Constrained estimation via the fused lasso and some generalizations
| dc.contributor.advisor | Scott, James (Statistician) | |
| dc.contributor.committeeMember | Caramanis, Constantine | |
| dc.contributor.committeeMember | Sarkar, Purnamrita | |
| dc.contributor.committeeMember | Zhou, Mingyuan | |
| dc.creator | Madrid Padilla, Oscar Hernan | |
| dc.date.accessioned | 2018-01-03T20:43:27Z | |
| dc.date.available | 2018-01-03T20:43:27Z | |
| dc.date.created | 2017-05 | |
| dc.date.issued | 2017-05-08 | |
| dc.date.submitted | May 2017 | |
| dc.date.updated | 2018-01-03T20:43:27Z | |
| dc.description.abstract | This dissertation studies structurally constrained statistical estimators. Two entwined main themes are developed: computationally efficient algorithms, and strong statistical guarantees of estimators across a wide range of frameworks. In the first chapter we discuss a unified view of optimization problems that enforces constrains, such as smoothness, in statistical inference. This in turn helps to incorporate spatial and/or temporal information about data. The second chapter studies the fused lasso, a non-parametric regression estimator commonly used for graph denoising. This has been widely used in applications where the graph structure indicates that neighbor nodes have similar signal values. I prove for the fused lasso on arbitrary graphs, an upper bound on the mean squared error that depends on the total variation of the underlying signal on the graph. Moreover, I provide a surrogate estimator that can be found in linear time and attains the same upper–bound. In the third chapter I present an approach for penalized tensor decomposition (PTD) that estimates smoothly varying latent factors in multiway data. This generalizes existing work on sparse tensor decomposition and penalized matrix decomposition, in a manner parallel to the generalized lasso for regression and smoothing problems. I present an efficient coordinate-wise optimization algorithm for PTD, and characterize its convergence properties. The fourth chapter proposes histogram trend filtering, a novel approach for density estimation. This estimator arises from looking at surrogate Poisson model for counts of observations in a partition of the support of the data. The fifth chapter develops a class of estimators for deconvolution in mixture models based on a simple two-step bin-and-smooth procedure, applied to histogram counts. The method is both statistically and computationally efficient. By exploiting recent advances in convex optimization,we are able to provide a full deconvolution path that shows the estimate for the mixing distribution across a range of plausible degrees of smoothness, at far less cost than a full Bayesian analysis. Finally, the sixth chapter summarizes my contributions and provides possible directions for future work. | |
| dc.description.department | Statistics | |
| dc.format.mimetype | application/pdf | |
| dc.identifier | doi:10.15781/T2T43JK27 | |
| dc.identifier.uri | http://hdl.handle.net/2152/63067 | |
| dc.language.iso | en | |
| dc.subject | Fused lasso | |
| dc.subject | Penalized likelihood. | |
| dc.title | Constrained estimation via the fused lasso and some generalizations | |
| dc.type | Thesis | |
| dc.type.material | text | |
| thesis.degree.department | Statistics | |
| thesis.degree.discipline | Statistics | |
| thesis.degree.grantor | The University of Texas at Austin | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy |