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姓名	段薇虹(Duong Viet Hang)  查詢紙本館藏	畢業系所	資訊工程學系
論文名稱	複數矩陣分解法及其應用 (Complex Matrix Factorization and Its Applications)
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摘要(中)	在這個工作中，我們構造了一個新的用於學習複雜領域的低秩投影的降維模型，以在實際應用中獲得直觀特徵和高性能，特別是面部表情識別和圖像聚類。複數矩陣分解（CMF）是一種將復數矩陣分解為兩個複矩陣因子的矩陣分解方法。建議的模型可以在不限制數據符號的情況下執行。這些建議的框架可以應用於負面和正面的數據，從而在實際應用中產生擴展性和有效性。因此，能夠提取諸如短時傅立葉變換之類的複雜特徵的多個操作將被直接使用，而不是其絕對值（幅度/功率譜圖）。 從基本框架CMF開始，我們通過添加圖和稀疏約束來獲得圖的正則化复矩陣分解（GraCMF）和具有稀疏約束的複矩陣分解（SpaCMF），從而開發了兩個約束CMF框架。 此外，我們修改了標準CMF的結構以提供更多的擴展。其中之一就是學習基礎在原始空間（示例）中的示例嵌入复矩陣分解（EE-CMF）。投影复矩陣分解（ProCMF）是作為一種新的學習子空間方法開發的，每個數據點的係數位於一個投影複數矩陣的列向量所跨越的子空間內。簡化的複雜矩陣分解（SiCMF）是一種新的模型，通過凸組合潛在的組件來重構原始實例。 為了滿足非負性要求，非負矩陣分解（NMF）通常使用各種策略來使函數最小化，這導致計算複雜度。相反，與所提出的複雜模型的NMF方法相比，其顯著的優越性在於構建一個無約束優化問題，簡化了提取基礎和內在特徵的框架。 Wirtinger的微積分被用來計算成本函數的導數。梯度下降法被用來解決複雜的優化問題。提出的算法被證明為人臉和麵部表情識別以及圖像聚類提供了有效的特徵。對這些任務的實驗表明，所提出的複雜域上的矩陣分解方法比標準NMF提供一致的更好的識別結果。
摘要(英)	In this work, we construct new dimension reduction models for learning low rank projection in the complex domain to obtain both intuitive features and high performance in real applications, particularly face, facial expression recognition and image clustering. Complex matrix factorization (CMF) is a matrix factorization method that decomposes a complex matrix into two complex matrix factors. The proposed models can be performed without limiting the sign of data. These proposed frameworks can be applied to both negative and positive data which yield extension and effectiveness on real-world applications. Therefore, several operations that can extract complex features, such as the short-time Fourier transform, are going to be utilized directly instead of their absolute values (magnitude/power spectrogram). From the basic framework, CMF, we developed two constrained CMF frameworks by adding graph and sparse constraints to obtain graph regularized complex matrix factorization (GraCMF) and complex matrix factorization with sparsity constraint (SpaCMF). Besides, we modified the structure of standard CMF to provide more extensions. One of them is exemplar-embed complex matrix factorization (EE-CMF) which a learned base lies within original space (exemplar). Projective complex matrix factorization (ProCMF) was developed as a new learning subspace method that the coefficient of each data point lies within the subspace spanned by the column vectors of one projection complex matrix. Simplical complex matrix factorization (SiCMF), a new model was figured out by convex combining the latent components to reconstruct the original instances. To satisfy nonnegative requirement, nonnegative matrix factorization (NMF) usually uses various strategies on minimizing a function, which lead to computational complexity. On the contrary, the significant superiority compared to NMF approaches of the proposed complex models is to construct an unconstraint optimization problem that simplified the framework of extracting the basis and intrinsic features. Wirtinger′s calculus is used to compute the derivative of the cost functions. The gradient descent method is used to solve complex optimization problems. The proposed algorithms are proved to provide effective features for a face and facial expression recognition as well as image clustering. Experiments on these tasks reveal that the proposed methods of matrix factorization on complex domain provide consistently better recognition results than standard NMFs.
關鍵字(中)	★ 計算機視覺 ★ 特徵提取 ★ 复矩陣分解	關鍵字(英)	★ Computer vision ★ Feature extraction ★ Complex matrix factorization
論文目次	摘要 I Abstract II Acknowledge IV List of symbols and abbreviations VIII List of Figures XI List of Tables XIII Chapter 1 Introduction 1 Chapter 2 Preliminaries 8 2.1 Matrix theory and linear algebra 8 2.2 Lagrangian duality and the Karush-Kuhn-Tucker (KKT) conditions 12 2.3 Complex analysis and optimization in the complex domain 13 2.4 Nonnegative Matrix Factorization 20 Chapter 3 Complex Matrix Factorization and Its Extensions with Hard Constraints 30 3.1 The cosine divergence 30 3.2 Euler’s formula and a space transformation 31 3.3 Complex matrix factorization problems 33 3.4 Complex matrix factorization by gradient descent method 35 3.5 CMFs for face recognition 38 3.6 Conclusion 48 Chapter 4 Structure Modified Complex Matrix Factorization 49 4.1 Introduction 49 4.2 Convex nonnegative matrix factorization and projective nonnegative matrix factorization 50 4.3 Examplar-embeded complex matrix factorization (EE-CMF) 51 4.4 Projective complex matrix factorization (PCMF) 52 4.5 Optimal solutions 52 4.6 EE-CMF and PCMF for facial expression recognition 54 4.7 Conclusion 69 Chapter 5 Bound Constrained Complex Matrix Factorization 71 5.1 Introduction 71 5.2 From NMF to siCMF 72 5.3 SiCMF for data representation 75 5.4 Experiments 76 5.5 Conclusion 80 Chapter 6 Conclusion and Future Works 81 6.1 Conclusion 81 6.2 Future Works 82 Bibliographies 84 Publication List 93
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指導教授	王家慶 范世寶(Jia-Ching Wang Pham The Bao)	審核日期	2017-11-15
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