Gene-Amortized Fields for Held-Out Gene Imputation in Spatial Transcriptomics

Spatial transcriptomics profiles gene expression across intact tissue but measures only a bounded gene panel. We study gene-level generalization within a tissue domain, where a gene that is described by domain-level side information but withheld entirely from model supervision must have its spatial expression reconstructed at every location from the remaining genes. A gene held out in full has no measurements of its own to fit, which makes the task out-of-column matrix completion. Completion is feasible only if genes carry side information relating masked columns to observed ones. Existing methods take that side information from an external single-cell reference in which the masked gene is measured, and none of the eight tissue domains studied here has a paired one. We instead learn a single gene-independent coordinate field, the shared field, and decode every gene from it through a map amortized from a fixed gene embedding. The embedding is computed from the panel's own co-expression rather than fitted to the gene it describes, so a gene the model was never supervised on is decoded by the same map with no code fitted to it. It need not appear in a single-cell atlas, though it must have been measured in spatial data of the domain, and a cell-type reference still anchors the count likelihood. The imputation is therefore reference-light rather than reference-free. Expression is structured at multiple non-stationary scales, so the field is multi-resolution, instantiated as a gene-amortized multi-scale wavelet representation. On eight HEST-1K tissue domains spanning two species and five organs under gene-level cross-validation, the model surpasses the strongest of ten reproduced baselines on average ($0.300$ against $0.265$) and on seven of eight domains. The per-domain differences are significant under a paired signed-rank test ($p=0.016$). A single-scale variant is worse on all eight. Over the two hundred genes whose measured expression is most spatially autocorrelated, the mean correlation is $0.47$. A low-rank spectral coupling added to the readout leaves accuracy unchanged but exposes a factorization over genes whose modes align with measured co-expression in all eight domains once the input embedding is controlled for, and recover programs recurring across organs and species.