CMPSCI 791BB: Advanced ML: Laplacian Learning

Transcription

1 CMPSCI 791BB: Advanced ML: Laplacian Learning Sridhar Mahadevan

2 Outline! Spectral graph operators! Combinatorial graph Laplacian! Normalized graph Laplacian! Random walks! Machine learning on graphs! Clustering! Regression

3 Operators on Graphs

4 Laplacian Spectra of Complete Graph K 6 Eigenvalues of Complete Graph K

5 Some basic results! Theorem: Given a connected graph G = (V, E), the eigenspace of the first eigenvalue has dimension 1! Proof:! We know that λ =! Let x be the associated eigenvector, so L x = λ x! We also know that x T L x = (u,v) ε E (x u x v ) 2 =! So, x must be a constant eigenvector, and hence of dimension 1! Corollary: The multiplicity of the first eigenvalue gives the number of connected components of G

6 Some basic results! Theorem: The Laplacian of the complete graph K n has n-1 eigenvalues equal to n and 1 eigenvalue =! Proof:! We already know that the constant vector 1 is an eigenvector of L! Consider any vector x such that x T 1 =! Note that L x i = (n-1) x i - j i x j = n x i! Any eigenvector associated with λ > must be perpendicular to 1! Naming convention: Spielman (and others) label K n to mean the complete graph on n vertices (each of degree n-1)

7 Laplacian as an Operator! Note that the Laplacian acts differently than the adjacency operator. L f(u) = v ~ u (f(u) - f(v)) w uv! Theorem: For any graph G, the number of spanning trees is given by where σ i are the eigenvalues of the combinatorial Laplacian

8 Combinatorial Laplacian on a Path Eigenfunctions Spectrum

9 Normalized Laplacian! The normalized Laplacian is L = D -1/2 (D-A) D -1/2! For a weighted graph, the normalized Laplacian is

10 Normalized Laplacian! L = D -1/2 L D -1/2 = I D -1/2 A D -1/2! Note that for a k-regular graph, L = I 1/k A! Unlike the combinatorial Laplacian, the normalized Laplacian takes the degree of each vertex into account

11 Random Walk on Graphs! For any graph G = (V, E), we can associate a natural random walk defined as P(u,v) = w(u,v)/ v w(u,v )! Let P * (u) be the long-term probability of being in vertex u of the random walk defined by P.! It can be shown that this invariant distribution is reversible, and expressed as P*(u) = w(u) / W, where w(u) = v w(u,v)! where W = u w(u)

12 Random Walks and the Normalized Laplacian! Another way to define the random walk on a graph = D -1 A! Note that this is not a symmetric matrix! However, its eigenvalues are all real, because it is closely related to the normalized Laplacian! Define two matrices A and B as similar if A = M B M -1! Here, M is any invertible matrix! Note that if x is an eigenvector of A, then we have A x = λ x = M B M -1 x! This implies that λ M -1 x = B M -1 x, so M -1 x is an eigenvector of B

13 Random Walks and Normalized Laplacian! Since L = I D -1/2 A D -1/2, we get I L = D -1/2 A D -1/2 D -1/2 (I L) D 1/2 = D -1 A! This shows that the random walk operator is spectrally similar to the normalized Laplacian!! The eigenvalues of the random walk operator are the same as that of I L! The eigenvectors of the random walk operator are???

14 Spectral Clustering using the Laplacian (Ng, Jordan, and Weiss, NIPS 22)! Given a set of instances D, compute the distance between each pair of points using some local metric! Gaussian kernel! Nearest neighbor kernel! Define a graph G whose edges are weighted by the distance between each pair of points! Compute the eigenvectors of the normalized graph Laplacian! Given any point, compute its embedding using k eigenvectors associated with the smallest eigenvalues! Use any standard clustering method on the embedded points (E.g, k-means)

15 Spectral Clustering using Graph Laplacian Embedding using the 2 nd and 3 rd eigenvector of the graph Laplacian Cluster: 1 Adler Barrington Immerman Kurose Rosenberg Shenoy Sitaraman Towsley Weems cluster: 2 Adrion Allan Avrunin Barto Brock Clarke Cohen Croft Grupen Hanson Jensen Lehnert Lesser Levine Mahadevan Manmatha McCallum Moll Moss Osterweil Riseman Rissland Schultz Utgoff Woolf Zilberstein

16 Partitioning Graphs using the Combinatorial Laplacian The sign of the second eigenvector can separate the vertices in Room 1 vs. Room 2 and Room3 Spatial Environment 2 nd Eigenvector of Graph Laplacian

17 Regression using the Graph Laplacian (Belkin and Niyogi, STOC 2; Mahadevan, ICML 2)

18 Regression using the Graph Laplacian

19 Comparison of Polynomial and Laplacian Basis Representations 8 Desired Function 6 Value 4 Mean squared error Polynomial Basis Approximation Laplacian Polynomial Number of basis functions

20 Approximation on a Grid Optimal Value Function 1 1 MEAN-SQUARED ERROR OF LAPLACIAN vs. POLYNOMIAL STATE ENCODING 8 7 LAPLACIAN POLYNOMIAL Least-Squares Approximation using automatically learned Proto-Value Functions 1 MEAN-SQUARED ERROR NUMBER OF BASIS FUNCTIONS

21 Nonlinear Function Approximation Laplacian Least Squares Function Approximation Target Function Polynomial Least Squares Function Approximation