Parseval Convolution Operators and Neural Networks
<|reference_start|>Parseval Convolution Operators and Neural Networks: We first establish a kernel theorem that characterizes all linear shift-invariant (LSI) operators acting on discrete multicomponent signals. This result naturally leads to the identification of the Parseval convolution operators as the class of energy-preserving filterbanks. We then present a constructive approach for the design/specification of such filterbanks via the chaining of elementary Parseval modules, each of which being parameterized by an orthogonal matrix or a 1-tight frame. Our analysis is complemented with explicit formulas for the Lipschitz constant of all the components of a convolutional neural network (CNN), which gives us a handle on their stability. Finally, we demonstrate the usage of those tools with the design of a CNN-based algorithm for the iterative reconstruction of biomedical images. Our algorithm falls within the plug-and-play framework for the resolution of inverse problems. It yields better-quality results than the sparsity-based methods used in compressed sensing, while offering essentially the same convergence and robustness guarantees.<|reference_end|>
@article{unser2024parseval,
    title={Parseval Convolution Operators and Neural Networks},
    author={Michael Unser, Stanislas Ducotterd},
    journal={arXiv preprint arXiv:2408.09981},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.09981},
    primaryClass={eess.SP cs.LG math.FA stat.ML}
}