Diffusion Operator Geometry of Feedforward Representations

Neural operators perform well on structured domains, yet their behaviour on irregular geometries remains poorly understood. We show that this limitation is not merely an encoding issue, but a depth-wise failure mode inherent to deep operator architectures. We formalise the Geometric Forgetting Hypothesis: due to the Markovian structure of operator layers and their reliance on global mixing mechanisms, neural operators progressively lose access to domain geometry as depth increases. Using layer-wise geometric probing, we demonstrate that both spectral and attention-based operators systematically lose geometric fidelity. We show that this geometric forgetting degrades accuracy, stability, and generalisation. To counteract it, we introduce a lightweight geometry memory injection mechanism that restores geometric constraints at intermediate depths with minimal architectural overhead. This simple intervention consistently mitigates forgetting and exposes a geometric shortcut instability in transformer-based operators, revealing that geometric retention is a structural requirement rather than a design choice.

Diffusion geometry is a manifold learning framework that uses random walks defined by Markov transition matrices to characterize the geometry of a dataset at multiple scales. We use diffusion geometry for neural representations, incorporating tools from multi-view learning into this field for the first time. Our key technical observation is that a broad class of similarity measures based on representational similarity matrices (RSMs) admits a closed-form equivalent formulation in terms of row-stochastic Markov matrices, opening the door to manipulations from diffusion geometry. As a first application, we develop multi-scale variants of Centered Kernel Alignment and Distance Correlation, which utilise the $t^{th}$ power of the underlying transition matrix to probe the data geometry at adjustable diffusion scales. Going further, we introduce variants of these measures which fuse the Markov matrices of several layers via alternating diffusion into a single operator that captures the network's joint sample geometry, allowing similarity to be computed across multiple layers and shifting the comparison from layer-to-layer to network-to-network. We perform extensive numerical experiments, evaluating our measures on the Representational Similarity (ReSi) benchmark comprising 14 architectures trained on 7 datasets across three different domains. Our methods achieve SoTA results in accuracy and output correlation for both language and vision tasks across different models. We furthermore show SoTA performance on an additional benchmark evaluating on out-of-distribution data.

Recurrent representations are trajectories, but representation geometry is often measured from static snapshots. We develop finite-lag operator geometry for recurrent hidden states from observed source-successor pairs $(X_t,X_{t+Δ})$. The primitive is the conditional transport law $Q_Δ(dy\mid x)$, estimated by a dense Gaussian source-smoothing operator. From this directed finite-lag law we derive a source-centered transport tensor $G_Δ$, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation $W_Δ^ρ$, which summarizes directed lagged flow. We prove affine covariance with explicit metric dependence of scalar summaries, dense estimator stability on bounded trajectory clouds, and a finite-lag separation result showing that source-centered transport detects deterministic recurrent motion not recorded by infinitesimal carre-du-champ geometry. A linear-Gaussian closed form calibrates the quantities in terms of the update $A_Δ$, source covariance, and innovation covariance. Controlled experiments validate the decomposition, circulation, covariance, and stability predictions. In performance matched repeat-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace, while coherent displacement fraction is metric and resolution dependent.