Measuring the Intrinsic Dimension of Earth Representations
Authors: Arjun Rao
Marc Rußwurm
Konstantin Klemmer
Esther Rolf
Venue: ICLR 2026 Poster (2026)
Keywords: Implicit Neural Representations
Location Encoding
Intrinsic Dimension
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We show that the intrinsic dimension of a geographic Implicit Neural Representation (INR) informs how (i) representative, (ii) task-aligned, and (iii) spatially-biased the INR is.

Abstract: Within the context of representation learning for Earth observation, geographic Implicit Neural Representations (INRs) embed low-dimensional location inputs (longitude, latitude) into high-dimensional embeddings, through models trained on geo-referenced satellite, image or text data. Despite the common aim of geographic INRs to distill Earth's data into compact, learning-friendly representations, we lack an understanding of how much information is contained in these Earth representations, and where that information is concentrated. The intrinsic dimension of a dataset measures the number of degrees of freedom required to capture its local variability, regardless of the ambient high-dimensional space in which it is embedded. This work provides the first study of the intrinsic dimensionality of geographic INRs. Analyzing INRs with ambient dimension between 256 and 512, we find that their intrinsic dimensions fall roughly between 2 and 10 and are sensitive to changing spatial resolution and input modalities during INR pre-training. Furthermore, we show that the intrinsic dimension of a geographic INR correlates with downstream task performance and can capture spatial artifacts, facilitating model evaluation and diagnostics. More broadly, our work offers an architecture-agnostic, label-free metric of information content that can enable unsupervised evaluation, model selection, and pre-training design across INRs.

Published as conference paper ICLR 2026
MEASURING THE INTRINSIC DIMENSION OF EARTH REPRESENTATIONS
Arjun Rao"
Marc Rußwurm" University of Bonn & Wageningen University
Konstantin Klemmer LGND Al & University College London
Department of Computer Science University of Colorado Boulder
Esther Rolf
Department of Computer Science University of Colorado Boulder
ABSTRACT
Within the context representation learning for Earth observation, geographic Implicit Neural Representations (INRs) embed low-dimensional location inputs (longitude, lati- tude) into high-dimensional embeddings, through models trained on geo-referenced satel- lite, image text data. Despite the common aim of geographic INRs to distill Earth's data into compact, forming-friendly representations, we lack an understanding of how much information is contained in these Earth representations, and where that information is con- centrated. The intrinsic dimension of dataset measures the number of degrees of freedom required capture its local variability, regardless of the ambient nigh-dimensional space in which it is embedded. This work provides the first study of the intrinsic dimensional- ity of geographic INRs. Analyzing INRs with ambient dimension between 256 and 5 11, we find that their intrinsic dimensions fall roughly between and 10 and are sensitive to changing spatial resolution and input modalities during INR pre-training. Furthermore, we show that the intrinsic dimension of geographic INR correlates with downstream task performance and can capture spatial artifacts, facilitating model evaluation and diag- nostics. More broadly, our work offers architecture-agnose label-free metric of infor- mation content that can enable unsupervised evaluation, model selection, and pre-training design across INRs.
1 INTRODUCTION
Across vision, audio, and other modalities, seemingly high-dimensional observations often vary along far fewer degrees of freedom. This especially true of geographic data, which is often characterized by strong spatio-temporal dependencies. For example, classical work in metcorology uses dimensionality reduction techniques since large-scale oscillations in climate trends can be explained by a handful of indices (van den Dool, 2006). This phenomenon leveraged by class of representation learning techniques aimed em- bedding signals in Earth's 's data into succinet, general purpose vector representations (Rolf et al., 2025). This is done either through direct embedding of geo-referenced data with image or text encoders, or through a new class of geographic implicit neural representations (INRs) that encode geospatial signals in the weights of location encoder network which takes geographic position (latitude and longitude) as input.
"Corresponding authors. Please email wwwarrjnncoooorerrro.....o. edu marc. russwurm@uni www.eeuuui-bonccccco

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Published as a conference paper ICLR 2026
Currently, the quality of Earth representation models is evaluated largely in terms of supervised model performance for specific downstream tasks. Pre-trained geographic INRs have driven state-of-the-art per- formance in tasks like land cover segmentation, object detection, and image geo-localization (Cepeda et al., 2023; Klemmer et al., 2025; Mai et al., 2023a; Liu et al., 2025). In addition, geographic INRs are increas- ingly used to improve geospatial data interpolation (Mac Aodha et al., 2019a; Chen al., 2025; Lange et al., 2025), for instance, in global species distribution modeling (Cole et al., 2023b; Dhakal et al., 2025). While sask-fpe metrics evaluate the "rrniinn-rendddnnesss" property of location encoder models (as outlined by Mai et al. (2022)), focusing only task-specific metrics prevents us from measuring progress fundamental aim of location embeddings: to generate rich, general purpose representations of Earth's 's data.
In this work, we study the intrinsic dimension (ID) of geographic INRs task-agnostic (unsupervised) metric to quantify information-richnes across space and where that capacity is concentrated. Defined in Section 2.2, the intrinsic dimension measures how many independent directions a learned representation actually uses. We compute point-wise, local estimates of ID to capture regional effects as well as global ID metrics for comparison between different location encoder models. Our results show that the intrinsic dimension reveals two important, and previously unexplored properties of geographic INRs: (i) represen- tativeness the amount of independent, non-redundant variation of the INRs and (ii) task-alignment indicating how well downstream predictors can compress the INRs onto a low-dimensional, target-aligned manifold. Our overall methodology summarized in Figure and our key findings can be summarized as follows:
Global ID estimates current geographic INRs are an order of magnitude lower than their ambient size, yet are competitive with ID estimates of Earth embeddings generated with large-scale image encoders.
ID estimates of geographic INRs increase with additional input modalities and increased spatial resolution of the location encoder, highlighting that ID captures increases in spectral and spatial representativeness of these embeddings.
Local ID estimates reveal spatial artifacts of pre-trained geographic INRs, which can arise from biases in the pre-training dataset coverage or properties of their model architectures.
Global ID) correlates with downstream task performance across encoders and tasks. Interestingly, correla- tion is positive when ID is calculated in the embedding space of frozen, pre-trained models, but is negative when ID is calculated in the activation space of supervised downstream models. This sheds light on the connections between the intrinsic dimension, representativennss,, and task-alignment of geographic INRs.
While much of our analysis extends naturally to general classes of INRs, geographic INRs are a particularly interesting case study because the underlying domain geometry is explicit. Inputs lie the sphere (S²), which makes it possible to separate the known 2D manifold from learned information content above it. Moreover, the explicit goal of many geographic Earth embedding models is compress and coalesce as much signal from Earth's is data as possible. The strategies we present for estimating different types of ID offer a new unsupervised evaluation strategy for representation learning of Earth data. Our code is available at https::////so.. / hut. com/ arrjuaaaaal99/eeooaia
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RELATED WORK
2.1 GEOGRAPHIC INRS AND LOCATION ENCODERS
Implicit neural representations (INRs) encode signals functions that map coordinates to values using neural networks (Chen & Zhang, 2019; Mildenhall et al., 2020). Analogously, geographic INRs encode ge- ographic signals the sphere using location encoder networks. These models take geographic coordinates (longitude, latitude) as input and return a corresponding value or vector embedding (Mai et al., 2022). Lo- cation encoders be trained directly using supervised loss to, for instance, interpolate between species
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Representativeness
Task-alignment
DDcluctiion Emberteings))
BX Arintiiee
P.dal
(1 (g.
Louation Encudar
Enccation Encechrr NN OM &
NN PEA
(Ag (uw)
Locstion Emnhsdding kx
Training Samples
Figure 1: Estimating the intrinsic dimension (ID) of geographic implicit neural representations (INRs). We compute the ID of geographic INRs in two ways, to measure model representativeness and task- alignment. Representativeness (left): We generate location embeddings with frozen pre-trained location encoders for coordinates across Earth's landmass. We calculate the global and local ID values the re- sulting embeddings. Task-alignment (right): We train a downstream task-specific model using location embeddings as input. We use a TwoNN ID estimator measure the ID of the activations of the tpec-ip model's 's last hidden layer.
observations (Cole et al., 2023b) or sea ice thickness measurements (Chen et al., 2025). Alternatielly, they be (pre)trained on unlabeled data, e.g., using contrastive objective, obtain embedding vector.
Location encoder architectures consist of a positional encoding (PE), projecting longitude/latitude inputs into a higher-dimensions feature space, and neural network projecting these features into the desired output space. Popular positional encodings include multi-scale sinusoidal functions (e.g. Vehere2Vec (Mai al., 2023b) and Space2Vec (Mai al., 2020)), multi-scale Random Fourier Features (RFFs) as used in GeoCLIP (Cepeda et al., 2023), and spherical harmonic functions that provide an orthogonal, sphere- native basis (Rußwurm et al., 2024). The "resolution" of the location embeddings is controlled by these positional encoding hyperparameters. The most common pre-training objective for location encoders is contrastive mocgellocaion matching. This way, encoder network and-as at inference time-can -can be accessed by providing solely coordinate inputs. Examples include SatCLIP (Klemmer et al., 2025) and GeoCLIP (Cepeda et al., 2023), which are available as pre- trained models and can be seamlessly integrated in other frameworks, for instance, in location-aware image synthesis (Sastry et al., 2024) or super-resolution (Panangian & Bittner, 2025).
image features are encoded in the location
An alternative way to obtain location embedding vectors is download data (c.g., satellite image) one location and use pre-trained vision model as an image encoder. Single-modality pre-trained ResNets and ViTs are available in repositories like TorchGeo (Stewart al., 2025) and SSL4EO (Wang al., 2023). Recent work on multi-modal remote sensing foundation models like DOFA (Xiong et al., 2024), CROMA (Fuller et al., 2023), MMEarth (Nedungadi et al., 2024) are other large-scale geospatial image encoders. In this work, we primarily study the intrinsic dimension of pre-trained location encoders as continuous geographic INRs, but also compare them quantitatively embeddings from image encoders.
2.2 INTRINSIC DIMENSION (ID)
The intrinsic dimension of dataset can be thought of nonlinear analogue of Principal Component Analysis (PCA). While PCA finds single global linear rank, captures the minimal number of degrees of freedom needed to describe data locally. ID is uniquely suited for measuring the true dimensionality of data representations since they occupy a curved manifold (Ansuini et al., 2019). Intrinsic dimension has been used in deep learning in several ways: to define normalized notion of task difficulty (Li al., 2018), explain generalization ability and sample efficiency (Pope et al., 2021; Ansuini et al., 2019; Gong et al., 2019), characterize adversarial examples (Ma al., 2018), detect Al-gererd content (Lorenz et al., 2023;
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Tulchinskii et al., 2023), or regularize local or joint input-feature dimensions improve self-supervised representations (Huang et al., 2024; Zhu et al., 2018). Learning theory treats intrinsic dimensionality as a key factor influencing learnability (Narayanan & Niyogi, 2009; Narayanan Mitter, 2010).
2.2.1 ESTIMATORS OF II
ID estimators quantify how the local neighborhood around an embedding = E D grows within small ball in feature space. The ID can be calculated with distance-bused or angle-based estimators. Distance-based methods assume that, in a sufficiently small neighborhood of 2, samples are drawn from an approximately uniform Poisson process to the & nearest neighbors of 2. The eevina-Bicke maximum likelihood estimator (MLE) (Levina Bickel, 2005) defines local ID estimate
d-dimensional manifold. Let R1 (2) .... I (xx2)) be the Euclidean distances
(x) Rx(x)
d2)
and aggregates these to global ID via a harmonic mean over embeddings 2, The TwoNN (Facco et al., 2017). MOM (Amsaleg et al., 2018), and TLE estimators are natural variants of this same Poisson-ball model. They replace the full set of log-ratios {In(Rx/ Rx/R,)) by alternative summaries of the k-NN configu- ration (for example. the first two radii or the mean radius R(2) relative to R&(=)), (z)), but still infer d from how quickly the number of neighbors grow with distance.
Angle-based estimators instead operate on directions after re-centering and whitening, FisherS (Albergante al., 2019) standardizes the embeddings } by mean-centering, retaining well-conditioned principal components, whitening them, and projecting each point onto the unit sphere to obtain z 2//l| At a cosine similarity margin a E (0, 1), it measures an empirical inseparability frequeney p(a). defined as the fraction of pairs (i.j) with (Ii, (g)) >> a. For points sampled uniformly on unit n-sphere, this frequency is well-approximated by
a² 1-22)(m-11/+
Paphere (oe; n) 23
av/2mn
and FisherS inverts this reference curve to recover an effective dimension
In(1 aa)
W p(a) a²(1 (2
A(a)
- Inn(1 a²)
where W() is the Lambert W function. Since angle-based estimators infer ID from the angular spread of neighbor directions by recentering the point's neighborhood and using only the unit directions to its neighbors, they are robust to local spatial variabilities.
To our knowledge, no prior work measures the intrinsic dimensionality of implicit neural representa- tions esspecillyy geographic INRs; our study is the first to provide these measurements and link ID to generalization and representativeness in geographical settings.
3 ESTIMATING THE ID OF GEOGRAPHIC INRS
We model a pre-trained location encoder as a map f S2 + RD that returns an embedding 2 = f(x) for a geographic location how many independent directions the embedding & varies when we perturb x slightly on Earth's 's surface. Equivalently, d(x: is the smallest number of coordinates needed to describe the support of the embedding
(A, Φ). The intrinsic dimension I, which denote d(x), summarizes
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distribution in a small neighborhood of z. The ambient dimension D of the embedding is fixed by the last layer of the location encoder architecture, whereas d(x:) I D reflects how much distinct geographic signal the encoder actually expresses at that location.
We estimate the geographic INR ID across two scales: The local ID d(x:) reveals where the representation is complex or compressive across Earth's 's surface, while the global ID aggregates d(x:) over a specified set of locations to provide single scalar value. We report both: local IID maps can diagnose spatial heterogeneity and the global ID allows us to compare between location encoders. High local ID values signal regions where embeddings vary along many independent directions whereas low values reveal compressive regimes where the representation is effectively one- or two-dimensional.
The choice of angle- versus distance-based ID estimators is also important in our analysis: Angle-based estimators' robustness to spatial heterogeneity make them a natural choice to estimate a global intrinsic di- mension over the Earth's 's surface. For global analyses, distance-based estimators can be disproportionately biased by local patterns as they read changes in spacing as changes in dimension. Within the context of heterogeneous representations of the Earth, these local patterns be induced by changing climate zones or terrain edge effects, for instance, in coastal areas. However, this sensitivity can be beneficial in lo- cally analyzing where these intrinsic dimensions change spatially. Thus, our measurements of local ID use distance-based estimators.
3.1 MEASURING REPRESENTATIVENESS AND TASK-ALIGNMENT WITH ID
As illustrated in Figure I, we estimate intrinsic dimension (ID) at two different stages:
1. Measuring representativeness in embedding space: With a frozen location encoder f and globally sampled N geographic coordinates o), we form embeddings Zgoo S(PF(A, Φ)) IR and estimate the global and local ID of these embeddings. When correlating global ID to downstream task performance, we use the angle-based FisherS estimator.
2. Measuring task-alignmen in activation space dlatase--conditionnel We then train shallow clas- sifier on task-specific locations and compute ID with the distence-based TwoNN estimator on the penul- timate ReLU activations evaluated only at the dataset's coordinates for each split. Concretely, for the vrain/validation/eess splits we pass their spatial coordinates through f. feed the resulting embeddings to the classifier, and estimate TwoNN ID jointly for the full dataset. This follows established practice of measuring the ID of neural network representations first introduced in Ansuini et al. (2019).
Unlike other interpretations of task alss-alignment that use statistical dependence metrics such as mutual infor- mation and estimate task-relevan dependence between embeddings and labels via variational bounds (Cheng et al., 2020; Hjelm et al., 2019; Liu et al., 2024), our notion of task-aligmment is explicitly geometric. We say that a representation is task-aligned when a shallow head can compress onto a low-dimensional manifold, which we probe directly through the intrinsic dimension of its activations.
3.2 DATASETS AND EXPERIMENTS
In our experiments, we correlate the estimated ID of different geographic INRs (detailed in section 2.1) with downstream task performance on several geospatial regression and classification tasks that use either (i) only location (X, Φ) context or (ii) location and additional context (e.g. an image). Location-based regression tasks include air temperature (Hooker et al., 2018), and elevation, population density, nightlights, and tree-cover prediction from Rolf et al. (2021). Location-basee classification tasks include biome and countries classification from Klemmer et al. (2025). We also measure the ID-performance relationship on several image-location regression tasks aimed measuring socioeconomic outcomes from the SustainBench (Yeh et al., 2021) benchmark. We use the task setup, including labels and precomputed Inception V3 image features provided through the TorchSpatial benchmark for location encoders (Wu et al., 2024).
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Our downstream classifcation/rregession heads trained on these tasks are either 3-layer MLP. For the SustainBench image-location regression tasks, we use one 2-layer MLP branch that receives image features imput, and 5-layer MLP which takes the geographic coordinates input. All fine-tuning experiments are trained for between 20-50 epochs with an early-stopping condition on a held-out validation loss. We use a grid search with the Optuna framework (Akiba al., 2019) to determine the optimal learning rate, best hidden dimension size of the MLP, and best values for weight decay. Mean performance metrics across 10 random seeds are reported.
When measuring the effeet additional input modalities have geographic INR ID and representativeness (Section 4.3), we use MMEarth, a multi-modal Earth observation corpus with Sentinel-2 optical, Sentinel-1 SAR, ASTER GDEM terrain,
canopy height, Dynamic World land cover, and ESA World- Cover data. Within SatCLIP we compare three variants: (i) an S2-only baseline with MoCo-pre-traanee ResNet-50 image encoder trained on MMEarth's Sentinel-2, (ii) SatCLIP location encoder pre-trained on Sentinel-1 and Sentinel-2 imagery, and (iii) SatCLIP pre-trained on all MMEarth pixel-level modalities. For each model we compute global FisherS ID on uniformly sampled land-only coordinates, then train a small 3-layer MLP on frozen embeddings to predict air temperature, elevation, and population-densit values.
4 RESULTS
4.1 GLOBAL AND LOCAL ID OF GEOGRAPHIC INRS
Global geographic INR ID is significantly lower than its ambient dimension but often greater than 2. Table 1 presents global ID esti- mates for geographic INRs from pre-trained loca- tion encoders and commonly used geospatial im- age encoders. We observe that the global IDs are substantially lower than the ambient embedding dimensions (D) for all geographic INRs. Sat- CLIP and CSP embeddings are 256-dimensional while GeoCLIP uses 512 dimensions yet all ID estimates for these models fall below 14. In Ap- pendix Table 2, we empirically verify the relia- bility ID over other dimensionality reduction techniques such as PCA and ICA. We observe that the number of retained components in both PCA and ICA sharply rise with increasing D, while ID remains relatively stable. This is veri- fied in Appendix Table 2b where we observe that increasing the D in SatCLIP does not accompany with an improvement in task performance.
Table I: Global intrinsic dimension of Earth embed- dings. Distance-based estimators use A: 20 nearest neighbors. Appendix Table shows results for more es- timators and sampling schemes.
Model
D FisherS MLE MOM
TLE
Location encoders, Land sampling (100k points)
SatCLIP L10
256
5,00
1.96
2.02
2.16
SatCLIP-L40
256
8.08
2.03
2.39
2.32
GeoCLIP
512
7.68
11.21
13.02
11.53
CSP-fMoW
256
1.70
5.18
5.23
6.25
CSP-iNat
256
0.92
3.37
4.64
4.14
SINR
256
3.19
2.19
3.36
2.74
TaxaBind-Loc
512
3.33
9.44
11.56
10.30
Image encoders S2-J00K (Klenmer et al.,
2025)
RCF
512
1.64
6.32
5.23
7.10
CROMA
768
19.57
17.00
20.30
DOFA
768
3.32
15.58
13.78
16.20
ResNet18
512
6.32
16.14
12.27
16.80
ResNet50
2048
6.42
16.27
13.18
17.00
The ID varies substantially by location encoder architecture. GeoCLIP shows Ifs of 11-13 across distance-based estimators, CSP variants range from 3-6, while SatCLIP remains lowest at 2-2.5. Estimator choice also matters: Angle-based Fish- erS produces notably different soieernns-iilddnnn 8.08 for SatCLIP-L40 (versus 2-2.4 for distance- based methods MLE, MOM, TLE) but drops be- low for CSP models.
ResNet152
2048
7.60
20.72
17.50
21.50
ViT-Small
384
3.33
18.53
15.80
19.20
AlphaFarth
64
4.67
7.80
9.48
8.61
TaxaBind-Sat (RGB)
512
3.39
10.04
14.98
12.80
ScaleMAE (RGB)
1024
2.96
10.16
8.90
11.00
ResNet18 (RGB)
512
0.92
10.85
8.70
11.70
ResNet50 (RGB)
2048
0.92
9.92
8.10
10.80
DINOv3-Sat (RGB)
4096
3.87
13.03
19.52
15.95
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GooLIP (Cepedaa al., 2123)
CSP-aNat (Mai al., 2025)
SeCLiP L-400 (Klemnee 2025)
20
2.5
Figure 2:
Local intrinsic dimension of geographic INRs reveal spatial artifacts. We the MLE
estimator on embeddings generated over Earth's 's landmass. = 100, 000 points sampled with & = 100 neighbors used in the MLE ID calculation. We plot the local ID of more INRs in Appendix Figure 11.
Global IDs of geographic INRs are similar that of embeddings derived from image encoders. The intrinsic dimension of geographic location encoders (purely with a (X, Φ) context) record similar intrinsic di- mension estimates compared large- -calee image encoders evaluated on S2- 100K Sentinel-2 tiles (Table 1). GeoCLIP's ID of 11-13 approaches that of foundation models like DOFA (ID: 14-16) (Xiong et al., 2024), CROMA (ID: 17-20) (Fuller et al., 2023), and is higher than the ID of AlphaEarth embeddings (4-9) (Brown et al., 2025), the pre-trained SINR model from Cole al. (2023a), and TaxaBind's multi-modal satellite im- age encoder (Sastry et al., 2025). This indicates that current pre-trained location encoders contain a similar amount of overall information content embeddings of multi-spectral satellite imagery obtained through specialized image encoders, as measured through these global ID estimators.
Local estimates of ID reveal spatial artifacts of pre-trained geographic location encoders. Figure 2 plots local ID estimates of pre-trained geographic INRs using the distance-based MLE estimator with k = 100 nearest neighbors. (Results are not very sensitive to the choice of k. which we further evaluate in Appendix B). For GeoCLIP, local ID is highest in the United States and western Europe, reflecting the spatial distribution of the social-media images on which it is pre-trained. The local IDs of CSP show a grid pattern because its positional encoding repeats at regular steps in longitude and latitude, so equally spaced locations look alike and form bands along meridians and parallels. For SatCLIP, local ID maps show no regional coverage bias (consistent with S2-100K's s global sampling). but they exhibit thin, periodic oscillations, reflecting the finite-order caherical-harmonee functions used in its location encoder.
4.2 GEOGRAPHIC INR ID CORRELATES WITH TASK PERFORMANCE
Geographic INRs with high global ID record higher task performance. In Figure 3a, we compute the global FisherS ID in embedding space to measure representativeness pictured in the left panel in Figure! 1. We then correlate these ID estimates with the performance of task-specific supervised learning models (small MLPs trained on top of embeddings from frozen geographic INRs). Across datasets and tasks, the scatter plots exhibit a clear positive linear trend: geographic INRs with larger global ID lead to higher downstream performance. higher global IDD implies a richer coverage of geographic variability iee., more task-relevant directions are available for shallow learner to exploit with limited supervision.
Lower global ID in activation space of supervised models corresponds to higher task performance. Still keeping the geographic INR frozen, we train a small task head and then measure the global TwoNN ID in activation space of this downstream task head (following the experimental settings in Ansuini et al. (2019)). This measures task-alignment as pictured in the right panel in Figure 1. In Figure 3b, observe a strong negative correlation between activation- spaconnssseeeceee ID and performance. indicating that supervised adaptation compresses the INR features onto task-aligned manifold with fewer effective degrees of freedom. This is consistent with past work, that found lower ID indicates more concentrated, linearly separable structure and thus better generalization (Ansuini et al., 2019; Pope et al., 2021; Zhu et al., 2018).
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Air Temperature
Elevation
Population
Biome
Countries
Test Accuracy
0.9
0.8
0.9
Test
0.6
0.5
0.6
0.8
0.8
0.4
10
0
2
10
0
2
10
6
10
10
FisherS Global ID (Representativennss)
(a) FisherS global ID of geographic INRs test R2 and top-1 accuracy using land-hased coordinate sampling.
Air Temperature
Elevation
Population
Biome
Countries
Test Accuracy
0.8
Test R2
0.9
0.6
0.9
0.9
0.7
0.4
0.8
0.8
0.8
0.5
0.7
TwoNN Global ID (Task-Allinmennt)
CSP-FMoW
CSP-iNat
GeoCLIP
SaICLIP-IL10
SatCI.IP-L40
SINR
TaxaBind-Loc
(b) TwoNN global ID of ReLU activations of 3-hidden-layer MLP's penultimate layer test R and top-1 accuracy.
Figure 3: Relationship between global ID of geographic INRs and downstream task performance mea- sured across five regression and classification tasks. In both rows, the location embeddings frozen while task-specific predictions heads (3 layer MLPs) are learned. In (a), ID (horizontal axis) is calculated on the frozen pre-trained embeddings in Table 1. In (b), ID measured in activation space using the TwoNN estimator on a learned classifier's penultimate layer.
Nightlights
Tree Cover
Asset Index
Sanitation Index
Women Education
0.35
0.75
0.6
0.6
0.65
0.3
0.7
0.05
0.6
Test
0.5
0.25
0.65
0.5
0.55
0.45
0.4
10
16
5
10
2.6
2.8
3
3.2
3.4
1.5
2.5
3.5
2.65
3
3.5
FisherS Global ID
Sphere2Vec-DFS
Sphere2Vec-M
Sphere2Vec-C
NeRF
Figure 4: FisherS global ID of task-specific location embeddings learned via supervised learning vs test R² of four continuous location encoders on five tasks from TorchSpatial (Wu et al., 2024) FisherS ID is calculated on the intermediate location embeddings from the location encoder, similar to Figure 3a. The asset index, sanitation index, and women education tasks are image-location regression tasks, as detailed Section 3.2.
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S2V-M S2V-M+ SPACE2V
SatCLIP
GeoCLIP
S2V-C
S2V-C+
Global FisherS ID
80
75.7
80
60
7.5
80
40
6
500
20
72
5.0
10
20
30
40
22
242
220
10
15
25
30
Legendre Polynomials
Max RFF Freq
Hicrarchy Levels (M)
Frequency/Seale Components (S)
Figure 5: Effect of location encoder spatial resolution global ID. (Left) for SatCLIP, we pre-train the location encoder with L = 10, 20, and 40 Legendre Polynomials. (Middle) For GeoCLIP, we increase both the maximum RFF frequency and the number of hierarchical levels (M) used by the location encoder by fine- tuning the new higher-frequency branches on a YFCC (Thomee et al., 2016) image geo-localization task. (Right) For Sphere2 (S2V) and Space2 (SPACE2V) encoders, we increase the number of frequency components (S) and train the location encoder with supervised learning on the MOSAIKS nightlights re- gression task.
Within a supervised learning setting, lower ID of task-specific location encoders corresponds to higher task performance. To further investigate task-alignment, we train continuous location encoders Sphere2 Vec and Vec end-to-end with supervised learning tion 3.2. We then compute the global FisherS ID on the task-specific learned embeddings, across training sample locations. In Figure 4, interestingly, we again find consistent negative relationship between ID and R², suggesting that direct supervision drives the representations toward a lower-ID, task-specific manifold that is easier separate regress on. This mirrors Figure 3b and reinforces the picture that the benefits of self-supervised pre-training stem from high global ID (expressivity/coverage)) while subsequent supervised models benefit from low global IDD coompressionnas--aliganeen
the image-location regression tasks outlined in Sec-
4.3 EFFECT OF RESOLUTION AND INPUT MODALITIES ON THE ID OF GEOGRAPHIC INRS
Having established that ID measures can capture the relative information content of geographic INRs across space and relate to downstream task-specific performance, we examine how ID values change when the properties of geographic location encoders change. We focus on two properties along which geographic location encoders routinely modified that should result in different information content in their em- beddings: the spatial resolution of the location encoder architecture, and the input modalities used during pre-training.
Increased spatial resolution of geographic INRs increases global ID and representativeness. The spatial resolution of several location encoder architectures is controlled by specific model hyperparameters: spher- ical harmonics by the number of Legendre polynomials used L. Random Fourier Features (RFF) by their maximum frequency Fmax and hierarchy depth M,, and Space2Vec/Sphere2W scale components S. In Figure 5, we plot how ID changes as we vary these hyperparameters. Intuitively, increasing resolution should allow the encoder to resolve finer geospatial phenomena and thus use more inde- pendent directions in representation space. For SatCLIP (equipped with spherical harmonic and sinusoidal representation network positional encoder), the global FisherS ID rises with L. Similarly, for GeoCLIP. increasing o max by appending high frequency branches to the original pre-trained set and increasing the hi- erarchy density M both increase global ID. Increasing max produces a sharp increase in global ID, whereas increasing M produces more gradual increases. Across Space2 variants, we vary the parameter, the rise in ID is steepest for the theory/grid formulation of Space2 moderate for the compositional variants, and smallest for the multiplicative variants. This supports the view that adding multi-scall components
via the number of multi-
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larges the set of independent directions the encoder can use, thereby increasing the representational capacity of the location embedding.
Additional pre-training data modalities increases global ID and representativennss. In Figure 6, we train SatCLIP models with different subsets of pre- pre-training data modalities from the MMEarth dataset. We find that increasing the number of geographic layers seen during training increases both the global ID and its downstream task performance. SatCLIP location encoders with the most number of input modalities (optical Sentinel-1, multispectral Sentinel-2 and all pixel-level modalities on MMFarth) record both the highest global ID and performance across air temperature, elevation, and population density regression task. This confirms that ID can capture differences in information content due to increasing the number modalities beyond multi-spectral Sentinel-2 imagery. The clear gains to downstream performance in Figure 6 echoes our results linking increased representativeness to increased downstream task performance in Figure 3.
5 DISCUSSION AND CONCLUSION
Intrinsic dimension (ID) offers an architecture- and task- agnostic metric measure the information content en- coded in geographic INRs. Despite being a lask-agmostic metric, global ID still informs the (Mai et al., 2022) property of a location encoder. When measured on frozen embeddings, higher global ID aligns with stronger downstream performance after fine-tuning, indicating greater representativeness. When measured on the activations of a supervised prediction head, lower global ID accompanies better generalization, consistent with task-aligned compression. Local ID maps expose spa- tial artifacts that could influence the performance of ge- ographic INRs on downstream applications. Thus, both global and local ID analysis can support model selection via label-free model evaluation at the pre-training stage.
Air Temperature
Elevation
Population
0.86
S1-S2
Mean Test R2
0.84
S1-32
0.82
0.8
All
$1-S2
0.78
0.76
0.74
7.5
8.5 FisherS Global ID
9.5
Figure 6: Using additional input modalities during pre-training increases ID and down- stream task performance of geographic INRs. Colors represent different tasks. Mod- els pre-trained on subsets of the MMEarth dataset: Sentinel-2 (S2), Sentinel-1 and 2 (S1+S2), and all available rasters (All).
Beyond this model selection rolc, our results suggest sev- eral concrete applications of ID for designing, auditing, and deploying geospatial machine learning models. First, global ID of frozen location encoders can act a label-free proxy for downstream performance when comparing archi- tectures, positional encodings, resolution hyperparameters. or input modality sets, reducing the cost of exploring new INR designs. Second, monitoring the divergence between encoder and head ID offers a signal for early stopping that complements standard validation met- rics. Third, spatially resolved local ID maps allow practitioners to audit geographic biases, guiding targeted data collection and region- -ware fine-tuning.
While these applications already make ID practically useful, we view it as one component of a broader eval- uation toolkit for geographic representation learning. Beyond the supervised regression and classification benchmarks we study here, ID-based diagnostics could be extended to geo-prior settings such as location- conditioned satellite image generation and fine-grained classification with geospatial priors (Feng et al., 2025; Mac Aodha et al., 2019b). Complementary measures could quantify data provenance by attributing information content to specific pre- traininiing corpora or regions, and could improve interpretability by local- izing that content to subspaces of the embedding. Future work could also develop more principled ways use patterns in local ID to drive task-specific fine-tuning or data acquisition strategies that adapt to regional variation in information content.
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ACKNOWLEDGEMENTS
A majority of training runs conducted in this work node provided by the University of Colorado Boulder's high performance computing system Alpine. We thank Brandon Reyes and the RC computing team at CU Boulder for allowing priority access to this re- source. Alpine is jointly funded by the University of Colorado Boulder, the University of Colorado An- schutz, Colorado State University, and the National Science Foundation (award 2201538). We thank Ben Aoki Sherwood and Nico Lang for providing valuable feedback during the writing stages of this work.
NVIDIA Grace-Hopper (GH200) GPU
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Table Intrinsic dimension compared to PCA and ICA as a function embedding size. (a) Global intrinsic dimension (ID) estimates for image encoder (RCF) and geospatial location encoder (SatCLIP, 10) are stable as the ambient embedding size D increases. (b) For SatCLIP, PCA and ICA retain many components D grows and closely track the ambient dimensionality, while ID and downstream task performance change stay relatively constant.
(a) ID stability across embedding sizes.
Encoder
MLE FisherS
MOM
TLE
5,56
1.68
5.58
7.09
128
5.57
1.68
5.48
7.11
RCF
256
5.64
1.68
5.56
7.12
512
6.32
1.64
5.23
7.10
64
1.958
4.91
2.12
2.15
128
1.958
5.00
2.10
2.13
SatCLIP
1.967
5.00
2.16
512
1.956
5.00
2.10
2.16
(b) PCA/ICA components and downstream performance.
D
ID (MLE)
PCA
ICA
Air temp R²
Elev. R²
Pop. R²
Countries Top-1
Biome
99% var.
comps.
Top-1
64
1.958
36
64
95.9
+ 0.18
81.5
2.13
78.4 + 0.51
93.8
0.16
92.0 + 0.47
128
1.958
46
127
95.8 1 0.28
81.8
0.45
78.1 + 0.76
94.1
= 0.17
91.6 1 0.35
256
1.967
84
256
95.1
0.15
80.8 = 1.88
79.0
1.03
93.9 : 0.16
92.23 + 0.25
512
1.956
59
512
95.9
1 0.27
81.2 = 1.52
78.4 + 0.44
94.4
0.00
91.8 + 0.29
A ESTIMATORS OF INTRINSIC DIMENSION
A.1 DISTANCE- BASED ESTIMATORS
MLE (Levina & Bickel, 2005) Let & E IRD an embedding (feature vector) and denote by radii R (2) < (2) the Euclidean distances its i nearest neighbors. A broad family of intrinsic- dimension estimators rests on the assumption that, in sufficiently small neighborhood of z, samples are drawn from approximately uniform Poisson process write down a closed-fore density for the joint distribution of the radii, which depends on d only through the volume of unit d-ball. The Levina Levina-Bickel MLE (Levina & Bickel, 2005) estimator leverages this fact define local estimator
d-dimensional manifold. Under this hypothesis
Ra (z)
(=)
whose global counterpart pools the local estimates using its harmonic mean across embeddings 2;.
1
dgtobal
i=1 dMLE(Zi)
k
R (2:)
k
Ry (z+)
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MOM (Amsaleg al., 2018) Under the same Poisson-ball model, conditioning on the outer radius (x) makes the inner neighbors approximately uniform inside the d-ball of radius (z).). In higher dimensions more mass concentrates near the boundary, so the average inner radius moves closer (2(2)) Let
R(=) = k
R2(2).
The method of moments estimator equates the empirical ratio R(z)/Rx(z) to its model value and solves for d, which yiclds the local estimator
R(2) Ra(2) - R(2)
d=)M(=)
Large (z) therefore corresponds to neighborhoods where most points lie close to the outer shell.
TLE (Amsaleg et al., 2019). Tight Local Estimation keeps the Levina-Bickel likelihood identity but ex- tracts logarithmic spacing information from the full k-NN configuration, not only the ordered radii. Inside the ball of radius (2) it constructs pairwise geometric surrogates Sty and Ti,(2) E (0, R& (2)] that summarize how neighbors are positioned relative to one another. Degenerate near zero terms are dis- carded by a fixed threshold to stabilize the estimate. Schematically, the MLE log-ratios (In( Rg. RRy) are replaced by (In( (1n(R/55 R5y)) In( Rg/Tas) and combined through the same harmonic-mean aggregation pro- duce (z). The result is less sensitive to very small spacings while remaining driven by how tightly the neighborhood fills the d-ball.
ESS (Johnsson et al., 2014). Expected Simplex Skewness uses the shape of the neighbor constellation rather than only radial spacings. After mean-centering the k-NN patch 2, one forms vectors to neighbors and builds many small simplices from these vectors. A normalized "skewness" statistic s(z) combines their volumes and edge lengths. For uniform set of finite-neighbor samples around in d-ball this statistic concentrates around dimension-specific reference curve. The local estimate dess (z) obtained by inverting the observed s=)) against that curve, using linear interpolation between consccutive integer dimensions. Because ESS is driven mainly by relative geometry and angles, it tends to be less sensitive to mild density gradients while still reflecting curvature and noise through systematic changes in the neighbor shape.
A.2 ANGLE-BASED ESTIMATORS
We provide detailed explanation of the angle-based FisherS estimator below (Albergante al., 2019).
Let ] C I denote embeddings. The FisherS estimator starts from concentration-ol measure assumption: in high dimensions, points sampled uniformly on sphere tend be almost orthogonal, so a typical point can be linearly separated from the rest by Fisher discriminant FisherS quantifies how separable each point is from all others at similarity margin a E (0, 1), then inverts a closed-form reference curve recover effective dimension.
-
Following Albergante et al. (2019), inputs are standardized to make the separability model comparable with a uniform n-sphere: (i) center Z by subtracting the sample mean; (ii) reduce dimension using PCA, retaining components with eigenvalues at least Amax/C (default C' 10), which removes ill-conditioned directions; (iii) whiten the retained coordinates the covariance is the identity; and (iv) project each vector to the unit sphere, I; = /:/ =>| These steps yield an array X E R x with rows Zi--
Fisher's lincar discriminant is the single direction (a line) that best separates two groups by making their projected means far apart relative to their spread.
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For each on grid, we form the normalized Gram matrix G XXx and test, for every pair (i,j) with i * J. whether
13) > a (1i,
Since ||=:||| 1, this reduces (cq, > a, that is, whether the cosine similarity exceeds a. Point is called inseparable at level if it has at least one neighbor whose similarity crosses this threshold. Empirically we record, for each i, the fraction
1 # ti: (z)) > aa} p.(a)
and then average over points to obtain p(a) = E Pi(a).
If points sampled uniformly on the unit n-sphere, the mean inseparability probability at margin is well-approximated by
(1 (1- 22)(n-1)/2
Dophere (or; n)
av2Tn
Given the measured p(a), FisherS solves for by inverting this expression with the Lambert W function:
In(1 a a²)
W
2T p(a)² (11/)22(1100 a²) - a2)
n(a))
Varying a yields a profile a F> M(ae): FisherS then picks single global estimate by selecting the largest a for which p(a) > 0 (i.e., not all points are fully separable) and reporting ñ(a*) for a x 0.9 max p(a) > 0}. The method also supports pointwise ID by plugging (a) into the same inversion at a chosen
FisherS does not rely on nearest-neighbor radii. It asks instead: "At a given angular margin a, how often do points fail to be linearly separable from the rest?" On true n-sphere, this failure rate decays at rate governed by n. the dataset is more clustered or locally low-dimensional, inseparability persists to larger a, inflating p(o:) and thus lowering the inferred n; if the dataset spreads more uniformly, p(a) drops and the inferred n rises. The PCA+whitening+spher projection makes this comparison meaningful by matching the spherical reference.
How DO THE NUMBER OF NEAREST NEIGHBORS CHANGE GEOGRAPHIC INR ID?
In distance-based estimators, the neighbor count k controls bias-variance tradeoff. With small k, the estimate is anchored to an extremely local neighborhood of so it is sensitive to the true local geometry and density but suffers high variance because averages (harmonic mean (Levina & Bickel, 2005) and arithmetic mean in (Amsaleg et al., 2019; 2018)) over few nearest neighbors. As k increases, variance decreases by pooling more radii. The effective neighborhood increases and begins to mix in curvature, density gradients, and boundary effects. These departures from local homogeneity induce bias whose sign and magnitude depend on the estimator. For the Levina-Bickel MLE estimator specifically, which is derived under a local homogeneous Poisson model, the estimator is approximately unbiased when the model holds and is small to moderate, and its variance scales like == dd/ /(k - 3) for fixed intrinsic dimension d (Levina & Bickel, 2005): pushing & larger suppresses variance further but also violates the locality assumptions, allowing global structure to pull the estimate upward or downward.
Across estimators, increasing k changes the balance between local detail and global structure, and results reflect this clearly. ESS grows rapidly with k most strikingly for GeoCLIP: under Fibonacci sampling it rises from about 12 at k= nearly 99 at 2-200. This pattern is consistent with ESS aggregating more
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Table 3: Global intrinsic dimensions of geographic INRs derived from SatCLIP (Klemmer et al., 2025), GeoCLIP (Cepeda et al., 2023), and CSP (Mai et al., 2023a). For all distance-based estimators, k = 20 nearest neighbors. Results shown for sampling schemes, estimated by six total methods. Variation containing N 50, 000 points each.
shows 1x standard deviation over 3 sub-sampling
Scheme
Model
Corrint
FisherS
MLE
MOM
TLE
TwoNN
SatCLIP-LIO
2.16 1.66:.0. 0.02
5.94 + 0.00
2.45 = 0.05
2.15 = 0.00
2.43 + 0.02
44.14 + 0.80
SaICLIP-L40
1.36:0000
9.95 0.03
2.58
0.05
- 0.10
0.03
0.03 1:
12.80 1 0.30
Fibonacci
GeoCLIP
22.70 0.05
0.10 11.16
13.17=0.10
23.70
= 0.30
13.20
24.15 + 0.08
CSP-FMoW
2.50
1
6.28
6.28
5.07 0.08
0.12
9.43 1 0.20
CSP-iNat
5.65
0.92
4.96
6.34 0.10
5.90
5.07 + 0.15
SaCCLI--LI
2.01 0.02
5.94 0.03
2.01 0.02
2.12 0.02
2.00 2.22 0.06
SaICLIP-L40
2.18 0.02
9.89 0.03
2.04
4.51 0.10
2.02 2.52 0.06
Sphere
GeoCLIP
20.79
11.10 0.04
11.50 +
22.48 + 0.03
12.08
15.07
0.08
CSP-FMoW
2.00
1.87 0.01
3.06
4.96 + 0.06
6.33
4.53 0.15
CSP-iNat
5.58
0.92 0.01
4.27 0.05
6.15 0.08
5.31
3.19 0.10
SatCLIP-LIO
0.00
5.00
1.96
2.02 + 0.00
2.16
1.98 + 0.01
SaCCIIP-144
2.05 0.00
+ 0.00
2.03
2.39 + 0.00
2.32 0.00
1.99 0.01
Land
GeoCLIP
12.07 0.05
11.22 0.10
13,02 = 0.20
11.53
11.25
: 0.50
CSP-FMoW
2.16 0.03
5.18 0.00
5.23 + 0.08
4.10
3.05 0.12
CSP-iNat
3.93 0.05
0.92
3.37 = 0.06
4.65 = 0.08
1.4
2.70
SaICLIP-LIP
4.4
2.00 0.00 0.00
2.15 = 0.00
2.20
2.61
SarCI.IP-140
+
8.76
0.00
3.40 = 0.00
2.56
2.58 0.03
Grid
GeoCLIP
18,87 0.08
+.08
13.00
22.22 = 0.00
1.05
1
CSP-FMoW
2.65 0.03
1.81
5.69 + 0.08
0.08
5.25 + 0.08
6.56
+
CSP-iNat
5.57
0.92
6.32 4.64 0.08
5.63
3.68
SarCLIP-LI0
1.76
4
2.01 0.02
2.12
2.22
2.01 1
SaCCIIPP-140
1.95 + 0.02
8.93 0.03
2.15 + 0.03
4.53 + 0.10
2.54+
2.02 + 0.08
Naive
GeoCLIP
18,02 + 0.08
0.08
7.09 0.00
12.08 = 0.00
21.53
0.03
12.76
13.63 0.07
CSP-FMoW
2.68 0.03
1.81 0.01
5.60 + 0.08
0.08
5.26 0.08
6.13
4.86 4.85
CSP-iNat
5
0.92
4.41 = 0.08
6.24
5.15
3.10
SarCLIP-LIC
1.75
4.47 0.00
2.01
2:00
2,00 2.22
SarCIIPP-140
1.92
8.91 0.01
2.15
2.49
0.54
2.01 +
Stratified
GeoCLIP
1.093
7.07 0.01
12.06
0.10
21.53
12.75
1.00
CSP-FMoW
2.67 0.03
1.81 0.01
5.59
+ 0.08
5.25 0.08
0.08
6.53
4.79 0.15
CSP-iNat
5.51
19220.01
4.42 = 0.08
6.24
5.45
0.10
directions and longer chords as the neighborhood widens, which inflates the measured simplex skewness and maps higher effective dimension. MOM and TLE typically drop from k=5 the k==20-50 range, then flatten or rebound slightly. The initial decrease suggests that adding a few more neighbors stabilizes local spacing statisties and trims extreme ratios, while very large neighborhoods begin to mix curvature and density variation, reintroducing upward bias. MLE is comparatively stable. For GeoCLIP it is often flat to gently increasing with & and occasionally U-shaped on land sampling. For SatCLIP encoders distance- based estimators remarkably steady, hovering near 2-3 across k which indicates tightly regular local geometry in those embeddings. Sampling also matters: fibonacci and spherical layouts amplify sensitivity for GeoCLIP and the CSP encoders, whereas land sampling tends to mute it. Overall, the table shows that methods that directly emphasize extreme interpoint ratios or long-range chords move the most with k. while estimators that average log-radii more evenly are less affected.
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Table 4: Global geographic INR ID versus number of nearest neighbors (k) for distance- based estima- tors on scikit-dimension. 100,000 points sampled with land, fibonacci, and spherical sampling.
(a) MLE (Levina & Bickel. 2005)
(b) MOM (Amsaleg et al., 2018)
k
Sampling Encoder
10
50
100
200
Sampling
Encoder
10
20
100
200
GeoCLIP
12.781
11.234
10.327
11.935
15,090
GeoCLIP
25.754
17,070
12.988
11.282
13.015
16.269
CSP-FMoW
4,050
4,770
5.169
5.109
4,736
4.245
CSP-FMoW
8.823
7.037
6.440
5.802
5.217
4.567
land
CSP-INat
3.151
3.375
3,930
CSP-iNat
5.752
4,250
4.396
4,886
SuCLIP-LIO
1,982
1,989
1,960
1.952
1,951
1.963
SaICLIP-L10
3.389
2.511
2.211
2.058
2.017
2.017
SACLIP-LL4
2,004
2,024
2.225
SuCLIIP-AA
3,407
2.565
2.296
2.248
2.804
GeoCLIP
17,070
13.041
13.173
17.795
22.188
25,560
GeoCLIP
26,250
15.685
14.256
19. 90.
23.700
27.114
CSP FMW
8.027
7.087
6,280
5.352
4,749
4.231
CSP-PMoW
13.145
8.991
7.191
5.808
5.069
4.476
libonscii
CSP-INat
4.787
4.792
4,964
5.269
5.457
5,588
Ebonacci
CSP-iNat
8.372
6.484
6,100
6.196
6,340
6.111
SaCLIP-LIO
4.490
2,610
2.445
2.207
2,161
2.170
SaCCLIP-L10
7,298
2.799
2.496
2.191
2.151
2.178
SaCLIP-L10
4.747
2.765
2.576
2.847
3,852
5.497
SuCLLI--L40
7.389
2.968
2.649
3.134
4.683
6.989
GeoCLIP
12,301
11,047
11.494
15.979
20.590
24.393
GeoCLIP
20.905
14.051
13.451
17.942
22.484
26.187
CSP-PMoW
5.158
5.483
5.432
5.016
4.591
4.151
CSP-FMoW
10.176
7.579
6.490
5.553
4.950
4.423
sphere
CSP-iNar
3,480
3,848
4.272
4.851
5.208
5,443
sphere
CSP-iNat
6.980
5.651
5.541
5.877
6.147
6.306
SaCLIP-LIO
2,007
2,009
2.013
2.068
SaCLII--LI
3.411
2.551
2.262
2.133
2.161
SaCLIP-L/0
2,013
2,072
2,140
2.553
3.529
5.135
SaCLII--L00
3.508
2.658
2.453
3.006
4.522
6,802
(c) TLE (Amsaleg et al., 2019)
(d) ESS (Johnsson et al., 2014)
Sampling Encuder
10
Sampling Encoder
20
GeoULIP
21,665
14,764
11.547
10.218
11,669
14.278
GeoCLIP
12,043
25 661
29,229
30.517
37.955
$0.230
CSP-FMoW
7,851
6.630
6.236
5.720
5,197
4.602
CSP-FMoW
5.621
7.107
7,523
7.239
6.570
5.718
land
CSP-iNat
5,070
4,259
4.149
4.326
land
CSP-iNat
3,800
4.781
5.418
SaCLIP-LIO
2.786
2.319
2.155
2.085
2.084
2.117
SaCCLIP-L11
2,148
2.097
2.072
2.097
2.166
2.290
SuCLIP-L40
2,892
2.446
2.319
2.355
2.523
2.923
SMCCLI--LA
2.222
2.249
2.356
2.632
2.970
3.682
GeoCLIP
23.199
14.034
13.196
16,840
20.173
22.568
GeoCLIP
12.031
25.870
39.545
63.910
$4,150
98.897
CSP-FMoW
12.282
8.618
7,023
5.777
5,100
4.546
CSP-FMoW
7,524
9.148
8,400
7,033
6.123
5,344
Nhooaccl
CSP-INar
7.756
6,209
5.898
5.962
6,049
6.057
Bhonacel
CSP-INat
5.158
5.878
6.179
6.590
6.848
7,008
SaCLIP-LIO
3,937
2.498
2.427
2.279
SaCLIP-LIO
2.726
2.326
2.293
2.326
2.622
SaCC 1P-1 40
5.858
2,970
2,784
3.305
4,779
6,884
SaCLIPP-AA0
2,852
2.753
2,868
4.075
6.557
10:122
GeoCLIP
18.149
12,514
12,064
15.750
19.257
21,906
GcoCLIP
11,801
24.231
34.033
$4.676
74,801
91.335
CSP-
7.206
6.330
4.985
CSP-EMoW
6.269
7,630
7.492
6.698
5.292
sphene
CSP-INat
6.180
5,301
5.305
5.637
5,856
5.952
spborc
CSP-iNal
4,438
5.116
5.596
6.236
6.627
6.881
SuCT IP--10
2.827
2.372
2.234
2.225
2.306
SACCLIP-L10
2.168
2.145
2.250
2.601
SaCT 1P:140
3.039
2.586
2.519
3.106
4.532
6.607
SaCCLIP-140
2.320
2.443
2.686
3.809
6.064
9.429
C EXPERIMENT DETAILS: INTRINSIC DIMENSION OF GEOSPATIAL IMAGE ENCODERS
For all image encoders displayed on Table 1, we compute intrinsic dimension (ID) on features extracted from the 10-1 Sentinel-2 multispectral image dataset (Klemmer et al., 2025). Models that accept multispectral inputs use all 13 Sentinel-2 bands; models that only support RGB receive bands B4/B3/B2 in that order. All inputs are resized/cropped to 224 x 224 pixels unless noted, and RGB models use standard ImageNet normalization. We estimate global ID from the resulting feature matrices using multiple estimators as in Table 1 (for distance-based methods, k- = 20 nearest neighbors). Below, list key experimental details for each image encoder.
Random Convolutional Filters (RCF) (Rolf et al., 2021). We use training- free RCFs as a simple base- line: 13-channel inputs convolved with random filter bank, passed through a nonlinearity, and spatially
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pooled into fixed-length descriptor. No external weights are loaded. The feature dimension is set 512 for accurate comparison with larger image encoders.
CROMA (Fuller et al., 2023). We evaluate the optical branch of CROMA's multimodal encoder (pre- trained Sentinel-1/2 pairs). Inputs are the 12 optical Sentinel-2 bands (with an excluded B10 cirrus band), resized to 120 x 120 pixels, and normalized to [0, 1] following official pre-trained model usage guidelines. Intrinsic dimension is calculated on the pooled optical embedding produced by the pre-trained encoder weights.
DOFA (Xiong et al., 2024). DOFA is a masked-autoencoodr foundation model that conditions its patch embedding channel wavelengths, enabling a single model to handle arbitrary band sets. We pass the 13 Sentinel-2 central wavelengths with the 13-band inputs and extract encoder features from the publicly released pre-trained weights.
ScaleMAE (Reed et al., 2023). ScaleMAB is a scale-aware masked autoencoder trained on RGB im- agery with boound-sapling-ddsttaeed cues and multiscale reconstruction objective. We feed RGB tiles (B4/B3/B2), apply ImageNet normalization, and use the pre-trained encoder's feature output.
ResNet baselines (He al., 2016). For multispectral ResNets, we use 13-band Sentinel-2 MoCo weights (ResNet-188)) 18/50) and extract the global-averaee-pooeed penultimate features. Pre-trained Sentinel-2 weights are loaded with TorchGeo package (Stewart et al., 2025). For RGB variants, we use the corresponding Sentinel-2 RGB MoCo weights with B4/B3/B2 inputs. For ResNet-152, we rely on ImageNet- K pre- trained weights (RGB only) and take the pooled backbone features.
ViT-Small (multispectral) (Dosovitskiy et al., 2021). We use the Sentinel-2 MoCo-pre-trained ViT-Small that accepts 13 bands. We run a forward pass to obtain patch tokens and mean-pool them (excluding any prefix/CLS token) to form a single feature vector per tile.
AlphaEarth (Brown et al., 2025). We extract embed- dings from the AlphaEarth embedding API, which pro- vides dense, pre-computed 64-dimensional feature vectors derived from Sentinel-2 satellite imagery at 10-meter spa- tial resolution. These embeddings are accessed through Google Earth Engine's image collection (GOOGLE/SATELL LITE
Table 5: Global ID estimates of AlphaEarth embeddings versus embedding sampling buffer size (meters).
Buffer (m) FisherS
MLE
MOM
TLB
with anmual composites
20
4.67
7.80
9.48
8.61
available from 2017 to 2024. Each embedding dimension rep- resents learned features from AlphaEarth's foundation model
100
4.43
7.23
8.78
8.04
200
4.40
6.96
8.45
7.77
trained on global Earth observation data. The results in Table use the single- sigel embedding (no averaging over space) and downloads embeddings for 2024. The embeddings are queried using geographic coordinates (longitude, latitude).
500
4.37
6.54
7.97
7.36
To investigate how spatial context affects the intrinsic dimensionality of AlphaEarth embeddings, we employ a systematic buffer-based aggregation strategy. For each coordinate point, we extract AlphaEarth embed- dings using circular buffers varying from 10 meters around the geographic coordinates to 500 meters (1,256 pixels approximately aggregated). Within each buffer zone, embeddings are aggregated using mean reduc- tion. From Table 5, we interestingly observe that increasing buffer sizes mildly decrease the global ID of AlphaEarth embeddings across all four estimators.
SINR (Cole et al., 2023a). The SINR encoder from Cole et al. (2023a) was originally developed for global- scale species distribution modeling from presence-only observation data. The SINR model uses sinusoidal encodings of geographic coordinates as input and learns implicit neural representations to predict species presence across 47k species. We extracted 256-dimensional feature vectors from approximately 100,000
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SatCLIP Europe
SaIC
Africa
Sar CLIP USA
50
Local Intrinsic Dimension
Local Intrinsic Dimension
Local Intrinsic Dimension
Figure 7: Local intrinsic dimension can identify pre-training dataset coverage of local models. We visualize local ID maps with the MLE estimator using 100 nearest neighbors. 100,000 geographic coordinates are sampled with the Fibonacci sampling scheme.
globally distributed locations in the S2-100K dataset from the pre-trained SINR model trained on 100 ob- servations/species.
BaxaBind (Sastry et al., 2025) We evaluate both the satellite image and location encoders from the Tax- aBind model. The satellite encoder is CLIP ViT-B/16 model. To process the 13-band Sentinel-2 inputs, we first select the RGB bands (B4, B3, B2). These 3-channel images are then resized to 224 x 224 and normal- ized using standard ImageNet statistics. The TaxaBind location encoder uses an identical model architecture as GeoCLIP (Cepeda et al., 2023) consisting of an Equal Earth projection, an RFF transform fed to an MLP.
DINO-SAT (Siméoni et al., 2025). We evaluate the DINOv3 (ViT-7B/16) model, which was pre-trained on the SAT-493M dataset of high-resolution Maxar imagery. To apply this model to our 13-band Sentinel- 2 dataset, we first select the visual-spectrum bands (B4, B3, and B2) create 3-channel RGB image. This RGB image is then passed to the model's official Hugging Face AutolmageProcessor which handles the resizing (to 224x224) and normalization that the model expects. We use the output CLS token as the final feature vector. Although the model was pre-trained
different sensor (Maxar) at different spatial resolution, this application a valid test of the model's generality. We are assessing the model's 's capability powerful, ceneral-purpon visual feature extractor new, retaining a fair comparison to other image encoders in Table 1.
satellite domain while
D EXPERIMENT DETAILS:
EFFECT OF SPATIAL RESOLUTION AND SPATIAL COVERAGE ON GEOGRAPHIC INR ID
We detail our experimental framework that measures changes in the geographic INR ID with changing spatial resolution of the location encoder. We also include an additional experiment where we show the relationship between local and global geographic INR ID and spatial coverage of the pre-training data.
D.1 MODIFYING THE SPATIAL RESOLUTION OF GEOGRAPHIC LOCATION ENCODERS
We study the effect of increasing spatial resolution in SaICLIP ID by pre-training variants that differ only in the spherical-harmonee harmonic band-limil L (higher allows finer spatial variation), while holding the location-emin embedding dimension and all other hyperparameters fixed. After pre-training, we compute location embeddings for ==100,00 land-only coordinates (uniform on 52 then restricted to land) and report the global intrinsic dimension using the FisherS estimator.
GeoCLIP aligns images with GPS using CLIP-style contrastive objective and a continuous location encoder built from hierarchical random Fourier features (RFF) (Cepeda et al., 2023). As the public codebase provides
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Global 10
10.04
0.03
0.00
Global
Europe
Afric
USA
Mountain
dooaal
Courtries
Stome
0.31
0.52
0.42
Locas Sways
Med. Iooam
Chtleuiigg (CA)
Local Intrinsic Dimension
Figure 8: Effect of pre-training dataset coverage on local and global geographic INR ID: We generate several variants of the S2-100K datasets constrained to (i) continent-scale, (ii) country-scale, and (iii) region- scale boundaries, and train SatCLIP with
Legendre polynomials on each regional dataset. (Top) Global geographic INR ID for each regional location encoder. (Bottom) Local ID with the MLE estimator using k = 100 neighbors. Marked green outlines indicate spatial coverage of pre-training dataset. (Right) Performance of regional SatCLIP location encoders on (top) global and (bottom) local evaluation tasks.
inference and pre-trained models rather than end-to-end pre-training, we fine-tune only the location encoder on an image-to-GPS retrieval loss, freezing the CLIP image tower and the temperature (logit scale), with early stopping on validation loss (patience= 20 epochs). To modulate spatial resolution we (i) densify the RFF hierarchy by increasing the number of log-spaced frequency levels M while holding min: max fixed, and (ii) append one or two higher-frequene, RFF branches to extend mmax while keeping existing with M, we instantiate the location encoder from
branches frozen. Note that in the serrarch--eevl-sweer scratch for each M (no warm-start), avoiding inherited bias from a previous hierarchy layout. When adding higher RFF frequency branches, however, we copy weights for overlapping branches and freeze them, so any ID change can be attributed to the new high- high-frequency branches. After fine-tuning, we estimate global intrinsic dimension with the FisherS estimator on embeddings of uniformly sampled land-only coordinates, and report all results under this land-only sampling.
We study the effect of increased spatial resolution ID in Sphere2 (Mai et al., 2023b) and Veace2Vec (Mai et al., 2020) by modifying the multi-scale resolution parameter S. All encoders are instantiated with trained weights from the MOSAIKS nightlights task (Rolf et al., 2021). For Sphere2 we modu- late spatial resolution by sweeping the number of scales (TorchSpatial: freq)-S E {2, 4, 8, 16, 32)- sembed_dim=200
We keep the output width fixed SPHEREC, and SPHEREC+. Increasing S (and decreasing min.radius) adds higher-frequency compo- nents, yielding finer spatial detail. For Space2 Vec we control resolution via either the grid sampling den- sity (e.g., W H E {1180, 90), (360, 180). (720, 360). (1440, 720)}) or the number of theory frequencies S [2, 4, 8, 16,32), 32). We sample -=100,000 coordinates uniformly on S compute embeddings (dedupli- cating exact repeats), and estimate global intrinsic dimension with the FisherS estimator. We report ID as a function of the resolution knob for each encoder family Figure 5.
We apply this to SPHEREM, SPHEREM+,
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D.2 ADDITIONAL EXPERIMENT: MODIFYING THE SPATIAL COVERAGE OF PRE-TRAINING DATA
Geospatial models with strong global results can still underperform in specific regions; recent studies show that locally trained models locally fine-tuned variants can match exceed the accuracy of glob- ally trained systems in the same area, underscoring a local-global divide in practice (Rolf et al., 2024). Motivated by this, ask how the spatial coverage of pre-training data shapes the geometry of location encoders as seen through their global and local IDs. We construct region-constrained pre-training corpora (conti- nent, country, sub-region) from Sentinel-2 using the Google Earth Engine API and train SatCLIP location encoders with fixed capacity and opti- mization budget, varying only the geographic extent. We then estimate a global FisherS ID on embeddings computed on global scale and a local point-wise ID (MLE) that we visualize on orthographic maps.
Table 6: Regional Sentinel- 2 datasets used for local Sat- CLIP pre-training (tiles 10 m, 256x256).
Region
# Samples
United States
100,000
Europe
100,000
To study how pre-training coverage shapes geographic INRs, we build region-specific Sentinel-2 datasets and pre-train regional SatLLIP mod- els, then measure the ID of location embeddings derived from each local encoder. Regions are defined by Natural Earth polygons; for each region we sample point locations (random or stratified; cached for determinism) and retrieve COPERNICUS/S2 SR HARMONIZED imagery via Google Earth Engine, selecting the least-cloudy scene in a fixed window between Regions we create local datasets for, along with the number of Sentinel-2 tiles
Asia
100,000
Africa
100,000
France
50,000
Mountain West
50,000
Denver Metro
20,000
2023-01-01 00-0001---202-22-- queried are displayed in Table 6.
Around each point we extract multi-band 256x256 tile at 10 m, stack bands (BI,.... B12} into single GeoTIFF, and resample needed to preserve exact size. Per region we release the images, an nndex csv of (Ion, lat), and the scripts used to generate them. For regional pre-training, we keep the SatCLIP pre- training hyperparameters fixed and set the spherical-harmonic band-limit to L-40 (locationeemeedding dimension and all other hyperparameters held constant). The only factor that varies across runs the pre- training dataset. For ID evaluation. we feed the trained location encoder (no classifier) with N coordinates sampled globally and report global FisherS ID.
From Figures 7 and 8, as pre-training coverage narrows from global to continental to regional, the global ID decreases. The local ID exhibits a complementary and highly diagnostic pattern: values remain modest inside the training extent but rise sharply in its immediate geodesic neighborhood, forming anisotropic "ha- los" that effectively delineate the pre-training footprint. This finding adds further evidence that the intrinsic dimension can identify the spatial coverage of pre-training data.
E VALIDATING OUR ESTIMATES OF INTRINSIC DIMENSION
We verify the reliability of where the "ground truth" ID is effectively known. Pope et al. (2021) validated ID estimators in vision by creating synthetic image manifolds with a known upper bound on ID via GAN latent dimension, then checking whether estimators recover the expected scaling. This established baseline of estimator reliability for image data. For spherical INRs without additional information augmentation (such as those done Klemmer et al. (2025); Cepeda et al. (2023); Mai et al. (2023a)), the underlying representation geometry can be described by the 2-sphere (S²). We validate whether common ID estimators behave sensibly under sphere-repecctng positional encodings and simple learned mappings (where the "true" ID should remain s 2).
(ID) estimators for geographic INRs by constructing a testbed
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Published as conference paper ICLR 2026
Our brief experimental framework first mea- sures the intrinsic dimension of raw geographic coordinates. This toy example where the ambient (extrinsic) dimension is equal to the intrinsic dimension. culty by passing raw longitude-latiuude coordi- nates through a sntericallhammonicss (SH) po- sitional encoder (RuBwurm et al., 2024). Con- cretely, the encoder maps (A, ()) E $² to the concatenated vector of real SH basis val-
MLE
MOM
TLE
FisherS
SH
SH + SIRENNET
increase the diffi-
72"-90FN
1
567-72'N
latitude hand
367-547N
187-367N
18'5-0"
36'-18'S
541-36"S
ues (), = (A, ()) (e.L--e))..L..eesns -em. E
72"-54'8
(2 where Yr,m are the (real) spheri- cal harmonics (Eqs. (3)-(5) in (RuBwurm et al., 2024)). While arbitrary nonlinear transforma- tions can alter ID. maps that are smooth and lo- cally invertible (bi-Lipschitz, more generally C with full-rank Jacobian-i.e., immersions) do not. They reparameterize the underlying manifold without adding degrees of freedom. The spherical harmooni encoder & and its compositions with linear layers or smooth point- wise nonlinearities (e.g.. Siren) satisfy these conditions generically (Jacobian rank al- most everywhere on 5²), so the intrinsic dimen- sion should remain 2.
90'-72'5
2.11
2.2
233
2.1
2:2
2.3
GRID
GRID SIRENNET
72"-90NN
5e"-72'N
latitude hand
36"-54FN
187-36/N
18'5-0"
36--1888
541-36'S
721-54'8
907-72'8
10
20
30
40
10 20
40
Intrinsic Dimension
Figure 9: Latitudinal variation of ID estimators with a Spherical Harmonics (SH) and Grid encoding of geographic co-ordinates. The FisherS estimator is the most robust latitudinal effects and different encodings of geographic coordinates.
Table 7 shows that the FisherS estimator is consistently closest to the ground-truth ID = 2 across all stages (raw, SH, SH+linear, SH+Siren), yielding the smallest MAE overall. In contrast, the distance-ratio estimators MILE-
MOM, and TLE biased especialy after applying SH and learned happings-iindcatti greater sensitivity to the embedding and neighborhood construction.
Fig. 10, interestingly, both the MLE and FisherS ID estimators produce a latitude-based striping with the raw geographic coordinates that persists when these geographic coordinates are passed to the SH positional encoder. A plausible explanation is periodicity and coordinate singularities of longitdee-latiude: although the points lie on S (A. Φ) imposes rectangular chart with a discontinuity at the international date line (A =180% and singular behavior the poles. As result, locations that are geodesically close on the sphere can appear far apart (or vice versa) when distances and neighhorhoods are constructed in raw (A, Φ) space across the wrap and near the poles. Non-linearities introduced by linear or siren-network change these distinct latitude-basee striping patterns for both the MLE and FisherS estimator. We find additional evidence here that local ID maps point to properties the positional encoding functions used to construct the geographic INRs: a spherical harmonic positional encoding followed by linear network creates crowding and striping at the poles (column SH + LINEAR NN in Figure 10) which is removed completely when we SirenNet (column SH SIRENNET).
Lastly, we find strong evidence from Figure that the FisherS is most robust to latitudinal variation intrin- sic dimension, and is significantly more robust when comparing IDD results between geographie INRs structed with different positional encoders. The Grid positional encoding (employed in Mai et al. (2023a))
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Published as conference paper ICLR 2026
Raw Coordinates
SH
SH + Linear NN
SH + SirenNet
Figure 10: Verifying local ID estimates of embeddings derived from S Top row shows local ID estimates of the MLE estimator besstance-baseds and bottom row shows estimates from the FisherS estimator (angle- based). Columns indicate the local ID plots with (i) raw geographic coordinates (X, Φ) (ii) pure spherical- harmonic-based embeddings (SH) of (A, Φ), and spherical harmonic embeddings passed through a randomly initialized (iii) linear network and (iv) Sinusoidal representation network (Sitzmann et al., 2020; Rußwurm et al., 2024).
row
Table 7: Intrinsic dimension estimates for spherical data (true ID) = 2.0) with (i) raw geographic coordinates (A, σ) (ii) pure Aaherica--haronnic mented with randomly initialized (iii) linear network and (iv) Sinusoidal representation network (Sitzmann et al., 2020; Rußwurm et al., 2024). Values show mean standard deviation. Best values (closest to 2.0) for each stage shown in bold.
embeddings of (A, d). and spherical harmonic embeddings aug-
Stage
MILE
MOM
TLE
FisherS
Raw Coordinates
2.113 0.056
2.097 0.069
2,038 0.039
2.024 = 0.008
Pure SH
2.189 # 0.032
2.183 # 0.046
2.344 0.034
2.025 + 0.011
SH Linear
2.186 0.040
2.179 # 0.055
2.340 0.039
2.026 = 0.008
SH + SIREN
2.201 # 0.075
2.194 - 0.090
2.367 = 0.056
2.036 + 0.073
MAE
0.172
0.163
0.272
0.028
causes distance-based estimators ID to present high values of ID, but FisherS stays close to the true of
2.
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Published as conference paper ICLR 2026
GeoCLIP (Copeda e sl., 2023)
(SP-iNal [Mai al. 2023)
Laca Int Insc Dimorsion 1.b
7.0
SaCLIPP 1-10 (Klemner el al. 2023)
CSP-IMoW (Mai et ed. 2023)
Dimension
Spawe2Veo- Vec- (Mai et al., 2020) [M
Vehere2Vee SphaeM (Mal el al. 2023b) [Mow]]
9.9
10.0
11.
16.0
.1.0
20.0
124
14.1
26.0
10.0
20
Local Incinsi Dimonsion
Local Intrinse Dimrension
Sphere2Ve: SphereC (Mai el al., 2023b) [MoW
NeRF (Mildenhall et al., 2020) [fMoW]
10.0
12.0
14.0
:60
30.0
12.0
16.0
22.0
Hntiisic
insin
Figure 11: Full Plot: Local intrinsic dimension of geographic INRs reveal spatial artifacts. MLE estimator with 100 nearest neighbors used. All Sphere2 Vec. Space2 and NeRF models are trained on the functional map of the world dataset (Christie et al., 2018) with supervised learning.
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Reviews:
[{"review_id": "X5XvSbUsk3", "reviewer": "Reviewer_F7dp", "summary": "This paper presents a framework for quantifying the intrinsic dimension of Earth representations and studying the correlation between ID with the model performance on downstream tasks.", "questions": "1. \"Global IDs of geographic INRs are similar to that of embeddings derived from image encoders\". However, compared with image encoders, location encoders have much less learnable parameters. Does this result indicate that the ID depends more on the dataset and task characteristics (e.g., spatial distribution of the class labels) instead of the model?\n2. Figrue 4 shows that \"Lower global ID in activation space of supervised models corresponds to higher task performance\". Why the results from location embeddings and activation features show a reverse pattern? \"past work, that found lower ID indicates more concentrated, linearly separable structure and thus better generalization\" Can you explain this in details?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "1. The  author proposed to use Intrinsic Dimension (ID) as a new metric to quantify the representativeness and task-lignment of the Earth representation. It gives us a way to interpret the location encoders and how the design can impact the model performance\n2. A thorough analysis has been carried out to analysis the relation between ID and the task performance in the context of Earth presentation learning.", "weaknesses": "Although I enjoy reading this paper, there are some weaknesses I need to point out:\n1. In the main results table (Table 1), the authors compared different pretrained location encoders with the global ID metric. However, different location encoders have different designs, different pretraining objectives, and different pretraining datasets and modalities. It is very hard to see any pattern here. A controlled experiment is needed in which the type of location encoders, the pretraining objectives, and the pretrained datasets need to be ablated to see how different parts impact the ID and performance of the location representations.\n2. Figure 6 has a similar problem: 3 models use different data modalities, different architectures, and different pretraining strategies. A controlled experiment is needed here.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "frdYQtIpLv", "reviewer": "Reviewer_zZ97", "summary": "This paper presents a novel research on the intrinsic dimension (ID) of geographic Implicit Neural Representations (INRs). While the measures used to quantify ID already exist in literature, it is important to bring the concept to the community of spatial representation learning. The authors also conducted comprehensive analysis on how ID affects the downstream performance of geographic INRs, revealing that ID can be a good index of representation quality.", "questions": "1. The non-linear correlations in Figure 3 and Figure 4 may have a theoretical implication. In the 4th chapter of https://escholarship.org/uc/item/5bs589v5, it is mathematically proven that the average information content in the samples drawn within a given geographical region decreases logarithmically when there is spatial dependency.\n\n2. Can you train some baseline models on a spatially balanced dataset (e.g. OSV5M) and compare their ID? This is more useful to prove that ID can be used to quantify the quality of an embedding method.\n\n3. Do you have any idea why in Figure 5, the Space2Vec method has such large increase in ID compared to Sphere2Vec variants?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. It is important to introduce the concept of ID to the community of spatial representation learning. Unlike image or text embedding where there is **compression** of information, spatial representation such as location encoding usually embeds very low dimensional data (e.g. two-dimensional latitudes and longitudes) into high dimensional spaces, which is not a compression but an expansion. The quality of the expansion, i.e., whether the embedding manifold maintains useful topological information, is a critical concern -- by the curse of dimensionality, higher dimensions tend to make embeddings less distinguishable and less informative.\n\n2. Local ID and global ID allow fine-grained analysis of the embedding manifold.\n\n3. The correlation analysis between ID and downstream task performance is very useful. The positive/negative correlations can be used to guide spatial representation learning or to select appropriate embedding methods for specific downstream tasks.", "weaknesses": "1. The linear correlations fitted in Figure 3 and Figure 4 can be misleading. From the authors' perspective, they only need to show that the ID and model performance are positively/negatively correlated; the correlation does not need to be linear. For example, the subplots in Figure 3a look more quadratic/logarithmic than linear.\n\n2. Table 1 can not be used to fairly compare the ID of baseline models. The models evaluated are trained on very different datasets and downstream tasks. For example, GeoCLIP having very high ID, from my experience, owes a lot to the spatial bias of the MP16 dataset it was trained on. This is also shown in Figure 2. In order to make the comparison between numbers meaningful, it is better to train different models on the same dataset(s).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iXF6QuQHDH", "reviewer": "Reviewer_1jrM", "summary": "The paper presents a study on the intrinsic dimensionality of geographic implicit neural representations (INRs), quantifying how much real information these learned location embeddings actually capture. The authors present some interesting findings including that global ID estimates of geographic INRs are competitive with those of large-scale image encoders, and that that global ID correlates positively with downstream performance when measured on frozen, pre-trained models (representativeness), but correlates negatively when measured in the activation space of supervised models (task-alignment). \n\nOverall, I find this work quite interesting and valuable, as their methodology offers a principled way to measure information content in location embedding techniques beyond traditional downstream performance metrics.", "questions": "[1] Please see W1 & W2. \n[2] I am wondering if the study can generally extended to study encoders of geospatial objects such as [a] [b] [c]\n\n[a] \"Towards general-purpose representation learning of polygonal geometries.\" GeoInformatica 2023.\n\n[b] \"Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications.\" ICML 2025.\n\n[c] \"Geo2Vec: Shape-and Distance-Aware Neural Representation of Geospatial Entities.\" arXiv 2025.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "[S1] The paper is the first to examine the intrinsic dimension of location embeddings, providing valuable insights into the representational limits and strengths of popular approaches. The study is especially useful given the large and growing number of works on location embeddings in GeoAI.\n\n[S2] The paper is well written and easy to follow. The methodology for estimating intrinsic dimension is clearly explained, and the results give interesting insights, including how intrinsic dimension varies with model architecture, input modalities, and spatial resolution.\n\n[S3] The work is clearly written, well-organized, and thoroughly evaluated across a diverse range of models and experimental cases.", "weaknesses": "[W1] The TwoNN estimator is used to measure task alignment, with low ID values observed for task-specific representations. Can the authors elaborate on whether this could be an artifact of estimator bias due to non-uniform data, rather than actual compression?\n\n[W2] We observe notable differences in Global ID estimates across different estimators, especially for SatCLIP (e.g., SatCLIP-L40: 8.08 for FisherS vs. 2–2.5 for MLE, TLE, and TwoNN). The authors primarily focus on FisherS, which consistently gives much higher values, but the reasons for this discrepancy are not fully discussed. Is such a large gap expected in practice? Can the authors provide further empirical or theoretical justification for prioritizing FisherS over other estimators for Global ID?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OCYAmqpgYN", "reviewer": "Reviewer_Ayws", "summary": "The paper presents techniques to measure intrinsic dimension and degree of task-alignment for various geospatial location and image encoders. The paper compares various global and local measures of intrinsic dimension. The paper conducts various experiments comparing the ID of location and image encoder against downstream task performance. This could prove valuable to the community for analyzing the informativeness of geospatial embeddings.", "questions": "Please see weaknesses.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper is well motivated and makes a novel contribution by analyzing the intrinsic dimension of various geospatial location and image encoders and comparing it with downstream task performance.\n2. Geospatial embeddings are increasingly being used to address various geospatial tasks. The paper presents a methodology to evaluate the \"goodness\" of these embeddings generated using existing geospatial encoders. This methodology can be very useful for the community in future.", "weaknesses": "1. It is difficult to comprehend clearly the methods used for computing global and local IDs. Theory is included in the supplementary material which is very useful but I highly suggest moving some important mathematical theory into the main paper.\n2. Although measuring IDs of geospatial models an interesting direction, the paper falls short of several experiments that could help strengthen the paper:\n* How do IDs relate to PCA/ICA and geospatial embedding visualization? It might be interesting to discuss how IDs can be used for possibly better visualization of geospatial embeddings than maybe using PCA/ICA which are not spatially explicit.\n* One important experiment is missing that compares ID with the embedding dimension for each model architectures. The paper should discuss how the intrinsic dimension is related to the overall dimensionality of the embeddings (for the same model architecture and training regime). For example, if GeoCLIP is retrained with an embedding dimension of 768 or 1024, how would the intrinsic dimension change? Would it increase, decrease or remain the same. The same goes for image encoders. How would the intrinsic dimension vary when ScaleMAE is trained with different embedding dimensions.\n* For measuring the IDs of image encoders, the authors should consider comparing the IDs across different spatial resolution of satellite images. **Comparing the IDs of image encoders using low resolution Sentinel-2 imagery is not enough to conclude the global ID of GeoCLIP approaches to that of various image encoders**. The authors should also experiment with high resolution imagery as that will possibly increase the IDs of image encoders.\n* Concrete measures for task-alignment need to be shown. The authors could compute a mutual information estimate such as CLUB [1] between the geospatial embeddings and downstream task variables and compare it against the ID values to truly understand if IDs are correlated with downstream task performance.\n* Can ID measure be made more task-specific since tasks vary in spatial frequency. Currently, ID seems to be independent of task and only dependent on the embedding model.\n* **How do IDs relate to other downstream tasks that go beyond memorization** especially where the location encoders are used as a geo-prior such as satellite image generation or fine-grained image classification? **I believe the results demonstrated in the paper can easily be gamed by overfitting on the task, the authors need to show some analysis on tasks that are more challenging such as generation.**\n* It should be interesting to see IDs for non-geospatial image encoders such as Jepa or Dinov2. Recently, webscale version of Dinov3 has shown to outperform the satellite-only Dinov3 model on various geospatial downstream tasks.\n* Limited location and image encoders are compared. There are recent task-aligned location encoder models such as SINR, TaxaBind and Climplicit. There are multimodal image encoders such as AnySat and Galileo and Range. It will be interesting to see the ID analysis on such models.\n\n**Suggestion**\n\nI appreciate the authors for exploring the IDs of various geospatial models. The idea and motivation behind the paper is novel and will surely be useful for the community. However, the paper requires a major revision as the **paper makes strong claims without a comprehensive and sound experimental setup**. I would suggest the authors to include the above suggested experiments to improve the paper. The paper should clearly state concrete applications of computing the IDs of geospatial models. The paper also has typos which need to be fixed. The paper could be structured better with more theory in the main paper and better explanations for technical terms. I am definitely willing to increase my score once the authors have addressed my concerns.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)