Measuring the Intrinsic Dimension of Earth Representations

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. Across vision, audio, and other modalities, seemingly high-dimensional observations often vary along far fewer degrees of freedom. This is especially true of geographic data, which is often characterized by strong spatio-temporal dependencies. For example, classical work in meteorology use dimensionality reduction techniques since large-scale oscillations in climate trends can be explained by a handful of indices (van den Dool, 2006). This phenomenon is leveraged by a class of representation learning techniques aimed at embedding signals in Earth's data into succinct, 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 a location encoder network which takes geographic position (latitude and longitude) as input.