Signature Distance: Generalizing Energy Statistics

Comparing empirical distributions is central to generative model evaluation, hypothesis testing and data augmentation in high-dimensional biological data. Established methods such as energy distance summarize each point's relationship to the opposing distribution through a single expected distance, providing sensitivity to location shifts. We introduce Signature Distance (SD), a statistical distance that compares empirical distributions through the mean absolute difference of their sorted pointwise distance profiles. SD is a structural generalization of energy distance and matches its quadratic pairwise-distance cost, with an additional sorting step. In controlled experiments and on TCGA pan-cancer transcriptomic data, we show that (1) SD detects density changes with greater sensitivity than energy distance in the tested scale-contraction scenarios; (2) per-point mean-distance and signature-profile landscapes reveal the geometric mechanisms behind their different penalties; (3) linearly interpolated biological samples that receive no increased penalty from energy distance are penalized by SD; (4) SD provides a direct differentiable potential energy for model-free Langevin data expansion, with a bootstrap resampling protocol to assess the stopping epoch; and (5) SD is directly usable as a differentiable generative training loss. Code to reproduce all experiments is available at github.com/lazzaronico/signature-distance.