Machine Learning for Data Science (CS4786) Lecture 11. Spectral Embedding + Clustering

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1 Machine Learning for Data Science (CS4786) Lecture 11 Spectral Embedding + Clustering

2 MOTIVATING EXAMPLE

3 What can you say from this network?

4 MOTIVATING EXAMPLE

5 How about now?

6 THOUGHT EXPERIMENT For each user i we specify embedding (location) y i How do we find good locations y 1,...,y n? What are good properties?

7 KEY PRINCIPLE Points are centered at 0 Keep your Friends close (sum of distances between linked nodes should be small) Variance or spread amongst the nodes should be large

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10 REPRESENTING THE GRAPH

11 REPRESENTING THE GRAPH A =

12 REPRESENTING THE GRAPH

13 REPRESENTING THE GRAPH D =

14 THE LAPLACIAN MATRIX L = D - A

15 THE LAPLACIAN MATRIX L = D - A

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sha1_base64="qebnebzoo/koim1bbfi3z3rctey=">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</latexit> <latexit sha1_base64="qebnebzoo/koim1bbfi3z3rctey=">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</latexit> KEY PRINCIPLE Points are centered at 0 Keep your Friends close y > 1 =0 minimize y > Ly Variance or spread should be large Maximize 1 n kyk2 2 Minimize y> Ly kyk 2 2 s.t. y? 1 Minimize y > Ly s.t. kyk 2 2 =1 y? 1

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22 SPECTRAL CLUSTERING ALGORITHM (UNNORMALIZED) 1 Given matrix A calculate diagonal matrix D s.t. D i,i = n j=1 A i,j 2 Calculate the Laplacian matrix L = D A 3 Find eigen vectors v 1,...,v n of L (ascending order of eigenvalues) 4 Pick the K eigenvectors with smallest eigenvalues to get y 1,...,y n R K 5 Use K-means clustering algorithm on y 1,...,y n

23 TROUBLE MAKERS

24 TROUBLE MAKERS

25 TROUBLE MAKERS

26 TROUBLE MAKERS

27 TROUBLE MAKERS

28 TROUBLE MAKERS

29 Variance is high TROUBLE MAKERS

30 TROUBLE MAKERS Variance is high Almost all connected nodes have same (small value)

31 Demo

32 What was the problem?

33 What was the problem? Pushing few nodes far away from the rest increased variance

34 What was the problem? Pushing few nodes far away from the rest increased variance But these nodes were ones with one or few links

35 What was the problem? Pushing few nodes far away from the rest increased variance But these nodes were ones with one or few links We want nodes with fewer links to account for lesser of the variance

36 NORMALIZED SPECTRAL EMBEDDING

37 NORMALIZED SPECTRAL EMBEDDING Nodes linked to each other are close to each other

38 NORMALIZED SPECTRAL EMBEDDING Nodes linked to each other are close to each other Variance or spread should be large

39 NORMALIZED SPECTRAL EMBEDDING Nodes linked to each other are close to each other Variance or spread should be large But variance under what distribution?

40 NORMALIZED SPECTRAL EMBEDDING Nodes linked to each other are close to each other Variance or spread should be large But variance under what distribution? Higher degree nodes are more important!

41 NORMALIZED SPECTRAL EMBEDDING Nodes linked to each other are close to each other Variance or spread should be large But variance under what distribution? Higher degree nodes are more important! Lets try a distribution where probability of picking node is proportional to its degree

42 NORMALIZED SPECTRAL EMBEDDING Nodes linked to each other are close to each other Variance or spread should be large But variance under what distribution? Higher degree nodes are more important! Lets try a distribution where probability of picking node is proportional to its degree so pushing loners away does not account for much variance

43 NORMALIZED SPECTRAL EMBEDDING Nodes linked to each other are close to each other Variance or spread should be large But variance under what distribution? Higher degree nodes are more important! Lets try a distribution where probability of picking node is proportional to its degree so pushing loners away does not account for much variance There is more incentive to push the famous people outwards

44 NORMALIZED SPECTRAL EMBEDDING

45 <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">aaacthicdzdnsxwxgmyza6t2+7xqszfguuhblkskrgdb1ekpflwvdrzdjptxsyazixmndyn5a7146k1/rs89tjrcm7qffr4qehie982b/pjksqeefexac/fuly4tp2g/fpt4ydpoyuqjk2vlxycxqrrnoxncssmgiegjs8okpnmltvolgyy/fsesk6u5hlklrpqdg1lizibawyenic7bao8prrhvwoo408q8bml8xsiehyqtebenhwxch2zebsgqfopqnfnplg1vdy7udgmawt9tv6+uqtbpkh4hhfkkg0g3x5iodnb6m7sparpf6qj5hwwdd+m45luwbrhizg0pqwdkmqxjlqjtthaiyvycnythlizp4ub+fkbaz6mzxkvp4zgab91fjzztzs10hjs1g4n7m2vmf2xdgor+yett1samv1tu1apdiruykzwvhnqscsatjg/ffmiidyj82xhcz5/i/4utzr4lpfpmq7u3p8exjj6hdfqcubsn9tbrdiqgiknr9al9qv+tm+rz8i35ftfasuyza+i3ai3+ad8wttc=</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">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</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">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</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">aaacthicdzdnsxwxgmyza6t2+7xqszfguuhblkskrgdb1ekpflwvdrzdjptxsyazixmndyn5a7146k1/rs89tjrcm7qffr4qehie982b/pjksqeefexac/fuly4tp2g/fpt4ydpoyuqjk2vlxycxqrrnoxncssmgiegjs8okpnmltvolgyy/fsesk6u5hlklrpqdg1lizibawyenic7bao8prrhvwoo408q8bml8xsiehyqtebenhwxch2zebsgqfopqnfnplg1vdy7udgmawt9tv6+uqtbpkh4hhfkkg0g3x5iodnb6m7sparpf6qj5hwwdd+m45luwbrhizg0pqwdkmqxjlqjtthaiyvycnythlizp4ub+fkbaz6mzxkvp4zgab91fjzztzs10hjs1g4n7m2vmf2xdgor+yett1samv1tu1apdiruykzwvhnqscsatjg/ffmiidyj82xhcz5/i/4utzr4lpfpmq7u3p8exjj6hdfqcubsn9tbrdiqgiknr9al9qv+tm+rz8i35ftfasuyza+i3ai3+ad8wttc=</latexit> NORMALIZED SPECTRAL EMBEDDING Define distribution with p i = D i,i P n j=1 D j,j = D i,i E

46 <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">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</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">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</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">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</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">aaacthicdzdnsxwxgmyza6t2+7xqszfguuhblkskrgdb1ekpflwvdrzdjptxsyazixmndyn5a7146k1/rs89tjrcm7qffr4qehie982b/pjksqeefexac/fuly4tp2g/fpt4ydpoyuqjk2vlxycxqrrnoxncssmgiegjs8okpnmltvolgyy/fsesk6u5hlklrpqdg1lizibawyenic7bao8prrhvwoo408q8bml8xsiehyqtebenhwxch2zebsgqfopqnfnplg1vdy7udgmawt9tv6+uqtbpkh4hhfkkg0g3x5iodnb6m7sparpf6qj5hwwdd+m45luwbrhizg0pqwdkmqxjlqjtthaiyvycnythlizp4ub+fkbaz6mzxkvp4zgab91fjzztzs10hjs1g4n7m2vmf2xdgor+yett1samv1tu1apdiruykzwvhnqscsatjg/ffmiidyj82xhcz5/i/4utzr4lpfpmq7u3p8exjj6hdfqcubsn9tbrdiqgiknr9al9qv+tm+rz8i35ftfasuyza+i3ai3+ad8wttc=</latexit> NORMALIZED SPECTRAL EMBEDDING Define distribution with p i = D i,i P n j=1 D j,j = D i,i E Keep your Friends close

47 <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">aaacthicdzdnsxwxgmyza6t2+7xqszfguuhblkskrgdb1ekpflwvdrzdjptxsyazixmndyn5a7146k1/rs89tjrcm7qffr4qehie982b/pjksqeefexac/fuly4tp2g/fpt4ydpoyuqjk2vlxycxqrrnoxncssmgiegjs8okpnmltvolgyy/fsesk6u5hlklrpqdg1lizibawyenic7bao8prrhvwoo408q8bml8xsiehyqtebenhwxch2zebsgqfopqnfnplg1vdy7udgmawt9tv6+uqtbpkh4hhfkkg0g3x5iodnb6m7sparpf6qj5hwwdd+m45luwbrhizg0pqwdkmqxjlqjtthaiyvycnythlizp4ub+fkbaz6mzxkvp4zgab91fjzztzs10hjs1g4n7m2vmf2xdgor+yett1samv1tu1apdiruykzwvhnqscsatjg/ffmiidyj82xhcz5/i/4utzr4lpfpmq7u3p8exjj6hdfqcubsn9tbrdiqgiknr9al9qv+tm+rz8i35ftfasuyza+i3ai3+ad8wttc=</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">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</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">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</latexit> <latexit sha1_base64="1tr4hnjlhtx0l/wji1mqpkoc1mg=">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</latexit> NORMALIZED SPECTRAL EMBEDDING Define distribution with p i = D i,i P n j=1 D j,j = D i,i E Keep your Friends close minimize y > Ly

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