As seen in tomography, sparsity constraints, solvers, variational inference, deconvolution…

Robert Ackroyd had some nice phrasing around the connections 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

- Buit12: Tan Bui-Thanh (2012)
*A Gentle Tutorial on Statistical Inversion using the Bayesian Paradigm* - Osul86: Finbarr O’Sullivan (1986) A Statistical Perspective on Ill-Posed Inverse Problems.
*Statistical Science*, 1(4), 502–518. DOI - WFCH17: Qi Wei, Kai Fan, Lawrence Carin, Katherine A. Heller (2017) An inner-loop free solution to inverse problems using deep neural networks.
*ArXiv:1709.01841 [Cs]*. - DaDD04: I. Daubechies, M. Defrise, C. De Mol (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.
*Communications on Pure and Applied Mathematics*, 57(11), 1413–1457. DOI - TrWr10: J. A. Tropp, S. J. Wright (2010) Computational Methods for Sparse Solution of Linear Inverse Problems.
*Proceedings of the IEEE*, 98(6), 948–958. DOI - FFPP13: J. L. Fernández-Martínez, Z. Fernández-Muñiz, J. L. G. Pallero, L. M. Pedruelo-González (2013) From Bayes to Tarantola: New insights to understand uncertainty in inverse problems.
*Journal of Applied Geophysics*, 98, 62–72. DOI - Stua10: A. M. Stuart (2010) Inverse problems: A Bayesian perspective.
*Acta Numerica*, 19, 451–559. DOI - AdÖk18: Jonas Adler, Ozan Öktem (2018) Learned Primal-dual Reconstruction.
*IEEE Transactions on Medical Imaging*, 37(6), 1322–1332. DOI - MoTa95: Klaus Mosegaard, Albert Tarantola (1995) Monte Carlo sampling of solutions to inverse problems.
*Journal of Geophysical Research*, 100(B7), 12431. - BoSc16: Mark Borgerding, Philip Schniter (2016) Onsager-Corrected Deep Networks for Sparse Linear Inverse Problems.
*ArXiv:1612.01183 [Cs, Math]*. - PuWe17: Patrick Putzky, Max Welling (2017) Recurrent Inference Machines for Solving Inverse Problems.
*ArXiv:1706.04008 [Cs]*. - ScSt12: C. Schwab, A. M. Stuart (2012) Sparse deterministic approximation of Bayesian inverse problems.
*Inverse Problems*, 28(4), 045003. DOI