Diffusion Operator Geometry of Feedforward Representations

• URL: http://arxiv.org/abs/2605.01107v1
• Date: Fri, 01 May 2026 21:27:26 GMT
• Title: Diffusion Operator Geometry of Feedforward Representations
• Abstract summary: Neural networks transform data through learned representations whose geometry affects separation, contraction, and generalization.<n>Recent work studies this geometry using discrete curvature on neighborhood graphs, suggesting Ricci-flow-like behavior across layers.<n>We develop a smooth operator-theoretic alternative for feedforward representation snapshots.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Neural networks transform data through learned representations whose geometry affects separation, contraction, and generalization. Recent work studies this geometry using discrete curvature on neighborhood graphs, suggesting Ricci-flow-like behavior across layers. We develop a smooth operator-theoretic alternative for feedforward representation snapshots. Each feature cloud induces a Gaussian-kernel diffusion Markov operator, and transport, spectral, label-boundary, and local-scale observables are derived from this single object via Bakry-Emery $Γ$-calculus. In a balanced Gaussian class-conditional snapshot model with shared covariance, the population operator has closed-form class affinities, leakage, and coarse spectra, all controlled by pairwise regularized Mahalanobis separations $c_\varepsilon^{(a,b)}$. We also prove that the resulting operator observables vary smoothly under feature perturbations, while hard neighborhood-graph diagnostics can change discontinuously. Synthetic experiments validate the closed-form Gaussian bridge, while learned MNIST experiments show that the same operator observables track training, width, and perturbation stability. Together, these results give a stable operator-geometric framework for analyzing feedforward representation geometry.

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