Inversion of Integral Models: a Neural Network Approach

Duration: 35 mins 52 secs

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Description:	Ms Cecile Della Valle (Université Paris Saclay) 30th September 2021 \| 11:00 - 11:40


Created:	2021-10-26 10:25
Collection:	Mathematics of deep learning
Publisher:	Isaac Newton Institute for Mathematical Sciences
Copyright:	Ms Cecile Della Valle
Language:	eng (English)


Abstract:	We introduce a neural network architecture to solve inverse problems linked to a one-dimensional integral operator. This architecture is built by unfolding a forward-backward algorithm derived from the minimization of an objective function, which consists of the sum of a data-fidelity function and a Tikhonov-type regularization function. The robustness of this inversion method with respect to a perturbation of the input is theoretically analyzed. Ensuring robustness is consistent with inverse problem theory since it guarantees both the continuity of the inversion method and its insensitivity to small noise. The latter is a critical property as deep neural networks have been shown to be vulnerable to adversarial perturbations. One of the main novelties of our work is to show that the proposed network is also robust to perturbations of its bias. We apply our method to the inversion of Abel integral operators, which define a fractional integration involved in wide range of physical processes.

Abstract:

We introduce a neural network architecture to solve inverse problems linked to a one-dimensional integral operator. This architecture is built by unfolding a forward-backward algorithm derived from the minimization of an objective function, which consists of the sum of a data-fidelity function and a Tikhonov-type regularization function. The robustness of this inversion method with respect to a perturbation of the input is theoretically analyzed. Ensuring robustness is consistent with inverse problem theory since it guarantees both the continuity of the inversion method and its insensitivity to small noise. The latter is a critical property as deep neural networks have been shown to be vulnerable to adversarial perturbations. One of the main novelties of our work is to show that the proposed network is also robust to perturbations of its bias. We apply our method to the inversion of Abel integral operators, which define a fractional integration involved in wide range of physical processes.

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