Spectral Clustering Lecture 16 David Sontag New York - PowerPoint PPT Presentation

Spectral ¡Clustering ¡ Lecture ¡16 ¡ David ¡Sontag ¡ New ¡York ¡University ¡ Slides adapted from James Hays, Alan Fern, and Tommi Jaakkola

Spectral ¡clustering ¡ K-means Spectral clustering twocircles, 2 clusters two circles, 2 clusters (K − means) 5 5 5 4.5 4.5 4.5 4 4 4 3.5 3.5 3.5 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 [Shi & Malik ‘00; Ng, Jordan, Weiss NIPS ‘01]

Spectral ¡clustering ¡ nips, 8 clusters lineandballs, 3 clusters fourclouds, 2 clusters 5 5 5 4.5 4.5 4.5 4 4 4 3.5 3.5 3.5 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 squiggles, 4 clusters threecircles − joined, 3 clusters twocircles, 2 clusters threecircles − joined, 2 clusters 5 5 5 5 4.5 4.5 4.5 4.5 4 4 4 4 3.5 3.5 3.5 3.5 3 3 3 3 2.5 2.5 2.5 2.5 2 2 2 2 1.5 1.5 1.5 1.5 − 1 1 1 1 − 0.5 0.5 0.5 0.5 − 0 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 [Figures from Ng, Jordan, Weiss NIPS ‘01]

Spectral ¡clustering ¡ ¡ ¡Group ¡points ¡based ¡on ¡links ¡in ¡a ¡graph ¡ B A [Slide from James Hays]

!"# � $" � %&'($' � $)' � *&(+) � , • -$ � ./ � 0"11"2 � $" � 3/' � ( � *(3//.(2 � 4'&2'5 � $" � 0"1+3$' � /.1.5(&.$6 � 7'$#''2 � "78'0$/ � � � � • 92' � 0"35: � 0&'($' � – ; � <3556 � 0"22'0$': � =&(+) – 4 � 2'(&'/$ � 2'.=)7"& � =&(+) � >'(0) � 2":' � ./ � "256 � 0"22'0$': � $" � .$/ � 4 � 2'(&'/$ � 2'.=)7"&/? B A [Slide from Alan Fern]

Spectral ¡clustering ¡for ¡segmenta>on ¡ [Slide from James Hays]

Can ¡we ¡use ¡minimum ¡cut ¡for ¡ clustering? ¡ ���� �� � ���� ����� ������� ��� ����� � ��� ���������� [Shi & Malik ‘00]

Graph � partitioning

Graph � Terminologies • Degree � of � nodes • Volume � of � a � set

Graph � Cut • Consider � a � partition � of � the � graph � into � two � parts � A � and � B • Cut(A, � B) : � sum � of � the � weights � of � the � set � of � edges � that � connect � the � two � groups • An � intuitive � goal � is � find � the � partition � that �� minimizes � the � cut

Normalized � Cut • Consider � the � connectivity � between � groups � relative � to � the � volume � of � each � group cut ( A , B ) cut ( A , B ) A � � Ncut ( A , B ) Vol ( A ) Vol ( B ) B � Vol ( A ) Vol ( B ) � Ncut ( A , B ) cut ( A , B ) Vol ( A ) Vol ( B ) Minimized � when � Vol(A) � and � Vol(B) � are � equal. � Thus � encourage � balanced � cut

Solving � NCut • How � to � minimize � Ncut ? � Let W be the similarity matrix, W ( i , j ) W ; i , j � � Let D be the diag. matrix, D ( i , i ) W ( i , j ); j � � � � N Let x be a vector in { 1 , 1 } , x ( i ) 1 i A . • With � some � simplifications, � we � can � show: � T y ( D W ) y � min Ncut ( x ) min x y T y Dy Rayleigh � quotient 1 � y T D 0 ( y takes discrete values ) Subject � to: NP � Hard!

Solving � NCut • Relax � the � optimization � problem � into � the � continuous � domain � by � solving � generalized � eigenvalue � system: ��� � � � � � � � subject � to � � � �� � � • Which � gives: � � � � � ��� • Note � that � � � � � � � , � so � the � first � eigenvector � is � � � � � with � eigenvalue � � . • The � second � smallest � eigenvector � is � the � real � valued � solution � to � this � problem!!

2 � way � Normalized � Cuts 1. Compute � the � affinity � matrix � W, � compute � the � degree � matrix � (D), � D � is � diagonal � and � ��� 2. Solve � , � where � is � called � the � Laplacian matrix 3. Use � the � eigenvector � with � the � second � smallest � eigen � value � to � bipartition � the � graph � into � two � parts.

Creating � Bi � partition � Using � 2 nd Eigenvector • Sometimes � there � is � not � a � clear � threshold � to � split � based � on � the � second � vector � since � it �� takes � continuous � values • How � to � choose � the � splitting � point? � a) Pick � a � constant � value � (0, � or � 0.5). b) Pick � the � median � value � as � splitting � point. Look � for � the � splitting � point � that � has � the � minimum � Ncut c) value: Choose � n possible � splitting � points. 1. Compute � Ncut value. 2. 3. Pick � minimum.

Spectral clustering: example 6 6 5 5 4 4 3 3 2 2 1 1 0 0 − 1 − 1 − 2 − 2 − 3 − 2 − 1 0 1 2 3 4 5 − 4 − 2 0 2 4 6 Tommi Jaakkola, MIT CSAIL 18

Spectral clustering: example cont’d 0.5 0.4 0.3 0.2 0.1 0 − 0.1 − 0.2 − 0.3 − 0.4 − 0.5 0 5 10 15 20 25 30 35 40 Components of the eigenvector corresponding to the second largest eigenvalue Tommi Jaakkola, MIT CSAIL 19

K � way � Partition? • Recursive � bi � partitioning � (Hagen � et � al.,^91) – Recursively � apply � bi � partitioning � algorithm � in � a � hierarchical � divisive � manner. – Disadvantages: � Inefficient, � unstable • Cluster � multiple � eigenvectors – Build � a � reduced � space � from � multiple � eigenvectors. – Commonly � used � in � recent � papers – A � preferable � approach` � its � like � doing � dimension � reduction � then � k � means