Structural pattern recognition

Romain Raveauxromain.raveaux@univ-tours.fr

Maître de conférences

Université de Tours

Laboratoire d’informatique (LIFAT)

Equipe RFAI

Optimisation combinatoire et apprentissage structurel pour

l'appariement et la classification de graphes

Contributions and perspectives

Contributions and perspectives on combinatorial optimization

and machine learning onto graph space

Application to : graph matching and graph classification

Presentation : Romain Raveaux

• CV

• Teaching

• Research

• Projects

• Administrative tasks

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• Diplomas and qualification

2010-2011

CNU Qualification : 27

2006-2010

Doctorat de l’Université de La Rochelle. Mention : Très honorable avec félicitations

Fouille et classification de graphes : Application à l’analyse d’images cadastrales couleurs.

2004-2006

Master Génie Informatique, Université de Rouen.

Option : Extraction et Indexation de l’information.

2004-2006 :

Master Génie Électrique et Informatique Industrielle,Université de Rouen.

Option : Réseaux et Télécoms.

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• Since 2012, teacher at Université de Tours• At Polytech’Tours (Engineering school)

• Computer Science department

• Since 2012, researcher at LIFAT• Computer science laboratory

• Team : Pattern recognition and image analysis

• 2011-2012 : Research engineers at SOOD company

• 2010-2011 : Attaché Temporaire d’Enseignement et de Recherche à l’IUT d’informatique de La Rochelle

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Teaching

• Teaching : 241 h EqTD

• Projects : 30 h EqTD

• Responsability : 25 EqTD

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Research

• Topic : Machine learning, discrete optimization, computer vision

• Supervisor of : 3 PhD students and 3 master students

• 15 journal papers and 25 conferences.

• Projects : members of 3 projets• ALPAGE, CARAMBA, VISIT

• Collaboration : National (GREYC, LITIS, LORIA). International (Campinas,Brazil, Griffith University, Australia)

• Scientific committee: GBR, past: GREC

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Content

• Introduction• Title explanation• Data• Graph-based applications• Applicative Tools

• Combinatorial optimization for graph matching and graph classification• Graph matching• Graph classification

• Machine learning in graph space for matching and classification• Graph matching• Graph classification

• Conclusion• Summary• Perspective

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Title explanation

• Graph matching in a single picture :

Introduction : Combinatorial Optimization : Machine Learning : ConclusionTitle explanation – Data- Graph-based applications - Applicative Tools

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Title explanation

• Graph classification in a single picture :

Class 4Classifier

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Title explanation

• Combinatorial optimization in a single picture :

1-A 1-B 1-C

2-A 2-B

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Title explanation

• Machine learning in a single picture : Decision processData

Data set 1

Decision

Learningprocess

Data set 2

Introduction : Combinatorial Optimization : Machine Learning : ConclusionTitle explanation – Data- Graph based applications - Applicative Tools

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Title explanation

• Graph space in a single picture :

Introduction : Combinatorial Optimization : Machine Learning : ConclusionTitle explanation – Data - Graph-based applications - Applicative Tools

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• Data are crucial for computer scientists

• Data driven :• problems

• models

• solution methods (solving methods)

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Type of Data

Taken from M. Bronstein. CVPR Tutorial 2017 16

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Focus on structured Data

• String

• Tree

• Graph

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String

Taken from: Marçal Rusiñol et al: Symbol spotting in vectorized technical drawingsthrough a lookup table of region strings. PAA 2010

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R. Raveaux et al:Structured representations in a content based image retrieval context. J. Visual Communication and Image Representation 24(8): 1252-1268 (2013)

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• Data are represented as graphs:• By nature (Social Network, Molecule, Protein interaction network)

• By construction (from images)

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A graph can be represented by 4-tuple 𝐺 = (𝑉, 𝐸, 𝜇, 𝜉), with:

V the set of vertices,

𝐸 ⊆ 𝐸 ∩ 𝑉 × 𝑉 the set of edges,

𝜇: 𝑉 → 𝐿𝑉 the function that assigns attributes to vertices,

𝜉: 𝑉 → 𝐿𝐸 the function that assigns attributes to edges,

𝐿𝑉 the set of all possible attributes for vertices,

and 𝐿𝐸 the set of all possible attributes for edges.

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Region Adjacency Graph

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Region Adjacency Graph

Impact of noise on Graph-Based Representation 24

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Neighborhood graph

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Skeleton Graph

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Graph of molecules : Chemoinformatics

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Domain structure vs Data on a domain

Taken from [Brontein 2016, CoRR] 28

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Fixed vs different domain

1D,2D, 3D shapes(Different graphs)

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Structured data

• We will focus on graphs:• Graph as a generalization of Euclidean data : vector, matrix, tensors ….

• Graphs as a generalization of strings and trees.

• By nature, data are more likely to be graphs.

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Graph-based applications

• Graph classification/clustering/regression

• Vertex classification/clustering/regression

• Graph matching

• Graph distance

• Graph-based search• Subgraph search• Subgraph spotting• Similarity search

• Graph prototypes• Median graphs/ Super graphs

Artificial Intelligence

is the set of tools to solve these problems : Modelisation, Machine learning, Optimization … …

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Graph classification

1. cancerous or not cancerous molecules

2. determination of the boiling point

Molecular graph

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Vertex classification

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Vertex classification/clustering

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Semi-supervised vertex classification

• Setting: • Some nodes are labeled (black circle)

• All other nodes are unlabeled

• Task: • Predict node label of unlabeled nodes

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Graph matching

• When parts of the object must be tracked or compared.

• Detect and recognize at once

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Graph matching (https://goo.gl/8dYCZb)

• Mettre vidéo graph matching

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Graph matching (https://goo.gl/wi5m1E)

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Graph distance/similarity

How similar are theses graphs ?

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Graph prototype

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Graph-based search

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What do we need to solve these problems ?

• Modelisation, Machine learning, Optimization, …:• Graph comparison :

• Graph matching

• Graph distance

• Learning capability with graphs :• Learn to match

• Learn to classify

Introduction : Combinatorial Optimization : Machine Learning : ConclusionTitle explanation – Data- Graph-based applications - Tools

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Two ways to solve the problems

• Graph/Node embedding : [Luqman et al., 2017, PR], [Kipf and Welling, 2017, ICLR].• The graphs/nodes are projected into a vector space

• Graph space : [Neuhaus and Bunke., 2007, MPAI], [Riesen, 2015, ACVPR].• The graphs are compared through graph matching methods.

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Graph space vs graph embedding

Graph space Graph embedding

Graphcomparison

Pros • Manage structure and features at once

• Preserve the structure of graphs

• Fast algorithms available• Many learning techniques

Cons • Slow (Often) a combinatorialproblem

• Lack of learning techniques

• Does not capture the combinatorialnature of the problem

• Loss of topological information

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Our positionning

• Many works have focus their interest on graph embeddings.• Review on graph embeddings : [Goyal and Ferrara, 2017, KBSyst][Cai et al,

2017, IEEE TKDE].

• We think it is unfortunate to reduce the graphs into vectors.

• We explore new avenues:• It is interesting to develop learning techniques in graph space.

• It is interesting to design optimization methods in graph space.

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Content

• Combinatorial optimization for graph matching and graph classification• Graph matching

• Graph classification

• Machine learning in graph space for matching and classification• Graph matching

• Graph classification

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Combinatorial optimization for graph matchingand graph classification• Learning free methods.

• Pure combinatorial problems

Introduction : Combinatorial Optimization : Machine Learning : ConclusionProblems – State of the art – Deadlocks – Contributions - Summary

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Graph matching

• Problems

• State of the art

• Deadlocks

• Contributions

Introduction : Combinatorial Optimization : Machine Learning : ConclusionGraph Matching : Problems – State of the art – Deadlocks – Contributions - Summary

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Graph matching : Problems

• Exact matching

• Error-Tolerent matching• Subgraph matching

• Error-correcting matching

• can be expressed as QAP, MAP-inference, GED problems.

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Graph matching : the Graph Edit Distance (GED) Problem

This solution costs:

𝑑 =

𝑜𝑝𝑖 ∈ 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠

𝐶(𝑜𝑝𝑖)

Deletion Substitution Substitution Both

Vertices operations

• OP1: 1 ϵ C1

• OP3: 2 a C2

• OP4: 3 b C3

• OP6: 4 c C4

Edges operations

• OP2: (1,2) ϵ C5

• OP5: (2,3) (a,b) C6

• OP7: (3,4) (b,c) C7

• OP8: (2,4) ϵ C8

The goal is to find 𝑑min the set of operations with the minimum cost.

Graph matching : the Qadratic AssignmentProblem

𝜙 𝑖 = 𝑘𝜙 𝑗 = 𝑙𝐷𝑖,𝑘,𝑗,𝑙 = 𝑐 𝑖𝑗, 𝑘𝑙𝐵𝑖,𝑘 = 𝑐 𝑖, 𝑘𝑁 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑑𝑒𝑠 = 2

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Graph matching : MAP-inference of a ConditionalRandom Field (CRF)Finding the most likely configuration of discrete CRF. CRF=G1

𝑥𝑖 𝑎 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑒𝑑 𝑡𝑜 𝑛𝑜𝑑𝑒 𝑖𝑥𝑖 𝑐𝑎𝑛 𝑡𝑎𝑘𝑒 𝑖𝑡𝑠 𝑣𝑎𝑙𝑢𝑒 ∈ 𝑘, 𝑙𝑥𝑖𝑗 𝑡𝑎𝑘𝑒𝑠 𝑖𝑡𝑠 𝑣𝑎𝑙𝑢𝑒 ∈ 𝑘, 𝑙 × {𝑘, 𝑙}

𝑉 = 𝑖, 𝑗 𝑎𝑛𝑑 𝐸 = 𝑖𝑗𝜃 𝑥𝑖 = 𝑐𝑜𝑠𝑡 𝑡𝑜 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑒 𝑖 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑙𝑎𝑏𝑒𝑙 𝑥𝑖𝜃 𝑥𝑖𝑗 = 𝑐𝑜𝑠𝑡 𝑡𝑜 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑒 𝑖𝑗 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑙𝑎𝑏𝑒𝑙 𝑥𝑖𝑗

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Graph matching : Related Problems

Errot-tolerant graph matching : NP-hard problem. [Zeng et al. 2009, PVLDB]: Combinatorial problem

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Graph matching models

• Graph matching can be expressed by : • Integer Quadratic Program

• Integer Linear Program

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Graph matching models : Integer QuadraticProgram

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Graph matching models : Integer Linear Program

• Linear constraints

• Linear objective function

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Graph matching : State of the art

Subgraph GM problem Methods

Models

Exact Heuristique

ILP #papers : 0 #papers : 0

IQP #papers : 0 #papers : 7IPFP[Leordeanuet al., 2009]

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Graph matching : State of the art

Error-correcting GM problem

Methods

Models

Exact Heuristique

ILP #papers : 0 #papers : 1ILP [Justice andHero, 2006]

IQP #papers : 0 #papers : 3mIPFP[Bougleuxet al., 2017b]

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Graph matching : State of the artMethods Error-correcting GM Subgraph GM

Constrained quadraticprogramming

#papers : 1 mIPFP [Bougleuxet al., 2017b]

#papers : 4 – IPFP [Leordeanuet al., 2009]

Spectral #papers : 0 #papers : 2 SM[Leordeanuand Hebert, 2005]

Branch and bound #papers : 2 A* [Riesenet al., 2007]

#papers : 0

Bio-inspired #papers : 1 GEDEVO [Ibragimovet al., 2013]

#papers : 2 AG [Cross et al., 1997]

Hungarian method #papers : 3 SFBP [Serratosa,2015]

#papers : 0

Probabilistic #papers : 1 BayesianGED [Myers et al.,2000]

#papers : 1 [Zass andShashua, 2008]

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Deadlocks :

• Facts:• Exact methods are rarely study

• Few works have paid attention to ILP models.

• Deadlocks :• To study exact methods

• To derive heuristics from exact methods

• There is a need to study heuristics according to speed and effectiveness criteria.

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Exact methods for error-correcting matching

• Integer Linear Program

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• Notations

Error-correcting matching : Integer Linear Program

[Lerouge et al., 2017, PR]R. Raveaux

• Objective function:

Vertex substitutions Edge substitutions vertex deletions

vertex insertions edge insertionsedge deletions

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substitutionsdeletion

Insertion

deletionsubstitutions

Vertices

mapping

constraints

mapping

constraints

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• Topological constraints

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Heuristic : LocBra

• A Matheuristic is derive from F1.

• Δ = 𝐻𝑎𝑚𝑚𝑖𝑛𝑔 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

• Adding branching constraints

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Heuristic : Anytime

• Export all the list of dUB

• Interruptability: At each iteration, we can

stop the algorithm if it exceeds CT

[Abu-Aisheh et al., 2016, PRL]

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Antyime

A A’B B’

C’C D

A-A’ A-B’ A-C’ A-ϵ

B-A’ B-C’ B-ϵ

C-C’ C-ϵ

B-A’ B-B’ B-ϵ

f=6B-B’

g: current cost

h: estimated cost

f : total cost

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https://goo.gl/xJn5nq

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https://goo.gl/CebSg2

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Benchmarking

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Conference paper: Z. Abu-Aisheh, R. Raveaux and J-Y Ramel: A Graph Database Repository and

Performance Evaluation Metrics for Graph Edit Distance. GbRPR 2015 : 138-147.

Database # graphs avg. (max) # nodes

Decomposition Overview Purpose

GREC 50 12.5 (20) MIX, 5, 10, 15 and 20 Classification

CMU 111 30 (30) 30 Matching

MUTA 80 40 (70) MIX ,10, 20, 30, … and

Classification

PAH 94 20.7 (28) - Classification

Acyclic 185 8.2 (11) - Classification

Alkane 150 8.9 (10) - Classification

Datasets

[Abu-Aisheh et al.,2015a, GBR]

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ICRP 2016 Contest

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Summary

• Error tolerent graph matching :• New ILP models : F1, F2, F3 :

• New exact methods : Branch-bound, mathematical solver : • Fast convergence, evaluation with heuristics

• New heuristics : Matheuristic and Anytime• Accurate and flexible

• New benchmarks

• A contest

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• Problems

• Deadlocks

• Contributions

Introduction : Combinatorial Optimization : Machine Learning : ConclusionGraph Classification : Problems – State of the art – Deadlocks – Contributions - Summary

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• Learning-free

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Problem

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State of the art : speeding up the kNN problem

Generic (dissimilarity space) Graph oriented methods

Dissimilarity functionMetric property [Uhlmann, 1991, IPL] Fast GED methods [Riesen,

2015, ACVPR ]

Dissimilarity spaceSpace partition, prototypes, hashtable, proximity graph, line search [Malkov and Yashunin, 2016, PAMI]

Graph prototypes [Musmanno and Ribeiro, 2016, EJOR]

• Structuring the dissimilarity space lies at the art of machine learning (out of scope here)

• Graph distance is a combinatorial problem.• Many methods have focused on : fast GED methods.

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Deadlocks

• Is there a way to specialize a line search method to operate on graph space instead of the generic dissimilarity space ?

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Is there a way to specialize a line search method to operate on graph space instead of the generic dissimilarity space ?

• Merging the GED problem and the kNN problem in a single problem:

Γ set of all possible matchings between 𝐺 ∈ 𝑇𝑒𝑆 and 𝐺𝑗 ∈ 𝑇𝑟𝑆

Merging

The set of all possible matchings between 𝐺 and all 𝐺𝑗 ∈ 𝑇𝑟𝑆

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Is there a way to specialize a line search method to operate on graph space instead of the generic dissimilarity space ?

• Merging the GED problem and the kNN problem in a single problem:

Merging

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Solving the MGED problem

• A Branch and bound method

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Data sets

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Methods

• Fast GED methods : BP, FBP, DF, BS-1

• Our proposal : one-tree (with different initialisation)

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Classification test

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Summary

• « Learning free » methods for classification

• Merging GED problem and kNN in a single problem (MGED)

• New algorithm for this problem

• Efficient when the data size increases

• Can be combined with other learning based-methods:• Learning the distance

• Learning prototypes

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Machine learning in graph space for matching and classification

Introduction : Combinatorial Optimization : Machine Learning : ConclusionProblems – State of the art – Deadlocks – Contributions - Summary

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Graph matching

• Problem

• Deadlocks

• Contributions

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Learning graph mathching problem

A possible loss :

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State of the art

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Deadlocks

• How to deal with insertion and deletion costs ?

• Can an heuristic output solutions closer to optimality thanks to machine learning?

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Parametrized GED

[Raveaux et al., 2017, GBR].

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Parametrized GED

Φ = 𝑑 ∗, 1 ; 𝑑 ∗, 2 ; 𝑑 ∗, ε ; 𝑑 ∗∗, 12 ; 𝑑 ∗∗, εε𝛽 = 𝛽1 ; 𝛽2 ; 𝛽ε ; 𝛽4 ; 𝛽εε

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Learning with parametrized GED algorithm

• The learning problem reformulated :

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Learning algorithm : gradient descent

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Data set

CMU-House

|TrS| 50

|TeS| 50

|V| 30

|E| 70

Attributes xy distance

Node matching cost 1

Edge matching cost L1 norm

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Methods

• GED solver : BP

• BP without learning

• BP with our learning scheme

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Learning graph matching testhttps://youtu.be/JrMR2-5mjA4

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Learning Graph matching test

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Summary

• Parametrized error correcting graph matching

• Learning scheme for graph matching

• Independent to the graph matching methods

[Raveaux et al., 2017, GBR].

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• Problem

• Deadlocks

• Contributions

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Problem

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State of the art

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State of the art

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State of the art

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State of the art

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Deadlocks

• Learning graph distance for classification with local parameters for nodes and edges.

• Learning graph matching and graph prototypes in a hierarchical manner.

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Learning graph distance for classification with local parameters for nodes and edges.

Learning rule

[Martineau et al., 2018, PRL].

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Learning graph distance : Learning algorithm

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Benchmark

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Data sets

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Methods

• Learning schemes for global parameters :• Grid search [Riesen et al, 2009, IVC] (R-1NN)

• Constraint quadratic programming [Cortes et al, 2015, PRL] (C-1NN)

• 1NN methods

• 1NN based on median graphs

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Results

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Learning graph matching and graph prototypes in a hierarchical manner.

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The basics of artificial neural networks

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The basics of graph neural networks

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Adjacency matrix

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Degree matrix

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Key idea and Intuition [Kipf and Welling, 2016]

• The key idea is to generate node embeddings based on local neighborhoods.

• The intuition is to aggregate node information from their neighbors using neural networks.

• Nodes have embeddings at each layer and the neural network can be arbitrary depth. “layer-0” embedding of node u is its input feature, i.e. Fu.

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GNN: a pictorial model

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Simple example of f

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2 issues of this simple example

• Issue 1:• for every node, f sums up all the feature vectors of all neighboring nodes but

not the node itself.

• Fix: simply add the identity matrix to A

• Issue 2:• A is typically not normalized and therefore the multiplication with A will

completely change the scale of the feature vectors.

• Fix: Normalizing A such that all rows sum to one:

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Altogether: [Kipf and Welling, 2016]

• The two patched mentioned before +

• A better (symmetric) normalization of the adjacency matrix

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Our idea : graph matching based neural network

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Graph convolution layer :

• A layer is composed of many filter graphs (GF)

• A layer outputs : A graph with node features and edge features

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GNN for graph classification

• A GNN outputs : node embeddings

• Adding a global average pooling layer

• Average pooling

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Benchmarking

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Data set : MNIST

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Data set

# classes 2

|TrS| 1 000 graphs

|TeS| 5 000 graphs

|V| 196 and 75 nodes

Node Attributes Pixel intensity (gray level)

Edge Attributes none

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Methods

• Lenet 5 [Lecun et al., 1998, IEEE]

• MoNet [Monti et al., 2016, CVPR]

• Our

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Results

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Summary

• Learning scheme for error-correcting graph matching in a classification context.

• Hierarchical leaning of filter graphs and graph similarity.

• Deep learning paradigm can be extended to graph space

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Conclusion

• Summary• Perspective

Introduction : Combinatorial Optimization : Machine Learning : ConclusionSummary - Perspective

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Summary

• Remember the cons:• Slow methods (Often) a combinatorial problem• Lack of learning techniques

• Our asnwsers:• Faster methods to compare/classify graphs

• Thanks to discrete optimization techniques.

• Learning algorithm operating in graph space to compare/classify graphs• Parametrized graph matching• Learning scheme merging combinatorial problems and gradient descent

• Deep learning paradigm can be extended to graph space

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Perspectives

• Combinatorial optimization• Application of optimization tools for pattern recognition problems

• Learning prototypes

• Machine learning• Theoretical understanding of what is learned • Common benchmark datasets

• Merging Machine learning and Combinatorial optimization• Integration of machine learning into combinatorial optimization methods

(branch and bounds, matheuristics)• Integration of combinatorial optimization into machine learning mehods

• Combinatorial layers in neural networks

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Thank you for your attention

• Any question ?

Colleagues from the lab

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Our references :

• [Martineau et al., 2018, PRL] : Maxime Martineau, Romain Raveaux, Donatello Conte, Gilles Venturini. Learning error-correcting graph matching with a multiclass neural network. Pattern Recognition Letters, Elsevier, 2018.

• [Raveaux et al., 2017, GBR] : Romain Raveaux, Maxime Martineau, Donatello Conte, Gilles Venturini: Learning Graph Matching with a Graph-Based Perceptron in a Classification Context. GbRPR 2017: 49-58

• [Abu-Aisheh et al., 2017, PRL] : Zeina Abu-Aisheh, Benoit Gaüzère, Sébastien Bougleux, Jean-Yves Ramel, Luc Brun, Romain Raveaux, Pierre Héroux, Sébastien Adam: Graph edit distance contest: Results and future challenges. Pattern Recognition Letters 100: 96-103 (2017)

• [Abu-Aisheh et al.,2015a, GBR] : Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel: A Graph DatabaseRepository and Performance Evaluation Metrics for Graph Edit Distance. GbRPR 2015: 138-147

• [Abu-Aisheh et al., 2016, PRL] : Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel: Anytime graph matching. Pattern Recognition Letters 84: 215-224 (2016)

• [Lerouge et al., 2017, PR] : Julien Lerouge, Zeina Abu-Aisheh, Romain Raveaux, Pierre Héroux, Sébastien Adam: New binary linear programming formulation to compute the graph edit distance. Pattern Recognition 72: 254-265 (2017)

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References

• [Brontein 2016, CoRR] Michael M. Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst. Geometric deep learning: goingbeyond euclidean data. CoRR, abs/1611.08097, 2016.

• [Luqman et al., 2017, PR] Ramzi Chaieb, Karim Kalti, Muhammad MuzzamilLuqman, Mickaël Coustaty, Jean-Marc Ogier, and Najoua Essoukri Ben Amara. Fuzzy generalized median graphs computation: Application to content-based document retrieval. Pattern Recognition, 72:266 – 284, 2017.

• [Kipf and Welling, 2017, ICLR] Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. CoRR, abs/1609.02907, 2016.

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References

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