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.

## Refs

- BoSc16
- Borgerding, M., & Schniter, P. (2016) Onsager-Corrected Deep Networks for Sparse Linear Inverse Problems.
*arXiv:1612.01183 [Cs, Math]*. - Buit12
- Bui-Thanh, T. (2012) A Gentle Tutorial on Statistical Inversion using the Bayesian Paradigm.
- DaDD04
- 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. - MoTa95
- Mosegaard, K., & Tarantola, A. (1995) Monte Carlo sampling of solutions to inverse problems.
*Journal of Geophysical Research*, 100(B7), 12431. - Osul86
- O’Sullivan, F. (1986) A Statistical Perspective on Ill-Posed Inverse Problems.
*Statistical Science*, 1(4), 502–518. DOI. - ScSt12
- Schwab, C., & Stuart, A. M.(2012) Sparse deterministic approximation of Bayesian inverse problems.
*Inverse Problems*, 28(4), 045003. DOI. - Stua10
- Stuart, A. M.(2010) Inverse problems: A Bayesian perspective.
*Acta Numerica*, 19, 451–559. DOI. - TrWr10
- 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.