A statistical perspective on sparse regularization and geometric modelling
Seminar Room 1, Newton Institute
AbstractConsider a typical inverse problem where we wish to reconstruct an unknown function from a set of measurements. When the function is discretized it is usual for the number of data points to be insufficient to uniquely determine the unknowns – the problem is ill-posed. One approach is to reduce the size of the set of eligible solutions until it contains only a single solution—the problem is regularized. There are, however, infinitely many possible restrictions each leading to a unique solution. Hence the choice of regularization is crucial, but the best choice, even amongst those commonly used, is still difficult. Such regularized reconstruction can be placed into a statistical setting where data fidelity becomes a likelihood function and regularization becomes a prior distribution. Reconstruction then becomes a statistical inference task solved, perhaps, using the posterior mode. The common regularization approaches then correspond to different choices of prior di stribution. In this talk the ideas of regularized estimation, including ridge, lasso, bridge and elastic-net regression methods, will be defined. Application of sparse regularization to basis function expansions, and other dictionary methods, such as wavelets will be discussed. Their link to smooth and sparse regularization, and to Bayesian estimation, will be considered. As an alternative to locally constrained reconstruction methods, geometric models impose a global structure. Such models are usually problem specific, compared to more generic locally constrained methods, but when the parametric assumptions are reasonable they will make better use of the data, provide simpler models and can include parameters which may be used directly, for example in monitoring or control, without the need for extra post-processing. Finally, the matching of modelling and estimation styles with numerical procedures, to produce efficient algorithms, will be discussed.
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