Learning or Modeling? Virtual XL-RIS Observation Reconstruction for Near-Field Sensing

I. INTRODUCTION

• The central comparison is whether paired deployment samples or calibrated propagation geometry provides the more effective prior for reconstructing the VOR from shared measurements.The comparison controls the sensing operator, BS samples, and RIS target, while retaining each route’s distinct information requirements.Source: §I · pp. 1–2

A. Virtual RIS Observation Model

• The VOR is the complex signal that would be sampled across the RIS, offering an aperture-domain representation without adding receivers to the passive surface.The nominal signal is represented by a small set of LoS and reflected-path gains, rather than an independent parameter for every element.Source: §I · p. 1, A. Virtual RIS Observation Model · p. 3

I. INTRODUCTION

• With Q < N measurements, different aperture signals can yield the same noiseless observation, so reconstruction requires a prior restricting the admissible signals.A spherical-wave signal is structured by source position and complex gain rather than N independent element values.Source: §I · pp. 1–2
• The comparisons separate position-distribution shift, incorrect reflector metadata, and propagation mismatch because each changes a different part of the inference problem.Incorrect coordinates alone test sensitivity to model side information, not robustness to a changed channel.Source: §I · p. 2
• OG-NF-SPR initializes source position using reflected-path projection, then estimates all path gains and continuously refines position to recover the full aperture signal.The method uses no supervised BS/RIS pairs, but depends on calibration; its causal extension uses previous position estimates.Source: §I · p. 2

B. Compressed BS Observation

• The BS measurement under profile ℓ is y(ℓ) = H diag(ω(ℓ))yr + n(ℓ), with profiles cycled while position and path gains remain constant within a snapshot.Stacking the L profiles forms the compressed observation used for reconstruction.Source: B. Compressed BS Observation · p. 3
• Direct inversion is underdetermined because the sensing matrix has a nontrivial null space, and the VOR is desirable but does not specify how to select an inverse.Conditioned on side information, either reconstructor remains post-processing of the BS observation.Source: B. Compressed BS Observation · p. 3
• Learning uses paired deployment samples as side information, whereas modeling uses calibrated geometry and path statistics; neither route is prior-free.Noiseless identification can still be possible when the deployment mapping and sensing operator are injective on the low-dimensional channel manifold.Source: B. Compressed BS Observation · p. 3

C. Modeling: Geometry-Constrained MAP Reconstruction

• For each candidate position, the physical MAP objective balances agreement with received samples against a penalty on gains relative to calibrated path-power scales.The gains have a closed-form quadratic subproblem, leaving only three real position variables in the nonconvex search; the physical route requires geometry, noise, path-power scales, and a search region.Source: C. Modeling: Geometry-Constrained MAP… · p. 4

D. Learning: Supervised VOR Reconstruction

• The supervised route trains on paired BS inputs and clean RIS targets, not position labels, and predicts the aperture signal without fitting an explicit source location.Compact VOR-DNN is an ordering-agnostic control; DenseNet-VOR uses a fixed 2 × 4 × 4 compressed-measurement layout.Source: D. Learning: Supervised VOR… · p. 4
• DenseNet’s 4×4 input is a computational layout of compressed measurements, so its convolutional structure alone does not establish generalization to another sensing operator.Measurement ordering is maintained during training and testing.Source: D. Learning: Supervised VOR… · p. 4

E. Common Target, Different Prior Information

• Learning requires paired training data, while modeling requires calibrated geometry and path statistics; neither route is prior-free.The network uses fixed preprocessing and a forward pass, whereas the physical route fits all gains and refines position per snapshot.Source: E. Common Target, Different Prior… · p. 4

F. Deep VOR-Assisted Tracking Formulation

• Either reconstructed sequence can feed the SSFE–BiTP tracking chain, which extracts spatial and angular features and predicts current 3D position from an available window.The causal T-OG-NF-SPR extension uses a motion prior without supervised trajectory labels.Source: F. Deep VOR-Assisted Tracking… · p. 4

III. MODELING ALGORITHM: OG-NF-SPR

• Projection initializes position seeds, but the full observation and all calibrated paths enter the subsequent gain fitting and continuous position refinement.Multiple seeds reduce sensitivity to an incorrect local maximum, while refinement removes finite-grid position constraints.Source: §III · pp. 4–5

A. Initialization by Multipath Projection

• Reflected-subspace projection suppresses components explainable by calibrated reflector atoms before scoring candidate source positions.The method retains multiple spatially separated high-score seeds to reduce local-minimum failures.Source: A. Initialization by Multipath… · p. 5
• Normalized matched-filter scores rank candidate positions by directional agreement rather than compressed-atom energy.The search uses the projected observation and retains C spatially separated seeds.Source: A. Initialization by Multipath… · p. 5

B. Closed-Form Complex Path-Gain Estimation

• At a fixed position, regularized linear regression yields a conditional MAP estimate of the complex path gains.The profile objective scores each position after fitting its gains, and the regularizer limits unstable gains at low SNR.Source: B. Closed-Form Complex Path-Gain… · p. 5
• The gain fit retains both line-of-sight and reflected atoms, although reflected components are removed during initialization.This lets gain differences be explained by refitting paths rather than moving the source.Source: B. Closed-Form Complex Path-Gain… · p. 5

C. Continuous Off-Grid Refinement

• The projected-residual Jacobian removes changes explainable by refitting gains, providing a stable variable-projection approximation rather than the exact Hessian.Every trial is checked against the exact profiled MAP objective.Source: C. Continuous Off-Grid Refinement · p. 6
• A refined step is accepted only if the re-estimated gains reduce the exact profile objective; otherwise, the trust radius is halved.The step is clipped to a trust radius and projected onto the feasible position region.Source: C. Continuous Off-Grid Refinement · p. 6
• The final output is the N-element aperture signal, while the refined source position is also evaluated as a localization estimate.The position grid supplies an initial guess, not a restriction on the final estimate.Source: C. Continuous Off-Grid Refinement · p. 6

D. Trajectory-Aware Extension

• The causal temporal extension uses a random-walk position prior to resolve ambiguities between plausible snapshot locations.A small Qp favors a smooth trajectory, while a large Qp approaches independent-snapshot reconstruction and permits rapid maneuvers.Source: D. Trajectory-Aware Extension · p. 6
• With a motion prior, coarse seeds are biased toward the predicted basin, but final seed selection still uses the exact objective.The temporal penalty is added to the normal matrix and corresponding right-hand side during refinement.Source: D. Trajectory-Aware Extension · p. 6

• Objective descent and source-position distinguishability are separate questions; descent alone does not imply accurate localization.Source: §IV · p. 6

A. Descent and Local Identifiability

• Accepted OG-NF-SPR trials produce a nonincreasing, convergent objective sequence, but nonconvexity prevents a global-optimality guarantee.Local identifiability additionally requires the projected line-of-sight component to remain nonzero and the position-to-residual map to have rank three.Source: A. Descent and Local Identifiability · pp. 6–7

B. Why Learning and Modeling Cross Over

• When several positions remain plausible, the conditional mean averages their aperture signals while OG-NF-SPR selects one position and fitted gains.The averaging advantage is not exclusive to deep learning; the comparison concerns evaluated estimators, since LMMSE and model-based posterior-mean estimation also average.Source: B. Why Learning and Modeling Cross Over · p. 7
• OG-NF-SPR can cover unseen positions within P under correct calibration, but inaccurate geometry changes the assumed signal family.Mismatch-aware training offers an alternative when the repeatable discrepancy is represented in the labels.Source: B. Why Learning and Modeling Cross Over · p. 7

C. Computational Complexity

• For a large grid, the parallelizable QG search dominates OG-NF-SPR, while its temporal extension adds only an O(G) distance penalty.Storage is O(QG).Source: C. Computational Complexity · p. 7
• OG-NF-SPR takes 0.214–1.419 ms in MATLAB on CPU as refinement increases from zero to eight iterations; Compact and DenseNet-VOR inference takes 0.73 and 4.92 µs CUDA batch latency.Because hardware and batching differ, these timings do not establish a platform-independent speedup; physical fitting repeats search and refinement per snapshot.Source: C. Computational Complexity · p. 7

A. Simulation Setup

• The simulations use a 10 × 10 half-wavelength UPA at 3.5 GHz, with 100 RIS elements, 16 compressed measurements, and 400 held-out snapshots unless otherwise specified.All reconstructors share a fixed operator built from the nominal channel’s position derivatives and phase-code columns; physical probing design is not optimized, and arbitrary realizability is not demonstrated.Source: A. Simulation Setup · p. 7
• The reconstruction comparisons use identical test channels and noise realizations; deep baselines use 5,000 labeled channels generated offline from the simulated deployment.The cost of acquiring real RIS labels is not evaluated; the position-shift sweep uses 200 locations and its zero-shift result need not match the 400-snapshot SNR result.Source: A. Simulation Setup · p. 8

B. Effect of SNR and Position-Distribution Shift

• At 20 dB, OG-NF-SPR reaches 6.553 × 10−3 NMSE, approximately 67% below the 1.97 × 10−2 achieved by either deep network.At −5 dB, Compact VOR-DNN reaches 0.494 NMSE versus 0.926 for OG-NF-SPR, while LMMSE performs better at 0.450.Source: B. Effect of SNR and… · pp. 8–9, Figure 2 · p. 8
• At high SNR, similar deep-network plateaus reflect the tested architectures and training budget, not an intrinsic limit of deep learning; LMMSE falls below both networks at 30 dB.Minimum-norm LS cannot recover components removed by compression, and KNN is limited by single-sample retrieval.Source: B. Effect of SNR and… · p. 9
• Continuous position refinement reduces Grid NF-SPR’s 30 dB NMSE from 1.890 × 10−3 to 7.219 × 10−4, a 62% reduction.Because both estimators use the same channel family, this comparison isolates continuous-position fitting; the remaining gap to the known-position genie reflects position estimation.Source: B. Effect of SNR and… · p. 9
• At a 0.4 m test-position shift, OG-NF-SPR attains 5.918 × 10−3 NMSE, below LMMSE, KNN, Compact VOR-DNN, and DenseNet-VOR.The reported alternatives are 0.0312, 0.127, 0.0257, and 0.0247, respectively; the result concerns position generalization within one deployment, not a changed propagation environment.Source: Figure 2 · p. 8, B. Effect of SNR and… · p. 9

C. Shared-Snapshot Estimates and Refinement Mechanism

• Across three shared snapshots, OG-NF-SPR complex NMSEs are (2.8, 6.9, 7.1)×10^-4 versus 5.2×10^-3 to 2.2×10^-2 for deep reconstructors.Its corresponding position errors are 0.051, 0.144, and 0.270 m; similar aperture recovery therefore does not ensure equally precise position recovery.Source: C. Shared-Snapshot Estimates and… · p. 10

D. Error Distribution and Calibration Sensitivity

• OG-NF-SPR’s median per-snapshot NMSE is 5.327 × 10^-3, compared with 1.320 × 10^-2 for Compact VOR-DNN, 1.349 × 10^-2 for DenseNet-VOR, and 1.812 × 10^-2 for LMMSE.The matched-channel advantage also appears near the center of the per-snapshot error distribution.Source: D. Error Distribution and Calibration… · p. 10
• With reflector-coordinate errors of 3 cm and 5 cm, OG-NF-SPR NMSE rises to 1.432 × 10^-2 and 2.857 × 10^-2, crossing the deep baselines.This sweep scales one fixed random coordinate-error realization, so the crossing is not a universal calibration threshold.Source: D. Error Distribution and Calibration… · p. 10

E. Structural Mismatch and Training Coverage

• At ρ = 1, DR-VOR-DNN attains 3.397 × 10^-2 versus OG-NF-SPR’s 3.622 × 10^-2, while nominal networks approach 5.0 × 10^-2.This test includes omitted weak paths and element gain/phase errors; the reversal is specific to the tested repeatable mismatch family.Source: E. Structural Mismatch and Training… · pp. 10–11
• Including the tested discrepancy in training reduces error for DR-VOR-DNN versus same-size Compact VOR-DNN, tying the gain to training coverage rather than network size.Nominal networks do not automatically benefit when the physical model is inaccurate.Source: E. Structural Mismatch and Training… · p. 11

F. Reconstruction Along a Shared Trajectory

• Temporal regularization reduces position RMSE from 0.653 m to 0.225 m on the shared trajectory.At 15 dB, trajectory NMSEs for Compact VOR-DNN, DenseNet-VOR, independent OG-NF-SPR, and causal T-OG-NF-SPR are (2.649, 2.706, 1.911, 1.332) × 10−2, respectively.Source: F. Reconstruction Along a Shared… · p. 11

G. Implications for Downstream Tracking

• At 30 dB, BiTP achieves 0.163 m2 versus 0.181 and 0.241 m2 for one-layer and stacked unidirectional LSTMs.Bidirectional processing uses both directions within a window ending at the current slot; these accuracy curves do not quantify inference latency.Source: G. Implications for Downstream Tracking · p. 11
• At 30 dB, Full VOR achieves 0.163 m2 position MSE, while Oracle RIS reaches 0.107 m2, showing reconstruction removal does not eliminate feature and prediction error.Full VOR is the complete VOR+SSFE+BiTP chain; this tracking protocol is distinct from the shared-data reconstructor comparisons and revisits results from.Source: G. Implications for Downstream Tracking · p. 11, Figure 6 · p. 12
• At 30 dB, VOR-assisted tracking lowers random-walk MSE from 0.745 to 0.163 m2 and wave MSE from 0.0684 to 0.0186 m2, but loses to BS-only LSTM on a smooth spiral at −5 dB.The spiral comparison is 0.0518 m2 for BS-only LSTM versus 0.103 m2 for the full chain; comparing reconstructors end-to-end requires a common tracker protocol and sequence dataset.Source: G. Implications for Downstream Tracking · p. 12

H. Deployment Implications

• Deployment choice depends on calibration and labelled coverage: reliable geometry supports label-free OG-NF-SPR, while representative labels can favor mismatch-aware learning when the nominal model is deficient.The observed crossovers guide assessment but are not universal switching thresholds; validation of reconstruction and tracking should be separate.Source: H. Deployment Implications · p. 12

VI. CONCLUSION

• The preferred estimator depends on measurement quality and prior validity: statistical averaging helps when noise leaves locations plausible, while continuous physical fitting helps when a valid model distinguishes them.Measured-channel validation and controlled combinations of learning and modeling remain future work.Source: §VI · p. 13

• Multiplying the logarithmic derivative by [a(u)]n yields equations (32)–(33).Source: Appendix A · p. 13