring the more informative delay–angular spectrum. It is interesting to observe that reducing the number of receivers toN R = 3 raises the mean error of GLocFM only slightly, whereas the two LocGPT baselines show increases of approximately 0.1 m. This robustness is consistent with the reflected paths providing geometric constraints that complement direct-path triangulation. Fig. 3(a) compares the empirical CDF forN R = 4, where the proposed scheme achieves an accuracy of 82.2%, surpassing the baseline schemes. The larger CDF of GLocFM throughout the low-error range indicates that its mean improvement is not driven solely by a few favorable samples. Fig. 4 further reports the mean localization error for each of the 22 scenes in the test dataset, where GLocFM consistently achieves lower errors than the baselines. For a qualitative comparison, we visualize a representative localization sample in Fig. 1. It can be seen that GLocFM achieves the best performance among all methods, whose localization error is small relative to the scene scale. 5.2.2 Evaluation on Real Measurements We next evaluate on the NeRF 2 Zhao et al. (2023) dataset, in which each sample contains a 3D transmitter location and its corresponding angular power spectrum from the fixed receiver. Note that the authors did not provide CSI at the OFDM subcarriers; thus, no delay information is available. Consequently, we modify the 9 S01S02S03S04S05S06S07S08S09S10S11S12S13S14S15S16S17S18S19S20S21S22 Scene (#) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Mean Error (m) ProposedLocGPTiARKDLoc Figure 4 Mean localization error for each of the 22 scenes in the synthetic test dataset. SettingMean Median RMSE b F e (0.1) b F e (0.2) iARK0.2849 0.2666 0.3193 6.9% 29.7% LocGPT0.1244 0.0799 0.1816 63.5% 85.6% Fine-tune using p% training data GLocFM (p = 100) 0.0637 0.0185 0.1429 86.6% 91.4% GLocFM (p = 50) 0.0854 0.0273 0.1686 78.8% 87.1% GLocFM (p = 25) 0.1304 0.0492 0.2233 67.3% 78.8% Table 2 Localization error on NeRF 2 dataset. original GLocFM model by applying minor changes to the predicted spectrum, ˆ C r (v,ℓ), and the inputs to the RF encoder. As shown in Table 2, the modified GLocFM model achieves a lower localization error than the baselines. Note that we train both the LocGPT and iARK baselines utilizing all samples in the training dataset. For the proposed scheme, however, we evaluate its generalizability by fine-tuning the model usingp% of the samples in the training dataset. We observe that reducing the number of training samples from 100% to 25% raises mean localization error from 0.0637 to 0.1304 m and lowers the CDF at 0.1 m from 86.6% to 67.3%. It is worth mentioning that the weakened generalizability of the proposed scheme may be due to the differences in the experimental setup, such as antenna spacing and antenna patterns; investigating these effects is left for future work. 5.3 Ablation Studies 5.3.1 Effects of Different GLocFM Designs Table 3 summarizes the impact of different components on the localization performance of the proposed GLocFM. The “Reference” scheme corresponds to the default configuration with grid size N = 12. As can be seen, the size of the candidate grid is of vital importance: reducing it fromN= 12 toN= 3 increases the mean error from 0.1429 m to 0.4494 m, whereas increasing it toN= 16 yields an additional gain of 0.7 cm but nearly doubles the computational complexity. The learned scoring function is essential, as removing it substantially worsens the error statistics, showing that simply adopting cosine similarity alone cannot resolve ambiguous candidates. We also demonstrate the effectiveness of the one bounce reflection path by showing a degradation of 7.5 cm when adopting the LoS-only GLocFM model. 5.3.2 Different System Configurations We further perform an ablation study to evaluate the robustness of the proposed GLocFM model with respect to different system bandwidths and UPA settings, as shown in Fig. 3(b). Note that we fine-tune the model for 10 DesignVariantMean Median RMSE b F e (0.2) Reference –0.1429 0.1032 0.1924 82.2% Grid size N = 30.4494 0.2825 0.7042 31.4% N = 160.1360 0.1012 0.1830 84.2% ScorerCosine only 0.3914 0.2977 0.5049 26.1% Path model LoS only 0.2285 0.1495 0.3515 66.0% Table 3 Ablation study of different GLocFM designs. only a single epoch when applying it to different configurations. WithN R = 4, reducing the UPA from 4×4 to 3×3 and 2×2 increases the test error from 0.1429 m to 0.1684 m and 0.2332 m, respectively. The results obtained withN R = 3 show the same trend. For both setups, reducing the bandwidth from 80 to 20 MHz increases the localization error by roughly 7 cm. This is plausible because a smaller array weakens spatial resolution, while a lower bandwidth reduces range resolution and makes nearby ranges harder to separate. 5.4 Robustness to ToF Offset We then evaluate the localization performance of the proposed ToF-robust GLocFM model and compare with the original one. The ToF-robust GLocFM model is trained withσ b = 4. Both schemes are evaluated with ToF offsets ranging from 0 to 4 ns. As shown in Fig. 3(c), the mean localization error of the original model increases dramatically from 0.1429 to 0.7141 m, demonstrating its vulnerability to ToF offset. The ToF-robust GLocFM model performs worse under perfect synchronization, i.e.,b= 0 ns. 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