Multilevel Geometric Optimization for Regularised Constrained Linear Inverse Problems

TL;DR

This paper introduces a multilevel geometric optimization method for box-constrained linear inverse problems, leveraging hierarchical models and Riemannian geometry to improve efficiency while maintaining feasibility.

Contribution

It extends multigrid techniques to Riemannian constrained optimization, enabling efficient and feasible solutions for regularized inverse problems.

Findings

Speeds up convergence by using coarser models for search directions

Preserves feasibility through geometric hierarchy

Extends multigrid components to Riemannian structures

Abstract

We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are accurate but expensive to compute, while coarser models are less accurate but cheaper to compute. When working at the fine level, multilevel optimisation computes the search direction based on a coarser model which speeds up updates at the fine level. Moreover, exploiting geometry induced by the hierarchy the feasibility of the updates is preserved. In particular, our approach extends classical components of multigrid methods like restriction and prolongation to the Riemannian structure of our constraints.