Bayesian Reimaging of Sparsity in Inverse Problems - SIAM-IS Virtual Seminar

Explore Bayesian approaches to sparse inverse problems in this SIAM-IS Virtual Seminar Series talk. Delve into efficient computational frameworks for Bayesian sparse inverse solvers, focusing on hierarchical prior models that promote sparsity. Learn about inner-outer iteration schemes, dynamic scaling weight updates, and the application of Conjugate Gradient methods for least squares problems. Discover how this approach can be applied to imaging and dictionary learning, with examples demonstrating its performance. Gain insights into sparsity as a belief, sparsity-promoting priors, and the Iterated Alternating Sequential (IAS) algorithm. Examine computed examples, including 1D tests, starry night reconstructions, and applications to MNIST handwritten digit datasets with varying noise levels.

Syllabus

Intro
Linear Sparse Inverse Problems
Sparsity as a Belief
Sparsity in a Bayesian Way
Sparsity promoting priors
Hierarchical prior model for sparsity
The Bayesian solution is the posterior
Iterated Alternating Sequential (IAS) algorithm
IAS algorithm for Generalized Gamma hyperpriors
Approximate IAS and reduced model: the x update
The Gamma hyperprior (r = 1)
From Gamma to Generalized Gamma hyperprior
Convexity region
Special Generalized Gamma hyperpriors
Global Hybrid IAS
Support of the signal the meaning of
Computed example: 1D test
Computed examples: starry night
Dictionary Learning
MNIST Data: Handwritten digits
MNIST Dataset example: mismatch noise 0.01
MNIST Dataset: mismatch noise 0.05