Stochastic Optimization Using Ricci Flow
Varsha Gupta
TL;DR
This work tackles stochastic optimization by encoding the objective landscape as a smooth manifold constructed via Fourier wave superposition. It then guides search using Ricci-flow inspired deformations of the induced Riemannian metric, exploiting curvature blow-ups to signal candidate optima. The framework provides error bounds for the surrogate and demonstrates the approach on five benchmark functions, with enhanced accuracy achieved through iterative refinement near promising regions. Overall, the method offers a topology-aware, geometry-guided pathway to locate global optima in highly non-convex settings, with practical improvements through hybrid re-sampling around identified optima.
Abstract
This paper proposes a theoretical framework for modeling and optimizing the bounded functions based on the Fourier series approximation and Ricci flow. Specifically, the initial manifold, $\mathcal{M}_0$ is approximated using Fourier series approximation in conjunction with the center and boundary sampling procedure introduced in the paper. The manifold is iteratively evolved using an algorithm that involves sampling along geodesic hyper-sphere defined by the Riemannian metric tensor. Thus obtained surrogate manifold is optimized by applying inverse Ricci flow i.e. instead of regularizing the manifold, flow allows for the high curvature regions to blow into finite time singularities. This allows for the singularities to occur at potential global optima assuming the deviation of the manifold at any point is smaller than the optimum. In addition, the error bound is established on the accuracy of the surrogate manifold. Finally, the proposed method is tested on stochastic sampling from five benchmark functions to illustrate the utility of this method.