A Cheeger-Type Inequality on Simplicial Complexes Scientific and Statistical Computing Seminar – U Chicago Sayan Mukherjee Departments of Statistical Science, Computer Science, Mathematics Institute for Genome Sciences & Policy, Duke University Joint work with: J. Steenbergen (Duke), Carly Klivans (Brown) November 9, 2012
Motivation Definitions Results Open problems Acknowledgements Dimension reduction algorithm Examples include: Isomap - 2000 1 Locally Linear Embedding (LLE) - 2000 2 Hessian LLE - 2003 3 Laplacian Eigenmaps - 2003 4 Diffusion Maps - 2004 5 Laplacian Eigenmaps is based directly on the graph Laplacian. 2,
Motivation Definitions Results Open problems Acknowledgements Laplacian eigenmaps R p which we wish to map into Given data points x 1 , x 2 , . . . , x n ∈ I R k , k ≪ p , I 1 Construct a graph (association matrix) A out of the data points. 3,
Motivation Definitions Results Open problems Acknowledgements Laplacian eigenmaps R p which we wish to map into Given data points x 1 , x 2 , . . . , x n ∈ I R k , k ≪ p , I 1 Construct a graph (association matrix) A out of the data points. 2 Compute L = D − A where D ii = � j A ij 4,
Motivation Definitions Results Open problems Acknowledgements Laplacian eigenmaps R p which we wish to map into Given data points x 1 , x 2 , . . . , x n ∈ I R k , k ≪ p , I 1 Construct a graph (association matrix) A out of the data points. 2 Compute L = D − A where D ii = � j A ij 3 Compute eigenvalues and eigenvectors of L 0 = λ 0 ≤ λ 1 ≤ λ 2 ≤ . . . ≤ λ n , f 0 = 1 , ..., f n 5,
Motivation Definitions Results Open problems Acknowledgements Laplacian eigenmaps R p which we wish to map into Given data points x 1 , x 2 , . . . , x n ∈ I R k , k ≪ p , I 1 Construct a graph (association matrix) A out of the data points. 2 Compute L = D − A where D ii = � j A ij 3 Compute eigenvalues and eigenvectors of L 0 = λ 0 ≤ λ 1 ≤ λ 2 ≤ . . . ≤ λ n , f 0 = 1 , ..., f n R k by the map 4 Map the data points into I x i �→ ( f 1 ( x i ) , f 2 ( i ) , . . . , f k ( x i )) 6,
Motivation Definitions Results Open problems Acknowledgements Laplacian eigenmaps example 7,
Motivation Definitions Results Open problems Acknowledgements Laplacian eigenmaps example 8,
Motivation Definitions Results Open problems Acknowledgements Near 0 homology Near-Connected Components? 9,
Motivation Definitions Results Open problems Acknowledgements Clustering Clusters 10,
Motivation Definitions Results Open problems Acknowledgements Near 0-homology for graphs Cut S S For a graph with vertex set V , the Cheeger number is defined to be | δ S | h = min � . � min | S | , | S | ∅ � S � V 11,
Motivation Definitions Results Open problems Acknowledgements Near 0-homology for graphs Cut S S For a graph with vertex set V , the Cheeger number is defined to be | δ S | h = min � . � min | S | , | S | ∅ � S � V h = 0 ⇔ graph is disconnected. 12,
Motivation Definitions Results Open problems Acknowledgements Near 0-homology for graphs ⇔ For a graph with vertex set V , the Fiedler number is defined to be u ∼ v ( f ( u ) − f ( v )) 2 � λ = min . � u ( f ( u )) 2 f : V → I R f ⊥ 1 13,
Motivation Definitions Results Open problems Acknowledgements Near 0-homology for graphs ⇔ For a graph with vertex set V , the Fiedler number is defined to be u ∼ v ( f ( u ) − f ( v )) 2 � λ = min . � u ( f ( u )) 2 f : V → I R f ⊥ 1 λ = 0 ⇔ graph is disconnected. 14,
Motivation Definitions Results Open problems Acknowledgements Cheeger inequality for graphs Theorem (Alon, Milman, Lawler & Sokal, Frieze & Kannan & Polson,...) For any graph with Cheeger number h and Fiedler number λ 2 h ≥ λ 1 > h 2 2 M , M = max u u d , maximum vertex degree. 15,
Motivation Definitions Results Open problems Acknowledgements Edge expansion Expander graphs are families of graphs that are sparse and strongly connected. 16,
Motivation Definitions Results Open problems Acknowledgements Edge expansion Expander graphs are families of graphs that are sparse and strongly connected. A family of expander graphs G has the property h ( G ) > ǫ > 0 for all G ∈ G . 17,
Motivation Definitions Results Open problems Acknowledgements Edge expansion Expander graphs are families of graphs that are sparse and strongly connected. A family of expander graphs G has the property h ( G ) > ǫ > 0 for all G ∈ G . Cheeger inequality lets us use λ as criteria λ ( G ) > 0 for all G ∈ G . 18,
Motivation Definitions Results Open problems Acknowledgements Higher-dimensional notions 19,
Motivation Definitions Results Open problems Acknowledgements Homology 0-Homology 1-Homology 2-Homology Hole Void Connected Components β 0 = 2 , β 1 = 0 , β 2 = 0 β 0 = 1 , β 1 = 1 , β 2 = 0 β 0 = 1 , β 1 = 0 , β 2 = 1 20,
Motivation Definitions Results Open problems Acknowledgements Near one homology 21,
Motivation Definitions Results Open problems Acknowledgements Near one homology 22,
Motivation Definitions Results Open problems Acknowledgements Simplicial complexes A k -simplex is the convex hull of k + 1 affinely independent points, σ = { u 0 , ..., u k } . 23,
Motivation Definitions Results Open problems Acknowledgements Simplicial complexes A k -simplex is the convex hull of k + 1 affinely independent points, σ = { u 0 , ..., u k } . A face τ of σ is the convex hull of a non-empty subset of the u i , τ ≤ σ . 24,
Motivation Definitions Results Open problems Acknowledgements Simplicial complexes A k -simplex is the convex hull of k + 1 affinely independent points, σ = { u 0 , ..., u k } . A face τ of σ is the convex hull of a non-empty subset of the u i , τ ≤ σ . A simplicial complex is a finite collection of simplices K such that σ ∈ K and τ ≤ σ implies τ ∈ K , and σ, σ 0 ∈ K implies σ ∩ σ 0 is either empty or a face of both. 25,
Motivation Definitions Results Open problems Acknowledgements Chains and cochains X = simplicial complex of dimension m , ( X ) = m . C k ( I F ) = { I F -linear combinations of oriented k -simplices } C k ( I F ) = { I F -valued functions on oriented k -simplices } 26,
Motivation Definitions Results Open problems Acknowledgements Chains and cochains X = simplicial complex of dimension m , ( X ) = m . C k ( I F ) = { I F -linear combinations of oriented k -simplices } C k ( I F ) = { I F -valued functions on oriented k -simplices } Chain Complex: ∂ 1 ( I F ) ∂ 2 ( I F ) ∂ m ( I F ) 0 ← − C 0 ← − C 1 ← − · · · ← − C m ← − 0 Cochain Complex: → C 0 δ 0 ( I → C 1 δ 1 ( I δ m − 1 ( I F ) F ) F ) C m − 0 − − − → · · · − → → 0 I F = I R , Z 2 27,
Motivation Definitions Results Open problems Acknowledgements Chains and cochains Given a simplicial complex σ = { v 0 , ..., v k } an orientation of [ v 0 , ..., v k ] is the equivalence class of even permutations of ordering. 28,
Motivation Definitions Results Open problems Acknowledgements Chains and cochains Given a simplicial complex σ = { v 0 , ..., v k } an orientation of [ v 0 , ..., v k ] is the equivalence class of even permutations of ordering. Boundary map ∂ k ( I F ) : C k ( I F ) → C k − 1 ( I F ) k � ( − 1) i [ v 0 , ..., v i − 1 , v i +1 , ..., v k ] . ∂ k [ v 0 , ..., v k ] = i =1 29,
Motivation Definitions Results Open problems Acknowledgements Chains and cochains Given a simplicial complex σ = { v 0 , ..., v k } an orientation of [ v 0 , ..., v k ] is the equivalence class of even permutations of ordering. Boundary map ∂ k ( I F ) : C k ( I F ) → C k − 1 ( I F ) k � ( − 1) i [ v 0 , ..., v i − 1 , v i +1 , ..., v k ] . ∂ k [ v 0 , ..., v k ] = i =1 Coboundary map δ k − 1 ( I F ) : C k − 1 ( I F ) → C k ( I F ) is the transpose of the boundary map. 30,
Motivation Definitions Results Open problems Acknowledgements ∂ ( I R ) v 2 v 2 1 2 ∂ 2 ( R ) v 4 v 4 v 1 v 1 2 3 1 1 2 v 3 v 3 ∂ 2 ( R ) [ v 1 , v 3 , v 2 ] + 2[ v 2 , v 4 , v 3 ] [ v 1 , v 3 ] + [ v 2 , v 1 ] + 3[ v 3 , v 2 ] +2[ v 2 , v 4 ] + 2[ v 4 , v 3 ] 31,
Motivation Definitions Results Open problems Acknowledgements ∂ ( Z 2 ) v 2 v 2 1 1 ∂ 2 ( Z 2 ) v 1 v 4 v 1 v 4 0 1 1 1 1 v 3 v 3 ∂ 2 ( Z 2 ) [ v 1 , v 2 , v 3 ] + [ v 2 , v 3 , v 4 ] [ v 1 , v 2 ] + [ v 1 , v 3 ] +[ v 2 , v 4 ] + [ v 3 , v 4 ] 32,
Motivation Definitions Results Open problems Acknowledgements δ ( Z 2 ) v 2 v 2 1 0 δ 1 ( Z 2 ) v 4 v 4 v 1 v 1 1 1 1 1 0 v 3 v 3 δ 1 ( Z 2 ) [ v 1 , v 2 ] + [ v 1 , v 3 ] + [ v 2 , v 3 ] [ v 1 , v 2 , v 3 ] + [ v 2 , v 3 , v 4 ] 33,
Motivation Definitions Results Open problems Acknowledgements δ ( I R ) v 2 v 2 3 0 δ 1 ( R ) v 4 v 4 v 1 v 1 2 2 1 4 0 v 3 v 3 δ 1 ( R ) 3[ v 2 , v 1 ] + 4[ v 3 , v 1 ] + 2[ v 3 , v 2 ] [ v 1 , v 3 , v 2 ] + 2[ v 2 , v 4 , v 3 ] 34,
Motivation Definitions Results Open problems Acknowledgements Cheeger numbers Z 2 homology and cohomology H k ( Z 2 ) = ker δ k H k ( Z 2 ) = ker ∂ k , im δ k +1 . im ∂ k +1 35,
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