Testing that a Local Optimum of the Likelihood is Globally Optimum using Reparameterized Embeddings
This addresses the challenge of global optimality verification in imaging and inverse problems, though it is incremental as it builds on existing statistical testing frameworks.
The authors tackled the problem of verifying whether a local optimum in non-convex optimization for imaging is globally optimal by proposing a statistical test for likelihood maximization in generalized location families, and they improved its accuracy using reparameterized embeddings, showing enhanced accuracy and reduced computation in a camera-blur estimation example.
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.