Continuous generative models for inverse problems

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• Matteo Santacesaria (Università degli Studi di Genova)
• Tuesday 28 March 2023, 16:10-17:00
• Seminar Room 1, Newton Institute.

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RNTW02 - Rich and non-linear tomography in medical imaging, materials and non destructive testing

Generative models are a large class of deep learning architectures, trained to describe a subset of a high dimensional space with a small number of parameters. Popular models include variational autoencoders, generative adversarial networks, normalizing flows and, more recently, score-based diffusion models. In the context of inverse problems, generative models can be used to model prior information on the unknown with a higher level of accuracy than classical regularization methods. In this talk we will present a new data-driven approach to solve inverse problems based on generative models. Taking inspiration from well-known convolutional architectures, we construct and explicitly characterize a class of injective generative models defined on infinite dimensional functions spaces. The construction is based on wavelet multi resolution analysis: one of the key theoretical novelties is the generalization of the strided convolution between discrete signals to an infinite dimensional setting. After an off-line training of the generative model, the proposed reconstruction method consists in an iterative scheme in the low-dimensional latent space. The main advantages are the faster iterations and the reduced ill-posedness, which is shown with new Lipschitz stability estimates. We also present numerical simulations validating the theoretical findings for linear and nonlinear inverse problems such as electrical impedance tomography. This is joint work in collaboration with G.S. Alberti, J. Hertrich and S. Sciutto.

This talk is part of the Isaac Newton Institute Seminar Series series.

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