Signal Processing - July 2017 - 39

to first come up with a hypothesis and then test it on the data
[22]. Yet, such models assume some prior knowledge (e.g.,
isometric shape deformation model) and often fail to correctly capture the full complexity and richness of the data.
In computer vision, departing from handcrafted features
toward generic models learnable from the data in a taskspecific manner has brought a breakthrough in performance
and led to an overwhelming trend in the community to favor
deep-learning methods. Such a shift has not occurred yet
in the fields dealing with geometric data due to the lack of
adequate methods, but there are the first indications of a
coming paradigm shift.

Generalization

affect signals. This could prove useful to tackle applications
like abnormal activity detection in social or financial networks. In the domain of computer graphics and vision, potential applications deal with dynamic shapes (e.g., 3-D video
captured by a range sensor).

Directed graphs
Dealing with directed graphs is also a challenging topic, as
such graphs typically have nonsymmetric Laplacian matrices that do not have orthogonal eigendecomposition allowing easily interpretable spectral-domain constructions.
Citation networks, which are directed graphs, are often treated as undirected graphs (including in our example in
"Three-Dimensional Shape Correspondence Application")
considering citations between two articles without distinguishing which article cites which. This obviously may lose
important information.

Generalizing deep-learning models to geometric data
requires not only finding non-Euclidean counterparts of basic
building blocks (such as convolutional and pooling layers)
but also generalization across different domains. Ge neralization capability is a key requirement in many applicaSynthesis problems
tions, including computer graphics, where a model is learned
Our main focus in this review was primarily on analysis probon a training set of non-Euclidean domains (3-D shapes) and
lems on non-Euclidean domains. Not less important is the
then applied to previously unseen ones. Spectral formulation
question of data synthesis. There have been several recent
of convolution allows designing CNNs on
attempts to try to learn a generative model
a graph, but the model learned this way on
allowing to synthesize new images [120]
One of the main reasons
one graph cannot be straightforwardly
and speech waveforms [121]. Extending
for the computational
applied to another one, because the specsuch methods to the geometric setting
efficiency of deep-learning seems a promising direction, though the
tral representation of convolution is domain
dependent. A possible remedy to the genkey difficulty is the need to reconstruct the
architectures is the
eralization problem of spectral methods is
geometric structure (e.g., an embedding of
assumption of regularly
the recent architecture proposed in [118],
a 2-D manifold in the 3-D Euclidean space
structured data on a 1-D
applying the idea of spatial transformer
modeling a deformable shape) from some
or 2-D grid.
networks [119] in the spectral domain.
intrinsic representation [122].
This approach is reminiscent of the construction of compatible orthogonal bases by means of joint
Computation
Laplacian diagonalization [75], which can be interpreted as
The final consideration is a computational one. All existing
an alignment of two Laplacian eigenbases in a k-dimensiondeep-learning software frameworks are primarily optimized
al space.
for Euclidean data. One of the main reasons for the computaThe spatial methods, on the other hand, allow generalizational efficiency of deep-learning architectures (and one of the
tion across different domains, but the construction of lowfactors that contributed to their renaissance) is the assumpdimensional local spatial coordinates on graphs turns out to
tion of regularly structured data on a 1-D or 2-D grid, allowbe rather challenging. In particular, the construction of anisoing to take advantage of modern GPU hardware. Geometric
tropic diffusion on general graphs is an interesting research
data, on the other hand, in most cases do not have a grid
direction. The spectrum-free approaches also allow generalstructure, requiring different ways to achieve efficient comization across graphs, at least in terms of their functional form.
putations. It seems that computational paradigms developed
However, if multiple layers of (38) are used with no nonlinearfor large-scale graph processing are more adequate frameity or learned parameters i , simulating a high power of the
works for such applications.
diffusion, the model may behave differently on different kinds
of graphs. Understanding under what circumstances and to
Acknowledgments
what extent these methods generalize across graphs is currently
We are grateful to Federico Monti, Davide Boscaini,
being studied.
Jonathan Masci, Emanuele Rodolà, Xavier Bresson, Thomas
Kipf, and Michaël Defferard for their comments and for proTime-varying domains
viding some of the figures used in this article. This work was
An interesting extension of geometric deep-learning problems
supported in part by the European Research Council grant
discussed in this review is coping with signals defined over a
numbers 307047 (COMET) and 724228 (LEMAN), a Google
dynamically changing structure. In this case, we cannot
Faculty Research Award, a Radcliffe fellowship, a Rudolf
assume a fixed domain and must track how these changes
Diesel fellowship, and Nvidia equipment grants.
IEEE SIGNAL PROCESSING MAGAZINE

July 2017

Table of Contents for the Digital Edition of Signal Processing - July 2017