Signal Processing - July 2017 - 32
g a (f) = a (I + D -1/2 WD -1/2) f.
domain with spectral multipliers as in (30), it is possible to
represent the filters via a polynomial expansion:
g a (T) = Ug a (K) U ,
<
(31)
Because the eigenvalues of I + D -1/2 WD -1/2 are now in the
range [0, 2], repeated application of such a filter can result in
numerical instability. This can be remedied by a renormalization
where
g a (m) =
r-1
/ ajm ,
j
u -1/2 WD
u u -1/2 f,
g a (f) = aD
(32)
j=0
a is the r-dimensional vector of polynomial coefficients, and
g a (K) = diag ^g a ^m 1h, f, g a ^m n hh, resulting in filter matrices
C l, ll = g al,ll ^K h whose entries have an explicit form in terms
of the eigenvalues.
An important property of this representation is that it automatically yields localized filters, for the following reason.
Because the Laplacian is a local operator (working on one-hop
neighborhoods), the action of its jth power is constrained to j
hops. Because the filter is a linear combination of powers of
the Laplacian, overall (32) behaves like a diffusion operator
limited to r hops around each vertex.
GCNN, also known as ChebNet
Defferrard et al. used the Chebyshev polynomials generated
by the recurrence relation [45]
T j ^mh = 2mT j - 1 ^mh - T j - 2 ^mh,
T0 ^mh = 1,
T1 ^mh = m.
(33)
u) =
g a (T
/ a j UT j (Ku ) U <
j=0
r-1
=
/ a j T j (Tu ),
(34)
j=0
u = 2m -n 1 K - I denotes a rescalu = 2m -n 1 T - I and K
where T
ing of the Laplacian mapping its eigenvalues from the interval
[0, m n] to [- 1, 1] (necessary because the Chebyshev polynomials form an orthonormal basis in [- 1, 1]).
u ) f, we can use the recurrence relaDenoting fr ( j) = T j (D
( j)
u fr ( j - 1) - fr ( j - 2) , with fr (0) = f and
r
f
= 2D
tion (33) to compute
(
1
)
u f. The computational complexity of this procedure is
fr = D
therefore O (rn) operations and does not require an explicit
computation of the Laplacian eigenvectors.
Graph convolutional network
Kipf and Welling [77] simplified this construction by further
assuming r . 2 and m n . 2, resulting in filters of the form
g a (f) = a 0 f + a 1 (D - I) f
= a 0 f - a 1 D -1/2 WD -1/2 f.
(35)
Further constraining a = a 0 = - a 1, one obtains filters represented by a single parameter,
32
(37)
u = diag `/ w
u = W + I and D
u j.
where W
j ! i ij
Note that though we arrived at the constructions of ChebNet and graph convolutional network (GCN) starting in the
spectral domain, they boil down to applying simple filters acting on the r- or one-hop neighborhood of the graph in the spatial domain. We consider these constructions to be examples
of the more general graph neural network (GNN) framework.
GNN
GNNs [78] generalize the notion of applying the filtering
operations directly on the graph via the graph weights.
Similarly as Euclidean CNNs learn generic filters as linear
combinations of localized, oriented bandpass and low-pass filters, a GNN learns at each layer a generic linear combination
of graph low-pass and high-pass operators. These are given,
respectively, by f 7 Wf and f 7 Df and are thus generated
by the degree matrix D and the diffusion matrix W. Given a
p-dimensional input signal on the vertices of the graph, represented by the n # p matrix F, the GNN considers a generic
nonlinear function h i: R p # R p " R q, parameterized by trainable parameters i that is applied to all nodes of the graph,
A filter (32) can thus be parameterized uniquely via an expansion of order r - 1 such that
r-1
(36)
g i = h i ^(Wf) i, (Df) ih .
(38)
In particular, choosing h (a, b) = b - a, one recovers the
Laplacian operator Df, but more general, nonlinear choices
for h yield trainable, task-specific diffusion operators.
Similarly as with a CNN architecture, one can stack the resulting GNN layers g = C i (f ) and interleave them with graph
pooling operators. Chebyshev polynomials Tr (D) can be
obtained with r layers of (38), making it possible, in principle,
to consider ChebNet and GCN as particular instances of the
GNN framework.
Historically, a version of GNN was the first formulation
of deep learning on graphs, proposed in [49] and [78]. These
works optimized over the parameterized steady state of some
diffusion process (or random walk) on the graph. This can be
interpreted as in (38) but using a large number of layers where
each C i is identical, as the forward propagation through the
C i approximate the steady state. Recent works [50], [51], [55],
[79], [80] relax the requirements of approaching the steady
state or using repeated applications of the same C i.
Because the communication at each layer is local to a vertex neighborhood, one may worry that it would take many layers to get information from one part of the graph to another,
requiring multiple hops (this was one of the reasons for the
use of the steady state in [78]). However, for many applications, it is not necessary for information to completely traverse
IEEE SIGNAL PROCESSING MAGAZINE
|
July 2017
|
Table of Contents for the Digital Edition of Signal Processing - July 2017
Signal Processing - July 2017 - Cover1
Signal Processing - July 2017 - Cover2
Signal Processing - July 2017 - 1
Signal Processing - July 2017 - 2
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Signal Processing - July 2017 - Cover3
Signal Processing - July 2017 - Cover4
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