Edwin R' Hancock

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The University of York
Pattern Analysis with Graphs with applications
from computer vision
Edwin R. Hancock With help from Richard Wilson,
Bai Xiao Bin Luo, Antonio Robles-Kelly and Andrea
Torsello. University of YorkComputer Science
DepartmentYORK Y010 5DD, UK. erh_at_cs.york.ac.uk
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Outline

• Motivation Background
• Graphs from images
• Matching graphs
• Spectral methods
• Pattern spaces for sets of graphs
• Embedding and characterising graphs

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Motivation
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Problem
In computer vision graph-structures are used to
abstract image structure. However, the algorithms
used to segment the image primatives are not
reliable. As a result there are both additional
and missing nodes (due to segmentation error) and
variations in edge-structure. Hence image
matching and recognition can not be reduced to a
graph isomorphism or even a subgraph isomrophism
problem. Instread inexact graph matching methods
are needed.
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Measuring similarity of graphs

• Early work on graph-matching is vision ( Barrow
  and Popplestone) introduced association graph and
  showed how it could be used to locate maximum
  common subgraph,
• Work on syntactic and structural pattern
  recognition in 1980s unearthed problems with
  inexact matching (SanfeliuEshera and Fu.
  Haralick and Shapiro, Wong etc) and extended
  concept of edit distance from strings to graphs.
• Recent work has aimed to develop probability
  distributions for graph matching (Christmas,
  Kittler and Petrou, Wilson and Hancock, Seratosa
  and Sanfeliu) and match using advanced
  optmisation methods(Simic, Gold and Rangarjan).
• Renewed interest in placing classical methods
  such as edit distance (Bunke) and max-clique
  (Pelillo) on a more rigorous footing.

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Viewed from the perspective of learning
This work has shown how to measure the similarity
of graphs. It can be used to locate inexact
matches when significant levels of structural
error are present. May also provide a means by
which modes of structural variation can be
assessed.
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Learning with graphs

• Learn class structure Assign graphs to classes.
  Need a distance measure. Central clustering is
  difficult since number of nodes and edges varies
  and correspondences are not known. Easier to
  perform pairwise clustering. (Bunke, Buhman).
• Embed graphs in a low dimensional space
  Correspondences are again needed, but spectral
  methods may offer a solution. Can apply standard
  statistical and geometric learning methods to
  graph-vectors.
• Learn modes of structural variation Understand
  how edge (connectivity) structure varies for
  graphs belonging to the same class.
  (Dickinson,Williams)
• Build generative model Borrow ideas from
  graphical models (Langley, Friedman, Koller).

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Why is structural learning difficult

• Graphs are not vectors There is no natural
  ordering of nodes and edges. Correspondences must
  be used to establish order.
• Structural variations Numbers of nodes and
  edges are not fixed. They can vary due to
  segmentation error.
• Not easily summarised Since they do not reside
  in a vector space, mean and covariance hard to
  characterise.

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Eigenvector methods for learning in vision

• Eigenvectors of image covariance matrix
  Eigenfaces (Turk and Pentland), parametric
  eigenspaces (Murase and Nayar).
• Point distribution model apply PCA to landmark
  position covariance matrix (Cootes and Taylor).
• Kernel PCA Tipping and Bishop
• Many, many others.

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Spectral Embedding
Graphs reside on a manifold whose
Laplace-Beltrami operator is the Laplacian of the
graph.
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Spectral Methods
Use eigenvalues and eigenvectors of adjacency
graph (or Laplacian matrix) - Biggs, Cvetokovic,
Fan Chung

• Singular value methods for exact graph-matching
  and point-set alignment). (Umeyama)
• Singular value methods for point-set
  correspondence (Scott and Longuet-Higgins,
  Shapiro and Brady).
• Use of eigenvalues for image segmentation (Shi
  and Malik) and for perceptual grouping (Freeman
  and Perona, Sarkar and Boyer).
• Graph-spectral methods for indexing shock-trees
  (Dickinson and Shakoufandeh)

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Spectral Graph Theory

• Read books by Cvetokovic, Biggs and Chung.
• Good web resources (and papers!) provided by Jon
  Kleinberg (Cornell) and Mark Jerrum (Edinburgh).
• Used in routing problems, Googlebot and many
  other practical applications.

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A taster.

• Eigenvector expansion of adjacency matrix
• Leading eigenvector is steady-state random walk
  on the graph.
• Number of paths of length L on graph

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Aims
Combine probabilistic modelling with spectral
graph theory to develop methods for
correspondence matching and learning. Apply to
shape analysis problems furnished by computer
vison (2D shape). Look at three problems

• Discover shape categories
• Embed graphs in a pattern space
• Learn modes of structural variations

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What this talk is about

• Develop a probabilistic error model that can be
  used to find node correspondences and measure
  the similarity of graphs.
• Potentially cumbersome, so recast the problem
  in a matrix setting. Develop a robust spectral
  matching method.
• Use this model to learn structural variations for
  sets of graphs.

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Graphs in computer vision

• Structural representations of shape

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Graph (structural) representations of shape

• Region adjacency graphs ( Popplestone etc,,
  Worthington, Pizlo, Rosenfeld)
• View graphs (Freeman, Ponce)
• Aspect graphs (Dickisnon)
• Trees (Forsyth, Geiger).
• Shock graphs (Siddiqi, Zucker, Kimia).

Idea is to segment shape primitives from image
data and to abstract them using a graph. Shape
recognition becomes a problem of graph matching.
However, statistical learning of modes of shape
variation becomes difficult since available
methodology is limited.
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Delaunay Graph
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CMU Sequence
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Constrained Delaunay Triangulation
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Gabriel Graph
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Relative Neighbourhood Graph
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Shock graphs
Type 1 shock(monotonically increasing radius)
Type 2 shock(minimum radius)
Type 3 shock(constant radius)
Type 4 shock(maximum radius)
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Measuring the similarity of graphs

• Develop probabilistic measure of graph errors and
  use this to measure similarity.

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Starting point

• Probabilistic Framework for Graph Matching

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Probabilistic modelling

• Assignment errors

• Structural errors

Due to misplacement of correspondence labels
corrected by re-configuring the match f.
Due to the addition of extraneous nodes and
edges corrected by modifying the node set V and
the edge-set E.
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Relational graph matching
Graph G(V,E) with node set V and edge set E
Find state of match fD-gtM between data graph
and a model graph.
Use constraints provided by edges of the two
graphs.
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Distribution of matching errors
Expand probability over dictionary of consistent
matching configurations
Assume individual matching errors are independent
an memoryless
Two component error model for assignment errors
and structural errors
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Probability distribution for matching errors
Three component model
Depends on Hamming distance (number of assignment
errors), size difference (number of structural
errors) and numbner of connecting edges
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Uses

• Optimisation Gradient ascent, naïve mean field,
  genetic search, meta-heuristic (tabu search).
• Problem Computational complexity grows
  exponentially with number of dummy node
  insertions.
• Solution 1 Treat neighbourhoods as strings and
  use Dijkstra algorithm to compute edit distance.
• Solution 2 Adopt graph spectral approach,

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Edit Distance
Avoid exponential complexity of dictionary
padding by treating configurations as strings and
comparing them using string edit distance
Use edit distance to compute probability of
configuration of correspondences
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Literature

• Bayesian Framework (Wilson and Hancock IEEE
  PAMI, July, 1997)
• Graph editing (Wilson, Cross and Hancock, CVIU,
  1998)
• Soft-assign (Finch, Wilson and Hancock, Neural
  Computation, 1998).
• Edit-distance (Myers, Wilson and Hancock, IEEE
  PAMI, June, 2000)
• Dual-step EM algorithm (Cross and Hancock, IEEE
  PAMI, Nov, 1998)
• Image retrieval (Huet and Hancock, IEEE PAMI,
  Dec, 1999)
• Factorisation (Carcassoni and Hancock, CVPR00
  Luo and Hancock ICPR00)

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Spectral Correspondence Matching

• Use EM algorithm to develop iterative form of
  Umeyama algorithm that is robust to differences
  in graph size and structural error

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Graph-spectral Methods for Correspondence

• Develop a probabilistic approach to graph
  matching by matrix factorisation.
• Idea is to match using eigenvectors of graph
  adjacency matrix.
• Several authors have used algorithms based on
  the singular vectors for graphs and point-sets
  (Umeyama, Scott and Longuet-Higgins, Shapiro and
  Brady).
• These methods can be viewed as drawing their
  inspiration from spectral graph theory (Chung).
• They are highly fragile to differences in
  graph-structure.
• Aim in this paper is to show how EM algorithm can
  be used to provide iterative version of Umeyamas
  method that is robust to size differences and
  structural error.

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Umeyamas Algorithm

• Find permutation matrix S which minimises
  Froebenius norm D-Transpose( S) M S.
• Perform singular value decompositions on
  adjacency matrices MU .Diag .Transpose(U) and
  D V. Diag .Transpose(V).
• Here Diag is a diagonal matrix of singular
  values, and, U and V are orthogonal matrices.
• Optimal correspondence matrix is given by the
  singular vectors of the two adjacency matrices
  S V Transpose (U)
• Does not work when D and M are of different size.

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Approach

• If we consider the model graph node to data graph
  node assignments as a set of hidden variables,
  then we can apply the expectation-maximization
  (EM) algorithm to compute the optimal assignment
  that maximizes the likelihood.

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Matrix Representation
Matrix of assignment variables
Data graph connection matrix
Model graph connection matrix
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Problem Formulation
Correspondence variables, S, are hidden variables
that arise through a noisy observation process.
If we assume that any data node can be
generated from any model node, we can set up a
mixture model, yielding the following likelihood
function But how do we model the observation
density p(xaya,S) ?
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Maximum Likelihood Framework
Find the maximum likelihood pattern of
correspondences which satisfies the condition
Construct mixture model over model graph labels
and assume factorial distribution over data graph
nodes
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Factorial observation density
Assume that dat-graph nodes and model-graph nodes
are conditionally independent given
correspondence indicators
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Bernoulli Distribution for Correspondences
Random variable is edge-consistency indicator
Assumed to follow a Bernoulli distribution
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Multiple Edge Constraints
xa
ya
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Probability distribution for correspondences
After algebra, obtain the following exponential
distribution for the correspondence indicators
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Log-likelihood function
Log-likelihood function that needs to be
maximised with respect to the correspondence
indicators S is
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Expected log-likelihood
Correspondences are hidden Work instead with
expected log-likelihood.

• Expand using Bernoulli model

Where
is the a posteriori correspondence
probability.
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EM Algorithm

• Maximisation step Find maximum likelihood
  correspondences.
• Expectation step Find probabilities of
  correspondence indicators.

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Maximisation Step
New configuration of correspondence indicators
maximises the expected log-likelihood subject to
previously available correspondence indicators
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Matrix Representation
Cast expected log-likeihood function into a
matrix representation using assignment matrix and
connection matrices
Maximum likelihood correspondence matrix
satisfies the condition
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Maximisation using SVD
Locate intermediate matrix which satisfies the
condition
Scott and Longuet-Higgins showed that this matrix
can be found using the singular value
decomposition
The correspondences are located by selecting the
elements which are both row and column maxima
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Expectation step
Update a posteriori correspondence probabilities
using the Bayes rule
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CMU House
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Distortions
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Luo EMSVD
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Luo
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Luo
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Luo
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Luo
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Results Summary
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Umeyama
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Shapiro and Brady
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Correspondences
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Convergence
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Edge Errors (Size constant)
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Edge Errors versus Positional Jitter
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Summary

• Have used EM algorithm to develop an SVD method
  for graph-matching which works with graphs of
  different size and edge structure it copes with
  the inexact case.
• Subsequent work has shown how to convert graphs
  to strings using leading eigenvector, and how
  similarity can be measure using string edit
  distance can be computed (PAMI 2005
  Robles-KellyHancock).

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Graph seriation

• convert graphs to strings using eigenvector
  methods and find correspondences using string
  matching methods

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Motivation

• Theoretic framework for graph edit distance much
  less developed or rigorous than that for string
  edit distance.
• Aim here is to use seriation of adjacency matrix
  to convert graphs to strings.
• Use a Bayes model of string matching to compute
  edit costs.
• Match by finding minimum cost path through
  Levenshtein distance matrix.

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Context and contribution
Graph spectral methods can be used to convert
graphs to strings graph seriation (Robles-Kelly
and Hancock PAMI05).
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Example
Eigen-vector components
Original graph
Seriation path
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Semidefinite programming

• Optimisation over positive semi-definite
  matrices.
• Linear cost function and linear constraints.
• Convex solutions.

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Spectral seriation problem definition

• Our aim is to use a permutation p of the nodes to
  find a path sequence so that the edge weight
  matrix W decreases as the path is traversed.
• This is governed by the penalty function.
• Unfortunately, minimizing g(p)is NP complete.

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Approximate solution relaxed version

• Instead seek a relaxed solution is sought by
  using.
• vector
• and minimise
• under the constraints
• and.

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Graph Laplacian

• Graph Laplacian
• Graph
• Combinatorial Laplacian matrix LD-A
• Eigenspectrum of the Laplacian matrix
• Eigenvalues
• Eigenvectors

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Spectral seriation Fiedler vector

• Atkinson, Bowman and Hendrikson showed that the
  solution to this minimization problem is given by
  the Fiedler vector.

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.this is not surprising

• Both the continuous and discrete time random
  walks on a graph are determined by the Fiedler
  vector of the Laplacian (see review by Lovasc).
• Unfortunately random walks teleport and do
  not preserve edge connectivity constraints.

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Improved use of edge connectvity
Robles-KellyHancock (PAMI 05).

• Seek relaxed vector
• that minimises path length
• under the constraints
• and.

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Solution

• Relaxed solution is the leading eigenvector
  of that minimises the Rayleigh
  quotient

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Rank order of nodes gives seriation order

• Sort nodes according to their eigenvector rank
  order satisfies
• Seriation path given by rank order of leading
  eigenvector of transition probability matrix

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Edit Distance
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• Edit path is the sequence of states
• Optimal edit path is the one that satisfies the
  condition
• Under pairwise dependence of states

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Edit Distance

• Log posterior probability of edit path
• Elementary transition cost

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Model

• Transition probabilities (Bayes treatment)
• Posterior state probabilities

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Comparative Study
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Retrieval Experiments
We have used a data-set of Delaunay graphs
obtained using the corners extracted from the
first ten frames of the CMU and MOVI image
sequences.
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CMU Sequence
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MOVI Sequence
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Example adjacency matrices and matrix y
Final edit distance matrix
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Distance matrices and clustering
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MDS Thumbnails
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Learning
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Why is structural learning difficult

• Graphs are not vectors There is no natural
  ordering of nodes and edges. Correspondences must
  be used to establish order.
• Structural variations Numbers of nodes and
  edges are not fixed. They can vary due to
  segmentation error.
• Not easily summarised Since they do not reside
  in a vector space, mean and covariance hard to
  characterise.

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Structural Variations
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Learning with graphs

• Learn class structure Assign graphs to classes.
  Need a distance measure. Central clustering is
  difficult since number of nodes and edges varies
  and correspondences are not known. Easier to
  perform pairwise clustering. (Bunke,
  Buhman,Luo_Torsello_Robles-Kelly).
• Embed graphs in a low dimensional space
  Correspondences are again needed, but spectral
  methods may offer a solution. Can apply standard
  statistical and geometric learning methods to
  graph-vectors (LuoWilson).
• Learn modes of structural variation Understand
  how edge (connectivity) structure varies for
  graphs belonging to the same class.
  (Dickinson,Williams,Torsello)
• Build generative model Borrow ideas from
  graphical models (Langley, Friedman,
  Koller,Torsello,Xaio).

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Pattern spaces sets of graphs
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Preliminary Study (ACCV 2002)

• Known correspondences OR weighted graphs

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Aim of this paper
Investigate whether it is possible to generate
view based eigenspaces using relational graphs.
Image features Objects in images are
first represented by extracted
corners Graphs Objects are then
represented by relational graphs of Delaunay
triangulations Pattern space embedding PCA M
DS
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Graph Representation
Adjacency matrices

Vector representation
Order of components of vector is determined by
known correspondences graphs are of same size
with no node errors but variation in edge
structure .
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Embedding using PCA
Embedding graphs in a pattern space by
storing each image as a feature vector,
calculating the covariance matrix of the vectors,
finding the eigenvalues and eigenvectors using
PCA and projecting the images onto the leading
principal directions. Murase and Nayar
1994
Vector Matrix
Covariance Matrix
Eigenvalue Equation
Eigenvector Equation
Eigenspace Projection
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Synthetic Sequence
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PCA Eigenspaces Synthetic Sequence - 1
Left column Eigenspaces Right column
Graph distances in eigenspace Top row
Unweighted graph Middle row Weighted graph
proximity weights Bottom row Fully connected
weighted graph (point proximity
matrix)
This wont work in practice Components of vectors
are ordered using ground truth correspondences
and no size differences.
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Algebraic graph theory (PAMI 2005)

• Use symmetric polynomials to construct
  permutation invariants from spectral matrix

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Spectral Representation

• Compute Laplacian matrix LD-A, where A is the
  adjacency matrix and D is the matrix with the
  node degree on the diagonal.
• Perform spectral decomposition on the Laplacian
  matrix
• Construct spectral matrix

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Properties of the Laplacian

• Eigenvalues are positive and smallest eigenvalue
  is zero
• Multiplicity of zero eigenvalue is number
  connected components of graph.
• Zero eigenvalue is associated with all-ones
  vector.
• Eigenvector associated with the second smallest
  eigenvector is Fiedler vector.
• Fiedler vector can be used to perform clustering
  of nodes of graph by recursive bisection .

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Eigenvalue spectrum
Vector of ordered eigevectors is permutation
invariant
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Eigenvalues are invariant to permutations of the
Laplacian.

• ..would like to construct family of permutation
  invariants from full spectral matrix.

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Why

• According to perturbation analysis eigenvalues
  are relatively stable to noise.
• Eigenvectors are not stable to noise and undergo
  large rotations for small additions of noise.

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Symmetric polynomials on spectral matrix

• Symmetric polynomials
• Power symmetric polynomials
• Newton Giraud formula

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Spectral Feature Vector

• Construct a matrix of permutation invariants by
  applying symmetric polynomials to elements in
  columns of the spectral matrix. Use entropy
  measure to flatten distribution
• Stack columns of F to form a long-vector B.
• Set of graphs represented by data-matrix

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extend to weighted attributed graphs.
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Complex Representation

• Encode attributes as complex numbers.
• Off-diagonal elements. Edge weights (W) as
  modulus and normalised attributes as phase (y)
• Diagonal elements encode node attributes (x) and
  ensure H is positive semi-definite

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Spectral analysis

• Perform spectral analysis on H. Real eigenvalues
  and complex eigenvectors
• Construct spectral matrix of scaled complex
  eigenvectors
• Complex Laplacian

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Manifold learning methods

• ISOMAP construct neighbourhood graph on pairwise
  geodesic distance between data-points. Low
  distortion embedding by applying MDS to weighted
  graph (Tennenbaum).
• Locally linear embedding apply variant of PCA to
  data (Roweiss Saul)
• Locally linear projection use interpoint
  distances to compute weighted covariance matrix,
  and apply PCA (HeNiyogi).

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Pattern Spaces

• PCA Project long vectors onto leading
  eigenvectors of covariance matrix
• MDS Embed graphs in low dimensional space
  spanned by eigenvectors of distance matrix
• LLP Locally linear projection (Niyogi) perform
  eigenvector analysis on weighted covariance
  matrix (mixture of PCA and MDS). PCA/MDS hybrid.

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Separation under structural error
Mahalanobis distance between feature vectors for
noise corrupted graph and remaining graphs
Distance between graph and edge-edited variants
Distance between graph and random graphs of same
size and edge density
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Distribution of spectral features
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Variation under structural error (MDS)
MDS applied to Mahalanobis distances between
feature vectors.
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CMU Sequence
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MOVI Sequence
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YORK Sequence
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Comparison
Laplacian eigenvalues
Adj. poly
Lap. Poly.
PCA
MDS
LLP
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Visualisation (LLPLaplacian Polynomials)
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View Trajectories (MOVI)
Adjacency matrix polynomials (top) versus
Laplacian polynomials (bottom) left-to-right
(PCA, MDS, LLP).
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View Trajectories (Chalet)
Adjacency matrix polynomials (top) versus
Laplacian polynomials (bottom) left-to-right
(PCA, MDS, LLP).
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Shock graphs
Type 1 shock(monotonically increasing radius)
Type 2 shock(minimum radius)
Type 3 shock(constant radius)
Type 4 shock(maximum radius)
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Shock Graph Attributes

• Nodes are skeletal branches. Edges indicate
  existence of a junction between a pair of
  branches.
• Edge attributes are angles between branches
• Node attribute is average rate of change of
  boundary length with skeleton length along a
  branch. This is related to rate of change of
  bitangent radius with skeleton distance

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MDS
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PCA
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LDA
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Geometric characterisation og graphs
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Idea

• Kernel PCA applied to heat-kernel of graph.
• Embed nodes of graph into a vector space using
  the kernel mapping.
• Characterise graph using the geometry of the
  kernel embedding.

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Characterisations

• Heat-kernel eigenvalues (eigenvalues of
  covariance matrix for the embedded node
  co-ordinates).
• Moments of point-set distribution.
• Sectional curvatures of edges associated with the
  embedding.

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Graph spectra

• .some introductory material

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Normalised Laplacian and its Spectrum

• Adjacency matrix
• Degree matrix
• Normalised Laplacian

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Normalised Laplacian and its Spectrum

• Spectral Decomposition of Laplacian
• Element-wise

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Heat Kernel Trace
Trace
Time (t)-gt
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Moments of the heat-kernel trace

• .can we characterise graph by the shape of its
  heat-kernel trace function?

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Zeta function

• Definition of zeta function

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Zeta function and heat-kernel moments

• Mellin transform
• Trace and number of connected components
• Zeta function

C is multiplicity of zero eigenvalue or number of
connected components in graph.
Zeta-function is related to moments of
heat-kernel trace.
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Heat kernel moments as a function of view
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Clusters
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Spectral Clustering
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Zeta derivative and Laplacian determinant

• Zeta function in terms of natural exponential
• Derivative
• Derivative at origin
• Torsion

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works quite well as a feature
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Heat-kernels and random walks
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Heat Kernels

• Solution of heat equation and measures
  information flow across edges of graph with time
• Solution found by exponentiating Laplacian
  eigensystem

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Heat kernel is distribution of path lengths

• In terms of number of paths of length
  k from node u to node v
• Geodesic distance

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Greens function

• Definition
• Spectral representation
• Meaning psuedo inverse of Laplacian

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Commute Time

• Commute time
• Hitting time and the Greens function
• Commute time and Laplacian eigen-spectrum
• For a regular graph

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Heat kernel and random walk

• State vector of continuous time random walk
  satisfies the differential equation
• Solution

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• Heat-kernel embedding

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Kernel Mapping

• Heat-kernel ht is a Gram matrix for the points
  embedded in the manifold
• Determines point positions up to isometry
• Coordinate matrix Y given by Young-Householder
  decomposition
• The kernel mapping is a mapping from graph
  vertices to the corresponding column of Y
• Vertex is a point in V dimensional space

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Covariance structure

• Uses the covariance matrix of the point-set from
  the kernel mapping
• Eigenvalues of covariance are
• Characterise graph using vector of eigenvalues

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Kernel Mapping

• Euclidean distance related to kernel

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Graph clustering COIL database
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Stability of mapping

• Projection onto eigenspace spanned by leading
  two heat-kernel eigenvectors.
• Elipsoids fitted to points closest to reference
  graph nodes (reference graph is largest in the
  set).

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Empirical observations

• Points corresponding to nodes relatively stable.
• Could construct a generative model that described
  point distribution associated with nodes.
• Use either a mixture of Gaussians or a linear
  point distribution model.

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Linear generative model
Establish correspondences between reference graph
(largest is set) and each remaining graph from
training set.

• Mean embedded point position
• Covariance matrix
• Eigendecomposition
• Projection of graph onto eigenspace

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Example of projection
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Sectional curvature

• we have geodesic distance from the properties of
  the random walk and Euclidean distance from the
  embedding. So could we compute curvatures?

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Approach

• Idea
• Nodes of graph reside on a manifold in
  low-dimensional-space.
• Edges are geodesics on the manifold.
• Sectional curvatures of edges depend on
  difference between geodesic and sectional
  curvatures
• Characterise graph using the histogram of
  sectional curvatures for the edges.

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Idea
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Spectral Geometry

• Spectral geometry Characterise differential
  geometry of manifold using information flow
  dictated by heat equation on manifold solution
  provided by eigenspectrum of Laplace-Beltrami
  operator (Yau, Gilkey).

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Spectral Geometry of the Laplacian of Manifold

• Heat kernel trace
• Polynomial co-efficients

• Volume of manifold
• Gauss curvature
• Ricci curvature

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Way forward
.too hard. Computing the co-efficients is time
consuming. Theoretical physicists involved in
brane-theory are struggling to go beyond the
fifth term in the series. Instead we use the
heat kernel to estimate the Euclidean and
geodesic distances of an implicit embedding, and
make numerical estimates of the geodesic
distances associated with edges.
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Approximating sectional curvature

Euclidean distance
Geodesic distance
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Sectional Curvature

• Maclaurin series of the Euclidean distance
• Substitute from the geodesic distance
• Approximate sectional curvature

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Sectional Curvature for Graph Clustering

• Histograms
• Construct normalised histogram of sectional
  curvatures over the edge-set of the graph.
• Use normalised bin-contents as component
  of a feature vector. Apply PCA to vectors for a
  set of graphs,

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Histograms of sectional curvature
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Clustering Sectional Curvature Histograms
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Clustering using the Laplacian spectrum
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Clustering using geodesic distance histograms
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Conclusions

• Shown how spectral features can be used to
  construct pattern spaces for sets of graphs.
• Shown how heat-kernel embedding can be used to
  characterise graphs in a geometric manner and the
  characterisation used for clustering.
• Future plans revolve around using embedding to
  construct generative model of graph-structure.