From Local Contractivity to Global Geometry: Poisson Potentials for Representation Analysis

Contractive autoencoders characterize representation geometry through local derivatives of an encoder, but a pointwise Jacobian statistic does not describe how these local responses are spatially organized across or away from the data distribution. Our aim is to endow learned representations with mathematical structure beyond observed samples, including regions between samples and regions of low data density. To this end, we introduce a Poisson-based framework that uses an aggregate encoder-Jacobian measure as the source of a domain-wide scalar potential. Green's identity relates the expected source to a boundary-flux term and an interior coupling between the potential gradient and the data score. Because the boundary of the effective data support is unavailable from finite samples, a corruption operator transports observations toward an off-data anchor region that serves as an operational boundary surrogate; the resulting displacements define corruption-induced measurement directions. For Gaussian corruption, the normalized denoising displacement estimates the score of the smoothed data density and aligns the bulk coupling with the denoising task. We examine whether the potential exposes non-local structure beyond its pointwise source, whether its geometry depends on the representation-learning objective, and how different corruption families condition its directional response. Two-dimensional field visualizations show that the potential is smoother and more spatially integrated than the Jacobian source, while matched autoencoder objectives produce distinguishable induced fields. Comparisons across corruption regimes suggest a contrast between diffuse responses and axis-aligned, coordinate-selective responses, while Gaussian and diffusion-style corruptions show visually similar multiscale behavior. The resulting framework provides a mathematical mechanism for organizing representation geometry across sampled, inter-sample, and low-density regions of the domain.