Graph Partitioning Algorithms and Laplacian Eigenvalues

Size: px
Start display at page:

Download "Graph Partitioning Algorithms and Laplacian Eigenvalues"

Transcription

1 Graph Partitioning Algorithms and Laplacian Eigenvalues Luca Trevisan Stanford Based on work with Tsz Chiu Kwok, Lap Chi Lau, James Lee, Yin Tat Lee, and Shayan Oveis Gharan

2 spectral graph theory Use linear algebra to understand/characterize graph properties design efficient graph algorithms

3 linear algebra and graphs Take an undirected graph G = (V, E), consider matrix A[i, j] := weight of edge (i, j)

4 linear algebra and graphs Take an undirected graph G = (V, E), consider matrix A[i, j] := weight of edge (i, j)

5 linear algebra and graphs Take an undirected graph G = (V, E), consider matrix A[i, j] := weight of edge (i, j) eigenvalues combinatorial properties eigenvenctors algorithms for combinatorial problems

6 fundamental thm of spectral graph theory G = (V, E) undirected, A adjacency matrix, d v degree of v, D diagonal matrix of degrees L := I D 1 2 AD 1 2

7 fundamental thm of spectral graph theory G = (V, E) undirected, A adjacency matrix, d v degree of v, D diagonal matrix of degrees L := I D 1 2 AD 1 2 L is symmetric, all its eigenvalues are real and

8 fundamental thm of spectral graph theory G = (V, E) undirected, A adjacency matrix, d v degree of v, D diagonal matrix of degrees L := I D 1 2 AD 1 2 L is symmetric, all its eigenvalues are real and 0 = λ 1 λ 2 λ n 2

9 fundamental thm of spectral graph theory G = (V, E) undirected, A adjacency matrix, d v degree of v, D diagonal matrix of degrees L := I D 1 2 AD 1 2 L is symmetric, all its eigenvalues are real and 0 = λ 1 λ 2 λ n 2 λ k = 0 iff G has k connected components λ n = 2 iff G has a bipartite connected component

10 0, 2 3, 2 3, 2 3, 2 3, 2 3, 5 3, 5 3, 5 3, 5 3

11 0, 2 3, 2 3, 2 3, 2 3, 2 3, 5 3, 5 3, 5 3, 5 3

12 0, 0,.69,.69, 1.5, 1.5, 1.8, 1.8

13 0, 0,.69,.69, 1.5, 1.5, 1.8, 1.8

14 ,,,,,,,

15 ,,,,,,,

16 eigenvalues as solutions to optimization problems If M R n n is symmetric, and λ 1 λ 2 λ n are eigenvalues counted with multiplicities, then λ k = min A k dim subspace of R V max x A x T Mx x T x and eigenvectors of λ 1,..., λ k are basis of opt A

17 why eigenvalues relate to connected components If G = (V, E) is undirected and L is Laplacian, then x T Lx x T x = (u,v) E x u x v 2 v d v x 2 v

18 why eigenvalues relate to connected components If G = (V, E) is undirected and L is Laplacian, then x T Lx (u,v) E x T x = x u x v 2 v d v xv 2 If λ 1 λ 2 λ n are eigenvalues of L with multiplicities, λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2

19 why eigenvalues relate to connected components If G = (V, E) is undirected and L is Laplacian, then x T Lx (u,v) E x T x = x u x v 2 v d v xv 2 If λ 1 λ 2 λ n are eigenvalues of L with multiplicities, λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2 Note: (u,v) E x u x v 2 = 0 iff x constant on connected components

20 if G has k connected components λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2

21 if G has k connected components λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2 Consider the space of all vectors that are constant in each connected component; it has dimension at least k and so it witnesses λ k = 0.

22 if λ k = 0 λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2

23 if λ k = 0 λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2 The eigenvectors x (1),..., x (k) of λ 1,..., λ k are all constant on connected components.

24 if λ k = 0 λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2 The eigenvectors x (1),..., x (k) of λ 1,..., λ k are all constant on connected components. The mapping F (v) := (x v (1), x v (2),..., x v (k) ) is constant on connected components

25 if λ k = 0 λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2 The eigenvectors x (1),..., x (k) of λ 1,..., λ k are all constant on connected components. The mapping F (v) := (x v (1), x v (2),..., x v (k) ) is constant on connected components It maps to at least k distinct points

26 if λ k = 0 λ k = min A k dim subspace of R V max x A (u,v) E x u x v 2 v d v xv 2 The eigenvectors x (1),..., x (k) of λ 1,..., λ k are all constant on connected components. The mapping F (v) := (x v (1), x v (2),..., x v (k) ) is constant on connected components It maps to at least k distinct points G has at least k connected components

27 conductance vol(s) := v S d v φ(s) := E(S, S) vol(s) φ(g) := min S V :0<vol(S) 1 2 vol(v ) φ(s) In regular graphs, same as edge expansion and, up to factor of 2 non-uniform sparsest cut

28 conductance 0, 0,.69,.69, 1.5, 1.5, 1.8, 1.8 φ(g) = φ({0, 1, 2}) = 0 6 = 0

29 conductance 0,.055,.055,.211,... φ(g) = 2 18

30 conductance 0, 2 3, 2 3, 2 3, 2 3, 2 3, 5 3, 5 3, 5 3, 5 3 φ(g) = 5 15

31 G is a social network finding S of small conductance

32 finding S of small conductance G is a social network S is a set of users more likely to be friends with each other than anyone else

33 finding S of small conductance G is a social network S is a set of users more likely to be friends with each other than anyone else G is a similarity graph on items of a data sets

34 finding S of small conductance G is a social network S is a set of users more likely to be friends with each other than anyone else G is a similarity graph on items of a data sets S are items more similar to each other than to other items [Shi, Malik]: image segmentation

35 finding S of small conductance G is a social network S is a set of users more likely to be friends with each other than anyone else G is a similarity graph on items of a data sets S are items more similar to each other than to other items [Shi, Malik]: image segmentation G is an input of a NP-hard optimization problem Recurse on S, V S, use dynamic programming to combine in time 2 O( E(S, S) )

36 conductance versus λ 2 Recall: λ 2 = 0 G disconnected

37 conductance versus λ 2 Recall: λ 2 = 0 G disconnected φ(g) = 0

38 conductance versus λ 2 Recall: λ 2 = 0 G disconnected φ(g) = 0 Cheeger inequality (Alon, Milman, Dodzuik, mid-1980s): λ 2 2 φ(g) 2λ 2

39 λ 2 2 φ(g) 2λ 2 Cheeger inequality

40 Cheeger inequality λ 2 2 φ(g) 2λ 2 Constructive proof of φ(g) 2λ 2 : given eigenvector x of λ 2. sort vertices v according to x v a set of vertices S occurring as initial segment or final segment of sorted order has φ(s) 2λ 2

41 Cheeger inequality λ 2 2 φ(g) 2λ 2 Constructive proof of φ(g) 2λ 2 : given eigenvector x of λ 2. sort vertices v according to x v a set of vertices S occurring as initial segment or final segment of sorted order has φ(s) 2λ 2 2 φ(g) sweep algorithm can be implemented in Õ( V + E ) time

42 λ 2 2 φ(g) 2λ 2 applications

43 applications λ 2 2 φ(g) 2λ 2 rigorous analysis of sweep algorithm (practical performance usually better than worst-case analysis)

44 applications λ 2 2 φ(g) 2λ 2 rigorous analysis of sweep algorithm (practical performance usually better than worst-case analysis) to certify φ Ω(1), enough to certify λ 2 Ω(1)

45 applications λ 2 2 φ(g) 2λ 2 rigorous analysis of sweep algorithm (practical performance usually better than worst-case analysis) to certify φ Ω(1), enough to certify λ 2 Ω(1) ( ) mixing time of random walk is O 1 log V and so also λ2 O ( ) 1 φ 2 log V (G)

46 Lee, Oveis-Gharan, T, 2012 From fundamental theorem of spectral graph theory: λ k = 0 G has k connected components

47 Lee, Oveis-Gharan, T, 2012 From fundamental theorem of spectral graph theory: λ k = 0 G has k connected components We prove: λ k small G has k disjoint sets of small conductance

48 order-k conductance Define order-k conductance φ k (G) := min S 1,...,S k disjoint max φ(s i) i=1,...,k Note: φ(g) = φ 2 (G) φ k (G) = 0 G has k connected components

49 0, 0,.69,.69, 1.5, 1.5, 1.8, 1.8

50 0, 0,.69,.69, 1.5, 1.5, 1.8, 1.8 { φ 3 (G) = max 0, 2 6, 2 } = 1 4 2

51 0,.055,.055,.211,... { } 2 φ 3 (G) = max 12, 2 12, 2 =

52 λ k = 0 φ k = 0 λ k

53 λ k λ k = 0 φ k = 0 (Lee,Oveis Gharan, T, 2012) λ k 2 φ k(g) O(k 2 ) λ k Upper bound is constructive: in nearly-linear time we can find k disjoint sets each of expansion at most O(k 2 ) λ k.

54 λ k 2 φ k(g) O(k 2 ) λ k λ k

55 λ k λ k 2 φ k(g) O(k 2 ) λ k φ k (G) O( (log k) λ 1.1k ) (Lee,Oveis Gharan, T, 2012) φ k (G) O( (log k) λ O(k) ) (Louis, Raghavendra, Tetali, Vempala, 2012) Factor O( log k) is tight

56 spectral clustering The following algorithm works well in practice. (But to solve what problem?) Compute k eigenvectors x (1),..., x (k) of k smallest eigenvalues of L Define mapping F : V R k F (v) := (x v (1), x v (2),..., x v (k) ) Apply k-means to the points that vertices are mapped to

57 k = 3 spectral embedding of a random graph

58 spectral clustering LOT algorithms and LRTV algorithm find k low-conductance sets if they exist First step is spectral embedding Then geometric partitioning but not k-means

59 λ 2 2 φ(g) 2λ 2 Cheeger inequality

60 Cheeger inequality λ 2 2 φ(g) 2λ 2 φ(g) O(k) λ 2 λk (Kwok, Lau, Lee, Oveis Gharan, T, 2013) and the upper bound holds for the sweep algorithm

61 Cheeger inequality λ 2 2 φ(g) 2λ 2 φ(g) O(k) λ 2 λk (Kwok, Lau, Lee, Oveis Gharan, T, 2013) and the upper bound holds for the sweep algorithm Nearly-linear time sweep algorithm finds S such that, for every k, ( ) k φ(s) φ(g) O λk

62 1 We find a 2k-valued vector y such that ( ) x y 2 λ2 O where x is eigenvector of λ 2 proof structure λ k

63 1 We find a 2k-valued vector y such that ( ) x y 2 λ2 O where x is eigenvector of λ 2 proof structure 2 For every k-valued vector y, applying the sweep algorithm to x finds a set S such that ( φ(s) O(k) λ 2 + ) λ 2 x y λ k

64 1 We find a 2k-valued vector y such that ( ) x y 2 λ2 O where x is eigenvector of λ 2 proof structure 2 For every k-valued vector y, applying the sweep algorithm to x finds a set S such that ( φ(s) O(k) λ 2 + ) λ 2 x y λ k So the sweep algorithm finds S φ(s) O(k) λ 2 1 λk

65 intuition for first result Given x, we find a 2k-valued vector y such that ( ) R(x) x y 2 O λ k

66 intuition for first result Given x, we find a 2k-valued vector y such that ( ) R(x) x y 2 O λ k Suppose R(x) = 0 and λ k > 0.

67 intuition for first result Given x, we find a 2k-valued vector y such that ( ) R(x) x y 2 O λ k Suppose R(x) = 0 and λ k > 0. Then x is constant on connected components (because R(x) = 0) G has < k connected components (because λ k > 0) So x is at distance zero from a vector (itself) that is (k 1)-valued

68 idea of proof of first result Consider x as an embedding of vertices to the line

69 idea of proof of first result Consider x as an embedding of vertices to the line Partition the points vertices are mapped to into 2k intervals, minimizing 2k-means with squared-distance

70 idea of proof of first result Consider x as an embedding of vertices to the line Partition the points vertices are mapped to into 2k intervals, minimizing 2k-means with squared-distance Round to closest interval boundary

71 idea of proof of first result Consider x as an embedding of vertices to the line Partition the points vertices are mapped to into 2k intervals, minimizing 2k-means with squared-distance Round to closest interval boundary If rounded vector is far from original vector, a lot of points are far from boundary; then we construct witness that λ k is small by finding k disjointly supported vectors of small Rayleigh quotient

72 intuition for second result For every k-valued vector y, applying the sweep algorithm to x finds a set S such that ( φ(s) O(k) R(x) + ) R(x) x y It was known: φ(s) 2R(x) x y = 0 φ(s) kr(x) Our proof blends the two arguments

73 eigenvalue gap if λ k = 0 and λ k+1 > 0 then G has exactly k connected components;

74 eigenvalue gap if λ k = 0 and λ k+1 > 0 then G has exactly k connected components; [Tanaka 2012] If λ k+1 > 5 k λ k, then vertices of G can be partitioned into k sets of small conductance, each inducing a subgraph of large conductance. [Non-algorithmic proof]

75 eigenvalue gap if λ k = 0 and λ k+1 > 0 then G has exactly k connected components; [Tanaka 2012] If λ k+1 > 5 k λ k, then vertices of G can be partitioned into k sets of small conductance, each inducing a subgraph of large conductance. [Non-algorithmic proof] [Oveis-Gharan, T 2013] Sufficient φ k+1 (G) > (1 + ɛ)φ k (G); if λ k+1 > O(k 2 ) λ k, then partition can be found algorithmically

76 bounded threshold rank [Arora, Barak, Steurer], [Barak, Raghavendra, Steurer], [Guruswami, Sinop],... If λ k is large, then a cut of approximately minimal conductance can be found in time exponential in k using spectral methods [ABS] or convex relaxations [BRS, GS]

77 bounded threshold rank [Arora, Barak, Steurer], [Barak, Raghavendra, Steurer], [Guruswami, Sinop],... If λ k is large, then a cut of approximately minimal conductance can be found in time exponential in k using spectral methods [ABS] or convex relaxations [BRS, GS] [Kwok, Lau, Lee, Oveis-Gharan, T]: the nearly-linear time sweep algorithm finds good approximation if λ k is large [Oveis-Gharan, T]: the Linial-Goemans relaxation (solvable in polynomial time independent of k) can be rounded better than in the Arora-Rao-Vazirani analysis if λ k is large

78 challenge: explain practical performance of sweep algorithm λ 2 2 φ(g) φ( output of algorithm ) 2λ 2

79 challenge: explain practical performance of sweep algorithm λ 2 2 φ(g) φ( output of algorithm ) 2λ 2 (Kwok, Lau, Lee, Oveis Gharan, T): φ( output of algorithm ) O(k) λ 2 λk

80 challenge: explain practical performance of sweep algorithm λ 2 2 φ(g) φ( output of algorithm ) 2λ 2

81 challenge: explain practical performance of sweep algorithm λ 2 2 φ(g) φ( output of algorithm ) 2λ 2 (Spielman, Teng) in bounded-degree planar graphs: ( ) 1 λ 2 O n so φ( output of algorithm ) O matching the planar separator theorem. ( 1 n )

82 explain practical performance of sweep algorithm λ 2 2 φ(g) φ( output of algorithm ) 2λ 2

83 explain practical performance of sweep algorithm λ 2 2 φ(g) φ( output of algorithm ) 2λ 2 Can we find interesting classes of graphs for which λ 2 << φ(g)?

84 lucatrevisan.wordpress.com/tag/expanders/

Lecture 17: Cheeger s Inequality and the Sparsest Cut Problem

Lecture 17: Cheeger s Inequality and the Sparsest Cut Problem Recent Advances in Approximation Algorithms Spring 2015 Lecture 17: Cheeger s Inequality and the Sparsest Cut Problem Lecturer: Shayan Oveis Gharan June 1st Disclaimer: These notes have not been subjected

More information

Lecture 12: Introduction to Spectral Graph Theory, Cheeger s inequality

Lecture 12: Introduction to Spectral Graph Theory, Cheeger s inequality CSE 521: Design and Analysis of Algorithms I Spring 2016 Lecture 12: Introduction to Spectral Graph Theory, Cheeger s inequality Lecturer: Shayan Oveis Gharan May 4th Scribe: Gabriel Cadamuro Disclaimer:

More information

Lecture 13: Spectral Graph Theory

Lecture 13: Spectral Graph Theory CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 13: Spectral Graph Theory Lecturer: Shayan Oveis Gharan 11/14/18 Disclaimer: These notes have not been subjected to the usual scrutiny reserved

More information

EIGENVECTOR NORMS MATTER IN SPECTRAL GRAPH THEORY

EIGENVECTOR NORMS MATTER IN SPECTRAL GRAPH THEORY EIGENVECTOR NORMS MATTER IN SPECTRAL GRAPH THEORY Franklin H. J. Kenter Rice University ( United States Naval Academy 206) orange@rice.edu Connections in Discrete Mathematics, A Celebration of Ron Graham

More information

Random Walks and Evolving Sets: Faster Convergences and Limitations

Random Walks and Evolving Sets: Faster Convergences and Limitations Random Walks and Evolving Sets: Faster Convergences and Limitations Siu On Chan 1 Tsz Chiu Kwok 2 Lap Chi Lau 2 1 The Chinese University of Hong Kong 2 University of Waterloo Jan 18, 2017 1/23 Finding

More information

Graph Clustering Algorithms

Graph Clustering Algorithms PhD Course on Graph Mining Algorithms, Università di Pisa February, 2018 Clustering: Intuition to Formalization Task Partition a graph into natural groups so that the nodes in the same cluster are more

More information

Markov chain methods for small-set expansion

Markov chain methods for small-set expansion Markov chain methods for small-set expansion Ryan O Donnell David Witmer November 4, 03 Abstract Consider a finite irreducible Markov chain with invariant distribution π. We use the inner product induced

More information

Lecture 14: Random Walks, Local Graph Clustering, Linear Programming

Lecture 14: Random Walks, Local Graph Clustering, Linear Programming CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 14: Random Walks, Local Graph Clustering, Linear Programming Lecturer: Shayan Oveis Gharan 3/01/17 Scribe: Laura Vonessen Disclaimer: These

More information

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 11 Luca Trevisan February 29, 2016

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 11 Luca Trevisan February 29, 2016 U.C. Berkeley CS294: Spectral Methods and Expanders Handout Luca Trevisan February 29, 206 Lecture : ARV In which we introduce semi-definite programming and a semi-definite programming relaxation of sparsest

More information

1 Matrix notation and preliminaries from spectral graph theory

1 Matrix notation and preliminaries from spectral graph theory Graph clustering (or community detection or graph partitioning) is one of the most studied problems in network analysis. One reason for this is that there are a variety of ways to define a cluster or community.

More information

Spectral Algorithms for Graph Mining and Analysis. Yiannis Kou:s. University of Puerto Rico - Rio Piedras NSF CAREER CCF

Spectral Algorithms for Graph Mining and Analysis. Yiannis Kou:s. University of Puerto Rico - Rio Piedras NSF CAREER CCF Spectral Algorithms for Graph Mining and Analysis Yiannis Kou:s University of Puerto Rico - Rio Piedras NSF CAREER CCF-1149048 The Spectral Lens A graph drawing using Laplacian eigenvectors Outline Basic

More information

Graph Partitioning Using Random Walks

Graph Partitioning Using Random Walks Graph Partitioning Using Random Walks A Convex Optimization Perspective Lorenzo Orecchia Computer Science Why Spectral Algorithms for Graph Problems in practice? Simple to implement Can exploit very efficient

More information

Spectral Graph Theory and its Applications. Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale Unviersity

Spectral Graph Theory and its Applications. Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale Unviersity Spectral Graph Theory and its Applications Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale Unviersity Outline Adjacency matrix and Laplacian Intuition, spectral graph drawing

More information

18.S096: Spectral Clustering and Cheeger s Inequality

18.S096: Spectral Clustering and Cheeger s Inequality 18.S096: Spectral Clustering and Cheeger s Inequality Topics in Mathematics of Data Science (Fall 015) Afonso S. Bandeira bandeira@mit.edu http://math.mit.edu/~bandeira October 6, 015 These are lecture

More information

ORIE 6334 Spectral Graph Theory November 22, Lecture 25

ORIE 6334 Spectral Graph Theory November 22, Lecture 25 ORIE 64 Spectral Graph Theory November 22, 206 Lecture 25 Lecturer: David P. Williamson Scribe: Pu Yang In the remaining three lectures, we will cover a prominent result by Arora, Rao, and Vazirani for

More information

Lecture 7. 1 Normalized Adjacency and Laplacian Matrices

Lecture 7. 1 Normalized Adjacency and Laplacian Matrices ORIE 6334 Spectral Graph Theory September 3, 206 Lecturer: David P. Williamson Lecture 7 Scribe: Sam Gutekunst In this lecture, we introduce normalized adjacency and Laplacian matrices. We state and begin

More information

Lecture 12 : Graph Laplacians and Cheeger s Inequality

Lecture 12 : Graph Laplacians and Cheeger s Inequality CPS290: Algorithmic Foundations of Data Science March 7, 2017 Lecture 12 : Graph Laplacians and Cheeger s Inequality Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Graph Laplacian Maybe the most beautiful

More information

Unique Games and Small Set Expansion

Unique Games and Small Set Expansion Proof, beliefs, and algorithms through the lens of sum-of-squares 1 Unique Games and Small Set Expansion The Unique Games Conjecture (UGC) (Khot [2002]) states that for every ɛ > 0 there is some finite

More information

Stanford University CS366: Graph Partitioning and Expanders Handout 13 Luca Trevisan March 4, 2013

Stanford University CS366: Graph Partitioning and Expanders Handout 13 Luca Trevisan March 4, 2013 Stanford University CS366: Graph Partitioning and Expanders Handout 13 Luca Trevisan March 4, 2013 Lecture 13 In which we construct a family of expander graphs. The problem of constructing expander graphs

More information

Unique Games Conjecture & Polynomial Optimization. David Steurer

Unique Games Conjecture & Polynomial Optimization. David Steurer Unique Games Conjecture & Polynomial Optimization David Steurer Newton Institute, Cambridge, July 2013 overview Proof Complexity Approximability Polynomial Optimization Quantum Information computational

More information

Testing Cluster Structure of Graphs. Artur Czumaj

Testing Cluster Structure of Graphs. Artur Czumaj Testing Cluster Structure of Graphs Artur Czumaj DIMAP and Department of Computer Science University of Warwick Joint work with Pan Peng and Christian Sohler (TU Dortmund) Dealing with BigData in Graphs

More information

Approximation & Complexity

Approximation & Complexity Summer school on semidefinite optimization Approximation & Complexity David Steurer Cornell University Part 1 September 6, 2012 Overview Part 1 Unique Games Conjecture & Basic SDP Part 2 SDP Hierarchies:

More information

Improved Cheeger s Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

Improved Cheeger s Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap Improved Cheeger s Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap Tsz Chiu Kwok CUHK tckwok@csecuhkeduhk Shayan Oveis Gharan Stanford University shayan@stanfordedu

More information

SDP Relaxations for MAXCUT

SDP Relaxations for MAXCUT SDP Relaxations for MAXCUT from Random Hyperplanes to Sum-of-Squares Certificates CATS @ UMD March 3, 2017 Ahmed Abdelkader MAXCUT SDP SOS March 3, 2017 1 / 27 Overview 1 MAXCUT, Hardness and UGC 2 LP

More information

CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University

CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University CS224W: Social and Information Network Analysis Jure Leskovec Stanford University Jure Leskovec, Stanford University http://cs224w.stanford.edu Task: Find coalitions in signed networks Incentives: European

More information

Mining of Massive Datasets Jure Leskovec, AnandRajaraman, Jeff Ullman Stanford University

Mining of Massive Datasets Jure Leskovec, AnandRajaraman, Jeff Ullman Stanford University Note to other teachers and users of these slides: We would be delighted if you found this our material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit

More information

1 Matrix notation and preliminaries from spectral graph theory

1 Matrix notation and preliminaries from spectral graph theory Graph clustering (or community detection or graph partitioning) is one of the most studied problems in network analysis. One reason for this is that there are a variety of ways to define a cluster or community.

More information

ORIE 6334 Spectral Graph Theory September 22, Lecture 11

ORIE 6334 Spectral Graph Theory September 22, Lecture 11 ORIE 6334 Spectral Graph Theory September, 06 Lecturer: David P. Williamson Lecture Scribe: Pu Yang In today s lecture we will focus on discrete time random walks on undirected graphs. Specifically, we

More information

Spectral and Electrical Graph Theory. Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale Unviersity

Spectral and Electrical Graph Theory. Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale Unviersity Spectral and Electrical Graph Theory Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale Unviersity Outline Spectral Graph Theory: Understand graphs through eigenvectors and

More information

A Schur Complement Cheeger Inequality

A Schur Complement Cheeger Inequality A Schur Complement Cheeger Inequality arxiv:8.834v [cs.dm] 27 Nov 28 Aaron Schild University of California, Berkeley EECS aschild@berkeley.edu November 28, 28 Abstract Cheeger s inequality shows that any

More information

arxiv: v4 [cs.ds] 23 Jun 2015

arxiv: v4 [cs.ds] 23 Jun 2015 Spectral Concentration and Greedy k-clustering Tamal K. Dey Pan Peng Alfred Rossi Anastasios Sidiropoulos June 25, 2015 arxiv:1404.1008v4 [cs.ds] 23 Jun 2015 Abstract A popular graph clustering method

More information

LOCAL CHEEGER INEQUALITIES AND SPARSE CUTS

LOCAL CHEEGER INEQUALITIES AND SPARSE CUTS LOCAL CHEEGER INEQUALITIES AND SPARSE CUTS CAELAN GARRETT 18.434 FINAL PROJECT CAELAN@MIT.EDU Abstract. When computing on massive graphs, it is not tractable to iterate over the full graph. Local graph

More information

Lecture 9: Low Rank Approximation

Lecture 9: Low Rank Approximation CSE 521: Design and Analysis of Algorithms I Fall 2018 Lecture 9: Low Rank Approximation Lecturer: Shayan Oveis Gharan February 8th Scribe: Jun Qi Disclaimer: These notes have not been subjected to the

More information

STA141C: Big Data & High Performance Statistical Computing

STA141C: Big Data & High Performance Statistical Computing STA141C: Big Data & High Performance Statistical Computing Lecture 12: Graph Clustering Cho-Jui Hsieh UC Davis May 29, 2018 Graph Clustering Given a graph G = (V, E, W ) V : nodes {v 1,, v n } E: edges

More information

Networks and Their Spectra

Networks and Their Spectra Networks and Their Spectra Victor Amelkin University of California, Santa Barbara Department of Computer Science victor@cs.ucsb.edu December 4, 2017 1 / 18 Introduction Networks (= graphs) are everywhere.

More information

Lower bounds on the size of semidefinite relaxations. David Steurer Cornell

Lower bounds on the size of semidefinite relaxations. David Steurer Cornell Lower bounds on the size of semidefinite relaxations David Steurer Cornell James R. Lee Washington Prasad Raghavendra Berkeley Institute for Advanced Study, November 2015 overview of results unconditional

More information

Data Analysis and Manifold Learning Lecture 7: Spectral Clustering

Data Analysis and Manifold Learning Lecture 7: Spectral Clustering Data Analysis and Manifold Learning Lecture 7: Spectral Clustering Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline of Lecture 7 What is spectral

More information

Ma/CS 6b Class 23: Eigenvalues in Regular Graphs

Ma/CS 6b Class 23: Eigenvalues in Regular Graphs Ma/CS 6b Class 3: Eigenvalues in Regular Graphs By Adam Sheffer Recall: The Spectrum of a Graph Consider a graph G = V, E and let A be the adjacency matrix of G. The eigenvalues of G are the eigenvalues

More information

ANNALES. SHAYAN OVEIS GHARAN Spectral Graph Theory via Higher Order Eigenvalues and Applications to the Analysis of Random Walks

ANNALES. SHAYAN OVEIS GHARAN Spectral Graph Theory via Higher Order Eigenvalues and Applications to the Analysis of Random Walks ANNALES DE LA FACULTÉ DES SCIENCES Mathématiques SHAYAN OVEIS GHARAN Spectral Graph Theory via Higher Order Eigenvalues and Applications to the Analysis of Random Walks Tome XXIV, n o 4 (2015), p. 801-815.

More information

University of Bristol - Explore Bristol Research. Publisher's PDF, also known as Version of record

University of Bristol - Explore Bristol Research. Publisher's PDF, also known as Version of record Zanetti, L., Sun, H., & Peng, R. 05. Partitioning Well-Clustered Graphs: Spectral Clustering Works!. Paper presented at Conference on Learning Theory, Paris, France. Publisher's PDF, also known as Version

More information

Spectral Clustering on Handwritten Digits Database

Spectral Clustering on Handwritten Digits Database University of Maryland-College Park Advance Scientific Computing I,II Spectral Clustering on Handwritten Digits Database Author: Danielle Middlebrooks Dmiddle1@math.umd.edu Second year AMSC Student Advisor:

More information

eigenvalue bounds and metric uniformization Punya Biswal & James R. Lee University of Washington Satish Rao U. C. Berkeley

eigenvalue bounds and metric uniformization Punya Biswal & James R. Lee University of Washington Satish Rao U. C. Berkeley eigenvalue bounds and metric uniformization Punya Biswal & James R. Lee University of Washington Satish Rao U. C. Berkeley separators in planar graphs Lipton and Tarjan (1980) showed that every n-vertex

More information

Spectra of Adjacency and Laplacian Matrices

Spectra of Adjacency and Laplacian Matrices Spectra of Adjacency and Laplacian Matrices Definition: University of Alicante (Spain) Matrix Computing (subject 3168 Degree in Maths) 30 hours (theory)) + 15 hours (practical assignment) Contents 1. Spectra

More information

Topic: Balanced Cut, Sparsest Cut, and Metric Embeddings Date: 3/21/2007

Topic: Balanced Cut, Sparsest Cut, and Metric Embeddings Date: 3/21/2007 CS880: Approximations Algorithms Scribe: Tom Watson Lecturer: Shuchi Chawla Topic: Balanced Cut, Sparsest Cut, and Metric Embeddings Date: 3/21/2007 In the last lecture, we described an O(log k log D)-approximation

More information

Laplacian Matrices of Graphs: Spectral and Electrical Theory

Laplacian Matrices of Graphs: Spectral and Electrical Theory Laplacian Matrices of Graphs: Spectral and Electrical Theory Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale University Toronto, Sep. 28, 2 Outline Introduction to graphs

More information

Maximum cut and related problems

Maximum cut and related problems Proof, beliefs, and algorithms through the lens of sum-of-squares 1 Maximum cut and related problems Figure 1: The graph of Renato Paes Leme s friends on the social network Orkut, the partition to top

More information

Linear algebra and applications to graphs Part 1

Linear algebra and applications to graphs Part 1 Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces

More information

Conductance, the Normalized Laplacian, and Cheeger s Inequality

Conductance, the Normalized Laplacian, and Cheeger s Inequality Spectral Graph Theory Lecture 6 Conductance, the Normalized Laplacian, and Cheeger s Inequality Daniel A. Spielman September 17, 2012 6.1 About these notes These notes are not necessarily an accurate representation

More information

Lecture Introduction. 2 Brief Recap of Lecture 10. CS-621 Theory Gems October 24, 2012

Lecture Introduction. 2 Brief Recap of Lecture 10. CS-621 Theory Gems October 24, 2012 CS-62 Theory Gems October 24, 202 Lecture Lecturer: Aleksander Mądry Scribes: Carsten Moldenhauer and Robin Scheibler Introduction In Lecture 0, we introduced a fundamental object of spectral graph theory:

More information

Lecture 8: Spectral Graph Theory III

Lecture 8: Spectral Graph Theory III A Theorist s Toolkit (CMU 18-859T, Fall 13) Lecture 8: Spectral Graph Theory III October, 13 Lecturer: Ryan O Donnell Scribe: Jeremy Karp 1 Recap Last time we showed that every function f : V R is uniquely

More information

Lecture 1. 1 if (i, j) E a i,j = 0 otherwise, l i,j = d i if i = j, and

Lecture 1. 1 if (i, j) E a i,j = 0 otherwise, l i,j = d i if i = j, and Specral Graph Theory and its Applications September 2, 24 Lecturer: Daniel A. Spielman Lecture. A quick introduction First of all, please call me Dan. If such informality makes you uncomfortable, you can

More information

Markov Chains and Spectral Clustering

Markov Chains and Spectral Clustering Markov Chains and Spectral Clustering Ning Liu 1,2 and William J. Stewart 1,3 1 Department of Computer Science North Carolina State University, Raleigh, NC 27695-8206, USA. 2 nliu@ncsu.edu, 3 billy@ncsu.edu

More information

CS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory

CS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory CS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory Tim Roughgarden & Gregory Valiant May 2, 2016 Spectral graph theory is the powerful and beautiful theory that arises from

More information

Lecture: Local Spectral Methods (3 of 4) 20 An optimization perspective on local spectral methods

Lecture: Local Spectral Methods (3 of 4) 20 An optimization perspective on local spectral methods Stat260/CS294: Spectral Graph Methods Lecture 20-04/07/205 Lecture: Local Spectral Methods (3 of 4) Lecturer: Michael Mahoney Scribe: Michael Mahoney Warning: these notes are still very rough. They provide

More information

Lecture 5. Max-cut, Expansion and Grothendieck s Inequality

Lecture 5. Max-cut, Expansion and Grothendieck s Inequality CS369H: Hierarchies of Integer Programming Relaxations Spring 2016-2017 Lecture 5. Max-cut, Expansion and Grothendieck s Inequality Professor Moses Charikar Scribes: Kiran Shiragur Overview Here we derive

More information

Geometry of Similarity Search

Geometry of Similarity Search Geometry of Similarity Search Alex Andoni (Columbia University) Find pairs of similar images how should we measure similarity? Naïvely: about n 2 comparisons Can we do better? 2 Measuring similarity 000000

More information

Lectures 15: Cayley Graphs of Abelian Groups

Lectures 15: Cayley Graphs of Abelian Groups U.C. Berkeley CS294: Spectral Methods and Expanders Handout 15 Luca Trevisan March 14, 2016 Lectures 15: Cayley Graphs of Abelian Groups In which we show how to find the eigenvalues and eigenvectors of

More information

Lecture 1 and 2: Random Spanning Trees

Lecture 1 and 2: Random Spanning Trees Recent Advances in Approximation Algorithms Spring 2015 Lecture 1 and 2: Random Spanning Trees Lecturer: Shayan Oveis Gharan March 31st Disclaimer: These notes have not been subjected to the usual scrutiny

More information

Partitioning Algorithms that Combine Spectral and Flow Methods

Partitioning Algorithms that Combine Spectral and Flow Methods CS369M: Algorithms for Modern Massive Data Set Analysis Lecture 15-11/11/2009 Partitioning Algorithms that Combine Spectral and Flow Methods Lecturer: Michael Mahoney Scribes: Kshipra Bhawalkar and Deyan

More information

Spectral Clustering. Guokun Lai 2016/10

Spectral Clustering. Guokun Lai 2016/10 Spectral Clustering Guokun Lai 2016/10 1 / 37 Organization Graph Cut Fundamental Limitations of Spectral Clustering Ng 2002 paper (if we have time) 2 / 37 Notation We define a undirected weighted graph

More information

Random Lifts of Graphs

Random Lifts of Graphs 27th Brazilian Math Colloquium, July 09 Plan of this talk A brief introduction to the probabilistic method. A quick review of expander graphs and their spectrum. Lifts, random lifts and their properties.

More information

MATH 567: Mathematical Techniques in Data Science Clustering II

MATH 567: Mathematical Techniques in Data Science Clustering II This lecture is based on U. von Luxburg, A Tutorial on Spectral Clustering, Statistics and Computing, 17 (4), 2007. MATH 567: Mathematical Techniques in Data Science Clustering II Dominique Guillot Departments

More information

Reductions Between Expansion Problems

Reductions Between Expansion Problems Reductions Between Expansion Problems Prasad Raghavendra David Steurer Madhur Tulsiani November 11, 2010 Abstract The Small-Set Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness

More information

Lecture 14: SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices) Lecturer: Sanjeev Arora

Lecture 14: SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices) Lecturer: Sanjeev Arora princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 14: SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices) Lecturer: Sanjeev Arora Scribe: Today we continue the

More information

Diffusion and random walks on graphs

Diffusion and random walks on graphs Diffusion and random walks on graphs Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Structural

More information

Nonlinear Eigenproblems in Data Analysis

Nonlinear Eigenproblems in Data Analysis Nonlinear Eigenproblems in Data Analysis Matthias Hein Joint work with: Thomas Bühler, Leonardo Jost, Shyam Rangapuram, Simon Setzer Department of Mathematics and Computer Science Saarland University,

More information

Sum-of-Squares Method, Tensor Decomposition, Dictionary Learning

Sum-of-Squares Method, Tensor Decomposition, Dictionary Learning Sum-of-Squares Method, Tensor Decomposition, Dictionary Learning David Steurer Cornell Approximation Algorithms and Hardness, Banff, August 2014 for many problems (e.g., all UG-hard ones): better guarantees

More information

ON THE QUALITY OF SPECTRAL SEPARATORS

ON THE QUALITY OF SPECTRAL SEPARATORS ON THE QUALITY OF SPECTRAL SEPARATORS STEPHEN GUATTERY AND GARY L. MILLER Abstract. Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators

More information

Lecture: Local Spectral Methods (1 of 4)

Lecture: Local Spectral Methods (1 of 4) Stat260/CS294: Spectral Graph Methods Lecture 18-03/31/2015 Lecture: Local Spectral Methods (1 of 4) Lecturer: Michael Mahoney Scribe: Michael Mahoney Warning: these notes are still very rough. They provide

More information

Spectral Clustering. Zitao Liu

Spectral Clustering. Zitao Liu Spectral Clustering Zitao Liu Agenda Brief Clustering Review Similarity Graph Graph Laplacian Spectral Clustering Algorithm Graph Cut Point of View Random Walk Point of View Perturbation Theory Point of

More information

Approximating the isoperimetric number of strongly regular graphs

Approximating the isoperimetric number of strongly regular graphs Electronic Journal of Linear Algebra Volume 13 Volume 13 (005) Article 7 005 Approximating the isoperimetric number of strongly regular graphs Sivaramakrishnan Sivasubramanian skrishna@tcs.tifr.res.in

More information

Convex and Semidefinite Programming for Approximation

Convex and Semidefinite Programming for Approximation Convex and Semidefinite Programming for Approximation We have seen linear programming based methods to solve NP-hard problems. One perspective on this is that linear programming is a meta-method since

More information

U.C. Berkeley Better-than-Worst-Case Analysis Handout 3 Luca Trevisan May 24, 2018

U.C. Berkeley Better-than-Worst-Case Analysis Handout 3 Luca Trevisan May 24, 2018 U.C. Berkeley Better-than-Worst-Case Analysis Handout 3 Luca Trevisan May 24, 2018 Lecture 3 In which we show how to find a planted clique in a random graph. 1 Finding a Planted Clique We will analyze

More information

Spectral Clustering. Spectral Clustering? Two Moons Data. Spectral Clustering Algorithm: Bipartioning. Spectral methods

Spectral Clustering. Spectral Clustering? Two Moons Data. Spectral Clustering Algorithm: Bipartioning. Spectral methods Spectral Clustering Seungjin Choi Department of Computer Science POSTECH, Korea seungjin@postech.ac.kr 1 Spectral methods Spectral Clustering? Methods using eigenvectors of some matrices Involve eigen-decomposition

More information

CSC2420 Spring 2017: Algorithm Design, Analysis and Theory Lecture 11

CSC2420 Spring 2017: Algorithm Design, Analysis and Theory Lecture 11 CSC2420 Spring 2017: Algorithm Design, Analysis and Theory Lecture 11 Allan Borodin March 27, 2017 1 / 1 Announcements and todays agenda Announcements 1 Assignment 2 was due today. I just saw some messages

More information

1 Review: symmetric matrices, their eigenvalues and eigenvectors

1 Review: symmetric matrices, their eigenvalues and eigenvectors Cornell University, Fall 2012 Lecture notes on spectral methods in algorithm design CS 6820: Algorithms Studying the eigenvalues and eigenvectors of matrices has powerful consequences for at least three

More information

Combinatorial Optimization

Combinatorial Optimization Combinatorial Optimization 2017-2018 1 Maximum matching on bipartite graphs Given a graph G = (V, E), find a maximum cardinal matching. 1.1 Direct algorithms Theorem 1.1 (Petersen, 1891) A matching M is

More information

1 T 1 = where 1 is the all-ones vector. For the upper bound, let v 1 be the eigenvector corresponding. u:(u,v) E v 1(u)

1 T 1 = where 1 is the all-ones vector. For the upper bound, let v 1 be the eigenvector corresponding. u:(u,v) E v 1(u) CME 305: Discrete Mathematics and Algorithms Instructor: Reza Zadeh (rezab@stanford.edu) Final Review Session 03/20/17 1. Let G = (V, E) be an unweighted, undirected graph. Let λ 1 be the maximum eigenvalue

More information

Inverse Power Method for Non-linear Eigenproblems

Inverse Power Method for Non-linear Eigenproblems Inverse Power Method for Non-linear Eigenproblems Matthias Hein and Thomas Bühler Anubhav Dwivedi Department of Aerospace Engineering & Mechanics 7th March, 2017 1 / 30 OUTLINE Motivation Non-Linear Eigenproblems

More information

THE HIDDEN CONVEXITY OF SPECTRAL CLUSTERING

THE HIDDEN CONVEXITY OF SPECTRAL CLUSTERING THE HIDDEN CONVEXITY OF SPECTRAL CLUSTERING Luis Rademacher, Ohio State University, Computer Science and Engineering. Joint work with Mikhail Belkin and James Voss This talk A new approach to multi-way

More information

1 Adjacency matrix and eigenvalues

1 Adjacency matrix and eigenvalues CSC 5170: Theory of Computational Complexity Lecture 7 The Chinese University of Hong Kong 1 March 2010 Our objective of study today is the random walk algorithm for deciding if two vertices in an undirected

More information

Dissertation Defense

Dissertation Defense Clustering Algorithms for Random and Pseudo-random Structures Dissertation Defense Pradipta Mitra 1 1 Department of Computer Science Yale University April 23, 2008 Mitra (Yale University) Dissertation

More information

Four graph partitioning algorithms. Fan Chung University of California, San Diego

Four graph partitioning algorithms. Fan Chung University of California, San Diego Four graph partitioning algorithms Fan Chung University of California, San Diego History of graph partitioning NP-hard approximation algorithms Spectral method, Fiedler 73, Folklore Multicommunity flow,

More information

Graph Partitioning by Spectral Rounding: Applications in Image Segmentation and Clustering

Graph Partitioning by Spectral Rounding: Applications in Image Segmentation and Clustering Graph Partitioning by Spectral Rounding: Applications in Image Segmentation and Clustering David A. Tolliver Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213. tolliver@ri.cmu.edu Gary

More information

Graph Sparsification I : Effective Resistance Sampling

Graph Sparsification I : Effective Resistance Sampling Graph Sparsification I : Effective Resistance Sampling Nikhil Srivastava Microsoft Research India Simons Institute, August 26 2014 Graphs G G=(V,E,w) undirected V = n w: E R + Sparsification Approximate

More information

On a Cut-Matching Game for the Sparsest Cut Problem

On a Cut-Matching Game for the Sparsest Cut Problem On a Cut-Matching Game for the Sparsest Cut Problem Rohit Khandekar Subhash A. Khot Lorenzo Orecchia Nisheeth K. Vishnoi Electrical Engineering and Computer Sciences University of California at Berkeley

More information

Graphs in Machine Learning

Graphs in Machine Learning Graphs in Machine Learning Michal Valko INRIA Lille - Nord Europe, France Partially based on material by: Ulrike von Luxburg, Gary Miller, Doyle & Schnell, Daniel Spielman January 27, 2015 MVA 2014/2015

More information

Tight Size-Degree Lower Bounds for Sums-of-Squares Proofs

Tight Size-Degree Lower Bounds for Sums-of-Squares Proofs Tight Size-Degree Lower Bounds for Sums-of-Squares Proofs Massimo Lauria KTH Royal Institute of Technology (Stockholm) 1 st Computational Complexity Conference, 015 Portland! Joint work with Jakob Nordström

More information

A local algorithm for finding dense bipartite-like subgraphs

A local algorithm for finding dense bipartite-like subgraphs A local algorithm for finding dense bipartite-like subgraphs Pan Peng 1,2 1 State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences 2 School of Information Science

More information

Space-Variant Computer Vision: A Graph Theoretic Approach

Space-Variant Computer Vision: A Graph Theoretic Approach p.1/65 Space-Variant Computer Vision: A Graph Theoretic Approach Leo Grady Cognitive and Neural Systems Boston University p.2/65 Outline of talk Space-variant vision - Why and how of graph theory Anisotropic

More information

Aditya Bhaskara CS 5968/6968, Lecture 1: Introduction and Review 12 January 2016

Aditya Bhaskara CS 5968/6968, Lecture 1: Introduction and Review 12 January 2016 Lecture 1: Introduction and Review We begin with a short introduction to the course, and logistics. We then survey some basics about approximation algorithms and probability. We also introduce some of

More information

Lecture: Flow-based Methods for Partitioning Graphs (1 of 2)

Lecture: Flow-based Methods for Partitioning Graphs (1 of 2) Stat260/CS294: Spectral Graph Methods Lecture 10-02/24/2015 Lecture: Flow-based Methods for Partitioning Graphs (1 of 2) Lecturer: Michael Mahoney Scribe: Michael Mahoney Warning: these notes are still

More information

Introduction to Spectral Graph Theory and Graph Clustering

Introduction to Spectral Graph Theory and Graph Clustering Introduction to Spectral Graph Theory and Graph Clustering Chengming Jiang ECS 231 Spring 2016 University of California, Davis 1 / 40 Motivation Image partitioning in computer vision 2 / 40 Motivation

More information

Convergence of Random Walks

Convergence of Random Walks Graphs and Networks Lecture 9 Convergence of Random Walks Daniel A. Spielman September 30, 010 9.1 Overview We begin by reviewing the basics of spectral theory. We then apply this theory to show that lazy

More information

Unsupervised Learning

Unsupervised Learning The Complexity of Unsupervised Learning Santosh Vempala, Georgia Tech Unsupervised learning } Data (imagine cool images here) } with no labels (or teachers) } How to } Understand it/find patterns in it?

More information

Machine Learning for Data Science (CS4786) Lecture 11

Machine Learning for Data Science (CS4786) Lecture 11 Machine Learning for Data Science (CS4786) Lecture 11 Spectral clustering Course Webpage : http://www.cs.cornell.edu/courses/cs4786/2016sp/ ANNOUNCEMENT 1 Assignment P1 the Diagnostic assignment 1 will

More information

approximation algorithms I

approximation algorithms I SUM-OF-SQUARES method and approximation algorithms I David Steurer Cornell Cargese Workshop, 201 meta-task encoded as low-degree polynomial in R x example: f(x) = i,j n w ij x i x j 2 given: functions

More information

Invertibility and Largest Eigenvalue of Symmetric Matrix Signings

Invertibility and Largest Eigenvalue of Symmetric Matrix Signings Invertibility and Largest Eigenvalue of Symmetric Matrix Signings Charles Carlson Karthekeyan Chandrasekaran Hsien-Chih Chang Alexandra Kolla Abstract The spectra of signed matrices have played a fundamental

More information

Positive Semi-definite programing and applications for approximation

Positive Semi-definite programing and applications for approximation Combinatorial Optimization 1 Positive Semi-definite programing and applications for approximation Guy Kortsarz Combinatorial Optimization 2 Positive Sem-Definite (PSD) matrices, a definition Note that

More information

Petar Maymounkov Problem Set 1 MIT with Prof. Jonathan Kelner, Fall 07

Petar Maymounkov Problem Set 1 MIT with Prof. Jonathan Kelner, Fall 07 Petar Maymounkov Problem Set 1 MIT 18.409 with Prof. Jonathan Kelner, Fall 07 Problem 1.2. Part a): Let u 1,..., u n and v 1,..., v n be the eigenvalues of A and B respectively. Since A B is positive semi-definite,

More information