Stochastic Optimization Using Ricci Flow

TL;DR

This paper introduces a novel stochastic optimization framework using Ricci flow to evolve and optimize manifolds approximated by Fourier series, with theoretical error bounds and validation on benchmark functions.

Contribution

It presents a new method combining Ricci flow with Fourier series approximation for manifold optimization, including error bounds and empirical testing on benchmark functions.

Findings

The method effectively optimizes functions with high curvature regions.

The approach provides theoretical error bounds on approximation accuracy.

Empirical results demonstrate utility on five benchmark functions.

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.