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

Johannes Hirn

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.