Diffusion Operator Geometry of Feedforward Representations

Recent work has studied the layerwise geometry of neural representations through curvature on hard neighborhood graphs. We instead use Gaussian-kernel diffusion operators, which provide a smooth description of class transport while retaining spectral, boundary, and local geometric information. Across CIFAR-10 and CIFAR-100 ResNet-18 representations, class transport becomes increasingly persistent with depth while retaining structured inter-class relations. Under matched perturbations, the directed $k$-nearest-neighbor class chain changes $4.2$--$6.8\times$ more than the diffusion class chain on CIFAR-10 and $27.6$--$40.4\times$ more on CIFAR-100 measured in total variation. We characterize when the empirical class chain is an exact Markov quotient and derive both its population counterpart and an affinity-based overlap chain. For balanced shared-covariance Gaussian class-conditionals, the affinities admit closed forms governed by regularized Mahalanobis separation, yielding explicit expressions for leakage and coarse spectral behaviour. These results provide an operator-based alternative to hard graph constructions for studying the geometry and stability of learned representations.