A Statistical Framework to Investigate the Optimality of Signal-Reconstruction Methods

TL;DR

This paper introduces a statistical framework for benchmarking the optimality of signal reconstruction algorithms, especially neural networks, in linear inverse problems using synthetic sparse signals and Gibbs sampling.

Contribution

It develops a novel framework that quantitatively assesses the optimality of various reconstruction methods, including neural networks, for signals modeled as sparse stochastic processes.

Findings

Neural networks outperform variational methods in many cases.

Performance of neural networks declines for heavy-tailed distributions.

Framework provides quantitative measures of reconstruction optimality.

Abstract

We present a statistical framework to benchmark the performance of reconstruction algorithms for linear inverse problems, in particular, neural-network-based methods that require large quantities of training data. We generate synthetic signals as realizations of sparse stochastic processes, which makes them ideally matched to variational sparsity-promoting techniques. We derive Gibbs sampling schemes to compute the minimum mean-square error estimators for processes with Laplace, Student's t, and Bernoulli-Laplace innovations. These allow our framework to provide quantitative measures of the degree of optimality (in the mean-square-error sense) for any given reconstruction method. We showcase our framework by benchmarking the performance of some well-known variational methods and convolutional neural network architectures that perform direct nonlinear reconstructions in the context of deconvolution and Fourier sampling. Our experimental results support the understanding that, while these neural networks outperform the variational methods and achieve near-optimal results in many settings, their performance deteriorates severely for signals associated with heavy-tailed distributions.