Motivation Learning QAPs Algebra of S n Algorithm Experiments Incorporating Domain Knowledge in Matching Problems via Harmonic Analysis Deepti Pachauri (joint work with Maxwell Collins, Risi Kondor, Vikas Singh) University of Wisconsin-Madison University of Chicago International Conference on Machine Learning 2012
Motivation Learning QAPs Algebra of S n Algorithm Experiments Matching Problems are Ubiquitous Photo Tourism
Motivation Learning QAPs Algebra of S n Algorithm Experiments Matching Problems are Ubiquitous Shape Matching
Motivation Learning QAPs Algebra of S n Algorithm Experiments Matching Problems are Ubiquitous Shape Matching General Strategy Write the functional form of the matching problem and then use an appropriate optimization engine to find a solution.
Motivation Learning QAPs Algebra of S n Algorithm Experiments Matching Problems are Ubiquitous Shape Matching General Strategy Write the functional form of the matching problem and then use an appropriate optimization engine to find a solution. Use past knowledge to make future instances easier . . . ?
Motivation Learning QAPs Algebra of S n Algorithm Experiments Overview Motivation Problem Setup Graph Matching and QAPs Why learn QAPs? Algebraic Structure of S n and Harmonic Analysis Learning in Fourier Space Evaluations
Motivation Learning QAPs Algebra of S n Algorithm Experiments
Motivation Learning QAPs Algebra of S n Algorithm Experiments 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 G ′ = G = 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0
Motivation Learning QAPs Algebra of S n Algorithm Experiments 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 G ′ = G = 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 Solution of matching problem is a permutation matrix y 0 0 0 0 1 1 0 0 0 0 y = σ := ( 51342 ) 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 such that yGy ⊤ = G ′
Motivation Learning QAPs Algebra of S n Algorithm Experiments Quadratic Assignment Problem (QAP) � � y ∗ = arg max c ii ′ y ii ′ + d ii ′ jj ′ y ii ′ y jj ′ y ii ′ ii ′ jj ′
Motivation Learning QAPs Algebra of S n Algorithm Experiments Quadratic Assignment Problem (QAP) � � y ∗ = arg max c ii ′ y ii ′ + d ii ′ jj ′ y ii ′ y jj ′ y ii ′ ii ′ jj ′ Computationally expensive: n ≥ 40 infeasible in general.
Motivation Learning QAPs Algebra of S n Algorithm Experiments Supervised Learning Given Training data : (( x 1 , y 1 ) , ..., ( x m , y m )) f ω ( x i ) ≈ y i ( x 1 , y 1 ) : f ω ( x 1 ) ≈ y 1 ( x 2 , y 2 ) : f ω ( x 2 ) ≈ y 2 ( x 3 , y 3 ) : f ω ( x 3 ) ≈ y 3 and so on ....... . . . and we will solve arg max f ω ( x 3 ) cheaply .
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning for QAPs? Given Training data : (( x 1 , σ 1 ) , ..., ( x m , σ m )) arg max f ω ( x i ) ≈ σ i ( x 1 , σ 1 ) : arg max f ω ( x 1 ) ≈ σ 1 ( x 2 , σ 2 ) : arg max f ω ( x 2 ) ≈ σ 2 ( x 3 , σ 3 ) : arg max f ω ( x 3 ) ≈ σ 3 and so on .......
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning for QAPs? Given Training data : (( x 1 , σ 1 ) , ..., ( x m , σ m )) arg max f ω ( x i ) ≈ σ i ( x 1 , σ 1 ) : arg max f ω ( x 1 ) ≈ σ 1 ( x 2 , σ 2 ) : arg max f ω ( x 2 ) ≈ σ 2 ( x 3 , σ 3 ) : arg max f ω ( x 3 ) ≈ σ 3 and so on ....... . . . and we want to solve arg max f ω ( x i ) cheaply .
Motivation Learning QAPs Algebra of S n Algorithm Experiments Inspired in part by Caetano et al., PAMI 2009 Structure learning approach to find most violated constraints using linear assignment. Xu et al., JMLR 2009 Use disciminative learning to acquire a domain–specific heuristic for controlling beam–search. Stobbe et al., AISTATS 2012 Fourier space sparsity to recover a set function from very few samples.
Motivation Learning QAPs Algebra of S n Algorithm Experiments Structure of σ ∈ S n Harmonic Analysis Fourier transform of a function f : R �→ C � ˆ f ( x ) e 2 π ix λ f ( λ ) = λ ∈ R , x ∈ R
Motivation Learning QAPs Algebra of S n Algorithm Experiments Structure of σ ∈ S n Harmonic Analysis on Symmetric Groups S n � ˆ f ( ρ λ ) = f ( σ ) ρ λ ( σ ) ρ λ ∈ R σ ∈ S n λ is the integer partition of n , λ ⊢ n ρ λ ( σ ) is the irreducible representation of S n ρ 1 , 1 · · ρ 1 , d λ ρ λ ( σ ) = · · · · ρ d λ , 1 · · ·
Motivation Learning QAPs Algebra of S n Algorithm Experiments Properties S n Convolution ( f ∗ g )( σ ) = � f ∗ g ( λ ) = ˆ � τ ∈ S n f ( στ − 1 ) g ( τ ) f ( λ )ˆ g ( λ ) Correlation ( f ⋆ g )( σ ) = � τ ∈ S n f ( στ ) g ( τ ) ∗ f ⋆ g ( λ ) = ˆ � g ( λ ) † f ( λ )ˆ S n − 1 is a subgroup of S n
Motivation Learning QAPs Algebra of S n Algorithm Experiments Properties S n Convolution ( f ∗ g )( σ ) = � f ∗ g ( λ ) = ˆ � τ ∈ S n f ( στ − 1 ) g ( τ ) f ( λ )ˆ g ( λ ) Correlation ( f ⋆ g )( σ ) = � τ ∈ S n f ( στ ) g ( τ ) ∗ f ⋆ g ( λ ) = ˆ � g ( λ ) † f ( λ )ˆ S n − 1 is a subgroup of S n The set σ S n − 1 is called a left coset of σ Two left (right) cosets are either disjoint or the same
Motivation Learning QAPs Algebra of S n Algorithm Experiments Coset Tree Cosets provide a partition of S n : S 3 S 2 S 2 S 2 231 321 132 312 123 213
Motivation Learning QAPs Algebra of S n Algorithm Experiments f : S n → C Graph function of G f A ( σ ) = A σ ( n ) ,σ ( n − 1 ) Properties: S n − 2 -invariant function on adjacency matrix A (Kondor, 2010) Band-limited in Fourier domain (Rockmore, 2002) Under relabeling, f A π = f π A
Motivation Learning QAPs Algebra of S n Algorithm Experiments Graph Matching Problem Standard QAP: Given a pair of graphs � n A i , j A ′ σ ∈ S n f ( σ ) = max σ ( i ) ,σ ( j ) i , j = 1 Graph Correlation: � 1 f ( σ ) = f A ( σπ ) f A ′ ( π ) ( n − 2 )! π ∈ S n ( A , A ′ ) could be weighted or unweighted adjacency matrices.
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning Graph Matching Given: A training set of related graph pairs with D encodings of adjacency matrices : ( G m , G ′ m ) , m = { 1 , · · · , M } . Goal: “Learn” parameters ω such that QAP procedure finds a good solution ( quickly ) for the test case (unseen graph pairs).
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning Graph Matching Given: A training set of related graph pairs with D encodings of adjacency matrices : ( G m , G ′ m ) , m = { 1 , · · · , M } . Goal: “Learn” parameters ω such that QAP procedure finds a good solution ( quickly ) for the test case (unseen graph pairs). Define parameter vector ω ∈ R D
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning Graph Matching Given: A training set of related graph pairs with D encodings of adjacency matrices : ( G m , G ′ m ) , m = { 1 , · · · , M } . Goal: “Learn” parameters ω such that QAP procedure finds a good solution ( quickly ) for the test case (unseen graph pairs). Define parameter vector ω ∈ R D � D ω d f d ( σ ) QAP Objective for Learning: f ω ( σ ) = d = 1 � � A d ij A ′ d where f d ( σ ) = 1 f A d ( σπ ) f A ′ d ( π ) = ( n − 2 )! σ ( i ) σ ( j ) π ∈ S n i , j
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning Correct bounds on Coset Tree S 3 S 2 S 2 S 2 231 321 132 312 123 213
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning Correct bounds on Coset Tree S 3 S 2 S 2 S 2 231 321 132 312 123 213
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning Correct bounds on Coset Tree
Motivation Learning QAPs Algebra of S n Algorithm Experiments Learning Correct bounds on Coset Tree
Motivation Learning QAPs Algebra of S n Algorithm Experiments Fourier Domain QAP Solver Fast Fourier Transform � n � d λ ˆ ˆ f ω ( λ ) = f ω ρ λ ([[ i , n ]]) i ( µ ) nd µ i = 1 µ ∈ λ ↓ n − 1 Fourier Space Bounds [Kondor et.al.] � � ˆ f ω B n → i = i ( µ ) � ∗ µ ⊢ n − 1
Motivation Learning QAPs Algebra of S n Algorithm Experiments Risk Minimization Loss Function � � + � n � � ˆ i ( µ ) � ∗ − � ˆ f ω f ω n − k ( µ ) � ∗ + 1 i ∗ k = 1 i ∈ children (( n − k + 1 ) ∗ ) i ∗ n − k is the correct node at level n − k in coset tree.
Motivation Learning QAPs Algebra of S n Algorithm Experiments Risk Minimization Jensen’s Inequality i ( µ ) = � D For parameterization: ˆ d = 1 ω d ˆ f ω f d i ( µ ) � D � D � ˆ ω d ˆ ω d � ˆ f ω f d f d i ( µ ) � ∗ = � i ( µ ) � ∗ ≤ i ( µ ) � ∗ d = 1 d = 1
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