Parseval Convolution Operators and Neural Networks
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913471947341824 |
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| author | Unser, Michael Ducotterd, Stanislas |
| author_facet | Unser, Michael Ducotterd, Stanislas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parseval Convolution Operators and Neural Networks Unser, Michael Ducotterd, Stanislas Signal Processing Machine Learning Functional Analysis 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. |
| title | Parseval Convolution Operators and Neural Networks |
| topic | Signal Processing Machine Learning Functional Analysis |
| url | https://arxiv.org/abs/2408.09981 |