applied hodge theory - university of wisconsin–madisontopology.cs.wisc.edu/yao.pdf · outline...

Outline Hodge Theory Applications Summary

Applied Hodge Theory

Yuan Yao

School of Mathematical SciencesPeking University

June 25th, 2014

Yuan Yao ICML’14: App. Hodge Theo.


1 What’s Hodge TheoryHodge Theory on Riemannian ManifoldsHodge Theory on Metric SpacesCombinatorial Hodge Theory on Cell Complexes

2 Applications in Game Theory and Statistical RankingComputer VisionStatistical Ranking via Paired Comparison Method

HodgeRank on GraphsRandom Graph Models for SamplingRobust RankingOnline Algorithms

Game TheoryHodge Decomposition of Finite GamesBimatrix Games

3 Summary



Topological & Geometric Methods in Data Analysis

Differential Geometric methods: manifolds• data distribution: manifold learning/NDR, etc.• model space: information geometry (high-order efficiency forparametric 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 viaoptimization/spectrum)



Helmholtz-Hodge Decomposition

Theorem (c.f. Marsden-Chorin 1992)

A vector field w on a simply-connected D can be uniquelydecomposed in the form

w = u + gradφ

where u has zero divergence and is parallel to ∂D.



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).



Hodge Theory on Riemannian Manifolds

Classical Hodge Theory on Riemannian Manifolds

de Rham complex: d2 = dk dk−1 = 0

0→ Ω0(M)d0−→ Ω1(M)

d1−→ · · · dn−1−−−→ Ωn(M)dn−→ 0

where M is a compact Riemannian manifold with k-differentialforms Ωk(M) and d is the exterior derivative operator whoseadjoint, codifferential operator δ satisfies 〈du, v〉 = 〈u, δv〉Laplacian ∆ = dδ + δd and Harmonic forms ker(∆)

Hodge decomposition (W.V.D. Hodge, 1903-1975)

Ωk(M) = im(dk−1)⊕ ker(∆k)⊕ im(δk)

where ker(∆k) is isomorphic to de Rham cohomology groupHk(M) = ker(dk)/ im(dk−1).



Hodge Theory on Metric Spaces


(Alexander-Spanier, Bartholdi-Schick-Smale-Smale, 2011)complex, d2 = 0

0→ L2(X )d0−→ L2(X 2)

d1−→ · · · dn−1−−−→ L2(X n)dn−→ ·

• L2(X ): square integral functions on metric space X• finite difference (Gilboa-Osher’08) d : L2(X k)→ L2(X k+1)

(df )(x0, . . . , xk) =k∑

i=1

(−1)i∏j 6=i

√K (xi , xj)f (x−i )

• adjoint operator δ : L2(X k+1)→ L2(X k)

δg(x) =k∑

i=0

(−1)i∫X

k−1∏j=0

√K (t, xj)g(x0, . . . , xi−1, t, xi , . . . , xk−1)dt




continued: Hodge Theory on Metric Spaces

(Bartholdi-Schick-Smale-Smale-Baker, 2011)

If X satisfies some regularity conditions, then Hodgedecomposition holds

L2(X k) = im(dk−1)⊕ ker(∆k)⊕ im(δk)

In particular, if X is a compact Riemannian manifold withregularity conditions on convexity and curvature, there is ascale/kernel such that ker(∆k) is isomorphic to the L2-cohomologyand de Rham cohomology.

∆ = dδ + δdfor finite X , it essentially builds up a Cech complex for pointcloud data at certain scale and applies combinatorial Hodgetheory



Combinatorial Hodge Theory on Cell Complexes


X is finite

χ(X ) ⊆ 2X is a simplicial complex formed by X , such thatτ ∈ χ(X ) and σ ⊆ τ , then σ ∈ χ(X )

k-forms or cochains as alternating functions

Ωk(X ) = u : χk+1(X )→ R, uiσ(0),...,iσ(k) = sign(σ)ui0,...,ik

where σ ∈ Sk+1 is a permutation on (0, . . . , k).

coboundary maps dk : Ωk(X )→ Ωk+1(X ) are defined as thealternating difference operator

(dku)(i0, . . . , ik+1) =k+1∑j=0

(−1)j+1u(i0, . . . , ij−1, ij+1, . . . , ik+1)




Example: graph and clique complex

G = (X ,E ) is a undirected graph

Clique complex χG ⊆ 2X collects all complete subgraph of G

k-forms or cochains Ωk(χG ) as alternating functions:• 0-forms: v : V → R ∼= Rn

• 1-forms as skew-symmetric functions: wij = −wji

• 2-forms as triangular-curl:zijk = zjki = zkij = −zjik = −zikj = −zkji

coboundary operators dk : Ωk(χG )→ Ωk(χG ) as alternatingdifference operators:• (d0v)(i , j) = vj − vi =: (grad v)(i , j)• (d1w)(i , j , k) = (±)(wij + wjk + wki ) =: (curl w)(i , j , k)

d1 d0 = curl(grad u) = 0




continued: Combinatorial Hodge Theory on Cell Complexes

So we have

0→ Ω0(X )d0−→ Ω1(X )

d1−→ · · · dn−1−−−→ Ωn(X )dn−→ · · ·

dk dk−1 = 0

combinatorial Laplacian ∆ = dk−1d∗k−1 + d∗kdk

• k = 0, ∆0 = d∗0d0 is the well-known graph Laplacian• k = 1, 1-Hodge Laplacian

∆1 = curl curl∗− div grad

Hodge decomposition holds for Ωk(X )• Ωk(X ) = im(dk−1)⊕ ker(∆k)⊕ im(δk)• dim(∆k) = βk(χ(X ))




Forgetful functors

Riemannian manifolds→ Metric spaces→ Cell complexes

From differentiable to combinatorial structures, Hodgedecomposition is functorial (invariant)

Topological invariants (homology) are preserved in suchcoarse-grained functors

Natural for data analysis, a connection between geometry andtopology: harmonic basis

More important than data itself, relations between data viafunctions, mappings, etc.



Applications of Hodge Decomposition

Boundary Value Problem (Schwarz, Chorin-Marsden’92)

Computer vision• Optical flow decomposition and regularization(Yuan-Schnorr-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)



Computer Vision

Optical Flow Decomposition and Regularization

Rudin-Osher-Fatemi’1992: piecewise constant flows

1

2‖v − u‖22 + TV (u), u, v ∈ R2

TV (u) :=

∫ √(grad u1)2 + (grad u2)2,

Yuan-Schnorr-Steidl’2007: piecewise harmonic flows

TV (u)→ R(u) =

∫ √(div u)2 + (curl u)2



Computer Vision

Example: periodic motions are harmonic

Figure: Better motion separation with Hodge decomposition



Computer Vision

Adelson’s iIlusion in Computer Vision

Figure: Adelson’s illusion: on the left the chess board is shadowed by acolumn such that the white square has the same illuminance intensity asthe black square, proved by the right picture.



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 fieldminX ‖ grad X − T (Y )‖1

Figure: Ma-Morel-Osher-Chien 2011



Statistical Ranking via Paired Comparison Method

Crowdsourcing QoE evaluation of Multimedia

Figure: (Xu-Huang-Y., et al. 11) Crowdsouring subjective Quality ofExperience evaluation




Learning relative attributes: age

2

2

3

1

2

Unintentional errorsIntentional errorsCorrect pairs

Ranking

scores:+10.6 -1.5

1

2

Figure: Age: a relative attribute estimated from paired comparisons(Fu-Y.-Xiang et al. 2013)




Collaborative Ideas Prioritization

10/18/13 CrowdRank | Your Ranking Engine with Real Consumer Reports - Consumers Report and Vote

www.crowdrank.net 1/3

Search

15.1 million votes cast

Insights Articles

Greatest AllTimeBasketball Player

Sexiest MAN Alive TV Brands Wireless Carriers

Sexiest Woman Alive Hotels MBA Best Dating Site

Colleges Airlines Beer Brewer Smartphone Brands

All Categories

CrowdRank

Read more

Last month, we shared an analysis of votes in our Sexiest Woman Alive category evaluating whether gentlemenprefer blondes. The overall answer was that globally men prefer brunettes but a slim 50.1% margin. But, theU.S. diverged from the global average and voters preferred blondes 50.9% of the time. The U.S. story gets moreinteresting, however, if we drill down to a state level. When we look at individual states, there is more parity: 21states show a preference for blondes, 18 prefer brunettes, and 7 prefer redheads. Meanwhile 4 states have noclear winner between blondes, brunettes, and redheads.

In the US, Do Gentlemen Prefer Blondes?

CrowdRank Insights

Brands Education Sports TV & Movies More

Nexus 7 from$229www.google.com/nexus

The 7" tablet fromGoogle with theworld's sharpestscreen. Buy now.

Flights fromChicago

The DepotRenaissanceMinneapolisHotelBeautifulChilean Girls

MBAMarketingDegree

Figure: Left: www.allourideas.org/wikipedia-banner-challenge, by Prof.Matt Salganik at Princeton; Right: www.crowdrank.net




Paired comparison data on graphs

Graph G = (V ,E )

V : alternatives to be ranked or rated

(iα, jα) ∈ E a pair of alternatives

yαij ∈ R degree of preference by rater α

ωαij ∈ R+ confidence weight of rater α

Examples: relative attributes, subjective QoE assessment,perception of illuminance intensity, sports, wine taste, etc.




Generalized Linear Models in Statistics: l2(E )

Majority voting (Condorcet’1785): inconsistency arises(Arrow’s impossibility theorem 1950s)

Statistical majority voting:• Yij = (

∑α ω

αij Y α

ij )/(∑

α ωαij ) = −Yji , ωij =

∑α ω

αij

Y from generalized linear models• Uniform model: Yij = 2πij − 1.

• Bradley-Terry model: Yij = logπij

1−πij .

• Thurstone-Mosteller model: Yij = Φ−1(πij).

Φ(x) =1√2π

∫ ∞−x/[2σ2(1−ρ)]1/2

e−12t2dt.

• Angular transform model: Yij = arcsin(2πij − 1).

Inner product induced on Y ∈ l2ω(E ), 〈u, v〉ω =∑

uijvijωij

where u, v skew-symmetric




Hodge Decomposition on Graphs [Jiang-Lim-Y.-Ye’11]

Paired comparison data Yij ∈ l2ω(E ) admits an orthogonaldecomposition,

Y = Y (g) + Y (h) + Y (c), (1)

whereY

(g)ij = βi − βj , for some θ ∈ RV , (2)

Y(h)ij + Y

(h)jk + Y

(h)ki = 0, for each i , j , k ∈ T , (3)∑

j∼iωij Y

(h)ij = 0, for each i ∈ V . (4)




Global ranking and Local vs. Global Inconsistencies

Y (g) = (δ0β)(i , j) := βi − βj where β solves

minβ∈R|V |

∑α,(i ,j)∈E

ωαij (βi − βj − Y αij )2 ⇔ ∆0β = δT0 Y

Residues Y (h) and Y (c) accounts for inconsistencies:

Y (c), the local inconsistency, triangular curls

• Y(c)ij + Y

(c)jk + Y

(c)ki 6= 0 , i , j , k ∈ T

Y (h), the global inconsistency, harmonic ranking• harmonic ranking leads to circular coordinates (MVJtutorial) on V ⇒ fixed tournament issue




Topological Constraints

To get a faithful ranking, two topological conditions on the cliquecomplex χ2

G = (V ,E ,T ) are important:

Connectivity: G is connected, then an unique global ranking ispossible;

Loop-free: harmonic ranking vanishes if χ2G is loop-free,

topology plays a role of obstruction of fixed-tournament• “Triangular arbitrage-free implies arbitrage-free”




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




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 underpermutations (relabeling)• Generalized de Finetti’s Theorem [Aldous 1983, Kallenberg2005]: 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• Erdos-Renyi: P(i , j) = P(edge) =

∫ 10

∫ 10 h(u, v)dudv =: p

• edge-independent process (Chung-Lu’06)




Phase Transitions of Large Random Graphs

For an Erdos-Renyi random graph G (n, p) with n vertices and eachedge independently emerging with probability p(n),

(Erdos-Renyi 1959) One phase-transition for β0• p << 1/n1+ε (∀ε > 0), almost always disconnected• p >> log(n)/n, almost always connected

(Kahle 2009) Two phase-transitions for βk (k ≥ 1)• p << n−1/k or p >> n−1/(k+1), almost always βk vanishes;• n−1/k << p << n−1/(k+1), almost always βk is nontrivial

For example: with n = 16, 75% distinct edges included in G , thenχG with high probability is connected and loop-free. In general,O(n log(n)) samples for connectivity and O(n3/2) for loop-free.




Other sampling models

Random k-regular graphs• Kim-Vu sandwich theorem/conjecture: coupling withErdos-Renyi 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




Robust Ranking with Sparse Outliers

For each (i , j) ∈ E ,

yαij = β0 + βi − βj + zαij (5)

where

βV : global ranking score on V

β0: head-advantage (home- in NBA, white- in chess)

zij errorzαij = γαij + εαij

• [A0a] γαij symmetric p-sparse (zero w.p. p and median 0)• [A0b] εαij ∼ N (0, σ2/wij)




Huber’s LASSO [Xiong-Cheng-Y.’13,Xu-Xiong-Huang-Y.’13]

Robust ranking can be formulated as a Huber’s LASSOproblem (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




Exact Recover against pure Sparse Outliers

Theorem (Xiong-Cheng-Y.’2013)

Let G (n, q) be an Erdos-Renyi Random Graph with n nodes andeach 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 solutionexactly recovers the global ranking β∗ if

p O

(√log n

nq

).

Note: no method can recover if

p O

(1√nq

).




Persistent Homology: online algorithm for topologytracking (e.g Edelsbrunner-Harer’08)

Figure: Persistent Homology Barcodes

vertice, edges, andtriangles etc.sequentially added

online update ofhomology

O(m) for surfaceembeddable complex;and O(m2.xx) ingeneral (m number ofsimplex)




Online HodgeRank as Stochastic Approximations

Robbins-Monro (1951) algorithm for Ax = b

xt+1 = xt − γt(Atxt − bt), E(At) = A, E(bt) = b

Now consider ∆0s = δ∗0Y , with new rating Yt(it+1, jt+1)

st+1(it+1) = st(it+1)− γt [st(it+1)− st(jt+1)− Yt(it+1, jt+1)]

st+1(jt+1) = st(jt+1) + γt [st(it+1)− st(jt+1)− Yt(it+1, jt+1)]

Note:

updates only occur locally on edge it+1, jt+1initial choice: s0 = 0 or any vector

∑i s0(i) = 0

step size• γt = a(t + b)−θ (θ ∈ (0, 1])• γt = const(T ), .e.g. 1/T where T is total sample size




Minimax Optimal Convergence Rates (Lim-Y.’13,Xu-Xiong-Huang-Y.’13)

Choose γt ∼ t−1/2 (e.g. a=1/λ1(∆0) and b large enough)

In this case, st converges to s∗ (population solution), withprobability 1− δ, in the (optimal) rate of t

‖st − s∗‖ ≤ O

(t−1/2 · κ3/2(∆0) · log1/2

1

δ

)Dependence on κ3/2 can be improved to κ by Ji Liu (UWisc-Madison) (optimal order of κ?)




Some reference

Random graph sampling models: Erdos-Renyi 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

l1-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



Game Theory

Strategic Simplicial Complex for Flow Games

O F

O 3, 2 0, 0

F 0, 0 2, 3

(a) Battle of the sexes

O F

O 4, 2 0, 0

F 1, 0 2, 3

(b) Modified battle ofthe sexes

It is easy to see that these two games have the same pairwise comparisons, which will lead toidentical equilibria for the two games: (O, O) and (F, F ). It is only the actual equilibrium payoffsthat would differ. In particular, in the equilibrium (O, O), the payoff of the row player is increasedby 1.

The usual solution concepts in games (e.g., Nash, mixed Nash, correlated equilibria) are definedin terms of pairwise comparisons only. Games with identical pairwise comparisons share the sameequilibrium sets. Thus, we refer to games with identical pairwise comparisons as strategicallyequivalent games.

By employing the notion of pairwise comparisons, we can concisely represent any strategic-formgame in terms of a flow in a graph. We recall this notion next. Let G = (N, L) be an undirectedgraph, with set of nodes N and set of links L. An edge flow (or just flow) on this graph is a functionY : N × N → R such that Y (p,q) = −Y (q,p) and Y (p,q) = 0 for (p,q) /∈ L [21, 2]. Note thatthe flow conservation equations are not enforced under this general definition.

Given a game G, we define a graph where each node corresponds to a strategy profile, andeach edge connects two comparable strategy profiles. This undirected graph is referred to as thegame graph and is denoted by G(G) (E, A), where E and A are the strategy profiles and pairsof comparable strategy profiles defined above, respectively. Notice that, by definition, the graphG(G) has the structure of a direct product of M cliques (one per player), with clique m havinghm vertices. The pairwise comparison function X : E × E → R defines a flow on G(G), as itsatisfies X(p,q) = −X(q,p) and X(p,q) = 0 for (p,q) /∈ A. This flow may thus serve as anequivalent representation of any game (up to a “non-strategic” component). It follows directlyfrom the statements above that two games are strategically equivalent if and only if they have thesame flow representation and game graph.

Two examples of game graph representations are given below.

Example 2.2. Consider again the “battle of the sexes” game from Example 2.1. The game graphhas four vertices, corresponding to the direct product of two 2-cliques, and is presented in Figure 2.

(O, O) (O, F )

(F, O) (F, F )

3 2

2

3

Figure 2: Flows on the game graph corresponding to “battle of the sexes” (Example 2.2).

Example 2.3. Consider a three-player game, where each player can choose between two strategiesa, b. We represent the strategic interactions among the players by the directed graph in Figure3a, where the payoff of player i is −1 if its strategy is identical to the strategy of its successor

7

O F

O 3, 2 0, 0

F 0, 0 2, 3

(a) Battle of the sexes

O F

O 4, 2 0, 0

F 1, 0 2, 3

(b) Modified battle ofthe sexes

It is easy to see that these two games have the same pairwise comparisons, which will lead toidentical equilibria for the two games: (O, O) and (F, F ). It is only the actual equilibrium payoffsthat would differ. In particular, in the equilibrium (O, O), the payoff of the row player is increasedby 1.

The usual solution concepts in games (e.g., Nash, mixed Nash, correlated equilibria) are definedin terms of pairwise comparisons only. Games with identical pairwise comparisons share the sameequilibrium sets. Thus, we refer to games with identical pairwise comparisons as strategicallyequivalent games.

By employing the notion of pairwise comparisons, we can concisely represent any strategic-formgame in terms of a flow in a graph. We recall this notion next. Let G = (N, L) be an undirectedgraph, with set of nodes N and set of links L. An edge flow (or just flow) on this graph is a functionY : N × N → R such that Y (p,q) = −Y (q,p) and Y (p,q) = 0 for (p,q) /∈ L [21, 2]. Note thatthe flow conservation equations are not enforced under this general definition.

Given a game G, we define a graph where each node corresponds to a strategy profile, andeach edge connects two comparable strategy profiles. This undirected graph is referred to as thegame graph and is denoted by G(G) (E, A), where E and A are the strategy profiles and pairsof comparable strategy profiles defined above, respectively. Notice that, by definition, the graphG(G) has the structure of a direct product of M cliques (one per player), with clique m havinghm vertices. The pairwise comparison function X : E × E → R defines a flow on G(G), as itsatisfies X(p,q) = −X(q,p) and X(p,q) = 0 for (p,q) /∈ A. This flow may thus serve as anequivalent representation of any game (up to a “non-strategic” component). It follows directlyfrom the statements above that two games are strategically equivalent if and only if they have thesame flow representation and game graph.

Two examples of game graph representations are given below.

Example 2.2. Consider again the “battle of the sexes” game from Example 2.1. The game graphhas four vertices, corresponding to the direct product of two 2-cliques, and is presented in Figure 2.

(O, O) (O, F )

(F, O) (F, F )

3 2

2

3

Figure 2: Flows on the game graph corresponding to “battle of the sexes” (Example 2.2).

Example 2.3. Consider a three-player game, where each player can choose between two strategiesa, b. We represent the strategic interactions among the players by the directed graph in Figure3a, where the payoff of player i is −1 if its strategy is identical to the strategy of its successor

7

n person game, each one has utility with strategy profileuk(s1, s2, . . . , sn);

Every strategy vector (s1, s2, . . . , sn) is a node

Comparable strategies connected by edge:(s−k , sk) = (s1, . . . , sk , . . . , sn) and(s−k , s

′k) = (s1, . . . , s

′k , . . . , sn)

Edge flow: uk(s−k , sk)− uk(s−k , s′k)



Game Theory

Hodge Decomposition of Finite Games

Note: Shapley-Monderer Condition ≡ Harmonic-free ≡quadrangular-curl free



Game Theory

Bimatrix Games

For bi-matrix game (A,B),

potential game is decided by ((A + A′)/2, (B + B ′)/2)

harmonic game is zero-sum ((A− A′)/2, (B − B ′)/2)

Computation of Nash Equilibrium:• each of them is tractable;• however direct sum is NP-hard;• approximate potential game leads to approximateNash-equilibrium;

a special case of Leontief Equilibrium for Exchange Market



Summary

Hodge Decomposition Theorem finds new applications in thefollowing fields

Statistical ranking: where every paired comparison data isdecomposed 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

Computer Vision: shade-removal, optical flow decompositionetc.

more are coming ...



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), YinyuYe (Stanford)Parrilo (MIT)


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