Graph matching through entropic manifold alignment

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GRAPH MATCHING THROUGH ENTROPIC MANIFOLD ALIGNMENT
Francisco Escolano, Edwin R. Hancock and Miguel A. Lozano
University of Alicante, Spain and University of York, UK
Objectives and Contributions
1. Cast Graph Matching as non-Rigid Manifold Alignment: Graphs as d-dimensional point-sets.
2. Low-dimensional Manifolds: Perform a low-pass filtering on Commute Times embeddings.
3. Information-theoretic Dissimilarity Measure: SNESV – computed through d-dimensional entropy.
4. A Novel Kernel Between Graphs: We proof that SNESV induces a positive definite kernel.
5. Commute Times Embedding is optimal: In terms of graph retrieval performance.
6. Entropic Alignment: improves state-of-the-art graph based algorithms for point matching.
Commute Times and Commute Times Embedding
Given an undirected and unweighted graph G = (V,E) with adjacency matrix A:
• Laplacian Matrix: L = D − A where D is the diagonal degree matrix and vol = trace(D).
• Laplacian Eigen-decomposition: L = ΦΛΦTwhere Λ is the diagonal eigenvalue matrix with
λ(1)= 0 < λ(2)< ... < λ(|V |)and Φ = [φ(1),...,φ(|V |)] is the eigenvector matrix.
• CommuteTime: CT(i,j)is theexpectedtimeforarandom walkto travelfromnodeitoreachnode
j and then return: CT(i,j) = O(i,j) + O(j,i) where O(.,.) is the hitting time. It is a distance [1].
CT(i,j) = vol
|V |
?
z=2
1
λ(z)
?
φ(z)(i) − φ(z)(j)
?2
(1)
• Commute Time Embedding: Θ =√volΛ−1/2ΦTso that CT(i,j) = ||Θ(i)− Θ(j)||2, where
?
Θ(i)=
√
vol
0
1
λ(2)φ(2)(i)...
?
1
?
λ(|V |)φ(|V |)(i)
?T
.
(2)
If we retain from the second component to the component d + 1 (low pass filter), we obtain a
d−dimensional approximationˆΘ(i)and then?
COMMUTE TIMES TAKE INTO ACCOUNT THE GLOBAL STRUCTURE OF THE GRAPH
CT(i,j) = ||ˆΘ(i)−ˆΘ(j)||2≤ CT(i,j).
CT=15.6703
CT=60.8719
COMMUTE TIMES EMBEDDING PRESERVES HIGH ORDER TOPOLOGICAL RELATIONS
||i - j|| <= CT(i,j)
2
i i
j j
R2
von Luxburg Law: when |V| high then
CT(i,j) = vol(1/D(i,i) +1/D(j,j))

NOT WORKING IN DELAUNAY TRIANGULATIONS!
Low -pass flter of the manifold
CT more global than degree and
NOT shortest path
INSTEAD OF MATCHING GRAPHS
WHE PERFORM A NON-RIGID ALIGNMENT OF EMBEDDINGS!
• Normalized Laplacian Matrix: L = D−1/2LD−1/2with Θ =√volΛ−1/2ΦTD−1/2.
Non-rigid Alignment of CT Embeddings
Let i ∈ VXand u ∈ VYbe nodes of graphs X and Y , where N = |VX|, M = |VY| and let T be a
non rigid transformation which aligns the manifoldˆΘYwithˆΘX. Then, we can define
?
CT∗in terms of non-rigid manifold alignment using Coherent-Point
Drift (CPD) [2]; we must minimize
CT∗(i,u) = min
T
||ˆΘ(i)
X− T (ˆΘ(u)
Y)||2
(3)
We pose the problem of finding?
E(W,σ2) =
1
2σ2
N,M
?
i=1,u=1
Piu?
CT(i,u) +NP
2
logσ2+ γ(W) .
(4)
where W are the parameters of a non-rigid transformation T , σ is the isotropic variance, NP=
1TP1, where P is a M × N probability matrix with elements Pui= P(ˆΘ(u)
?
?
Y|i) given by
Pui=
exp
−1
2
????
2
????
ˆΘ(i)
X−T (ˆΘ(u)
Y;W)
σ
????
????
2?
????
?M
v=1exp
−1
????
????
ˆΘ(i)
X−T (ˆΘ(v)
Y;W)
σ
????
2?
+ c
=
exp
?
−1
?
2σ2?
CT(i,u)
?
?M
v=1exp
−1
2σ2?
CT(i,v)
?
+ c
.
(5)
• Function γ(W) = (λ/2)trace(WTGW) is a regularization term following the motion-coherence
theory (the kernel G is a Green’s function as in thin-plates splines)
• The problem is solve through an EM-like process: In the E-step we estimate P from the current T ,
and in the M-step we update W (and, thus, T ) and σ2.
Symmetrized Normalized Entropy Square Variation
The Symmetrized Normalized-Entropy-Square Variation SNESV is defined by
SV(ˆΘX,ˆΘY) =(H(T∗(ˆΘY)) − H(ˆΘX))2
H(T∗(ˆΘY)) + H(ˆΘX)
=
I(T∗(ˆΘY);ˆΘX) + H(T∗(ˆΘY),ˆΘX)+
where H(.) and H(.,.) are respectively the Shannon entropy and joint entropy, and I(.;.) denotes the
mutual information.
• Normalization is key when comparing graphs (manifolds) with a significantly different number of
nodes (points) and is also consistent with mutual information maximization.
• Given probability distributions of point sets after alignment, SNESV induces a p.d. kernel:
e−βy(H(T∗(ˆΘY))−H(ˆΘX))2
H(T∗(ˆΘY)) + H(ˆΘX) + ay
where βy, βx, ay, ax> 0. With a training set, we can optimize a graph kernel machine!
+(H(T∗(ˆΘX)) − H(ˆΘY))2
H(T∗(ˆΘX)) + H(ˆΘY)
I(T∗(ˆΘX);ˆΘY) + H(T∗(ˆΘX),ˆΘY).
(6)
(H(T∗(ˆΘY)) − H(ˆΘX))2
(H(T∗(ˆΘX)) − H(ˆΘY))2
KSV(p∗
X,p∗
Y) =
+
e−βx(H(T∗(ˆΘX))−H(ˆΘY))2
H(T∗(ˆΘX)) + H(ˆΘY) + ax
,
(7)
Experimental Results
Use the challenging GatorBait database. Four types of experiments for optimal d = 5 (we use Leo-
nenko et al. entropy estimator):
• Compare SNESV retrieval-recall behavior with respect to other IT-dissimilarity measures (Henze-
Penrose, KD-partition, Symmetrized KL, Jensen-Tsallis).
• Compare CT with other state-of-the-art embeddings (ISOMAP, Difussion Map (DM), Heat-Kernel
(HK) and Laplacian Eigenmap (LEM)).
• Compare EA with state-of-the-art graph and hypergraph-based point matching methods (tensor-
based (TB), Reweighted Random Walks (RRW), and classical Graduated Assignment (GA)). We
only use topological information (NOT POINTS!)
• Check that our graphs (Delaunay triangulations) are not prone to von Luxburg Law [3]. We ana-
lyze the ratio |R(i,j) − 1/D(i,i) − 1/D(j,j)|/R(i,j). If we plot the log(.) of the median of the
ratio versus the size of the graphs, this should be monotonically decreasing with the size of the
graphs. However, this is not the case of Delaunay triangulation representations of shapes since the
edge density is relatively low. In fact, for the GatorBait database the median of edge densities is
0.3409(34%)
Conclusions
In this work, there are four contributions: a) casting (purely topological) graph matching problem in
terms of non-rigid manifold alignment, b) find that CT is the optimal choice for building the mani-
fold, c)proposed a novel dissimilarity measurewith good retrieval-recallbehavior, d) EA outperforms
other state-of-the-art point matching algorithms exploiting graphsa.
Funding: F. Escolano and M. Lozano: project TIN2008-04416 of the Spanish Government. E.R. Hancoc: EU FET
project SIMBAD and a Royal Society Wolfson Research Merit Award.
References
[1] H. Qiu and E.R. Hancock: Clustering and Embedding using Commute Times. IEEE Transactions
on PAMI 29(11), 1873–1890, (2007)
[2] A. Myronenko and X. B. Song: Point-Set Registration: Coherent Point Drift. IEEE Transactions
on PAMI 32(12), 2262–2275, (2010)
[3] U. von Luxburg and A. Radl and M. Hein: Getting Lost in Space: Large Sample Analysis of the
Commute Distance. Proc. of NIPS, 2010.
aMATLAB code and data for reproducing all the experiments in this paper can be found in http://sites.google.com/site/scohomepage/