Spherical and Hyperbolic Embeddings of Data

Alippi

2019

Applied Soft Computing

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.

Section: Priors On Ccmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Adversarial autoencoders with constant-curvature latent manifolds

Grattarola

Alippi

2019

Applied Soft Computing

“…, x t ∈ M k . By following [11], we consider a symmetric distance measure d(·, ·) operating in G and propose to adopt an embedding aiming at preserving the distances, i.e., ρ κ (f (g 1 ), f (g 2 )) ≈ d(g 1 , g 2 ) ∀g 1 , g 2 ∈ G.…”

Section: Embedding Based On Dissimilarity Matrixmentioning

confidence: 99%

Anomaly and Change Detection in Graph Streams through Constant-Curvature Manifold Embeddings

Zambon

2018 International Joint Conference on Neural Networks (IJCNN)

Alippi

2018

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.

“…Many relevant machine learning applications require to go beyond conventional Euclidean geometry, as in the case of data described by attributed graphs [1], [2]. When studying problems on graphs, one of the key issues is to find representations that allow dealing with their underlying geometry, which is usually defined by application-specific distances that often do not satisfy the identity of indiscernibles or the triangular inequality [3], [4]. The use of metric distances, like graph alignment distances [5], only mitigates the problem, as they are computationally intractable and hence not useful in practical applications.…”

Section: Introductionmentioning

confidence: 99%

Change Detection in Graph Streams by Learning Graph Embeddings on Constant-Curvature Manifolds

Grattarola

Zambon

IEEE Trans. Neural Netw. Learning Syst.

et al. 2020

The space of graphs is often characterised by a nontrivial 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 realworld 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.