簡易檢索 / 詳目顯示

回結果列表
研究生： 許哲睿
Hsu, Zhe-Ruei
論文名稱： 奇異值分解與其應用
Singular Value Decomposition and its Application
指導教授： 劉育佑
Liu, Yu-Yu
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2025
畢業學年度： 113
語文別： 英文
論文頁數： 63
中文關鍵詞： 奇異值分解 、低秩近似 、主成分分析 、推薦系統 、隨機化奇異值分解
外文關鍵詞： singular value decomposition, low-rank approximation, principal component analysis, recommendation system, randomized SVD
相關次數： 點閱：22  下載：10 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 本論文探討奇異值分解的理論與應用。我們介紹其數學基礎，包括奇異向量的構造、基本子空間的結構以及 Eckart–Young 定理。接著，我們展示了 SVD 在圖像壓縮、主成分分析以及推薦系統等多種應用。此外，我們也介紹了一種用於快速近似 SVD 的隨機化演算法。

    In this paper, we explore the theory and applications of singular value decomposition. We present its mathematical foundations, including the construction of singular vectors, the structure of fundamental subspaces and the Eckart–Young theorem. Then we present various applications such as image compression, principal component analysis and recommendation systems. In addition, we introduce the randomized algorithm for fast approximation of SVD.

    摘要 i Abstract ii 誌謝 iii Contents iv 1 Introduction 1 2 Theoretical Background 3 2.1 Existence of SVD 3 2.2 Computation of SVD 5 2.3 The Fundamental Subspaces with SVD 7 2.4 Geometry of SVD 9 2.5 Matrix Norms and Singular Values 10 3 Matrix Approximation 13 3.1 Low-Rank Approximation 14 3.2 Eckart-Young Theorem and Relative Error 15 3.3 Application: Image Compression 16 3.4 Application: Recommendation System 21 3.4.1 SVD-Based Collaborative Filtering 21 3.4.2 Example 23 4 Linear Regression and Pseudo-Inverse 25 4.1 Least Square Problem 26 4.2 Least Norm Problem 28 4.3 Moore-Penrose Pseudoinverse 29 4.3.1 Application: Moore-Penrose Pseudoinverse 31 5 Principal Component Analysis 33 5.1 Linear Regression 34 5.2 Data Centering and Normalization 35 5.3 PCA with One Component 36 5.4 Theory of Principal Component Analysis 38 5.5 Computation 38 5.6 Application: Principal Component Analysis 40 6 Randomized SVD 45 6.1 Randomized Linear Algebra 46 6.2 Randomized SVD Algorithm 46 6.3 Constructing the Matrix Q 47 6.4 Error Bound 48 6.5 Oversampling and Power Iteration 49 6.6 Application of Randomized SVD 51 6.6.1 Image Compression 51 6.6.2 Principal Component Analysis 52 7 Conclusion 54 References 55

    [1]Richard W. Hamming. Numerical Methods for Scientists and Engineers. McGraw-Hill, Inc., USA, 1973.
    [2]Gilbert Strang. Linear Algebra and Learning from Data. Wellesley-Cambridge Press, Philadelphia, PA, 2019.
    [3]Mark H. Holmes. Introduction to Scientific Computing and Data Analysis. Springer, 2 edition, 2023.
    [4]Gilbert Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press, 5 edition, 2016.
    [5]Steven L. Brunton and J. Nathan Kutz. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, 2019.
    [6]Gene H. Golub and Charles F. Van Loan. Matrix Computations - 4th Edition. Johns Hopkins University Press, Philadelphia, PA, 2013.
    [7]Yuefeng Zhang. An introduction to matrix factorization and factorization machines in recommendation system, and beyond. ArXiv, abs/2203.11026, 2022.
    [8]Cerdeira A. Almeida F. Matos T. Cortez, Paulo and J. Reis. Wine Quality. UCI Machine Learning Repository, 2009. DOI: https://doi.org/10.24432/C56S3T.
    [9]Mangasarian Olvi Street Nick Wolberg, William and W. Street. Breast Cancer Wisconsin (Diagnostic). UCI Machine Learning Repository, 1993. DOI: https://doi.org/10.24432/C5DW2B.
    [10]Petros Drineas and Michael W. Mahoney. Randnla: randomized numerical linear algebra. Commun. ACM, 59(6):80–90, May 2016.
    [11]N. Halko, P. G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53(2):217–288, 2011.
    [12]Margherita Guido, Daniel Kressner, and Paolo Ricci. Subspace acceleration for a sequence of linear systems and application to plasma simulation. Journal of Scientific Computing, 99, 04 2024.

    下載圖示 校內：立即公開
    校外：立即公開
    QR CODE