交互擴散映射(Alternating diffusion map)是一種降維(dimension reduction)方法。對同一現象同時具有數個觀測資料時，可用於整合並從中抽取出資料特徵。Lederman, Talmon, Wu, Lo and Coifman提出一種交互擴散映射過程用於找出共同來源(common sources)的距離關係。Shnitzer, Ben-Chen, Guibas, Talmon and Wu基於交互擴散映射引入了兩個算子，可用於分辨資料的相同與相異結構。本文的目的是驗證兩種交互擴散映射方法，並比較兩方法在睡眠階段分類中的績效。此外，我們也針對Lederman等人提出的交互擴散映射過程給出離散版本證明。

Alternating diffusion map is a dimension reduction method which is used to integrate and extract features of the observations measured from multiple sensors. Lederman, Talmon, Wu, Lo and Coifman [1] proposed a process to find out a distance relation of common sources. Shnitzer, Ben-Chen, Guibas, Talmon and Wu [2] introduced two operator based on alternating diffusion method to distinguish common and difference structure of the datasets. The goal of this paper is to validate two alternating diffusion methods and compare the performance of two different alternating diffusion method on sleep stage assessment. In addition, we give a proof of discrete version for alternating diffusion process proposed by Lederman et. al.

Table of Contents
1. Preliminary 1
1.1 Kronecker Product 1
1.2 Lederman’s method 1
1.3 Shnitzer’s method 5
2. Discrete version of Lederman’s method 8
3. Validation 14
3.1 Validation on Lederman’s method 14
3.2 Validation on Shnitzer’s method 16
4. Sleep Stage Assessment 18
4.1 Feature extraction through Lederman’s method 20
4.2 Feature extraction through Shnitzer’s method 22
4.3 Cluster based on density of eigenvectors 24
4.4 Regrouped by subjects 25
5. Conclusion 28
6. Algorithm 29
References 38

[1] Lederman, Roy R., Ronen Talmon, Hau-Tieng Wu, Yu-Lun Lo, and Ronald R. Coifman.
"Alternating Diffusion for Common Manifold Learning with Application to Sleep Stage Assessment."
2015 IEEE International Conference on Acoustics, Speech and Signal Processing
(ICASSP), 2015. doi:10.1109/icassp.2015.7179075.
[2] Shnitzer, T., Ben-Chen, M., Guibas, L., Talmon, R., andWu, H. T. "Recovering hidden components
in multimodal data with composite diffusion operators." arXiv preprint arXiv:1808.07312
[3] Coifman, Ronald R., and Stéphane Lafon. "Diffusion Maps." Applied and Computational
Harmonic Analysis 21, no. 1 (2006): 5-30. doi:10.1016/j.acha.2006.04.006.
[4] Froyland, Gary. "Dynamic Isoperimetry and the Geometry of Lagrangian Coherent Structures."
Nonlinearity 28, no. 10 (2015): 3587-622. doi:10.1088/0951-7715/28/10/3587.
[5] Froyland, Gary, and Eric Kwok. "A Dynamic Laplacian for Identifying Lagrangian Coherent
Structures on Weighted Riemannian Manifolds." Journal of Nonlinear Science, 2017.
doi:10.1007/s00332-017-9397-y.