Emergent Sparsity: How Many Effective Variables Survive Coarse-Graining in AlphaEarth Embeddings?

Johannes Hirn

cs.LGOct 18, 2026

Emergent Sparsity: How Many Effective Variables Survive Coarse-Graining in AlphaEarth Embeddings?

Authors: Johannes Hirn

Organizations: Image Processing Laboratory (IPL), Universitat de València, 46980 Paterna, Valencia, Spain · Universidad Cardenal Herrera-CEU, 46115 Alfara del Patriarca, Valencia, Spain

Abstract

AlphaEarth encodes each 10 m land pixel as a 64-component vector intended to support diverse global mapping tasks. Yet many scientific applications work at coarser spatial supports, for which fewer than 64 effective variables may suffice. We ask whether these vectors can be reorganized into ranked global nonlinear coordinates, how many suffice at a chosen reconstruction accuracy, and whether that number decreases with spatial scale. We thus describe effective dimension as a two-control surface over tolerance and spatial scale, rather than one intrinsic number. At each scale, a $β$ -VAE scan with a common architecture produces a rank--distortion envelope, giving the smallest retained count achieved within the scanned family at each reconstruction tolerance. At 3% validation tolerance, this number falls from 24 to 7 when the scale is increased from 10 m to 78.9 km, whereas the corresponding PCA rank falls only from 46 to 38. Repeating the analysis on unit-norm targets preserves this trend.

Figures & tables

Figure 1: Validation scale–tolerance contours for ball (a) and unit-norm (b) targets. Each labeled

K

curve gives the lowest relative distortion attained with at most

K

coordinates. Colored solid curves are VAE envelopes over the

\beta

scan, and gray dashed curves show PCA. Distortions are normalized by the training-mean predictor’s validation distortion at each support.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A1 : Embedding radius and angle from the matched native vector across scales. Boxes span the interquartile range with a line at the median. Whiskers extend to the most extreme observations within 1.5 interquartile ranges of the box, and individual outliers are omitted.

Figure A2 : Validation rank–distortion curves at native 10 m support for the geography-aware split. Color denotes encoder/decoder depth and line style denotes hidden width. Every model has 128 latent coordinates.

Figure A3 : Held-out relative distortion for ball targets across normalized $\widetilde{\beta}$ at native 10 m support (left) and for 78.9 km polygon means (right). Color denotes retained ranks $K=1$ –63, from red to dark violet. Black shows the full model with all 128 coordinates retained across the entire scan.

Figure A4 : Held-out rank–distortion curves for ball targets using the common VAE with eight 256-unit hidden layers in each of the encoder and decoder and 128 latent coordinates (solid), compared with PCA (dotted). Color denotes scale. Ranks $K>64$ show the overcomplete VAE tail.

Model	Support	Retained $K$	Validation $D/D_{0}$	Test $D/D_{0}$
A	78.9 km	3	9.81%	12.55%
B	78.9 km	7	2.60%	2.75%
C	10 m	8	9.26%	9.65%

Appendix

Table A1 : Relative distortions of the validation-selected ball-target models, with no selection on test.

Figure A5 : Minimum validation-selected dimension across scale for (a) ball and (b) unit-norm targets. Solid lines show the VAE, dotted lines show PCA, and color denotes tolerance. Dots mark model A, model B, and model C, the ball-target descriptions used in subsequent figures.

Figure A6 : Posterior means of the three retained coordinates for model A (10% validation tolerance at 78.9 km). All 21,299 anchor centers are shown, in the original KL-rate order. Each panel has its own symmetric color limits, clipped using the outer 2% tails. Latent signs are arbitrary. Blank areas have no retained anchors.

Figure A7 : Absolute Pearson and $\phi_{K}$ associations between the three retained posterior means of model A and polygon-averaged WorldClim bioclimatic variables. Model A was selected at 78.9 km and 10% tolerance. T denotes temperature and P precipitation. Ranks follow validation KL rate. Correlations use all dataset splits. Rank 1 associates mainly with temperature, and rank 3 with annual and wet-season precipitation. Rank 2 has weak Pearson correlations but appreciable $\phi_{K}$ associations across both groups.

Figure A8 : Associations between the three retained posterior means of model A and the twelve biosphere climatologies. Missing-data filtering is applied independently for each probe. Rank 1 associates with evaporative stress and heat fluxes, while rank 3 associates with water and carbon-cycle quantities. The probes overlap environmentally and do not identify separate physical variables.

Figure A9 : Associations between model A (three coordinates, 10% tolerance) and model B (seven coordinates, 3% tolerance), both at 78.9 km. Columns and rows follow each model’s own frozen KL-rate order. Both panels use all 21,299 matched locations and a fixed zero-to-one color scale.

Figure A10 : Associations between model A (three retained coordinates at 78.9 km) and model C (eight at native resolution), both selected at 10% tolerance. Each native embedding is sampled at the corresponding polygon’s anchor center. All 21,299 matched locations enter both panels. Each model retains its own frozen KL-rate order, and color limits match Figure A9 .

Figure A11 : Associations between model B (seven retained coordinates at 78.9 km and 3% tolerance) and model C (eight at native resolution and 10% tolerance). All 21,299 matched locations enter both panels. The pairing uses the native embedding at each anchor center and the corresponding polygon-mean embedding, as in Figure A10 . Coordinates retain each model’s frozen KL-rate order, and color limits match the other comparisons.

Figure A12 : Validation scale curves for

K=1

–64 before (top) and after (bottom) division by the measured native 10 m relative distortion at each

K

. Solid colored curves span the empirical-profile interval

K=5

–30, and other ranks are gray dashed. The bottom row shows the geometric-mean profile at all twelve measured supports and its separate free-amplitude spatial power law, fitted over 100 m–10 km and extended across the displayed scale range. Portions outside the fitting interval are extrapolations.

Targets	Spatial $\gamma$	Native $A$	Native $p$	Native $K_{c}$
Ball	0.18	0.49	0.77	42
Unit-norm	0.16	0.49	0.77	42

Appendix

Table A2 : Spatial exponent fitted to the empirical

K=5

–30 profile over 100 m–10 km, followed by separate native-Ansatz parameters fitted over

K=2

–64.

K_{c}

is rounded to two significant digits.

Figure A13 : Validation rank–distortion curves for

K=1

–64 before (top) and after (bottom) division by the empirical

K=5

–30 spatial profile, with

g(10\,\mathrm{m})=1

. Color denotes support, from native violet to red at 78.9 km. Vertical guides bound the profile-estimation interval. The native reference and its black dashed native Ansatz are unchanged between rows because the native spatial factor is exactly one. The fit uses native

K=2

–64 only, with the dashed extension to

K=1

shown as extrapolation.

Figure A14 : Measured validation scale–tolerance contours (solid) and the combined analytic approximation (dotted), for ball targets (left) and unit-norm targets (right). The

K=3

comparison exposes departures at low rank, while

K=50

and

60

expose departures in the high-rank tail.

Dimensionality Reduction

Dimensionality reduction maps high-dimensional data to compact representations for visualization, prediction, compression, and scientific modeling.

See all 145 papers

What on Earth is AlphaEarth? Hierarchical structure and functional interpretability for global land cover
Mar 8, 2026Ivan Felipe Benavides-Martinez, Justin Guthrie, Jhon Edwin Arias +6Geospatial Foundation Models Geospatial Representation Learning
Artificial Intelligence for Climate and Sustainability The Institute for Experiential Artificial Intelligence Northeastern University, Portland, ME, USA · Gulf of Maine Research Institute Portland, ME, USA · Sustainability and Data Sciences Laboratory Northeastern University, Boston, MA, USA +2
Characterizing AlphaEarth Embedding Geometry for Agentic Environmental Reasoning
Apr 20, 2026Mashrekur Rahman, Samuel J. Barrett, Christina LastGeospatial Foundation Models Geospatial Reasoning
Dartmouth Libraries, Dartmouth College, Hanover, 03755, NH, USA. · LGND AI, Canarias, Spain. · TipplyAI, London, UK.
Earth Embeddings
Aug 4, 2026Adam J. Stewart, Heng Fang, Isaac A. Corley +1Geospatial Representation Learning Land Cover Classification
Chair of Data Science in Earth Observation, Technical University of Munich, Munich, Germany · Munich Center for Machine Learning, Munich, Germany · KTH Royal Institute of Technology, Stockholm, Sweden +1

Mar 8, 2026J/K move · Enter open · S save

What on Earth is AlphaEarth? Hierarchical structure and functional interpretability for global land cover

Ivan Felipe Benavides-Martinez, Justin Guthrie, Jhon Edwin Arias, Yeison Alberto Garces-Gomez+5

Artificial Intelligence for Climate and Sustainability The Institute for Experiential Artificial Intelligence Northeastern University, Portland, ME, USA · Gulf of Maine Research Institute Portland, ME, USA · Sustainability and Data Sciences Laboratory Northeastern University, Boston, MA, USA+2

Geospatial Foundation Models Geospatial Representation Learning Neural Network Interpretability Hierarchical Representation Learning Land Cover Classification Feature Selection

Open paper AskPDF arXiv

Geospatial foundation models generate high-dimensional embeddings that achieve strong predictive performance, yet their internal organization remains obscure, limiting their scientific use. Recent interpretability studies relate Google AlphaEarth Foundations (GAEF) embeddings to continuous environmental variables, but it is still unclear whether the embedding space exhibits a functional or hierarchical organization, in which some dimensions act as specialized representations while others encode shared or broader geospatial structure. In this work, we propose a functional interpretability framework that reverse-engineers the role of embedding dimensions by characterizing their contribution to land cover structure from observed classification behavior. The approach combines large-scale experimentation with a structural analysis of embedding-class relationships based on feature importance patterns and progressive ablation. Our results show that embedding dimensions exhibit consistent and non-uniform functional behavior, allowing them to be categorized along a hierarchical functional spectrum: specialist dimensions associated with specific land cover classes, low- and mid-generalist dimensions capturing shared characteristics between classes, and highgeneralist dimensions reflecting broader environmental gradients. Critically, we find that accurate land cover classification (98% of baseline performance) can be achieved using as few as 2 to 12 of the 64 available dimensions, depending on the class. This demonstrates substantial redundancy in the embedding space and offers a pathway toward significant reductions in computational cost. Together, these findings reveal that AlphaEarth embeddings are not only physically informative, but also functionally organized into a hierarchical structure, providing practical guidance for dimension selection in operational classification tasks.

Abstract

Dimensionality Reduction

Explore similar work

What on Earth is AlphaEarth? Hierarchical structure and functional interpretability for global land cover

Characterizing AlphaEarth Embedding Geometry for Agentic Environmental Reasoning

Earth Embeddings