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Digital Twin-Guided Wireless Localization via Probabilistic Inverse Learning

Mahmoud Saad Abouamer, Henrik C. M. Frederiksen, Robin J. Williams, Shashi Raj Pandey, Petar Popovski

arXiv:2610.36905v1cs.ITeess.SPmath.IT

TL;DR

Wireless localization from limited distributed single-antenna AP observations must contend with noise, multipath, and hardware complexity, while site-specific DT use need not assume exact channels or differentiability. PRISM learns a probabilistic location proposal and uses DT consistency to reweight sampled candidates; experiments report lower localization error than benchmark methods, including about 0.7 m median error with 10 APs.

  • Problem

    Wireless localization from channel observations remains challenging under noise, complex multipath, and requirements for low hardware complexity and cost efficiency.

    Source: §I · p. 1

  • Method

    PRISM learns a conditional proposal over UE locations from multi-node CIRs, regularizes its candidate spread during training, and uses an approximate DT forward-consistency check to reweight candidates at inference.

    Source: §I · p. 2, C. DT-Informed Probabilistic… · p. 4, §VI · p. 12

  • Results

    About 0.7 m median localization error was achieved using 10 distributed single-antenna APs, with up to 63% lower median error relative to a Transformer-based benchmark.

    Source: §VI · p. 12

  • Takeaways & Limitations

    The experiments indicate that DT-guided probabilistic refinement can improve localization over classical fingerprinting and data-driven neural benchmarks in realistic site-specific ray-traced environments.

    Source: §VI · p. 12

Abstract

from arXiv · show

This work investigates how site-specific digital twins (DTs) can support wireless localization using simplified distributed infrastructure. We propose PRISM (Probabilistic Refinement using Inverse learning and a Site-specific Model), a DT-guided probabilistic localization scheme with single-antenna access points (APs). Localization is formulated as a probabilistic inverse problem in which a conditional proposal distribution over UE locations is learned from multi-node channel impulse responses (CIRs). The proposal generates candidate locations, with its spread regularized during training and DT consistency used to reweight candidates and refine the estimate at inference. The DT serves as an approximate forward-consistency model, avoiding assumptions of exact channel fidelity or differentiability with respect to user position. Numerical experiments demonstrate gains over data-driven and classical fingerprinting benchmarks, while achieving a median error of approximately $0.7$ m using 10 single-antenna APs. Without retraining, PRISM remains effective under degraded inference-time DT features. PRISM also yields uncertainty estimates from its proposal covariance, with empirical $\chi^2$ calibration approaching ideal behavior as APs increase.

I. INTRODUCTION

The paper studies whether site-specific propagation structure can support localization with distributed single-antenna APs, avoiding the cost of richer sensing hardware. PRISM combines a learned probabilistic inverse proposal with a DT forward-consistency check to refine candidate locations.

I. INTRODUCTION

  • Conventional higher-resolution sensing, including massive MIMO and large arrays, can separate multipath more effectively but adds hardware, energy, and deployment costs.These costs can limit scalability in practical industrial environments.Source: §I · p. 1
  • The setting isolates how site-specific physical knowledge can aid localization from limited observations without relying on exact DT accuracy or position differentiability.The work uses a conservative distributed setup with a limited number of single-antenna APs, rather than arrays or dense fingerprint infrastructure.Source: §I · p. 1
  • Direct learning-based localization can capture environment-dependent structure but typically lacks an explicit mechanism to enforce consistency with site-specific propagation physics.Such predictions may be statistically plausible yet inconsistent with the site-specific propagation geometry.Source: §I · p. 1
  • Earlier DT localization approaches mainly use twins for synthetic fingerprints, training augmentation, or propagation surrogates, while differentiable ray tracing can require useful spatial derivatives.The paper positions inference-time channel-consistency refinement as an alternative to those uses and to iterative gradient-based point optimization.Source: §I · p. 2
  • PRISM formulates localization as a probabilistic inverse problem and uses the DT as an approximate forward-consistency check, without requiring exact channel fidelity or differentiability.The inverse proposal generates plausible locations; the DT evaluates their channel-domain consistency.Source: §I · p. 2
  • PRISM regularizes proposal spread during training and reweights sampled candidates using DT consistency during inference.This finite-candidate procedure controls candidate-search spread while applying the DT check at inference.Source: §I · p. 2
  • Experiments report improvements over data-driven and classical fingerprinting benchmarks, robustness to degraded inference-time DT information without retraining, and better empirical uncertainty calibration as APs increase.These findings are summarized as the paper’s numerical contributions.Source: §I · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system model considers uplink localization from observed multi-AP CIR features and a site-specific DT that predicts features from hypothesized locations. The DT is approximate, and the formulation does not require it to be differentiable with respect to location.

A. Multi-AP Channel Observation Model

  • The localization input is the observed multi-AP CIR feature associated with an unknown UE position in the deployment domain.The system has distributed single-antenna APs at known positions in a site-specific propagation environment.Source: A. Multi-AP Channel Observation Model · p. 3
  • Each AP contributes a sampled delay-domain CIR with K retained taps, and the CIRs from all APs are stacked into a joint feature.The nominal real-valued feature concatenates real and imaginary components and has dimension Dy = 2NK.Source: A. Multi-AP Channel Observation Model · p. 3
  • The nominal site-specific mapping produces the processed multi-AP CIR feature, while the observed feature includes an effective observation-domain error term.The errors include channel-estimation noise, receiver noise, and preprocessing mismatch.Source: A. Multi-AP Channel Observation Model · p. 3

B. Site-Specific Digital Twin Forward Model

  • The DT provides an approximate forward operator mapping a hypothesized UE location to a predicted processed CIR feature in the observed feature space.Ray tracing uses site geometry, AP locations, and modeled electromagnetic interactions before applying the same feature construction.Source: B. Site-Specific Digital Twin Forward… · p. 3
  • The DT need not exactly replicate the physical environment; its predictions may differ because of geometry, materials, calibration, ray-tracing resolution, or unmodeled propagation effects.The formulation also does not require differentiability with respect to UE location.Source: B. Site-Specific Digital Twin Forward… · p. 3
  • The model’s intended role is approximate site-specific forward consistency, not exact inversion or a differentiable trainable layer.This limits the required DT capability to predicting channel features for hypothesized locations.Source: B. Site-Specific Digital Twin Forward… · p. 3

C. DT-Informed Probabilistic Localization

The section frames localization as inference from channel observations and adds a DT mismatch to assess candidate locations. A learned conditional proposal restricts that assessment to plausible candidates, with training shaping the proposal and inference reweighting candidates by DT consistency.

C. DT-Informed Probabilistic Localization

  • A conventional deterministic estimator maps each processed channel observation to a point estimate, whereas the DT supplies a forward response for checking hypothesized locations.The conventional minimum mean-squared-error criterion is stated for the deterministic estimator.Source: C. DT-Informed Probabilistic… · p. 3
  • DT mismatch measures discrepancy between observed and DT-predicted channel features, but does not itself specify an efficient inverse search.Multipath can leave multiple locations plausible, while evaluating the full deployment domain with the DT would undermine the efficiency of a learned inverse model.Source: C. DT-Informed Probabilistic… · p. 3
  • A learned conditional proposal distribution samples candidate UE locations from the observed CIR feature, restricting DT evaluation to a data-informed plausible set.The proposal addresses the gap between having a consistency score and performing an efficient inverse search.Source: C. DT-Informed Probabilistic… · p. 3
  • The proposal must both assign high likelihood to the ground-truth location and retain enough spatial spread for finite sampling to produce useful alternatives.An overly concentrated proposal can omit corrective candidates, while an excessively diffuse one reduces finite-sampling efficiency.Source: C. DT-Informed Probabilistic… · pp. 3–4
  • The objective combines supervised probabilistic localization with proposal shaping, and λ controls their relative contribution.The supervised term encourages high likelihood at the labeled UE location; the shaping term targets candidate generation for DT refinement.Source: C. DT-Informed Probabilistic… · p. 4

III. PROPOSED PROBABILISTIC INVERSE LEARNING FRAMEWORK

  • At inference, the DT mismatch scores and reweights candidates sampled from the learned proposal to produce a refined location estimate.The framework evaluates a finite candidate set rather than the full deployment domain.Source: C. DT-Informed Probabilistic… · p. 4, §III · p. 4

A. Neural Inverse Proposal

The inverse model represents the proposal as a Gaussian with predicted mean and covariance, using the covariance to set the candidate region. The DT then scores a finite set of sampled candidates and assigns relative consistency weights for refinement.

A. Neural Inverse Proposal

  • The proposal is a conditional Gaussian whose predicted mean is the nominal location estimate and whose covariance determines the candidate-sampling region.A broader proposal makes a larger region available for DT-guided refinement.Source: A. Neural Inverse Proposal · p. 4
  • The proposal’s supervised term is the empirical negative log-likelihood of the ground-truth UE location.For the Gaussian proposal, the paper gives its corresponding form; μ and Σ denote predicted mean and covariance, and du is the localization dimension.Source: A. Neural Inverse Proposal · p. 4

B. DT-Based Consistency Weighting

  • The DT consistency energy scores agreement between the observed CIR feature and the feature predicted at a candidate, with lower energy indicating stronger agreement.The mismatch supplies the forward-consistency criterion for comparing locations.Source: B. DT-Based Consistency Weighting · p. 4
  • Rather than querying the DT across the full deployment domain, inference samples S candidate locations independently from the learned proposal.Each sampled candidate is evaluated using the DT consistency energy.Source: B. DT-Based Consistency Weighting · p. 4
  • Normalized DT-consistency weights express relative preference among sampled candidates, favoring smaller mismatches; decreasing τ concentrates weight more strongly on the most consistent candidates.τ controls sensitivity to the mismatch.Source: B. DT-Based Consistency Weighting · p. 4
  • The overview separates offline proposal learning with finite-candidate regularization from online DT-based reweighting and estimate refinement.The Transformer maps the observed channel feature to a proposal distribution from which candidates are sampled.Source: Figure 2 · p. 5

C. Proposal Regularization

The proposal regularizer uses DT consistency scores on sampled candidates to incorporate site-specific forward information into learning. It complements the supervised objective without requiring DT derivatives or treating the finite-candidate term as an exact posterior objective.

C. Proposal Regularization

  • For each training observation, candidate locations are sampled from the current proposal and assigned relative weights according to DT consistency.Those weighted samples provide the finite-candidate mechanism for incorporating forward-consistency information into proposal learning.Source: C. Proposal Regularization · p. 5
  • The finite-candidate regularizer gives greater contribution to sampled locations with stronger agreement between observed and DT-predicted channel features.It incorporates relative DT preferences among candidates generated by the current proposal.Source: C. Proposal Regularization · p. 5
  • Computing the regularizer requires numerical DT consistency scores for sampled candidates, but not derivatives of the DT forward model.Source: C. Proposal Regularization · p. 5
  • The regularization term is combined with a supervised objective that anchors the proposal to the labeled UE location, with λ controlling regularization strength.The two terms support an observation-informed proposal and candidate generation that can benefit from DT guidance.Source: C. Proposal Regularization · p. 5

D. Closed-Loop Inference

PRISM restricts inference to a data-informed candidate region and uses DT consistency to refine sampled locations. Its probabilistic Transformer proposal encodes multi-AP CIR observations and represents uncertainty with a diagonal Gaussian in the reported implementation.

D. Closed-Loop Inference

  • At inference, the DT scores and reweights candidates sampled from the learned proposal, and the final estimate is their DT-consistency-weighted mean.This is a refinement over the sampled candidate set, not an exact posterior mean; at small temperature, weights concentrate on candidates with smallest DT mismatch.Source: D. Closed-Loop Inference · pp. 5–6
  • The DT evaluates only the finite, data-informed candidate region supplied by the learned inverse model, rather than searching the complete deployment domain.Source: D. Closed-Loop Inference · pp. 5–6

A. Neural Proposal Architecture

  • The proposal encoder processes multi-AP CIRs with AP-wise tokenization, learned AP-ID embeddings, a masked Transformer, and attention pooling before distributional output heads.The AP mask prevents unavailable APs from contributing to attention.Source: A. Neural Proposal Architecture · p. 6
  • The reported implementation uses a diagonal Gaussian proposal, whose heads predict its mean and log-variance.The paper reports no substantial preliminary gains from full-covariance or mixture-based alternatives in the considered scenarios.Source: A. Neural Proposal Architecture · p. 6
  • Proposal candidates are generated by reparameterized sampling using the predicted mean and standard deviations with Gaussian noise.The resulting candidates are evaluated by the DT forward model.Source: A. Neural Proposal Architecture · p. 6

B. Training and DT-Guided Inference

Training balances a supervised probabilistic localization objective with proposal regularization, while inference scores proposal candidates using DT consistency. The reported consistency measure is selected by validation performance.

B. Training and DT-Guided Inference

  • The supervised diagonal-Gaussian loss scales its log-variance term by α, where α = 1 recovers the standard negative log-likelihood and α < 1 relaxes the penalty on larger variances.This is used to avoid overly concentrated proposals when DT-guided refinement benefits from a broader candidate set.Source: B. Training and DT-Guided Inference · p. 6
  • The SSIM score is computed over local windows of structured multi-AP CIR representations and averaged across windows.The structured tensor uses AP index, delay, and real/imaginary components.Source: B. Training and DT-Guided Inference · p. 6
  • SSIM provides the best validation performance among the considered DT-consistency measures and is used in the reported PRISM experiments.The alternatives evaluated are cosine similarity and SSIM.Source: B. Training and DT-Guided Inference · p. 6
  • At inference, the Transformer predicts proposal parameters, samples candidates, scores them with DT consistency, and forms the final estimate as a weighted candidate mean.The architecture supplies proposal mean and log-variance estimates from multi-AP CIR input.Source: B. Training and DT-Guided Inference · p. 7, Figure 3 · p. 7

C. Benchmark Schemes

The benchmark suite separates direct neural localization from classical fingerprint matching and includes a denser fingerprint database to test performance at higher spatial resolution. The comparisons use CNN and Transformer predictors alongside three fingerprinting variants.

C. Benchmark Schemes

  • The benchmark suite includes CNN-Direct and Transformer-Direct as single-location neural predictors, plus C-FP, Dense-FP, and RFMap-FP fingerprinting schemes.Transformer-Direct shares PRISM's encoder structure but uses a point-estimate head; CNN-Direct is adapted from, and RFMap-FP is based on.Source: C. Benchmark Schemes · p. 7
  • Transformer-Direct isolates the Transformer CIR representation from probabilistic sampling and DT-guided refinement.It uses the proposal network's encoder structure with a point-estimate output.Source: C. Benchmark Schemes · p. 7
  • Dense-FP uses the C-FP cosine-similarity rule with a denser database, increasing spatial resolution at the cost of a larger exhaustive search.It tests PRISM against a more computationally demanding fingerprinting baseline.Source: C. Benchmark Schemes · p. 7
  • The Munich scene figure shows AP positions and shades the candidate outdoor UE locations.Source: Figure 4 · p. 8
  • RFMap-FP uses the same database as C-FP in this simulation setup, so their comparison primarily reflects the fingerprint-matching rule.Source: C. Benchmark Schemes · p. 8

V. NUMERICAL EXPERIMENTS

Numerical experiments use simulated site-specific channels and distributed single-antenna AP configurations to evaluate localization under observation noise and inference-time DT-feature distortion. Training and inference use a held-out test set and finite-candidate DT-guided reweighting.

1) Scenario, Channel Generation, and AP Configurations:

  • The Munich urban-outdoor scenario evaluates five nested configurations of 5, 7, 10, 12, and 15 single-antenna APs, using 1 m UE-grid spacing; Dense-FP also uses 0.75 m and 0.5 m grids.Additional San Francisco experiments showed similar qualitative performance trends, but their results were omitted due to space limitations.Source: 1) Scenario, Channel Generation, and… · p. 8

2) Training and Inference:

  • Models train for 500 epochs with the lowest-validation-error checkpoint evaluated on a held-out test set; PRISM inference uses 200 proposal samples and SSIM-based DT consistency scoring.The inference temperature is 10^-4, concentrating the estimate on the most DT-consistent candidates.Source: 2) Training and Inference · p. 8

3) Observation Noise and DT-Feature Distortion:

  • Observation noise is applied to observed CIR features at 25 dB SNR by default, while robustness tests separately distort inference-time DT-predicted features without retraining.SDR_DT reports the power ratio for DT-feature distortion, distinct from observation SNR.Source: 3) Observation Noise and DT-Feature… · p. 8

B. Localization Accuracy and Distribution of Errors

Across the evaluated AP configurations, PRISM improves localization accuracy over learning-based and fingerprinting benchmarks, including when fingerprint resolution is increased. Its gains also appear in median error, spatial error patterns, and NLOS test locations.

Localization accuracy

  • At 10 APs, PRISM achieves 1.11 m MAE versus 2.09 m for Transformer-Direct, a 47% reduction; similar gains occur across AP configurations.Both use the same encoder architecture, so the additional gain is not attributable to the Transformer representation alone.Source: Figure 5 · p. 9, B. Localization Accuracy and… · p. 9
  • At 10 APs, PRISM achieves 1.11 m MAE versus 2.17 m for Dense-FP with 0.5 m spacing, an approximately 49% reduction.PRISM remains more accurate across configurations even as finer fingerprint grids improve accuracy, at the cost of larger databases and exhaustive search.Source: Figure 6 · p. 9, B. Localization Accuracy and… · p. 9
  • At 10 APs, PRISM reduces median error to approximately 0.70 m from 1.90 m for Transformer-Direct, a 63% reduction; at 15 APs its median is 0.66 m.PRISM has the lowest mean and median errors; fingerprinting has competitive median accuracy but larger mean errors.Source: Figure 7 · p. 9, B. Localization Accuracy and… · p. 9

Distribution of errors

  • In the 10-AP deployment, PRISM reduces the intensity and spatial extent of Transformer-Direct’s high-error regions, especially near scene boundaries and building edges.Challenging regions remain for both schemes.Source: Figure 8 · p. 10, B. Localization Accuracy and… · p. 10
  • For NLOS test locations, PRISM’s median error is approximately 0.79 m versus 2.07 m for Transformer-Direct, 2.24 m for C-FP, and 3.14 m for CNN-Direct.Its 75th-percentile error of approximately 1.76 m is below every benchmark’s median error.Source: Figure 9 · p. 10, B. Localization Accuracy and… · p. 10

C. Robustness to Observation SNR and DT Imperfections

PRISM remains accurate under imperfect DT information and observation-SNR shifts, with trained models reused without retraining. The results show graceful degradation as mismatches increase.

1) Robustness to Imperfect DT Information:

  • At 10 APs, magnitude-only DT features yielded 1.51 m MAE, about 28% below Transformer-Direct’s 2.09 m.With full DT features, MAE was 1.11 m; with SDRDT of 40, 25, and 15 dB, it was 1.17 m, 1.36 m, and 1.47 m, respectively.Source: 1) Robustness to Imperfect DT… · pp. 10–11
  • 0.99 m MAE with the full DT rose to 1.04, 1.08, 1.21, and 1.31 m at 25%, 50%, 75%, and 100% material replacement (15 APs).Even with full material replacement, PRISM remained more accurate than Transformer-Direct across the considered AP configurations; preserved geometry and relative channel consistency still provided guidance.Source: 1) Robustness to Imperfect DT… · p. 11
  • Trucks caused mainly localized DT mismatch with no noticeable change in aggregate mean or median error, but systematic evaluation under more dynamic scene changes remains future work.The reported nearby-location result was also limited to the considered setting.Source: 1) Robustness to Imperfect DT… · p. 11

2) Robustness to Observation SNR Mismatch:

  • 1.25 m to 1.94 m MAE under 15-to-25 dB mismatch and 1.11 m to 1.59 m under 25-to-40 dB mismatch; PRISM stayed below 2 m in every tested condition.It also had the smallest absolute degradation in both mismatch cases. Higher test SNR did not necessarily improve localization: CNN-Direct MAE rose from 3.56 m to 12.56 m under 15-to-25 dB mismatch and from 3.30 m to 8.68 m under 25-to-40 dB mismatch.Source: Table I · p. 11, 2) Robustness to Observation SNR… · p. 11

D. Uncertainty Quantification

PRISM’s proposal covariance is assessed against localization errors using squared Mahalanobis statistics and coverage. Empirical calibration approaches the χ2(2) reference as AP count increases.

D. Uncertainty Quantification

  • The predicted mean and covariance define a squared Mahalanobis statistic that should follow χ2(2) if the two-dimensional Gaussian proposal is calibrated.The assessment compares empirical statistics with the reference distribution across AP counts.Source: D. Uncertainty Quantification · p. 11
  • 2.87, 2.14, and 1.99 were the mean squared Mahalanobis statistics for 5, 10, and 15 APs, approaching the χ2(2) reference mean of 2.The 5-AP distribution was overconfident on average with a heavier upper tail; the 15-AP case was well calibrated in the mean.Source: D. Uncertainty Quantification · pp. 11–12, Figure 12 · p. 12
  • 93.36%, 96.88%, and 98.63% empirical coverage at 15 APs were close to or slightly above the nominal 90%, 95%, and 99% levels.The mean statistic was 1.992 versus the χ2(2) reference mean of 2, although some deviation remained across the full distribution.Source: Table II · p. 12, D. Uncertainty Quantification · p. 12

VI. CONCLUSION

  • The conclusion reports useful predictive uncertainty estimates, with empirical calibration improving as more AP observations are available.The uncertainty assessment is part of the framework’s probabilistic formulation.Source: §VI · p. 12
  • Multi-antenna sensing, three-dimensional localization, and online adaptation in dynamic environments remain future directions.The conclusion also identifies federated or distributed data collection as a possible direction.Source: §VI · p. 12
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