資料降維技術使得人們能夠一窺高維度資料的結構與相關性；近
年來，資料降維技術被運用在許多資料分析上，例如：資料探勘、影
像辨識、機器學習等。多年來已有多篇相關論文探討資料降維技術的
改善，但卻較少相關探討運用平行計算於資料降維技術方面的研究，
本論文的主要研究則是運用平行計算方法來加快資料降維技術運算
速度。
由於多核心處理器的普及，圖形運算處理器(GPU)也邁入了多核
心架構。新一代的圖形運算處理器結合中央處理器以異質性多核心架
構，強調高平行度以及優越的運算能力。本文提出最鄰近圖形建立平
行方法，並將該方法實作於多核心圖形運算處理器上。除此之外，本
文也提出兩種最短距離搜尋平行演算法，同時也將該平行計算方式實
作於多核心圖形運算處理器，最後將我們提出的平行計算方式結合資
料降維技術，提升資料降維計算效率。
我們將平行計算的結果結合MATLAB 實現於 Nvidia 的 CUDA 平台
上，我們發現最鄰近圖形建立法經由平行運算後速度提升了 10倍，
至於最短距離搜尋法在平行計算之後也加快了2-3 倍速度。最後我們
將平行的結果實現於資料降維技術上，發現經過平行計算處理後的資
料降維技術運算效率提升約20%-50%。

Data dimensionality reduction techniques let people understand the
structure of the multi-dimensional data. Data mining, pattern recognition
or machine learning take it to analyze data and retrieve the implicit
information from data in the high dimensional space. A lot of algorithms
have proposed to process the linear data or non-linear data dimensionality
reduction. In this thesis, we try to propose the parallel data dimensionality
reduction formulations and implement them on the Chip of the Multi-core
Processor (CMP).
Multi-core microprocessor has been the main stream in the
computer. Graphic Processing Unit (GPU) is emerging as the many core
architecture with the multi-thread programming platform. The new
generation GPU with powerful computing capacities that is not only
suitable for dealing with graphic processing but also solving the problems
in various applications.
Most of data dimensionality reduction techniques take the nearest
neighbor (NN) graph construction or take the all pairs shortest paths
(APSP) algorithm to approximate data. We conduct these two methods in
parallel on the GPU. We can speed up NN about 10X and APSP
algorithm about 2X. At last, we upgrade 20% to 50% manifold learning
computation performance.

[1] L.J.P. van der Maaten. An Introduction to Dimensionality Reduction Using Matlab. Technical Report MICC 07-07. Maastricht University, Maastricht, The Netherlands, 2007
[2] L.J.P. van der Maaten, E.O. Postma, and H.J. van den Herik. Dimensionality reduction: A comparative review. Preprint, 2008
[3] J. Tenenbaum, V. deSilva, J. Langford, “A Global Geometric Framework
for Nonlinear Dimensionality Reduction,”Science, vol. 290, pp. 2319-
2323, Dec. 2000.
[4] S. Roweis, L. Saul, Nonlinear Dimensionality Reduction by Locally
Linear Embedding , Science, vol. 290, pp. 2323-2326, Dec. 2000.
[5] M. Belkin, P. Niyogi, Laplacian Eigenmaps for Dimensionality Reduction and Data Representation , Neural Computation, vol. 15, no. 2, pp.
1373-1396, 2003.
[6] Tsai, F.S., Kap Luk Chan, Dimensionality reduction techniques for data exploration, Information, Communications & Signal Processing, 2007 6th International Conference, Dec. 2007.
[7] V de Silva, JB Tenenbaum , Global versus local methods in nonlinear dimensionality reduction, advances in neural information processing systems, 2003
[8] Yoshua Bengio, Jean-françois Paiement, Pascal Vincent, Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering, In Annual Advances in Neural Information Processing Systems 16: Proceedings of the 2003 Conference, 2003.
[9] Cayton, “Algorithms for Manifold Learning”. Technical Report, 2005.
[10] Vin de Silva and Joshua B. Tenenbaum. Sparse multidimensional scaling using landmark points. Stanford Mathematics Technical Report, 2004.
[11] T.R. Halfhill, “Parallel Processing With CUDA,” Microprocessor Report, Jan. 2008.
[12] Buck, I., “GPU Computing: Programming a Massively Parallel Processor.” International Symposium on Code Generation and Optimization, San José, ca, 2007.
[13] P. Harish and P.J.Narayanan. Accelerating large graph algorithms on the GPU using CUDA. In International Conference on High Performance Computing (HiPC 2007), 2007.
[14] J. D. Owens, D. Luebke, N. Govindaraju, M. Harris, J. Kru¨ger, A. E. Lefohn, and T. Purcell, BA survey of general-purpose computation on graphics hardware, Comput. Graph. Forum, vol. 26, no. 1, pp. 80–113, 2007.
[15] Y. Han, V. Pan, and J. Reif. Efficient parallel algorithms for computing all pair shortest paths in directed graphs. In Proc. 4th annual ACM Symposium on Parallel Algorithms and Architectures, pages 353-362. ACM, 1997.
[16] Ananth Grama, Anshul Gupta, George Karypis and Vipin Kumar, Introduction to parallel computing, Addison Wesley, 2003.
[17] Nvidia Corp. , CUDA programming guide 2.0, Nvidia, 2008.
[18] Nvidia Corp. , Accelerating MATLAB with CUDA Using MEX Files white paper, Nvidia, 2007.
[19] T. F. Cox and M. A. A. Cox, Multidimensional Scaling. London: Chapman & Hall, 1994.
[20] Nvidia Corp. www.nvidia.com, 2008
[21] Floyd, Robert W. Algorithm 97: Shortest Path. Communications of the ACM 5 (6): 345, 1962.
[22] David L. Donoho, Carrie Grimes, Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data, Proceedings of the National Academy of Sciences, 2003.
[23] Sam T. Roweis, Lawrence K. Saul, Local Linear Embedding Homepage, http://www.cs.toronto.edu/~roweis/lle/related.html, 2008.
[24] J. B. Tenenbaum, V. de Silva and J. C. Langford, Isomap Homepage, http://waldron.stanford.edu/~isomap/, 2008.
[25] L.J.P. van der Maaten, Matlab Toolbox for Dimensionality Reduction, http://ticc.uvt.nl/~lvdrmaaten/Laurens_van_der_Maaten/Matlab_Toolbox_for_Dimensionality_Reduction.html, 2008.
[26] Martin H. C. and Anil K. Jain, Incremental Nonlinear Dimensionality Reduction By Manifold Learning, IEEE Transactions of Pattern Analysis and Intelligence, 2003.
[27] V. Garcia, E. Debreuve, and M. Barlaud, “Fast k nearest neighbor search using gpu,” in Proceedings of Computer Vision and Pattern Recognition Workshops, June 2008, pp. 1–6.
[28] B. Bustos, O. Deussen, S. Hiller, and D. Keim. A Graphics Hardware Accelerated Algorithm for Nearest Neighbor Search, in Proc. 6th Int. Conf. Comput. Sci. Vol. 3994, pp. 196–199, 2006.
[29] Fraleigh Beauregard, Linear Algebra, Addison-Wesley, 1995.