Local Influence Inference: Learning Shared Feature Rankings from Spatial Neighborhoods

Spatial analysis often seeks to identify which features of a local environment influence an outcome at a focal point. The target is a single feature ranking shared across locations, with influence that decays over distance. Predictive accuracy alone does not ensure recovery of this ranking. In a distance-weighted linear model, we show that bag-averaged permutation importance can reverse two positive influence coefficients even at the true predictor when neighborhood size co-varies with within-neighborhood feature variation. We formulate Local Influence Inference and propose LION, which represents the ranking as a shared model parameter. LION combines a learnable distance kernel, a feature-preserving normalizer, and a global influence vector through bilinear aggregation, using G + 4 trainable parameters. Under correct specification and identification conditions, the reduced maximum-likelihood estimator consistently recovers all strict feature comparisons. Across seven ranking benchmarks spanning synthetic data and real tissue geometry, LION leads or ties distance-weighted logistic regression in mean Spearman on every benchmark, with the largest gains on morphology-interaction (+0.22) and Dense (+0.11). LION ranks first in AUROC on five of seven benchmarks among the evaluated deep and interpretable MIL baselines, with 12 to 64 parameters against thousands for the deep baselines. On three Xenium/TCR-profiled tumors, LION increases the mean reactive fraction among recovered T cells and the mean bystander fraction among missed T cells over every baseline at matched predicted-positive rates. These results support a shared feature ranking as a useful model output for spatial inference, alongside bag-level prediction.