HodgeDepth: Spatial Representation Learning through Global Geometry

A central challenge in spatial representation learning is to connect local feature patterns with global geometric patterns. For example, understanding layered tissues requires connecting local molecular variation to global spatial organization. We introduce HodgeDepth, a global-geometry-aware framework that learns an interpretable depth coordinate describing progression across layers and corresponding positions along curved tissue contours. The method aligns noisy local directions of molecular change and integrates them into a coherent coordinate, making global geometry an explicit representation-learning target. The learned depth provides a spatial axis for inspecting tissue contours and gene-expression profiles, with a complementary construction for recurrent layers. Our theory characterizes geometric bias and contour recovery, establishes conditions for correct global ordering from local estimates, and bounds recovery of a canonical depth scale based on cumulative spatial mass. We evaluate HodgeDepth on 23 human cortical Visium sections across two cohorts. On the 12-section DLPFC benchmark, mean absolute depth–layer Spearman correlation increases from 0.690 for GASTON to 0.885. Downstream clustering achieves mean ARI 0.618 versus 0.533 for SEDR, the strongest of nine tested baselines, a 16% relative improvement. Controlled synthetic examples further illustrate recovery of layered geometry. Together, these results support global spatial depth as an interpretable representation of tissue organization with measurable benefits for downstream analysis.