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REVIEW 4 major objections 4 minor 52 references

No Location Left Behind: Measuring and Improving the Fairness of Implicit Representations for Earth Data

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Spherical wavelets end Earth models' island blind spot.

desk verdict A useful fairness benchmark for Earth INRs, with a plausible wavelet mitigation that is oversold by the title and a correlation analysis that needs more statistical care. read the letter →

arxiv 2502.06831 v1 pith:ZAOYOQBP submitted 2025-02-05 cs.LG cs.AI

Daniel Cai , Randall Balestriero This is my paper · ORCID
classification cs.LGcs.AI
keywords implicitneuralrepresentationsEarthdatafairnesssphericalwaveletsharmonicencodingslocationencodingsubgroupperformancedisparitymulti-scalerepresentationbenchmarkdataset
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Earth-data implicit neural representations (INRs) are usually judged by global average error, which hides where they fail. This paper claims that those failures are systematic: islands, coastlines, and other high-frequency, localized regions are consistently modeled worse than large landmasses, and the global spherical-harmonic bases used by state-of-the-art encodings force a tradeoff between global and local accuracy. To make the problem measurable, it introduces FAIR-Earth, a benchmark dataset with land-sea, temperature, precipitation, CO2, and population signals, plus metadata for stratifying by landmass size, coast distance, population density, and country. To fix the bias, it proposes spherical wavelet encodings, which use multi-scale Morlet wavelets on the sphere and show positive land-island correlation with competitive global performance. The paper's caveat is that the wavelet construction is ill-defined at the poles, where its error grows.

What carries the argument

The load-bearing object is a spherical Morlet mother wavelet, $\psi_M(\theta,\phi) = [\Pi^{-1}\psi_{\mathbb{R}^2}](\theta,\phi) = e^{i\tan(\theta/2)\cos\phi}\,e^{-\frac{1}{2}\tan^2(\theta/2)}/(1+\cos\theta)$, obtained by inverse stereographic projection of the Euclidean Morlet wavelet. From it, the encoding builds an orthogonal family by rotations $R(\rho)$ over a Fibonacci lattice $F_N$ and dyadic dilations $D(a)$ with $a \in \{2^{i/Q}\}$, giving $O(NM)$-size embeddings that localize signals in both dilation and rotation. This multi-scale localization is what lets the encoding represent islands and coastlines at coarse resolutions without aliasing the global land-sea structure. The same construction is what breaks near the poles: the inverse stereographic projection is ill-defined there, and the paper's latitudinal analysis shows error rising from equator to poles.

What would settle it

On a FAIR-Earth task, restrict evaluation to latitudes above $80^\circ$ and compare spherical-wavelet and spherical-harmonic error; if the wavelet encoding's polar error exceeds the harmonic encoding's by more than the island error dropped, the claim that no location is left behind is not supported for polar regions.

Watch

Extended reading notes

Core claim

The paper's central claim is that optimizing Earth INRs for global loss is not fairness-neutral: on FAIR-Earth, spherical-harmonic, theory, and grid/sphere encodings all exhibit much higher loss on islands and coastlines than on land or sea, and when trained at finer resolution the harmonic encodings trade off land accuracy against island accuracy, producing a negative correlation. The authors attribute this to the global support of Fourier bases, which alias around sharp localized discontinuities. Their proposed spherical wavelet encoding replaces global harmonics with a multi-scale basis built from the inverse stereographic projection of the Morlet wavelet, discretized over a Fibonacci lattice with dyadic dilations; this encoding maintains competitive or better average loss while producing a positive correlation between land and island performance, meaning local and global accuracy improve together. The authors also report that spherical wavelets are computationally cheaper at large encoding sizes than spherical harmonics, though they degrade at high latitudes and underperform harmonics on a coarse single-scale checkerboard task.

Load-bearing premise

The encoding is built from a projection that is not defined at the poles and that visibly degrades near them; the paper assumes those polar regions are a small enough share of Earth-data tasks that the fairness gain on islands and coastlines is not offset by a newly introduced high-latitude bias.

Editorial extensions

If this is right

  • Spherical wavelet encodings turn the land-island performance correlation from negative to positive across training resolutions, so optimizing for global accuracy no longer sacrifices island accuracy.
  • On FAIR-Earth's land-sea, temperature, precipitation, and CO2 tasks, spherical wavelet encodings match or beat spherical harmonic encodings in average loss while narrowing island and coastline gaps.
  • Spherical wavelet encodings scale better in generation time at larger sizes than spherical harmonic encodings, which suffer factorial-related numerical instability at high Legendre degree.
  • The method does not dominate everywhere: spherical harmonics remain better on the coarse single-scale checkerboard task, and spherical wavelets degrade toward the poles.
  • FAIR-Earth enables country-level and population-stratified fairness audits that were not possible with previously available geospatial datasets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the multi-scale property transfers, combining a global harmonic branch with a localized wavelet branch could remove the polar degradation while keeping the island gain; the paper does not test such a hybrid.
  • Because FAIR-Earth's grid is uniform in latitude and longitude, polar cells cover much smaller areas than equatorial cells, so the fairness metrics may under-weight polar surface area and soften the polar caveat.
  • The same wavelet-encoding idea could be evaluated on other INR domains with multi-scale structure, such as 3D scene or medical signal reconstruction, to see whether the positive land-island correlation generalizes beyond Earth data.
  • The reported negative land-island correlation for spherical harmonics could be used as a cheap diagnostic in other geospatial benchmarks: compute per-subgroup loss on land versus small landmasses before deploying an Earth INR.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. This paper introduces FAIR-Earth, an open-source benchmark package for measuring subgroup-level performance of implicit neural representations (INRs) on Earth data, with metadata stratifications for islands, coastlines, population density, and countries. Using this benchmark, the authors evaluate spherical harmonic, theory, grid-and-sphere, and their proposed Spherical Wavelet (SW) encodings, and report that existing encodings show strong biases against islands and coastlines. They propose SW encodings built from inverse stereographic projections of Morlet wavelets over a Fibonacci lattice, and argue that SW produces positive correlations between land and island performance across training resolutions, reduces island/coastline losses on the land-sea task, and remains competitive with spherical harmonics on global metrics. The paper also includes external validation on checkerboard and Rußwurm land-sea tasks and reports computational comparisons.

Significance. The FAIR-Earth dataset and accompanying open-source package are a useful contribution: they provide a concrete way to stratify Earth INR evaluation by geographic and demographic subgroups, and the empirical documentation of island/coastline bias in spherical harmonic and related encodings is valuable and credible. The SW encoding is a reasonable proposal, and the paper includes external tasks, hyperparameter sweeps, and error analyses that go beyond a single benchmark. However, the central claim that SW 'corrects localized biases' is currently supported mainly by the land-sea classification task; the correlation evidence in Table 2 is under-specified, and the temperature subgroup results in Table 15 show SW worse than SH on every subgroup. The theoretical claim of an orthogonal wavelet basis is also not established. With these points clarified, the paper could be a solid contribution.

major comments (4)
  1. [Section 4.3, Table 2] The claim that SW 'induces a positive trend between global (land) and localized (island) performance' rests on Pearson correlations whose sample size is not reported. If each correlation is computed across hyperparameter configurations at a fixed training resolution, the number of pairs needs to be stated and confidence intervals or significance tests provided; if it is computed across the four resolutions, n=4 and the sequence 0.51, 0.42, 0.04, 0.31 is not consistently positive and the conclusion would be fragile to a single resolution. The post hoc omission of SPHERE C+ from Table 2 'due to abnormally high bias' should be replaced by a prespecified exclusion criterion or by an appendix table including all baselines.
  2. [Section 4.2, after Eq. (2)] The text states that the rotated and dilated Morlet wavelets form 'a set of orthogonal basis functions.' Inverse stereographic projection of an admissible Euclidean wavelet yields an admissible spherical wavelet (zero-mean), but admissibility does not imply orthogonality, and Morlet wavelets are typically frames rather than orthogonal bases; discretization over a Fibonacci lattice with dyadic dilations does not by itself restore orthogonality. Please either prove the claim, replace it with the correct frame terminology, or provide a numerical check (e.g., Gram matrix condition) if orthogonality is used in the reconstruction formula.
  3. [Section 4.3 and Fig. 23] The paper acknowledges that the inverse stereographic construction is ill-defined at the poles and that SW performance degrades at high latitudes. Because the title and framing promise that 'no location is left behind,' the paper should quantify the affected area or population and state explicitly that the fairness improvement applies away from polar regions. As written, the polar degradation is a known limitation but its scope is not assessed.
  4. [Section 4.3 and Table 15] On the surface temperature regression task, SW has higher subgroup loss than SH for every subgroup (land 0.087 vs. 0.076, sea 0.043 vs. 0.028, island 0.049 vs. 0.041, coast 0.101 vs. 0.083). Thus the claimed fairness improvement over baselines is demonstrated only on the land-sea binary classification task, not across the modalities promised in the abstract. The paper should either restrict the fairness claim to the land-sea task or provide subgroup-level evidence showing improvement on other FAIR-Earth modalities.
minor comments (4)
  1. [Table 1] The sea loss at resolution 30000 is reported as 0.80 ± 0.03, while all other sea entries are 0.16 or below; this appears to be a typo for 0.08 and should be corrected.
  2. [Section 3.2] The phrase 'data specifications available in Appendix A.1 and Appendix A.1' should reference the correct appendix sections, as the duplicate reference is a typo.
  3. [Section 4.1] The sentence 'particularly struggle with to demarcate the Spain's fine Mediterranean coastline' contains a grammatical error and should be rephrased.
  4. [Section 3.2] The description of the 0.1° grid as 'state-of-the-art resolution' is an overstatement for some of the constituent datasets and should be qualified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SW fairness claim is empirical and externally benchmarked, and the dataset construction and wavelet derivation do not reduce to the measured outcomes.

full rationale

The paper's claimed derivation chain is self-contained rather than circular. FAIR-Earth is assembled from external geophysical sources (IMERG, OCO-2, CHELSA, GPWv4) with metadata thresholds such as islands defined as landmasses under 30,000 square miles, and the fairness findings are empirical measurements on that benchmark, not consequences of the benchmark's definition. The spherical wavelet encoding is built on external wavelet theory: 'As proven in Sanz et al. (2006), this inverse stereographic projection of an admissible Euclidean wavelet yields an admissible spherical wavelet'; the Morlet mother wavelet is selected by an ablation (Fig. 25), not by the fairness metric. The claim that SW mitigates subgroup bias is tested under 'identical settings' against SH, THEORY, and GRID AND SPHERE baselines over a shared hyperparameter grid (Section 4.3, Tables 2 and 4), and additionally checked on external checkerboard and Rußwurm land-sea tasks (Tables 13-14), including an acknowledged polar failure mode (Fig. 23). The only self-citations (Balestriero et al. 2022; Kirichenko et al. 2023) motivate subgroup-level evaluation in general and do not appear in the derivation of the SW encoder or in the definition of the fairness statistics. No equation in the paper defines the SW correlation or the fairness metric in terms of the SW construction, and no fitted parameter is relabeled as a prediction; the positive-trend claim is an empirical correlation over trained configurations, however statistically fragile it may be. Hence no circular step is exhibited.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central fairness results depend on four parametric choices in the SW encoding (wave number K, scale factor, maximum dilation scale, maximum rotations) that are tuned on the FAIR-Earth validation tasks. The wavelet construction itself is inherited from prior work and is treated as axiomatic. The island and coastline subgroup definitions are explicitly ad hoc. No new physical entities are introduced.

free parameters (4)
  • K Value (wave number) = grid-searched, range 4-10, default 6
    Controls the central frequency of the Morlet wavelet envelope (Fig 6c, Table 6, Table 7). Tuned on FAIR-Earth validation tasks.
  • Scale Factor = grid-searched, range 0.75-1.25, default 1
    Controls energy concentration and dilation of the mother wavelet (Fig 6d, Table 7). Tuned per task.
  • Maximum Scale (dilations) = grid {2,3,4,5} or {3,4,5}
    Number of dyadic dilation levels M in the basis {ψ_{a,ρ}}; controls multi-resolution capacity (Table 6, Table 7).
  • Maximum Rotations = grid {20..200 step 40} or {50,90,130,170}
    Number of rotation points N on the Fibonacci lattice; controls angular resolution of the encoding (Table 6, Table 7).
assumptions (4)
  • standard math Inverse stereographic projection of an admissible Euclidean wavelet yields an admissible spherical wavelet.
    Invoked in Section 4.2 to justify the Morlet mother wavelet construction; cited to Sanz et al. (2006), not proven in the paper.
  • domain assumption The discretized set {R(ρ)D(a)ψ_M} over a Fibonacci lattice and dyadic scales forms an orthogonal basis or well-behaved frame of L2(S2).
    Stated in Section 4.2 as 'a set of orthogonal basis functions' without proof. Rotated and dilated wavelets generally form frames, not orthonormal bases.
  • domain assumption The FAIR-Earth 0.1° gridded sampling introduces only negligible temporal and polar bias.
    Acknowledged in Section 3.3 as a limitation: 'temporal and polar biases may manifest from the gridded format'. The fairness claims rely on this bias being minor enough not to reverse subgroup conclusions.
  • ad hoc to paper Islands are defined as landmasses under 30,000 square miles, and coastline distance thresholds are set by the authors.
    Section 3.2 and A.1 state these are 'ad-hoc definitions'. The subgroup fairness conclusions depend on these thresholds.

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Pith. "Pith review of No Location Left Behind: Measuring and Improving the Fairness of Implicit Representations for Earth Data." pith.science (2026). https://pith.science/paper/ZAOYOQBP

@misc{pith2026250206831,
  author       = {Pith},
  title        = {Pith review of: No Location Left Behind: Measuring and Improving the Fairness of Implicit Representations for Earth Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZAOYOQBP}},
  note         = {Machine review of arXiv:2502.06831}
}
read the original abstract

Implicit neural representations (INRs) exhibit growing promise in addressing Earth representation challenges, ranging from emissions monitoring to climate modeling. However, existing methods disproportionately prioritize global average performance, whereas practitioners require fine-grained insights to understand biases and variations in these models. To bridge this gap, we introduce FAIR-Earth: a first-of-its-kind dataset explicitly crafted to examine and challenge inequities in Earth representations. FAIR-Earth comprises various high-resolution Earth signals and uniquely aggregates extensive metadata along stratifications like landmass size and population density to assess the fairness of models. Evaluating state-of-the-art INRs across the various modalities of FAIR-Earth, we uncover striking performance disparities. Certain subgroups, especially those associated with high-frequency signals (e.g., islands, coastlines), are consistently poorly modeled by existing methods. In response, we propose spherical wavelet encodings, building on previous spatial encoding research. Leveraging the multi-resolution capabilities of wavelets, our encodings yield consistent performance over various scales and locations, offering more accurate and robust representations of the biased subgroups. These open-source contributions represent a crucial step towards the equitable assessment and deployment of Earth INRs.

Figures

Figures reproduced from arXiv: 2502.06831 by the authors.

Figure 1
Figure 1. Heatmap of the spatial distribution of approximation errors using existing INRs to model land-sea data of the Earth. Leveraging the resolution of FAIR-EARTH, we uncover clear bias against islands, where error magnitude is significantly higher. Against the same task, SPHERICAL WAVELETS resolves these issues by reconciling global signals with fine and localized signals. Details and plots available in Appendix A.3. 1Av… view at source ↗
Figure 2
Figure 2. FAIR-EARTH is the first Earth INR-framework for fairness assessment, and moreover is an open-source package that allows for agile testing and analysis of subgroup-level performance. Extensive experiments against FAIR-EARTH reveal significant disparities in the performance of state-of-the-art INR methods across different subgroups (Section 4.1). In particular, our analysis uncovers a strong negative correlation betwe… view at source ↗
Figure 3
Figure 3. Visualization of existing encoding mechanisms. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Model behavior at different resolutions and regularizations. Smaller models fail to capture [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Qualitative performance of SPHERICAL HARMONIC on surface temperature regression task. SH exhibit bias against high-frequency details, with loss concentrated in areas of abrupt change (e.g., coastlines). Biases Across Multiple Modalities and Subgroups While the land-sea…
Figure 6
Figure 6. Figure 6: SPHERICAL WAVELET parameterization. All filters are identical to (Eq. (2)) except for one change in the specified parameter. necessary zero-mean condition (Demanet & Vandergheynst, 2003). Empirically, we find that the Morlet mother wavelet shows consistently superior p…
Figure 7
Figure 7. Figure 7: Comparisons of SPHERICAL HARMONIC and SPHERICAL WAVELET behavior in local￾ized regions. Full figures available in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Binary land-sea Continued from previous page Label Description Misc. Source Temperature Air surface temperature, based on satel￾lite measurements and reflectance mod￾eling. Monthly Resolu￾tion (Karger et al., 2017) Emissions Measured and modeled CO2 emissions from the …
Figure 9
Figure 9. Figure 9: Coastline [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Islands (in dark purple) 17 [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Air surface temperature plot (Jan. 2018) [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Cumulative precipitation plot (Jan. 2018) [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Global population plot [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: CO2 emissions plot 19 [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: Color-coded country plot 20 [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 16
Figure 16. Figure 16: Coverage of different landmass sizes using uniform sampling. At [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: Zoomed-in inset plots for main diagram [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]
Figure 18
Figure 18. Figure 18: At higher frequencies (L = 30), the closed-form calculations provided in Rußwurm et al. (2024) run into numerical error [PITH_FULL_IMAGE:figures/full_fig_p022_18.png]
Figure 19
Figure 19. Figure 19: SPHERICAL HARMONIC encoding prediction plot A.3 MISCELLANEOUS FIGURES 22 [PITH_FULL_IMAGE:figures/full_fig_p022_19.png]
Figure 20
Figure 20. Figure 20: SPHERICAL WAVELET encoding prediction plot 23 [PITH_FULL_IMAGE:figures/full_fig_p023_20.png]
Figure 21
Figure 21. Figure 21: Fine tuning dynamics for different training resolutions, showing 5-iteration median win [PITH_FULL_IMAGE:figures/full_fig_p026_21.png]
Figure 22
Figure 22. Figure 22: Land and island loss for SH, SW representations, trained on the same resolution with [PITH_FULL_IMAGE:figures/full_fig_p027_22.png]
Figure 23
Figure 23. Figure 23: Differences in latitudinal loss on land-sea (Rußwurm et al. (2024)) classification, where [PITH_FULL_IMAGE:figures/full_fig_p028_23.png]
Figure 24
Figure 24. Figure 24: Visualizations of various wavelet filters on the sphere. Of the three, only Gabor consis [PITH_FULL_IMAGE:figures/full_fig_p029_24.png]
Figure 25
Figure 25. Figure 25: A comparison of different wavelet filter performance on the FAIR-E [PITH_FULL_IMAGE:figures/full_fig_p029_25.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.