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Radiomap Blind Prediction under Incomplete Observation: Error Characterization and Correctable Propagation-Prior Learning

Xiaojie Li, Yu Han, Han Fang, Shangqing Liu, Guangxu Zhu, Shi Jin, Chao-Kai Wen

arXiv:2609.32836v1eess.SPcs.AIcs.LG

TL;DR

Radiomap blind prediction must infer maps from incomplete environment and configuration inputs, leaving the target underdetermined. The paper analyzes this uncertainty and propagation-prior bias, then proposes residual refinement of prior-guided predictors. Its RadioFR and RadioLSR instantiations improve on their respective bases across the reported settings.

  • Problem

    Observable environment and base-station representations omit propagation-relevant factors, so the same input may correspond to different radiomaps and the deterministic predictor's target under incomplete observation needs clarification.

    Source: §I · pp. 1–2

  • Method

    The paper identifies the conditional mean as the squared-loss target, decomposes prediction risk, analyzes propagation-prior bias, and proposes residual refinement of a prior-guided base.

    Source: §I · p. 2

  • Results

    RadioFR and RadioLSR improve upon their respective bases under random, distribution-shifted, and limited-data settings.

    Source: §VII · p. 12

  • Takeaways & Limitations

    Observation enrichment and correctable propagation-prior learning provide complementary approaches to radiomap prediction under incomplete observation.

    Source: §VII · p. 12

Abstract

from arXiv · show

Radiomap blind prediction aims to infer radiomaps from observable representations of the propagation environment and base station configuration without field measurements. In practice, the observable representations are inherently incomplete. Thus, the target radiomap is not fully determined by the inputs when generalizing to unseen configurations or environments. Under incomplete observation, we establish a population-level theory of deterministic radiomap blind prediction that identifies the conditional mean as its optimal target and separates prediction error into reducible predictor approximation and irreducible uncertainty caused by missing physical information. The framework further characterizes the train-test risk gap and the uncertainty reduction enabled by observation enrichment. Building on it, we reveal the dual role of propagation priors: they provide physically grounded guidance, yet their implementable forms may bias the attainable predictor. This motivates RadioDecomp, which treats a prior-guided predictor as a correctable base and learns its remaining predictable discrepancy through residual refinement. To evaluate RadioDecomp across distinct propagation-prior designs, we instantiate it with a feature-guided monolithic base and a LoS-Shadow structured base, yielding RadioFR and RadioLSR, respectively. Across random, cross-configuration, and cross-environment settings, experiments confirm the benefit of propagation-related representations and show that both instantiations improve upon their respective bases. Further controlled studies on base capacity, training-support coverage, and observation coarsening corroborate the proposed analysis.

I. INTRODUCTION

Radiomap blind prediction must infer maps from incomplete environment and configuration representations, so deterministic predictions face unresolved physical ambiguity. The paper develops an error framework, analyzes propagation-prior benefits and biases, and proposes residual correction for prior-guided predictors.

A. From Physical Generation to Blind Prediction

  • Incomplete representations can map to multiple physical states and radiomaps, while deterministic blind predictors return one estimate per observable input.The paper asks what statistical target such predictors approximate and how to understand their error and generalization under incomplete observation.Source: §I · pp. 1–2

B. Propagation Priors for Radiomap Prediction

  • Propagation priors expose useful physical regularities, but selected or simplified implementations may be inconsistent with the optimal prediction relation and introduce bias.Prior injection occurs at feature, model, and loss levels; existing work emphasizes empirical benefits, leaving the benefit-bias tradeoff under incomplete observation insufficiently understood.Source: §I · p. 2, B. Propagation Priors for Radiomap… · p. 3

I. INTRODUCTION

  • The paper identifies the conditional mean as the squared-loss target and separates predictor approximation error from irreducible uncertainty caused by hidden propagation factors.It also characterizes the train-test risk gap and proves that additional observations reduce uncertainty when they change the conditional-mean radiomap with positive probability.Source: §I · p. 2, A. From Physical Generation to Blind… · p. 3
  • RadioDecomp retains a propagation-prior-guided predictor as its base and learns its remaining predictable discrepancy through a residual branch.RadioFR and RadioLSR address, respectively, errors after information-preserving feature augmentation and discrepancies under a more restrictive LoS-Shadow base.Source: §I · p. 2

A. Learning-Based Radiomap Prediction

  • Early learning-based approaches included image-to-image prediction, conditional GANs, and conditional diffusion for radiomap prediction.These methods addressed blind prediction from observable environment and base-station representations.Source: §I · p. 1, A. Learning-Based Radiomap Prediction · p. 2

B. Domain-Wise Risk Decomposition and Train-Test Gap

The section decomposes population risk into predictor approximation error and irreducible uncertainty, then explains how both terms shape transfer between training and test domains.

B. Domain-Wise Risk Decomposition and Train-Test Gap

  • For either domain, population risk separates into domain-specific approximation error and irreducible uncertainty induced by incomplete observation.The approximation term measures how well the predictor captures the conditional-mean target; the uncertainty term reflects radiomap variation left unresolved by hidden propagation factors.Source: B. Domain-Wise Risk Decomposition and… · p. 4
  • Improving capacity or training can reduce approximation error, but cannot resolve propagation differences that the available inputs do not distinguish.Broader training coverage helps learn relations in the training domain; it does not itself remove uncertainty due to missing physical information.Source: B. Domain-Wise Risk Decomposition and… · p. 4
  • The train-test risk gap reflects changes in both approximation and uncertainty, as domain shifts can alter conditional targets and unresolved hidden-factor variability.Limited coverage of relevant layouts or configurations can increase the approximation gap, while different hidden propagation conditions can change the conditional mean and unresolved uncertainty.Source: B. Domain-Wise Risk Decomposition and… · pp. 3–4

C. Observation Enrichment and Conditional Uncertainty

The section shows that adding information while retaining the original observation cannot increase conditional uncertainty, and can lower it when the added information changes the conditional-mean radiomap.

C. Observation Enrichment and Conditional Uncertainty

  • Additional propagation information can distinguish conditions previously indistinguishable from the original input, lowering the minimum achievable prediction error.Examples include adding material properties to geometry or providing a detailed antenna pattern alongside a coarse beam descriptor.Source: C. Observation Enrichment and… · pp. 4–5
  • When the enriched observation retains the original input, conditional uncertainty is no greater than under the original observation.Uncertainty falls strictly when the enrichment changes the conditional-mean radiomap with positive probability.Source: C. Observation Enrichment and… · pp. 4–5
  • Training coverage and predictor learning address learning the predictable relation, whereas richer observations can make previously unresolved radiomap variations predictable.With observations fixed, propagation priors guide the use of their available physical information.Source: C. Observation Enrichment and… · p. 5

IV. PROPAGATION-PRIOR GUIDANCE AND BIAS

Propagation knowledge can guide learning toward test-domain relations, but prior representations may compress information or constrain the attainable predictor. The section distinguishes those effects across prior-injection forms.

A. Propagation Guidance for Predictor Approximation

  • Partial knowledge of the shared propagation mechanism can guide learning relations that transfer from training samples to unseen test inputs.The paper treats propagation priors as partial structural knowledge, incorporable through features, model structures, losses, or combinations.Source: A. Propagation Guidance for Predictor… · p. 5

1) Feature-level prior:

  • Feature-level priors make selected propagation relations explicit in inputs, either replacing the original observations or supplementing them.Examples include geometry-derived blockage indicators, distance-based attenuation maps, and beam-gain maps.Source: 1) Feature-level prior · p. 5
  • When original observations are retained alongside deterministic prior features, those features reorganize information rather than enrich it.The predictor can still retain finite-sample optimization discrepancy even though concatenation does not alter the population conditional target.Source: 1) Feature-level prior · p. 5

1) Prior-only input:

  • With prior-only input, minimum risk includes the original observation's irreducible uncertainty and an additional information-compression bias.That bias is zero only when the transformed input retains enough information to determine the original conditional target; otherwise, even its population-optimal predictor incurs the additional error.Source: 1) Prior-only input · pp. 5–6

2) Prior-augmented input:

Prior augmentation retains the original observation and therefore preserves the conditional target, whereas model-level priors constrain the predictor’s available functions. These approaches can increase finite-sample search demands or introduce approximation bias.

2) Prior-augmented input:

  • Retaining the original observation alongside prior-derived features preserves the optimal radiomap target but increases input dimensionality, making finite-sample learning more demanding.The prior-derived representation is deterministic from the original observation, so it adds no population-level information.Source: 2) Prior-augmented input · p. 6

2) Model-level prior:

  • A model-level prior restricts the functions available to approximate the conditional-mean target, producing persistent approximation error if its hypothesis class cannot represent that mapping.The bias vanishes when functions in the induced class can approximate the target arbitrarily well under the training distribution.Source: 2) Model-level prior · p. 6

3) Loss-level prior:

Loss-level priors softly favor predictors consistent with propagation, but an incompatible penalty can shift the population solution away from the conditional target. RadioDecomp instead keeps a prior-guided base and learns its predictable discrepancy through a residual branch.

3) Loss-level prior:

  • An incompatible loss-level penalty with fixed nonzero weight can shift the selected solution away from the conditional target, even with unlimited data and ideal optimization.The bias is zero when the loss-guided solution remains a best approximation within the hypothesis class; examples include physical-consistency penalties and smoothness or boundary constraints.Source: 3) Loss-level prior · p. 6

A. RadioDecomp Formulation

  • RadioDecomp preserves the prior-guided base when the residual is zero and, under squared loss, targets the domain-specific conditional-mean discrepancy.The residual branch uses observable cues and the base prediction as an anchor; its population target is the conditional mean minus the base estimate.Source: A. RadioDecomp Formulation · p. 7

B. Representative Instantiations

RadioDecomp is instantiated with complementary prior-guided bases, and both use propagation-related representations derived from the observable environment and base-station configuration. Training freezes each base before optimizing its residual branch against the remaining error.

B. Representative Instantiations

  • RadioFR and RadioLSR examine residual correction for feature-augmented and more restrictive LoS-Shadow bases, respectively.RadioFR jointly processes available representations; RadioLSR uses a LoS-Shadow partition with restricted branch inputs.Source: B. Representative Instantiations · p. 7

1) Common Representations and Two-Stage Training:

  • The common representations include a beam map exposing configuration-dependent direct coverage, directional edge maps exposing geometry, and a blockage score derived from observable geometry and base-station position.These are deterministic transformations of the available observations, reorganizing them as propagation-related features rather than adding genuine observation information.Source: 1) Common Representations and… · pp. 7–8
  • Both instantiations train the prior-guided base first, freeze it, then optimize only the residual branch against the radiomap difference using masked squared L2 loss.Prediction is restricted to the valid non-building region.Source: 1) Common Representations and… · p. 8

2) RadioFR (Feature-Residual Prediction):

RadioFR uses a monolithic feature-guided base and a residual U-Net to correct errors that can remain despite information-preserving feature augmentation. Its base is frozen before the residual is trained on the remaining radiomap error.

2) RadioFR (Feature-Residual Prediction):

  • RadioFR tests whether residual correction helps when a monolithic U-Net uses augmented features that preserve available information but may leave predictable errors.Finite capacity, limited training data, and imperfect optimization may contribute to those remaining errors.Source: 2) RadioFR (Feature-Residual Prediction) · p. 8
  • The residual branch receives the same propagation-related features as the base together with the base radiomap prediction, and is implemented using another U-Net.The base is frozen before residual optimization.Source: 2) RadioFR (Feature-Residual Prediction) · p. 8
  • RadioFR’s final prediction combines the feature-guided base prediction with the residual prediction.The residual target is defined from the frozen base and training radiomap.Source: 2) RadioFR (Feature-Residual Prediction) · p. 8

3) RadioLSR:

RadioLSR uses a LoS-Shadow structured base that models direct-coverage and blockage trends with restricted inputs, then adds a residual branch to recover predictable spatial details those inputs miss.

3) RadioLSR:

  • The LoS branch uses the beam map, while the Shadow branch uses the beam map and blockage score to form a structured base.The restricted inputs emphasize direct-coverage and blockage trends, leaving geometry-dependent variations such as reflection and diffraction for correction.Source: 3) RadioLSR · p. 8
  • RadioLSR’s residual branch receives height and edge maps alongside the beam map and base prediction to recover predictable spatial details unavailable to the base.The residual prediction is implemented using a U-Net.Source: 3) RadioLSR · p. 8
  • The prediction branches use a common U-Net backbone.The LoS and Shadow branches are each implemented with a U-Net.Source: 3) RadioLSR · p. 8, Table I · p. 9
  • The LoS and Shadow branches are trained on their corresponding regions, then frozen while only the residual branch is optimized.The LoS and Shadow objectives are weighted by λLoS and λShd, respectively.Source: 3) RadioLSR · p. 9

VI. EXPERIMENTS

Experiments evaluate radiomap prediction on the U6G XL-MIMO dataset under random, cross-configuration, and cross-environment splits, comparing standalone and prior-guided predictors with and without residual correction.

1) Dataset and Evaluation Protocols:

  • The U6G XL-MIMO Radiomap dataset contains 78,400 radiomaps from 800 urban scenes and 98 BS configurations, with 128 × 128 received-power maps.Unless otherwise specified, the data use a 6/1/3 training/validation/test split with random seed 42.Source: 1) Dataset and Evaluation Protocols · p. 9
  • The protocols test random samples, disjoint BS configurations, or disjoint urban scenes between training and test sets.These are the random, cross-config, and cross-env protocols, respectively.Source: 1) Dataset and Evaluation Protocols · p. 9

2) Compared Predictors:

  • The comparisons include Vanilla, feature-guided BaseFR and RadioFR, and LoS-Shadow BaseLSR and RadioLSR.The Base variants omit residual correction; RadioFR and RadioLSR are the corresponding complete instantiations.Source: 2) Compared Predictors · p. 9

3) Training and Metrics:

Across evaluation protocols, propagation-related representations improve on Vanilla, and residual correction improves both prior-guided bases. Limited-supervision and capacity studies show that the correction’s benefits depend on training support and base capacity.

3) Training and Metrics:

  • Training used AdamW with batch size 32 and initial learning rate 10^-3; RMSE measured squared-error performance, while MAE complemented it with average-error reporting less sensitive to large local errors.RadioFR and RadioLSR used two training stages, with the base trained first and frozen before residual optimization.Source: 3) Training and Metrics · p. 9

B. Propagation-Prior Guidance and Residual Correction

  • BaseFR reduced Vanilla’s test RMSE by 2.3339, 4.8974, and 3.4660 dB under random, cross-config, and cross-env evaluation, respectively.Vanilla was consistently outperformed by BaseFR, RadioFR, and RadioLSR in test RMSE and MAE across all three protocols.Source: B. Propagation-Prior Guidance and… · p. 9
  • RadioFR reduced test RMSE versus BaseFR by 0.0487, 0.4684, and 0.3178 dB; RadioLSR versus BaseLSR by 3.8386, 1.8635, and 2.0240 dB across protocols.The protocol order is random, cross-config, then cross-env.Source: B. Propagation-Prior Guidance and… · p. 9, Table II · p. 10

C. Propagation-Prior Learning under Extremely Limited Supervision

  • With only 10 training radiomaps, RadioFR reduced test RMSE versus BaseFR by 1.2091 dB to 14.4285 dB; RadioLSR reduced BaseLSR’s test MAE by 2.5665 dB.RadioLSR achieved the lowest test MAE, 9.3178 dB, while its test RMSE remained relatively high because localized large errors weigh more heavily in RMSE.Source: C. Propagation-Prior Learning under… · p. 10, Table III · p. 10

D. Residual Correction across Feature-Guided Base Capacities

  • RadioFR’s largest test-RMSE reductions over BaseFR were 0.1707 dB under random evaluation and 0.4684 dB under cross-config; at width 96, differences were only 0.0031 and 0.0038 dB.Under cross-env, RadioFR reduced test RMSE at every width by 0.2245 to 0.3315 dB; RadioFR-64 also beat BaseFR-96, 7.6678 versus 8.0248 dB, despite fewer parameters.Source: D. Residual Correction across… · p. 10

E. Training-Support Coverage under a Fixed Test Domain

With a fixed test domain, training-data coverage matters: environment coverage was more beneficial than configuration coverage in the tested RadioLSR setting, and covering both yielded the best fixed-budget results. Coarsening studies and the conclusion also support complementary roles for richer observations and predictor refinement.

E. Training-Support Coverage under a Fixed Test Domain

  • With the common 17,600-sample budget, Both Covered achieved the best fixed-test results: 4.6924 dB test RMSE and 2.5305 dB test MAE.Environment Only reduced errors more than Configuration Only relative to Neither Covered in this RadioLSR dataset setting.Source: E. Training-Support Coverage under a… · pp. 11–12, Table V · p. 12
  • Expanding Both Covered from 17,600 to 37,200 training samples yielded additional reductions of 0.2384 dB in RMSE and 0.1738 dB in MAE.The reported acquisition priority is to cover target-relevant environments and configurations first, with particular attention to environment coverage; denser sampling within covered support provides a smaller further benefit.Source: E. Training-Support Coverage under a… · p. 12

F. Effect of Observable-Representation Coarsening

  • The continuous beam map reduced training RMSE versus the 3-dB mask by 2.2894 dB with the complete height map and 2.1410 dB with 10% height missingness.Removing 10% of height observations changed training RMSE by only 0.1356 dB with the full beam map and by -0.0128 dB with the 3-dB mask.Source: F. Effect of Observable-Representation… · p. 12, Table VI · p. 12

VII. CONCLUSION

  • The conclusion identifies the conditional-mean radiomap as the squared-loss population optimum and decomposes risk into reducible approximation error and irreducible uncertainty from hidden propagation factors.It presents observation enrichment and correctable propagation-prior learning as complementary routes to reducing uncertainty and predictor-approximation error, respectively.Source: §VII · p. 12
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