The End Of Knowledge - Vault 6/14 - CVPR - 2016 - Laplacian Matrices of Graphs: Algorithms and Applications

graph LR classDef main fill:#f9d4d4, font-weight:bold, font-size:14px classDef intro fill:#d4f9d4, font-weight:bold, font-size:14px classDef laplacian fill:#d4d4f9, font-weight:bold, font-size:14px classDef solvers fill:#f9f9d4, font-weight:bold, font-size:14px classDef applications fill:#f9d4f9, font-weight:bold, font-size:14px classDef future fill:#d4f9f9, font-weight:bold, font-size:14px Main[Laplacian Matrices of
Graphs: Algorithms and
Applications] Main --> A[Introduction to Laplacian Matrices] A --> A1[Spielman: Yale researcher, algorithms
contributions 1] A --> A2[Laplacian matrices: numerous applications,
connections 2] A --> A3[Interpolation problems solved using
Laplacians 3] A --> A4[Laplacian quadratic form minimizes
interpolation 4] A --> A5[Laplacians analyzed physical systems
for centuries 5] A --> A6[Spring networks minimize potential
energy 6] Main --> B[Laplacian Properties and Applications] B --> B1[Tutte: planar drawing via
spring settling 7] B --> B2[Laplacians measure graph node
set boundaries 8] B --> B3[Spectral clustering uses small
Laplacian eigenvectors 9] B --> B4[Laplacian matrix structure and
representation 10] B --> B5[Sparsification approximates graphs, preserves
quadratics 16] B --> B6[ϵ-sparsifiers with n/ϵ^2 edges exist 17] Main --> C[Laplacian Solvers] C --> C1[Nearly linear time Laplacian
solvers 11] C --> C2[Solving Ax=b to ϵ-accuracy defined 12] C --> C3[Fast solvers enable various
computations 13] C --> C4[Simple, fast Laplacian solver
algorithm 20] C --> C5[Approximate Gaussian elimination sparsifies
cliques 21] C --> C6[Analysis uses random matrix
theory 22] Main --> D[Applications and Extensions] D --> D1[Laplacian solvers in optimization
problems 14] D --> D2[Interior point methods use
Laplacians 15] D --> D3[Sparsifiers computed by sampling
edges 18] D --> D4[Current best sparsification algorithms
described 19] D --> D5[Solvers extend to generalized
Laplacians 25] D --> D6[Julia package integrates Laplacian
methods 26] Main --> E[Future Directions and Challenges] E --> E1[Resources available on Spielmans
webpage 27] E --> E2[Challenges: fast solvers for
systems 28] E --> E3[Relation to graphical model
inference 29] E --> E4[Extending methods to asymmetric
matrices 30] class Main main class A,A1,A2,A3,A4,A5,A6 intro class B,B1,B2,B3,B4,B5,B6 laplacian class C,C1,C2,C3,C4,C5,C6 solvers class D,D1,D2,D3,D4,D5,D6 applications class E,E1,E2,E3,E4 future

Resume:

1.- Dan Spielman is a prominent researcher in the theory of computing from Yale University who has made fundamental contributions in algorithms.

2.- Laplacian matrices of graphs have numerous applications and connections to fields like machine learning.

3.- Garamani and Lafferty used Laplacian matrices to solve interpolation problems on graphs, like estimating probability of proteins being involved in diseases.

4.- Minimizing the Laplacian quadratic form allows interpolating values on graphs by solving a system of linear equations in the Laplacian matrix.

5.- Laplacian matrices have been studied for over 100 years to analyze physical systems like spring networks.

6.- In a spring network, vertices settle at positions minimizing the total potential energy, which relates to the Laplacian quadratic form.

7.- Tutte proved that nailing down an outer face of a planar graph and letting springs settle yields a planar drawing.

8.- Laplacian matrices provide a way to measure boundaries of sets of nodes in graphs using the quadratic form of characteristic vectors.

9.- Fiedler suggested spectral clustering by finding small Laplacian eigenvectors. Thresholding the Fiedler vector partitions graphs into good clusters.

10.- The Laplacian matrix has -1 for edges, degrees on diagonal. It equals B^T W B for edge-vertex incidence matrix B, weights W.

11.- Spielman and Teng developed nearly linear time algorithms for solving Laplacian systems. Current best runtime is about m log n log(1/ϵ).

12.- Solving Ax=b to ϵ-accuracy means finding x with ||x-x*||_A ≤ ϵ ||x*||_A, where ||x||_A = sqrt(x^T A x), x* is optimal.

13.- Fast Laplacian solvers enable quickly computing eigenvalues/vectors and are used as subroutines in many graph optimization problems.

14.- King, Rao, Sachdeva used Laplacian solvers in interior point methods to get fast algorithms for isotonic regression and other graph problems.

15.- Interior point methods for graph problems solve a Laplacian system in each iteration; constant accuracy suffices for convergence.

16.- Sparsification aims to approximate graphs by sparser graphs while preserving Laplacian quadratic forms. Effective resistance governs importance of edges.

17.- Spielman and Srivastava proved every graph has an ϵ-sparsifier with about n/ϵ^2 edges, computable in nearly linear time.

18.- Sparsifiers can be computed by randomly sampling each edge with probability proportional to its effective resistance and rescaling weights.

19.- Current best sparsification algorithms take about m log^2 n or min(m log n, n^(1+α)) time to produce sparsifiers with about n log n/ϵ^2 edges.

20.- Recent work gives a simple, nearly linear time algorithm for solving Laplacian systems by approximate Gaussian elimination.

21.- Approximate Gaussian elimination modifies standard Gaussian elimination by randomly sparsifying the cliques created during each elimination step.

22.- Analysis relies on writing Gaussian elimination as a sum of random matrices and applying tools from random matrix theory.

23.- Key aspects are using a random vertex permutation, duplicating edges, controlling variances, and applying matrix concentration inequalities.

24.- Simple implementation takes about m log^2 n time; m log n seems possible. More complex variant may improve log factors.

25.- Laplacian solvers extend to some generalized Laplacians like connection Laplacians arising in applications like cryo-EM and computer vision.

26.- Spielman's group is developing a Julia package at Laplacians.jl integrating Laplacian solvers with interior point methods for ease of use.

27.- Surveys, papers, and course materials on Laplacian methods are available on Spielman's webpage. Vishnoi's manuscript also covers the material.

28.- Ongoing challenges include developing fast solvers for other linear systems like those arising in structural mechanics or multicommodity flows.

29.- Sachdev and King's randomized Gaussian elimination may relate to approximate inference in graphical models, a promising research direction.

30.- Laplacian methods may extend to asymmetric matrices, an active area of current research aiming to efficiently solve general M-matrices.

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