Aggregation of regularized solutions from multiple observation models

Abstract

Joint inversion of multiple observation models has important applications in many disciplines including geoscience, image processing and computational biology. One of the methodologies for joint inversion of ill-posed observation equations naturally leads to multi-parameter regularization, which has been intensively studied over the last several years. However, problems such as the choice of multiple regularization parameters remain unsolved. In the present study, we discuss a rather general approach to the regularization of multiple observation models, based on the idea of the linear aggregation of approximations corresponding to different values of the regularization parameters. We show how the well-known linear functional strategy can be used for such an aggregation and prove that the error of a constructive aggregator differs from the ideal error value by a quantity of an order higher than the best guaranteed accuracy from the most trustable observation model. The theoretical analysis is illustrated by numerical experiments with simulated data.

Galaxy And Mass Assembly: automatic morphological classification of galaxies using statistical learning

“…The methods that we have chosen construct classifiers with different mathematical structures.Therefore each constructed classifier may capture different aspects of the ideal classifier effectively. Using a combination of classifiers constructed using different statistical learning methods may give rise to a new classifier with better accuracy (and closer to an ideal classifier) than each classifier taken individually (Chen et al 2015;Kriukova et al 2016). The design of appropriate combination strategies is another avenue that we may explore in the future.…”

Adaptive multi-parameter regularization approach to construct the distribution function of relaxation times

“…Note that this kind of endeavor has not been reported in the aforementioned literature. In addition, in order to enable an automatic regularization in the present study, we use the idea (Chen et al 2015) of an aggregation of regularized solutions corresponding to different values of multiple regularization parameters.…”

The quasi-optimality criterion in the linear functional strategy

“…Note that x s agg,y δ can be effectively computed because it only uses access to T and y δ . Then by the same arguments as in the proof of Theorem 3.7 in [9], it follows from (5.2) that then the accuracy of x s agg may only be better than the one of x δ α(y δ ) . Moreover, from (5.1), (5.4), it follows that the error of the effectively computed aggregator x s agg,y δ differs from the error of x s agg by a quantity of higher order than the accuracy guaranteed by the standard quasi-optimality criterion.…”

“…At the same time, as follows from our results, in the linear functional strategy, the above-mentioned convergence rate gap can be essentially reduced. This hints at an opportunity to use the linear functional strategy equipped with the quasi-optimality criterion for aggregating the constructed regularized approximants in a way described in [9]. Then from [9], it follows that such aggregation by the linear functional strategy can improve the accuracy compared to the aggregated regularized approximations, and this can be seen as a way to use the quasi-optimality criterion for mildly and severely ill-posed problems.…”