A convergent relaxation of the Douglas-Rachford algorithm

This paper proposes an algorithm for solving structured optimization problems, which covers both the backward-backward and the Douglas-Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the algorithm is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point operators are established. When applying to nonconvex feasibility including the inconsistent case, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets which need not intersect. In this special case, we refine known linear convergence criteria for the Douglas-Rachford algorithm (DR). As a consequence, for feasibility with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems

What Is In Shape?

I sat apprehensively on the foam floor mats of the group fitness classroom. Those first day jitters swirled in my stomach as I waited for the instructor. My friends had persuaded me to give fitness classes a try. “The teachers are just such great motivation,” they promised. A thirtysomething, rather large woman entered the room, bulging bags in tow, with a take-charge look on her face. “That lady definitely needs a good workout,” I remember thinking, smugly. But when she took her place at the front of the room, it suddenly became clear—she was the instructor

About regularity properties in variational analysis and applications in optimization

Regularity properties lie at the core of variational analysis because of their importance for stability analysis of optimization and variational problems, constraint qualications, qualication conditions in coderivative and subdierential calculus and convergence analysis of numerical algorithms. The thesis is devoted to investigation of several research questions related to regularity properties in variational analysis and their applications in convergence analysis and optimization. Following the works by Kruger, we examine several useful regularity properties of collections of sets in both linear and Holder-type settings and establish their characterizations and relationships to regularity properties of set-valued mappings. Following the recent publications by Lewis, Luke, Malick (2009), Drusvyatskiy, Ioe, Lewis (2014) and some others, we study application of the uniform regularity and related properties of collections of sets to alternating projections for solving nonconvex feasibility problems and compare existing results on this topic. Motivated by Ioe (2000) and his subsequent publications, we use the classical iteration scheme going back to Banach, Schauder, Lyusternik and Graves to establish criteria for regularity properties of set-valued mappings and compare this approach with the one based on the Ekeland variational principle. Finally, following the recent works by Khanh et al. on stability analysis for optimization related problems, we investigate calmness of set-valued solution mappings of variational problems.Doctor of Philosoph

Slice-level Detection of Intracranial Hemorrhage on CT Using Deep Descriptors of Adjacent Slices

The rapid development in representation learning techniques such as deep neural networks and the availability of large-scale, well-annotated medical imaging datasets have to a rapid increase in the use of supervised machine learning in the 3D medical image analysis and diagnosis. In particular, deep convolutional neural networks (D-CNNs) have been key players and were adopted by the medical imaging community to assist clinicians and medical experts in disease diagnosis and treatment. However, training and inferencing deep neural networks such as D-CNN on high-resolution 3D volumes of Computed Tomography (CT) scans for diagnostic tasks pose formidable computational challenges. This challenge raises the need of developing deep learning-based approaches that are robust in learning representations in 2D images, instead 3D scans. In this work, we propose for the first time a new strategy to train \emph{slice-level} classifiers on CT scans based on the descriptors of the adjacent slices along the axis. In particular, each of which is extracted through a convolutional neural network (CNN). This method is applicable to CT datasets with per-slice labels such as the RSNA Intracranial Hemorrhage (ICH) dataset, which aims to predict the presence of ICH and classify it into 5 different sub-types. We obtain a single model in the top 4% best-performing solutions of the RSNA ICH challenge, where model ensembles are allowed. Experiments also show that the proposed method significantly outperforms the baseline model on CQ500. The proposed method is general and can be applied to other 3D medical diagnosis tasks such as MRI imaging. To encourage new advances in the field, we will make our codes and pre-trained model available upon acceptance of the paper.Comment: Accepted for presentation at the 22nd IEEE Statistical Signal Processing (SSP) worksho

About uniform regularity of collection sets

We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving feasibility problems