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標題: 交替方向乘子法在降維問題之應用
Application of Alternating Direction of Multiplier Method on dimensional reduction
作者: 蘇郁舜
Yu-Shun Su
關鍵字: 可視化技術;Principal components analysis(PCA);Locally linear embedding(LLE);Maximum variance unfolding(MVU);交替方向乘子法(ADMM);Visualization techniques;Principle Component Analysis(PCA);Locally Linear Embedding(LLE);Maximum Variance Unfolding(MVU);Alternating Direction method of Multipliers(ADMM)
引用: [1] Robert J. Mason, MD, V. Courtney Broaddus, MD, Thomas Martin, Talmadge King, Jr., MD, Dean MD Schraufnagel, John F. Murray, MD, DSc(Hon), FRCP and Jay A. Nadel, MD, DSc(HON) “Murray and Nadel’s textbook of respiratory medicine.”Chapter 38. ISBN 1-4160-4710-7. [2] Xin Liu, Duygu Tosun, Michael W Weiner, Norbert Schuff,“Locally linear embedding (LLE) for MRI based Alzheimer’s disease classification”, Published 2013 in NeuroImage. [3] Ardeshir Mohammad Ebtehaj, Rafael Luis Bras, Efi Foufoula-Georgiou, “Shrunken Locally Linear Embedding for Passive Microwave Retrieval of Precipitation ”, Submitted on 2 May 2014 (v1), last revised 6 Sep 2014 (this version, v2). [4] Pearson, K. (1901). “On Lines and Planes of Closest Fit to Systems of Points in Space”. Philosophical Magazine 2 (11): 559–572. [5] Hotelling, H. (1933).“Analysis of a complex of statistical variables into principal components.”Journal of Educational Psychology. [6] component_ analysis [7]博客/2016/11/12/pca_ svd_ recall/ [8] S. T. Roweis and L. K. Saul, “Nonlinear Dimensionality Reduction by Locally Linear Embedding”, Science Vol 290, 22 December 2000, 2323–2326. [9] Chun-An Chen, “Dimension reduction on analysis asthma lung deformation”,January 2016. [10] S. T. Roweis and L. K. Saul, “An Introduction to Locally Linear Embedding”,January 2001. [11] S. T. Roweis and L. K. Saul, “An Introduction to Locally Linear Embedding”, roweis/lle/,January 2001 [12] Kilian Q. Weinberger and Lawrence K. Saul,“Unsupervised Learning of Image Manifolds by Semidefinite Progranning”, International Journal of Computer Vision 70(1),77-90,2006.
資料降維化不僅讓我們壓縮數據而使用較少的計算機內存磁碟空間, 也讓我們加快學習算法。最重要的是,降維可以幫助我們能將數據可視化, 我們便能尋找到數據的低維架構及其中所隱含的重要資訊。

在所有的數據可視化技術中,Principal components analysis(PCA) 是揭示線性結構中最常用的方法,Maximum variance unfolding(MVU) 和Locally linear embedding(LLE) 則是非線性結構降維最典型的方法。在本文中,我們對MVU和LLE 的鄰近點選取概念應用使用在交替方向乘子法(The alternating direction
method of multipliers,簡稱ADMM) 中,並透過最大化資料點的F-norm 來達到降維的效果。

透過代入LLE 的輸出作為ADMM 迭代初始值,可以用來改善LLE 降維後會扭曲變形的問題,讓降維的結果達到更好的效果。

另外,我們做出了一系列的實驗足以證明他們的性能,包含swissroll 和自製九維模型,以及探討如何選擇較好的ADMM演算法參數β。

Dimensionality reduction helps us not only reduce the use of capacity of computer disk, but also speed up the machine learning algorithms.

Most importantly, dimensionality reduction helps us visualize the data so that we can find the structure in low rank structure and the underlying information lurking
behind it.

Among all the data visualization techniques, Principle Component Analysis(PCA) is the most popular method in uncovering the linear low rank structure. Maximum Variance Unfolding(MVU) and Locally Linear Embedding(LLE) are the classical methods for dimensionality reduction of nonlinear structures. In this paper, we use the concept of neighbor selection of MVU and LLE in the Alternating Direction method of Multipliers(ADMM) and accomplish dimensionality reduction by maximizing Frobenius norm of the data.

The output of LLE can be used as an initial value of ADMM iteration to improve the result of distortion and lost distant by LLE to get a better result.

In addition, we make a series of experiments to demonstrate their performance, including swissroll and nine-dimensional models, as well as exploring how to choose the better parameters beta of ADMM algorithm.
Rights: 同意授權瀏覽/列印電子全文服務,2018-01-31起公開。
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