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NESTANets: stable, accurate and efficient neural networks for analysis-sparse inverse problems

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Abstract

Solving inverse problems is a fundamental component of science, engineering and mathematics. With the advent of deep learning, deep neural networks have significant potential to outperform existing state-of-the-art, model-based methods for solving inverse problems. However, it is known that current data-driven approaches face several key issues, notably hallucinations, instabilities and unpredictable generalization, with potential impact in critical tasks such as medical imaging. This raises the key question of whether or not one can construct deep neural networks for inverse problems with explicit stability and accuracy guarantees. In this work, we present a novel construction of accurate, stable and efficient neural networks for inverse problems with general analysis-sparse models, termed NESTANets. To construct the network, we first unroll NESTA, an accelerated first-order method for convex optimization. The slow convergence of this method leads to deep networks with low efficiency. Therefore, to obtain shallow, and consequently more efficient, networks we combine NESTA with a novel restart scheme. We then use compressed sensing techniques to demonstrate accuracy and stability. We showcase this approach in the case of Fourier imaging, and verify its stability and performance via a series of numerical experiments. The key impact of this work is demonstrating the construction of efficient neural networks based on unrolling with guaranteed stability and accuracy.

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Notes

  1. The full quote reads: “Such hallucinatory features are not acceptable and especially problematic if they mimic normal structures that are either not present or actually abnormal.”

  2. For example, in the work On hallucinations in tomographic image reconstruction [21] the authors write “The potential lack of generalization of deep learning-based reconstruction methods as well as their innate unstable nature may cause false structures to appear in the reconstructed image that are absent in the object being imaged”.

  3. Specifically, in [84, Thm. 2] the objective error bound is given in terms of \(\hat{x}_{\mu }\), where \(\hat{x}_{\mu }\) is an optimal solution of the smoothed problem (4). But the proof does not use the optimality property in any way, only feasibility. It therefore applies to any feasible vector x. This minor modification is convenient for later developments.

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Acknowledgements

BA acknowledges support from NSERC through Grant R611675. MNN acknowledges support from an NSERC CGS-M scholarship. Both authors would like to thank Vegard Antun and Matthew Colbrook for helpful advice and feedback.

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  1. Department of Mathematics, Simon Fraser University, Burnaby, Canada

    Maksym Neyra-Nesterenko & Ben Adcock

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Communicated by Ron Levie.

The original online version of this article was revised: The first equation line on page 6 has been corrected. The correct version reads: “\(\sigma _s(z)_{\ell ^1} = \min \left\{ \Vert z - u_S \Vert _{\ell ^1} \ : \ u \in \mathbb {C}^M, \ S \subseteq [M], \ |S| \le s \right\} \)”

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Neyra-Nesterenko, M., Adcock, B. NESTANets: stable, accurate and efficient neural networks for analysis-sparse inverse problems. Sampl. Theory Signal Process. Data Anal. 21, 4 (2023). https://doi.org/10.1007/s43670-022-00043-5

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