Maxwell Institute Afternoon on Inverse Problems

Title: Challenges in high dimensional filtering

Abstract: In various application areas it is crucial to make predictions or decisions based on sequentially incoming observations and previous existing knowledge on the system of interest. Here we will focus on estimating space weather and thermodynamical modeling of asteroids. 

The prior knowledge is often given in the form of evolution equations (e.g., ODEs/PDEs derived via first principles or fitted based on previously collected data), from here on referred to as model. Despite the available observation and prior model information, accurate predictions of the “true“ reference dynamics can be very difficult. Common reasons that make this problem so challenging are: (i) the underlying system is extremely complex (e.g., highly nonlinear) and chaotic (i.e., crucially dependent on the initial conditions), (ii) the associate state and/or parameter space is very high dimensional (e.g., worst case 10^8) (iii) Observations are noisy, partial in space and discrete in time.

In practice these obstacles are combated with a series of approximations (the most important ones being based on assuming Gaussian densities and using Monte Carlo type estimations) and numerical tools that work surprisingly well in some settings. Yet the mathematical understanding of the signal tracking ability of a lot of these methods is still lacking. Additionally, solutions of some of the more complicated problems that require simultaneous state and parameter estimation (including control parameters that can be understood as decisions/actions performed) can still not be approximated in a computationally feasible fashion. Here we will try to address the first layer of these issues step by step and outline the next advances that need to be made in these many layered problems.

Title: On posterior consistency in non-linear Bayesian data assimilation problems

Abstract: Bayesian methods are widely used in statistical data assimilation tasks. They model infinite-dimensional aspects of possibly non-linear dynamical systems — such as initial conditions, or diffusivity parameters -- by Gaussian process (or similar) prior probability measures. The posterior distribution is obtained from updating the system from discrete and noisy measurements of the observed dynamics, and can often be approximately computed by filtering or MCMC methods. In non-linear settings, such posterior distributions are not themselves Gaussian any longer, and very little is known rigorously about the statistical behaviour of the updated non-linear systems. In this talk we will explain how recent developments in the theory of nonlinear Bayesian inverse problems can be used to prove the statistical large sample consistency of posterior inferences in two representative PDE models arising with a) discretely sampled multi-dimensional SDEs as well as b) Eulerian measurements of the (2D) Navier Stokes equations.

Title: A Bayesian model for localizing radiation sources with the Compton Imager

Abstract: In this work, we study a Compton Imager, made of an array of scintillation crystals. This imaging system differs from a more classical Compton camera since the sensor array acts simultaneously as a set of scatterers and absorbers. From the recorded data, the objective is to localize the positions of point-like sources responsible for the emission of the measured radiation. The inverse problem is formulated within a Bayesian framework, and a Markov chain Monte Carlo method is investigated to infer the source locations.

Title: Learning to Image

Abstract: Today modern machine learning offers the state-of-the art solutions for compressed and computational imaging, exploiting the sophisticated statistical dependencies within images. However, such solutions usually require unrealistic access to a large quantity of ground truth images for training. This suggests that we need to have "solved" the imaging problem before it is amenable to machine learning technniques and has led to a big drive to try to train such systems without ground truth image data.

In the talk, I will consider dimension-reducing linear inverse problems, such as accelerated (undersampled) MRI or sparse angle CT imaging, and will discuss a new theoretical and algorithmic framework for learning the image reconstruction mapping in such ill-posed inverse problems using only measurement data for training. The theory suggests that in many cases with mild prior assumptions we are able to learn to image from incomplete measurements. Furthermore it is not dependent on any particular machine learning architecture. I will conclude by discussing some initial results using a new class of self-supervised learning algorithms, called equivariant imaging, that have demonstrated performance for MRI and CT imaging tasks that is on a par with solutions trained in a fully supervised manner.

 Title: Expectation-Propagation for uncertainty quantification with low-rank constraints

Abstract: Bayesian inference in high-dimensional problems beyond first-order statistics is challenging and exact methods remain generally computationally expensive. Variational inference thus stands as a promising alternative, provided that scalable algorithms  can be adopted. 

In this talk, we will consider the problem of scalable approximate inference and covariance estimation for linear inverse problems using Expectation-Propagation (EP). Traditional EP methods rely on Gaussian approximations with either diagonal or full covariance structures. Full covariance matrices can capture correlation but do not scale well as the dimension of the problem increases, while diagonal matrices scale better but omit potentially important correlations. Here we will consider low-rank decompositions within EP for linear regression. The potential benefits of such covariance structures will be illustrated via a sparse linear regression problem and a more challenging spectral unmixing problem where the sparse mixing coefficients are, in addition, subject to positivity constraints. We will conclude with preliminary results obtained when using EP to train spiking neural networks.