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研究生:洪承郁
研究生(外文):Cheng-Yu Hung
論文名稱:穩定主成份分析以及其延伸
論文名稱(外文):Robust PCA and its Extension
指導教授:杜憶萍杜憶萍引用關係
口試委員:姚怡慶陳定立陳素雲陳宏
口試日期:2018-06-21
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:應用數學科學研究所
學門:數學及統計學門
學類:其他數學及統計學類
論文種類:學術論文
論文出版年:2018
畢業學年度:106
語文別:英文
論文頁數:35
中文關鍵詞:穩定估計隨機抽樣隨機分群生物影像代理函數
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主成份分析 (Principal Component Analysis) 已經被廣泛運用在各種 影像處理上, 但是越來越複雜的影像導致主成份分析的假設已被破壞。 所以 Candés et al. (2011) 提出了穩定主成份分析,來應對這些新的挑 戰,例如 sensor failure 以及 corrupted sample。在這篇碩士論文裡,我 們針對穩定主成份分析做了一些調整,以擴展其應用。我們運用了策 略抽樣的方法,讓數據可以滿足 RPCA 。
Principal Component Analysis (PCA) has been used in an overwhelming manner for data analysis. However, PCA did not perform well when data did not follow the model well like sensor failure or corrupted sample. Can- dés et al. (2011) proposed Robust Principal Component Analysis (RPCA) to recover the data and proved that it can perform very well when data has the sparsity property for the signal with a low rank background. Unfortunately, the FRET data set does not satisfy the working condition. Here, we employ a sampling scheme to enable the application for the FRET data. For extremely large number of pixel image application, RPCS may suffer from computation loading. Thus, we also extend RPCA to a high order SVD version.
口試委員會審定書 iii
誌謝 v
摘要 vii
Abstract ix
1 Introduction 1
2 Theoretical Literature Review of Robust PCA 3
2.1 Fundamental Review ............................ 3
2.2 nuclear norm heuristic ........................... 4
2.3 Vector case: l1-norm minimization..................... 6
2.4 PCP problem of Robust PCA........................ 7
2.5 ‹Incoherence of L0 ............................ 7
2.6 ‹Support of the Sparse component S0 .................. 8
2.7 Main Result of RPCA .......................... 8
2.8 Algorithm of RPCA ............................ 9
3 Application challenges and Our Solutions 13
3.1 Application Challenges........................... 13
3.2 A Resampling Scheme ........................... 15
3.3 Refinement Scheme............................. 16
3.4 Higher-order RPCA............................. 17
4 Numerical Examples 19
4.1 Simple Examples .............................. 19
4.2 Localization of singlenano-sized light emitter . . . . . . . . . . . . . . . 21
4.3 Surveillance video ............................. 23
4.4 Examples of S0 notsatisfythe requirement of RPCA . . . . . . . . . . . 26
4.5 smFRET experiments............................ 29
5 Summary 33
Bibliography 35
1 D. Bertsekas. Constrained optimization and lagrange multiplier method. Academic Press, 1982.
2 J.-F. Cai, E. J. Candés, and Z. Shen. A singular value thresholding algorithm for matrix completion. SIAM J. on Optimization, 2010.
3 E. Candés, X. Li, Y. Ma, and J. Wright. Robust principal component analysis? Journal of the ACM (JACM), 2011.
4 M. Fazel. Matrix rank minimization with applications. Ph.D. dissertation, Stanford University, 2002.
5 Z. Lin, M. Chen, and Y. Ma. The augmented lagrange multiplier method for exact re- covery of corrupted low-rank matrices. arXiv preprint, 2010.
6 R. Roy, S. Hohng, and T. Ha. A practical guide to single-molecule fret. Nature methods, 2008.
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