# Testing that a Local Optimum of the Likelihood is Globally Optimum using Reparameterized Embeddings

## TL;DR

This paper introduces a statistical test to verify if a local maximum of the likelihood function is actually the global maximum, using reparameterization to improve accuracy and computational efficiency in non-convex imaging problems.

## Contribution

It proposes a novel reparameterization approach for likelihood functions to enhance global maximum testing in non-convex optimization problems.

## Key findings

- Improved accuracy of global maximum testing through reparameterization.
- Reduced computational cost in practical imaging applications.
- Validated method on camera-blur estimation problem.

## Abstract

Many mathematical imaging problems are posed as non-convex optimization problems. When numerically tractable global optimization procedures are not available, one is often interested in testing ex post facto whether or not a locally convergent algorithm has found the globally optimal solution. When the problem is formulated in terms of maximizing the likelihood function under a statistical model for the measurements, one can construct a statistical test that a local maximum is in fact the global maximum. A one-sided test is proposed for the case that the statistical model is a member of the generalized location family of probability distributions, a condition often satisfied in imaging and other inverse problems. We propose a general method for improving the accuracy of the test by reparameterizing the likelihood function to embed its domain into a higher dimensional parameter space. We show that the proposed global maximum testing method results in improved accuracy and reduced computation for a physically-motivated joint-inverse problem arising in camera-blur estimation.