Incidence-Structured Spatial Factors for Latent Causal Discovery

Latent causal discovery compresses many spatial sites into a few latent time series and learns the causal graph among them. Existing methods link latents to sites through kernels over site coordinates, learned site partitions or principal components. Spatial data often carry memberships such as regions, and the hypergraph lift of a kernel sees them only through their pairwise operator. We show that this operator does not in general determine the latent coordinates that the memberships determine. Two incidences with the same pairwise operator admit models with the same observed law and inequivalent latent laws. We therefore propose incidence-structured spatial factors, which build each latent mode as a weighted union of structural units. For the unsmoothed factor, we prove the latents identifiable up to permutation, scaling and translation under site-specific smooth invertible observation maps when every mode owns an anchor site and bridge sites connect the modes. On a public grid benchmark, the factor recovers the latent causal graph more accurately than kernel, single-parent, two-stage and sequence-model baselines. On random committee hypergraphs with membership-following footprints, hyperedge atoms, given or reconstructed from the pairwise operator, recover the latents more accurately than pair atoms and heat kernels built on that operator.