Do Location Encoders Capture Spatial Effects? A GeoShapley Benchmark Across Scales [experiments] Daniel Kiv
Shaowen Wang
Siebel School of Computing & Data Science, University of Illinois Urbana-Champaign Urbana, IL, USA [email protected]
Department of Geography & GIS and Siebel School of Computing & Data Science, University of Illinois Urbana-Champaign Urbana, IL, USA [email protected]
arXiv:2606.23453v1 [cs.LG] 22 Jun 2026
Abstract Location encoders transform geographic coordinates into high dimensional embeddings for downstream machine learning, but it is unclear how well these representations capture interpretable spatial effects. We benchmark whether GeoShapley, a game-theoretic explainer that treats all location features as a single joint player, can recover spatially varying coefficients from models built on locationencoder embeddings. Eleven encoders from the TorchSpatial framework are evaluated against a synthetic process with known coefficients, across three scales (grid, county, global), with and without raw coordinates alongside the embedding, and under untrained and contrastively trained conditions. Measuring recovery as the correlation between estimated and true coefficients, we report how it varies with scale and encoder architecture and compare the embeddings against a raw-coordinate baseline. Recovery of the primary coefficient is consistently high across encoders, whereas recovery of a secondary coefficient is more scale-dependent, differing most at the global scale; the raw-coordinate baseline remains competitive throughout.
CCS Concepts • Computing methodologies → Supervised learning by regression; Spatial and physical reasoning; • Information systems → Data mining.
Keywords Location encoding, GeoAI, explainable AI, GeoShapley, spatially varying coefficients, spatial interpretability
1
Introduction
Spatial heterogeneity is a fundamental property of geographic processes. It refers to the way relationships between variables can change from place to place, so that a single global model often fails to explain geographic data well. Spatially varying coefficient (SVC) models, such as Geographically Weighted Regression (GWR) and its multiscale extension, capture this by letting model coefficients change with location [Anselin 1988; Fotheringham et al. 2017]. These models are interpretable, but they impose strong assumptions about distribution and linearity and can be expensive to fit. Machine learning offers a more flexible alternative that learns nonlinear spatial relationships with fewer assumptions. A common way to bring location into a machine learning model is location encoder, a deep learning method that turns raw coordinates into a high dimensional embedding for a downstream task
[Aodha et al. 2019; Mai et al. 2022, 2023]. Raw latitude and longitude are awkward inputs. They are angular measures on a sphere, so a model may not realize that two points on opposite sides of the antimeridian are actually close together. A location encoder can address this with a hand-designed transform of the coordinates followed by a neural network. The neural network is effective but opaque, and this creates a deeper interpretability problem: when the spatial input is itself produced by a neural network, location becomes a black box inside an already black-box model. The geographic information is distributed across all 𝐷 embedding dimensions, with no single dimension corresponding to an interpretable geographic quantity, so standard feature-wise post-hoc explainers, which score one feature at a time, spread the attribution across all 𝐷 dimensions and never recover a single, coherent location effect. GeoShapley [Li 2024] addresses this by treating all of the location features as a single joint player in the Shapley value framework [Harris et al. 2022], collapsing the entire location representation into one intrinsic location effect from which spatially varying coefficients can be recovered. Li [2024] notes that a high dimensional embedding could in principle be grouped into this player exactly as raw coordinates are, but evaluates GeoShapley only on coordinates and analytic spatial bases [Li 2022; Liu 2024]. Closest to our idea, Li and Peng [2025] run the same template, synthetic data with known SVCs and GeoShapley recovery across two geometries, but represent location with an analytic Moran-eigenvector basis. Whether GeoShapley recovers spatial effects from learned, task-agnostic neural encoders that span planar and spherical geometry has not been tested [Rußwurm et al. 2024]. We fill that gap. We benchmark eleven encoders from the TorchSpatial framework [Wu et al. 2025] against a synthetic process with known coefficients across three spatial scales, asking two questions: can GeoShapley recover interpretable spatial effects from encoder-based models, and does the choice of encoder architecture matter [Rao et al. 2026]?
2
Method
GeoShapley. GeoShapley extends the classical Shapley value by grouping location features into a single joint player “GEO”. The intrinsic location effect is 𝜙 GEO =
∑︁
𝑠! (𝑝 − 𝑠 − 𝑔)! 𝑓 (𝑆 ∪ {GEO}) − 𝑓 (𝑆) , (𝑝 − 𝑔 + 1)!
(1)
𝑆 ⊆𝑀\{GEO}
where 𝑀 is the feature set, 𝑝 the number of features, 𝑠 = |𝑆 |, and 𝑔 the number of grouped location features; spatially varying interaction effects 𝜙 (GEO,𝑗 ) between GEO and a covariate 𝑗 yield the SVCs.
SIGSPATIAL ’26, November 3–6, 2026, Riverside, CA, USA
Kiv and Wang
and encoder. The downstream model is an MLP (scikit-learn [Pedregosa et al. 2011]) with ReLU activations trained by Adam, tuned per run with 5-fold randomized search (20 candidates) over hiddenlayer sizes, the ℓ2 penalty, and the initial learning rate. Each (scale, encoder, condition) is run for 25 seeds. AI coding assistants (Claude Code) were used to write and refactor the experiment code, run and manage the compute-cluster jobs, and assist with the data analysis; all experimental design, results, and conclusions were verified by the authors.
2 SVC Recovery (Pattern, Pearson r)
Sphere2Vec-dfs
0.93
0.98*
0.98*
0.93
0.92
0.91
0.94
0.94
0.64
Sphere2Vec-sphereM+
0.93
0.99*
0.99*
0.93
0.93
0.91
0.94
0.94
0.56
1.0
wrap_ffn
0.93
0.99*
0.99*
0.93
0.93
0.92
0.94
0.93
0.59
0.9
Sphere2Vec-sphereM
0.93
0.92
0.76
0.93
0.94*
0.76
0.94
0.95*
0.87
Sphere2Vec-sphereC
0.93
0.92
0.66
0.93
0.94*
0.78
0.94
0.95*
0.87
Sphere2Vec-sphereC+
0.93
0.92
0.64
0.93
0.94*
0.74
0.94
0.95*
0.88
Space2Vec-grid
0.93
0.92
0.69
0.93
0.94*
0.80
0.94
0.95*
0.87
Space2Vec-theory
0.93
0.92
0.70
0.93
0.94*
0.80
0.94
0.95*
0.86
NeRF
0.93
0.92
0.72
0.93
0.94*
0.79
0.94
0.95*
0.86
0.4
rff
0.93
0.94*
0.49
0.93
0.94*
0.51
0.94
0.95*
0.85
0.3
0.93
0.92
0.41
0.93
0.92
0.40
0.94
0.94
0.53
Global Global Global County County County BL EC EO BL EC EO
Grid BL
Grid EC
Grid EO
tile_ffn
0.8 0.7 0.6 0.5
Evaluation. Trained models are explained with GeoShapley over all 𝑁 observations, recovering 𝛽ˆ0, 𝛽ˆ1, 𝛽ˆ2 . Pearson correlation 𝑟 = ˆ 𝛽) against the ground truth is the primary metric for Pearson( 𝛽, spatial-pattern recovery; we report mean 𝑟 over the 25 seeds.
3
BL = Baseline (coords only) EC = Embedding + Coordinates EO = Embedding Only * significantly > coords-only baseline (p<0.05)
Figure 1: 𝛽 2 recovery (Pearson 𝑟 ) per encoder across the three scales and the baseline (BL), embedding+coordinates (EC), and embedding-only (EO) conditions defined in the panel (MLP, untrained). ∗ marks values significantly above the coords-only baseline (paired Wilcoxon, 𝑝 < 0.05); 𝛽 1 (𝑟 > 0.98 in nearly all conditions) is omitted (Table 1). The GeoShapley package [Li 2024] estimates these with Kernel SHAP via weighted least squares [Covert and Lee 2021]. Data. Following Li [2024] and Fotheringham et al. [2017], the response is linear in two non-spatial covariates with spatially varying coefficients, 𝑦𝑖 = 𝛽 0 (𝑠𝑖 ) + 𝛽 1 (𝑠𝑖 )𝑋 1𝑖 + 𝛽 2 (𝑠𝑖 )𝑋 2𝑖 + 𝜀𝑖 ,
𝜀𝑖 ∼ N (0, 𝜎 2 ),
(2)
with 𝑋 1, 𝑋 2 ∼ Uniform(−2, 2) and 𝜎 = 0.1; we refer to 𝛽 1 as the primary coefficient and 𝛽 2 as the secondary coefficient. Three scales are used: a 25 × 25 grid (𝑁 =625); U.S. county centroids (𝑁 ≈3,000); and a global sample (𝑁 =10,000) over (−180, 180) × (−90, 90). At grid/county scales 𝛽 0 is parabolic, 𝛽 1 a linear gradient, and 𝛽 2 sinusoidal; at global scale the surfaces are built on the sphere, with 𝛽 2 a multi-scale oscillation continuous across the antimeridian. Data are split 80/20 train/test. Encoders and training. Coordinates are transformed into a 𝐷=8 embedding by one of eleven TorchSpatial encoders [Wu et al. 2025]: Space2Vec-grid/-theory [Mai et al. 2020], tile_ffn [Tang et al. 2015], wrap_ffn [Aodha et al. 2019], NeRF [Mildenhall et al. 2021], five Sphere2Vec variants [Mai et al. 2023], and random Fourier features [Rahimi and Recht 2007]; a none baseline uses only normalized coordinates. The feature vector concatenates the covariates, the 𝐷 embedding dimensions, and, in the default condition, normalized lon/lat. To separate the embedding’s standalone contribution from coordinates available in every model, we also run an embeddings-only condition that omits lon/lat. Two encoder conditions are compared: untrained (weights frozen at random init, testing architectural bias) and contrastively trained (weights updated by a spatial contrastive objective, then frozen). Crossed with the untrained/trained axis, this gives four configurations per scale
Results
Across all three scales, GeoShapley recovers 𝛽 1 with high fidelity regardless of encoder, with 𝑟 > 0.98 in nearly all conditions (Table 1). Recovery of 𝛽 2 is more variable and reveals meaningful architectural differences (Figure 1). At grid and county scales most encoders reach 𝑟 ≈ 0.92–0.95 for 𝛽 2 and the raw-coordinate baseline is competitive with every embedding; the small grid and county gains over baseline, though occasionally significant, are negligible in magnitude (≤ 0.02). This suggests that at these scales any reasonable location representation supplies enough signal for the MLP. At global scale the spread across encoders widens (Table 1). Sphere2Vec-sphereM+ (𝑟 =0.991), wrap_ffn (𝑟 =0.988), and Sphere2Vecdfs (𝑟 =0.982) score highest, while the remaining encoders and the baseline (𝑟 =0.926) cluster near 𝑟 =0.92–0.94. The three highest scorers do not correspond to a single model family: wrap_ffn is not a Sphere2Vec model, and three other Sphere2Vec variants score no higher than the baseline. Contrastive training generally does not improve, and often slightly degrades, recovery: the Δ column in Table 1 is negative for most encoders, indicating that the frozen random initialization already provides sufficient positional signal; the clearest exception is Sphere2VecsphereM, where training improves global 𝛽 2 by 0.030. Training was occasionally unstable (e.g. wrap_ffn at grid scale). The baseline stays competitive throughout. This tells us that embeddings do not automatically improve SVC recovery over plain normalized coordinates, which echoes Li and Peng [2025], who likewise found coordinate-only models hard to beat with an analytic eigenvector basis. The highest-scoring encoders at global scale are the exception, pulling ahead of the baseline there. Embeddings without coordinates. Because lon/lat are present in every condition above, that comparison captures only the embedding’s marginal value over raw coordinates, which is small. Removing the coordinates isolates the embedding as the sole location representation (Table 2), and large architectural differences emerge that invert with scale. At global scale Sphere2Vec-sphereM+, wrap_ffn, and Sphere2Vec-dfs retain near-full 𝛽 2 recovery while the others drop sharply (Figure 2); at grid scale this ordering reverses, with NeRF, Space2Vec-grid, and rff holding up while those three global leaders fall to 𝑟 ≈ 0.56–0.64. Without coordinates, then, the encoders that recover global structure are largely the ones that recover the local grid least well, and vice versa.
Do Location Encoders Capture Spatial Effects? A GeoShapley Benchmark Across Scales [experiments]
SIGSPATIAL ’26, November 3–6, 2026, Riverside, CA, USA
Global 2 Recovery: Representative Encoders (rep 0) Baseline (coords only)
Sphere2Vec-dfs (emb only)
3.0
tile_ffn (emb only)
Space2Vec-grid (emb only, trained)
2.5 2.0 2
1.5 1.0 0.5 0.0
1.0
0.0
Error
0.5
-0.5 -1.0
Figure 2: Global 𝛽 2 surfaces for representative encoders (top row: ground truth): the coordinates-only baseline, and Sphere2Vecdfs, Space2Vec-grid, and tile_ffn in the embedding-only condition. Table 1: Pearson correlation (𝑟 ) by encoder at global scale, untrained vs. contrastively trained (Δ = trained − untrained). Bold marks the highest 𝛽 2 recovery. 𝛽1
𝛽2
Encoder
Untr.
Tr.
Δ
Untr.
Tr.
Δ
NeRF Space2Vec-grid Space2Vec-theory Sphere2Vec-dfs Sphere2Vec-sphereC Sphere2Vec-sphereC+ Sphere2Vec-sphereM Sphere2Vec-sphereM+ none (baseline) rff tile_ffn wrap_ffn
0.996 0.996 0.996 0.999 0.997 0.996 0.997 0.999 0.998 0.995 0.998 0.999
0.996 0.996 0.995 0.999 0.995 0.994 0.996 0.999 0.998 0.994 0.997 0.997
− .000 − .001 − .001 − .000 − .001 − .002 − .000 − .000 .000 − .002 − .001 − .002
0.919 0.921 0.920 0.982 0.921 0.922 0.922 0.991 0.926 0.938 0.921 0.988
0.897 0.918 0.914 0.979 0.912 0.922 0.952 0.984 0.926 0.928 0.910 0.989
− .022 − .004 − .006 − .003 − .009 +.000 +.030 − .007 .000 − .010 − .011 +.001
Table 2: Recovery of 𝛽 2 from embeddings only (lon/lat removed) vs. embeddings+coordinates, untrained. With coordinates, encoders are near-interchangeable; without them, recovery varies widely and the strongest encoders at global scale are the weakest at grid scale, and vice versa. Bold marks the strongest embeddings-only recovery per scale. Grid
4 Robustness to nonlinearity. Because Eq. (2) is linear in the covariates, we repeated all experiments with a nonlinear process that adds a global quadratic term and offset, 𝑦𝑖 = 𝛽 0 (𝑠𝑖 ) + [𝛽 1 (𝑠𝑖 )𝑋 1𝑖 + 𝑋 1𝑖2 ] + [𝛽 2 (𝑠𝑖 )𝑋 2𝑖 + 2𝑋 2𝑖 ] + 𝜀𝑖 . Recovery of 𝛽 2 was essentially unchanged at every scale (mean |Δ𝑟 | ≤ 0.005), and the global-scale ordering was preserved (Sphere2Vec-sphereM+ 0.990, wrap_ffn 0.984, Sphere2Vec-dfs 0.979 vs. baseline 0.916). Recovery of 𝛽 1 degraded modestly (mean Δ𝑟 from −0.02 to −0.07), concentrated on the coefficient sharing a feature with the quadratic term, as GeoShapley absorbs the global nonlinearity into that main effect. The globalscale differences between encoders are thus not an artifact of a linear data-generating process.
Global
Encoder
+coord
emb-only
+coord
emb-only
Sphere2Vec-sphereM+ Sphere2Vec-dfs wrap_ffn NeRF Space2Vec-grid rff tile_ffn none (baseline)
0.935 0.935 0.935 0.950 0.951 0.949 0.935 0.938
0.561 0.641 0.592 0.856 0.875 0.852 0.528 –
0.991 0.982 0.988 0.919 0.921 0.938 0.921 0.926
0.988 0.984 0.985 0.715 0.689 0.485 0.407 –
Discussion and Conclusion
The results show a clear scale-dependence in where encoder architecture matters. At grid and county scales, differences between encoders are small and normalized coordinates suffice. At global scale, where 𝛽 2 varies across the ±180◦ antimeridian, the differences between encoders become substantial. This complements intrinsicdimensionality analyses of the same encoder zoo [Rao et al. 2026] and benchmarks of GeoShapley against GWR/MGWR [Liu 2024], here extended to learned high dimensional encoders rather than analytic spatial bases. The same scale-dependence appears in the embeddings-only results: the encoders that recover global effects unaided are largely those that recover the local grid least well, and vice versa. Untrained encoders also match or beat contrastively
SIGSPATIAL ’26, November 3–6, 2026, Riverside, CA, USA
trained ones, suggesting the contrastive objective is not aligned with SVC recovery. Limitations: the data-generating process is synthetic and smooth, so the encoders that performed best here need not suit sharp or discontinuous effects; the embedding dimension is fixed at 𝐷=8 for tractability; and results are specific to the MLP and explainer settings. Overall, GeoShapley drops into a location-encoder pipeline as is, without an interpretable embedding space, and reliably recovers spatially structured effects. The practical takeaways are that encoder choice matters most at global scale and that contrastive pre-training should not be assumed to help. A safe default follows from the embeddings-only reversal: keep raw coordinates alongside any embedding so recovery degrades gracefully when the embedding alone is a poor match for the scale, and reserve specialized encoders for settings where coordinates alone underperform. The ability to audit whether a model has learned spatial effects is a step toward spatial-fairness frameworks [Cai and Balestriero 2025; Saxena et al. 2024]. Future work should evaluate higher embedding dimensions, better-aligned contrastive objectives, discontinuous spatial-effect regimes, and real-world data with partially known spatial processes. Code and data availability. Code to reproduce all experiments, including the synthetic data generators, the encoder wrappers, and the GeoShapley evaluation, is available at https://github.com/ cybergis/loc-enc-svcs.
Acknowledgments This research used the TGI RAILS compute resource (NSF award OAC-2232860, Taylor Geospatial Institute) and Virtual ROGER, supported by the CyberGIS Center for Advanced Digital and Spatial Studies and the School of Earth, Society and Environment at the University of Illinois Urbana-Champaign. This project was supported by Award No. 15PNIJ-24-GG-01573-RESS, awarded by the National Institute of Justice, Office of Justice Programs, U.S. Department of Justice. The opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect those of the Department of Justice.
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