Recovering data from indirect and incoherent observations is a core task in fields like computational imaging, communications and information processing, group testing and others. Such inverse problems are ill-posed and therefore prior structural assumptions are necessary to restrict solutions.
As prototypical examples, compressed sensing and low-rank recovery deal with the problem of recovering a sparse vector or a low-rank matrix from very few compressive observations, far less than its ambient dimension. Fundamental works show that in many cases this can be provably achieved in a robust and stable manner with computationally tractable algorithms.
However, sparsity and low-rankness are simple priors and recovery algorithms often require tuning. It is difficult and often impossible to treat detailed structure and optimal tuning in real-world problems analytically. Recovery approaches, well-understood in theory, perform often sub-optimal in practice. Algorithms converge slowly and increased acquisition time and sampling rates are necessary to achieve a given target resolution.
On the other hand, in many cases neural networks can be trained to empirically achieve high expressivity and the question is how make these ideas accessible to the inverse problem setting.
In this talk I will discuss potential links between the compressed sensing methodology, data-driven approaches for inverse problems and tuning of algorithms. I will first present some recent tuning-free compressed sensing results with applications in communication and group testing showing that strict guarantees are can be obtained in non-standard settings. Then I will focus on how structure and tuning can be incorporated data-driven into recovery algorithms. I will discuss here some ideas and recent results for compressed sensing and phase retrieval which show that substantial improvements in terms of recovery quality and run-time are possible.