As seen in tomography, sparsity constraints, solvers, variational inference, deconvolution…
Robert Ackroyd had some nice phrasing around the connections (indeed isomorphisms) between statistical estimation theory and inverse problem solving.
I thought I had something to say about this general perspective on inverse problems, but I don’t yet.
Interesting specific framings
Leaning to reconstruct has an interesting partly-learned, partly designed reconstruction operator trick.
- Borgerding, M., & Schniter, P. (2016) Onsager-Corrected Deep Networks for Sparse Linear Inverse Problems. arXiv:1612.01183 [Cs, Math].
- Bui-Thanh, T. (2012) A Gentle Tutorial on Statistical Inversion using the Bayesian Paradigm.
- Daubechies, I., Defrise, M., & De Mol, C. (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 57(11), 1413–1457. DOI.
- Mosegaard, K., & Tarantola, A. (1995) Monte Carlo sampling of solutions to inverse problems. Journal of Geophysical Research, 100(B7), 12431.
- O’Sullivan, F. (1986) A Statistical Perspective on Ill-Posed Inverse Problems. Statistical Science, 1(4), 502–518. DOI.
- Schwab, C., & Stuart, A. M.(2012) Sparse deterministic approximation of Bayesian inverse problems. Inverse Problems, 28(4), 045003. DOI.
- Stuart, A. M.(2010) Inverse problems: A Bayesian perspective. Acta Numerica, 19, 451–559. DOI.
- Tropp, J. A., & Wright, S. J.(2010) Computational Methods for Sparse Solution of Linear Inverse Problems. Proceedings of the IEEE, 98(6), 948–958. DOI.