Recognizable Series on Hypergraphs el Bailly 1 cois Denis 2 Guillaume Rabusseau 2 Rapha¨ Fran¸ 1 Universit´ e de Technologie de Compi` egne 2 LIF, CNRS, Aix-Marseille Universit´ e LATA’2015 March 5, 2015 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 1 / 18
Outline Objective and Method 1 Graph Weighted Model 2 Main Results 3 Towards Learning GWMs 4 Conclusion 5 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 2 / 18
Objective and Method Grammatical Inference: estimate probability distributions on string/trees from samples → Lot of works rely on the notion of recognizable/rational series : ֒
Objective and Method Grammatical Inference: estimate probability distributions on string/trees from samples → Lot of works rely on the notion of recognizable/rational series : ֒ A string series r : Σ ∗ → R is recognizable ⇔ There exists a finite weighted automaton computing r Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 3 / 18
Objective and Method Grammatical Inference: estimate probability distributions on string/trees from samples → Lot of works rely on the notion of recognizable/rational series : ֒ A string series r : Σ ∗ → R is recognizable ⇔ There exists a finite weighted automaton computing r ⇔ r has a linear representation � ι ∈ R d , τ ∈ R d , { M σ ∈ R d × d } σ ∈ Σ � r ( w ) = ι ⊤ M w 1 M w 2 · · · M w n τ for all w ∈ Σ ∗ Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 3 / 18
Objective and Method Grammatical Inference: estimate probability distributions on string/trees from samples → Lot of works rely on the notion of recognizable/rational series : ֒ A string series r : Σ ∗ → R is recognizable ⇔ There exists a finite weighted automaton computing r ⇔ r has a linear representation � ι ∈ R d , τ ∈ R d , { M σ ∈ R d × d } σ ∈ Σ � r ( w ) = ι ⊤ M w 1 M w 2 · · · M w n τ for all w ∈ Σ ∗ Objective Extend the notion of recognizable series to graphs and hypergraphs. ֒ → by directly aiming for an algebraic characterization similar to linear representations of string/tree series. Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 3 / 18
Outline Objective and Method 1 Graph Weighted Model 2 Main Results 3 Towards Learning GWMs 4 Conclusion 5 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 4 / 18
Graphs A graph G = ( V , E , ℓ ) on a ranked alphabet F = (Σ , ♯ ) Vertices V , Labeling function ℓ : V → Σ, Set of ports P = { ( v , j ) : v ∈ V , 1 ≤ j ≤ ♯ℓ ( v ) } , Edges E ⊂ P × P (partition of P ) . 1 2 3 1 g f a 1 2 2 1 f Figure : A graph on the ranked alphabet F = { a ( · ) , f ( · , · ) , g ( · , · , · ) } . V = { 1 , 2 , 3 , 4 } , ℓ (1) = l (2) = f , ℓ (3) = g , ℓ (4) = a , � � E = { (1 , 1) , (3 , 2) } , { (1 , 2) , (2 , 1) } , { (2 , 2) , (3 , 1) } , { (3 , 3) , (4 , 1) } Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 5 / 18
Tensors Tensor T ∈ � k R d = R d ⊗ · · · ⊗ R d ≃ Multi-array ( T i 1 ... i k ) ∈ R d ×···× d . Let e 1 , . . . , e d be the canonical basis of V = R d , T can be expressed as � T = T i 1 ... i k e i 1 ⊗ · · · ⊗ e i k i 1 ,..., i k ∈ [ d ] k = 1: vector v i (1 ≤ i ≤ d ) k = 2: matrix M i 1 i 2 (1 ≤ i 1 , i 2 ≤ d ) k = 3: higher order tensor T i 1 i 2 i 3 (1 ≤ i 1 , i 2 , i 3 ≤ d ) Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 6 / 18
Graph Weighted Models (GWM) A graph G on the ranked alphabet F = { g ( · , · , · ) , f ( · , · ) , a ( · ) } : 1 2 3 1 g f a 1 2 2 1 f Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 7 / 18
Graph Weighted Models (GWM) A graph G on the ranked alphabet F = { g ( · , · , · ) , f ( · , · ) , a ( · ) } : 1 2 3 1 g f a 1 2 2 1 f Graph Weighted Model: � d , { T x ∈ � # x R d } x ∈F � . Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 7 / 18
Graph Weighted Models (GWM) A graph G on the ranked alphabet F = { g ( · , · , · ) , f ( · , · ) , a ( · ) } : 1 2 3 1 g f a 1 2 2 1 f Graph Weighted Model: � d , { T x ∈ � # x R d } x ∈F � . Computation of a GWM: 1 Tensor product of all tensors associated to vertices in G : i 3 i 4 T g T f i 1 i 2 T f i 5 i 6 i 7 T a i 8 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 7 / 18
Graph Weighted Models (GWM) A graph G on the ranked alphabet F = { g ( · , · , · ) , f ( · , · ) , a ( · ) } : 1 2 3 1 g f a 1 2 2 1 f Graph Weighted Model: � d , { T x ∈ � # x R d } x ∈F � . Computation of a GWM: 1 Tensor product of all tensors associated to vertices in G : i 3 i 4 T g T f i 1 i 2 T f i 5 i 6 i 7 T a i 8 2 Contractions directed by the edges of G : � i 3 i 4 T g T f i 1 i 2 T f i 5 i 1 i 7 T a i 8 i 1 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 7 / 18
Graph Weighted Models (GWM) A graph G on the ranked alphabet F = { g ( · , · , · ) , f ( · , · ) , a ( · ) } : 1 2 3 1 g f a 1 2 2 1 f Graph Weighted Model: � d , { T x ∈ � # x R d } x ∈F � . Computation of a GWM: 1 Tensor product of all tensors associated to vertices in G : i 3 i 4 T g T f i 1 i 2 T f i 5 i 6 i 7 T a i 8 2 Contractions directed by the edges of G : � i 2 i 4 T g T f i 1 i 2 T f i 5 i 1 i 7 T a i 8 i 1 i 2 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 7 / 18
Graph Weighted Models (GWM) A graph G on the ranked alphabet F = { g ( · , · , · ) , f ( · , · ) , a ( · ) } : 1 2 3 1 g f a 1 2 2 1 f Graph Weighted Model: � d , { T x ∈ � # x R d } x ∈F � . Computation of a GWM: 1 Tensor product of all tensors associated to vertices in G : i 3 i 4 T g T f i 1 i 2 T f i 5 i 6 i 7 T a i 8 2 Contractions directed by the edges of G : � i 2 i 4 T g T f i 1 i 2 T f i 4 i 1 i 7 T a i 8 i 1 i 2 i 4 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 7 / 18
Graph Weighted Models (GWM) A graph G on the ranked alphabet F = { g ( · , · , · ) , f ( · , · ) , a ( · ) } : 1 2 3 1 g f a 1 2 2 1 f Graph Weighted Model: � d , { T x ∈ � # x R d } x ∈F � . Computation of a GWM: 1 Tensor product of all tensors associated to vertices in G : i 3 i 4 T g T f i 1 i 2 T f i 5 i 6 i 7 T a i 8 2 Contractions directed by the edges of G : � i 2 i 4 T g T f i 1 i 2 T f i 4 i 1 i 7 T a i 7 i 1 i 2 i 4 i 7 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 7 / 18
Graph Weighted Models (GWM) A graph G on the ranked alphabet F = { g ( · , · , · ) , f ( · , · ) , a ( · ) } : 1 2 3 1 g f a 1 2 2 1 f Graph Weighted Model: � d , { T x ∈ � # x R d } x ∈F � . Computation of a GWM: 1 Tensor product of all tensors associated to vertices in G : i 3 i 4 T g T f i 1 i 2 T f i 5 i 6 i 7 T a i 8 2 Contractions directed by the edges of G : � i 2 i 4 T g T f i 1 i 2 T f i 4 i 1 i 7 T a r ( G ) = i 7 i 1 i 2 i 4 i 7 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 7 / 18
GWM: Examples F = { ι ( · ) , τ ( · ) , a ( · , · ) , b ( · , · ) , c ( · , · ) } , GWM { ι , M a , M b , M c , τ } 1 1 2 1 2 1 2 1 G = ι a c τ b Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 8 / 18
GWM: Examples F = { ι ( · ) , τ ( · ) , a ( · , · ) , b ( · , · ) , c ( · , · ) } , GWM { ι , M a , M b , M c , τ } 1 1 2 1 2 1 2 1 G = ι a c τ b 1 ι i 1 M a i 2 i 3 M b i 4 i 5 M c i 6 i 7 τ i 8 � 2 r ( G ) = ι i 1 M a i 1 i 3 M b i 3 i 5 M c i 5 i 7 τ i 7 = ι ⊤ M a M b M c τ i 1 i 3 i 5 i 7 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 8 / 18
GWM: Examples F = { ι ( · ) , τ ( · ) , a ( · , · ) , b ( · , · ) , c ( · , · ) } , GWM { ι , M a , M b , M c , τ } 1 1 2 1 2 1 2 1 G = ι a c τ b 1 ι i 1 M a i 2 i 3 M b i 4 i 5 M c i 6 i 7 τ i 8 � 2 r ( G ) = ι i 1 M a i 1 i 3 M b i 3 i 5 M c i 5 i 7 τ i 7 = ι ⊤ M a M b M c τ i 1 i 3 i 5 i 7 c 2 1 F = { a ( · , · ) , b ( · , · ) , c ( · , · ) } , G = 1 2 2 1 a b Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 8 / 18
GWM: Examples F = { ι ( · ) , τ ( · ) , a ( · , · ) , b ( · , · ) , c ( · , · ) } , GWM { ι , M a , M b , M c , τ } 1 1 2 1 2 1 2 1 G = ι a c τ b 1 ι i 1 M a i 2 i 3 M b i 4 i 5 M c i 6 i 7 τ i 8 � 2 r ( G ) = ι i 1 M a i 1 i 3 M b i 3 i 5 M c i 5 i 7 τ i 7 = ι ⊤ M a M b M c τ i 1 i 3 i 5 i 7 c 2 1 F = { a ( · , · ) , b ( · , · ) , c ( · , · ) } , G = 1 2 2 1 a b 1 M a i 1 i 2 M b i 3 i 4 M c i 5 i 6 � 2 r ( G ) = M a i 6 i 2 M b i 2 i 4 M c i 4 i 6 = Tr ( M a M b M c ) i 2 i 4 i 6 Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 8 / 18
Recognizable graph series A series r : G F → R is recognizable iff it can be computed by a GWM. Bailly, Denis, Rabusseau (UTC - LIF) Recognizable Series on Hypergraphs March 5, 2015 9 / 18
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