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High-Dimensional Statistical Inference: from Vector to Matrix University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2015 High-dimensional Statistical Inference: from Vector to Matrix Anru Zhang University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Applied Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Zhang, Anru, "High-dimensional Statistical Inference: from Vector to Matrix" (2015). Publicly Accessible Penn Dissertations. 1172. https://repository.upenn.edu/edissertations/1172 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1172 For more information, please contact [email protected]. High-dimensional Statistical Inference: from Vector to Matrix Abstract Statistical inference for sparse signals or low-rank matrices in high-dimensional settings is of significant interest in a range of contemporary applications. It has attracted significant ecentr attention in many fields including statistics, applied mathematics and electrical engineering. In this thesis, we consider several problems in including sparse signal recovery (compressed sensing under restricted isometry) and low-rank matrix recovery (matrix recovery via rank-one projections and structured matrix completion). The first part of the thesis discusses compressed sensing and affineank r minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while leads to sharp results. It is shown that, in compressed sensing, $\delta_k^A<1/3$, $\delta_k^A+\theta_{k,k}^A <1$, or $\delta_{tk}^A < \sqrt{(t-1)/t}$ for any given constant $t\ge {4/3}$ guarantee the exact recovery of all $k$ sparse signals in the noiseless case through the constrained $\ell_1$ minimization, and similarly in affineank r minimization $\delta_r^\mathcal{M}<1/3$, $\delta_r^{\mathcal{M}}+\theta_{r, r}^{\mathcal{M}}<1$, or $\delta_{tr}^\mathcal{M}< \sqrt{(t-1)/t}$ ensure the exact reconstruction of all matrices with rank at most $r$ in the noiseless case via the constrained nuclear norm minimization. Moreover, for any $\epsilon>0$, $\delta_{k}^A < 1/3+\epsilon$, $\delta_k^A+\theta_{k,k}^A<1+\epsilon$, or $\delta_{tk}^A<\sqrt{\frac{t-1}{t}}+\epsilon$ are not sufficient to guarantee the exact recovery of all $k$- sparse signals for large $k$. Similar result also holds for matrix recovery. In addition, the conditions $\delta_k^A<1/3$, $\delta_k^A+\theta_{k,k}^A<1$, $\delta_{tk}^A < \sqrt{(t-1)/t}$ and $\delta_r^\mathcal{M}<1/3$, $\delta_r^\mathcal{M}+\theta_{r,r}^\mathcal{M}<1$, $\delta_{tr}^\mathcal{M}< \sqrt{(t-1)/t}$ are also shown to be sufficientespectiv r ely for stable recovery of approximately sparse signals and low-rank matrices in the noisy case. For the second part of the thesis, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is adaptive to the rank and robust against small perturbations. Both upper and lower bounds for the estimation accuracy under the Frobenius norm loss are obtained. The proposed estimator is shown to be rate-optimal under certain conditions. The estimator is easy to implement via convex programming and performs well numerically. The techniques and main results developed in the chapter also have implications to other related statistical problems. An application to estimation of spiked covariance matrices from one-dimensional random projections is considered. The results demonstrate that it is still possible to accurately estimate the covariance matrix of a high-dimensional distribution based only on one-dimensional projections. For the third part of the thesis, we consider another setting of low-rank matrix completion. Current literature on matrix completion focuses primarily on independent sampling models under which the individual observed entries are sampled independently. Motivated by applications in genomic data integration, we propose a new framework of structured matrix completion (SMC) to treat structured missingness by design. Specifically, our proposed method aims at efficient matrixeco r very when a subset of the rows and columns of an approximately low-rank matrix are observed. We provide theoretical justification for the proposed SMC method and derive lower bound for the estimation errors, which together establish the optimal rate of recovery over certain classes of approximately low-rank matrices. Simulation studies show that the method performs well in finite sample under a variety of configurations. The method is applied to integrate several ovarian cancer genomic studies with different extent of genomic measurements, which enables us to construct more accurate prediction rules for ovarian cancer survival. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Applied Mathematics First Advisor T. Tony Cai Keywords Constrained l_1 minimization, Constrained nuclear norm minimization, Genomic data integration, Low- rank matrix recovery, Optimal rate of convergence, Sparse signal recovery Subject Categories Applied Mathematics | Statistics and Probability This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/1172 HIGH-DIMENSIONAL STATISTICAL INFERENCE: FROM VECTOR TO MATRIX Anru Zhang A DISSERTATION in Applied Mathematics and Computational Science Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2015 Supervisor of Dissertation T. Tony Cai Dorothy Silberberg Professor of Statistics Graduate Group Chairperson Charles L. Epstein Thomas A. Scott Professor of Mathematics Dissertation Committee T. Tony Cai, Professor Lawrance D. Brown Dorothy Silberberg Professor of Statistics Miers Busch Professor of Statistics Mark G. Low, Professor Lawrance Shepp (in Memoriam) Walter C. Bladstrom Professor of Statistics Professor of Statistics HIGH-DIMENSIONAL STATISTICAL INFERENCE: FROM VECTOR TO MATRIX COPYRIGHT © 2015 Anru Zhang Acknowledgments I would like to first and foremost thank my advisor Professor Tony Cai. His sharp thinking and extensive knowledge have continuously been a great source of inspiration for my research. With his kindness, enthusiasm, patience, he guided me through a lot of hurdles during my PhD study. He is truly the epitome of an excellent scholar and mentor. My sincere thanks go to Professor Larry Brown and Professor Mark Low for their kindly help and support for my research, teaching and job searching. Their brilliant comments and suggestions have been extremely helpful and will be valuable for my entire career. I want to thank the late Professor Larry Shepp. His smiles, jokes and encouragement will always be well remembered. I am also very grateful to Professor Charles Epstein, who admitted me to this program and gave me such a great opportunity to study at UPenn. I want to thank my collaborators, Professor Tianxi Cai, Professor Dylan Small, Professor Hongzhe Li, Hyunseung Kang for the inspiring discussions I had with them. They have all given me generous support over the past five years. I want to thank all the faculty members, staffs and my fellow students in both Department of Statistics and Program of Applied Mathematics and Computational iii Science. Their accompany makes the past five years a very enjoyable and memorable experience. My final, and most genuine acknowledgment goes to my family. Words cannot express how grateful I am to my parents and my wife Pixu, as they are my source of strength. They always gave me unconditional love and support whenever I met with difficulties. I would never be able to finish this thesis without them. iv ABSTRACT HIGH-DIMENSIONAL STATISTICAL INFERENCE: FROM VECTOR TO MATRIX Anru Zhang T. Tony Cai Statistical inference for sparse signals or low-rank matrices in high-dimensional settings is of significant interest in a range of contemporary applications. It has at- tracted significant recent attention in many fields including statistics, applied math- ematics and electrical engineering. In this thesis, we consider several problems in including sparse signal recovery (compressed sensing under restricted isometry) and low-rank matrix recovery (matrix recovery via rank-one projections and structured matrix completion). The first part of the thesis discusses compressed sensing and affine rank mini- mization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while leads to sharp results. It is shown A A A A p that, in compressed sensing, δk < 1=3, δk + θk;k < 1, or δtk < (t − 1)=t for any given constant t ≥ 4=3 guarantee the exact recovery of all k sparse signals in the noiseless case through the constrained `1 minimization, and similarly in affine rank v M M M M p minimization δr < 1=3, δr + θr;r < 1, or δtr < (t − 1)=t ensure the exact recon- struction of all matrices with rank at most r in the noiseless case via the constrained A A A nuclear norm minimization. Moreover, for any > 0, δk < 1=3 + , δk + θk;k < 1 + , A q t−1 or δtk < t + are not sufficient to guarantee the exact recovery of all k-sparse signals for large k. Similar result also holds for matrix recovery. In addition, the A A A A p M M M conditions δk < 1=3, δk + θk;k < 1, δtk < (t − 1)=t and δr < 1=3, δr + θr;r < 1, M p δtr < (t − 1)=t are also shown to be sufficient respectively for stable recovery of approximately sparse signals and low-rank matrices in the noisy case. For the second part of the thesis, we introduce a rank-one projection model for low- rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case.
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