A Data-Dependent Weighted LASSO

Sparse linear inverse problems appear in a variety of settings, butoften the noise contaminating observations cannot accurately bedescribed as bounded or arising from a Gaussian distribution. Poissonobservations in particular are a characteristic feature of severalreal-world applications, including photon-limited imaging systems,network flow tracking, and genetic motif analysis. Previous work onsparse Poisson inverse problems encountered several limiting technicalhurdles. I will describe an alternative, streamlined analysis approachfor sparse Poisson inverse problems based on a weighted LASSOestimator. This approach (a) sidesteps the technical challengespresent in previous work, (b) admits estimators that can readily becomputed using off-the-shelf LASSO algorithms, and (c) hints at ageneral weighted LASSO framework for broader classes ofheteroscedastic problems. At the heart of this new approach lies aweighted LASSO estimator for which data-dependent weights are based onPoisson concentration inequalities. Unlike previous analyses of theweighted LASSO, the proposed analysis admits data-dependent weights,relies on standard conditions on the sensing or design matrix, andallows signal-dependent noise. This is joint work with Xin Jiang,Patricia Reynaud-Bouret, Vincent Rivoirard, and Laure Sansonnet.refreshments served