Measuring the Intrinsic Dimension of Earth Representations¶

Conference: ICLR 2026 arXiv: 2511.02101 Code: GitHub Area: Remote Sensing / Representation Learning Keywords: intrinsic dimension, geographic implicit neural representations, Earth observation, representation learning, unsupervised evaluation

TL;DR¶

This work presents the first systematic measurement of the intrinsic dimension (ID) of Geographic Implicit Neural Representations (Geographic INR), finding that 256–512-dimensional embeddings have true IDs of only 2–10. Higher ID in frozen embedding spaces correlates positively with downstream performance, while lower ID in supervised task-head activation spaces correlates positively with performance, revealing a dual mechanism of "representativeness vs. task alignment."

Background & Motivation¶

Geographic INRs map latitude–longitude coordinates \((λ, ϕ)\) to high-dimensional embedding vectors \(z = f(λ, ϕ) \in \mathbb{R}^D\) (typically \(D = 256\) or \(512\)) via contrastive pre-training on satellite imagery, ground-level photos, or text. Models such as SatCLIP, GeoCLIP, and CSP have been widely adopted for downstream tasks including land-cover segmentation, object detection, and image geolocalization.

Core Problem: How much effective information do these high-dimensional representations actually encode? Existing evaluation relies entirely on downstream task labels, lacking label-free, architecture-agnostic measures of information content.

Key Insight: The Earth's surface is inherently a two-dimensional sphere \(S^2\), so the input manifold dimension of any INR is known to be 2. If the embedding ID substantially exceeds 2, the model genuinely encodes geographic signals beyond raw coordinates; if the ID approaches the ambient dimension \(D\), redundancy may be present. This "known input dimension + measurable output ID" setting makes Geographic INRs an ideal testbed for studying ID.

Method¶

Overall Architecture¶

The methodology follows two parallel tracks corresponding to the paper's dual mechanism:

Representativeness Measurement: The pre-trained INR is frozen; 100,000 coordinates are uniformly sampled across global land, producing embeddings \(Z_{geo} \in \mathbb{R}^{N \times D}\). The angular estimator FisherS is then applied to compute the global ID. FisherS is robust to spatial heterogeneity and is not confounded by local density variations such as climate-zone boundaries.
Task-Alignment Measurement: INR embeddings are frozen while a shallow MLP classification/regression head is trained. The distance estimator TwoNN is applied to the penultimate ReLU activations to measure the ID of the task-aligned manifold onto which supervised learning compresses the embeddings.

Key Designs¶

Complementary Use of Angular and Distance Estimators: Angular estimators (FisherS) eliminate local density differences via whitening and spherical projection, making them suitable for global cross-model comparisons. Distance estimators (MLE/TwoNN/MOM/TLE) are sensitive to local neighborhood distances, making them suitable for generating spatial ID maps that reveal artifacts. This design insight is the foundation of the paper's analytical reliability.
Spatial Visualization of Local ID: MLE (\(k=100\) neighbors) is used to compute per-point IDs and render global maps, directly revealing that GeoCLIP's ID peaks over the United States and Western Europe (reflecting social-media image distribution bias), that CSP exhibits grid-stripe artifacts (from periodic repetition in positional encodings), and that SatCLIP shows subtle oscillations (from finite-order truncation of spherical harmonics).
Causal Resolution–ID Relationship: The paper systematically controls resolution hyperparameters—Legendre polynomial order \(L\) for SatCLIP, maximum RFF frequency \(\sigma_{max}\) and hierarchy level \(M\) for GeoCLIP, and frequency component count \(S\) for Space2Vec—and observes monotonically increasing ID with resolution, establishing a causal rather than merely correlational relationship.

Key Experimental Results¶

Global Intrinsic Dimension per Model¶

Model	Type	\(D\)	FisherS	MLE	MOM	TLE
SatCLIP-L10	Location encoder	256	5.00	1.96	2.02	2.16
SatCLIP-L40	Location encoder	256	8.08	2.03	2.39	2.32
GeoCLIP	Location encoder	512	7.68	11.21	13.02	11.53
CSP-fMoW	Location encoder	256	1.70	5.18	5.23	6.25
CSP-iNat	Location encoder	256	0.92	3.37	4.64	4.14
SINR	Location encoder	256	3.19	2.19	3.36	2.74
TaxaBind-Loc	Location encoder	512	3.33	9.44	11.56	10.30
CROMA	Image encoder	768	9.79	19.57	17.00	20.30
DOFA	Image encoder	768	3.32	15.58	13.78	16.20
ResNet152	Image encoder	2048	7.60	20.72	17.50	21.50

All location encoders exhibit IDs one to two orders of magnitude below their ambient dimensions. GeoCLIP's distance-estimated ID (11–13) approaches that of the large image encoder DOFA (14–16), indicating that coordinate-only inputs can encode rich geographic information.

Effect of Input Modality on ID and Performance¶

Pre-training modality	Global FisherS ID	Temperature R²	Elevation R²	Population R²
Sentinel-2	~7.5	~0.76	~0.74	~0.78
S1 + S2	~8.5	~0.80	~0.82	~0.82
All modalities	~9.5	~0.84	~0.86	~0.86

More input modalities → higher ID → better downstream performance, with all three quantities increasing monotonically.

Key Findings¶

Embedding-space ID positively correlates with performance: Higher global FisherS ID in frozen INR embeddings consistently yields better downstream regression/classification performance across all five tasks (temperature, elevation, population, biome, and country classification). Higher ID implies greater representativeness, providing more independent directions for shallow learners to exploit.
Activation-space ID negatively correlates with performance: Lower TwoNN ID in the penultimate layer of supervised MLPs corresponds to better performance. Supervised adaptation compresses INR features onto a lower-dimensional task-aligned manifold, consistent with the findings of Ansuini et al. (2019) for classification networks.
Resolution controls ID: Increasing SatCLIP's Legendre order from 10 to 40 raises FisherS ID from 5.0 to 8.1; increasing GeoCLIP's RFF maximum frequency causes ID to surge from 7.7 to 75.7.
Local ID exposes data bias: GeoCLIP's ID is highest over the US and Western Europe (dense training-data regions); CSP exhibits grid artifacts from periodic positional encodings—both directly usable for model diagnostics.

Highlights & Insights¶

The dual mechanism of representativeness vs. task alignment is the paper's most central contribution: the same ID measure exhibits opposite correlation directions in embedding space vs. activation space, elegantly unifying the intuitions that "pre-training should be broad" and "fine-tuning should be narrow."
The practical utility of ID as a label-free metric is clearly demonstrated: it can substitute for expensive downstream evaluation in model selection, hyperparameter search, and early stopping.
Local ID maps constitute an intuitive and effective model diagnostic tool, capable of revealing pre-training data coverage bias and spatial artifacts introduced by architectural choices.
The ID of Geographic INRs (2–10) is far below the ambient dimension (256–512), suggesting that current representations are heavily redundant and offer significant room for compression.

Effect of Resolution on ID¶

Model	Resolution parameter	Value	Global FisherS ID
SatCLIP	Legendre order \(L\)	10	5.0
SatCLIP	Legendre order \(L\)	20	~6.5
SatCLIP	Legendre order \(L\)	40	8.1
GeoCLIP	RFF max frequency \(\sigma_{max}\)	\(2^8\)	7.7
GeoCLIP	RFF max frequency \(\sigma_{max}\)	\(2^{16}\)	75.7

SatCLIP's ID grows nearly linearly with spherical harmonic order; GeoCLIP's ID increases nearly tenfold upon raising the RFF frequency, indicating that high-frequency positional encodings substantially expand the effective degrees of freedom in the embedding.

Limitations & Future Work¶

Different ID estimators yield substantially different numerical values (e.g., SatCLIP-L40: FisherS = 8.08 vs. MLE = 2.03), requiring careful estimator selection depending on context.
The analysis is limited to static INRs with 2D coordinate inputs; spatiotemporal representations incorporating a time dimension are not considered.
ID is a scalar summary and cannot characterize the directional structure or semantic organization of the embedding space.
The representativeness–task-alignment correlation analysis is based on a limited set of seven location encoders and five downstream tasks; statistical significance depends on sample size.
The paper does not explore how ID analysis could inversely guide INR architecture design, such as adaptive dimension allocation based on local ID or region-weighted fine-tuning.
Representation learning evaluation: Conventional evaluation relies on downstream task probing; this work provides a label-free alternative. The ID analysis framework generalizes to pre-trained representation evaluation in other domains (e.g., language model representations in NLP, medical image representations).

Rating¶

Novelty: ⭐⭐⭐⭐ (novel perspective, though analytical tools are existing)
Experimental Thoroughness: ⭐⭐⭐⭐ (comprehensive multi-model, multi-dimensional analysis)
Writing Quality: ⭐⭐⭐⭐ (27 pages with detailed appendices)
Value: ⭐⭐⭐⭐ (provides an important analytical tool for Earth observation representation learning)