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I presented my paper on approximation properties of vectorvalued neural networks at the Analytic and Geometric Approaches to Machine Learning Symposium in Bath (online).
Slides are available here.
Analytic and Geometric Approaches to Machine Learning Symposium 
I submitted a preprint on twolayer neural networks that take values in a Banach space. It extends the results of Bach (2017), E, Ma and Wu (2019), E and Wojtowytsch (2020) and others on approximation rates of infinitely wide scalarvalued twolayer neural networks and establishes MonteCarlo rates in Bochner spaces. The most unexpected result (for me) is that in the vectorvalued case continuity of such neural networks can only be established with respect to the weak$^*$ topology in the target space. This turns out to be a significant restriction for networks with the ReLU activation function.
The preprint can be found here: arxiv:2105.02095
Neural networks with values in a Banach space 
Our paper with Leon Bungert, Martin Burger and Carola Schönlieb on Variational regularisation for inverse problems with imperfect forward operators and general noise models has appeared in Inverse Problems. We analyse inverse problems where the operator contains errors that can be described by an interval in a Banach lattice (e.g., kernel of an integral operator with pointwise errors) and the data are corrupted by noise that can be described using some data fidelity function. Our results apply to, e.g., Gaussian, saltandpepper, Poisson and mixed noise.
The paper is in open acces and can be found here: DOI:10.1088/13616420/abc531
Inverse problems, Banach lattices and general fidelity functions 
Our paper with Tamara Grossmann, Guy Gilboa and Carola Schönlieb on Deeply Learned Spectral Total Variation Decomposition has been accepted to NeurIPS 2020. Congratulations Tamara on your great work!
We compute a nonlinear Total Variation decomposition of an image 10000 faster then classical methods.
Check it out here: arXiv:2006.10004
Deep learning and nonlinear spectral analysis  NeurIPS 2020 
I was awarded an EPSRC Fellowship! The title is “Regularisation theory in the data driven setting”. My goal is to extend regularisation theory to the setting where there is no direct access to the forward operator at the time of solving the inverse problem and only inputoutput training pairs are available. Such pairs can be either collected experimentally or obtained from a computationally expensive model prior to solving the inverse problem. The latter scenario is relevant for timesensitive applications where near realtime reconstructions are required.
Furthermore, the model free setting is the natural habitat of neural networks, and my longterm goal is to better understand their regularisation properties in the context of illposed inverse problems in infinite dimensions.
The fellowship is due to start in April 2021.
EPSRC Postdoctoral Fellowship 
Our paper with Andrea Aspri and Otmar Scherzer on Data Driven Regularization by Projection has appeared in Inverse Problems! We show that regularisation can be defined and rigorously studied in the setting when there is no numerical access to the forward operator and the operator is given only via inputoutput training pairs. Such pairs can be either collected experimentally or obtained from a computationally expensive model prior to solving the inverse problem (which may be useful in timesensitive applications). We show connections to classical regularisation methods such as regularisation by projection and variational regularisation, study stability of the reconstructions and present numerical experiments on data driven inversion of the Radon transform.
The paper is in open acces and can be found here: DOI:10.1088/13616420/abb61b
Data driven regularisation 
I gave a talk about our paper on variational regularisation in Banach lattices at the SIAM Imaging Science conference. Thank you Nicolai Riis for inviting me to your minisympsium! And also many thanks to the organisers Michael Elad and Stacey Levine for organising the conference under current difficult circumstances.
Slides are available here.
SIAM Imaging Science 2020 
Our preprint with Tamara Grossmann, Guy Gilboa and Carola Schönlieb on Deeply Learned Spectral Total Variation Decomposition is online! We compute a nonlinear Total Variation decomposition of an image 10000 faster then classical methods.
Check it out here: arXiv:2006.10004
Deep learning and nonlinear spectral analysis 
Our preprint with Leon Bungert, Martin Burger and Carola Schönlieb on Variational regularisation for inverse problems with imperfect forward operators and general noise models is now online! We analyse inverse problems where the operator contains errors that can be described by an interval in a Banach lattice (e.g., kernel of an integral operator with pointwise errors) and the data are corrupted by noise that can be described using some data fidelity function. Our results apply to, e.g., Gaussian, saltandpepper, Poisson and mixed noise.
Check it out here: arXiv:2005.14131
Inverse problems, Banach lattices and general fidelity functions 
Our preprint with Leon Bungert and Martin Burger on Linfinity variational problems and relations to distance functions is now online! We show that the distance function is the unique ground state of a certain Linfinity variational problem and devise an efficient numerical algorithm for computing distance functions on graphs.
Check it out here: arXiv:2001.07411
Computing distance functions using gradient flows
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