A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery

This paper studies a recovery task of finding a low multilinear-rank tensor that fulfills some linear constraints in the general settings, which has many applications in computer vision and graphics. This problem is named as the low multilinear-rank tensor recovery problem. The variable splitting technique and convex relaxation technique are used to transform this problem into a tractable constrained optimization problem. Considering the favorable structure of the problem, we develop a splitting augmented Lagrangian method to solve the resulting problem. The proposed algorithm is easily implemented and its convergence can be proved under some conditions. Some preliminary numerical results on randomly generated and real completion problems show that the proposed algorithm is very effective and robust for tackling the low multilinear-rank tensor completion problem

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

An introduction to continuous optimization for imaging

International audienceA large number of imaging problems reduce to the optimization of a cost function , with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification

A System Centric View of Modern Structured and Sparse Inference Tasks

University of Minnesota Ph.D. dissertation.June 2017. Major: Electrical/Computer Engineering. Advisor: Jarvis Haupt. 1 computer file (PDF); xii, 140 pages.We are living in the era of data deluge wherein we are collecting unprecedented amount of data from variety of sources. Modern inference tasks are centered around exploiting structure and sparsity in the data to extract relevant information. This thesis takes an end-to-end system centric view of these inference tasks which mainly consist of two sub-parts (i) data acquisition and (ii) data processing. In context of the data acquisition part of the system, we address issues pertaining to noise, clutter (the unwanted extraneous signals which accompany the desired signal), quantization, and missing observations. In the data processing part of the system we investigate the problems that arise in resource-constrained scenarios such as limited computational power and limited battery life. The first part of this thesis is centered around computationally-efficient approximations of a given linear dimensionality reduction (LDR) operator. In particular, we explore the partial circulant matrix (a matrix whose rows are related by circular shifts) based approximations as they allow for computationally-efficient implementations. We present several theoretical results that provide insight into existence of such approximations. We also propose a data-driven approach to numerically obtain such approximations and demonstrate the utility on real-life data. The second part of this thesis is focused around the issues of noise, missing observations, and quantization arising in matrix and tensor data. In particular, we propose a sparsity regularized maximum likelihood approach to completion of matrices following sparse factor models (matrices which can be expressed as a product of two matrices one of which is sparse). We provide general theoretical error bounds for the proposed approach which can be instantiated for variety of noise distributions. We also consider the problem of tensor completion and extend the results of matrix completion to the tensor setting. The problem of matrix completion from quantized and noisy observations is also investigated in as general terms as possible. We propose a constrained maximum likelihood approach to quantized matrix completion, provide probabilistic error bounds for this approach, and numerical algorithms which are used to provide numerical evidence for the proposed error bounds. The final part of this thesis is focused on issues related to clutter and limited battery life in signal acquisition. Specifically, we investigate the problem of compressive measurement design under a given sensing energy budget for estimating structured signals in structured clutter. We propose a novel approach that leverages the prior information about signal and clutter to judiciously allocate sensing energy to the compressive measurements. We also investigate the problem of processing Electrodermal Activity (EDA) signals recorded as the conductance over a user's skin. EDA signals contain information about the user's neuron ring and psychological state. These signals contain the desired information carrying signal superimposed with unwanted components which may be considered as clutter. We propose a novel compressed sensing based approach with provable error guarantees for processing EDA signals to extract relevant information, and demonstrate its efficacy, as compared to existing techniques, via numerical experiments

Riemannian Optimization for Convex and Non-Convex Signal Processing and Machine Learning Applications

The performance of most algorithms for signal processing and machine learning applications highly depends on the underlying optimization algorithms. Multiple techniques have been proposed for solving convex and non-convex problems such as interior-point methods and semidefinite programming. However, it is well known that these algorithms are not ideally suited for large-scale optimization with a high number of variables and/or constraints. This thesis exploits a novel optimization method, known as Riemannian optimization, for efficiently solving convex and non-convex problems with signal processing and machine learning applications. Unlike most optimization techniques whose complexities increase with the number of constraints, Riemannian methods smartly exploit the structure of the search space, a.k.a., the set of feasible solutions, to reduce the embedded dimension and efficiently solve optimization problems in a reasonable time. However, such efficiency comes at the expense of universality as the geometry of each manifold needs to be investigated individually. This thesis explains the steps of designing first and second-order Riemannian optimization methods for smooth matrix manifolds through the study and design of optimization algorithms for various applications. In particular, the paper is interested in contemporary applications in signal processing and machine learning, such as community detection, graph-based clustering, phase retrieval, and indoor and outdoor location determination. Simulation results are provided to attest to the efficiency of the proposed methods against popular generic and specialized solvers for each of the above applications

Scaling Up Large-scale Sparse Learning and Its Application to Medical Imaging

abstract: Large-scale $\ell_1$-regularized loss minimization problems arise in high-dimensional applications such as compressed sensing and high-dimensional supervised learning, including classification and regression problems. In many applications, it remains challenging to apply the sparse learning model to large-scale problems that have massive data samples with high-dimensional features. One popular and promising strategy is to scaling up the optimization problem in parallel. Parallel solvers run multiple cores on a shared memory system or a distributed environment to speed up the computation, while the practical usage is limited by the huge dimension in the feature space and synchronization problems. In this dissertation, I carry out the research along the direction with particular focuses on scaling up the optimization of sparse learning for supervised and unsupervised learning problems. For the supervised learning, I firstly propose an asynchronous parallel solver to optimize the large-scale sparse learning model in a multithreading environment. Moreover, I propose a distributed framework to conduct the learning process when the dataset is distributed stored among different machines. Then the proposed model is further extended to the studies of risk genetic factors for Alzheimer's Disease (AD) among different research institutions, integrating a group feature selection framework to rank the top risk SNPs for AD. For the unsupervised learning problem, I propose a highly efficient solver, termed Stochastic Coordinate Coding (SCC), scaling up the optimization of dictionary learning and sparse coding problems. The common issue for the medical imaging research is that the longitudinal features of patients among different time points are beneficial to study together. To further improve the dictionary learning model, I propose a multi-task dictionary learning method, learning the different task simultaneously and utilizing shared and individual dictionary to encode both consistent and changing imaging features.Dissertation/ThesisDoctoral Dissertation Computer Science 201