Laplacian Paradigm 2.0
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1 Laplacian Paradigm 2.0 8:40-9:10: Merging Continuous and Discrete(Richard Peng) 9:10-9:50: Beyond Laplacian Solvers (Aaron Sidford) 9:50-10:30: Approximate Gaussian Elimination (Sushant Sachdeva) 10:30-11:00: coffee break 11:00-12:00: Analysis using matrix Martingales (Rasmus Kyng) 12:00-14:00 lunch 14:00-15:00 Graph Structure via Eliminations (Aaron Schild) Website: bit.ly/laplacian2
2 Merging the Continuous and Discrete Richard Peng Oct 6, 2018
3 Outline Graphs and Laplacians Building Blocks Laplacian Paradigm 2.0
4 Large Networks Data mining: centrality, clustering Image/video processing: segmentation, denoising Scientific computing: stress, fluids, waves \ (linear system solve) CVX (convex optimization) Eigenvector solvers
5 Graphs and Matrices High performance computing: non-zeros ó edges, design / analyze matrix algorithms using graph theory n vertices m edges n rows / columns O(m) non-zeros
6 Graphs and Matrices High performance computing: non-zeros ó edges, design / analyze matrix algorithms using graph theory n vertices m edges n rows / columns O(m) non-zeros graph Laplacian matrix L Diagonal: degrees Off-diagonal: -edge weights
7 Source of Laplacians graph Laplacian matrix L Diagonal: degrees Off-diagonal: -edge weights d-regular graphs: L = di A, A: adjacency matrix
8 Source of Laplacians graph Laplacian matrix L Diagonal: degrees Off-diagonal: -edge weights d-regular graphs: L = di A, A: adjacency matrix Graph cuts: x T Lx = u~v w uv (x u - x v ) 2 (1-0) 2 =1 x a =1 (1-1) 2 =0 x b =0 x c =1 x indicator vector of cut è weight of cut
9 Source of Laplacians graph Laplacian matrix L Diagonal: degrees Off-diagonal: -edge weights d-regular graphs: L = di A, A: adjacency matrix Graph cuts: x T Lx = u~v w uv (x u - x v ) 2 (1-0) 2 =1 x a =1 (1-1) 2 =0 x b =0 x c =1 x indicator vector of cut è weight of cut L = B T WB where B is edge-vertex incidence matrix
10 Origin of the Laplacian Paradigm [Spielman Teng `04] Input: graph Laplacian L vector b Output: vector x s.t. Lx b Runtime: O(mlog O(1) nlog(1/ε))
11 Origin of the Laplacian Paradigm [Spielman Teng `04] Input: graph Laplacian L vector b Output: vector x s.t. Lx b Runtime: O(mlog O(1) nlog(1/ε)) [Cohen-Kyng-Miller-Pachocki-P-Rao-Xu `14]: 1/2
12 Origin of the Laplacian Paradigm [Spielman Teng `04] Input: graph Laplacian L vector b Output: vector x s.t. Lx b Runtime: O(mlog O(1) nlog(1/ε)) [Cohen-Kyng-Miller-Pachocki-P-Rao-Xu `14]: 1/2 Wall clock: m 10 7 in 20s
13 The Laplacian Paradigm Directly related: Elliptic systems Few iterations: Eigenvectors, Heat kernels Many iterations / modify algorithm Graph problems Image processing
14 Outline Graphs and Laplacians Building Blocks Laplacian Paradigm 2.0
15 Lx = b as a graph problem x: voltage vectors Dual: electrical flow f Unified formulation: min f with resdiual b f p : p = 2: solving Lx = b p = 1: shortest path / transshipment p = : max-flow/min-cut
16 Direct Methods (combinatorial) Repeatedly remove vertices by creating equivalent graphs on their neighborhoods M (2) ß Eliminate(M (1), i 1 ) M (3) ß Eliminate(M (2), i 2 )
17 Direct Methods (combinatorial) Repeatedly remove vertices by creating equivalent graphs on their neighborhoods M (2) ß Eliminate(M (1), i 1 ) M (3) ß Eliminate(M (2), i 2 ) Parallel graph algorithms Matrix multiplication / dense solves Sparsified squaring
18 Iterative Methods (numerical) Solve Ax = b by x ß x - (Ax b) Fixed point: Ax b = 0
19 Iterative Methods (numerical) Preconditioning: Solve Ax = b by Solve x ß Bx -1 - Ax (Ax = Bb) -1 b by: x ß x - B -1 (Ax b) Fixed point: Ax b = 0 Simple B: B = I, many iterations B = A: 1 iteration, but same problem
20 Iterative Methods (numerical) Preconditioning: Solve Ax = b by Solve x ß Bx -1 - Ax (Ax = Bb) -1 b by: x ß x - B -1 (Ax b) Fixed point: Ax b = 0 Simple B: B = I, many iterations B = A: 1 iteration, but same problem Krylov space methods / PCG Convex optimization algorithms
21 Hard instances Direct methods create too much fill on highly connected graphs Iterative methods take too many iterations paths
22 Hard instances Direct methods create too much fill on highly connected graphs Iterative methods take too many iterations paths Still `easy by themselves Easy for iterative methods Easy for direct methods
23 Hard instances Direct methods create too much fill on highly connected graphs Iterative methods take too many iterations paths Still `easy by themselves Easy for iterative methods Easy for direct methods Must handle both simultaneously, but avoid paying n iterations m per iteration
24 Hybrid algorithms (aka. v1.0) Scientific computing: ichol, multigrid [Vaidya `89] precondition with graphs
25 Hybrid algorithms (aka. v1.0) Scientific computing: ichol, multigrid [Vaidya `89] precondition with graphs
26 Hybrid algorithms (aka. v1.0) Scientific computing: ichol, multigrid [Vaidya `89] precondition with graphs Focus: how to combine [Gemban-Miller `96]: spectral graph theory [Spielman-Teng `04]: spectral (ultra-)sparsify
27 Key glue : sparsification [Spielman-Teng `04]: for any G, can find H with O(nlog O(1) n) edges s.t. x T L G x x T L H x x
28 Key glue : sparsification [Spielman-Teng `04]: for any G, can find H with O(nlog O(1) n) edges s.t. x T L G x x T L H x x Combinatorial parameter: #edges Numerical parameter : approximations
29 Key glue : sparsification [Spielman-Teng `04]: for any G, can find H with O(nlog O(1) n) edges s.t. x T L G x x T L H x x Combinatorial parameter: #edges Numerical parameter : approximations [Spielman-Srivatava`08]: sample by effective resistances gives H with O(nlogn) edges
30 Outline Graphs and Laplacians Building Blocks Laplacian Paradigm 2.0
31 Max-Flow problem Maximum number of disjoint s-t paths Applications: Routing Scheduling s t Recall: min f with resdiual b f p : p = 2: solving Lx = b p = : max-flow/min-cut
32 Max-Flow problem Maximum number of disjoint s-t paths Dual: separate s and t by removing fewest edges Applications: Applications: Routing Scheduling s t Partitioning Clustering Recall: min f with resdiual b f p : p = 2: solving Lx = b p = : max-flow/min-cut
33 Hybrid Algorithms for Max-Flow [Daitch-Spielman `08][Christiano-Kelner-Madry-Spielman-Teng `10]: [Lee-Sidford `14] Max-flow/Min-cut via (several) electrical flows s t Repeat about m 1/3 iters Solve linear systems Re-adjust edge weights
34 Hybrid Algorithms for Max-Flow [Daitch-Spielman `08][Christiano-Kelner-Madry-Spielman-Teng `10]: [Lee-Sidford `14] Max-flow/Min-cut via (several) electrical flows s t Repeat about m 1/3 iters Solve linear systems Re-adjust edge weights [Madry `10] [Racke-Shah-Taubig `14]: cut approximator / oblivious routing O(n o(1) )-approx. in O(m 1+o(1) )
35 Hybrid Algorithms for Max-Flow [Daitch-Spielman `08][Christiano-Kelner-Madry-Spielman-Teng `10]: [Lee-Sidford `14] Max-flow/Min-cut via (several) electrical flows s t [Lee-Rao-Srivastava `13][Sherman `13, `17][Kelner-Lee-Orecchia-Sidford `14]: Preconditioning, (1+ε)-approx Repeat about m 1/3 iters Solve linear systems Re-adjust edge weights [Madry `10] [Racke-Shah-Taubig `14]: cut approximator / oblivious routing O(n o(1) )-approx. in O(m 1+o(1) )
36 Hybrid Algorithms for Max-Flow [Daitch-Spielman `08][Christiano-Kelner-Madry-Spielman-Teng `10]: [Lee-Sidford `14] Max-flow/Min-cut via (several) electrical flows s t [Lee-Rao-Srivastava `13][Sherman `13, `17][Kelner-Lee-Orecchia-Sidford `14]: Preconditioning, (1+ε)-approx Repeat about m 1/3 iters Solve linear systems Re-adjust edge weights [P`16]: recurse them into each other: O(mlog 41 n), optimistically mlog 6 n [Madry `10] [Racke-Shah-Taubig `14]: cut approximator / oblivious routing O(n o(1) )-approx. in O(m 1+o(1) )
37 Laplacian Paradigm 2.0 Motivated by the goal of hybrid algorithms, modify direct and iterative methods New Intermediate structures / theorems motivated by the overall algorithms
38 Examples Directed graphs / asymmetric matrices 1 b 2 a 1 1 c Sparsified/Approximate Gaussian Elimination F A[FF] A[FC] SC(A, C) C A[CF] A[CC]
39 Under the hood Matrix (martingale) concentration Partitioning / Localizations of Random Walks G[V 1 ] Sc(G, V 2 ) V 1 V 2
40 Not covered L Matrix Zoo from Scientific Computing [Boman-Hendrickson-Vavasis `04] [Kyng-Lee-P-Sachdeva-Spielman `16] [Kyng-Zhang `17][Kyng-P-Schweiterman-Zhang `18] Interactions with data structures G events [Kelner-Orecchia-Sidford-Zhu `13] [Nanongkai-Saranuk `17][Wulff-Nilsen `17] [Durfee-Kyng-Peebles-Rao-Sachdeva `17] H
41 Questions Wist list Direct Iterative Hybrid Convex functions? J JJ?? Arbitrary values J??LJ! Dynamic/streaming J L J??? Approximate eliminations beyond spectral condition # Unreasonable effectiveness of pcg(ichol(a), b), multigrid Non-linear (preconditioned) iterative methods [Adil-Kyng-P-Sachdeva `19]: p-norm iterative refinment
![Solvers in Practice [Kyng-Rao-Sachdeva `15] we suggest rerunning the program a few](/docs-images/94/121840912/images/42-1.jpg "times An alternate solver based onicholisprovided Questions: Precision (pseudo)")
42 Solvers in Practice [Kyng-Rao-Sachdeva `15] we suggest rerunning the program a few times An alternate solver based onicholisprovided Questions: Precision (pseudo) deterministic
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