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

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 ma...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2023-01, Vol.71, p.1-14
Main Authors: Bohra, Pakshal, Pla, Pol del Aguila, Giovannelli, Jean-Francois, Unser, Michael
Format: Article
Language:English
Subjects:
Algorithms
Artificial neural networks
Biomedical measurement
Computational modeling
Computer architecture
convolutional neural networks
Engineering Sciences
Error analysis
Image reconstruction
Imaging
Inverse problems
minimum mean-square error
Neural networks
Optimization
Reconstruction
Sampling
sparse stochastic processes
Stochastic processes
Training
Training data
Variational methods
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Summary: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.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2023.3282062