Develop a general theory for nonlinear forward operators in self-supervised inverse problems

Develop a general theoretical framework that characterizes self-supervised learning losses and guarantees for inverse problems with nonlinear forward operators A:R^n→R^m, including problems such as phase retrieval, inverse scattering, and quantized sensing, extending beyond analyses restricted to linear operators.

Background

Many real-world inverse problems are nonlinear, e.g., quantized sensing, phase retrieval, and inverse scattering governed by PDEs. While the self-supervised losses introduced in the manuscript can, in principle, be applied to nonlinear forward models, most existing analyses and guarantees treat only the linear case.

The authors identify the lack of a unified theoretical understanding of self-supervised learning in nonlinear inverse problems as a core gap.

References

While, in principle, most of the self-supervised losses presented in~\Cref{chap: multioperators} can be applied with non-linear forward models, most of the theoretical analyses associated with these losses are restricted to the linear case and the development of a general theoretical framework for nonlinear operators is an open problem.

— Self-Supervised Learning from Noisy and Incomplete Data  (2601.03244 - Tachella et al., 6 Jan 2026) in Chapter 5 (Extensions and open problems), Section "Non-linear inverse problems"

(iii) Phase retrieval as a stochastic unrolled solver with a jointly-trained differentiable selector, where ``correct'' is defined by the data consistency the solver itself enforces rather than by a label---the one avenue the non-uniqueness analysis leaves open.

— CNet: A Complex-Valued Deep Learning Framework with Wirtinger Autodifferentiation and FFT--Hadamard Convolution  (2610.08592 - Crasmaru, 6 Oct 2026) in Section 1, subsection “The Fourier phase problem (coherent diffraction / crystallography)”; Section “Outlook: a research program,” paragraph “Concrete follow-on experiments”