Signal Processing - July 2017 - 19

and [13] for many additional examples of successful applications of deep learning). Today, deep learning
has matured into a technology that is widely used in commercial applications, including Siri speech recognition in Apple iPhone, Google text translation, and Mobileye vision-based technology for autonomously
driving cars.
One of the key reasons for the success of deep neural networks is their ability to leverage statistical properties of the data, such as stationarity and compositionality through local statistics,
which are present in natural images, video, and speech [14], [15]. These statistical properties
have been related to physics [16] and formalized in specific classes of convolutional neural
networks (CNNs) [17]-[19]. In image analysis applications, one can consider images as
functions on the Euclidean space (plane), sampled on a grid. In this setting, stationarity
is owed to shift invariance, locality is due to the local connectivity, and compositionality stems from the multiresolution structure of the grid. These properties are exploited
by convolutional architectures [20], which are built of alternating convolutional and
downsampling (pooling) layers. The use of convolutions has a twofold effect. First, it
allows extracting local features that are shared across the image domain and greatly reduces the number of parameters in the network with respect to generic deep
architectures (and thus also the risk of overfitting), without sacrificing the expressive capacity of the network. Second, the convolutional architecture itself imposes
some priors about the data, which appear very suitable especially for natural images
[17]-[19], [21].
While deep-learning models have been particularly successful when dealing
with speech, image, and video signals, in which there are an underlying Euclidean structure, recently there has been a growing interest in trying to apply learning
on non-Euclidean geometric data. Such kinds of data arise in numerous applications. For instance, in social networks, the characteristics of users can be modeled
as signals on the vertices of the social graph [22]. Sensor networks are graph models
of distributed interconnected sensors, whose readings are modeled as time-dependent signals on the vertices. In genetics, gene expression data are modeled as signals
defined on the regulatory network [23]. In neuroscience, graph models are used to represent anatomical and functional structures of the brain. In computer graphics and vision,
three-dimensional (3-D) objects are modeled as Riemannian manifolds (surfaces) endowed
with properties such as color texture.
The non-Euclidean nature of such data implies that there are no such familiar properties as
global parameterization, common system of coordinates, vector space structure, or shift
invariance. Consequently, basic operations like convolution that are taken for granted in
the Euclidean case are even not well defined on non-Euclidean domains. The purpose
of this article is to show different methods of translating the key ingredients of successful deep-learning methods, such as CNNs, to non-Euclidean data.

Geometric learning problems
Broadly speaking, we can distinguish between two classes of geometric
learning problems. In the first class of problems, the goal is to characterize
the structure of the data. The second class of problems deals with analyzing functions defined on a given non-Euclidean domain. These two classes are related, because understanding the properties of functions defined
on a domain conveys certain information about the domain, and vice
versa, the structure of the domain imposes certain properties on the functions on it.

Structure of the domain
As an example of the first class of problems, assume to be given a set of
data points with some underlying low-dimensional structure embedded into a
high-dimensional Euclidean space. Recovering that low-dimensional structure
is often referred to as manifold learning or nonlinear dimensionality reduction
and is an instance of unsupervised learning (note that the notion of manifold in this
setting can be considerably more general than a classical smooth manifold; see, e.g.,
IEEE SIGNAL PROCESSING MAGAZINE

July 2017

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