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Interpretable Physics Informed WiFi Indoor Localization: Learning an Effective Access Point Geometry and Using It to Prune

Arshia Eftekhari zadeh, Rezvan Nasiri, Hadi Moradi

arXiv:2609.32539v1cs.AIcs.LG

TL;DR

Deep WiFi fingerprinting can localize accurately, but its black-box character limits interpretability and reuse of learned information. This paper learns effective AP geometry jointly with location, then uses it to rank and prune APs. On UJIIndoorLoc, mean 3D error is 7.07 m, and the geometry also supports AP selection without surveyed coordinates.

  • Problem

    Deep fingerprinting models can be black boxes, limiting interpretability and reuse of learned information for decisions such as which APs to keep.

    Source: §I · p. 1

  • Method

    A physics-informed hierarchical network predicts location while learning effective AP geometry from RSSI and labelled positions, then uses that geometry to rank APs.

    Source: §I · p. 2

  • Results

    7.07 m mean 3D error, a 26.4% improvement over WKNN (k=7) on UJIIndoorLoc; the full model also outperforms confirmed published results on the official split.

    Source: §I · p. 2, A. Comparison with Baselines and… · p. 7

  • Takeaways & Limitations

    Learned geometry supplies coordinates needed for Fisher-information AP ranking without surveyed AP positions and supports selection on two further datasets.

    Source: §VII · p. 11

    The path-loss exponent is shared per building with only small per-AP deviations, so it cannot represent APs whose propagation genuinely departs from the building average.

    Source: §VI · p. 11

Abstract

from arXiv · show

Deep learning models can achieve high accuracy for indoor localization, but their black-box nature limits interpretability and the reuse of learned information. We propose a hierarchical deep learning framework for WiFi fingerprint-based indoor localization that jointly predicts user location and learns an effective geometry of the surrounding access points (APs). Physics-informed decoders infer this geometry directly from RSSI measurements and labelled user positions, without requiring the true AP coordinates during training. The learned geometry is then used to rank and prune APs. On the UJIIndoorLoc dataset, the proposed chained model achieves a mean 3D localization error of 7.07 m, reducing error by 26% to 36% compared with baseline models. Previously published methods evaluated on the same official split report errors 10.6% to 31.0% higher. Pruning 35% or 50% of the APs causes only a small loss in localization accuracy. The inferred geometry also enables Fisher-information-based AP ranking even when fingerprint databases do not contain surveyed AP coordinates. Experiments on the Tampere/TUT and UTSIndoorLoc datasets show that geometry-guided AP selection performs comparably to selectors built directly from labelled data. These results show that physics-informed interpretability can improve indoor localization while also supporting effective feature selection.

I. INTRODUCTION

WiFi fingerprinting offers an inexpensive indoor-localization approach, but deep models can be opaque and buildings may contain hundreds of APs. This paper learns effective AP geometry alongside location, then uses that geometry to rank and prune APs.

I. INTRODUCTION

  • Deep fingerprinting models are black boxes, limiting inspection and reuse of physically meaningful information for engineering decisions such as AP selection.WiFi fingerprinting matches online RSSI vectors against a database collected at known positions.Source: §I · p. 1
  • The model learns effective AP geometry from RSSI patterns and training positions, without using routers’ true coordinates as labels.The geometry represents where the network believes each signal originates, based on how it changes with position.Source: §I · pp. 1–2
  • The learned geometry supports AP ranking and pruning, addressing the need to reduce survey and computation costs in buildings with many APs.The paper evaluates Fisher-information-based ranking from geometry against other pruning methods across four prune levels, five seeds, and three datasets.Source: §I · pp. 1–2
  • 7.07 m versus 7.21–7.29 m for the ablations: the interpretable configuration costs no considerable amount of accuracy.The paper also describes the recovered geometry as faithful to real radio geometry and useful for geometry-driven AP selection.Source: §I · p. 2

II. RELATED WORK

Prior fingerprinting research has pursued accuracy, statistical AP selection, or physics-informed localization, while this paper makes learned AP geometry itself an interpretable artifact for downstream selection. The problem setting maps RSSI and detection masks to building, floor, and planar position.

II. RELATED WORK

  • Classic AP selection uses statistical criteria computed from RSSI, while Fisher-information selection applies a geometry-driven sensor-layout perspective.The paper compares against detection count and mutual information and applies Fisher information to learned geometry.Source: §II · p. 2
  • Deep fingerprinting work cited here emphasizes architectural accuracy, while this paper evaluates interpretability by whether a downstream selection decision can use the learned artifact.The paper positions physics-informed learning as intrinsic interpretability, with physically meaningful parameters constrained by construction rather than attributed post hoc.Source: §II · pp. 2–3
  • The paper distinguishes its approach from prior physics-informed localization by jointly learning effective AP geometry with a differentiable decoder, without supplied AP coordinates.Earlier approaches typically provided coordinates and channel parameters, calibrated them separately, or recovered locations through other procedures.Source: §II · p. 3

A. Problem Setting

  • Each sample pairs an RSSI vector and AP detection mask with building, floor, and planar-position labels.Evaluation uses UJIIndoorLoc, Tampere/TUT, and UTSIndoorLoc; UJIIndoorLoc has 520 APs and an official split of 19,937 training and 1,111 test samples.Source: A. Problem Setting · p. 3

B. Physics Informed Model

PhysicsChainedNet combines hierarchical building–floor–position prediction with a physics decoder that reconstructs RSSI using learned access-point coordinates and bounded path-loss parameters.

B. Physics Informed Model

  • The encoder maps normalized RSSI and detection-mask channels through a residual MLP stack with dimensions 512, 256, and 64 to a latent vector.The network’s three components are an encoder, hierarchical prediction heads, and two RSSI decoders.Source: B. Physics Informed Model · p. 3
  • Three chained heads predict building, floor, and position sequentially, conditioning each later head on a 32-dimensional embedding of the preceding prediction.The sequence uses building information to predict floor, then building and floor information to predict position.Source: B. Physics Informed Model · p. 3
  • During training, teacher forcing uses the true-label embedding at the current ratio; otherwise, differentiable softmax-weighted embeddings carry uncertainty forward, and testing always uses the blend.The blend avoids a hard argmax so downstream heads continue to receive gradients.Source: B. Physics Informed Model · p. 3
  • The physics decoder computes expected AP RSSI from predicted position, building, and floor, using distance to learned AP coordinates, reference power, floor attenuation, and path-loss exponent.The path-loss exponent is learned per building and adjusted per AP because obstruction differs between them.Source: B. Physics Informed Model · p. 3
  • The physics parameters are learned with the encoder under bounded ranges: P0 [−110, 0] dBm, neff,j [1.1, 5.0], β [0, 30] dB per floor, and σj dBm.Smooth sigmoid-style mappings keep the parameters within these ranges.Source: B. Physics Informed Model · p. 3
  • The latent decoder reconstructs RSSI from z without physical parameters; AP-drop and Gaussian-noise corruption train the encoder-decoder pair as a denoising autoencoder against uncorrupted RSSI.Each AP is dropped with probability 0.15, and surviving channels receive Gaussian noise of scale 0.02.Source: B. Physics Informed Model · p. 3

C. Training Procedure

Training first fits the physics decoder to true training positions, then progressively trains the chained model and releases AP coordinates. The objective combines localization and RSSI reconstruction with constraints that keep learned geometry physically plausible.

C. Training Procedure

  • The three-stage procedure begins with a physics prefit against true training positions, followed by curriculum rounds that release AP coordinates only in the final round.The prefit is repeated on the retained WAP set before retraining after pruning.Source: C. Training Procedure · pp. 3–4
  • Teacher forcing decays linearly from 1.0 to 0.0, reaching zero as AP coordinates become trainable at the start of round 3.This gradually shifts the chained heads from true labels to their own predictions.Source: C. Training Procedure · pp. 4–5

D. Training Objective

  • The localization loss combines position, floor, and building errors, with an auxiliary expected 3D term that penalizes uncertainty across candidate floors.The expected floor error uses a probability distribution over floors and a 4 m floor-height constant.Source: D. Training Objective · p. 5
  • A latent decoder reconstructs RSSI from z alongside the physics decoder, providing an easier reconstruction signal and pressure for z to retain RSSI information.Its reconstruction weight decays but does not reach zero.Source: D. Training Objective · pp. 5–6
  • Physics reconstruction uses a Gaussian likelihood for detected APs and a censored likelihood for non-detections, with learned per-AP shadowing to down-weight noisy APs.A non-detection means the signal is below the −110 dBm threshold, so the model is penalized only for predictions clearly above it.Source: D. Training Objective · p. 6
  • Three round-3 penalties constrain AP drift, floor coordinates, and positions outside the training footprint plus a 50 m margin.The drift penalty allows adaptation while discouraging large departures from prefit coordinates.Source: D. Training Objective · p. 6

E. Hyperparameter Search

Hyperparameters are selected with a multi-fidelity Gaussian Process search that screens configurations at increasing epoch budgets. The search tunes optimization settings on training data and confirms the best configuration across five seeds.

E. Hyperparameter Search

  • A multi-fidelity Gaussian Process search uses training epochs as an input dimension so low-epoch trials can screen configurations before full-length runs.The regressor uses a Matern kernel plus a white-noise term.Source: E. Hyperparameter Search · p. 6
  • Configurations advance through 100, 300, 600, and 1000 epochs across 48 training runs, and the best configuration is confirmed across all five seeds.Source: E. Hyperparameter Search · p. 6
  • The search tunes optimization hyperparameters, not architecture or ablation choices, and scores configurations on a held-out slice of training data rather than the official validation split.Source: E. Hyperparameter Search · p. 6

F. Probing the Learned Geometry with D Optimal Design

The learned geometry is evaluated by applying a classical Fisher-information criterion to the learned AP coordinates. Greedy forward selection ranks APs, after which the model is retrained from scratch on the retained subset.

F. Probing the Learned Geometry with D Optimal Design

  • The Fisher-information criterion tests whether learned AP geometry supports the same geometric reasoning as true coordinates and can be computed without surveyed AP locations.Source: F. Probing the Learned Geometry with D… · p. 6
  • A larger log determinant of the position Fisher information matrix means an AP subset constrains position better, using learned rather than known AP geometry.Detection probability comes from the learned detection-threshold model.Source: F. Probing the Learned Geometry with D… · p. 6
  • Greedy forward selection adds the AP with the greatest gain in average log determinant, producing a nested ranking whose shallower subsets are prefixes.The objective averages over sample positions.Source: F. Probing the Learned Geometry with D… · p. 6
  • After AP selection, PhysicsChainedNet is retrained from scratch on the retained WAPs, including a new physics prefit, rather than merely masking APs at inference.Source: F. Probing the Learned Geometry with D… · p. 6

IV. EXPERIMENTAL SETUP

The experimental setup specifies the dataset and split, comparison baselines, AP-pruning methods, and evaluation metrics.

IV. EXPERIMENTAL SETUP

  • The setup fixes the dataset and split used for evaluation.Source: §IV · p. 7
  • It specifies the baselines used for comparison and the pruning methods used to assess geometry-based selection.Source: §IV · p. 7
  • It also identifies the metrics used to score the experiments.Source: §IV · p. 7

A. Dataset and Protocol

The evaluation uses UJIIndoorLoc’s harder official split with all 520 WAPs retained, then compares localization baselines and AP-selection strategies across common pruning levels and metrics.

A. Dataset and Protocol

  • 19,937 training and 1,111 test samples form the official UJIIndoorLoc split; all 520 WAPs are retained without detection-rate prefiltering.The samples were collected at different times.Source: A. Dataset and Protocol · p. 7

B. Baselines

  • WKNN (k=7), KNN (k=1), and a feed-forward MLP are the classic baselines, alongside published results evaluated on the same official split.The MLP uses default hyperparameters.Source: B. Baselines · p. 7

C. Pruning Methods Compared

  • Five selection strategies are compared at 20%, 35%, 50%, and 65% pruning, with the full model as reference, across five random seeds.Fisher and MSCP use learned geometry; random, detection count, and mutual information do not.Source: C. Pruning Methods Compared · p. 7

D. Metrics

  • Localization is evaluated by mean, median, P90, and P95 3D error, plus building and floor accuracy; paired tests include Holm correction and Cohen’s dz.Building height is converted using a fixed floor height.Source: D. Metrics · p. 7

V. RESULTS

  • The results are organized to test task competitiveness, the accuracy cost of physics, geometry faithfulness, and geometry usefulness for selection, robustness, and transfer.The transfer evaluation uses two additional datasets.Source: §V · p. 7

A. Comparison with Baselines and Published Results

The full model improves mean 3D error against classic baselines and confirmed official-split comparators. Ablations and geometry diagnostics show a small accuracy difference, physical signal patterns, and limits on interpreting some learned parameters.

A. Comparison with Baselines and Published Results

B. Physics Ablation Study

  • 7.07 m for the full model compares with 7.21–7.29 m across four ablations; physics terms produce learned geometry and path-loss parameters rather than a large accuracy change.The ablations remove physics loss, censored likelihood, or learned AP geometry, or randomize the geometry.Source: Table IV · p. 8, B. Physics Ablation Study · p. 8

C. Geometry Diagnostics

  • Across 232 evaluable APs, median RSSI–distance correlation is −0.43 and 94.8% are negative; randomized geometry controls collapse the median correlation to near zero.The learned geometry is faithful in aggregate, but not in every particular.Source: C. Geometry Diagnostics · pp. 8–9
  • Across five seeds, median pairwise AP displacement is 6.17 m, while x/y coordinate correlations are 0.89/0.89, indicating broad-shape reproducibility without pixel-level identity.The passage characterizes the embedding as constrained but not fully identifiable.Source: C. Geometry Diagnostics · p. 8
  • 32.1% of AP exponents saturate the [1.1, 5.0] bound; the exponent should be interpreted physically only for the unsaturated majority.The proposed explanation involving sparse detections or noisy readings is an untested assumption.Source: C. Geometry Diagnostics · p. 8

D. WAP Selection Across Three Datasets

The section compares geometry-only Fisher ranking with alternative AP selectors at four pruning levels across three independently collected datasets. Its performance depends on dataset and pruning budget, with strong results at moderate pruning and limits at the deepest UJIIndoorLoc pruning level.

Setup

  • The geometry-only Fisher score uses recovered AP coordinates and physics-layer uncertainty alone, and all three datasets are evaluated at 20%, 35%, 50%, and 65% input-feature removal.Random pruning is redrawn three times per prune level and averaged.Source: D. WAP Selection Across Three Datasets · p. 9

UJIIndoorLoc

  • At 35% and 50% pruning, geometry-only selection records UJIIndoorLoc’s lowest 3D error; its small MSCP margins at 20% and 65% are not Holm-significant.It beats random pruning by 5% at 20% and 35% at 65%.Source: D. WAP Selection Across Three Datasets · p. 9

Tampere/TUT

  • From 35% pruning onward, Fisher leads every alternative on Tampere/TUT, with 7.30–7.51 m errors across its range, never more than 0.15 m below the full model.Table VI compares best 3D error and floor accuracy by method and pruned input features.Source: Table VI · p. 9, D. WAP Selection Across Three Datasets · p. 9

UJIIndoorLoc

  • At 65% UJIIndoorLoc pruning, geometry-only reaches 8.12 m, behind MSCP at 7.57 m and mutual information at 7.71 m.The MSCP difference is significant after Holm correction (p=0.032, Cohen’s dz=5.10).Source: D. WAP Selection Across Three Datasets · p. 9

UTSIndoorLoc

  • On UTSIndoorLoc, geometry-only selection has the best floor accuracy at 35% and 50% pruning (88.4%, 88.1%) and reaches 85.6% at 65%, versus alternatives’ 75.0%–82.5%.Table VII compares best 3D error and floor accuracy by method and pruned input features.Source: Table VII · p. 10, D. WAP Selection Across Three Datasets · p. 10

Cross-dataset comparison

  • Across three sites, learned coordinates can substitute for surveyed ones; at 65% pruning geometry-only is overtaken on 3D error on UJIIndoorLoc and UTSIndoorLoc, but leads on Tampere/TUT from 35% onward.The sites differ in building count, floor count, footprint shape, and device population.Source: D. WAP Selection Across Three Datasets · p. 10

E. Robustness to AP Removal and RSSI Noise

The test-time study stresses already trained full and 35%-pruned models with further AP removal and synthetic RSSI noise. All degrade smoothly, but the relative robustness of Fisher and mutual-information pruning depends on the stressor.

Further AP removal

  • Under further AP removal, all three models degrade smoothly without a sudden collapse, while the already-pruned models degrade somewhat faster than the full model.Mutual-information pruning degrades less than Fisher pruning at every removal level except 20%, ending at 49.7% versus 50.9% relative degradation after 50% removal.Source: E. Robustness to AP Removal and RSSI… · p. 10

Interpretation

  • The relative-robustness ranking depends on the stressor: Fisher’s pruning-time advantage does not carry over cleanly to further unplanned AP loss.The authors describe the high-noise result as an independent hint that a well-chosen smaller AP subset can act as a light regularizer against corrupted input.Source: E. Robustness to AP Removal and RSSI… · p. 10

RSSI noise

  • At the highest tested RSSI noise (σ = 0.15), Fisher pruning has the smallest relative degradation (36.9%), below the full model (42.8%) and mutual-information pruning (44.1%).At low to moderate noise (σ ≤0.05), the three conditions remain within about a percentage point of one another.Source: E. Robustness to AP Removal and RSSI… · p. 10

VI. DISCUSSION AND LIMITATION

The discussion identifies limits in greedy AP ranking, path-loss parameterization, and the paper’s scope for locating blind spots. The conclusion emphasizes the learned geometry’s selection value while noting that its performance and robustness have operating boundaries.

VI. DISCUSSION AND LIMITATION

  • At heavy pruning, greedy one-at-a-time log-det selection can end in a locally good rather than globally optimal AP order.The authors propose Shapley-value and Ollivier-Ricci-curvature scores as candidate future alternatives.Source: §VI · p. 11
  • A building-level path-loss exponent with small AP deviations cannot describe an AP whose propagation genuinely departs from its building average; roughly a third of AP exponents hit the permitted boundary.The authors plan to learn a fully free per-AP exponent, which would make the decoder individually invertible and permit trilateration over learned geometry.Source: §VI · p. 11
  • Using the Fisher matrix to guide new fingerprint collection, potentially alongside position-head uncertainty, remains a future design and evaluation task.The authors also leave comparison with spatial coverage rules such as k-centre selection for future work.Source: §VI · p. 11

VII. CONCLUSION

  • The conclusion reports that the geometry-only selector matches labelled-data scores on UJIIndoorLoc and leads on Tampere/TUT from 35% pruning, while retaining the highest floor accuracy at deeper pruning on the 16-floor UTS building.The learned geometry also supplies coordinates required by classical D-optimal design when fingerprint databases lack surveyed AP positions.Source: §VII · p. 11
  • The artifact has defined operating limits: parity with the strongest classic method fails at 65% UJIIndoorLoc pruning, some per-AP exponents reach their bound, and Fisher pruning is not uniformly most robust to further AP loss.The conclusion points to these limitations as motivations for the discussion’s future work.Source: §VII · p. 11

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