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Reconstruction and Clustering with Graph optimization and Priors on Gene Networks and Images Aurlie Pirayre Frdrique BIDARD-MICHELOT IFP Energies nouvelles Camille COUPRIE Facebook A.I. Research PhD supervisors: Laurent DUVAL IFP

  1. I NTRODUCTION Our BRANE strategy BRANE : Biologically Related A priori Network Enhancement Extend classical thresholding Integrate biological priors into the functional to be optimized Enforce modular networks Additional knowledge: Transcription factors (TFs): regulators Non transcription factors (TFs): targets Method a priori Formulation Algorithm BRANE Cut Gene co-regulatiton Discrete Maximal flow Inference BRANE Relax TF-connectivity Continuous Proximal method Joint inference BRANE Clust Gene grouping Mixed Alternating scheme and clustering July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 9 / 45

  2. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A discrete method: BRANE Cut We look for a discrete solution for x ⇔ x ∈ { 0 , 1 } E July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 10 / 45

  3. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori A priori : modular structure and gene co-regulation � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 11 / 45

  4. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori A priori : modular structure and gene co-regulation � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E Modular network: favors links between TFs and TFs  ∈ T 2 2 η if ( i , j ) /   if ( i , j ) ∈ T 2 λ i , j = 2 λ TF   λ TF + λ TF otherwise. July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 11 / 45

  5. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori A priori : modular structure and gene co-regulation � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E Modular network: favors links between TFs and TFs  ∈ T 2 2 η if ( i , j ) /   if ( i , j ) ∈ T 2 λ i , j = 2 λ TF   λ TF + λ TF otherwise. with: T : the set of TF indices η > max { ω i , j | ( i , j ) ∈ V 2 } λ TF > λ TF A linear relation is sufficient: λ TF = βλ TF with β = |V| |T | July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 11 / 45

  6. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori A priori : modular structure and gene co-regulation � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 12 / 45

  7. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori A priori : modular structure and gene co-regulation � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E Gene co-regulation: favors edge coupling � Ψ( x i , j ) = ρ i , j , j ′ | x i , j − x i , j ′ | ( j , j ′ ) ∈ T 2 i ∈ V \ T ρ i , j , j ′ : co-regulation probability July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 12 / 45

  8. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori A priori : modular structure and gene co-regulation � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E Gene co-regulation: favors edge coupling � Ψ( x i , j ) = ρ i , j , j ′ | x i , j − x i , j ′ | ( j , j ′ ) ∈ T 2 i ∈ V \ T ρ i , j , j ′ : co-regulation probability with � 1 ( min { ω j , j ′ , ω j , k , ω j ′ , k } > γ ) k ∈V\ ( T ∪{ i } ) ρ i , j , j ′ = |V\T |− 1 γ : the ( |V| − 1 ) th of the normalized weights ω July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 12 / 45

  9. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  10. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  11. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i s t July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  12. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i x 1 , 2 x 1 , 3 x 1 , 4 s t x 2 , 3 x 2 . 4 x 3 , 4 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  13. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i x 1 , 2 x 1 , 3 8 5 x 1 , 4 5 s 10 t x 2 , 3 5 1 x 2 . 4 x 3 , 4 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  14. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i x 1 , 2 x 1 , 3 v 1 8 5 x 1 , 4 v 2 5 s 10 t x 2 , 3 v 3 5 1 x 2 . 4 v 4 x 3 , 4 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  15. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i x 1 , 2 x 1 , 3 v 1 8 ∞ 5 x 1 , 4 v 2 5 ∞ s 10 ∞ t x 2 , 3 v 3 5 ∞ 1 x 2 . 4 v 4 x 3 , 4 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  16. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i η = 6 λ TF = 3 λ TF = 1 x 1 , 2 x 1 , 3 v 1 8 ∞ 5 x 1 , 4 v 2 5 ∞ s 10 ∞ t x 2 , 3 v 3 5 ∞ 1 x 2 . 4 v 4 x 3 , 4 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  17. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i η = 6 λ TF = 3 λ TF = 1 x 1 , 2 x 1 , 3 v 1 8 ∞ 5 x 1 , 4 v 2 5 ∞ s 10 ∞ t x 2 , 3 v 3 5 ∞ 1 x 2 . 4 v 4 3 x 3 , 4 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  18. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i η = 6 λ TF = 3 λ TF = 1 x 1 , 2 x 1 , 3 v 1 8 ∞ 5 x 1 , 4 v 2 5 ∞ s 10 ∞ t x 2 , 3 v 3 5 ∞ 1 x 2 . 4 v 4 3 x 3 , 4 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  19. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i η = 6 λ TF = 3 λ TF = 1 x 1 , 2 x 1 , 3 v 1 8 ∞ 5 x 1 , 4 v 2 ∞ 5 s = 1 t = 0 10 ∞ x 2 , 3 v 3 ∞ 5 1 x 2 . 4 v 4 3 x 3 , 4 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  20. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i η = 6 λ TF = 3 λ TF = 1 1 v 1 8 1 ∞ 5 v 2 5 0 ∞ s = 1 t = 0 ∞ 10 v 3 1 ∞ 5 1 v 4 0 3 0 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  21. BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution A maximal flow for a minimum cut formulation � � ω i , j | x i , j − 1 | + λ i , j x i , j + ρ i , j , j ′ | x i , j − x i , j ′ | minimize x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T ( j , j ′ ) ∈ T 2 , j ′ > j j > i η = 6 λ TF = 3 λ TF = 1 1 v 1 8 1 ∞ 5 v 2 5 0 ∞ s = 1 t = 0 ∞ 10 v 3 1 ∞ 5 1 v 4 0 3 0 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 13 / 45

  22. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM A continuous method: BRANE Relax We look for a continuous solution for x ⇔ x ∈ [ 0 , 1 ] E July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 14 / 45

  23. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM A priori A priori : modular structure and TF connectivity � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 15 / 45

  24. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM A priori A priori : modular structure and TF connectivity � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E Modular network: favors links between TFs and TFs  ∈ T 2 2 η if ( i , j ) /   if ( i , j ) ∈ T 2 λ i , j = 2 λ TF   λ TF + λ TF otherwise. July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 15 / 45

  25. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM A priori A priori : modular structure and TF connectivity � ( i , j ) ∈ V 2 ω i , j ϕ ( x i , j − 1 ) + λ i , j ϕ ( x i , j ) + µ Ψ( x i , j ) minimize x ∈{ 0 , 1 } E Modular network: favors links between TFs and TFs  ∈ T 2 2 η if ( i , j ) /   if ( i , j ) ∈ T 2 λ i , j = 2 λ TF   λ TF + λ TF otherwise. TF connectivity: constraint TF node degree   � � Ψ( x i , j ) = φ x i , j − d  j ∈ V i ∈ V \ T φ ( · ) : a convex distance function with β -Lipschitz continuous gradient July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 15 / 45

  26. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Formulation A convex relaxation for a continuous formulation � � � � � ω i , j ( 1 − x i , j ) + λ i , j x i , j + µ x i , j − d minimize φ x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T j ∈ V j > i July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 16 / 45

  27. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Formulation A convex relaxation for a continuous formulation � � � � � ω i , j ( 1 − x i , j ) + λ i , j x i , j + µ x i , j − d minimize φ x ∈{ 0 , 1 } E ( i , j ) ∈ V 2 i ∈ V \ T j ∈ V j > i Relaxation and vectorization: � E � E P � � � ω l ( 1 − x l ) + λ l x l + µ φ Ω i , k x k − d , minimize x ∈ [ 0 , 1 ] E l = 1 i = 1 k = 1 where Ω ∈ { 0 , 1 } P × E encodes the degree of the P TFs nodes in the complete graph. � 1 if j is the index of an edge linking the TF node v i in the complete graph, Ω i , j = 0 otherwise. July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 16 / 45

  28. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Formulation Distance function in BRANE Relax � E � E P � � � ω l ( 1 − x l ) + λ l x l + µ φ Ω i , k x k − d minimize x ∈ [ 0 , 1 ] E l = 1 i = 1 k = 1 Choice of φ : node degree distance function, with respect to d E � z i = Ω i , k x k − d k = 1 squared ℓ 2 norm: φ ( z ) = || z || 2 � z 2 if | z i | ≤ δ i Huber function: φ ( z i ) = 2 δ ( | z i | − 1 2 δ ) otherwise July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 17 / 45

  29. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Resolution Optimization strategy via proximal methods Splitting ω ⊤ ( 1 E − x ) + λ ⊤ x + µ Φ( Ω x − d ) + ι [ 0 , 1 ] E ( x ) minimize x ∈ R E � �� � � �� � f 2 f 1 f 1 ∈ Γ 0 ( R E ) : proper, convex, and lower semi-continuous f 2 : convex, differentiable with an L − Lipschitz continuous gradient July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 18 / 45

  30. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Resolution Optimization strategy via proximal methods Splitting ω ⊤ ( 1 E − x ) + λ ⊤ x + µ Φ( Ω x − d ) + ι [ 0 , 1 ] E ( x ) minimize x ∈ R E � �� � � �� � f 2 f 1 f 1 ∈ Γ 0 ( R E ) : proper, convex, and lower semi-continuous f 2 : convex, differentiable with an L − Lipschitz continuous gradient Algorithm 1: Forward-Backward Fix x 0 ∈ R E for k = 0 , 1 , . . . do Select the index j k ∈ { 1 , . . . , J } of a block of variables z ( j k ) = x ( j k ) − γ k A − 1 j k ∇ j k f 2 ( x k ) k k x ( j k ) ( z ( j k ) k + 1 = prox γ k − 1 , A jk f ) ( jk ) k 1 x (¯ k + 1 = x (¯ j k ) j k ) ¯ j k = { 1 , . . . , J }\{ j k } , k July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 18 / 45

  31. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Resolution Optimization strategy via proximal methods Splitting ω ⊤ ( 1 E − x ) + λ ⊤ x + µ Φ( Ω x − d ) + ι [ 0 , 1 ] E ( x ) minimize x ∈ R E � �� � � �� � f 2 f 1 f 1 ∈ Γ 0 ( R E ) : proper, convex, and lower semi-continuous f 2 : convex, differentiable with an L − Lipschitz continuous gradient Algorithm 2: Preconditioned Forward-Backward Fix x 0 ∈ R E for k = 0 , 1 , . . . do Select the index j k ∈ { 1 , . . . , J } of a block of variables z ( j k ) = x ( j k ) − γ k A − 1 j k ∇ j k f 2 ( x k ) k k x ( j k ) ( z ( j k ) k + 1 = prox γ k − 1 , A jk , f ) ( jk ) k 1 x (¯ k + 1 = x (¯ j k ) j k ) ¯ j k = { 1 , . . . , J }\{ j k } , k July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 18 / 45

  32. BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Resolution Optimization strategy via proximal methods Splitting ω ⊤ ( 1 E − x ) + λ ⊤ x + µ Φ( Ω x − d ) + ι [ 0 , 1 ] E ( x ) minimize x ∈ R E � �� � � �� � f 2 f 1 f 1 ∈ Γ 0 ( R E ) : proper, convex, and lower semi-continuous f 2 : convex, differentiable with an L − Lipschitz continuous gradient Algorithm 3: Block Coordinate + Preconditioned Forward-Backward Fix x 0 ∈ R E for k = 0 , 1 , . . . do Select the index j k ∈ { 1 , . . . , J } of a block of variables z ( j k ) = x ( j k ) − γ k A − 1 j k ∇ j k f 2 ( x k ) k k x ( j k ) ( z ( j k ) k + 1 = prox γ k − 1 , A jk , f ) ( jk ) k 1 x (¯ k + 1 = x (¯ j k ) j k ) ¯ j k = { 1 , . . . , J }\{ j k } , k July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 18 / 45

  33. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING A mixed method: BRANE Clust We look for a discrete solution for x and a continuous one for y July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 19 / 45

  34. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING A priori A priori : gene grouping and modular structure � ( i , j ) ∈ V 2 f ( y i , y j ) ω i , j x i , j + λ ( 1 − x i , j ) + Ψ( y i ) maximize x ∈{ 0 , 1 } E y ∈ N G July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 20 / 45

  35. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING A priori A priori : gene grouping and modular structure � ( i , j ) ∈ V 2 f ( y i , y j ) ω i , j x i , j + λ ( 1 − x i , j ) + Ψ( y i ) maximize x ∈{ 0 , 1 } E y ∈ N G Clustering-assisted inference Node labeling y ∈ N G Weight ω i , j reduction if nodes v i and v j belong to distinct clusters Cost function: f ( y i , y j ) = β − 1 ( y i � = y j ) β July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 20 / 45

  36. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING A priori A priori : gene grouping and modular structure � ( i , j ) ∈ V 2 f ( y i , y j ) ω i , j x i , j + λ ( 1 − x i , j ) + Ψ( y i ) maximize x ∈{ 0 , 1 } E y ∈ N G Clustering-assisted inference Node labeling y ∈ N G Weight ω i , j reduction if nodes v i and v j belong to distinct clusters Cost function: f ( y i , y j ) = β − 1 ( y i � = y j ) β TF-driven clustering promoting modular structure � Ψ( y i ) = µ i , j 1 ( y i = j ) i ∈ V j ∈ T µ i , j : modular structure controlling parameter July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 20 / 45

  37. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Alternating optimization strategy � β − 1 ( y i � = y j ) � ω i , j x i , j + λ ( 1 − x i , j ) + maximize µ i , j 1 ( y i = j ) β x ∈{ 0 , 1 } n ( i , j ) ∈ V 2 i ∈ V y ∈ N G j ∈ T July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 21 / 45

  38. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Alternating optimization strategy Alternating optimization � β − 1 ( y i � = y j ) � ω i , j x i , j + λ ( 1 − x i , j ) + maximize µ i , j 1 ( y i = j ) β x ∈{ 0 , 1 } n ( i , j ) ∈ V 2 i ∈ V y ∈ N G j ∈ T July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 21 / 45

  39. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Alternating optimization strategy Alternating optimization � β − 1 ( y i � = y j ) � ω i , j x i , j + λ ( 1 − x i , j ) + maximize µ i , j 1 ( y i = j ) β x ∈{ 0 , 1 } n ( i , j ) ∈ V 2 i ∈ V y ∈ N G j ∈ T At y fixed and x variable: β − 1 ( y i � = y j ) � maximize ω i , j x i , j + λ ( 1 − x i , j ) β x ∈{ 0 , 1 } n ( i , j ) ∈ V 2 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 21 / 45

  40. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Alternating optimization strategy Alternating optimization � β − 1 ( y i � = y j ) � ω i , j x i , j + λ ( 1 − x i , j ) + maximize µ i , j 1 ( y i = j ) β x ∈{ 0 , 1 } n ( i , j ) ∈ V 2 i ∈ V y ∈ N G j ∈ T At y fixed and x variable: β − 1 ( y i � = y j ) � maximize ω i , j x i , j + λ ( 1 − x i , j ) β x ∈{ 0 , 1 } n ( i , j ) ∈ V 2 At x fixed and y variable: ω i , j x i , j � � 1 ( y i � = y j ) + µ i , j 1 ( y i � = j ) minimize β y ∈ N G i ∈ V , j ∈ T ( i , j ) ∈ V 2 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 21 / 45

  41. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Alternating optimization strategy Alternating optimization � β − 1 ( y i � = y j ) � ω i , j x i , j + λ ( 1 − x i , j ) + maximize µ i , j 1 ( y i = j ) β x ∈{ 0 , 1 } n ( i , j ) ∈ V 2 i ∈ V y ∈ N G j ∈ T At y fixed and x variable: β − 1 ( y i � = y j ) � maximize ω i , j x i , j + λ ( 1 − x i , j ) β x ∈{ 0 , 1 } n ( i , j ) ∈ V 2 � λβ if ω i , j > 1 β − 1 ( y i � = y j ) Explicit form: x ∗ i , j = 0 otherwise. At x fixed and y variable: ω i , j x i , j � � 1 ( y i � = y j ) + µ i , j 1 ( y i � = j ) minimize β y ∈ N G i ∈ V , j ∈ T ( i , j ) ∈ V 2 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 21 / 45

  42. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Clustering optimization strategy At x fixed and y variable: ω i , j x i , j � � 1 ( y i � = y j ) + µ i , j 1 ( y i � = j ) minimize β y ∈ N G i ∈ V , j ∈ T ( i , j ) ∈ V 2 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 22 / 45

  43. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Clustering optimization strategy At x fixed and y variable: ω i , j x i , j � � 1 ( y i � = y j ) + µ i , j 1 ( y i � = j ) minimize ( NP ) β y ∈ N G i ∈ V , j ∈ T ( i , j ) ∈ V 2 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 22 / 45

  44. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Clustering optimization strategy At x fixed and y variable: ω i , j x i , j � � 1 ( y i � = y j ) + µ i , j 1 ( y i � = j ) minimize ( NP ) β y ∈ N G i ∈ V , j ∈ T ( i , j ) ∈ V 2 discrete problem ⇒ quadratic relaxation T -class problem ⇒ T binary sub-problems label restriction to T : { s ( 1 ) , . . . , s ( T ) } such that s ( t ) = 1 if j = t and 0 otherwise. j Y = { y ( 1 ) , . . . , y ( T ) } such that y ( t ) ∈ [ 0 , 1 ] G July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 22 / 45

  45. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Clustering optimization strategy At x fixed and y variable: ω i , j x i , j � � 1 ( y i � = y j ) + µ i , j 1 ( y i � = j ) minimize ( NP ) β y ∈ N G i ∈ V , j ∈ T ( i , j ) ∈ V 2 discrete problem ⇒ quadratic relaxation T -class problem ⇒ T binary sub-problems label restriction to T : { s ( 1 ) , . . . , s ( T ) } such that s ( t ) = 1 if j = t and 0 otherwise. j Y = { y ( 1 ) , . . . , y ( T ) } such that y ( t ) ∈ [ 0 , 1 ] G Problem re-expressed as:   T ω i , j x i , j � � 2 � � 2 �  � � y ( t ) − y ( t ) y ( t ) − s ( t ) + µ i , j minimize  i j i j β Y t = 1 ( i , j ) ∈ V 2 i ∈ V , j ∈ T July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 22 / 45

  46. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Clustering optimization strategy   T � � 2 � � 2 ω i , j x i , j �  � � y ( t ) − y ( t ) y ( t ) − s ( t ) + µ i , j minimize  i j i j β Y t = 1 ( i , j ) ∈ V 2 i ∈ V , j ∈ T This is the Combinatorial Dirichlet problem Minimization via solving a linear system of equations [Grady, 2006] July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 23 / 45

  47. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Clustering optimization strategy   T � � 2 � � 2 ω i , j x i , j �  � � y ( t ) − y ( t ) y ( t ) − s ( t ) + µ i , j minimize  i j i j β Y t = 1 ( i , j ) ∈ V 2 i ∈ V , j ∈ T This is the Combinatorial Dirichlet problem Minimization via solving a linear system of equations [Grady, 2006] Final labeling: node i is assigned to label t for which y ( t ) is maximal i y ( t ) y ∗ i = argmax i t ∈ T July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 23 / 45

  48. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Random walker in graphs y 1 6 12 10 3 We want to obtain the optimal labeling y ∗ based on y 5 y 4 5 a weighted graph ⇒ Random Walker algorithm 10 7 3 5 y 3 y 2 9 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 24 / 45

  49. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Random walker in graphs 1 y 1 6 10 3 12 We want to obtain the optimal labeling y ∗ based on y 5 y 4 5 a weighted graph ⇒ Random Walker algorithm 10 7 3 5 y 3 y 2 9 3 2 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 24 / 45

  50. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Random walker in graphs 1 y 1 6 10 3 12 We want to obtain the optimal labeling y ∗ based on y 5 y 4 5 a weighted graph ⇒ Random Walker algorithm 10 7 3 5 y 3 y 2 9 3 2 1 0.95 0.35 0.46 0.03 0.01 0 0 y ( 1 ) July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 24 / 45

  51. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Random walker in graphs 1 y 1 6 10 3 12 We want to obtain the optimal labeling y ∗ based on y 5 y 4 5 a weighted graph ⇒ Random Walker algorithm 10 7 3 5 y 3 y 2 9 3 2 1 0 0.95 0.01 0.35 0.46 0.19 0.28 0.03 0.01 0.02 0.97 0 0 0 1 y ( 1 ) y ( 2 ) July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 24 / 45

  52. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Random walker in graphs 1 y 1 6 10 3 12 We want to obtain the optimal labeling y ∗ based on y 5 y 4 5 a weighted graph ⇒ Random Walker algorithm 10 7 3 5 y 3 y 2 9 3 2 1 0 0 0.95 0.01 0.03 0.35 0.46 0.19 0.28 0.46 0.26 0.03 0.01 0.02 0.97 0.95 0.02 0 0 0 1 1 0 y ( 1 ) y ( 2 ) y ( 3 ) July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 24 / 45

  53. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution Random walker in graphs 1 y 1 6 10 3 12 We want to obtain the optimal labeling y ∗ based on y 5 y 4 5 a weighted graph ⇒ Random Walker algorithm 10 7 3 5 y 3 y 2 9 3 2 1 0 0 0.95 0.95 0.01 0.03 1 0.35 0.46 0.46 0.19 0.28 0.46 0.46 0.26 3 1 0.03 0.01 0.02 0.97 0.97 0.95 0.95 0.02 3 2 0 0 0 1 1 0 y ∗ = { 1 , 1 , 2 , 3 , 3 } y ( 1 ) y ( 2 ) y ( 3 ) July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 24 / 45

  54. BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution hard- vs soft- clustering in BRANE Clust � � T � � 2 � � 2 � � ω i , j x i , j y ( t ) − y ( t ) � y ( t ) − s ( t ) + µ i , j minimize i j i j β Y t = 1 ( i , j ) ∈ V 2 i ∈ V , j ∈ T hard -clustering soft -clustering # clusters = # TF # clusters ≤ # TF  α if i = j � → ∞ if i = j   µ i , j = µ i , j = α 1 ( ω i , j > τ ) if i � = j and i ∈ T 0 otherwise.  ω i , j 1 ( ω i , j > τ ) if i � = j and i / ∈ T  July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 25 / 45

  55. BRANE RESULTS It’s time to test the BRANE philosophy... July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 26 / 45

  56. BRANE RESULTS Methodology Numerical evaluation strategy AUPR AUPR BRANE Ref Reference BRANE Precision-Recall curve Precision-Recall curve Classical thresholding BRANE edge selection Gene-gene interaction scores | TP | P = (ND)-CLR or (ND)-GENIE3 | TP | + | FP | | TP | R = Gene expression data | TP | + | FN | DREAM4 or DREAM5 challenges July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 27 / 45

  57. BRANE RESULTS DREAM4 synthetic results BRANE performance on in-silico data DREAM4 [Marbach et al., 2010] July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 28 / 45

  58. BRANE RESULTS DREAM4 synthetic results BRANE performance on in-silico data DREAM4 [Marbach et al., 2010] Network 1 2 3 4 5 Average Gain CLR 0.256 0.275 0.314 0.313 0.318 0.295 10 . 9 % BRANE Cut 0.282 0.308 0.343 0.344 0.356 0.327 7 . 8 % BRANE Relax 0.278 0.293 0.336 0.333 0.345 0.317 12 . 2 % BRANE Clust 0.275 0.337 0.360 0.335 0.342 0.330 GENIE3 0.269 0.288 0.331 0.323 0.329 0.308 BRANE Cut 0.298 0.316 0.357 0.344 0.352 0.333 8 . 4 % BRANE Relax 0.293 0.320 0.356 0.345 0.354 0.334 8 . 5 % BRANE Clust 0.287 0.348 0.364 0.371 0.367 0.347 12 . 8 % Network 1 2 3 4 5 Average Gain ND-CLR 0.254 0.250 0.324 0.318 0.331 0.295 BRANE Cut 0.271 0.277 0.334 0.335 0.343 0.312 5 . 9 % 3 . 1 % BRANE Relax 0.270 0.264 0.327 0.325 0.332 0.304 2 . 5 % BRANE Clust 0.258 0.251 0.327 0.337 0.342 0.303 ND-GENIE3 0.263 0.275 0.336 0.328 0.354 0.309 BRANE Cut 0.275 0.312 0.367 0.346 0.368 0.334 7 . 2 % BRANE Relax 0.276 0.307 0.369 0.347 0.371 0.334 7 . 3 % BRANE Clust 0.273 0.311 0.354 0.373 0.370 0.336 8 . 1 % July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 28 / 45

  59. BRANE RESULTS DREAM4 synthetic results BRANE performance on in-silico data DREAM4 [Marbach et al., 2010] CLR GENIE3 ND-CLR ND-GENIE3 10 . 9 % 8 . 4 % 5 . 9 % 7 . 2 % BRANE Cut 7 . 8 % 8 . 5 % 3 . 1 % 7 . 3 % BRANE Relax 12 . 2 % 12 . 8 % 2 . 5 % 8 . 1 % BRANE Clust BRANE approaches validated on small synthetic data BRANE methodologies outperform classical thresholding First and second best performers: BRANE Clust and BRANE Cut ⇒ Validation on more realistic synthetic data July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 29 / 45

  60. BRANE RESULTS DREAM5 synthetic results BRANE performance on in-silico data DREAM5 [Marbach et al., 2012] July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 30 / 45

  61. BRANE RESULTS DREAM5 synthetic results BRANE performance on in-silico data DREAM5 [Marbach et al., 2012] AUPR Gain AUPR Gain CLR 0.252 GENIE3 0.283 6 . 3 % 4 . 2 % BRANE Cut 0.268 BRANE Cut 0.295 BRANE Relax 0.272 7 . 9 % BRANE Relax 0.294 3 . 8 % 19 . 4 % 18 . 6 % BRANE Clust 0.301 BRANE Clust 0.336 AUPR Gain AUPR Gain ND-CLR 0.272 ND-GENIE3 0.313 1 . 9 % 1 . 1 % BRANE Cut 0.277 BRANE Cut 0.317 0.274 0 . 6 % 0.314 0 . 3 % BRANE Relax BRANE Relax 6 . 2 % 10 . 2 % BRANE Clust 0.289 BRANE Clust 0.345 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 30 / 45

  62. BRANE RESULTS DREAM5 synthetic results BRANE performance on in-silico data DREAM5 [Marbach et al., 2012] AUPR Gain AUPR Gain CLR 0.252 GENIE3 0.283 6 . 3 % 4 . 2 % BRANE Cut 0.268 BRANE Cut 0.295 BRANE Relax 0.272 7 . 9 % BRANE Relax 0.294 3 . 8 % 19 . 4 % 18 . 6 % BRANE Clust 0.301 BRANE Clust 0.336 AUPR Gain AUPR Gain ND-CLR 0.272 ND-GENIE3 0.313 1 . 9 % 1 . 1 % BRANE Cut 0.277 BRANE Cut 0.317 0.274 0 . 6 % 0.314 0 . 3 % BRANE Relax BRANE Relax 6 . 2 % 10 . 2 % BRANE Clust 0.289 BRANE Clust 0.345 BRANE approaches validated on realistic synthetic data and outperform classical thresholding First and second best performer: BRANE Clust and BRANE Cut ⇒ Validation of BRANE Cut and BRANE Clust on real data July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 30 / 45

  63. BRANE RESULTS Escherichia coli results BRANE Clust performance on real data Escherichia coli dataset AUPR Gain AUPR Gain CLR 0.0378 GENIE3 0.0488 5 . 5 % 9 . 8 % BRANE Clust 0.0399 BRANE Clust 0.0536 July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 31 / 45

  64. BRANE RESULTS Escherichia coli results BRANE Clust performance on real data Escherichia coli dataset AUPR Gain AUPR Gain CLR 0.0378 GENIE3 0.0488 5 . 5 % 9 . 8 % BRANE Clust 0.0399 BRANE Clust 0.0536 BRANE Clust predictions using GENIE3 weights July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 31 / 45

  65. BRANE RESULTS Escherichia coli results BRANE Clust performance on real data Escherichia coli dataset AUPR Gain AUPR Gain CLR 0.0378 GENIE3 0.0488 5 . 5 % 9 . 8 % BRANE Clust 0.0399 BRANE Clust 0.0536 BRANE Clust predictions using GENIE3 weights BRANE Clust validated on real dataset July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 31 / 45

  66. BRANE RESULTS Trichoderma results results BRANE Cut in the real life GRN of T. reesei obtained with BRANE Cut using CLR weights July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 32 / 45

  67. BRANE RESULTS Trichoderma results results BRANE Cut in the real life GRN of T. reesei obtained with BRANE Cut using CLR weights July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 32 / 45

  68. BRANE RESULTS Trichoderma results results BRANE Cut in the real life GRN of T. reesei obtained with BRANE Cut using CLR weights July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 32 / 45

  69. BRANE RESULTS Trichoderma results results BRANE Cut in the real life GRN of T. reesei obtained with BRANE Cut using CLR weights July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 32 / 45

  70. C ONCLUSIONS It’s time to conclude... July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 33 / 45

  71. C ONCLUSIONS Conclusions Inference: BRANE Cut and BRANE Relax Joint inference and clustering: BRANE Clust July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 34 / 45

  72. C ONCLUSIONS Conclusions Inference: BRANE Cut and BRANE Relax Joint inference and clustering: BRANE Clust The BRA- in BRANE : integrating biological a priori constrains the search of relevant edges The -NE in BRANE : proposed graph inference methods lead to promising results and outperforms state-of-the-art methods July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 34 / 45

  73. C ONCLUSIONS Conclusions Inference: BRANE Cut and BRANE Relax Joint inference and clustering: BRANE Clust The BRA- in BRANE : integrating biological a priori constrains the search of relevant edges The -NE in BRANE : proposed graph inference methods lead to promising results and outperforms state-of-the-art methods ⇒ Average improvements around 10 % ⇒ Biological relevant inferred networks ⇒ Negligible time complexity with respect to graph weight computation July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 34 / 45

  74. C ONCLUSIONS Conclusions Inference: BRANE Cut and BRANE Relax Joint inference and clustering: BRANE Clust The BRA- in BRANE : integrating biological a priori constrains the search of relevant edges The -NE in BRANE : proposed graph inference methods lead to promising results and outperforms state-of-the-art methods ⇒ Average improvements around 10 % ⇒ Biological relevant inferred networks ⇒ Negligible time complexity with respect to graph weight computation Biological a priori relevance for network inference BRANE Clust ≻ BRANE Cut ≻ BRANE Relax July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 34 / 45

  75. C ONCLUSIONS Perspectives From biological graphs... GRN July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 35 / 45

  76. C ONCLUSIONS Perspectives From biological graphs... TF-based a priori GRN July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 35 / 45

  77. C ONCLUSIONS Perspectives From biological graphs... TF-based a priori Clustering GRN July 3 th , 2017 Recons. and Clust. with Graph Optim. and Priors on GRN and Images 35 / 45

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