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.
Adler, Jonas, and Ozan Öktem. 2018. “Learned Primal-Dual Reconstruction.” IEEE Transactions on Medical Imaging 37 (6): 1322–32. https://doi.org/10.1109/TMI.2018.2799231.
Borgerding, Mark, and Philip Schniter. 2016. “Onsager-Corrected Deep Networks for Sparse Linear Inverse Problems,” December. http://arxiv.org/abs/1612.01183.
Bui-Thanh, Tan. 2012. “A Gentle Tutorial on Statistical Inversion Using the Bayesian Paradigm.” http://users.ices.utexas.edu/~tanbui/PublishedPapers/BayesianTutorial.pdf.
Daubechies, I., M. Defrise, and C. De Mol. 2004. “An Iterative Thresholding Algorithm for Linear Inverse Problems with a Sparsity Constraint.” Communications on Pure and Applied Mathematics 57 (11): 1413–57. https://doi.org/10.1002/cpa.20042.
Fernández-Martínez, J. L., Z. Fernández-Muñiz, J. L. G. Pallero, and L. M. Pedruelo-González. 2013. “From Bayes to Tarantola: New Insights to Understand Uncertainty in Inverse Problems.” Journal of Applied Geophysics 98 (November): 62–72. https://doi.org/10.1016/j.jappgeo.2013.07.005.
Mosegaard, Klaus, and Albert Tarantola. 1995. “Monte Carlo Sampling of Solutions to Inverse Problems.” Journal of Geophysical Research 100 (B7): 12431. http://www.gfy.ku.dk/~klaus/ip/MT-1995.pdf.
Murray-Smith, Roderick, and Barak A. Pearlmutter. 2005. “Transformations of Gaussian Process Priors.” In Deterministic and Statistical Methods in Machine Learning, edited by Joab Winkler, Mahesan Niranjan, and Neil Lawrence, 110–23. Lecture Notes in Computer Science. Springer Berlin Heidelberg. http://bcl.hamilton.ie/~barak/papers/MLW-Jul-2005.pdf.
O’Callaghan, Simon Timothy, and Fabio T. Ramos. 2011. “Continuous Occupancy Mapping with Integral Kernels.” In Twenty-Fifth AAAI Conference on Artificial Intelligence. https://www.aaai.org/ocs/index.php/AAAI/AAAI11/paper/view/3784.
O’Sullivan, Finbarr. 1986. “A Statistical Perspective on Ill-Posed Inverse Problems.” Statistical Science 1 (4): 502–18. https://doi.org/10.1214/ss/1177013525.
Putzky, Patrick, and Max Welling. 2017. “Recurrent Inference Machines for Solving Inverse Problems,” June. http://arxiv.org/abs/1706.04008.
Schwab, C., and A. M. Stuart. 2012. “Sparse Deterministic Approximation of Bayesian Inverse Problems.” Inverse Problems 28 (4): 045003. https://doi.org/10.1088/0266-5611/28/4/045003.
Stuart, A. M. 2010. “Inverse Problems: A Bayesian Perspective.” Acta Numerica 19: 451–559. https://doi.org/10.1017/S0962492910000061.
Tropp, J. A., and S. J. Wright. 2010. “Computational Methods for Sparse Solution of Linear Inverse Problems.” Proceedings of the IEEE 98 (6): 948–58. https://doi.org/10.1109/JPROC.2010.2044010.
Wei, Qi, Kai Fan, Lawrence Carin, and Katherine A. Heller. 2017. “An Inner-Loop Free Solution to Inverse Problems Using Deep Neural Networks,” September. http://arxiv.org/abs/1709.01841.