Riemannian pullback-metric curvature analysis

methods / Theoretical / Analytical / Geometric analysis / Riemannian pullback-metric curvature analysis

Riemannian pullback-metric curvature analysis

Techniqueadvanced

Derives a Riemannian pullback metric on a dynamical system's state-space manifold from its own local Jacobian, then computes intrinsic quantities (Gaussian curvature, metric eigenvalue spectrum) at different points/times to show the manifold's shape itself dynamically warps during task execution — distinct from point-cloud curvature estimators (e.g. MAPC), which fit curvature to a static sample of activations rather than deriving it from the system's own dynamics.

Used in (3 observations)

Toy MLP's pullback metric localizes at class boundariesmeasured

structure: Decision boundary (as a codimension-1 hypersurface) · models: Small tanh MLP (AND/OR/XOR on synthetic toroidal/planar input manifolds, rich vs. lazy regimes) · paper: Emergent Riemannian Geometry over Learning Discrete Computations on Continuous Manifolds

A Riemannian metric derived directly from Stable Diffusion's own score-function Jacobian -- not pulled back from any assumed linear latent -- and geodesic-guided interpolation outperforms Lerp/Slerp baselinesmeasured

structure: Score-Jacobian pullback Riemannian metric · models: Stable Diffusion v1.5, Custom diffusion model (MNIST, T=1000 steps) · paper: Image Interpolation with Score-based Riemannian Metrics of Diffusion Models

Task-trained RNN manifolds warp with sign-changing curvature and eigenvalue collapsemeasured

structure: Torus · models: Continuous-time rate RNN (contextual evidence-integration task, SDE-trained), Continuous-time rate RNN (sequential circular-manifold working-memory task, SDE-trained) · paper: RNNs Perform Task Computations by Dynamically Warping Neural Representations