Geometric flow regularization in latent spaces for smooth dynamics with the efficient variations of curvature

Abstract

We design strategies in nonlinear geometric analysis to temper the effects of adversarial learning for sufficiently smooth data of numerical method-type dynamics in encoder-decoder methods, variational and deterministic, through the use of geometric flow regularization. We augment latent spaces with geometric flows to control structure, relying on adaptations of curvature and Ricci flow. All of our flows are solved using physics-informed learning. Traditional geometric meaning is traded for computing ability, but we maintain key geometric invariants, the primary of which are maintained, intrinsically-low structure, nontriviality due to sufficient lower bounds on curvature, distortion of volume element, that develop quality in the inference stage. We instill representations that are canonical, smooth, curvature-aware, geodesic-aware, and non-topologically void or sparse. The primary bottleneck of a Ricci curvature flow is that Ricci curvature is high order, thus expensive to compute, so we will attempt to overcome this with properly justified proxies. Our primary contributions are fourfold. We develop a loss based on Gaussian curvature using closed path circulation integration for surfaces, bypassing automatic differentiation of the Christoffel symbols through use of Stokes' theorem. We invent a new parametric flow valid under a Taylor expansion derived from the Gauss equation. We develop two strategies based on time differentiation of functionals, one with a special case of scalar curvature for conformally-changed metrics, and another with harmonic maps, their energy, and induced metrics. Our losses are diminished analytically and mostly heuristic but maintain overall integral latent structure. We showcase that curvature flows and the formulation of geometric structure in intermediary encoded settings enhance learning and overall zero-shot and adversarial fidelity.