Applied Hodge Theory

Transcription

1 Applied Hodge Theory School of Mathematical Sciences Peking University Oct 14th, 2014

2 1 What s Hodge Theory Hodge Theory on Riemannian Manifolds Hodge Theory on Metric Spaces Combinatorial Hodge Theory on Cell Complexes 2 Applications of Hodge Decomposition Computer Vision Statistical Ranking via Paired Comparison Method HodgeRank on Graphs Random Graph Models for Sampling Robust Ranking Online Algorithms Game Theory Hodge Decomposition of Finite Games 3 Summary

3 Topological & Geometric Methods in Data Analysis Differential Geometric methods: manifolds data distribution: manifold learning/ndr, etc. model space: information geometry (high-order efficiency for parametric statistics) Algebraic Geometric methods: polynomials/varieties tensor (matrices etc.) algebraic statistics polynomial optimization (SOS) Algebraic Topological methods: complexes (graphs, etc.) persistent homology (robust, slow) Euler calculus (non-stable, fast) Hodge theory (geometry topology via optimization/spectrum)

4 Outline Hodge Theory Applications Helmholtz-Hodge Decomposition Theorem (c.f. Marsden-Chorin 1992) A vector field w on a simply-connected D can be uniquely decomposed in the form w = u + grad φ where u has zero divergence and is parallel to D. Summary

5 Algebraic Elements of Hodge Decomposition For inner product spaces X, Y, and Z, consider X A Y B Z. and = AA + B B : Y Y where ( ) is adjoint operator of ( ). If B A = 0, then ker( ) = ker(a) ker(b ) and orthogonal decomposition Y = im(a) + ker( ) + im(b ) Note: ker(b)/ im(a) ker( ) is the (real) (co)-homology group (R rings; vector spaces module).

6 Hodge Theory on Riemannian Manifolds Classical Hodge Theory on Riemannian Manifolds de Rham complex: d 2 = d k d k 1 = 0 0 Ω 0 (M) d 0 Ω 1 (M) d 1 dn 1 Ω n (M) dn 0 where M is a compact Riemannian manifold with k-differential forms Ω k (M) and d is the exterior derivative operator whose adjoint, codifferential operator δ satisfies du, v = u, δv Laplacian = dδ + δd and Harmonic forms ker( ) Hodge decomposition (W.V.D. Hodge, ) Ω k (M) = im(d k 1 ) ker( k ) im(δ k ) where ker( k ) is isomorphic to de Rham cohomology group H k (M) = ker(d k )/ im(d k 1 ).

7 Hodge Theory on Metric Spaces Hodge Theory on Metric Spaces (Alexander-Spanier, Bartholdi-Schick-Smale-Smale, 2011) complex, d 2 = 0 0 L 2 (X ) d 0 L 2 (X 2 ) d 1 dn 1 L 2 (X n ) dn L 2 (X ): square integral functions on metric space X finite difference (Gilboa-Osher 08) d : L 2 (X k ) L 2 (X k+1 ) k (df )(x 0,..., x k ) = i=1( 1) i j i adjoint operator δ : L 2 (X k+1 ) L 2 (X k ) δg(x) = k ( 1) i i=0 k 1 X j=0 K(x i, x j )f (x i ) K(t, x j )g(x 0,..., x i 1, t, x i,..., x k 1 )dt

8 Hodge Theory on Metric Spaces continued: Hodge Theory on Metric Spaces (Bartholdi-Schick-Smale-Smale-Baker, 2011) If X satisfies some regularity conditions, then Hodge decomposition holds L 2 (X k ) = im(d k 1 ) ker( k ) im(δ k ) In particular, if X is a compact Riemannian manifold with regularity conditions on convexity and curvature, there is a scale/kernel such that ker( k ) is isomorphic to the L 2 -cohomology and de Rham cohomology. = dδ + δd for finite X, it essentially builds up a Čech complex for point cloud data at certain scale and applies combinatorial Hodge theory

9 Combinatorial Hodge Theory on Cell Complexes Combinatorial Hodge Theory on Cell Complexes X is finite χ(x ) 2 X is a simplicial complex formed by X, such that τ χ(x ) and σ τ, then σ χ(x ) k-forms or cochains as alternating functions Ω k (X ) = {u : χ k+1 (X ) R, u iσ(0),...,i σ(k) = sign(σ)u i0,...,i k } where σ S k+1 is a permutation on (0,..., k). coboundary maps d k : Ω k (X ) Ω k+1 (X ) are defined as the alternating difference operator k+1 (d k u)(i 0,..., i k+1 ) = ( 1) j+1 u(i 0,..., i j 1, i j+1,..., i k+1 ) j=0

10 Combinatorial Hodge Theory on Cell Complexes Example: graph and clique complex G = (X, E) is a undirected graph Clique complex χ G 2 X collects all complete subgraph of G k-forms or cochains Ω k (χ G ) as alternating functions: 0-forms: v : V R = R n 1-forms as skew-symmetric functions: w ij = w ji 2-forms as triangular-curl: z ijk = z jki = z kij = z jik = z ikj = z kji coboundary operators d k : Ω k (χ G ) Ω k (χ G ) as alternating difference operators: (d 0 v)(i, j) = v j v i =: (grad v)(i, j) (d 1 w)(i, j, k) = (±)(w ij + w jk + w ki ) =: (curl w)(i, j, k) d 1 d 0 = curl(grad u) = 0

11 Combinatorial Hodge Theory on Cell Complexes continued: Combinatorial Hodge Theory on Cell Complexes So we have 0 Ω 0 (X ) d 0 Ω 1 (X ) d 1 dn 1 Ω n (X ) dn d k d k 1 = 0 combinatorial Laplacian = d k 1 d k 1 + d k d k k = 0, 0 = d 0 d 0 is the (unnormalized) graph Laplacian k = 1, 1-Hodge Laplacian (Helmholtzian) 1 = curl curl div grad Hodge decomposition holds for Ω k (X ) Ω k (X ) = im(d k 1 ) ker( k ) im(δ k ) dim( k ) = β k (χ(x ))

12 Combinatorial Hodge Theory on Cell Complexes Forgetful functors Riemannian manifolds Metric spaces Cell complexes From differentiable to combinatorial structures, Hodge decomposition is functorial (invariant) Topological invariants (homology) are preserved in such coarse-grained functors Natural for data analysis, a connection between geometry and topology: harmonic basis More important than data itself, relations between data via functions, mappings, etc.

13 Applications of Hodge Decomposition Boundary Value Problem (Schwarz, Chorin-Marsden 92) Computer vision Optical flow decomposition and regularization (Yuan-Schnörr-Steidl 2008, etc.) Retinex theory and shade-removal (Ma-Morel-Osher-Chien 2011) Relative attributes (Fu-Xiang-Y. et al. 2014) Sensor Network coverage (Jadbabai et al. 10) Statistical Ranking or Preference Aggregation (Jiang-Lim-Y.-Ye 2011, etc.) Decomposition of Finite Games (Candogan-Menache-Ozdaglar-Parrilo 2011)

14 Computer Vision Optical Flow Decomposition and Regularization Rudin-Osher-Fatemi 1992: piecewise constant flows 1 2 v u TV (u), u, v R 2 TV (u) := (grad u 1 ) 2 + (grad u 2 ) 2, Yuan-Schnörr-Steidl 2007: piecewise harmonic flows TV (u) R(u) = (div u) 2 + (curl u) 2

15 Computer Vision Example: periodic motions are harmonic Figure: Better motion separation with Hodge decomposition

16 Computer Vision Adelson s iilusion in Computer Vision Figure: Adelson s illusion: on the left the chess board is shadowed by a column such that the white square has the same illuminance intensity as the black square, proved by the right picture.

17 Computer Vision Retinex Theorey based on Approximation of Gradient Flows The edge information is a gradient field of intensity grad I Shade adds sparse noise Y = grad I + E Find sparse approximation of de-noised gradient field min X grad X T (Y ) 1 Figure: Ma-Morel-Osher-Chien 2011

18 Statistical Ranking via Paired Comparison Method Crowdsourcing QoE evaluation of Multimedia Figure: (Xu-Huang-Y., et al. 11) Crowdsouring subjective Quality of Experience evaluation

19 Statistical Ranking via Paired Comparison Method Learning relative attributes: age Ranking scores: Unintentional errors Intentional errors Correct pairs Figure: Age: a relative attribute estimated from paired comparisons (Fu-Y.-Xiang et al. 2013)

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21 Statistical Ranking via Paired Comparison Method Paired comparison data on graphs Graph G = (V, E) V : alternatives to be ranked or rated (i α, j α ) E a pair of alternatives y α ij R degree of preference by rater α ω α ij R + confidence weight of rater α Examples: relative attributes, subjective QoE assessment, perception of illuminance intensity, sports, wine taste, etc.

22 Statistical Ranking via Paired Comparison Method Statistical Paired Preference Aggregation: l 2 (E) Majority voting (Condorcet 1785): inconsistency arises (Arrow s impossibility theorem 1950s) Statistical majority voting: Ŷ ij = ( α ωα ij Y α ij )/( α ωα ij ) = Ŷ ji, ω ij = α ωα ij Ŷ from generalized linear models Uniform model: Ŷ ij = 2ˆπ ij 1. Bradley-Terry model: Ŷ ij = log ˆπ ij 1 ˆπ ij. Thurstone-Mosteller model: Ŷ ij = Φ 1 (ˆπ ij ). Φ(x) = 1 2π e 1 x/[2σ 2 (1 ρ)] 1/2 2 t2 dt. Angular transform model: Ŷ ij = arcsin(2ˆπ ij 1). Inner product induced on Ŷ l 2 ω(e), u, v ω = u ij v ij ω ij where u, v skew-symmetric

23 Statistical Ranking via Paired Comparison Method Hodge Decomposition on Graphs [Jiang-Lim-Y.-Ye 11] Ŷ ij = Ŷ ji l 2 ω(e) admits an orthogonal decomposition, where Ŷ (h) ij In other words + Ŷ (h) jk Ŷ = Ŷ (g) + Ŷ (h) + Ŷ (c), (1) Ŷ (g) ij = ˆβ i ˆβ j, for some ˆβ R V, (2a) j i + Ŷ (h) ki = 0, for each {i, j, k} T, (2b) ω ij Ŷ (h) ij = 0, for each i V. (2c) im(grad) ker( 1 ) im(curl )

24 Statistical Ranking via Paired Comparison Method Global ranking and Local vs. Global Inconsistencies Ŷ (g) = (δ 0 β)(i, j) := ˆβ i ˆβ j where ˆβ solves Graph Laplacian equation min ωij α (β i β j Yij α ) 2 0 ˆβ = δ 0 Ŷ β R V α,(i,j) E Residues Ŷ (h) and Ŷ (c) accounts for inconsistencies: Ŷ (c), the local inconsistency, triangular curls Ŷ (c) ij + Ŷ (c) jk + Ŷ (c) ki 0, {i, j, k} T Ŷ (h), the global inconsistency, harmonic ranking harmonic ranking leads to circular coordinates on V fixed tournament issue it creates all chaotic voting results

25 Statistical Ranking via Paired Comparison Method Topological Obstructions To get a faithful ranking, two topological conditions on the clique complex χ 2 G = (V, E, T ) are important: Connectivity: G is connected, then an unique global ranking is possible; Loop-free: harmonic ranking vanishes if χ 2 G is loop-free, topology plays a role of obstruction of fixed-tournament Triangular arbitrage-free implies arbitrage-free

26 Statistical Ranking via Paired Comparison Method Basic Problems in HodgeRank sampling method for crowdsourcing passive, active, random graph theory, etc. reliability of data: inconsistency outlier detection and robust ranking sequential or streaming data: online algorithms persistent homology, online ranking

27 Statistical Ranking via Paired Comparison Method Random Graph Models for Crowdsourcing Recall that in crowdsourcing ranking on internet, unspecified raters compare item pairs randomly online, or sequentially sampling random graph models for experimental designs P a distribution on random graphs, invariant under permutations (relabeling) Generalized de Finetti s Theorem [Aldous 1983, Kallenberg 2005]: P(i, j) (P ergodic) is an uniform mixture of h(u, v) = h(v, u) : [0, 1] 2 [0, 1], h unique up to sets of zero-measure Erdös-Rényi: P(i, j) = P(edge) = h(u, v)dudv =: p edge-independent process (Chung-Lu 06)

28 Statistical Ranking via Paired Comparison Method Phase Transitions of Large Random Graphs For an Erdos-Renyi random graph G(n, p) with n vertices and each edge independently emerging with probability p(n), (Erdös-Rényi 1959) One phase-transition for β 0 p << 1/n 1+ɛ ( ɛ > 0), almost always disconnected p >> log(n)/n, almost always connected (Kahle 2009) Two phase-transitions for β k (k 1) p << n 1/k or p >> n 1/(k+1), almost always β k vanishes; n 1/k << p << n 1/(k+1), almost always β k is nontrivial For example: with n = 16, 75% distinct edges included in G, then χ G with high probability is connected and loop-free. In general, O(n log(n)) samples for connectivity and O(n 3/2 ) for loop-free.

29 Statistical Ranking via Paired Comparison Method Other sampling models [Xu et al. 2012] Random k-regular graphs Kim-Vu sandwich theorem/conjecture: coupling with Erdös-Rényi if edges are dense enough Preferential-attachment random graphs online but dependent (active) sampling coupling with edge-independent process (Chung-Lu 06) Geometric random graphs ranking items from Euclidean feature space Active sampling? Osting, Brune, and Osher, ICML 2013 Osting, Xiong, Xu, and Y., 2014

30 Statistical Ranking via Paired Comparison Method Three sampling methods Uniform sampling with replacement (i.i.d.) (G 0 (n, m)). Each edge is sampled from the uniform distribution on ( n 2) edges, with replacement. This is a weighted graph and the sum of weights is m. Uniform sampling without replacement (G(n, m)). Each edge is sampled from the uniform distribution on the available edges without replacement. For m ( n 2), this is an instance of the Erdös-Rényi random graph model G(n, p) with p = m/ ( n 2). Greedy sampling (G (n, m)). Each pair is sampled to maximize the algebraic connectivity of the graph in a greedy way: the graph is built iteratively; at each iteration, the Fiedler vector is computed and the edge (i, j) which maximizes (ψ i ψ j ) 2 is added to the graph.

31 Statistical Ranking via Paired Comparison Method Asymptotic Estimates for Fiedler Values [Braxton-Xu-Xiong-Y. 14] Key Estimates of Fiedler Value near Connectivity Threshold. λ 2 G 0 (n, m): np a 2 1(p 0, n) := (3) p 0 n λ 2 G(n, m): np a 2 2(p 0, n) := 1 1 p (4) p 0 where p 0 := 2m/(n log n) 1, p = p 0 log n n and a(p 0 ) = 1 2/p 0 + O(1/p 0 ), for p 0 1.

32 Statistical Ranking via Paired Comparison Method Without-replacement as good as Greedy! Figure: A comparison of the Fiedler value, minimal degree, and estimates a(p 0 ), a 1 (p 0 ), and a 2 (p 0 ) for graphs generated via random sampling with/without replacement and greedy sampling at n = 64.

33 Statistical Ranking via Paired Comparison Method Robust Ranking with Sparse Outliers For each (i, j) E, where β V : global ranking score on V y αij = β 0 + β i β j + z αij (5) β 0 : head-advantage (home- in NBA, white- in chess) z ij error z αij = γ αij + ε αij [A0a] γ αij symmetric p-sparse (zero w.p. p and median 0) [A0b] ε αij N (0, σ 2 /w ij )

34 Statistical Ranking via Paired Comparison Method Huber s LASSO [Xiong-Cheng-Y. 13, Xu-Xiong-Huang-Y. 13] Robust ranking can be formulated as a Huber s LASSO problem (Gannaz 07, She-Owen 09, Fan-Tang-Shi 12) Sparse outliers are sparse approximation of cyclic rankings (curl+harmonic) Exact recovery is possible without Gaussian noise Outlier detection is possible against Gaussian noise, provided Irrepresentable condition (e.g. random graph) Outliers have large enough magnitudes

35 Statistical Ranking via Paired Comparison Method Exact Recover against pure Sparse Outliers Theorem (Xiong-Cheng-Y. 2013) Let G(n, q) be an Erdös-Rényi Random Graph with n nodes and each edge drawn independently with probability q (0, 1]. (A) Suppose that paired comparison data y is collected on G(n, q) subject to the linear model with symmetric p-sparse outliers (p [0, 1]). Then with probability tending to one the L1 solution exactly recovers the global ranking β if ( ) log n p O. nq Note: no method can recover if p O ( 1 nq ).

36 Statistical Ranking via Paired Comparison Method Persistent Homology: online algorithm for topology tracking (e.g Edelsbrunner-Harer 08) Figure: Persistent Homology Barcodes vertice, edges, and triangles etc. sequentially added online update of homology O(m) for surface embeddable complex; and O(m 2.xx ) in general (m number of simplex)

37 Statistical Ranking via Paired Comparison Method Online HodgeRank as Stochastic Approximations Robbins-Monro (1951) algorithm for Āx = b x t+1 = x t γ t (A t x t b t ), E(A t ) = Ā, E(b t ) = b Now consider 0 s = δ 0Ŷ, with new rating Y t(i t+1, j t+1 ) s t+1 (i t+1 ) = s t (i t+1 ) γ t [s t (i t+1 ) s t (j t+1 ) Y t (i t+1, j t+1 )] s t+1 (j t+1 ) = s t (j t+1 ) + γ t [s t (i t+1 ) s t (j t+1 ) Y t (i t+1, j t+1 )] Note: updates only occur locally on edge {i t+1, j t+1 } initial choice: s 0 = 0 or any vector i s 0(i) = 0 step size γ t = a(t + b) θ (θ (0, 1]) γ t = const(t ),.e.g. 1/T where T is total sample size

38 Statistical Ranking via Paired Comparison Method Minimax Optimal Convergence Rates (Lim-Y. 13, Xu-Xiong-Huang-Y. 13) Choose γ t t 1/2 (e.g. a=1/λ 1 ( 0 ) and b large enough) In this case, s t converges to s (population solution), with probability 1 δ, in the (optimal) rate of t ( s t s O t 1/2 κ 3/2 ( 0 ) log 1/2 1 ) δ Dependence on κ 3/2 can be improved to κ by Ji Liu (U Wisc-Madison) (optimal order of κ?)

39 Statistical Ranking via Paired Comparison Method Some reference Random graph sampling models: Erdös-Rényi and beyond Xu, Jiang, Yao, Huang, Yan, and Lin, ACM Multimedia, 2011, IEEE Trans Multimedia, 2012 Online algorithms Xu, Huang, and Yao, ACM Multimedia 2012 l 1 -norm ranking Osting, Darbon, and Osher, 2012 Robust ranking: Huber s Lasso Xiong, Cheng, and Yao, 2013 Xu, Xiong, Huang, and Yao, ACM Multimedia 2013 Active sampling Osting, Brune, and Osher, ICML 2013 Osting, Xiong, Xu, and Yao, 2014

40 same flow representation and game graph. Two examples of game graph representations are given below. Outline Hodge Theory Applications Summary Game Theory Strategic Simplicial Complex for Games Example 2.2. Consider again the battle of the sexes game from Example 2.1. has four vertices, corresponding to the direct product of two 2-cliques, and is pres 2 (O, O) (O, F) Figure 2: Flows on the game graph corresponding to battle of the sexes (E O F O F O 3, 2 0, 0 3 O 4, 2 0, 2 0 F 0, 0 2, 3 F 1, 0 2, 3 3 (a) Battle of the sexes (F, O) (b) Modified (F, battle F) of the sexes t is easy Extension to see that to multiplayer these two games: haveg the = same (V, E) pairwise comparisons, which will tical equilibria V = for{(x the 1, two..., games: x n ) =: (O, (x i, O) x i and )} = (F, F). n It is only the actual equilibrium i=1 S i, n person game; would Example differ In particular, Considerina the three-player equilibrium game, (O, O), where thepayoff each player of thecan rowchoose player is betw in. undirected edge: {(x {a, b}. We represent the strategic i, x i ), (x interactions i, x i)} = E among the players by the directe 3a, whereeach the player payoff has of player utility ifunction is 1 ifuits i (x i strategy, x i ); is identical to the strateg he usual solution concepts in games (e.g., Nash, mixed Nash, correlated equilibria) are erms of pairwise Edgecomparisons flow (1-form): only. ugames i (x i, x i with ) identical u i (x i, x i) pairwise comparisons share th librium sets. Thus, we refer to games with identical 7 pairwise comparisons as strat valent games.

41 Game Theory Nash and Correlated Equilibrium π(x i, x i ), a joint distribution tensor on i S i, satisfies x i, x i, x i π(x i, x i )(u i (x i, x i ) u i (x i, x i )) 0, i.e. expected flow (E[ x i ]) is nonnegative. Then, tensor π is a correlated equilibrium (CE, Aumann 1974); if π is a rank-one tensor, π(x) = i µ(x i ), then it is a Nash equilibrium (NE, Nash 1951); fully decided by the edge flow data.

42 Game Theory Why Correlated Equilibria? Players are never independent in reality, e.g. Bayesian decision process (Aumman 87) Finding NE is NP-hard, e.g. solving polynomial equations (Sturmfels 02, Datta 03) Finding CE is linear programming, easy for graphical games (Papadimitriou-Roughgarden 08) Some natural learning processes converges to CE (Foster-Vohra 97)

43 Game Theory More... CE contains the convex hull of NE, nonnegative tensor rank #(NE) (e.g. 2 pure NE in BoS) Theorem (Landsberg-Manivel 03, Raicu) A tensor is of rank 2 if and only if all its matrix flattenings are of rank 2. Theorem (Allman-Rhodes-Sturmfels-Zwiernik 13) A nonnegative tensor has nonnegative rank 2 if and only if it is of rank 2 and supermodular. It is possible to minimize matrix nuclear or max norm to search rank-1 NE within CE (linear inequality constraints).

44 Game Theory Hodge Decomposition of Finite Games Theorem (Candogan-Menache-Ozdaglar-Parrilo,2011) Every finite game admits a unique decomposition: Furthermore: Potential Games Harmonic Games Neutral Games Shapley-Monderer Condition: Potential games quadrangular-curl free Extending G = (V, E) to complex by adding quadrangular cells, harmonic games can be further decomposed into curl games

45 Game Theory Bimatrix Games For bi-matrix game (A, B), potential game is decided by ((A + A )/2, (B + B )/2) pure Nash equilibria are the set of local maxima Correlated equilibria: the convex hull of such maxima (e.g. Neyman 97) harmonic game is zero-sum ((A A )/2, (B B )/2) only mixed Nash equilibria Computation of Nash Equilibrium: each of them is tractable however direct sum is NP-hard approximate potential game leads to approximate NE

46 Game Theory Example: Hodge Decomposition of Prisoner s Dilemma Note: Shapley-Monderer Condition Harmonic-free quadrangular-curl free

47 Game Theory Graphical Games n-players live on a network of n-nodes player i utility only depends on its neighbor players N(i) strategies correlated equilibria allows a concise representation with parameters linear to the size of the network (Kearns et al. 2001; 2003) π(x) = 1 n ψ i (x Z N(i) ) i=1 this is not rank-one, but low-order interaction reduce the complexity from O(e 2n ) to O(ne 2d ) (d = max i N(i) ) polynomial algorithms for CE in tree and chodal graphs.

48 Summary Hodge Decomposition Theorem finds new applications: Statistical ranking: where every paired comparison data is decomposed into gradient flow (global ranking) harmonic flow (global inconsistency) curl flow (local inconsistency) Game Theory: every finite game can be decomposed into potential game harmonic game CE and NE are preserved, tractable in some settings Computer Vision: shade-removal, optical flow decomposition, subjective visual attributes... more are coming...

49 Acknowledgement Multimedia group: Qianqian Xu, Postdoc at BICMR, PKU Qingming Huang, GUCAS; Bowei Yan, Tingting Jiang, PKU Methodology: Jiechao Xiong, Stat PhD student in PKU Xiuyuan Cheng, Princeton & ENS-Paris Relative attribute group: Yanwei Fu, EECS PhD student at University of London Tao Xiang, Tim Hospedales, Shaogang Gong, QMUL Yizhou Wang, PKU Other collaborators Lek-Heng Lim(U Chicago), Osher, Ostings (UCLA), Yinyu Ye (Stanford)Parrilo (MIT)