3 research outputs found
Anomaly and Change Detection in Graph Streams through Constant-Curvature Manifold Embeddings
Mapping complex input data into suitable lower dimensional manifolds is a
common procedure in machine learning. This step is beneficial mainly for two
reasons: (1) it reduces the data dimensionality and (2) it provides a new data
representation possibly characterised by convenient geometric properties.
Euclidean spaces are by far the most widely used embedding spaces, thanks to
their well-understood structure and large availability of consolidated
inference methods. However, recent research demonstrated that many types of
complex data (e.g., those represented as graphs) are actually better described
by non-Euclidean geometries. Here, we investigate how embedding graphs on
constant-curvature manifolds (hyper-spherical and hyperbolic manifolds) impacts
on the ability to detect changes in sequences of attributed graphs. The
proposed methodology consists in embedding graphs into a geometric space and
perform change detection there by means of conventional methods for numerical
streams. The curvature of the space is a parameter that we learn to reproduce
the geometry of the original application-dependent graph space. Preliminary
experimental results show the potential capability of representing graphs by
means of curved manifold, in particular for change and anomaly detection
problems.Comment: To be published in IEEE IJCNN 201
Adversarial Autoencoders with Constant-Curvature Latent Manifolds
Constant-curvature Riemannian manifolds (CCMs) have been shown to be ideal
embedding spaces in many application domains, as their non-Euclidean geometry
can naturally account for some relevant properties of data, like hierarchy and
circularity. In this work, we introduce the CCM adversarial autoencoder
(CCM-AAE), a probabilistic generative model trained to represent a data
distribution on a CCM. Our method works by matching the aggregated posterior of
the CCM-AAE with a probability distribution defined on a CCM, so that the
encoder implicitly learns to represent data on the CCM to fool the
discriminator network. The geometric constraint is also explicitly imposed by
jointly training the CCM-AAE to maximise the membership degree of the
embeddings to the CCM. While a few works in recent literature make use of
either hyperspherical or hyperbolic manifolds for different learning tasks,
ours is the first unified framework to seamlessly deal with CCMs of different
curvatures. We show the effectiveness of our model on three different datasets
characterised by non-trivial geometry: semi-supervised classification on MNIST,
link prediction on two popular citation datasets, and graph-based molecule
generation using the QM9 chemical database. Results show that our method
improves upon other autoencoders based on Euclidean and non-Euclidean
geometries on all tasks taken into account.Comment: Submitted to Applied Soft Computin
Change Detection in Graph Streams by Learning Graph Embeddings on Constant-Curvature Manifolds
The space of graphs is often characterised by a non-trivial geometry, which
complicates learning and inference in practical applications. A common approach
is to use embedding techniques to represent graphs as points in a conventional
Euclidean space, but non-Euclidean spaces have often been shown to be better
suited for embedding graphs. Among these, constant-curvature Riemannian
manifolds (CCMs) offer embedding spaces suitable for studying the statistical
properties of a graph distribution, as they provide ways to easily compute
metric geodesic distances. In this paper, we focus on the problem of detecting
changes in stationarity in a stream of attributed graphs. To this end, we
introduce a novel change detection framework based on neural networks and CCMs,
that takes into account the non-Euclidean nature of graphs. Our contribution in
this work is twofold. First, via a novel approach based on adversarial
learning, we compute graph embeddings by training an autoencoder to represent
graphs on CCMs. Second, we introduce two novel change detection tests operating
on CCMs. We perform experiments on synthetic data, as well as two real-world
application scenarios: the detection of epileptic seizures using functional
connectivity brain networks, and the detection of hostility between two
subjects, using human skeletal graphs. Results show that the proposed methods
are able to detect even small changes in a graph-generating process,
consistently outperforming approaches based on Euclidean embeddings.Comment: 14 pages, 8 figure