Bayesian sparse reconstruction: a brute-force approach to astronomical imaging and machine learning

TL;DR

This paper introduces a Bayesian framework for signal reconstruction that automatically determines the optimal basis functions and their parameters, improving efficiency and applicability to astronomical images and neural networks.

Contribution

It presents a novel Bayesian approach for sparse reconstruction that treats the number and type of basis functions as parameters, enabling data-driven model selection and increased computational efficiency.

Findings

Order-of-magnitude speed-up in Bayesian evidence calculation

Effective application to noisy astronomical images

Potential extension to neural network architecture optimization

Abstract

We present a principled Bayesian framework for signal reconstruction, in which the signal is modelled by basis functions whose number (and form, if required) is determined by the data themselves. This approach is based on a Bayesian interpretation of conventional sparse reconstruction and regularisation techniques, in which sparsity is imposed through priors via Bayesian model selection. We demonstrate our method for noisy 1- and 2-dimensional signals, including astronomical images. Furthermore, by using a product-space approach, the number and type of basis functions can be treated as integer parameters and their posterior distributions sampled directly. We show that order-of-magnitude increases in computational efficiency are possible from this technique compared to calculating the Bayesian evidences separately, and that further computational gains are possible using it in combination with dynamic nested sampling. Our approach can also be readily applied to neural networks, where it allows the network architecture to be determined by the data in a principled Bayesian manner by treating the number of nodes and hidden layers as parameters.