The Living Thing / Notebooks :

Inverse problems

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.