Uncertainty Quantification for high-dimensional inverse problems

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Jason sent me the following a few months ago: 

Hi Igor,
Just wanted to draw your attend to a couple of papers we had published recently on uncertainty quantification for high-dimensional inverse problems:
Uncertainty quantification for radio interferometric imaging: I. proximal MCMC methods
Xiaohao Cai, Marcelo Pereyra, Jason D. McEwen
https://arxiv.org/abs/1711.04818
http://dx.doi.org/10.1093/mnras/sty2004


Uncertainty quantification for radio interferometric imaging: II. MAP estimation
Xiaohao Cai, Marcelo Pereyra, Jason D. McEwen
https://arxiv.org/abs/1711.04819
http://dx.doi.org/10.1093/mnras/sty2015


These articles target radio interferometric imaging but the techniques are general so I thought they might be of interest to your readers.

Many thanks!

Best,
Jason
--
www.jasonmcewen.org

Thanks Jason !

Uncertainty quantification for radio interferometric imaging: I. proximal MCMC methods by Xiaohao Cai, Marcelo Pereyra, Jason D. McEwen

Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Since radio interferometric imaging requires solving a high-dimensional, ill-posed inverse problem, uncertainty quantification is difficult but also critical to the accurate scientific interpretation of radio observations. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from which uncertainties can then be quantified. However, traditional high-dimensional sampling methods are generally limited to smooth (e.g. Gaussian) priors and cannot be used with sparsity-promoting priors. Sparse priors, motivated by the theory of compressive sensing, have been shown to be highly effective for radio interferometric imaging. In this article proximal MCMC methods are developed for radio interferometric imaging, leveraging proximal calculus to support non-differential priors, such as sparse priors, in a Bayesian framework. Furthermore, three strategies to quantify uncertainties using the recovered posterior distribution are developed: (i) local (pixel-wise) credible intervals to provide error bars for each individual pixel; (ii) highest posterior density credible regions; and (iii) hypothesis testing of image structure. These forms of uncertainty quantification provide rich information for analysing radio interferometric observations in a statistically robust manner.

Uncertainty quantification for radio interferometric imaging: II. MAP estimation by Xiaohao Cai, Marcelo Pereyra, Jason D. McEwen

Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from which uncertainties can then be quantified. However, for massive data sizes, like those anticipated from the Square Kilometre Array (SKA), it will be difficult if not impossible to apply any MCMC technique due to its inherent computational cost. We formulate Bayesian inference problems with sparsity-promoting priors (motivated by compressive sensing), for which we recover maximum a posteriori (MAP) point estimators of radio interferometric images by convex optimisation. Exploiting recent developments in the theory of probability concentration, we quantify uncertainties by post-processing the recovered MAP estimate. Three strategies to quantify uncertainties are developed: (i) highest posterior density credible regions; (ii) local credible intervals (cf. error bars) for individual pixels and superpixels; and (iii) hypothesis testing of image structure. These forms of uncertainty quantification provide rich information for analysing radio interferometric observations in a statistically robust manner. Our MAP-based methods are approximately 105 times faster computationally than state-of-the-art MCMC methods and, in addition, support highly distributed and parallelised algorithmic structures. For the first time, our MAP-based techniques provide a means of quantifying uncertainties for radio interferometric imaging for realistic data volumes and practical use, and scale to the emerging big-data era of radio astronomy.

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