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Learning Low-Dimensional Models for Heterogeneous Data by David

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Learning Low-Dimensional Models for Heterogeneous Data by David Learning Low-Dimensional Models for Heterogeneous Data by David Hong A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering: Systems) in The University of Michigan 2019 Doctoral Committee: Professor Laura Balzano, Co-Chair Professor Jeffrey A. Fessler, Co-Chair Professor Rina Foygel Barber, The University of Chicago Professor Anna Gilbert Professor Raj Rao Nadakuditi David Hong [email protected] ORCID iD: 0000-0003-4698-4175 © David Hong 2019 To mom and dad. ii ACKNOWLEDGEMENTS As always, no list of dissertation acknowledgements can hope to be complete, and I have many people to thank for their wonderful influence over the past six years of graduate study. To start, I thank my advisors, Professor Laura Balzano and Professor Jeffrey Fessler, who introduced me to the exciting challenges of learning low-dimensional image models from heterogeneous data and taught me the funda- mental ideas and techniques needed to tackle it; this dissertation is the direct result. Laura and Jeff, I am always amazed by your deep and broad technical insights, your sincere care for students and your great patience (with me!). Working with you has changed the way I think, write and teach, and I cannot say how thankful I am to have had the opportunity to learn from you. I hope to one day be as wonderful an advisor to students of my own. I am also thankful for my excellent committee. Professor Rina Barber has many times provided deep statistical insight into my work, and her wonderful suggestions can be found throughout this dissertation. Professor Anna Gilbert has always chal- lenged me to strive for work that is both deep in theory and significant in practice, and she continues to be a role model for me. Professor Raj Nadakuditi guided me when I first arrived in Ann Arbor, encouraged me during my early difficulties, and taught me many of the random matrix theory tools I used in Chapters III andIV.I am indebted to all three and am honored to have had them form my committee. I also had the distinct pleasure of spending the summer of 2017 interning at Sandia National Laboratories in Livermore, California, where I worked on the Gen- eralized CP tensor decomposition in ChapterV. I am so thankful for the wonderful mentorship of Dr. Tammy Kolda and Dr. Cliff Anderson-Bergman, who proposed this work and taught me nearly everything I now know about tensor decomposition and its numerous applications. I continue to learn from their many insights into the practice and theory of data analysis. At Sandia, I also had the pleasure of working alongside Dr. Jed Duersch and of sharing a cubicle with John Kallaugher, who both made my time there truly delightful. I have also been fortunate to interact with many great postdoctoral scholars (all in fact hired by members of my committee!), who made Ann Arbor intellectually fertile grounds: Dr. Sai Ravishankar, Dr. Il Yong Chun, Dr. Greg Ongie and Dr. Lalit Jain. I am thankful for our many fun conversations that have shaped how I think about choosing and tackling research problems. Chapters III andIV of the dissertation also iii benefitted from discussions with Professor Edgar Dobriban and Professor Romain Couillet, and I thank them for sharing their insights about spiked covariance models and random matrix theory tools for analyzing their asymptotic properties. My many wonderful peers have also created a great learning environment. I thank Wooseok Ha (who in fact introduced me to Rina), Audra McMillan, and Yan Shuo Tan for our many stimulating discussions over the years; I am so glad we got to meet at the 2016 Park City Mathematics Institute. Likewise, I am thankful for Mohsen Heidari, Matthew Kvalheim, Aniket Deshmukh, John Lipor and Dejiao Zhang, who began the Ph.D. program with me and have been wonderful co-voyagers. A special thank you goes to John { J.J., you have often been a true encouragement and I have learned so much from you about life and research. I am also so thankful for all the students of the SPADA Lab and Lab of Jeff (LoJ) who make coming to campus a joy; I will miss our many fun chats. I thank Donghwan Kim, Madison McGaffin, Mai Le and Gopal Nataraj, who welcomed me and my many questions about optimization and imaging, as well as Curtis Jin, Raj Tejas Suryaprakash, Brian Moore, Himanshu Nayar and Arvind Prasadan, who welcomed my random matrix theory questions. I also had the pleasure of working with Robert Malinas while he was an undergraduate on the Sequences of Unions of Subspaces (SUoS) in Chapter VII; I am so excited that you are continuing in research and look forward to all that you will accomplish! My graduate studies were supported by a Rackham Merit Fellowship, a National Science Foundation (NSF) Graduate Research Fellowship (DGE #1256260) and NSF Grant ECCS-1508943. I am thankful to these sources that afforded me great intel- lectual freedom and made this dissertation possible. Hyunkoo and Yujin Chung, Chad Liu and Anna Cunningham, Hal Zhang, Ruiji Jiang and Max Li, thank you for your friendship over the years. Thanks also to the Campredon's { you are dear friends (and have fed me many times). Special thanks to David Allor, David Nesbitt and Chris Dinh, who were not only roommates but also true brothers in Christ, and to Pastor Shannon Nielsen and my church, who so faithfully point me to Christ and my ultimate hope in Him. Finally and most importantly, I thank my family. My older sisters always took excellent care of their little brother, and I have followed in their footsteps all my life (since, you know, I want to be like you both). Pauline and Esther, thank you for being such loving sisters and incredible examples to me. Josh and Martin, it is wonderful to finally have older brothers(-in-law) too! Mirabelle, my super cute niece, thanks for all your awesome smiles! Mom and Dad, you introduced me to an exciting world full of wonder and adventure, taught me to have fun working hard, and lovingly picked me up all those times I fell down. Thank you. This dissertation is dedicated to you. iv TABLE OF CONTENTS DEDICATION ::::::::::::::::::::::::::::::::::: ii ACKNOWLEDGEMENTS :::::::::::::::::::::::::::: iii LIST OF TABLES ::::::::::::::::::::::::::::::::: viii LIST OF FIGURES :::::::::::::::::::::::::::::::: ix LIST OF ALGORITHMS ::::::::::::::::::::::::::::: xx ABSTRACT :::::::::::::::::::::::::::::::::::: xxi CHAPTER I. Introduction ................................1 1.1 Low-dimensional models for high-dimensional data.........1 1.2 Data with heterogeneous quality...................2 1.3 Data with heterogeneous statistical assumptions..........2 1.4 Data with heterogeneous linear structure..............3 1.5 Organization of dissertation.....................4 II. Background ................................6 2.1 Notations and Conventions......................6 2.2 Principal Component Analysis (PCA)................7 2.3 Canonical Polyadic Tensor Decomposition............. 14 2.4 Unions of subspaces.......................... 19 2.5 Low-dimensional models and medical imaging........... 22 III. Asymptotic performance of PCA for high-dimensional het- eroscedastic data ............................. 26 3.1 Introduction.............................. 27 3.2 Main results.............................. 30 3.3 Impact of parameters......................... 38 3.4 Numerical simulation......................... 43 v 3.5 Proof of Theorem 3.4......................... 45 3.6 Proof of Theorem 3.9......................... 52 3.7 Discussion............................... 53 3.8 Supplementary material....................... 55 IV. Optimally weighted PCA for high-dimensional heteroscedastic data ..................................... 73 4.1 Introduction.............................. 73 4.2 Model for heteroscedastic data.................... 77 4.3 Asymptotic performance of weighted PCA............. 78 4.4 Proof sketch for Theorem 4.3..................... 83 4.5 Optimally weighted PCA....................... 85 4.6 Suboptimal weighting......................... 89 4.7 Impact of model parameters..................... 90 4.8 Optimal sampling under budget constraints............. 94 4.9 Numerical simulation......................... 97 4.10 Discussion............................... 98 4.11 Supplementary material....................... 101 V. Generalized canonical polyadic tensor decomposition for non- Gaussian data ............................... 123 5.1 Introduction.............................. 124 5.2 Background and notation....................... 126 5.3 Choice of loss function........................ 128 5.4 GCP decomposition.......................... 135 5.5 Experimental results......................... 141 5.6 Discussion............................... 154 5.7 Supplementary material....................... 155 VI. Ensemble K-subspaces for data from unions of subspaces ... 158 6.1 Introduction.............................. 159 6.2 Problem Formulation & Related Work............... 160 6.3 Ensemble K-subspaces........................ 164 6.4 Recovery Guarantees......................... 168 6.5 Experimental Results......................... 177 6.6 Discussion............................... 181 6.7 Proofs of Theoretical Results..................... 181 6.8 Implementation Details........................ 192 VII. Sequences of unions of subspaces for data with heterogeneous complexity ................................. 195 vi 7.1 Introduction.............................. 196 7.2 Related Work............................. 197 7.3 Learning a Sequence of Unions of Subspaces...........
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