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Uncertainty-Guided UAV Spectrum Cartography with Deep-Unfolded Online Tensor Decomposition

Shangjie Zhuang, Jiahui Liang, Shijian Gao

arXiv:2609.40280v1eess.SP

TL;DR

UAV spectrum cartography must reconstruct PSD maps from spatially sparse and frequency-incomplete measurements while accounting for limited sensing bandwidth and energy. The paper combines rapid online deep-unfolded tensor decomposition with uncertainty-guided target selection and learned movement and bandwidth decisions. Experiments report faster reconstruction and lower final NMSE than evaluated active-sampling baselines.

Abstract

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Spectrum cartography is crucial for spectrum-aware resource management in low-altitude networks, where uncrewed aerial vehicles (UAVs) collect spectrum measurements to reconstruct power spectral density (PSD) maps. However, limited energy and sensing bandwidth make measurements sparse in space and incomplete in frequency. To collect these measurements efficiently, the UAV actively plans its next move based on the latest map and its uncertainty, which requires rapid online reconstruction. We therefore propose an active online spectrum cartography framework. We first develop online deep-unfolded tensor decomposition (ODU-TD) for rapid map updates. An ensemble of reconstructors then estimates uncertainty to select informative sensing targets, and a learning-based policy determines the UAV movement and sensing bandwidth under the energy budget. Experiments show that ODU-TD achieves an approximately 27-fold speedup over online tensor decomposition and the lowest normalized mean square error (NMSE) among the compared reconstructors under sparse spatial and spectral observations, and the proposed framework reduces the NMSE by at least 55% compared with the representative baselines.

I. INTRODUCTION

The introduction motivates online PSD-map reconstruction for UAVs that must choose informative spatial and frequency measurements under energy and bandwidth constraints. It presents a closed-loop framework combining rapid online reconstruction, uncertainty-guided target selection, and learned movement and bandwidth decisions.

I. INTRODUCTION

  • Existing active radio-map approaches mainly choose where to measure, while PSD mapping also requires choosing which and how many frequency bands to sense under limited bandwidth and energy.These choices rely on rapidly updated PSD maps and their uncertainty.Source: §I · p. 1
  • The framework updates the PSD map from cumulative measurements, uses ensemble uncertainty to select a location and frequency band, and learns UAV movement and sensing bandwidth under the energy budget.The measurement is added to the map before the cycle repeats.Source: §I · p. 1
  • The proposed ODU-TD unfolds alternating updates into fixed stages and replaces the computational bottleneck with a learnable module for timely, accurate updates under partial observations.The framework also uses ensemble uncertainty and a learned policy to select targets, movement, and sensing bandwidth under energy constraints.Source: §I · p. 1
  • Experiments report that ODU-TD accelerates online tensor decomposition and has the lowest error under partial observations, while the framework lowers reconstruction error relative to representative baselines.The supplied introduction gives no numerical values for these comparisons.Source: §I · p. 2

A. PSD Radio Map Model

  • The PSD radio map is represented as a tensor over spatial grid cells and frequency bands, with the tensor-decomposition model pairing each emitter's spatial propagation field with its power spectrum.The grid and monitored spectrum provide the map's spatial and frequency dimensions.Source: A. PSD Radio Map Model · p. 2

B. Partial Spectrum Observation Model

At each time slot, the UAV senses a selected subset of consecutive frequency bands at its current location. Its energy model accounts for movement and sensing costs, with sensing cost increasing as more bands are sensed.

B. Partial Spectrum Observation Model

  • At slot t, the UAV senses q_t consecutive frequency bands centered on a target band, represented by a binary mask over the K bands.The number of selected bands equals q_t.Source: B. Partial Spectrum Observation Model · p. 2
  • The full PSD vector at the UAV's current position is H(u_t, :), and the measurements are affected by small-scale fading and disturbance.The observation model describes partial frequency-band sensing at each slot.Source: B. Partial Spectrum Observation Model · p. 2

C. Energy Consumption Model

  • The energy model includes both UAV movement and sensing costs, and sensing cost increases with the number of bands q_t.The total energy consumption at slot t is E_t, subject to an energy budget E_max.Source: C. Energy Consumption Model · p. 2

D. Problem Formulation

The active online cartography problem chooses feasible UAV movements and sensed-band counts to minimize final NMSE under a constrained optimization formulation.

D. Problem Formulation

  • The objective is to minimize the final NMSE after T sensing slots.The final error is measured between the reconstructed and ground-truth PSD maps.Source: D. Problem Formulation · p. 2
  • Each slot’s action consists of a feasible UAV movement and a number of sensed bands.The band count is selected from the available choices.Source: D. Problem Formulation · p. 2
  • The active online cartography task is formulated as a constrained optimization problem.Source: D. Problem Formulation · p. 2

III. ONLINE SPECTRUM CARTOGRAPHY

The section reviews tensor decomposition for reconstructing PSD maps from partial measurements, then motivates its online extension and deep-unfolded acceleration. The offline model jointly estimates source spectra and spatial propagation fields using local polynomial fits and regularized optimization.

III. ONLINE SPECTRUM CARTOGRAPHY

  • Online cartography needs rapid, accurate map updates under partial observations to support uncertainty-guided sensing.The section reviews offline tensor decomposition before developing sequential online reconstruction and a deep-unfolded version.Source: §III · p. 2

A. Recap of the Tensor Decomposition

  • Offline TD jointly estimates source power spectra and spatial propagation fields from a fixed set of PSD measurements.It approximates each source’s propagation field locally with polynomial features and promotes low-rank fields through nuclear-norm regularization.Source: A. Recap of the Tensor Decomposition · pp. 2–3
  • A frequency-availability mask and spatial kernel weights define the weighted regression residuals used in the offline reconstruction problem.The objective also enforces consistency between local estimates and propagation fields, with nonnegative source spectra.Source: A. Recap of the Tensor Decomposition · pp. 2–3

B. Spectrum Cartography via Online TD

Online TD reuses previous estimates as measurements arrive in batches, updating only affected local coefficients while retaining block-alternating updates. Repeated dense SVDs in the propagation-field update are its main computational bottleneck.

B. Spectrum Cartography via Online TD

  • Online TD processes new measurements in batches and recomputes local coefficients only for cells affected since the previous reconstruction.Between reconstruction updates, estimates remain unchanged; unaffected local coefficients are reused.Source: B. Spectrum Cartography via Online TD · p. 3
  • The propagation-field update uses iterative SVT initialized from the previous estimate, reusing the prior solution to reduce convergence iterations.The method’s alternating updates also update shared spectra using residual information aggregated across the grid.Source: B. Spectrum Cartography via Online TD · p. 3
  • Repeated dense SVDs make the propagation-field update the main computational bottleneck in online TD.The update cost scales as O(RJ_svtN_g^3).Source: B. Spectrum Cartography via Online TD · p. 3

C. Deep-Unfolded Online Spectrum Cartography

ODU-TD unfolds online tensor decomposition’s alternating updates into trainable stages and replaces iterative SVT with a learned module. Its spatial-field network refines propagation estimates, while training combines map, source-map, and observation-level supervision.

C. Deep-Unfolded Online Spectrum Cartography

  • ODU-TD unfolds alternating updates into trainable stages and replaces iterative SVT while retaining model-based interpolation and spectral estimation.These retained components preserve explicit relationships among measurements, source spectra, and local estimates.Source: C. Deep-Unfolded Online Spectrum… · p. 3
  • A convolutional residual network refines spatial propagation fields using neighborhood patterns beyond global nuclear-norm regularization.Because spatial fields are shared across bands, refinement also improves estimates at unobserved bands, particularly under partial observations.Source: C. Deep-Unfolded Online Spectrum… · pp. 3–4
  • Training combines PSD-map NMSE, source-map supervision, and penalties for mismatches at observed entries.The observation-level term discourages refinements that conflict with supplied measurements.Source: C. Deep-Unfolded Online Spectrum… · p. 4
  • Replacing SVT with a learned module reduces source-map refinement from cubic to quadratic complexity in N_g.For fixed iteration and stage counts, the fixed stage count also removes convergence-dependent repetition of alternating updates.Source: C. Deep-Unfolded Online Spectrum… · p. 4

IV. UNCERTAINTY-GUIDED ACTIVE SAMPLING

The scheme prioritizes informative measurements under limited energy and sensing bandwidth. Ensemble uncertainty selects a target location and frequency band, while a learned UAV policy chooses movement and sensing width within the energy budget.

  • Limited energy and sensing bandwidth make prioritizing informative measurements central to the active sampling scheme.Source: §IV · p. 4
  • Ensemble uncertainty selects an informative target location and frequency band.Source: §IV · p. 4
  • A learned policy chooses UAV movement and the number of bands to sense under the energy budget.Source: §IV · p. 4

A. Uncertainty-Guided Target Selection

The target-selection method uses ensemble uncertainty to assess candidate locations and frequency bands. It combines spatial and frequency-specific informativeness, with visit and search constraints shaping selection.

B. Energy-Constrained Movement and Bandwidth Planning

Movement and sensing width are jointly planned because sensing more bands yields more observations per slot but consumes energy for later measurements. An actor–critic policy uses UAV status and target context to select actions, with shaped rewards and PPO training.

V. EXPERIMENTAL RESULTS

The evaluation uses synthetic PSD scenes built from RadioMapSeer and ARM-Omni fields and compares reconstruction methods under sequential online measurements. ODU-TD cuts cumulative solver time over the measurement sequence to 1.58 s, an approximately 27-fold speedup over online TD.

A. Dataset and Settings

  • PSD scenes comprise dense 100 × 100 × 30 tensors, with eight synthesized power spectra per base scene and an 80%/10%/10% train/validation/test split.Source: A. Dataset and Settings · p. 5

B. Online Reconstruction Efficiency

  • Methods receive measurements one at a time in the same order and update the PSD map after each new measurement; runtimes are recorded per module and cumulatively.Source: B. Online Reconstruction Efficiency · p. 5
  • 1.58 s cumulative solver time over the measurement sequence for ODU-TD, an approximately 27-fold speedup over online TD.Online TD's SVT-based update remains the bottleneck at 183.16 ms per call; ODU-TD replaces it with a learned module costing 0.73 ms per call.Source: B. Online Reconstruction Efficiency · p. 5, Table II · p. 5

C. Reconstruction Accuracy under Partial Observations

ODU-TD maintains low reconstruction error under partial observations, particularly when spatial and spectral observations are sparse. In active sampling, the proposed framework outperforms the evaluated baselines, and the conclusion emphasizes timely updates under the UAV energy budget.

D. Active Sampling Performance

  • 55.0% lower final NMSE than random sampling; with PPO held fixed, the proposed framework lowers final NMSE by 58.2% versus SPC-QV-based and 79.5% versus LRTC-Smooth-based variants.Fig. 4 compares NMSE versus flight steps under active sampling.Source: Figure 4 · p. 6, D. Active Sampling Performance · p. 6

C. Reconstruction Accuracy under Partial Observations

  • ODU-TD achieves the lowest NMSE across settings, with its clearest advantage when spatial and spectral observations are sparse, and its NMSE varies only slightly with observed-band proportion for M ≥150.The evaluation compares offline and online TD, SPC-QV, LRTC-Smooth, and BNN-SF reconstructors.Source: C. Reconstruction Accuracy under… · pp. 5–6

VI. CONCLUSIONS

  • The framework combines timely online map updates with ensemble-based uncertainty to select sensing locations and frequency bands under the UAV energy budget.The conclusion describes the resulting uncertainty-guided feedback loop as determining UAV movement and the number of sensed bands.Source: §VI · p. 6

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