page.1Introduction Graph-based Methods Conclusions

Graph-based Pattern Recognition

Nicola Strisciuglio

University of Groningen

n.strisciuglio@rug.nl

28/10/2013

page.2Introduction Graph-based Methods Conclusions

Statistical vs. Graph-based PR

Statistical vs. Graph-based Pattern Recognition

Statistical PR

I =(x1, x2, . . . , xn

) Graph-based PR

page.3Introduction Graph-based Methods Conclusions

Statistical vs. Graph-based PR

Statistical vs. Graph-based Pattern Recognition

Statistical PR

I =(y1, y2, . . . , yn

) Graph-based PR

page.4Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Graph-based Methods

Pure MethodsLearning and classification problems are faced directly in the graphspace.

Impure Methods

Transposition of the methods of Statistical Pattern Recognition tograph space.

Extreme MethodsTransformation of graphs into vectors. Use of the well estabilishedlearning and classification techniques.

page.5Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Pure Methods: Graph Matching

I Exact Matching: find an exact correspondence betweengraphs (or sub-graphs)

I Inexact Matching: deals with distortions in findinf acorrespondence between graphs

I It needs a metric to define dissimilarities

page.6Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Graph edit distance

We need a distance metric between graphs: we have Graph editdistance. Cheapest sequence of operations to trasform one graphin another graph.

page.7Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Impure Methods: k-NN

I The reference set is made by a collection of graphs, instead ofpoints in a m-dimensional space

I For a graph to classify, the graph edit distance is computedwith respect to each of the graphs in the reference set

I The decision is taken as majority voting on the K nearestgraphs

I Generally, the time needed for a classification is very high

page.8Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Impure Methods: k-Means

For the classical k-Means algorithm a centroid is computed as theaverage of the vectors around it, while for the graph a mediangraph should be computed.

Median Graph

S = arg minGi∈S

∑Gj∈S

Dg (Gi ,Gj)

Generalized Median Graph

S = arg minGi∈U

∑Gj∈S

Dg (Gi ,Gj)

page.9Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Impure Methods: k-Means

For the classical k-Means algorithm a centroid is computed as theaverage of the vectors around it, while for the graph a mediangraph should be computed.

Median Graph

S = arg minGi∈S

∑Gj∈S

Dg (Gi ,Gj)

Generalized Median Graph

S = arg minGi∈U

∑Gj∈S

Dg (Gi ,Gj)

page.10Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Impure Methods: k-Means

For the classical k-Means algorithm a centroid is computed as theaverage of the vectors around it, while for the graph a mediangraph should be computed.

Median Graph

S = arg minGi∈S

∑Gj∈S

Dg (Gi ,Gj)

Generalized Median Graph

S = arg minGi∈U

∑Gj∈S

Dg (Gi ,Gj)

page.11Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Impure Methods: LVQ

I Given a pattern x , the winner prototype Pk moves towards xby ∆ = α(x − Pk) if the class of x is the same of Pk (by∆ = −α(x − Pk) otherwise)

I Updating a prototype requires its transformation in anothergraph more similar to the input pattern x by a fraction α ofthe distance

I We need to compute a fraction of edit distance!!!I Pk → Gx needs D = 7 operations on graphI If α = 0.3, we do bD ∗ αc = 2 operations to move Pk to an

intermediate graph

page.12Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Impure Methods: LVQ

I Given a pattern x , the winner prototype Pk moves towards xby ∆ = α(x − Pk) if the class of x is the same of Pk (by∆ = −α(x − Pk) otherwise)

I Updating a prototype requires its transformation in anothergraph more similar to the input pattern x by a fraction α ofthe distance

I We need to compute a fraction of edit distance!!!I Pk → Gx needs D = 7 operations on graph

I If α = 0.3, we do bD ∗ αc = 2 operations to move Pk to anintermediate graph

page.13Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Impure Methods: LVQ

I Given a pattern x , the winner prototype Pk moves towards xby ∆ = α(x − Pk) if the class of x is the same of Pk (by∆ = −α(x − Pk) otherwise)

I Updating a prototype requires its transformation in anothergraph more similar to the input pattern x by a fraction α ofthe distance

I We need to compute a fraction of edit distance!!!I Pk → Gx needs D = 7 operations on graphI If α = 0.3, we do bD ∗ αc = 2 operations to move Pk to an

intermediate graph

page.14Introduction Graph-based Methods Conclusions

Pure Impure and Extreme Methods

Extreme methods: Graph Embedding

I Represent a graph as a point in a suitable feature space

I Use of the classical statistical pattern recognition tools

I The similarity of graph in graph space should be preserved inthe vector space

I The translation of a graph (including all the attributes andrelations) into a fixed-lenght vector is difficult

page.15Introduction Graph-based Methods Conclusions

Conclusions

Some Applications

I Structural description and matching of moleculesI Segmentation of shapes (e.g. letters)I Stereo vision (e.g. for robot navigation)I Learning and recognising objects in scenesI Data mining

page.16Introduction Graph-based Methods Conclusions

Conclusions

Some Applications

I Structural description and matching of moleculesI Segmentation of shapes (e.g. letters)I Stereo vision (e.g. for robot navigation)I Learning and recognising objects in scenesI Data mining

page.17Introduction Graph-based Methods Conclusions

Conclusions

Statistical vs. Graph-based Pattern Recognition

Statistical PRAdvantages

I Theoretically estabilished

I Many powerful algorithms

Disadvantages

I Size of the feature vectorfixed

I Unary features: no relations

Graph-based PRAdvantages

I Variable representation size

I More description power(relationships)

Disadvantages

I Lack of algorithms

I Less mathematicalfoundations

page.18Introduction Graph-based Methods Conclusions

Conclusions

References

Mario Vento (2013)

A long trip in the charming world of graphs for Pattern Recognition

Pattern Recognition

page.19Introduction Graph-based Methods Conclusions

Conclusions

Thank you!!!