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**Clustering High Dimensional Data Using SVM**

Tsau Young Lin and Tam Ngo Department of Computer Science San José State University

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**Overview Introduction Support Vector Machine (SVM)**

What is SVM How SVM Works Data Preparation Using SVD Singular Value Decomposition (SVD) Analysis of SVD The Project Conceptual Exploration Result Analysis Conclusion Future Work

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**Introduction World Wide Web Project’s Goals**

No. 1 place for information contains billions of documents impossible to classify by humans Project’s Goals Cluster documents Reduce documents size Get reasonable results when compared to humans classification

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**Support Vector Machine (SVM)**

a supervised learning machine outperforms many popular methods for text classification used for bioinformatics, signature/hand writing recognition, image and text classification, pattern recognition, and spam categorization

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**Motivation for SVM How do we separate these points? with a hyperplane**

Source: Author’s Research

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SVM Process Flow Feature Space Input Space Input Space Source: DTREG

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Convex Hulls Source: Bennett, K. P., & Campbell, C., 2000

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**Simple SVM Example How would SVM separates these points?**

Class X1 +1 -1 1 2 3 How would SVM separates these points? use the kernel trick Φ(X1) = (X1, X12) It becomes 2-deminsional Source: Author’s Research

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**Simple Points in Feature Space**

Class X1 X12 +1 -1 1 2 4 3 9 All points here are support vectors. Source: Author’s Research

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**SVM Calculation Positive: w x + b = +1 Negative: w x + b = -1**

Hyperplane: w x + b = 0 find the unknowns, w and b Expending the equations: w1x1 + w2x2 + b = +1 w1x1 + w2x2 + b = -1 w1x1 + w2x2 + b = 0

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**Use Linear Algebra to Solve w and b**

w1x1 + w2x2 + b = +1 w10 + w20 + b = +1 w13 + w29 + b = +1 w1x1 + w2x2 + b = -1 w11 + w21 + b = -1 w12 + w24 + b = -1 Solution is w1 = -3, w2 = 1, b = 1 SVM algorithm can find the solution that returns a hyperplane with the largest margin

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**Use Solutions to Draw the Planes**

Positive Plane: w x + b = +1 w1x1 + w2x2 + b = +1 -3x1 + 1x2 + 1 = +1 x2 = 3x1 Negative Plane: w x + b = -1 w1x1 + w2x2 + b = -1 -3x1 + 1x2 + 1 = -1 x2 = x1 Hyperplane: w x + b = 0 w1x1 + w2x2 + b = 0 -3x1 + 1x2 + 1 = 0 x2 = x1 X1 X2 1 3 2 6 9 X1 X2 -2 1 2 4 3 7 X1 X2 -1 1 2 5 3 8 Source: Author’s Research

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**Simple Data Separated by a Hyperplane**

Source: Author’s Research

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**LIBSVM and Parameter C LIBSVM: A Java Library for SVM**

C is very small: SVM only considers about maximizing the margin and the points can be on the wrong side of the plane. C value is very large: SVM will want very small slack penalties to make sure that all data points in each group are separated correctly.

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Choosing Parameter C Source: LIBSVM

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4 Basic Kernel Types LIBSVM has implemented 4 basic kernel types: linear, polynomial, radial basis function, and sigmoid 0 -- linear: u'*v 1 -- polynomial: (gamma*u'*v + coef0)^degree 2 -- radial basis function: exp(-gamma*|u-v|^2) 3 -- sigmoid: tanh(gamma*u'*v + coef0) We use radial basis function with large parameter C for this project.

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**Data Preparation Using SVD**

SVM is excellent for text classification, but requires labeled documents to use for training Singular Value Decomposition (SVD) separates a matrix into three parts; left eigenvectors, singular values, and right eigenvectors decompose data such as images and text. reduce data size We will use SVD to cluster

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**SVD Example of 4 Documents**

D1: Shipment of gold damaged in a fire D2: Delivery of silver arrived in a silver truck D3: Shipment of gold arrived in a truck D4: Gold Silver Truck Source: Garcia, E., 2006

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**Matrix A = U*S*VT D1 D2 D3 D4 a 1 arrived damaged delivery fire gold**

arrived damaged delivery fire gold in of shipment silver 2 truck Given a matrix A, we can factor it into three parts: U, S, and VT. Source: Garcia, E., 2006

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**Using JAMA to Decompose Matrix A**

S = Source: JAMA (MathWorks and the National Institute of Standards and Technology (NIST))

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**Using JAMA to Decompose Matrix A**

V = VT = Matrix A can be reconstructed by multiplying matrices: U*S*VT Source: JAMA

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**Rank 2 Approximation (Reduced U, S, and V Matrices)**

S’ = V’ =

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**Use Matrix V to Calculate Cosine Similarities**

calculate cosine similarities for each document. sim(D’, D’) = (D’• D’) / (|D’| |D’|) example, Calculate for D1’: sim(D1’, D2’) = (D1’• D2’) / (|D1’| |D2’|) sim(D1’, D3’) = (D1’• D3’) / (|D1’| |D3’|) sim(D1’, D4’) = (D1’• D4’) / (|D1’| |D4’|)

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**Result for Cosine Similarities**

Example result for D1’: sim(D1’, D2’) = (( * ) + ( * )) = ( (0.4652)2 + ( )2 ) * ( (0.6406)2 + (0.6401) 2 ) sim(D1’, D3’) = (( * ) + ( * )) = ( (0.4652)2 + ( )2 ) * ( (0.5622)2 + ( )2 ) sim(D1’, D4’) = (( * ) + ( * )) = ( (0.4652)2 + ( )2 ) * ( (0.2391)2 + (0.2450)2 ) D3 returns the highest value, pair D1 with D3 Do the same for D2, D3, and D4.

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**Result of Simple Data Set**

label 1: 1 3 label 2: 2 4 label 1: D1: Shipment of gold damaged in a fire D3: Shipment of gold arrived in a truck label 2: D2: Delivery of silver arrived in a silver truck D4: Gold Silver Truck

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**Check Cluster Using SVM**

Now we have the label, we can use it to train with SVM SVM input format on original data: 1 1:1.00 2:0.00 3:1.00 4:0.00 5:1.00 6:1.00 7:1.00 8:1.00 9: : :0.00 2 1:1.00 2:1.00 3:0.00 4:1.00 5:0.00 6:0.00 7:1.00 8:1.00 9: : :1.00 1 1:1.00 2:1.00 3:0.00 4:0.00 5:0.00 6:1.00 7:1.00 8:1.00 9: : :1.00 2 1:0.00 2:0.00 3:0.00 4:0.00 5:0.00 6:1.00 7:0.00 8:0.00 9: : :1.00

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**Results from SVM’s Prediction**

Results from SVM’s Prediction on Original Data Documents use for Training Predict the Following Document SVM Prediction Result SVD Cluster Result D1, D2, D3 D4 1.0 2 D1, D2, D4 D3 1 D1, D3, D4 D2 2.0 D2, D3, D4 D1 Source: Author’s Research

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**Using Truncated V Matrix**

We want to reduce data size, more practical to use truncated V matrix SVM input format (truncated V matrix): 1 1: : 2 1: :0.6401 1 1: : 2 1: :0.2450

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**SVM Result From Truncated V Matrix**

Results from SVM’s Prediction on Reduced Data Documents use for Training Predict the Following Document SVM Prediction Result SVD Cluster Result D1, D2, D3 D4 2.0 2 D1, D2, D4 D3 1.0 1 D1, D3, D4 D2 D2, D3, D4 D1 Using truncated V matrix gives better results. Source: Author’s Research

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**Vector Documents on a Graph**

Source: Author’s Research

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**Analysis of the Rank Approximation**

Cluster Results from Different Ranking Approximation Rank 1 Rank 2 Rank 3 Rank 4 D1: 4 D2: 4 D3: 4 D4: 3 D1: 3 D3: 1 D4: 2 D2: 3 D1: 2 D3: 2 label 1: label 1: 1 3 label 2: 2 4 label 1: label 1: Source: Author’s Research

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**Program Process Flow use the previous methods on larger data sets**

compare the results with that of humans classification Program Process Flow

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**Conceptual Exploration**

Reuters-21578 a collection of newswire articles that have been human-classified by Carnegie Group, Inc. and Reuters, Ltd most widely used data set for text categorization

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**First Data Set from Reuters-21578 (200 x 9928)**

Result Analysis Clustering with SVD vs. Humans Classification First Data Set First Data Set from Reuters (200 x 9928) # of Naturally Formed Cluster using SVD SVD Cluster Accuracy SVM Prediction Accuracy Rank 002 80 75.0% 65.0% Rank 005 66 81.5% 82.0% Rank 010 60.5% 54.0% Rank 015 64 52.0% 51.5% Rank 020 67 38.0% 46.5% Rank 030 72 60.0% Rank 040 62.5% 58.5% Rank 050 73 54.5% Rank 100 75 45.5% Source: Author’s Research

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**Second Data Set from Reuters-21578 (200 x 9928)**

Result Analysis Clustering with SVD vs. Humans Classification Second Data Set Second Data Set from Reuters (200 x 9928) # of Naturally Formed Cluster using SVD SVD Cluster Accuracy SVM Prediction Accuracy Rank 002 76 67.0% 84.5% Rank 005 73 Rank 010 64 70.0% 85.5% Rank 015 63.0% 81.0% Rank 020 67 59.5% 50.0% Rank 030 69 68.5% 83.5% Rank 040 59.0% 79.0% Rank 050 44.5% 25.5% Rank 100 71 52.0% 47.0% Source: Author’s Research

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Result Analysis highest percentage accuracy for SVD clustering is 81.5% lower rank value seems to give better results SVM predicts about the same accuracy as SVD cluster

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**Result Analysis: Why results may not be higher?**

humans classification is more subjective than a program reducing many small clusters to only 2 clusters by computing the average may decrease the accuracy

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**Conclusion Showed how SVM works**

Explore the strength of SVM Showed how SVD can be used for clustering Analyzed simple and complex data the method seems to cluster data reasonably Our method is able to: reduce data size (by using truncated V matrix) cluster data reasonably classify new data efficiently (based on SVM) By combining known methods, we created a form of unsupervised SVM

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Future Work extend SVD to very large data set that can only be stored in secondary storage looking for more efficient kernels of SVM

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Thank You!

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References Bennett, K. P., & Campbell, C. (2000). Support Vector Machines: Hype or Hellelujah?. ACM SIGKDD Explorations. VOl. 2, No. 2, 1-13 Chang, C & Lin, C. (2006). LIBSVM: a library for support vector machines, Retrived November 29, 2006, from Cristianini, N. (2001). Support Vector and Kernel Machines. Retrieved November 29, 2005, from Cristianini, N., & Shawe-Taylor, J. (2000). An Introduction to Support Vector Machines. Cambridge UK: Cambridge University Press Garcia, E. (2006). SVD and LSI Tutorial 4: Latent Semantic Indexing (LSI) How-to Calculations. Retrieved November 28, 2006, from Guestrin, C. (2006). Machine Learning. Retrived November 8, 2006, from Hicklin, J., Moler, C., & Webb, P. (2005). JAMA : A Java Matrix Package. Retrieved November 28, 2006, from

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References Joachims, T. (1998). Text Categorization with Support Vector Machines: Learning with Many Relevant Features. Joachims, T. (2004). Support Vector Machines. Retrived November 28, 2006, from Reuters Text Categorization Test Collection. Retrived November 28, 2006, from SVM - Support Vector Machines. DTREG. Retrived November 28, 2006, from Vapnik, V. N. (2000, 1995). The Nature of Statistical Learning Theory. Springer-Verlag New York, Inc.

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