多線性主成分分析

多線性主成分分析方法^[1]（英語：Multilinear Principal Component Analysis，MPCA），可將高維度空間映射到低維空間中去，降維的過程就是捨棄不重要的特徵向量縮減維度，相較於一般的主成分分析，多線性主成分分析保留了資料的結構性且有較佳的解釋比例。多線性主成分分析(MPCA)是主成分分析(PCA)到多維的一個延伸。PCA是投影向量(Vector)到向量，而MPCA是投影張量(Tensor)到張量，投影的結構相對簡單，另外運算在較低維度的空間進行，因此處理高維度數據時有低運算量的優勢。舉例來說，給一個100x100的圖片，主成分分析運做在1000x1的向量上，而多線性主成分分析則是在二階模式上運作100x1的向量。對於等量的降維來說，主成分分析需要估算的變數量為多線性主成分分析的49((10000/(100x2)-1))倍，因此在實用面上多線性主成分分析可以比主成分分析更有效率。

演算法

多線性主成分分析(MPCA)定義一個多重子空間，此子空間擷取了大部分正交多維的輸入變異量，藉此達到特徵提取的效果。如同主成分分析，多線性主成分分析可運用在已中央化的資料上。多線性主成分分析的計算遵照交替最小次方(Alternating Least Square，ALS^[2])方法。因此會有迭代動作，並且以分解原本的空間至一系列的多為映射子空間。每一個子空間都是一個經典的主成分空間，很容易被解析。

延伸

多種MPCA的延伸算法已被開發:^[3]
Boosting (meta-algorithm)+MPCA^[4]
Non-negative MPCA (NMPCA) ^[5]
Robust MPCA (RMPCA) ^[6]

資源

Matlab 原始碼: MPCA （页面存档备份，存于互联网档案馆）.
Matlab 原始碼: UMPCA (including data) （页面存档备份，存于互联网档案馆）.

參考

^ H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, (2008) "MPCA: Multilinear principal component analysis of tensor objects" （页面存档备份，存于互联网档案馆）, IEEE Trans. Neural Netw., 19 (1), 18–39
^ P. M. Kroonenberg and J. de Leeuw, Principal component analysis of three-mode data by means of alternating least squares algorithms^{[永久失效連結]}, Psychometrika, 45 (1980), pp. 69–97.
^ Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. A Survey of Multilinear Subspace Learning for Tensor Data (PDF). Pattern Recognition. 2011, 44 (7): 1540–1551 [2013-07-03]. doi:10.1016/j.patcog.2011.01.004. （原始内容存档 (PDF)于2019-07-10）.
^ H. Lu, K. N. Plataniotis and A. N. Venetsanopoulos, "Boosting Discriminant Learners for Gait Recognition using MPCA Features^{[失效連結]}", EURASIP Journal on Image and Video Processing, Volume 2009, Article ID 713183, 11 pages, 2009. doi:10.1155/2009/713183.
^ Y. Panagakis, C. Kotropoulos, G. R. Arce, "Non-negative multilinear principal component analysis of auditory temporal modulations for music genre classiﬁcation", IEEE Trans. on Audio, Speech, and Language Processing, vol. 18, no. 3, pp. 576–588, 2010.
^ K. Inoue, K. Hara, K. Urahama, "Robust multilinear principal component analysis", Proc. IEEE Conference on Computer Vision, 2009, pp. 591–597.

[MPCA-1] H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, (2008) "MPCA: Multilinear principal component analysis of tensor objects" （页面存档备份，存于互联网档案馆）, IEEE Trans. Neural Netw., 19 (1), 18–39

[2] P. M. Kroonenberg and J. de Leeuw, Principal component analysis of three-mode data by means of alternating least squares algorithms^{[永久失效連結]}, Psychometrika, 45 (1980), pp. 69–97.

[3] Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. A Survey of Multilinear Subspace Learning for Tensor Data (PDF). Pattern Recognition. 2011, 44 (7): 1540–1551 [2013-07-03]. doi:10.1016/j.patcog.2011.01.004. （原始内容存档 (PDF)于2019-07-10）.

[4] H. Lu, K. N. Plataniotis and A. N. Venetsanopoulos, "Boosting Discriminant Learners for Gait Recognition using MPCA Features^{[失效連結]}", EURASIP Journal on Image and Video Processing, Volume 2009, Article ID 713183, 11 pages, 2009. doi:10.1155/2009/713183.

[5] Y. Panagakis, C. Kotropoulos, G. R. Arce, "Non-negative multilinear principal component analysis of auditory temporal modulations for music genre classiﬁcation", IEEE Trans. on Audio, Speech, and Language Processing, vol. 18, no. 3, pp. 576–588, 2010.

[6] K. Inoue, K. Hara, K. Urahama, "Robust multilinear principal component analysis", Proc. IEEE Conference on Computer Vision, 2009, pp. 591–597.

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