Structure Detection in High Dimensional Graphical Models

Graphical models are powerful tools to study the dependency structures among random variables. In this thesis, we study structure detection problems in high dimensional graphical models. Given independent samples generated from a graphical model, the goal of structure detection is to distinguish whether the underlying graph is empty, or contains a subgraph of a certain structure. In the first part of the thesis, we study the information-theoretic limits of structure detection problems in high temperature ferromagnetic (positive interaction only) Ising models. First, based on the high temperature expansion of Ising models, we derive a general information-theoretic lower bound for arbitrary alternative hypothesis. We then propose a general framework of linear tests on the sample covariance matrix that matches our minimax lower bounds. Our results reveal that a key quantity called graph arboricity drives the testability of the problem. In the second part of the thesis, we discuss the computational limits of structure detection problems. First, based on a computational hardness conjecture of sparse principal component analysis, we prove that, unless the signal is strong enough, no polynomial time linear tests on the sample covariance matrix are capable of testing this problem. We also give another computational lower bound based on the oracle computational model, where the computational complexity is measured by the times an algorithm queries an oracle. Our results show that, for problems with high graph arboricity, the information-theoretic limits are unachievable by tractable algorithms. In the third part of the thesis, we propose a unified framework to quantify local and global inferential uncertainty for high dimensional nonparanormal graphical models. In particular, we consider the problems of testing the presence of a single edge and constructing a uniform confidence subgraph. We develop pseudo likelihood based score and Wald tests to eliminate the unknown nuisance transformations. A U-statistic multiplier bootstrap method is then proposed to construct the confidence subgraph. i Acknowledgements First and foremost, I would like to express my deepest gratitude to my advisor, Professor Han Liu. Without his guidance and encouragement, I could not possibly overcome the obstacles I have been facing through my research. I really appreciate all the days and nights we work together, from which I gradually learned how to conduct research. His wisdom, patience and passion are always examplary to me. I owe special thanks to Professor Weinan E, for the tremendous help he offered me. It is very fortunate for me to have had the chance to interact closely with one of the greatest mathematicians. His critical thinking and enlightening insight can always inspire me deeply. Thank you Professor E, for your invaluable advice on research and effective communication! My sincere thanks also goes to Professor Tong Zhang, for offering me the summer internship opportunities in his group and guiding my work on all the exciting projects. It is my greatest honor to have Professor Mengdi Wang on my final public oral exam committee. I would also like to thank Professor Weinan E and Professor Yuxin Chen for being the readers of my thesis. Thank you for all the valuable comments and encouragement. I am deeply grateful to all my other collaborators, including Professor Quanquan Gu, Professor Matey Neykov, Professor Junwei Lu, Professor Zhaoran Wang, Professor Yang Ning, Mr. Zhuoran Yang and Mr. Hao Lu. It has been an absolute pleasure working with them. I wish to say thank you to all my friends in ORFE and PACM: Xingyuan Fang, Tuo Zhao, Tianqi Zhao, Yuyan Wang, Ziwei Zhu, Qiang Sun, Wenxin Zhou, Junchi Li, Huanran Lu, Carson Eisenach, Jason Ge, Yiqiao Zhong, Cong ma, Suqi Liu, Jiacheng Zhang, Yongyi Guo, Kun Lu, Qianxiao Li, Patrick Chan, Yuehaw Khoo, Afonso Bandeira, Borui Liu, Chu Wang, Cheng Tai, Jiequn Han. I would like to let you know how grateful I am to to all of you. ii I would also like to acknowledge my undergraduate advisor Professor Weiguo Gao. Professor Gao has always been supportive over the past ten years. Words cannot describe how thankful I am. Last but not the least, I would like to say thank you to my parents, for their unconditional love and support in every step of my journey. To them I dedicate this dissertation.