Overcomplete representation in a hierarchical Bayesian framework

06/24/2020
by   Monica Pragliola, et al.
0

A common task in inverse problems and imaging is finding a solution that is sparse, in the sense that most of its components vanish. In the framework of compressed sensing, general results guaranteeing exact recovery have been proven. In practice, sparse solutions are often computed combining ℓ_1-penalized least squares optimization with an appropriate numerical scheme to accomplish the task. A computationally efficient alternative for finding sparse solutions to linear inverse problems is provided by Bayesian hierarchical models, in which the sparsity is encoded by defining a conditionally Gaussian prior model with the prior parameter obeying a generalized gamma distribution. An iterative alternating sequential (IAS) algorithm has been demonstrated to lead to a computationally efficient scheme, and combined with Krylov subspace iterations with an early termination condition, the approach is particularly well suited for large scale problems. Here the Bayesian approach to sparsity is extended to problems whose solution allows a sparse coding in an overcomplete system such as composite frames. It is shown that among the multiple possible representations of the unknown, the IAS algorithm, and in particular, a hybrid version of it, is effectively identifying the most sparse solution. Computed examples show that the method is particularly well suited not only for traditional imaging applications but also for dictionary learning problems in the framework of machine learning.

READ FULL TEXT

page 9

page 10

page 11

page 13

research
03/29/2023

Computationally efficient sampling methods for sparsity promoting hierarchical Bayesian models

Bayesian hierarchical models have been demonstrated to provide efficient...
research
05/19/2022

Hierarchical Ensemble Kalman Methods with Sparsity-Promoting Generalized Gamma Hyperpriors

This paper introduces a computational framework to incorporate flexible ...
research
02/07/2023

Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform

Compressed sensing allows for the recovery of sparse signals from few me...
research
03/14/2020

Sparsity promoting hybrid solvers for hierarchical Bayesian inverse problems

The recovery of sparse generative models from few noisy measurements is ...
research
06/15/2010

Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in ...
research
11/05/2019

Penalized least squares and sign constraints with modified Newton-Raphson algorithms: application to EEG source imaging

We propose a modified Newton-Raphson (MNR) algorithm to estimate multipl...
research
03/14/2020

Hybrid solver for hierarchical Bayesian inverse problems

The recovery of sparse generative models from few noisy measurements is ...

Please sign up or login with your details

Forgot password? Click here to reset